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# Polynomial algorithms for p-dispersion problems in a 2d Pareto Front Nicolas Dupin ###### Abstract Having many best compromise solutions for bi-objective optimization problems, this paper studies p-dispersion problems to select $p\geqslant 2$ representative points in the Pareto Front(PF). Four standard variants of p-dispersion are considered. A novel variant, denoted Max-Sum-Neighbor p-dispersion, is introduced for the specific case of a 2d PF. Firstly, it is proven that $2$-dispersion and $3$-dispersion problems are solvable in $O(n)$ time in a 2d PF. Secondly, dynamic programming algorithms are designed for three p-dispersion variants, proving polynomial complexities in a 2d PF. The Max-Min p-dispersion problem is proven solvable in $O(pn\log n)$ time and $O(n)$ memory space. The Max-Sum-Min p-dispersion problem is proven solvable in $O(pn^{3})$ time and $O(pn^{2})$ space. The Max-Sum-Neighbor p-dispersion problem is proven solvable in $O(pn^{2})$ time and $O(pn)$ space. Complexity results and parallelization issues are discussed in regards to practical implementation. Keywords : Optimization, Operational Research, Computational Geometry ; Dynamic programming ; p-dispersion problems ; complexity ; bi-objective optimization ; Pareto front ; Parallel Computing ## 1 Introduction This paper is motivated by real-life applications of multi-objective optimization (MOO) [5, 25]. Many best compromise solutions may exist considering the Pareto dominance [11]. A Pareto Front (PF) denotes the projection of these solutions in the space of the objectives. This work aims to select $p$ solutions from $n\gg p$ non dominated solutions, while maximizing the diversity of these $p$ solutions in the objective space. Firstly, such problem occurs when selecting alternatives for decision makers, $p$ is small in such applications. Secondly, a similar problem occurs inside MOO meta-heuristics to archive diversified solutions of the PF, $p$ is larger in such applications [32, 34]. For such problem, one can use the hypervolume measure as in [2, 21], or clustering algorithms as studied in [6, 7, 8]. In this paper, we consider (discrete) p-dispersion problems, dispersing the selected points as much as possible, as in [12]. Actually, four variants of discrete p-dispersion problems are defined [13]. Max-Min and Max-Sum p-dispersion problems, the most studied variants, are proven NP-complete in the general case ([12, 16]). Although p-dispersion is mentioned to have relevant applications for MOO [13, 27], no specific studies concerned the p-dispersion in a PF to the best of our knowledge. This paper studies the four variants of p-dispersion problems in the case of a 2-dimensional (2d) PF. Another problem, denoted Max-Sum-Neighbor p-dispersion, is also introduced in a 2d PF. For the five p-dispersion problems, the cases $p=2$ and $p=3$ are proven to be solvable in $O(n)$ time. Generally, the Max-Min, Max-Sum-Min and Max-Sum-Neighbor p-dispersion problems are proven to be solvable in polynomial time in a 2d PF thanks to dynamic programming (DP) algorithms. This paper is organized as following. In Section 2, we formally describe the problem. In Section 3, we discuss related state-of-the-art elements to appreciate our contributions. In Section 4, intermediate results are presented, allowing to define the Max-Sum-Neighbor p-dispersion problem. In Section 5, a DP algorithm is presented for the Max-Sum-Neighbor variant with a complexity proven in $O(pn^{2})$ time and $O(pn)$ memory space. In section 6, the previous DP algorithm is adapted for the Max-Min variant with a complexity in $O(pn\log n)$ time and $O(n)$ space. In section 7, a DP algorithm is presented for the Max-Min-Sum variant with a complexity in $O(pn^{3})$ time and $O(pn^{2})$ space. In Section 8, practical applications are discussed. In Section 9, our contributions are summarized, discussing also future directions of research. ## 2 Problem statement and notation We suppose in this paper having a set $E$ of $n$ elements of a 2d PF, where the minimization of two objectives is considered (without loss of generality, transforming objectives to maximize $f$ into $-f$). Similarly with [7], this can be formalized $E=\\{x_{1},\dots,x_{n}\\}$, having $n$ elements of ${\mathbb{R}}^{2}$, such that for all $i\neq j$, $x_{i}\phantom{0}\mathcal{I}\phantom{0}x_{j}$, defining the binary relations $\mathcal{I},\prec$ for all $y=(y^{1},y^{2}),z=(z^{1},z^{2})\in{\mathbb{R}}^{2}$ with: $\displaystyle y\prec z$ $\displaystyle\Longleftrightarrow$ $\displaystyle y^{1}<z^{1}\phantom{2}\mbox{and}\phantom{2}y^{2}>z^{2}$ (1) $\displaystyle y\preccurlyeq z$ $\displaystyle\Longleftrightarrow$ $\displaystyle y\prec z\phantom{2}\mbox{or}\phantom{2}y=z$ (2) $\displaystyle y\phantom{1}\mathcal{I}\phantom{1}z$ $\displaystyle\Longleftrightarrow$ $\displaystyle y\prec z\phantom{2}\mbox{or}\phantom{2}z\prec y$ (3) To measure distances between points $x_{i},x_{j}\in E$, we consider $d_{ij}=d(x_{i},x_{j})^{\alpha}$ where $\alpha>0$ and $d$ is the Euclidian distance: $\forall y=(y^{1},y^{2}),z=(z^{1},z^{2})\in{\mathbb{R}}^{2},\phantom{2}d(y,z)=\sqrt{\left(y^{1}-z^{1}\right)^{2}+\left(y^{2}-z^{2}\right)^{2}}$ (4) The p-dispersion problems select $p\geqslant 2$ out of $n$ given candidate points, while maximizing a dispersion function $f$: ${\mathcal{P}}_{disp}(E,p)=\max_{(z_{1},z_{2},\dots,z_{p})\in D_{p}}f(z_{1},z_{2},\dots,z_{p})$ (5) where $D_{p}$ denotes the set of all the distincts points of $E$: $D_{p}=\\{(z_{1},z_{2},\dots,z_{p})\in E^{p}|\forall 1\leqslant i<j\leqslant p,z_{i}\neq z_{j}\\}$ (6) The most standard p-dispersion problem, also denoted Max-Min p-dispersion problem or p-dispersion-MM, the dispersion function is the minimum of distances $d_{ij}$ between pairs of the selected points. Formally, the Max-Min p-dispersion problem for $p\geqslant 2$ can be written as: ${\mathcal{P}}_{disp}^{MM}(E,p)=\max_{(z_{1},z_{2},\dots,z_{p})\in D_{p}}\min_{i,j:1\leqslant i<j\leqslant p}d_{ij}$ (7) Another known variant is the Max-Sum(-Sum) dispersion problem, denoted p-dispersion-MS, with: ${\mathcal{P}}_{disp}^{MS}(E,p)=\max_{(z_{1},z_{2},\dots,z_{p})\in D_{p}}\sum_{i=1}^{p-1}\sum_{j=i+1}^{p}d_{ij}$ (8) We consider also in this paper the Max-Sum-Min p-dispersion problem, denoted p-dispersion-MSM: ${\mathcal{P}}_{disp}^{MSM}(E,p)=\max_{(z_{1},z_{2},\dots,z_{p})\in D_{p}}\sum_{i=1}^{p}\min_{j\neq i:1\leqslant j\leqslant p}d_{ij}$ (9) Another specific variant of Max-Sum-Min p-dispersion in a 2d PF will be defined in the Section 4, using specific properties of a 2d PF. Lastly, the Max-Min-Sum p-dispersion problem, denoted p-dispersion-MMS, is defined as following: ${\mathcal{P}}_{disp}^{MMS}(E,p)=\max_{(z_{1},z_{2},\dots,z_{p})\in D_{p}}\min_{i\in[\\![1,p]\\!]}\sum_{j\in[\\![1,p]\\!]-\\{i\\}}d_{ij}$ (10) ## 3 Related works This section describes related works to appreciate the contributions of this paper. ### 3.1 Complexity results for p-dispersion problems Max-Min and Max-Sum p-dispersion problems are NP-hard problems[12, 16]. It is proven in both cases by a polynomial reduction to the maximum independent set problem [12, 16]. Max-Min and Max-Sum p-dispersion problems are still NP-hard problems when distances fulfill the triangle inequality [12, 16]. The planar (2d) Max-Min p-dispersion problem is also NP-hard [33], the NP-hardness of the planar Max-Sum p-dispersion problem is still an open question to the best of our knowledge. The one-dimensional (1d) cases of Max-Min and Max-Sum p-dispersion problems are solvable in polynomial time, with a similar DP algorithm running in $O(\max\\{pn,n\log n\\})$ time [27, 33]. Complexity results have also been proven for the APX classes of several variants of p-dispersion problems. Firstly, for all $\alpha>1$, any $\alpha$-approximation for the Max-Min p-dispersion problem is NP-hard in the general case [27]. In two specific cases, 2d or having triangle inequality, it exists a $2$-approximation for the Max-Min p-dispersion problem [27]. For all $\alpha<2$, any $\alpha$-approximation for the Max-Min p-dispersion problem with triangle inequality is NP-hard [27]. Although, p-dispersion is mentioned to have relevant applications for MOO [13, 27], no specific studies concerned the p-dispersion in a PF to the best of our knowledge. We note that an affine 2d PF is a line in ${\mathbb{R}}^{2}$, such case is equivalent to the 1d case. Hence, Max-Min and Max-Sum p-dispersion problems are solvable in $O(\max\\{pn,n\log n\\})$ time thanks to [27, 33]. General planar cases of p-dispersion problems can also be seen as specific cases of three-dimensional (3d) PF, affine 3d PF. Having a NP-hard complexity proven for the planar cases of Max-Min p-dispersion, it implies that the Max- Min p-dispersion problem is also NP-hard for 3d PF. ### 3.2 Exact methods to solve p-dispersion problems The Max-Sum p-dispersion problem can be formulated as a quadratic optimization problem, defining binary variables $z_{j}\in\\{0,1\\}$ with $z_{j}=1$ if and only if the point $x_{j}$ is selected: $\begin{array}[]{llrllr}{\mathcal{P}}_{disp}^{MS}(E,p)&=&\max\displaystyle\sum_{i=1}^{n}\sum_{j=1}^{n}d_{i,j}&z_{i}z_{j}&&(\ref{mathProgMaxSumDispersion}.1)\\\ &s.t:&\sum_{j=1}^{n}z_{j}&=p&&(\ref{mathProgMaxSumDispersion}.2)\\\ &&z_{j}&\in\\{0,1\\}\hskip 19.91684pt&\forall j\in[\\![1,n]\\!],&(\ref{mathProgMaxSumDispersion}.3)\end{array}$ (11) The linearization of (11.3) leads to the Integer Linear Programming (ILP) formulation provided by [20]. Mathematical programming formulations are also used in exact Branch&Bound (B& B) algorithms iterating with the computations of upper and lower bounds, with a Lagrangian relaxation in [1] or with upper bounds computable in $O(n^{3}$) in [26]. Similarly, the Max-Min p-dispersion problem can be modeled using non linear optimlization ([26]): $\begin{array}[]{llrllr}{\mathcal{P}}_{disp}^{MM}(E,p)&=\displaystyle\max_{d\geqslant 0}&d&&&(\ref{mathProgDispersion}.1)\\\ &s.t:&\sum_{j=1}^{n}z_{j}&=p&&(\ref{mathProgDispersion}.2)\\\ &&dz_{i}z_{j}&\geqslant d_{i,j}&\forall 1\leqslant i<j\leqslant n,&(\ref{mathProgDispersion}.3)\\\ &&z_{j}&\in\\{0,1\\}\hskip 19.91684pt&\forall j\in[\\![1,n]\\!],&(\ref{mathProgDispersion}.4)\end{array}$ (12) The standard linearization of constraints (12.3) leads to the Mixed Integer Linear Programming (MILP) formulation provided by [20]. Another MILP formulation was proposed, to speed-up the resolution using straightforward B& B solving and implementing specific cuts in a Branch&Cut algorithm [28]. Decomposition schemes from [1] and [26]. can also be extended for the Max-Min p-dispersion problem. Similarly, MILP formulations were designed for the Max-Sum-Min and Max-Min-Sum p-dispersion variants [13]. Such variants were less studied, a recent work proposed a unified MILP formulation and B&B algorithm for the four variants of p-dispersion problems [22]. ### 3.3 Clustering/selecting points in Pareto frontiers We summarize here results related to the selection or the clustering of points in PF, with applications to MOO algorithms. Maximizing the quality of discrete representations of Pareto sets was studied with the hypervolume measure in the Hypervolume Subset Selection (HSS) problem [2, 29]. The HSS problem, maximizing the representativity of $k$ solutions among a PF of size $n$ initial ones, is known to be NP-hard in dimension 3 (and greater dimensions) since [3]. An exact algorithm in $n^{O(\sqrt{k})}$ and a polynomial-time approximation scheme for any constant dimension $d$ are also provided in [3]. The 2d case is solvable in polynomial time thanks to a DP algorithm with a complexity in $O(kn^{2})$ time and $O(kn)$ space provided in [2]. The time complexity of the DP algorithm was improved in $O(kn+n\log n)$ by [4] and in $O(k(n-k)+n\log n)$ by [21]. Some similar results were also proven for clustering problems. k-median and k-medoid problems are known to be NP hard in dimension 2 since [24]. The specific case of 2d PF were proven to be solvable in $O(n^{3})$ time with DP algorithms [9, 7]. The 1d case are solvable in $O(nk)$ time [17]. K-means, one of the most famous unsupervised learning problem, is also NP-hard for 2d cases [23]. The restriction to 2d PF would be also solvable in $O(n^{3})$ time with a DP algorithm if a conjecture is proven [6]. The 1d case of k-means is also solvable by a DP algorithm, with a complexity in $O(kn)$ using memory space in $O(n)$ [15]. For k-means, k-medoids, k-median, significant differences in the complexities exist between the 2d PF and 1d cases. Lastly, p-center problems present also similar results. The discrete and continuous p-center problems are NP-hard in general, the discrete p-center problem in ${\mathbb{R}}^{2}$ with a Euclidian distance is also NP-hard [24]. The 1d case of continuous p-center is solvable in $O(pn\log n)$ time and $O(n)$ space [18], which is the same complexity for continuous p-center problems in a 2d PF [8]. The discrete p-center problem in a 2d PF is solvable $O(pn\log^{2}n)$ time and $O(n)$ space [8]. For p-center problems, the complexity for 2d PF problems is similar to the one from 1d cases, i.e. affine 2d PF. ## 4 Intermediate results This section presents intermediate results required for the following developments. Firstly, Lemma 1 extends trivially the properties of $\leqslant$ and $<$ in ${\mathbb{R}}$: ###### Lemma 1 $\preccurlyeq$ is an order relation, and $\prec$ is a transitive relation: $\forall x,y,z\in{\mathbb{R}}^{2},\phantom{3}x\prec y\phantom{1}\mbox{and}\phantom{1}y\prec z\Longrightarrow x\prec z$ (13) Lemma 1 implies an order among the points of $E$, for a reindexation in $O(n.\log n)$ time: ###### Proposition 1 (Total order) Points $(x_{i})$ can be indexed such that: $\displaystyle\forall(i_{1},i_{2})\in[\\![1;n]\\!]^{2},$ $\displaystyle i_{1}<i_{2}\Longrightarrow$ $\displaystyle x_{i_{1}}\prec x_{i_{2}}$ (14) $\displaystyle\forall(i_{1},i_{2})\in[\\![1;n]\\!]^{2},$ $\displaystyle i_{1}\leqslant i_{2}\Longrightarrow$ $\displaystyle x_{i_{1}}\preccurlyeq x_{i_{2}}$ (15) This property is stronger than the property that $\preccurlyeq$ induces a total order in $E$. Furthermore, the complexity of the sorting reindexation is in $O(n\log n)$ Proof: We reindex $E$ such that the first coordinate is increasing: $\forall(i_{1},i_{2})\in[\\![1;n]\\!]^{2},i_{1}<i_{2}\Longrightarrow x_{i_{1}}^{1}<x_{i_{2}}^{1}$ This sort has a time complexity in $O(n\log n)$. Let $(i_{1},i_{2})\in[\\![1;n]\\!]^{2}$, with $i_{1}<i_{2}$. We have thus $x_{i_{1}}^{1}<x_{i_{2}}^{1}$. Having $x_{i_{1}}\mathcal{I}x_{i_{2}}$ implies $x_{i_{1}}^{2}>x_{i_{2}}^{2}$. $x_{i_{1}}^{1}<x_{i_{2}}^{1}$ and $x_{i_{1}}^{2}>x_{i_{2}}^{2}$ is by definition $x_{i_{1}}\prec x_{i_{2}}$. $\square$ $\emph{Obj}_{1}$$\emph{Obj}_{2}$$x_{1}$$\bullet$$x_{2}$$\bullet$$x_{3}$$\bullet$$x_{4}$$\bullet$$x_{5}$$\bullet$$x_{6}$$\bullet$$x_{7}$$\bullet$$x_{8}$$\bullet$$x_{9}$$\bullet$$x_{10}$$\bullet$$x_{11}$$\bullet$$x_{12}$$\bullet$$x_{13}$$\bullet$$x_{14}$$\bullet$$x_{15}$$\bullet$ Figure 1: Illustration of a 2d PF with 15 points and the indexation implied by Proposition 1 The re-indexation of Proposition 1 implies a monotony for the distances in the 2d PF: ###### Proposition 2 We suppose that points $(x_{i})$ are sorted following Proposition 1. $\displaystyle\forall(i_{1},i_{2},i_{3})\in[\\![1;n]\\!]^{3},\;\;\;$ $\displaystyle i_{1}\leqslant i_{2}<i_{3}\Longrightarrow d_{i_{1},i_{2}}<d_{i_{1},i_{3}}$ (16) $\displaystyle\forall(i_{1},i_{2},i_{3})\in[\\![1;n]\\!]^{3},\;\;\;$ $\displaystyle i_{1}<i_{2}\leqslant i_{3}\Longrightarrow d_{i_{2},i_{3}}<d_{i_{1},i_{3}}$ (17) Proof: We index $E$ following Proposition 1. Let’s prove it implies (16), (17) is similar with (16). Let $i_{1}<i_{2}<i_{3}$. We note $x_{i_{1}}=(x^{1}_{i_{1}},x^{2}_{i_{1}})$, $x_{i_{2}}=(x^{1}_{i_{2}},x^{2}_{i_{2}})$ and $x_{i_{3}}=(x^{1}_{i_{3}},x^{2}_{i_{3}})$ . (14) implies $x^{1}_{i_{1}}<x^{1}_{i_{2}}<x^{1}_{i_{3}}$ and $x^{2}_{i_{1}}>x^{2}_{i_{2}}>x^{2}_{i_{3}}$. Hence, $(x^{1}_{i_{1}}-x^{1}_{i_{2}})^{2}<(x^{1}_{i_{1}}-x^{1}_{i_{3}})^{2}$ and $(x^{2}_{i_{1}}-x^{2}_{i_{2}})^{2}<(x^{2}_{i_{1}}-x^{2}_{i_{3}})^{2}$. $d(x_{i_{1}},x_{i_{2}})^{2}={(x^{1}_{i_{1}}-x^{1}_{i_{2}})^{2}+(x^{2}_{i_{1}}-x^{2}_{i_{2}})^{2}}<d(x_{i_{1}},x_{i_{3}})^{2}$. Having $\alpha>0$, it implies $d(x_{i_{1}},x_{i_{2}})^{\alpha}<d(x_{i_{1}},x_{i_{3}})^{\alpha}$ which proves (16). $\hfill\square$ Proposition 2 allows to reformulate Max-Min and Max-Sum-Min p-dispersion problems: ###### Proposition 3 Let $E=\\{z_{1},\dots,z_{n}\\}$ a subset of $n$ points of ${\mathbb{R}}^{2}$, such that for all $i\neq j$, $x_{i}\prec x_{j}$. The Max-Min and Max-Sum-Min p-dispersion problems in $E$ are also defined with: ${\mathcal{P}}_{disp}^{MM}(E,p)=\max_{1\leqslant i_{1}<\dots<i_{p}\leqslant p}\phantom{2}\min_{j\in[\\![1;p-1]\\!]}d_{i_{j},i_{j+1}}$ (18) ${\mathcal{P}}_{disp}^{MSM}(E,p)=\max_{1\leqslant i_{1}<\dots<i_{p}\leqslant p}\phantom{2}\sum_{i=1}^{p}\min(d_{i_{j},i_{j+1}},d_{i_{j},i_{j-1}})$ (19) Proof: In the last minimization of problems (7) and (9), distances $d_{i_{j},i_{j^{\prime}}}$ are considered. Such distances are minimized by $d_{i_{j},i_{j+1}}$ and $d_{i_{j^{\prime}},i_{j^{\prime}+1}}$ using Proposition 2, and it remains only distances among consecutive points. $\hfill\square$ Furthermore, Proposition 2 allows to define a new variant of the Max-Sum-Min p-dispersion problem, denoted as the Max-Sum-Neighbor p-dispersion problem or p-dispersion-MSN: ###### Definition 1 Let $E=\\{z_{1},\dots,z_{n}\\}$ a subset of $n$ points of ${\mathbb{R}}^{2}$, such that for all $i\neq j$, $x_{i}\prec x_{j}$. The Max-Sum-Neighbor p-dispersion, also denoted p-dispersion-MSN, can be defined in the 2d PF $E$ summing only the distances between neighbor points: ${\mathcal{P}}_{disp}^{MSN}(E,p)=\max_{1\leqslant i_{1}<i_{2}<\dots<i_{p}\leqslant p}\sum_{i=1}^{p-1}d_{i_{j},i_{j+1}}$ (20) The dispersion definitions after the Proposition 3 are illustrated in Table 1. With the order of Proposition 1 and the monotony of Proposition 2, p-dispersion-MSM is not a symmetric expression of the distances, inducing more importance to extreme distances AB and CD. On the contrary, p-dispersion-MSN induces symmetrical expressions. This motivated the introduction of p-dispersion-MSN in the specific case of a 2d PF. Table 1: Illustration of the Proposition 3 and the difference of the dispersion functions: we consider the 4-dispersion problems, with four selected points, A,B,C and D such that $A\prec B\prec C\prec D$. Dispersion type | | Dispersion of A,B,C,D ---|---|--- p-dispersion-MM | : | $\min$(AB,BC,CD) p-dispersion-MS | : | AB+AC+AD+BC+BD+CD p-dispersion-MSN | : | AB+BC+CD p-dispersion-MSM | : | AB+min(AB,BC) +min(BC,CD) +CD p-dispersion-MMS | : | min(AB+AC+AD,AB+BC+BD,AD+BD+CD) Propositions 1 and 2 allow to get first polynomial results in Propositions 4, 6 and 7 present some polynomial complexity results. A key element is that the extreme points of the 2d PF are natural candidates for p-dipersion problems, as analyzed in Propositions 4 and 5: ###### Proposition 4 (2-dispersion problems) Let $E=\\{z_{1},\dots,z_{n}\\}$ a subset of $n$ points of ${\mathbb{R}}^{2}$, such that for all $i\neq j$, $z_{i}{\mathcal{I}}z_{j}$. 2-dispersion problems are solvable in $O(n)$ time using $O(1)$ additional memory space in the 2d PF $E$, considering variants (7), (8), (9), (10) or (20). There is a unique optimal solution, selecting $z_{1}$ and $z_{n}$. Proof: Considering variants (7), (8), (9), (10) or (20), the same problem must be solved: ${\mathcal{P}}_{2}(E)=\max_{1\leqslant i<j\leqslant p}d(z_{i},z_{j})$ (21) Indeed, ${\mathcal{P}}_{disp}^{MM}(E,2)={\mathcal{P}}_{disp}^{MS}(E,2)={\mathcal{P}}_{disp}^{MMS}(E,2)={\mathcal{P}}_{disp}^{MSN}(E,2)={\mathcal{P}}_{2}(E)$ and ${\mathcal{P}}_{disp}^{MSM}(E,2)=2{\mathcal{P}}_{2}(E)$. Using Proposition 2, it is trivial that there is a unique optimal solution to (21), selecting the two extreme points $z_{1}$ and $z_{n}$ after the re-indexation of proposition 1, with a optimal cost $d_{1,n}$. The complexity, once having computed $z_{1}$ and $z_{n}$ is in $O(1)$ time and additional space. Re- indexing the whole PF would induce a complexity in $O(n\log n)$ time. Actually, there is no need for a full re-indexation, computing the extreme points can be done in one traversal of $E$, inducing a complexity in $O(n)$ time. $\hfill\square$ ###### Proposition 5 (p-dispersion and extreme points) Considering variants (7), (8), (9), (10) or (20), an optimal solution of p-dispersion can be considered selecting $x_{1}$ and $x_{n}$ for $p\geqslant 2$ once points $(x_{i})$ are re-indexed as in Proposition 1. Reciprocally, any optimal solution contains the extreme points in the case of the Max-Sum and Max-Sum-Neighbor variants, contrary to the Max-Min, Max-Sum-Min and Max-Min- Sum variants where counter-examples exist. Proof: Considering variants (7), (8), (9), (10) or (20), let $1\leqslant i_{1}<i_{2}<\dots<i_{p}\leqslant p$ the indexes defining an optimal solution. Considering new indexes $i_{1}^{\prime}=1,i_{2}^{\prime}=i_{2},\dots i_{p-1}^{\prime}=i_{p-1},i_{p}^{\prime}=p$, we have thanks to Proposition 2 $d(i_{j_{1}},i_{j_{2}})^{\alpha}\leqslant d(i_{j_{1}}^{\prime},i_{j_{2}}^{\prime})^{\alpha}$. It implies that the points defined with new indexes have at least the same dispersion than the original points. The new points define necessarily an optimal solution of the considered variant of p-dispersion. In the case of Max-Sum and Max-Sum-Neighbor p-dispersion problems, having an optimal solution with $1<i_{1}$ or $i_{p}<p$ induces that we have furthermore $d(i_{1},i_{2})^{\alpha}+d(i_{p-1},i_{p})^{\alpha}<d(i_{1}^{\prime},i_{j_{2}}^{\prime})^{\alpha}+d(i_{p-1}^{\prime},i_{p}^{\prime})^{\alpha}$, and the new construction of points with indexes $i^{\prime}$ induces a strictly better solution. So, an optimal solution of Max-Sum and Max-Sum- Neighbor p-dispersion problems contains necessarily the extreme points. Lastly, a counter example of the reciprocity with Max-Min 3-dispersion and 4 points in a 2d PF is given with: $z_{1}=(0,10)$, $z_{2}=(1,9)$, $z_{3}=(3,7)$, $z_{4}=(5,5)$.$\hfill\square$ Remark: This property of p-dispersion is relevant in the selection of points of a PF for human decision makers. Indeed, it is natural to present the extreme points to evaluate the preferences of the decision maker. This is a justification to use p-dispersion in that goal instead of clustering measure like k-medoids, which would not furnish the extreme points [7]. Proposition 5 is useful to determine the complexity of $3$-dispersion problems, and also improve the general complexity enumerating all the possibilities, that would be in $O(n^{p})$. ###### Proposition 6 (3-dispersion) $3$-dispersion problems in a 2d PF are solvable in $O(n)$ time using $O(1)$ additional space. Proof:Considering variants (7), (8), (9), (10) or (20), we consider the two extreme points, which can be found in $O(n)$ time with one traversal of $E$ like in Proposition 4. Then, there are $n-2$ cases to enumerate the last point, each cost computation of 3-dispersion being in $O(1)$ time, this last naive enumeration is in $O(n)$ time. $3$-dispersion problems are thus solved in $O(n)$ time using $O(1)$ additional space with two traversals of $E$.$\hfill\square$ ###### Proposition 7 (p-dispersion) The $p$-dispersion problems in a 2d PF are solvable in $O(n^{p-2})$ time using $O(1)$ additional space. Proof: Similarly with Proposition 6, once the extreme points are found in $O(n)$ time, the naive enumeration of the other $p-2$ selected points induces ${n-2}\choose{p-2}$ computations, requiring $O(p)$ or $O(p^{2})$ time computations.$\hfill\square$ ## 5 p-dispersion-MSN is polynomially solvable Proposition 2 allows to prove Bellman equations for the p-dispersion-MSN problem in a 2d PF, which is the key ingredient to design a DP algorithm: ###### Proposition 8 (Bellman equations for p-dispersion-MSN) Defining $C_{k,i}^{MSN}$ as the optimal cost of $k$-dispersion-MSN among the points indexed in $[\\![1,i]\\!]$ for all $k\in[\\![2,p]\\!]$ and $i\in[\\![k,n]\\!]$, we have: $\forall i\in[\\![1,n]\\!],\>\>\>C^{MSN}_{2,i}=d_{1,i}$ (22) $\forall i\in[\\![1,n]\\!],\>\forall k\in[\\![3,p]\\!],\>\>\>C^{MSN}_{k,i}=\max_{j\in[\\![k-1,i-1]\\!]}(C^{MSN}_{k-1,j}+d_{j,i})$ (23) Proof: (22) is given by Proposition 4. We suppose $k\geqslant 3$ and prove (23). Let $i\in[\\![k,n]\\!]$. Selecting for each $j\in[\\![k-1,i-1]\\!]$ an optimal solution of $(k-1)$-dispersion-MSN among points indexed in $[\\![1,j]\\!]$, and adding point $i$, it makes a feasible solution for $(k-1)$-dispersion-MSN among points indexed in $[\\![1,i]\\!]$ with a cost $C^{MSN}_{k-1,j}+d_{j,i}$. This last cost is lower than the optimal $k$ dispersion cost, thus $C^{MSN}_{k,i}\geqslant C^{MSN}_{k-1,j}+d_{j,i}$. $C^{MSN}_{k,i}\geqslant\max_{j\in[\\![k-1,i-1]\\!]}(C^{MSN}_{k-1,j}+d_{j,i})$ (24) Let $j_{1},j_{2},\dots,j_{k-1},j_{k}$ indexes such that $1=j_{1}<j_{2}<\dots<j_{k-1}<j_{k}=i$ defines an optimal solution of $k$-dispersion-MSN, its cost is $C^{MSN}_{k,i}$. Necessarily, $j_{1},j_{2},\dots,j_{k-1}$ defines an optimal solution of $(k-1)$-dispersion- MSN among points indexed in $[\\![1,j_{k-1}]\\!]$. On the contrary, a strictly better solution for $C^{MSN}_{k,i}$ would be constructed adding the index $i$. We have thus: $C^{MSN}_{k,i}=C^{MSN}_{k-1,j_{k-1}}+d_{j_{k-1},i}$. Combined with (24), it proves : $C^{MSN}_{k,i}=\max_{j\in[\\![k-1,i-1]\\!]}(C^{MSN}_{k-1,j}+d_{j,i})$. $\hfill\square$ It allows to design a DP algorithm for p-dispersion-MSN in Algorithm 1. The first phase constructs by induction the matrix of optimal costs $C^{MSN}_{k,i}$ with $k$ increasing. $C^{MSN}_{p,n}$ is the optimal value of MSN p-dispersion. Then, backtracking operations in the matrix $C^{MSN}_{k,i}$ return an optimal solution. Algorithm 1: p-dispersion-MSN in a 2d PF with $p>3$ --- Input: \- $n$ points of ${\mathbb{R}}^{2}$, $E=\\{x_{1},\dots,x_{n}\\}$ such that for all $i_{1}<i_{2}$, $x_{i_{1}}\phantom{0}\mathcal{I}\phantom{0}x_{i_{2}}$ ; \- an integer $p$ with $3<p\leqslant N$. initialize matrix $C$ with $C_{k,i}:=0$ for all $i\in[\\![1;N]\\!],k\in[\\![2;p-1]\\!]$ for $i=1$ to $N-1$: $C_{2,i}:=d_{1,i}$ end for for $k=3$ to $p-1$ : for $i=1$ to $N-1$: $C_{k,i}:=\max_{j\in[\\![k-1,i-1]\\!]}(C_{k-1,j}+d_{j,i})$ end for end for $OPT:=\max_{j\in[\\![p-1,N-1]\\!]}(C_{p-1,j}+d_{j,N})$ $j:=\mbox{argmax}_{j\in[\\![p-1,N-1]\\!]}(C_{p-1,j}+d_{j,N})$ initialize $i:=j$ and ${\mathcal{I}}:=\\{1,j,N\\}$. for $k=p-1$ to $3$ with increment $k\leftarrow k-1$: $j:=\mbox{argmax}_{j\in[\\![k-1,i-1]\\!]}(C_{p-1,j}+d_{j,i})$ add $j$ in ${\mathcal{I}}$ $i:=j$ end for return $OPT$ the optimal cost and the set of selected indexes ${\mathcal{I}}$. ###### Theorem 1 Let $E=\\{x_{1},\dots,x_{n}\\}$ a subset of $n$ points of ${\mathbb{R}}^{2}$, such that for all $i\neq j$, $x_{i}\phantom{0}\mathcal{I}\phantom{0}x_{j}$. Max-Sum-Neighbor p-dispersion is solvable in polynomial time in the 2d PF $E$. The cases $p=2,3$ are solvable with a complexity in $O(n)$ time using an additional memory space in $O(1)$. Algorithm 1 solves the cases $p>3$ with a complexity in $O(pn^{2})$ time and $O(pn)$ memory space. Proof: The cases $p=2,3$ are given by Propositions 4, and 6, so that we suppose $p\geqslant 4$ and we consider the Algorithm 1. The induction formula (23) uses only values $C_{i,j}$ with $j<k$. Hence, $C_{k,n}$ is at the end of each loop in $k$ the optimal value of the $k$-dispersion-MSN among the $n$ points of $E$, and the optimal cost. The remaining operations after the computation of $C_{p,n}$ is a standard backtrack algorithm to return an optimal solution. This proves the validity of Algorithm 1 to solve optimally $p$-dispersion-MSN. Let us analyze the complexity of Algorithm 1. Re-indexing $E$ following Proposition 1 has a time complexity in $O(n\log n)$. Computing the line $k=2$ of the DP matrix has also a time complexity in $O(n)$. Computing $\max_{j\in[\\![k-1,i-1]\\!]}(C_{k-1,j}+d_{j,i})$ is in $O(i-k)$ and thus in $O(n)$ enumerating all the $i-k$ possibilities. It induces time complexities in $O(pn^{2})$ for the construction of the DP matrix, and in $O(pn)$ for the backtracking operations. Finally, the time complexity is given by the construction of the DP matrix, in $O(pn^{2})$ time. The space complexity is in $O(pn)$, storing the DP matrix $C$, the complexity is proven. $\hfill\square$ Remark: In Algorithm 1, the DP matrix $C$ is computed line by line, with the index $k$ increasing. The computation of line $k+1$ requires only the line $k$ and computations of distances using $O(1)$ additional memory space. If one wishes to compute only the optimal value $C_{p,n}$, it is possible to delete the line $k-1$ once the line $k$ is completed. Such implementation has at most $2n$ elements in the memory, and thus a complexity in $O(n)$ memory space. Theorem 1 states a complexity in $O(pn)$ memory space as the backtracking operations to compute an optimal solution, as written in Algorithm 1, use the whole DP matrix. ## 6 Max-Min p-dispersion is polynomially solvable Proposition 2 allows also to prove Bellman equations for the Max-Min p-dispersion problem in a 2d PF: ###### Proposition 9 (Bellman equations) Let $E=\\{x_{1},\dots,x_{n}\\}$ a subset of $n$ points of ${\mathbb{R}}^{2}$, such that for all $i\neq j$, $x_{i}\prec x_{j}$. Let $E=\\{x_{1},\dots,x_{n}\\}$ a subset of $n$ points of ${\mathbb{R}}^{2}$, such that for all $i\neq j$, $x_{i}\prec x_{j}$. Defining $C^{MM}_{k,i}$ as the optimal cost of Max-Min $k$-dispersion among the points indexed in $[\\![1,i]\\!]$ for all $k\in[\\![2,p]\\!]$ and $i\in[\\![k,n]\\!]$ , we have the following relations: $\forall i\in[\\![1,n]\\!],\>\>\>C_{2,i}^{MM}=d_{1,i}$ (25) $\forall k\in[\\![3,p]\\!],\>\forall i\in[\\![k,n]\\!],\>\>\>C_{k,i}^{MM}=\max_{j\in[\\![k-1,i-1]\\!]}\min(C_{k-1,j}^{MM},d_{j,i})$ (26) Proof: The proof is similar with the one of Proposition 8. $\hfill\square$ Remark: Similarly with Algorithm 1, equations (26) allows to design a DP algorithm with a complexity in $O(pn^{2})$ time and $O(pn)$ space. Following developments improve this complexity. Algorithm 2: Computation of $\mbox{max}_{j\in[\\![k-1,i-1]\\!]}\min(C_{k-1,j}^{MM},d_{j,i})$ --- input: indexes $k<i$ `` `` define $a:=k-1$, $b:=i-1$ `` while $b-a\geqslant 2$ `` Compute $j=\left\lfloor\frac{a+b}{2}\right\rfloor$ `` if $C_{k-1,j}^{MM}-d_{j,i}>0$ ` `then :` ` $b:=j$ `` else ` `:` ` $a:=j$ `` end while `` `` return $\max(\min(C_{k-1,a}^{MM},d_{a,i}),\min(C_{k-1,b}^{MM},d_{b,i}))$ ###### Proposition 10 Let $k\in[\\![3,p]\\!]$ and $i\in[\\![k,n]\\!]$. Algorithm 2 computes $C_{k,i}^{MM}=\max_{j\in[\\![k-1,i-1]\\!]}\min(C_{k-1,j}^{MM},d_{j,i})$ with a time complexity in $O(\log(i+1-k))$ once the $C_{j-1,k-1}$ are computed. Proof: Let $k\in[\\![3,p]\\!]$ and $i\in[\\![k,n]\\!]$. Reformulating Proposition 2, the application $j\in[\\![k-1,i-1]\\!]\mapsto d_{j,i}$ is strictly decreasing. The application $j\in[\\![k-1,i-1]\\!]\mapsto C_{k-1,j}^{MM}$ is increasing: any feasible solution of $(k-1)$-dispersion-MM among the $j$ first points, is a feasible solution for $(k-1)$-dispersion-MM considering the $j+1$ first points, and the optimal value $C_{k-1,j}^{MM}$ is increasing. Hence, $\varphi_{i,k}:j\in[\\![k-1,i-1]\\!]\mapsto C_{k-1,j}^{MM}-d_{j,i}$ is strictly increasing. Let $\psi_{i,k}:j\in[\\![k-1,i-1]\\!]\mapsto\min(C_{k-1,j}^{MM},d_{j,i})$ . $\varphi_{i,k}(i-1)=C_{i,k-1}^{MM}>0$. Let $\alpha=\min\\{j\in[\\![k-1,i-1]\\!],\varphi_{i,k}(j)\geqslant 0\\}$. For $j\geqslant\alpha$, $\psi_{i,k}(j)=C_{k-1,j}^{MM}$, and $\psi_{i,k}$ is increasing for $j\geqslant\alpha$. For $j<\alpha$, $\psi_{i,k}(j)=C_{k-1,j}^{MM}$, and $\psi$ is strictly decreasing for $j<\alpha$. Hence, $\psi_{i,k}$ reaches a minimum for $j=\alpha$ or $j=\alpha-1$. The computation of $\alpha$, as the minimal value such that $\varphi_{i,k}(j)\geqslant 0$, can be solved with a dichotomic search presented in Algorithm 2, for a time complexity in $O(\log(i+1-k))$. $\hfill\square$ Proposition 10 allows to improve the time complexity of the DP algorithm for Max-Min p-dispersion problem. The following developments improve the space complexity. Similarly with the remark page 1 for p-dispersion-MSN, the final cost $C_{p,n}^{MM}$ can be constructed using a memory space in $O(n)$, deleting the line $k$ of the DP matrix when the line $k+1$ is fully computed. The point here is to provide backtracking algorithms which do not require to have stored the whole DP matrix $C_{k,i}^{MM}$, with a complexity in at most $O(n)$ memory space and $O(pn\log n)$ time. Algorithm 3 and 3’ compute optimal solutions of p-dispersion-MM knowing the optimal cost OPT, with greedy strategies: Algorithm 3: Backtracking algorithm using $O(n)$ memory space --- input: ``\- $n$ points of a 2d PF, $E=\\{z_{1},\dots,z_{n}\\}$, sorted such that for all $i<j$, $z_{i}\prec z_{j}$ ; ``\- $p\in{\mathbb{N}},p>2$; ``\- $OPT$, the optimal cost of Max-Min $p$-dispersion; `` `` initialize $M:=1$, $m:=1$, ${\mathcal{I}}=\\{1,n\\}$. `` for $k=2$ to $p-1$ with increment $k\leftarrow k+1$ `` $M$ := the smallest index such that $d(x_{m},x_{M})\geqslant OPT$ `` add $M$ in ${\mathcal{I}}$ `` $m:=M$ `` end for return ${\mathcal{I}}$ Algorithm 3’: Backtracking algorithm using $O(n)$ memory space --- input: ``\- $n$ points of a 2d PF, $E=\\{z_{1},\dots,z_{n}\\}$, sorted such that for all $i<j$, $z_{i}\prec z_{j}$ ; ``\- $p\in{\mathbb{N}},p>2$; ``\- $OPT$, the optimal cost of Max-Min $p$-dispersion; `` `` initialize $M:=n$, $m:=n$, ${\mathcal{I}}=\\{1,n\\}$. `` for $k=p-1$ to $2$ with increment $k\leftarrow k-1$ `` $m$ := the biggest index such that $d(x_{m},x_{M})\geqslant OPT$ `` add $m$ in ${\mathcal{I}}$ `` $M:=m$ `` end for return ${\mathcal{I}}$ ###### Proposition 11 Let $p\in{\mathbb{N}},p\geqslant 3$. Let $E=\\{z_{1},\dots,z_{n}\\}$, sorted such that for all $i<j$, $z_{i}\phantom{0}\prec\phantom{0}z_{j}$. Once the optimal cost of Max-Min p-dispersion problem is computed, Algorithm 3 and 3’ computes an optimal partition in $O(p\log n)$ time using $O(1)$ additional memory space. Furthermore, let $j_{1}=1,j_{2},\dots,j_{p-1},j_{p}=n$ the indexes of an optimal solution, let $1,i_{2},\dots,i_{p-1},n$ (resp $1,i_{2}^{\prime},\dots,i_{p-1}^{\prime},n$) the indexes given by Algorithm 3 (resp 3’). We have: for all $k\in[\\![2,p-1]\\!],i_{k}\leqslant j_{k}\leqslant i^{\prime}_{k}$. In other words, the indexes given by Algorithm 3 (resp 3’) are lower (resp upper) bounds of the indexes of any optimal solution of Max- Min p-dispersion considering the extreme points $z_{1}$ and $z_{n}$. Proof: We prove the result for Algorithm 3, the proof is similar for Algorithm 3’. Let $j_{1}=1,j_{2},\dots,j_{p-1},j_{p}=n$ the indexes of an optimal solution, let $i_{1}=1,i_{2},\dots,i_{p-1},i_{p}=n$ the indexes given by Algorithm 3. Firstly, We prove by induction on $k$ that for all $k\in[\\![1,p-1]\\!],i_{k}\leqslant j_{k}$. The case $k=1$ is given by $j_{1}=i_{1}=1$. We suppose $k>1$ and that the induction hypothesis is true or $k-1$, i.e. $i_{k-1}\leqslant j_{k-1}$. $i_{k}$ is the smallest index such that $d(x_{i_{k}},x_{i_{k-1}})\geqslant OPT$. Using Proposition 2, $d(x_{j_{k}},x_{j_{k-1}})\leqslant d(x_{j_{k}},x_{j_{i-1}})$. $i_{k-1}>j_{k-1}$ would be in contradiction with $d(x_{j_{k}},x_{j_{k-1}})\geqslant OPT$ and the definition of $i_{k}$ as the smallest index such that $d(x_{i_{k}},x_{i_{k-1}})\geqslant OPT$. We have also $i_{k}\leqslant j_{k}$, which terminates the induction proof that indexes $i_{k}$ are lower bounds of indexes $j_{k}$. let us prove that indexes $i_{1}=1,i_{2},\dots,i_{p-1},i_{p}=n$ define an optimal solution. We have by construction $d(x_{i_{k}},x_{i_{k-1}})\geqslant OPT$ for all $k\in[\\![1,p-1]\\!]$, we have just to prove that $d(x_{n},x_{i_{p-1}})\geqslant OPT$. $i_{p-1}\leqslant j_{p-1}\leqslant n=j_{p}=i_{p}$. Using Proposition 2, $d(x_{n},x_{i_{p-1}})\geqslant d(x_{n},x_{j_{p-1}})$. The optimality implies $d(x_{n},x_{j_{p-1}})\geqslant OPT$, and thus $d(x_{n},x_{i_{p-1}})\geqslant OPT$. Algorithm 3 returns the optimal solution with the minimal indexes, let’s analyze the complexity. Algorithm 4 calls at most $p-2$ times the computation of smallest index $i_{k}$ such that $d(x_{i_{k}},x_{i_{k-1}})\geqslant OPT$, which can be proceeded with a dichotomic search. Hence, Algorithm 3 runs in $O(p\log n)$ time. $\hfill\square$ Proposition 11 allows to define in Algorithm 4 a valid DP algorithm for Max-Min p-dispersion running in $O(n)$ memory space. Theorem 2 summarizes the complexity results for Max-Min p-dispersion: Algorithm 4: Max-Min p-dispersion in a 2d-PF with $p>3$ --- Input: \- $n$ points of ${\mathbb{R}}^{2}$, $E=\\{x_{1},\dots,x_{n}\\}$ such that for all $i_{1}<i_{2}$, $x_{i_{1}}\phantom{0}\mathcal{I}\phantom{0}x_{i_{2}}$ ; \- an integer $p$ with $3<p\leqslant N$. re-index $E$ following the order of Proposition 1 initialize line $2$ of the matrix $C$ with $C_{2,i}=0$ for all $i\in[\\![1;n]\\!]$ for $i=1$ to $n-1$: $C_{2,i}:=d_{1,i}$ end for for $k=3$ to $p$ : initialize line $k$ of the matrix $C$ with $C_{k,i}=0$ for all $i\in[\\![k;n]\\!]$ for $i=1$ to $n-1$: $C_{k,i}:=\displaystyle\max_{j\in[\\![k-1,i-1]\\!]}\min(C_{k-1,j},d_{j,i})$ with Algorithm 2 end for delete line $k-1$ of the matrix $C$ end for delete line $p$ of the matrix $C$ $OPT:=C_{p,n}$ return $OPT$ and a solution of Algorithm 3 (or Algorithm 3’) ###### Theorem 2 Let $E=\\{x_{1},\dots,x_{n}\\}$ a subset of $n$ points of ${\mathbb{R}}^{2}$, such that for all $i\neq j$, $x_{i}\mathcal{I}x_{j}$. The Max-Min p-dispersion problem is polynomially solvable to optimality in the 2d PF $E$. The cases $p=2,3$ are solvable with a complexity in $O(n)$ time using an additional memory space in $O(1)$ . With $p>3$, Algorithm 4 solves the Max-Min p-dispersion problem with a complexity in $O(pn\log n)$ time and $O(n)$ memory space. Proof: The cases $p=2,3$ are given by Propositions 4 and 6, so that we suppose $p\geqslant 4$ and we consider the Algorithm 4. The induction formula (23) in Proposition 9 uses only values $C_{i,j}$ with $j<k$ in Algorithm 4. $C_{k,n}$ is at the end of each loop in $k$ the optimal value of the Max-Min $k$-dispersion among the $n$ points of $E$, and the optimal cost is given by $C_{k,n}$. The validity of the backtracking procedures is proven in Proposition 11. Let us analyze the complexity of Algorithm 4. The space complexity is in $O(n)$, storing at most two lines of the DP matrix $C$. Sorting and indexing the elements of $E$ following Proposition 1 has a time complexity in $O(n\log n)$. Computing the line $k=2$ of the DP matrix has also a time complexity in $O(n)$. The other lines are computed in $O(n\log n)$ time with Proposition 10, for a total computation of the DP matrix in $O(pn\log n)$ time. The backtracking operations run in $O(p\log n)$ time, so that the time complexity is given by the computation of the DP matrix in $O(pn\log n)$ time. $\hfill\square$ ## 7 Max-Sum-Min p dispersion is polynomially solvable Proposition 2 allows also to design Bellman equations for the p-dispersion-MSM problem in a 2d PF: ###### Proposition 12 (Bellman equations) Let $E=\\{x_{1},\dots,x_{n}\\}$ a subset of $n$ points of ${\mathbb{R}}^{2}$, such that for all $i\neq j$, $x_{i}\prec x_{j}$. Let $p\in{\mathbb{N}},p\geqslant 4$. For all $k\in[\\![3,p]\\!]$, we define $C_{k,i,i^{\prime}}^{MSM}$ as the optimal cost of Max-Sum-Min $k$-dispersion among the points indexed in $[\\![k,i^{\prime}]\\!]$ with $i<i^{\prime}$ being last selected point before $i^{\prime}$. We have the following relations: $\forall i\in[\\![1,n]\\!],\>\>\>C_{3,i,i^{\prime}}^{MSM}=d_{i,i^{\prime}}+d_{1,i^{\prime}}+\min(d_{1,i^{\prime}},d_{i^{\prime},i})$ (27) $\forall k\in[\\![4,p]\\!],\>\forall i^{\prime}\in[\\![1,n]\\!],\>\forall i<i^{\prime}\>\>\>C_{k,i,i^{\prime}}^{MSM}=\max_{j\in[\\![1,i-1]\\!]}C_{k-1,j,i}^{MSM}+\min(d_{j,i},d_{i^{\prime},i})$ (28) Proof: $C_{3,i,i^{\prime}}^{MSM}$ is not an optimization problem with Proposition 5, there is a unique solution to consider: selecting, $1,i^{\prime},i$. This makes the dispersion given in (27). Let $k\in[\\![4,p]\\!]$, let $i\in[\\![k,n-1]\\!]$, let $i^{\prime}\in[\\![i+1,n]\\!]$. Selecting for each $j\in[\\![k-1,i-1]\\!]$ an optimal solution of $(k-1)$-dispersion-MSM among points indexed in $[\\![1,i]\\!]$ with $j$ as last selected point before $i$, and adding point $i$, it makes a feasible solution for $(k-1)$-dispersion-MSM among points indexed in $[\\![1,i]\\!]$ with a cost $C_{k-1,j,i}^{MSM}+\min(d_{j,i},d_{i^{\prime},i})$. This last cost is lower than the optimal $k$ dispersion cost, thus $C_{k,i,i^{\prime}}^{MSM}\geqslant C_{k-1,j,i}^{MSM}+\min(d_{j,i},d_{i^{\prime},i})$. $C_{k,i,i^{\prime}}^{MSM}\geqslant\max_{j\in[\\![1,i-1]\\!]}C_{k-1,j,i}^{MSM}+\min(d_{j,i},d_{i^{\prime},i})$ (29) Let $j_{1},\dots,j_{k}$ indexes such that $1\leqslant j_{1}<j_{2}<\dots<j_{k-1}=i<j_{k}=i^{\prime}$ defining an optimal solution of $k$-dispersion-MSM in $[\\![1,i^{\prime}]\\!]$ with $i$ as last selected point before $i^{\prime}$, its cost is $C_{k,i,i^{\prime}}^{MSM}$. Necessarily, $j_{1},j_{2},\dots,j_{k-1}$ defines an optimal solution of $(k-1)$-dispersion- MSM among points indexed in $[\\![1,i]\\!]$ with $j_{k-2}$ as last selected point before $i$. On the contrary, a strictly better solution for $C_{k,i,i^{\prime}}^{MSM}$ would be constructed adding the index $i^{\prime}$. We have thus: $C_{k,i,i^{\prime}}^{MSM}=C_{k-1,j_{k-2},i}^{MSM}+\min(d_{j_{k-2},i},d_{i^{\prime},i})$. Combined with (29), it proves : $C^{MSN}_{k,i}=\max_{j\in[\\![k-1,i-1]\\!]}C_{k-1,j,i}^{MSM}+\min(d_{j,i},d_{i^{\prime},i})$. $\hfill\square$ Algorithm 5: Max-Sum-Min p-dispersion in a 2d-PF with $p>5$ --- Input: \- $n$ points of ${\mathbb{R}}^{2}$, $E=\\{x_{1},\dots,x_{n}\\}$ such that for all $i_{1}<i_{2}$, $x_{i_{1}}\prec x_{i_{2}}$ ; \- an integer with $5<p\leqslant N$. initialize matrix $C$ with $C_{k,i,i^{\prime}}:=0$ for all $i\in[\\![0;n]\\!],k\in[\\![2;p-1]\\!]$ re-index $E$ following the order of Proposition 1 for $i=1$ to $n-1$ for $i^{\prime}=1$ to $i-1$ $C_{3,i,i^{\prime}}^{MSM}:=d_{i,i^{\prime}}+d_{1,i^{\prime}}+\min(d_{1,i^{\prime}},d_{i^{\prime},i})$ for $k=4$ to $p-1$ $C_{k,i,i^{\prime}}:=\max_{j\in[\\![1,i-1]\\!]}C_{k-1,j,i}+\min(d_{j,i},d_{i^{\prime},i})$ end for end for $OPT:=\max_{j\in[\\![p-1,N-1]\\!]}C_{p,j,N}$ $j:=\mbox{argmax}_{j\in[\\![p-1,N-1]\\!]}C_{p,j,N}$ initialize $i^{\prime}:=N$, $i:=j$ and ${\mathcal{I}}:=\\{1,j,N\\}$. for $k=p-1$ to $3$ with increment $k\leftarrow k-1$ find $j\in[\\![1,i-1]\\!]$ such that $C_{k,i,i^{\prime}}=\max_{j\in[\\![1,i-1]\\!]}C_{k-1,j,i}+\min(d_{j,i},d_{i^{\prime},i})$ add $j$ in ${\mathcal{I}}$ $i^{\prime}:=i$ $i:=j$ end for return $OPT$ the optimal cost and the set of selected indexes ${\mathcal{I}}$. ###### Theorem 3 Let $E=\\{x_{1},\dots,x_{n}\\}$ a subset of $n$ points of ${\mathbb{R}}^{2}$, such that for all $i\neq j$, $x_{i}\mathcal{I}x_{j}$. Max-Sum-Min p-dispersion is polynomially solvable to optimality in the 2d PF $E$. The cases $p=2,3$ are solvable with a complexity in $O(n)$ time using an additional memory space in $O(1)$ . The cases $p=4$ (resp $p=5$) are solvable with a complexity in $O(n^{2})$ (resp $O(n^{3})$) time using an additional memory space in $O(1)$. In the cases $p>5$, Algorithm 5 solves Max-Sum-Min p-dispersion with a complexity in $O(pn^{3})$ time and $O(pn^{2})$ memory space. Proof: The cases $p=2,3,4,5$ are given by Propositions 4, and 6, so that we suppose $p\geqslant 6$ and we consider the Algorithm 5. The proof of the validity of Algorithm 5 to compute the optimal value and an solution for Max- Sum-Min p-dispersion is similar with Theorems 1 and 2. Algorithm 5 computes optimal values of $C_{k,i,i^{\prime}}^{MSM}$ with $k$ increasing requiring only $C_{k-1,i,i^{\prime}}^{MSM}$ values. By induction, it proves that for all $k$, $C_{k,i,i^{\prime}}^{MSM}$ has the wished optimal value at the end of the loop $k$. The optimal value of Max-Sum-Min p-dispersion in $E$ is then given by $\max_{j\in[\\![p-1,N-1]\\!]}C_{p,j,N}$. The remaining operations define a standard backtrack algorithm. This proves the validity of Algorithm 5 to return an optimal solution of $k$-dispersion-MSM. Let us analyze the complexity. The space complexity is in $O(pn^{2})$, storing the DP matrix $C$. The time complexity is given by the construction of the matrix $C_{k,i,i^{\prime}}^{MSM}$, ie $O(pn^{2})$ operations running in $O(n)$ time with naive enumerations to compute $C_{k,i,i^{\prime}}=\max_{j\in[\\![1,i-1]\\!]}C_{k-1,j,i}+\min(d_{j,i},d_{i^{\prime},i})$. Algorithm 5 has thus a time complexity in $O(pn^{3})$. $\hfill\square$ ## 8 Discussions This section discusses some practical applications of Theorems 1,2,3 and Algorithms 1,4,5. ### 8.1 Comparison of DP algorithms and complexity results Max-Min p-dispersion was proven NP-hard in 2d and solvable in $O(\max\\{pn,n\log n\\})$ time in 1d (and thus for affine cases of 2d PF) [33]. Theorem 2 illustrates that the PF hypothesis is crucial for the complexity results. Max-Min p-dispersion is polynomial in a 2d PF thanks to Proposition 1: a 2d PF can be projected in a 1d structure. The complexity in $O(pn\log n)$ time and $O(n)$ space is not very different from the 1d case using Max-Min p-dispersion, contrary to k-medoids and k-means problems [6, 7]. The difference in complexity from 2d PF to 1d cases are due to the triangle inequalities in 2d PF cases, whereas 1d cases allow to use an additivity of distances. Bellman equations are similar for p-dispersion-MSN and p-dispersion-MM, changing the inner minimization into a summation. A generic implementation of Algorithm 1 for p-dispersion-MSN and p-dispersion-MM problems is possible, with a complexity in $O(pn^{2})$ and $O(pn)$ space, which may be efficient for practical applications. The improvements presented in section 6 are only valid for p-dispersion-MM. On the contrary, the Bellman equations for p-dispersion- MSM in Proposition 12 require to store the previous indexes to compute an additive cost. The complexity in $O(pn^{3})$ time and $O(pn^{2})$ space is a limiting factor when $n$ is large. Complexity results induces to prefer the variant p-dispersion-MSN to the original p-dispersion-MSM for a practical application. This was also the preference illustrated Table 1. ### 8.2 Equivalent solutions and hierarchic p-dispersion Proposition 11 gives tight bounds on the indexes of optimal solutions of Max- Min p-dispersion in a 2d PF. Many optimal solutions may exist for p-dispersion-MM. Having an optimal solution, one can identify the bottleneck distance and rearrange other selected points without changing the Max-Min dispersion. Such situations occur when p is large enough and when the points in the 2d PF are not well distributed. We note that the solutions of Algorithms 3 and 3’ are very unbalanced, leading to the highest values or the terminating distances. For a practical application, one seeks a selection of well distributed. To solve this issue, a bi-objective hierarchic optimization can be considered, ranking the optimal solutions of Max-Min p-dispersion with the MSN-dispersion function. Bellman equations of Propositions 8 and 9 can be extended to design a DP algorithm for this hierarchic optimization. Indeed, we can define DP matrix pairs $(C_{k,i}^{{}^{\prime}MM},C_{k,i}^{{}^{\prime}MSN})$ denoting the optimal costs with the lexicographic order, optimizing firstly the Max-Min k-dispersion, and then the k-dispersion-MSN among points indexed in $[\\![1,i]\\!]$. The costs $C_{k,i}^{{}^{\prime}MM}$ are the same with the mono-objective optimization of Max-Min p-dispersion. Such DP matrices are constructed in $O(pn^{2})$, enumerating for each computation $i,k$ the possible costs with an intermediate index $j$, and sorting the best current value with the lexicographic order. The backtracking algorithm use similar computations, as in Algorithm 1, and the hierarchic optimization is also running in $O(pn^{2})$ time and $O(pn)$ space. One may also speed-up such cost computations reusing the Algorithm 2 and Proposition 10, and showing that the optimal and equivalent solutions leading to $C_{k,i}^{MM}$ are in a plateau, ending at $\alpha$ or $\alpha-1$. If one wishes to have an algorithm with a complexity in $O(pn\log n)$ time and $O(n)$ space, one may try to polish the solutions given by Algorithms 3 and 3’ to have more evenly distributed distances among the consecutive selected points. One polishing procedure is to polish locally a solution, considering 3-dispersion-MM optimizations in the consecutive points. Each 3-dispersion-MM computation is running in $O(\log n)$ using also Proposition 10 as the points are already sorted. This induces a quick and efficient polishing local search. ### 8.3 Parallelization issues The time complexity in $O(pn\log n)$ or $O(pn^{2})$ can be satisfactory for large scale computations of p-dispersion problems. For a practical speed-up, Algorithms 1,4 and 5 have also useful properties for a parallel implementation in multi or many-core environments. As mentioned in Theorems 1,2 and 3, the construction of the DP matrix is the bottleneck in the time complexity, it is crucial to parallelize carefully this phase. The backtracking algorithm is essentially sequential, but with a low complexity, so that a parallelization of this phase is not crucial. The initial sorting algorithm is significant in the computation times only for the Max-Min p-dispersion problem and small values of $p$. In this case, parallelization on General Purpose Graphical Processing Units (GPGPU) is available [31]. In the computation of each line of the DP matrix in Algorithms 1, 4 and 5, many operations are independent using the previous line. In Algorithm 1 (and for the hierarchic optimization discussed in section 8.2), there are $O(n^{2})$ independent operations. In Algorithm 5, there are $O(n^{3})$ independent operations. After the specific improvements for Max-Min p-dispersion with Algorithms 2 and 4, there are $O(n)$ independent operations to compute each value $C_{i,k}^{MM}$ for $i\in[\\![k,n]\\!]$ in $O(\log i)$ time. These $O(n)$ operations shall be computed from the highest values of $i$ with $i=n$ to the lowest with $i=k$, for a better load balancing following the results of LPT (Lowest Processing Times) heuristic and approximation results [14]. In all cases, the parallel implementation is straightforward in a shared memory environment like OpenMP, inducing only $p-3$ synchronizations, which is useful for the practical efficiency of parallelization. Lastly, we note that $p$-dispersion-MM and MSN (and the hierarchic optimization discussed in section 8.2) can be parallelized under GPGPU, with the enumerations running in $O(pn^{2})$ time. However, the dichotomic search in Algorithm 2 is not compatible with a GPGPU parallelization. For Max-Min p-dispersion, one may parallelize the $O(pn\log n)$ time version with OpenMP, or the $O(pn^{2})$ time and massively parallelized version under GPGPU. Practical efficiency becomes dependent on the characteristic of the GPU hardware. We note that the computation of the DP matrix line by line is an useful point for GPGPU parallelization, using less memory space in the GPU hardware, which may be a limiting factor. ### 8.4 Application to k-medoids/k-means clustering in a 2d PF The exact DP algorithms for k-medoids and k-means run in $O(n^{3})$ time [6, 7]. Such complexity may be a bottleneck for the practical applications dealing with large values of $n$. Classical heuristics for k-means/k-medoids problems may be used in such cases, improving in quick local search procedures an initialization of solutions [19, 30]. Such heuristics have no guarantee of optimality, and depends heavily on the initialization strategies [19, 30]. One may initialize these local searches with solutions of Max-Min p-dispersion to have a quick initialization procedure using the Algorithm 4. Actually, several initialization strategies are possible. Firstly, one can initialize the $k$ centroids using $k$-dispersion. Secondly, one can solve $2k+1$-dispersion, and select the $k$ intermediate points following order of Proposition 1. For both strategies, one can compute the optimal DP algorithms, or also use solutions build by local search iterations of 3-dispersion as discussed in section 8.2. Having many different initialization is useful for the final quality of solutions, implementing several local searches with different initializations in parallel environments (using Message Passing Interface protocols for a coarse grain parallelization), as in [10]. ## 9 Conclusion and perspectives This paper examined properties of the four p-dispersion problems in a 2d PF using Euclidian distances. A novel variant, namely Max-Sum-Neighbor p-dispersion, is also designed specifically. The cases $p=2$ and $p=3$ induce a complexity in $O(n)$ time with an additional memory space in $O(1)$. The cases $p=4$ and $p=5$ have respectively time complexities in $O(n^{2})$ and $O(n^{3})$ time with an additional memory space in $O(1)$. Such results can be useful in the application to select a small number of representative solutions for a decision maker. This offers another perspective, comparing the different variants of p-dispersion in large 2d PF with small values $p$, allowing to compare the results from [13]. Three variants are proven to be solvable in polynomial time, designing similar DP algorithms. The Max-Sum-Min p-dispersion problem is solvable in $O(pn^{3})$ time and $O(pn^{2})$ memory space. The Max-Sum-Neighbor, is proven solvable in $O(pn^{2})$ time and $O(pn)$ space. The standard Max-Min p-dispersion problem is solvable in $O(pn\log n)$ time and $O(n)$ space. Furthermore, the DP algorithms can be implemented with quasi linear speed-up with parallelization. These results offer practical perspectives for large 2d PF, as demanded by population meta-heuristics in bi-objective optimization. Heuristic speed-up and quality of solutions were also discussed for such applications. The efficiency of Max-Min p-dispersion in a 2d PF is also useful for an application to heuristics for k-means and k-medoids clustering in a 2d PF, where the DP algorithm has a complexity in $O(n^{3})$. Initialization strategies can use the work on p-dispersion problems, as discussed in this paper. Max-Min p-dispersion is NP hard in the planar case, it is thus also the case for three dimensional PF, there is no hope to extend the results of this paper in dimension 3. The perspective is only to design quick heuristics for PF in dimensions greater than 3. ## References * [1] S. Ağca, B. Eksioglu, and J. Ghosh. Lagrangian solution of maximum dispersion problems. 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2024-09-04T02:54:54.812757

2002.11832

arxiv-papers

spacing=nonfrench # On the quasicompactness of the moduli stack of logarithmic $G$-connections over a curve Andres Fernandez Herrero ###### Abstract Fix a smooth projective curve over a field of characteristic zero and a finite set of punctures. Let $G$ be a connected linear algebraic group. We prove that the moduli of $G$-bundles with logarithmic connections having fixed residue classes at the punctures is an algebraic stack of finite type. ###### Contents 1. 1 Introduction 2. 2 Preliminaries 1. 2.1 Notation 2. 2.2 Two representability lemmas 3. 2.3 Change of group for $G$-bundles 3. 3 The moduli stack of regular $G$-connections 1. 3.1 Regular $G$-connections 2. 3.2 Functoriality for regular $G$-connections 3. 3.3 The stack of regular $G$-connections 4. 4 The moduli stack of logarithmic $G$-connections 1. 4.1 Logarithmic $G$-connections and their residues 2. 4.2 Indecomposable example over elliptic curves 3. 4.3 Functoriality for logarithmic $G$-connections 4. 4.4 Stacks of logarithmic $G$-connections ## 1 Introduction Fix a curve $C$ that is smooth projective and geometrically irreducible over a field $k$ of characteristic $0$. Let $G$ be a connected linear algebraic group over $k$. If $k=\mathbb{C}$, then the curve $C$ is a Riemann surface. The Riemann-Hilbert correspondence establishes an equivalence between the groupoid of regular $G$-connections on $C$ and the groupoid of homomorphism from the topological fundamental group $\pi_{1}(C)$ into $G(\mathbb{C})$. One would like to view the moduli stack $\text{Conn}_{G}(C)$ of regular $G$-connections on $C$ as an algebraic avatar of the moduli of representations of $\pi_{1}(C)$. Motivated by this, we show that $\text{Conn}_{G}(C)$ is an algebraic stack of finite type for an arbitrary connected linear algebraic group $G$ over any field of characteristic $0$. More generally, the Riemann-Hilbert correspondence can be established for a noncompact Riemann surface. Fix a finite set $\\{x_{i}\\}$ of $k$-points $x_{i}\in C$. Set $D=\sum_{i}x_{i}$ to be the corresponding reduced divisor. In this case one would like to study representations of the fundamental group of $C\setminus D$. The corresponding objects considered in this context are logarithmic connections with poles at $D$ [Del70]. This logarithmic version of the Riemann-Hilbert correspondence gives an analytic description of the monodromy around a puncture $x_{i}$, meaning the conjugacy class of the element of $G(\mathbb{C})$ determined by a loop around $x_{i}$. Namely, the monodromy around $x_{i}$ is related to the residue $O_{i}$ of the corresponding logarithmic connection at $x_{i}$. Here $O_{i}$ is an adjoint orbit in the Lie algebra of $G$. The notion of logarithmic connection is algebraic; there is an algebraic stack $\text{Conn}_{G}^{D}(C)$ parametrizing logarithmic $G$-connections on $C$ with poles at $D$. In view of the Riemann-Hilbert correspondence, it is natural to view this stack as an algebraic avatar of representations of the fundamental group of $C\setminus D$. The main goal of this paper is to study some geometric properties of $\text{Conn}_{G}^{D}(C)$. To this end, we prove the following theorem. ###### Main Theorem (=Theorem 4.16). Let $k$ be a field of characteristic $0$. Let $G$ be a connected linear algebraic group over $k$. Let $\text{Conn}_{G}^{D,O_{i}}(C)$ denote the moduli stack of $G$-logarithmic connections with poles at $D$ and fixed residue class $O_{i}$ at each $x_{i}$. Then $\text{Conn}_{G}^{D,O_{i}}(C)$ is of finite type over $k$. The boundedness of a related moduli problem for $G=\text{GL}_{n}$ was proven by Inaba, Iwasaki and Saito [IIS06]. In that paper they assume that the logarithmic connection comes equipped with a compatible parabolic vector bundle that is semistable with respect to a fixed set of weights. If $D$ is not empty, we prove in Proposition 4.14 that $\text{Conn}_{G}^{D}(C)$ is quasicompact if and only if $G$ is unipotent. This shows that in most cases of interest it is necessary to restrict to a substack of $\text{Conn}_{G}^{D}(C)$ in order to expect a bounded moduli problem. Theorem 4.16 says that the substack $\text{Conn}^{D,O_{i}}_{G}(C)$ with fixed residues is quasicompact. In a different direction, Nitsure [Nit93][Prop. 3.1] proved that the substack of semistable connections inside $\text{Conn}_{G}^{D}(C)$ is quasicompact when $G=\text{GL}_{n}$. He uses this to construct a quasiprojective coarse moduli space for semistable logarithmic connections [Nit93][Thm. 3.5]. The first part of this paper deals with the case when the set of punctures $\\{x_{i}\\}$ is empty. Then the relevant geometric object is the stack $\text{Conn}_{G}(C)$ of regular $G$-connection. We prove Proposition 3.7, which is a special case of Theorem 4.16 above. The Harder-Narasimhan filtration for vector bundles is our main tool to show quasicompactness. The proof Proposition 3.7 is different than for the more general Theorem 4.16; it yields better bounds for the Harder-Narasimhan type of the underlying bundle. In the case when $G=\text{GL}_{n}$, Simpson also proved that $\text{Conn}_{G}(C)$ is of finite type [Sim94a][Cor. 3.4]. This was one of the main ingredients for his construction of the deRham moduli space in [Sim94a] [Sim94b]. See pages 15-16 for a comparison of our approach with Simpson’s in the case of $\text{GL}_{n}$. The second part of the paper deals with the case when the set of punctures $\\{x_{i}\\}$ is nonempty. This part is where we prove our main theorem (Theorem 4.16). We also prove an auxiliary result that could be of independent interest. This is Proposition 4.2. It gives an explicit description of the cohomological obstruction for a $G$-bundle to admit a logarithmic connection with some given set of residues. The proof of Proposition 4.2 is completely algebraic and applies to families over an arbitrary base without any assumptions on the characteristic of the ground field. It generalizes [BDP18][Prop. 3.1], which was proven for the group $\text{GL}_{n}$ over $\mathbb{C}$ using transcendental methods. Acknowledgements: I would like to thank my advisor Nicolas Templier for his help in improving the manuscript, and to Indranil Biswas and Arideep Saha for helpful comments. I acknowledge support by NSF grants DMS-1454893 and DMS-2001071. ## 2 Preliminaries ### 2.1 Notation We work over a fixed perfect ground field $k$. Some results will hold only when the characteristic of $k$ is $0$; we will explicitly mention when this is the case. Unless otherwise stated, all schemes will be understood to be schemes over $k$. An undecorated product of $k$-schemes (e.g. $X\times S$) should always be interpreted as a fiber product over $k$. We will sometimes write $X_{S}$ instead of $X\times S$. If $R$ is a $k$-algebra and $S$ is a $k$-scheme, we may use the notation $S_{R}$ to denote the fiber product $S\times\text{Spec}(R)$. If a scheme $t$ is Spec of a field, we will write $\kappa(t)$ for the corresponding coordinate field. We fix once and for all a curve $C$ that is smooth, projective and geometrically connected over $k$. We let $g$ denote the genus of $C$. We also fix an ample line bundle $\mathcal{O}_{C}(1)$ on $C$. Choose a a finite set $\\{x_{i}\\}_{i\in I}$ of $k$-points in $C$. We will denote by $q_{i}:x_{i}\rightarrow C$ the closed immersion of $x_{i}$ into $C$. Let $G$ be a smooth connected linear algebraic group over $k$. Write $\mathfrak{g}=\text{Lie}(G)$ for the Lie algebra of $G$. There is a representation $Ad:G\longrightarrow\text{GL}(\mathfrak{g})$ called the adjoint representation. For $G=\text{GL}_{n}$ it is given by matrix conjugation. G-bundles will play a prominent role in this paper. Let $X$ be a $k$-scheme. A $G$-bundle over $X$ is a scheme $\pi:\mathcal{P}\rightarrow X$ equipped with a right $G$-action. This action is required to make $\mathcal{P}$ a $G$-torsor in the étale topology. This means that for some étale cover $T\rightarrow X$ the base-change $\mathcal{P}\times_{X}T$ is $G$-equivariantly isomorphic to the trivial $G$-torsor $G_{T}$. An isomorphism between $G$-bundles is an isomorphism of $X$-schemes that intertwines the $G$-actions. We will often deal with quasicoherent sheaves on schemes. Let $X$ and $Y$ be schemes and let $\mathcal{Q}$ be quasicoherent sheaf on $Y$. If there is a clear implicit choice of morphism $f:X\rightarrow Y$, we will write $\mathcal{Q}|_{X}$ to denote the pullback $f^{*}\mathcal{Q}$ of the quasicoherent sheaf $\mathcal{Q}$ by $f$. The same convention will be used for pullbacks of $G$-bundles. ### 2.2 Two representability lemmas In this subsection we state two technical lemmas. We first fix some notation. Let $X,S$ be $k$-schemes. Suppose that we have a morphism $f:X\longrightarrow S$. ###### Definition 2.1. Let $\mathcal{E}$ be a vector bundle on $X$. Define $\Gamma_{X/S}(\mathcal{E})$ to be the functor from $S$-schemes to sets given as follows. For any $S$-scheme $g:T\longrightarrow S$, we set $\Gamma_{X/S}(\mathcal{E})\,(T)\vcentcolon=H^{0}\left(X\times_{S}T,\,(\text{id}_{X}\times_{S}g)^{*}\,\mathcal{E}\,\right)$ ###### Lemma 2.2. Let $X\longrightarrow S$ be a proper flat morphism of finite presentation. Let $\mathcal{E}$ be a vector bundle on $X$. The functor $\Gamma_{X/S}(\mathcal{E})$ is represented by a scheme that is relatively affine and of finite presentation over $S$. ###### Proof. This is a special case of [Sta19, Tag 08K6]. In this case we set $\mathcal{G}=\mathcal{O}_{X}$ and $\mathcal{F}=\mathcal{E}$. ∎ ###### Lemma 2.3. Let $f:X\longrightarrow S$ be a proper flat morphism of finite presentation. Let $\mathcal{E}$ be a vector bundle on $X$. Fix a cohomology class $\sigma\in H^{i}(X,\,\mathcal{E})$. Let $Z$ be the functor from $k$-schemes to sets defined as follows. For any $k$-scheme $T$, we let $Z(T)\vcentcolon=\;\left\\{\begin{matrix}\text{morphisms $g:T\rightarrow S$ such that}\\\ \text{$(\text{id}_{X}\times_{S}g)^{*}\sigma=0$}\end{matrix}\right\\}$ Then $Z$ is represented by a closed subscheme of $S$. ###### Proof. After passing to a Zariski cover of $S$, we can assume that $S$ is affine. Let $F^{\bullet}$ be the Grothendieck complex of $\mathcal{E}$ with respect to the morphism $X\longrightarrow S$ [Sta19, Tag 0B91]. Choose a section $\overline{v}\in F^{i}\,/\,F^{i-1}$ corresponding to the cohomology class $\sigma\in H^{i}\left(X,\,\mathcal{E}\right)$. For any $k$-scheme $T$, we have $Z(T)=\;\left\\{\begin{matrix}\text{morphisms $g:T\rightarrow S$ such that}\\\ \text{$g^{*}\overline{v}=0$}\end{matrix}\right\\}$ This functor is represented by the scheme theoretic support of the section $\overline{v}$ of the coherent $\mathcal{O}_{S}$-sheaf $F^{i}\,/\,F^{i-1}$. Hence it is a closed subscheme of $S$, as desired. ∎ ### 2.3 Change of group for $G$-bundles Let $\text{B}G$ denote the classifying stack of $G$. This is the pseudofunctor from $k$-schemes to groupoids given as follows. For any $k$-scheme $S$, $\text{B}G\,(S)\;\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid of $G$-bundles $\mathcal{P}$}\;\text{over $S$}\end{matrix}\right\\}$ Let $\text{Bun}_{G}(C)$ denote the moduli stack of $G$-bundles on $C$. This is defined by $\text{Bun}_{G}(C)\,(S)\vcentcolon=\text{B}G(C\times S)$ for any $k$-scheme $S$. ###### Proposition 2.4. ([Beh91][Prop. 4.4.4]). Suppose that $G$ is a connected reductive group over $k$. 1. (i) Let $\rho:G\longrightarrow\text{GL}_{n}$ be a faithful representation. Let $\rho_{*}:\text{Bun}_{G}(C)\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$ be the morphism induced by extension of structure groups. Then, $\rho_{*}$ is schematic, affine and of finite type. 2. (ii) $\text{Bun}_{G}(C)$ is an algebraic stack locally of finite type over $k$. ###### Proof. 1. (i) Let $S$ be a $k$-scheme. Let $\mathcal{P}:S\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$ be a $\text{GL}_{n}$-bundle over $C\times S$. Set $L\vcentcolon=\text{Bun}_{G}(C)\times_{\text{Bun}_{\text{GL}_{n}}(C)}S$. We want to show that $L$ is represented by a scheme that is relatively affine and of finite type over $S$. Matsushima’s criterion [Ric77] implies that the homogeneous space $\text{GL}_{n}\,/\,G$ is affine. Form the associated fiber bundle $X\vcentcolon=\mathcal{P}\times^{\text{GL}_{n}}\left(\text{GL}_{n\,C\times S}\,/\,G_{C\times S}\right)$. For any $S$-scheme $f:T\longrightarrow S$, we have $L(T)=\left\\{\;\text{sections of the morphism}\;X\times_{S}T\longrightarrow C\times T\;\right\\}$ The action of $\text{GL}_{n}$ on the affine scheme $\text{GL}_{n}\,/\,G$ induces a representation of $\text{GL}_{n}$ on the coordinate ring $\mathcal{O}_{\text{GL}_{n}\,/\,G}$ via translations. Since $\text{GL}_{n}\,/\,G$ is of finite type over $k$, we can choose finitely many generators for $\mathcal{O}_{\text{GL}_{n}\,/\,G}$ as a $k$-algebra. The first theorem in page 24 of [Wat79] implies that there is a finite dimensional $k$-subrepresentation $V\subset\mathcal{O}_{\text{GL}_{n}\,/\,G}$ such that $V$ contains the generators. The inclusion $V\hookrightarrow\mathcal{O}_{\text{GL}_{n}\,/\,G}$ induces a $\text{GL}_{n}$-equivariant ring homomorphism $\text{Sym}(V)\rightarrow\mathcal{O}_{\text{GL}_{n}\,/\,G}$. Since $V$ contains a set of generators of $\mathcal{O}_{\text{GL}_{n}\,/\,G}$, the homomorphism is surjective. Applying the functor $\text{Spec}(-)$ we get a $\text{GL}_{n}$-equivariant closed immersion $\text{GL}_{n}/G\hookrightarrow V^{*}$. Let $\mathcal{E}$ denote the vector bundle on $C\times S$ given by the associated bundle $\mathcal{P}\times^{\text{GL}_{n}}V^{*}_{C\times S}$. By étale descent for morphisms [Sta19, Tag 040L], the morphism $\text{GL}_{n}/G\hookrightarrow V^{*}$ induces a map $h:X\longrightarrow\mathcal{E}$. It is a closed immersion of finite presentation, because these properties can be checked étale locally [Sta19, Tag 02L6] [Sta19, Tag 02L0]. By definition, the functor $\Gamma_{C\times S/S}(\mathcal{E})$ sends a $S$-scheme $g:T\longrightarrow S$ to the set $\begin{split}\Gamma_{C\times S/S}(\mathcal{E})&\vcentcolon=\;\;\;\;\;\;H^{0}\left(C\times T,\,(\text{id}_{C}\times g)^{*}\mathcal{E}\,\right)\\\ &=\,\left\\{\;\text{sections of the morphism}\;\mathcal{E}_{C\times T}\longrightarrow C\times T\;\right\\}\end{split}$ Let $s$ be a section of the morphism $X\times_{S}T\longrightarrow C\times T$. We can compose with $h\times_{S}\,\text{id}_{T}:X\times_{S}T\longrightarrow\mathcal{E}_{C\times T}$ in order to obtain a section $(h\times_{S}\,\text{id}_{T})\circ s$ of the map $\mathcal{E}_{C\times T}\longrightarrow C\times T$. This induces a morphism of functors $\psi:L\rightarrow\Gamma_{C\times S/S}(\mathcal{E})$. By Lemma 2.2, $\Gamma_{C\times S/S}(\mathcal{E})$ is represented by a scheme that is relatively affine and of finite type over $S$. We shall prove that $\psi:L\rightarrow\Gamma_{C\times S/S}(\mathcal{E})$ is a closed immersion, which shows that $L$ is relatively affine and of finite type. Let $g:T\longrightarrow S$ be a $S$-scheme. For our purposes we can assume that $T$ is affine. Choose a morphism $T\longrightarrow\Gamma_{C\times S/S}(\mathcal{E})$ represented by a section $s$ of $\mathcal{E}_{C\times T}\longrightarrow C\times T$. We want to show that the fiber product $L\times_{\Gamma_{C\times S/S}(\mathcal{E})}T$ is represented by a closed subscheme of $T$. For any $k$-scheme $Y$, we have $\displaystyle L\times_{\Gamma_{C\times S/S}(\mathcal{E})}T\,(Y)=\left\\{\begin{matrix}\text{morphisms $j:Y\rightarrow T$ such that}\\\ \;\;\text{$(\text{id}_{C}\times j)^{*}s:C\times Y\longrightarrow\mathcal{E}_{C\times Y}$ \; \text{factors through} \; $X\times_{S}Y$}\;\;\end{matrix}\right\\}$ Define $Z$ to be the fiber product ${Z}$${C\times T}$${X\times_{S}T}$${\mathcal{E}_{C\times T}}$$\hookrightarrow$$\scriptstyle{s}$$\scriptstyle{h\times\text{id}_{T}}$ The morphism $\iota:Z\hookrightarrow C\times T$ is a closed immersion of finite presentation. For any $k$-scheme $Y$, we have $L\times_{\Gamma_{C\times S/S}(\mathcal{E})}T\,(Y)=\left\\{\begin{matrix}\text{morphisms $j:Y\rightarrow T$ such that}\\\ \;\;\text{$(\text{id}_{C}\times j)^{*}\iota:Z_{C\times Y}\hookrightarrow C\times Y$}\;\text{is an isomorphism}\end{matrix}\right\\}$ Let us denote by $\mathcal{J}$ the finitely presented sheaf of ideals corresponding to the closed immersion $\iota:Z\hookrightarrow C\times T$. Let $\varphi:\mathcal{J}\hookrightarrow\mathcal{O}_{C\times T}$ denote the inclusion. We have $\displaystyle L\times_{\Gamma_{C\times S/S}(\mathcal{E})}T\,(Y)=\left\\{\begin{matrix}\text{maps $j:Y\rightarrow T$ such that the morphism}\\\ \;\;\text{$(\text{id}_{C}\times j)^{*}\varphi:(\text{id}_{C}\times j)^{*}\mathcal{J}\longrightarrow\mathcal{O}_{C\times Y}$}\;\text{is identically $0$}\;\;\end{matrix}\right\\}$ There exists some positive integer $N$ such that the twist $\mathcal{J}(N)$ is globally generated. This means that there is a surjective morphism $\mathcal{O}^{\oplus l}_{C\times T}\twoheadrightarrow\mathcal{J}(N)$ for some positive integer $l$. Let $\sigma$ denote the composition $\sigma:\mathcal{O}^{\oplus l}_{C\times T}\xtwoheadrightarrow{\;\;\;\;\;\;}\mathcal{J}(N)\xrightarrow{\;\varphi\,\otimes\,\text{id}_{\mathcal{O}_{C\times T}(N)}\;}\mathcal{O}_{C\times T}(N)$ $\sigma$ can be viewed as a section of the vector bundle $\text{Hom}\left(\mathcal{O}_{C\times T}^{\oplus l},\,\mathcal{O}_{C\times T}(N)\right)\cong\mathcal{O}_{C\times T}^{\oplus l}(N)$. We have $L\times_{\Gamma_{C\times S/S}(\mathcal{E})}T\,(Y)=\left\\{\text{ \, maps $j:Y\rightarrow T$ such that $(\text{id}_{C}\times j)^{*}\sigma=0$ \,}\right\\}$ By Lemma 2.3, this functor is represented by a closed subscheme of $T$. 2. (ii) By part $(i)$, it suffices to show that $\text{Bun}_{\text{GL}_{n}}(C)$ is an algebraic stack that is locally of finite type over $k$. The stack $\text{Bun}_{\text{GL}_{n}}(C)$ classifies rank $n$ vector bundles on $C$. This is an algebraic stack locally of finite type over $k$, see [Neu09][Thm. 2.57], [Hei10][Example 1.14] or [LMB00][4.6.2.1]. ∎ We now return to the general case when $G$ is a smooth connected linear algebraic group. Let $U$ denote the unipotent radical of $G$. Recall that this is the biggest smooth connected normal unipotent subgroup of $G$. The quotient $G/U$ is a reductive linear algebraic group. Let $\rho_{G}:G\rightarrow G/U$ denote the quotient morphism. ###### Proposition 2.5. Suppose that the characteristic of $k$ is $0$. Let $(\rho_{G})_{*}:\text{Bun}_{G}(C)\rightarrow\text{Bun}_{G/U}(C)$ denote the morphism induced by extension of structure groups. 1. (i) For all $k$-schemes $T$ and morphisms $T\rightarrow\text{Bun}_{G/U}(C)$, the fiber product $\text{Bun}_{G}(C)\times_{\text{Bun}_{G/U}(C)}T$ is an algebraic stack of finite type over $T$. 2. (ii) $\text{Bun}_{G}(C)$ is an algebraic stack locally of finite type over $k$. 3. (iii) The morphism of algebraic stacks $(\rho_{G})_{*}$ is of finite type. 4. (iv) If the group $G$ is unipotent, then $\text{Bun}_{G}(C)$ is an algebraic stack of finite type over $k$. ###### Proof. 1. (i) We will induct on the length $l(U)$ of the nilpotent series for the unipotent group $U$. The base case is $l(U)=0$. Then $U$ is trivial and so $(\rho_{G})_{*}$ is the identity. Hence $(i)$ holds trivially for the base case. Suppose that (i) holds whenever $l(U)\leq n$. Assume that $l(U)=n+1$. Let $Z_{U}$ denote the neutral component of the center of $U$. The algebraic group $Z_{U}$ is a smooth connected normal subgroup of $G$. Let $\rho_{z}$ denote the quotient morphism $\rho_{z}:G\rightarrow G/Z_{U}$. The unipotent radical of $G/Z_{G}(U)$ is $U/Z_{G}(U)$. The morphism $(\rho_{G})_{*}$ factors as $(\rho_{G})_{*}:\text{Bun}_{G}(C)\xrightarrow{(\rho_{z})_{*}}\text{Bun}_{G/Z_{U}}(C)\xrightarrow{(\rho_{G/Z_{U}})_{*}}\text{Bun}_{G/U}(C)$ Note that $l(U/Z_{U})=n$ by construction. By the induction hypothesis, (i) holds for the morphism $(\rho_{G/Z_{U}})_{*}$. Therefore it suffices to show that the morphism $(\rho_{z})_{*}$ satisfies (i). Let $T$ be a $k$-scheme. Choose a morphism $T\rightarrow\text{Bun}_{G/Z_{U}}(C)$ represented by a $G/Z_{U}$-bundle $\mathcal{P}$ on $C_{T}$. We want to show that the fiber product $\text{Bun}_{G}(C)\times_{\text{Bun}_{G/Z_{U}}(C)}T$ is an algebraic stack of finite type over $T$. After passing to a Zariski cover of $T$, we can assume that $T$ is affine. Let $(Z_{U})^{\mathcal{P}}$ denote the unipotent commutative group scheme over $C_{T}$ obtained by twisting $Z_{U}$ by $\mathcal{P}$ via the natural conjugation action of $G/Z_{U}$ on $Z_{U}$. Since the characteristic of $k$ is $0$, the commutative unipotent group $Z_{U}$ is isomorphic to a product of additive groups $(\mathbb{G}_{a})^{r}$. Furthermore $G/Z_{U}$ acts by linear automorphisms on $Z_{U}\cong(\mathbb{G}_{a})^{r}$. By étale descent for vector bundles, it follows that the twisted group $(Z_{U})^{\mathcal{P}}$ is the total space of a vector bundle $\mathcal{V}$ on $C_{T}$. By [Hof10][Prop. 3.1], there is an obstruction in $H^{2}_{fppf}(C_{T},\mathcal{V})$ for lifting the $G/Z_{U}$-bundle $\mathcal{P}$ to a $G$-bundle. Since $\mathcal{V}$ is a vector bundle, the fppf cohomology group $H^{2}_{fppf}(C_{T},\mathcal{V})$ is the same as the usual sheaf cohomology group $H^{2}(C_{T},\mathcal{V})$ in the small Zariski site (cf. [Hof10][Remark 3.3]). Since the projective morphism $C_{T}\rightarrow T$ has relative dimension $1$ and $T$ is affine, the Leray spectral sequence implies that $H^{2}(C_{T},\mathcal{V})=0$. Now [Hof10][Prop. 3.1] shows that the fiber product $\text{Bun}_{G}(C)\times_{\text{Bun}_{G/Z_{U}}(C)}T$ is isomorphic to the stack $\text{Bun}_{(Z_{U})^{\mathcal{P}}}(C_{T})$ classifying $(Z_{U})^{\mathcal{P}}$-bundles on $C_{T}$. We are reduced to showing that $\text{Bun}_{(Z_{U})^{\mathcal{P}}}(C_{T})$ is an algebraic stack of finite type over $T$. Let $F^{\bullet}$ be the Grothendieck complex of $\mathcal{V}$ with respect to the morphism $C_{T}\longrightarrow T$ [Sta19, Tag 0B91]. Since the relative dimension of the projective morphism $C_{T}\rightarrow T$ is $1$, we can choose $F^{\bullet}$ to be a two term complex $\left[F^{0}\rightarrow F^{1}\right]$, where $F^{0}$ and $F^{1}$ are finite free modules on the affine scheme $T$. We will also denote by $F^{0}$ and $F^{1}$ the corresponding (constant) vector group schemes over $T$. Note that $F^{0}$ acts on $F^{1}$ via the differential morphism $F^{0}\rightarrow F^{1}$. A similar reasoning as in [Hof10][Remark 2.2] shows that $\text{Bun}_{(Z_{U})^{\mathcal{P}}}(C_{T})$ is isomorphic to the quotient stack $\left[F^{1}/\,F^{0}\right]$. This is an algebraic stack of finite type over $T$, as desired. 2. (ii) In order to show that $\text{Bun}_{G}(C)$ is an algebraic stack locally of finite type over $k$, we are allowed to check after passing to a smooth cover of the pseudofuctor $\text{Bun}_{G}(C)$. By Proposition 2.4[(ii)], $\text{Bun}_{G/U}(C)$ is an algebraic stack locally of finite type over $k$. This implies that $\text{Bun}_{G/U}(C)$ admits an atlas $A\rightarrow\text{Bun}_{G/U}(C)$ such that $A$ is locally of finite type over $k$. Consider the fiber product: ${A\times_{\text{Bun}_{G/U}(C)}\text{Bun}_{G}(C)}$${\text{Bun}_{G}(C)}$${A}$${\text{Bun}_{G/U}(C)}$ The morphism $A\times_{\text{Bun}_{G/U}(C)}\text{Bun}_{G}(C)\rightarrow\text{Bun}_{G}(C)$ is a smooth cover of $\text{Bun}_{G}(C)$. By part (i), $A\times_{\text{Bun}_{G/U}(C)}\text{Bun}_{G}(C)$ is an algebraic stack of finite type over $A$. Since $A$ is locally of finite type over $k$, this implies that $A\times_{\text{Bun}_{G/U}(C)}\text{Bun}_{G}(C)\rightarrow\text{Bun}_{G}(C)$ is locally of finite type over $k$, as desired. 3. (iii) This is just a restatement of (i). 4. (iv) If $G$ is unipotent, then $G/U$ is the trivial group. Therefore $\text{Bun}_{G/U}(C)=Spec(k)$, and $(\rho_{G})_{*}$ is the structure morphism of $\text{Bun}_{G}(C)$. By part (iii), it follows that $\text{Bun}_{G}(C)$ is of finite type over $k$. ∎ ## 3 The moduli stack of regular $G$-connections ### 3.1 Regular $G$-connections There is a canonical left-invariant $\mathfrak{g}$-valued $1$-form on $G$ called the Maurer-Cartan form. It is defined as follows. Left translation induces an isomorphism $T_{G}\cong\mathfrak{g}\otimes\mathcal{O}_{G}$ for the tangent sheaf $T_{G}$ of $G$. This yields a chain of isomorphisms $\Omega^{1}_{G/k}\otimes\mathfrak{g}=\text{Hom}_{\mathcal{O}_{G}}(T_{G},\mathcal{O}_{G})\otimes\mathfrak{g}\cong\text{Hom}_{k}(\mathfrak{g},\mathfrak{g})\otimes\mathcal{O}_{G}$. The invariant $\mathfrak{g}$-valued 1-form on $G$ that corresponds to $\text{id}_{\mathfrak{g}}\otimes 1$ under this isomorphism is the Maurer- Cartan form. We denote it by $\omega\in H^{0}\left(G,\,\Omega_{G/k}^{1}\otimes\mathfrak{g}\right)$. Let $S$ be a $k$-scheme. The same construction can be applied to define the relative Maurer-Cartan form $\omega_{S}\in H^{0}\left(G_{S},\,\Omega^{1}_{G\times S/S}\otimes\mathfrak{g}\right)$ for the group scheme $G_{S}$ over the base $S$. This can be used to define a homomorphism $\phi:G_{C\times S}\longrightarrow\text{Aff}\left(\Omega^{1}_{C\times S/S}\otimes\mathfrak{g}\right)$ of $C\times S$-group schemes from $G_{C\times S}$ to the group scheme of affine linear transformations of the locally free $\mathcal{O}_{C\times S}$-module $\Omega^{1}_{C\times S/S}\otimes\mathfrak{g}$. We describe this map in terms of functors of points. Given a $C\times S$-scheme $U$ and an element $g\in G_{S}(U)$, we can pull back the relative Maurer-Cartan form in order to obtain $g^{*}\omega_{S}\in H^{0}\left(U,\,\Omega_{U/S}^{1}\otimes\mathfrak{g}\right)$. We define $\phi(U)\,(g)\vcentcolon=Id_{\Omega^{1}_{U/S}}\underset{\mathcal{O}_{U}}{\otimes}Ad(g)\,+\,(g^{-1})^{*}\omega_{S}$. The fact that this is a homomorphism follows from the left-invariance of the Maurer-Cartan form. Let $\mathcal{P}$ be a $G$-bundle on $C\times S$. Suppose that $X$ is a relatively affine $C\times S$-scheme equipped with an action of $G_{C\times S}$. The group scheme $G_{C\times S}$ acts diagonally on the product $\mathcal{P}\times_{C\times S}X$. We define the associated bundle $\mathcal{P}\times^{G}X$ to be the quotient $(\mathcal{P}\times_{C\times S}X)\,/\,G_{C\times S}$. Étale descent for affine morphisms [Sta19, Tag 0245] implies that $\mathcal{P}\times^{G}X$ is represented by a scheme. One such example is the vector bundle $\mathcal{P}\times^{G}\mathfrak{g}_{C\times S}$ associated to the adjoint representation $Ad:G_{C\times S}\longrightarrow\text{GL}(\mathfrak{g}_{C\times S})$. This is called the adjoint bundle of $\mathcal{P}$, and will henceforth be denoted by $Ad\,\mathcal{P}$. We now recall the definition of $G$-connection on $\mathcal{P}$ relative to $S$. The homomorphism $\phi$ induces an action of $G_{C\times S}$ on the total space of $\Omega^{1}_{C\times S/S}\otimes\mathfrak{g}$ by affine linear transformations. A $G$-connection on $\mathcal{P}$ relative to $S$ is defined to be a section of the associated affine bundle $\mathcal{P}\times^{G}\left(\Omega^{1}_{C\times S/S}\otimes\mathfrak{g}\right)$. By construction the bundle of $G$-connections $\mathcal{P}\times^{G}\left(\Omega^{1}_{C\times S/S}\otimes\mathfrak{g}\right)$ is an étale torsor for the $C\times S$-sheaf $Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}$, which is viewed as an abelian sheaf under addition. This torsor represents a cohomology class $\gamma_{\mathcal{P}}\in H^{1}\left(C\times S,\,Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)$ called the Atiyah class of $\mathcal{P}$ relative to $S$. Here $H^{1}\left(C\times S,\,Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)$ is the ordinary sheaf cohomology group in the Zariski site. It coincides with the corresponding étale cohomology group because $Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}$ is quasicoherent [Sta19, Tag 03P2]. We can think of the Atiyah class $\gamma_{\mathcal{P}}\in H^{1}\left(C\times S,\,Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)$ as an element of $\text{Ext}^{1}\left(\mathcal{O}_{C\times S},\,\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)$. It corresponds to an extension of $\mathcal{O}_{C\times S}$-modules $0\longrightarrow Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\longrightarrow At(\mathcal{P})\longrightarrow\mathcal{O}_{C\times S}\longrightarrow 0$ We call $At(\mathcal{P})$ the Atiyah bundle. The short exact sequence above will be referred to as the Atiyah sequence. By definition a $G$-connection on the $G$-bundle $\mathcal{P}$ is the same as a $\mathcal{O}_{C\times S}$-linear splitting of the Atiyah sequence. Such splitting exists if and only if the Atiyah class $\gamma_{\mathcal{P}}$ vanishes. ###### Remark 3.1. In the literature (e.g. [Bis10]) the Atiyah sequence differs from the one we have defined. It is usually the twist of the sequence above by $\left(\Omega^{1}_{C\times S/S}\right)^{\vee}$. ###### Example 3.2. Suppose that $S=\text{Spec}\,k$ and $G=\mathbb{G}_{m}$. A $\mathbb{G}_{m}$-torsor is the same thing as a line bundle on $C$. Let $\mathcal{L}$ be such a line bundle. By definition $\gamma_{\mathcal{L}}\in H^{1}\left(C,\,Ad\,\mathcal{L}\otimes\Omega_{C/k}^{1}\right)$. Since $\mathbb{G}_{m}$ is abelian, there is a canonical identification $Ad\,\mathcal{L}\cong\mathcal{O}_{C}\otimes\mathfrak{gl}_{1}$. Hence $\gamma_{\mathcal{L}}\in H^{1}\left(C,\,\Omega_{C/k}^{1}\otimes\mathfrak{gl}_{1}\right)$. Serre duality yields an isomorphism $H^{1}\left(C,\,\Omega_{C/k}^{1}\otimes\mathfrak{gl}_{1}\right)\cong\left(H^{0}\left(C,\,\mathcal{O}_{C}\right)\otimes\mathfrak{gl}_{1}\right)^{\vee}$. Since $C$ is geometrically irreducible and proper, there is a canonical isomorphism of $k$-algebras $H^{0}\left(C,\,\mathcal{O}_{C}\right)\cong k$. Under this identification we have $\gamma_{\mathcal{L}}\in\mathfrak{gl}_{1}^{\vee}$. Finally, there is a canonical isomorphism $\mathfrak{gl}_{1}\cong k$ obtained via the natural inclusion $\mathbb{G}_{m}\hookrightarrow\mathbb{A}^{1}_{k}$. So we can view $\gamma_{\mathcal{L}}$ as an element of $k$. ###### Lemma 3.3. Let $\mathcal{L}$ be as in Example 3.2. Under the identification above, we have $\gamma_{\mathcal{L}}=-\text{deg}\,\mathcal{L}$. ###### Proof. This follows from a Cech cohomology computation similar to the one in [Ati57a] Section 3. ∎ Part (i) in the following proposition is one of the directions in a theorem of Weil. See [BR08] Section 7 for an exposition. ###### Lemma 3.4. Suppose that the characteristic of $k$ is $0$. Let $\mathcal{E}$ be a vector bundle on $C$ equipped with a connection. 1. (i) Every direct summand $\mathcal{F}$ of $\mathcal{E}$ satisfies $\text{deg}\,\mathcal{F}=0$. In particular $\mathcal{E}$ itself has degree $0$. 2. (ii) If $\mathcal{F}$ is a subbundle of $\mathcal{E}$ that is stable under the connection, then $\text{deg}\,\mathcal{F}=0$. ###### Proof. 1. (i) Suppose that $\mathcal{E}$ is a direct sum $\mathcal{E}=\mathcal{F}\oplus\mathcal{G}$ of vector bundles. Let $i:\mathcal{F}\hookrightarrow\mathcal{E}$ and $\text{pr}:\mathcal{E}\rightarrow\mathcal{F}$ denote the inclusion and projection morphisms induced by the direct sum decomposition. We can think of the connection on $\mathcal{E}$ as a morphism of sheaves $\theta:\mathcal{E}\rightarrow\mathcal{E}\otimes\Omega^{1}_{C/k}$ that satisfies the Leibniz rule. Then, the composition $\mathcal{F}\xrightarrow{i}\mathcal{E}\xrightarrow{\theta}\mathcal{E}\otimes\Omega^{1}_{C/k}\xrightarrow{\text{pr}\otimes\text{id}}\mathcal{F}\otimes\Omega^{1}_{C/k}$ also satisfies the Leibniz rule, so it is a connection on $\mathcal{F}$. Suppose that the rank of $\mathcal{F}$ is $r$. Using the functoriality of connections for the determinant morphism $\text{det}:\text{GL}_{r}\rightarrow\mathbb{G}_{m}$, we conclude that the determinant bundle $\text{det}\,\mathcal{F}$ admits a connection. This means that $\gamma_{\text{det}\,\mathcal{F}}=0$. By Lemma 3.3, this is equivalent to $-\text{deg}\,\mathcal{F}=0$ when viewed as an element of $k$. Since the characteristic of $k$ is $0$, it follows that $\text{deg}\,\mathcal{F}=0$. 2. (ii) Suppose that $\mathcal{F}\subset\mathcal{E}$ is stable under the connection $\theta$. Then, we can restrict $\theta|_{\mathcal{F}}$ in order to obtain a connection defined on $\mathcal{F}$. Now part (i) shows that $\text{deg}\,\mathcal{F}=0$. ∎ ### 3.2 Functoriality for regular $G$-connections Connections are contravariant with respect to morphisms of base schemes. To see why this is the case, let $f:T\longrightarrow S$ be a morphism of $k$-schemes. Let $\mathcal{P}$ be a $G$-bundle on $C\times S$. By definition the bundle of $G$-connections $(Id_{C}\times f)^{*}\mathcal{P}\times^{G}\left(\Omega^{1}_{C\times T/T}\otimes\mathfrak{g}\right)$ is the pullback of the $Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}$-torsor $\mathcal{P}\times^{G}\left(\Omega^{1}_{C\times S/S}\otimes\mathfrak{g}\right)$. This means that the Atiyah class behaves well under pullbacks, meaning $\gamma_{(Id_{C}\times f)^{*}\mathcal{P}}=(Id_{C}\times f)^{*}\gamma_{\mathcal{P}}$. Here we write $(Id_{C}\times f)^{*}$ on the right-hand side to denote the natural map on sheaf cohomology groups $\displaystyle H^{1}\left(C\times S,\,Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)\,\longrightarrow\,H^{1}\left(C\times T,\,(Id_{c}\times f)^{*}\left(Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)\,\right)\,\xlongequal{\;\;}\,H^{1}\left(C\times T,\,Ad\,(Id_{C}\times f)^{*}\mathcal{P}\otimes\Omega^{1}_{C\times T/T}$ As a consequence, the Atiyah sequence for $(Id_{C}\times f)^{*}\mathcal{P}$ is the $(Id_{C}\times f)$-pullback of the Atiyah sequence for $\mathcal{P}$. We can therefore pullback connections by interpreting them as splittings of the Atiyah sequence. For a $G$-connection $\theta$ on $\mathcal{P}$, we denote by $f^{*}\theta$ the corresponding connection on $(Id_{C}\times f)^{*}\mathcal{P}$. On the other hand, connections are covariant with respect to morphisms of structure groups. Fix a $k$-scheme $S$. Let $H$ be a linear algebraic group over $k$ with Lie algebra $\mathfrak{h}$. Let $\varphi:G\longrightarrow H$ be a homomorphism of algebraic groups. For any $G$-bundle $\mathcal{P}$ over $C\times S$, there exists an associated $H$-bundle $\varphi_{*}\mathcal{P}\vcentcolon=\mathcal{P}\times^{G}H$ obtained via extension of structure group. Observe that there is an induced morphism of tangent spaces at the identity $\text{Lie}(\varphi):\mathfrak{g}\longrightarrow\mathfrak{h}$. By étale descent for morphisms of quasicoherent sheaves [Sta19, Tag 023T], this yields a vector bundle morphism $Ad\,\varphi:Ad\,\mathcal{P}\longrightarrow Ad\,\varphi_{*}\mathcal{P}$. This induces the following map of sheaf cohomology groups. $\varphi^{1}_{*}:H^{1}\left(C\times S,\,Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)\longrightarrow H^{1}\left(C\times S,\,Ad\,\varphi_{*}\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)$ By construction $\varphi_{*}^{1}\left(\gamma_{\mathcal{P}}\right)=\gamma_{\varphi_{*}\mathcal{P}}$. Recall that the functoriality of $\text{Ext}^{1}(\mathcal{O}_{C\times S},-)$ can be described concretely in terms of pushouts [Sta19, Tag 010I]. This description shows that there is a canonical commutative diagram of short exact sequences between the extension defined by $\gamma_{\mathcal{P}}$ and the extension defined by $\varphi_{*}^{1}(\gamma_{\mathcal{P}})$. Since $\varphi_{*}^{1}\left(\gamma_{\mathcal{P}}\right)=\gamma_{\varphi_{*}\mathcal{P}}$, there is a commutative diagram of Atiyah sequences ${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}}$${At(\mathcal{P})}$${\mathcal{O}_{C\times S}}$${0}$${0}$${\text{Ad}\,\varphi_{*}\mathcal{P}\otimes\Omega^{1}_{C\times S/S}}$${At(\varphi_{*}\mathcal{P})}$${\mathcal{O}_{C\times S}}$${0}$$\scriptstyle{Ad\,\varphi\,\otimes\,id}$$\scriptstyle{At(\varphi)}$$\xlongequal{}$ We denote the middle vertical morphism by $\text{At}(\varphi)$. Let $\theta$ be a $G$-connection on $\mathcal{P}$, viewed as a splitting of the top row. We compose $\theta$ with $At(\varphi)$ in order to define a $H$-connection $\varphi_{*}\theta\vcentcolon=At(\varphi)\circ\theta$ on the $H$-bundle $\varphi_{*}\mathcal{P}$. ### 3.3 The stack of regular $G$-connections The contravariant functoriality for connections with respect maps of base schemes implies that the $G$-connections form a pseudofunctor into groupoids. This allows to define the moduli stack of flat $G$-bundles over $C$. ###### Definition 3.5. The moduli stack $\text{Conn}_{G}(C)$ of flat $G$-bundles over $C$ is the pseudofunctor from $k$-schemes to groupoids defined as follows. For every $k$-scheme $S$, we define $\text{Conn}_{G}(C)\,(S)\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid of $G$-torsors}\;\mathcal{P}\text{ over }C\times S\\\ $+$\\\ \text{ a $G$-connection $\theta$ on $\mathcal{P}$ relative to $S$}\end{matrix}\right\\}$ The isomorphisms are required to be compatible with the connections. There is a morphism of pseudofunctors $Forget:\text{Conn}_{G}(C)\longrightarrow\text{Bun}_{G}(C)$ given by forgetting the connection. ###### Proposition 3.6. The morphism $Forget:\text{Conn}_{G}(C)\longrightarrow\text{Bun}_{G}(C)$ is schematic, affine and of finite type. In particular $\text{Conn}_{G}(C)$ is an algebraic stack that is locally of finite type over $k$. ###### Proof. Let $Z$ be the subfunctor of $\text{Bun}_{G}(C)$ defined as follows. For all $k$-schemes $S$, we set $Z(S)\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid of $\mathcal{P}\in\text{Bun}_{G}(C)\,(S)$}\\\ \text{such that $\gamma_{\mathcal{P}}=0$}\end{matrix}\right\\}$ We claim that $Z$ is a closed substack of $\text{Bun}_{G}(C)$. Choose a $k$-scheme $S$ and a morphism $S\longrightarrow\text{Bun}_{G}(C)$ represented by a $G$-bundle $\mathcal{P}$ on $C\times S$. We want to show that the morphism $Z\times_{\text{Bun}_{G}(C)}S\xrightarrow{pr_{2}}S$ is a closed immersion. The functor $Z\times_{\text{Bun}_{G}(C)}S$ can be described as follows. For all $k$-schemes $T$, we have $Z\times_{\text{Bun}_{G}(C)}S\,(T)=\;\left\\{\begin{matrix}\text{morphisms $f:T\rightarrow S$ such that}\\\ \text{$(Id_{C}\times f)^{*}\gamma_{\mathcal{P}}=0$}\end{matrix}\right\\}$ We are implicitly using the compatibility of the Atiyah class with pullbacks. Lemma 2.3 applied to the proper flat morphism $C\times S\longrightarrow S$ implies that $Z\times_{\text{Bun}_{G}(C)}S$ is represented by a closed subscheme of $S$. This concludes our proof of the claim. Consider now the morphism $Forget:\text{Conn}_{G}(C)\longrightarrow\text{Bun}_{G}(C)$. Since the vanishing of the Atiyah class is a necessary condition for the existence of a $G$-connection, the morphism $Forget$ factors through the closed immersion $Z\hookrightarrow\text{Bun}_{G}(C)$. It suffices to show that $\text{Conn}_{G}(C)\longrightarrow Z$ is schematic, affine and of finite type. Let $S$ be a $k$-scheme and $\mathcal{P}:S\longrightarrow Z$ be a $G$-bundle on $C\times S$ with $\gamma_{\mathcal{P}}=0$. By definition the fiber product $\text{Conn}_{G}(C)\times_{Z}S$ is the functor from $S$-schemes $g:T\longrightarrow S$ to sets given by $\text{Conn}_{G}(C)\times_{Z}S\,(T)=\left\\{\;\text{$G$-connections on}\;(\text{id}_{C}\times g)^{*}\mathcal{P}\;\text{relative to $T$}\;\right\\}$ We want to show that $\text{Conn}_{G}(C)\times_{Z}S$ is represented by a scheme that is relatively affine and of finite type over $S$. Since $\gamma_{\mathcal{P}}=0$, the Atiyah sequence for $\mathcal{P}$ is split. Choose a splitting $\theta$, i.e. a $G$-connection on $\mathcal{P}$ relative to $S$. Any other $G$-connection on $\mathcal{P}$ is of the form $\theta+s$ where $s\in H^{0}\left(C\times S,\,Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\,\right)$. This induces an isomorphism of presheaves of sets $\psi:\Gamma_{C\times S/S}\left(\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}\right)\xrightarrow{\sim}\text{Conn}_{G}(C)\times_{Z}S$ To be precise, $\psi$ is defined as follows. For any $S$-scheme $g:T\longrightarrow S$ and section $s\in H^{0}\left(C\times T,\,(\text{id}_{C}\times g)^{*}\,Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times T/T}\,\right)$ we define $\psi(T)\,(s)=(\text{id}_{C}\times g)^{*}\theta+s$ By Lemma 2.2, we know that $\Gamma_{C\times S/S}\left(\text{Ad}\,\mathcal{P}\,\otimes\,\Omega_{C\times S/S}^{1}\right)$ is represented by a scheme that is relatively affine and of finite type over $S$. The proposition follows. ∎ We now use the $Forget$ map to prove that the stack $\text{Conn}_{G}(C)$ is quasicompact whenever $\text{char}\,k=0$. ###### Proposition 3.7. Suppose that the characteristic of $k$ is $0$. The algebraic stack $\text{Conn}_{G}(C)$ is of finite type over $k$. ###### Proof. By Proposition 3.6, it suffices to show that that the map $Forget:\text{Conn}_{G}(C)\longrightarrow\text{Bun}_{G}(C)$ factors through an open substack $W\subset\text{Bun}_{G}(C)$ that is of finite type over $k$. We shall prove this. Let $U$ be the unipotent radical of $G$. Denote by $\rho_{G}:G\rightarrow G/U$ the correspoding quotient morphism. Choose a faithful representation $\rho:G/U\longrightarrow\text{GL}_{n}$ of the reductive group $G/U$. For any $G$-bundle $\mathcal{P}$ we can form the associated $\text{GL}_{n}$-bundle $(\rho\circ\rho_{G})_{*}\mathcal{P}$ via extension of structure group. This construction defines a morphism $(\rho\circ\rho_{G})_{*}:\text{Bun}_{G}(C)\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$. For any $G$-connection $\theta$ on $\mathcal{P}$ we get an induced $\text{GL}_{n}$-connection $(\rho\circ\rho_{G})_{*}\theta$ on $\rho_{*}\mathcal{P}$. There is a commutative diagram $\textstyle{\text{Conn}_{G}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Forget}$$\textstyle{\text{Conn}_{G/U}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Forget}$$\textstyle{\text{Conn}_{\text{GL}_{n}}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Forget}$$\textstyle{\text{Bun}_{G}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\rho_{G})_{*}}$$\textstyle{\text{Bun}_{G/U}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{*}}$$\textstyle{\text{Bun}_{\text{GL}_{n}}(C)}$ Propositions 2.4 and 2.5 show that the two horizontal morphisms at the bottom of the diagram above are of finite type. Therefore the composition $(\rho\circ\rho_{G})_{*}$ is of finite type. The claim will follow if we can show that the forgetful map $Forget:\text{Conn}_{\text{GL}_{n}}(C)\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$ factors through an open substack $W\subset\text{Bun}_{\text{GL}_{n}}(C)$ of finite type over $k$. For this purpose, we will use the Harder-Narasimhan stratification of $\text{Bun}_{\text{GL}_{n}}$ [Sch15]. For any rational cocharacter $\lambda\in\mathbb{Q}^{n}$, let $\text{Bun}_{\text{GL}_{n}}^{\leq\lambda}(C)$ be the quasicompact open substack of $\text{Bun}_{GL_{n}}(C)$ consisting of the strata with Harder-Narasimhan polygon smaller than $\lambda$. We will show that $Forget:\text{Conn}_{\text{GL}_{n}}(C)\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$ factors through a finite union $\bigcup_{j}\text{Bun}_{\text{GL}_{n}}^{\leq\lambda_{j}}(C)$ for some $\lambda_{j}$. This can be checked at the level of field-valued points. Hence it suffices to check that the set of Harder-Narasimhan polygons of geometric points $\overline{t}\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$ with nonempty fiber $\text{Conn}_{\text{GL}_{n}}(C)_{\overline{t}}$ is a finite set. Let $\mu=(\mu_{1}\geq\mu_{2}\geq...\geq\mu_{n})$ be a tuple of rational numbers in $\frac{1}{n!}\mathbb{Z}^{n}$. Let $\mathcal{E}:\overline{t}\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$ be a vector bundle on $C\times\overline{t}$, where $\overline{t}$ is Spec of an algebrically closed field. Suppose that the Harder-Narasimhan polygon of $\mathcal{E}$ is given by $\mu$. Assume that the fiber $\text{Conn}_{\text{GL}_{n}}(C)_{\overline{t}}$ is nonempty. This means that $\mathcal{E}$ admits a connection. We claim that $\mu$ satisfies the following two conditions 1. (a) $\sum_{j=1}^{n}\mu_{j}=0$. 2. (b) $\mu_{j}-\mu_{j+1}\leq\text{max}\\{2g-2,0\\}$ for all $1\leq j\leq n-1$. The claim implies that $-\left(\text{max}(\\{g-1,0\\}\right)\cdot(n-1)\leq\mu_{n}\leq\mu_{1}\leq\left(\text{max}\\{g-1,0\\}\right)\cdot(n-1)$. This would show that there are finitely many possibilities for $\mu\in\frac{1}{n!}\mathbb{Z}^{n}$. We are left to proving the claim. 1. (a) By Lemma 3.4, $\text{deg}\,\mathcal{E}=0$. This is equivalent to the condition (a) in the claim above. 2. (b) For the sake of contradiction, assume that $\mu_{j+1}-\mu_{j}>\text{max}\\{2g-2,0\\}$ for some $1\leq j\leq n-1$. Suppose that $\mathcal{E}$ has Harder-Narasimhan filtration $0\subset\mathcal{F}_{1}\subset\mathcal{F}_{2}\subset...\subset\mathcal{F}_{l}=\mathcal{E}$ There is an index $1\leq k\leq l-1$ such that $\mu\left(\mathcal{F}_{k}\,/\,\mathcal{F}_{k-1}\right)=\mu_{j}$. We have a short exact sequence $0\longrightarrow\mathcal{F}_{k}\longrightarrow\mathcal{E}\longrightarrow\mathcal{E}\,/\,\mathcal{F}_{k}\longrightarrow 0$ An application of Serre duality plus the definition of semistability and the hypothesis $\mu_{j+1}-\mu_{j}>\text{max}(2g-2,0)$ implies that $\text{Ext}^{1}\left(\mathcal{E}\,/\,\mathcal{F}_{k}\,,\,\mathcal{F}_{k}\right)=0$. We conclude that $\mathcal{E}=\mathcal{F}_{k}\,\oplus\,\mathcal{E}\,/\,\mathcal{F}_{k}$. But by assumption we must have $\text{deg}\,\mathcal{F}_{k}\,>\,\text{deg}\,\mathcal{E}\,>\,\text{deg}\,\mathcal{E}\,/\,\mathcal{F}_{k}$ Hence $\text{deg}\,\mathcal{F}_{k}\,>\,0$. This contradicts Lemma 3.4 (i), because $\mathcal{F}_{k}$ is a direct summand of $\mathcal{E}$. ∎ We record another proof that $\text{Conn}_{\text{GL}_{n}}(C)$ is of finite type as a consequence of Simpson’s work [Sim94a]. The stack $\text{Conn}_{\text{GL}_{n}}(C)$ classifies rank $n$ vector bundles equipped with a connection. These are $\Lambda$-modules of pure dimension $1$ on $C$, where $\Lambda$ is the sheaf of differential operators on $C$ (using the notation in [Sim94a][pg. 85-86]). If $\mathcal{E}$ is a vector bundle with a connection, then $\text{deg}(\mathcal{E})=0$ by Lemma 3.4. If furthermore $\mathcal{E}$ has rank $n$, then we can use Riemann-Roch to see that the Hilbert polynomial of $\mathcal{E}$ is given by $p(\mathcal{E},x)=\left(n\cdot\text{deg}\,\mathcal{O}_{C}(1)\right)x+n\cdot(1-g)$ In particular the slope of $\mathcal{E}$ with respect to $\mathcal{O}_{C}(1)$ is given by $\widehat{\mu}(\mathcal{E})=\frac{1-g}{\text{deg}\,\mathcal{O}_{C}(1)}$. Suppose that $\mathcal{G}\subset\mathcal{E}$ is a $\Lambda$-submodule. This means that $\mathcal{G}$ is a subbundle that is stable under the connection. Lemma 3.4 (ii) implies that $\text{deg}\,\mathcal{G}=0$. The same Riemann-Roch computation can be applied to see that $\widehat{\mu}(\mathcal{G})=\frac{1-g}{\text{deg}\,\mathcal{O}_{C}(1)}$. This shows that $\mathcal{E}$ is a $\mu$-semistable $\Lambda$-module in the sense of [Sim94a][§3]. It follows that $\text{Conn}_{\text{GL}_{n}}(C)$ classifies $\mu$-semistable $\Lambda$-modules with fixed Hilbert polynomial $P(x)=\left(n\cdot\text{deg}\,\mathcal{O}_{C}(1)\right)x+n\cdot(1-g)$. We can therefore apply [Sim94a][Cor. 3.4] to conclude that the moduli stack $\text{Conn}_{\text{GL}_{n}}(C)$ is quasicompact. In fact [Sim94a][Lemma 3.3] provides a bound on the Harder-Narasimhan polygons of points in the image of the morphism $\text{Forget}:\text{Conn}_{\text{GL}_{n}}(C)\rightarrow\text{Bun}_{\text{GL}_{n}}(C)$. We assume that the ample line bundle $\mathcal{O}_{C}(1)$ has degree $1$. Using the notation of [Sim94a][Lemma 3.3], we have that $\text{Gr}_{1}(\Lambda)$ is the tangent bundle of $C$. A Riemann-Roch computation shows that we can take $m=4g-3$ in [Sim94a][Lemma 3.3]. Let $\mathcal{E}\in\text{Forget}\left(\text{Conn}_{\text{GL}_{n}}(C)\right)$ be a vector bundle with Harder-Narasimhan type $\mu=(\mu_{1}\geq\mu_{2}\geq...\geq\mu_{n})$. We have $\mu_{1}=\mu(\mathcal{F}_{1})=\widehat{\mu}(\mathcal{F}_{1})+(1-g)$, where $\mathcal{F}_{1}$ is the first step in the Harder-Narasimhan filtration. [Sim94a][Lemma 3.3] shows that $\mu_{1}\leq\left(\text{max}\\{4g-3,0\\}\right)\cdot n$. Using this inequality and the fact that $\sum_{j=1}^{n}\mu_{j}=0$, we get a bound on the Harder- Narasimhan polygon of $\mathcal{E}$. Hence there are finitely many possibilities for $\mu$. This bound on the Harder-Narasimhan type is weaker than the one obtained in the proof of Proposition 3.7. ## 4 The moduli stack of logarithmic $G$-connections ### 4.1 Logarithmic $G$-connections and their residues We start this section by recalling the necessary preliminaries on logarithmic connections. Our initial exposition is adapted from the holomorphic case considered in [BDPS17][2.2, 2.3]. Let $D=\sum_{i\in I}x_{i}$ be the reduced divisor of $C$ supported at the $x_{i}$’s. There is a canonical inclusion of sheaves $\mathcal{O}_{C\times S}(-D)\hookrightarrow\mathcal{O}_{C\times S}$ induced by $x_{i}$. The logarithmic Atiyah sequence $\text{At}^{D}(\mathcal{P})$ with poles at $D$ is defined by the following pullback diagram. ${At^{D}(\mathcal{P})}$${\mathcal{O}_{C\times S}(-D)}$${At(\mathcal{P})}$${\mathcal{O}_{C\times S}}$$\hookrightarrow$$\hookrightarrow$ By definition, there is a commutative diagram of short exact sequences ${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}}$${At^{D}(\mathcal{P})}$${\mathcal{O}_{C\times S}(-D)}$${0}$${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}}$${At(\mathcal{P})}$${\mathcal{O}_{C\times S}}$${0}$$\xlongequal{}$$\hookrightarrow$$\hookrightarrow$ The top row is called the logarithmic Atiyah sequence with poles at $D$. A splitting of this sequence is called a logarithmic connection with poles at $D$. By construction it makes sense to ask whether a logarithmic connection $\theta$ of $\mathcal{P}$ extends to a $G$-connection of $\mathcal{P}$. This just means that the composition $\mathcal{O}_{C\times S}(-D)\xrightarrow{\theta}\text{At}^{D}(\mathcal{P})\rightarrow\text{At}(\mathcal{P})$ (uniquely) extends to $\mathcal{O}_{C\times S}$. ###### Example 4.1. Suppose that $S=\text{Spec}\,k$ and $G=\mathbb{G}_{m}$. Let $\mathcal{L}$ be a line bundle on $C$. If the divisor $D$ is nonzero, the logarithmic Atiyah sequence for $\mathcal{L}$ splits. Therefore all line bundles admit logarithmic connections. We tensor the logarithmic Atiyah sequence with $\mathcal{O}_{C\times S}(D)$ to obtain the following diagram ${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}(D)}$${At^{D}(\mathcal{P})(D)}$${\mathcal{O}_{C\times S}}$${0}$ Figure 1: Diagram 1 Let $i\in I$. Base-changing Diagram 1 to the closed subscheme $x_{i}\times S$ we get a diagram with exact rows and columns ${0}$${0}$${0}$${0}$${0}$${\left(At^{D}(\mathcal{P})\,/\,At(\mathcal{P})\right)(D)\,|_{x_{i}\times S}}$${\mathcal{O}_{x_{i}\times S}}$${0}$${0}$${Ad(\mathcal{P})\otimes\Omega^{1}_{C\times S/S}(D)\,|_{x_{i}\times S}}$${At^{D}(\mathcal{P})(D)|_{x_{i}\times S}}$${\mathcal{O}_{x_{i}\times S}}$${0}$${0}$${Ad(\mathcal{P})\otimes\Omega^{1}_{C\times S/S}(D)\,|_{x_{i}\times S}}$${At(\mathcal{P})(D)|_{x_{i}\times S}}$${\mathcal{O}_{x_{i}\times S}}$${0}$${0}$${0}$${\left(At^{D}(\mathcal{P})\,/\,At(\mathcal{P})\right)(D)\,|_{x_{i}\times S}}$${\mathcal{O}_{x_{i}\times S}}$${0}$${0}$${0}$${0}$$\scriptstyle{\psi_{i}}$$\scriptstyle{Id}$$\scriptstyle{Id}$$\scriptstyle{0}$$\scriptstyle{Id}$$\scriptstyle{\psi_{i}}$ The first two rows yield a canonical splitting of the base-change of Diagram 1 to $x_{i}\times S$ ${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}(D)\,|_{x_{i}\times S}}$${At^{D}(\mathcal{P})(D)|_{x_{i}\times S}}$${\mathcal{O}_{x_{i}\times S}}$${0}$${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}(D)\,|_{x_{i}\times S}}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}(D)\,|_{x_{i}\times S}\oplus\mathcal{O}_{x_{i}\times S}}$${\mathcal{O}_{x_{i}\times S}}$${0}$$\xlongequal{}$$\xrightarrow{\sim}$$\xlongequal{}$ Figure 2: Diagram 2 Consider the stalk $\Omega^{1}_{C/k}(D)|_{\mathcal{O}_{C,x_{i}}}$. It is a free $\mathcal{O}_{C,x_{i}}$-module of rank $1$. There is a canonical generating element given as follows. Choose a uniformizer $z_{i}$ of the discrete valuation ring $\mathcal{O}_{C,x_{i}}$. Then, the meromorphic $1$-form $\frac{dz_{i}}{z_{i}}$ is a generator of $\Omega^{1}_{C/k}(D)|_{\mathcal{O}_{C,x_{i}}}$ and does not depend on the choice of uniformizer. This induces a canonical trivialization $\Omega^{1}_{C/k}(D)|_{\mathcal{O}_{C,x_{i}}}\cong\mathcal{O}_{C,x_{i}}$. This in turn yields an isomorphism $\Omega^{1}_{C\times S/S}(D)|_{x_{i}\times S}\cong\mathcal{O}_{x_{i}\times S}$ after base-changing to $x_{i}\times S$. This canonical isomorphism $\Omega^{1}_{C\times S/S}(D)|_{x_{i}\times S}\cong\mathcal{O}_{x_{i}\times S}$ and the splitting in Diagram 2 can be used to identify the $x_{i}\times S$ fiber of Diagram 1 with the following split short exact sequence ${0}$${\text{Ad}\left(\mathcal{P}|_{x_{i}\times S}\right)}$${\text{Ad}\left(\mathcal{P}|_{x_{i}\times S}\right)\oplus\mathcal{O}_{x_{i}\times S}}$${\mathcal{O}_{x_{i}\times S}}$${0}$ Figure 3: Diagram 3 Let $\theta$ be a logarithmic connection on $\mathcal{P}$ with poles at $D$, viewed as a splitting of the logarithmic Atiyah sequence. We can tensor with $\mathcal{O}_{C\times S}(D)$ and pass to the $x_{i}\times S$ fiber to obtain a splitting of Diagram 3. Such splittings are in natural correspondence with global sections of the vector bundle $\text{Ad}\left(\mathcal{P}|_{x_{i}\times S}\right)$. The section corresponding to the splitting induced by $\theta$ is called the residue of $\theta$ at $x_{i}$, and will be denoted $\text{Res}_{x_{i}}\theta\in H^{0}\left(x_{i}\times S,\,\text{Ad}\left(\mathcal{P}|_{x_{i}\times S}\right)\,\right)$. Our next goal is to define a version of the logarithmic Atiyah sequence with prescribed residues. Let $S$ be a $k$-scheme. Let $\mathcal{P}$ be a $G$-bundle on $C\times S$. For each $i\in I$, fix a section $s_{i}\in H^{0}\left(x_{i}\times S,\,Ad(\mathcal{P}|_{x_{i}\times S})\,\right)$. Set $W_{i}$ to be the cokernel of the map $s_{i}\oplus Id_{\mathcal{O}_{x_{i}\times S}}:\,\mathcal{O}_{x_{i}\times S}\longrightarrow Ad(\mathcal{P}|_{x_{i}\times S})\oplus\mathcal{O}_{x_{i}\times S}$. We have seen that there is an isomorphism $At^{D}(\mathcal{P})(D)|_{x_{i}\times S}\xrightarrow{\sim}\text{Ad}\left(\mathcal{P}|_{x_{i}\times S}\right)\oplus\mathcal{O}_{x_{i}\times S}$ Define $At^{D,s_{i}}(\mathcal{P})$ to be the kernel of the composition $\displaystyle At^{D}(\mathcal{P})(D)\xrightarrow{\underset{i\in I}{\oplus}unit}\bigoplus_{i\in I}(q_{i}\times id_{S})_{*}\left(At^{D}(\mathcal{P})(D)|_{x_{i}\times S}\right)\xrightarrow{\sim}\bigoplus_{i\in I}(q_{i}\times id_{S})_{*}\,\left(Ad(\mathcal{P}|_{x_{i}\times S})\oplus\mathcal{O}_{x_{i}\times S}\right)\twoheadrightarrow\bigoplus_{i\in I}(q_{i}\times id_{S})_{*}\,W_{i}$ Consider the pullback diagram ${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}}$${At^{D,s_{i}}(\mathcal{P})}$${\mathcal{O}_{C\times S}}$${0}$${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}(D)}$${At^{D}(\mathcal{P})(D)}$${\mathcal{O}_{C\times S}}$${0}$$\hookrightarrow$$\hookrightarrow$$\xlongequal{}$ Figure 4: Diagram 4 By construction, there is a bijection between logarithmic $G$-connections $\theta$ on $\mathcal{P}$ such that $\text{Res}_{x_{i}}\theta=s_{i}$ for all $i\in I$ and splittings of the top row in Diagram 4. We refer to this short exact sequence as the logarithmic Atiyah sequence with prescribed residues. ${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}}$${At^{D,s_{i}}(\mathcal{P})}$${\mathcal{O}_{C\times S}}$${0}$ Figure 5: Diagram 5 We set $\gamma^{D,s_{i}}_{\mathcal{P}}\in H^{1}\left(C\times S,\,Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\,\right)$ to be the cohomology class associated to the extension in Diagram 5. It is the analogue of the Atiyah class for logarithmic connections with prescribed residues. The $G$-bundle $\mathcal{P}$ admits a logarithmic connections with prescribed residues $s_{i}$ if and only if $\gamma^{D,s_{i}}_{\mathcal{P}}=0$. Proposition 4.2 gives an explicit description $\gamma^{D,s_{i}}_{\mathcal{P}}$. In order to understand this description we need some setup. For each $i\in I$, we have a short exact sequence $0\longrightarrow\mathcal{O}_{C\times S}\longrightarrow\mathcal{O}_{C\times S}(x_{i})\xrightarrow{unit}(q_{i}\times id_{S})_{*}\left(\mathcal{O}_{C}(x_{i})|_{x_{i}\times S}\right)\longrightarrow 0$ We tensor this sequence with $\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}$ in order to obtain $\displaystyle 0\longrightarrow\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\longrightarrow\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}(x_{i})\longrightarrow(q_{i}\times id_{S})_{*}\left(\text{Ad}\,\mathcal{P}|_{x_{i}\times S}\otimes\Omega^{1}_{C\times S/S}(x_{i})|_{x_{i}\times S}\right)\longrightarrow 0$ The canonical trivialization $\Omega^{1}_{C\times S/S}(x_{i})|_{x_{i}\times S}\cong\mathcal{O}_{x_{i}\times S}$ described earlier can be used rewrite the short exact sequence above. $0\longrightarrow\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\longrightarrow\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}(x_{i})\longrightarrow(q_{i}\times id_{S})_{*}\left(\text{Ad}\,\mathcal{P}|_{x_{i}\times S}\right)\longrightarrow 0$ Denote by $\delta_{i}:H^{0}\left(x_{i}\times S,\,\text{Ad}\,\mathcal{P}|_{x_{i}\times S}\right)\longrightarrow H^{1}\left(C\times S,\,\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)$ the corresponding connecting homomorphism of sheaf cohomology groups. The following proposition is proven in [BDP18][Prop. 3.1] in the case when $k=\mathbb{C}$, $G=\text{GL}_{n}$ and $S=\text{Spec}\,\mathbb{C}$. The argument found there uses transcendental methods. We give an algebraic proof of the analogous result in the case of an arbitrary base $S$, arbitrary ground field $k$, and any smooth connected linear algebraic group $G$. ###### Proposition 4.2. Fix a $k$-scheme $S$. Let $\mathcal{P}$ be a $G$-bundle over $C\times S$. Let $s_{i}\in H^{0}\left(x_{i}\times S,\,\text{Ad}(\mathcal{P}|_{x_{i}\times S})\,\right)$ be sections as above. Then, we have $\gamma^{D,s_{i}}_{\mathcal{P}}=\gamma_{\mathcal{P}}-\sum_{i\in I}\delta_{i}(s_{i})$. ###### Proof. Consider the short exact sequence $0\longrightarrow\mathcal{O}_{C\times S}\longrightarrow\mathcal{O}_{C\times S}(D)\xrightarrow{unit}\bigoplus_{i\in I}(q_{i}\times id_{S})_{*}\left(\mathcal{O}_{C}(x_{i})|_{x_{i}\times S}\right)\longrightarrow 0$ By tensoring with $\text{Ad}\,\mathcal{P}\,\otimes\,\Omega_{C\times S/S}^{1}$ and using the identifications $\Omega^{1}_{C\times S/S}(x_{i})|_{x_{i}\times S}\cong\mathcal{O}_{x_{i}\times S}$ as described above, we obtain a short exact sequence $\displaystyle 0\longrightarrow\text{Ad}\,\mathcal{P}\,\otimes\,\Omega^{1}_{C\times S/S}\,\xrightarrow{\;\;j\;\;}\,\text{Ad}\,\mathcal{P}\,\otimes\,\Omega^{1}_{C\times S/S}(D)\,\xrightarrow{\;\;u\;\;}\,\bigoplus_{i\in I}(q_{i}\times id_{S})_{*}\left(\text{Ad}\,\mathcal{P}|_{x_{i}\times S}\right)\longrightarrow 0$ The morphisms $j$ and $u$ above are labeled for future use. The construction of the connecting homomorphism in Cech cohomology implies that the connecting homomorphism for this short exact sequence is given by the sum $\sum_{i\in I}\delta_{i}\,:\;\bigoplus_{i\in I}H^{0}\left(x_{i}\times S,\,\text{Ad}\,\mathcal{P}|_{x_{i}\times S}\right)\longrightarrow H^{1}\left(C\times S,\,\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)$ We recall how to describe the extension corresponding to the cohomology class $-\sum_{i\in I}\delta_{i}(s_{i})\in H^{1}\left(C\times S,\,\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)=\text{Ext}^{1}\left(\mathcal{O}_{C\times S},\,\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)$. Define a global section $v_{i}:\mathcal{O}_{C\times S}\longrightarrow(q_{i}\times id_{S})_{*}\left(\text{Ad}\,\mathcal{P}|_{x_{i}\times S}\right)$ to be the composition $v_{i}:\;\mathcal{O}_{C\times S}\,\xrightarrow{unit}\,(q_{i}\times id_{S})_{*}\,\mathcal{O}_{x_{i}\times S}\,\xrightarrow{(q_{i}\times id_{S})_{*}s_{i}}\,(q_{i}\times id_{S})_{*}\left(\text{Ad}\,\mathcal{P}|_{x_{i}\times S}\right)$ Let $\mathcal{F}$ be the $\mathcal{O}_{C\times S}$-sheaf given by the pullback diagram ${\mathcal{F}}$${\mathcal{O}_{C\times S}}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}(D)}$${\bigoplus_{i\in I}(q_{i}\times id_{S})_{*}\left(\text{Ad}\,\mathcal{P}|_{x_{i}\times S}\right)}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{\left(\,-v_{i}\,\right)_{i}}$$\scriptstyle{u}$ This means that $\mathcal{F}$ is the kernel of the morphism $\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}(D)\,\oplus\,\mathcal{O}_{C\times S}\,\xrightarrow{\;\;\;u\,+\,\left(v_{i}\right)_{i}\;\;\;}\,\bigoplus_{i\in I}\,(q_{i}\times id_{S})_{*}\left(\text{Ad}\,\mathcal{P}|_{x_{i}\times S}\right)$ Here the maps $p_{l}$ for $l=1,2$ are the natural projections. The morphism $(j,0):\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}\longrightarrow\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}(D)\,\oplus\,\mathcal{O}_{C\times S}$ factors through the subsheaf $\mathcal{F}\subset\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}(D)\,\oplus\,\mathcal{O}_{C\times S}$. By construction $(j,0):\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}\longrightarrow\mathcal{F}$ is a kernel for the morphism $p_{2}:\mathcal{F}\longrightarrow\mathcal{O}_{C\times S}$. The extension corresponding to the cohomology class $-\sum_{i\in I}\delta_{i}(s_{i})$ is given by $0\longrightarrow\,\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}\,\xrightarrow{\;\;\;\ (j,0)\;\;\;}\,\mathcal{F}\,\xrightarrow{\;\;\;\;p_{2}\;\;\;\;}\,\mathcal{O}_{C\times S}\,\longrightarrow\,0$ We describe now the extension corresponding to the cohomology class $\gamma_{\mathcal{P}}-\sum_{i\in I}\delta_{i}(s_{i})$. Suppose that the Atiyah sequence for $\mathcal{P}$ is given by $0\,\longrightarrow\,Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\,\xrightarrow{\;\;\;\;b\;\;\;\;}\,At(\mathcal{P})\,\xrightarrow{\;\;\;\;p\;\;\;\;}\,\mathcal{O}_{C\times S}\,\longrightarrow\,0$ Define $\mathcal{G}$ to be the kernel of the morphism $p-p_{2}\,:\,\text{At}(\mathcal{P})\oplus\mathcal{F}\longrightarrow\mathcal{O}_{C\times S}$. Consider the map $(b,\,(j,0)\,):\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}\longrightarrow\text{At}(\mathcal{P})\oplus\mathcal{F}$. By construction it factors through the subsheaf $\mathcal{G}\subset\text{At}(\mathcal{P})\oplus\mathcal{F}$. Recall that in the Yoneda group $\text{Ext}^{1}\left(\mathcal{O}_{C\times S},\,\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)$ addition is given by the Baer sum [Sta19, Tag 010I]. The Baer sum $\gamma_{\mathcal{P}}-\sum_{i\in I}\delta_{i}(s_{i})$ is given by $0\,\longrightarrow\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}\,\xrightarrow{\;\;\;(b,\,0)\;\;\;}\,\mathcal{G}\,/\,\text{Im}(b,\,(j,0)\,)\,\xrightarrow{\;\;\;\;p\;\;\;\;}\mathcal{O}_{C\times S}\,\longrightarrow\,0$ Figure 6: Diagram 6 The proposition amounts to showing that the extension in Diagram 6 is isomorphic to the logarithmic Atiyah sequence with prescribed residues in Diagram 5. We will explicitly construct an isomorphism. There is a canonical inclusion $k:\text{At}(\mathcal{P})\hookrightarrow\text{At}(\mathcal{P})(D)$. Define the map $w:\text{At}(\mathcal{P})\,\oplus\,\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}(D)\longrightarrow\text{At}(\mathcal{P})(D)$ to be the composition $\text{At}(\mathcal{P})\,\oplus\,\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}(D)\,\xrightarrow{(k,\,-b(D))}\text{At}(\mathcal{P})(D)\oplus\text{Ad}(\mathcal{P})(D)\,\xrightarrow{\;\;+\;\;}\,\text{At}(\mathcal{P})(D)$ Let $a:\mathcal{G}\longrightarrow\text{At}(\mathcal{P})(D)$ denote the composition $\mathcal{G}\,\hookrightarrow\,\text{At}(\mathcal{P})\oplus\mathcal{F}\,\xrightarrow{\;\;id\oplus p_{1}\;\;}\,\text{At}(\mathcal{P})\,\oplus\,\text{Ad}\,\mathcal{P}\otimes\Omega_{C\times S/S}^{1}(D)\,\xrightarrow{\;\;\;w\;\;\;}\,\text{At}(\mathcal{P})(D)$ By construction $\text{Im}(b,\,(j,0)\,)$ is the kernel of $a$. We are left to show the following two claims. 1. (C1) The morphism $a$ factors through the subsheaf $\text{At}^{D,s_{i}}(\mathcal{P})\subset\text{At}(\mathcal{P})(D)$. 2. (C2) The induced map $a:\mathcal{G}\,/\,\text{Im}(b,\,(j,0)\,)\,\longrightarrow\,\text{At}^{D,s_{i}}(\mathcal{P})$ yields an isomorphism of short exact sequences ${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}}$${\mathcal{G}\,/\,\text{Im}(b,\,(j,0)\,)}$${\mathcal{O}_{C\times S}}$${0}$${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}}$${\text{At}^{D,s_{i}}(\mathcal{P})}$${\mathcal{O}_{C\times S}}$${0}$$\xlongequal{}$$\scriptstyle{(b,0)}$$\scriptstyle{p}$$\scriptstyle{a}$$\xlongequal{}$ These two claims can be checked at the level of stalks. Let $x$ be a topological point of $C\times S$. Suppose first that $x$ does not belong to any of the divisors $x_{i}\times S$ for $i\in I$. Then, the stalk at $x$ of the logarithmic Atiyah sequence in Diagram 5 becomes canonically isomorphic to the stalk of the regular Atiyah sequence. Similarly, the stalk at $x$ of Diagram 6 becomes canonically isomorphic to the regular Atiyah sequence. This is because all of the operations we have performed above are trivial outside of the divisors $x_{i}\times S$. Under these identifications, the induced morphism on stalks $a_{x}$ in (C2) becomes the identity. This concludes the proof when $x$ is not contained in any of the divisors $x_{i}\times S$. Assume now that $x$ is contained in $x_{i}\times S$ for some $i\in I$. The stalks at $x$ of the short exact sequences we have considered are just extensions of finite free modules over the local ring $\mathcal{O}_{C\times S,x}$. It follows that all of these extensions split. Choose a splitting of the $x$-stalk of the regular Atiyah sequence ${0}$${\left(\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)_{x}}$${At(\mathcal{P})_{x}}$${\mathcal{O}_{C\times S,x}}$${0}$${0}$${\left(\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)_{x}}$${\left(\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)_{x}\oplus\mathcal{O}_{C\times S,x}}$${\mathcal{O}_{C\times S,x}}$${0}$$\xlongequal{}$$\scriptstyle{b_{x}}$$\scriptstyle{p_{x}}$$\xrightarrow{\sim}$$\xlongequal{}$ Such splitting induces an identification $\text{At}^{D}(\mathcal{P})(D)_{x}\cong\left(\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}(D)\right)_{x}\oplus\mathcal{O}_{C\times S,x}$. Furthermore, we can choose the splitting so that this identification is compatible with Diagram 2 after base-changing to the residue field of $x$. The residue isomorphism $\Omega^{1}_{C/k}(D)|_{\mathcal{O}_{C,x_{i}}}\cong\mathcal{O}_{C,x_{i}}$ induces by base-change an isomorphism $\Omega^{1}_{C\times S/S}(D)_{x}\cong\mathcal{O}_{C\times S,x}$. We can use this to identify the stalks $\text{At}(\mathcal{P})(D)_{x}\cong\text{Ad}(\mathcal{P})_{x}\oplus\mathcal{O}(D)_{x}$. Let $\overline{s}_{i}\in\text{Ad}(\mathcal{P})_{\kappa(x)}$ be the restriction of the section $s_{i}$ to the residue field of $x$. The submodule $\text{At}^{D,s_{i}}(\mathcal{P})_{x}\subset\text{At}(\mathcal{P})(D)_{x}$ corresponds to $\displaystyle\text{At}^{D,s_{i}}(\mathcal{P})_{x}=\left\\{\,(c,r)\in\text{Ad}(\mathcal{P})_{x}\oplus\mathcal{O}_{C\times S,x}\;\mid\;c_{\kappa(x)}=r_{\kappa(x)}\,\overline{s}_{i}\,\right\\}\;\subset\;\text{Ad}(\mathcal{P})_{x}\oplus\mathcal{O}(D)_{x}$ Diagram 5 is identified with $0\,\longrightarrow\text{Ad}\,\mathcal{P}(-D)_{x}\,\xrightarrow{\;\;\;\left(\,k(-D)_{x},\,0\,\right)\;\;\;}\,\text{At}^{D,s_{i}}(\mathcal{P})_{x}\,\xrightarrow{\;\;\;\;\text{pr}_{2}\;\;\;\;}\mathcal{O}_{C\times S,x}\,\longrightarrow\,0$ By the construction of $\mathcal{F}$, the canonical isomorphism $\Omega^{1}_{C\times S/S}(D)_{x}\cong\mathcal{O}_{C\times S,x}$ induces an identification $\mathcal{F}_{x}=\left\\{\,(c,r)\in\text{Ad}(\mathcal{P})_{x}\oplus\mathcal{O}_{C\times S,x}\;\mid\;c_{\kappa(x)}=-r_{\kappa(x)}\,\overline{s}_{i}\,\right\\}$ The submodule $\mathcal{G}_{x}\subset\text{At}(\mathcal{P})_{x}\oplus\mathcal{F}_{x}$ is given by $\displaystyle\mathcal{G}_{x}=\left\\{\,(c,r,d,r)\in\text{Ad}(\mathcal{P})(-D)_{x}\oplus\mathcal{O}_{C\times S,x}\oplus\text{Ad}(\mathcal{P})_{x}\oplus\mathcal{O}_{C\times S,s}\;\mid\;d_{\kappa(x)}=-r_{\kappa(x)}\,\overline{s}_{i}\,\right\\}$ The map $a_{x}:\mathcal{G}_{x}\longrightarrow\text{At}(\mathcal{P})(D)_{x}$ corresponds under the identification $\text{At}(\mathcal{P}(D)_{x}\cong\text{Ad}(\mathcal{P})_{x}\oplus\mathcal{O}_{C\times S,x}$ to the composition $\displaystyle a_{x}:\mathcal{G}_{x}\,\hookrightarrow\,\text{Ad}(\mathcal{P})(-D)_{x}\oplus\mathcal{O}_{C\times S,x}\oplus\text{Ad}(\mathcal{P})_{x}\oplus\mathcal{O}_{C\times S,x}\xrightarrow{\left(\,k(-D)_{x}\circ\text{pr}_{1}-\text{pr}_{3},\,\text{pr}_{2}\right)}\,\text{Ad}(\mathcal{P})_{x}\oplus\mathcal{O}_{C\times S,x}$ In other words, $a_{x}(c,r,d,r)=(c-d,\,r)$. This implies that the image of $a_{x}$ is the submodule $\text{At}^{D,s_{i}}(\mathcal{P})_{x}$. Part (C1) of the claim follows. On the other hand the commutativity of the induced diagram ${0}$${\text{Ad}(\mathcal{P})(-D)_{x}}$${\mathcal{G}_{x}\,/\,\text{Im}(b,\,(j,0)\,)_{x}}$${\mathcal{O}_{x}}$${0}$${0}$${\text{Ad}(\mathcal{P})(-D)_{x}}$${\text{At}^{D,s_{i}}(\mathcal{P})_{x}}$${\mathcal{O}_{x}}$${0}$$\xlongequal{}$$\scriptstyle{(\text{id},\,0,0,0)}$$\scriptstyle{p_{x}}$$\xrightarrow{\sim}$$\xlongequal{}$$\scriptstyle{(\,k(-D)_{x},\,0\,)}$ follows plainly from the concrete description of $a_{x}$ we have given. This concludes the proof of the claims. ∎ ###### Remark 4.3. Using the notation of [BDP18][Prop. 3.1] when $G=\text{GL}_{n}$, we have $\gamma_{\mathcal{P}}=-\phi^{0}_{E}$, $s_{i}=A(x_{i})$ and $\delta_{i}=\gamma_{x_{i}}$. ###### Example 4.4. Set $S=\text{Spec}\,k$ and $G=\mathbb{G}_{m}$. Let $\mathcal{L}$ be a line bundle on $C$. The identifications explained in Example 3.2 induce a canonical isomorphisms $H^{1}\left(C,\,Ad\,\mathcal{L}\otimes\Omega_{C/k}^{1}\right)\cong k$ and $H^{0}\left(x_{i},\,Ad\,\mathcal{L}|_{x_{i}}\right)\cong k$. Under these identifications we have that $\delta_{i}$ is the identity for all $i\in I$. Lemma 3.3 and Proposition 4.2 imply that $\gamma_{\mathcal{L}}^{D,s_{i}}=-\text{deg}\,\mathcal{L}-\sum_{i\in I}s_{i}$. ### 4.2 Indecomposable example over elliptic curves Let $C$ be an elliptic curve over a field $k$ of characteristic not equal to $2$. Let $x\in C$ be a $k$-point, and set $D=x$. We will work with $G=SL_{2}$. Since the $k$-vector space $H^{1}(C,\mathcal{O}_{C})$ is one-dimensional, there exists a unique nonsplit extension of vector bundles $0\to\mathcal{O}_{C}\to\mathcal{E}\to\mathcal{O}_{C}\to 0$. Here $\mathcal{E}$ is a vector bundle of rank $2$, which we can think of as a $GL_{2}$-bundle. It follows from [Ati57b][Lemma 16] that $\mathcal{E}$ is indecomposable as a vector bundle. There is a canonical trivialization of the determinant $\text{det}(\mathcal{E})\xrightarrow{\sim}\text{det}(\mathcal{O}_{C})\otimes\text{det}(\mathcal{O}_{C})\xrightarrow{\sim}\mathcal{O}_{C}$. This yields a reduction of structure group to $SL_{2}$. We denote the corresponding $SL_{2}$-bundle by $\mathcal{P}$. ###### Lemma 4.5. Suppose that the characteristic of $k$ is $0$. 1. (i) $\mathcal{P}$ is $L$-indecomposable in the sense of [BBN05][Def. 2.1]. 2. (ii) Let $\text{Aut}(\mathcal{P})$ denote the automorphism group of $\mathcal{P}$ (as defined in [BBN05]). Then all tori contained in $\text{Aut}(\mathcal{P})$ are trivial. 3. (iii) The Atiyah class $\gamma_{\mathcal{P}}$ is $0$. ###### Proof. 1. (i) We use [BBN05][Prop. 2.4]. We have to check that $\mathcal{P}$ does not admit a reduction of structure group to any proper Levi sugroup of $SL_{2}$. Such Levi reduction would induce a nontrivial direct sum decomposition of the vector bundle $\mathcal{E}$, contradicting the fact that $\mathcal{E}$ is indecomposable. 2. (ii) Since the center $Z_{G}$ of $SL_{2}$ is finite, [BBN05][Defn. 2.1] implies that any torus $T\subset\text{Aut}(\mathcal{P})$ is trivial. 3. (iii) It follows from [BBN05][Remark 3.6] that the Atiyah class $\gamma_{\mathcal{P}}$ vanishes. ∎ By definition, the extension bundle $0\to\mathcal{O}_{C}\to\mathcal{E}\to\mathcal{O}_{C}\to 0$ comes from a $\mathbb{G}_{a}$-bundle. The Cech cocycle of $\mathcal{E}$ is obtained by choosing local splittings of the extension. This shows that the bundle $\mathcal{P}$ admits a reduction of structure group to the unipotent radical $U=\mathbb{G}_{a}$ of the Borel group of upper triangular matrices inside $SL_{2}$. Restrict the adjoint representation $\text{Ad}:SL_{2}\to GL(\mathfrak{sl}_{2})$ to the subgroup $U$. As a $U$-representation, $\mathfrak{sl}_{2}$ admits a filtration $0\subset\mathfrak{u}\subset\mathfrak{u}\oplus\mathfrak{t}\subset\mathfrak{sl}_{2}$ where $\mathfrak{u}$ is the Lie algebra of $U$ and $\mathfrak{t}$ is the toral Lie algebra of diagonal matrices inside $\mathfrak{sl}_{2}$. Note that $U$ acts trivially on $\mathfrak{u}$, because $U$ is commutative. Therefore the associated line bundle $\mathcal{P}\times^{U}\mathfrak{u}$ is trivial. On the other hand, a matrix computation shows that the $U$-representations $\mathfrak{u}\oplus\mathfrak{t}$ and $\mathfrak{sl}_{2}/\mathfrak{u}$ are isomorphic to the composition of the inclusion of $U\hookrightarrow SL_{2}$ and the standard representation $SL_{2}\to GL_{2}$ (here we are using the fact that the characteristic of $k$ is not $2$). It follows that $\mathcal{P}\times^{U}(\mathfrak{u}\oplus\mathfrak{t})\cong\mathcal{E}$ and $\mathcal{P}\times^{U}(\mathfrak{sl}_{2}/\mathfrak{u})\cong\mathcal{E}$. Therefore we have an induced filtration of $\text{Ad}(\mathcal{P})$ $0\subset\mathcal{O}_{C}\subset\mathcal{E}\subset\text{Ad}(\mathcal{P})$ with $\text{Ad}(\mathcal{P})/\mathcal{O}_{C}\cong\mathcal{E}$ and $\text{Ad}(\mathcal{P})/\mathcal{E}\cong\mathcal{O}_{C}$. The restriction of the filtration to $x$ yields full flag of the $3$-dimensional fiber $\text{Ad}(\mathcal{P})|_{x}$ by vector subspaces $0\subset V_{1}\subset V_{2}\subset\text{Ad}(\mathcal{P})|_{x}$ Let $\delta:H^{0}(C,\text{Ad}(\mathcal{P})|_{x})\to H^{1}(C,\text{Ad}(\mathcal{P})\otimes\Omega_{C}^{1})$ denote the boundary morphism for the short exact sequence $0\to\text{Ad}(\mathcal{P})\otimes\Omega_{C}^{1}\to\text{Ad}(\mathcal{P})\otimes\Omega_{C}^{1}(x)\to\text{Ad}(\mathcal{P})|_{x}\to 0$ as defined before Proposition 4.2. The following lemma gives an explicit description of $\delta$. ###### Lemma 4.6. With notation as above, we have: 1. (i) The $k$-vector space of obstructions $H^{1}(C,\text{Ad}(\mathcal{P})\otimes\Omega^{1})$ has dimension $1$. 2. (ii) The boundary morphism $\delta:H^{0}(C,\text{Ad}(\mathcal{P})|_{x})\to H^{1}(C,\text{Ad}(\mathcal{P})\otimes\Omega^{1})$ is surjective. 3. (iii) $V_{2}$ is the kernel of $\delta$. ###### Proof. Consider the exact sequence in cohomology $\displaystyle H^{0}(C,\text{Ad}(\mathcal{P})\otimes\Omega^{1}_{C}(x))\to H^{0}(C,\text{Ad}(\mathcal{P})|_{x})\xrightarrow{\delta}H^{1}(C,\text{Ad}(\mathcal{P})\otimes\Omega^{1}_{C})\to H^{1}(C,\text{Ad}(\mathcal{P})\otimes\Omega^{1}_{C}(x))$ Since $C$ is an elliptic curve, we have $\Omega^{1}_{C}\cong\mathcal{O}_{C}$. Using this identification plus an application of Serre duality yields the following exact sequence $\displaystyle H^{0}(C,\text{Ad}(\mathcal{P})(x))\to H^{0}(C,\text{Ad}(\mathcal{P})|_{x})\xrightarrow{\delta}H^{0}(C,\text{Ad}(\mathcal{P})^{\vee})^{*}\to H^{0}(C,\text{Ad}(\mathcal{P})^{\vee}(-x))^{*}$ Since the characteristic of $k$ is not $2$, the trace form on $\mathfrak{sl}_{2}$ is nondegenerate. This shows that the adjoint representation is self-dual, and hence $\text{Ad}(\mathcal{P})\cong\text{Ad}(\mathcal{P})^{\vee}$. Therefore we can rewrite $H^{0}(C,\text{Ad}(\mathcal{P})(x))\to H^{0}(C,\text{Ad}(\mathcal{P})|_{x})\xrightarrow{\delta}H^{0}(C,\text{Ad}(\mathcal{P}))^{*}\to H^{0}(C,\text{Ad}(\mathcal{P})(-x))^{*}$ (1) We now proceed with the proof of the lemma. 1. (i) The discussion above implies that $\text{dim}\,H^{1}(C,\text{Ad}(\mathcal{P})\otimes\Omega_{C}^{1})=\text{dim}\,H^{0}(C,\text{Ad}(\mathcal{P}))$. We know that we have filtration $0\subset\mathcal{O}_{C}\subset\mathcal{E}\subset\text{Ad}(\mathcal{P})$ with $\text{Ad}(\mathcal{P})/\mathcal{O}_{C}\cong\mathcal{E}$ and $\text{Ad}(\mathcal{P})/\mathcal{E}\cong\mathcal{O}_{C}$. This can be used to prove that $0\to\mathcal{O}_{C}\to\text{Ad}(\mathcal{P})\to\mathcal{E}\to 0$ is the unique indecomposable extension of $\mathcal{E}$ by $\mathcal{O}_{C}$ as in [Ati57b][Lemma 16]. It follows by direct computation (or [Ati57b][Lemma 15(i)]) that $\text{dim}\,H^{0}(C,\text{Ad}(\mathcal{P}))=1$. 2. (ii) We show that $H^{0}(C,\text{Ad}(\mathcal{P})(-x))=0$ in the exact sequence (1). Equivalently, we need to prove that there are no nontrivial morphisms $\mathcal{O}_{C}\to\text{Ad}(\mathcal{P})(-x)$. The subquotients of the filtration $0\subset\mathcal{O}_{C}\subset\mathcal{E}\subset\text{Ad}(\mathcal{P})$ are all isomorphic to the (stable) line bundle $\mathcal{O}_{C}$ of slope $\mu=0$. It follows that $\text{Ad}(\mathcal{P})$ is semistable of slope $0$. Therefore $\text{Ad}(\mathcal{P})(-x)$ is a semistable vector bundle of negative slope $\mu=-1$. A standard argument now shows that there are no notrivial morphisms $\mathcal{O}_{C}\to\text{Ad}(\mathcal{P})$ from the semistable bundle $\mathcal{O}_{C}$ of slope $0$ to the semistable bundle $\text{Ad}(\mathcal{P})(-x)$ of negative slope (see e.g. [HL97][Lemma 1.3.3]). 3. (iii) By the exact sequence (1), we need to prove that the image of the following composition is $V_{2}$. $H^{0}(C,\text{Ad}(\mathcal{P})(x))\to H^{0}(C,\text{Ad}(\mathcal{P})(x)|_{x})\xrightarrow{\sim}H^{0}(C,\text{Ad}(\mathcal{P})|_{x})$ By dimension count, it suffices to show that $\text{Im}(H^{0}(C,\text{Ad}(\mathcal{P})(x)))\subset V_{2}$. In other words, we want to show that the following composition is trivial. $\displaystyle H^{0}(C,\text{Ad}(\mathcal{P})(x))\to H^{0}(C,\text{Ad}(\mathcal{P})(x)|_{x})\to H^{0}(C,(\text{Ad}(\mathcal{P})/\mathcal{E})(x)|_{x})\xrightarrow{\sim}H^{0}(C,\text{Ad}(\mathcal{P})|_{x}))/V_{2}$ Choose a global section $f$ in $H^{0}(C,\text{Ad}(\mathcal{P})(x))$. We need to prove that the image in $(\text{Ad}(\mathcal{P})/\mathcal{E})(x)|_{x}$ is $0$. By reduction, $f$ yields a global section $\overline{f}$ of the quotient $(\text{Ad}(\mathcal{P})/\mathcal{E})(x)\cong\mathcal{O}_{C}(x)$. A Riemann- Roch computation shows that $\text{dim}\,H^{0}(C,\mathcal{O}_{C}(x))=\text{dim}\,H^{0}(C,\mathcal{O}_{C})=1$. This implies that the global section $\overline{f}$ must necessarily come from the subsheaf $\text{Ad}(\mathcal{P})/\mathcal{E}\cong\mathcal{O}_{C}$ of $(\text{Ad}(\mathcal{P})/\mathcal{E})(x)$. Therefore $\overline{f}$ becomes $0$ when restricted to the fiber $(\text{Ad}(\mathcal{P})/\mathcal{E})(x)|_{x}$, as desired. ∎ Now we use our previous discussion to characterize the residues that can arise from a logarithmic connection on $\mathcal{P}$. ###### Proposition 4.7. Suppose that the characteristic of $k$ is $0$. The $SL_{2}$-bundle $\mathcal{P}$ admits a logarithmic connection with residue $s\in\text{Ad}(\mathcal{P})|_{x}$ at $x$ if and only if $s$ belongs to the vector subspace $V_{2}\subset\text{Ad}(\mathcal{P})|_{x}$. ∎ ###### Proof. Let $s\in H^{0}(C,\text{Ad}(\mathcal{P})|_{x})$. Recall from Diagram 5 that the $SL_{2}$-bundle $\mathcal{P}$ admits a logarithmic connection with residue $s$ if and only if the cohomological obstruction $\gamma_{\mathcal{P}}^{D,s}$ vanishes. By Proposition 4.2, we have $\gamma_{\mathcal{P}}^{D,s}=\gamma_{\mathcal{P}}-\delta(s)$. Lemma 4.5(iii) shows that $\gamma_{\mathcal{P}}=0$. Hence $\gamma_{\mathcal{P}}^{D,s}=-\delta(s)$. We now conclude by Lemma 4.6, which says that $\delta(s)=0$ if and only if $s\in V_{2}$. ∎ We consider now the setup of [BDPS17][§1]. On [BDPS17][pg. 2] we have a maximal torus $T\subset\text{Aut}(\mathcal{P})$ and an associated Levi subgroup $H\subset G$. In our example of $SL_{2}$-bundle $\mathcal{P}$, Lemma 4.5(i) implies that $H=SL_{2}$ (the identity reduction to $SL_{2}$ satisfies both conditions at the end of page 12 in [BDPS17]). Furthermore, it follows from Lemma 4.5(ii) that $T\subset\text{Aut}(\mathcal{P})$ is trivial. In particular every residue $s\in\text{Aut}(\mathcal{P})|_{x}$ is $T$-rigid. Choose a residue $s\in\text{Ad}(\mathcal{P})|_{x}$ that does not lie in the $2$-dimensional vector subspace $V_{2}$. Then [BDPS17][Thm. 1.1] is false for the $SL_{2}$-bundle $\mathcal{P}$ and the residue $w_{x}=s$. Indeed, property (1) in [BDPS17][Thm. 1.1] is not satisfied: by Proposition 4.7 the bundle $\mathcal{P}$ does not admit a logarithmic connection with residue $s$ at $x$. On the other hand property (2) of [BDPS17][Thm. 1.1] is satisfied. This is because the only character $\chi$ of $H=SL_{2}$ is the trivial character, which satisfies $d\chi=0$ and $\text{deg}\,E_{H}(\chi)=\text{deg}\,\mathcal{O}_{C}=0$. This contradicts [BDPS17] which says that properties (1) and (2) are equivalent. [BDPS17][Thm. 1.2] does not hold in this case either, because of the same reason. The above error originates from the use of [BDPS17][Cor. 3.1] during the second half of the proof of [BDPS17][Prop. 5.1]. Let $S$ be the quotient $H/T_{G}$ of $H$ by its maximal central torus $T_{G}$, and let $E_{S}$ the $S$-bundle obtained from $E_{H}$ via the morphism $H\to S$. The reduction of structure group $E_{P}\subset E_{S}$ constructed in page 16 of [BDPS17] need not be compatible with the residues $w_{x}^{S}$ obtained from $w_{x}$ under the induced morphism of adjoint bundles $\text{Ad}(E_{H})\to\text{Ad}(E_{S})$. However, the application of [BDPS17][Cor. 3.1] requires compatibility of the reduction with the residues. One way to repair [BDPS17][Thm. 1.1, 1.2] is to impose the above condition that $w_{x_{i}}^{S}\in\text{Ad}(E_{P})|_{x_{i}}$ for all $i\in I$. Indeed the proof of [BDPS17][Prop. 5.1] then goes through. As an example, consider our $SL_{2}$-bundle $\mathcal{P}$ over an elliptic curve $C$. In this case we have $S=SL_{2}$. The space of sections $H^{0}(C,\text{Ad}(\mathcal{P}))$ is one- dimensional, and hence there is a unique nonzero adjoint section up to scaling. This section is given by the inclusion $\mathcal{O}_{C}\subset\text{Ad}(\mathcal{P})$ in the first step of the filtration described above, so it does not vanish anywhere. The corresponding parabolic subgroup $P$ constructed in [BDPS17][pg. 16] is a Borel subgroup of $S$. The Borel reduction $E_{P}$ comes from the filtration $0\subset\mathcal{O}_{C}\subset\mathcal{E}$, and the subspace $\text{Ad}(E_{P})|_{x}$ of compatible residues coincides with the two- dimensional subspace $V_{2}\subset\text{Ad}(\mathcal{P})|_{x}$. This is consistent with our Proposition 4.7. Another possible way to repair [BDPS17][Thm. 1.1, 1.2] would be to impose the additional requirement that appears in [BDPS17][pg. 15] that $\beta=0$. In our notation, this translates to the condition that $\gamma_{E_{S}}^{D,w^{S}_{x_{i}}}=-\sum_{i\in I}\delta_{i}(w^{S}_{x_{i}})$ is $0$ in $H^{1}(C,\text{Ad}(E_{S})\otimes\Omega_{C}^{1})$. Note that the bundle $E_{S}$ is $L$-indecomposable by [BBN05][Thm. 3.2], and hence $\gamma_{E_{S}}=0$. ### 4.3 Functoriality for logarithmic $G$-connections Logarithmic connections enjoy the same functoriality properties as regular connections. Let $f:T\longrightarrow S$ be a morphism of $k$-schemes. Let $\mathcal{P}$ be a $G$-bundle on $C\times S$. The same reasoning as for regular connections shows that the logarithmic Atiyah class $\gamma^{D}_{\mathcal{P}}$ behaves well under pullback. Explicilty, $\gamma^{D}_{(Id_{C}\times f)^{*}\mathcal{P}}=(Id_{C}\times f)^{*}\gamma^{D}_{\mathcal{P}}$. Here we are abusing notation and writing $(Id_{C}\times f)^{*}$ on the right-hand side to denote the natural map on the corresponding sheaf cohomology groups. It follows that the logarithmic Atiyah sequence for $(Id_{C}\times f)^{*}\mathcal{P}$ is the $(Id_{C}\times f)$-pullback of the Atiyah sequence for $\mathcal{P}$. We can therefore pullback logarithmic connections by interpreting them as splittings of the logarithmic Atiyah sequence. Let $\theta$ be a logarithmic $G$-connection on $\mathcal{P}$ with poles on $D$. We denote by $f^{*}\theta$ the corresponding logarithmic connection on $(Id_{C}\times f)^{*}\mathcal{P}$. This construction is compatible with taking residues. This implies that the obstruction classes $\gamma^{D,s_{i}}_{\mathcal{P}}$ and the corresponding Atiyah sequences with prescribed residues are compatible with pullback. We leave the precise formulation of these statements to the interested reader. Logarithmic connections have covariant functoriality with respect to morphisms of structure groups. Fix a $k$-scheme $S$. Let $H$ be a linear algebraic group over $k$ with Lie algebra $\mathfrak{h}$. Let $\varphi:G\longrightarrow H$ be a homomorphism of algebraic groups. Let $\mathcal{P}$ be a $G$-bundle over $C\times S$. Fix a set of sections $s_{i}\in H^{0}\left(x_{i}\times S,\,Ad\,\mathcal{P}|_{x_{i}\times S}\right)$. Recall that there is a naturally defined morphism of vector bundles $Ad\,\varphi:Ad\,\mathcal{P}\longrightarrow Ad\,\varphi_{*}\mathcal{P}$. Set $\varphi_{*}s_{i}\vcentcolon=Ad\,\varphi|_{x_{i}\times S}\,(s_{i})\in H^{0}\left(x_{i}\times S,\,Ad\,\mathcal{P}|_{x_{i}\times S}\right)$. The description of $\gamma_{\mathcal{P}}^{D,s_{i}}$ in Proposition 4.2 and the discussion in Subsection 3.2 imply that $\gamma_{\varphi_{*}\mathcal{P}}^{D,\,\varphi_{*}s_{i}}=\varphi_{*}^{1}(\gamma_{\mathcal{P}}^{D,s_{i}})$. Here recall that $\varphi^{1}_{*}:H^{1}\left(C,\,Ad\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)\longrightarrow H^{1}\left(C,\,Ad\,\varphi_{*}\mathcal{P}\otimes\Omega^{1}_{C\times S/S}\right)$ is the map on cohomology groups induced by $Ad\,\varphi$. The concrete description of $\varphi_{*}^{1}(\gamma_{\mathcal{P}}^{D,s_{i}})$ in terms of pushouts [Sta19, Tag 010I] shows that there is a commutative diagram of Atiyah sequences with prescribed residues ${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}}$${At^{D,s_{i}}(\mathcal{P})}$${\mathcal{O}_{C\times S}}$${0}$${0}$${\text{Ad}\,\varphi_{*}\mathcal{P}\otimes\Omega^{1}_{C\times S/S}}$${At^{D,\,\varphi_{*}s_{i}}(\varphi_{*}\mathcal{P})}$${\mathcal{O}_{C\times S}}$${0}$$\scriptstyle{Ad\,\varphi\,\otimes\,id}$$\scriptstyle{At^{D,\,s_{i}}(\varphi)}$$\xlongequal{}$ We denote by $\text{At}^{D,\,s_{i}}(\varphi)$ the middle vertical arrow in the diagram above. Let $\theta$ be a logarithmic $G$-connection on $\mathcal{P}$ with residue $s_{i}$ at each $x_{i}$, viewed as a splitting of the top row. We compose it with $At^{D,\,s_{i}}(\varphi)$ in order to define a logarithmic $H$-connection $\varphi_{*}\theta\vcentcolon=At^{D,\,s_{i}}(\varphi)\circ\theta$ on the $H$-bundle $\varphi_{*}\mathcal{P}$ with residue $\varphi_{*}s_{i}$ at $x_{i}$. ###### Example 4.8. Let $S=\text{Spec}\,k$ and $G=\text{GL}_{n}$. Consider the determinant character $\text{det}:\text{GL}_{n}\longrightarrow\mathbb{G}_{m}$. The corresponding Lie algebra map $\text{Lie}(\text{det}):\mathfrak{gl}_{n}\longrightarrow\mathfrak{gl}_{1}$ is given by taking the trace of a matrix. Let $\mathcal{E}$ be a vector bundle of rank $n$ on $C$, viewed as a $\text{GL}_{n}$-bundle. Fix an endomorphisms $s_{i}\in\text{End}(\mathcal{E}|_{x_{i}})$ for each $i\in I$. The corresponding element $\text{det}_{*}s_{i}\in\mathfrak{gl}_{1}$ is given by the trace $\text{tr}\,s_{i}$. Supose that $\mathcal{E}$ admits a logarithmic connection $\theta$ with residue $s_{i}$ at $x_{i}$. By functoriality, there is an induced connection $\text{det}_{*}\,\theta$ on $\text{det}\,\mathcal{E}$ with residue $\text{tr}\,s_{i}$ at $x_{i}$. Therefore the Atiyah class with prescribed residues $\gamma_{\text{det}\,\mathcal{E}}^{D,\,\text{tr}\,s_{i}}$ must vanish. It follows from Example 4.4 that $\text{deg}\,\mathcal{E}=-\sum_{i\in I}\text{tr}\,s_{i}$. ### 4.4 Stacks of logarithmic $G$-connections We describe the stack of logarithmic $G$-connections with a fixed set of poles. Let $D$ be a reduced divisor $D=\sum_{i\in I}x_{i}$, where $x_{i}\in C(k)$. ###### Definition 4.9. The moduli stack $\text{Conn}_{G}^{D}(C)$ is the pseudofunctor from $k$-schemes $S$ to groupoids given as follows. For every $k$-scheme $S$, we define $\text{Conn}_{G}^{D}(C)\,(S)\vcentcolon=\;\left\\{\begin{matrix}\text{groupoid of $G$-torsors}\;\mathcal{P}\text{ over }C\times S\\\ $+$\\\ \text{ a logarithmic $G$-connection $\theta$ on $\mathcal{P}$ with poles at $D$ }\end{matrix}\right\\}$ We will want to keep track of the residues of the logarithmic connection. In order to do this we define another stack over $\text{Bun}_{G}(C)$. For every $G$-bundle $\mathcal{P}$ on a $k$-scheme $S$, we can form the adjoint bundle of $Ad\,\mathcal{P}$. This construction allows us to define a vector bundle $\mathcal{V}$ over $\text{B}G$. The pullback of $\mathcal{V}$ under a map $\mathcal{P}:S\rightarrow\text{B}G$ is the associated bundle $Ad\,\mathcal{P}$ on $S$. The total space of the vector bundle is represented by the map of stacks $\pi:\left[G\,\backslash\,\mathfrak{g}\right]\longrightarrow\left[G\,\backslash\,\text{Spec}\,k\right]=\text{B}G$ Here the action in the left-most quotient stack is given by the adjoing representation. For each $i\in I$, there is a morphism of algebraic stacks $\psi_{i}:\text{Bun}_{G}(C)\longrightarrow\text{B}G$ defined by taking the fiber at $x_{i}$. More precisely, for every $k$-scheme $S$ and every $G$-bundle $\mathcal{P}$ over $C\times S$, set $\psi_{i}(\mathcal{P})=\mathcal{P}|_{x_{i}\times S}$. ###### Definition 4.10. The stack $\text{Bun}_{G}^{Ad,\,D}(C)$ is defined to be the fiber product $\textstyle{\text{Bun}_{G}^{Ad,\,D}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{Bun}_{G}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod_{i\in I}\psi_{i}}$$\textstyle{\prod_{i\in I}\left[G\,\backslash\,\mathfrak{g}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;\;\;\prod_{i\in I}\pi\;\;}$$\textstyle{\prod_{i\in I}\text{B}G}$ The definition implies that $\text{Bun}_{G}^{Ad,\,D}(C)$ is an algebraic stack that is locally of finite type over $k$. For each $k$-scheme $S$, the groupoid $\text{Bun}_{G}^{Ad,\,D}(C)\,(S)$ is naturally equivalent to the groupoid of $G$-bundles $\mathcal{P}$ on $C\times S$ along with a global section of the associated bundle $Ad\left(\mathcal{P}|_{x_{i}\times S}\right)$ for each $i\in I$. There is a natural map $Forget^{D}:\text{Conn}^{D}_{G}(C)\longrightarrow\text{Bun}_{G}^{Ad,\,D}(C)$ given by forgetting the connection and remembering only the $G$-bundle and the residue at each point $x_{i}$. We have the following result, analogous to Proposition 3.6. ###### Proposition 4.11. The map $Forget^{D}:\text{Conn}^{D}_{G}(C)\longrightarrow\text{Bun}_{G}^{Ad,\,D}(C)$ is schematic, affine and of finite type. In particular, $\text{Conn}^{D}_{G}(C)$ is an algebraic stack that is locally of finite type over $k$. ###### Proof. Let $S$ be a $k$-scheme. Let $(\mathcal{P},s_{i}):\,S\longrightarrow\text{Bun}_{G}^{Ad,\,D}(C)$ be a morphism. It consists of the data of a $G$-bundle $\mathcal{P}$ on $C\times S$ and a section $s_{i}\in H^{0}\left(x_{i}\times S,\,Ad(\mathcal{P}|_{x_{i}\times S})\,\right)$ for all $i\in I$. We want to show that the projection $\text{Conn}^{D}_{G}(C)\times_{\text{Bun}_{G}^{Ad,\,D}(C)}\,S\,\longrightarrow\,S$ is schematic, affine and of finite type. There is a short exact sequence as in Diagram 5 above. ${0}$${\text{Ad}\,\mathcal{P}\otimes\Omega^{1}_{C\times S/S}}$${At^{D,s_{i}}(\mathcal{P})}$${\mathcal{O}_{C\times S}}$${0}$ Let $\gamma_{\mathcal{P}}^{D,s_{i}}\in H^{1}\left(C\times S,\,Ad\,\mathcal{P}\times\Omega^{1}_{C\times S/S}\,\right)$ be the corresponding cohomology class. Let $Z$ denote the subfunctor of $S$ defined as follows. For any $k$-scheme $T$, $Z(T)\vcentcolon=\;\left\\{\begin{matrix}\text{morphisms $f:T\rightarrow S$ such that}\\\ \text{$(Id_{C}\times f)^{*}\,\gamma_{\mathcal{P}}^{D,s_{i}}=0$}\end{matrix}\right\\}$ Lemma 2.3 shows that $Z$ is represented by a closed subscheme of $S$. The vanishing of $\gamma_{\mathcal{P}}^{D,s_{i}}$ is a necessary condition for the existence of a logarithmic $G$-connection with residue $s_{i}$ at $x_{i}$. Therefore, the map $\text{Conn}^{D}_{G}(C)\times_{\text{Bun}_{G}^{Ad,\,D}(C)}\,S\longrightarrow S$ factors through the closed subscheme $Z$. It suffices to show that the morphism $\text{Conn}^{D}_{G}(C)\times_{\text{Bun}_{G}^{Ad,\,D}(C)}\,S\longrightarrow Z$ is schematic, affine and of finite type. The proof of this follows from similar arguments as in Proposition 3.6, using Diagram 5 instead of the Atiyah sequence. ∎ It is not necessarily true that the moduli stack of logarithmic connections $\text{Conn}_{G}^{D}(C)$ is quasicompact, even when $\text{char}\,k=0$. We illustrate this with an example of an unbounded family on $\mathbb{P}^{1}$. ###### Example 4.12. Let $C=\mathbb{P}^{1}_{k}=\text{Proj}(k[x_{0},x_{1}])$. Set $G=\mathbb{G}_{m}$. We choose a single puncture $\infty$ in $\mathbb{P}^{1}_{k}$. For each $n\geq 1$, we equip the line bundle $\mathcal{O}(-n)$ with a logarithmic connection as follows. There is a natural inclusion $\mathcal{O}(-n)\xrightarrow{x_{1}^{n}}\mathcal{O}_{\mathbb{P}^{1}_{k}}$ that identifies $\mathcal{O}(-n)$ with the ideal sheaf of the divisor $n\infty$. We compose this with the universal derivation $d:\mathcal{O}_{\mathbb{P}^{1}_{k}}\rightarrow\Omega_{\mathbb{P}^{1}_{k}/k}^{1}$ and the natural inclusion $\Omega_{\mathbb{P}^{1}_{k}/k}^{1}\hookrightarrow\Omega_{\mathbb{P}^{1}_{k}/k}^{1}(\infty)$ in order to obtain a morphism $\partial_{n}:\mathcal{O}(-n)\hookrightarrow\mathcal{O}_{\mathbb{P}^{1}_{k}/k}\xrightarrow{d}\Omega_{\mathbb{P}^{1}_{k}/k}^{1}\hookrightarrow\Omega_{\mathbb{P}^{1}/k}^{1}(\infty)$ This can be described in coordinates in the following way. Over the affine open $\mathbb{A}^{1}_{\frac{x_{0}}{x_{1}}}$ we have $\partial_{n}\left(f\cdot\frac{1}{x_{1}^{n}}\right)=df$ On the other hand, over the affine open $\mathbb{A}^{1}_{\frac{x_{1}}{x_{0}}}$ we have $\partial_{n}\left(f\cdot\frac{1}{x_{0}^{n}}\right)=\left(\frac{x_{1}}{x_{0}}\right)^{n}\otimes df+nf\left(\frac{x_{1}}{x_{0}}\right)^{n}\otimes\frac{d\left(\frac{x_{1}}{x_{0}}\right)}{\left(\frac{x_{1}}{x_{0}}\right)}$ This description shows that $\partial_{n}$ factors through the subsheaf $\mathcal{O}(-n)\otimes\Omega_{\mathbb{P}_{1}/k}^{1}(\infty)\subset\Omega_{\mathbb{P}^{1}/k}^{1}(\infty)$. This way we obtain a logarithmic connection $\partial_{n}:\mathcal{O}(-n)\rightarrow\mathcal{O}(-n)\otimes\Omega_{\mathbb{P}^{1}/k}^{1}(\infty)$ defined on $\mathcal{O}(-n)$. The residue at the pole is $n$. This example generalizes to the case when $G$ is an arbitrary nontrivial connected reductive group. ###### Lemma 4.13. Let $G$ be a nontrivial connected reductive group over $k$. Suppose that $D$ is nonempty. The stack of logarithmic connections $\text{Conn}_{G}^{D}(C)$ is not of finite type over $k$. ###### Proof. Since the property of being of finite type is stable under base-change, we can replace the ground field $k$ with an algebraic closure. For the rest of the proof we assume that $k$ is algebraically closed. Let $T$ be a maximal torus in $G$. By assumption, $T$ is a nontrivial split torus. We fix an identification $T\xrightarrow{\sim}\mathbb{G}_{m}^{n}$ for some $m\geq 1$. A $T$-bundle on $C$ is equivalent to a $n$-tuple $\vec{\mathcal{L}}=(\mathcal{L}_{j})_{j=1}^{n}$ of line bundles on $C$. For such $n$-tuple $\vec{\mathcal{L}}$, we define a $n$-tuple of integers $\vec{\text{deg}}(\vec{\mathcal{L}})\vcentcolon=(\text{deg}\,\mathcal{L}_{j})_{j=1}^{n}$. The tuple $\vec{\text{deg}}(\vec{\mathcal{L}})$ can be interpreted more canonically as a cocharacter in $X_{*}(T)$. This is the degree of the $T$-bundle $\vec{\mathcal{L}}$ as in [Sch15][2.2.2]. Choose a Borel subgroup of $G$ containing $T$. Let $\lambda\in X_{*}(T)$ be a cocharacter of $T$ contained in the interior of the cone of $B$-dominant cocharacters. Using the identification $T\xrightarrow{\sim}\mathbb{G}_{m}^{n}$ we think of $\lambda$ as a tuple of integers $(m_{j})_{j=1}^{n}$. Since $k$ is algebraically closed, there exist line bundles $\mathcal{L}_{j}^{\lambda}$ such that $\text{deg}\,\mathcal{L}_{j}^{\lambda}=m_{j}$ for each $j$. We can think of the tuple $\vec{\mathcal{L}}^{\lambda}=(\mathcal{L}_{j})_{j=1}^{n}$ as a $T$-bundle. Let $i\in I$. By Example 4.4, there is a canonical identification $H^{0}(x_{i},\text{Ad}(\mathcal{L}_{j}^{\lambda}|_{x_{i}})\cong k$. Set $s_{i,j}=-\frac{1}{|I|}\text{deg}\,\mathcal{L}^{\lambda}_{j}$. Example 4.4 shows that the obstruction $\gamma_{\mathcal{L}_{j}^{\lambda}}^{D,s_{i,j}}$ satisfies $\gamma_{\mathcal{L}_{j}^{\lambda}}^{D,s_{i,j}}=0$. Therefore the line bundle $\mathcal{L}_{j}^{\lambda}$ admits a logarithmic connection $\theta_{j}^{\lambda}$ with residue $s_{i,j}$ at $x_{i}$. The pair of $n$-tuples $\left(\vec{\mathcal{L}}^{\lambda},\vec{\theta}^{\lambda}\right)\vcentcolon=\left(\mathcal{L}_{j}^{\lambda},\theta_{j}^{\lambda}\right)_{j=1}^{n}$ is a $T$-bundle equipped with a logarithmic connection with poles at $D$. Let $\rho$ denote the inclusion $\rho:T\hookrightarrow G$. By functoriality of logarithmic connections, there is an associated $G$-bundle with logarithmic connection $(\mathcal{G}^{\lambda},\nu^{\lambda})\vcentcolon=\rho_{*}\left(\,\left(\vec{\mathcal{L}}^{\lambda},\vec{\theta}^{\lambda}\right)\,\right)$ This way we get an infinite set of $G$-bundles $\\{\mathcal{G}^{\lambda}\\}_{\lambda}$ indexed by the cocharacters $\lambda\in X_{*}(T)$ that are contained in the interior of the $B$-dominant cone. Each $\mathcal{G}^{\lambda}$ admits a logarithmic connection $\nu^{\lambda}$ with poles at $D$. In order to conclude the proposition, we shall show that the set $\\{\mathcal{G}^{\lambda}\\}_{\lambda}$ is unbounded in $\text{Bun}_{G}(C)$. For each $\lambda$, we have $\mathcal{G}^{\lambda}=\rho_{*}(\vec{\mathcal{L}}^{\lambda})$. Let $\psi$ denote the inclusion $\psi:T\hookrightarrow B$. The $T$-bundle $\vec{\mathcal{L}}^{\lambda}$ is semistable in the sense of [Sch15][2.2.3], because $T$ does not have nontrivial proper parabolic subgroups. By construction, $\lambda$ is the degree of $\psi_{*}(\vec{\mathcal{L}}^{\lambda})$ as in [Sch15][2.2.2]. Since $\lambda$ is in the interior of the $B$-dominant cone, it follows that $\psi_{*}(\vec{\mathcal{L}}^{\lambda})$ is the canonical reduction of $\mathcal{G}^{\lambda}$ [Sch15][2.4.1]. Therefore the cocharacter $\lambda$ is the Harder-Narasimhan type of $\mathcal{G}^{\lambda}$. Hence the $G$-bundles $\\{\mathcal{G}^{\lambda}\\}_{\lambda}$ have distinct Harder-Narasimhan types. Since the image of each Harder-Narasimhan stratum is constructible in $\text{Bun}_{G}(C)$ [Sch15][Thm. 2.3.3(a)], it follows that the set $\\{\mathcal{G}^{\lambda}\\}_{\lambda}$ must be unbounded. ∎ Let’s return to the case when $G$ is an arbitrary smooth connected linear algebraic group. Using Lemma 4.13 we can prove the following. ###### Proposition 4.14. Suppose that $D$ is nonempty and that the characteristic of $k$ is $0$. Then the stack $\text{Conn}_{G}^{D}(C)$ is of finite type over $k$ if and only if the group $G$ is unipotent. ###### Proof. Assume that the group $G$ is unipotent. Consider the composition $h_{G}:\text{Conn}^{D}_{G}(C)\xrightarrow{\text{Forget}^{D}}\text{Bun}_{G}^{Ad,D}(C)\rightarrow\text{Bun}_{G}(C)$ By definition, $\text{Bun}_{G}^{Ad,D}(C)\rightarrow\text{Bun}_{G}(C)$ is of finite type. Proposition 4.11 shows that $\text{Forget}^{D}$ is also of finite type. Therefore, the composition $h_{G}$ is of finite type. By Proposition 2.5 (iv), $\text{Bun}_{G}(C)$ is of finite type over $k$. We conclude that $\text{Conn}_{G}^{D}(C)$ is of finite type over $k$. Conversely, suppose that $G$ is not unipotent. Let $U$ denote the unipotent radical of $G$. We have a short exact sequence of algebraic groups $1\rightarrow U\rightarrow G\rightarrow\overline{G}\rightarrow 1$ where $\overline{G}$ is a nontrivial connected reductive group. Since the characteristic of $k$ is $0$, the short exact sequence above admits a splitting. Therefore we can view $G=U\rtimes\overline{G}$. Consider the chain of morphisms $\overline{G}\xhookrightarrow{\iota}U\rtimes\overline{G}=G\xtwoheadrightarrow{q}\overline{G}$ By functoriality of logarithmic connections, this induces a chain of morphisms of stacks $\text{Conn}^{D}_{\overline{G}}(C)\xrightarrow{\iota_{*}}\text{Conn}_{G}^{D}(C)\xrightarrow{q_{*}}\text{Conn}_{\overline{G}}^{D}(C)$ By definition, the composition $q_{*}\circ\iota_{*}$ is the identity. Hence $\iota_{*}$ exhibits $\text{Conn}^{D}_{\overline{G}}(C)$ as a subfunctor of $\text{Conn}^{D}_{G}(C)$. Assume for the sake of contradiction that $\text{Conn}^{D}_{G}(C)$ is quasicompact. This would imply that the substack $\text{Conn}^{D}_{G}(C)$ is quasicompact, thus contradicting Lemma 4.13. ∎ For each $i\in I$, choose an orbit $O_{i}$ for the adjoint action of $G$ on its Lie algebra $\mathfrak{g}$. Each orbit $O_{i}$ is a smooth locally closed subscheme of $\mathfrak{g}$. After quotienting by the action of $G$, we get a locally closed substack $\prod_{i\in I}\left[G\,\backslash\,O_{i}\right]\,\hookrightarrow\,\prod_{i\in I}\left[G\,\backslash\,\mathfrak{g}\right]$ ###### Definition 4.15. $\text{Conn}_{G}^{D,O_{i}}(C)$ is defined to be the locally closed substack of $\text{Conn}_{G}^{D}(C)$ given by the following cartesian diagram $\textstyle{\text{Conn}_{G}^{D,O_{i}}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{Conn}^{D}_{G}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Forget^{D}}$$\textstyle{\prod_{i\in I}\left[G\,\backslash\,O_{i}\right]\times_{\prod_{i\in I}\left[G\,\backslash\,\mathfrak{g}\right]}\text{Bun}_{G}^{\text{Ad},D}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{Bun}_{G}^{Ad,\,D}(C)}$ The stack $\text{Conn}_{G}^{D,O_{i}}(C)$ parametrizes $G$-bundles with logarithmic connections whose residue at $x_{i}$ lies in the orbit $O_{i}$. ###### Theorem 4.16. Suppose that $\text{char}\,k=0$. The algebraic stack $\text{Conn}_{G}^{D,O_{i}}(C)$ is of finite type over $k$. ###### Proof. By Proposition 4.11, the map $Forget^{D}$ is of finite type. By definition the map $\text{Bun}^{Ad,\,D}_{G}(C)\longrightarrow\text{Bun}_{G}(C)$ is of finite type. Hence it suffices to show that the composition $h_{G}:\text{Conn}_{G}^{D,O_{i}}(C)\hookrightarrow\text{Bun}_{G}^{D}(C)\xrightarrow{Forget^{D}}\text{Bun}_{G}^{Ad,\,D}(C)\longrightarrow\text{Bun}_{G}(C)$ factors through a quasicompact open substack of $\text{Bun}_{G}(C)$. Let $U$ be the unipotent radical of $G$. We denote by $\rho_{G}:G\rightarrow G/U$ the correspoding quotient morphism. Choose a faithful representation $\rho:G/U\longrightarrow\text{GL}_{n}$ of the reductive group $G/U$. Let $(\rho\circ\rho_{G})_{*}O_{i}$ denote the unique $\text{GL}_{n}$-orbit in $\mathfrak{gl}_{n}$ containing the image $(\rho\circ\rho_{G})(O_{i})$. The functoriality properties described in Subsection 4.3 imply that there is a commutative diagram $\textstyle{\text{Conn}_{G}^{D,O_{i}}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{G}}$$\textstyle{\text{Conn}_{G/U}^{D,\,(\rho_{G})_{*}(O_{i})}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{G/U}}$$\textstyle{\text{Conn}_{\text{GL}_{n}}^{D,\,(\rho\circ\rho_{G})_{*}(O_{i})}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\text{GL}_{n}}}$$\textstyle{\text{Bun}_{G}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\rho_{G})_{*}}$$\textstyle{\text{Bun}_{G/U}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{*}}$$\textstyle{\text{Bun}_{\text{GL}_{n}}(C)}$ The bottom horizontal maps are of finite type by Propositions 2.4 and 2.5. Therefore it suffices to show that $\text{Conn}^{D,\,(\rho\circ\rho_{G})_{*}O_{i}}_{\text{GL}_{n}}(C)$ is quasicompact. We shall establish that $h_{\text{GL}_{n}}$ factors through a quasicompact open substack of $\text{Bun}_{\text{GL}_{n}}(C)$. Just as in the proof of Proposition 3.7, it suffices to check that the set of Harder- Narasimhan polygons of geometric points $\overline{t}\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$ with nonempty fiber $\text{Conn}^{D,O_{i}}_{\text{GL}_{n}}(C)_{\overline{t}}$ is a finite set. Let $\mu=(\mu_{1}\geq\mu_{2}\geq...\geq\mu_{n})$ be a tuple of rational numbers in $\frac{1}{n!}\mathbb{Z}$. Let $\mathcal{E}:\overline{t}\longrightarrow\text{Bun}_{\text{GL}_{n}}(C)$ be a vector bundle on $C\times\overline{t}$, where $\overline{t}$ is Spec of an algebrically closed field. Suppose that the Harder-Narasimhan polygon of $\mathcal{E}$ is given by $\mu$. Assume that the fiber $\text{Conn}^{D,O_{i}}_{\text{GL}_{n}}(C)_{\overline{t}}$ is nonempty. This means that $\mathcal{E}$ admits a logarithmic connection $\theta$ with residue $\text{Res}_{x_{i}}\theta$ in the conjugacy class $O_{i}|_{\kappa(\overline{t})}$. Set $s_{i}\vcentcolon=\text{Res}_{x_{i}}\theta$. After choosing a trivialization of $\mathcal{E}|_{x_{i}\times\overline{t}}$, the endomorphism $s_{i}$ is represented by a matrix in $\mathfrak{gl}_{n}\otimes\kappa(\overline{t})$. Let $(\lambda_{i}^{(l)})_{l=1}^{n}$ be the tuple of eigenvalues of $s_{i}$. This tuple does not depend up to permutation on the choice of trivialization of $\mathcal{E}|_{x_{i}\times\overline{t}}$. Let $B(\\{1,2,...,n\\})$ denote the power set of $\\{1,2,...,n\\}$. Define the set of $I$-tuples $A\vcentcolon=\left\\{(J_{i})_{i\in I}\in B(\\{1,2,...,n\\})^{I}\,\mid\,J_{i}\neq\emptyset\;\text{for all $i\in I$}\;\;\,\text{and}\;\,\sum_{i\in I}\sum_{j\in J_{i}}\lambda_{i}^{(j)}\in\mathbb{Z}\right\\}$ Set $M\vcentcolon=\underset{(J_{i})\in A}{\text{max}}\,\left\lvert\sum_{i\in I}\sum_{j\in J_{i}}\lambda_{i}^{(j)}\right\rvert$. The quantity $M$ does not depend on the choice of $\mathcal{E}$, because the eigenvalues are completely determined by the conjugacy classes $(\rho\circ\rho_{G})_{*}(O_{i})$ of matrices. We claim that $\mu$ satisfies the following two conditions 1. (a) $\sum_{j=1}^{n}\mu_{j}=-\sum_{i\in I}\text{tr}\,s_{i}$. 2. (b) $\mu_{1}\leq\left(\text{max}\\{4g-3-|I|,0\\}\right)\cdot n+M$. The claim implies that there are finitely many possibilities for the Harder- Narasimhan polygon $\mu\in\frac{1}{n!}\mathbb{Z}^{n}$. Hence we are reduced to showing the claim. 1. (a) This follows from the discussion in Example 4.8, because $\sum_{j=1}^{n}\mu_{j}=\text{deg}\,\mathcal{E}$. 2. (b) Set $m\vcentcolon=\text{max}\\{4g-3-|I|,0\\}$. Suppose that the Harder- Narasimhan filtration of $\mathcal{E}$ is given by $0\subset\mathcal{F}_{1}\subset\mathcal{F}_{2}\subset...\subset\mathcal{F}_{l}=\mathcal{E}$ We use the interpretation of logarithmic $\text{GL}_{n}$-connections as $\Lambda$-modules, as in [Sim94a][§2]. Let $\mathcal{D}_{C\times\overline{t}/\,\overline{t}}$ denote the usual sheaf of rings of differential operators on $C\times\overline{t}$ [Sim94a][§2, pg. 85]. Define $\Lambda$ to be the subsheaf of rings of $\mathcal{D}_{C\times\overline{t}/\,\overline{t}}$ generated by $\Omega^{1}_{C\times\overline{t}/\overline{t}}(D)^{\vee}\subset T_{C\times\overline{t}/\overline{t}}$ (see [Sim94a][§2, pg. 87]). Recall that there is a filtration of $\mathcal{D}_{C\times\overline{t}/\,\overline{t}}$ given by the order of the differential operator. This induces a filtration on the subsheaf $\Lambda\subset\mathcal{D}_{C\times\overline{t}/\,\overline{t}}$, which endows $\Lambda$ with the structure of a sheaf ring of differential operators as defined in [Sim94a][§2, pg.77]. The first graded piece $\text{Gr}_{1}\Lambda$ associated to the filtration is $\text{Gr}_{1}\Lambda=\Omega_{C\times\overline{t}/\,\overline{t}}^{1}(D)^{\vee}$. Observe that the twist $\Omega_{C\times\overline{t}/\,\overline{t}}^{1}(D)^{\vee}(m)$ is generated by global sections. This follows from a standard cohomology argument using Serre duality. The proof of Lemma 3.3 in [Sim94a][§3] shows that $\mu_{1}=\mu(\mathcal{F}_{1})\leq mn+\mu(\mathcal{G})$, where $\mathcal{G}$ is the smallest subbundle containing $\mathcal{F}_{1}$ and preserved by the logarithmic connection $\theta$. We will show that any subsheaf $\mathcal{G}\subset\mathcal{E}$ preserved by the logarithmic connection satisfies $\mu(\mathcal{G})\leq M$. Restrict the logarithmic connection $\theta$ to the subbundle $\mathcal{G}$ in order to obtain a logarithmic connection $\theta^{\mathcal{G}}$ on $\mathcal{G}$. By assumption the residue $s_{i}\in\text{End}(\mathcal{E}|_{x_{i}\times\overline{t}})$ preserves the subspace $\mathcal{G}|_{x_{i}\times\overline{t}}$. The residue $s_{i}^{\mathcal{G}}\vcentcolon=\text{Res}_{x_{i}}\,\theta^{\mathcal{G}}$ is the restriction of $s_{i}$ to $\mathcal{G}|_{x_{i}\times\overline{t}}$. For all $i\in I$, there is a nonempty subset $J_{i}\subset\\{1,2,...,n\\}$ such that the eigenvalues of $s_{i}^{\mathcal{G}}$ are $(\lambda_{i}^{(j)})_{j\in J_{i}}$. The discussion in Example 4.8 applied to $\mathcal{G}$ shows that $\text{deg}\,\mathcal{G}=-\sum_{i\in I}\text{tr}\,s_{i}^{\mathcal{G}}=-\sum_{i\in I}\sum_{j\in J_{i}}\lambda_{i}^{(j)}$ Since $\text{deg}\,\mathcal{G}\in\mathbb{Z}$, we have $(J_{i})_{i\in I}\in A$. By definition, the right hand side is bounded by $M$. This implies that $\mu(\mathcal{G})\leq M$, thus concluding the proof of the claim. ∎ ## References * [Ati57a] M. F. Atiyah. Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc., 85:181–207, 1957. * [Ati57b] M. F. Atiyah. Vector bundles over an elliptic curve. Proc. London Math. Soc. (3), 7:414–452, 1957. * [BBN05] V. Balaji, Indranil Biswas, and D. S. Nagaraj. Krull-Schmidt reduction for principal bundles. J. Reine Angew. Math., 578:225–234, 2005. * [BDP18] Indranil Biswas, Ananyo Dan, and Arjun Paul. Criterion for logarithmic connections with prescribed residues. Manuscripta Math., 155(1-2):77–88, 2018. * [BDPS17] Indranil Biswas, Ananyo Dan, Arjun Paul, and Arideep Saha. Logarithmic connections on principal bundles over a riemann surface. International Journal of Mathematics, 28(12):1750088, Nov 2017. * [Beh91] Kai Achim Behrend. The Lefschetz trace formula for the moduli stack of principal bundles. 1991\. 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Introduction to affine group schemes, volume 66 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1979. Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, New York 14853, USA E-mail address<EMAIL_ADDRESS>

2024-09-04T02:54:54.830041

2002.11837

arxiv-papers

# Full Duplex Hybrid A/D Beamforming with Reduced Complexity Multi-Tap Analog Cancellation George C. Alexandropoulos1, Md Atiqul Islam2, and Besma Smida2 This work was partially funded by the National Science Foundation CAREER award #1620902. 1Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Greece 2Department of Electrical and Computer Engineering, University of Illinois at Chicago, USA emails<EMAIL_ADDRESS>{mislam23<EMAIL_ADDRESS> ###### Abstract Although the hardware complexity of the analog self-interference canceller in full duplex Multiple Input Multiple Output (MIMO) designs does not necessarily scale with the number of transceiver antennas, exploiting the benefits of analog cancellation in massive MIMO systems with hundreds of antenna elements is still quite impractical. Hybrid Analog and Digital (A/D) beamforming architectures have been lately considered as a candidate technology for realizing massive MIMO transceivers with very large number of antenna elements, but with much fewer numbers of Radio Frequency (RF) chains. In this paper, we present a novel architecture for full duplex hybrid A/D beamforming transceivers including multi-tap analog cancellation with reduced number of taps and simple multiplexers for efficient signal routing among the transceiver RF chains. Capitalizing on the proposed transceiver architecture, we present a joint design of analog cancellation and A/D beamforming with the objective to maximize the achievable full duplex rate performance. Representative millimeter wave simulation results demonstrate the effectiveness of the proposed architecture and algorithmic framework for enabling simultaneous uplink and downlink communications with reduced complexity analog self-interference cancellation. ###### Index Terms: Analog cancellation, full duplex, hybrid beamforming, joint optimization, multi-user communication, massive MIMO. ## I Introduction In band full duplex, also known shortly as Full Duplex (FD), is a candidate technology for the Release $17$ of the fifth Generation (5G) New Radio (NR) standard enabling simultaneous UpLink (UL) and DownLink (DL) communication within the entire frequency band [1]. An FD radio can transmit and receive at the same time and frequency resource units, consequently, it can double the spectral efficiency achieved by a Half Duplex (HD) radio. Current wireless systems exploit Multiple Input Multiple Output (MIMO) communication, where increasing the number of Transmitter (TX) and Receiver (RX) antennas can increase the spatial Degrees of Freedom (DoF), hence boosting rate performance. Combining FD with MIMO operation can provide further spectral efficiency gains [2, 3, 4, 5, 6, 7, 8]. FD radios suffer from Self Interference (SI), a term referring to the signal transmitted by the FD radio TX that leaks to the FD radio RX. At the RX of the FD radio, the SI power can be many times stronger than the power of the received signal of interest. Consequently, SI can severely degrade the reception of the signal of interest, and thus SI mitigation is required in order to maximize the spectral efficiency gain of the FD operation. As the number of antennas increases, mitigating SI becomes more challenging, since more antennas naturally result in more SI components. Conventional SI suppression techniques in Single-Input Single-Output (SISO) systems include propagation domain isolation, analog domain suppression, and digital cancellation[4, 9]. Although analog SI cancellation in FD MIMO systems can be implemented through SISO replication, its hardware requirements scale with the number of TX/RX antennas. The authors in [2, 5] presented spatial suppression techniques that alleviate the need for analog SI cancellation, which was replaced solely by digital TX/RX beamforming. In [10], a joint design of multi-tap analog cancellation and TX/RX beamforming, where the number of taps does not scale with the product of TX and RX antenna elements, was proposed. The FD technology has been lately theoretically combined with Hybrid analog and digital BeamForming (HBF) [11] to enable simultaneous UL and DL communications in massive MIMO systems operating in the millimeter wave frequency bands [12, 13, 14, 15]. These works mainly assigned the role of SI mitigation to the hybrid beamformers and/or deployed analog SI cancellation that scales with the number of TX/RX antennas. In this paper, we present a novel hardware architecture for FD HBF systems enabling the joint design of A/D TX/RX beamformers with reduced complexity tap-based analog cancellation. The proposed analog canceller interconnects a subset of the outputs of the TX Radio Frequency (RF) chains to a subset of the inputs to the RX RF chains in order to ensure that the residual SI signal after A/D TX precoding and analog RX combining remains below the noise floor. Our indicative simulation results with the proposed architecture and an example FD HBF algorithmic framework showcase a $1.7$ times rate improvement over HD HBF communication. Figure 1: The considered bidirectional communication system with the proposed FD HBF architecture in the MIMO node $k$ including $N$-tap analog cancellation and A/D TX/RX beamforming. The HD multi-antenna nodes $q$ and $m$ communicate with node $k$ in the DL and UL directions, respectively. Each TX RF chain consists of a Digital to Analog Converter (DAC), a mixer upconverting the signal from BaseBand (BB) to RF, and a Power Amplifier (PA). A RX RF chain consists of a Low Noise Amplifier (LNA), a mixer downconverting the signal from RF to BB, and an Analog to Digital Converter (ADC). Upsampling and pulse shaping are used to prepare the BB signal for DAC and RF transmission at the TX side, whereas matched filtering and downsampling are used at the RX side before BB processing of the received RF signal. Notation: Vectors and matrices are denoted by boldface lowercase and boldface capital letters, respectively. The transpose and Hermitian transpose of $\mathbf{A}$ are denoted by $\mathbf{A}^{\rm T}$ and $\mathbf{A}^{\rm H}$, respectively, and $\det(\mathbf{A})$ is the determinant of $\mathbf{A}$, while $\mathbf{I}_{n}$ ($n\geq 2$) is the $n\times n$ identity matrix and $\mathbf{0}_{n\times m}$ ($n,m\geq 2$) is a $n\times m$ matrix with zeros. $\|\mathbf{A}\|_{\rm F}$ is the Frobenius norm of $\mathbf{A}$, $\|\mathbf{a}\|$ stands for $\mathbf{a}$’s Euclidean norm, and ${\rm diag}\\{\mathbf{a}\\}$ denotes a square diagonal matrix with $\mathbf{a}$’s elements in its main diagonal. $[\mathbf{A}]_{i,j}$ represents $\mathbf{A}$’s $(i,j)$-th element, while $[\mathbf{a}]_{i}$ denotes the $i$-th element of $\mathbf{a}$. $\mathbb{R}$ and $\mathbb{C}$ represent the real and complex number sets, respectively, and $|\cdot|$ denotes the amplitude of a complex number. ## II System Model and Proposed Architecture ### II-A System Model We consider the $3$-user bidirectional communication system in Fig. 1 comprising of a FD MIMO node $k$ equipped with $N_{k}$ TX and $M_{k}$ RX antenna elements, and two HD multi-antenna nodes $q$ and $m$ having $M_{q}$ and $N_{m}$ antennas, respectively. It is assumed that node $k$ communicates simultaneously (in the same time and frequency resources) with node $q$ in the DL and node $m$ in the UL. All nodes are considered capable of performing digital beamforming, which for simplicity we assume to be realized with linear filters. Node $k$ is also capable of analog TX/RX beamforming using the partially connected HBF architecture [11], as will be detailed in the sequel. It is assumed that node $m$ makes use of the digital precoding matrix $\mathbf{V}_{m}^{(\mathrm{BB})}\in\mbox{$\mathbb{C}$}^{N_{m}\times d_{m}}$ for processing in BaseBand (BB) its unit power symbol vector $\mathbf{s}_{m}\in\mbox{$\mathbb{C}$}^{d_{m}\times 1}$ (chosen in practice from a discrete modulation set) before UL transmission. The dimension of $\mathbf{s}_{m}$ satisfies $d_{m}\leq\min\\{M_{k}^{(\mathrm{RF})},N_{m}\\}$ with $M_{k}^{(\mathrm{RF})}$ denoting the number of RX RF chains at node $k$. It holds $M_{k}^{(\mathrm{RF})}\leq M_{k}$, although in practical systems it can be $M_{k}^{(\mathrm{RF})}\ll M_{k}$. The constraint for $d_{m}$ certifies data decodability for the considered UL communication. It additionally holds that $\mathbb{E}\\{\|\mathbf{V}_{m}^{(\mathrm{BB})}\mathbf{s}_{m}\|^{2}\\}\leq{\rm P}_{m}$, where ${\rm P}_{m}$ is the total TX power of node $m$. On the DL, the reception node $q$ applies the digital combining matrix $\mathbf{U}_{q}^{(\mathrm{BB})}\in\mbox{$\mathbb{C}$}^{N_{q}\times d_{k}}$ in the BB received signal that includes the unit power symbol vector $\mathbf{s}_{k}\in\mbox{$\mathbb{C}$}^{d_{k}\times 1}$ (again chosen from a discrete modulation set) transmitted from node $k$ such that $d_{k}\leq\min\\{M_{q},N_{k}^{(\mathrm{RF})}\\}$ with $N_{k}^{(\mathrm{RF})}$ ($N_{k}^{(\mathrm{RF})}\leq N_{k}$, but practically it can be $N_{k}^{(\mathrm{RF})}\ll N_{k}$) denoting the number of TX RF chains at node $k$. Similarly, the latter constraint verifies the spatial DoF of the effective $M_{q}\times N_{k}^{(\mathrm{RF})}$ DL MIMO channel between the TX RF chains of node $k$ and the RX RF chains of node $q$. It is noted that each antenna at node $q$ is connected to a dedicated RF chain. ### II-B Proposed FD HBF Hardware Architecture The proposed FD HBF hardware architecture, comprising of $N$-tap analog cancellation for the SI signal as well as A/D precoding and combining for the outgoing and incoming signals, is adopted for the MIMO node $k$, as depicted in the left part of Fig. 1. Differently from [10]’s architecture that considered only fully digital TX/RX beamforming, node $k$ is capable of HBF through its partially connected beamforming architecture. As shown in the figure, the analog canceller interconnects the $N_{k}^{(\mathrm{RF})}$ inputs of the analog TX precoder to the $M_{k}^{(\mathrm{RF})}$ outputs of the analog RX combiner. The complexity of the analog canceller expressed in the number of taps $N$ is independent of the numbers $N_{k}$ and $M_{k}$ of the TX and RX antennas, respectively, and as it will be shown next, it scales with the product $\xi N_{k}^{(\mathrm{RF})}M_{k}^{(\mathrm{RF})}$ with $\xi<1$. This is in contrast to [10] where the analog canceller interconnects the $N_{k}$ TX antenna inputs to the $M_{k}$ RX antenna outputs. #### II-B1 Partially Connected HBF Each of the $N_{k}^{(\mathrm{RF})}$ TX RF chains of node $k$ is connected to a separate subset of the available TX antenna elements. As shown in Fig. 1, the $i$-th TX RF chain with $i=1,2,\ldots,N_{k}^{(\mathrm{RF})}$ is connected via phase shifters with $N_{k}^{({\rm A})}$ TX antenna elements, each denoted as ${\rm TX}(i,j)$ $\forall$$j=1,2,\ldots,N_{k}^{({\rm A})}$. Clearly, it holds $N_{k}=N_{k}^{(\mathrm{RF})}N_{k}^{({\rm A})}$ for the total number of TX antennas at node $k$. Stacking the values of the $N_{k}^{({\rm A})}$ phase shifters that connect each $i$-th TX RF chain with its antenna elements in a complex-valued $N_{k}^{({\rm A})}\times 1$ vector $\mathbf{v}_{i}$, we can formulate the complex-valued $N_{k}\times N_{k}^{(\mathrm{RF})}$ analog TX precoder, as follows: $\mathbf{V}_{k}^{(\mathrm{RF})}=\left[\begin{matrix}\mathbf{v}_{1}&\mathbf{0}_{N_{k}^{({\rm A})}\times 1}&\cdots&\mathbf{0}_{N_{k}^{({\rm A})}\times 1}\\\ \mathbf{0}_{N_{k}^{({\rm A})}\times 1}&\mathbf{v}_{2}&\cdots&\mathbf{0}_{N_{k}^{({\rm A})}\times 1}\\\ \vdots&\vdots&\ddots&\vdots\\\ \mathbf{0}_{N_{k}^{({\rm A})}\times 1}&\mathbf{0}_{N_{k}^{({\rm A})}\times 1}&\cdots&\mathbf{v}_{N_{k}^{({\rm RF})}}\end{matrix}\right].$ (1) The elements of each $\mathbf{v}_{i}$ are assumed to have constant magnitude, i.e., $|[\mathbf{v}_{i}]_{n}|^{2}=1/N_{k}^{({\rm A})}$ $\forall$$n=1,2,\ldots,N_{k}^{({\rm A})}$. We also assume that $\mathbf{v}_{i}\in\mathbb{F}_{\rm TX}$ $\forall$$i=1,2,\ldots,N_{k}^{(\mathrm{RF})}$, which means that all analog TX precoding vectors belong in a predefined beam codebook $\mathbb{F}_{\rm TX}$ including ${\rm card}(\mathbb{F}_{\rm TX})$ distinct vectors (or analog beams). Apart from applying $\mathbf{V}_{k}^{(\mathrm{RF})}$ in the analog domain to the signal before transmission, the symbol vector $\mathbf{s}_{k}$ is also processed in BB with the digital TX precoder $\mathbf{V}_{k}^{(\mathrm{BB})}\in\mbox{$\mathbb{C}$}^{N_{k}^{(\mathrm{RF})}\times d_{k}}$ (recall that $d_{k}\leq\min\\{M_{q},N_{k}^{(\mathrm{RF})}\\}$) before entering into the $N_{k}^{(\mathrm{RF})}$ TX RF chains, as shown in Fig. 1. Similar to the UL communication from node $m$ to $k$, we assume that the DL transmission from node $k$ to $q$ is power limited according to $\mathbb{E}\\{\|\mathbf{V}_{k}^{(\mathrm{RF})}\mathbf{V}_{k}^{(\mathrm{BB})}\mathbf{s}_{k}\|^{2}\\}\leq{\rm P}_{k}$ with ${\rm P}_{k}$ being the total available TX power at node $k$. The RX of node $k$ is composed of an analog combiner connecting the RX antenna elements to the inputs of the RX RF chains, and a digital combiner that processes the outputs of the RX RF chains in BB before signal decoding. In particular, the $n$-th RX RF chain with $n=1,2,\ldots,M_{k}^{(\mathrm{RF})}$ is connected through phase shifters with $M_{k}^{({\rm A})}$ distinct RX antennas; these phase shifters are denoted as ${\rm RX}(n,\ell)$ $\forall$$\ell=1,2,\ldots,M_{k}^{({\rm A})}$. It should hold that $M_{k}=M_{k}^{(\mathrm{RF})}M_{k}^{({\rm A})}$ for the total number of RX antennas at node $k$. We define the complex-valued $M_{k}\times M_{k}^{(\mathrm{RF})}$ analog RX combiner $\mathbf{U}_{k}^{(\mathrm{RF})}$ having a similar block diagonal structure to (1). In particular, $\mathbf{U}_{k}^{(\mathrm{RF})}$ contains $\mathbf{u}_{n}$’s with $n=1,2,\ldots,M_{k}^{(\mathrm{RF})}$ in the diagonal, where each $\mathbf{u}_{n}$ contains the constant magnitude values of the $M_{k}^{({\rm A})}$ phase shifters (i.e., $|[\mathbf{u}_{n}]_{j}|^{2}=1/M_{k}^{({\rm A})}$ $\forall$$j=1,2,\ldots,M_{k}^{({\rm A})}$) connecting each $n$-th RX RF chain with its antenna elements. We also assume that $\mathbf{u}_{n}\in\mathbb{F}_{\rm RX}$ $\forall$$n$, i.e., all analog RX combiners belong in a predefined beam codebook $\mathbb{F}_{\rm RX}$ having ${\rm card}(\mathbb{F}_{\rm RX})$ vectors. Finally, $\mathbf{U}_{k}^{(\mathrm{BB})}\in\mbox{$\mathbb{C}$}^{M_{k}^{(\mathrm{RF})}\times d_{m}}$ with $d_{m}\leq\min\\{M_{k}^{(\mathrm{RF})},N_{m}\\}$ represents the digital RX combiner at node $k$. #### II-B2 Multi-Tap Analog Cancellation The analog canceller at node $k$ consists of $N$ taps with each tap connected via a $N_{k}^{\mathrm{(RF)}}$-to-$1$ MUltipleXer (MUX) to all $N_{k}^{\mathrm{(RF)}}$ outputs of the respective TX RF chains. A tap includes a fixed delay, a variable phase shifter, and a variable attenuator [16, 10]. To route the cancellation signal to one of the adders located just before the RX RF chains, the output of each tap is connected to a $1$-to-$M_{k}^{\mathrm{(RF)}}$ DEMUltipleXer (DEMUX). There is a total of $NM_{k}^{\mathrm{(RF)}}$ such adders and we use the notation “Adder ($i,j$)” to label the adder that connects DEMUX $j$ to RX RF chain $i$, where $i=1,2,\ldots,M_{k}^{\mathrm{(RF)}}$ and $j=1,2,\ldots,N$. The adders before the RX RF chains can be implemented via power combiners or directional couplers, while the analog RF MUXs/DEMUXs can be implemented with RF switches. Clearly, the proposed analog canceller interconnects the outputs of some of the available TX RF chains to the inputs of some of the RX RF chains, and in contrast to [10], the size of each MUX/DEMUX depends on the number of TX/RF RF chains and not on the number of TX/RX antennas. Similar to [10], we model in BB the analog processing realized by the analog canceller as $\mathbf{C}_{k}\triangleq\mathbf{L}_{3}\mathbf{L}_{2}\mathbf{L}_{1}\in\mbox{$\mathbb{C}$}^{M_{k}^{\mathrm{(RF)}}\times N_{k}^{\mathrm{(RF)}}}$, where $\mathbf{L}_{1}\in\mbox{$\mathbb{R}$}^{N\times N_{k}^{\mathrm{(RF)}}}$, $\mathbf{L}_{2}\in\mbox{$\mathbb{C}$}^{N\times N}$, and $\mathbf{L}_{3}\in\mbox{$\mathbb{R}$}^{M_{k}^{\mathrm{(RF)}}\times N}$. The elements $[\mathbf{L}_{1}]_{j,\ell}$ and $[\mathbf{L}_{3}]_{i,j}$ with $j=1,2,\ldots,N$, $\ell=1,2,\ldots,N_{k}^{\mathrm{(RF)}}$, and $i=1,2,\ldots,M_{k}^{\mathrm{(RF)}}$ take the binary values $0$ or $1$, and it must hold that $\sum_{\ell=1}^{N_{k}^{\mathrm{(RF)}}}[\mathbf{L}_{1}]_{j,\ell}=\sum_{i=1}^{M_{k}^{\mathrm{(RF)}}}[\mathbf{L}_{3}]_{i,j}=1\,\,\,\forall j=1,2,\ldots,N.$ (2) The $\mathbf{L}_{2}$ in $\mathbf{C}_{k}$ is a diagonal matrix whose complex entries represent the attenuation and phase shift of the canceller taps; the magnitude and phase of the element $[\mathbf{L}_{2}]_{i,i}$ with $i=1,2,\ldots,N$ specify the attenuation and phase of the $i$-th tap. Recall that the tap delays in each canceller tap are fixed, hence, we model the effects of the $i$-th tap delay as a phase shift that is incorporated to the phase of $[\mathbf{L}_{2}]_{i,i}$. ### II-C Received Signal Models Using the previously described system configuration, the BB received signal $\mathbf{y}_{q}\in\mbox{$\mathbb{C}$}^{M_{q}\times 1}$ at node $q$ in the DL communication can be mathematically expressed as $\mathbf{y}_{q}\triangleq\mathbf{H}_{q,k}\mathbf{V}_{k}^{(\mathrm{RF})}\mathbf{V}_{k}^{(\mathrm{BB})}\mathbf{s}_{k}+\mathbf{H}_{q,m}\mathbf{V}_{m}\mathbf{s}_{m}+\mathbf{n}_{q},$ (3) where $\mathbf{H}_{q,k}\in\mbox{$\mathbb{C}$}^{M_{q}\times N_{k}}$ is the DL channel gain matrix (i.e., between nodes $q$ and $k$), $\mathbf{H}_{q,m}\in\mbox{$\mathbb{C}$}^{M_{q}\times N_{m}}$ denotes the channel gain matrix for inter-node interference (i.e., between nodes $q$ and $m$), and $\mathbf{n}_{q}\in\mbox{$\mathbb{C}$}^{M_{q}\times 1}$ represents the Additive White Gaussian Noise (AWGN) at node $q$ with variance $\sigma_{q}^{2}$. In the UL communication, the symbol vector $\hat{\mathbf{s}}_{m}\in\mbox{$\mathbb{C}$}^{d_{m}\times 1}$ used for the estimation of $\mathbf{s}_{m}$ at the FD HBF node $k$ is derived as $\displaystyle\hat{\mathbf{s}}_{m}\triangleq$ $\displaystyle\left(\mathbf{U}_{k}^{(\mathrm{BB})}\right)^{\rm H}\left(\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm H}\mathbf{H}_{k,k}\mathbf{V}_{k}^{(\mathrm{RF})}+\mathbf{C}_{k}\right)\mathbf{V}_{k}^{(\mathrm{BB})}\mathbf{s}_{k}$ $\displaystyle+\left(\mathbf{U}_{k}^{(\mathrm{BB})}\right)^{\rm H}\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm H}\left(\mathbf{H}_{k,m}\mathbf{V}_{m}\mathbf{s}_{m}+\mathbf{n}_{k}\right),$ (4) where $\mathbf{H}_{k,k}\in\mbox{$\mathbb{C}$}^{M_{k}\times N_{k}}$ denotes the SI channel seen at the RX antennas of node $k$ due to its own DL transmission, $\mathbf{H}_{k,m}\in\mbox{$\mathbb{C}$}^{M_{k}\times N_{m}}$ is the UL channel gain matrix (i.e., between nodes $k$ and $m$), and $\mathbf{n}_{k}\in\mbox{$\mathbb{C}$}^{M_{k}\times 1}$ denotes the received AWGN at node $k$ with variance $\sigma_{k}^{2}$. The first term in (II-C) describes the residual SI signal after analog cancellation and A/D TX/RX beamforming, while its second term contains the A/D RX combined signal transmitted from node $m$ plus AWGN. In contrast to [10], $\mathbf{C}_{k}$ needs to cancel the SI channel $(\mathbf{U}_{k}^{(\mathrm{RF})})^{\rm H}\mathbf{H}_{k,k}\mathbf{V}_{k}^{(\mathrm{RF})}$, which is a matrix of dimension $M_{k}^{\mathrm{(RF)}}\times N_{k}^{\mathrm{(RF)}}$ and not the actual $M_{k}\times N_{k}$ SI channel $\mathbf{H}_{k,k}$. ## III Joint Design Problem Formulation We focus on the FD HBF node $k$ in Fig. 1 and present a sum-rate optimization framework for the joint design of $\mathbf{C}_{k}$, $\mathbf{V}_{k}^{(\mathrm{RF})}$, $\mathbf{V}_{k}^{(\mathrm{BB})}$, $\mathbf{U}_{k}^{(\mathrm{RF})}$, and $\mathbf{U}_{k}^{(\mathrm{BB})}$. Using the notation $\mathbf{V}_{k}\triangleq\mathbf{V}_{k}^{(\mathrm{RF})}\mathbf{V}_{k}^{(\mathrm{BB})}$ and assuming Gaussian signaling and capacity-achieving combining at node $q$, the achievable DL rate that is a function of the A/D TX precoding matrices $\mathbf{V}_{k}^{(\mathrm{RF})}$ and $\mathbf{V}_{k}^{(\mathrm{BB})}$ of node $k$ as well as the digital TX precoder $\mathbf{V}_{m}$ of node $m$, is given by $\mathcal{R}_{\rm DL}=\log_{2}\left({\rm det}\left(\mathbf{I}_{M_{q}}+\mathbf{H}_{q,k}\mathbf{V}_{k}\mathbf{V}_{k}^{\rm H}\mathbf{H}_{q,k}^{\rm H}\mathbf{Q}_{q}^{-1}\right)\right),$ (5) where $\mathbf{Q}_{q}\in\mathbb{C}^{M_{q}\times M_{q}}$ denotes the covariance matrix of the Interference-plus-Noise (IpN) at node $q$ that is obtained as $\mathbf{Q}_{q}\triangleq\mathbf{H}_{q,m}\mathbf{V}_{m}\mathbf{V}_{m}^{\rm H}\mathbf{H}_{q,m}^{\rm H}+\sigma_{q}^{2}\mathbf{I}_{M_{q}}.$ (6) We hereinafter assume that there is no inter-node interference between the HD multi-antenna nodes $q$ and $m$ due to, for example, appropriate node scheduling [7] for the FD operation at node $k$. The latter assumption translates to setting the channel matrix between those involved nodes in (3) as $\mathbf{H}_{q,m}=\mathbf{0}_{M_{q}\times N_{k}}$, which means that (6) simplifies to $\mathbf{Q}_{q}=\sigma_{q}^{2}\mathbf{I}_{M_{q}}$. For the computation of the achievable UL rate, we use the notation $\mathbf{U}_{k}\triangleq\mathbf{U}_{k}^{(\mathrm{RF})}\mathbf{U}_{k}^{(\mathrm{BB})}$ to express this rate as a function of the A/D RX combiners $\mathbf{U}_{k}^{(\mathrm{RF})}$ and $\mathbf{U}_{k}^{(\mathrm{BB})}$, the A/D TX precoders $\mathbf{V}_{k}^{(\mathrm{RF})}$ and $\mathbf{V}_{k}^{(\mathrm{BB})}$, and the analog cancellation matrix $\mathbf{C}_{k}$ of node $k$ as well as of the digital TX precoder $\mathbf{V}_{m}$ of node $m$. Using (II-C), the UL rate is given by $\mathcal{R}_{\rm UL}=\log_{2}\left({\rm det}\left(\mathbf{I}_{d_{m}}+\mathbf{U}_{k}^{\rm H}\mathbf{H}_{k,m}\mathbf{V}_{m}\mathbf{V}_{m}^{\rm H}\mathbf{H}_{k,m}^{\rm H}\mathbf{U}_{k}\mathbf{Q}_{k}^{-1}\right)\right).$ (7) where $\mathbf{Q}_{k}\in\mathbb{C}^{d_{m}\times d_{m}}$ denotes the IpN covariance matrix after A/D RX combining at node $q$, which can be expressed as $\begin{split}\mathbf{Q}_{k}\triangleq&\left(\mathbf{U}_{k}^{(\mathrm{BB})}\right)^{\rm H}\tilde{\mathbf{H}}_{k,k}\mathbf{V}_{k}^{(\mathrm{BB})}\left(\mathbf{V}_{k}^{(\mathrm{BB})}\right)^{\rm H}\tilde{\mathbf{H}}_{k,k}^{\rm H}\mathbf{U}_{k}^{(\mathrm{BB})}\\\ &+\sigma_{k}^{2}\left(\mathbf{U}_{k}^{(\mathrm{BB})}\right)^{\rm H}\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm H}\mathbf{U}_{k}^{(\mathrm{RF})}\mathbf{U}_{k}^{(\mathrm{BB})}.\end{split}$ (8) In the latter expression, $\tilde{\mathbf{H}}_{k,k}\in\mbox{$\mathbb{C}$}^{M_{k}^{\mathrm{(RF)}}\times N_{k}^{\mathrm{(RF)}}}$ denotes the effective SI channel after performing analog TX/RX beamforming and analog cancellation, which is defined as $\tilde{\mathbf{H}}_{k,k}\triangleq\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm H}\mathbf{H}_{k,k}\mathbf{V}_{k}^{(\mathrm{RF})}+\mathbf{C}_{k}.$ (9) Using the expressions (5) and (7) for the achievable DL and UL rates, respectively, the sum-rate optimization problem for the joint design of the analog canceller and the A/D TX/RX beamformers is mathematically expressed as $\begin{split}&\mathcal{OP}:\max_{\mathbf{C}_{k},\mathbf{V}_{k}^{(\mathrm{RF})},\mathbf{V}_{k}^{(\mathrm{BB})},\mathbf{U}_{k}^{(\mathrm{RF})},\mathbf{U}_{k}^{(\mathrm{BB})}}\mathcal{R}_{\rm DL}+\mathcal{R}_{\rm UL}\\\ &\textrm{s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{tr}\\{\mathbf{V}_{k}^{(\mathrm{RF})}\mathbf{V}_{k}^{(\mathrm{BB})}\left(\mathbf{V}_{k}^{(\mathrm{BB})}\right)^{\rm H}\left(\mathbf{V}_{k}^{(\mathrm{RF})}\right)^{\rm H}\\}\leq{\rm P}_{k},\,\,({\rm C1})\\\ &\mathbf{C}_{k}=\mathbf{L}_{3}\mathbf{L}_{2}\mathbf{L}_{1}\,\,{\rm with}\,\,\eqref{Eq:L_1_L_3}\,\,{\rm and}\,\,[\mathbf{L}_{2}]_{i,j}=0\,\,{\rm for}\,\,i\neq j,\hskip 5.97527pt({\rm C2})\\\ &\left\|\left[\tilde{\mathbf{H}}_{k,k}\mathbf{V}_{k}^{(\mathrm{BB})}\right]_{(j,:)}\right\|^{2}\leq\rho_{\rm A}\,\,\forall j=1,2,\ldots,M_{k}^{\mathrm{(RF)}},\hskip 4.97931pt({\rm C3})\\\ &\mathbf{u}_{j}\in\mathbb{F}_{\rm RX}\,\,\forall j\,\,{\rm and}\,\,\mathbf{v}_{n}\in\mathbb{F}_{\rm TX}\,\,\forall n=1,2,\ldots,N_{k}^{({\rm RF})},({\rm C4})\end{split}$ where constraint $({\rm C1})$ relates to the average TX power at node $k$ and constraint $({\rm C2})$ refers to the hardware capabilities of the analog canceller. Constraint $({\rm C3})$ imposes the threshold $\rho_{\rm A}\in\mbox{$\mathbb{R}$}$ on the average power of the residual SI signal after analog cancellation and analog TX/RX beamforming. Finally, constraint $({\rm C4})$ refers to the predefined TX and RX beam codebooks. To tackle $\mathcal{OP}$, which is a nonconvex problem with nonconvex constraints, we adopt a similar to [10] decoupled way that in this case requires at most $\alpha_{\max}\triangleq N_{k}^{(\mathrm{RF})}-1$ iterations including closed form expressions for the design parameters. We first solve for $\mathbf{C}_{k}$, $\mathbf{V}_{k}^{(\mathrm{RF})}$, $\mathbf{V}_{k}^{(\mathrm{BB})}$, and $\mathbf{U}_{k}^{(\mathrm{RF})}$ maximizing the DL rate, and then find $\mathbf{U}_{k}^{(\mathrm{BB})}$ maximizing the UL rate. Specifically, we formulate the following optimization subproblem for the design of $\mathbf{C}_{k}$, $\mathbf{V}_{k}^{(\mathrm{RF})}$, $\mathbf{V}_{k}^{(\mathrm{BB})}$, and $\mathbf{U}_{k}^{(\mathrm{RF})}$: $\begin{split}\mathcal{OP}1:&\max_{\mathbf{C}_{k},\mathbf{V}_{k}^{(\mathrm{RF})},\mathbf{V}_{k}^{(\mathrm{BB})},\mathbf{U}_{k}^{(\mathrm{RF})}}\mathcal{R}_{\rm DL}\leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\ ({\rm C1}),\,({\rm C2}),\,\text{and}\,({\rm C3}).\end{split}$ Algorithm 1 Digital TX Precoder Design 1:Input: ${\rm P}_{k}$, $\mathbf{V}_{k}^{(\mathrm{RF})}$ and $\mathbf{U}_{k}^{(\mathrm{RF})}$ solving $\mathcal{OP}2$, $\mathbf{H}_{k,k}$, and $\mathbf{H}_{q,k}$ as well as a realization of $\mathbf{C}_{k}$ for a given $N$ satisfying constraint $({\rm C2})$. 2:Set $\tilde{\mathbf{H}}_{k,k}=\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm H}\mathbf{H}_{k,k}\mathbf{V}_{k}^{(\mathrm{RF})}+\mathbf{C}_{k}$. 3:Obtain $\mathbf{D}_{k}$ with the $N_{k}^{\mathrm{(RF)}}$ right-singular vectors of $\widetilde{\mathbf{H}}_{k,k}$ corresponding to the singular values in descending order. 4:for $\alpha=\alpha_{\max},\alpha_{\max}-1,\ldots,2$ do 5: Set $\mathbf{F}_{k}=[\mathbf{D}_{k}]_{(:,N_{k}^{\mathrm{(RF)}}-\alpha+1:N_{k}^{\mathrm{(RF)}})}$. 6: Set $\mathbf{G}_{k}$ as the optimum precoding for the effective 7: DL MIMO channel $\mathbf{H}_{q,k}\mathbf{V}_{k}^{(\mathrm{RF})}\mathbf{F}_{k}$ given ${\rm P}_{k}$. 8: if $\|[\widetilde{\mathbf{H}}_{k,k}\mathbf{F}_{k}\mathbf{G}_{k}]_{(i,:)}\|^{2}\leq\rho_{\rm A}$ $\forall i=1,\ldots,M_{k}^{\mathrm{(RF)}}$, then 9: Output $\mathbf{V}_{k}^{(\mathrm{BB})}=\mathbf{F}_{k}\mathbf{G}_{k}$ and stop the algorithm. 10: end if 11:end for 12:Set $\mathbf{F}_{k}=[\mathbf{D}_{k}]_{(:,N_{k}^{\mathrm{(RF)}})}$ and $\mathbf{G}_{k}={\rm P}_{k}^{1/2}$. 13:if $|[\widetilde{\mathbf{H}}_{k,k}\mathbf{F}_{k}\mathbf{G}_{k}]_{i}|^{2}\leq\rho_{\rm A}$ $\forall i=1,\ldots,M_{k}^{\mathrm{(RF)}}$, then 14: Output $\mathbf{V}_{k}^{(\mathrm{BB})}=\mathbf{F}_{k}\mathbf{G}_{k}$ and stop the algorithm. 15:else 16: Output that the $\mathbf{C}_{k}$ realization does not meet 17: the residual SI constraint. 18:end if For the solution of the latter problem we use an alternating optimization approach. First, we find $\mathbf{V}_{k}^{(\mathrm{RF})}$ and $\mathbf{U}_{k}^{(\mathrm{RF})}$ constrained as $({\rm C4})$ that boost the DL rate, while minimizing the SI signal before any other form of cancellation. Particularly, we perform the following exhaustive search: $\begin{split}\mathcal{OP}2:&\max_{\mathbf{V}_{k}^{(\mathrm{RF})},\mathbf{U}_{k}^{(\mathrm{RF})}}\frac{\left\|\mathbf{H}_{q,k}\mathbf{V}_{k}^{(\mathrm{RF})}\right\|_{\rm F}}{\left\|\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm H}\mathbf{H}_{k,k}\mathbf{V}_{k}^{(\mathrm{RF})}\right\|_{\rm F}}\leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\ ({\rm C4}).\end{split}$ In the sequel, given the solution for $\mathcal{OP}2$ and supposing that the available number of analog canceller taps $N$ and a realization of $\mathbf{C}_{k}$ satisfying $({\rm C2})$ are given, we seek for $\mathbf{V}_{k}^{(\mathrm{BB})}$ maximizing the DL rate while meeting $({\rm C1})$ and $({\rm C3})$. The latter procedure is repeated for all allowable realizations of $\mathbf{C}_{k}$ for the given $N$ in order to find the best $\mathbf{V}_{k}^{(\mathrm{BB})}$ solving $\mathcal{OP}1$; this procedure is summarized in Algorithm 1. The values for $\mathbf{C}_{k}$, $\mathbf{V}_{k}^{(\mathrm{RF})}$, $\mathbf{V}_{k}^{(\mathrm{BB})}$, and $\mathbf{U}_{k}^{(\mathrm{RF})}$ solving $\mathcal{OP}1$ are finally substituted into the achievable UL rate expression $\mathcal{R}_{\rm UL}$ in (7). The $\mathbf{U}_{k}^{(\mathrm{BB})}$ maximizing this point-to-point MIMO rate is obtained in closed form using [17, Sec. 4.2]. ## IV Simulation Results and Discussion In this section, we investigate the performance of the considered $3$-user bidirectional communication system for the case where $N_{k}^{(\mathrm{RF})}=M_{q}=4$, $M_{k}^{(\mathrm{RF})}=2$, $N_{m}=1$, and $N_{k}^{(\mathrm{A})}=M_{k}^{(\mathrm{A})}=\\{16,128\\}$. We have assumed that the antennas at the FD HBF node $k$ are arranged in Uniform Linear Arrays (ULAs) with $\lambda/2$ space separation between adjacent elements, where $\lambda$ is the wavelength. The distance and angle between the TX and RX ULAs at node $k$ were set respectively to $d=2\lambda$ and $\omega=\pi/6$ [12]. The DL and UL channels were simulated as millimeter wave clustered channels, as described in [13, eq. (2)] with pathloss $110$dB. In addition, the SI channel was modeled as Rician [13, eq. (7)] with $K$-factor $35$dB and pathloss $40$dB. The RX noise floors at both nodes $k$ and $q$ were set to $-110$dBm, resulting in an effective dynamic range of $62$dB for $14$-bit ADCs with a $10$dB peak-to-average power ratio. Hence, to avoid saturation, the residual SI power after analog cancellation at the input of each RX RF chain needs to be below $-47$dBm. Non-ideal $N$-tap analog cancellation has been considered as in [10], and for the analog TX/RX beamformers, we have used beam codebooks based on the discrete Fourier transform matrix. The achievable FD rate as a function of the transmit powers of nodes $k$ and $m$ is illustrated in Fig. 2 for $N=4$ taps for the proposed analog canceller, which translates to $50\%$ reduction in the number of taps compared to a case that connects all outputs of the TX RF chains to every input to the RX RF chains. For the rate results, we averaged over $1000$ independent channel realizations and calculated the FD rate with the proposed Algorithm 1, as well as the achievable HD rate. It is shown in the figure that all rates increase with increasing transmit power, and no rate saturation is exhibited for the proposed FD HBF technique. The latter trend witnesses the effective adaptability of Algorithm 1 to the SI conditions for both considered pairs of $N_{k}^{(\mathrm{A})}$ and $M_{k}^{(\mathrm{A})}$. As an indicative example, it is shown that for $40$dBm transmit powers, the proposed approach results in a $52$bps/Hz achievable rate, which is around $1.7$ times more than the achievable HD rate. Figure 2: Average FD and HD rates vs the transmit powers in dBm for $N_{k}^{(\mathrm{RF})}=M_{q}=N=4$, $M_{k}^{(\mathrm{RF})}=2$, $N_{m}=1$, and different values for $N_{k}^{(\mathrm{A})}$ and $M_{k}^{(\mathrm{A})}$ at node $k$. ## References * [1] A. Sabharwal _et al._ , “In-band full-duplex wireless: Challenges and opportunities,” _IEEE J. Sel. Areas Commun._ , vol. 32, no. 9, pp. 1637–1652, Sep. 2014. * [2] T. Riihonen _et al._ , “Mitigation of loopback self-interference in full-duplex MIMO relays,” _IEEE Trans. Signal Proces._ , vol. 59, no. 12, pp. 5983–5993, Dec. 2011. * [3] D. Nguyen _et al._ , “Precoding for full duplex multiuser MIMO systems: Spectral and energy efficiency maximization,” _IEEE Trans. Signal Process._ , vol. 61, no. 13, pp. 4038–4050, Aug. 2013. * [4] D. Bharadia and S. Katti, “Full duplex MIMO radios,” in _Proc. USENIX NSDI_ , Seattle, WA, 2-4 Apr. 2014, pp. 359–372. * [5] E. Everett _et al._ , “SoftNull: Many-antenna full-duplex wireless via digital beamforming,” _IEEE Trans. Wireless Commun._ , vol. 12, no. 15, pp. 8077–8092, Dec. 2016. * [6] H. H. M. Tam _et al._ , “Successive convex quadratic programming for quality-of-service management in full-duplex MU-MIMO multicell networks,” _IEEE Trans. Commun._ , vol. 64, no. 6, pp. 2340–2353, Jun. 2016. * [7] G. C. Alexandropoulos _et al._ , “User scheduling and optimal power allocation for full-duplex cellular networks,” in _Proc. IEEE SPAWC_ , Edinburgh, UK, 3-6 Jul. 2016, pp. 1–6. * [8] M. A. Islam _et al._ , “A unified beamforming and A/D self-interference cancellation design for full duplex MIMO radios,” in _Proc. IEEE PIMRC_ , Istanbul, Turkey, 8-11 Sep. 2019, pp. 1–6. * [9] D. Korpi _et al._ , “Widely linear digital self-interference cancellation in direct-conversion full-duplex transceiver,” _IEEE J. Sel. Areas Commun._ , vol. 32, no. 9, pp. 1674–1687, Sep. 2014. * [10] G. C. Alexandropoulos and M. Duarte, “Joint design of multi-tap analog cancellation and digital beamforming for reduced complexity full duplex MIMO systems,” in _Proc. IEEE ICC_ , Paris, France, May 2017, pp. 1–7. * [11] A. F. Molisch _et al._ , “Hybrid beamforming for massive MIMO: A survey,” _IEEE Commun. Mag._ , vol. 55, no. 9, pp. 134–141, Sep. 2017\. * [12] Z. Xiao _et al._ , “Full-duplex millimeter-wave communication,” _IEEE Wireless Commun._ , vol. 24, no. 6, pp. 136–143, Dec. 2017. * [13] K. Satyanarayana _et al._ , “Hybrid beamforming design for full-duplex millimeter wave communication,” _IEEE Trans. Veh. Technol._ , vol. 68, no. 2, pp. 1394–1404, Dec. 2018. * [14] Y. Cai _et al._ , “Robust joint hybrid transceiver design for millimeter wave full-duplex mimo relay systems,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 2, pp. 1199–1215, Jan. 2019. * [15] I. P. Roberts and S. Vishwanath, “Beamforming cancellation design for millimeter-wave full-duplex,” _[online] https://arxiv.org/pdf/1908.06505.pdf_ , 2019. * [16] K. E. Kolodziej _et al._ , “Multitap RF canceller for in-band full-duplex wireless communications,” _IEEE Trans. Wireless Commun._ , vol. 15, no. 6, pp. 4321–4334, Jun. 2016. * [17] G. C. Alexandropoulos and C. B. Papadias, “A reconfigurable iterative algorithm for the $K$-user MIMO interference channel,” _Signal Process. (Elsevier)_ , vol. 93, no. 12, pp. 3353–3362, Dec. 2013.

2024-09-04T02:54:54.840394

2002.11855

arxiv-papers

# Computational molecular field theory for nematic liquid crystals Cody D. Schimming<EMAIL_ADDRESS>School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Jorge Viñals School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA ###### Abstract Nematic liquid crystals exhibit configurations in which the underlying ordering changes markedly on macroscopic length scales. Such structures include topological defects in the nematic phase and tactoids within nematic- isotropic coexistence. We discuss a computational study of inhomogeneous configurations that is based on a field theory extension of the Maier-Saupe molecular model of a uniaxial, nematic liquid crystal. A tensor order parameter is defined as the second moment of an orientational probability distribution, leading to a free energy that is not convex within the isotropic-nematic coexistence region, and that goes to infinity if the eigenvalues of the order parameter become non-physical. Computations of the spatial profile of the order parameter are presented for an isotropic-nematic interface in one dimension, a tactoid in two dimensions, and a nematic disclination in two dimensions. We compare our results to those given by the Landau de-Gennes free energy for the same configurations and discuss the advantages of such a model over the latter. ## I Introduction Liquid crystals represent an interesting opportunity to study a unique interplay between topology, anisotropy, and elasticity in materials. The entropy driven local ordering of rod-like molecules accounts for anisotropic optical and transport properties even in homogeneous nematics. Furthermore, external fields or topological defects can distort the local ordering of the molecules giving rise to several elastic modes de Gennes (1975); Selinger (2018). The ability to quantitatively model these complex features of liquid crystals is imperative to address recent applications, including electrokinetics of colloidal particles or biological materials Lazo _et al._ (2014); Peng _et al._ (2015, 2018), surface and texture generation and actuation in nematic surfaces Mostajeran (2015); Babakhanova _et al._ (2018), systems of living nematics Genkin _et al._ (2017), and stabilization of liquid shells Hokmabad _et al._ (2019). Liquid crystals generally belong to one of two main classes: Thermotropics are short molecules that undergo ordering through changes in temperature, while lyotropics are more complex molecules or assemblies of molecules in solvent that order through changes in concentration. Thermotropics have been extensively studied, both theoretically and experimentally, due to their applications in displays de Gennes (1975); Yeh and Gu (2009). However, because of their small characteristic length scale, the fine structure of defects and two phase domains (commonly referred to as tactoids) are generally beyond the resolution of standard optical techniques. On the other hand experimental studies of defect core structures and tactoids have been recently undertaken in so called lyotropic chromonic liquid crystals. These materials are composed of disc-like molecules that stack to form rod-like structures Collings _et al._ (2010, 2015). The characteristic length scale that determines the size of defects and tactoid interfacial thickness in chromonics are thousands of times larger than those in thermotropics, and hence are readily observable with conventional optical techniques. Such experiments have revealed anisotropic geometries of the order parameter near the core of defects, and “cusp-like” features on the interface of tactoids Kim _et al._ (2013); Zhou _et al._ (2017). To mathematically model a liquid crystal in its nematic phase a unit vector $\mathbf{n}$, the director, is typically defined to characterize the local orientation of the molecules. Because the molecules are apolar, any model involving $\mathbf{n}$ must be symmetric with respect to $\mathbf{n}\to-\mathbf{n}$. Distorted nematic configurations are described by three independent elastic modes: splay, twist, and bend. The energy cost of each mode is associated with three elastic constants $K_{1}$, $K_{2}$, and $K_{3}$ in the Oseen-Frank free energy Selinger (2018); Frank (1958). Models and computations often assume that these constants are equal, though it has been shown for chromonics that the values of all three constants are widely different for the relevant range of temperatures and molecular concentrations Zhou _et al._ (2014). Additionally, topological defects and tactoids lead to large distortions of the underlying order. To model defected configurations using the Oseen-Frank free energy either a short distance cutoff is introduced, and the defect core treated separately, or a new variable representing the degree of order of the molecules is added to the free energy Leslie (1966); Ericksen (1991). This new variable also has the effect of regularizing singularities at the core of defects. The method has recently allowed the study of tactoids within the coexistence region Zhang _et al._ (2018). Resolving the degree of orientational order and the orientation poses several challenges computationally, however. The director is undefined both at the core of defects and in the isotropic phase, and half-integer disclinations (the stable line defects in liquid crystals) cannot be adequately described computationally with a polar vector. Therefore, the model that is widely used to describe either disclinations or tactoids is the phenomenological Landau-de Gennes (LdG) free energy Meiboom _et al._ (1983); Golovaty _et al._ (2019); Popa-Nita _et al._ (1997). In the LdG framework, the order parameter is defined to be a traceless and symmetric tensor, $\mathbf{Q}$, typically proportional to a macroscopic quantity, e.g. the magnetic susceptibility Gramsbergen _et al._ (1986); Lubensky (1970). The free energy is then assumed to be an analytic function in powers of $\mathbf{Q}$. To model spatial inhomogeneity, an expansion in gradients of $\mathbf{Q}$ is typically added to the free energy. Such an expansion in gradients can be mapped to the elastic modes in the director $\mathbf{n}$ in the Oseen-Frank elastic energy Selinger (2018). The validity of the LdG free energy in regions of large variation of the order is not well understood, and it has been shown that the simplest LdG elastic expansions that capture differences in the Oseen-Frank constants result in unbounded free energies Longa _et al._ (1987); Ball and Majumdar (2010). Therefore, when working in the LdG framework, one must introduce more computationally complex assumptions to bound the free energy. In this work, we present an alternative field theoretic model of a nematic liquid crystal that is based on a microscopic description, and that allows for anisotropic elastic energy functionals that can capture the elasticity observed in chromonics. The model presented here is a computational implementation of the model introduced by Ball and Majumdar Ball and Majumdar (2010), which itself is a continuum extension of the well known Maier-Saupe model for the nematic-isotropic phase transition Maier and Saupe (1959). The Maier-Saupe model is a mean field molecular theory in which the orientation of the molecules of the liquid crystal is described by a probability distribution function, so that each molecule interacts only with the average of its neighbors. Below, we define $\mathbf{Q}$ microscopically, based on a probability distribution that is allowed to vary spatially (as in the hypothesis of local equilibrium in nonequilibrium thermodynamics). Our ultimate goal is to develop a computationally viable implementation of the model for fully anisotropic systems. We present below the results of several proof of concept computations on various prototypical liquid crystal configurations, albeit in the one elastic constant approximation. All our results are compared with those from the LdG free energy for analogous configurations. In Section II we briefly summarize the model as put forth in Ref. Ball and Majumdar (2010) with minor adjustments to notation and conceptual understanding. In Section III we present the computational implementation of the model and derive the equations that are solved numerically. We also briefly discuss the conventions used to compare to the LdG free energy. In Section IV we compare the free energies of the model presented here with that given by LdG and show that they are both non-convex. We then present computational results from the model for a one dimensional nematic-isotropic interface, a two-dimensional tactoid, and a two-dimensional disclination. All of these are compared to results given by LdG. Finally, in Section V we summarize and discuss the computational model and results, and discuss future potential for the model. ## II Model Following Ref. Ball and Majumdar (2010), we consider a tensor order parameter defined over a small volume at $\mathbf{r}$ $\mathbf{Q}(\mathbf{r})=\int_{S^{2}}\big{(}\bm{\xi}\otimes\bm{\xi}-\frac{1}{3}\mathbf{I}\big{)}p(\bm{\xi};\mathbf{r})\,d\bm{\xi}$ (1) where $\bm{\xi}$ is a unit vector in $S^{2}$, $\mathbf{I}$ is the identity tensor, and $p(\bm{\xi};\mathbf{r})$ is the canonical probability distribution of molecular orientation in local equilibrium at some temperature $T$ at $\mathbf{r}$. Due to the symmetry of the molecules, $p(\bm{\xi};\mathbf{r})$ must have a vanishing first moment; hence, $\mathbf{Q}$ is defined as the second moment of the orientational probability distribution. With this definition, the order parameter is symmetric, traceless, and, most importantly, has eigenvalues that are constrained to lie in the range $-1/3\leq q\leq 2/3$. The situation where $q=-1/3,\,2/3$ represents perfect ordering of the molecules (i.e. the variance of the distribution goes to zero), and is therefore interpreted as unphysical. We note that Eq. (1) can be generalized to biaxial molecules, that is, molecules that are microscopically plate-like, by appropriately changing the domain of the probability distribution to three Euler angles, and considering the second moment of the extended probability distribution. Such a description may be useful in studying similar defects and domains for biaxial molecules, as in Ref. Chiccoli _et al._ (2019). A mean field free energy functional of $\mathbf{Q}(\mathbf{r})$ is defined by $F[\mathbf{Q}(\mathbf{r})]=H[\mathbf{Q}(\mathbf{r})]-T\Delta S$ (2) where $H$ is the energy of a configuration, and $\Delta S$ its entropy relative to the uniform distribution. The energy is chosen to be $H[\mathbf{Q}(\mathbf{r})]=\int_{\Omega}\Big{(}-\alpha\operatorname{Tr}[\mathbf{Q}^{2}]+f_{e}(\mathbf{Q},\nabla\mathbf{Q})\Big{)}\,d\mathbf{r}$ (3) where $\alpha$ is an interaction parameter, and $f_{e}$ is an elastic energy. The term $-\alpha\operatorname{Tr}[\mathbf{Q}^{2}]$ originates from the Maier- Saupe model, and incorporates an effective contact interaction that promotes alignment Maier and Saupe (1959); Selinger (2016). In the spatially homogeneous case $f_{e}=0$. The entropy is the usual Gibbs entropy $\Delta S=-nk_{B}\int_{\Omega}\bigg{(}\int_{S^{2}}p(\bm{\xi};\mathbf{r})\ln\Big{(}4\pi p(\bm{\xi};\mathbf{r})\Big{)}\,d\bm{\xi}\bigg{)}\,d\mathbf{r}$ (4) where $n$ is the number density of molecules. It should be noted that the outer integral is on the physical domain of the system, and the inner integral is on the unit sphere, the domain of the probability distribution. This model, with these definitions, is equivalent to the Maier-Saupe model in the spatially homogeneous case Maier and Saupe (1959). We extend the Maier-Saupe treatment to spatially nonuniform configurations by minimization of Eq. (2) subject to boundary conditions that lead to topological defects in the domain, or two-phase configurations at coexistence. We then find configurations $\mathbf{Q}(\mathbf{r})$ that are not uniform, and that minimize Eq. (2) subject to the constraint (1). Figure 1: Examples of the probability distribution, $p(\bm{\xi})$ of Eq. (5), on the sphere spanned by $\bm{\xi}$ for (a) a uniaxial configuration and (b) a biaxial configuration. Note that the probability distribution involves a uniaxial molecule, but a biaxial order parameter can occur for a probability distribution with biaxial second moment. Only northern hemispheres are displayed since the probability distribution is symmetric about the equator due to the symmetry of the molecules. For these plots, (a) $\bm{\Lambda}=4\operatorname{diag}(-1,\,-1,\,0.5)$ and (b) $\bm{\Lambda}=10\operatorname{diag}(-0.25,\,-1,\,0.25)$. The entropy, Eq. (4), can be maximized, subject to the constraint (1), by introducing a tensor of Lagrange multipliers, $\bm{\Lambda}(\mathbf{r})$, for each component of the constraint Ball and Majumdar (2010); Katriel _et al._ (1986). The resulting probability that maximizes the entropy is given by $\displaystyle p(\bm{\xi};\mathbf{r})$ $\displaystyle=\frac{\exp[\bm{\xi}^{T}\bm{\Lambda}(\mathbf{r})\bm{\xi}]}{Z[\bm{\Lambda}(\mathbf{r})]}$ (5) $\displaystyle Z[\bm{\Lambda}(\mathbf{r})]$ $\displaystyle=\int_{S^{2}}\exp[\bm{\xi}^{T}\bm{\Lambda}(\mathbf{r})\bm{\xi}]\,d\bm{\xi}$ (6) where $Z$ can be interpreted as a single particle partition function. Fig. 1 shows graphical examples of the probability distribution on the unit sphere. We mention that the single particle partition function can only be computed numerically, and hence the minimization procedure described next has to be carried out numerically in its entirety. The minimization of $F$ in Eq. (2) with $p(\bm{\xi};\mathbf{r})$ given by Eqs. (5) and (6) is therefore reformulated in terms of two tensor fields on the domain, $\mathbf{Q}(\mathbf{r})$ and $\bm{\Lambda}(\mathbf{r})$ (from here on the dependence on $\mathbf{r}$ will be dropped for brevity). $\bm{\Lambda}$ acts as an effective interaction field which mediates interactions among molecules. Substituting Eq. (5) into the constraint, Eq. (1), leads to a relation between $\mathbf{Q}$ and $\bm{\Lambda}$: $\mathbf{Q}+\frac{1}{3}\mathbf{I}=\frac{\partial\ln Z[\bm{\Lambda}]}{\partial\bm{\Lambda}}.$ (7) It has been shown that if the eigenvalues of $\mathbf{Q}$ approach the endpoints of their physically admissible values, both $\bm{\Lambda}$ and the free energy diverge. This feature is not present in the LdG theory, which can lead to nonphysical configurations for certain choices of the elastic energy, $f_{e}$, in Eq. (3) Ball and Majumdar (2010); Bauman and Phillips (1986). The fields $\mathbf{Q}$ and $\bm{\Lambda}$ that minimize Eq. (2) and satisfy Eq. (7) are the equilibrium configuration for a given set of boundary conditions. In the next section we describe a computational implementation of the model presented here. ## III Computational Method ### III.1 Molecular Theory To find the configuration $\mathbf{Q}$ that minimizes the free energy of the molecular field theory we numerically solve the differential equations $\delta F/\delta\mathbf{Q}=0$. This, in principle, is a system of nine equations. However, since $\mathbf{Q}$ is traceless and symmetric, there are only five degrees of freedom. The eigenvalues of $\mathbf{Q}$ describe two degrees of freedom since $\mathbf{Q}$ is traceless. The eigenvectors of $\mathbf{Q}$ form an orthonormal frame (since $\mathbf{Q}$ is symmetric) which accounts for the other three degrees of freedom: the first vector has two degrees of freedom since it is a unit vector, the second vector has one degree of freedom since it is a unit vector and must be orthogonal to the first vector, and the third vector is determined from the other two vectors since it must be orthogonal to both. The eigenvalues are related to the amount of order in the system, while the eigenvector which corresponds to the largest eigenvalue is the director, $\mathbf{n}$. This is illustrated in Fig. 1 which shows the probability distribution for molecules with a director along the z-axis. Fig. 1a shows a uniaxial configuration in which two of the eigenvalues are degenerate, leading to arbitrary eigenvectors in the xy-plane. It is possible for the probability distribution to be of the form in Fig. 1b in which the director is still along the z-axis, but all three eigenvalues are distinct. In this case, we call the probability distribution biaxial since it leads to a second moment, $\mathbf{Q}$, that is biaxial. It is known that biaxiality of the order parameter is important near defects and at interfaces in systems of uniaxial molecules as modeled by the LdG free energy Pismen (1999); Popa-Nita _et al._ (1997); Mottram and Newton (2014). Despite the uniaxial character of the molecules, Eq. (1), the molecular theory detailed here can accommodate biaxial order. Local biaxial order will be parametrized as $\mathbf{Q}=S(\mathbf{n}\otimes\mathbf{n}-\frac{1}{3}\mathbf{I})+P(\mathbf{m}\otimes\mathbf{m}-\bm{\ell}\otimes\bm{\ell})$ (8) where $\\{\mathbf{n},\mathbf{m},\bm{\ell}\\}$ are an orthonormal triad of vectors. This representation explicitly includes the five degrees of freedom of $\mathbf{Q}$, namely, three for the orthonormal set of vectors and two for the amplitudes $S$ and $P$. In addition to $\mathbf{n}$ being the director, $S$ represents the amount of uniaxial order, and $P$ the amount of biaxial order. That is, $S=(3/2)\,q_{1}$ and $|P|=(1/2)\,(q_{2}-q_{3})$ where $q_{i}$ are the eigenvalues of $\mathbf{Q}$, and $q_{3}\leq q_{2}\leq q_{1}$. Because we are primarily concerned with experiments in thin nematic films, we further reduce the degrees of freedom of $\mathbf{Q}$ by only considering spatial variation in at most two dimensions. If we write $\mathbf{n}=(\cos\phi,\,\sin\phi,\,0)$, $\mathbf{m}=(-\sin\phi,\,\cos\phi,\,0)$, and $\bm{\ell}=(0,\,0,\,1)$, where $\phi$ is the angle the director makes with the x-axis, we need only one degree of freedom to describe the eigenframe of $\mathbf{Q}$. We can then further simplify the computations by transforming to the auxiliary variables Sen and Sullivan (1987) $\displaystyle\eta$ $\displaystyle=S-\frac{3}{2}(S-P)\sin^{2}\phi$ $\displaystyle\mu$ $\displaystyle=P+\frac{1}{2}(S-P)\sin^{2}\phi$ (9) $\displaystyle\nu$ $\displaystyle=\frac{1}{2}(S-P)\sin 2\phi.$ This transformation is equivalent to expressing $\mathbf{Q}$ in terms of a new basis for traceless, symmetric matrices. While we do this for ease of computation, we can transform back to the original parametrization after calculating the eigenvalues and eigenvectors of $\mathbf{Q}$. Although all of our calculations are conducted with the set $\\{\eta,\mu,\nu\\}$, we will present our results in terms of the more physically intuitive $S$, $P$, and $\phi$. The tensor order parameter in this representation is $\mathbf{Q}=\begin{bmatrix}\frac{2}{3}\eta&\nu&0\\\ \nu&-\frac{1}{3}\eta+\mu&0\\\ 0&0&-\frac{1}{3}\eta-\mu\end{bmatrix}.$ (10) We can now substitute Eq. (10) into Eq. (1) to write the constraint in terms of $\eta$, $\mu$, and $\nu$. Following the procedure of Section II, we introduce three Lagrange multipliers $\Lambda_{1}$, $\Lambda_{2}$, and $\Lambda_{3}$ corresponding to $\eta$, $\mu$ and $\nu$ respectively, and a partition function $Z[\Lambda_{1},\Lambda_{2},\Lambda_{3}]=\int_{S^{2}}\exp\bigg{[}\frac{3}{2}\Lambda_{1}\xi_{1}^{2}+\Lambda_{2}(\frac{1}{2}\xi_{1}^{2}+\xi_{2}^{2})+\Lambda_{3}\xi_{1}\xi_{2}\bigg{]}\,d\bm{\xi}$ (11) while the relation from Eq. (7) manifests itself as the three equations $\displaystyle\frac{\partial\ln Z}{\partial\Lambda_{1}}$ $\displaystyle=\eta+\frac{1}{2}$ $\displaystyle\frac{\partial\ln Z}{\partial\Lambda_{2}}$ $\displaystyle=\mu+\frac{1}{2}$ (12) $\displaystyle\frac{\partial\ln Z}{\partial\Lambda_{3}}$ $\displaystyle=\nu$ that implicitly relate the variables $\eta$, $\mu$, and $\nu$ to the Lagrange multipliers. Note that since $Z[\Lambda_{1},\Lambda_{2},\Lambda_{3}]$ cannot be obtained analytically, relation (12) can only be solved numerically. The free energy, Eq. (2), is rewritten as $F=\int_{\Omega}\Big{(}f_{b}(\eta,\mu,\nu,\Lambda_{1},\Lambda_{2},\Lambda_{3})+f_{e}(\eta,\mu,\nu,\nabla\eta,\nabla\mu,\nabla\nu)\Big{)}\,d\mathbf{r}$ (13) where $f_{b}$ is a bulk free energy density that does not depend on gradients of the fields. Written explicitly, $f_{b}=-2\alpha\big{(}\frac{1}{3}\eta^{2}+\mu^{2}+\nu^{2}\big{)}+\\\ nk_{B}T\Big{(}\Lambda_{1}\big{(}\eta+\frac{1}{2}\big{)}+\Lambda_{2}\big{(}\mu+\frac{1}{2}\big{)}+\Lambda_{3}\nu+\ln(4\pi)-\ln Z[\Lambda_{1},\Lambda_{2},\Lambda_{3}]\Big{)}.$ (14) We will focus in this paper on an isotropic elastic energy $f_{e}=L\partial_{k}Q_{ij}\partial_{k}Q_{ij}$ where repeated indices are summed, and $L$ is the elastic constant. This is the ‘one constant approximation’ so that mapping this elastic energy to the Oseen-Frank elastic energy yields the same value for all three elastic constants Longa _et al._ (1987). Written in terms of the auxiliary variables we have $f_{e}=2L\big{(}\frac{1}{3}|\nabla\eta|^{2}+|\nabla\mu|^{2}+|\nabla\nu|^{2}\big{)}.$ (15) Before deriving the differential equations to be solved we redefine quantities in a dimensionless way: $\tilde{f_{b}}=\frac{f_{b}}{nk_{B}T},\quad\tilde{f_{e}}=\frac{f_{e}}{nk_{B}T},\quad\tilde{x}=\frac{x}{\xi_{MS}},\quad\tilde{L}=\frac{L}{\xi_{MS}^{2}nk_{B}T}$ (16) where $\xi_{MS}$ is a length scale which we set by defining the value of the dimensionless parameter $\tilde{L}$ instead. For the rest of the paper the tildes are omitted for brevity. To derive the equilibrium equations, we note that Eq. (12) relates $\eta$, $\mu$, and $\nu$ as functions of $\\{\Lambda_{i}\\}$ through the unknown single particle partition function. It has been shown that these relations are invertible when $\eta$, $\mu$, and $\nu$ give physical eigenvalues of $\mathbf{Q}$ Katriel _et al._ (1986). We can then regard $\Lambda_{1}$, $\Lambda_{2}$, and $\Lambda_{3}$ as functions of $\eta$, $\mu$, and $\nu$ via the inverse of Eq. (12). Although an analytic inverse does not exist we can numerically invert this equation using a Newton-Raphson method. We create a MATLAB scattered interpolant from values given by the Newton-Raphson method. We select interpolant points from the values $0\leq S\leq 0.7$, $0\leq P\leq 0.1$, and $-\pi/2\leq\phi\leq\pi/2$ with $\Delta S=\Delta P=0.05$ and $\Delta\phi=0.0245$. These values are then transformed to $\eta$, $\mu$, and $\nu$ through Eqs. (9) and the Newton-Raphson method is run using these values to find $\Lambda_{i}$ for the chosen interpolant points. The MATLAB scattered interpolant is then created and used in the numerical minimization procedure. The Euler-Lagrange equations are derived by taking the variations of Eqs. (14) and (15) with respect to $\eta$, $\mu$, and $\nu$ while using Eqs. (12) to simplify. The dimensionless equations are $\displaystyle\frac{4}{3}L\nabla^{2}\eta=\Lambda_{1}-\frac{4}{3}\frac{\alpha}{nk_{B}T}\eta$ $\displaystyle 4L\nabla^{2}\mu=\Lambda_{2}-4\frac{\alpha}{nk_{B}T}\mu$ (17) $\displaystyle 4L\nabla^{2}\nu=\Lambda_{3}-4\frac{\alpha}{nk_{B}T}\nu$ where, again, $\Lambda_{i}$ are numerically calculated as functions of $\eta$, $\mu$, and $\nu$. Eqs. (17) are the central equations of this study and are solved numerically in the following section for various cases of interest. To numerically solve Eqs. (17) we use a finite differencing scheme. For one- dimensional configurations, an implicit backward Euler method is used with 129 discrete points and time step $\Delta t=0.1\Delta x^{2}$. For two-dimensional configurations a Gauss-Seidel relaxation method with $257^{2}$ discrete points is used Press _et al._ (2002). We iterate until the calculated energy of a configuration fails to change to within $10^{-7}$. We check that the calculated energy of the initial condition is larger than the energy of the final configuration. In all cases we use Dirichlet boundary conditions that depend on the case being studied, as described in the relevant section. The MATLAB code used for the numerical solutions can be found in Ref. Schimming (2020). ### III.2 Landau-de Gennes Theory Here, we summarize the conventions and notation used in the calculations to compare the LdG free energy with the molecular field theory presented in the previous section. The bulk energy density is of the form $f_{LdG}=\frac{1}{2}a(T-T^{*})\operatorname{Tr}[\mathbf{Q}^{2}]-\frac{1}{3}B\operatorname{Tr}[\mathbf{Q}^{3}]+\frac{1}{4}C\big{(}\operatorname{Tr}[\mathbf{Q}^{2}]\big{)}^{2}$ (18) where $a$, $B$, and $C$ are material parameters, and $T^{*}$ is the temperature at which the isotropic phase loses its stability. We use the same elastic free energy defined above when comparing to the molecular field theory as well. For the sake of computation, we define the following dimensionless quantities: $\tilde{f}_{LdG}=\frac{f_{LdG}}{C},\quad\tilde{f_{e}}=\frac{f_{LdG}}{C},\quad\tilde{x}=\frac{x}{\xi_{LdG}},\quad\tilde{L}=\frac{L}{\xi_{LdG}^{2}C}$ (19) which leaves $a(T-T^{*})/C$, $B/C$, and $\tilde{L}$ as dimensionless parameters for the model. $\xi_{LdG}$ here is a length scale for the model defined by the value of $\tilde{L}$ similar to $\xi_{MS}$ in Eq. (16). As before, the tilde is subsequently dropped for brevity. Computations are done using the same auxiliary variables defined in Eq. (9) with the same finite difference scheme outlined above to solve the Euler- Lagrange equations resulting from $f_{LdG}$. ## IV Results Figure 2: Equilibrium value of the uniaxial order, $S$, versus the parameter $\alpha/(nk_{B}T)$. At high $T$, the system is in an isotropic phase, while at low $T$ the system is in a uniaxial nematic phase. A first order phase transition occurs at $\alpha/(nk_{B}T)\approx 3.4049$. ### IV.1 Uniform Configuration and Bulk Free Energy We first check our numerical method and methodology with known results for the Maier-Saupe free energy. As mentioned above, this model should be equivalent to the Maier-Saupe model in the case of a uniform system, $f_{e}=0$. In this case, it has been shown that minimizers of the bulk free energy, Eq. (14), will be uniaxial states Ball and Majumdar (2010). Thus, because we are considering a uniform system, the choice of director is arbitrary. We choose $\phi=0$ for this analysis so the auxiliary variables defined by Eq. (9) give $\eta=S$, $\mu=P$, and $\nu=0$. Further, since we know the system will be uniaxial we can take $\mu=P=0$. One can show that this implies $\Lambda_{2}=\Lambda_{3}=0$ from Eq. (12). Because the system is uniform, $S$ is constant, and hence $\nabla^{2}S=0$. Defining $S_{N}$ as the value of $S$ in uniform equilibrium, we find, from Eq. (17): $\Lambda_{1}=\frac{4}{3}\frac{\alpha}{nk_{B}T}S_{N}$ (20) which is a well known result for the Maier-Saupe model when $\Lambda_{1}$ is regarded as an effective interaction strength de Gennes (1975); Maier and Saupe (1959); Selinger (2016). We then substitute Eq. (20) into Eq. (14) and numerically minimize it to find the value of $S$ in equilibrium for a uniform system. Fig. 2 shows $S_{N}$ as a function of $\alpha/(nk_{B}T)$. At high temperatures, the equilibrium phase is isotropic with $S=0$. At low temperatures a uniaxial nematic phase is stable with $S=S_{N}$. A first order phase transition occurs at $\alpha/(nk_{B}T)\approx 3.4049$ with $S_{N}=0.4281$. The diagram of Fig. 2 agrees with previous studies of the Maier-Saupe model which has been used successfully to describe phase transitions in experiments Selinger (2016). We can further elucidate the nature of the molecular field theory by examining the bulk free energy density, Eq. (14), restricted to a uniaxial configuration. For a uniform, uniaxial system, the free energy density is $f_{b}(S)=-\frac{2}{3}\frac{\alpha}{nk_{B}T}S^{2}+\Lambda_{1}\Big{(}S+\frac{1}{2}\Big{)}-\ln Z[\Lambda_{1}]+\ln(4\pi)$ (21) where $\Lambda_{1}$ is calculated as a function of $S$ through Eq. (12). This function is plotted in Fig. 3 for three different values of $\alpha/(nk_{B}T)$. As $\alpha/(nk_{B}T)$ increases we find that $f_{b}$ becomes non-convex, leading to a coexistence region in the phase diagram, and a first order phase transition. It is well known that these features are also present in the LdG free energy of Eq. (18) Gramsbergen _et al._ (1986). The primary difference between LdG and the Maier-Saupe theory is that in the latter $f_{b}$ diverges when $S=-1/2$ or $S=1$, that is, when the eigenvalues leave the physical range. The non-convexity obtained agrees with similar plots for the Maier-Saupe free energy in Ref. Selinger (2016). Figure 3: Bulk free energy density as a function of the uniaxial order, $S$ for three values of the parameter $\alpha/(nk_{B}T)$. As $\alpha/(nk_{B}T)$ increases, the free energy becomes non-convex, leading to coexistence between the isotropic and nematic phases. The non-convexity and similarity of the bulk free energy to LdG suggest that there should exist stable interfacial configurations at coexistence as well as stable solutions for topological defects in the nematic phase. In the following three subsections we demonstrate just this and compare to results given by LdG theory. ### IV.2 Planar Isotropic-Nematic Interface We consider a one-dimensional configuration with a planar interface in which the order parameter $\mathbf{Q}(\mathbf{r})=\mathbf{Q}(x)$. We solve Eqs. (17) on a domain of size $\mathcal{L}=100\xi_{MS}$ with Dirichlet boundary conditions where $S=S_{N}$ at $x=-50\xi_{MS}$ and $S=0$ at $x=50\xi_{MS}$. We set $\alpha/(nk_{B}T)=3.4049$ and $S_{N}=0.4281$ so that the isotropic and nematic bulk phases coexist. An important note is that since we are using the “one-constant approximation” for the elastic free energy there are no anisotropic effects, such as anchoring, in our analysis. It is known that anisotropy changes the width of an interface for different director orientations, however, because we are only considering isotropic terms here the structure of the interfacial profile should not change if the angle of the director in the nematic phase, $\phi$, is changed Popa-Nita _et al._ (1997). Fig. 4 shows the equilibrium uniaxial order parameter $S$ for $\phi=0$. We find a smooth, diffuse interface with $P=0$, that is, no biaxiality. We also find that changing the angle of the director does not change the solution, as expected. We can calculate the width of the interface by finding the points where $S=0.1S_{N}$ and $S=0.9S_{N}$ and define them as $x_{1}$ and $x_{2}$ respectively. Then we define the width as $x_{1}-x_{2}$. Figure 4: $S$ as a function of position for a one-dimensional interface. Dirichlet boundary conditions maintain $S=S_{N}$ at the left boundary while $S=0$ at the right boundary. $L=1$ for this configuration. In order to compare with the LdG free energy, Eq. (18), we recall that the interfacial profile for this configuration is known exactly $S_{LdG}(x)=\frac{S_{N}}{2}\bigg{(}1-\tanh\Big{(}\frac{x}{w_{LdG}}\Big{)}\bigg{)}$ (22) with $w_{LdG}=\frac{6\sqrt{6}}{B/C}\sqrt{L}$ (23) which sets the width of the interface. This implies that $(x_{1}-x_{2})\propto\sqrt{L}$. One can similiarly show that the bulk energy contribution, i.e. the bulk contribution to the surface tension, $\sigma\propto\sqrt{L}$. With this in mind, we compare the scaling of the molecular field theory solutions that we obtain with $\sqrt{L}$. To this end, we find the interface widths and bulk surface tensions for solutions to Eqs. (17) for a variety of values of $L$. The bulk surface tension is found by numerically integrating the bulk free energy density, Eq. (14). Interface widths and bulk surface tensions are plotted in Fig. 5 for both the molecular field theory and LdG. We find both $(x_{1}-x_{2})\propto\sqrt{L}$ and $\sigma\propto\sqrt{L}$ for the molecular field theory. Note that the LdG solution allows additional tuning via the parameter $B/C$, which we have set to 9 in Fig. 5. In Fig. 5b the discrepency between the LdG solution and the molecular field theory computations highlights that even if the widths of LdG interfaces are tuned to be similar to those of the molecular field theory, the surface tensions cannot be, and vice versa. Figure 5: (a) Interface width and (b) bulk surface tension versus $\sqrt{L}$. Dots represent the molecular field theory (MFT) computations while the solid lines are derived from the analytical solution for LdG, Eq. (22), with $B/C=9$. Both the interface width and excess free energy (i.e. surface tension) scale linearly with the parameter $\sqrt{L}$, the same scaling relationship as that of Landau-de Gennes. We note that the similarity in bulk free energy landscape likely leads to the similarity in solutions for LdG and the molecular field theory. Anisotropic effects have yet to be analyzed for our model, for which it is known for LdG there is nonzero biaxiality at interfaces Popa-Nita _et al._ (1997). This will be the subject of a future study. Figure 6: Plots of $S(x,y)$ for (a) a tactoid with $m=1$ director configuration at the outer boundary and (b) tactoid with $m=-1/2$ director configuration at the outer boundary. The radius in (a) is $R/\xi_{MS}=19.92\pm 0.2$ and the radius in (b) is $R/\xi_{MS}=4.59\pm 0.2$. The smaller size of the $m=-1/2$ tactoid is due to the director distortion energy’s $m^{2}$ dependence. For both computations $L=1$. ### IV.3 Tactoids We consider a two-dimensional square domain of size $\mathcal{L}=100\xi_{MS}$. We set $S=S_{N}$, $P=0$, and $\phi=m\theta$ at the outer boundary, where $\theta$ is the polar angle and $m$ is the winding number of $\phi$. We set $\alpha/(nk_{B}T)=3.4049$ and $L=1$. As initial conditions we set $S=0$ within a disc centered at the origin of radius $R=15\xi_{MS}$. By “tactoid” we refer to a two-phase domain separated by an interface. In the isotropic region $S=P=0$. We consider distorted boundary conditions to ensure an interface forms in the simulation. Because the director can vary as a function of position in two dimensions, the boundary conditions imposed will change the size and shape of the object under consideration. Since we are only considering isotropic gradients in the elastic free energy, there is no anchoring term at the interface, i.e. there is not a difference in energy based on the orientation of the molecules relative to the interface. Thus, we expect the tactoids to be cylindrical. The topology of the boundary conditions does impact the size of the tactoids, however. This is due to a balance between two energies: the surface tension, which in two dimensions is proportional to $R$, the radius of the tactoid, and the elastic energy in the nematic region from Oseen-Frank which is proportional to $m^{2}\ln(\mathcal{L}/R)$. Due to the symmetry of the molecules, half integer $m$ is allowed and costs four times less director distortion energy than integer $m$. Hence, we expect that tactoids with integer boundary conditions should be approximately four times larger than those with half integer boundary conditions. In Fig. 6, we show equilibrium configurations for boundary conditions with $m=1$ and $m=-1/2$. In both cases an isotropic region with $S=P=0$ is present at the center of the computational domain. As expected, both configurations are cylindrical in shape and we find that $R/\xi_{MS}=19.92\pm 0.2$ for the $m=1$ configuration and $R/\xi_{MS}=4.59\pm 0.2$ for the $m=-1/2$ configuration. To find the radii we take a cut from the center of the tactoid to the outer boundary and find the point where $S=0.5S_{N}$. It should be noted that LdG, in the one-constant approximation in elastic energy, gives similar results in terms of the size and shape of tactoids. It is known for the LdG bulk free energy with anisotropic elastic free energies that the shape of the tactoids also changes due to anchoring at the interface Golovaty _et al._ (2019). Anisotropic effects on the shape of tactoids in the molecular field theory will be the subject of a future study. ### IV.4 Nematic Disclinations We consider next the case of disclination lines in thin films. We consider a two-dimensional square of size $\mathcal{L}=10\xi_{MS}$. For all calculations $L=1$ and $\alpha/(nk_{B}T)>3.4049$, so nematic ordering is energetically advantageous. At the outer boundary we fix the system to be uniaxial ($P=0$) and fix the director orientation, $\phi=(-1/2)\theta$. The initial configuration is $S(r)=S_{N}\big{(}1-\exp(r/2)\big{)}$ with $P=0$ everywhere. In Fig. 7 we show the director profile, and the radial profile of equilibrium $S$ and $P$ from the center of a disclination to the boundary of the domain for the parameter $\alpha/(nk_{B}T)=4$. For the director, $\phi=-(1/2)\theta$ outside the core. Much like solutions for the LdG free energy, we see a disclination core that is biaxial Meiboom _et al._ (1983); Schopohl and Sluckin (1987). The biaxiality of the core was explained topologically by Lyuksyutov, assuming a LdG bulk free energy Lyuksyutov (1978). Using this free energy for analysis, one can define a “biaxial length” scale for the disclinations, $R_{b}\approx\sqrt{K/(BS^{3})}$, where $K$ is on the order of the Frank constants and $B$ is the parameter associated with the cubic term in the LdG bulk energy, Eq. (18). For distances from the core smaller than $R_{b}$, the elastic energy becomes comparable to the cubic term in the LdG free energy and the system can remove the elastic singularity by becoming biaxial, since a biaxial order parameter can remove the singularity. We note that at the core, $S=P$ in both models. Using the parametrization from Eq. (8), one can show that this is interpreted as a uniaxial order parameter, but for a disc if $S>0$ or a rod aligned with the z-axis if $S<0$. For both models, $S>0$ at the core. Thus, we interpret the biaxial solution as a macroscopic “transformation” of rods far away from the core to discs at the core. Microscopically, the probability distribution describing individual molecules becomes more and more spread out in the x-y plane in an attempt to alleviate the elastic energy singularity. Figure 7: (a) Director profile and (b) radial plots of the uniaxial order $S$ and the biaxial order $P$ for a nematic disclination. The spatial extent of biaxiality is on the order of the radius of the disclination core. Here, $\alpha/(nk_{B}T)=4$ and $L=1$. We emphasize that it is not obvious that the molecular field theory should give biaxial core solutions for the disclinations since, by construction, the model is markedly different from LdG. While LdG is an expansion of a macroscopic order parameter, the model here is based on a microscopic description. Because of this, it is difficult to quantitatively compare the solutions for the disclinations given by the two models. While we note that the spatial extent of the biaxiality for the disclinations is on the order of the radius of the defects, there is not a cubic term in the free energy to define a length such as $R_{b}$. Instead, this behavior is induced by the single particle partition function which appears in Eq. (14) since the Maier- Saupe energy is purely quadratic in $\mathbf{Q}$. Another aspect of the disclinations that we can compare, at least qualitatively, to the LdG model is the scaling of the radius of disclinations with temperature. To find the radius, we take a cut from the center of the disclination to the boundary and find the point where $S-P=S_{N}(1-e^{-1})$. The results are plotted in Fig. 8. We show both the scaling for the molecular field theory and for results given by LdG. It can be seen that the scaling is similar for both models in a wide range of temperatures up to the coexistence temperature, where the isotropic phase becomes energetically favorable. We are currently investigating the effects of anisotropic elastic free energies on disclinations. It is known that the director structure becomes less symmetric away from the disclination core if the Frank constants for bend and splay are not equal, and recent experiments have found anisotropic core structures Zhou _et al._ (2017). Figure 8: Radius of disclinations plotted as a function of temperature for (a) the molecular field theory of section II and (b) the Landau-de Gennes model. $T^{*}$ is the temperature where the isotropic phase loses its metastability, while the dotted line on the plots indicate where coexistence between phases is for the respective model. For the molecular field theory we use $L=1$, and for Landau-de Gennes $L=1$ and $B/C=4$ for all simulations. ## V Conclusion In this work, we have presented a computational implementation of the model of reference Ball and Majumdar (2010). We show that the model can be interpreted as replacing direct interactions between molecules via an effective interaction field $\bm{\Lambda}$ in the mean field approximation. Further, we investigate the similarity between the free energy of this molecular field theory and the LdG free energy and compare solutions given by both for the cases of interfaces, tactoids, and topological defects. We find that all have qualitatively similar results which is an interesting result given that the construction of the two models is very different. This model allows for a more fundamental understanding of the underlying microscopic and mesoscopic physics at play, and can serve as an alternative to the LdG free energy when describing systems with inhomogeneous ordering. The extension of the Maier-Saupe model to a field theory allows us to understand not just the phase transition but also inhomogeneous configurations, and can possibly be used to describe experiments like those of Refs. Zhou _et al._ (2017); Kim _et al._ (2013). Moving forward, we are currently investigating the results of adding anisotropy to the elastic free energy, which has been done to some extent for the LdG model Golovaty _et al._ (2019). 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2024-09-04T02:54:54.857771

2002.11882

arxiv-papers

# A Visual Communication Map for Multi-Agent Deep Reinforcement Learning Ngoc Duy Nguyen, Thanh Thi Nguyen, Doug Creighton, Saeid Nahavandi Ngoc Duy Nguyen, Doug Creighton and Saeid Nahavandi are with the Institute for Intelligent Systems Research and Innovation, Deakin University, Waurn Ponds Campus, Geelong, Victoria, Australia (e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>and [email protected]). Thanh Thi Nguyen is with the School of Information Technology, Deakin University, Burwood Campus, Melbourne, Victoria, Australia (e-mail: [email protected]). ###### Abstract Deep reinforcement learning has been applied successfully to solve various real-world problems and the number of its applications in the multi-agent settings has been increasing. Multi-agent learning distinctly poses significant challenges in the effort to allocate a concealed communication medium. Agents receive thorough knowledge from the medium to determine subsequent actions in a distributed nature. Apparently, the goal is to leverage the cooperation of multiple agents to achieve a designated objective efficiently. Recent studies typically combine a specialized neural network with reinforcement learning to enable communication between agents. This approach, however, limits the number of agents or necessitates the homogeneity of the system. In this paper, we have proposed a more scalable approach that not only deals with a great number of agents but also enables collaboration between dissimilar functional agents and compatibly combined with any deep reinforcement learning methods. Specifically, we create a global communication map to represent the status of each agent in the system visually. The visual map and the environmental state are fed to a shared-parameter network to train multiple agents concurrently. Finally, we select the _Asynchronous Advantage Actor-Critic_ (A3C) algorithm to demonstrate our proposed scheme, namely _Visual communication map for Multi-agent A3C_ (VMA3C). Simulation results show that the use of visual communication map improves the performance of A3C regarding learning speed, reward achievement, and robustness in multi-agent problems. ###### Index Terms: reinforcement learning, deep learning, learning systems, multi-agent systems. ## I Introduction Applications of deep reinforcement learning (RL) methods in the multi-agent environments have attracted much attention recently because of their capability of solving various complex problems. Multi-agent learning is to learn the interaction between multiple distributed self-operated agents to balance a shared interest or collectively achieve a designated objective [1, 2]. A vital form of such interaction involves the ability to communicate among agents, which basically ensures agents to function as a team rather than as individuals [3]. Examples of communication activities in multi-agent systems include the scheduling problem in wireless sensor networks [4], strategic planning in robot soccer [5], traffic junction problem [6], elevator control [7], and multi-agent games [8]. Therefore, learning to communicate in multi- agent systems has drawn a great deal of attention from the research community in recent years [9, 10]. Figure 1: A traffic light can be used as a visual indicator to control the level of cooperation between vehicles in a road intersection. Deep RL [11] is known as a normative approach to deal with the _curse of dimensionality_ in domains where the observation space and action space are highly dimensional [12]. It is possible because deep RL approximately estimates the value function by the use of neural networks, which basically include convolutional layers to process raw graphical data directly. In this paper, we leverage the potential of _Convolutional Neural Networks_ [13] (ConvNets) to examine the effect of graphical features to the gradual cooperation between interacting agents during training. To make it feasible, we define a set of visual indicators so that each indicator represents the current status of an agent. The visual indicators are globally visible to all agents and plays as a virtual communication medium. For this reason, we call a set of visual indicators a visual communication map. To clearly present the motivation of the study, we consider the following example. Fig. 1 describes a scenario in a road intersection. Vehicles in the intersection must follow the traffic light to make subsequent decisions and avoid possible collisions. When the light turns red, the vehicles are notified to stop and when the light turns green, the vehicles are allowed to move forward. In this case, the traffic light plays as a virtual communication medium among vehicles. Because the traffic light is visible to every vehicle in the intersection, it can be used as visual indicators to enable cooperation between vehicles. Therefore, the drivers are only required to learn the traffic light rules. The example implies that visual indicators can, somehow, help cooperation in a multi-agent system. Figure 2: Visual indicators are used to enable cooperation between multiple agents. By analyzing the previous example meticulously, we design a visual communication map for multi-agent environments, as shown in Fig. 2. Initially, we create a visual map that includes a set of visual indicators. Each indicator represents the current status of an agent. The visual map is combined with the environmental state to feed into a ConvNet. The output of the ConvNet is connected with a fully-connected network followed by a control layer where subsequent actions are predicted. Finally, each agent uses the augmented knowledge of other agents via the visual communication map to learn the optimal distributed policy on its own and, at the same time, maintain a certain level of cooperation with other agents. Deep RL brings in a great deal of success in many complex real-world problems such as Atari games [12, 14, 15, 16], the game of Go [17], robotics control tasks [18, 19, 20, 21], autonomous driving [22], and surgical robotics [23, 24]. As opposed to previous studies, we examine deep RL in multi-agent domains. Multi-agent learning is more sophisticated than the single-agent counterpart as the action space exponentially increases with the number of agents. Furthermore, we face the challenge of _moving target problem_ [26, 27], which is a primary difficulty in finding the optimal policy of every agent. In other words, the rewarding signal of an agent depends on the decisions of other agents and causes the seeking policies to become _non- stationary_ [27, 28]. The use of _experience replay_ memory even makes the situation worse as a large portion of transitions in the memory could become deprecated [26, 29, 30]. For this reason, we choose the A3C [31] method, which is a policy-based algorithm, to demonstrate our proposed scheme. There are different variants of A3C such as [32, 33, 34], but we use the A3C version based on GPU [32] due to various beneficial factors: high performance with multiple concurrent actor-learners, efficiency within a short period of training, robustness with a wide range of learning rate, and easy to implement due to many open-source codes available in Python. Figure 3: A gameplay interface of Milk Factory. Last but not least, we develop a game named _Milk Factory_ that is used to benchmark our proposed scheme. Milk Factory simulates a task in a warehouse where robots are used to automate the production chain. Fig. 3 describes the gameplay interface of Milk Factory. There are two kinds of robots in the factory: pick-up robots and mechanic robots. At first, a pick-up robot waits for the milk bottle that is on the running conveyor belt and picks it when the bottle is in front of the robot. The robot can only pick one milk bottle at a time. It then brings the bottle to the box that is placed far away from the conveyor belt. After placing the milk bottle in the box, the robot is free and can go back to the conveyor belt to wait for another milk bottle. However, the pick-up robot may stop to work (e.g. due to battery depletion) during the operation period. In this case, the mechanic robot goes to the pick-up robot’s position and fixes the error. After fixing, the pick-up robot can continue to operate. A pick-up robot is rewarded a score of 10 whenever it picks a milk bottle or puts the bottle in the box. The mechanic robot is also given a score of 10 whenever it fixes the pick-up robot successfully. Therefore, the game requires the cooperation of two robots to maximize reward achievement. Finally, the paper contributes to the following highlights: * • The study conducts an interesting investigation of the relationship between visual indicators and interacting agents. Experimental results show that visual indicators play as a communication bridge to enable the cooperation between multiple agents. Therefore, the study provides a practical method for the development of multi-agent systems. * • We develop a multi-agent environment named Milk Factory. Milk Factory is developed in Python, which follows the standard interface of _Arcade Learning Environment_ (ALE) [35, 36] and _OpenAI Gym_ [37]. Therefore, previous deep RL methods, which work with ALE and OpenAI Gym, adapt well with Milk Factory without significant modifications. Furthermore, Milk Factory is a configurable game that provides unlimited potentials to suit with any research purposes. * • A3C is not a well-known method for multi-agent learning [8]. In this study, however, we show that the combination of visual communication map and A3C can work properly with multi-agent problems. * • Although we use A3C to demonstrate our proposed scheme, it can be used with any deep RL algorithms. Furthermore, the method is scalable in terms of the number of agents and compatible with heterogeneous environments where agents can function different tasks. The paper is organized as follows. Section II examines recent studies in multi-agent domains. Section III presents the concept of a visual communication map and the proposed scheme VMA3C. The performance evaluation is conducted based on Milk Factory in Section IV. Finally, to conclude the paper, we present potential extension directions and the future work in Section V. ## II Related Work Since the first development of deep RL, namely _Deep Q-Network_ (DQN) [12], there have been numerous variants of DQN such as _Double Q-Learning_ [30], _Dueling Network Architecture_ [38], _Recurrent Deep Q-Learning_ [39], _Deep Attention Recurrent Q-Network_ [40], and _Prioritized Experience Replay_ [29]. However, these approaches concentrate on approximating the value function, which requires a large amount of memory to store historical transitions. Furthermore, the experience replay is known to amplify the non-stationary problem in multi-agent systems [26]. Therefore, a policy-based approach such as A3C [31] and its variants, e.g. _UNsupervised REinforcement and Auxiliary Learning_ (UNREAL) [41], are developed to hasten the learning speed by allowing multiple actors-learners to be trained at the same time. These methods are more efficient than the value-based methods as they do not require an experience replay. Simulation results show that the policy-based methods outperform the value-based ones concerning learning speed and total reward achievement in the Atari domain [31]. Besides, _Deep Deterministic Policy Gradient_ (DDPG) [42] and _Trust Region Policy Optimization_ (TRPO) [43] are proposed to deal with continuous action domains. In the first case, DDPG combines DQN with the actor-critic architecture to find the optimal policy. Whereas, TRPO is a policy gradient method that allows controlling the policy improvement in every training step. In this paper, we select A3C as the baseline method to demonstrate our proposed scheme because Milk Factory has a discrete action space. There are notable methods that use communication to control multiple agents such as Foerster _et al._ [3] formulate two approaches _Reinforced Inter-Agent Learning_ (RIAL) and _Differentiable Inter-Agent Learning_ (DIAL). In the first scheme, RIAL combines DQN with a recurrent network to independently train every agent through a shared-parameter network. Whereas, DIAL additionally sends messages across agents during the centralized learning. These real-valued messages are mapped to a limited number of communication actions. In this paper, we also adopt the centralized learning by training multiple agents at the same time but operating in a distributed nature when deployed. It is feasible because the visual indicators can be used as a virtual communication medium. Agents are trained to learn the cooperation through the visual indicators. Because the channel is virtual, there is no cost for exchanging messages between agents and hence energy efficiency. Sukhbaatar _et al._ [6] propose a communication model that allows multiple agents to communicate before deciding subsequent actions. The authors create a deep neural network that has an open access to a shared communication channel. Agents receive aggregated transmissions from other agents via a continuous vector embedded in the channel. Because the communication channel carries continuous data, it is possible to combine the approach with a standard single-agent RL method. However, the scheme is not scalable due to the complication of the neural network structure. In this study, we use the visual communication map, which is basically separated from the network configuration. Therefore, our proposed method is more robust and scalable. Finally, Gupta _et al._ [8] examine a set of cooperative control tasks and compare the performance of three different deep RL approaches policy gradient, temporal-difference error, and actor-critic method. The authors combine various disciplines to efficiently train multiple agents to learn a wide range of control tasks from discrete to continuous action domains. Initially, the agents are trained to learn a simple task and move to the harder ones afterward. The knowledge of the previous task is accumulated over time rather than acquiring new knowledge from scratch. This approach is called _curriculum learning_ [44, 45]. The authors also show that the use of a shared-parameter network is scalable in terms of the number of agents. However, the paper examines only homogeneous domains where agents function similar jobs. In our study, we extend the investigation to heterogeneous systems where agents can be in different task domains. Last but not least, we though evaluate our proposed framework in Milk Factory, a fully observation problem, the presented approach can be applied to partially observation domains such as [46] similarly. ## III Proposed Schemes In this section, we have first presented preliminary terminologies of RL and the A3C method. We then introduce the use of a visual communication map with A3C as well as the network architecture for multi-agent problems. Finally, we propose two implementation approaches for the visual communication map. ### III-A Preliminary #### III-A1 RL Figure 4: The relationship between an agent and the environment in RL. RL involves the interaction between an agent and the environment, as shown in Fig. 4. At time-step $t$, the agent perceives an environmental state $s_{t}$. It reacts with an action $a_{t}$ and the environment returns a reward $r_{t}$. From time to time, the agent obtains a set of transitions: $S^{T}=\\{s_{0},a_{0},r_{1},s_{1},a_{1},r_{2},...,s_{T},a_{T},r_{T+1}\\}$, where $T$ is the terminal state of the episode. The initial state $s_{0}$ is generated randomly from the observation space $S$ and the terminal reward $r_{T+1}$ is assigned to 0. On the agent’s side, a policy $\pi$ is defined as a mapping function $f$ from the observation space $S$ to the action space $A$, _i.e._ , $\pi=f:S\longrightarrow A$. In stochastic environments, it is convenient to define the policy $\pi$ as a conditional probability distribution of action $a_{t}$, given the state $s_{t}$, as follows: $\pi(a_{t}|s_{t})=\left\\{\ p(a_{t}|s_{t})\ \bigg{|}\ \forall a_{t}\in A,\forall s_{t}\in S\right\\}.$ In this definition, the next state $s_{t+1}$ is assumed to depend only on the previous state $s_{t}$ and the corresponding action $a_{t}$. A problem that satisfies this condition is called a _Markovian decision process_ (MDP). Milk Factory is an MDP. In real-world problems, the agent only senses a limited view of the surrounding environment. Therefore, it is more practical to formulate the problem as a _partially observable MDP_ (POMDP) [47]. In this paper, for concise, we only consider the MDP, but the same procedure can be inferred for the POMDP. On the other hand, we define the discounted return at time-step $t$ as $R_{t}=r_{t+1}+\gamma r_{t+2}+\gamma^{2}r_{t+3}+...$, where $\gamma$ is a discounted factor so that $\gamma\in[0,1]$. The goal of RL is to maximize the discounted return $R_{t}$. Finally, to evaluate the _value_ of a state, we define the value function $V$ of a state $s_{t}$ under a policy $\pi$ as the expected value of the discounted return $R_{t}$, _i.e._ , $V(s_{t})|_{\pi}=\mathbf{E}\left\\{R_{t}\middle|\pi\right\\}=\mathbf{E}\left\\{\sum_{i=0}^{T-t-1}\gamma^{i}r_{t+i+1}\middle|\pi\right\\}.$ (1) #### III-A2 A3C A3C is a policy-based method that uses the actor-critic architecture [48] and the advantage function [49] to estimate the optimal policy $\pi*$. Furthermore, it enables concurrent learning by creating multiple actors- learners to update the parameters of the neural network asynchronously. In particular, as shown in Fig. 5, the A3C’s network structure consists of two essential sub-networks: one actor network $N^{a}(\theta)$ and one critic network $N^{c}(\theta^{\prime})$, where $\theta$ and $\theta^{\prime}$ denote $N^{a}$’s weight parameters and $N^{c}$’s weight parameters respectively. The actor network estimates the policy $\pi(a_{t}|s_{t},\theta)$ of the agent while the critic network evaluates the value function $V(s_{t},\theta^{\prime})$ (1). Therefore, the A3C method optimizes two loss functions: the actor loss function $L^{a}(\theta)$ and the critic loss function $L^{c}(\theta^{\prime})$: Figure 5: Actor-critic architecture in A3C. $L^{a}(\theta)\simeq\log(\pi(a_{t}|s_{t},\theta))\left(A_{t}(\theta^{\prime})-V(s_{t},\theta^{\prime})\right)$ (2) and $L^{c}(\theta^{\prime})\simeq\frac{1}{2}\left(R_{t}(s_{t},\theta^{\prime})-V(s_{t},\theta^{\prime})\right)^{2},$ (3) where $A_{t}$ is calculated by the following equation: $A_{t}(\theta^{\prime})=\sum_{k=0}^{T_{max}-t-1}\gamma^{k}r_{t+k+1}+\gamma^{T_{max}-t}V(s_{T_{max}},\theta^{\prime}),$ (4) where $T_{max}$ denotes the maximum number of observation steps. If $T_{max}=T$, we have $V(s_{T_{max}},\theta^{\prime})=V(s_{T})=0$. Finally, to enable the agent’s exploration during the training process, the entropy regularization factor $H(s_{t},\theta)$ is added to the total loss. Finally, the total loss is the summation of the actor loss, the critic loss, and the entropy regularization: $L^{total}=L^{a}+L^{c}+\beta H,$ where $\beta$ denotes the control parameter so that $\beta\in[0,1]$. ### III-B VMA3C #### III-B1 Multi-agent A3C Figure 6: Multiple-actor-single-critic architecture in A3C. In this section, we extend the use of A3C to multi-agent learning, as shown in Fig. 6. The procedure can be applied to any standard deep RL methods. It is even more straightforward to derive from a value-based method such as DQN because its network has a single output while the A3C’s network includes two outputs (actor-critic), as explained in the previous subsection. We consider an MDP (or a POMDP) that consists of $N$ agents $\\{1,2,...,N\\}$. Because A3C has two factors in the network configuration (actor and critic), we create $N$ actor networks $N^{a}_{i}(\theta_{i})$ for $N$ agents ($i=1..N$) while using a single critic network $N^{c}(\theta^{\prime})$. To efficiently train multiple agents at the same time, we use a shared-parameter network for $N$ actors, _i.e._ , $\theta_{1}=\theta_{2}=...=\theta_{N}=\theta$. Let $\pi_{i}(a_{i,t}|s_{t})$ be a policy of the agent $i$, where $a_{i,t}\in A_{i}$, $s_{t}\in S$, and $A_{i}$ is the action space of the agent $i$. We consider that the intermediate reward $r_{t}$, given $s_{t}$ and $a_{i,t}(i=1..N)$, is the summation of individual reward, _i.e._ , $r_{t}=\sum_{i=1}^{N}r_{i,t}$. Because the state space $S$ is shared among all agents and because we use a single critic network, the critic loss function for a multi-agent scenario is the same as in equation (3), _i.e._ , $L^{c}(\theta^{\prime})\simeq\frac{1}{2}\left(R_{t}(s_{t},\theta^{\prime})-V(s_{t},\theta^{\prime})\right)^{2}$. The actor loss $L^{a}_{i}(\theta)$ of the agent $i$ is also calculated as in equation (2) as below: $L^{a}_{i}(\theta)\simeq\log(\pi_{i}(a_{i,t}|s_{t},\theta))\left(A_{t}(\theta^{\prime})-V(s_{t},\theta^{\prime})\right).$ However, the entropy regularization $H$ in this case is the summation of $N$ individual factors, _i.e._ , $H=\sum_{i=1}^{N}H_{i}(s_{t},\theta)$. Finally, we have the total loss for multi-agent problems using A3C as follows: $L^{total}=\sum_{i=1}^{N}L_{i}^{a}+\alpha L^{c}+\beta H,$ (5) where $\alpha$ ($\alpha\geq 1$) is added to balance the importance of the critic among multiple actors. #### III-B2 Visual communication map For each agent, we define a set of status $S^{stat}_{i}=\\{c_{i1},c_{i2},...,c_{ij}\\}$, where $i=1..N$, $j\geq 0$, and $c_{il}\neq c_{ip}(l\neq p)$. We also define a mapping function $F_{i}$ that maps $S^{stat}_{i}$ to a set of graphical representations $G=\\{g_{1},g_{2},...,g_{m}\\}$ so that * • $\forall x\in s_{t}\Rightarrow x\not\in g_{i}$, * • $\forall x\in g_{i}\Rightarrow x\not\in g_{j}(i\neq j$ and $i,j\in\\{1,...,m\\})$, where $m=\sum_{i=1}^{N}||S^{stat}_{i}||$. In other words, we have $F_{i}:S^{stat}_{i}\rightarrow G$ $(i=1..N)$. Let $C_{i}^{t}$ be the status of the agent $i$ at time-step $t$. We have $C_{i}^{t}\in S^{stat}_{i}$. A visual communication map at time-step $t$ is defined as a set of graphical representations of $N$ agents’ status at time- step $t$: $M_{t}=\\{F_{1}(C_{1}^{t}),F_{2}(C_{2}^{t}),...,F_{N}(C_{N}^{t})\\}$. Finally, we define an operator $\oplus$ that combines the environmental state $s_{t}$ with $M_{t}$. We assume that $S_{t}=s_{t}\oplus M_{t}$. Then, $S_{t}$ is used to feed to the neural network (instead of $s_{t}$). To maximize the learning efficiency, we define the operator $\oplus$ so as to satisfy the following conditions: ###### Premise 1. The environmental state $s_{t}$ is a strict subset of $S_{t}$, i.e., $s_{t}\subset S_{t}$. ###### Premise 2. Every element of the visual communication map $M_{t}$ is a strict subset of $S_{t}$, i.e., $F_{i}(C_{i}^{t})\subset S_{t}(i=1..N)$. ###### Premise 3. Given $E$=$\\{s_{t},F_{1}(C_{1}^{t}),F_{2}(C_{2}^{t}),...,F_{N}(C_{N}^{t})\\}$ and an arbitrary $x$ so that $x\in S_{t}$, we have that $x$ exclusively belongs to one element of $E$. Premises 1 and 2 make sure that $S_{t}$ maintains the integrity information of the environmental state and the current status of all agents in the system. Premise 3 is defined to maximize learning efficiency. We can infer the following rules based on these conditions: ###### Lemma 1. Let $s_{t}\neq\emptyset$ be the environmental state at time-step $t$ and $G_{i}^{t}$ be the graphical representation of the status of agent $i$ at time-step $t$, i.e., $G_{i}^{t}=F_{i}(C_{i}^{t})(i=1..N)$ so that $\exists k\in\\{1,...,N\\}$ and $G_{k}^{t}\neq\emptyset$. The following operator $S_{t}=s_{t}\oplus M_{t}$ $=$ $s_{t}\cup\left(\cup_{i=1}^{N}G_{i}^{t}\right)$ satisfies the three premises. _Proof_ : Because $S_{t}=s_{t}\cup\left(\cup_{i}^{N}G_{i}^{t}\right)$, we have $s_{t}\subseteq S_{t}$ and $G_{i}^{t}\subseteq S_{t}(i=1..N)$. However, the _equal_ sign does not occur because $s_{t}\neq\emptyset$ and $\exists k\in\\{1,...,N\\}$ so that $G_{k}^{t}\neq\emptyset$ and $s_{t}\neq G_{i}^{t}(i=1..N)$. This implies that the operator satisfies premises 1 and 2. Secondly, we get an element $x\in S_{t}$. We assume that $x$ belongs to at least two elements of $E$. There are two possible cases: * • If $x\in s_{t}$, we have $\exists k\in\\{1,...,N\\}$ so that $x\in G_{k}^{t}$ and $G_{k}^{t}\neq\emptyset$. This contradicts with the definition of the visual communication map, _i.e._ , $x\not\in G_{k}^{t}(\forall x\in s_{t})$. * • If $x\not\in s_{t}$, we have $\exists i,j\in\\{1,...,N\\}$ so that $i\neq j,x\in G_{i}^{t}$ and $x\in G_{j}^{t}$, which also contradicts with the definition of the visual communication map. The lemma is completely proved. $\hfill\blacksquare$ ###### Lemma 2. Let $s_{t}\neq\emptyset$ be the environmental state at time-step $t$, $G_{i}^{t}$ be the graphical representation of the status of agent $i$ at time-step $t$, and $\Phi$ be an arbitrary 1-to-1 mapping function from set to set. We assume that $G_{i}^{t}\neq\emptyset(i=1..N)$. The following operator $S_{t}=s_{t}\oplus M_{t}$ $=$ $\Phi(s_{t})\cup\left(\cup_{i=1}^{N}\Phi(G_{i})\right)$ satisfies the three premises. _Proof_ : Because $\Phi:A\rightarrow B$ is a 1-to-1 mapping function from set $A$ to set $B$, _i.e._ , $\forall x\in A,\exists!y\in B\Rightarrow y=\Phi(x)$ and vice versa. Therefore, the problem can be converted to Lemma 1, which is completely proved. $\hfill\blacksquare$ Lemma 1 and Lemma 2 are important as they can be used to suggest different implementation approaches for the visual communication map. In this paper, we suggest the following approaches: Figure 7: Visual aggregation by creating a holder. 1. 1. We create a graphical holder (a mask) that embeds both the environmental state and the visual communication map, as shown in Fig. 7. The communication map is a graphical representation that contains the current status information of $N$ agents. The holder is used as $S_{t}$ and is fed to the neural network. This approach is inferred from Lemma 1. 2. 2. We create $N+1$ ConvNets: one for the environmental state and the other $N$ ConvNets for $N$ agents, as shown in Fig. 8. The output features of $N+1$ ConvNets are concatenated to form $S_{t}$. This approach is inferred from Lemma 2. Figure 8: Concatenating output features of $N+1$ ConvNets. In this paper, we use the first method to benchmark our proposed scheme because it is more efficient (using a single shared-parameter ConvNet). The detailed algorithm is described in Algorithm 1. Algorithm 1 Visual communication Map for Multi-agent A3C (VMA3C) 1:$N\leftarrow 0$ $\triangleright$ Global shared counter 2:procedure VMA3C $\triangleright$ A learner procedure 3: $t\leftarrow 0$ 4: repeat 5: $s\leftarrow s_{t}\oplus M_{t}$ 6: $c\leftarrow 0$ 7: Reset experience memory $E\leftarrow\\{\\}$ 8: repeat 9: $a\leftarrow\\{a_{1,t},a_{2,t},...,a_{N,t}\\}$ $\triangleright$ Using $\pi_{1},\pi_{2},...,\pi_{N}$ 10: $r\leftarrow r_{t+1}=\sum_{i}^{N}r_{i,t+1}$ 11: $s^{\prime}\leftarrow s_{t+1}$ 12: Save transition $(s,a,r,s^{\prime})$ to $E$ 13: $s\leftarrow s^{\prime}$ 14: $t\leftarrow t+1,N\leftarrow N+1,c\leftarrow c+1$ 15: until $s=s_{T}\text{ or }c=T_{\text{max}}$ 16: Retrieve all $(s_{j},a_{j},r_{j},s^{\prime}_{j})$ from $E$ 17: Calculate $A_{t}$ based on (4) 18: Calculate gradients $\frac{\partial L^{total}}{\partial\theta}$, $\frac{\partial L^{total}}{\partial\theta^{\prime}}$ based on (5) 19: Perform asynchronous update of $\theta$ and $\theta^{\prime}$ 20: if $s=s_{T}$ then 21: $t\leftarrow 0$ 22: until $N>N_{\text{end}}$ In a real-world application, it is not scalable if the number of visual representations increases with the number of agents. Therefore, it is more efficient to use the following _condensed_ rule to reduce the number of visual representations. Condensed rule: Let $c_{ij}$ be the status $j$-th of an agent $i$. If there exists $K=\\{k_{1},k_{2},...,k_{N-1}\\}$ so that the conditional probability $P(c_{ij}|c_{mk_{p}})>0$ $(\forall m\in\\{1,2,...,N\\}-\\{i\\},k_{p}\in K)$ then $F_{i}(c_{ij})=\emptyset$. For example, in the road intersection problem, we only need three visual indicators (red, green, orange) to control all vehicles. However, this rule requires extra information about the problem to estimate the conditional probability $P(c_{ij}|c_{mk_{p}})$. ## IV Performance Evaluation ### IV-A Parameter settings In this section, we describe the parameter settings that are used in Milk Factory and the proposed scheme (VMA3C) for performance evaluation. The algorithm is run in a computer with an octa-core processor and a GTX 1080Ti graphics card. In Milk Factory, an episode terminates when the number of steps reaches 200. The pick-up robot is rewarded a score of 10 whenever it successfully picks a milk bottle or puts it into the box. Whereas, the mechanic robot is rewarded a score of 10 if it successfully fixes the pick-up robot. The error rate of the pick-up robot is represented as a random variable _ER_ , _i.e._ , the probability of an error per action of the pick-up robot equals to $1\%$. Moreover, the pick-up robot has 5 decision actions: moving up, moving down, moving left, moving right, and picking/dropping a milk bottle. The mechanic robot also has 5 decision actions: moving up, moving down, moving left, moving right, and fixing the pick-up robot. A history of 4 game frames is used as a single state to feed into the neural network. Finally, based on the _condensed rule_ , we only map the status of the pick-up robot to visual representations because the state of the mechanic robot depends on the pick-up robot. The pick-up robot has two states: busy state (bringing the milk bottle) and failed state (because of errors). A milk bottle represents the visual indicator of a busy state and a question mark represents the visual indicator of a failed state. We compare the performance of the VMA3C method with the A3C method. We keep the parameter settings of A3C same as the previous study [31] except the following changes. The learning rate starts from 0.001 and anneals to 0 during the course of training. The discounted factor $\gamma$ equals to 0.99. The maximum observation step $T_{max}$ equals to 5. The encouraging factor $\beta$ equals to 0.01. Moreover, we perform the gradient clipping by using the following formula [50]: $\delta_{i}^{G}=\delta_{i}^{G}*\frac{\varpi}{\max(\omega,\varpi)},$ where $\varpi=40$, $\forall i:\delta_{i}^{G}\in\Delta_{G}$, and $\forall j:\delta_{i,j}^{G}\in\delta_{i}^{G}$ we have: $\omega=\sqrt{\sum_{i,j}|\delta_{i,j}^{G}|^{2}}.$ Finally, we use the RMSProp [51] optimizer with a decay of 0.99 and an epsilon of 0.1. The network of the VMA3C method includes 4 layers: 2 convolutional layers (one has 16 filters of 8 $\times$ 8 with a stride of 4 followed by another one that has 32 filters of 4 $\times$ 4 with a stride of 2), one fully-connected layer with a size of 256 and ReLU activation functions, and one output layer that includes a single critic and and set of $N$ actors ($N=2$ for two-agent setting and $N=3$ for three-agent setting). The critic is connected into a linear activation function while each actor is connected into a _softmax_ activation function. Figure 9: The reward distribution of VMA3C in the two-agent setting of Milk Factory. We use A3C and VMA3C to train agents in 4 million steps. It takes approximately 12 training hours for each variant. During the training process, we create 40 checkpoints in different training steps. For each checkpoint, we perform a 10,000-step evaluation and record the mean of total reward and its standard error. We divide the performance evaluation into three stages. Firstly, we compare the performance of VMA3C with A3C in the two-agent setting of Milk Factory (one pickup robot and one mechanic robot). Subsequently, we add an additional pick-up robot to examine the scalability of the proposed scheme. Finally, we re-evaluate the previous two stages with different values of _ER_ to examine the robustness of VMA3C in stochastic environments. ### IV-B Two-agent setting In this setting, two robots are operated in Milk Factory: one pick-up robot and one mechanic robot. We examine the cooperation between the robots to maximize the total reward. Because the maximum number of episode steps is set to 200, it is feasible to calculate the optimal score that the two robots can achieve. In an ideal case, it takes 8 steps to pick a milk bottle, go to the box, and put the bottle into the box. The robot optimally achieves a reward of 20 in each round. Therefore, the maximum reward that the robot can achieve is $200*20/8=500$ without any errors. If an error occurs during the operation, the total reward must be smaller because the robot misses the current milk on the conveyor belt. Therefore, the reward of 500 is also the optimal total reward of two robots. Fig. 9 describes the reward distribution of VMA3C during the training. We can see that VMA3C takes only three training hours to establish an optimal policy that approximately reaches a reward of 500. It is possible due to the visual map aids the mechanic robot to fix the pick-up robot immediately when an error occurs. Figure 10: Comparison of the mean of total rewards between two methods (A3C and VMA3C) in the two-agent setting of Milk Factory. Fig. 10 shows the performance of two methods, A3C and VMA3C, regarding the mean of total rewards. In the figure, A3C performs poorly without any improvements after 12 hours of training. Furthermore, the pick-up robot gets stuck after picking a milk bottle. The mechanic robot is located next to it, as shown in the following video _https://youtu.be/J0qusnfyrr0_. As opposed to A3C, VMA3C achieves an optimal policy after 3 hours of training. This marks the importance of visual communication map in multi-agent problems. The pick- up robot in VMA3C is operated as expected: it picks a milk bottle, goes to the box by the shortest path, puts the milk bottle into the box, and goes back to the conveyor belt. Moreover, the mechanic robot fixes the pick-up robot if necessary. ### IV-C Three-agent setting Figure 11: The reward distribution of VMA3C in the three-agent setting of Milk Factory. To examine the scalability of VMA3C, we add a pick-up robot into the gameplay. Therefore, we have two pick-up robots and one mechanic robot. We also add an additional box and a milk bottle on the conveyor to increase the productivity. Fig. 11 shows the reward distribution of the VMA3C method in the three-agent setting. It can estimate an optimal policy that reaches a reward of 900. However, the results vary wildly because it is impossible to fix two pick-up robots at the same time. In this case, the environment is more stochastic than the previous one. Figure 12: Comparison of the mean of total rewards between two methods (A3C and VMA3C) in the three-agent setting of Milk Factory. Fig. 12 compares the performance of VMA3C and A3C in the three-agent setting of Milk Factory. The A3C method obtains a maximum reward of 300 after 12 hours of training while the VMA3C has a 200% higher performance than the A3C method. As shown in the following video, two pick-up robots in VMA3C operate concurrently while the mechanic robot is located in the middle of the screen, as shown in the following video _https://youtu.be/eoud2D0nW1k_. ### IV-D Robustness Figure 13: The performance of A3C in Milk Factory with four different error rates: $2\%$, $3\%$, $4\%$, and $5\%$. The left figure is conducted in the two-agent setting and the right figure is conducted in the three-agent setting of Milk Factory. Figure 14: The performance of VMA3C in Milk Factory with four different error rates: $2\%$, $3\%$, $4\%$, and $5\%$. The left figure is conducted in the two-agent setting and the right figure is conducted in the three-agent setting of Milk Factory. In this subsection, we vary the values of _ER_ to examine the robustness of the proposed method. We record the performance of both approaches by using four different error rates: $2\%$, $3\%$, $4\%$, and $5\%$. Fig. 13 shows the performance of A3C in the two-agent setting and the three-agent setting of Milk Factory. In the two-agent setting, A3C performs poorly, as explained in the previous subsection. We notice that the mean of total reward increases with the error rate. This implies that the total reward is calculated by the mechanic robot. Finally, Fig. 14 presents the performance of VMA3C in the two and three-agent setting of Milk Factory. We conclude that VMA3C is robust in the stochastic environment because it still achieves a high reward regardless of the error rate. ## V Conclusions Communications between multiple agents are difficult to implement, especially when agents are characterized by deep RL algorithms. The paper conducts an interesting investigation of using virtual communication channel via visual indicators to establish a cooperative policy for multi-agent problems. The method has practical meaning and can be applied widely in real-world applications due to many beneficial factors. Firstly, it can be used with any standard RL methods. Secondly, the proposed method is scalable in terms of the number of agents and the type of agents. Finally, the method is robust in stochastic environments and plays a vital role to solve the non-stationary problem in multi-agent domains. By using visual communication map, the agents learn a cooperative policy on their own to maximize the reward in a short period of training time. We continue to work on the effect of visual indicators to multi-agent domains in different directions: * • The method requires human knowledge to reduce the number of visual indicators, which prevents the automation of the approach. 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2024-09-04T02:54:54.869523

2002.11883

arxiv-papers

# Review, Analysis and Design of a Comprehensive Deep Reinforcement Learning Framework Ngoc Duy Nguyen, Thanh Thi Nguyen, Hai Nguyen, Doug Creighton, Saeid Nahavandi Ngoc Duy Nguyen, Doug Creighton and Saeid Nahavandi are with the Institute for Intelligent Systems Research and Innovation, Deakin University, Waurn Ponds Campus, Geelong, Victoria, Australia (e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>and [email protected]). Thanh Thi Nguyen is with the School of Information Technology, Deakin University, Burwood Campus, Melbourne, Victoria, Australia (e-mail: [email protected]). Hai Nguyen is with Khoury College of Computer Science, Notheastern University, Boston, USA (e-mail<EMAIL_ADDRESS> ###### Abstract The integration of deep learning to reinforcement learning (RL) has enabled RL to perform efficiently in high-dimensional environments. Deep RL methods have been applied to solve many complex real-world problems in recent years. However, development of a deep RL-based system is challenging because of various issues such as the selection of a suitable deep RL algorithm, its network configuration, training time, training methods, and so on. This paper proposes a comprehensive software framework that not only plays a vital role in designing a connect-the-dots deep RL architecture but also provides a guideline to develop a realistic RL application in a short time span. We have designed and developed a deep RL-based software framework that strictly ensures flexibility, robustness, and scalability. By inheriting the proposed architecture, software managers can foresee any challenges when designing a deep RL-based system. As a result, they can expedite the design process and actively control every stage of software development, which is especially critical in agile development environments. To enforce generalization, the proposed architecture does not depend on a specific RL algorithm, a network configuration, the number of agents, or the type of agents. Using our framework, software developers can develop and integrate new RL algorithms or new types of agents, and can flexibly change network configuration or the number of agents. ###### Index Terms: reinforcement learning, deep learning, software architecture, learning systems, multi-agent systems, human-machine interactions, framework. ## I Introduction Recent development of deep learning has been applied to solve various complex real-world problems. Notably, its integration into reinforcement learning (RL) has attracted a great deal of research attention. RL conducts a learning procedure by allowing agents to directly interact with the environment. An RL agent can imitate human learning process to achieve a designated goal, _i.e._ , the agent conducts trial-and-error learning (exploration) and draws on “experience” (exploitation) to improve its behaviors [1, 2]. Therefore, RL is used in countless domains, such as IT resources management [3], cyber-security [4], robotics [5, 6, 7, 8], surgical robotics [9, 10], control systems [11, 12], recommendation systems [13], bidding and advertising campaigns [14], and video games [15, 16, 17, 18, 19]. However, traditional RL methods and dynamic programming [20], which use a _bootstrapping_ mechanism to approximate the objective function, cease to work in high-dimensional environments due to memory and processing power limitations. This so-called issue, _the curse of dimensionality_ , creates a major challenge in RL literature. Figure 1: Using a UML sequential diagram to describe an RL problem. Fig. 1 describes an RL problem by using a _Unified Modeling Language_ (UML) [21] sequential diagram. Specifically, the problem includes two entities: a decision maker and the environment. The environment can be an artificial simulator or a wrapper of the real-world environment. While the environment is a _passive_ entity, the decision maker is an _active_ entity that periodically interacts with the environment. In the RL context, a decision maker and an agent can be interchangeable, though they can be two identified objects from a software design perspective. At first, the decision maker perceives a state $s_{t}$ from the environment. Then it uses its internal model to select the corresponding action $a_{t}$. The environment interacts with the chosen action $a_{t}$ by sending a numerical reward $r_{t+1}$ to the decision maker. The environment also brings the decision maker to a new state $s_{t+1}$. Finally, the decision maker uses the current transition $\vartheta=\\{a_{t},s_{t},r_{t+1},s_{t+1}\\}$ to update its decision model. This process is iterated until $t$ equals $T$, where $s_{T}$ denotes the terminal state of an episode. There are different methods to develop a decision model, such as fuzzy logic [22], genetic algorithms [23, 24], or dynamic programming [25]. In this paper, however, we consider a deep neural network as the decision model. The previous diagram infers that RL is _online_ learning because the model is updated with incoming data. However, RL can be performed offline via a _batch learning_ [26] technique. In particular, the current transition $\vartheta$ can be stored in an _experience replay_ [27] and retrieved later to train the decision model. Finally, the goal of an RL problem is to maximize the expected sum of discounted reward $R_{t}$, _i.e._ , $R_{t}=r_{t+1}+\gamma r_{t+2}+\gamma^{2}r_{t+3}+...+\gamma^{T-t-1}r_{T},$ where $\gamma$ denotes the discounted factor and $0<\gamma\leq 1$. In 2015, Google DeepMind [28] announced a breakthrough in RL by combining it with deep learning to create an intelligent agent that can beat a professional human player in a series of 49 Atari games. The idea was to use a deep neural network with convolutional layers [29] to directly process raw images (states) of the game screen to estimate subsequent actions. The study is highly valued because it opened a new era of RL with deep learning, which partially solves the curse of dimensionality. In other words, deep learning is a great complement for RL in a wide range of complicated applications. For instance, Google DeepMind created a program, AlphaGo, which beat the Go grandmaster, Lee Sedol, in the best-of-five tournament in 2016 [30]. AlphaGo is a full-of-tech AI that is based on Monte Carlo Tree Search [31], a hybrid network (policy network and value network), and a self-taught training strategy [32]. Other applications of deep RL can be found in self-driving cars [33, 34], helicopters [35], or even NP-hard problems such as _Vehicle Routing Problem_ [36] and combinatorial graph optimization [37]. As stated above, deep RL is crucial owing to its appealing learning mechanism and widespread applications in the real world. In this study, we further delve into practical aspects of deep RL by analyzing challenges and solutions while designing a deep RL-based system. Furthermore, we consider a real-world scenario where multiple agents, multiple objectives, and human-machine interactions are involved. Firstly, if we can take advantage of using multiple agents to accomplish a designated task, we can shorten the _wall time_ , _i.e._ , the computational time to execute the assigned task. Depending on the task, the agents can be _cooperative_ or _competitive_. In the cooperative mode, agents work in _parallel_ or in _pipeline_ to achieve the task [38]. In the case of competition, agents are scrambled, which basically raises the _resource hunting_ problem [39]. However, in contrast to our imagination, competitive learning can be fruitful. Specifically, the agent is trained continually to place the opponent into a disadvantaged position and the agent is improved over time. Because the opponent is also improved over time, this phenomenon eventually results in the _Nash equilibrium_ [40]. Moreover, competitive learning originates the self-taught strategy (_e.g._ AlphaGo) and a series of techniques such as the _Actor-Critic architecture_ [41, 42], opponent modeling [43, 44], and _Generative Adversarial Networks_ [45]. Finally, we notice the problem of _moving target_ [46, 47] in multi-agent systems, which describes a scenario when the decision of an agent depends on other agents, thus the optimal policy becomes _non-stationary_ [47, 48]. Secondly, a real-world objective is often complicated as it normally comprises of multiple sub-goals. It is straightforward if sub-goals are _non- conflicting_ because they can be seen as a single composite objective. The more difficult case is when there are _conflicting objectives_. One solution is to convert a multi-objective problem into a single objective counterpart by applying _scalarization_ through an application of a linear weighted sum for individual objective [49] or non-linear methods [50]. These approaches are categorized as _single-policy methods_. In contrast, the _multi-policy methods_ [51] seek multiple optimal policies at the same time. Although the number of multi-policy methods is restricted, it can be powerful. For instance, the _Convex Hull Value Iteration_ algorithm [52] computes a set of objective combinations to retrieve all deterministic optimal policies. Finally, to benchmark a multi-objective method, we find or approximate a boundary surface, namely _Pareto dominance_ , which presents the maximum performance of different weights (if scalarization is used) [53]. Recent studies have tried to integrate multi-objective mechanisms into deep RL [54, 55, 56]. TABLE I: Key RL terminologies Term | Description | Pros | Cons ---|---|---|--- Model-free RL | The environment is a black box. Agents mostly | The algorithm does not | Requires a large amount of conduct a trial-and-error procedure to learn on its own. | need a model of the | samples They use rewards to update their decision models | environment | Model-based RL | Agents construct a model that | Speed up learning | Having an accurate simulates the environment and use it to generate | and improve sample | and useful model is often future episodes. By using the model, agents can | efficiency | challenging estimate not only actions but also future states | | Temporal difference learning | Use TD error to estimate the value function. | Fast convergence as it | Estimates can be For example, in Q-learning, | does not need to wait until | biased $Q(s_{t},a_{t})=Q(s_{t},a_{t})+\beta(r_{t}+\gamma\max_{a}Q(s_{t+1},a))$ | the episode ends | Monte-Carlo method | Estimate the value function by obtaining the average | The values are non-biased | Slow convergence and the of the same values in different episodes | estimates | estimates have high $Q(s_{t},a_{t})=\lim_{N->\infty}\sum_{i=1}^{N}Q(s_{t}^{i},a_{t}^{i})$ | | variances. Has to wait until | | | episode ends to do updates Continuous action space | The number of control actions is continuous | A policy-based method | Cannot use a | can be used | value-based method Discrete action space | The number of control actions is discrete | Both policy-based method | Intractable if the number and finite | and value-based method | of actions is large | can be used | Deterministic policy | The policy maps each state to a specific action | Reduce data sampling | Vulnerable to noise and | | stochastic environments Stochastic policy | The policy maps each state to a probability distribution | Better exploration | Requires a large amount of over actions | | samples On-policy method | Improve the current policy that the agent | Safer to explore | May be stuck in local is using to make decisions | | minimum solutions Off-policy method | Learn the optimal policy (while samples are generated | Instability. Often used with | Might be unsafe because by the behavior policy) | an experience replay | the agent is free to explore Fully observable | All agents can observe the complete states of the | Easier to solve than | The number of state environment | environment | partially observable | can be large | | environments | Partially observable | Each agent only observes a limited observation | More practical in | More difficult to solve environment | of the environment | real-world applications | as the agents require | | | to remember past states Last but not least, human-machine interaction is another key factor in a deep RL-based system. A self-driving car, for example, should accept human intervention in emergency cases [57, 58]. Therefore, it is critical to ensure a certain level of safety while designing a hybrid system in which humans and machines can work together. Due to its importance, Google DeepMind and OpenAI have presented novel ways that have encouraged a number of inventions in recent years [59]. For instance, Christiano _et al._ [60] propose a novel scheme that accepts human feedback during the training process. However, the method requires an operator to constantly observe the agent’s behavior, which is an onerous and error-prone task. Recent work [61] provides a more practical approach by introducing a behavioral control system. The system is used to control multiple agents in real time via human dictation. Table I summarizes key terminologies that are widely used in RL contexts. In summary, the study contributes the following key factors: * • The paper presents an overall picture of contemporary deep RL. We briefly overview state-of-the-art deep RL methods considering three key factors of a real-world application such as multi-agent learning, multi-objective problems, and human-machine interactions. Thereafter, the paper offers a checklist for software managers, a guideline for software designers, and a technical document for software programmers. * • We analyze challenges and difficulties while designing a deep RL-based system and hence mitigate possible mistakes during the development process. In other words, software designers can inherit the proposed design, foresee difficulties, and eventually expedite the entire development procedure, especially in agile software development. * • Finally, the source code of the proposed framework can be found in [62]. Based on this template, RL beginners can prototype an RL method and develop an RL- based application in a short time span. As a result, the paper is a key factor to share deep RL to a wider community. The paper has the following sections. Section II conducts a brief survey of state-of-the-art deep RL methods in different research directions. Section III presents our proposed system architecture, which supports multiple agents, multiple objectives, and human-machine interactions. Finally, we conclude the paper in Section IV. ## II Literature Review TABLE II: Key deep RL methods in literature Method | Description and Advantage | Technical Requirements | Drawbacks | Implementation ---|---|---|---|--- Value-based method DQN | Use a deep convolutional network to | $\bullet$ Experience replay | $\circ$ Excessive memory usage | [85] | directly process raw graphical data and | $\bullet$ Target network | $\circ$ Learning instability | [86] | approximate the action-value function | $\bullet$ Q-learning | $\circ$ Only for discrete action | [87] | | | space | [88] Double DQN | Mitigate the DQN’s maximization bias | $\bullet$ Double Q-learning | $\circ$ Inherit DQN’s drawbacks | [88] | problem by using two separate networks: | | | | one for estimating the value, one for | | | | selecting action. | | | Prioritized | Prioritize important transitions so | $\bullet$Importance sampling | $\circ$ Inherit DQN’s drawbacks | [85] Experience Replay | that they are sampled more frequently. | | $\circ$ Slower than non-prioritized | [88] | Improve sample efficiency | | experience replay (speed) | Dueling Network | Separate the DQN architecture into | $\bullet$ Dueling network | $\circ$ Inherit DQN’s drawbacks | [88] | two streams: one estimates state-value | architecture | | | function and one estimates the advantage | $\bullet$ Prioritized replay | | | of each action | | | Recurrent DQN | Integrate recurrency into DQN | $\bullet$ Long Short Term | $\circ$ Inherit DQN’s drawbacks | [88] | Extend the use of DQN in | Memory | | | partially observable environments | | | Attention Recurrent | Highlight important regions of the | $\bullet$ Attention mechanism | $\circ$ Inherit DQN’s drawbacks | [68] DQN | environment during the training process | $\bullet$ Soft attention | | | | $\bullet$ Hard attention | | Rainbow | Combine different techniques in DQN | $\bullet$ Double Q-learning | $\circ$ Inherit DQN’s drawbacks | [85] | variants to provide the state-of-the-art | $\bullet$ Prioritized replay | | | performance on Atari domain | $\bullet$Dueling network | | | | $\bullet$Multi-step learning | | | | $\bullet$Distributional RL | | | | $\bullet$Noisy Net | | Policy-based method A3C/A2C | Use actor-critic architecture to estimate | $\bullet$ Multi-step learning | $\circ$ Policy updates exhibit | [86] | directly the agent policy. A3C enables | $\bullet$ Actor-critic model | high variance | [87] | concurrent learning by allowing multiple | $\bullet$ Advantage function | | [88] | learners to operate at the same time | $\bullet$ Multi-threading | | UNREAL | Use A3C and multiple unsupervised | $\bullet$ Unsupervised reward | $\circ$ Policy updates exhibit | [89] | reward signals to improve learning efficiency | signals | high variance | | in complicated environments | | | DDPG | Concurrently learn a deterministic policy | $\bullet$ Deterministic | $\circ$ Support only continuous | [87] | and a Q-function in DQN’s fashion | policy gradient | action space | [88] TRPO | Limit policy update variance by | $\bullet$ Kullback-Leibler | $\circ$ Computationally expensive | [87] | using the conjugate gradient to estimate | divergence | $\circ$ Large batch of rollouts | [88] | the natural gradient policy | $\bullet$ Conjugate gradient | $\circ$ Hard to implement | | TRPO is better than DDPG in terms of | $\bullet$ Natural policy gradient | | | sample efficiency | | | ACKTR | Inherit the A2C method | $\bullet$ Kronecker-factored | $\circ$ Still complex | [87] | Use Kronecker-Factored approximation to | approximate curvature | | | reduce computational complexity of TRPO | | | | ACKTR outperforms TRPO and A2C | | | ACER | Integrate an experience replay into A3C | $\bullet$ Importance weight | $\circ$ Excessive memory usage | [87] | Introduce a light-weight version of TRPO | truncation & bias correction | $\circ$ Still complex | [88] | ACER outperforms TRPO and A3C | $\bullet$ Efficient TRPO | | PPO | Simplify the implementation of TRPO | $\bullet$ Clipped objective | $\circ$ Require network tuning | [86] | by using a surrogate objective function | $\bullet$ Adaptive KL penalty | | [87] | Achieve the best performance in continuous | coefficient | | [88] | control tasks | | | ### II-A Single-Agent Method The first advent of deep RL, _Deep Q-Network_ (DQN) [28, 63], basically uses a deep neural network to estimate values of state-action pairs via a _Q-value function_ (_a.k.a._ , _action-value function_ or $Q(s,a)$). Thereafter, a number of variants based on DQN were introduced to improve the original algorithm. Typical extensions can be examined such as _Double DQN_ [64], _Dueling Network_ [65], _Prioritized Experience Replay_ [66], _Recurrent DQN_ [67], _Attention Recurrent DQN_ [68], and an ensemble method named _Rainbow_ [69]. These approaches use an experience replay to store historical transitions and retrieve them in batches to train the network. Moreover, a separate network _target network_ can be used to mitigate the correlation of the sequential data and prevent the training network from overfitting. Instead of estimating the action-value function, we can directly approximate the agent’s policy $\pi(s)$. This approach is known as the _policy gradient_ or _policy-based_ method. _Asynchronous Advantage Actor-Critic_ (A3C) [70] is one of the first policy-based deep RL methods to appear in the literature. In particular, A3C includes two networks: an actor network that is used to estimate the agent policy $\pi(s)$ and a critic network that is used to estimate the _state-value function_ $V(s)$. Additionally, to stabilize the learning process, A3C uses the _advantage function_ , _i.e._ , $A(s,a)=Q(s,a)-V(s)$. There is a synchronous version of A3C, namely A2C [70], which has the advantage of being simpler but with comparable or better performance. A2C mitigates the risk of multiple learners which might be overlapping when updating the weights of the global networks. There have been a great number of policy gradient methods since the development of A3C. For instance, _UNsupervised REinforcement and Auxiliary Learning_ (UNREAL) [71] uses multiple unsupervised pseudo-reward signals at the same time to improve the learning efficiency in complicated environments. Rather than estimating a stochastic policy, _Deterministic Policy Gradient_ [72] (DPG) finds a deterministic policy, which significantly reduces data sampling. Moreover, _Deep Deterministic Policy Gradient_ [73] (DDPG) combines DPG with DQN to enable the learning of a deterministic policy in a continuous action space using the actor-critic architecture. The authors in [74] even propose _Multi-agent DDPG_ (MADDPG), which employs DDPG in multi-agent environments. To further stabilize the training process, the authors in [75] introduce the _Trust Region Policy Optimization_ (TRPO) method, which integrates the _Kullback–Leibler divergence_ [76] into the training procedure. However, the implementation of the method is complicated. In 2017, Wu _et al._ [77] proposed _Actor-Critic using Kronecker-Factored Trust Region_ (ACKTR), which applies Kronecker-factored approximation curvature into gradient update steps. Additionally, the authors in [78] introduced an efficient off-policy sampling method based on A3C and an experience replay, namely _Actor-Critic with Experience Replay_ (ACER). To simplify the implementation of TRPO, ACKTR, and ACER, _Proximal Policy Optimization_ (PPO) [79] is introduced by using a clipped “surrogate” objective function together with stochastic gradient ascent. Finally, some studies combine a policy-based and value-based method such as [80, 81, 82] or an on-policy and off-policy method such as [83, 84]. Table II summarizes key deep RL methods and their reliable implementation repositories. Based on specific application domains, software managers can select a suitable deep RL method to act as a baseline for the target system. ### II-B Multi-Agent Method In multi-agent learning, there are two widely used schemes in the literature: _individual_ and _mutual_. In the first case, each agent in the system can be considered as an independent decision maker and other agents as a part of the environment. In this way, any deep RL methods in the previous subsection can be used in multi-agent learning. For instance, Tampuu _et al._ [90] used DQN to create an independent policy for each agent. The authors analyze the behavioral convergence of the involved agents with respect to cooperation and competition. Similarly, Leibo _et al._ [91] introduced a sequential social dilemma, which basically uses DQN to analyze the agent’s strategy in Markov games such as Prisoner’s Dilemma, Fruit Gathering, and Wolfpack. However, the approach limits the number of agents because the computational complexity increases with the number of policies. To overcome this obstacle, Nguyen _et al._ developed a behavioral control system [61] in which homogeneous agents can share the same policy. As a result, the method is robust and scalable. Another problem in multi-agent learning is the use of an experience replay, which amplifies the non-stationary problem that occurs due to asynchronous data sampling of different agents [92, 93]. A lenient approach [94] can subdue the problem by mapping transitions into decaying temperature values, which basically controls the magnitude of updating different policies. In mutual scheme, agents can “speak” with each other via a settled communication channel. Moreover, agents are often trained in a centralized manner but eventually operate in a decentralized fashion when deployed [95]. In other words, a multi-agent RL problem can be divided into two sub-problems: a goal-directed problem and a communication problem. Specifically, Foerster _et al._ [96] introduced two communication schemes based on the centralized- decentralized rationale: _Reinforced Inter-Agent Learning_ (RIAL) and _Differentiable Inter-Agent Learning_ (DIAL). While RIAL reinforces agents’ learning by sharing parameters, DIAL allows _inter-communication_ between agents via a shared medium. Both methods, however, operate with a discrete number of communication actions. As opposed to RIAL and DIAL, the authors in [97] introduce a novel network architecture, namely _Communication Neural Net_ (CommNet), which enables communication by using a continuous vector. As a result, the agents are trained to learn to communicate by backpropagation. However, CommNet limits the number of agents due to the increase of computational complexity. To make it scalable, Gupta _et al._ [98] introduced a parameter sharing method that can handle a large number of agents. However, the method only works with homogeneous systems. Finally, Nguyen _et al._ [61] extended the Gupta’s study to heterogeneous systems by designing a behavioral control system. For rigorous study, a complete survey on multi-agent RL can be found in [99, 100, 101]. In summary, it is critical to address the following factors in multi-agent learning because they have a great impact on the target software architecture: * • It is preferable to employ the centralized-decentralized rationale in a multi- agent RL-based system because the training process is time-consuming and computationally expensive. A working system may require hundreds to thousands of training sessions by searching through the hyper-parameter space to find the optimal solution. * • The communication between agents can be _realistic_ or _imaginary_. In realistic communication, agents “speak” with each other using an established communication protocol. However, there is no actual channel in imaginary communication. The agents are trained to collaborate using a specialized network architecture. For instance, OpenAI [102] proposes an actor-critic architecture where the critic is augmented with other agents’ policy information. As a result, the two methods can differentiate how to design an RL-based system. * • A partially observable environment has a great impact on designing a multi- agent system because each agent has its own unique perspective of the environment. Therefore, it is important to first carefully examine the environment and application type to avoid any malfunction in the design. ### II-C RL Challenges In this subsection, we briefly review major challenges while designing a deep RL-based system and corresponding solutions. To remain concise, the proposed framework is not concerned with these techniques but it is straightforward to extend the architecture to support these rectifications. _Catastrophic forgetting_ is a problem that occurs in _continual learning_ and _multi-task learning_ , _i.e._ , an another task is trained after learning the first task by using the same neural network. In this case, the neural network gradually forgets the knowledge of the first task to adopt the new one. One solution is to use regularization [103, 104] or a dense neural network [105, 106]. However, these approaches are only feasible with a limited number of tasks. Recent studies introduce more scalable approaches such as _Elastic Weight Consolidation_ (EWC) [107] or _PathNet_ [108]. While EWC finds a configuration of the network that yields the best performance in different tasks, PathNet uses a “super” neural network to fulfill the knowledge of different tasks in different paths. _Policy distillation_ [109] or _transfer learning_ [110, 111] can be used to train an agent to learn individual tasks and collectively transfer the knowledge to a single network. Transfer learning is often used when the actual experiment is expensive and intractable. In this case, the network is trained with simulations and later is deployed into the target experiment. However, a _negative transfer_ may occur when the performance of the learner is lower than the trainer. The authors in [112] introduced _Hierarchical Prioritized Experience Replay_ that uses high-level features of the task and selects important data from the experience replay to mitigate the negative transfer. One recent study [113] aligned the mutual learning to achieve a comparable performance between the actual experiment and simulations. Another obstacle in RL is dealing with _long-horizon_ environments with _sparse rewards_. In such tasks, the agent hardly receives any reward and easily gets stuck in local minimum solutions. One straightforward solution is to use _reward shaping_ [114] that continuously instructs the agent to achieve the objective. The problem can also be divided into a hierarchical tree of sub-problems where the parent problem has a higher abstraction than the child problem (_Hierarchical RL_) [115]. To encourage self-exploration, the authors in [116] introduced intrinsic reward signals to reinforce the agent to make a generic decision. State-of-the-art methods of _intrinsic motivation_ can be found in [117, 118, 119]. Finally, Andrychowicz _et al._ [120] propose _Hindsight Experience Replay_ that implicitly simulates _curriculum learning_ [121] by creating imagined trajectories in the experience replay with positive rewards. In this way, the agent can learn from failures and automatically generalize a solution in success cases. Finally, a variety of RL-related techniques are proposed to make RL feasible in large-scale applications. One approach is to augment the neural network with a “memory” to enhance sample efficiency in complicated environments [122, 123]. Additionally, to enforce scalability, many distributed methods have been proposed such as _Distributed Experience Replay_ [124], deep RL acceleration [125], and distributed deep RL [126]. Finally, _imitation learning_ can be used together with _inverse RL_ to speed up training by directly learning from expert demonstrations and extracting the expert’s cost function [127]. ### II-D Deep RL Framework In this subsection, we discuss the latest deep RL frameworks in the literature. We select the libraries based on different factors including Python-based implementation, clear documentation, reliability, and active community. Based on our analysis, software managers can select a suitable framework depending on project requirements. Figure 2: A “pyramid” software architecture. * • _Chainer_ – Chainer [88] is a powerful and flexible framework for neural networks. The framework is currently supported by IBM, Intel, Microsoft, and Nvidia. It provides an easy way to manipulate neural networks such as by creating a customized network, visualizing a computational graph, and supporting a debug mode. It also implements a variety of deep RL methods. However, the Chainer’s architecture is complicated and requires a great effort to develop a new deep RL method. The number of integrated environments is also limited, _e.g._ , Atari [128], OpenAI Gym [129], and Mujoco [130]. * • _Keras-RL_ – Keras-RL [131] is a friendly deep RL library, which is recommended for deep RL beginners. However, the library provides a limited number of deep RL methods and environments. * • _TensorForce_ – TensorForce [133] is an ambitious project that targets both industrial applications and academic research. The library has the best modular architecture we have reviewed so far. Therefore, it is convenient to use the framework to integrate customized environments, modify network configurations, and tweak deep RL algorithms. However, the framework has a deep software stack (“pyramid” model) that includes many abstraction layers, as shown in Fig. 2. This hinders novice readers from prototyping a new deep RL method. * • _OpenAI Baselines_ – OpenAI Baselines [87] is a high-quality framework for contemporary deep RL. In contrast to TensorForce, the library is suitable for researchers who want to reproduce original results. However, OpenAI Baselines is unstructured and incohesive. * • _RLLib_ – RLLib [86] is a well-designed deep RL framework that provides a means to deploy deep RL in distributed systems. Therefore, the usage of RLLib is not friendly for RL beginners. * • _RLLab_ – RLLab [136] provides a diversity of deep RL methods including TRPO, DDPG, Cross-Entropy Method, and Evolutionary Strategy. The library is friendly to use but not straightforward for modifications. In summary, most deep RL frameworks focus on the performance of deep RL methods. As a result, those frameworks limits code legibility, which basically restricts RL beginners from readability and modifications. In this paper, we propose a comprehensive framework that has the following properties: * • Allow new users to prototype a deep RL method in a short period of time by following a modular design. As opposed to TensorForce, we limit the number of abstraction layers and avoid the pyramid structure. * • The framework is friendly with a simplified user interface. We provide an API based on three key concepts: policy network, network configuration, and learner. * • Enforce scalability and generalization while supporting multiple agents, multiple objectives, and human-machine interactions. * • Finally, we introduce a concept of unification and transparency by creating plugins. Plugins are gateways that extract learners from other libraries and plug them into our proposed framework. In this way, users can interact with different frameworks using the same interface. ## III Software Architecture In this section, we examine core components towards designing a comprehensive deep RL framework, which basically employs generality, flexibility, and interoperability. We aim to support a broad range of RL-related applications that involve multiple agents, multiple objectives, and human-agent interaction. We use the following pseudocode to describe a function signature: $\bullet\textbf{function\\_name}([param1,param2,...])\rightarrow\\\ [return1,return2,...]\text{ or }\\{return1,return2,...\\}\text{ or }A$ where $\rightarrow$ denotes a return operation, $A$ is a scalar value, $[...]$ denotes an array, and $\\{...\\}$ denotes a list of possible values of a single variable. ### III-A Environment First, we create a unique interface for the _environment_ to establish a communication channel between the framework and agents. However, to reduce complexity, we put any human-related communication into the environment. As a result, human interactions can be seen as a part of the environment and are hidden from the framework, _i.e._ , the environment provides two interfaces: one for the framework and one for human, as shown in Fig. 3. While the framework interface is often in programming level (functions), the human interface has a higher abstraction mostly in human understanding forms such as voice dictation, gesture recognition, or control system. Figure 3: A conceptual model of the environment with a human interface. Essentially, the environment’s framework interface should provide the following functions: * • clone(): the environment can duplicate itself. The function is useful when an RL algorithm requires multiple learners at the same time (_e.g._ A3C). * • reset(): reset the environment to the initial state. The function must be called after or before an episode. * • step($[a_{1},a_{2},...,a_{N}]$) $\rightarrow$ [$r_{1},r_{2},...,r_{M}$]: executes $N$ specified actions of $N$ agents in the environment. The function returns $M$ rewards, each of which represents an objective function. * • get_state() $\rightarrow$ [$s_{1},s_{2},...,s_{N}$]: retrieves the current states of the environment. If the environment is a partially observable MDP, the function returns $N$ states, which each presents the current state of an agent. However, if the environment is a fully observable MDP, we have $s_{1}=s_{2}=...=s_{N}=s$. * • is_terminal() $\rightarrow$ {True, False}: checks if the episode is terminated. * • get_number_of_objectives() $\rightarrow$ $M$: is a helper function that indicates the number of objectives in the environment. * • get_number_of_agents() $\rightarrow$ $N$: is a helper function that indicates the number of agents in the environment. Finally, it is important to consider the following questions while designing an environment component as they have a great impact on subsequent design stages: * • _Is it a simulator or a wrapper?_ In the case of a wrapper, the environment is already developed and configured. Our duty is to develop a wrapper interface that can compatibly interact with the framework. In contrast to the wrapper, developing a simulator is complicated and requires expert knowledge. In real- time applications, we may first develop a simulator in C/C++ (for better performance) and then create a Python wrapper interface (for easier integration). In this case, we need to develop both a simulator and a wrapper. * • _Is it stochastic or deterministic?_ Basically, a stochastic environment is more challenging to implement than a deterministic one. There are potential factors that are likely to contribute to the randomness of the environment. For example, a company intends to open a bike rental service. _N_ bikes are equally distributed into _M_ potential places. However, at a specific time, place _A_ still has a plenty of bikes because there is no customer. As a result, bikes in place _A_ are delivered to other places where the demand is higher. The company seeks development of an algorithm that can balance the number of bikes in each place over time. In this example, the bike rental service is a stochastic environment. We can start building a simple stochastic model based on Poisson distribution to represent the rental demand in each place. We end up with a complicated model based on a set of observable factors such as rush hour, weather, weekend, festival, etc. Depending on the stochasticity of the model, we can decide to use a model-based or model-free RL method. * • _Is it complete or incomplete?_ A complete environment at any time provides sufficient information to construct a branch of possible moves in the future (_e.g._ Chess or Go). The completeness can help to decide an effective RL method later. For instance, a complete environment can be solved with a careful planning rather than a trial-and-error approach. * • _Is it fully observable or partially observable?_ The observability of environment is essential when designing a deep neural network. Partially observable environments might require recurrent layers or an attention mechanism to enhance the network’s capacity during the training. A self- driving scenario is partially observable while a board game is fully observable. * • _Is it continuous or discrete?_ As described in Table I, this factor is important to determine the type of methods used such as policy-based or value- based methods or the network configuration such as actor-critic architectures. * • _How many objectives does it have?_ Real-world applications often have multiple objectives. If the importance weights between objectives can be identified in the beginning, it is reasonable to use a single-policy RL method. Alternatively, a multi-policy RL method can prioritize the importance of an objective in real time. * • _How many agents does it have?_ A multi-agent RL-based system is much more complicated than a single-agent counterpart. Therefore, it is essential to analyze the following factors of a multi-agent system before delving into the design: the number of agents, the type of agents, communication abilities, cooperation strategies, and competitive potentials. ### III-B Network The _neural network_ is also a key module of our proposed framework, which includes a network configuration and a policy network, as illustrated in Fig. 4. A network configuration defines the deep neural network architecture (_e.g._ CNN or LSTM), loss functions (_e.g._ Mean Square Error or Cross Entropy Loss), and optimization methods (_e.g._ Adam or SGD). Depending on the project’s requirements, a configuration can be divided to different abstraction layers, where the lower abstraction layer is used as a mapping layer for the higher abstraction layer. In the lowest abstraction level (programming language), a configuration is implemented by a deep learning library, such as Pytorch [134] (with dynamic graph) or TensorFlow (with static graph). The next layer is to use a scripting language, such as _xml_ or _json_ , to describe the network configuration. This level is useful because it provides a faster and easier way to configure a network setting. For those who do not have much knowledge of implementation details such as system analysts, a graphical user interface can assist them. However, there is a trade-off here: the higher abstraction layer achieves better usability and productivity but has a longer development cycle. A policy network is a composite component that includes a number of network configurations. However, the dependency between a policy network and a configuration can be weak, _i.e._ , an _aggregation_ relationship. The policy network’s objective is twofold. It provides a high-level interface that maintains connectivity with other modules in the framework, and it initializes the network, saves the network’s parameters into checkpoints, and restores the network’s parameters from checkpoints. Finally, the neural network interface should provide the following functions: Figure 4: A neural network module includes a network configuration and a policy network. * • create_network() $\rightarrow$ [$\theta_{1},\theta_{2},...,\theta_{K}$]: instantiates a deep neural network by using a set of network configurations. The function returns the network’s parameters (references) $\theta_{1},\theta_{2},...,\theta_{K}$. * • save_model(): saves the current network’s parameters into a checkpoint file. * • load_model([chk]): restores the current network’s parameters from a specified checkpoint file _chk_. * • predict([$s_{1},s_{2},...,s_{N}$]) $\rightarrow$ [$a_{1},a_{2},...,a_{N}$]: given the current states of $N$ agents $s_{1},s_{2},..._{,}s_{N}$, the function uses the network to predict the next $N$ actions $a_{1},a_{2},...,a_{N}$. * • train_network([data_dict]): trains the network by using the given data dictionary. The data dictionary often includes the current states, current actions, next states, terminal flags, and miscellaneous information (global time step or objective weights) of $N$ agents. ### III-C Learner Figure 5: A high-level design of a learner module. The last key module of our proposed framework is a _learner_ , as shown in Fig. 5. While the environment module and the network module create the application’s shell, the learner plays as an engine that allows the system to operate properly. The three modules jointly create the backbone of the system. In particular, the learner uses the environment module to generate episodes. It manages the experience replay memory and defines the RL implementation details, such as multi-step learning, multi-threading, or reward shaping. The learner is often created together with a monitor. The monitor is used to manage multiple learners (if multi-threading is used) and collect any data from the learners during training, such as performance information for debugging purposes and post-evaluation reports. Finally, the learner collects necessary data, packs it into a dictionary, and sends it to the network module for training. Additionally, a _factory_ pattern [135] can be used to hide the operating details between the monitor and the learners. As a result, the factory component promotes higher abstraction and usability through a simplified user API as below: * • create_learner([monitor_dict, learner_dict]) $\rightarrow$ obj: The factory creates a learner by using the monitor’s data dictionary (batch size, the number of epochs, report frequency, etc) and the learner’s data dictionary (the number of threads, epsilon values, reward clipping thresholds, etc.). * • train(): trains the generated learner. * • evaluate(): evaluates the generated learner. ### III-D Plugin There have been a great number of RL methods in the literature. Therefore, it is impractical to implement all of them. However, we can reuse the implementation from existing libraries such as TensorForce, OpenAI Baselines, or RLLab. To enforce flexibility and interoperability, we introduce a concept of unification by using plugins. A plugin is a piece of program that extracts learners or network configurations from third party libraries and plugs them into our framework. As a result, the integrated framework provides a unique user API but supports a variety of RL methods. In this way, users do not need to learn different libraries. The concept of unification is described in Fig. 6. Figure 6: A unification of different RL libraries by using plugins. A plugin can also act as a conversion program that converts the environment’s interface of this library into the environment’s interface of other libraries. As a result, the proposed framework can work with any environments in third party libraries and vice versa. Therefore, a plugin should include the following functions: * • convert_environment([source_env]) $\rightarrow$ target_env: converts the environment’s interface from the source library to the environment’s interface defined in the target library. * • extract_learner([param_dict]) $\rightarrow$ learner: extracts the learner from the target library. * • extract_configuration([param_dict]) $\rightarrow$ config: extracts the network configuration from the target library. ### III-E Overall Structure Figure 7: A UML sequential diagram of the training process. Figure 8: A UML sequential diagram of the evaluation process. TABLE III: Demonstration codes of different use cases [62]. Use case | Description | Source Code ---|---|--- 1\. How to inherit an existing learner? | Develop a Monte-Carlo learner that inherits the | fruit/learners/mc.py | existing Q-Learning learner | 2\. Develop a new environment | Create a Grid World [1] environment that follows | fruit/envs/games/grid_world/ | the framework’s interface | 3\. Develop a multi-agent environment | Create a Tank Battle [61] game in which | fruit/envs/games/tank_battle/ with human-agent interaction | humans and AI agents can play together | 4\. Multi-objective environment and | Use a multi-objective learner (MO Q-Learning) | fruit/samples/basic/multi_objectives_test.py multi-objective RL | to train an agent to play Mountain Car [51] | 5\. Multi-agent learner with | Create a multi-agent RL method based on A3C [98] | fruit/samples/tutorials/chapter_6.py human-agent interaction | and apply it to Tank Battle | 6\. How to use a plugin? | Develop a TensorForce plugin, extract the PPO learner, | fruit/plugins/quick_start.py | and train an agent to play Cart Pole [1] | Putting everything together, we have a sequential diagram of the training process, as described in Fig. 6. The workflow breaks down the training process into smaller procedures. Firstly, the factory instantiates a specified learner (or a plugin) and sends its reference to the monitor. The monitor clones the learner into multiple learner threads. Each learner thread is run until the number of epochs exceeds a predefined value, $K$. The second loop within the learner thread is used to generate episodes. In each episode, a learner thread perceives the current states of the environment and predicts next actions using the policy network and configuration network. The next actions are applied to the environment. The environment returns next states and a terminal flag. Finally, the policy network is trained for every $L$-step. There are minor changes in the evaluation process, as shown in Fig. 8. First, the policy network’s parameters are restored from a specified checkpoint file while initializing the learner. Second, all training procedure calls are discarded while generating episodes. To enhance usability and reduce redundancy, it is advisable to implement the framework in _Object-Oriented Programming_ (OOP). In this way, a new learner (configuration) can be easily developed by inheriting existing learners (configurations) in the framework, as shown in Fig. 9. Figure 9: An inheritance relationship between learners and configurations. ## IV Conclusions It is promising to see many successful applications of deep RL methods in recent years. This paper has briefly reviewed recent advances in the RL literature with respect to multi-agent learning, multi-objective learning, and human-machine interactions. We also examine different deep RL libraries and analyze their limitations. Finally, we propose a novel design of a deep RL framework that exhibits great potential in terms of usability, flexibility, and interoperability. We highlight any considerable notices during the design so that software managers can avoid possible mistakes while designing an RL- based application. The proposed framework can be considered as a _template_ to design a real- world RL-based application. Because the framework is developed in OOP, it is beneficial to utilize OOP principles, such as inheritance, polymorphism, and encapsulation to expedite the whole development process. We advisedly created a “soft” software layer stack, where the number of modules is minimal while maintaining a certain level of cohesion. As a result, the learning curve is not steep. By providing a simplified API, the framework is suitable for novice readers who are new to deep RL, especially software engineers. Finally, the framework acts as a bridge to connect different RL communities around the world. Our ultimate goal is to build an educational software platform for deep RL. The next development milestone includes three steps: * • Implement a variety of plugins for the proposed framework. * • Develop a GUI application that can configure the neural network, modify the learner, and visualize the RL workflow. * • Complete documentation including tutorials and sample codes. ## Appendix To keep the paper brief, we provide documentation of the proposed framework as online materials [136]. These include an installation guide, code samples, benchmark scores, tutorials, an API reference guide, a class diagram, and a package diagram. Table III lists demonstration codes of different use cases (codebase [62]). ## Acknowledgement The authors wish to thank our colleagues in the Institute for Intelligent Systems Research and Innovation for their comments and helpful discussions. We truly appreciate Nguyen Chau, a principle IT product manager at Atlassian, who shared his expertise in the field to eradicate any misunderstandings in this paper. 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2024-09-04T02:54:54.884194

2002.11912

arxiv-papers

# Max-Affine Spline Insights into Deep Generative Networks Randall Balestriero Sebastien Paris Richard G. Baraniuk ###### Abstract We connect a large class of Generative Deep Networks (GDNs) with spline operators in order to derive their properties, limitations, and new opportunities. By characterizing the latent space partition, dimension and angularity of the generated manifold, we relate the manifold dimension and approximation error to the sample size. The manifold-per-region affine subspace defines a local coordinate basis; we provide necessary and sufficient conditions relating those basis vectors with disentanglement. We also derive the output probability density mapped onto the generated manifold in terms of the latent space density, which enables the computation of key statistics such as its Shannon entropy. This finding also enables the computation of the GDN likelihood, which provides a new mechanism for model comparison as well as providing a quality measure for (generated) samples under the learned distribution. We demonstrate how low entropy and/or multimodal distributions are not naturally modeled by DGNs and are a cause of training instabilities. Generative Deep Networks, Machine Learning, Manifold, Dropout, Multimodal Distributions ## 1 Introduction Deep Generative Networks (DGNs), which map a low-dimensional latent variable $\bm{z}$ to a higher-dimensional generated sample $\bm{x}$, have made enormous leaps in capabilities in recent years. Popular DGNs include Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) and their variants (Dziugaite et al., 2015; Zhao et al., 2016; Durugkar et al., 2016; Arjovsky et al., 2017; Mao et al., 2017; Yang et al., 2019); Variational Autoencoders (Kingma & Welling, 2013) and their variants (Fabius & van Amersfoort, 2014; van den Oord et al., 2017; Higgins et al., 2017; Tomczak & Welling, 2017; Davidson et al., 2018); and flow based models such as NICE (Dinh et al., 2014), Normalizing Flow (Rezende & Mohamed, 2015), and their variants (Dinh et al., 2016; Grathwohl et al., 2018; Kingma & Dhariwal, 2018). While DGNs are easy to describe and analyze locally in terms of simple affine operators and scalar nonlinearities, a general framework for their global structure has remained elusive. In this paper, we take a step in the direction of a better theoretical understanding of DGNs constructed using continuous, piecewise affine nonlinearities by leveraging recent progress on max-affine spline operators (MASOs) (Balestriero & Baraniuk, 2018b). Our main contributions are as follows; [C1] We characterize the piecewise-affine manifold structure of the generated samples, including its intrinsic dimension (Section 3), which sheds new light on the impact of techniques like dropout (Section 3.3) and provides practionioners with sensible design principles for constructing a desired manifold. [C2] We characterize the local coordinates of the generated manifold and the inverse mapping from data points $\bm{x}$ back to latent variables $\bm{z}$ (Section 4.1), which provides new necessary and sufficient conditions for disentanglement, interpretability and new links between DGNs and adaptive basis methods (Section 4.2). By characterizing the angles between adjacent local affine regions, we demonstrate how weight sharing in a DGN heavily constrains the curvature of the generated manifold despite the fact that the DGN might be tremendously overparameterized (Section 4.3). [C3] We provide a DGN input-output formula that enables us to derive the analytical probability density on the generated manifold that is induced by the latent space (Section 5.1). We use this result to derive Normalizing Flows (NMFs) form first principles and highlight how the DGN design, $\bm{x}\mapsto\bm{z}$ (most NMFs) versus $\bm{z}\mapsto\bm{x}$ (DGNs) allow for either fast likelihood computation and slow sampling or vice-versa (Section 5.2). Finally, the Shannon entropy of the output density provide a new lens through which to study the difficulty of generating multidimensional, low-entropy distributions using DGNs (Section 5.3). Reproducible code for the various experiments and figures is be provided on Github111https://github.com/RandallBalestriero/GAN.git. ## 2 Background Deep Networks. A deep (neural) network (DN) is an operator $f_{\Theta}$ with parameters $\Theta$ that maps the input $\bm{z}\in{\mathbb{R}}^{S}$ to the output $\bm{x}\in{\mathbb{R}}^{D}$ by composing $L$ intermediate layer mappings $f_{\ell}$, $\ell=1,\dots,L$, that combine affine and simple nonlinear operators such as the fully connected operator (simply the affine transformation defined by the weight matrix $\bm{W}_{\ell}$ and bias vector $\bm{b}_{\ell}$), convolution operator (with circulent $\bm{W}_{\ell}$), activation operator (applying a scalar nonlinearity such as the ubiquitous ReLU), or pooling operator. Precise definitions of these operators can be found in (Goodfellow et al., 2016). We will omit $\Theta$ for conciseness unless it is needed. We precisely define a layer $f_{\ell}$ as comprising a single nonlinear operator composed with any (if any) preceding linear operators that lie between it and the preceding nonlinear operator. Each layer $f_{\ell}$ transforms its input feature map $\bm{v}_{\ell-1}\in{\mathbb{R}}^{D_{\ell-1}}$ into an output feature map $\bm{v}_{\ell}\in{\mathbb{R}}^{D_{\ell}}$ with the initializations $\bm{v}_{0}:=\bm{z}$, $D_{0}=S$, and $\bm{v}_{L}:=\bm{x},D_{L}=D$. In this paper, we focus on DGNs, where $S<D$, $\bm{z}$ is interpreted as a latent representation, and $\bm{x}$ is the generated data, e.g, a time-serie or image. The feature maps $\bm{v}_{\ell}$ can be viewed equivalently as signals, flattened column vectors, or tensors depending on the context. Max-Affine Spline Deep Networks. A $K$-dimensional max-affine spline operator (MASO) concatenates $K$ independent max-affine spline (MAS) functions, with each MAS formed from the point-wise maximum of $R$ affine mappings (Magnani & Boyd, 2009; Hannah & Dunson, 2013). Given an input vector $\bm{u}$, the output of a MASO is given by $\displaystyle{\rm MASO}(\bm{u};\\{\bm{A}_{r},\bm{b}_{r}\\}_{r=1}^{R})=\max_{r=1,\dots,R}\bm{A}_{r}\bm{u}+\bm{b}_{r},$ (1) where $\bm{A}_{r}\in\mathbb{R}^{D_{\ell}\times D_{\ell-1}}$ are the slopes and $\bm{b}_{r}\in\mathbb{R}^{D_{\ell}}$ are the offset/bias parameters $\forall r$ and the maximum is taken coordinate-wise. Note that a MASO is a continuous piecewise-affine (CPA) operator (wang2005generalization). The key background result for this paper is that the layers of DNs (DGNs) constructed from piecewise affine operators (e.g., convolution, ReLU, and max- pooling) are MASOs Balestriero & Baraniuk (2018b, a); hence a DN (DGN) is a composition of MASOs. For example, a layer comprising a fully connected operator with weights $\bm{W}_{\ell}$ and biases $\bm{b}_{\ell}$ followed by a ReLU activation operator has parameters $R=2,\bm{A}_{1}=\bm{W}_{\ell},\bm{A}_{2}=\mathbf{0},\bm{b}_{1}=\bm{b}_{\ell},\bm{b}_{2}=\mathbf{0}$. The piecewise-affine spline interpretation provides a powerful global geometric interpretation of a DN (DGN) in which it partitions its input space ${\mathbb{R}}^{L}$ into polyhedral regions (the set $\Omega$) and then assigns a different, fixed affine transformation to each region. The partition regions are built up over the layers via a subdivision process and are closely related to Voronoi and power diagrams (Balestriero et al., 2019). ## 3 The Generated Manifold of a DGN In this section we study the properties of the mapping $\bm{G}_{\Theta}:\mathbb{R}^{S}\rightarrow\mathbb{R}^{D}$ of a deep generative network (DGN) comprising $L$ piecewise-affine MASO layers. ### 3.1 Input Space Partition and Region Mapping While our approach holds for arbitrary piecewise affine layers, for concreteness of exposition, we will focus on nonlinearities with $R=2$ (e.g., ReLU, leaky ReLU, absolute value). In all such cases, the state of the nonlinearity can be encoded as a value from $\\{\alpha,1\\}$ with $\alpha=0$ for ReLU, $\alpha=-1$ for absolute value and $\alpha>0$ for leaky-ReLU. At layer $\ell$, observing an input $\bm{v}_{\ell-1}$ defines the state of the layer nonlinearities and in turn defines the “piece” of the layer MASO used to produce the output $\bm{v}_{\ell}$. We call the nonlinearity’s state its code $\bm{q}_{\ell}(\bm{v}_{\ell-1})\in\\{\alpha,1\\}^{D_{\ell}},\bm{v}_{\ell-1}\in\mathbb{R}^{D_{\ell-1}}$ with $\displaystyle[\bm{q}_{\ell}(\bm{v}_{\ell-1})]_{i}=\begin{cases}\alpha,&[\bm{W}_{\ell}\bm{v}_{\ell-1}+\bm{b}_{\ell}]_{i}\leq 0\\\ 1,&[\bm{W}_{\ell}\bm{v}_{\ell-1}+\bm{b}_{\ell}]_{i}>0\end{cases}$ (2) leading to the simple forward formula for the layer $\displaystyle f_{\ell}(\bm{v}_{\ell-1})=\text{diag}(\bm{q}_{\ell}(\bm{v}))(\bm{W}_{\ell}\bm{v}+\bm{b}_{\ell}).$ (3) We concatenate the per-layer codes into $\bm{q}(\bm{z})=[\bm{q}_{1}(\bm{z})^{T},\dots,\bm{q}_{L}(\bm{z})^{T}]^{T}\in\\{\alpha,1\\}^{\prod_{l=1}^{L}D_{l}}$ with $\bm{z}\in\mathbb{R}^{S}$ the DGN input and $\bm{q}_{\ell}(\bm{v}_{\ell-1})$ abbreviated as $\bm{q}_{\ell}(\bm{z})$. ###### Definition 1. A partition region $\omega_{\mathbf{k}}$ of the DGN input space partition $\Omega$ is defined as the input space region for which the MASO states $\bm{q}(\cdot)$ are identical $\displaystyle\omega_{\mathbf{k}}=$ $\displaystyle\left\\{\bm{z}\in\mathbb{R}^{S}:\bm{q}(\bm{z})={\mathbf{k}}\right\\},\forall\mathbf{k}\in\\{\alpha,1\\}^{\prod_{\ell=1}^{L}D_{\ell}},$ (4) $\displaystyle\Omega=$ $\displaystyle\left\\{\omega_{\mathbf{k}},\mathbf{k}\in\\{\alpha,1\\}^{\prod_{\ell=1}^{L}D_{\ell}}\right\\}\setminus\emptyset.$ (5) Note that $\cup_{\omega\in\Omega}\;\omega=\mathbb{R}^{S}$ and $\forall(\omega,\omega^{\prime})\in\Omega^{2},\omega\not=\omega^{\prime},\omega^{\circ}\cap\omega^{{}^{\prime}\circ}=\emptyset$, with $(\cdot)^{\circ}$ the interior operator (Munkres, 2014). Since $\bm{G}$ is formed from a composition of MASOs, it is itself a CPA with a fixed affine mapping over each region $\omega\in\Omega$. As a result, the generator $\bm{G}$ maps each $S$-dimensional convex latent space region $\omega\in\Omega$ to the convex affine subspace $\bm{G}(\omega)\subset\mathbb{R}^{D}$ as $\displaystyle\forall\omega\in\Omega,\;\;\bm{G}(\omega)=\\{\bm{A}_{\omega}\bm{z}+\bm{b}_{\omega},\bm{z}\in\omega\\}\subseteq\mathbb{R}^{D},$ (6) with $\bm{A}_{\omega}$ and $\bm{b}_{\omega}$ obtained by composing (3) and distributing the terms; we will call this the generated manifold. ###### Proposition 1. A DGN $\bm{G}$ comprised of MASO layers is a CPA operator with input space partition $\Omega$ given by (5). Each region $\omega\in\Omega$ is a convex polytope in $\mathbb{R}^{S}$ that is affinely mapped to the affine subspace $\bm{G}(\omega)$ (recall (6), a convex polytope in $\mathbb{R}^{D}$ given by (6). (Proof in Appendix C.1.) Analytically, the form of (6) can be obtained directly by composing the per layer mappings from (3), distributing and rearranging the terms into the slope and bias terms of the affine mapping. Computationally, the affine parameters $\bm{A}_{\omega},\bm{b}_{\omega}$ can be obtained efficiently in one of the two following ways. On the one hand, if one possesses an input $\bm{z}$ belonging to the desired region $\omega$, then we simply perform $\displaystyle\bm{A}_{\omega}=\nabla_{\bm{z}}\bm{G}(\bm{z}),\;\;\bm{b}_{\omega}=\bm{G}(\bm{z})-\bm{A}_{\omega}\bm{z}.$ (7) On the other hand, in the case where one has access to the code $\bm{q}(\omega)$ of the region (as opposed to a point in the region), one can directly impose the nonlinearity states (defined by $\bm{q}(\omega)$) on the DGN mapping. Once the nonlinearities are fixed, one can feed an arbitrary input $\bm{z}\in\mathbb{R}^{S}$ and compute the affine parameters as in (7) on the fixed DGN. We can extend (6) to the entire domain of the generator via $\displaystyle\bm{G}(\text{supp}(\bm{p}_{\bm{z}}))=\bigcup_{\omega\in\Omega}\bm{G}(\omega\cap\text{supp}(\bm{p}_{\bm{z}})),$ (8) with $\bm{p}_{\bm{z}}$ the probability distribution on the latent space that generates $\bm{z}$, and where $\bm{G}(\text{supp}(p_{\bm{z}}))$ denotes the image of $\bm{G}$ by considering all inputs $\bm{z}\in\text{supp}(p_{\bm{z}})$ with nonzero probability, e.g., $\mathbb{R}^{S}$ if the latent distribution is a Gaussian, and the $S$-dimensional hypercube if it is a standard Uniform distribution. Thus, the generated manifold (8) combines the per-region affine transformations of the input space partition per (6). With this formulation, we now characterize the intrinsic dimension of the per-region and overall manifold mapping of $\bm{G}$. ### 3.2 Generated Manifold Intrinsic Dimension We now turn into the intrinsic dimension of the per-region affine subspaces $\bm{G}(\omega))$ that comprise the generated manifold. In fact, as per (8), its dimension depends not only on the latent dimension $S$ but also on the per layer parameters. ###### Lemma 1. The intrinsic dimension of the affine subspace $\bm{G}(\omega))$ (recall (8)) has the following upper bound $\displaystyle\dim(\bm{G}(\omega))\leq\min\Big{(}S,\min_{\ell}\big{(}{\rm rank}\left({\rm diag}(\bm{q}_{\ell}(\omega))\bm{W}_{\ell}\right)\big{)}\Big{)}.$ (Proof in Appendix C.2.) We make three observations. First, we see that the choice of the nonlinearity (i.e., the choice of $\alpha$) and/or the choice of the per-layer dimensions (i.e., the “width” of the DGN) are the key elements controlling the upper bound of $\dim(\bm{G})$. For example, in the case of ReLU ($\alpha=0$) then $\dim(\bm{G}(\omega))$ is directly impacted by the number of $0$s in the codes $\bm{q}_{\ell}(\omega)$ of each layer in addition of the rank of $\bm{W}_{\ell}$; this sensitivity does not occur when using other nonlinearities ($\alpha\not=0$). Second, “bottleneck layers” impact directly the dimension of the subspace and thus should also be carefully picked based on the a priori knowledge of the target manifold intrinsic dimension. Third, we obtain the following condition relating the per-region dimension to the bijectivity and surjectivity of the mapping. The latter should be avoided in DGNs, since it implies that multiple different latent vectors will generate the same output sample. ###### Proposition 2. A DGN is bijective on $\omega$ iff $\dim(\bm{G}(\omega))=S,\forall\omega\in\Omega$ and surjective iff $\exists\omega\in\Omega\text{ s.t. }\dim(\bm{G}(\omega))<S$. A DGN is bijective on $\text{supp}(\bm{p}_{\bm{z}})$ iff it is bijective for each region $\omega\in\Omega$ and $\bm{G}(\omega)\cap\bm{G}(\omega^{\prime})=\emptyset,\forall\omega\not=\omega^{\prime}$. (Proof in Appendix C.3.) ### 3.3 Application: Effect of Dropout/Dropconnect Figure 1: Demonstration of a GAN DGN trained on a circle dataset. Each line is the learned piecewise linear manifold generated by the DGN; each color corresponds to a different realization of dropout noise. Dropout turns a DGN into a finite ensemble of DGNs with the same or lower intrinsic dimension. Noise techniques, such as dropout (Wager et al., 2013) and dropconnect (Wan et al., 2013), alter the per-region affine mapping in a very particular way that we now characterize. Dropout and dropconnect techniques apply a multiplicative binary noise onto the feature maps and/or the weights; the multiplicative noise $\epsilon_{\ell}\in\\{0,1\\}^{D_{\ell}}$ is typically an iid Bernoulli random variable. (Isola et al., 2017). To characterize how this noise impacts the DGN mapping, denote the generator $\bm{G}$ equipped with random dropout/dropconnect by $\widetilde{\bm{G}}$, and the generator in which the noise realization is fixed by $\bm{G}(\bm{z}|\bm{\epsilon})=\bm{A}_{\omega}(\bm{\epsilon})\bm{z}+\bm{b}_{\omega}(\bm{\epsilon})$ where $\bm{\epsilon}$ concatenates the random variables of each layer. Given the mapping form (recall (3)) with per layer parameters $\bm{W}_{\ell},\bm{b}_{\ell}$, the presence of dropout noise leads to the following noisy input-output mapping on a region $\omega$ (recall (6)) as $\bm{G}(\bm{z}|\bm{\epsilon})=\left(\prod_{\ell=L}^{1}\text{diag}(\bm{q}_{\ell}(\omega)\odot\epsilon_{\ell})\bm{W}_{\ell}\right)\bm{z}\\\ +\sum_{\ell=1}^{L}\left(\prod_{\ell^{\prime}=L}^{\ell+1}\text{diag}(\bm{q}_{\ell^{\prime}}(\omega)\odot\epsilon_{\ell^{\prime}})\bm{W}_{\ell^{\prime}}\right)\bm{b}_{\ell},$ (9) where $\odot$ is the Hadamard product and $\bm{z}\in\omega$. (See Appendix G for the dropconnect formula.) Thus, the noisy generator actually combines all the above mappings for each noise realisation via $\displaystyle\widetilde{\bm{G}}(\text{supp}(p_{\bm{z}}))=\bigcup_{\bm{\epsilon}\in\text{supp}(p_{\bm{\epsilon}})}\bm{G}(\text{supp}(p_{\bm{z}})|\bm{\epsilon}).$ (10) Dropout 0.1 Dropout 0.3 Dropconnect 0.1 Dropconnect 0.3 $\dim(\bm{G}(\omega))$ Figure 2: Impact of dropout and dropconnect on the dimension of the noisy generator affine subspaces $\bm{G}(\omega|\epsilon),\forall\omega$ (recall (10)). We depict two “drop” probabilities $0.1$ and $0.3$ for a generator $\bm{G}$ with $S=6$, $D=10$, $L=3$ and varying width $D_{1}=D_{2}$ ranging in $\\{6,8,10,12,24,48\\}$ (x-axis); note that the architecture limits the dimension to $S=6$. The boxplot represents the distribution of the per-region affine subspace dimensions for $2000$ sampled regions over $2000$ different noise realizations $\epsilon$. We make two observations. First, dropconnect tends to preserve the latent dimension $S$ even when the width $D_{1},D_{2}$ is close to $S$. Second, the dropout-induced collection of generators tends to have degenerate dimension (much smaller than $S$) until the width is twice the latent space dimension $(D_{\ell}\geq 2S)$. As a result, while dropout turns a generator into a collection of generators, those generators will have degenerate dimension unless $\bm{G}$ is much wider that $S$. ###### Proposition 3. Multiplicative binary dropout/dropconnect transforms the generator $\bm{G}$ into the union of generators given by (10), each with per-region dimension between $0$ and $\max_{\omega}\text{dim}(\bm{G}(\omega))$. (Proof in Appendix C.4.) From the above, we see that the multiplicative binary noise does not make the generator $\widetilde{\bm{G}}$ dense in its output space, but rather turns $\bm{G}$ into a collection of piecewise linear generators, each corresponding to a different realization of $\bm{\epsilon}$ as depicted in Fig. 1. Furthermore, each noise realization produces a generator with a possibly different per-region dimension being upper bounded by the dimension of the original generator $\bm{G}$. Also, each induced generator has a possibly different input space partition based on the noise realisation. On the one hand, this highlights a potential limitation of those techniques for narrow models ($D_{\ell}\approx S$) for which the noisy generators will tend to be degenerate (per-region dimension smaller than $S$), implying surjectivity (recall Prop. 2). On the other hand, when used with wide DGNs ($D_{\ell}\gg S$) much more noisy generators will maintain the same affine subspace dimension that the original generator. The latter is crucial when $S$ is picked a priori to match exactly the true intrinsic dimension. We illustrate the above in Fig. 2. ### 3.4 Application: Optimal Dimension and Training Error Increase We now emphasize how the DGN dimension $\tilde{S}\triangleq\max_{\omega}\dim(\bm{G}(\omega))$ impacts the training error loss and training sample recovery. We answer the following question: Can a generator generate $N$ training samples from a continuous $S^{*}$ dimensional manifold if $\tilde{S}<S^{*}$? Denote the empirical error measuring the ability to generate the data samples by $E^{*}_{N}=\min_{\Theta}\frac{1}{N}\sum_{n=1}^{N}\min_{\bm{z}}\|\bm{G}_{\Theta}(\bm{z})-\bm{x}_{n}\|$. We now demonstrate and empirically validate that if $\tilde{S}<S^{*}$ then $E^{*}_{N}$ increases with $N$ for any data manifold. ###### Theorem 1. Given the true intrinsic dimension of an arbitrary manifold $S^{*}$, any finite DGN with $\tilde{S}<S^{*}$ will have increasing error $E^{*}$ with the dataset size $N$ as in $\exists\ell\in\\{0,\dots,L\\}:D_{\ell}<S^{*}\implies\forall N>0,\exists N^{\prime}>N:E^{*}_{N^{\prime}}>E^{*}_{N}$. (Proof in Appendix C.11.) In general, it is clear that, for smooth manifolds, $E^{*}$ increases with $N$ since the DGN is piecewise linear. However, the above result extends this to any manifold, even when the data manifold is as simple as a linear manifold (translated subspace). We empirically validate this phenomenon in Fig. 3 for the simple case of a linear manifold. (The experimental details are given in Appendix F.) Minimum error $E^{*}$ DGN latent dimension $S$, ($S^{*}$ in red) Figure 3: Error $E^{*}$ (y-axis) for a linear manifold with $S^{*}=5$, for increasing dataset size $N\in\\{100,120,140,160,180,200,300,400,500,1000\\}$ (blue to green) for different latent space dimension $S\in\\{1,2,3,4,5,6,7\\}$ (x-axis) which forces $\tilde{S}<S$, the $E^{*}=0$ line is depicted in black. This demonstrates, as per Thm. 1, that whenever $\tilde{S}<S^{*}$, the training error $E^{*}$ increases with the dataset size $N$ and is $0$ otherwise whenever $\tilde{S}\geq S^{*}$. It is clear that a direct implication of Thm. 1 is that there does not exist a finite architecture with $D_{\ell}<D$ for some $\ell$ and parameter $\Theta$ such that a MASO DGN would be bijective with $\mathbb{R}^{D}$. The above results are key to understand the challenges and importance of the design of DGN starting with the width of the hidden layers and latent dimensions in conjunction with the choice of nonlinearities and constraints on $\bm{W}_{\ell}$ of all layers. ## 4 Manifold Local Coordinates and Curvature We now turn to the study of the local coordinates of the affine mappings comprising a DGN’s generated manifold. We then study the coupling between the affine mappings of adjacent regionsto characterize the curvature/angularity of the generated manifold. ### 4.1 Local Coordinate Systems and Inverse Mapping Recall from (8) that a DGN is a CPA operator. Inside region $\omega\in\Omega$, points are mapped to the output space affine subspace which is itself governed by a coordinate system or basis. For the remaining of the section we assume that $\dim(\bm{G}(\omega))=S,\forall\omega\in\Omega$ and thus the columns of $\bm{A}_{\omega}$ are linearly independent. For cases where $\dim(\bm{G}(\omega))<S$ then the following analysis also applies by considering a lower dimensional latent space $S^{\prime}<S$ and the corresponding sub-network that only depend on the kept latent space dimensions. ###### Lemma 2. A basis for the affine subspace $\bm{G}(\omega)$ (recall (6)) is given by the columns of $\bm{A}_{\omega}$. In other words, the columns of $\bm{A}_{\omega}$ form the local coordinate space, and each latent space dimension moves a point in this region by adding to it the corresponding slope column. Prior leveraging this result for latent space characterization, we derive an inverse of the generator $\bm{G}$ that maps any point from the generated manifold to the latent space. This inverse is well-defined as long as the generator is injective, preventing that $\exists\bm{z}_{1}\not=\bm{z}_{2}$ s.t. $\bm{G}(\bm{z}_{1})=\bm{G}(\bm{z}_{2})$. Assuming injectivity, the inverse of $\bm{G}$ on a region $\bm{G}(\omega)$ in the output space is obtained by $\displaystyle\bm{G}_{\omega}^{-1}(\bm{x})=\left(\bm{A}^{T}_{\omega}\bm{A}_{\omega}\right)^{-1}\bm{A}_{\omega}^{T}(\bm{x}-\bm{b}_{\omega}),\forall\bm{x}\in\bm{G}(\omega),$ (11) leading to $\bm{G}_{\omega}^{-1}(\bm{G}(\omega))=\omega,\forall\omega\in\Omega$. Note that the inverse $\left(\bm{A}^{T}_{\omega}\bm{A}_{\omega}\right)^{-1}$ is well defined as $\bm{A}_{\omega}$ is full column rank since we only consider a generator with $\tilde{S}=S$. We can then simply combine the region- conditioned inverses to obtain the overall generator inverse. ###### Lemma 3. The inverse mapping of an injective DGN is the CPA operator mapping $\bm{G}(\text{supp}(\bm{p}_{\bm{z}}))\mapsto\text{supp}(\bm{p}_{\bm{z}})$ given by $\bm{G}^{-1}(\bm{x})=\sum_{\omega\in\Omega}\bm{G}_{\omega}^{-1}(\bm{x})\mathbbm{1}_{\\{\bm{x}\in\bm{G}(\omega)\\}}.$ (Proof in Appendix C.5.) ### 4.2 Application: Adaptive Basis and Disentenglement As mentioned in the above section, $\bm{A}_{\omega}$ forms a basis of the affine subspace $\bm{G}(\omega)$. The latent vector $\bm{v}$ combines them to obtain the subspace which is then shifted by the bias $\bm{b}_{\omega}$. This process is performed locally for each region $\omega$, in a manner similar to an “adaptive basis” (Donoho et al., 1994). learned initial FC GAN CONV GAN FC VAE CONV VAE Figure 4: Visualization of a single basis bector $[\bm{A}_{\omega}]_{.,k}$ with $\omega$ at initialization and after learning obtained from a region $\omega$ containing the digits $7,5,9$, and $0$ respectively, and this for GAN and VAE models made of fully connected or convolutional layer (see Appendix B for details). By visual inspection, one can observe that the depicted basis vector encodes right rotation, cedilla extension, left rotation, and upward translation respectively. In addition, we observe how the basis vectors are smoother for VAE-based models which why they tend to generate blurred samples. In this context, we aim to characterize the subspace basis in term of disentanglement, i.e., the alignment of the basis vectors with respect to each other. While there is not a unique definition, a disentangled basis should provide a “compact” and interpretable latent representation $\bm{z}$ for the associated $\bm{x}=\bm{G}(\bm{z})$. In particular, it should ensure that a small perturbation of dimension $d$ of $\bm{z}$ implies a transformation independent from a small perturbation of dimension $d^{\prime}\not=d$ (Schmidhuber, 1992; Bengio et al., 2013). That is, $\langle\bm{G}(\bm{z})-\bm{G}(\bm{z}+\epsilon\delta_{d}),\bm{G}(\bm{z})-\bm{G}(\bm{z}+\epsilon\delta_{d^{\prime}})\rangle\approx 0$ with $\delta_{d}$ a one-hot vector at position $d$ and length $Z$ (Kim & Mnih, 2018). A disentangled representation is thus considered to be most informative as each latent dimension imply a transformation that leaves the others unchanged (Bryant & Yarnold, 1995). For example, rotating an object should not alter its vertical or horizontal translation and vice-versa ###### Proposition 4. A necessary condition for disentanglement is to have “near orthogonal” columns as $\langle[\bm{A}_{\omega}]_{.,i},[\bm{A}_{\omega}]_{.,j}\rangle\approx 0,\forall i,\not=j,\forall\omega\in\Omega$. (Proof in Appendix C.6.) Table 1: Depiction of the cosine similarity summed over pairwise different columns of $\bm{A}_{\omega}$. Measure of $0$ means that the basis vectors are orthogonal, improving disentanglement (recall Prop. 4). The first line represent the maximum of this quantity over $10000$ sampled regions $\omega$, the second line represents the average; the std of those quantities is given for $8$ runs. We see that training increases disentanglement, and fully connected models offer increased disentanglement as compared to convolutional models. | FC GAN | CONV GAN | FC VAE | CONV VAE ---|---|---|---|--- init. | 8.84 $\pm$ 0.07 | 3.2 $\pm$ 0.33 | 5.23 $\pm$ 0.29 | 3.5 $\pm$ 0.27 4.41 $\pm$ 0.26 | 1.84 $\pm$ 0.08 | 2.25 $\pm$ 0.08 | 1.74 $\pm$ 0.06 learn | 1.36 $\pm$ .08 | 1.72 $\pm$ 0.07 | 1.32 $\pm$ 0.07 | 1.77 $\pm$ 0.11 0.9 $\pm$ 0.03 | 1.12 $\pm$ 0.03 | 0.89 $\pm$ 0.03 | 1.15 $\pm$ 0.03 Figure 4 visualizes one of the basis vectors of four different DGNs trained on the MNIST dataset with $S=10$. Interpretability of the transformation encoded by the dimension of the basis vector can be done as well as model comparison such as blurriness of VAE samples that is empirically observed across datasets (zhao2017towards; huang2018introvae). We also provide in Table 1 the value of $\|Q_{\omega}-I\|_{2}$ with $Q_{\omega}\in[0,1]^{S\times S}$ the matrix of cosine angles between basis vector of $\bm{A}_{\omega}$ for 10,000 regions sampled randomly and where we report the average over the regions and the maximum. Finally, this process is performed over $8$ runs, the mean and standard deviation are reported in the table. We observe that there does not seem to be a difference in the degree of disentanglement different GAN and VAE; however, the topology, fully connected vs. convolution, plays an important part, favoring the former. To visually control the quality of the DGN, randomly generated digits are given in Fig. 13 in the Appendix; we also provide more background on disentanglement in Appendix E. ### 4.3 Generated Manifold Curvature We now study the curvature or angularity of the generated mapping. That is, whenever $\widetilde{S}<D$, the per-region affine subspace of adjacent region are continuous, and joint at the region boundaries with a certain angle that we now characterize. ###### Definition 2. Two regions $\omega,\omega^{\prime}$ are adjacent whenever they share part of their boundary as in $\overline{\omega}\cap\overline{\omega^{\prime}}\not=\emptyset$. The angle between adjacent affine subspace is characterized by means of the greatest principal angle (Afriat, 1957; Bjorck & Golub, 1973) and denote $\theta$. Denote the per-region projection matrix of the DGN by $\displaystyle P(\bm{A}_{\omega})=\bm{A}_{\omega}(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-1}\bm{A}_{\omega}^{T}$ (12) where $\bm{A}_{\omega}^{T}\bm{A}_{\omega}\in\mathbb{R}^{S\times S}$ and $P(\bm{A}_{\omega})\in\mathbb{R}^{D\times D}$. We now assume that $\dim(\bm{G})=Z$ ensuring that $\bm{A}_{\omega}^{T}\bm{A}_{\omega}$ is invertible.222The derivation also applies if $\dim(\bm{G})<Z$ by replacing $\bm{A}_{\omega}$ with $\bm{A}_{\omega}^{\prime}$ (recall Lemma 2). ###### Theorem 2. The angle between adjacent (recall Def. 2) region mappings $\theta(\bm{G}(\omega),\bm{G}(\omega^{\prime}))$ is given by $\displaystyle\sin\big{(}\theta(\bm{G}(\omega),\bm{G}($ $\displaystyle\omega^{\prime}))\big{)}=\|P(\bm{A}_{\omega})-P(\bm{A}_{\omega^{\prime}})\|_{2},$ (13) $\forall\omega\in\Omega,\omega^{\prime}\in\text{adj}(\omega)$. (Proof in Appendix C.7.) Notice that two special cases of the above theorem emerge. When $S=1$, the angle is given by the cosine similarity between the vectors $\bm{A}_{\omega}$ and $\bm{A}_{\omega^{\prime}}$ of adjacent regions. When $S=D-1$ the angle is given by the cosine similarity between the normal vectors of the $D-1$ subspace spanned by $\bm{A}_{\omega}$ and $\bm{A}_{\omega^{\prime}}$ respectively. We illustrate the angles in a simple case $D=2$ and $Z=1$ in Fig. 5. It can be seen how a DGN with few parameters produces angles mainly at the points of curvature of the manifold. We also provide many additional figures with different training settings in Fig. 10 in the Appendix as well as repetitions of the same experiment with different random seeds. Figure 5: Piecewise linear continuous 1-D manifold learned by a GAN DGN (in black) from a collection of data points (in blue). The breakpoints between adjacent regions are depicted by dots of color proportional to the angle. The manifold starts at the green box and proceeds clockwise. Figure 11 in the Appendix contains additional examples. ### 4.4 Application: Angle Distribution of a DGN with Random Weights We can use the above result to study the distribution of angles of different DGNs with random weights and study the impact of depth, width, as well $Z$ and $D$, the latent and output dimensions respectively. Figure 6 summarizes the distribution of angles for several different settings. Two key observations emerge. First, the codes of adjacent regions $\bm{q}(\omega),\bm{q}(\omega^{\prime})$ share a large number of their values (see Appendix D for details) implying that most of the DGN parameters are shared in their composition to produce $\bm{A}_{\omega}$ and $\bm{A}_{\omega^{\prime}}$. In turn, this weight sharing correlates adjacent hyperplanes, such that their angles are much smaller than if randomly picked form one another. The random case (in blue in Fig. 6) favors aggressively large angles as opposed to the ones of DGNs. Second, the distribution moments depend on the ratio $S/D$ rather that those values taken independently. In particular, as this ratio gets smaller, as the angle distribution becomes bi- modal with an emergence of high angles. That is, the manifold is “flat” overall except in some parts of the space where high angularity is present. This effect is strengthened with wider DGNs. Notice that this large ratio $S/D$ is the one encountered in practice where it is common to have $S\approx 100$ and $D>800$. Figure 6: Histograms of the largest principal angles for DGNs with two hidden layers, $S=16$ and $D=17,D=32,D=64$ respectively and varying width $D_{\ell}$ on the y-axis. Three trends to observe: When the width becomes large, the distribution becomes more bimodal and greatly favors near $0$ angles; when the output space dimension becomes large, there is an increase in the number of angles near orthogonal; the amount of weight sharing between the parameters $\bm{A}_{\omega}$ and $\bm{A}_{\omega^{\prime}}$ of adjacent regions $\omega$ and $\omega^{\prime}$ allow to greatly constrain the angles to be small, as opposed to the distribution of angles between random subspaces (Absil et al., 2006) depicted in blue. Hence despite the large amount of regions, most will be aligned with each other leading to an overall well behave manifold. Additional figures are available in Fig. 10 in the Appendix. The above experiment demonstrates the impact of width and latent space dimension into the angularity of the DGN output manifold and how to pick its architecture based on a priori knowledge of the target manifold. Under the often-made assumptions that the weights of overparametrized DGN do not move far from their initialization during training (Li & Liang, 2018), these results also hint at the distribution of angles after training. ## 5 Density on the Generated Manifold The study of DGNs would not be complete without considering that the latent space is equipped with a density distribution $\bm{p}_{\bm{z}}$ from which $\bm{z}$ are sampled in turn leading to sampling of $\bm{G}(\bm{z})$. Thus, we now study how this density is spread over the output space, covering the generated manifold and highlighting some key properties such as density conentration, entropy computation and training instabilities. ### 5.1 Analytical Output Density Given a distribution $\bm{p}_{\bm{z}}$ over the latent space, we can explicitly compute the output distribution after the application of $\bm{G}$, which lead to an intuitive result exploiting the piecewise affine property of the generator. ###### Lemma 4. Denote by $\sigma_{i}(\bm{A}_{\omega})$ the $i^{\rm th}$ singular value of $\bm{A}_{\omega}$. Then, the volume of a region $\omega\in\Omega$ denoted by $\mu(\omega)$ is related to the volume of $\bm{G}(\omega)$ by $\displaystyle\mu(\bm{G}(\omega))=\sqrt{\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})}\mu(\omega)=\prod_{i:\sigma_{i}(\bm{A}_{\omega})>0}\sigma_{i}(\bm{A}_{\omega})\mu(\omega).$ (Proof in Appendix C.8.) ###### Theorem 3. The generator probability density $p_{\bm{G}}(\bm{x})$ given $\bm{p}_{\bm{z}}$ and a injective generator $\bm{G}$ with per-region inverse $\bm{G}^{-1}_{\omega}$ from (11) is given by $\displaystyle p_{\bm{G}}(\bm{x})=\sum_{\omega\in\Omega}\frac{\bm{p}_{\bm{z}}\left(\bm{G}^{-1}_{\omega}(\bm{x})\right)}{\sqrt{\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})}}\mathbbm{1}_{\\{\bm{x}\in\bm{G}(\omega)\\}}.$ (14) (Proof in Appendix C.9.) That is, the distribution obtained in the output space naturally corresponds to a piecewise affine transformation of the original latent space distribution, weighted by the change in volume of the per-region mappings. From the analytical derivation of the generator density distribution, we obtain its differential entropy, i.e., the Shannon entropy for continuous distributions. ###### Corollary 1. The differential entropy of the output distribution $\bm{p}_{\bm{G}}$ of the DGN is given by $\displaystyle E(\bm{p}_{\bm{G}})=E(\bm{p}_{\bm{z}})+\sum_{\omega\in\Omega}P(\bm{z}\in\omega)\log(\sqrt{\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})}).$ (Proof in Appendix C.10.) As the result, the differential entropy of the output distribution $\bm{p}_{\bm{G}}$ corresponds to the differential entropy of the latent distribution $\bm{p}_{\bm{z}}$ plus a convex combination of the per-region volume change. Two results emerge directly. First, it is possible to optimize the latent distribution $\bm{p}_{\bm{z}}$ to better fit the target distribution entropy as been done for example in (Ben-Yosef & Weinshall, 2018). Second, whenever this distribution is fixed, any gap between the latent and output distribution entropy imply the need for high change in volumes between $\omega$ and $\bm{G}(\omega)$. Gaussian Case. We now demonstrate the use of the above derivation by considering practical examples for which we are able to gain ingights into the DGN data modeling and generation. First, consider that the latent distribution is set as $\bm{z}\sim\mathcal{N}(0,1)$ we obtain the following result directly from Thm. 3. ###### Corollary 2. The generator density distribution $p_{\bm{G}}(\bm{x})$ given $\bm{z}\sim\mathcal{N}(\mathbf{0},I)$ is $\displaystyle p_{\bm{G}}(\bm{x})=\sum_{\omega\in\Omega}\frac{e^{-\frac{1}{2}(\bm{x}-\bm{b}_{\omega})^{T}(\bm{A}_{\omega}^{+})^{T}\bm{A}_{\omega}^{+}(\bm{x}-\bm{b}_{\omega})}}{\sqrt{(2\pi)^{S}\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})}}\mathbbm{1}_{\\{\bm{x}\in\bm{G}(\omega)\\}}.$ (Proof in Appendix C.12.) The above formula is reminiscent of Kernel Density Estimation (KDE) (Rosenblatt, 1956) and in particular adaptive KDE (Breiman et al., 1977), where a partitioning of the data manifold is performed on each cell ($\omega$ in our case) different kernel parameters are used. Uniform Case. We now turn into the uniform latent distribution case. Consider the following question: Suppose we start from a uniform distribution $\bm{z}\sim\mathcal{U}(0,1)$ on the hypercube in $\mathbb{R}^{S}$, will the samples be uniformly distributed on the manifold of $\bm{G}$? ###### Proposition 5. Given a uniform latent distribution $\bm{v}\sim\mathcal{U}(0,1)$, the sampling of the manifold $\bm{G}(\text{supp}(\bm{p}_{\bm{z}}))$ will be uniform iff $\det(\bm{A}^{T}_{\omega}\bm{A}_{\omega})=c,\forall\omega:\omega\cap\text{supp}(\bm{p}_{\bm{z}})\not=\emptyset,c>0$. ### 5.2 Generative Deep Network Likelihood and Normalizing Flows Note from Thm. 3 that we obtain an explicit density distribution. One possibility for learning thus corresponds to minimizing the negative log- likelihood (NLL) between the generator output distribution and the data. Recall from Thm. 3 that $\sqrt{\det{((\bm{A}^{+}_{\omega})^{T}\bm{A}^{+}_{\omega})}}=(\sqrt{\det{(\bm{A}_{\omega}^{T}\bm{A}_{\omega})}})^{-1}$; thus we can write the log density from (14) over a sample $\bm{x}_{n}$ as $\mathcal{L}(\bm{x}_{n})=\log(\bm{p}_{\bm{z}}(\bm{G}^{-1}(\bm{x}_{n})))+\log(\sqrt{\det(J(\bm{x}_{n}))})$, where $J(\bm{x}_{n})=J_{\bm{G}^{-1}}(\bm{x}_{n})^{T}J_{\bm{G}^{-1}}(\bm{x}_{n})$, $J$ the Jacobian operator. Learning the weights of the DGN by minimization of the NLL given by $-\sum_{n=1}^{N}\mathcal{L}(\bm{x}_{n})$, corresponds to the normalizing flow model. The practical difference between this formulation and most NF models comes from having either a mapping from $\bm{x}\mapsto\bm{z}$ (NF) or from $\bm{z}\mapsto\bm{x}$ (DGN case). This change only impacts the speed to either sample points or to compute the probability of observations. In fact, the forward pass of a DGN is easily obtained as opposed to its inverse requiring a search over the codes $\bm{q}$ itself requiring some optimization. Thus, the DGN formulation will have inefficient training (slow to compute the likelihood) but fast sampling while NMFs will have efficient training but inefficient sampling. ### 5.3 On the Difficulty of Generating Low entropy/Multimodal Distributions We conclude this section with the study of the instabilities encountered when training DGNs on multimodal densities or other atypical cases. $\sigma_{1}=0,\sigma_{2}\in\\{1,2,3\\}$ $\sigma_{1}=1,\sigma_{2}\in\\{1,2,3\\}$ $\sigma_{1}=2,\sigma_{2}\in\\{1,2,3\\}$ Figure 7: Distribution of $\log(\sqrt{\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})})$ for $2000$ regions $\omega$ with a DGN with $L=3,S=6,D=10$ and weights initialized with Xavier; then, half of the weights’ coefficients (picked randomly) are rescaled by $\sigma_{1}$ and the other half by $\sigma_{2}$. We observe that greater variance of the weights increase the spread of the log-determinants and increase the mean of the distribution. # regions data Figure 8: Distribution of the log-determinant of the per-region mappings for different true distributions: two Gaussian with increasing standard deviation (left to right). We observe how the concentration and inter-mode distance impacts greatly the distribution of the log-determinant to allow the generator to fit the distribution, this in turns increases the variance of the weights $\bm{W}_{\ell}$. Figure 9: Example of a GAN DGN trained on a mixture of $25$ Gaussians. On the left is depicted the latent space per-region log-determinant (color coded) with values ranging from $-6$ to $4$ in log-scale ($0.002$ to $55$ in linear scale). On the middle are depicted the true sample (blue) and generated ones (oranges), and on the right are depicted points sampled from the generator from regions with low determinant (green) and large determinant (red). We can observe that this failure case (poor approximation of the true distribution) is due to the continuous properties of MASO DGNs, which makes the generator move continuously between modes while not being able to reduce enough the sampling probability $\bm{p}_{\bm{G}}$ in between the modes. Additional examples are contained in Fig. 14 We demonstrated in Thm. 3 and Cor. 1 that the product of the nonzero singular values of $\bm{A}_{\omega}$ plays the central role to concentrate or disperse the density on $\bm{G}(\omega)$. Let now consider a simple case of mixture of Gaussians. It becomes clear that the standard deviation of the modes and the inter-mode distances will put constraints on the singular values of the slope matrix $\bm{A}_{\omega}$. However, imposing a large variance in the singular values of $\bm{A}_{\omega}$ for different regions $\omega$ directly stress the parameters $\bm{W}_{\ell}$ as they compose the slope matrix. This is highlighted in Fig. 7 where we depict the distribution of the log-determinants for different bimodal distribution for the weights $\bm{W}_{\ell}$ showing the correlation between the variance of those parameters and the variance of log- determinant over different regions. We also illustrate the variation of the learned generator log-determinant distribution across regions in Fig. 8, where we trained a GAN DGN on two Gaussians for different scalings. This further highlights the importance of the distribution of the determinants that is reachable by a DGN which depends on the architecture and parameter space. In conclusion, for multimodal and low entropy distribution, the required log-determinant for the DGN to approximate the true distribution goes against some standard regularization techniques such as Tikhonov, which (recall Fig. 7) pushes the generator output density $\bm{p}_{\bm{G}}$ to be more uniform with higher entropy. Supplementary Material ## Appendix A Extra Figures All the below pictures have been compressed to be uploaded on Arxiv. ### A.1 Angles Histogram S=2, D=3 S=8, D=9 S=2, D=4 S=8, D=16 S=2, D=8 S=8, D=32 S=4, D=5 S=16, D=17 S=4, D=8 S=16, D=32 S=4, D=16 S=16, D=64 Figure 10: Reproduction of Fig. 6. Histograms of the largest principal angles for DGNs with one hidden layer (first two rows) and two hidden layers (last two rows). In each case the latent space dimension and width of the hidden layers is in the top of the column. The observations reinforce the claims on the role of width and $S$ versus $D$ dimensions. ### A.2 Angles Manifold Figure 11: Reproduction of Fig. 5 for various GDNs topologies. The columns represent different widths $D_{\ell}\in\\{6,8,16,32\\}$ and the rows correspond to repetition of the learning for different random initializations of the GDNs for consecutive seeds. ### A.3 More on MNIST Disentanglement Figure 12: Randomly generated digits from the trained GAN (top) and trained VAE(bottom) models for the experiment from Fig. 4. Each row represents a model that was training on a different random initialization (8 runs in total) which produced the result in Table 1. Figure 13: Randomly generated digits from the trained CONV GAN (top) and trained CONV VAE(bottom) models for the experiment from Fig. 4. Each row represents a model that was training on a different random initialization (8 runs in total) which produced the result in Table 1. ### A.4 More on Determinant Figures Figure 14: Reproduction of Fig. 8 for multiple standard deviations and multiple random seeds. Each column represent a different standard deviation of the two Gaussians $\sigma\in\\{0.002,0.01,0.05,0.1,0.3,1,2\\}$ and each row is a run with a different seed. As can be seen in all cases (except when lack of convergence) the distribution of the determinants support the claim and relate with the Entropy of the true distribution (blue points). ## Appendix B Architecture Details We describe the used models below. The Dense(T) represents a fully connected layer with $T$ units (activation function not included). The Conv2D(I, J, K) represent $I$ filters of spatial shape $(J,K)$ and the input dilation and padding follow the standard definition. For the VAE models the encoder is given below and for the GAN models the discriminator is given below as well. FC GAN model means that the FC generator is used in conjonction with the discriminator, the CONV GAN means that the CONV generator is used in conjonction with the discriminator and similarly for the VAE case. FC generator | CONV generator | Encoder | Discriminator ---|---|---|--- Dense(256) | Dense(256) | Dense(512) | Dense(1024) leaky ReLU | leaky ReLU | Dropout(0.3) | Dropout(0.3) Dense(512) | Dense(8 * 6 * 6) | leaky ReLU | leaky ReLU leaky ReLU | leaky ReLU | Dense(256) | Dense(512) Dense(1024) | Reshape(8, 6, 6) | leaky ReLU | Dropout(0.3) leaky ReLU | Conv2D(8, 3, 3, inputdilation=2, pad=same) | Dense(2*S) | leaky ReLU Dense(28*28) | leaky ReLU | | Dense(256) | Conv2D(8, 4, 4, inputdilation=3, pad=valid) | | Dropout(0.3) | Reshape(28*28) | | leaky ReLU | | | Dense(2) all the training procedures employ the Adam optimizer with a learning of $0.0001$ which stays constant until training completion. In all cases trainig is done on $300$ epochs, an epoch consisting of viewing the entire image training set once. ## Appendix C Proofs ### C.1 Proof of Proposition 1 ###### Proof. The result is a direct application of Corollary 3 in (Balestriero et al., 2019) adapted to GDNs (and not classification based DNs). The input regions are proven to be convex polytopes. Then by linearity of the per region mapping, conexity is preserved and with form given by (6). ∎ ### C.2 Proof of Lemma 1 ###### Proof. First recall the standard result that $\displaystyle\text{rank}(AB)\leq\min(\text{rank}(A),\text{rank}(B)),$ for any matrix $A\in\mathbb{R}^{N\times K}$ and $B\in\mathbb{R}^{K\times D}$ (see for example (banerjee2014linear) chapter 5). Now, noticing that $\min(\min(a,b),\min(c,d))=\min(a,b,c,d)$ leads to the desired result by unrolling the product of matrices that make up the $\bm{A}_{\omega}$ matrix to obtain the desired result. ∎ ### C.3 Proof of Proposition 2 ###### Proof. First notice that there can only be two major cases. First for the dimension of the affinely mapped region $\bm{G}(\omega)$ to be $S$ or to be smaller than $S$. Let first consider the bijective case. For the GDN to be bijective on the region we need a one-to-one mapping from $\omega$ to $\bm{G}(\omega)$. If the dimension of the subsapce $\bm{G}(\omega)$ is $S$, then it means that the matrix $\bm{A}_{\omega}$ is full-rank, with rank $S$. In turn, this means that the columns of the matrix are linearly independent. This implies bijectivity on the region as each point in $\omega$ is mapped to an unique point in $\bm{G}(\omega)$ and vice-versa. The surjectivity is direct as if the dimension is smaller than $S$, then the matrix $\bm{A}_{\omega}$ is not full- rank and all the points in the region $\omega$ that leave in the kernel of the matrix (lifted with the bias $\bm{b}_{\omega}$) will be mapped to the same output points. This means that there exists different points in $\omega$ s.t. they are mapped to the same point in $\bm{G}(\omega)$ which gives surjectivity. For global bijectivity, we need an additional condition. In fact, to ensure that the entire GDN preserves a one-to-one mapping, we need per region bijectivity coupled with the fact that the mapping for different region do not intersect. In fact, we know look at bijectivity between $\text{supp}(\bm{p}_{\bm{z}})$ and $\bm{G}(\text{supp}(\bm{p}_{\bm{z}}))$. Thus if the regions do not intersection after affine projections then there does not exist different latent vectors $\bm{z}$ and $\bm{z}^{\prime}$ that would be mapped to the same output point. Yet because we have bijectivity on between $\omega$ and $\bm{G}(\omega),\forall\omega$ it means that each point in $\text{supp}(\bm{p}_{\bm{z}})$ is mapped to an unique point in $\bm{G}(\text{supp}(\bm{p}_{\bm{z}}))$ which gives global bijectivity. ∎ ### C.4 Proof of Proposition 3 ###### Proof. The first bound is obtained by taking the realization of the noise where $\bm{r}=\mathbf{0}$, in that case the input space partition is the entire space as any input is mapped to the same VQ code. As such, the mapping associated to this trivial partition has $\mathbf{0}$ slope (matrix filled with zeros) and a possibly nonzeros bias; as such the mapping is zero- dimensional (any points is mapped to the same point). This gives the lower bound stating that in the mixture of GDNs, one will have dimension $0$. For the other case, simply take the trivial case of $\bm{r}=\mathbf{1}$ which gives the result. ∎ ### C.5 Proof of Lemma 3 ###### Proof. First, as we impose injectivity, we can not have multiple regions of the input space, say $\omega$ and $\omega^{\prime}$ such that $\bm{G}(\omega)\cap\bm{G}(\omega^{\prime})\not=\emptyset$. Second, a region of the input space is mapped to another region in the output space by means of the affine transformation, thus even though the ambiant space $D$ might be of greater dimension that $\dim(\bm{G})$, the injectivity implies that points in $\omega$ are mapped to at most one point in $\bm{G}(\omega)$. They are affinely mapped meaning that the inverse is given by removing the bias and inverting the linear mapping which is given by the pseudo inverse. Recalling the above result on surjectivity, we see that for the GDN to be injective the per region dimension msut be $S$ showing existence of the pseudo inverse. ∎ ### C.6 Proof of Proposition 4 ###### Proof. The proof is straightforward from the used definition of disentenglement. In fact, recall that we aim to have $\langle\bm{G}(\bm{z})-\bm{G}(\bm{z}+\epsilon\delta_{d}),\bm{G}(\bm{z})-\bm{G}(\bm{z}+\epsilon\delta_{d^{\prime}})\rangle\approx 0$. In our case, consider only small transformation such that $\bm{z}+\epsilon\delta_{d}$ and $\bm{z}+\epsilon\delta_{d^{\prime}}$ remain the in the same region $\omega$ in which was $\bm{z}$. Then it is clear that for any positive constant $\epsilon$ fulfilling this condition, the above disentangelement definition translates into $\displaystyle\langle\bm{G}(\bm{z})-\bm{G}(\bm{z}+\epsilon\delta_{d}),\bm{G}(\bm{z})-\bm{G}(\bm{z}+\epsilon\delta_{d^{\prime}})\rangle\approx 0\iff\langle[\bm{A}_{\omega}]_{.,d},[\bm{A}_{\omega}]_{.,d^{\prime}}\rangle\approx 0,$ This gives a necessary condition which is not sufficient as this alone does not guarantee that each dimension of the latent space only impacts a single transformation of the output. But a disentangled representation must have near orthogonal columns for the slope matrices $\bm{A}_{\omega}$. ∎ ### C.7 Proof of Theorem 2 ###### Proof. First, notice that $P(\bm{A}_{\omega})=\bm{A}_{\omega}(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-1}\bm{A}_{\omega}^{T}$ defines a projection matrix. In fact, we have that $\displaystyle P(\bm{A}_{\omega})^{2}$ $\displaystyle=\bm{A}_{\omega}(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-1}\bm{A}_{\omega}^{T}\bm{A}_{\omega}(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-1}\bm{A}_{\omega}^{T}$ $\displaystyle=\bm{A}_{\omega}(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-1}\bm{A}_{\omega}^{T}$ $\displaystyle=P(\bm{A}_{\omega})$ and we have that $(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-1}$ is well defined as we assume injectivity ($\text{rank}(\bm{A}_{\omega})=S$) making the $S\times S$ matrix $\bm{A}_{\omega}^{T}\bm{A}_{\omega}$ full rank. Now it is clear that this projection matrix maps an arbitrary point $\bm{x}\in\mathbb{R}^{D}$ to the affine subspace $\bm{G}(\omega)$ up to the bias shift. As we are interested in the angle between two adjacent subspaces $\bm{G}(\omega)$ and $\bm{G}(\omega^{\prime})$ it is also clear that the biases (which do not change the angle) can be omited. Hence the task simplifies to finding the angle between $P(\bm{A}_{\omega})$ and $P(\bm{A}_{\omega^{\prime}})$. This can be done by means of the greatest principal angle (proof can be found in Stewart (1973)) with the result being $\sin\big{(}\theta(\bm{G}(\omega),\bm{G}(\omega^{\prime}))\big{)}=\|P(\bm{A}_{\omega})-P(\bm{A}_{\omega^{\prime}})\|_{2}$ as desired. ∎ ### C.8 Proof of Lemma 4 ###### Proof. In the special case of an affine transform of the coordinate given by the matrix $A\in\mathbb{R}^{D\times D}$ the well known result from demonstrates that the change of volume is given by $|\det(A)|$ (see Theorem 7.26 in (rudin2006real)). However in our case the mapping is a rectangular matrix as we span an affine subspace in the ambiant space $\mathbb{R}^{D}$ making $|\det(A)|$ not defined. However by applying Sard’s theorem (spivak2018calculus) we obtain that the change of volume from the region $\omega$ to the affine subspace $\bm{G}(\omega)$ is given by $\sqrt{\det(A^{T}A)}$ which can also be written as follows with $USV^{T}$ the svd-decomposition of the matrix $A$: $\displaystyle\sqrt{\det(A^{T}A)}=\sqrt{\det((USV^{T})^{T}(USV^{T}))}=$ $\displaystyle\sqrt{\det((VS^{T}U^{T})(USV^{T}))}$ $\displaystyle=$ $\displaystyle\sqrt{\det(VS^{T}SV^{T})}$ $\displaystyle=$ $\displaystyle\sqrt{\det(S^{T}S)}$ $\displaystyle=$ $\displaystyle\prod_{i:\sigma_{i}\not=0}\sigma_{i}(A)$ ∎ ### C.9 Proof of Theorem 3 ###### Proof. We will be doing the change of variables $\bm{z}=(\bm{A}^{T}_{\omega}\bm{A}_{\omega})^{-1}\bm{A}_{\omega}^{T}(\bm{x}-\bm{b}_{\omega})=\bm{A}_{\omega}^{+}(\bm{x}-\bm{b}_{\omega})$, also notice that $J_{\bm{G}^{-1}}(\bm{x})=A^{+}$. First, we know that $P_{\bm{G}(\bm{z})}(\bm{x}\in w)=P_{\bm{z}}(\bm{z}\in\bm{G}^{-1}(w))=\int_{\bm{G}^{-1}(w)}p_{\bm{z}}(\bm{z})d\bm{z}$ which is well defined based on our full rank assumptions. We then proceed by $\displaystyle P_{\bm{G}}(\bm{x}\in w)=$ $\displaystyle\sum_{\omega\in\Omega}\int_{\omega\cap w}p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\sqrt{\det(J_{\bm{G}^{-1}}(\bm{x})^{T}J_{\bm{G}^{-1}}(\bm{x}))}d\bm{x}$ $\displaystyle=$ $\displaystyle\sum_{\omega\in\Omega}\int_{\omega\cap w}p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\sqrt{\det((\bm{A}_{\omega}^{+})^{T}\bm{A}_{\omega}^{+})}d\bm{x}$ $\displaystyle=$ $\displaystyle\sum_{\omega\in\Omega}\int_{\omega\cap w}p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))(\prod_{i:\sigma_{i}(\bm{A}_{\omega}^{+})>0}\sigma_{i}(\bm{A}_{\omega}^{+}))d\bm{x}$ $\displaystyle=$ $\displaystyle\sum_{\omega\in\Omega}\int_{\omega\cap w}p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))(\prod_{i:\sigma_{i}(\bm{A}_{\omega})>0}\sigma_{i}(\bm{A}_{\omega}))^{-1}d\bm{x}\;\;\;\text{Etape 1}$ $\displaystyle=$ $\displaystyle\sum_{\omega\in\Omega}\int_{\omega\cap w}p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\frac{1}{\sqrt{\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})}}d\bm{x}$ Let now prove the Etape 1 step by proving that $\sigma_{i}(A^{+})=(\sigma_{i}(A))^{-1}$ where we lighten notations as $A:=\bm{A}_{\omega}$ and $USV^{T}$ is the svd-decomposition of $A$: $\displaystyle A^{+}=(A^{T}A)^{-1}A^{T}=$ $\displaystyle((USV^{T})^{T}(USV^{T}))^{-1}(USV^{T})^{T}$ $\displaystyle=$ $\displaystyle(VS^{T}U^{T}USV^{T})^{-1}(USV^{T})^{T}$ $\displaystyle=$ $\displaystyle(VS^{T}SV^{T})^{-1}VS^{T}U^{T}$ $\displaystyle=$ $\displaystyle V(S^{T}S)^{-1}S^{T}U^{T}$ $\displaystyle\implies$ $\displaystyle\sigma_{i}(A^{+})=(\sigma_{i}(A))^{-1}$ with the above it is direct to see that $\sqrt{\det((\bm{A}_{\omega}^{+})^{T}\bm{A}_{\omega}^{+})}=\frac{1}{\sqrt{\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})}}$ as follows $\displaystyle\sqrt{\det((\bm{A}_{\omega}^{+})^{T}\bm{A}_{\omega}^{+})}=\prod_{i:\sigma_{i}\not=0}\sigma_{i}(\bm{A}_{\omega}^{+})=$ $\displaystyle\prod_{i:\sigma_{i}\not=0}\sigma_{i}(\bm{A}_{\omega})^{-1}$ $\displaystyle=$ $\displaystyle\left(\prod_{i:\sigma_{i}\not=0}\sigma_{i}(\bm{A}_{\omega})\right)^{-1}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})}}$ which gives the desired result. ∎ ### C.10 Proof of Cor. 1 ###### Proof. The derivation of the Entropy will consist in rewritting the Entropy w.r.t the distribution in the output space of the GDN and performing the change of coordinates leveragign the abvoe result to finally obtain the desired result as follows: $\displaystyle E(\bm{p}_{\bm{G}})=$ $\displaystyle-\sum_{\omega\in\Omega}\int_{\bm{G}(\omega)}\bm{p}_{\bm{G}}(\bm{x})\log(\bm{p}_{\bm{G}}(\bm{x}))d\bm{x}$ $\displaystyle=$ $\displaystyle-\sum_{\omega\in\Omega}\int_{\bm{G}(\omega)}p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-\frac{1}{2}}\log\left(p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-\frac{1}{2}}\right)$ $\displaystyle=$ $\displaystyle-\sum_{\omega\in\Omega}\int_{\bm{G}(\omega)}p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-\frac{1}{2}}\left(\log\left(p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\right)+\log\left(\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-\frac{1}{2}}\right)\right)$ $\displaystyle=$ $\displaystyle-\sum_{\omega\in\Omega}\int_{\bm{G}(\omega)}\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-\frac{1}{2}}p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\log\left(p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\right)$ $\displaystyle-\sum_{\omega\in\Omega}\int_{\bm{G}(\omega)}\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-\frac{1}{2}}p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\log\left(\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-\frac{1}{2}}\right)$ $\displaystyle=$ $\displaystyle E(\bm{p}_{\bm{z}})\;\;\;\;\;\;\;\text{(apply the change of coordinate $\bm{z}=G(\bm{x})$)}$ $\displaystyle-\sum_{\omega\in\Omega}\int_{\bm{G}(\omega)}p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-\frac{1}{2}}\log\left(\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-\frac{1}{2}}\right)$ $\displaystyle=$ $\displaystyle E(\bm{p}_{\bm{z}})+\frac{1}{2}\sum_{\omega\in\Omega}P(\bm{z}\in\omega)\log\left(\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})\right)\;\;\;\;\;\;\;\text{(apply the change of coordinate $\bm{z}=G(\bm{x})$)}$ which gives the desired result. For a complete review of integrals on manifold please see (Cover & Thomas, 2012). ∎ ### C.11 Proof of Theorem 1 ###### Proof. For very small $N$ it is clear than in general, even if $S<S^{*}$, the memorization capacity of the generator will be s.t. it can fit through those points. Just imagine a couple of points sampled from a $2D$ linear manifold, even though $S=1$, the GDN can go through those two points and thus have $E^{*}=0$ for $N=2$. We now consider the case where $N$ is large enough. Two cases have to be studied. * • Case $S<S^{*}$: if $S<S^{*}$, the generated manifold can never be dense in the true linear manifold. This means that the newly introduced point will almost surely not lie in the span of the current generated manifold. Thus, $E^{*}(N+1)>E^{*}(N)$. * • Case $S\geq S^{*}$: in that case, it is clear the there always exist a setting of the parameters $\Theta$ s.t. the DGN spans the linear manifold. For example if using ReLU, consider any weights for the first $L-1$ layers s.t. the ReLU ae always “on” and use the last layer affine mapping to rotate and translate the affine subspace to the true one. That is, $E^{*}(N)=0,\forall N>0$. The above demonstrates how for the simplest target manifold (linear) an too narrow DN leading to $S<S^{*}$ will haev a training error $E^{*}$ increasing with $N$ or $0$ if $S\geq S^{*}$ for any $N$. ∎ ### C.12 Proof of Corollary 2 ###### Proof. First, by applying the above results on the general density formula and setting $\bm{p}_{\bm{z}}$ a standard Normal distribution we obtain that $\displaystyle\bm{p}_{\bm{G}}(\bm{x}\in w)=$ $\displaystyle\sum_{\omega\in\Omega}\int_{\omega\cap w}\mathbbm{1}_{\bm{x}\in\bm{G}(\omega)}p_{\bm{z}}(\bm{G}^{-1}(\bm{x}))\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})^{-\frac{1}{2}}d\bm{x}$ $\displaystyle=$ $\displaystyle\sum_{\omega\in\Omega}\int_{\omega\cap w}\mathbbm{1}_{\bm{x}\in\bm{G}(\omega)}\frac{1}{(2\pi)^{S/2}\sqrt{\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})}}e^{-\frac{1}{2}\|\bm{G}^{-1}(\bm{x})\|_{2}^{2}}d\bm{x}$ $\displaystyle=$ $\displaystyle\sum_{\omega\in\Omega}\int_{\omega\cap w}\mathbbm{1}_{\bm{x}\in\bm{G}(\omega)}\frac{1}{(2\pi)^{S/2}\sqrt{\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})}}e^{-\frac{1}{2}((\bm{A}^{+}_{\omega}(\bm{x}-\bm{b}_{\omega}))^{T}((\bm{A}^{+}(\bm{x}-\bm{b}_{\omega}))}d\bm{x}$ $\displaystyle=$ $\displaystyle\sum_{\omega\in\Omega}\int_{\omega\cap w}\mathbbm{1}_{\bm{x}\in\bm{G}(\omega)}\frac{1}{(2\pi)^{S/2}\sqrt{\det(\bm{A}_{\omega}^{T}\bm{A}_{\omega})}}e^{-\frac{1}{2}(\bm{x}-\bm{b}_{\omega})^{T}(\bm{A}^{+}_{\omega})^{T}\bm{A}^{+}_{\omega}(\bm{x}-\bm{b}_{\omega})}d\bm{x}$ giving the desired result. ∎ ## Appendix D Codes of neighbour regions Each code is equivalent to a system of inequalities that define the regions. In fact, a code depends on the signs of the feature map pre activation (recall (3)). This defines a polytope in the input space and also in the output space. Now, when traveling from a point $\bm{z}$ to another point $\bm{z}^{\prime}$ of a neighbouring region (recall Def. 2), we ask the question on how many indices of the code will change. That is, what is the Hamming distance between $\bm{q}(\bm{z})$ and $\bm{q}(\bm{z}^{\prime})$. As a neighbouring region is defined as a region which shares some of its boundary with another (their interior is disjoint) we can see that the degree of the face that is shared between the two regions define the amount of changes in their corresponding codes. If two regions share a $S-1$ dimensional face, then only $1$ value of the code changes. If they share in general a $S-r$ dimensional face, then the code will change by $r$ values. As most adjacent regions will share a high dimensional face, we see that $r$ tends to be small and thus codes are similar. For details and analytical study of the above please see (lautensack_zuyev_2008). ## Appendix E More on Disentangled Latent Representations It has been coined that providing interpretable and practical generators lies in the ability to learn a disentangled representation of an input $\bm{x}=\bm{G}(\bm{z})$ (Schmidhuber, 1992; Bengio et al., 2013). The code $\bm{z}$ should contain all the information present in $\bm{x}$ in a compact and interpretable structure where distinct, informative factors of variations are encoded by different dimensions. Such motivations orginitated from the (non-)linear independent component analysis focusing on recovering independent factors from observed data (Comon, 1994; Hyvarinen & Morioka, 2016). In fact, even in recent GAN/VAE based models, disentengled representations are associated to independent transformations of the input such as pose, hair color, eye color and so on which should behave independently form each other (yim2015rotating; tran2017disentangled). For a more in-depth review of learning disentangled representation, see (Locatello et al., 2018). ## Appendix F Details on Training Procedure The experiment aims at depicting the training error being $E^{*}$ on the training set for varying latent dimensions $S$ in the simple case of a linear true data manifold approximation. In order to prevent any optimization unlucky degeneracy we repeat the training procedure $30$ times and compute for each poch the error $E^{*}$ and report the minimum over the $30$ trials and training epochs. We also set a very large number of epochs: $2000$. Due to the large number of trials and epochs the reported results are not due to some random initialization settings and convey the point of the result which is that even for such a simple data model (linear manifold) if $S<S^{*}$ then the training error $E^{*}$ will increase with $N$. Finally, the minimization over $z$ is replacer by an autoencoder with a very wide encoder s.t. it has the capacity for each training point to memorize the optimum $z$ that minimizes $E$. That is, when minimizing $\displaystyle\min_{\Theta}\min_{\Theta^{\prime}}\|\bm{G}_{\Theta}(E_{\Theta^{\prime}}(\bm{x}))-\bm{x}\|\approx\min_{\Theta}\min_{\bm{z}}\|\bm{G}_{\Theta}(\bm{z})-\bm{x}\|,$ got a large enough encoder network $E$. In our case given that we used $S^{*}=6$ we used an encoder with $D_{\ell}=256$ units and $L=3$. ## Appendix G More on Effect of Dropout/Dropconnect As opposed to the Dropout case which applies the binary noise ont the feature maps $\bm{v}_{\ell}$, Dropconnect (Wan et al., 2013) applies this binary noise onto the slope matrices $\bm{W}_{\ell}$ making the mapping noise become $\displaystyle\bm{G}(\bm{z})=\left(\prod_{\ell=L}^{1}\text{diag}(\bm{q}_{\ell})(\bm{W}_{\ell}\odot\bm{R}_{\ell})\right)\bm{z}+\sum_{\ell=1}^{L}\left(\prod_{\ell^{\prime}=L}^{\ell+1}\text{diag}(\bm{q}_{\ell^{\prime}})(\bm{W}_{\ell^{\prime}}\odot\bm{R}_{\ell^{\prime}})\right)\bm{b}_{\ell},$ where the binary noise matrices are denoted by $\bm{R}_{\ell}$. Despite this change of nosei application, the exact same result applies and Prop. 3 also holds. That is the dropconnect equipped GDN becomes a mixture of GDNs with varying dimensions and parameters. 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2024-09-04T02:54:54.899547

2002.11942

arxiv-papers

# A Lower Bound of the Number of Rewrite Rules Obtained by Homological Methods Mirai Ikebuchi Massachusetts Institute of Technology, Cambridge, MA, USA <EMAIL_ADDRESS> ###### Abstract. It is well-known that some equational theories such as groups or boolean algebras can be defined by fewer equational axioms than the original axioms. However, it is not easy to determine if a given set of axioms is the smallest or not. Malbos and Mimram investigated a general method to find a lower bound of the cardinality of the set of equational axioms (or rewrite rules) that is equivalent to a given equational theory (or term rewriting systems), using homological algebra. Their method is an analog of Squier’s homology theory on string rewriting systems. In this paper, we develop the homology theory for term rewriting systems more and provide a better lower bound under a stronger notion of equivalence than their equivalence. The author also implemented a program to compute the lower bounds, and experimented with 64 complete TRSs. ###### Key words and phrases: Term rewriting systems, Equational logic, Homological algebra The author thanks Keisuke Nakano for comments that greatly improved the manuscript and Aart Middeldorp for his suggestion about prime critical pairs. ## 1\. Introduction The purpose of this paper is to find a lower bound of the number of axioms that are equivalent to a given equational theory. For example, the theory of groups is given by the following axioms: $\begin{array}[]{lll}G_{1}.\ m(m(x_{1},x_{2}),x_{3})=m(x_{1},m(x_{2},x_{3})),&G_{2}.\ m(x_{1},e)=x_{1},&G_{3}\ m(e,x_{1})=x_{1},\\\ G_{4}.\ m(i(x_{1}),x_{1})=e,&G_{5}.\ m(x_{1},i(x_{1}))=e.&\end{array}$ (1) It is well-known that $G_{2}$ and $G_{5}$ can be derived from only $\\{G_{1},G_{3},G_{4}\\}$. Moreover, the theory of groups can be given by two axioms: the axiom $m(x_{1},i(m(m(i(m(i(x_{2}),m(i(x_{1}),x_{3}))),x_{4}),i(m(x_{2},x_{4})))))=x_{3}$ together with $G_{4}$ is equivalent to the group axioms [11]. If we use the new symbol $d$ for division instead of multiplication, a single axiom, $d(x_{1},d(d(d(x_{1},x_{1}),x_{2}),x_{3}),d(d(d(x_{1},x_{1}),x_{1}),x_{3}))=x_{2},$ is equivalent to the group axioms [4]. However, no single axiom written in symbols $m,i,e$ is equivalent to the group axioms. This is stated without proof by Tarski [19] and published proofs are given by Neumann [11] and Kunen [7]. Malbos and Mimram developed a general method to calculate a lower bound of the number of axioms that are “Tietze-equivalent” to a given complete term rewriting system (TRS) [9, Proposition 23]. We state the definition of Tietze equivalence later (Definition 4), but roughly speaking, it is an equivalence between equational theories (or TRSs) $(\Sigma_{1},R_{1})$, $(\Sigma_{2},R_{2})$ where signatures $\Sigma_{1}$ and $\Sigma_{2}$ are not necessarily equal to each other, while the usual equivalence between TRSs is defined for two TRSs $(\Sigma,R_{1})$, $(\Sigma,R_{2})$ over the same signature (specifically, by ${\xleftrightarrow{*}_{R_{1}}}={\xleftrightarrow{*}_{R_{2}}}$). For string rewriting systems (SRSs), a work was given earlier by Squier [16], and Malbos and Mimram’s work is an extension of Squier’s work. Squier provided a rewriting view for “homology groups of monoids”, and proved the existence of an SRS that does not have any equivalent SRSs that are finite and complete. In this paper, we will develop Malbos and Mimram’s theory more, and show an inequality which gives a better lower bound of the number of axioms with respect to the usual equivalence between TRSs over the same signature. For the theory of groups, our inequality gives that the number of axioms equivalent to the group axioms is greater than or equal to 2, so we have another proof of Tarski’s theorem above as a special case. Our lower bound is algorithmically computable if a complete TRS is given. We will first give the statement of our main theorem and some examples in Section 2. Then, we will see Malbos-Mimram’s work briefly. The idea of their work is to provide an algebraic structure to TRSs and extract information of the TRSs, called homology groups, which are invariant under Tietze equivalence. The basics of such algebraic tools are given in Section 3. We will explain how resolutions of modules, a notion from abstract algebra, is related to rewriting systems, which is not written in usual textbooks. and we will see the idea of the construction of the homology groups of TRSs in Section 4. After that, in Section 5, we will prove our main theorem. In Section 6, we show prime critical pairs are enough for our computation at the matrix operation level and also at the abstract algebra level. In Section 7, we study the number of rewrite rules in a different perspective: the minimum of $\\#R-\\#\Sigma$ over all equivalent TRSs $(\Sigma,R)$ is called the _deficiency_ in group theory and we show that deciding whether the deficiency is less than a given integer or not is computationally impossible. ## 2\. Main Theorem In this section, we will see our main theorem and some examples. Throughout this paper, we assume that terms are over the set of variables $\\{x_{1},x_{2},\dotsc\\}$ and all signatures we consider are unsorted. For a signature $\Sigma$, let $T(\Sigma)$ denote the set of terms over the signature $\Sigma$ and the set of variables $\\{x_{1},x_{2},\dotsc\\}$. Let $(\Sigma,R)$ be a TRS. The degree of $R$, denoted by $\deg(R)$, is defined by $\deg(R)=\gcd\\{\\#_{i}l-\\#_{i}r\mid l\rightarrow r\in R,i=1,2,\dotsc\\}$ where $\\#_{i}t$ is the number of occurrences of $x_{i}$ in $t$ for $t\in T(\Sigma)$ and we define $\gcd\\{0\\}=0$ for convenience. For example, $\deg(\\{f(x_{1},x_{2},x_{2})\rightarrow x_{1},\ g(x_{1},x_{1},x_{1})\rightarrow e\\})=\gcd\\{0,2,3\\}=1$. Let $(\Sigma,R=\\{l_{1}\rightarrow r_{1},\dotsc,l_{n}\rightarrow r_{n}\\})$ be a TRS and $\operatorname{CP}(R)=\\{(t_{1},s_{1}),\dots,(t_{m},s_{m})\\}$ be the set of the critical pairs of $R$. For any $i\in\\{1,\dots,m\\}$, let $a_{i},b_{i}$ be the numbers in $\\{1,\dots,n\\}$ such that the critical pair $(t_{i},s_{i})$ is obtained by $l_{a_{i}}\rightarrow r_{a_{i}}$ and $l_{b_{i}}\rightarrow r_{b_{i}}$, that is, $t_{i}=r_{a_{i}}\sigma\leftarrow l_{a_{i}}\sigma=C[l_{b_{i}}\sigma]\rightarrow C[r_{b_{i}}\sigma]=s_{i}$ for some substitution $\sigma$ and single-hole context $C$. Suppose $R$ is complete. We fix an arbitrary rewriting strategy and for a term $t$, let $\operatorname{nr}_{j}(t)$ be the number of times $l_{j}\rightarrow r_{j}$ is used to reduce $t$ into its $R$-normal form with respect to the strategy. To state our main theorem, we introduce a matrix $D(R)$ and a number $e(R)$: Suppose $d=\deg(R)$ is prime or $0$. If $d=0$, let $\mathfrak{R}$ be $\mathbb{Z}$, and if $d$ is prime, let $\mathfrak{R}$ be $\mathbb{Z}/d\mathbb{Z}$ (integers modulo $d$). For $1\leq i\leq m$, $1\leq j\leq n$, let $D(R)_{ij}$ be the integer $\operatorname{nr}_{j}(s_{i})-\operatorname{nr}_{j}(t_{i})+\delta(b_{i},j)-\delta(a_{i},j)$ where $\delta(x,y)$ is the Kronecker delta. The matrix $D(R)$ is defined by $D(R)=(D(R)_{ji})_{j=1,\dots,n,i=1,\dots,m}$. Let $\mathfrak{R}$ be $\mathbb{Z}$ or $\mathbb{Z}/p\mathbb{Z}$ for any prime $p$. If an $m\times n$ matrix $M$ over $\mathfrak{R}$ is of the form $\left(\begin{matrix}e_{1}&0&\dots&\dots&\dots&\dots&\dots&0\\\ 0&e_{2}&0&\dots&\dots&\dots&\dots&0\\\ \vdots&0&\ddots&0&\dots&\dots&\dots&\vdots\\\ \vdots&\vdots&0&e_{r}&0&\dots&\dots&\vdots\\\ \vdots&\vdots&\vdots&0&0&\dots&\dots&\vdots\\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\dots&\vdots\\\ 0&0&\dots&\dots&\dots&\dots&\dots&0\end{matrix}\right)$ and $e_{i}$ divides $e_{i+1}$ for every $1\leq i<r$, we say $M$ is in _Smith normal form_. We call $e_{i}$s the _elementary divisors_. It is known that every matrix over $\mathfrak{R}$ can be transformed into Smith normal form by elementary row/column operations, that is, (1) switching a row/column with another row/column, (2) multiplying each entry in a row/column by an invertible element in $\mathfrak{R}$, and (3) adding a multiple of a row/column to another row/column [14, 9.4]. (If $d=0$, the invertible elements in $\mathfrak{R}\cong\mathbb{Z}$ are $1$ and $-1$, and if $d$ is prime, any nonzero element in $\mathfrak{R}=\mathbb{Z}/d\mathbb{Z}$ is invertible. So, $e(R)$ is equal to the rank of $D(R)$ if $d$ is prime.) In general, the same fact holds for any principal ideal domain $\mathfrak{R}$. We define $e(R)$ as the number of invertible elements in the Smith normal form of the matrix $D(R)$ over $\mathfrak{R}$. We state the main theorem. ###### Theorem 1. Let $(\Sigma,R)$ be a complete TRS and suppose $d=\deg(R)$ is 0 or prime. For any set of rules $R^{\prime}$ equivalent to $R$, i.e., $\xleftrightarrow{*}_{R^{\prime}}\,=\,\xleftrightarrow{*}_{R}$, we have $\\#R^{\prime}\geq\\#R-e(R).$ (2) We shall see some examples. Consider the signature $\Sigma=\\{{\sf 0}^{(0)},{\sf s}^{(1)},{\sf ave}^{(2)}\\}$ and the set $R$ of rules $\begin{array}[]{lll}A_{1}.{\sf ave}({\sf 0},{\sf 0})\rightarrow{\sf 0},&A_{2}.{\sf ave}(x_{1},{\sf s}(x_{2}))\rightarrow{\sf ave}({\sf s}(x_{1}),x_{2}),&A_{3}.{\sf ave}({\sf s}({\sf 0}),{\sf 0})\rightarrow{\sf 0},\\\ A_{4}.{\sf ave}({\sf s}({\sf s}({\sf 0})),{\sf 0})\rightarrow s({\sf 0}),&A_{5}.{\sf ave}({\sf s}({\sf s}({\sf s}(x_{1}))),x_{2})\rightarrow{\sf s}({\sf ave}({\sf s}(x_{1}),x_{2})).&\end{array}$ $R$ satisfies $\deg(R)=0$ and has one critical pair $C$: We can see the matrix $D(R)$ is the $5\times 1$ zero matrix. The zero matrix is already in Smith normal form and $e(R)=0$. Thus, for any $R^{\prime}$ equivalent to $R$, $\\#R^{\prime}\geq\\#R=5$. This means there is no smaller TRS equivalent to $R$. Also, Malbos-Mimram’s lower bound, denoted by $s(H_{2}(\Sigma,R))$, is equal to 3, though we do not explain how to compute it in this paper. (We will roughly describe $s(H_{2}(\Sigma,R))$ in Section 4.) As a generalization of this example, we have an interesting corollary of our main theorem: ###### Corollary 2. Let $(\Sigma,R)$ be a complete TRS. If for any critical pair $u\leftarrow t\rightarrow v$, two rewriting paths $t\rightarrow u\rightarrow\dots\rightarrow\hat{t}$ and $t\rightarrow v\rightarrow\dots\rightarrow\hat{t}$ contain the same number of $l\rightarrow r$ for each $l\rightarrow r\in R$, then there is no $R^{\prime}$ equivalent to $R$ which satisfies $\\#R^{\prime}<\\#R$. We compute the lower bound for the theory of groups, (1). A complete TRS $R$ for the theory of groups is given by $\begin{array}[]{ll}G_{1}.\ m(m(x_{1},x_{2}),x_{3})\rightarrow m(x_{1},m(x_{2},x_{3}))&G_{2}.\ m(e,x_{1})\rightarrow x_{1}\\\ G_{3}.\ m(x_{1},e)\rightarrow x_{1}&G_{4}.\ m(x_{1},i(x_{1}))\rightarrow e\\\ G_{5}.\ m(i(x_{1}),x_{1})\rightarrow e&G_{6}.\ m(i(x_{1}),m(x_{1},x_{2}))\rightarrow x_{2}\\\ G_{7}.\ i(e)\rightarrow e&G_{8}.\ i(i(x_{1}))\rightarrow x_{1}\\\ G_{9}.\ m(x_{1},m(i(x_{1}),x_{2}))\rightarrow x_{2}&G_{10}.\ i(m(x_{1},x_{2}))\rightarrow m(i(x_{2}),i(x_{1})).\end{array}$ Since $\deg(R)=2$, we set $\mathfrak{R}=\mathbb{Z}/2\mathbb{Z}$. $R$ has 48 critical pairs and we get the $10\times 48$ matrix $D(R)$ given in Appendix A. The author implemented a program which takes a complete TRS as input and computes its critical pairs, the matrix $D(R)$, and $e(R)$. The program is available at https://github.com/mir-ikbch/homtrs. The author checked $e(R)=\operatorname{rank}(D(R))=8$ by the program, and also by MATLAB’s gfrank function (https://www.mathworks.com/help/comm/ref/gfrank.html). Therefore we have $\\#R-e(R)=2$. This provides a new proof that there is no single axiom equivalent to the theory of groups. Malbos-Mimram’s lower bound is given by $s(H_{2}(\Sigma,R))=0$. Let $\Sigma=\\{-^{(1)},f^{(1)},+^{(2)},\cdot^{(2)}\\}$ and $R$ be $\begin{array}[]{ll}A_{1}.\ -(-x_{1})\rightarrow x_{1},&A_{2}.\ -f(x_{1})\rightarrow f(-x_{1}),\\\ A_{3}.\ -(x_{1}+x_{2})\rightarrow(-x_{1})\cdot(-x_{2}),&A_{4}.\ -(x_{1}\cdot x_{2})\rightarrow(-x_{1})+(-x_{2}).\end{array}$ Figure 1. The critical pairs of $R$ We have $\deg(R)=0$ and $R$ has four critical pairs (Figure 1). The corresponding matrix $D(R)$ and its Smith normal form are computed as $D(R)=\left(\begin{matrix}0&0&1&1\\\ 2&0&0&0\\\ 0&0&1&1\\\ 0&0&1&1\\\ \end{matrix}\right)\rightsquigarrow\left(\begin{matrix}0&0&1&1\\\ 2&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{matrix}\right)\rightsquigarrow\left(\begin{matrix}0&0&1&0\\\ 2&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{matrix}\right)\rightsquigarrow\left(\begin{matrix}1&0&0&0\\\ 0&2&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{matrix}\right).$ Thus, $\\#R-e(R)=3$. This tells $R$ does not have any equivalent TRS with 2 or fewer rules, and it is not difficult to see $R$ has an equivalent TRS with 3 rules, $\\{A_{1},A_{2},A_{3}\\}$. Malbos-Mimram’s lower bound for this TRS is given by $s(H_{2}(\Sigma,R))=1$. Although the equality of (2) is attained for the above three examples, it is not guaranteed the equality is attained by some TRS $R^{\prime}$ in general. For example, the TRS with only the associative rule $\\{f(f(x_{1},x_{2}),x_{3})\rightarrow f(x_{1},f(x_{2},x_{3}))\\}$ satisfies $\\#R-e(R)=0$ and it is obvious that no TRSs with zero rule is equivalent. Also, in Appendix B, Malbos-Mimram’s and our lower bounds for various examples are given. ## 3\. Preliminaries on Algebra In this section, we give a brief introduction to module theory, homological algebra, and Squier’s theory of homological algebra for string rewriting systems (SRSs) [16]. Even though Squier’s theory is not directly needed to prove our theorem, it is helpful to understand the homology theory for TRSs, which is more complicated than SRSs’ case. ### 3.1. Modules and Homological Algebra We give basic definitions and theorems on module theory and homological algebra without proofs. For more details, readers are referred to [14, 13] for example. Modules are the generalization of vector spaces in which the set of scalars form a ring, not necessarily a field. Let $\mathfrak{R}$ be a ring and $(M,+)$ be an abelian group. For a map $\cdot:\mathfrak{R}\times M\rightarrow M$, $(M,+,\cdot)$ is a _left $\mathfrak{R}$-module_ if for all $r,s\in\mathfrak{R}$ and $x,y\in M$, we have $r\cdot(x+y)=r\cdot x+r\cdot y,\ (r+s)\cdot x=r\cdot x+s\cdot x,\ (rs)\cdot x=r\cdot(s\cdot x)$ where $rs$ denotes the multiplication of $r$ and $s$ in $\mathfrak{R}$. We call the map $\cdot$ _scalar multiplication_. For a map $\cdot:M\times\mathfrak{R}\rightarrow M$, $(M,+,\cdot)$ is a _right $\mathfrak{R}$-module_ if for any $r,s\in\mathfrak{R}$ and $x,y\in M$, $(x+y)\cdot r=x\cdot r+y\cdot r,\ x\cdot(r+s)=x\cdot r+x\cdot s,\ x\cdot(sr)=(x\cdot s)\cdot r.$ If ring $\mathfrak{R}$ is commutative, we do not distinguish between left $\mathfrak{R}$-modules and right $\mathfrak{R}$-modules and simply call them $\mathfrak{R}$-modules. Linear maps and isomorphisms of modules are also defined in the same way as for vector spaces. For two left $\mathfrak{R}$-modules $(M_{1},+_{1},\cdot_{1}),(M_{2},+_{2},\cdot_{2})$, a group homomorphism $f:(M_{1},+_{1})\rightarrow(M_{2},+_{2})$ is an _$\mathfrak{R}$ -linear map_ if it satisfies $f(r\cdot_{1}x)=r\cdot_{2}f(x)$ for any $r\in\mathfrak{R}$ and $x\in M_{1}$. An $\mathfrak{R}$-linear map $f$ is an _isomorphism_ if it is bijective, and two modules are called _isomorphic_ if there exists an isomorphism between them. Any abelian group $(M,+)$ is a $\mathbb{Z}$-module under the scalar multiplication $n\cdot x=\underbrace{x+\dotsb+x}_{n}$. For any ring $\mathfrak{R}$, the direct product $\mathfrak{R}^{n}=\underbrace{\mathfrak{R}\times\dotsb\times\mathfrak{R}}_{n}$ forms a left $\mathfrak{R}$-module under the scalar multiplication $r\cdot(r_{1},\dotsc,r_{n})=(rr_{1},\dotsc,rr_{n})$. Let $\mathfrak{R}$ be a ring and $X$ be a set. $\mathfrak{R}\underline{X}$ denotes the set of formal linear combinations $\sum_{x\in X}r_{x}\underline{x}\quad(r_{x}\in\mathfrak{R})$ where $r_{x}=0$ except for finitely many $x$s. The underline is added to emphasize a distinction between $r\in\mathfrak{R}$ and $x\in X$. $\mathfrak{R}\underline{X}$ forms a left $\mathfrak{R}$-module under the addition and the scalar multiplication defined by $\left(\sum_{x\in X}r_{x}\underline{x}\right)+\left(\sum_{x\in X}s_{x}\underline{x}\right)=\sum_{x\in X}(r_{x}+s_{x})\underline{x},\quad s\cdot\left(\sum_{x\in X}r_{x}\underline{x}\right)=\sum_{x\in X}(sr_{x})\underline{x}.$ If $X$ is the empty set, $\mathfrak{R}\underline{X}$ is the left $\mathfrak{R}$-module $\\{0\\}$ consisting of only the identity element. We simply write $0$ for $\\{0\\}$. $\mathfrak{R}\underline{X}$ is called the _free left $\mathfrak{R}$-module generated by $X$_. If $\\#X=n\in\mathbb{N}$, $\mathfrak{R}\underline{X}$ can be identified with $\mathfrak{R}^{n}$. A left $\mathfrak{R}$-module $M$ is said to be _free_ if $M$ is isomorphic to $\mathfrak{R}\underline{X}$ for some $X$. Free modules have some similar properties to vector spaces. If a left $\mathfrak{R}$-module $F$ is free, then there exists a basis (i.e., a subset that is linearly independent and generating) of $F$. If a free left $\mathfrak{R}$-module $F$ has a basis $(v_{1},\dotsc,v_{n})$, any $\mathfrak{R}$-linear map $f:F\rightarrow M$ is uniquely determined if the values $f(v_{1}),\dotsc,f(v_{n})$ are specified. Suppose $F_{1}$, $F_{2}$ are free left $\mathfrak{R}$-modules and $f:F_{1}\rightarrow F_{2}$ is an $\mathfrak{R}$-linear map. If $F_{1}$ has a basis $(v_{1},\dotsc,v_{n})$ and $F_{2}$ has a basis $(w_{1},\dotsc,w_{m})$, the matrix $(a_{ij})_{i=1,\dotsc,n,j=1,\dotsc,m}$ where $a_{ij}$s satisfy $f(v_{i})=a_{i1}w_{1}+\dotsb+a_{im}w_{m}$ for any $i=1,\dotsc,n$ is called a _matrix representation_ of $f$. We define submodules and quotient modules, as in linear algebra. Let $(M,+,\cdot)$ be a left (resp. right) $\mathfrak{R}$-module. A subgroup $N$ of $(M,+)$ is a _submodule_ if for any $x\in N$ and $r\in\mathfrak{R}$, the scalar multiplication $r\cdot x$ (resp. $x\cdot r$) is in $N$. For any submodule $N$, the quotient group $M/N$ is also an $\mathfrak{R}$-module. $M/N$ is called the _quotient module_ of $M$ by $N$. For submodules and quotient modules, the following basic theorem is known: ###### Theorem 3 (First isomorphism theorem). [14, Theorem 8.8] Let $(M,+,\cdot),(M^{\prime},+^{\prime},\cdot^{\prime})$ be left (or right) $\mathfrak{R}$-modules, and $f:M\rightarrow M^{\prime}$ be an $\mathfrak{R}$-linear map. 1. (1) The inverse image of $0$ by $f$, $\ker f=\\{x\in M\mid f(x)=0\\}$, is a submodule of $M$. 2. (2) The image of $M$ by $f$, $\operatorname{im}f=\\{f(x)\mid x\in M\\}$, is a submodule of $M^{\prime}$. 3. (3) The image $\operatorname{im}f$ is isomorphic to $M/\ker f$. ###### Theorem 4 (Third isomorphism theorem). [14, Theorem 7.10] Let $M$ be a left (or right) $\mathfrak{R}$-module, $N$ be a submodule of $M$, and $L$ be a submodule of $N$. Then $(M/L)/(N/L)$ is isomorphic to $M/N$. ###### Theorem 5. [14, Theorem 9.8] Let $\mathfrak{R}$ be $\mathbb{Z}$ or $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$. Every submodule of a free $\mathfrak{R}$-module is free. Moreover, if an $\mathfrak{R}$-module $M$ is isomorphic to $\mathfrak{R}^{n}$, then every submodule $N$ of $M$ is isomorphic to $\mathfrak{R}^{m}$ for some $m\leq n$. (In general, this holds for any principal ideal domain $\mathfrak{R}$.) Let $M$ be a left $\mathfrak{R}$-module. For $S\subset M$, the set $\mathfrak{R}S$ of all elements in $M$ of the form $\sum_{i=1}^{k}r_{i}s_{i}$ $(k\in\mathbb{Z}_{\geq 0},r_{i}\in\mathfrak{R},s_{i}\in S)$ is a submodule of $M$. If $\mathfrak{R}S=M$, $S$ is called a _generating set_ of $S$ and the elements of $S$ are called _generators_ of $M$. Let $S=\\{s_{i}\\}_{i\in I}$ be a generating set of $M$ for some indexing set $I$. For a set $X=\\{x_{i}\\}_{i\in I}$, the linear map $\epsilon:\mathfrak{R}\underline{X}\ni x_{i}\mapsto s_{i}\in M$ is a surjection from the free module $\mathfrak{R}\underline{X}$. The elements of $\ker\epsilon$, that is, elements $\sum_{x_{i}\in X}r_{i}\underline{x_{i}}$ satisfying $\epsilon(\sum_{x_{i}\in X}r_{i}\underline{x_{i}})=\sum_{x_{i}\in X}r_{i}s_{i}=0$, are called _relations_ of $M$. Now, we introduce one of the most important notions to develop the homology theory of rewriting systems, free resolutions. We first start from the following example. Let $M$ be the $\mathbb{Z}$-module defined by $\mathbb{Z}\underline{\\{a,b,c,d,e\\}}/\mathbb{Z}\\{\underline{a}+\underline{b}+\underline{c}-\underline{d}-\underline{e},\ 2\underline{b}-\underline{c},\ \underline{a}+2\underline{c}-\underline{b}-\underline{d}-\underline{e}\\}.$ We consider the $\mathbb{Z}$-linear map between free $\mathbb{Z}$-modules $f_{0}:\mathbb{Z}^{3}\rightarrow\mathbb{Z}\underline{\\{a,b,c,d,e\\}}$ defined by $f_{0}(1,0,0)=\underline{a}+\underline{b}+\underline{c}-\underline{d}-\underline{e},\ f_{0}(0,1,0)=2\underline{b}-\underline{c},\ f_{0}(0,0,1)=\underline{a}+2\underline{c}-\underline{b}-\underline{d}-\underline{e}.$ We can see that the image of $f_{0}$ is the set of relations of $M$. In other words, $\operatorname{im}f_{0}=\ker\epsilon$ for the linear map $\epsilon:\mathbb{Z}\underline{\\{a,b,c,d,e\\}}\rightarrow M$ which maps each element to its equivalence class. Then, we consider the “relations between relations”, that is, triples $(n_{1},n_{2},n_{3})$ which satisfy $f_{0}(n_{1},n_{2},n_{3})=n_{1}(\underline{a}+\underline{b}+\underline{c}-\underline{d}-\underline{e})+n_{2}(2\underline{b}-\underline{c})+n_{3}(\underline{a}+2\underline{c}-\underline{b}-\underline{d}-\underline{e})=0$, or equivalently, elements of $\ker f_{0}$. We can check $\ker f_{0}=\\{m(-1,1,1)\mid m\in\mathbb{Z}\\}$. This fact can be explained in terms of rewriting systems. If we write relations in the form of rewrite rules $A_{1}.\ \underline{a}+\underline{b}+\underline{c}\rightarrow\underline{d}+\underline{e},\ A_{2}.\ 2\underline{b}\rightarrow\underline{c},\ A_{3}.\ \underline{a}+2\underline{c}\rightarrow\underline{b}+\underline{d}+\underline{e},$ we see $\\{A_{1},A_{2},A_{3}\\}$ is a complete rewriting system (over the signature $\\{\underline{a},\underline{b},\underline{c},\underline{d},\underline{e},+\\}$) with two joinable critical pairs We associate these critical pairs with an equality between formal sums $A_{2}+A_{3}=A_{1}$, and it corresponds to $f_{0}(-1,1,1)=\underbrace{-(\underline{a}+\underline{b}+\underline{c}-\underline{d}-\underline{e})}_{-A_{1}}+\underbrace{(2\underline{b}-\underline{c})}_{A_{2}}+\underbrace{(\underline{a}+2\underline{c}-\underline{b}-\underline{d}-\underline{e})}_{A_{3}}=0.$ In fact, this correspondence between critical pairs and “relations between relations” is a key to the homology theory of TRSs. We define a linear map $f_{1}:\mathbb{Z}\rightarrow\mathbb{Z}^{3}$ by $f_{1}(1)=(-1,1,1)$ and then $f_{1}$ satisfies $\operatorname{im}f_{1}=\ker f_{0}$. We can go further, that is, we can consider $\ker f_{1}$, but it clearly turns out that $\ker f_{1}=0$. We encode the above information in the following diagram: $\mathbb{Z}\xrightarrow{f_{1}}\mathbb{Z}^{3}\xrightarrow{f_{0}}\mathbb{Z}\underline{\\{a,b,c,d,e\\}}\xrightarrow{\epsilon}M$ (3) where $\operatorname{im}f_{1}=\ker f_{0},\operatorname{im}f_{0}=\ker\epsilon$ and $\epsilon$ is surjective. Sequences of modules and linear maps with these conditions are called free resolutions: A sequence of left $\mathfrak{R}$-modules and $\mathfrak{R}$-linear maps $\dotsb\xrightarrow{f_{i+1}}M_{i+1}\xrightarrow{f_{i}}M_{i}\xrightarrow{f_{i-1}}\dotsb$ is called an _exact sequence_ if $\operatorname{im}f_{i}=\ker f_{i-1}$ holds for any $i$. Let $M$ be a left $\mathfrak{R}$-module. For infinite sequence of free modules $F_{i}$ and linear maps $f_{i}:F_{i+1}\rightarrow F_{i}$, $\epsilon:F_{0}\rightarrow M$, if the sequence $\dotsb\xrightarrow{f_{1}}F_{1}\xrightarrow{f_{0}}F_{0}\xrightarrow{\epsilon}M$ is exact and $\epsilon$ is surjective, the sequence above is called a _free resolution_ of $M$. If the sequence is finite, it is called a _partial free resolution_. (Exact sequences and free resolutions are defined for right $\mathfrak{R}$-modules in the same way.) Notice that the exact sequence (3) can be extended to the infinite exact sequence $\dotsb\rightarrow 0\rightarrow\dotsb\rightarrow 0\rightarrow\mathbb{Z}\xrightarrow{f_{1}}\mathbb{Z}^{3}\xrightarrow{f_{0}}\mathbb{Z}\underline{\\{a,b,c,d,e\\}}\xrightarrow{\epsilon}M$ since $\ker f_{1}=0$. Thus, the sequence (3) is a free resolution of $M$. As there are generally several rewriting systems equivalent to a given equational theory, free resolutions of $M$ are not unique. However, we can construct some information of $M$ from a (partial) free resolution which does not depend on the choice of the free resolution. The information is called homology groups. To define the homology groups, we introduce the tensor product of modules. Let $N$ be a right $\mathfrak{R}$-module and $M$ be a left $\mathfrak{R}$-module. Let $F(N\times M)$ be the free abelian group generated by $N\times M$. The _tensor product_ of $N$ and $M$, denoted by $N\otimes_{\mathfrak{R}}M$, is the quotient group of $F(N\times M)$ by the subgroup generated by the elements of the form $(x,y)+(x,y^{\prime})-(x,y+y^{\prime}),\ (x,y)+(x^{\prime},y)-(x+x^{\prime},y),\ (x\cdot r,y)-(x,r\cdot y)$ where $x,x^{\prime}\in N$, $y,y^{\prime}\in M$, $r\in R$. The equivalence class of $(x,y)$ in $N\otimes_{\mathfrak{R}}M$ is written as $x\otimes y$. For a right $\mathfrak{R}$-module $N$ and a $\mathfrak{R}$-linear map $f:M\rightarrow M^{\prime}$ between left $\mathfrak{R}$-modules $M,M^{\prime}$, we write $N\otimes f:N\otimes_{\mathfrak{R}}M\rightarrow N\otimes_{\mathfrak{R}}M^{\prime}$ for the map $(N\otimes f)(a\otimes x)=a\otimes f(x)$. $N\otimes f$ is known to be well-defined and be a group homomorphism. Let $\dotsb\xrightarrow{f_{1}}F_{1}\xrightarrow{f_{0}}F_{0}\xrightarrow{\epsilon}M$ be a free resolution of a left $\mathfrak{R}$-module $M$. For a right $\mathfrak{R}$-module $N$, we consider the sequence $\dotsb\xrightarrow{N\otimes f_{1}}N\otimes_{\mathfrak{R}}F_{1}\xrightarrow{N\otimes f_{0}}N\otimes_{\mathfrak{R}}F_{0}.$ (4) Then, it can be shown that $\operatorname{im}(N\otimes f_{i})\subset\ker(N\otimes f_{i-1})$ for any $i=1,2,\dotsc$. In general, a sequence $\dotsb\xrightarrow{f_{i+1}}M_{i+1}\xrightarrow{f_{i}}M_{i}\xrightarrow{f_{i-1}}\dotsb$ of left/right $\mathfrak{R}$-modules satisfying $\operatorname{im}f_{i}\subset\ker f_{i-1}$ for any $i$ is called a chain complex. The homology groups of a chain complex are defined to be the quotient group of $\ker f_{i-1}$ by $\operatorname{im}f_{i}$: Let $(C_{\bullet},f_{\bullet})$ denote the pair $(\\{C_{i}\\}_{i=0,1,\dots},\\{f_{i}:C_{i+1}\rightarrow C_{i}\\}_{i=0,1,\dots})$. For a chain complex $\dotsb\xrightarrow{f_{i+1}}C_{i+1}\xrightarrow{f_{i}}C_{i}\xrightarrow{f_{i-1}}\dotsb$, the abelian group $H_{j}(C_{\bullet},f_{\bullet})$ defined by $H_{j}(C_{\bullet},f_{\bullet})=\ker f_{j-1}/\operatorname{im}f_{j}$ is called the _$j$ -th homology groups_ of the chain complex $(C_{\bullet},f_{\bullet})$. The homology groups of the chain complex (4) depend only on $M$, $N$, and $\mathfrak{R}$: ###### Theorem 6. [13, Corollary 6.21] Let $M$ be a left $\mathfrak{R}$-module and $N$ be a right $\mathfrak{R}$-module. For any two resolutions $\dotsb\xrightarrow{f_{1}}F_{1}\xrightarrow{f_{0}}F_{0}\xrightarrow{\epsilon}M$, $\dotsb\xrightarrow{f^{\prime}_{1}}F^{\prime}_{1}\xrightarrow{f^{\prime}_{0}}F^{\prime}_{0}\xrightarrow{\epsilon}M$, we have a group isomorphism $H_{j}(N\otimes_{\mathfrak{R}}F_{\bullet},N\otimes f_{\bullet})\cong H_{j}(N\otimes_{\mathfrak{R}}F_{\bullet}^{\prime},N\otimes f_{\bullet}^{\prime}).$ We end this subsection by giving some basic facts on exact sequences. ###### Proposition 7. [14, Proposition 7.20 and 7.21] 1. (1) $M_{1}\xrightarrow{f}M_{2}\rightarrow 0$ is exact if and only if $\ker f=0$. 2. (2) $0\rightarrow M_{1}\xrightarrow{f}M_{2}$ is exact if and only if $\operatorname{im}f=M_{2}$. 3. (3) If $M_{1}$ is a submodule of $M_{2}$, the sequence $0\rightarrow M_{2}\xrightarrow{\iota}M_{1}\xrightarrow{\pi}M_{1}/M_{2}\rightarrow 0$ is exact where $\iota$ is the inclusion map $\iota(x)=x$ and $\pi$ is the projection $\pi(x)=[x]$. ###### Proposition 8. Suppose we have an exact sequence of $\mathfrak{R}$-modules $0\rightarrow M_{1}\rightarrow M_{2}\rightarrow M_{3}\rightarrow 0$. If $M_{3}$ is free, then $M_{2}$ is isomorphic to $M_{1}\times M_{3}$. The proof is given by using [14, Proposition 7.22]. ### 3.2. String Rewriting Systems and Homology Groups of Monoids For an alphabet $\Sigma$, $\Sigma^{*}$ denotes the set of all strings of symbols over $\Sigma$. The set $\Sigma^{*}$ forms a monoid under the operation of concatenation with the empty string serving as the identity, and we call $\Sigma^{*}$ the free monoid generated by $\Sigma$. For a string rewriting system (SRS) $(\Sigma,R)$, we write $\mathcal{M}_{(\Sigma,R)}$ for the set defined by $\mathcal{M}_{(\Sigma,R)}=\Sigma^{*}\delimiter 84079374\mathopen{}\xleftrightarrow{*}_{R}$. We can see $\mathcal{M}_{(\Sigma,R)}$ is a monoid under the operations $[u]\cdot[v]=[uv]$ where $[w]$ denotes the equivalence class of $w\in\Sigma^{*}$ with respect to $\xleftrightarrow{*}_{R}$. We say that two SRSs $(\Sigma_{1},R_{1}),(\Sigma_{2},R_{2})$ are _Tietze equivalent_ if the monoids $\mathcal{M}_{(\Sigma_{1},R_{1})}$, $\mathcal{M}_{(\Sigma_{2},R_{2})}$ are isomorphic. It is not difficult to show that for any two SRSs $(\Sigma,R_{1}),(\Sigma,R_{2})$ with the same signature, if $R_{1}$ and $R_{2}$ are equivalent (i.e., $\xleftrightarrow{*}_{R_{1}}\,=\,\xleftrightarrow{*}_{R_{2}}$), then $(\Sigma,R_{1})$ and $(\Sigma,R_{2})$ are isomorphic. Roughly speaking, the notion that two SRSs are isomorphic means that the SRSs are equivalent but their alphabets can be different. For example, let $\Sigma_{1}$ be $\\{a,b,c\\}$ and $R_{1}$ be $\\{abb\rightarrow ab,\ ba\rightarrow c\\}$. Then, $(\Sigma_{1},R_{1})$ is isomorphic to $(\Sigma_{2},R_{2})$ where $\Sigma_{2}=\\{a,b\\}$ and $R_{2}=\\{abb\rightarrow ab\\}$. Intuitively, since $c$ is equivalent to $ba$ with respect to the congruence $\xleftrightarrow{*}_{R_{1}}$, $c$ is redundant as long as we consider strings modulo $\xleftrightarrow{*}_{R_{1}}$ and $(\Sigma_{2},R_{2})$ is the SRS made by removing $c$ from $(\Sigma_{1},R_{1})$. If a monoid $S$ is isomorphic to $\mathcal{M}_{(\Sigma,R)}$ for an SRS $(\Sigma,R)$, we call $(\Sigma,R)$ a _presentation_ of the monoid $S$. Let $S$ be a monoid and consider the free $\mathbb{Z}$-module $\mathbb{Z}\underline{S}$. The module $\mathbb{Z}\underline{S}$ can be equipped with a ring structure under the multiplication $\left(\sum_{w\in S}n_{w}\underline{w}\right)\left(\sum_{w\in S}m_{w}\underline{w}\right)=\sum_{w,v\in S}n_{w}m_{v}\underline{wv}$ where $n_{w}m_{v}$ is the usual multiplication of integers and $wv$ is the multiplication of the monoid $S$. $\mathbb{Z}\underline{S}$ as a ring is called the _integral monoid ring_ of $S$. When we think of $\mathbb{Z}\underline{S}$ as a ring, we write $\mathbb{Z}\langle S\rangle$ instead of $\mathbb{Z}\underline{S}$. We consider $\mathbb{Z}\langle S\rangle$-modules. The group of integers $\mathbb{Z}$ forms a left (resp. right) $\mathbb{Z}\langle S\rangle$-module under the scalar multiplication $(\sum_{w\in S}n_{w}\underline{w})\cdot m=\sum_{w\in S}n_{w}m$ (resp. $m\cdot(\sum_{w\in S}n_{w}\underline{w})=\sum_{w\in S}n_{w}m\underline{w}$). Let $\dotsb\xrightarrow{\partial_{1}}F_{1}\xrightarrow{\partial_{0}}F_{0}\xrightarrow{\epsilon}\mathbb{Z}$ be a free resolution of $\mathbb{Z}$ over the ring $\mathbb{Z}\langle S\rangle$. The abelian group $H_{i}(S)$ is defined as the $i$-th homology group of the chain complex $(\mathbb{Z}\otimes_{\mathbb{Z}\langle S\rangle}F_{\bullet},\mathbb{Z}\otimes\partial_{\bullet})$, i.e., $H_{i}(S)=H_{i}(\mathbb{Z}\otimes_{\mathbb{Z}\langle S\rangle}F_{\bullet},\mathbb{Z}\otimes\partial_{\bullet})=\ker\mathbb{Z}\otimes\partial_{i-1}/\operatorname{im}\mathbb{Z}\otimes\partial_{i}.$ If $S$ is isomorphic to $\mathcal{M}_{(\Sigma,R)}$ for some SRS $(\Sigma,R)$, it is known that there is a free resolution in the form of $\dotsb\rightarrow(\mathbb{Z}\langle S\rangle)\underline{P}\xrightarrow{\partial_{2}}(\mathbb{Z}\langle S\rangle)\underline{R}\xrightarrow{\partial_{1}}(\mathbb{Z}\langle S\rangle)\underline{\Sigma}\xrightarrow{\partial_{0}}(\mathbb{Z}\langle S\rangle)\underline{\\{\star\\}}\xrightarrow{\epsilon}\mathbb{Z}$ for some set $P$. Squier [16] showed that if the SRS $(\Sigma,R)$ is complete and reduced111An SRS $(\Sigma,R)$ is reduced if for each $l\rightarrow r\in R$, $r$ is normal w.r.t. $\rightarrow_{R}$ and there does not exist $l^{\prime}\rightarrow r^{\prime}\in R$ such that $l^{\prime}=ulv\neq l$ for some $u,v\in\Sigma^{*}$, there is $\partial_{2}:(\mathbb{Z}\langle S\rangle)\underline{P}\rightarrow(\mathbb{Z}\langle S\rangle)\underline{R}$ for $P=(\text{the critical pairs of $R$})$ so that we can compute $H_{2}(S)=\ker\partial_{1}/\operatorname{im}\partial_{2}$ explicitly. This is an analog of Example 5, but we omit the details here. For an abelian group $G$, let $s(G)$ denote the minimum number of generators of $G$ (i.e., the minimum cardinality of the subset $A\subset G$ such that any element $x\in G$ can be written by $x=a_{1}+\dotsb+a_{k}-a_{k+1}-\dotsb-a_{m}$ for $a_{1},\dotsc,a_{m}\in A$). Then, we have the following theorem: ###### Theorem 9. [21] Let $(\Sigma,R)$ be an SRS and $S=\mathcal{M}_{(\Sigma,R)}$. Then $\\#\Sigma\geq s(H_{1}(S))$, $\\#R\geq s(H_{2}(S))$. To prove this theorem, we use the following lemma: ###### Lemma 10. Let $X$ be a set. The group homomorphism $\mathbb{Z}\otimes_{\mathbb{Z}\langle S\rangle}(\mathbb{Z}\langle S\rangle)\underline{X}\rightarrow\mathbb{Z}\underline{X}$, $n\langle w\rangle\underline{x}\mapsto n\underline{x}$ is an isomorphism. This lemma is proved in a straightforward way. ###### Proof 3.1 (Proof of Theorem 9). Since $\mathbb{Z}\otimes_{\mathbb{Z}\langle S\rangle}(\mathbb{Z}\langle S\rangle)\underline{X}\cong\mathbb{Z}\underline{X}$ by the above lemma, $s(\mathbb{Z}\otimes_{\mathbb{Z}\langle S\rangle}(\mathbb{Z}\langle S\rangle)\underline{X})=s(\mathbb{Z}\underline{X})=\\#X$. For any set $Y$ and group homomorphism $f:\mathbb{Z}\underline{X}\rightarrow\mathbb{Z}\underline{Y}$, since $\ker f$ is a subgroup of $\mathbb{Z}\underline{X}$, we have $\\#X\geq s(\ker f)$. For any subgroup $H$ of $\ker f$, $\ker f/H$ is generated by $[x_{1}],\dotsc,[x_{k}]$ if $\ker f$ is generated by $x_{1},\dotsc,x_{k}$. Thus $\\#\Sigma\geq s(\ker\partial_{0}/\operatorname{im}\partial_{1})=s(H_{1}(S))$, $\\#R\geq s(\ker\partial_{1}/\operatorname{im}\partial_{2})=s(H_{2}(S))$. Note that $H_{i}(S)$ does not depend on the choice of presentation $(\Sigma,R)$ by Theorem 6. Therefore, Theorem 9 can be restated as follows: Let $(\Sigma,R)$ be an SRS. For any SRS $(\Sigma^{\prime},R^{\prime})$ isomorphic to $(\Sigma,R)$, the number of symbols $\\#\Sigma^{\prime}$ is bounded below by $s(H_{1}(\mathcal{M}_{(\Sigma,R)}))$ and the number of rules $\\#R^{\prime}$ is bounded below by $s(H_{2}(\mathcal{M}_{(\Sigma,R)}))$. ## 4\. An Overview of the Homology Theory of Algebraic Theories In this section, we will briefly see the homology theory of algebraic theories, which is the main tool to obtain our lower bounds. We fix a signature $\Sigma$. Let $t=\langle t_{1},\dotsc,t_{n}\rangle$ be a $n$-tuple of terms and suppose that for each $t_{i}$, the set of variables in $t_{i}$ is included in $\left\\{x_{1},\dots,x_{m}\right\\}$. For an $m$-tuple of term $s=\langle s_{1},\dotsc,s_{m}\rangle$, we define the composition of $t$ and $s$ by $t\circ s=\langle t_{1}[s_{1}/x_{1},\dotsc,s_{m}/x_{m}],\dotsc,t_{n}[s_{1}/x_{1},\dotsc,s_{m}/x_{m}]\rangle$ where $t_{i}[s_{1}/x_{1},\dotsc,s_{m}/x_{m}]$ denotes the term obtained by substituting $s_{j}$ for $x_{j}$ in $t_{i}$ for each $j=1,\dotsc,m$ in parallel. (For example, $f(x_{1},x_{2})[g(x_{2})/x_{1},g(x_{1})/x_{2}]=f(g(x_{2}),g(x_{1}))$.) By this definition, we can think of any $m$-tuple $\langle s_{1},\dotsc,s_{m}\rangle$ of terms as a (parallel) substitution $\left\\{x_{1}\mapsto s_{1},\dotsc,x_{m}\mapsto s_{m}\right\\}$. Recall that, for a TRS $R$, the reduction relation $\rightarrow_{R}$ between terms is defined as $t_{1}\rightarrow_{R}t_{2}\iff t_{1}=C[l\circ s],\ t_{2}=C[r\circ s]$ for some single-hole context $C$, $m$-tuple $s$ of terms, and rewrite rule $l\rightarrow r\in R$ whose variables are included in $\\{x_{1},\dotsc,x_{m}\\}$. This definition suggests that the pair of a context $C$ and an $m$-tuple of terms (or equivalently, substitution) $s$ is useful to think about rewrite relations. Malbos and Mimram [9] called the pair of a context and an $m$-tuple of terms a bicontext. For a bicontext $(C,t)$ and a rewrite rule $A$, we call the triple $(C,A,t)$ a rewriting step. The pair of two rewriting steps $(\square,l_{1}\rightarrow r_{1},s),(C,l_{2}\rightarrow r_{2},t)$ is called a critical pair if the pair $(r_{1}\circ s,C[r_{2}\circ t])$ of terms is a critical pair in the usual sense given by $l_{1}\rightarrow r_{1}$, $l_{2}\rightarrow r_{2}$. The composition of two bicontexts $(C,t),(D,s)$ ($t=\langle t_{1},\dotsc,t_{n}\rangle$, $s=\langle s_{1},\dotsc,s_{m}\rangle$) is defined by $(C,t)\circ(D,s)=(C[D\circ t],s\circ t)$ where $D\circ t=D[t_{1}/x_{1},\dotsc,t_{n}/x_{n}]$ and note that the order of composition is reversed in the second component. We write $\mathbb{K}(n,m)$ ($n,m\in\mathbb{N}$) for the set of bicontexts $(C,t)$ where $t=\langle t_{1},\dotsc,t_{n}\rangle$ and each $t_{i}$ and $C$ have variables in $\\{x_{1},\dotsc,x_{m}\\}$ (except $\square$ in $C$). To apply homological algebra to TRSs, we construct an algebraic structure from bicontexts. For two natural numbers $n,m$, we define $\mathbb{Z}\langle\mathbb{K}\rangle(n,m)$ to be the free abelian group generated by $\mathbb{K}(n,m)$ (i.e., any element in $\mathbb{Z}\langle\mathbb{K}\rangle(n,m)$ is written in the form of formal sum $\sum_{(C,t)\in\mathbb{K}(n,m)}\lambda_{(C,t)}(C,t)$ where each $\lambda_{(C,t)}$ is in $\mathbb{Z}$ and is equal to $0$ except for finitely many $(C,t)$s.) Then, the composition $\circ:\mathbb{K}(n,m)\times\mathbb{K}(k,n)\rightarrow\mathbb{K}(k,m)$ can be extended to $\circ:\mathbb{Z}\langle\mathbb{K}\rangle(n,m)\times\mathbb{Z}\langle\mathbb{K}\rangle(k,n)\rightarrow\mathbb{Z}\langle\mathbb{K}\rangle(k,m)$ by $\left(\sum_{(C,t)}\lambda_{(C,t)}(C,t)\right)\circ\left(\sum_{(D,s)}\mu_{(D,s)}(D,s)\right)=\sum_{(C,t)}\sum_{(D,s)}\lambda_{(C,t)}\mu_{(D,s)}((C,t)\circ(D,s)).$ This family of free abelian groups forms a structure called _ringoid_. Suppose an abelian group $(\mathcal{R}(i,j),+_{i,j},0_{i,j})$ is defined for each $i,j\in\mathbb{N}$. If for each $i,j,k\in\mathbb{N}$, a map $\circ_{i,j,k}:\mathcal{R}(j,k)\times\mathcal{R}(i,j)\rightarrow\mathcal{R}(i,k)$ is defined and satisfies the following conditions, $\mathcal{R}$ is called a _ringoid_ (also called a _small $\mathbf{Ab}$-enriched category_). 1. (1) For each $i$, there exists an element $1_{i}\in\mathcal{R}(i,i)$ such that $a\circ_{i,i,j}1_{i}=a$, $1_{i}\circ_{j,i,i}b=b$ ($j\in\mathbb{N}$, $a\in\mathcal{R}(i,j)$, $b\in\mathcal{R}(j,i)$), 2. (2) $(a\circ_{j,k,l}b)\circ_{i,j,l}c=a\circ_{i,k,l}(b\circ_{i,j,k}c)$ ($a\in\mathcal{R}(k,l)$, $b\in\mathcal{R}(j,k)$, $c\in\mathcal{R}(i,j)$), 3. (3) $(a+_{j,k}b)\circ_{i,j,k}c=a\circ_{i,j,k}c+_{i,k}b\circ_{i,j,k}c$ ($a,b\in\mathcal{R}(j,k)$, $c\in\mathcal{R}(i,j)$), 4. (4) $a\circ_{i,j,k}(b+_{i,j}c)=a\circ_{i,j,k}b+_{i,k}a\circ_{i,j,k}c$ ($a\in\mathcal{R}(j,k)$, $b,c\in\mathcal{R}(i,j)$), 5. (5) $a\circ_{i,j,k}0_{i,j}=0_{i,k}=0_{j,k}\circ_{i,j,k}b$ ($a\in\mathcal{R}(j,k)$, $b\in\mathcal{R}(i,j)$). We will omit subscripts of $+,\circ,0,1$ if there is no confusion. The notion of modules over a ring is extended to modules over a ringoid. Let $\mathcal{R}$ be a ringoid. Suppose that for each $i\in\mathbb{N}$, an abelian group $(M(i),+_{i},0_{i})$ is defined. If there is a map $\cdot_{i,j}:\mathcal{R}(i,j)\times M(i)\rightarrow M(j)$ satisfying the following conditions, $M$ is called a _left $\mathcal{R}$-module_. 1. (1) $(a\circ_{i,j,k}b)\cdot_{i,k}x=a\cdot_{j,k}(b\cdot_{i,j}x)$ ($a\in\mathcal{R}(j,k)$, $b\in\mathcal{R}(i,j)$, $x\in M(i)$), 2. (2) $1_{i}\cdot_{i,i}x=x$ ($x\in M(i))$), 3. (3) $(a+_{i,j}b)\cdot_{i,j}x=(a\cdot_{i,j}x)+_{j}(b\cdot_{i,j}x)$ ($a,b\in\mathcal{R}(i,j)$, $x\in M(i)$), 4. (4) $a\cdot_{i,j}(x+_{i}y)=(a\cdot_{i,j}x)+_{j}(a\cdot_{i,j}y)$ ($a\in\mathcal{R}(i,j)$, $x,y\in M(i)$), 5. (5) $0_{i,j}\cdot_{i,j}x=0_{j}$ ($x\in M(i)$). A _right $\mathcal{R}$-module_ $M$ is also defined with a map $\cdot_{i,j}:M(i)\times\mathcal{R}(i,j)\rightarrow M(j)$ in the same manner with right modules over a ring. An _$\mathcal{R}$ -linear map_ $f:M\rightarrow M^{\prime}$ between left $\mathcal{R}$-modules $M,M^{\prime}$ is a collection of group homomorphisms $f_{i}:M(i)\rightarrow M^{\prime}(i)$ ($i\in\mathbb{N}$) that satisfy $f_{j}(a\cdot_{i,j}x)=a\cdot_{i,j}f_{i}(x)\quad(a\in\mathcal{R}(i,j),x\in M(i)).$ Ringoids and modules over ringoids are originally defined in a category theoretic way (cf. [10, 9]). (A ringoid is a small Ab-enriched category, and a module over a ringoid is an additive functor.) Our definitions here are obtained by unfolding the category theoretic terminology in the original definitions so that those who are not familiar with category theory can understand them more easily. Let $\mathcal{R}$ be a ringoid and $P$ be a family of sets $P_{i}$ ($i\in\mathbb{N}$). The free left $\mathcal{R}$-module generated by $P$, denoted by $\mathcal{R}\underline{P}$ is defined as follows. For each $i\in\mathbb{N}$, $(\mathcal{R}\underline{P})(i)$ is the abelian group of formal finite sums $\sum_{p_{j}\in P_{j},\ j\in\mathbb{N}}a_{p_{j}}\underline{p_{j}},\quad(a_{p_{j}}\in\mathcal{R}(j,i))$ and for each $r\in\mathcal{R}(i,k)$, $r\cdot\left(\sum_{p_{j}\in P_{j},\ j\in\mathbb{N}}a_{p_{j}}\underline{p_{j}}\right)=\sum_{p_{j}\in P_{j},\ j\in\mathbb{N}}(r\circ a_{p_{j}})\underline{p_{j}}.$ If a left $\mathcal{R}$-module $M$ is isomorphic to $\mathcal{R}\underline{P}$ for some $P$, we say that $M$ is free. For $\mathbb{Z}\langle\mathbb{K}\rangle$, we write $C\underline{x}t$ for elements of $((\mathbb{Z}\langle\mathbb{K}\rangle)\underline{P})(X)$ instead of $(C,t)\underline{x}$, and $(D+C)\underline{x}t$ for $D\underline{x}t+C\underline{x}t$. The tensor product of two modules over a ringoid is also defined. [9] Let $\mathcal{R}$ be a ringoid, $M_{1}$ be a right $\mathcal{R}$-module, and $M_{2}$ be a left $\mathcal{R}$-module. For a family of groups $\\{G_{X}\mid X\in P\\}$ for some indexing set $P$, its direct sum, denoted by $\bigoplus_{X\in P}G_{X}$, is the subset of the direct product defined by $\\{(g_{X})_{X\in P}\in\prod_{X\in P}G_{X}\mid\text{$g_{X}=0$ except for finite $X$s}\\}$. The direct sum of groups also forms a group. The tensor product $M_{1}\otimes_{\mathcal{R}}M_{2}$ is the quotient abelian group of $\bigoplus_{X\in\mathcal{R}}M_{1}(X)\otimes_{\mathcal{R}(X,X)}M_{2}(X)$ by relations $(x\cdot a)\otimes y-x\otimes(a\cdot y)$ for all $a\in\mathcal{R}(Y,X)$, $x\in M(X),y\in M(Y)$. We define an equivalence between two TRSs $(\Sigma,R)$, $(\Sigma^{\prime},R^{\prime})$, called _Tietze equivalence_. Two TRSs are _Tietze equivalent_ if one is obtained from the other by applying a series of _Tietze transformations_ defined as follows: 1. (1) If $f^{(n)}$ is a symbol not in $\Sigma$ and $t\in T(\Sigma)$ has variables in $\\{x_{1},\dots,x_{n}\\}$, then $(\Sigma,R)$ can be transformed into $(\Sigma\cup\\{f\\},R\cup\\{t\rightarrow f(x_{1},\dots,x_{n})\\})$. 2. (2) If $t\rightarrow f(x_{1},\dots,x_{n})\in R$, $t\in T(\Sigma\setminus\\{f\\})$, and $f$ does not occur in any rule in $R^{\prime}=R\setminus\\{t\rightarrow f(x_{1},\dots,x_{n})\\}$, then $(\Sigma,R)$ can be transformed into $(\Sigma\setminus\\{f\\},R^{\prime})$. 3. (3) If $t\xleftrightarrow{*}_{R}s$, then $(\Sigma,R)$ can be transformed into $(\Sigma,R\cup\\{t\rightarrow s\\})$. 4. (4) If $t\rightarrow s\in R$ and $t\xleftrightarrow{*}_{R^{\prime}}s$ for $R^{\prime}=R\setminus\\{t\rightarrow s\\}$, then $(\Sigma,R)$ can be transformed into $(\Sigma,R^{\prime})$. We can see that any two TRSs $(\Sigma,R_{1})$,$(\Sigma,R_{2})$ are Tietze equivalent if they are equivalent in the usual sense, $\xleftrightarrow{*}_{R_{1}}\,=\,\xleftrightarrow{*}_{R_{2}}$. Tietze equivalence is originally introduced in group theory [20, §11] and is also defined for monoids [2, 7.2]. Consider the signature $\Sigma=\\{+^{(2)},S^{(1)},0^{(0)}\\}$ and the set $R$ of four rules $0+x\rightarrow x,\ x+0\rightarrow x,\ S(x)+y\rightarrow S(x+y),\ (x+y)+z\rightarrow x+(y+z).$ We can see $(\Sigma,R)$ is Tietze equivalent to $(\Sigma^{\prime},R^{\prime})$ where $\Sigma^{\prime}=\\{+^{(2)},0^{(0)},1^{(0)}\\},\ R^{\prime}=\\{0+x\rightarrow x,\ x+0\rightarrow x,\ (x+y)+z\rightarrow x+(y+z)\\}$ as follows: $\displaystyle(\Sigma,R)$ $\displaystyle\xlongrightarrow{\rm(1)}(\Sigma\uplus\\{1^{(0)}\\},R\uplus\\{S(0)\rightarrow 1\\})$ $\displaystyle\xlongrightarrow{\rm(3)}(\Sigma\uplus\\{1^{(0)}\\},R\uplus\\{S(0)\rightarrow 1,\ 1+x\rightarrow S(x)\\})$ $\displaystyle\xlongrightarrow{\rm(4)}(\Sigma\uplus\\{1^{(0)}\\},R\uplus\\{1+x\rightarrow S(x)\\})$ $\displaystyle\xlongrightarrow{\rm(4)}(\Sigma\uplus\\{1^{(0)}\\},R\uplus\\{1+x\rightarrow S(x)\\}\setminus\\{S(x)+y\rightarrow S(x+y)\\})$ $\displaystyle\xlongrightarrow{\rm(2)}(\Sigma\uplus\\{1^{(0)}\\}\setminus\\{S^{(1)}\\},R\setminus\\{S(x)+y\rightarrow S(x+y)\\})=(\Sigma^{\prime},R^{\prime}).$ Now, we outline Malbos-Mimram’s construction of the homology groups of TRSs. Let $d=\deg(R)$. 1. (1) We begin by defining a new ringoid from $\mathbb{Z}\langle\mathbb{K}\rangle$. That ringoid, denoted by $\overline{\mathbb{Z}\langle\mathbb{K}\rangle}^{(\Sigma,R)}$, depends only on the Tietze equivalence class of $(\Sigma,R)$. $\overline{\mathbb{Z}\langle\mathbb{K}\rangle}^{(\Sigma,R)}$ corresponds to $\mathbb{Z}\langle\mathcal{M}_{(\Sigma,R)}\rangle$ in the case $(\Sigma,R)$ is an SRS. 2. (2) From this step, we write $\mathcal{R}$ for $\overline{\mathbb{Z}\langle\mathbb{K}\rangle}^{(\Sigma,R)}$. It can be shown that we have a partial free resolution $\mathcal{R}\underline{\mathbf{P}_{3}}\xrightarrow{\partial_{2}}\mathcal{R}\underline{\mathbf{P}_{2}}\xrightarrow{\partial_{1}}\mathcal{R}\underline{\mathbf{P}_{1}}\xrightarrow{\partial_{0}}\mathcal{R}\underline{\mathbf{P}_{0}}\xrightarrow{\epsilon}\mathcal{Z}$ (5) where every $\mathbf{P}_{i}$ is a family of sets $(\mathbf{P}_{i})_{j}$ given by $(\mathbf{P}_{0})_{1}=\\{1\\}$, $(\mathbf{P}_{0})_{j}=\emptyset$ ($j\neq 1$), $(\mathbf{P}_{1})_{j}=\Sigma^{(j)}=\\{f\in\Sigma\mid\text{$f$ is of arity $j$}\\}$, $(\mathbf{P}_{2})_{j}=\\{l\rightarrow r\in R\mid\text{$l$ is of arity $j$}\\}$, $(\mathbf{P}_{3})_{j}=\\{((\square,A,s),(C,B,t))\text{ : critical pair}\mid\text{one of $A,B$ is in $(\mathbf{P}_{2})_{j}$, and the other is in }(\mathbf{P}_{2})_{k}\\\ \text{ for }k\leq j\\}$. 3. (3) By taking the tensor product $\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}-$, we have the chain complex $\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{3}}\xrightarrow{\mathbb{Z}/d\mathbb{Z}\otimes\partial_{2}}\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{2}}\xrightarrow{\mathbb{Z}/d\mathbb{Z}\otimes\partial_{1}}\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{1}}\xrightarrow{\mathbb{Z}/d\mathbb{Z}\otimes\partial_{0}}\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{0}}$ (6) where $\mathbb{Z}/d\mathbb{Z}$ above is the $\mathcal{R}$-module defined by $\mathbb{Z}/d\mathbb{Z}(i)=\mathbb{Z}/d\mathbb{Z}$ (the abelian group of integers) for each object $i$, and the scalar multiplication is given by $(C,t)\cdot k=k$. 4. (4) The homology groups can be defined by $H_{i}(\Sigma,R)=\ker(\mathbb{Z}/d\mathbb{Z}\otimes\partial_{i-1})/\operatorname{im}(\mathbb{Z}/d\mathbb{Z}\otimes\partial_{i}).$ It is shown that the homology groups of TRS depend only on the Tietze equivalence class of $(\Sigma,R)$. Thus, we have the following: $\xleftrightarrow{*}_{R_{1}}\,=\,\xleftrightarrow{*}_{R_{2}}\implies H_{i}(\Sigma,R_{1})\cong H_{i}(\Sigma,R_{2}).$ For the step 1, we define the relations of $\overline{\mathbb{Z}\langle\mathbb{K}\rangle}^{(\Sigma,R)}$. We identify elements in $\mathbb{Z}\langle\mathbb{K}\rangle$ as follows. (a) For two $m$-tuples $t=\langle t_{1},\dotsc,t_{m}\rangle,s=\langle s_{1},\dotsc,s_{m}\rangle$ of terms, we identify $t$ and $s$ if $t\xleftrightarrow{*}_{R}s$. (b) Similarly, for two single-hole contexts $C,D$, we identify $C$ and $D$ if $C\xleftrightarrow{*}_{R}D$. For the last identification, we introduce operator $\kappa_{i}$ which takes a term $t$ and returns the formal sum of single-hole contexts $C_{1}+\dotsb+C_{m}$ where $C_{j}$ $(j=1,\dotsc,m)$ is obtained by replacing the $j$-th occurrence of $x_{i}$ with $\square$ in $t$, and $m$ is the number of the occurrences of $x_{i}$ in $t$. For example, we have $\displaystyle\kappa_{1}(f(g(x_{1},x_{2}),x_{1}))$ $\displaystyle=f(g(\square,x_{2}),x_{1})+f(g(x_{1},x_{2}),\square),$ $\displaystyle\kappa_{2}(f(g(x_{1},x_{2}),x_{1}))$ $\displaystyle=f(g(x_{1},\square),x_{1}),$ $\displaystyle\kappa_{2}(h(x_{1}))$ $\displaystyle=0.$ The definition of $\kappa_{i}$ can be stated inductively as follows: $\displaystyle\kappa_{i}(x_{i})$ $\displaystyle=\square,\ \kappa_{i}(x_{j})=0\quad(j\neq i),$ $\displaystyle\kappa_{i}(f(t_{1},\dotsc,t_{n}))$ $\displaystyle=\sum_{k=1}^{n}f(t_{1},\dotsc,t_{k-1},\kappa_{i}(t_{k}),t_{k+1},\dotsc,t_{n}).$ Then, (c) we identify formal sums of bicontexts $(C_{1},t)+\dots+(C_{k},t)$ and $(D_{1},t)+\dots+(D_{l},t)$ if $\kappa_{i}(u)=C_{1}+\dots+C_{k}$, $\kappa_{i}(v)=D_{1}+\dots+D_{l}$ for some positive integer $i$ and terms $u,v$ such that $u\xleftrightarrow{*}_{R}v$. $\overline{\mathbb{Z}\langle\mathbb{K}\rangle}^{(\Sigma,R)}$ is defined as the quotient of $\mathbb{Z}\langle\mathbb{K}\rangle$ by the equivalence class generated by the identifications (a), (b), and (c). We omit the definitions of the $\mathcal{R}$-linear maps $\epsilon,\partial_{i}$ ($i=0,1,2$) in the step 2, but we describe the group homomorphisms $\mathbb{Z}/d\mathbb{Z}\otimes\partial_{i}:\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{i+1}}\rightarrow\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{i}}$. Let $\tilde{\partial}_{i}$ denote $\mathbb{Z}/d\mathbb{Z}\otimes\partial_{i}$ for simplicity. For the step 2, we define the $\mathcal{R}$-linear maps $\epsilon,\partial_{i}$ ($i=0,1,2$). For $f^{(n)}\in\Sigma$, the homomorphism $\tilde{\partial}_{0}:\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{1}}\rightarrow\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{0}}$ is given by $\tilde{\partial}_{0}(\underline{f})=(n-1)\underline{1}.$ For a term $t$, we define $\varphi(t)$ as the linear combinaton of symbols $\sum_{f\in\Sigma}n_{f}\underline{f}$ where $n_{f}$ is the number of occurrences of $f$ in $t$. Using this, for $l\rightarrow r\in R$, the homomorphism $\tilde{\partial}_{1}:\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{2}}\rightarrow\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{1}}$ is given by $\tilde{\partial}_{1}(\underline{l\rightarrow r})=\varphi(r)-\varphi(l).$ For a critical pair $((\square,l\rightarrow r,s),(C,u\rightarrow v,t))$, let $(D_{i},l_{i}\rightarrow r_{i},s_{i})$, $(C_{j},u_{j}\rightarrow v_{j},t_{j})$ ($i=1,\dots,k,j=1,\dots,l$) be rewriting steps such that $r\circ s=D_{1}[l_{1}\circ s_{1}],D_{1}[r_{1}\circ s_{1}]=D_{2}[l_{2}\circ s_{2}],\dots,D_{k-1}[r_{k-1}\circ s_{k-1}]=D_{k}[l_{k}\circ s_{k}]$, $C[v\circ t]=C_{1}[u_{1}\circ t_{1}],C_{1}[v_{1}\circ t_{1}]=C_{2}[u_{2}\circ t_{2}],\dots,C_{l-1}[v_{l-1}\circ t_{l-1}]=C_{l}[u_{l}\circ t_{l}]$, $D_{k}[r_{k}\circ s_{k}]=C_{l}[v_{l}\circ t_{l}]$. Then the map $\tilde{\partial}_{2}((\square,l\rightarrow r,s),(C,u\rightarrow v,t))$ is defined by $\underline{u\rightarrow v}-\underline{l\rightarrow v}-\sum_{i=1}^{k}\underline{u_{i}\rightarrow v_{i}}-\sum_{j=1}^{l}\underline{l_{j}\rightarrow r_{j}}.$ Malbos-Mimram’s lower bound for the number of rewrite rules is given by $s(H_{2}(\Sigma,R))$. (Recall that $s(G)$ denotes the minimum number of generators of an abelian group $G$.) More precisely, $\\#\Sigma^{\prime}\geq s(H_{1}(\Sigma,R))$ and $\\#R^{\prime}\geq s(H_{2}(\Sigma,R))$ hold for any TRS $(\Sigma^{\prime},R^{\prime})$ that is Tietze equivalent to $(\Sigma,R)$. These inequalities are shown in a similar way to the proof of Theorem 9. ## 5\. Proof of Main Theorem Let $(\Sigma,R)$ be a complete TRS. We first simplify the tensor product $\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{i}}$ in (6). ###### Lemma 11. Let $d=\deg(R)$ and $P$ be a family of sets $P_{0},P_{1},\dots$. Then, we have $\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{P}\cong(\mathbb{Z}/d\mathbb{Z})\underline{\biguplus_{i}P_{i}}$. Especially, if $d=0$, $\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{P}\cong\mathbb{Z}\underline{\biguplus_{i}P_{i}}$. ###### Proof 5.1. We define a group homomorphism $f:\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{P}\rightarrow(\mathbb{Z}/d\mathbb{Z})\underline{\biguplus_{i}P_{i}}$ by $f((w_{n})_{n\geq 0})=\sum_{n\geq 0}f_{n}(w_{n})$ where $f_{n}:\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}(n,n)}\mathcal{R}\underline{P}(n)\rightarrow(\mathbb{Z}/d\mathbb{Z})\underline{P_{n}}$ is defined by $f_{n}([k]\otimes C\underline{a}t)=[k]\underline{a}$ for $a\in P_{n}$. Since each $f_{n}$ is an isomorphism, $f$ is also an isomorphism. As special cases of this lemma, we have $\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{0}}\cong(\mathbb{Z}/d\mathbb{Z})\underline{\Sigma}$, $\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{1}}\cong(\mathbb{Z}/d\mathbb{Z})\underline{R}$, and $\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{2}}\cong(\mathbb{Z}/d\mathbb{Z})\underline{\operatorname{CP}(R)}$. Additionally, we can see each group homomorphism $\tilde{\partial}_{i}$ ($i=0,1,2$) is a $\mathbb{Z}/d\mathbb{Z}$-linear map. To prove Theorem 1, we show the following lemma. ###### Lemma 12. Let $d=\deg(R)$. If $d=0$ or $d$ is prime, $\\#R-e(R)=s(H_{2}(\Sigma,R))+s(\operatorname{im}\tilde{\partial}_{1})$. (Recall that $s(G)$ is the minimum number of generators of a group $G$.) ###### Proof 5.2. By definition, $D(R)$ defined in Section 2 is a matrix representation of $\tilde{\partial}_{2}$. Suppose $d$ is prime. In this case, $s(H_{2}(\Sigma,R))$ is equal to the dimension of $H_{2}(\Sigma,R)$ as a $\mathbb{Z}/d\mathbb{Z}$-vector space. By the rank-nullity theorem, we have $\displaystyle\dim(H_{2}(\Sigma,R))$ $\displaystyle=\dim(\ker\tilde{\partial}_{1})-\dim(\operatorname{im}\tilde{\partial}_{2})$ $\displaystyle=\dim(\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{1}})-\dim(\operatorname{im}\tilde{\partial}_{1})-\dim(\operatorname{im}\tilde{\partial}_{2})$ $\displaystyle=\dim((\mathbb{Z}/d\mathbb{Z})\underline{R})-\dim(\operatorname{im}\tilde{\partial}_{1})-\operatorname{rank}(D(R))$ $\displaystyle=\\#R-\dim(\operatorname{im}\tilde{\partial}_{1})-e(R).$ Suppose $d=0$. We show $H_{2}(\Sigma,R)\cong\mathbb{Z}^{\\#R-r-k}\times\mathbb{Z}/e_{1}\mathbb{Z}\times\dotsb\times\mathbb{Z}/e_{r}\mathbb{Z}$ where $r=\operatorname{rank}(D(R))$, $k=s(\operatorname{im}\tilde{\partial}_{1})$, and $e_{1},\dotsc,e_{r}$ are the elementary divisors of $D(R)$. Let $\overline{\partial}_{1}:\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{1}}/\operatorname{im}\tilde{\partial}_{2}\rightarrow\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{0}}$ be the group homomorphism defined by $[x]\mapsto\tilde{\partial}_{1}(x)$. $\overline{\partial}_{1}$ is well-defined since $\operatorname{im}\tilde{\partial}_{2}\subset\ker\tilde{\partial}_{1}$, and $\ker\overline{\partial}_{1}$ is isomorphic to $\ker\tilde{\partial}_{1}/\operatorname{im}\tilde{\partial}_{2}=H_{2}(\Sigma,R)$. By taking the basis $v_{1},\dotsc,v_{\\#R}$ of $\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{1}}\cong\mathbb{Z}\underline{R}$ such that $D(R)$ is the matrix representation of $\tilde{\partial}_{2}$ under the basis $v_{1},\dotsc,v_{\\#R}$ and some basis of $\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{2}}$, we can see $\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{1}}/\operatorname{im}\tilde{\partial}_{2}\cong\mathbb{Z}^{\\#R-r}\times\mathbb{Z}/e_{1}\mathbb{Z}\times\dotsb\times\mathbb{Z}/e_{k}\mathbb{Z}$. Suppose $\overline{\partial}_{1}(e_{i}[x])=0$ for some $x$ and $i=1,\dotsc,r$. Since $\overline{\partial}_{1}$ is a homomorphism, $\overline{\partial}_{1}(e_{i}[x])=e_{i}\overline{\partial}_{1}([x])\in\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{0}}\cong\mathbb{Z}\underline{\Sigma}$ holds. Since $\mathbb{Z}\underline{\Sigma}$ is free, we have $[x]=0$. Therefore, $\ker\overline{\partial}_{1}$ is included in the subset of $\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{1}}/\operatorname{im}\tilde{\partial}_{2}$ isomorphic to $\mathbb{Z}^{\\#R-r}\times\\{0\\}\times\dotsb\times\\{0\\}$. Thus, $\ker\overline{\partial}_{1}\cong\mathbb{Z}^{\\#R-r-k}\times\mathbb{Z}/e_{1}\mathbb{Z}\times\dotsb\times\mathbb{Z}/e_{r}\mathbb{Z}$. Since $\mathbb{Z}/e\mathbb{Z}\cong 0$ if $e$ is invertible, $\mathbb{Z}^{\\#R-r-k}\times\mathbb{Z}/e_{1}\mathbb{Z}\times\dotsb\times\mathbb{Z}/e_{k}\mathbb{Z}\cong\mathbb{Z}^{\\#R-r-k}\times\mathbb{Z}/e_{e(R)+1}\mathbb{Z}\times\dotsb\mathbb{Z}/e_{r}\mathbb{Z}=:G$. The group $G$ is generated by $(\underbrace{1,0,\dotsc,0}_{\\#R-r-k},\underbrace{[0],\dotsc,[0]}_{r-e(R)})$, $\dotsc$, $(0,\dotsc,0,1,[0],\dotsc,[0])$, $\dotsc$, $(0,\dotsc,0,[1],[0],\dotsc,[0])$, $\dotsc$, $(0,\dotsc,0,[0],\dotsc,[0],[1])$, so we have $s(G)\leq\\#R-r-k+r-e(R)=\\#R-k-e(R)$. Let $p$ be a prime number which divides $e_{e(R)+1}$. We can see $G/pG\cong(\mathbb{Z}/p\mathbb{Z})^{\\#R-k-e(R)}$. It is not hard to see $s(G)\geq s(G/pG)$, and since $G/pG$ is a $\mathbb{Z}/p\mathbb{Z}$-vector space, $s(G/pG)=\dim(G/pG)=\\#R-k-e(R)$. Thus, $s(H_{2}(\Sigma,R))=s(G)=\\#R-s(\operatorname{im}\tilde{\partial}_{1})-e(R)$. By Lemma 12, Theorem 1 is implied by the following theorem: ###### Theorem 13. Let $(\Sigma,R)$ be a TRS and $d=\deg(R)$. If $d=0$ or $d$ is prime, $\\#R\geq s(H_{2}(\Sigma,R))+s(\operatorname{im}\tilde{\partial}_{1}).$ (7) ###### Proof 5.3. By the first isomorphism theorem, we have an isomorphism between $\mathbb{Z}/d\mathbb{Z}$-modules $\operatorname{im}\tilde{\partial}_{1}\simeq\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbb{P}_{2}}\delimiter 84079374\mathopen{}\ker\tilde{\partial}_{1}$ and by the third isomorphism theorem, the right hand side is isomorphic to $\displaystyle\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbb{P}_{2}}\delimiter 84079374\mathopen{}\ker\tilde{\partial}_{1}$ $\displaystyle\simeq\left(\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbb{P}_{2}}\delimiter 84079374\mathopen{}\operatorname{im}\tilde{\partial}_{2}\right)\delimiter 84079374\mathopen{}\left(\ker\tilde{\partial}_{1}\delimiter 84079374\mathopen{}\operatorname{im}\tilde{\partial}_{2}\right)$ $\displaystyle\simeq\left(\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbb{P}_{2}}\delimiter 84079374\mathopen{}\operatorname{im}\tilde{\partial}_{2}\right)\delimiter 84079374\mathopen{}H_{2}(\Sigma,R).$ Thus, we obtain the following exact sequence by Proposition 7: $0\rightarrow H_{2}(\Sigma,R)\rightarrow\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbb{P}_{2}}\delimiter 84079374\mathopen{}\operatorname{im}\tilde{\partial}_{2}\rightarrow\operatorname{im}\tilde{\partial}_{1}\rightarrow 0.$ By Theorem 5, since $\operatorname{im}\tilde{\partial}_{1}\subset\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbb{P}_{1}}\cong(\mathbb{Z}/d\mathbb{Z})\underline{R}$ and $(\mathbb{Z}/d\mathbb{Z})\underline{R}$ is a free $\mathbb{Z}/d\mathbb{Z}$-module, $\operatorname{im}\tilde{\partial}_{1}$ is also free and by Proposition 8, we have $\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbb{P}_{2}}\delimiter 84079374\mathopen{}\operatorname{im}\tilde{\partial}_{2}\cong H_{2}(\Sigma,R)\times\operatorname{im}\tilde{\partial}_{1}$. Therefore, $s(\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbb{P}_{2}}\delimiter 84079374\mathopen{}\operatorname{im}\tilde{\partial}_{2})=s(H_{2}(\Sigma,R))+s(\operatorname{im}\tilde{\partial}_{1})$. Since $\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbb{P}_{2}}\delimiter 84079374\mathopen{}\operatorname{im}\tilde{\partial}_{2}$ is generated by $[l_{1}\rightarrow r_{1}],\dots,[l_{k}\rightarrow r_{k}]$ if $R=\\{l_{1}\rightarrow r_{1},\dots,l_{k}\rightarrow r_{k}\\}$, we obtain $k=\\#R\geq s(\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbb{P}_{2}}\delimiter 84079374\mathopen{}\operatorname{im}\tilde{\partial}_{2})=s(H_{2}(\Sigma,R))+s(\operatorname{im}\tilde{\partial}_{1}).$ Thus, we get (7). Now, we prove our main theorem, Theorem 1. ###### Proof 5.4 (Proof of Theorem 1). As we stated, $H_{2}(\Sigma,R)$ depends only on the Tietze equivalence class of $(\Sigma,R)$ and particularly, $H_{2}(\Sigma,R^{\prime})$ is isomorphic to $H_{2}(\Sigma,R)$ if $R^{\prime}$ is equivalent to $R$ (in the sense ${\xleftrightarrow{*}_{R}}={\xleftrightarrow{*}_{R^{\prime}}}$). Let us show $s(\operatorname{im}\tilde{\partial}_{1})$ depends only on the equivalence class of $R$. For a left $\mathfrak{R}$-module $M$, $\operatorname{rank}(M)$ denotes the cardinality of a minimal linearly independent generating set of $M$, that is, a minimal generating set $S$ of $G$ such that any element $s_{1},\dotsc,s_{k}\in\Gamma$, and $r_{1}s_{1}+\dotsb+r_{k}s_{k}=0\implies r_{1}=\dotsb=r_{k}=0$ for any $r_{1},\dotsc,r_{k}\in\mathfrak{R}$, $s_{1},\dotsc,s_{k}\in S$. It can be shown that $\operatorname{rank}(M)=s(M)$ if $M$ is free. Especially, $s(\operatorname{im}\tilde{\partial}_{1})=\operatorname{rank}(\operatorname{im}\tilde{\partial}_{1})$ since $\operatorname{im}\tilde{\partial}_{1}\subset\mathbb{Z}\underline{R}$ if $\deg(R)=0$. Also, $\operatorname{rank}(\operatorname{im}\tilde{\partial}_{1})=\operatorname{rank}(\ker\tilde{\partial}_{0})-\operatorname{rank}(\ker\tilde{\partial}_{0}/\operatorname{im}\tilde{\partial}_{1})$ is obtained by a general theorem [14, Ch 10, Lemma 10.1]. By definition, $\tilde{\partial}_{0}$ does not depend on $R$. Since $\ker\tilde{\partial}_{0}/\operatorname{im}\tilde{\partial}_{1}=H_{1}(\Sigma,R)$ depends only on the Tietze quivalence class of $R$, two sets of rules $R,R^{\prime}$ with ${\xleftrightarrow{*}_{R}}={\xleftrightarrow{*}_{R^{\prime}}}$ give the same $\operatorname{rank}(\operatorname{im}\tilde{\partial}_{1})$. In conclusion, for any TRS $R^{\prime}$ equivalent to $R$, we obtain $\\#R^{\prime}\geq s(H_{2}(\Sigma,R))+s(\operatorname{im}\tilde{\partial}_{1})=\\#R-e(R)$. ## 6\. Prime Critical Pairs in a Homological Perspective Let $(\Sigma,R)$ be a complete TRS. It is known that in confluence tests (and then in Knuth-Bendix completion), it suffices to consider only prime critical pairs [6]. (A critical pair $r_{1}\sigma\leftarrow l_{1}\sigma=C[l_{2}\sigma]\rightarrow C[r_{2}\sigma])$ is prime if no proper subterm of $l_{2}\sigma$ is reducible by $R$.) We have defined the matrix $D(R)$ using the critical pairs of $R$, but in fact, we can restrict the critical pairs to the prime ones and obtain the same $e(R)$. In other words, we have the following theorem. ###### Theorem 14. In the matrix $D(R)$, all columns corresponding to critical pairs that are not prime can be transformed into the zero vectors by elementary column operations. ###### Proof 6.1. For any terms $t,s$ and position $p\in\operatorname{Pos}(t)$, we write $t[s]_{p}$ for the term obtained from $t$ by replacing the subterm at $p$ with $s$. Suppose that $R=\\{l_{1}\rightarrow r_{1},\dots,l_{n}\rightarrow r_{n}\\}$ and that there is a non-prime critical pair $(r_{i}\sigma,l_{i}\sigma[r_{j}\sigma]_{p})$ for some position $p\in\operatorname{Pos}(l_{i})$. Then, there is a rule $l_{k}\rightarrow r_{k}$ such that $l_{k}$ matches $(l_{j}\sigma)|_{p}$ for some position $p^{\prime}\in\operatorname{Pos}(l_{j}\sigma)$. So, we have the following three paths. ${l_{i}\sigma}$${{l_{i}\sigma[l_{j}\sigma]_{p}}}$${{l_{i}\sigma[l_{j}\sigma[l_{k}\sigma]_{p^{\prime}}]_{p}}}$${r_{i}\sigma}$${{l_{i}\sigma[r_{j}\sigma]_{p}}}$${{l_{i}\sigma[l_{j}\sigma[r_{k}\sigma]_{p^{\prime}}]_{p}}}$${\vdots}$${\vdots}$${\vdots}$${t}$ (8) We show that the column for $(r_{i}\sigma,l_{i}\sigma[r_{j}\sigma]_{p})$ in $D(R)$ can be transformed into the zero vector by elementary column operations. Let $\operatorname{Pos}_{\mathcal{F}}(s)$ be the set $\operatorname{Pos}(s)\setminus\\{\text{variable positions in $s$}\\}$ for a term $s$. Consider the following four cases: (1) $p^{\prime}\in\operatorname{Pos}_{\mathcal{F}}(l_{j})$ and $pp^{\prime}\in\operatorname{Pos}_{\mathcal{F}}(l_{i})$, (2) $p^{\prime}\in\operatorname{Pos}_{\mathcal{F}}(l_{j})$ and $pp^{\prime}\notin\operatorname{Pos}_{\mathcal{F}}(l_{i})$, (3) $p^{\prime}\notin\operatorname{Pos}_{\mathcal{F}}(l_{j})$ and $pp^{\prime}\in\operatorname{Pos}_{\mathcal{F}}(l_{i})$, (4) $p^{\prime}\notin\operatorname{Pos}_{\mathcal{F}}(l_{j})$ and $pp^{\prime}\notin\operatorname{Pos}_{\mathcal{F}}(l_{i})$. 1. (1) Case where $p^{\prime}\in\operatorname{Pos}_{\mathcal{F}}(l_{j})$ and $pp^{\prime}\in\operatorname{Pos}_{\mathcal{F}}(l_{i})$: In this case, for some substitutions $\sigma^{\prime},\sigma^{\prime\prime}$ more general than $\sigma$, the pairs $P_{j,k}=(r_{j}\sigma^{\prime},l_{j}\sigma^{\prime}[r_{k}\sigma^{\prime}]_{p^{\prime}})$ and $P_{i,k}=(r_{i}\sigma^{\prime\prime},l_{i}\sigma^{\prime\prime}[l_{j}\sigma^{\prime\prime}[r_{k}\sigma^{\prime\prime}]_{p^{\prime}}]_{p})$ are critical. Let $P_{i,j}=(r_{i}\sigma,l_{i}\sigma[r_{j}\sigma]_{p})$. Also, we write $v_{i,m},v_{j,m},v_{k,m}$ for the numbers of $l_{m}\rightarrow r_{m}$ appears in $l_{i}\sigma\rightarrow r_{i}\sigma\rightarrow\dots\rightarrow t$, $l_{i}\sigma[l_{j}\sigma]_{p}\rightarrow l_{i}\sigma[r_{j}\sigma]_{p}\rightarrow\dots\rightarrow t$, $l_{i}\sigma[l_{j}\sigma[l_{k}\sigma]_{p^{\prime}}]_{p}\rightarrow l_{i}\sigma[l_{j}\sigma[r_{k}\sigma]_{p^{\prime}}]_{p}\rightarrow\dots\rightarrow t$, respectively. Then, the column of the matrix $D(R)$ for the critical pair $P_{\alpha,\beta}$ ($\alpha,\beta\in\\{i,j,k\\}$) can be given by $V_{\alpha,\beta}:=(v_{\alpha,1}-v_{\beta,1}~{}v_{\alpha,2}-v_{\beta,2}~{}\dots~{}v_{\alpha,n}-v_{\beta,n})^{T}$. So, we have $V_{i,j}=V_{i,k}-V_{j,k}$ and this means that the column for $P_{i,j}$ can be transformed into the zero vector by an elementary column operation. 2. (2) Case where $p^{\prime}\in\operatorname{Pos}_{\mathcal{F}}(l_{j})$ and $pp^{\prime}\notin\operatorname{Pos}_{\mathcal{F}}(l_{i})$: In this case, $(r_{j}\sigma^{\prime},l_{j}\sigma^{\prime}[r_{k}\sigma^{\prime}]_{p^{\prime}})$ is a critical pair for some $\sigma^{\prime}$ and $l_{k}\rightarrow r_{k}$ can rewrite $x_{a}\sigma$ for some variable $x_{a}$ in $l_{i}$ into some term $s$. Then, we have the two paths: ${{l_{i}\sigma[l_{j}\sigma[l_{k}\sigma]_{p^{\prime}}]_{p}}}$${{l_{i}\sigma[l_{j}\sigma[r_{k}\sigma]_{p^{\prime}}]_{p}}}$${\dots}$${l_{i}\sigma^{\prime\prime}}$${l_{i}\sigma}$${r_{i}\sigma}$${\dots}$${r_{i}\sigma^{\prime\prime}}$$\scriptstyle{l_{k}\rightarrow r_{k}}$$\scriptstyle{\\#_{a}l_{i}\text{ times}}$$\scriptstyle{l_{k}\rightarrow r_{k}}$$\scriptstyle{l_{k}\rightarrow r_{k}}$$\scriptstyle{l_{i}\rightarrow r_{i}}$$\scriptstyle{l_{i}\rightarrow r_{i}}$$\scriptstyle{l_{k}\rightarrow r_{k}}$$\scriptstyle{\\#_{a}r_{i}\text{ times}}$$\scriptstyle{l_{k}\rightarrow r_{k}}$ where $\sigma^{\prime\prime}$ is the substitution that is the same as $\sigma$ but $x_{a}\sigma^{\prime\prime}=s$. Then, $v_{i,m}-v_{k,m}=0$ for any $m\neq k$ and $v_{i,k}-v_{k,k}=\\#_{a}l_{i}-\\#_{a}r_{i}=0\mod\deg(R)$. Therefore, we have $V_{i,j}=-V_{j,k}\mod\deg(R)$. 3. (3) Case where $p^{\prime}\notin\operatorname{Pos}_{\mathcal{F}}(l_{j})$ and $pp^{\prime}\in\operatorname{Pos}_{\mathcal{F}}(l_{i})$: We can show $V_{i,j}=V_{i,k}$ in a way similar to (2). 4. (4) Case where $p^{\prime}\notin\operatorname{Pos}_{\mathcal{F}}(l_{j})$ and $pp^{\prime}\notin\operatorname{Pos}_{\mathcal{F}}(l_{i})$: Also in a similar way, we can see that $V_{i},j$ is the zero vector. Thus, we can remove the column for $P_{i,j}$ in $D(R)$. Repeating this proccess for all non-prime pairs, we obtain the desired result. For case (1) in the proof, we can actually remove the column for $P_{i,k}$ or $P_{j,k}$ instead of $P_{i,j}$ by symmetry. We shall call a triple of critical pairs like (8) a _critical triple_. Recall that we mentioned $D(R)$ is a matrix presentation of $\tilde{\partial}_{2}=\mathbb{Z}/d\mathbb{Z}\otimes\partial_{2}:\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{3}}\rightarrow\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{2}}$ in Section 5. If we define $\mathbf{P}_{4}$ to be a collection of critical triples and $\tilde{\partial}_{3}:\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{4}}\rightarrow\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{3}}$ to be $\tilde{\partial}_{3}(\underline{(P_{i,j},P_{i,k},P_{j,k})})=\underline{P_{i,j}}-\underline{P_{i,k}}+\underline{P_{j,k}}$, then we have $\tilde{\partial}_{2}\circ\tilde{\partial}_{3}(\underline{(P_{i,j},P_{i,k},P_{j,k})})=\tilde{\partial}_{2}(\underline{P_{i,j}}-\underline{P_{i,k}}+\underline{P_{j,k}})=0$. (This corresponds to $V_{i,k}=V_{i,k}-V_{j,k}$.) Therefore $\ker\tilde{\partial}_{2}\supset\operatorname{im}\tilde{\partial}_{3}$ holds, so we can extend our chain complex (6): $\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{4}}\xrightarrow{\tilde{\partial}_{3}}\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{3}}\xrightarrow{\tilde{\partial}_{2}}\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{2}}\xrightarrow{\tilde{\partial}_{1}}\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{1}}\xrightarrow{\tilde{\partial}_{0}}\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{0}}.$ Note that since we have not defined $\partial_{3}:\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{3}}\to\mathbb{Z}/d\mathbb{Z}\otimes_{\mathcal{R}}\mathcal{R}\underline{\mathbf{P}_{2}}$, the third homology $H_{3}$ is meaningless unless we define $\partial_{3}$ extending resolution (5) and show $\tilde{\partial}_{3}=\mathbb{Z}_{d}\otimes\partial_{3}$. However, this suggests that the next term of our partial resolution could be generated by critical triples. ## 7\. Deficiency and Computability We consider the case where every symbol in $\Sigma$ is of arity 1. Notice that any TRS $(\Sigma,R)$ can be seen as an SRS and $\deg(R)=0$ in this case. We have $\operatorname{rank}(\ker\tilde{\partial}_{0})=\\#\Sigma$ since $\tilde{\partial}_{0}(\underline{f})=0$ for any $f\in\Sigma$. Therefore, (7) can be rewritten to $\\#R-\\#\Sigma\geq s(H_{2}(\Sigma,R))-\operatorname{rank}(H_{1}(\Sigma,R)).$ (9) So, for SRSs, we have a lower bound of the difference between the number of rewrite rules and the number of symbols. For groups, in fact, this inequality is proved in terms of group homology [3] without using the notion of a rewriting system. In group theory, a group presentation $(\Sigma,R)$ is called _efficient_ if the equality of (9) holds and a group is called efficient if it has an efficient presentation. It is known that inefficient groups exist [18]. Let us move back to the case of general TRSs. We have already seen that there exists a TRS such that none of its equivalent TRS satisfies the equality of (7) in the last paragraph of Section 2. The _deficiency_ of (the equivalence class of) a TRS $(\Sigma,R)$, denoted by $\operatorname{def}\langle\Sigma,R\rangle$, is the minimum of $\\#R^{\prime}-\\#\Sigma^{\prime}$ over all TRSs $(\Sigma^{\prime},R^{\prime})$ Tietze equivalent to $(\Sigma,R)$. We pose the problem to decide inequalities of the deficiency for TRSs and see its undecidability is shown by using powerful facts from group theory. ###### Problem 15. Given an integer and a TRS $(\Sigma,R)$, does $\operatorname{def}\langle\Sigma,R\rangle\leq n$ hold? We will prove that Problem 15 is undecidable. It suffices to restrict the problems to the case where $n$ is negative and $(\Sigma,R)$ presents a group of finite index, that is, $(\Sigma,R)$ is an SRS and $\mathcal{M}_{(\Sigma,R)}=\Sigma^{*}/{\xleftrightarrow{*}_{R}}$ forms a group of finite index. ###### Problem 16. Given a negative integer $n$ and an SRS $(\Sigma,R)$ whose corresponding monoid $\mathcal{M}_{(\Sigma,R)}$ forms a group of finite index, does $\operatorname{def}\langle\Sigma,R\rangle\leq n$ hold? ###### Theorem 17. Problem 16 is undecidable, and then so is Problem 15 To prove the theorem, we will apply one of the most useful tools on computability in group theory called Adian-Rabin theorem which states every “Markov property” is undecidable. Let $P$ be a property of finitely presented groups which is preserved under group isomorphism. The property $P$ is said to be a _Markov property_ if 1. (1) there exists a finitely presented group $G_{+}$ with $P$, and 2. (2) there exists a finitely presented group $G_{-}$ such that there is no injective group homomorphism from $G_{-}$ to a finitely presented group $G$ with $P$. The condition (2) is equivalent to the case where there exists a finitely presented group $G_{-}$ which does not have $P$ and whenever a finitely generated group $G$ has $P$, all subgroups of $G$ also have $P$. ###### Theorem 18. [1][12][8, Theorem 4.1] Markov properties are undecidable. ###### Proof 7.1 (Proof of Theorem 17). Let $n$ be a negative integer. If a group $G$ is presented by $(\Sigma,R)$, we write $\operatorname{def}G$ for $\operatorname{def}\langle\Sigma,R\rangle$. We show that $\operatorname{def}G\leq n$ is a Markov property. Since the free group $F_{k}$ satisfies $\operatorname{def}F_{k}=0-k=-k$ for any $k\geq 0$, there always exist $G_{+}$ with $\operatorname{def}G_{+}\leq n$ and $G_{-}$ with $G_{-}>n$. Therefore it is enough to show that for any finitely presented group $G$ of finite index and a subgroup $H$ of $G$, $\operatorname{def}G\leq n$ implies $\operatorname{def}H\leq n$. Let $G$ be a finitely presented group with $\operatorname{def}G\leq n$ and $H$ be a subgroup of $G$. Given a finite presentation $(\Sigma,R)$ of $G$, it is known that we can construct a presentation $(\Sigma^{\prime},R^{\prime})$ of $H$ satisfying $\\#R^{\prime}-\\#\Sigma^{\prime}+1=[G:H](\\#R-\\#\Sigma+1)$ where $[G:H]$ is the index of $H$ in $G$. (See [8, Ch. II. Proposition 4.1], for example.) The way of construction is known as _Reidemeister-Schreier method_. Thus, we have $\operatorname{def}H+1\leq[G:H](\operatorname{def}G+1)\leq\operatorname{def}G+1\leq n+1.$ ## 8\. Conclusions We have seen that the number of rewrite rules is bounded below by a computable number defined using homology groups of TRSs. The computation is by simple term rewriting and matrix transformation. 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In Contributions to Mathematical Logic, volume 50 of Studies in Logic and the Foundations of Mathematics, pages 275 – 288. Elsevier, 1968. * [20] Heinrich Tietze. Über die topologischen invarianten mehrdimensionaler mannigfaltigkeiten. Monatshefte für Mathematik und Physik, 19(1):1–118, Dec 1908\. * [21] V. A. Ufnarovskij. Combinatorial and asymptotic methods in algebra. 1995\. * [22] Ian Wehrman and Aaron Stump. Mining propositional simplification proofs for small validating clauses. Electronic Notes in Theoretical Computer Science, 144(2):79 – 91, 2006. Proceedings of the Third Workshop on Pragmatics of Decision Procedures in Automated Reasoning (PDPAR 2005). ## Appendix A The matrix $D(R)$ for The Theory of Groups For the TRS $R$ defined in Example 2, $D(R)$ is given by the transpose of $\left(\begin{array}[]{cccccccccc}1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&1&1\\\ 0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0&1\\\ 1&1&0&0&1&1&0&0&0&0\\\ 1&1&0&1&0&0&0&0&1&0\\\ 1&0&0&0&0&0&0&0&1&1\\\ 1&1&1&0&0&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0&0&0\\\ 0&1&1&0&0&0&1&0&0&1\\\ 0&0&0&0&0&0&1&0&1&0\\\ 0&0&0&0&0&1&1&0&0&0\\\ 0&1&0&1&0&0&1&0&0&0\\\ 0&1&1&0&0&0&0&0&0&0\\\ 0&1&1&0&0&0&1&0&0&1\\\ 0&0&1&1&0&0&0&0&1&0\\\ 0&0&1&0&1&1&0&0&0&0\\\ 0&0&1&0&1&0&1&0&0&0\\\ 1&0&0&0&0&0&0&0&1&1\\\ 0&0&0&1&1&0&1&0&0&1\\\ 0&0&1&1&0&0&0&1&1&0\\\ 0&0&0&1&1&0&0&1&0&0\\\ 0&1&0&1&0&0&1&0&0&0\\\ 0&0&1&1&0&1&0&0&0&0\\\ 1&0&0&0&0&1&0&0&0&1\\\ 0&0&0&1&1&0&1&0&0&1\\\ 0&0&1&0&1&0&0&0&1&0\\\ 0&0&0&1&1&0&0&1&0&0\\\ 0&1&0&0&1&0&1&0&0&0\\\ 0&0&1&0&1&1&0&1&0&0\\\ 0&0&0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&1&0&0&0&1\\\ 1&0&1&0&1&1&0&1&0&0\\\ 0&0&0&0&0&1&0&0&1&0\\\ 0&0&0&0&0&1&0&0&1&0\\\ 0&0&0&0&0&1&0&1&1&0\\\ 0&0&0&0&0&1&1&0&0&0\\\ 0&0&0&0&0&0&1&0&1&0\\\ 0&0&0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&1&0&1&1&0\\\ 0&0&0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&0&1&1\\\ 1&0&1&0&1&0&0&0&1&0\\\ \end{array}\right)$ where the $i$-th column corresponds to the rule $G_{i}$, and the $j$-th row corresponds to the critical pair $C_{j}$ shown in the next two pages. $\displaystyle C_{1}:\ $ $\displaystyle m(m(x_{1},x_{2}),x_{3})\rightarrow m(x_{1},m(x_{2},x_{3})),\quad m(m(x_{4},x_{5}),x_{6})\rightarrow m(x_{4},m(x_{5},x_{6})),\quad m(\square,x_{3}),$ $\displaystyle\\{x_{6}\mapsto x_{2},x_{1}\mapsto m(x_{4},x_{5})\\}$ $\displaystyle C_{2}:\ $ $\displaystyle i(m(x_{1},x_{2}))\rightarrow m(i(x_{2}),i(x_{1})),\quad m(m(x_{3},x_{4}),x_{5})\rightarrow m(x_{3},m(x_{4},x_{5})),\quad i(\square),$ $\displaystyle\\{x_{5}\mapsto x_{2},x_{1}\mapsto m(x_{3},x_{4})\\}$ $\displaystyle C_{3}:\ $ $\displaystyle m(m(x_{1},x_{2}),x_{3})\rightarrow m(x_{1},m(x_{2},x_{3})),\quad m(x_{4},m(i(x_{4}),x_{5}))\rightarrow x_{5},\quad m(\square,x_{3}),$ $\displaystyle\\{x_{2}\mapsto m(i(x_{1}),x_{5}),x_{4}\mapsto x_{1}\\}$ $\displaystyle C_{4}:\ $ $\displaystyle m(x_{1},m(i(x_{1}),x_{2}))\rightarrow x_{2},\quad m(m(x_{3},x_{4}),x_{5})\rightarrow m(x_{3},m(x_{4},x_{5})),\quad\square,$ $\displaystyle\\{x_{5}\mapsto m(i(m(x_{3},x_{4})),x_{2}),x_{1}\mapsto m(x_{3},x_{4})\\}$ $\displaystyle C_{5}:\ $ $\displaystyle m(m(x_{1},x_{2}),x_{3})\rightarrow m(x_{1},m(x_{2},x_{3})),\quad m(i(x_{4}),m(x_{4},x_{5}))\rightarrow x_{5},\quad m(\square,x_{3}),$ $\displaystyle\\{x_{2}\mapsto m(x_{4},x_{5}),x_{1}\mapsto i(x_{4})\\}$ $\displaystyle C_{6}:\ $ $\displaystyle m(i(x_{1}),m(x_{1},x_{2}))\rightarrow x_{2},\quad m(m(x_{3},x_{4}),x_{5})\rightarrow m(x_{3},m(x_{4},x_{5})),\quad m(i(x_{1}),\square),$ $\displaystyle\\{x_{5}\mapsto x_{2},x_{1}\mapsto m(x_{3},x_{4})\\}$ $\displaystyle C_{7}:\ $ $\displaystyle m(m(x_{1},x_{2}),x_{3})\rightarrow m(x_{1},m(x_{2},x_{3})),\quad m(i(x_{4}),x_{4})\rightarrow e,\quad m(\square,x_{3}),$ $\displaystyle\\{x_{4}\mapsto x_{2},x_{1}\mapsto i(x_{2})\\}$ $\displaystyle C_{8}:\ $ $\displaystyle m(m(x_{1},x_{2}),x_{3})\rightarrow m(x_{1},m(x_{2},x_{3})),\quad m(x_{4},i(x_{4}))\rightarrow e,\quad m(\square,x_{3}),$ $\displaystyle\\{x_{2}\mapsto i(x_{1}),x_{4}\mapsto x_{1}\\}$ $\displaystyle C_{9}:\ $ $\displaystyle m(x_{1},i(x_{1}))\rightarrow e,\quad m(m(x_{2},x_{3}),x_{4})\rightarrow m(x_{2},m(x_{3},x_{4})),\quad\square,$ $\displaystyle\\{x_{4}\mapsto i(m(x_{2},x_{3})),x_{1}\mapsto m(x_{2},x_{3})\\}$ $\displaystyle C_{10}:\ $ $\displaystyle m(m(x_{1},x_{2}),x_{3})\rightarrow m(x_{1},m(x_{2},x_{3})),\quad m(x_{4},e)\rightarrow x_{4},\quad m(\square,x_{3}),\quad\\{x_{2}\mapsto e,x_{4}\mapsto x_{1}\\}$ $\displaystyle C_{11}:\ $ $\displaystyle m(x_{1},e)\rightarrow x_{1},\quad m(m(x_{2},x_{3}),x_{4})\rightarrow m(x_{2},m(x_{3},x_{4})),\quad\square,\quad\\{x_{4}\mapsto e,x_{1}\mapsto m(x_{2},x_{3})\\}$ $\displaystyle C_{12}:\ $ $\displaystyle m(m(x_{1},x_{2}),x_{3})\rightarrow m(x_{1},m(x_{2},x_{3})),\quad m(e,x_{4})\rightarrow x_{4},\quad m(\square,x_{3}),\quad\\{x_{4}\mapsto x_{2},x_{1}\mapsto e\\}$ $\displaystyle C_{13}:\ $ $\displaystyle i(m(x_{1},x_{2}))\rightarrow m(i(x_{2}),i(x_{1})),\quad m(e,x_{3})\rightarrow x_{3},\quad i(\square),\quad\\{x_{3}\mapsto x_{2},x_{1}\mapsto e\\}$ $\displaystyle C_{14}:\ $ $\displaystyle m(x_{1},m(i(x_{1}),x_{2}))\rightarrow x_{2},\quad m(e,x_{3})\rightarrow x_{3},\quad\square,\quad\\{x_{3}\mapsto m(i(e),x_{2}),x_{1}\mapsto e\\}$ $\displaystyle C_{15}:\ $ $\displaystyle m(i(x_{1}),m(x_{1},x_{2}))\rightarrow x_{2},\quad m(e,x_{3})\rightarrow x_{3},\quad m(i(x_{1}),\square),\quad\\{x_{3}\mapsto x_{2},x_{1}\mapsto e\\}$ $\displaystyle C_{16}:\ $ $\displaystyle m(x_{1},i(x_{1}))\rightarrow e,\quad m(e,x_{2})\rightarrow x_{2},\quad\square,\quad\\{x_{2}\mapsto i(e),x_{1}\mapsto e\\}$ $\displaystyle C_{17}:\ $ $\displaystyle m(x_{1},e)\rightarrow x_{1},\quad m(e,x_{2})\rightarrow x_{2},\quad\square,\quad\\{x_{2}\mapsto e,x_{1}\mapsto e\\}$ $\displaystyle C_{18}:\ $ $\displaystyle i(m(x_{1},x_{2}))\rightarrow m(i(x_{2}),i(x_{1})),\quad m(x_{3},e)\rightarrow x_{3},\quad i(\square),\quad\\{x_{2}\mapsto e,x_{3}\mapsto x_{1}\\}$ $\displaystyle C_{19}:\ $ $\displaystyle m(x_{1},m(i(x_{1}),x_{2}))\rightarrow x_{2},\quad m(x_{3},e)\rightarrow x_{3},\quad m(x_{1},\square),\quad\\{x_{2}\mapsto e,x_{3}\mapsto i(x_{1})\\}$ $\displaystyle C_{20}:\ $ $\displaystyle m(i(x_{1}),m(x_{1},x_{2}))\rightarrow x_{2},\quad m(x_{3},e)\rightarrow x_{3},\quad m(i(x_{1}),\square),\quad\\{x_{2}\mapsto e,x_{3}\mapsto x_{1}\\}$ Figure 2. The critical pairs of the complete TRS $R$ (1) ($C_{j}:\ l\rightarrow r,\ l^{\prime}\rightarrow r^{\prime},\ C,\ \sigma$ means $C_{j}$ is the critical pair $(r\sigma,C[r^{\prime}\sigma])$.) $\displaystyle C_{21}:\ $ $\displaystyle m(i(x_{1}),x_{1})\rightarrow e,\quad m(x_{2},e)\rightarrow x_{2},\quad\square,\quad\\{x_{1}\mapsto e,x_{2}\mapsto i(e)\\}$ $\displaystyle C_{22}:\ $ $\displaystyle m(x_{1},i(x_{1}))\rightarrow e,\quad i(m(x_{2},x_{3}))\rightarrow m(i(x_{3}),i(x_{2})),\quad m(x_{1},\square),\quad\\{x_{1}\mapsto m(x_{2},x_{3})\\}$ $\displaystyle C_{23}:\ $ $\displaystyle i(m(x_{1},x_{2}))\rightarrow m(i(x_{2}),i(x_{1})),\quad m(x_{3},i(x_{3}))\rightarrow e,\quad i(\square),\quad\\{x_{2}\mapsto i(x_{1}),x_{3}\mapsto x_{1}\\}$ $\displaystyle C_{24}:\ $ $\displaystyle m(x_{1},m(i(x_{1}),x_{2}))\rightarrow x_{2},\quad m(x_{3},i(x_{3}))\rightarrow e,\quad m(x_{1},\square),\quad\\{x_{2}\mapsto i(i(x_{1})),x_{3}\mapsto i(x_{1})\\}$ $\displaystyle C_{25}:\ $ $\displaystyle m(x_{1},i(x_{1}))\rightarrow e,\quad i(i(x_{2}))\rightarrow x_{2},\quad m(x_{1},\square),\quad\\{x_{1}\mapsto i(x_{2})\\}$ $\displaystyle C_{26}:\ $ $\displaystyle m(x_{1},i(x_{1}))\rightarrow e,\quad i(e)\rightarrow e,\quad m(x_{1},\square),\quad\\{x_{1}\mapsto e\\}$ $\displaystyle C_{27}:\ $ $\displaystyle m(i(x_{1}),m(x_{1},x_{2}))\rightarrow x_{2},\quad m(x_{3},i(x_{3}))\rightarrow e,\quad m(i(x_{1}),\square),\quad\\{x_{2}\mapsto i(x_{1}),x_{3}\mapsto x_{1}\\}$ $\displaystyle C_{28}:\ $ $\displaystyle m(i(x_{1}),x_{1})\rightarrow e,\quad i(m(x_{2},x_{3}))\rightarrow m(i(x_{3}),i(x_{2})),\quad m(\square,x_{1}),\quad\\{x_{1}\mapsto m(x_{2},x_{3})\\}$ $\displaystyle C_{29}:\ $ $\displaystyle i(m(x_{1},x_{2}))\rightarrow m(i(x_{2}),i(x_{1})),\quad m(i(x_{3}),x_{3})\rightarrow e,\quad i(\square),\quad\\{x_{3}\mapsto x_{2},x_{1}\mapsto i(x_{2})\\}$ $\displaystyle C_{30}:\ $ $\displaystyle m(x_{1},m(i(x_{1}),x_{2}))\rightarrow x_{2},\quad m(i(x_{3}),x_{3})\rightarrow e,\quad m(x_{1},\square),\quad\\{x_{1}\mapsto x_{2},x_{3}\mapsto x_{2}\\}$ $\displaystyle C_{31}:\ $ $\displaystyle m(i(x_{1}),x_{1})\rightarrow e,\quad i(i(x_{2}))\rightarrow x_{2},\quad m(\square,x_{1}),\quad\\{x_{1}\mapsto i(x_{2})\\}$ $\displaystyle C_{32}:\ $ $\displaystyle m(i(x_{1}),x_{1})\rightarrow e,\quad i(e)\rightarrow e,\quad m(\square,x_{1}),\quad\\{x_{1}\mapsto e\\}$ $\displaystyle C_{33}:\ $ $\displaystyle m(i(x_{1}),m(x_{1},x_{2}))\rightarrow x_{2},\quad m(i(x_{3}),x_{3})\rightarrow e,\quad m(i(x_{1}),\square),\quad\\{x_{3}\mapsto x_{2},x_{1}\mapsto i(x_{2})\\}$ $\displaystyle C_{34}:\ $ $\displaystyle m(i(x_{1}),m(x_{1},x_{2}))\rightarrow x_{2},\quad m(i(x_{3}),m(x_{3},x_{4}))\rightarrow x_{4},\quad m(i(x_{1}),\square),\quad\\{x_{2}\mapsto m(x_{3},x_{4}),x_{1}\mapsto i(x_{3})\\}$ $\displaystyle C_{35}:\ $ $\displaystyle m(i(x_{1}),m(x_{1},x_{2}))\rightarrow x_{2},\quad i(m(x_{3},x_{4}))\rightarrow m(i(x_{4}),i(x_{3})),\quad m(\square,m(x_{1},x_{2})),\quad\\{x_{1}\mapsto m(x_{3},x_{4})\\}$ $\displaystyle C_{36}:\ $ $\displaystyle i(m(x_{1},x_{2}))\rightarrow m(i(x_{2}),i(x_{1})),\quad m(i(x_{3}),m(x_{3},x_{4}))\rightarrow x_{4},\quad i(\square),\quad\\{x_{2}\mapsto m(x_{3},x_{4}),x_{1}\mapsto i(x_{3})\\}$ $\displaystyle C_{37}:\ $ $\displaystyle m(i(x_{1}),m(x_{1},x_{2}))\rightarrow x_{2},\quad m(x_{3},m(i(x_{3}),x_{4}))\rightarrow x_{4},\quad m(i(x_{1}),\square),\quad\\{x_{2}\mapsto m(i(x_{1}),x_{4}),x_{3}\mapsto x_{1}\\}$ $\displaystyle C_{38}:\ $ $\displaystyle m(x_{1},m(i(x_{1}),x_{2}))\rightarrow x_{2},\quad m(i(x_{3}),m(x_{3},x_{4}))\rightarrow x_{4},\quad m(x_{1},\square),\quad\\{x_{2}\mapsto m(x_{1},x_{4}),x_{3}\mapsto x_{1}\\}$ $\displaystyle C_{39}:\ $ $\displaystyle m(i(x_{1}),m(x_{1},x_{2}))\rightarrow x_{2},\quad i(i(x_{3}))\rightarrow x_{3},\quad m(\square,m(x_{1},x_{2})),\quad\\{x_{1}\mapsto i(x_{3})\\}$ $\displaystyle C_{40}:\ $ $\displaystyle m(i(x_{1}),m(x_{1},x_{2}))\rightarrow x_{2},\quad i(e)\rightarrow e,\quad m(\square,m(x_{1},x_{2})),\quad\\{x_{1}\mapsto e\\}$ $\displaystyle C_{41}:\ $ $\displaystyle m(x_{1},m(i(x_{1}),x_{2}))\rightarrow x_{2},\quad i(e)\rightarrow e,\quad m(x_{1},m(\square,x_{2})),\quad\\{x_{1}\mapsto e\\}$ $\displaystyle C_{42}:\ $ $\displaystyle i(i(x_{1}))\rightarrow x_{1},\quad i(e)\rightarrow e,\quad i(\square),\quad\\{x_{1}\mapsto e\\}$ $\displaystyle C_{43}:\ $ $\displaystyle i(i(x_{1}))\rightarrow x_{1},\quad i(i(x_{2}))\rightarrow x_{2},\quad i(\square),\quad\\{x_{1}\mapsto i(x_{2})\\}$ $\displaystyle C_{44}:\ $ $\displaystyle i(i(x_{1}))\rightarrow x_{1},\quad i(m(x_{2},x_{3}))\rightarrow m(i(x_{3}),i(x_{2})),\quad i(\square),\quad\\{x_{1}\mapsto m(x_{2},x_{3})\\}$ $\displaystyle C_{45}:\ $ $\displaystyle m(x_{1},m(i(x_{1}),x_{2}))\rightarrow x_{2},\quad i(i(x_{3}))\rightarrow x_{3},\quad m(x_{1},m(\square,x_{2})),\quad\\{x_{1}\mapsto i(x_{3})\\}$ $\displaystyle C_{46}:\ $ $\displaystyle m(x_{1},m(i(x_{1}),x_{2}))\rightarrow x_{2},\quad m(x_{3},m(i(x_{3}),x_{4}))\rightarrow x_{4},\quad m(x_{1},\square),$ $\displaystyle\\{x_{2}\mapsto m(i(i(x_{1})),x_{4}),x_{3}\mapsto i(x_{1})\\}$ $\displaystyle C_{47}:\ $ $\displaystyle m(x_{1},m(i(x_{1}),x_{2}))\rightarrow x_{2},\quad i(m(x_{3},x_{4}))\rightarrow m(i(x_{4}),i(x_{3})),\quad m(x_{1},m(\square,x_{2})),\quad\\{x_{1}\mapsto m(x_{3},x_{4})\\}$ $\displaystyle C_{48}:\ $ $\displaystyle i(m(x_{1},x_{2}))\rightarrow m(i(x_{2}),i(x_{1})),\quad m(x_{3},m(i(x_{3}),x_{4}))\rightarrow x_{4},\quad i(\square),\quad\\{x_{2}\mapsto m(i(x_{1}),x_{4}),x_{3}\mapsto x_{1}\\}$ Figure 3. The critical pairs of the complete TRS $R$ (2) ($C_{j}:\ l\rightarrow r,\ l^{\prime}\rightarrow r^{\prime},\ C,\ \sigma$ means $C_{j}$ is the critical pair $(r\sigma,C[r^{\prime}\sigma])$.) ## Appendix B Experimental Results We present our experimental data in Table 1 and Table 2. The data set of complete TRSs is taken from experimental results of MKBtt [15], which include benchmark problems [17],[22],[5]. The column headed “degree” shows the degree of the TRS, the column $\\#R_{\textit{before}}$ the number of rules, the column $\\#R_{\textit{after}}$ the number of rules after completion, the column $s(H_{2})$ Malbos-Mimram’s lower bound, and the column $\\#R_{\textit{after}}-e(R)$ our lower bound. The table is also available at https://mir-ikbch.github.io/homtrs/experiment/result.html which has links to TRS files. Table 1. Malbos-Mimram’s and our lower bounds (1) name | degree | $\\#R_{\mathit{before}}$ | $\\#R_{\mathit{after}}$ | $s(H_{2})$ | $\\#R_{\mathit{after}}-e(R)$ ---|---|---|---|---|--- ASK93_1 | 0 | 2 | 2 | 0 | 2 ASK93_6 | 0 | 11 | 11 | 0 | 9 BD94_collapse | 1 | 5 | 5 | – | – BD94_peano | 1 | 4 | 4 | – | – BD94_sqrt | 2 | 3 | 4 | 0 | 3 BGK94_D08 | 2 | 6 | 21 | 2 | 5 BGK94_D10 | 2 | 6 | 21 | 1 | 4 BGK94_D12 | 2 | 6 | 20 | 2 | 5 BGK94_D16 | 2 | 6 | 20 | 2 | 5 BH96_fac8_theory | 1 | 6 | 6 | – | – Chr89_A2 | 2 | 5 | 18 | 0 | 4 Chr89_A3 | 2 | 7 | 16 | 0 | 6 KK99_linear_assoc | 0 | 2 | 2 | 0 | 1 LS94_G0 | 2 | 8 | 13 | 1 | 4 Les83_fib | 1 | 9 | 9 | – | – Les83_subset | 1 | 12 | 12 | – | – OKW95_dt1_theory | 1 | 11 | 11 | – | – SK90_3.01 | 2 | 4 | 11 | 0 | 3 SK90_3.02 | 0 | 3 | 3 | 1 | 2 SK90_3.03 | 2 | 5 | 11 | 0 | 3 SK90_3.04 | 1 | 4 | 8 | – | – SK90_3.05 | 1 | 4 | 13 | – | – SK90_3.06 | 1 | 5 | 12 | – | – SK90_3.07 | 1 | 5 | 15 | – | – SK90_3.08 | 2 | 5 | 4 | 0 | 2 SK90_3.10 | 2 | 4 | 8 | 0 | 3 SK90_3.11 | 0 | 4 | 3 | 0 | 3 SK90_3.12 | 2 | 4 | 9 | 0 | 2 SK90_3.13 | 0 | 6 | 6 | 0 | 3 SK90_3.14 | 0 | 7 | 8 | 1 | 5 SK90_3.15 | 2 | 8 | 7 | 1 | 4 SK90_3.16 | 1 | 4 | 4 | – | – SK90_3.17 | 1 | 3 | 5 | – | – Table 2. Malbos-Mimram’s and our lower bounds (2) name | degree | $\\#R_{\mathit{before}}$ | $\\#R_{\mathit{after}}$ | $s(H_{2})$ | $\\#R_{\mathit{after}}-e(R)$ ---|---|---|---|---|--- SK90_3.18 | 0 | 5 | 6 | 2 | 4 SK90_3.19 | 0 | 9 | 7 | 1 | 4 SK90_3.20 | 1 | 10 | 11 | – | – SK90_3.21 | 1 | 9 | 4 | – | – SK90_3.23 | 0 | 4 | 8 | 1 | 4 SK90_3.24 | 0 | 3 | 2 | 0 | 2 SK90_3.25 | 0 | 1 | 2 | 0 | 1 SK90_3.27 | 0 | 8 | 3 | 0 | 3 SK90_3.28 | 0 | 9 | 18 | 0 | 6 SK90_3.29 | 0 | 7 | 8 | 2 | 7 SK90_3.30 | 1 | 3 | 3 | – | – SK90_3.31 | 1 | 3 | 3 | – | – SK90_3.32 | 1 | 3 | 2 | – | – SK90_3.33 | 0 | 3 | 3 | 0 | 2 TPTP-BOO027-1_theory | 1 | 5 | 5 | – | – TPTP-COL053-1_theory | 0 | 1 | 1 | 0 | 1 TPTP-COL056-1_theory | 0 | 3 | 3 | 0 | 3 TPTP-COL060-1_theory | 0 | 2 | 2 | 0 | 2 TPTP-COL085-1_theory | 0 | 1 | 1 | 0 | 1 TPTP-GRP010-4_theory | 2 | 4 | 11 | 1 | 3 TPTP-GRP011-4_theory | 2 | 4 | 11 | 1 | 3 TPTP-GRP012-4_theory | 2 | 4 | 10 | 0 | 2 slothrop_ackermann | 1 | 3 | 3 | – | – slothrop_cge | 2 | 6 | 20 | 0 | 4 slothrop_cge3 | 2 | 9 | 28 | 0 | 5 slothrop_endo | 2 | 4 | 14 | 0 | 3 slothrop_equiv_proofs | 1 | 12 | 23 | – | – slothrop_fgh | 1 | 4 | 3 | – | – slothrop_groups | 2 | 3 | 10 | 0 | 2 slothrop_groups_conj | 2 | 5 | 10 | 0 | 2 slothrop_hard | 0 | 2 | 2 | 1 | 2

2024-09-04T02:54:54.913412

2002.11945

arxiv-papers

# Is my Neural Network Neuromorphic? Taxonomy, Recent Trends and Future Directions in Neuromorphic Engineering Sumon Kumar Bose School of EEE Nanyang Technological University Jyotibdha Acharya School of EEE Nanyang Technological University Arindam Basu School of EEE Nanyang Technological University ###### Abstract In this paper, we review recent work published over the last 3 years under the umbrella of Neuromorphic engineering to analyze what are the common features among such systems. We see that there is no clear consensus but each system has one or more of the following features:(1) Analog computing (2) Non von- Neumann Architecture and low-precision digital processing (3) Spiking Neural Networks (SNN) with components closely related to biology. We compare recent machine learning accelerator chips to show that indeed analog processing and reduced bit precision architectures have best throughput, energy and area efficiencies. However, pure digital architectures can also achieve quite high efficiencies by just adopting a non von-Neumann architecture. Given the design automation tools for digital hardware design, it raises a question on the likelihood of adoption of analog processing in the near future for industrial designs. Next, we argue about the importance of defining standards and choosing proper benchmarks for the progress of neuromorphic system designs and propose some desired characteristics of such benchmarks. Finally, we show brain-machine interfaces as a potential task that fulfils all the criteria of such benchmarks. ###### Index Terms: Neuromorphic, Low-power, Machine learning, Spiking neural networks, Memristor ## I Introduction The rapid progress of Machine Learning (ML) fuelled by Deep Neural Networks (DNN) in the last several years has created an impact in a wide variety of fields ranging from computer vision, speech analysis, natural language processing etc. With the progress in software, there has been a concomitant push to develop better hardware architectures to support the deployment as well as training of these algorithms[1, 2]. This has rekindled an interest in “Neuromorphic Engineering”–a term coined in 1990 by Carver Mead in his seminal paper [3] where he claimed that hardware implementations of algorithms like pattern recognition (where relative values are of more importance than absolute ones e.g. is this image more likely to be a cat or a dog?) would be more energy and area efficient if it adopts biological strategies of analog processing.S (a) (b) Figure 1: (a) Google search trends over the last 15 years for the topic “Neuromorphic Engineering” shows a decline around 2010 followed by a renewed interest in the last 5 years. (b) Number of neuromorphic papers published in journals from the Nature series have shown a steady increase in the last 10 years. Data for 2019 is till the month of October. While the above idea of brain-inspired analog processing is very appealing and showed initial promise with several interesting sensory prototypes, it failed to gain increased traction over time possibly due to the potential difficulties of creating robust, programmable, large-scale analog designs that can benefit from technology scaling in an easy manner. However, in the last 5 years, there has been renewed interest in this topic, albeit with a slightly expanded connotation of the term “neuromorphic”. Figure 1(a) shows a history of google searches of the term “neuromorphic engineering” over the past 15 years (obtainable from Google Trends). Data points are plotted for every month with the maximum search number normalized to $100$. It can be seen that there was a decline in interest about neuromorphic research around 2010. However, it has again gained momentum in the last five years with a slightly broadened scope which we refer to as version 2 (while referring to the Meadian definition as version 1). A similar trend (plotted in Figure 1(b)) is obtained also by analyzing the number of papers published in relevant journals (Nature, Nature Communications, Nature Electronics, Nature Machine Intelligence, Nature Materials, Nature Nanotechnology) from the Nature journal series over the last $\approx 10$ years that are on the topic of neuromorphic research. It can be seen that there is a rapid increase in the number of such papers over the last 5 years. The rest of the paper is organized as follows: the next section introduces the new connotation of the term “neuromorphic” followed by an analysis of some recent research trends in this field. Section IV describes the need for neuromorphic benchmarks and some desired criteria of such benchmarks while Section V proposes brain-machine interfaces as a potentially good benchmark. ## II Neuromorphic v2.0: A Taxonomy As discussed in the last section, the renaissance in Neuromorphic research over the last 5 years has seen the term being used in a wider sense than the original definition[3]. This is partially due to the fact that scientists from different communities (not only circuits or neuroscientists) ranging from material science to computer architects have now become involved. Based on the recent work, we describe next the key characteristic features of this new version of neuromorphic systems as: * • Use of analog or physics based processing as opposed to conventional digital circuits–this is same as the original version of neuromorphic system from a circuits perspective. * • From the viewpoint of computer architecture, usage of non von-Neumann architecture (independent of analog or digital compute) and low-precision digital datapath are hallmarks of neuromorphic systems. In other words, conventional computers using von-Neumann architectures read from memory, compute and write back the result–this is very different from brain-inspired systems where memory and computing are interspersed [4]. * • Computer scientists and algorithm developers on the other hand consider a system neuromorphic if it uses a spiking neural network (SNN) as opposed to a traditional artificial neural network (ANN). Neurons in an SNN inherently encode time and output a 1-bit digital pulse called a spike or action potential. Figure 2: Survey of neuromorphic systems reported over 2017-2019 in Nature, Science, Science Advances, Nature Nanotechnology, Nature Electronics, Nature Materials, Nature Communications . A large majority use SNN in their work. Details of all papers used in the survey are in [5]. Figure 3: Survey of IC implementations of non von-Neumann architecture over the same period in ISSCC, SOVC, JSSC however shows very few work uses the term “neuromorphic”. Details of all papers used in the survey are in [5]. We next illustrate how frequently each type of viewpoint is expressed in neuromorphic research. Figure 2 categorizes the neuromorphic research papers published between 2017-2019 in the Nature series of journals surveyed in Figure 1(b) along with the journals Science and Science Advances. The papers are categorized according to the neuromorphic aspect they primarily focus on–(1) Analog processing, (2) non von-Neumann architecture or (3) SNN. It can be seen that a large majority of the work focussed on the SNN aspect (details of papers used in the survey are available at [5]). Most of these work focus on new materials or device fabrication and then present SNN simulations using the novel device properties bypassing the circuit level. Hence, we also decided to create a survey of ML accelerator integrated circuits (IC) published in IEEE ISSCC and IEEE SOVC conferences inspired by the ADC survey[6]. In addition, we also considered papers published in the IEEE Journal of Solid State Circuits (JSSC). Figure 3 plots the result of categorizing all ML accelerators adopting non von-Neumann architecture published between 2017-2019 (details in [5]). Surprisingly, it can be seen that only 5 papers have used the term “neuromorphic” to describe their work! This clearly shows a stark difference in terminology used across different research communities. Figure 4: A new taxonomy that has non von-Neumann architecture as the overarching topic with neuromorphic v2.0 and ML accelerators as two sub-topics under it. Figure 5: Machine learning hardware trends: Peak energy efficiency in TOPS/W plotted against memory size and categorized by (a) bit-width of datapath and (b) digital vs mixed-signal analog approaches. (c) Throughput at peak energy efficiency and (d) Area efficiency plotted against peak energy efficiency for recent ASIC implementations reported over 2017-2019 in ISSCC, SOVC, and JSSC. The larger dots in (d) indicate ASIC area without pad. This leads us to propose a new taxonomy for neuromorphic systems as shown in Figure 4. It is possibly better to use the term non von-Neumann architecture as the overarching topic. Under its ambit, neuromorphic v2.0 can refer to systems using analog or mixed-signal circuits, implementing SNN algorithms or the extremely quantized version (1-bit) of ANNs. On the other hand, ML accelerators can refer to digital circuits with non von-Neumann architecture implementing multi-bit ANN. With this in mind, we look at some recent performance trends in ML accelerators using non von-Neumann architectures that were reviewed in Figure 3. ## III Trends in Machine Learning Hardware There are several important metrics to quantify the performance of ML accelerators such as energy efficiency, throughput and area efficiency. To identify some trends, we plot several combinations of these quantities in Figure 5. First, we expect bigger chips to have lower energy efficiency in general due to cost of moving data around large areas that dissipates more energy charging and discharging interconnects. Since the area of these ICs are dominated by the static random access memory (SRAM) required to store weights and activations, we use the SRAM size as a proxy for chip area. The energy efficiency in Tera operations (TOPS) per Watt are plotted against SRAM size for these designs in Fig. 5(a) and (b) and indeed show an inverse relation between energy efficiency and SRAM size or chip size. Figure 5(a) further uses different colours to categorize the data points according to bit width of datapath. As expected, it can be seen that the extremely quantized 1-bit designs[7, 8, 9] show best energy efficiency and are located significantly ($\approx 10X$) above the trend line. The same data is plotted in Fig. 5(b) but colour coded according to the design approach of digital versus analog mixed-signal. It is interesting to note that the mixed signal designs indeed exhibit higher energy efficiencies, but they are in general much smaller than the digital ones. Thus, in general we can see that the neuromorphic v2.0 principles of non von- Neumann architecture coupled with low data precision and analog computing (described earlier in Section II) do indeed provide great energy efficiencies. However, it can be seen that the energy efficiencies of pure digital approaches using only the principles of non von-Neumann architecture and low bit-width are at least much higher ($\approx 500X$) than the energy efficiency wall of $\approx 10$ GMACs/W for traditional von-Neumann processors[10, 11]. Hence, this raises an interesting question–given the scalability, testability and ease of porting across nodes offered by digital designs, is it reasonable to expect large scale industrial adoption of analog neuromorphic designs for an extra $10X$ in energy efficiency? Next, we analyze the trade-offs in throughput at peak energy efficiency by plotting it against peak energy efficiency in Fig. 5(c). Interestingly, these two quantities are positively correlated with a majority of designs exhibiting throughput $\approx 100$ GOPS. Higher throughput would generally mean the static power is better amortized across the operations leading to higher energy efficiency. Also, in general reduced bit precision designs that increases energy efficiency would also reduce critical path delays increasing throughput. Lastly, we analyze area efficiency of the designs (measured in GOPS/mm2) by plotting it against energy efficiency in Fig. 5(d). Again, these two quantities show a positive correlation implying again that good design practices of reduced bit precision and analog design positively impact both the quantities. This is also clarified in Figure 5 by demarcating the designs according to bit precision and design styles. These plots show that apart from energy efficiency, analog mixed signal design styles also provide $\approx 10X$ improvement in throughput and area efficiency. Coupled with energy efficiency advantages, these points might be sufficient to suggest that in the longer term, there is reason for large scale interest in neuromorphic designs following the principles outlined earlier. However, all of these comparisons are not very relevant unless they can all run a common set of benchmark problems. This is discussed next in the following section. ## IV Neuromorphic Benchmarks The comparison between all the hardware designs in the earlier section are not fair unless they can all at least report performance on a minimum set of benchmark algorithms. While there are at least some common benchmarks for the ANN community such as MNIST[12], CIFAR[13] and Imagenet[14] for image recognition, there is not much consensus about good benchmarks for neuromorphic SNN algorithms. Hence, while advocating usage of benchmarks from the ML community for neuromorphic hardware ANNs, we discuss more details about what might constitute desired criteria for SNN benchmarks. While this topic is deemed important, there has been very few dedicated efforts in this area[15]. Given the role the Imagenet benchmark played in catalysing progress in ANN research, we believe it is of utmost importance that the neuromorphic community spend more effort immediately on devising good benchmarks for SNN. Some recent work on SNN has focussed on converting images from ANN benchmarks to spike trains and then classifying them[16, 17]. While being great pieces of research, we feel that this is fundamentally not a good application for SNN since the original input signal is static and does not change with time. Instead, it might be more natural to use SNN as dynamical systems to track moving objects in video streams[18, 19] or classify signals that vary over time such as speech[20]. With this in mind, we propose the following desired characteristics for neuromorphic benchmarks: 1. 1. The signal being processed should be encoded in time naturally so that the continuous time dynamics of SNN can be more effective than ANN to process it. Signals such as speech, video, etc are good examples. From the biomedical domain, EEG signals are another good example. 2. 2. There should be a need for real-time response of the system such as in closed- loop systems such as sensori-motor loops in robotics. The rapid response time of neuromorphic sensors and SNN processing should be useful in such cases. 3. 3. There should be need for the system to adapt or learn frequently. This necessitates learning from few samples, a common complaint with current deep learning based ANNs that require many thousands of examples to train. 4. 4. Ideally, the example applications should be ones that require low-power operation so that the energy efficiency of neuromorphic hardware meets an important design requirement. 5. 5. There would potentially be different benchmarks for different scales of the problem–edge deployment (sensory information processing) or cloud based analytics (large scale search, creativity etc). We argue in the next section that brain-machine interfaces provide a benchmark application that meets all of the above criteria. Figure 6: Example of a BMI experimental setup where the NHP is using his thoughts to move a wheelchair (adopted from [21] under CC-BY license). The decoder to convert brain signals to a command provides ideal opportunity for low-power, real-time neuromorphic machine learners. ## V Brain-Machine Interfaces The aim of intra-cortical Brain Machine Interfaces (iBMIs) is to substantially improve the lives of patients afflicted by spinal cord injury or debilitating neurodegenerative disorders such as tetraplegia, amyotrophic lateral sclerosis. These systems take neural activity as an input and drive effectors such as a computer cursor [22], wheelchair [21] and prosthetic [23], paralysed [24] limbs for the purposes of communication, locomotion and artificial hand control respectively. While early work focussed on non-invasive EEG based systems, invasive neural interfaces are needed for fine grained motor control as well as for advancing fundamental knowledge about the brain due to higher signal quality obtainable. Figure 6 demonstrates a typical experimental setup involving a non-human primate (NHP) where an implanted micro-electrode array is interfaced with amplifiers to readout neural activity at the level of single cells[21]. This neural data is collected while the primate is doing different types of tasks according to a given cue (typically visual). Based on the recorded data, a machine learner or decoder is trained to convert the neural recording to an action that affects the physical world and provides feedback to the NHP (again typically visual feedback is used). We argue that a decoder in iBMI satisfies all the conditions required for a neuromorphic system described in Section IV as explained below: 1. 1. Neural data recorded from the brain are indeed a streaming signal arriving continuously over time. Further, the data are naturally in the form of spikes avoiding the question for the need of spike conversion and how to do it. 2. 2. Due to the visual feedback provided to the NHP, decoding has to be done in real-time. In this case, typical update frequencies of $10$ Hz are used[25]. 3. 3. There is a need to frequently adapt the weights of the decoder since the neural data is non-stationary[26]. The statistics can change due to micro- motion of the electrode or scar tissue formation. 4. 4. The decoder must consume very little energy to prolong the battery life of the system[27]. If included within the implant, its area must be very small as well. There has been some initial work on neuromorphic decoders[28, 29, 30, 31, 25]. While [28, 29] performed software simulations, [30, 31, 25] have shown results from custom low-power neuromorphic ICs. Further, closed-loop decoding results from NHP have so far been demonstrated only in [25]. One of the issues behind lack of results in this domain is the difficulty and cost of creating a NHP based experiment. Open-source datasets are just beginning to be available in this field[32, 33]. While these definitely will provide a good starting point, they cannot be used to simulate closed-loop settings. We envision that setting up AI based models to mimic closed-loop BMI experimental settings could be a good research direction for this area. ## Conclusion In this paper, we reviewed the recent trend in papers published on the topic of neuromorphic engineering or computing and showed that the connotation of the term has broadened beyond its original definition of brain-inspired analog computing. Neuromorphic v2.0, as we call it in this paper, includes the concept of non von-Neumann and low precision digital computing from computer architecture and spiking neural networks from the computer science and algorithm community. However, there are differences in the way different scientific communities have used the term and a potential better taxonomy is to consider non von-Neumann computing as an umbrella under which a sub-concept is neuromorphic computing. Trends in recently published ML accelerator ICs indeed show that using the above neuromorphic concepts lead to $\approx 10X$ benefit in energy efficiency, area efficiency and throughput over digital non von-Neumann architectures. We also pointed out the need for benchmarks in SNN research and suggested some potential characteristics of such benchmarks. Finally, we pointed out that brain-machine interfaces (BMI) have all these desired characteristics of real-time response, processing time varying signals, need for quick re-training as well as strict requirement for low- power dissipation. We envision generation of BMI based benchmarks in the future for testing and standardization of different neuromorphic systems. ## References * [1] A. Basu, J. Acharya, and et. al., “Low-power, adaptive neuromorphic systems: Recent progress and future directions,” _IEEE Journal of Emerging Topics in Circuits and Systems_ , vol. 8, no. 1, pp. 6–27, 2018. * [2] C.-Y. Chen, B. Murmann, J.-S. 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Available: https://zenodo.org/record/583331

2024-09-04T02:54:54.929108

2002.11986

arxiv-papers

11institutetext: University Autonoma Madrid, Department of Theoretical Physics, 28049 Madrid, Spain22institutetext: University of Bern, Albert Einstein Center for Fundamental Physics, Laboratory for High Energy Physics (LHEP), Bern, Switzerland33institutetext: Boston University, Department of Physics, Boston, Massachusetts, U.S.A.44institutetext: University of British Columbia, Department of Physics and Astronomy, Vancouver, British Columbia, Canada55institutetext: University of California, Irvine, Department of Physics and Astronomy, Irvine, California, U.S.A.66institutetext: IRFU, CEA Saclay, Gif-sur-Yvette, France77institutetext: University of Colorado at Boulder, Department of Physics, Boulder, Colorado, U.S.A.88institutetext: Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A.99institutetext: Duke University, Department of Physics, Durham, North Carolina, U.S.A.1010institutetext: Ecole Polytechnique, IN2P3-CNRS, Laboratoire Leprince-Ringuet, Palaiseau, France 1111institutetext: ETH Zurich, Institute for Particle Physics and Astrophysics, Zurich, Switzerland1212institutetext: CERN European Organization for Nuclear Research, CH-1211 Genéve 23, Switzerland1313institutetext: University of Geneva, Section de Physique, DPNC, Geneva, Switzerland1414institutetext: University of Glasgow, School of Physics and Astronomy, Glasgow, United Kingdom1515institutetext: H. Niewodniczanski Institute of Nuclear Physics PAN, Cracow, Poland1616institutetext: High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan1717institutetext: University of Houston, Department of Physics, Houston, Texas, U.S.A.1818institutetext: Institut de Fisica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra (Barcelona) Spain1919institutetext: IFIC (CSIC & University of Valencia), Valencia, Spain2020institutetext: Institute For Interdisciplinary Research in Science and Education (IFIRSE), ICISE, Quy Nhon, Vietnam2121institutetext: Imperial College London, Department of Physics, London, United Kingdom2222institutetext: INFN Sezione di Bari and Università e Politecnico di Bari, Dipartimento Interuniversitario di Fisica, Bari, Italy2323institutetext: INFN Sezione di Napoli and Università di Napoli, Dipartimento di Fisica, Napoli, Italy2424institutetext: INFN Sezione di Padova and Università di Padova, Dipartimento di Fisica, Padova, Italy2525institutetext: INFN Sezione di Roma and Università di Roma “La Sapienza”, Roma, Italy2626institutetext: Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia2727institutetext: International Centre of Physics, Institute of Physics (IOP), Vietnam Academy of Science and Technology (VAST), 10 Dao Tan, Ba Dinh, Hanoi, Vietnam2828institutetext: Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, University of Tokyo, Kashiwa, Chiba, Japan2929institutetext: Keio University, Department of Physics, Kanagawa, Japan3030institutetext: King’s College London, Department of Physics, Strand, London WC2R 2LS, United Kingdom3131institutetext: Kobe University, Kobe, Japan3232institutetext: Kyoto University, Department of Physics, Kyoto, Japan3333institutetext: Lancaster University, Physics Department, Lancaster, United Kingdom3434institutetext: University of Liverpool, Department of Physics, Liverpool, United Kingdom3535institutetext: Louisiana State University, Department of Physics and Astronomy, Baton Rouge, Louisiana, U.S.A.3636institutetext: Michigan State University, Department of Physics and Astronomy, East Lansing, Michigan, U.S.A.3737institutetext: Miyagi University of Education, Department of Physics, Sendai, Japan3838institutetext: National Centre for Nuclear Research, Warsaw, Poland3939institutetext: State University of New York at Stony Brook, Department of Physics and Astronomy, Stony Brook, New York, U.S.A.4040institutetext: Okayama University, Department of Physics, Okayama, Japan4141institutetext: Osaka City University, Department of Physics, Osaka, Japan4242institutetext: Oxford University, Department of Physics, Oxford, United Kingdom4343institutetext: University of Pennsylvania, Department of Physics and Astronomy, Philadelphia, PA, 19104, USA.4444institutetext: University of Pittsburgh, Department of Physics and Astronomy, Pittsburgh, Pennsylvania, U.S.A.4545institutetext: Queen Mary University of London, School of Physics and Astronomy, London, United Kingdom4646institutetext: University of Regina, Department of Physics, Regina, Saskatchewan, Canada4747institutetext: University of Rochester, Department of Physics and Astronomy, Rochester, New York, U.S.A.4848institutetext: Royal Holloway University of London, Department of Physics, Egham, Surrey, United Kingdom4949institutetext: RWTH Aachen University, III. Physikalisches Institut, Aachen, Germany5050institutetext: University of Sheffield, Department of Physics and Astronomy, Sheffield, United Kingdom5151institutetext: University of Silesia, Institute of Physics, Katowice, Poland5252institutetext: SLAC National Accelerator Laboratory, Stanford University, Menlo Park, California, USA5353institutetext: Sorbonne Université, Université Paris Diderot, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Paris, France5454institutetext: STFC, Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington, United Kingdom5555institutetext: University of Tokyo, Department of Physics, Tokyo, Japan5656institutetext: University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan5757institutetext: University of Tokyo, Institute for Cosmic Ray Research, Research Center for Cosmic Neutrinos, Kashiwa, Japan5858institutetext: Tokyo Institute of Technology, Department of Physics, Tokyo, Japan5959institutetext: Tokyo Metropolitan University, Department of Physics, Tokyo, Japan6060institutetext: Tokyo University of Science, Faculty of Science and Technology, Department of Physics, Noda, Chiba, Japan6161institutetext: University of Toronto, Department of Physics, Toronto, Ontario, Canada6262institutetext: TRIUMF, Vancouver, British Columbia, Canada6363institutetext: University of Victoria, Department of Physics and Astronomy, Victoria, British Columbia, Canada6464institutetext: University of Warsaw, Faculty of Physics, Warsaw, Poland6565institutetext: Warsaw University of Technology, Institute of Radioelectronics and Multimedia Technology, Warsaw, Poland6666institutetext: University of Warwick, Department of Physics, Coventry, United Kingdom6767institutetext: University of Winnipeg, Department of Physics, Winnipeg, Manitoba, Canada6868institutetext: Wroclaw University, Faculty of Physics and Astronomy, Wroclaw, Poland6969institutetext: Yokohama National University, Faculty of Engineering, Yokohama, Japan7070institutetext: York University, Department of Physics and Astronomy, Toronto, Ontario, Canadaaainstitutetext: Also at INFN-Laboratori Nazionali di Legnarobbinstitutetext: Also at J-PARC, Tokai, Japanccinstitutetext: Affiliated member at Kavli IPMU (WPI), the University of Tokyo, Japanddinstitutetext: Also at National Research Nuclear University "MEPhI" and Moscow Institute of Physics and Technology, Moscow, Russiaeeinstitutetext: Also at the Graduate University of Science and Technology, Vietnam Academy of Science and Technologyffinstitutetext: Also at JINR, Dubna, Russiagginstitutetext: Also at Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP)hhinstitutetext: Also at BMCC/CUNY, Science Department, New York, New York, U.S.A.‡‡institutetext: Deceased # Measurement of the charged-current electron (anti-)neutrino inclusive cross- sections at the T2K off-axis near detector ND280 K. Abe 45 N. Akhlaq, 57 R. Akutsu 32 A. Ali 11 C. Alt 54,34 C. Andreopoulos 21 L. Anthony 19 M. Antonova 31 S. Aoki 2 A. Ariga 59 T. Arihara 69 Y. Asada 32 Y. Ashida 21 E.T. Atkin 59 Y. Awataguchi 32 S. Ban 46 M. Barbi 66 G.J. Barker 42 G. Barr 42 D. Barrow 34 C. Barry 15 M. Batkiewicz-Kwasniak 26 A. Beloshapkin 34 F. Bench 22 V. Berardi 58 L. Berns 70 S. Bhadra, 53 S. Bienstock 53,13 A. Blondel 6 S. Bolognesi 68 T. Bonus 18 B. Bourguille 66 S.B. Boyd 33 D. Brailsford 13 A. Bravar 1 D. Bravo Berguño 56 C. Bronner 13 S. Bron 51 A. Bubak 10 M. Buizza Avanzini 36 J. Calcutt 7 T. Campbell 16 S. Cao 50 S.L. Cartwright 22 M.G. Catanesi 19 A. Cervera 66 A. Chappell 24 C. Checchia 17 D. Cherdack 55 N. Chikuma 12 G. Christodoulou 24,a M. Cicerchia, 34 J. Coleman 24 G. Collazuol 42,28 L. Cook 42 D. Coplowe 7 A. Cudd 15 A. Dabrowska 23 G. De Rosa 33 T. Dealtry 66 P.F. Denner 34 S.R. Dennis 54 C. Densham 30 F. Di Lodovico 39 N. Dokania 12 S. Dolan 33 T.A. Doyle 10 O. Drapier 53 J. Dumarchez 21 P. Dunne 55 A. Eguchi 14 L. Eklund 6 S. Emery-Schrenk 2 A. Ereditato 19 P. Fernandez 4,62 T. Feusels 33 A.J. Finch, 70 G.A. Fiorentini 23 G. Fiorillo 2 C. Francois 16,b M. Friend 16,b Y. Fujii 55 R. Fujita 40 D. Fukuda 60 R. Fukuda 37 Y. Fukuda 11 K. Fusshoeller 53 C. Giganti 68 T. Golan 10 M. Gonin 26 A. Gorin 53 M. Guigue 66 D.R. Hadley 66 J.T. Haigh 49 P. Hamacher-Baumann 62,28 M. Hartz 16,b T. Hasegawa 6 S. Hassani 16 N.C. Hastings 32 T. Hayashino 56,28 Y. Hayato 32 A. Hiramoto, 8 M. Hogan 51 J. Holeczek 20,27 N.T. Hong Van 41 T. Honjo 24 F. Iacob 32 A.K. Ichikawa 56 M. Ikeda 16,b T. Ishida 16,b T. Ishii 60 M. Ishitsuka 55 K. Iwamoto 26 A. Izmaylov 60 N. Izumi 16 M. Jakkapu 67 B. Jamieson 50 S.J. Jenkins 18 C. Jesús-Valls 32 M. Jiang 7 S. Johnson 21 P. Jonsson 39,c C.K. Jung 57 X. Junjie 21 P.B. Jurj 42 M. Kabirnezhad 48,54 A.C. Kaboth, 57,c T. Kajita 59 H. Kakuno 56 J. Kameda 63,62 D. Karlen 35 S.P. Kasetti 56 Y. Kataoka 69 Y. Katayama 30 T. Katori 56 Y. Kato 3,28,c E. Kearns 26 M. Khabibullin 26 A. Khotjantsev 32 T. Kikawa 55 H. Kikutani 41 H. Kim 30 S. King 51 J. Kisiel 66 A. Knight 33 A. Knox 41 T. Kobata 16,b T. Kobayashi 42 L. Koch 55 T. Koga 62 A. Konaka 33 L.L. Kormos, 40,c Y. Koshio 26 A. Kostin 38 K. Kowalik 32 H. Kubo 26,d Y. Kudenko 41 N. Kukita 32 S. Kuribayashi 65 R. Kurjata 35 T. Kutter 58 M. Kuze 1 L. Labarga 38 J. Lagoda 24 M. Lamoureux 43 D. Last 24 M. Laveder 33 M. Lawe 10 M. Licciardi 62 T. Lindner 14 R.P. Litchfield 39 S.L. Liu 39 X. Li 24 A. Longhin 25 L. Ludovici 42 X. Lu 18 T. Lux, 23 L.N. Machado 22 L. Magaletti 36 K. Mahn 50 M. Malek 47 S. Manly 13 L. Maret 7 A.D. Marino 56,28 L. Marti-Magro 61 J.F. Martin 16,b T. Maruyama 16 T. Matsubara 55 K. Matsushita 26 V. Matveev 43 C. Mauger 34 K. Mavrokoridis 6 E. Mazzucato 70 M. McCarthy 34 N. McCauley 50 J. McElwee 47 K.S. McFarland 39 C. McGrew 26 A. Mefodiev 34 C. Metelko 24 M. Mezzetto 69 A. Minamino, 26 O. Mineev 5 S. Mine 56,c M. Miura 11 L. Molina Bueno 56,c S. Moriyama 36 J. Morrison 10 Th.A. Mueller 6 L. Munteanu 11 S. Murphy 7 Y. Nagai 16,b T. Nakadaira 56,28 M. Nakahata 56 Y. Nakajima 40 A. Nakamura 32 K.G. Nakamura 28,16,b K. Nakamura 31 Y. Nakano 56,28 S. Nakayama 32,28 T. Nakaya 16,b K. Nakayoshi 61 C. Nantais 21 C.E.R. Naseby 20,e T.V. Ngoc 68 K. Niewczas 16,‡ K. Nishikawa, 29 Y. Nishimura 13 E. Noah 21 T.S. Nonnenmacher 54 F. Nova 19 P. Novella 33 J. Nowak 14 J.C. Nugent 33 H.M. O’Keeffe 50 L. O’Sullivan 32 T. Odagawa 16 T. Ogawa 40 R. Okada 57,28 K. Okumura 41 T. Okusawa 4,62 S.M. Oser 45 R.A. Owen 16,b Y. Oyama 23 V. Palladino 39 J.L. Palomino 44 V. Paolone 24 M. Pari 48 W.C. Parker 13 S. Parsa 21 J. Pasternak 34 P. Paudyal, 62 M. Pavin 34 D. Payne 34 G.C. Penn 36 L. Pickering 50 C. Pidcott 69 G. Pintaudi 70 E.S. Pinzon Guerra 2 C. Pistillo 53,f B. Popov 51 K. Porwit 64 M. Posiadala-Zezula 34 A. Pritchard 10 B. Quilain 49 T. Radermacher 22 E. Radicioni 11 B. Radics 33 P.N. Ratoff 8 E. Reinherz-Aronis 39 C. Riccio 38 E. Rondio 49 S. Roth 11 A. Rubbia 23 A.C. Ruggeri 14 C.A. Ruggles 65 A. Rychter, 16,b K. Sakashita 13 F. Sánchez 70 G. Santucci 11 C.M. Schloesser 9,c K. Scholberg 8 J. Schwehr 21 M. Scott 41,g Y. Seiya 16,b T. Sekiguchi 56,28,c H. Sekiya 11 D. Sgalaberna 54,42 R. Shah 26 A. Shaikhiev 67 F. Shaker 26 A. Shaykina 56,28 M. Shiozawa 21 W. Shorrock 26 A. Shvartsman 26 A. Smirnov 5 M. Smy 68 J.T. Sobczyk 5,28 H. Sobel 14 F.J.P. Soler 56 Y. Sonoda 49 J. Steinmann, 26,6 S. Suvorov 31 A. Suzuki 16,b S.Y. Suzuki 28 Y. Suzuki 21 A.A. Sztuc 16,b M. Tada 32 M. Tajima 56 A. Takeda 31,28 Y. Takeuchi 56,c H.K. Tanaka 52,61 H.A. Tanaka 41 S. Tanaka 69 Y. Tanihara 32 M. Tani 41 N. Teshima 50 L.F. Thompson 8 W. Toki 34 C. Touramanis 61 T. Towstego 34 K.M. Tsui 16,b T. Tsukamoto 35 M. Tzanov 21 Y. Uchida 28,5 M. Vagins 66 S. Valder, 39 Z. Vallari 18 D. Vargas 6 G. Vasseur 39 C. Vilela 66 W.G.S. Vinning 54 T. Vladisavljevic 26 V.V. Volkov 15 T. Wachala 67 J. Walker 33 J.G. Walsh 39 Y. Wang 54,42 D. Wark 21 M.O. Wascko 54,42 A. Weber 32,c R. Wendell 39 M.J. Wilking 2 C. Wilkinson 30 J.R. Wilson 8 R.J. Wilson 39 K. Wood 47 C. Wret 16,‡ Y. Yamada 41,g K. Yamamoto 39,h C. Yanagisawa 39 G. Yang, 56 T. Yano 32 K. Yasutome 62 S. Yen 26 N. Yershov 55,c M. Yokoyama 58 T. Yoshida 70 M. Yu 15 A. Zalewska 38 J. Zalipska 65 K. Zaremba 38 G. Zarnecki 65 M. Ziembicki 7 E.D. Zimmerman 53 M. Zito 30 S. Zsoldos 26 and A. Zykova <EMAIL_ADDRESS> ###### Abstract The electron (anti-)neutrino component of the T2K neutrino beam constitutes the largest background in the measurement of electron (anti-)neutrino appearance at the far detector. The electron neutrino scattering is measured directly with the T2K off-axis near detector, ND280. The selection of the electron (anti-)neutrino events in the plastic scintillator target from both neutrino and anti-neutrino mode beams is discussed in this paper. The flux integrated single differential charged-current inclusive electron (anti-)neutrino cross-sections, $d\sigma/dp$ and $d\sigma/d\cos(\theta)$, and the total cross-sections in a limited phase-space in momentum and scattering angle ($p>300$ MeV/c and $\theta\leq 45^{\circ}$) are measured using a binned maximum likelihood fit and compared to the neutrino Monte Carlo generator predictions, resulting in good agreement. ###### Keywords: neutrino cross-section ††arxiv: 2002.11986 ## 1 Introduction The measurement of the $\nu_{\mu}\rightarrow\nu_{e}$ (and $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$) oscillations - which is the main goal of the T2K experiment T2KExperiment \- is affected by two main background sources. The first is the intrinsic $\nu_{e}$ and $\bar{\nu}_{e}$ beam contaminations and the second is the neutral current (NC) $\pi^{0}$ production, where the $\pi^{0}$ can mimic an electron from a charged-current (CC) $\nu_{e}$ or $\bar{\nu}_{e}$ interaction at the far detector, Super- Kamiokande. In addition, the $\nu_{\mu}\rightarrow\nu_{e}$ ($\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$) appearance signal is predicted by using a predominantly $\nu_{\mu}$ ($\bar{\nu}_{\mu}$) sample, which relies on the knowledge of the $\nu_{e}$ ($\bar{\nu}_{e}$) cross-section relative to the $\nu_{\mu}$ ($\bar{\nu}_{\mu}$). The modelling of signal and backgrounds is strongly depending on the neutrino cross-sections and the near detector is crucial for measuring them. The electron (anti-)neutrino flux arises from the decay of kaons, muons and pions produced when the proton beam impinges upon a graphite target. Kaons can decay to electron (anti-)neutrinos through the decay channels, $K^{\pm}\rightarrow\pi^{0}+e^{\pm}+\nu_{e}(\bar{\nu}_{e})$ and $K^{0}_{e3}\rightarrow\pi^{\pm}+e^{\mp}+\bar{\nu}_{e}(\nu_{e})$. Muons, mainly produced from pion decay, can also decay to electron (anti-)neutrinos through $\mu^{\pm}\rightarrow e^{\pm}+\bar{\nu}_{\mu}(\nu_{\mu})+\nu_{e}(\bar{\nu}_{e})$. The direct contribution of the pion decays to the electron (anti-)neutrino flux is tiny. Together these combinations provide the $\nu_{e}$ and $\bar{\nu}_{e}$ flux at the near detector. In general, the electron (anti-)neutrinos from kaon decays are more energetic than those from muon decays and populate the high energy tail of the neutrino energy spectrum. The CC electron (anti-)neutrino selection at the near detector is challenging for two reasons. Firstly, there is a small number of electrons (positrons) produced from CC $\nu_{e}$ ($\bar{\nu}_{e}$) interactions, compared to the much larger number of muons, pions and protons produced in the final states of CC and NC $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ interactions. The particle identification (PID) must work extremely well to obtain a pure electron selection. The second reason is the large number of background electrons from sources such as $\pi^{0}$, which can be produced either inside or outside the target detectors. Rejection of background electrons (positrons) is vital for the measurement of the CC $\nu_{e}$ ($\bar{\nu}_{e}$) interactions. Electron (anti-)neutrino cross-section measurements in the GeV region are rare since the (anti-)neutrino beams primarily produce muon (anti-)neutrinos. The first CC-$\nu_{e}$ inclusive cross-section measurement and the only CC-$\bar{\nu}_{e}$ inclusive cross-section measurement so far were made by the Gargamelle bubble chamber experiment in 1978 Gargamelle . Thirty-six years later, in 2014, T2K measured the CC-$\nu_{e}$ inclusive cross-section ND280NueXs and in 2016 MINERvA performed the first CC-$\nu_{e}$ cross-section measurement without pions in the final state MinervaNue . Measurements of the electron (anti-)neutrino cross-sections will have a pivotal role for the precision measurements of neutrino oscillations for the current and next generation of long-baseline neutrino oscillation experiments DUNETDR ; HYPERKTDR . Compared to the 2014 results, the work in this paper follows a different approach to measure the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ cross-sections. Following the developments in the T2K muon neutrino cross-sections measurements ND280cc0pi ; ND280numucc4pi ; ND280nucleff , the differential cross-sections are measured in a model independent way as a function of electron and positron kinematics (momentum and scattering angle), the quantities which are measured in the near detector. Although cross-section results were calculated in $Q^{2}$ in the 2014 work, such measurements could introduce model dependencies and are not included in this work. Each $Q^{2}$ bin contains contributions from events with different electron kinematics leading to model dependencies when correcting for the efficiencies since our acceptance for backward and high angle events is very poor. Similarly, cross- section measurements in momentum, scattering angle and neutrino energy which are extrapolated to regions with no or very little acceptance are also model dependent since they depend on the underlying model for the efficiency corrections. Such results are not produced in this paper. For the differential cross-section extraction, following the experience from T2K muon neutrino cross-section measurements ND280cc0pi ; ND280numucc4pi ; ND280nucleff , this work uses a binned likelihood fit with control samples to tune the backgrounds instead of an iterative matrix inversion method dagostini . The likelihood fit method is preferred as the correction of detector smearing effects is independent of the signal model used in the simulation and it allows in-depth validation of the background tuning and of the extracted results. Finally, events with momentum below 200 MeV/c were not considered in the 2014 results. This background enriched region can be used for fit validation studies and it is used in the current work. Since the CC-$\nu_{e}$ inclusive cross-section measurement in 2014, T2K has doubled the neutrino data and collected a significant amount of anti-neutrino data. With these new datasets, T2K performs new measurements of the CC-$\nu_{e}$ inclusive cross-sections in neutrino and anti-neutrino modes. In addition, the first CC-$\bar{\nu}_{e}$ inclusive cross-section in anti- neutrino mode, since Gargamelle, is measured. ## 2 Experimental Setup ### 2.1 T2K beam The T2K neutrino beam is produced at the Japan Proton Accelerator Research Complex (J-PARC) by colliding 30 GeV protons with a graphite target. The pions and kaons produced in the target are focused by three magnetic horns and decay in flight to produce neutrinos. T2K can run with either forward horn current (FHC) or with reverse horn current (RHC) producing beams in neutrino or anti- neutrino enhanced mode, respectively. The T2K beamline T2Kbeamline is simulated using FLUKA2011 fluka1 ; fluka2 , GEANT3 geant3 and GCALOR gcalor . The simulated yields of hadronic particles are tuned using the NA61/SHINE na61shine1 ; na61shine2 ; na61shine3 thin target measurements. The neutrino fluxes at the off-axis near detector ND280 in FHC and RHC are shown in Figure 1. The off-axis position of the near detector, from the neutrino beam direction, results in a narrow-band $\nu_{\mu}$ or $\bar{\nu}_{\mu}$ beam, however, the same does not occur with $\nu_{e}$ and $\bar{\nu}_{e}$ fluxes due to their production via three-body decays, resulting in broader $\nu_{e}$ and $\bar{\nu}_{e}$ spectra. The mean of the $\nu_{e}$ energy spectrum at ND280 is 1.28 GeV in FHC and 1.98 GeV in RHC. The mean of the $\bar{\nu}_{e}$ energy spectrum in RHC is 0.99 GeV. The total integrated $\nu_{e}$ flux at ND280 in FHC is $\Phi_{\nu_{e}}^{FHC}=\left(2.67\pm 0.24\right)\times 10^{11}$ $\rm neutrinos/cm^{2}$ and in RHC is $\Phi_{\nu_{e}}^{RHC}=\left(2.65\pm 0.21\right)\times 10^{10}$ $\rm neutrinos/cm^{2}$. The total integrated $\bar{\nu}_{e}$ flux at ND280 in RHC is $\Phi_{\bar{\nu}_{e}}^{RHC}=\left(1.00\pm 0.10\right)\times 10^{11}$ anti- neutrinos$\rm/cm^{2}$. Figure 1: The neutrino and anti-neutrino fluxes at ND280 in neutrino (FHC) mode (left) and in anti-neutrino (RHC) mode (right). ### 2.2 T2K off-axis near detector ND280 The $2.5^{\circ}$ off-axis near detector, ND280, is located 280 metres from the proton target. The main goal of ND280 is to constrain the neutrino flux and the interaction cross-sections. It is composed of several sub-detectors located inside a 0.2 T magnet, as depicted in Figure 2. The front part is the $\pi^{0}$ detector (P0D) ND280P0D and is optimised to measure neutrino interactions with $\pi^{0}$ production. The rear part is the tracker and it is optimised to measure charged particles produced in neutrino interactions. It consists of two Fine-Grained Detectors ND280FGD , the first of which is composed of layers of plastic scintillator (FGD1) and the second has alternating layers of plastic scintillators and water (FGD2). Figure 2: An exploded view of the T2K near detector, ND280. The neutrino beam enters ND280 from the left. The P0D, FGD1 and FGD2 provide the target mass for neutrino interactions and each is followed by a Time Projection Chamber (TPC1, TPC2 and TPC3) ND280TPC . The TPCs are filled with a gas mixture based on argon and provide excellent track reconstruction with a momentum resolution of roughly 8% for 1 GeV/c tracks. This can be combined with energy loss ($dE/dx$) measurements in order to perform PID of tracks crossing the TPCs. The measured and the expected $dE/dx$ are used to define the "pull" (the difference between the measured mean ionization and the expected one divided by the resolution) of each particle species. The TPC energy loss for negatively and positively charged tracks originating in FGD1 is shown in Figure 3. Notice the region below 200 MeV/c where the electron $dE/dx$ curve crosses with the muon and pion dE/dx curves, and the region around 1 GeV/c where the proton $dE/dx$ curve crosses with the electron $dE/dx$ curve. Figure 3: TPC energy loss for tracks in data originating in FGD1. Left: negatively charged tracks. Right: positively charged tracks. The expected energy loss curves for electrons, muons, pions and protons are also shown. The P0D and the tracker are surrounded by the lead-scintillator Electromagnetic Calorimeter (ECal) ND280ECal and a Side Muon Range Detector (SMRD) ND280SMRD . The ECal measures the energy of photon and electrons (EM energy) and provides additional PID for minimum ionizing particles (MIP), electromagnetic showers (EM) and highly ionizing stopping particles (HIP) like protons. The ECal EM energy is reconstructed under the hypothesis that the energy deposit is due to an electromagnetic shower. Comparing the TPC momentum with the ECal EM energy, electrons can be separated from muons and protons. The ratio of the TPC momentum over the ECal EM energy peaks at unity for electrons and at lower values for muons and protons. The ECal EM energy resolution is approximately 10% at 1 GeV. The ECal PID is based on the longitudinal and lateral profile of ECal clusters to generate probability density functions (PDFs). These are combined for each particle type and PID variable to form a likelihood from the products of the PDFs, see ND280NueSel for details. $R_{MIP/EM}$ is the log-likelihood ratio of the MIP and electron hypothesis and $R_{EM/HIP}$ is the log-likelihood ratio of the electron and proton hypothesis. The $R_{MIP/EM}$ for high purity control samples (90% or better) is shown in Figure 4, where the muon sample comprises cosmic muons and muons produced by neutrino interactions outside ND280 that cross the detector (through-going muons), the electron sample is formed from electron-positron pairs from photon conversions and the protons are from neutrino interactions. The ECal can provide supplementary PID to the TPC, especially in the region around 1 GeV/c where the TPC energy loss curves of electrons and protons cross. Figure 4 also shows the $R_{EM/HIP}$ for showers (classified by $R_{MIP/EM}>0$) with $p>600$ MeV/c only. Although there are some shape differences in data and simulation for $R_{MIP/EM}$ and $R_{EM/HIP}$, the data and simulation efficiencies to select electron (or positrons) and reject muons and protons are similar. The PID efficiencies in the simulation are corrected using the data control samples. Figure 4: Performance of the ECal PID using high purity control samples of cosmic and through-going muons, electrons and positrons from gamma conversions and protons from neutrino interactions. Left: Log-likelihood ratio of the ECal track-shower ($R_{MIP/EM}$) PID. Right: Log-likelihood ratio of the ECal electron-proton ($R_{EM/HIP}$) PID for showers with $R_{MIP/EM}>0$ and $p>600$ MeV/c. Plots are normalised to unity. ## 3 Data samples and MC simulation For FHC, $11.92\times 10^{20}$ protons-on-target (POT) are analysed corresponding to data collected in the periods 2010-2013 and 2016-2017. For RHC, $6.29\times 10^{20}$ POT are analysed corresponding to data collected from 2014 to 2016. The ND280 flux is simulated as described in section 2.1. The (anti-)neutrino interactions with the ND280 detector materials, including nuclear effects, are simulated using NEUT 5.3.2 neutmc and GENIE 2.8.0 geniemc Monte Carlo (MC) generators. The neutrino generators account for differences in the lepton mass for the muon and electron neutrino cross-section computations. However, other effects like radiative corrections, modifications of the pseudoscalar form factors and the effect of form factors to second class vector and axial currents are not considered NueNumuCCQE . NEUT 5.3.2 uses the Llewellyn-Smith formalism LlewellynSmith to describe the CC quasi-elastic neutrino-nucleon cross sections. The spectral function is used as the nuclear model SpectralFunction . The axial mass used for the CC quasi-elastic process is set to 1.21 $\rm GeV/c^{2}$. The simulation of multi- nucleon interactions, where the neutrino interacts with a correlated pair of nucleons, is described using the Nieves et al. model Nieves2p2h . The resonant pion production process with an invariant mass $W\leq 2$ $\rm GeV/c^{2}$ is described by the Rein-Sehgal model ReinSehgal . The resonant axial mass set to 0.95 $\rm GeV/c^{2}$. The deep inelastic scattering (DIS) is calculated for $W>1.3$ $\rm GeV/c^{2}$ and is modeled using the GRV98 parton distribution function GRV98 including the Bodek and Yang corrections BodekYangNeut . Single pion production with $W\leq 2$ $\rm GeV/c^{2}$ is suppressed to avoid double counting with resonant production. Final state interactions describe the transportation of the hadrons produced from neutrino interaction through the nucleus and are simulated using a semi-classical intra-nuclear cascade model. GENIE 2.8.0 uses a different value for the axial mass for quasi-elastic process of 0.99 $\rm GeV/c^{2}$. It relies on a different nuclear model using a relativistic Fermi gas with Bodek and Ritchie modifications BodekRitchie . Resonant production is based on Rein-Sehgal model, same as NEUT. In GENIE the resonant model is not restricted to the single pion decay channel. To avoid double counting with the DIS model, the resonant model is switched off when $W>1.7$ $\rm GeV/c^{2}$. The resonant axial mass is set to 1.12 $\rm GeV/c^{2}$. DIS is simulated similar to NEUT but using slightly different Bodek-Yang corrections BodekYangGenie . A parametrized model of final state interactions (GENIE "hA" model) is used. Detail description of the NEUT and GENIE models can be found in previous T2K publications ND280numucc4pi ; T2KOscLong . GEANT 4.9.4 geant4 is used to transport the final state particles through the ND280 detector. Nominal MC is produced by simulating approximately 10 times the data POT for both NEUT and GENIE. Data-driven reconstruction efficiency corrections are applied to the nominal MC. These corrections are estimated using high-purity ($>$ 90%) control samples of cosmic and through-going muons, electrons and positrons from photon conversions and protons from neutrino interactions. The nominal ND280 MC simulates only the neutrino interactions that occur within the ND280 detector. In reality, neutrino interactions also occur in the surrounding material (sand interactions) and these produce particles that enter ND280. These particles can then affect the event selection by triggering one of the three veto cuts 111Section 4.2 describes the event selection and the veto cuts in the TPC, ECal and P0D sub-detectors. during a beam bunch time window (i.e. an ND280 event) and hence causing an ND280 event to fail the selection cut. This is the sand pile-up effect, which is inherently present in the data, but not in the nominal ND280 MC. To simulate the effect, a second MC simulation is generated (sand MC) to estimate the rate at which sand interactions trigger these veto cuts in coincidence with an ND280 event. To propagate this effect to the nominal ND280 MC there is a pile-up correction, which is a weight that is applied to all ND280 events, for each of the veto cuts. If sand interactions are estimated to trigger a given veto for X% of ND280 events, then a weight of (1 - X/100) is applied to all ND280 events. Since the pile-up rate depends on the beam intensity and on the beam mode (FHC or RHC), the corrections are computed separately for each data period. For the high intensity neutrino beam in 2017, the total pile-up correction is approximately 5%. ## 4 Selection of electron (anti-)neutrino interactions at ND280 The selection of electron (anti-)neutrinos in FGD1 closely follows the steps described in the 2014 FHC CC-$\nu_{e}$ analysis ND280NueSel and is summarised below. There are several reconstruction improvements since the 2014 analysis and additional selection criteria are applied to improve purities. The RHC CC-$\nu_{e}$ selection is identical to the FHC selection, but for the CC-$\bar{\nu}_{e}$ additional selection criteria are applied to remove the proton background. Details are described in section 4.2. ### 4.1 Signal and background definitions A MC event is defined as signal if the selected primary track is an electron (positron) from a CC-$\nu_{e}$ (CC-$\bar{\nu}_{e}$) interaction with the vertex inside the FGD1 fiducial volume, which has a total mass of 919.5 kg, corresponding to $\left(5.54\pm 0.04\right)\times 10^{29}$ nucleons. Backgrounds are separated into four categories: photon, muon, proton and other backgrounds. The photon background category considers events where the selected primary track is an electron or positron from a photon conversion and the true conversion point is inside the FGD1 fiducial volume. Events where the selected primary track is a muon (proton), but misidentified as electron enter the muon (proton) background category. Any other backgrounds including misidentified pions, electrons from photons converting outside of the fiducial volume but reconstructed inside the fiducial volume, electrons from $\pi^{0}$ Dalitz decay and Michel electrons go into the other background category. ### 4.2 Event selection The event selection for CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ events is described in the following: 1. (i) Only events during periods of good beam and detector quality are used. The event time has to be reconstructed within one of the eight distinct beam bunches. 2. (ii) The highest momentum negatively charged (leading negatively charged) FGD1-TPC track, for the CC-$\nu_{e}$ selection, or the highest momentum positively charged (leading positively charged) FGD1-TPC track, for CC-$\bar{\nu}_{e}$ selection, with a vertex in the FGD1 fiducial volume is selected. The leading positively charged track in the CC-$\bar{\nu}_{e}$ selection must also be the highest momentum track (from all negatively and positively charged tracks). 3. (iii) To ensure reliable PID and momentum measurements, the selected leading track is required to have at least 18 TPC hits if it enters the ECal or 36 TPC hits if it does not enter the ECal. The momentum spectra of the selected leading negatively charged and leading positively charged tracks with the minimum number of TPC hits are shown in Figure 5. Notice the large number of protons selected as the leading positively charged track in the RHC CC-$\bar{\nu}_{e}$ selection. Some data-MC discrepancies are visible in the low momentum region which contains the poorly modelled photon and other backgrounds. Figure 5: Momentum distribution of the selected leading negatively charged track with a vertex in the FGD1 fiducial volume for (a) FHC CC-$\nu_{e}$, (b) RHC CC-$\nu_{e}$ and (c) leading positively charged track for RHC CC-$\bar{\nu}_{e}$. The number of MC events is normalized to the data POT. The last bin is the overflow bin. 4. (iv) TPC PID is applied to select electrons and remove minimum-ionizing tracks. Using the electron TPC pull, the leading track must agree with the electron TPC $dE/dx$ hypothesis. If the leading track does not enter the ECal, then additional cuts on the TPC PID are applied using the muon and pion TPC pulls. The event is rejected if the leading track agrees with the muon or pion TPC hypothesis; events around 150 MeV/c, including electrons, are rejected where the only information (TPC) is unable to distinguish them. 5. (v) Additional PID is applied using either the ECal EM energy or the ECal PID depending on the momentum of the leading track as it enters the ECal. To maximize the efficiency, if the leading track has $p>1$ GeV/c and is fully contained in the ECal, the reconstructed ECal EM energy is used to separate EM showers from MIPs and it is required to be larger than 1 GeV. Otherwise the ECal MIP/EM shower PID discriminator $R_{MIP/EM}$ has to agree with the EM shower PID hypothesis. Events that pass the TPC and ECal PID are shown in Figure 6. For the CC-$\bar{\nu}_{e}$ selection a complication arises since the TPC energy loss curves for positrons and protons cross around 1 GeV/c (see Figure 3) leaving a significant amount of proton background. Figure 6: Momentum distribution after the TPC and ECal PID cuts for (a) FHC CC-$\nu_{e}$, (b) RHC CC-$\nu_{e}$ and (c) RHC CC-$\bar{\nu}_{e}$ candidates. The number of MC events is normalized to the data POT. Notice the significant proton background around 1 GeV/c in the CC-$\bar{\nu}_{e}$ selection due to the weakness of the TPC PID to separate positrons from protons, see the text and Figure 3 for details. Additional PID is applied to remove this proton background, see the text for more details. The last bin is the overflow bin. 6. (vi) Search for the paired FGD1-TPC electron or positron track from a potential photon conversion. The paired track must have opposite charge than the leading track, start within 5 cm from the leading track and agree with the electron TPC dE/dx hypothesis. If several paired tracks are found, the pair with the lowest invariant mass is considered since it is more likely to come from a photon conversion. Pairs with invariant mass less than 110 $\rm MeV/c^{2}$ are removed. 7. (vii) Veto P0D, TPC and ECal activity upstream of the vertex and remove events with additional vertices in FGD1. Events with multiple vertices more likely to come from a $\nu_{\mu}$ interaction with one or more $\pi^{0}$ in the final state. 8. (viii) For the CC-$\bar{\nu}_{e}$ selection, additional selection criteria are applied if the leading positively charged track has $p>600$ MeV/c, the region which is contaminated by the proton background. If the leading positively charged track produce shower activity in FGD2 then it is selected. If the leading positively charged track enters the ECal, the proton background can be removed by comparing the ECal EM energy ($E$) and the TPC momentum ($p$) using a cut $E/p>0.65$. In addition, the $R_{EM/HIP}$ shower PID discriminator has to agree with the EM shower hypothesis. 9. (ix) For the CC-$\bar{\nu}_{e}$ selection, if the leading positively charged track stops in FGD2, the FGD2 energy loss must not agree with the proton hypothesis. 10. (x) Remove external background by comparing the time stamps of the leading track between FGD1 and ECal. This cut aims to remove tracks originating in the ECal and stop in FGD1 but are mis-reconstructed with the wrong direction. 11. (xi) Check if the leading track is broken inside FGD1. A track is broken if it originates in FGD1 and is not reconstructed as a single track, but is broken into two or more components. In such pathological cases the leading track could originate outside the fiducial volume but mis-reconstructed within it. If the leading track follows an isolated FGD1 track then the event is removed. Figure 7 summarises the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections. Figure 7: Summary of the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections in FGD1. See the text for the details of each cut. ### 4.3 Final selection The momentum and angular (with respect to the neutrino direction) distributions of all selected CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ candidates are shown in Figures 8 and 9, respectively. These plots also show the total systematic uncertainty on the MC event yields, which is discussed in section 6. A significant data deficit is observed at low momentum ($p<600$ MeV/c) in the FHC CC-$\nu_{e}$ channel. In this region the photon background is dominant which has significant systematic uncertainties associated with the $\pi^{0}$ production. Roughly a third of the photon background comes from neutrino interactions outside of FGD1. These external backgrounds could originate from neutrino interactions on heavy targets, like iron, copper or lead with significant final state interaction systematic uncertainties. In addition, roughly another third of the photon background come from NC interactions which are poorly measured. The final third of the photon background come from CC-$\nu_{\mu}$ and CC-$\bar{\nu}_{\mu}$ interactions, usually when the muon is emitted in high angles and it is lost. In such occasions the most energetic of the other tracks is selected as the leading track. A similar data deficit is also observed in the statistically poorer RHC CC-$\bar{\nu}_{e}$ channel. In addition, an excess of events has been observed in the RHC channels at high momenta (more visible in the RHC CC-$\bar{\nu}_{e}$ channel). For the photon background produced from $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ interactions in FGD1, roughly 10% is coming from NC DIS interactions in all three selections. The relevant fraction of FGD1 CC DIS events entering the photon background is approximately 4% in CC-$\bar{\nu}_{e}$ selection, 16% in RHC CC-$\nu_{e}$ selection and 20% in FHC CC-$\nu_{e}$ selection. The differences are due to the additional selection criteria applied to CC-$\bar{\nu}_{e}$ and the presence of protons which can be selected as the leading track instead of the primary muon or background positron. Figure 8: Momentum distribution of the selected electron and positron candidates for (a) FHC CC-$\nu_{e}$, (b) RHC CC-$\nu_{e}$ and (c) RHC CC-$\bar{\nu}_{e}$. The number of MC events is normalized to the data POT. The effect of the total systematic uncertainty on the MC event yields (see section 6 for details) is also shown on these plots. The last bin is the overflow bin. Figure 9: Angular distribution of the selected electron and positron candidates for (a) FHC CC-$\nu_{e}$, (b) RHC CC-$\nu_{e}$ and (c) RHC CC-$\bar{\nu}_{e}$. The number of MC events is normalized to the data POT. The effect of the total systematic uncertainty on the MC event yields (see section 6 for details) is also shown on these plots. The last bin includes all backward-going candidates. Most of the efficiency loss is observed at low momentum since the electron and muon/pion $dE/dx$ energy loss curves cross around 150 MeV/c (see Figure 3). In addition, high angle tracks that do not enter the TPC are not selected and the events are lost. Another important source of efficiency loss is due to electron shower or bremsstrahlung in FGD1. As a result the primary electron track does not enter the TPC or another track is selected as the leading track. As estimated from the MC, 35-45% of the signal electrons or positrons are lost because the primary electron track does not enter the TPC. The efficiency loss is larger in the FHC CC-$\nu_{e}$ channel since the electron momentum spectrum is softer and at higher angles. The true vs reconstructed momentum and angular distributions in the MC for the selected signal electrons and positrons are shown in Figure 10. The effect of bremsstrahlung is visible as the reconstructed momentum spectrum is biased towards lower momenta. A summary of efficiencies and purities is shown in Table 1. Figure 10: Distribution of the true vs reconstructed values of momentum (left) and angle (right) for signal electrons and positrons that passed all cuts in the MC. The effect of bremsstrahlung is visible on the left plot, see the text for details. The muon mis-identification probability (probability of a muon to be mis- identified as an electron after applying the PID) was studied in previous T2K publications ND280NueSel with very good agreement between data and MC. Similarly, the proton mis-identification probability is important for the CC-$\bar{\nu}_{e}$ selection. A high-purity independent sample of protons has the same PID criteria as the CC-$\bar{\nu}_{e}$ selection applied and the number of protons that survive is checked. An independent control sample that can be used is the FHC CC-$\bar{\nu}_{e}$ selection. This channel has a tiny signal contribution and a much larger proton background and it is not used in the cross-section measurements. Before applying the proton rejection cuts (viii) and (ix), approximately 94% of the leading tracks selected with $p>600$ MeV/c and not entering the ECal are protons. The measured proton mis- identification probability is the fraction of protons that survive from these independent proton enriched samples and is $\left(4.6\pm 0.8\right)$% for the data compared to $\left(5.0\pm 0.3\right)$% in the MC. The errors are statistical only. The proton purity is lower in the case where the leading track enters the ECal and is approximately 70% with the rest to be mostly positrons. Due to the relatively low proton purity of this sample, only an approximate proton mis-identification probability can be measured in this case, $\left(9.4\pm 0.1\right)$% in the data compared to $\left(11.9\pm 0.05\right)$% in the MC. ### 4.4 Event selection using alternative MC The CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections in the MC are repeated using GENIE (2.8.0) instead of NEUT (5.3.2) MC. There are some differences between these two neutrino generators, see section 3 and for more details the description in ND280numucc4pi . One of the most important is that the neutrino multi-nucleon interaction simulations are turned-off in this version of GENIE. Efficiencies and purities for NEUT and GENIE agree quite well. Compared to the selected events, both NEUT and GENIE predictions disagree with data at low momenta with the FHC CC-$\nu_{e}$. The prediction of the photon background in particular is similar in both neutrino generators. Tables 1 and 2 summarize the event selections using NEUT and GENIE. Table 1: Summary of efficiency, purity and number of MC events normalised to the $11.92\times 10^{20}$ POT in the FHC beam and $6.29\times 10^{20}$ POT in the RHC beam for the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ channels using NEUT (5.3.2) and GENIE (2.8.0) MC, in addition to the number of data events that survive all cuts in each channel. Channel | Efficiency | Purity | MC Events | Data events ---|---|---|---|--- NEUT FHC CC-$\nu_{e}$ | 0.26 | 0.54 | 797.07 | 697 GENIE FHC CC-$\nu_{e}$ | 0.27 | 0.53 | 769.17 | 697 NEUT RHC CC-$\nu_{e}$ | 0.33 | 0.48 | 175.92 | 176 GENIE RHC CC-$\nu_{e}$ | 0.33 | 0.44 | 168.10 | 176 NEUT RHC CC-$\bar{\nu}_{e}$ | 0.31 | 0.54 | 99.99 | 95 GENIE RHC CC-$\bar{\nu}_{e}$ | 0.30 | 0.51 | 99.21 | 95 Table 2: Breakdown of the number of CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ events selected in FGD1 according to their category for NEUT (5.3.2) and GENIE (2.8.0) MC. The number of events is normalized to data POT. The photon background is separated to events with a true vertex in FGD1 (In-FGD) and to events with a true vertex out of FGD1 (OO-FGD). Channel | Signal (%) | In-FGD $\gamma$ (%) | OO-FGD $\gamma$ (%) | $\mu^{\pm}$ (%) | Proton (%) | Other (%) ---|---|---|---|---|---|--- NEUT FHC CC-$\nu_{e}$ | 429.16 (53.9) | 162.23 (20.4) | 78.09 (9.8) | 35.67 (4.5) | - | 91.92 (11.4) GENIE FHC CC$-\nu_{e}$ | 409.23 (53.5) | 152.56 (20.0) | 78.00 (10.2) | 33.29 (4.4) | - | 96.10 (12.0) NEUT RHC CC-$\nu_{e}$ | 83.62 (47.5) | 42.41 (24.1) | 20.23 (11.5) | 6.38 (3.6) | - | 23.28 (13.2) GENIE RHC CC-$\nu_{e}$ | 73.28 (43.6) | 43.46 (25.9) | 21.67 (12.9) | 6.33 (3.8) | - | 23.35 (13.9) NEUT RHC CC-$\bar{\nu}_{e}$ | 53.85 (53.9) | 18.76 (18.8) | 12.47 (12.5) | 1.22 (1.2) | 6.52 (6.5) | 7.17 (7.2) GENIE RHC CC-$\bar{\nu}_{e}$ | 50.49 (51.2) | 21.28 (21.5) | 11.43 (11.5) | 1.74 (1.7) | 7.20 (7.3) | 7.07 (7.1) ## 5 Photon background control samples Since the photon background is the most important in the electron (anti-)neutrino selections, a dedicated photon control sample of electrons and positrons from photon conversions is selected to constrain this background. Photon candidates are selected from two nearby electron-like FGD1-TPC tracks of opposite charge with low invariant mass that start in the FGD1 fiducial volume. ### 5.1 Selection of photon candidates The steps to select photon candidates are: 1. (i) Only events during periods of good beam and detector quality are used. The event time has to be reconstructed within one of the eight distinct beam bunches. 2. (ii) The highest momentum negatively charged or highest momentum positively charged FGD1-TPC track (leading track) with a vertex in the FGD1 fiducial volume is selected. 3. (iii) The leading track must be compatible with the electron TPC $dE/dx$ hypothesis. If the leading track enters the ECal, and has momentum $p>1$ GeV/c and ECal energy E, then $E/p>0.5$ is required in order to clean up the high momentum tail. 4. (iv) Require a second track with opposite charge to the leading track, also compatible with the electron TPC $dE/dx$ hypothesis and with a starting position within 5 cm from the primary track. 5. (v) The invariant mass calculated from the leading and paired tracks must be less than 55 $\rm MeV/c^{2}$. The distributions of the invariant mass of the selected $e^{-}e^{+}$ pairs are shown in Figure 11. The invariant mass cut is very effective to remove backgrounds from misidentified muons, protons and electrons from CC-$\nu_{e}$ interactions. Figure 11: Invariant mass of electron-like FGD1-TPC pairs with opposite charge for (a) FHC selecting electron as the leading track, (b) RHC selecting electron as the leading track and (c) RHC selecting positron as the leading track. The number of MC events is normalized to the data POT. Last bin is the overflow bin. The arrow at 55 MeV/$c^{2}$ indicates the final photon to $e^{-}e^{+}$ conversion cut. 6. (vi) Although the photon selection at this stage is very pure, it is contaminated by external photons (photons from neutrino interactions outside FGD1). To remove external photons the same veto cuts used in the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections are applied. The signal and background categories are the same as for the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections. The momentum and angular distributions of the selected photon candidates are shown in Figures 12 and 13, respectively. The systematic uncertainties on the MC event yields are also shown in these plots, see section 6 for details. A MC excess below 300 MeV/c is visible. In the angular distributions a significant MC excess is observed at high angles in the FHC CC-$\nu_{e}$ selection but not in the photon control selection (Figures 9 and 13). Figure 12: Momentum distribution of the selected photon candidates for (a) FHC selecting electron as the leading track, (b) RHC selecting electron as the leading track and (c) RHC selecting positron as the leading track. The number of MC events is normalized to the data POT. The effect of the total systematic uncertainty on the MC event yields (see section 6 for details) is also shown on these plots. Last bin is the overflow bin. Figure 13: Angular distribution of the selected photon candidates for (a) FHC selecting electron as the leading track, (b) RHC selecting electron as the leading track and (c) RHC selecting positron as the leading track. The number of MC events is normalized to the data POT. The effect of the total systematic uncertainty on the MC event yields (see section 6 for details) is also shown on these plots. The last bin includes all backward-going candidates. The purity of the photon control samples is approximately 80% when selecting electrons and 85% when selecting positrons. A significant fraction of the selected photon candidates is classified in the other background category where the photons are coming from a true conversion point outside the FGD1 fiducial volume, but are mis-reconstructed inside of it. Including these events in the photon category definition increases the purity to approximately 90%. The rest of the other background contributes (5 - 6)% in the photon control samples and comes from $\pi^{0}$ Dalitz decay, general mis- reconstructions like broken tracks and accidental matching when at least one of the two tracks selected in the pair is not electron or positron. The signal leakage (CC-$\nu_{e}$ or CC-$\bar{\nu}_{e}$) in the photon control samples is around (3 - 4)% when the selected leading track is an electron. The leakage is otherwise negligible when the selected leading track is a positron. The muon background entering the photon control samples is less than 1% in all of the cases. When selecting electrons as the leading track in the photon control selections, approximately 40% of the photon candidates come from external photons, approximately 30% come from NC interactions in FGD1 and the other 30% come from CC interactions in FGD1. When selecting positrons as the leading track in the photon control selections the contributions are slightly different. Approximately 45% of the photon candidates come from external photons, approximately 35% come from NC interactions in FGD1 and 20% come from CC interactions in FGD1. Often the event is rejected if the selected highest positively charged momentum track is a proton. However, since the protons are invisible when selecting negatively charged tracks the same event could be selected when searching for the highest momentum negatively charged track. This explains the difference in the number of photon candidates in RHC when the leading track selected is the electron or the positron. ### 5.2 Comparisons with the photon background in the standard selections Although the photon control samples are of high purity they have some differences compared to the photon background entering the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections. The main reason is that the photon control selection requires both the electron and positron to be reconstructed in the TPC, while the photon background is mostly related to events where either the electron or positron is lost, usually when it is not very energetic or emitted at high angles. As a result, the photon background consists mostly of highly asymmetric events where most of the energy of the photon goes into one of two electrons. For high angle events it is predominantly due to one of the two electrons being lost, resulting in more high angle photon background in the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections. This angular dependence will introduce different external photons to the photon background and the photon control selection. Most of the external photons entering the photon control samples come from neutrino interactions in the P0D or in the aluminium frame of TPC1. For the photon background, however, a significant population of external photons are also from neutrino interactions in the ECals. The photons mostly come from $\pi^{0}$ decays and Table 3 shows the different contributions to the photon background and the photon control selections from CC and NC interactions and from external photons. Despite the differences discussed, the origin of the photon background entering the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections and the photon control selections is similar. This provides confidence that the photon control samples can be used to constrain the photon background in the signal channels. Additional simulation studies are also performed to check for shape variations in momentum and angle in the photon selections and in the photon background in the signal selections. These studies include the variation of the relevant fraction of CC/NC photon events by a factor of 2, weighting the nominal MC by varying the Delta resonance width by $\pm 1\sigma$ and varying the external photon background between (40 - 75)% based on the target material the neutrino interaction occurred. In all the cases the effect on the momentum and angular shapes is found to be very small. Table 3: Comparison of the photon background entering the CC electron (anti-)neutrino selections and the photon control selections split down to different $\pi^{0}$ contributions from CC and NC interactions in FGD1 and to external photons. Out of fiducial volume (OOFV) photons are separated into events where the true neutrino vertex is in FGD1 (In-FGD) and into events where the true neutrino vertex is out of FGD1 (OO-FGD). Interaction Type | FHC CC-$\nu_{e}$ (%) | Photon Selection (%) | RHC CC-$\nu_{e}$ | Photon Selection (%) | RHC CC-$\bar{\nu}_{e}$ | Photon Selection (%) ---|---|---|---|---|---|--- CC $0\pi^{0}$ | 4.5 | 4.3 | 4.8 | 6.9 | 1.1 | 5.4 CC $1\pi^{0}$ | 15.7 | 14.6 | 14.7 | 12.8 | 6.7 | 11.8 CC $>1\pi^{0}$ | 6.1 | 4.7 | 5.4 | 4.5 | 1.9 | 3.8 NC $0\pi^{0}$ | 3.6 | 3.6 | 2.6 | 3.0 | 1.9 | 2.3 NC $1\pi^{0}$ | 24.8 | 28.5 | 26.7 | 30.5 | 35.1 | 31.1 NC $>1\pi^{0}$ | 4.3 | 5.1 | 4.7 | 4.2 | 2.8 | 3.6 OOFV (In-FGD) | 8.5 | 7.4 | 8.8 | 7.8 | 10.7 | 8.9 OOFV (OO-FGD) | 32.5 | 31.8 | 32.3 | 30.2 | 39.9 | 33.1 ## 6 Systematic uncertainties Systematic uncertainties affecting the MC prediction on event yields are separated into five main categories: cross-section modelling, final state interactions, detector, external backgrounds and flux. Cross-section modelling. The cross-section interaction modelling in NEUT and GENIE is briefly described in section 3 and in detail in previous T2K publications ND280numucc4pi ; T2KOscLong . In this section, the systematic uncertainties relevant to cross-section modelling parameters will be briefly discussed. Neutrino cross-section parameters in NEUT relevant to charged- current quasi-elastic interactions are the axial mass ($M_{A}^{QEL}$ = 1.21 $\pm$ 0.41 $\rm GeV/c^{2}$), binding energy ($E_{B}^{C}$ = 25.0 $\pm$ 9.0 MeV) and Fermi momentum ($p_{F}^{C}$ = 223.0 $\pm$ 31.0 MeV/c). Binding energy and Fermi momentum are target dependent, for this analysis only those relevant to carbon are considered. For multi-nucleon interactions, a 100% normalization uncertainty is assumed. The CC resonant production model has three parameters in NEUT: the axial mass ($M_{A}^{RES}$ = 0.95 $\pm$ 0.15 $\rm GeV/c^{2}$), the normalization of the axial form factor for resonant pion production ($CA_{5}^{RES}$ = 1.01 $\pm$ 0.12) and the normalisation of the isospin non- resonant component ($I_{\frac{1}{2}}$ = 1.3 $\pm$ 0.2). For the CC DIS process an energy dependent normalisation uncertainty (10% at 4 GeV) is considered. For CC coherent interactions a 100% normalisation uncertainty is considered. For neutral-current interactions, due to poor constraints from external data, a 30% normalisation uncertainty is applied. The effect of the cross-section uncertainties on the event yields is evaluated by shifting each cross-section parameter by $\pm 1\sigma$ and shifting the nominal MC. Final State Interactions The pion final state interaction systematic uncertainties include the effects of absorption, inelastic scattering, charge exchange and quasi-elastic scattering inside the nucleus. A full description can be found in previous T2K publications ND280numucc4pi ; T2KOscLong . Similarly with the cross-section uncertainties, the effect of final state interaction systematic uncertainties on the event yields is evaluated by varying simultaneously the final state interaction effects by $\pm 1\sigma$ and shifting the nominal MC. Detector. Detector systematic uncertainties encapsulate the performance of each ND280 sub-detector (FGDs, TPCs and ECals). They are applied to simulated events and are separated in three categories: normalization, selection efficiency and variation of the observable. Normalization systematics are applied as a single weight to all events. Efficiency systematics are applied as a weight that depends on one or more observables. Variation systematics are treated by varying the observables and redoing the event selections. Detector systematic uncertainties considered and their treatment are summarised in Table 4. Detector systematics are evaluated using high purity ($>$ 90%) control samples from cosmic and through-going muons, electrons and positrons from photon conversions and protons from neutrino interactions. ECal related uncertainties are evaluated using the same methodology described in ND280NueSel . All other detector systematics, except FGD2 shower efficiency, are evaluated in the same way as explained in ND280NueSel ; ND280numucc4pi ; T2KOscLong . The FGD2 shower efficiency describes the probability of electrons and protons originating in FGD1 to shower in FGD2. Since FGD2 is a thin detector and cannot contain showers, a shower is defined when multiple FGD2-TPC3 tracks are produced when the leading track passes through FGD2. Since this systematic is only relevant for the CC-$\bar{\nu}_{e}$ channel, the uncertainty is evaluated using events with single electron or proton tracks in the neutrino beam originating in FGD1, passing through FGD2 and comparing the FGD2 shower efficiencies for data and MC. Table 4: List of detector systematic uncertainties and their treatment for simulated events. Normalization systematics are applied as a single weight to all events. Efficiency systematics are applied as a weight that depends on one or more observables. Variation systematics are treated by varying the observables and redoing the event selection. Systematic | Treatment | Comment ---|---|--- TPC tracking efficiency | efficiency | TPC charge mis-identification | efficiency | TPC momentum resolution and scale | variation | B-field distortions | variation | TPC PID | variation | FGD-TPC matching efficiency | efficiency | TPC-ECal matching efficiency | efficiency | FGD2 PID | variation | Only applied to CC-$\bar{\nu}_{e}$ FGD2 shower efficiency | efficiency | Only applied to CC-$\bar{\nu}_{e}$ FGD1 mass | normalisation | TPC, P0D and ECal pile-up | normalisation | ECal $R_{MIP/EM}$ PID | efficiency | ECal $R_{EM/HIP}$ PID | efficiency | Only applied to CC-$\bar{\nu}_{e}$ ECal EM energy resolution and scale | variation | Pion and proton secondary interactions | efficiency | Sand interactions | efficiency | FGD1-ECal time resolution | variation | External backgrounds. These are related to the uncertainties associated with photons (or other particles) produced outside of the FGD1, either in other sub-detectors or outside of ND280, that propagate inside FGD1. A large number of these neutrino interactions are on heavier nuclear targets (aluminium, iron and lead) with considerable cross-section modelling uncertainties. A detailed study of the external photon propagation in ND280 was performed in ND280SingleGamma but only in limited angular regions. Outside these angular regions there are large data/MC differences due to the poor simulation of inactive material. Since the method developed in ND280SingleGamma is very sensitive to the material density and composition (small changes can cause large variation in systematic uncertainties), conservatively a 100% systematic uncertainty on the external photon production and propagation is assumed. The effect on the momentum and angular shapes for both the photon background in the signal selections and the photon selections is studied with additional simulations. The external photon events in the simulation are varied between (40 - 75)% based on the target material the neutrino interaction occurred. The effect on both momentum and angular shapes is found to be negligible. Flux. Flux systematic uncertainties are calculated as a function of the neutrino energy and they are correlated between the neutrino flavours and between the neutrino and anti-neutrino beams. Flux systematic uncertainties are larger at the high energy tail of the neutrino spectrum and for the $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ fluxes are in the range (7.0 - 14)%. The $\nu_{e}$ and $\bar{\nu}_{e}$ flux systematic uncertainties are shown in Figure 14 and are dominated by the systematic uncertainties on hadron production. The evaluation of the flux systematic uncertainties can be found in previous T2K publications T2KOscLong ; t2kfluxerror . Figure 14: Flux systematic uncertainties for the FHC $\nu_{e}$ flux (top left), RHC $\nu_{e}$ flux (top right) and RHC $\bar{\nu}_{e}$ flux (bottom). ### 6.1 Effect of systematic uncertainties on the event yields A summary of systematic uncertainties on signal and background MC event yields for the CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ selections is shown in Table 5. The systematic uncertainties on signal yields are dominated by the flux (8 – 10%) and cross-section modelling (13 – 14%). The larger cross-section systematic uncertainties come from the large uncertainties considered on the quasi-elastic axial mass $M_{A}^{QEL}$ and multi-nucleon interactions, each contributing (6.5 – 8.5)% to the total cross-section systematic uncertainty. Detector systematic uncertainties on signal yields are (2 – 4)% with the most important being the TPC PID and TPC-ECal matching efficiencies. For CC-$\bar{\nu}_{e}$, the ECal PID and FGD2 shower efficiency, which are related to the proton background rejection, are also important. For an inclusive CC selection, final state interaction systematic uncertainties on signal yields are small. They are only considered if a charged pion, after final state interactions, becomes more or less energetic than the primary electron or when there is a $\pi^{0}$ involved as the secondary electrons can be more or less energetic than the primary electron. The total systematic uncertainty on the signal yields is approximately (16 – 17)% in all the channels. The systematic uncertainties on the MC background event yields are separated into photon background and all other backgrounds. The total systematic uncertainties on the MC photon background event yields are approximately (23 – 26)% in all channels and are dominated by the cross-section and external systematic uncertainties. Cross-section systematic uncertainties (16 – 19)% are dominated by the charged-current and neutral-current resonant and DIS production models. The flux systematic uncertainties are around 8% and the final state interaction systematic uncertainties are (1.5 – 3.0)%. Detector systematic uncertainties are (3 – 6)%, with TPC PID, FGD1 and ECal time resolutions, TPC-ECal matching efficiency and pion secondary interactions being the most important. Approximately a third of the photon background comes from neutrino interactions outside FGD1, either in other sub-detectors or outside the ND280 and the majority of these events populate the low momentum and/or high angle regions. The systematic uncertainties on the other backgrounds MC event yields vary from (19 – 33)% since different sources of backgrounds contribute to each channel. The biggest difference comes from the external background which dominates the systematic uncertainties on the other background event yields and is different in each channel since the neutrino flux is different. Flux systematic uncertainties are around 8% and the cross-section systematic uncertainties are around (11 – 12)%. Detector systematic uncertainties are (4.0 – 6.5)%, which are larger than the corresponding detector systematic uncertainties for signal and photon background event yields. Table 5: Summary of systematic uncertainties on MC signal and background event yields. The total systematic uncertainty is the quadratic sum of all the systematic sources. Possible correlations between the different systematic sources are ignored. | Source of uncertainty | Signal (%) | $\gamma$ background (%) | Other backgrounds (%) ---|---|---|---|--- FHC CC-$\nu_{e}$ | Detector | 2.96 | 3.02 | 3.91 External background | 0.00 | 17.25 | 29.07 Flux | 8.92 | 7.61 | 7.60 Final State Interactions | 0.52 | 2.78 | 3.72 Cross-section | 13.60 | 16.54 | 11.18 Total | 16.54 | 25.41 | 32.51 RHC CC-$\nu_{e}$ | Detector | 2.12 | 3.09 | 5.12 External background | 0.00 | 12.71 | 17.56 Flux | 8.11 | 8.28 | 8.23 Final State Interactions | 0.98 | 1.48 | 4.97 Cross-section | 13.45 | 17.71 | 10.67 Total | 15.88 | 23.57 | 23.26 RHC CC-$\bar{\nu}_{e}$ | Detector | 3.46 | 5.68 | 6.46 External background | 0.00 | 14.90 | 7.20 Flux | 9.95 | 8.33 | 8.01 Final State Interactions | 0.39 | 1.95 | 7.94 Cross-section | 12.98 | 18.88 | 12.01 Total | 16.72 | 26.15 | 19.11 ### 6.2 Effect of systematic uncertainties on the photon control samples The systematic uncertainties on the photon control samples are roughly (20 – 23)% and are summarised in Table 6. The dominant sources are coming from the external background and cross-section modelling. Table 6: Effect of the systematic uncertainties on the photon control sample MC event yields selecting either electron as the leading track ($\gamma$-Elec.) or positron as the leading track ($\gamma$-Posi.). The total systematic uncertainty is the quadratic sum of all the systematic sources. Possible correlations between the different systematic sources are ignored. Systematic uncertainty | FHC $\gamma$-Elec. (%) | RHC $\gamma$-Elec. (%) | RHC $\gamma$-Posi. (%) ---|---|---|--- Detector | 2.35 | 1.81 | 1.72 External background | 14.24 | 9.57 | 11.10 Flux | 7.62 | 8.29 | 8.26 Final State Interactions | 2.62 | 1.49 | 1.93 Cross-section | 16.49 | 15.28 | 15.67 Total | 23.35 | 19.98 | 21.06 ## 7 Fit model The flux integrated single differential cross-section as a function of the electron or positron true momentum $p$ or true scattering angle $\rm\cos(\theta)$ is expressed as $\frac{d\sigma_{i}}{dk_{i}}=\frac{N_{i}}{\epsilon_{i}}\times\frac{1}{T\Phi\Delta k_{i}},$ (1) where $k$ is either $p$ or $\cos(\theta)$, $N_{i}$ is the number of signal events in bin $i$, $\epsilon_{i}$ is the efficiency in bin $i$, $\Phi$ is the neutrino flux, $T$ the number of target nucleons and $\Delta k_{i}$ is the true momentum or true scattering angle bin interval. The number of signal events in each bin is calculated using an extended, binned maximum likelihood fit. The PDFs are constructed from histogram templates using ROOT’s histogram-based HistFactory fit package histfactory , which is based on the RooStats roostats and RooFit roofit packages. The fit is performed simultaneously on all the signal channels (FHC and RHC CC-$\nu_{e}$ and RHC CC-$\bar{\nu}_{e}$) and their corresponding photon control channel. Each channel is broken down to angular regions and each angular region is broken down to one dimensional templates in momentum for signal, photon background and other backgrounds. For the photon control channels the small signal contribution is merged in the other backgrounds. A likelihood is constructed from all the signal and background templates, nuisance parameters $\vec{\theta}$ and their constraints $C\left(\theta_{\kappa}^{0},\theta_{\kappa}\right)$ and a set of scaling parameters $c$ and $g$ controlling the signal and photon background respectively, given the observed data $\vec{N}$ $\displaystyle L\left(\vec{N}|c,g,\vec{\theta}\right)=$ (2) $\displaystyle\left[\prod_{i=1}^{N_{region}}\prod_{j=1}^{N_{bin}}\frac{\left[c_{ij}S_{ij}(\vec{\theta})+g_{i}B_{ij}^{\gamma}(\vec{\theta})+B_{ij}^{other}(\vec{\theta})\right]^{n_{ij}}}{n_{ij}!}e^{-\left[c_{ij}S_{ij}(\vec{\theta})+g_{i}B_{ij}(\vec{\theta})+B_{ij}^{other}(\vec{\theta})\right]}\right]$ $\displaystyle\times\left[\prod_{i=1}^{N_{region}}\prod_{l=1}^{N_{bin;PC}}\frac{\left[g_{i}B_{il;PC}^{\gamma}(\vec{\theta})+B_{il;PC}^{other}(\vec{\theta})\right]^{m_{il}}}{m_{il}!}e^{-\left[g_{i}B_{il;PC}^{\gamma}(\vec{\theta})+B_{il;PC}^{other}(\vec{\theta})\right]}\right]$ $\displaystyle\times\prod_{k=1}^{N_{syst}}C\left(\theta_{\kappa}^{0},\theta_{\kappa}\right),$ where $N_{region}$ is the number of angular regions which is the same for signal and photon control channels, $N_{bin}$ ($N_{bin;PC}$) is the number of bins in signal (photon control) region, $S_{ij}$ are the signal templates contributing to reconstructed bin $j$ for region $i$, $B_{ij}^{\gamma}$ ($B_{il;PC}^{\gamma}$) is the number of photon events in reconstructed bin $j$ ($l$) for signal (photon control) region $i$, $B_{ij}^{other}$ ($B_{il;PC}^{other}$) is the number of other background events in reconstructed bin $j$ ($l$) for signal (photon control) region $i$, $n_{ij}$ ($m_{il}$) are the number of entries in each bin in signal (photon control) region and $N_{syst}$ is the number of nuisance parameters. ### 7.1 Propagation of systematic uncertainties Systematic uncertainties are included in the fit as nuisance parameters and are calculated as $\pm 1\sigma$ variations of the nominal samples in the signal and photon control samples, $S_{ij}(\vec{\theta})$, $B_{ij}^{\gamma}(\vec{\theta})$, $B_{ij}^{other}(\vec{\theta})$, $B_{il;PC}^{\gamma}(\vec{\theta})$ and $B_{il;PC}^{other}(\vec{\theta})$. These variations can either change the normalisation or produce bin-dependent shape differences or have a correlated effect on shape and normalisation. For each variation (or set of variations) a nuisance parameter is used to interpolate between the $\pm 1\sigma$ uncertainties with a Gaussian constraint. Systematic uncertainties that are common between samples or channels are fully correlated in the fit. A summary of the nuisance parameters included in the fit is shown in Table 7. Variations from the cross-section uncertainties are calculated by varying each cross-section parameter by $\pm 1\sigma$ and changing the nominal samples. Some of the cross-section uncertainties may produce asymmetric variations and these are considered in the fit. Variations related to the final state interaction systematic uncertainties, including their correlations, are estimated following the methodology described in T2KOscLong . Variations from the flux uncertainties are calculated using the full beam covariance taking into account all the correlations between the neutrino beams, neutrino flavours and energy bins. Variations of the nominal samples arising from the detector, pile-up and external background systematics are evaluated using MC simulations varying the systematics to change the number of events in each reconstructed bin. Three nuisance parameters are used for the three pile-up systematics in each beam mode (FHC or RHC). Four nuisance parameters in each beam mode are used to describe the external background systematic uncertainties. The external backgrounds are separated based on their origin (ND280 or sand interactions), their background category (photon or other backgrounds) and their beam mode (FHC or RHC). MC statistical uncertainties, describing the finite size of the simulated events in each sample, are also included as nuisance parameters in the fit following the Barlow-Beeston barlowbeeston approach considering one nuisance parameter per channel and bin. Table 7: Summary of nuisance parameters related to systematic uncertainties considered in the fit. Source of uncertainty | Number of parameters | Constraint | Variation type ---|---|---|--- MC statistical | 29 | Poissonian | One per bin Pile-up | 6 | Gaussian | Normalisation External backgrounds | 8 | Gaussian | Shape and normalisation Detector and flux | 15 | Gaussian | Shape and/or normalisation Cross-section and final state interactions | 13 | Gaussian | Shape and normalisation ### 7.2 Binning choice The choice of the binning depends on a number of factors, some of the most important are: sufficient number of signal events in each bin, isolation of the backgrounds in specific $p-\theta$ regions, event migration due to bremsstrahlung and flat efficiency corrections. The first criterion for the binning choice is to not consider high angle events ($\theta>45^{\circ}$) since the acceptance due to detector effects is almost zero. In addition, the photon background is large and the statistics in the photon control channels is poor. The high angle regions and the low momentum ($p<300$ MeV/c) bins are background enriched and are kept in the fit as background validation regions. Approximately 75% of the external photon background is contained in the low momentum and high angle regions. The signal contribution in the low momentum bins ($p<300$ MeV/c) is tiny and, to help the fit performance, it is kept constant in the fit. Two angular regions are considered to better describe the middle and forward angular cases. The momentum bins are identical in each angular region. The momentum bins are optimised to minimize the effect of bremsstrahlung. Since bremsstrahlung is not a detector smearing effect, but a physics effect depending on the initial electron kinematics and the material propagated, special requirements are considered to minimize this effect. The (anti-)correlations between the momentum bins introduced by bremsstrahlung are studied with MC simulations requiring them to be less than 50%. If the chosen momentum binning fails this requirement, the momentum bins are expanded to reduce the migration of events due to bremsstrahlung and the MC simulations are repeated. Due to the large momentum bins chosen in this analysis, the effect of bremsstrahlung can be efficiently handled in the fit. Signal efficiencies are also a significant factor for the binning choice as they should be flat to avoid model dependencies. The efficiencies in the two angular regions in each signal channel are shown in Figure 15 and are relatively flat with some small fluctuations observed between NEUT and GENIE and in the low statistics bins. Although the cross-section measurements are calculated in one dimension, fitting in $p-\theta$ is important to check for model dependencies due to the efficiency corrections. After the total statistical and systematic uncertainties are applied on signal efficiencies, the efficiency errors are artificially inflated to cover differences between NEUT and GENIE and variations between momentum bins. The efficiencies with statistical, systematic and inflation uncertainties are shown in Figure 16. The binning choice for each signal channel is shown in Table 8. In total there are 9 free parameters controlling the photon background (one for each angular region) and 17 free parameters controlling the signal (one for each bin in the table, except for the six lowest momentum bins which are kept constant in the fit since the number of signal events is negligible). Figure 15: Signal efficiencies for NEUT and GENIE MC using a finer binning than used in the cross-section measurements. Errors are statistical only. Figure 16: Signal efficiencies in different angular regions for the three samples for NEUT MC with statistical, systematics and inflation uncertainties. Table 8: Summary of the binning for CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ inclusive channels included in the fit. Validation bins are background enriched and are used as extra fit validation regions. These bins are excluded from the cross-section measurements. | Angular region ($\cos(\theta)$) | Momentum bin (GeV/c) | Comment ---|---|---|--- FHC CC-$\nu_{e}$ | -1.00 - 0.7071 | 0 - 30 | Validation bin 0.7071 - 0.88 | 0 - 0.3 | Validation bin 0.7071 - 0.88 | 0.3 - 1.6 | 0.7071 - 0.88 | 1.6 - 3.2 | 0.7071 - 0.88 | 3.2 - 30 | 0.88 - 1.00 | 0 - 0.3 | Validation bin 0.88 - 1.00 | 0.3 - 1.6 | 0.88 - 1.00 | 1.6 - 3.2 | | 0.88 - 1.00 | 3.2 - 30 | RHC CC-$\nu_{e}$ | -1.00 - 0.7071 | 0 - 30 | Validation bin 0.7071 - 0.95 | 0 - 0.3 | Validation bin 0.7071 - 0.95 | 0.3 - 1.6 | 0.7071 - 0.95 | 1.6 - 30 | 0.95 - 1.00 | 0 - 0.3 | Validation bin 0.95 - 1.00 | 0.3 - 1.6 | | 0.95 - 1.00 | 1.6 - 30 | RHC CC-$\bar{\nu}_{e}$ | -1.00 - 0.7071 | 0 - 30 | Validation bin 0.7071 - 0.92 | 0 - 0.3 | Validation bin 0.7071 - 0.92 | 0.3 - 1.6 | 0.7071 - 0.92 | 1.6 - 30 | 0.92 - 1.00 | 0 - 0.3 | Validation bin 0.92 - 1.00 | 0.3 - 1.6 | | 0.92 - 1.00 | 1.6 - 30 | ## 8 Cross-section results The fit is used to measure the number of signal events in all channels including all systematic uncertainties as described in section 7. The best fit results and the fit covariance matrix are used to measure the flux-integrated single differential cross-sections $d\sigma/dp$ and $d\sigma/d\cos(\theta)$ using eq. (1). Prior to fitting the data, the signal and background events are varied under different model assumptions to create pseudo datasets generated from variations of nominal MC (toy experiments). These pseudo datasets are used to check the fit performance, possible biases, over-constraining the nuisance parameters and the impact of nuisance parameters to the signal normalisation parameters and understand the dependencies on signal and background models. In addition, the cross-sections are measured using two generators to test different model assumptions. The results are in good agreement with all the tests providing confidence that our measurements are free from model dependencies. The differential cross-section results in electron and positron momentum, $\rm d\sigma/dp$, using NEUT (5.3.2) or GENIE (2.8.0) as input MC are shown in the top plot in Figure 17 and they are in agreement with the predictions. The CC-$\nu_{e}$ cross-sections are expected to be larger in RHC since the neutrino energy spectrum peaks at higher energy and it is much broader with larger contribution from higher energy neutrinos. Differences between the results using either NEUT or GENIE simulations are expected due to small differences in the efficiency corrections (Figure 15) and small differences in the muon, proton and other backgrounds (Table 2) which are kept constant in the fit. The cross-section results are dominated by the statistical uncertainty, especially in RHC. The statistical uncertainty is estimated by fixing all the nuisance parameters to their post-fit nominal values and repeating the fit. The differential cross-sections are also calculated in electron and positron scattering angles, $\rm d\sigma/d\cos(\theta)$, for both NEUT and GENIE. They are calculated in the same angular regions defined in Table 8 and for $p>300$ MeV/c. The results are shown in the bottom plot in Figure 17 and they are in agreement with the NEUT and GENIE predictions. Figure 17: CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ inclusive cross-section results in $d\sigma/dp$ (top) and $d\sigma/d\cos(\theta)$ (bottom) in a limited phase- space ($p>300$ MeV/c and $\theta\leq 45^{\circ}$). The statistical uncertainty is computed by fixing all the nuisance parameters to their post-fit nominal values and redoing the fit. The systematic uncertainty is computed by subtracting in quadrature the statistical uncertainty from the total uncertainty. The systematic uncertainties are propagated in the final cross-section measurements using toy experiments. For each toy experiment the best-fit values and the post-fit covariance are used to vary the number of signal events. Simultaneously, the flux, efficiency and the number of targets are also varied for each toy resulting in a new measurement of the cross-section using equation 1. For N toy experiments the covariance, in a fractional form, is computed from $V_{ij}=\frac{1}{N}\sum_{i=1}^{N}\frac{\left(\frac{d\sigma_{i}^{variation}}{dk_{i}}-\frac{d\sigma_{i}^{meas.}}{dk_{i}}\right)\left(\frac{d\sigma_{j}^{variation}}{dk_{j}}-\frac{d\sigma_{j}^{meas.}}{dk_{j}}\right)}{\frac{d\sigma_{i}^{meas.}}{dk_{i}}\frac{d\sigma_{j}^{meas.}}{dk_{j}}},$ (3) where $k$ is either $p$ or $\cos(\theta)$, $\frac{d\sigma_{i}^{meas.}}{dk_{i}}$ is the measured differential cross- section in bin $i$ and $\frac{d\sigma_{i}^{variation}}{dk_{i}}$ is the differential cross-section in bin $i$ calculated from a toy experiment variation. The single differential cross-sections in momentum and $\cos(\theta)$ are calculated using the same two dimensional fit and the covariance matrix should include the correlations between $d\sigma/dp$ and $d\sigma/d\cos(\theta)$. The full fractional covariance matrix as calculated from Equation 3 and is shown in Figure 18. Figure 18: Cross-section fractional covariance matrix for $d\sigma/dp$ (bottom left area) and $d\sigma/d\cos(\theta)$ (top right area) measurements for NEUT (5.3.2). The top left and bottom right areas show the covariance between $d\sigma/dp$ and $d\sigma/d\cos(\theta)$ measurements. ### 8.1 Total cross-sections in limited phase-space The total cross-sections in the measured phase-space ($p>300$ MeV/c and $\theta\leq 45^{\circ}$) using NEUT and GENIE MC are shown in Table 9. The results are compatible with the NEUT and GENIE predictions, although larger cross-sections are measured in RHC, but with large statistical uncertainties. Table 9: Measurement of the flux integrated CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ inclusive total cross-sections in a limited phase-space ($p>300$ MeV/c and $\theta\leq 45^{\circ}$) obtained using NEUT (5.3.2) and GENIE (2.8.0) MC. The statistical uncertainty is computed by fixing all nuisance parameters to their post-fit nominal and redoing the fit. The systematic uncertainty is computed by subtracting in quadrature the statistical uncertainty from the total uncertainty. The mean of the neutrino energy, $<E>$, in each beam mode is also shown. Selection | Measured $\sigma$ | Nominal $\sigma$ | $<E>$ ---|---|---|--- | $[/10^{-39}\rm cm^{2}/nucleon]$ | $[/10^{-39}\rm cm^{2}/nucleon]$ | GeV FHC CC-$\nu_{e}$ NEUT | $6.62\pm 1.32(\rm stat)\pm 1.30(\rm syst)$ | 7.18 | 1.28 GENIE | $6.93\pm 1.40(\rm stat)\pm 1.33(\rm syst)$ | 6.87 | RHC CC-$\nu_{e}$ NEUT | $14.56\pm 4.90(\rm stat)\pm 2.31(\rm syst)$ | 12.96 | 1.98 GENIE | $14.73\pm 5.06(\rm stat)\pm 2.01(\rm syst)$ | 11.44 | RHC CC-$\bar{\nu}_{e}$ NEUT | $3.01\pm 1.36(\rm stat)\pm 0.57(\rm syst)$ | 2.61 | 0.99 GENIE | $3.10\pm 1.46(\rm stat)\pm 0.53(\rm syst)$ | 2.51 | ### 8.2 Comparisons to additional models Using the NUISANCE framework nuisance , the fit results are compared to cross- section predictions from recent neutrino generator models in NEUT (5.4.0), GENIE (2.12.10) and also from NuWro (19.02) nuwro . NEUT 5.4.0 uses a local Fermi gas (instead of spectral function). Other interaction modelling and final state interactions are similar to NEUT 5.3.2 (detailed in section 3). GENIE 2.12.10 interaction modelling is similar to 2.8.0 (detailed in section 3), with the "empirical MEC" model for the description of multi-nucleon interactions enabled. NuWro simulates the CC quasi-elastic process with the Llewellyn-Smith model with axial mass value of 1.03 $\rm GeV/c^{2}$. The nuclear model is simulated using the relativistic Fermi gas including random phase approximation corrections rpa . Multi-nucleon interactions are simulated similar to NEUT using the model from Nieves2p2h . For pion production a single $\Delta$-model by Adler-Rarita-Schwinger adler is used for the hadronic mass W $<$ 1.6 $\rm GeV/c^{2}$ with axial mass value of 0.94 $\rm GeV/c^{2}$. A smooth transition to DIS processes is made for W between 1.3 and 1.6 $\rm GeV/c^{2}$. The total cross section is based on the Bodek and Yang approach BodekYangNeut . Similar to NEUT, final state interactions are simulated using a semi-classical cascade model. The comparisons of the data to NEUT 5.4.0, GENIE 2.12.10 and NuWro 19.02 are shown in Figure 19. A $\chi^{2}$ between the data measurements and each neutrino generator model predictions is defined as $\chi^{2}=\sum_{i}\sum_{j}\left(\frac{d\sigma_{i}^{meas.}}{dk_{i}}-\frac{d\sigma_{i}^{model}}{dk_{i}}\right)V_{ij}^{-1}\left(\frac{d\sigma_{j}^{meas.}}{dk_{j}}-\frac{d\sigma_{j}^{model}}{dk_{j}}\right),$ (4) where $k$ is either $p$ or $\cos(\theta)$, $\frac{d\sigma_{i}^{meas.}}{dk_{i}}$ is the differential cross-section measurement in bin $i$, $\frac{d\sigma_{i}^{model}}{dk_{i}}$ is the differential cross-section model prediction in bin $i$ and $V_{ij}$ is the covariance matrix as defined in equation 3 and shown in Figure 18. The $\chi^{2}$ is measured for each neutrino generator individually and is summarised in Table 10. NEUT 5.4.0 has the lowest $\chi^{2}$ compared to our data. GENIE 2.12.10 has a slightly larger $\chi^{2}$. The $\chi^{2}$ for NuWro 19.02 is significantly larger. The $\chi^{2}$ is also calculated individually for the single differential cross-sections $d\sigma/dp$ and $d\sigma/d\cos(\theta)$. A reduced covariance is used considering only the momentum or $\cos(\theta)$ part of the full covariance in Figure 18. In these cases the $\chi^{2}$, in both momentum and $\cos(\theta)$ measurements, are smaller and similar for all neutrino generators. This highlights the importance of using the combined cross-section measurements in momentum and $\cos(\theta)$ when doing model comparisons, rather than using each cross- section measurement in momentum or $\cos(\theta)$ individually. Figure 19: Flux integrated CC-$\nu_{e}$ and CC-$\bar{\nu}_{e}$ inclusive cross-section results in a limited phase-space ($p>300$ MeV/c and $\theta\leq 45^{\circ}$) with comparisons to neutrino generator models from NEUT 5.4.0, GENIE 2.12.10 and NuWro 19.02 obtained using the NUISANCE framework. The top plot shows the results in momentum and the bottom plot the results in scattering angle. The $\chi^{2}$ is the total from the combined measurements in momentum and $\cos(\theta)$. Table 10: The $\chi^{2}$ comparing data with neutrino generator models. The $\chi^{2}$ is calculated using Equation 4. The full covariance, as shown in Figure 18, is used for $p-\cos(\theta)$ $\chi^{2}$ calculation. A reduced covariance considering only the momentum and $\cos(\theta)$ part of the full covariance is used to calculate the $p$-only and $\cos(\theta)$-only $\chi^{2}$ respectively. The number of degrees of freedom (ndof) for each $\chi^{2}$ is also shown. Generator | $p-\cos(\theta)$ $\chi^{2}$ | $p$-only $\chi^{2}$ | $\cos(\theta)$-only $\chi^{2}$ ---|---|---|--- | (ndof = 13) | (ndof = 7) | (ndof = 6) NEUT 5.4.0 | 14.63 | 5.82 | 5.34 GENIE 2.12.10 | 16.32 | 4.16 | 4.55 NuWro 19.02 | 32.08 | 4.52 | 5.08 ## 9 Summary and conclusions Electron-like neutrino and anti-neutrino events are selected in the T2K off- axis near detector ND280, using both FHC and RHC modes. A significant amount of photon background populates the low momentum and high angle regions, constrained by an independent photon control selection. The regions dominated by the photon background also show significant data and MC discrepancies and are dominated by large systematic uncertainties. The flux integrated single differential cross-sections, as a function of momentum and scattering angle, are measured by fitting simultaneously the CC inclusive selections and their corresponding photon control selections. To minimize detector effects, the cross-sections are measured in a limited phase-space, $p>300$ MeV/c and $\theta\leq 45^{\circ}$. The results are consistent from the two fits with both NEUT 5.3.2 and GENIE 2.8.0 predictions. The cross-section results are also compared with more recent neutrino generator models using NEUT 5.4.0, GENIE 2.12.10 and NuWro 19.02. The best agreement is observed with NEUT 5.4.0. These are the first CC-$\nu_{e}$ cross-section measurements using both FHC and RHC fluxes and the first CC-$\bar{\nu}_{e}$ cross-section measurement since the Gargamelle measurements in 1978. The data release from this paper can be found here nuedata . ###### Acknowledgements. We thank the J-PARC staff for superb accelerator performance. We thank the CERN NA61/SHINE Collaboration for providing valuable particle production data. We acknowledge the support of MEXT, Japan; NSERC (Grant No. SAPPJ-2014-00031), NRC and CFI, Canada; CEA and CNRS/IN2P3, France; DFG, Germany; INFN, Italy; National Science Centre (NCN) and Ministry of Science and Higher Education, Poland; RSF (Grant #19-12-00325) and Ministry of Science and Higher Education, Russia; MICINN and ERDF funds, Spain; SNSF and SERI, Switzerland; STFC, UK; and DOE, USA. We also thank CERN for the UA1/NOMAD magnet, DESY for the HERA-B magnet mover system, NII for SINET4, the WestGrid and SciNet consortia in Compute Canada, and GridPP in the United Kingdom. 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2024-09-04T02:54:54.948286

2002.11998

arxiv-papers

l3regexThis package is obsolete # A Quantum Money Solution to the Blockchain Scalability Problem Andrea Coladangelo Computing and Mathematical Sciences, Caltech Or Sattath Computer Science Department, Ben-Gurion University ###### Abstract We put forward the idea that classical blockchains and smart contracts are potentially useful primitives not only for classical cryptography, but for quantum cryptography as well. Abstractly, a smart contract is a functionality that allows parties to deposit funds, and release them upon fulfillment of algorithmically checkable conditions, and can thus be employed as a formal tool to enforce monetary incentives. In this work, we give the first example of the use of smart contracts in a quantum setting. We describe a simple hybrid classical-quantum payment system whose main ingredients are a classical blockchain capable of handling stateful smart contracts, and quantum lightning, a strengthening of public-key quantum money introduced by Zhandry [Zha19]. Our hybrid payment system employs quantum states as banknotes and a classical blockchain to settle disputes and to keep track of the valid serial numbers. It has several desirable properties: it is decentralized, requiring no trust in any single entity; payments are as quick as quantum communication, regardless of the total number of users; when a quantum banknote is damaged or lost, the rightful owner can recover the lost value. Note: This work supersedes two previous independent works, [Col19] and [Sat19]. ###### Contents 1. 1 Introduction 2. 2 Preliminaries 1. 2.1 Notation 2. 2.2 Quantum money and quantum lightning 3. 2.3 Universal Composability 3. 3 Blockchains and smart contracts 4. 4 A payment system based on quantum lightning and a classical blockchain 1. 4.1 The payment system and its components 5. 5 Security 6. 6 Practical issues in a less idealized setting 1. 6.1 Attacks outside of the idealized setting 2. 6.2 Trading a Bolt for a Signature 3. 6.3 Security against Sabotage 4. 6.4 A resolution of the practical issues 7. 7 A practical implementation on Bitcoin and space optimization 1. 7.1 Bitcoin to Quantum Money: The Simple Approach 2. 7.2 Bitcoin to Quantum Money: Space Optimization 8. 8 Comparing our scheme to classical alternatives 9. 9 Conclusion 10. A Appendix 1. A.1 Proof of Proposition 2 2. A.2 Standard Security Definitions ## 1 Introduction Cryptocurrencies, along with blockchains and smart contracts, have recently risen to popular attention, the most well-known examples being Bitcoin and Ethereum [Nak08, But14]. Informally, a blockchain is a public ledger consisting of a sequence of blocks. Each block typically contains information about a set of transactions, and a new block is appended regularly via a consensus mechanism that involves the parties of a network and no a priori trusted authority. A blockchain is endowed with a native currency which is employed in transactions, and whose basic unit is a “coin”. The simplest type of transaction is a payment, which transfers coins from one party to another. However, more general transactions are allowed, which are known as smart contracts. These can be thought of as contracts stored on a blockchain, and whose consequences are executed upon fulfilment of algorithmically checkable conditions. A central issue that needs to be resolved for blockchains to achieve mass- adoption is scalability. This refers to the problem of increasing the throughput of transactions (i.e. transactions per second) while maintaining the resources needed for a party to participate in the consensus mechanism approximately constant, and while maintaining security against adversaries that can corrupt constant fractions of the parties in the network. For example, Bitcoin and Ethereum, can currently handle only on the order of $10$ transactions per second (Visa for a comparison handles about $3500$ per second). In this work, we show that quantum information is inherently well-suited to tackle this problem. We show that a classical blockchain can be leveraged using tools from quantum cryptography (in particular, quantum money) to provide a simple solution to the scalability problem.111We clarify that this solution only solves the scalability problem for payment transactions, and not for the more general smart contracts transactions. The main quantum ingredient that we employ is a primitive called quantum lightning, formally introduced by Zhandry [Zha19] and inspired by Lutomirski et al.’s notion of collision resistant quantum money [LAF+09]. In a public-key quantum money scheme, a bank is entrusted with generating quantum states (we refer to these as quantum banknotes) with an associated serial number, and a public verification procedure allows anyone in possession of the banknote to check its validity. Importantly, trust is placed in the fact that the central bank will not create multiple quantum banknotes with the same serial number. A quantum lightning scheme has the additional feature that no generation procedure, not even the honest one (and hence not even a bank!), can produce two valid money states with the same serial number, except with negligible probability. This opens to the possibility of having a completely decentralized quantum money scheme. However, if the system ought to be trust- less, some issues need to be addressed: most importantly, who is allowed to mint money? Who decides which serial numbers are valid? Our solution leverages a (classical) blockchain to address these questions. #### Our contributions We design a hybrid classical-quantum payment system that uses quantum states as banknotes and a classical blockchain to settle disputes and to keep track of the valid serial numbers. This is, to the best of our knowledge, the first example of the use of a classical blockchain in combination with quantum cryptographic tools. Our payment system has the following desirable features: * (i) It is decentralized, requiring no trust in any single entity. * (ii) Payments involve the exchange of quantum banknotes, and enjoy many of the properties of cash, which cryptocurrencies do not have. For example, transactions are not recorded on the blockchain, and they involve only the payer and the payee. Thus the throughput is unbounded. Payments are as quick as quantum communication, and they do not incur transaction fees. * (iii) The rightful owner of a quantum banknote can recover the original value, even if the quantum banknote is damaged or lost; an adversary who tried to abuse this feature would be penalized financially. Our contribution is primarily conceptual, but our treatment is formal: we work within the Generalized Universal Composability framework (GUC) of Canetti et al. [CDPW07], and we formulate an ideal functionality for a blockchain that supports smart contracts. Note that we do not prove composable security of our payment system. Instead, we prove a “one-shot” version of security, assuming parties have access to such an ideal functionality. Nonetheless, we find it desirable to work within the GUC framework, and we discuss the reason for this choice in more detail below. We also provide an informal construction of our payment system on the Bitcoin blockchain. Its treatment is less rigorous, but it addresses some ways in which the payment system could be optimized. As a further technical contribution, in order to achieve part (iii), we formalize a novel property of quantum lightning schemes, which we call “bolt- to-signature” capability and is reminiscent of one-time digital signatures. We provide a provably secure construction of a lightning scheme with such a property (assuming a secure lightning scheme which might not satisfy this). The construction is based on the hash-and-sign paradigm as well as Lamport signatures scheme. It allows a user holding a valid quantum lightning state with serial number $s$ to sign a message $\alpha$ with respect to $s$. The security guarantees are that no one who does not possess a lightning state with serial number $s$ can forge signatures with respect to $s$, and once the lightning state is utilized to produce even a single signature, it will no longer pass the lightning verification procedure. We envision that such a primitive could find applications elsewhere and is of independent interest. #### How to model a blockchain? The essential properties of blockchains and smart contracts can be abstracted by modeling them as ideal functionalities in the Universal Composability (UC) framework of Canetti [Can01]. Such a framework provides both a formal model for multiparty computation, and formal notions of security with strong composability properties. The approach of studying blockchains and smart contracts within such a framework was first proposed by Bentov al. in [BK14], and explored further in [BKM17]. The main reason why such an approach is desirable is that it abstracts the features of blockchains and smart contracts into building blocks that can be utilized to design more complex protocols in a modular way. The limitation of the works [BK14, BKM17] is that they modify the original model of computation of UC in order to incorporate coins, but they do not prove that a composition theorem holds in this variant. A more natural approach was proposed by Kiayias et al. in [KZZ16], which uses the Generalized Universal Composability (GUC) framework by Canetti et al. [CDPW07]. In this work, we define a ledger functionality in the GUC framework which supports universal smart contracts. We use this as an abstraction layer which significantly simplifies exposition and the security analysis. The downside of this approach is that, to the best of our knowledge, there is no known proof of a GUC-secure realization of an ideal functionality for a ledger supporting smart contracts by any cryptocurrency. Proving that a complicated system such as Bitcoin or Ethereum GUC-securely realizes a simple ledger functionality, even without the support for smart contracts, requires already substantial work [BMTZ17]. The upside is that, as soon as one provides a GUC secure realization on some cryptocurrency of our ideal ledger functionality, one can replace the latter in our payment system with its real world implementation, and the security of our payment system would hold verbatim, by the composition properties of the GUC framework. For this reason, we find the approach very desirable. Other approaches that do not fall into the GUC framework include [KMS+16] and [DEF18]. We emphasize that we do not prove that our payment system can be composed securely, i.e. we do not define an ideal functionality for our payment system and prove a secure realization of it. Rather, the security we prove is only “one-shot”. #### A sketch of our payment system The main ingredient that we employ is quantum lightning. As suggested by Zhandry, this primitive seems well-suited for designing a decentralized payment system, as it is infeasible for anyone to copy banknotes (even a hypothetical bank). However, since the generation procedure is publicly known, there needs to be a mechanism that regulates the generation of new valid banknotes (to prevent parties from continuously generating banknotes). We show how stateful smart contracts on a classical blockchain can be used to provide such a mechanism. For instance, they allow to easily keep track of a publicly trusted list of valid serial numbers. We elaborate on this. In our payment system, the native coin of a classical blockchain is used as a baseline classical currency. Any party can spend coins on the classical blockchain to add a serial number of their choice to the list of valid serial numbers. More precisely, they can deposit any amount (of their choice) $d$ of coins into an appropriately specified smart contract and set the initial value of a serial number state variable to whatever they wish (presumably the serial number of a quantum banknote that they have just generated locally). They have thus effectively added to the blockchain a serial number associated to a quantum banknote that, in virtue of this, we can think of as having “acquired” value $d$. Payments are made by transferring quantum banknotes: party $A$, the payer, transfers his quantum banknote with serial number $s$ to party $B$, the payee, and references a smart contract whose serial number state variable is $s$. Party $B$ then _locally_ verifies that the received quantum banknote is valid. This completes the transaction. Notice that this does not involve interaction with any other third party, and only requires read access to the blockchain. Of course, the same quantum banknote can be successively spent an unlimited number of times in the same manner, without ever posting any new transaction on the blockchain. The latter is only invoked when a banknote is first generated, and in case of a dispute. Thus, throughput of transactions is no longer a concern. Likewise, long waiting times between when a payment is initiated and when it is confirmed (which are typically in the order of minutes – for example, it is recommended to accept a Bitcoin transaction after 6 confirmations, which takes an hour in expectation) are also no longer a concern since payments are verified immediately by the payee. One can think of the quantum banknotes as providing an off-chain layer that allows for virtually unlimited throughput. We give an extended comparison of our payment system with other off-chain solutions, like Bitcoin’s Lightning Network, in Section 8. The full payment system includes two additional optional features: * (i) A mechanism that allows any party in possession of a quantum banknote to recover the coins deposited in the corresponding smart contract. Together with the mechanism outlined earlier, this makes quantum banknotes and coins on the blockchain in some sense interchangeable: one can always convert one into the other by publishing a single transaction on the blockchain. * (ii) A mechanism that allows any honest party who has lost or damaged a valid quantum banknote to change the serial number state variable of the corresponding smart contract to a fresh value of their choice. These two additional desirable features are realizable as long as the quantum lightning scheme employed satisfies an additional property. We formalize this property, which we call “bolt-to-certificate capability”, and show that Zhandry’s proposed constructions satisfy this property. The latter asserts informally that it is possible to measure a valid quantum banknote and obtain a classical certificate, but it is impossible to simultaneously hold both a valid banknote and a valid certificate. In other words, the classical certificate acts as a “proof of destruction”: a certificate which guarantees that no quantum lightning state with the corresponding serial number exists. We also introduce a novel implementation of a one-time signature scheme based on the “bolt-to-certificate capability”, and inspired by Lamport’s signature scheme. This protects honest parties who broadcast a classical certificate to the network from having their classical certificate stolen before it is registered on the blockchain. We emphasize that the bolt-to-certificate capability of the lightning scheme is only required in order to achieve the two additional features (i) and (ii) above, but is not necessary in order to achieve the basic functionality of our payment system described above. #### Limitations of our solution We see two main limitations to our payment system: * • The technologies needed to implement our payment system will not be available in the foreseeable future. For instance, our payment system would require the ability of each party to store large entangled quantum states for extended periods of time. It would also require that each party be able to send quantum states to any party it wishes to make payments to. The latter could be achieved, for example, in a network in which any two parties have the ability to request joint EPR pairs (i.e. maximally entangled pairs of qubits), which is one of the primary components of a “quantum internet” [WEH18]. One can view our scheme as a possible use case of a quantum internet. * • The only known concrete constructions of a quantum lightning scheme rely on non-standard assumptions for security. The construction of Farhi et al. [FGH+12a] contains no security proof. The construction of Zhandry [Zha19] is secure based on an assumption related to the multi-collision resistance of certain degree-2 hash functions. Arguably, neither of these assumptions is well-studied. In Section 8, we extensively discuss the disadvantages (as well as the advantages) of our payment system vis-à-vis with other classical alternatives – namely, a standard crypto-currency such as Bitcoin and second layer solutions such as the Lightning Network. #### Outline Section 2 covers preliminaries: 2.1 covers basic notation; 2.2 introduces quantum lightning; 2.3 gives a concise overview of the Universal Composability framework of Canetti [Can01]. Section 3 gives first an informal description of blockchains and smart contracts, followed by a formal definition of our global ideal functionality for a transaction ledger that handles stateful smart contracts. Section 4 describes our payment system. In Section 5, we describe an adversarial model, and then prove security guarantees with respect to it. ## 2 Preliminaries ### 2.1 Notation For a function $f:\mathbb{N}\rightarrow\mathbb{R}$, we say that $f$ is negligible, and we write $f(n)=negl(n)$, if for any positive polynomial $p(n)$ and all sufficiently large $n$’s, $f(n)<\frac{1}{p(n)}$. A binary random variable is a random variable over $\\{0,1\\}$. We say that two ensembles of binary random variables $\\{X_{n}\\}$ and $\\{Y_{n}\\}$ are indistinguishable if, $\left|\,\Pr[X_{n}=1]-\Pr[Y_{n}=1]\,\right|=negl(n).$ We use the terms PPT and QPT as abbreviations of probabilistic polynomial time and quantum polynomial time respectively. ### 2.2 Quantum money and quantum lightning Quantum money is a theoretical form of payment first proposed by Wiesner [Wie83], which replaces physical banknotes with quantum states. In essence, a quantum money scheme consists of a generation procedure, which mints banknotes, and a verification procedure, which verifies the validity of minted banknotes. A banknote consists of a quantum state together with an associated serial number. The appeal of quantum money comes primarily from a fundamental theorem in quantum theory, the No-Cloning theorem, which informally states that there does not exist a quantum operation that can clone arbitrary states. A second appealing property of quantum money, which is not celebrated nearly as much as the first, is that quantum money can be transferred almost instantaneously (by quantum teleportation for example). The first proposals for quantum money schemes required a central bank to carry out both the generation and verification procedures. The idea of public key quantum money was later formalized by Aaronson [Aar09]. In public-key quantum money, the verification procedure is public, meaning that anyone with access to a quantum banknote can verify its validity. In this section, we focus on quantum lightning, a primitive recently proposed by Zhandry [Zha19], and we enhance this to a decentralized quantum payment system. Informally, a quantum lightning scheme is a strengthening of public- key quantum money. It consists of a public generation procedure and a public verification procedure which satisfy the following two properties: * • Any quantum banknote generated by the honest generation procedure is accepted with probability negligibly close to $1$ by the verification procedure. * • No adversarial generation procedure (not even the honest one) can generate two banknotes with the same serial number which both pass the verification procedure with non-negligible probability. As mentioned earlier, there is only one known construction of quantum lightning, by Zhandry [Zha19], who gives a construction which is secure under a computational assumption related to the multi-collision resistance of some degree-2 hash function. Zhandry also proves that any non-collapsing hash function can be used to construct quantum lightning. However, to the best of our knowledge, there are no known hash functions that are proven to be non- collapsing. In this section, we define quantum lightning formally, but we do not discuss any possible construction. Rather, in Section 4, we will use quantum lightning as an off-the-shelf primitive. ###### Definition 1 (Quantum lightning [Zha19]). A quantum lightning scheme consists of a PPT algorithm $\textsf{QL.Setup}(1^{\lambda})$ (where $\lambda$ is a security parameter) which samples a pair of polynomial-time quantum algorithms $(\textsf{gen- bolt}$, $\textsf{verify-bolt})$. gen-bolt outputs pairs of the form $\left(\ket{\psi}\in\mathcal{H}_{\lambda},s\in\\{0,1\\}^{\lambda}\right)$. We refer to $\ket{\psi}$ as a “bolt” and to $s$ as a “serial number”. ver-bolt takes as input a pair of the same form, and outputs either “accept” (1) or “reject” (0) (together with a post-measurement state). They satisfy the following: * • $\displaystyle\Pr[\textsf{verify-bolt}(\ket{\psi},s)$ $\displaystyle=1:(\ket{\psi},s)\leftarrow\textsf{gen-bolt},(\textsf{gen- bolt},\textsf{verify-bolt})\leftarrow\textsf{QL.Setup}]$ (1) $\displaystyle=1-negl(\lambda)$ (2) * • For a state $\ket{\psi}$ let $\mathcal{M}_{\ket{\psi}}$ be the two outcome measurement which accepts $\ket{\psi}$ and rejects all states orthogonal to it. $\displaystyle\Pr[\mathcal{M}_{\ket{\psi}}(\ket{\psi^{\prime}})=1$ $\displaystyle:(b,\ket{\psi^{\prime}})\leftarrow\textsf{verify- bolt}(\ket{\psi},s),(\ket{\psi},s)\leftarrow\textsf{gen-bolt},$ (3) $\displaystyle(\textsf{gen-bolt},\textsf{verify- bolt})\leftarrow\textsf{QL.Setup}(\lambda)]=1-negl(\lambda)$ (4) Here $\ket{\psi^{\prime}}$ is the post-measurement state upon running $\textsf{verify-bolt}(\ket{\psi},s)$. * • For all $s^{\prime}\in\\{0,1\\}^{\lambda}$, $\displaystyle\Pr[\textsf{verify-bolt}(\ket{\psi},s^{\prime})=$ $\displaystyle 1\,\,\land\,\,s^{\prime}\neq s:((\ket{\psi},s)\leftarrow\textsf{gen- bolt},(\textsf{gen-bolt},\textsf{verify-bolt})\leftarrow\textsf{QL.Setup}]$ (5) $\displaystyle=negl(\lambda)$ (6) The three requirements simply ask that, with overwhelming probability, for any validly generated bolt $\ket{\psi}$, there is a single serial number $s$ such that $(\ket{\psi},s)$ is accepted by the verification procedure, and that the verification does not perturb the bolt, except negligibly. For security, we require that no adversarial generation procedure can produce two bolts with the same serial number. Formally, we define security via the following game between a challenger and an adversary $\mathcal{A}$. * • The challenger runs $(\textsf{gen-bolt},\textsf{verify- bolt})\leftarrow\textsf{QL.Setup}(\lambda)$ and sends $(\textsf{gen- bolt},\textsf{verify-bolt})$ to $\mathcal{A}$. * • $\mathcal{A}$ produces a pair $\ket{\Psi_{12}}\in\mathcal{H}_{\lambda}^{\otimes 2},s\in\\{0,1\\}^{\lambda}$. * • The challenger runs $\textsf{verify-bolt}(\cdot,s)$ on each half of $\ket{\Psi_{12}}$. The output of the game is $1$ if both outcomes are “accept”. We let $\textsf{Counterfeit}(\lambda,\mathcal{A})$ be the random variable which denotes the output of the game. ###### Definition 2 (Security [Zha19]). A quantum lightning scheme is secure if, for all polynomial-time quantum adversaries $\mathcal{A}$, $\Pr[\textsf{Counterfeit}(\lambda,\mathcal{A})=1]=\textnormal{negl}(\lambda).$ We define an additional property of a quantum lightning scheme, which in essence establishes that one can trade a quantum banknote for some useful classical certificate. Intuitively, this is meant to capture the fact that in the proposed construction of quantum lightning by Zhandry, one can measure a bolt with serial number $y$ in the computational basis to obtain a pre-image of $y$ under some hash function. However, doing so damages the bolt so that it will no longer pass verification. In order to define this additional property, we change the procedure $\textsf{QL.Setup}(1^{\lambda})$ slightly, so that it outputs a tuple $(\textsf{gen-bolt},\textsf{verify-bolt},\textsf{gen- certificate},\textsf{verify-certificate})$, where gen-certificate is a QPT algorithm that takes as input a quantum money state and a serial number and outputs a classical string of some fixed length $l(\lambda)$ for some polynomially bounded function $l$, which we refer to as a certificate, and verify-certificate is a PPT algorithm which takes as input a serial number and a certificate, and outputs “accept” ($1$) or “reject” ($0$). The additional property is defined based on the following game Forge-certificate between a challenger and an adversary $\mathcal{A}$: * • The challenger runs $(\textsf{gen-bolt},\textsf{verify-bolt},\textsf{gen- certificate},\textsf{verify- certificate})\leftarrow\textsf{QL.Setup}(1^{\lambda})$ and sends the tuple to $\mathcal{A}$. * • $\mathcal{A}$ returns $c\in\\{0,1\\}^{l(\lambda)}$ and $(\ket{\psi},s)$. * • The challenger runs $\textsf{verify-certificate}(s,c)$ and $\textsf{verify- bolt}(\ket{\psi},s)$. Outputs $1$ if they both accept. Let $\textsf{Forge-certificate}(\mathcal{A},\lambda)$ be the random variable for the output of the challenger in the game above. ###### Definition 3 (Trading the bolt for a classical certificate). Let $\lambda\in\mathbb{N}$. We say that a quantum lightning scheme has “bolt- to-certificate capability” if: * (I) $\displaystyle\Pr[\textsf{verify-certificate}(s,c)=1:$ $\displaystyle(\textsf{gen-bolt},\textsf{verify-bolt},\textsf{gen- certificate},\textsf{verify- certificate})\leftarrow\textsf{QL.Setup}(1^{\lambda}),$ (7) $\displaystyle(\ket{\psi},s)\leftarrow\textsf{gen-bolt},$ (8) $\displaystyle c\leftarrow\textsf{gen-certificate}(\ket{\psi},s)]=1-negl(\lambda)$ (9) * (II) For all polynomial-time quantum algorithms $\mathcal{A}$, $\Pr[\textsf{Forge-certificate}(\mathcal{A},\lambda)=1]=negl(\lambda).$ Notice that property $(II)$ also implies that for most setups and serial numbers $s$ it is hard for any adversary to find a $c$ such that $\textsf{verify-certificate}(s,c)=1$ without access to a valid state whose serial number is $s$. In fact, if there was an adversary $\mathcal{A}$ which succeeded at that, this could clearly be used to to construct an adversary $\mathcal{A^{\prime}}$ that succeeds in Forge-certificate: Upon receiving $(\textsf{gen-bolt},\textsf{verify-bolt},\textsf{gen- certificate},\textsf{verify-certificate})$ from the challenger, $\mathcal{A}^{\prime}$ computes $(\ket{\psi},s)\leftarrow\textsf{gen-bolt}$; then runs $\mathcal{A}$ on input $s$ to obtain some $c$. $\mathcal{A}^{\prime}$ returns $c$,$\ket{\psi}$ and $s$ to the challenger. We emphasize that in game Forge-certificate it is the adversary himself who generates the state $\ket{\psi}$. This is important because when we employ the quantum lightning scheme later on parties are allowed to generate their own quantum banknotes. ###### Proposition 1. Any scheme that uses Zhandry’s construction instantiated with a non-collapsing hash function [Zha19] satisfies the property of Definition 3. ###### Proof. Zhandry already proved that his scheme satisfies Definitions 1 and 2. As we will see, since our construction does not change $\textsf{QL.Setup},\ \textsf{gen-bolt}$, and verify-bolt, we only need to prove that it satisfies Definition 3. We refer the reader to [Zha19] for a definition of non- collapsing hash function. In Zhandry’s construction based on a non-collapsing hash function, $\textsf{QL.Setup}(1^{\lambda})$ outputs $(\textsf{gen- bolt},\textsf{verify-bolt})$. A bolt generated from gen-bolt has the form $\ket{\Psi}=\bigotimes_{i=1}^{n}\ket{\psi_{y_{i}}}$, where $y_{i}\in\\{0,1\\}^{\lambda}$ for all $i$ and $\ket{\psi_{y_{i}}}=\sum_{x:H(x)=y_{i}}\ket{x}$, where $H$ is a non-collapsing hash-function, and $n\in\mathbb{N}$ is polynomial in the security parameter $\lambda$. verify-bolt has the form of $n$-fold product of a verification procedure “Mini-Ver” which acts on a single one of the $n$ registers, though it is not crucial to understand how Mini-Ver for this work. In Zhandry’s construction, the serial number associated to the bolt $\ket{\Psi}$ above is $s=(y_{1},\ldots,y_{n})$. We define $\mathsf{gen\textit{-}certificate}$ as the QPT algorithm that measures $\ket{\psi}$ in the standard basis, and outputs the outcome. When applied to an honestly generated bolt, the outcomes are pre-images $x_{i}$ of $y_{i}$, for $i=1,..,n$. We define $\mathsf{verify\textit{-}certificate}$ as the deterministic algorithm which receives a serial number $s=(y_{1},\ldots,y_{n})$ and a certificate $c=(x_{1},\ldots,x_{n})$ and checks that for all $i\in[n]$, $H(x_{i})=y_{i}$. Is is clear that $(I)$ holds. For property $(II)$, suppose there exists $\mathcal{A}$ such that $\Pr[\textsf{Forge-certificate}(\mathcal{A},\lambda)=1]$ is non-negligible. We use $\mathcal{A}$ to construct an adversary $\mathcal{A}^{\prime}$ that breaks collision-resistance of a non-collapsing hash function as follows: $\mathcal{A}^{\prime}$ runs $(\textsf{gen-bolt},\textsf{verify- bolt})\leftarrow\textsf{QL.Setup}(1^{\lambda})$. Let $H$ be the non-collapsing hash function hard-coded in the description of verify-bolt. $\mathcal{A^{\prime}}$ sends the tuple $(\textsf{gen-bolt},\textsf{verify- bolt},\textsf{gen-certificate},\textsf{verify-certificate})$ to $\mathcal{A}$, where the latter two are defined as above in terms of $H$. $\mathcal{A}$ returns $(c,\ket{\psi})$, where $c$ is parsed as $c=(x_{1},..,x_{n})$. $\mathcal{A}^{\prime}$ then measures each of the $n$ registers of $\ket{\psi}$ to get $c^{\prime}=(x^{\prime}_{1},..,x^{\prime}_{n})$. If $x_{i}\neq x_{i}^{\prime}$, then $\mathcal{A}^{\prime}$ outputs $(x_{i},x_{i}^{\prime})$. We claim that with non-neglibile probability $\mathcal{A^{\prime}}$ outputs a collision for $H$. To see this, notice that since $\mathcal{A}$ wins Forge- certificate with non-negligible probability, then $\ket{\psi}$ must pass verify-bolt with non-negligible probability; and from the analysis of [Zha19], any such state must be such that at least one of the registers is in a superposition which has non-negligible weight on at least two pre-images. For more details, see the proof of Theorem 5 in [Zha19], and specifically the following claim: > If the bolts are measured, two different pre-images of the same y, and hence > a collision for $H^{\otimes r}$, will be obtained with probability at least > 1/200. ∎ ###### Proposition 2. Zhandry’s construction based on the multi-collision resistance of certain degree-2 hash functions (from section 6 of [Zha19]) satisfies the property of Definition 3. The proof is similar to the proof of Proposition 1. We include it for completeness in Appendix A.1. ###### Fact 4. Zhandry’s construction from 2 only requires a common random string (CRS) for QL.Setup. This fact will be relevant when we discuss a concrete implementation in a system such as Bitcoin. For more details, see Section 7. #### Other Quantum Lightning Schemes? Farhi et al. [FGH+12b] constructed a public quantum money scheme which they conjectured to have the following property: > A rogue mint running the same algorithm as the mint can produce a new set of > money pairs, but (with high probability) none of the serial numbers will > match those that the mint originally produced. This is less formal, but we believe captures Zhandry’s security definition of a quantum lightning (though it was introduced roughly 7 years earlier). Very recently, Peter Shor gave a talk on ongoing (unpublished) work for a lattice-based quantum lightning scheme (see https://youtu.be/8fzLByTn8Xk). Here, we briefly compare the two existing published constructions, and we omit Shor’s scheme from this discussion. * • Farhi et al. only provide several plausibility arguments supporting the security of their scheme, but their work does not contain any security proof. In a follow-up work, Lutomirski [Lut11] showed a security reduction from a problem _related_ to counterfeiting Farhi et al.’s money scheme. Even though it does not have a (proper) security proof, Farhi et al.’s scheme was first published 8 years ago, and was not broken since then. Zhandry’s construction is proved to be secure under a non-standard hardness assumption, which was first introduced in that work. In his Eurocrypt talk, Zhandry reports that the construction is “broken in some settings” – see https://youtu.be/fjumbNTZSik?t=1302.222Additionally, the hardness assumption was changed between the second and third version on the IACR e-print. This might suggest that the original hardness assumption was also broken. The most updated e-print version does not discuss that discrepancy. * • The scheme of Farhi et al. does not require a trusted setup. * • The scheme of Farhi et al. does not have a bolt-to-certificate capability. This property is required in our payment system to transform quantum banknotes back to coins on the blockchain – see Section 4. ### 2.3 Universal Composability This section is intended as a concise primer about the Universal Composability (UC) model of Canetti [Can01]. We refer the reader to [Can01] for a rigorous definition and treatment of the UC model and of composable security, and to the tutorial [Can06] for a more gentle introduction. At the end, we provide a brief overview of the Generalized UC model (GUC) [CDPW07]. While introducing UC and GUC, we also setup some of the notation that we will employ in the rest of the paper. As mentioned in the Introduction, we elect to work in the GUC, because this provides a convenient formal abstraction layer for modeling a ledger functionality that supports smart-contracts, and allows for a clean security proof in the idealized setting in which parties have access to this functionality. The reader familiar with the UC and GUC framework may wish to skip ahead to the next section. In the universal composability framework (UC), parties are modelled as Interactive Turing Machines (ITM) who can communicate by writing on each other’s externally writable tapes, subject to some global constraints. Informally, a protocol is specified by a code $\pi$ for an ITM, and consists of various rounds of communication and local computation between instances of ITMs (the parties), each running the code $\pi$ on their machine, on some private input. Security in the UC model is defined via the notion of emulation. Informally, we say that a protocol $\pi$ emulates a protocol $\phi$ if whatever can be achieved by an adversary attacking $\pi$ can also be achieved by some other adversary attacking $\phi$. This is formalized by introducing simulators and environments. Given protocols $\pi$ and $\phi$, we say that $\pi$ emulates (or “is as secure as”) $\phi$ in the UC model, if for any polynomial-time adversary $\mathcal{A}$ attacking protocol $\pi$, there exists a polynomial-time simulator $\mathcal{S}$ attacking $\phi$ such that no polynomial-time distinguisher $\mathcal{E}$, referred to as the environment, can distinguish between $\pi$ running with $\mathcal{A}$ and $\phi$ running with $\mathcal{S}$. Here, the environment $\mathcal{E}$ is allowed to choose the protocol inputs, read the protocol outputs, including outputs from the adversary or the simulator, and to communicate with the adversary or simulator during the execution of the protocol (without of course being told whether the interaction is with the adversary or with the simulator). In this framework, one can formulate security of a multiparty cryptographic task by first defining an ideal functionality $\mathcal{F}$ that behaves exactly as intended, and then providing a “real-world” protocol that emulates, or “securely realizes”, the ideal functionality $\mathcal{F}$. We give more formal definitions for the above intuition. To formulate precisely what it means for an environment $\mathcal{E}$ to tell two executions apart, one has to formalize the interaction between $\mathcal{E}$ and the protocols in these executions. Concisely, an execution of a protocol $\pi$ with adversary $\mathcal{A}$ and environment $\mathcal{E}$ consists of a sequence of activations of ITMs. At each activation, the active ITM runs according to its code, its state and the content of its tapes, until it reaches a special wait state. The sequence of activations proceeds as follows: The environment $\mathcal{E}$ gets activated first and chooses inputs for $\mathcal{A}$ and for all parties. Once $\mathcal{A}$ or a party is actived by an incoming message or an input, it runs its code until it produces an outgoing message for another party, an output for $\mathcal{E}$, or it reaches the wait state, in which case $\mathcal{E}$ is activated again. The execution terminates when $\mathcal{E}$ produces its output, which can be taken to be a single bit. Note that each time it is activated, $\mathcal{E}$ is also allowed to invoke a new party, and assign a unique PID (party identifier) to it. Allowing the environment to invoke new parties will be particularly important in Section 5, where we discuss security. There, the fact that the environment has this ability implies that our security notion captures realistic scenarios in which the set of parties is not fixed at the start, but is allowed to change. Moreover, each invocation of a protocol $\pi$ is assigned a unique session identifier SID, to distinguish it from other invocations of $\pi$. We denote by $\textrm{EXEC}_{\pi,\mathcal{A},\mathcal{E}}(\lambda,z)$ the output of environment $\mathcal{E}$ initialized with input $z$, and security parameter $\lambda$ in an execution of $\pi$ with adversary $\mathcal{A}$. We are ready to state the following (slightly informal) definition. ###### Definition 5. A protocol $\pi$ UC-emulates a protocol $\phi$ if, for any PPT adversary $\mathcal{A}$, there exists a PPT simulator $\mathcal{S}$ such that, for any PPT environment $\mathcal{E}$, the families of random variables $\\{\textrm{EXEC}_{\pi,\mathcal{A},\mathcal{E}}(\lambda,z)\\}_{\lambda\in\mathbb{N},z\in\\{0,1\\}^{poly(\lambda)}}$ and $\\{\text{EXEC}_{\phi,\mathcal{S},\mathcal{E}}(\lambda,z)\\}_{\lambda\in\mathbb{N},z\in\\{0,1\\}^{poly(\lambda)}}$ are indistinguishable. Then, given an ideal functionality $\mathcal{F}$ which captures the intended ideal behaviour of a certain cryptographic task, one can define the ITM code $I_{\mathcal{F}}$, which behaves as follows: the ITM running $I_{\mathcal{F}}$ simply forwards any inputs received to the ideal functionality $\mathcal{F}$. We then say that a “real-world” protocol $\pi$ securely realizes $\mathcal{F}$ if $\pi$ emulates $I_{\mathcal{F}}$ according to Definition 5. #### A composition theorem The notion of security we just defined is strong. One of the main advantages of such a security definition is that it supports composition, i.e. security remains when secure protocols are executed concurrently, and arbitrary messages can be sent between executions. We use the notation $\sigma^{\pi}$ for a protocol $\sigma$ that makes up to polynomially many calls to another protocol $\pi$. In a typical scenario, $\sigma^{\mathcal{F}}$ is a protocol that makes use of an ideal functionality $\mathcal{F}$, and $\sigma^{\pi}$ is the protocol that results by implementing $\mathcal{F}$ through the protocol $\pi$ (i.e. replacing calls to $\mathcal{F}$ by calls to $\pi$). It is natural to expect that if $\pi$ securely realizes $\mathcal{F}$, then $\sigma^{\pi}$ securely realizes $\sigma^{\mathcal{F}}$. This is the content of the following theorem. ###### Theorem 6 (Universal Composition Theorem). Let $\pi$, $\phi$, $\sigma$ be polynomial-time protocols. Suppose protocol $\pi$ UC-emulates $\phi$. Then $\sigma^{\pi}$ UC-emulates $\sigma^{\phi}$. Replacing $\phi$ by $I_{\mathcal{F}}$ for some ideal functionality $\mathcal{F}$ in the above theorem yields the composable security notion discussed above. #### Generalized UC model The formalism of the original UC model is not able to handle security requirements in the presence of a “global trusted setup”. By this, we mean some global information accessible to all parties, which is guaranteed to have certain properties. Examples of this are a public-key infrastructure or a common reference string. Emulation in the original UC sense is not enough to guarantee composability properties in the presence of a global setup. Indeed, one can construct examples in which a UC-secure protocol for some functionality interacts badly with another UC-secure protocol and affects its security, if both protocols make reference to the same global setup. For more details and concrete examples see [CDPW07]. The generalized UC framework (GUC) of Canetti et al. [CDPW07] allows for a “global setup”. The latter is modelled as an ideal functionality which is allowed to interact not only with the parties running the protocol, but also with the environment. GUC formulates a stronger security notion, which is sufficient to guarantee a composition theorem, i.e. ideal functionalities with access to a shared global functionality $\mathcal{G}$ can be replaced by protocols that securely realize them in the presence of $\mathcal{G}$. Further, one can also replace global ideal functionalities with appropriate protocols realizing them. This kind of replacement does not immediately follow from the previous composition theorem and requires a more careful analysis, as is done in [CSV16], where sufficient conditions for this replacement are established. #### Universal Composability in the quantum setting In our setting, we are interested in honest parties, adversaries and environments that are quantum polynomial-time ITMs. The notion of Universal Composability has been studied in the quantum setting in [BOM04], [Unr04] and [Unr10]. In particular, in [Unr10], Unruh extends the model of computation of UC and its composition theorems to the setting in which polynomial-time classical ITMs are replaced by polynomial-time quantum ITMs (and ideal functionalities are still classical). The proofs are essentially the same as in the classical setting. Although the quantum version of the Generalized UC framework has not been explicitly studied in [Unr10], one can check that the proofs of the composition theorems for GUC from [CDPW07] and [CSV16] also go through virtually unchanged in the quantum setting. ## 3 Blockchains and smart contracts In this section, we start by describing blockchains and smart contracts informally. We follow this by a more formal description. As mentioned in the introduction, the essential features of blockchains and smart contracts can be abstracted by modeling them as ideal functionalities in the Universal Composability framework of Canetti [Can01]. In this section, we introduce a global ideal functionality that abstracts the properties of a transaction ledger capable of handling stateful smart contracts. We call this $\mathcal{F}_{Ledg}$, which we describe in Fig. 1. We remark that $\mathcal{F}_{Ledg}$ is somewhat of an idealized functionality which allows us to obtain a clean description and security proof for our payment system. However, $\mathcal{F}_{Ledg}$ does not capture, for example, attacks that stem from miners possibly delaying honest parties’ messages from being recorded on the blockchain. We discuss this and other issues extensively in Section 6. We also discuss ways to resolve such issues in detail, but we do not formally incorporate these in the description of our our payment system in Section 4: since we view our contribution as primarily conceptual, we elect to keep the exposition and security proof of the basic payment system as clean and accessible as possible. Informally, a blockchain is a public ledger consisting of a sequence of blocks. Each block typically contains information about a set of transactions, and a new block is appended regularly via a consensus mechanism that involves the nodes of a network. A blockchain is equipped with a native currency which is employed in transactions, and whose basic unit is referred to as a coin. Each user in the network is associated with a public key (this can be thought of as the user’s address). A typical transaction is a message which transfers coins from a public key to another. It is considered valid if it is digitally signed using the secret key corresponding to the sending address. More precisely, in Bitcoin, parties do not keep track of users’s accounts, but rather they just maintain a local copy of a set known as “unspent transaction outputs set” (UTXO set). An unspent output is a transaction that has not yet been “claimed”, i.e. the coins of these transactions have not yet been spent by the receiver. Each unspent output in the UTXO set includes a circuit (also known as a “script”) such that any user that can provide an input which is accepted by the circuit (i.e. a witness) can make a transaction that spends these coins, thus creating a new unspent output. Hence, if only one user knows the witness to the circuit, he is effectively the owner of these coins. For a standard payment transaction, the witness is a signature and the circuit verifies the signature. However, more complex circuits are also allowed, and these give rise to more complex transactions than simple payments: smart contracts. A smart contract can be thought of as a transaction which deposits coins to an address. The coins are released upon fulfillment of certain pre- established conditions. In [BK14] and [BKM17], smart contracts are defined as ideal functionalities in a variant of the Universal Composability (UC) model [Can01]. The ideal functionality that abstracts the simplest smart contracts was formalized in [BK14], and called “Claim or Refund”. Informally, this functionality specifies that a sender $P$ locks his coins and chooses a circuit $\phi$, such that a receiver $Q$ can gain possession of these coins by providing a witness $w$ such that $\phi(w)=1$ before an established time, and otherwise the sender can reclaim his coins. The “Claim or Refund” ideal functionality can be realized in Bitcoin as long as the circuit $\phi$ can be described in Bitcoin’s scripting language. On the other hand, Ethereum’s scripting language is Turing-complete, and so any circuit $\phi$ can be described. “Claim or refund” ideal functionalities can be further generalized to “stateful contracts”. In Ethereum, each unspent output also maintains a state. In other words, each unspent output comprises not only a circuit $\phi$, but also state variables. Parties can claim partial amounts of coins by providing witnesses that satisfy the circuit $\phi$ in accordance with the current state variables. In addition, $\phi$ also specifies an update rule for the state variables, which are updated accordingly. We refer to these type of transactions as stateful contracts, as opposed to the “stateless” contract of “Claim or Refund”. Stateful contracts can be realized in Ethereum, but not in Bitcoin. From now onwards, we will only work with stateful contracts. We will use the terms “smart contracts” and “stateful contracts” interchangeably. We emphasize that our modeling is inspired by [BK14] and [BKM17], but differs in the way that coins are formalized. One difference from the model of Bentov et al. is that there, in order to handle coins, the authors augment the original UC model by endowing each party with a wallet and a safe, and by considering coins as atomic entities which can be exchanged between parties. To the best of our knowledge, this variant of the UC framework is not subsumed by any of the previously studied variants, and thus it is not known whether a composition theorem holds for it. On the other hand, we feel that a more natural approach is to work in the Generalized UC model [CDPW07], and to define a global ideal functionality $\mathcal{F}_{Ledg}$ which abstracts the essential features of a transaction ledger capable of handling stateful smart contracts. This approach was first proposed by Kiayias et al. [KZZ16]. The appeal of modeling a transaction ledger as a global ideal functionality is that composition theorems are known in the Generalized UC framework. In virtue of this, any secure protocol for some task that makes calls to $\mathcal{F}_{Ledg}$ can be arbitrarily composed while still maintaining security. This means that one need not worry about composing different concurrent protocols which reference the same transaction ledger. One would hope that it is also the case that a secure protocol that makes calls to $\mathcal{F}_{Ledg}$ remains secure when the latter are replaced by calls to secure real-world realizations of it (on Ethereum for example). This requires a more careful analysis, and Canetti et al. provide in [CSV16] sufficient conditions for this replacement to be possible. We do not prove that a secure real-world realization of $\mathcal{F}_{Ledg}$ on an existing blockchain exists, but we believe that $\mathcal{F}_{Ledg}$, or a close variant of it, should be securely realizable on the Ethereum blockchain. In any case, we work abstractly by designing our payment system, and proving it secure, assuming access to such an ideal functionality. The appeal of such an approach is that the security of the higher-level protocols is independent of the details of the particular real-world implementation of $\mathcal{F}_{Ledg}$. Next, we describe our global ideal functionality $\mathcal{F}_{Ledg}$. In doing so, we establish the notation that we will utilize in the rest of the paper. #### Global ledger ideal functionality We present in Fig. 1 our global ledger ideal functionality $\mathcal{F}_{Ledg}$. In a nutshell, this keeps track of every registered party’s coins, and allows any party to transfer coins in their name to any other party. It also allows any party to retrieve information about the number of coins of any other party, as well as about any previous transaction. The initial amount of coins of a newly registered party is determined by its PID (recall that in a UC-execution the PID of each invoked party is specified by the environment; see Section 2.3 for more details). Moreover, $\mathcal{F}_{Ledg}$ handles (stateful) smart contracts: it accepts deposits from the parties involved in a contract and then pays rewards appropriately. Recall that in stateful smart contracts a party or a set of parties deposit an amount of coins to the contract. The contract is specified by a circuit $\phi$, together with an initial value for a state variable st. A state transition is triggered by any party $P$ with PID $pid$ sending a witness $w$ which is accepted by $\phi$ in accordance with the current state and the current time $t$. More precisely, the contract runs $\phi(pid,w,t,\textsf{st})$, which outputs either “$\perp$” or a new state (stored in the variable st) and a number of coins $d\in\mathbb{N}$ that is released to $P$. Each contract then repeatedly accepts state transitions until it has distributed all the coins that were deposited into it at the start. Notice that $\phi$ can accept different witnesses at different times (the acceptance of a witness can depend on the current time $t$ and the current value of the state variable st). Information about existing smart contracts can also be retrieved by any party. The stateful-contract portion of $\mathcal{F}_{Ledg}$ resembles closely the functionality $\mathcal{F}_{StCon}$ from [BKM17]. Our approach differs from that of [BKM17] in that the we make the smart contract functionality part of the global ideal functionality $\mathcal{F}_{Ledg}$ which also keeps track of party’s coins and transactions. In [BKM17] instead, coins are incorporated in the computation model by augmenting the ITMs with wallets and safes (this changes the model of computation in a way that is not captured by any of the the previously studied variants of the UC framework). We implicitly assume access to an ideal functionality for message authentication $\mathcal{F}_{auth}$ which all parties employ when sending their messages, and also to a global ideal functionality for a clock that keeps track of time. We assume implicitly that $\mathcal{F}_{Ledg}$ makes calls to the clock and keeps track of time. Alternatively, we could just have $F_{Ledg}$ maintain a local variable that counts the number of transactions performed, and a local time variable $t$, which is increased by $1$ every time the number of transactions reaches a certain number, after which the transaction counter is reset (this mimics the process of addition of blocks in a blockchain, and time simply counts the number of blocks). From now onwards, we do not formally reference calls to $\mathcal{F}_{auth}$ or to the clock to avoid overloading notation. We are now ready to define $\mathcal{F}_{Ledg}$. Global ledger ideal functionality Initialize the sets $\textsf{parties}=\\{\\}$, $\textsf{contracts}=\\{\\}$, $\textsf{AllTransactions}=\\{\\}$. (Throughout, the variable $t$ denotes the current time.) #### Add party: Upon receiving a message AddParty from a party with PID $pid=(id,d)$, send (AddedParty, $pid$) to the adversary; upon receiving a message ok from the adversary, and if this is the first request from $pid$, add $pid$ to the set parties. Set $pid.\textsf{id}\leftarrow id$ and $pid.\textsf{coins}\leftarrow d$. #### Retrieve party: Upon receiving a message (RetrieveParty, $pid$) from some party $P$ (or the adversary), output (RetrieveParty, $pid$, $d$) to $P$ (or to the adversary), where $d=\perp$ if $pid\notin\textsf{parties}$, else $d=pid.\textsf{coins}$. (We abuse notation here in that, when taken as part of a message, $pid$ is treated as a string, but, when called by the functionality, it is a variable with attributes $pid.\textsf{id}$ and $pid.\textsf{coins}$). #### Add transaction: Upon receiving a message (AddTransaction, $pid^{\prime}$, $d$) from some party $P$ with PID $pid$, and $pid\in\textsf{parties}$, do the following: * • If $pid^{\prime}\in\textsf{parties}$ and $pid.\textsf{coins}>d$, update $pid^{\prime}.\textsf{coins}\leftarrow pid^{\prime}.\textsf{coins}+d$ and $pid.\textsf{coins}\leftarrow pid.\textsf{coins}-d$. Set $trId=|\textsf{AllTransactions}|+1$. Add a variable named $trId$ to AllTransactions, with attribute $trId.\textsf{transaction}=(pid,pid^{\prime},d,t)$. Send a message (Executed, $trId$) to $P$. * • Else, return $\perp$ to $P$. #### Retrieve transaction: Upon receiving a message (RetrieveTransaction, $trId$) from some party $P$ (or the adversary), output (RetrieveTransaction, $trId$, $s$), where $s=\perp$ if $trId\notin\textsf{allTransactions}$, and $s=trId.\textsf{transaction}$ otherwise. #### Add/trigger smart contract: Upon receiving a message (AddSmartContract, Params=$(I,D,\phi,\textsf{st}_{0})$), where $I$ is a set of PID’s, $D$ is a set $\\{(pid,d_{pid}):pid\in I\\}$ of “initial deposits”, with $d_{pid}$ being the amount required initially from the party with PID $pid$, $\phi$ is a circuit, and $\textsf{st}_{0}$ is the initial value of a state variable st, check that $I\subseteq\textsf{parties}$. If not, ignore the message; if yes, set $ssid=|\textsf{contracts}|+1$. Add a variable named $ssid$ to contracts with attributes $ssid.\textsf{Params}=(I,D,\phi,\textsf{st}_{0})$, $ssid.\textsf{state}=\textsf{st}$ and $ssid.\textsf{coins}\leftarrow 0$. Send a message (RecordedContract, $ssid$) to $P$. Then, do the following: * • Initialization phase: Wait to get message $(\textsf{InitializeWithCoins},\textit{ssid},\textsf{Params}=(I,D,\phi,\textsf{st}_{0}))$ from party with PID $pid$ for all $pid\in I$. When all messages are received, and if, for all $pid\in I$, $pid.\textsf{coins}\geq d_{pid}$, then, for all $pid\in I$, update: $pid.\textsf{coins}\leftarrow pid.\textsf{coins}-d_{pid}$ and $ssid.\textsf{coins}\leftarrow ssid.\textsf{coins}+d_{pid}$. Set $\textsf{st}\leftarrow\textsf{st}_{0}$ (We assume that $ssid.\textsf{state}$ changes dynamically with st). * • Execution phase: Repeat until termination: Upon receiving a message of the form $(\textsf{Trigger},\textit{ssid},w,d)$ at time $t$ from some party with PID $pid\in\textsf{parties}$ (where it can also be $d=0$) such that $\phi(pid,w,t,\textsf{st},d)\neq\perp$, do the following: * – If $d>0$, update $pid.\textsf{coins}\leftarrow pid.\textsf{coins}-d$ and $ssid.\textsf{coins}\leftarrow ssid.\textsf{coins}+d$ * – Update $(\textsf{st},e)\leftarrow\phi(pid,w,t,\textsf{st},d)$. * – If $e=\textnormal{``all coins''}$, let $q:=ssid.\textsf{coins}$. Send the message (Reward, $ssid$, $q$) to the party with PID $pid$ and update $pid.\textsf{coins}\leftarrow pid.\textsf{coins}+q$ and $ssid.\textsf{coins}\leftarrow 0$. If $e>0$ and $ssid.\textsf{coins}>e$, send the message (Reward, $ssid$, $e$) to the party with PID $pid$, and update $pid.\textsf{coins}\leftarrow pid.\textsf{coins}+e$ and $ssid.\textsf{coins}\leftarrow ssid.\textsf{coins}-e$. Else, if $ssid.\textsf{coins}=e^{\prime}\leq e$, send the message (Reward, $ssid$, $e^{\prime}$) to the party with PID $pid$, and update $pid.\textsf{coins}\leftarrow pid.\textsf{coins}+e^{\prime}$ and $ssid.\textsf{coins}\leftarrow 0$. Then, terminate. #### Retrieve smart contract: Upon receiving a message (RetrieveContract, $ssid$) from some party $P$ (or the adversary), output (RetrieveContract, $ssid$, $z$), where $z=\perp$ if $ssid\notin\textsf{contracts}$, and $z=(ssid.\textsf{Params},ssid.\textsf{state},ssid.\textsf{coins})$ otherwise. Figure 1: Global ledger ideal functionality $\mathcal{F}_{Ledg}$ We think of “Retrieve” operations as being fast, or free, as they do not alter the state of the ledger. We think of “Add” and “Trigger” operations as being slow, as they alter the state of the ledger. From now on, we will often refer to the number of coins $ssid.\textsf{coins}$ of a contract with session identifier $ssid$ as the coins deposited in the contract. When we say that a contract releases some coins to a party $P$ with PID $pid$, we mean more precisely that $\mathcal{F}_{Ledg}$ updates its local variables and moves coins from $ssid.\textsf{coins}$ to $pid.\textsf{coins}$. ## 4 A payment system based on quantum lightning and a classical blockchain In this section, we describe our payment system. We give first an informal description, and in Section 4.1 we give a formal description. The building block of our payment system is a quantum lightning scheme, reviewed in detail in Section 2.2. Recall that a quantum lightning scheme consists of a generation procedure which creates quantum banknotes, and a verification procedure that verifies them and assigns serial numbers. The security guarantee is that no generation procedure (not even the honest one) can create two banknotes with the same serial number except with negligible probability. As mentioned earlier, this property is desirable if one wants to design a decentralized payment system, as it prevents anyone from cloning banknotes (even the person who generates them). However, this calls for a mechanism to regulate generation of new valid quantum banknotes. In this section, we describe formally a proposal that employs smart contracts to provide such a mechanism. As we have described informally in the introduction, the high-level idea is to keep track of the valid serial numbers using smart contracts. Any party is allowed to deposit any amount of coins $d$ (of their choice) into a smart contract with specific parameters (see definition 7 below), and with an initial value of his choice for a serial number state variable. We can think of the quantum banknote with the chosen serial number as having “acquired” value $d$. A payment involves only two parties: a payer, who sends a quantum banknote, and a payee who receives it and verifies it locally. As anticipated in the introduction, the full payment system includes the following additional features, which we describe here informally in a little more detail (all of these are described formally in Section 4.1): * • Removing a serial number from the list of valid serial numbers in order to recover the amount of coins deposited in the corresponding smart contract. This makes the two commodities (quantum banknotes and coins on the blockchain) interchangeable. This is achieved by exploiting the additional property of of some quantum lightning scheme from Definition 3. Recall that, informally, this property states that there is some classical certificate that can be recovered by measuring a valid quantum banknote, which no efficient algorithm can recover otherwise. The key is that once the state is measured to recover this certificate, it is damaged in a way that it only passes verification with negligible probability (meaning that it can no longer be spent). We allow users to submit this classical certificate to a smart contract, and if the certificate is consistent with the serial number stored in the contract, then the latter releases all of the coins deposited in the contract to the user. * • Allowing a party to replace an existing serial number with a new one of their choice in case they lose a valid quantum state (they are fragile after all!). We allow a user $P$ to file a “lost banknote claim” by sending a message and some fixed amount of coins $d_{0}$ to a smart contract whose serial number is the serial number of the lost banknote. The idea is that if no one challenges this claim, then after a specified time $t_{tr}$ user $P$ can submit a message which changes the value of the serial number state variable to a value of his choice and recovers the previously deposited $d_{0}$ coins. On the other hand, if a user $P$ maliciously files a claim to some contract with serial number $s$, then any user $Q$ who possesses the valid banknote with serial number $s$ can recover the classical certificate from Definition 3, and submit it to the contract. This releases all the coins deposited in the contract to $Q$ (including the $d_{0}$ deposited by $P$ to make the claim). As you might notice, this requires honest users to monitor existing contracts for “lost banknote claims”. This, however, is not much of a burden if $t_{tr}$ is made large enough (say a week or a month). The requirement of being online once a week or once a month is easy to meet in practice. ### 4.1 The payment system and its components In this section, we describe in detail all of the components of the payment system. It consists of the following: a protocol to generate valid quantum banknotes; a protocol to make a payment; a protocol to file a claim for a lost banknote; a protocol to prevent malicious attempts at filing claims for lost banknotes; and a protocol to trade a valid quantum banknote in exchange for coins. Let $\lambda\in\mathbb{N}$. From now onwards, we assume that $(\textsf{gen-bolt},\textsf{verify-bolt},\textsf{gen- certificate},\textsf{verify- certificate})\leftarrow\textsf{QL.Setup}(\lambda),$ where the latter is the setup procedure of a quantum lightning scheme with bolt-to-certificate capability (i.e. a quantum lightning scheme that satisfies the additional property of Definition 3). We are thus assuming a trusted setup for our payment system (note there are practical ways to ensure that such a setup, which is a one-time procedure, is performed legitimately even in a network where many parties might be dishonest). We also assume that all parties have access to an authenticated and ideal quantum channel. Recall from the description of $\mathcal{F}_{Ledg}$ that each smart contract is specified by several parameters: $I$ is a set of PIDs of parties who are expected to make the initial deposit, with $\\{d_{pid}:pid\in I\\}$ being the required initial deposit amounts; a circuit $\phi$ specifies how the state variables are updated and when coins are released; an initial value $\textsf{st}_{0}$ for the state variable st; a session identifier $ssid$. In Definition 7 below, we define an instantiation of smart contracts with a particular choice of parameters, which we refer to as banknote-contracts. Banknote-contracts are the building blocks of the protocols that make up our payment system. We describe a banknote-contract informally before giving a formal definition. A banknote-contract is a smart contract initialized by a single party, and it has a state variable of the form $\textsf{st}=(\textsf{serial},\textsf{ActiveLostClaim})$. The party initializes the banknote-contract by depositing a number of coins $d$ and by setting the initial value of serial to any desired value. The banknote- contract handles the following type of requests: * • As long as $\textsf{ActiveLostClaim}=\text{``No active claim''}$ (which signifies that there are no currently active lost-banknote claims), any party $P$ can send the message BanknoteLost, together with a pre-established amount of coins $d_{0}$ to the contract. This will trigger an update of the state variable ActiveLostClaim to reflect the active lost-banknote claim by party $P$. * • As long as there is an active lost-banknote claim, i.e. $\textsf{ActiveLostClaim}=\text{``Claim by $pid$ at time $t$''}$, any party $Q$ can challenge that claim by submitting a message $(\textsf{ChallengeClaim},c,s^{\prime})$ to the contract, where $s^{\prime}$ is a proposed new serial number. We say that $c$ is a valid classical certificate for the current value $s$ of serial if $\textsf{verify- certificate}(s,c)=1$. Such a $c$ can be thought of as a proof that whoever is challenging the claim actually possessed a quantum banknote with serial number $s$, and destroyed it in order to obtain the certificate $c$, and thus that the current active lost-banknote claim is malicious. If $c$ is a valid classical certificate for $s$, then serial is updated to the new value $s^{\prime}$ submitted by $Q$, who also receives all of the coins deposited in the contract (including the $d_{0}$ coins deposited by the malicious claim). * • If party $P$ has previously submitted a lost-banknote claim, and his claim stays unchallenged for time $t_{tr}$, then party $P$ can send a message $(\textsf{ClaimUnchallenged},s^{\prime})$ to the contract, where $s^{\prime}$ is a proposed new serial number. Then the contract returns to $P$ the $d_{0}$ coins he initially deposited when making the claim, and updates serial to $s^{\prime}$. * • Any party $P$ can submit to the contract a message $(\textsf{RecoverCoins},c)$. If $c$ is a valid classical certificate for the current value of $s$ of serial, then the contract releases to $P$ all the coins currently deposited in the contract. This allows party $P$ to “convert” back his quantum banknote into coins. Next, we will formally define banknote-contracts, and then formally describe all of the protocols that make the payment system. $\phi_{\$}\left(pid,w,t,(\textsf{serial},\textsf{ActiveLostClaim}),d\right)$ takes as input strings $pid$ and $w$, where $pid$ is meant to be the PID of some party $P$, and we refer to $w$ as the “witness”, $t\in\mathbb{N}$ denotes the “current time” mantained by $\mathcal{F}_{Ledg}$, $(\textsf{serial},\textsf{ActiveLostClaim})$ is the current value of the state variable, and $d\in\mathbb{N}$ is the number of coins that are being deposited to the smart contract with the current message. $\phi_{\$}$ has hardcoded parameters: $d_{0}\in\mathbb{N}$ the amount of coins needed to file a claim for a lost money state, $t_{tr}\in\mathbb{N}$ the time after which an unchallenged claim can be settled ($d_{0}$ and $t_{tr}$ are fixed constants agreed upon by all parties, and they are the same for all banknote-contracts), $l(\lambda)$ the length of a certificate in the lightning scheme, a description of verify-certificate. The circuit $\phi_{\$}$ outputs new values for the state variables and an amount of coins as follows: On input $(pid,w,t,(\textsf{serial}=s,\textsf{ActiveLostClaim}),d)$, $\phi_{\$}$ does the following: * • If $\textsf{ActiveLostClaim}=\text{``No active claim''}$: * – If $w=\textsf{BanknoteLost}$ and $d=d_{0}$, then $\phi$ outputs $0((\textsf{serial}=s,\textsf{ActiveLostClaim}=\text{``Claim by $pid$ at time $t$''}),00)$ (to symbolize that at time $t$ party with PID $pid$ has claimed to have lost the money state with serial number $s$, and that zero coins are being released). * – If $w=(\textsf{RecoverCoins},c)$, where $c\in\\{0,1\\}^{l(\lambda)}$ and $\textsf{verify-certificate}(s,c)=1$, then $\phi$ outputs $0((\textsf{serial}=\perp,\textsf{ActiveLostClaim}=\perp),\textnormal{``all coins''}0)$ * • If $\textsf{ActiveLostClaim}=\text{``Claim by $pid^{\prime}$ at time $t_{0}$''}$ for some $pid^{\prime},t_{0}$: * – If $w=(\textsf{ChallengeClaim},c,s^{\prime})$, where $c\in\\{0,1\\}^{l(\lambda)}$ and $\textsf{verify-certificate}(s,c)=1$, and $s^{\prime}\in\\{0,1\\}^{\lambda}$ then $\phi$ outputs $0((\textsf{serial}=s^{\prime},\textsf{ActiveLostClaim}=\text{``No active claim''}),d_{0}0)$. * – If $w=(\textsf{ClaimUnchallenged},s^{\prime})$, $pid=pid^{\prime}$ and $t-t_{0}>t_{tr}$, then $\phi$ outputs $0((\textsf{serial}=s^{\prime},\textsf{ActiveLostClaim}=\text{``No active claim''}),d_{0}0)$. Figure 2: Circuit $\phi_{\$}$ for banknote-contracts ###### Definition 7. (Banknote-contract) A banknote-contract, is a smart contract on $\mathcal{F}_{Ledg}$ specified by parameters of the following form: $I=\\{pid\\}$ for some $pid\in[n]$, $D=\\{(pid,d_{pid})\\}$ for some $d_{pid}\in\mathbb{N}$, $\textsf{st}_{0}=(s,\text{``No active claim''})$ for some $s\in\\{0,1\\}^{\lambda}$, and circuit $\phi=\phi_{\$}$, where $\phi_{\$}$ is defined as in Fig. 2. For convenience, we denote by serial and ActiveLostClaim respectively the first and second entry of the state variable of a banknote-contract. #### Generating valid quantum banknotes We describe the formal procedure for generating a valid quantum banknote. Protocol carried out by some party $P$ with PID $pid$. Input of $P$: An integer $d$ such that $pid.\textsf{coins}>d$ in $\mathcal{F}_{Ledg}$ ($d$ is the “value” of the prospective banknote). * • Run $(\ket{\psi},s)\leftarrow\textsf{gen-bolt}$. * • Send $0(\textsf{AddSmartContract},\textsf{Params}0)$ to $\mathcal{F}_{Ledg}$, where $\textsf{Params}=(\\{pid\\},\\{(pid,d)\\},\phi,(s,\text{``No active claim''}))$. Upon receipt of a message of the form (RecordedContract, $ssid$), send the message (InitializeWithCoins, $ssid$, Params) to $\mathcal{F}_{Ledg}$. Figure 3: Generating a valid banknote #### Making a payment We describe formally the protocol for making a payment in Fig. 4. Informally, the protocol is between a party $P$, the payer, and a party $Q$, the payee. In order to pay party $Q$ with a bolt whose serial number is $s$, party $P$ sends the valid bolt to party $Q$, the payee, together with the $ssid$ of a smart contract with $\textsf{serial}=s$. Party $Q$ verifies that $ssid$ corresponds to a banknote-contract with $\textsf{serial}=s$, and verifies that the banknote passes verification and has serial number $s$. The protocol is between some party $P$ with PID $pid$(the payer) and a party $Q$ with PID $pid^{\prime}$ (the payee): Input of $P$: $\ket{\Psi}$, a valid bolt with serial number $s$. $ssid$ the session identifier of a smart contract on $\mathcal{F}_{Ledg}$ such that $ssid.\textsf{state}=(s,\text{``No active claim''})$, and $ssid.\textsf{coins}=d$. * • $P$ sends $(\ket{\Psi},s,ssid,d)$ to $Q$. * • $Q$ sends a message (RetrieveContract, $ssid$) to $\mathcal{F}_{Ledg}$. Upon receiving a message (RetrieveContract, $ssid$, $z$) from $\mathcal{F}_{Ledg}$ (where $z=(ssid.\textsf{Params},ssid.\textsf{state},ssid.\textsf{coins})$ if $P$ is honest), $Q$ does the following: * – If $z=(\textsf{Params},(s,\text{``No active claim''}),d)$, then $Q$ checks that the parameters Params are of the form of a banknote-contract (from Definition 7). If so, runs $\textsf{verify-bolt}(\ket{\Psi},s)$ and checks that the outcome is $1$. If so, sends the message accept to $P$. * – Else, $Q$ sends the message reject and the state $\ket{\Psi}$ back to $P$. Figure 4: Protocol for making and verifying a payment #### Recovering lost banknotes As much as we can hope for experimental progress in the development of quantum memories, for the foreseeable future we can expect quantum memories to only be able to store states for a time on the order of days. It is thus important that any payment system involving quantum money is equipped with a procedure for users to recover the value associated to quantum states that get damaged and become unusable. Either users should be able to “convert” quantum money states back to coins on the blockchain, or they should be able, upon losing a quantum banknote, to change the serial number state variable of the associated smart contract to a new serial number (presumably of freshly generated quantum banknote). Here, we describe a protocol for the latter. After this, we will describe a protocol for the former. Informally, a party $P$ who has lost a quantum banknote with serial number $s$ associated to a smart contract with session identifier $ssid$, makes a “lost banknote claim” at time $t$ by depositing a number of coins $d_{0}$ to that banknote-contract. Recall the definition of banknote-contracts from Definition 7, and in particular of the circuit $\phi_{\$}$: * • If party $P$ is honest, then after a time $t_{tr}$ has elapsed, he will be able to update the state variable serial of the banknote-contract from $s$ to $s^{\prime}$ (where $s^{\prime}$ is presumably the serial number of a new valid bolt that party $P$ has just generated). * • If party $P$ is dishonest, and he is claiming to have lost a banknote with serial number $s$ that someone else possesses, then the legitimate owner can run $\textsf{gen-certificate}(\ket{\psi},s)$ where $\ket{\Psi}$ is the legitimate bolt, and obtain a valid certificate $c$. He can then send $c$ to the contract and a new serial number $s^{\prime}$ (presumably of a freshly generate bolt) and obtain $d_{0}$ coins from the contract (the $d_{0}$ coins deposited by $P$ in his malicious claim). We describe the protocol formally in Fig. 5. One might wonder whether, in practice, an adversary can instruct a corrupt party to make a “lost banknote claim”, and then intercept an honest party’s classical certificate $c$ before this is posted to the blockchain, and have a second corrupt party post it instead. This attack would allow the adversary to “steal” the honest party’s value. Or alternatively, an adversary could monitor the network for lost-banknote claims by honest parties, and whenever he sees one he will delay this claim, and instruct a corrupt party to make the same claim so that it is registered first on the ledger. In our analysis, we do not worry about such attacks, as we assume access to the ideal functionality $\mathcal{F}_{Ledg}$, which, by definition, deals with incoming messages in the order that they are received. We also assume in our adversarial model, specified more precisely in Section 5, that the adversary does not have any control over the delivery of messages (and their timing). If one assumes a more powerful adversary (with some control over the timing of delivery of messages), then the first issue can still be resolved elegantly. The second issue has a satisfactory resolution but it is trickier to analyze formally. We discuss this in more detail in Section 6. Protocol carried out by party $P$ with PID $pid$ for changing the serial number of a smart contract. $P$’s input: $s$ the serial number of a (lost) quantum banknote. $ssid$ the session identifier of a banknote-contract such that $ssid.\textsf{state}=(s,\text{``No active claim''})$. * • $P$ sends $(\textsf{Trigger},ssid,\textsf{BanknoteLost},d_{0}$) to $\mathcal{F}_{Ledg}$. This updates $ssid.\textsf{state}$ to $(s,\text{``Active claim by $pid$ at time $t$''})$ (where $t$ is the current time mantained by $\mathcal{F}_{Ledg}$), and deposits $d_{0}$ coins into the contract. * • After time $t_{tr}$, $P$ sends $(\textsf{Trigger},ssid,(\textsf{ClaimUnchallenged},s^{\prime}),0)$ to $\mathcal{F}_{Ledg}$. If $P$ was honest then $ssid.\textsf{state}$ is updated to $(s^{\prime},\text{``No active claim''})$, and $d_{0}$ coins are released to $P$. Figure 5: Protocol for changing the serial number of a smart contract Next, we give a protocol carried out by all parties to prevent malicious attempts at changing the state variable serial of a smart contract. Informally, this involves checking the blockchain regularly for malicious attempts at filing lost-banknote claims. Recall that $t_{tr}$ was defined in Definition 7. Protocol carried out by a party $P$ to prevent malicious attempts at changing the state variable serial of a smart contract. Input of $P$: A triple $(\ket{\Psi},s,ssid)$, where $\ket{\Psi}$ is a quantum banknote with serial number $s$, and $ssid$ is the session identifier of a banknote-contract such that $ssid.\textsf{state}=(s,\text{``No active claim''})$ At regular intervals of time $t_{r}-1$, do the following: * • Send a message (RetrieveContract, $ssid$) to $\mathcal{F}_{Ledg}$. Upon receiving a message (RetrieveContract, $ssid$, $z$) from $\mathcal{F}_{Ledg}$, if $z=(\textsf{Params},(s,\text{``Claim by $pid^{\prime}$ at time $t$ ''}),d)$ for some $pid^{\prime},t,d$ and for some banknote-contract parameters Params: * – Run $c\leftarrow\textsf{gen-certificate}(\ket{\Psi},s)$. * – Sample $(\ket{\Psi^{\prime}},s^{\prime})\leftarrow\textsf{gen-bolt}$. * – Send $(\textsf{Trigger},ssid,(\textsf{ChallengeClaim},c,s^{\prime}),0)$ to $\mathcal{F}_{Ledg}$. (If $P$ was honest, this updates $ssid.\textsf{state}\leftarrow(s^{\prime},\text{``No active claim''})$ and releases $d_{0}$ coins to $P$). Figure 6: Protocol for preventing malicious attempts at changing the state variable serial of a smart contract. #### Trading a quantum banknote for coins: Finally, we describe a protocol for trading a quantum banknote to recover all the coins deposited in its associated banknote-contract. Protocol carried out by a party $P$. Input of $P$: A tuple $(\ket{\Psi},s,ssid,d)$, where $\ket{\Psi}$ is a quantum banknote with serial number $s$, and $ssid$ is the session identifier of a banknote-contract such that $ssid.\textsf{state}=(z,\text{``No active claim''})$ and $ssid.\textsf{coins}=d$. * • Run $c\leftarrow\textsf{gen-certificate}(\ket{\Psi},s)$. * • Send message $(\textsf{Trigger},ssid,(\textsf{RecoverCoins},c),0)$ to $\mathcal{F}_{Ledg}$. This releases $d$ coins to $P$. Figure 7: Protocol for trading a quantum banknote for coins. ## 5 Security We first specify an adversarial model. Security with respect to this adversarial model is formally captured by Theorem 8. At a high-level, Theorem 8 establishes that, within this adversarial model, no adversary can increase his “value” beyond what he has legitimately spent or received to and from honest parties. This captures, for example, the fact that the adversary will not be able to double-spend his banknotes, or successfully file a “lost banknote claim” for banknotes he does not legitimately possess. #### Adversarial model We assume that all the messages of honest parties are sent using the ideal functionality for authenticated communication $\mathcal{F}_{auth}$, and that the adversary sees all messages that are sent (in UC language, we assume that the adversary is activated every time a party sends a message) but has no control over the delivery of messages (whether they are delivered or not) and their timing. Our payment system can be made to work also if we assume that the adversary can delay delivery of honest parties’ messages by a fixed amount of time (see the remark preceding Fig. 5 for more details), but, for simplicity, we do not grant the adversary this power. The adversary can corrupt any number of parties, and it may do so adaptively, meaning that the corrupted parties are not fixed at the start, but rather an honest party can become corrupted, or a corrupted party can return honest, at any point. The process of corruption is modeled analogously as in the original UC framework, where the adversary simply writes a corrupt message on the incoming tape of an honest party, upon which the honest party hands all of its information to the adversary, who can send messages on the corrupted party’s behalf. Our setting is slightly more involved in that corrupted parties also possess some quantum information, in particular the quantum banknotes. We assume that when an adversary corrupts a party he takes all of its quantum banknotes. Importantly, we assume that these are not returned to the party once the party is no longer corrupted. It might seem surprising that we do not upper bound the fraction of corrupted parties. Indeed, such a bound would only be needed in order to realize securely the ideal functionality $\mathcal{F}_{Ledg}$ (any consensus-based realization of $F_{Ledg}$ would require such a bound). Here, we assume access to such an ideal functionality, and we do not worry about its secure realization. Naturally, when replacing the ideal functionalities with real-world realizations one would set the appropriate bound on the corruption power of the adversary, but we emphasize that our schemes are independent of the particular real-world realization. Note that we do not fix a set of parties at the start, but rather new parties can be created (see below for more details). We assume that (ITMs of) honest parties run the code $\pi$. This represents the “honest” code which executes the protocols from Section 4 as specified. The input to $\pi$ then specifies when and which protocols from Section 4 are to be executed. As part of $\pi$, we specify that, upon invocation, a party sends a message AddParty to $\mathcal{F}_{Ledg}$ to register itself. We also specify as part of $\pi$ that an honest party runs the protocol of Fig. 6 (to prevent malicious claims for lost banknotes). Moreover, for notational convenience, we specify as part of $\pi$ that each party maintains a local variable banknoteValue, which keeps track of the total value of the quantum banknotes possessed by the party. banknoteValue is initialized to $0$, and updated as follows. Whenever a party $P$ successfully receives a quantum banknote (i.e. $P$ is the payee in the protocol from Fig. 4 and does not abort) of value $d$ (i.e. the associated smart contract has $d$ coins deposited), then $P$ updates $\textsf{banknoteValue}\leftarrow\textsf{banknoteValue}+d$. Similarly, when $P$ sends a quantum banknote of value $d$, it updates $\textsf{banknoteValue}\leftarrow\textsf{banknoteValue}-d$. Finally, we specify also as part of $\pi$, that whenever a party that was corrupted is no longer corrupted, it resets $\textsf{banknoteValue}=0$ (this is because we assumed that quantum banknotes are not returned by the adversary). The following paragraph leads up to a notion of security and a security theorem. Let $\mathcal{A}$ be a quantum polynomial-time adversary and $\mathcal{E}$ a quantum polynomial-time environment. Consider an execution of $\pi$ with adversary $\mathcal{A}$ and environment $\mathcal{E}$ (see Section 2.3 for more details on what an “execution” is precisely). We keep track of two quantities during the execution, which we denote as AdversaryValueReceived and AdversaryValueCurrentOrSpent (These quantities are not computed by any of the parties, adversary or environment. Rather, they are just introduced for the purpose of defining security). The former represents the amount of value, coins or banknotes, that the adversary has received either by virtue of having corrupted a party, or by having received a payment from an honest party. The latter counts the total number of coins currently possessed by corrupted parties, as recorded on $\mathcal{F}_{Ledg}$, and the total amount spent by the adversary to honest parties either via coins or via quantum banknotes (it does not count the value of quantum banknotes currently possessed; these only count once they are successfully spent). Both quantities are initialized to $0$, and updated as follows throughout the execution: * (i) When $\mathcal{A}$ corrupts a party $P$: let $d$ be the number of coins of $P$ according to the global functionality $\mathcal{F}_{Ledg}$ and $d^{\prime}$ be $P$’s banknoteValue just before being corrupted. Then, $\textsf{AdversaryValueReceived}\leftarrow\textsf{AdversaryValueReceived}+d+d^{\prime}$, and $\textsf{AdversaryValueCurrentOrSpent}\leftarrow\textsf{AdversaryValueCurrentOrSpent}+d$. * (ii) When a corrupted party $P$ with $d$ coins and $\textsf{banknoteValue}=d^{\prime}$ ceases to be corrupted and returns honest, $\textsf{AdversaryValueReceived}\leftarrow\textsf{AdversaryValueReceived}-d$. * (iii) When an honest party pays $d$ coins to a corrupted party, $\textsf{AdversaryValueReceived}\leftarrow\textsf{AdversaryValueReceived}+d$. Likewise, when an honest party sends a quantum banknote of value $d$ to a corrupted party, through the protocol of Fig. 4, then (even if the corrupted party does not return accept) $\textsf{AdversaryValueReceived}\leftarrow\textsf{AdversaryValueReceived}+d$. * (iv) When $\mathcal{A}$ succesfully spends a quantum banknote of value $d$ to an honest party $P$, i.e. a corrupted party is the payer in the protocol from Fig. 4 and $P$ is the payee and returns accept, or when $\mathcal{A}$ pays $d$ coins to an honest party, then $\textsf{AdversaryValueCurrentOrSpent}\leftarrow\textsf{AdversaryValueCurrentOrSpent}+d$. * (v) When a corrupted party receives $d$ coins from a banknote-contract, then $\textsf{AdversaryValueCurrentOrSpent}\leftarrow\textsf{AdversaryValueCurrentOrSpent}+d$. Notice that this can happen only in two ways: $\mathcal{A}$ successfully converts a quantum banknote of value $d$ to coins on $\mathcal{F}_{Ledg}$ (via the protocol of Fig. 7), or a corrupted party successfully challenges a BanknoteLost claim (in this case $d=d_{0}$). Intuitively, if our payment scheme is secure, then at no point in time should the adversary be able to make $\textsf{AdversaryValueCurrentOrSpent}-\textsf{AdversaryValueReceived}>0$. This would mean that he has successfully spent/stolen value other than the one he received by virtue of corrupting a party or receiving honest payments. The following theorem formally captures this notion of security. First, we denote by $\mathcal{F}_{Ledg}\text{-}\textrm{EXEC}^{(MaxNetValue)}_{\pi,\mathcal{A},\mathcal{E}}(\lambda,z)$ the maximum value of $\textsf{AdversaryValueCurrentOrSpent}-\textsf{AdversaryValueReceived}$ during an execution of $\pi$ with adversary $\mathcal{A}$ and environment $\mathcal{E}$, with global shared functionality $\mathcal{F}_{Ledg}$. ###### Theorem 8 (Security). For any quantum polynomial-time adversary $\mathcal{A}$ and quantum polynomial-time environment $\mathcal{E}$, $\Pr[\mathcal{F}_{Ledg}\text{-}\textrm{EXEC}^{(MaxNetValue)}_{\pi,\mathcal{A},\mathcal{E}}(\lambda,z)>0]=negl(\lambda).$ The rationale behind considering executions of $\pi$ and quantifying over all possible adversaries and environments is that doing so captures all possible ways in which a (dynamically changing) system of honest parties running our payment system alongside an adversary can behave (where the adversary respects our adversarial model). Recall that, in an execution of $\pi$, the environment has the ability to invoke new parties and assign to them new unique PIDs. Since in $\mathcal{F}_{Ledg}$ the PIDs are used to register parties and initialize their number of coins, this means that the environment has the ability to pick the initial number of coins of any new party that it invokes. Moreover, by writing inputs to the parties input tapes, the environment can instruct honest parties to perform the honest protocols from Section 4 in any order it likes. Quantifying over all adversaries and environments, in the statement of Theorem 8 means that the adversary and the environment can intuitively be thought of as one single adversary. The statement of the theorem thus captures security against realistic scenarios in which new parties can be adversarially created with an adversarially chosen number of coins, and they can be instructed to perform the honest protocols of the payment system from Section 4, in whatever sequence is convenient to the adversary. ###### Proof of Theorem 8. Suppose for a contradiction that there exists $\mathcal{A}$ and $\mathcal{E}$ such that $\Pr[\mathcal{F}_{Ledg}\text{-}\textrm{EXEC}^{(MaxNetValue)}_{\pi,\mathcal{A},\mathcal{E}}(\lambda,z)>0]\neq negl(\lambda).$ (10) Then, we go through all of the possible ways that an adversary can increase its net value, i.e. increase the quantity $\textsf{AdversaryValueCurrentOrSpent}-\textsf{AdversaryValueReceived}$: the adversary can do so through actions from items (ii), (iv) and (v) above. Amongst these, it is easy to see that action (ii) never results in $\textsf{AdversaryValueCurrentOrSpent}-\textsf{AdversaryValueReceived}>0$. Thus, in order for (10) to hold, it must be the case that one of the following happens with non-negligible probability within an execution of $\pi$ with adversary $\mathcal{A}$ and environment $\mathcal{E}$. * • An action from item (iv) resulted in a positive net value for $\mathcal{A}$, i.e. $\textsf{AdversaryValueCurrentOrSpent}-\textsf{AdversaryValueReceived}>0$. Notice that for this to happen it must be the case that $\mathcal{A}$ has double-spent a banknote, i.e. $\mathcal{A}$ has produced two banknotes with the same serial number that have both been accepted by honest parties in a payment protocol of Fig. 4, and so they have both passed verification. But then, it is straightforward to see that we can use this adversary, together with $\mathcal{E}$ to construct an adversary $\mathcal{A}^{\prime}$ that breaks the security of the quantum lightning scheme (i.e. game Counterfeit): $\mathcal{A}^{\prime}$ simply simulates an execution of protocol $\pi$ with adversary $\mathcal{A}$ and environment $\mathcal{E}$, and with non-negligible probability the adversary $\mathcal{A}$ in this execution produces two banknotes with the same serial number. $\mathcal{A}^{\prime}$ uses these banknotes to win the security game of quantum lightning. * • An action from item (v) resulted in a positive net value for $\mathcal{A}$. Then, notice that for this to happen it must be that either: * – $\mathcal{A}$ has sent a message $(\textsf{Trigger},ssid,(\textsf{RecoverCoins},c),0)$ to $\mathcal{F}_{Ledg}$ for some $ssid$ and $c$ such that $\textsf{verify-certificate}(s,c)=1$, where $ssid.\textsf{state}=(s,\text{``No active claim''})$, and the last “make a payment” protocol (from Fig. 4) referencing $ssid$ had an honest party as payee which remained honest at least up until after $\mathcal{A}$ sent his message (or the banknote-contract was initialized by an honest user and the banknote was never spent). But then, one of the following must have happened: * * $\mathcal{A}$ possessed a bolt $\ket{\Psi}$ with serial number $s$ at some point, before $\ket{\Psi}$ was spent to the honest user. Then, this adversary would have recovered a valid $c$ and also spent a bolt with serial number $s$ successfully to an honest user. But such an $\mathcal{A}$, together with $\mathcal{E}$, can be used to win game Forge-certificate from Definition 3 with non-negligible probability, with a similar reduction to the one above, thus violating the property of Definition 3. * * $\mathcal{A}$ recovered $c$ such that $\textsf{verify-certificate}(s,c)=1$ without ever possessing a valid bolt with serial number $s$. Again, such an adversary could be used, together with $\mathcal{E}$ to win Forge-certificate from Definition 3). * * $\mathcal{A}$ has successfully changed the serial number of contract $ssid$ to $s$ from some previous $s^{\prime}$ without possessing a bolt $\ket{\Psi}$ with serial number $s^{\prime}$. This cannot happen since any honest user who possesses the valid bolt with serial number $s^{\prime}$ performs the protocol of Fig. 6. * – $\mathcal{A}$ has sent a message $(\textsf{Trigger},ssid,(\textsf{ChallengeClaim},c),0)$ to $\mathcal{F}_{Ledg}$ for some $ssid$ such $ssid.\textsf{state}=(s,\text{``Claim by $pid$ at time $t$''})$ for some $s,pid,t$ with $pid$ honest and $c$ such that $\textsf{verify- certificate}(s,c)=1$. Since $pid$ is honest, he must be the last to have possessed a valid bolt with serial number $s$. Then, there are two possibilities: * * $\mathcal{A}$ never possessed a valid bolt with serial number $s$, and succeeded in recovering $c$ such that $\textsf{verify-certificate}(s,c)=1$. Analogously to earlier, this adversary, together with $\mathcal{E}$, can be used to win Forge-certificate. * * $\mathcal{A}$ possessed a bolt $\ket{\Psi}$ with serial number $s$ at some point, before $\ket{\Psi}$ was spent to an honest user. Analogously to earlier, this means such an $\mathcal{A}$ both recovered a $c$ with $\textsf{verify-certificate}(s,c)=1$ and spent a bolt with serial number $s$ successfully. Such an $\mathcal{A}$ can be used, together with $\mathcal{E}$, to win Forge-certificate. ∎ #### Remark: The security guarantee of Theorem 8 establishes that an adversary cannot end up with more value than he started with (after taking into account the amount he received from honest parties and the amount he succesfully spent to honest parties). However, we do not analyze formally attacks which do not make the adversary gain value directly, but which “sabotage” honest parties, making them lose value. We believe that it should be possible to capture such attacks within our model by modifying the way we keep track of the adversary’s value, but we leave this analysis for future work. We discuss and formalize the notion of “sabotage” in detail in Section 6.3. ## 6 Practical issues in a less idealized setting The ideal functionality $\mathcal{F}_{Ledg}$ defined in Section 3 does not capture adversaries that are allowed to see messages sent by honest parties to $\mathcal{F}_{Ledg}$ before they are registered on the ledger, and who could try to use this information to their advantage: by definition of $\mathcal{F}_{Ledg}$, messages are processed and registered on the ledger in exactly the order that they are sent, and are not seen by the adversary until they make it onto the ledger. While such a definition of $\mathcal{F}_{Ledg}$ makes for a clear exposition and clean proof of security, it is in practice unrealistic. In typical real-world implementations of blockchains, miners (and hence potentially adversaries) can see a pool of pending messages which have not yet been processed and registered on the blockchain, and can potentially delay the processing of certain messages, while speeding up the processing of others. This makes the system susceptible to the attacks described in the following subsection. ### 6.1 Attacks outside of the idealized setting 1. (i) An adversary could file a malicious “lost banknote claim” (running the protocol of Fig. 5) corresponding to a serial number $s$ of a quantum banknote that he does not possess. This would prompt an honest party who possesses a valid quantum banknote with serial number $s$ to publish the corresponding classical certificate $x$, in order to stop the malicious claim. If the adversary could read this message before it is published on the ledger, it could instruct a corrupt party to also publish $x$, and attempt to have this message appear first on the ledger. This would effectively result in the adversary having stolen the honest party’s value associated to the serial number $s$. 2. (ii) Suppose an honest party wants to trade their quantum banknote with serial number $s$ (registered on the ledger) for the coins deposited in the corresponding contract. The honesty party executes the protocol of Fig. 7. This includes publishing the classical certificate $x$ associated to $s$. An adversary who sees $x$ before it is registered on the ledger, could instruct a corrupt party to make the same claim for for the coins in the contract associated to $s$. If the corrupt party’s message is processed faster than the honest party’s message, the adversary has succeeded in stealing the coins deposited in the contract. 3. (iii) Suppose an honest party has lost a valid quantum banknote with serial number $s$ associated to some contract on the ledger. The honest party files a “lost banknote claim” by executing the protocol of Fig. 5. An adversary who hears this before the claim is registered on the ledger could instruct a corrupt party to make a similar claim, and have it appear on the ledger before the honest claim. This would result in the corrupt party obtaining a valid quantum banknote associated to the above contract. 4. (iv) The unforgeability property alone is not enough for _public_ quantum money, as it allows sabotage: an attacker might be able to burn other people’s money, without a direct gain from the attack. Consider an adversary that wants to harm its competitor. The adversary does not follow the honest protocol that generates the quantum money state. Instead, it could create a tweaked quantum money which has a noticeable probability to pass verification once, and fail the second time. This way, the adversary could buy some merchandise using this tweaked quantum money. When the merchant will try to pay to others using this quantum money, the verification will fail – the receiver will run the verification (and this is exactly the second verification), which will cause it to fail. This is not an issue with schemes in which the verification is a projective (or very close to being projective), but, for example, the scheme by Farhi et al. [FGH+12b] is _not_ projective. Indeed, even though our security proof (see Theorem 8) guarantees that an adversary cannot end up with more money that he was legitimately given, it does not rule out sabotage. 5. (v) Similarly to the sabotage attack on the verification of quantum money, an attacker might sabotage the ability to produce a valid certificate from the verified money. In the next Section 6.2, we will introduce a novel property of quantum lightning schemes, which we call bolt-to-signature capability, we give a provably-secure construction of it. We will employ this in Section 6.4 to give a somewhat elegant resolution to issues (i) and (ii) above. In Section 6.3, we show that our scheme is secure against sabotage attacks, hence resolving issues (iv) and (v). In Section 6.4 we discuss a practical approach to resolving issue (iii) above, though it does not contain a formal analysis for reasons which are made clear there. ### 6.2 Trading a Bolt for a Signature We define a new property of a quantum lightning scheme which we call “trading a bolt for a signature” (and we say that a lightning scheme has bolt-to- signature capability). This is an extension of the property of “trading a bolt for a certificate” defined in Section 2.2. The known concrete constructions of quantum lightning do not possess this property, but we will show (Fig. 8) that a lightning scheme with bolt-to-certificate capability can be bootstrapped to obtain a lightning scheme with bolt-to-signature capability. We envision that this primitive could find application elsewhere, and is of independent interest. We start with an informal description of this property, highlighting the difference between the certificate and signature properties. Suppose Alice shows Bob a certificate with a serial number $s$. Bob can conclude that Alice cannot hold a bolt with the same serial number. In particular, if she held such bolt, it means she must have measured and destroyed it to produce the certificate. For the bolt-to-signature property, we ask the following: * • There is a way for Alice, who holds a bolt with serial number $s$, to produce a signature of any message $\alpha$ of her choice, with respect to serial number $s$. * • The signature should be verifiable by anyone who knows $s$. * • Just like a certificate, anyone who accepts the signature of $\alpha$ with respect to $s$ can conclude that Alice can no longer hold a quantum money with serial number $s$; * • (one-time security) As long as Alice signs a single message $\alpha$, no one other than Alice should be able to forge a signature for a message $\alpha^{\prime}\neq\alpha$ (even though her state is no longer a valid bolt, Alice can still sign more than one message, but the unforgeability guarantee no longer holds). A quantum lightning scheme with bolt-to-signature capability is a quantum lightning scheme with two additional algorithms: gen-sig is a QPT algorithm which receives as input a quantum state, a serial number, and a message of any length, and outputs a classical signature. verify-sig is a PPT algorithm which receives a serial number, a message of any length and a signature, and either accepts or rejects. Thus, we modify the setup procedure of the lightning scheme so that QL.Setup outputs a tuple $(\textsf{gen-bolt},\textsf{verify- bolt},\textsf{gen-sig},\textsf{verify-sig})$. The definition of the property has a completeness and a soundness part. The latter is formally defined through the following game Forge-sig. The game Forge-sig is similar in spirit to the game for _onetime_ security of a standard digital signature scheme – see, e.g. [KL14, Definition 12.14], [Gol04, Definition 6.4.2]. * • The challenger runs $(\textsf{gen-bolt},\textsf{verify-bolt},\textsf{gen- sig},\textsf{verify-sig})\leftarrow\textsf{QL.Setup}(1^{\lambda})$ and sends this tuple to $\mathcal{A}$. * • $\mathcal{A}$ sends $(\ket{\psi},s)$ and $\alpha$ to the challenger. * • The challenger runs $\textsf{verify-bolt}(\ket{\psi},s)$. If this rejects, the challenger outputs “$0$”. Else it proceeds to the next step. * • Let $\ket{\psi^{\prime}}$ be the leftover state. The challenger runs $\sigma\leftarrow\textsf{gen-sig}(\ket{\psi^{\prime}},s,\alpha)$ and sends $\sigma$ to $\adv$. * • $\adv$ returns a pair $(\alpha^{\prime},\sigma^{\prime})$. * • The challenger checks that $\alpha\neq\alpha^{\prime}$ and runs $\textsf{verify-sig}(s,\alpha^{\prime},\sigma^{\prime})$. If the latter accepts, the challenger outputs “$1$”. Let $\textsf{Forge-sig}(\adv,\lambda)$ be the random variable for the outcome of the game. ###### Definition 9 (Trading a bolt for a signature). We say that a quantum lightning scheme has bolt-to-signature capability if the following holds: * (I) For every $\alpha$: $\displaystyle\Pr[\textsf{verify-sig}(s,\sigma,\alpha)=1:$ $\displaystyle(\textsf{gen-bolt},\textsf{verify-bolt},\textsf{gen- sig},\textsf{verify-sig})\leftarrow\textsf{QL.Setup}(1^{\lambda}),$ (11) $\displaystyle(\ket{\psi},s)\leftarrow\textsf{gen-bolt},$ (12) $\displaystyle\sigma\leftarrow\textsf{gen- sig}(\ket{\psi},s,\alpha)]=1-negl(\lambda)$ (13) * (II) For all polynomial-time quantum algorithms $\mathcal{A}$, $\Pr[\textsf{Forge-sig}(\mathcal{A},\lambda)=1]=negl(\lambda).$ Informally, the security definition based on game Forge-sig guarantees that: * • As long as Alice signs only one message with respect to $s$, no one except her can forge a signature of another message with respect to $s$. This property is very similar to one-time unforgeability for digital signatures, if one views the bolt as a secret key, and the serial number $s$ as a public key. The difference is that the “secret key” in this case has meaning beyond enabling the signing of messages: it can be spent as quantum money. This is what make the next property important. * • Signing a message destroys the bolt, i.e. it is infeasible to simultaneously produce both a valid signature with respect to a serial number $s$ and a bolt with serial number $s$ which passes verification (an adversary who can succeed at this is easily seen to imply an adversary who wins Forge-sig). This property is unique to the quantum lightning setting. It says that signing a message with respect to serial number $s$ inevitably destroys the bolt with serial number $s$. We remark that it is possible to sign more messages with the leftover state, but such state will no longer pass the quantum lightning verification procedure, i.e. it can no longer be spent. One can think of the bolt with serial number $s$ as being “burnt” once the owner decides to use it to sign a message. We are now ready to present our construction of a quantum lightning scheme with bolt-to-signature capability. The construction is based on the hash-and- sign paradigm (see, e.g., [KL14]), as well as Lamport signatures [Lam79] (familiarity with those is helpful, although our presentation is self- contained). For convenience, we use $\mathcal{H}$ to denote a family of fixed- length hash functions, and we use the notation $H\leftarrow\mathcal{H}(1^{n})$ to denote an efficent random sampling of a hash function $H$ with output length $n$ from the family $\mathcal{H}$. Given: A quantum lightning scheme with bolt-to-certificate capability with setup procedure QLC.Setup. A family $\mathcal{H}$ of fixed-length collision- resistant hash functions. QLDS.Setup: takes as input a security parameter $\lambda$, and outputs a tuple of (descriptions of) algorithms $(\textsf{QLDS.Gen},\textsf{QLDS.Ver},\textsf{QLDS.gen- sig},\textsf{QLDS}.\mathsf{verify\textit{-}sig})$. * • Let $n=\poly$. Sample $(\textsf{gen-bolt},\textsf{verify-bolt},\textsf{gen- bolt},\textsf{verify-certificate})\leftarrow\textsf{QLC.Setup()}$. Sample $H:\\{0,1\\}^{*}\rightarrow\\{0,1\\}^{n}\leftarrow\mathcal{H}(1^{n})$. * • QLDS.Gen: Run gen-bolt $2n$ times to obtain bolts $(\ket{\psi_{i}}\in\mathcal{H}_{\lambda},s_{i})$, for $i=1,..,2n$. Let $\ket{\Psi}=\bigotimes_{i=1,..,2n}\ket{\psi_{i}}\in\mathcal{H}_{\lambda}^{\otimes 2n}$. Let $s=s_{1}||..||s_{2n}$. Output $(\ket{\Psi},s)$. * • QLDS.Ver: Takes as input a state $\ket{\Psi}\in\mathcal{H}_{\lambda}^{\otimes 2n}$. Applies QLDS.Ver to each of the $2n$ factors, and outputs “accept” if all $2n$ verifications output “accept”. * • QLDS.gen-sig: Takes as input a state $\ket{\Psi}\in\mathcal{H}_{\lambda}^{\otimes 2n}$, a serial number $s$ and a message $\alpha$ of any length. Lets $\beta=H(\alpha)\in\\{0,1\\}^{n}$. For $i=1,..,n$: Run gen-certificate on the $(\beta_{i}\cdot n+i)$-th factor to obtain a certificate $x_{i}$. Outputs $\sigma=x_{1}||..||x_{n}$. * • QLDS.verify-sig: Takes as input a serial number $s$, a message $\alpha$, a signature $\sigma$. Parses $s$ as $s=s_{1}||..||s_{2n}$, $\sigma$ as $\sigma=x_{1}||..||x_{n}$. Computes $\beta=H(\alpha)$. For $i=1,..,n$: Let $r_{i}=\textsf{verify-certificate}(s_{\beta_{i}\cdot n+i},x_{i})$. Output “accept” if $r_{i}=1$ for all $i$. Figure 8: Our construction of a quantum lightning scheme with bolt-to- signature capability $QLDS$ We state and prove the main theorem of this section. ###### Theorem 10. If there exists a secure quantum lightning scheme which has a bolt-to- certificate capability, and a family of fixed length quantum-secure collision- resistant hash functions, then there exists a quantum lightning scheme with bolt-to-signature capability. Specifically, under the assumptions that QLC is a secure quantum lightning scheme with a bolt-to-certificate capability and that $\mathcal{H}$ is a fixed-length family of collision-resistant hash functions, the construction QLDS of Fig. 8 is a secure quantum lightning scheme with bolt-to-signature capability. ###### Proof. First of all, it is clear that QLDS is still a secure quantum lightning scheme according to Definition 2. The fact that QLDS satisfies the correctness requirement of a quantum lightning scheme with bolt-to-signature capability ($(I)$ in Definition 9) follows from the correctness property of QLC (namely $(I)$ in Definition 3) and that $n$ is at most polynomial in $\lambda$. Assume towards a contradiction that there exists a QPT adversary $\adv$ that wins the $\mathsf{Sig\textit{-}Forge}$ game with non-negligible probability $\epsilon(\secpar)$. We construct an adversary $\mathcal{B}$ that wins the game Forge-certificate (from Definition 3) with probability at least $\epsilon(\lambda)$. $\mathcal{B}$ runs as follows: * • $\mathcal{B}$ receives a tuple $(\textsf{gen-bolt},\textsf{verify- bolt},\textsf{gen-certificate},\textsf{verify-certificate})$ from the challenger. * • $\mathcal{B}$ constructs algorithms gen-sig and verify-sig from gen- certificate and verify-certificate. Sends the tuple $(\textsf{gen- bolt},\textsf{verify-bolt},\textsf{gen-sig},\textsf{verify-sig})$ to $\mathcal{A}$. * • $\mathcal{A}$ returns a pair $(\ket{\psi},s)$ and a message $\alpha$. Let $\beta=H(\alpha)$. $\mathcal{B}$ simulates the next steps of the challenger in Forge-sig with the following modification: it runs $\textsf{verify- bolt}(\ket{\psi},s)$ but only measures the registeres corresponding to indexes associated with $\beta$ (i.e. $\beta_{i}\cdot n+i$ for $i\in[n]$). It then runs $\sigma\leftarrow\textsf{gen-sig}(\ket{\psi^{\prime}},s,\alpha)$, where $\ket{\psi^{\prime}}$ is the leftover state after verification. $\mathcal{B}$ sends $\sigma$ to $\adv$, and $\mathcal{A}$ returns a pair $(\alpha^{\prime},\sigma^{\prime})$, where $\sigma^{\prime}=x^{\prime}_{1}||..||x^{\prime}_{n}$. * • $\mathcal{B}$ computes $\beta=H(\alpha)$ and $\beta^{\prime}=H(\alpha^{\prime})$. If $\beta\neq\beta^{\prime}$, let $i$ be the first index such that $\beta_{i}\neq\beta_{i}^{\prime}$. $\mathcal{B}$ outputs $(\ket{\psi_{\beta_{i}^{\prime}\cdot n+i}},s_{\beta_{i}^{\prime}\cdot n+i})$ and $x^{\prime}_{i}$. We analyze the winning probability of $\mathcal{B}$ in game Forge-certificate. With probability at least $\epsilon(\lambda)$ it is $\alpha\neq\alpha^{\prime}$ (otherwise $\mathcal{A}$ simply loses). Moreover, with overwhelming probability, it must be $\beta\neq\beta^{\prime}$, otherwise it is immediate such an $\mathcal{A}$ could be used to break collision- resistance of $H$. Hence, with non-negligible probability, there is an index $i$ such that $\beta_{i}\neq\beta_{i}^{\prime}$. Using the definition of QLDS.verify-sig, we deduce that with probability at least $\epsilon$ it must be that $\textsf{verify-certificate}(s_{\beta_{i}^{\prime}\cdot n+i},x^{\prime}_{i})=1$. Moreover, the state $(\ket{\psi_{\beta_{i}^{\prime}\cdot n+i}}$ was not measured, and it must pass verification with probability at least $\epsilon(\lambda)$. ∎ ### 6.3 Security against Sabotage To address Item (iv) in Section 6.1, we define two security games that capture the notion of sabotage; the first is denoted $\mathsf{Sabotage\textit{-}Money}$: * • The challenger runs $(\textsf{gen-bolt},\textsf{verify- bolt})\leftarrow\textsf{QL.Setup}(\lambda)$ and sends $(\textsf{gen- bolt},\textsf{verify-bolt})$ to $\mathcal{A}$. * • $\adv$ outputs a quantum state $\ket{\psi}$ and sends it to the challenger. * • The challenger runs verify-bolt two consecutive times on the quantum state $\ket{\psi}$. * • The adversary wins if the first verification accepts with a serial number $s$ and the second rejects, or accepts with a serial number $s^{\prime}\neq s$. Let $\mathsf{Sabotage\textit{-}Money}(\adv,\lambda)$ be the random variable that is $1$ if the adversary $\adv$ wins, and is $0$ otherwise. ###### Definition 11 (Security against sabotage). A quantum lighting scheme is secure against sabotage if for every QPT $\adv$ there exists a negligible function $\negl[]$ such that: $\Pr(\mathsf{Sabotage\textit{-}Money}(\adv,\lambda)=1)=\negl$ (14) The security against sabotage was first defined in the context of quantum money in [BS16] (though, the term sabotage was not used). We extend the notion of sabotage in the natural way for schemes with bolt-to- certificate or bolt-to-signature capability. Our goal is to avoid a scenario in which an adversary gives a user a quantum lightning state which passes verification, but later fails to produce a valid certificate or signature. We define the following experiment, $\mathsf{Sabotage\textit{-}Certificate}$: 1. 1. The challenger runs $(\textsf{gen-bolt},\textsf{verify- bolt},\mathsf{gen\textit{-}certificate},\mathsf{verify\textit{-}certificate})\leftarrow\textsf{QL.Setup}(\lambda)$ and sends that tuple to $\mathcal{A}$. 2. 2. $\adv$ sends $\ket{\psi},s$ to the challenger. 3. 3. The challenger runs $\textsf{verify-bolt}(\ket{\psi},s)$. If verification fails, set $r=1$. Otherwise, the challenger uses the post-measured state $\ket{\psi^{\prime}}$ to generate a certificate $c\leftarrow\mathsf{gen\textit{-}certificate}(\ket{\psi^{\prime}},s)$, and checks whether it is a valid certificate: $r\leftarrow\mathsf{gen\textit{-}certificate}(\ket{\psi^{\prime}},s)$. 4. 4. $\adv$ wins if $r=0$. Let $\mathsf{Sabotage\textit{-}Certificate}(\adv,\lambda)$ be the random variable that is $1$ if the adversary $\adv$ wins, and is $0$ otherwise. ###### Definition 12. A quantum lighting scheme with a bolt-to-certificate capability is secure against sabotage if, in addition to the requirement in Eq. 14, for every QPT $\adv$ there exists a negligible function $\negl[]$ such that: $\Pr(\mathsf{Sabotage\textit{-}Certificate}(\adv,\lambda)=1)=\negl$ (15) We suspect that Zhandry’s construction based on non-collapsing hash functions, as well as the construction by Farhi et al. (see p. 2.2) does not satisfy the security against sabotage. Fortunately, the construction based on the multi- collision resistance is secure against sabotage: ###### Proposition 3. The quantum lighting construction with the bolt to certificate capability discussed in 2 is secure against sabotage. ###### Proof. Zhandry’s scheme discussed in 2 has the property that $\textsf{verify- bolt}(s,\cdot)$, is (exponentially close to) a rank-1 projector, and therefore upon one successful verification, it will continue to pass verifications and therefore satisfy Eq. 14. In fact, it holds even against unbounded adversaries. This rank-1 projector is such that the state that it accepts only has support on $x$’s that are valid certificates. Since the $\mathsf{gen\textit{-}certificate}$ algorithm is simply a measurement in the standard basis, we conclude that Eq. 15 holds. ∎ The $\mathsf{Sabotage\textit{-}Signature}$ experiment is defined in an analogous fashion: 1. 1. The challenger runs $(\textsf{gen-bolt},\textsf{verify- bolt},\mathsf{sign},\mathsf{verify\textit{-}sig})\leftarrow\textsf{QL.Setup}(\lambda)$ and sends $(\textsf{gen-bolt},\textsf{verify- bolt},\mathsf{sign},\mathsf{verify\textit{-}sig})$ to $\mathcal{A}$. 2. 2. $\adv$ sends $\ket{\psi}$, and (a document to be signed) $\alpha$ to the challenger. 3. 3. The challenger runs $\textsf{verify-bolt}(\ket{\psi})$. If verification fails, set $r=1$. If verification accepts the challenger uses the post-measured state $\ket{\psi^{\prime}}$ to generate a signature of $\alpha$: $\sigma\leftarrow\mathsf{sign}(\ket{\psi^{\prime}},s,\alpha)$. The challenger runs $r\leftarrow\mathsf{verify\textit{-}sig}(s,\sigma,\alpha)$. 4. 4. $\adv$ wins if $r=0$. Let $\mathsf{Sabotage\textit{-}Signature}(\adv,\lambda)$ be the random variable that is $1$ if the adversary $\adv$ wins, and is $0$ otherwise. ###### Definition 13. A quantum lighting scheme with a bolt-to-signature capability is secure against sabotage if, in addition to the requirement in Eq. 14, for every QPT $\adv$ there exists a negligible function $\negl[]$ such that: $\Pr(\mathsf{Sabotage\textit{-}Signature}(\adv,\lambda)=1)=\negl$ (16) ###### Proposition 4. The construction QLDS in Fig. 8 is secure against sabotage, assuming the underlying quantum lightning scheme with a bolt-to-certificate capability QLC which it uses is secure against sabotage. ###### Proof. Given a QPT adversary $\adv$ that wins the $\mathsf{Sabotage\textit{-}Signature}$ experiment with probability $\epsilon$ with respect to the QLDS scheme, we can construct ano adversary $\bdv$ that wins the $\mathsf{Sabotage\textit{-}Certificate}$ experiment with probability $\frac{\epsilon(\lambda)}{2n}$. By our assumption that QLC is secure against sabotage, we conclude that $\epsilon(\lambda)$ is necessarily a negligible function. The adversary $\bdv$ receives from his challenger $(\mathsf{Gen},\textsf{verify- bolt},\mathsf{gen\textit{-}certificate},\mathsf{verify\textit{-}certificate})$. $\bdv$ will sample $H:\\{0,1\\}^{*}\rightarrow\\{0,1\\}^{n}\leftarrow\mathcal{H}^{1^{n}}$. $\bdv$ will play the role of the challenger in the $\mathsf{Sabotage\textit{-}Signature}$ experiment, and will also simulate $\adv$. $\adv$ will receive the tuple above and $H$ as part of the setup, and will produce a state $\ket{\psi}$ over $2n$ registers, serial number $s_{1},\ldots,s_{n}$ a document $\alpha$. $\bdv$ will sample $i\in[2n]$ uniformly at random, and send the $i$’th register of $\ket{\psi}$ to his challenger. Next we show that indeed $\bdv$ wins with probability at least $\frac{\epsilon}{2n}$. We know that with probability $\epsilon(\lambda)$, $\adv$’s challenger will verify all the $2n$ states, generate a certificate from $n$ of these states, and at least one of these certificates will verification. Recall that $i$ was sampled uniformly at random. $\bdv$’s challenger preforms exactly the same procedure – the challenger generates a certificate from the state it receives and checks whether it it passes the verification as a certificate. The probability that $i$ is one of the failed certificates is therefore at least $\frac{\epsilon}{2n}$. ∎ ### 6.4 A resolution of the practical issues We first employ a quantum lightning scheme with bolt-to-signature capability to resolve issues (i) and (ii) of Section 6.1. We make a simple modification to the protocols of Fig. 6 (preventing and challenging malicious “lost banknote claims”) and Fig. 7 (trading quantum banknotes for coins). We upgrade the quantum lightning scheme with bolt-to-certificate capability to one with bolt-to-signature capability. To deal with issue (i), we make two modifications to our payment system of Section 4. * • We modify the protocol of Fig. 6 as follows: when party $P$ notices a malicious “lost banknote claim” for a serial number $s$ associated to a quantum banknote $\ket{\psi}$ that he possesses, he does not simply compute the classical certificate associated to $s$ and send it in clear to $\mathcal{F}_{Ledg}$. Rather, $P$ computes a signature $\sigma\leftarrow\textsf{gen-sig}(\ket{\psi},s,\alpha)$ where $\alpha$ is a message saying “Party $P$ challenges the claim”. * • We modify the definition of banknote-contract (Definition 7) so that whenever there is an active lost claim, the coins deposited in a contract with state variable $\textsf{serial}=s$ are released to a party $P$ only upon receipt of a signature $\sigma$ with respect to $s$ of a message “Party $P$ challenges the claim”. We do not give a formal proof of security of the modified scheme, as this would require first defining formally the modified ledger functionality. Instead we argue informally: it is straightforward to see that the attack described in point (i) of Section 6.1 is impossible. Any adversary that is able to carry out that attack could be used to create an adversary that is able to forge a signature with respect to a serial number $s$ of a bolt that they do not possess. This violates the security of the bolt-to-signature property of the lightning scheme. To deal with issue (ii), we make a similar simple modification: * • We modify the protocol of Fig. 7 as follows: in order to trade a valid quantum banknote with serial number $s$ for the coins deposited in a smart-contract with state variable $\textsf{serial}=s$, party $P$ does not simply compute the classical certificate associated to $s$ and send it in clear to $\mathcal{F}_{Ledg}$. Instead, $P$ computes a signature $\sigma\leftarrow\textsf{gen-sig}(\ket{\psi},s,\alpha)$ where $\alpha$ is a message saying “Party $P$ wishes to recover the coins in the contract”. * • We modify the definition of banknote-contract (Definition 7) so that whenever there is no active claim, the coins deposited in a contract with state variable $\textsf{serial}=s$ are released to a party $P$ only upon receipt of a signature $\sigma$ with respect to $s$ of a message “Party $P$ wishes to recover the coins in the contract”. For a similar reasoning as for point (i), the attack in point (ii) of Section 6.1 is no longer possible. Dealing with issue (iii) of Section 6.1 is trickier. In this case, an honest party $P$ has lost a valid quantum banknote with serial number $s$, and there is no way for $P$ to recover any certificate or signature proving possession of the banknote. The only difference between $P$ and everyone else, as far as owning the coins deposited in the associated contract, is that $P$ knows that the banknote has been damages and lost, and no one else does. The “lost banknote claim” protocol of Fig. 5 requires $P$ to send a message to $\mathcal{F}_{Ledg}$ declaring the loss, and the only reason why $P$ is be able to recover the coins deposited in the contract in the idealized setting is that he is the first to make this claim, and no has the ability to challenge it. The situation changes dramatically if we allow the adversary to delay the processing of honest parties’ messages to $\mathcal{F}_{Ledg}$ in favour of its own. The adversary could simply notice a “lost banknote claim” and take advantage of this by making its own claim and ensuring that it is registered first on the ledger. We propose the following modification to handle this issue: * • Instead of directly making a “lost banknote claim”, a party $P$ posts a commitment to a message of the form “$P$ is filing a lost banknote claim associated to the smart contract with identifier $ssid$”. The commitment contains a deposit of $d_{0}$ coins. * • $P$ has to reveal that commitment after no more than $t_{0}$ blocks. * • The coins are released to user $P$ only $t_{1}$ blocks after the reveal phase, and provided no reclaim was made during that time. * • In case there are two or more reveals to two or more commitments to the same $ssid$ – the one which was committed to the earliest (i.e., the commitment appears in a an earlier block) receives the coins. Intuitively, this modification resolves issue (iii). This is because the commitment is hiding, and hence the adversary does not learn anything about the claim it contains prior to the reveal. After the reveal phase, it is too late for the adversary to start the commitment phase. On the other hand, if an adversary simply tries to guess what the content of a claim is (i.e. which quantum banknote was lost), and tries to make a claim of his own, the adversary will most likely lose coins, assuming the frequency at which “lost banknote claims” are made is low (recall that making a claim requires staking some coins, which are lost if the claim is successfully challenged). As much as this resolution to issue (iii) seems to work in theory, the adversary could in practice possess some side information regarding the claim that is hidden in the commitment, and it becomes difficult to model this side information in a way that is both rigorous and faithful to what is possible in practice. We illustrate this with some practical examples. First, it might be hard to know, for an honest user who lost their quantum banknote, whether the latter was indeed lost, or whether it was stolen. Therefore, an honest user who wrongfully believes that his quantum banknote was lost might end up losing an extra $d_{0}$ coins, to the thief who stole it, when trying to recover it. Additionally, if an adversary could guess a serial number of a quantum money state that was or will be lost, and post it before the honest user, the adversary will be the one who will eventually receive these coins. Such a setting might be plausible, for example, during (or even before) power outages. Another attack vector is the following. Indeed, the commitment does not reveal which quantum banknote was lost. However, each commitment reveals that 1 banknote was lost. Suppose there is a claim for 1743 coins at some time $t$. There is a good chance that these $1743$ coins belong to one person who lost all his $1743$ coins. An adversary that can figure out who owns exactly $1743$ coins that were recently lost, and what are the serial numbers of these coins, can effectively steal these coins. Since our construction does not claim any guarantees about the privacy of users, this information might be readily available to the adversary. Various attacks on the users’ privacy are known in the “classical” Bitcoin literature – see Item 6 in Section 8. Therefore, we elect to stir away from formal security claims, and we leave this as a proposed resolution which requires further investigation. ## 7 A practical implementation on Bitcoin and space optimization In this section, we informally discuss a practical implementation of our payment system using Bitcoin. In Section 4, we have elected to describe our payment system in a model with access to a ledger functionality that supports stateful universal smart contracts. This was for two main reasons: clarity of exposition and flexibility of the framework. Nevertheless, it is possible to implement our payment system on the Bitcoin blockchain. Bitcoin uses a special purpose scripting language for transactions. The basic building block of a a script is called an _opcode_. The opcodes that Bitcoin supports are extremely limited. For example, the opcode OP_ADD, which adds two inputs, is supported, but even a simple operation such as multiplication or raising a number to a power are _not_ supported – as they are not strictly needed, yet they increase the attack surface (for example, they may be used to preform a memory-based attack). More information about the scripting language and opcodes can be found in Ref. [NBF+16]. The only required adjustment to the Bitcoin protocol needed to implement our payment system is to add two opcodes to the Bitcoin scripting language. These opcodes are utilized once when transforming Bitcoin to quantum banknotes, and once when the quantum banknotes are transformed back to Bitcoin. We provide more detail about this implementation by describing a possible improvement to the space efficiency of our payment system. In Section 7.1, we informally describe an implementation on Bitcoin of the original payment system. In Section 7.2, we informally describe a modification that drastically improves space efficiency. ### 7.1 Bitcoin to Quantum Money: The Simple Approach In order to use a quantum lightning scheme, of course, we need to first run $\textsf{QL.Setup}(\secparam)$. In some scenarios, where we can assume some trusted setup, this is not an issue. But in an existing distributed system, such as Bitcoin, this could be considered contentious. One of Zhandry’s construction only requires a Common Random String (CRS) – see Fact 4. In the context of Bitcoin, such a CRS could be generated by a multiple-source randomness extractor, applied on the headers of the first $n$ blocks, where $n$ is determined by the amount of randomness required for the quantum lightning scheme, and the min-entropy each header contains. More details regarding using Bitcoin as a source of randomness can be found in Ref. [BCG15]333Though, for our purposes, we do not need a recurring form of randomness usually called a randomness beacon, which is the focus of their work. In this context the source of randomness is only used once.. Bonneau et al. have argued that the min-entropy in each block header is at least 32 bits [BCG15, Table 1]. To transform $x$ bitcoins associated to a signing key $sk$, to quantum money, the user acts as follows. First, the user mints a quantum lightning state, together with some serial number: $(\ket{\$},s)\leftarrow\mathsf{mint}_{pk}$. Second, the user signs the concatenation444We use $||$ to denote concatenation. of a special “quantum money” opcode, the serial number, and the value $y$ using the secret key: $\sigma\leftarrow\mathsf{sign}_{sk}(\text{OP\\_BITCOIN\\_TO\\_QUANTUM\\_MONEY}||s||y).$ (17) Third, the user propagates the signed message $(\text{OP\\_BITCOIN\\_TO\\_QUANTUM\\_MONEY}||s||y,\sigma)$ to the Bitcoin miners. Here $y$ is the value which the quantum money holds, and $x-y$ represents the fee which incentivizes miners to include that message in the blockchain, in line with the current fee mechanism in Bitcoin (which we have not covered in this work – see [NBF+16] for more details). A miner would include the message $(\text{OP\\_BITCOIN\\_TO\\_QUANTUM\\_MONEY}||s||y,\sigma)$ in their block, as long as (i) the signature is legitimate, i.e., $\mathsf{verify}_{vk}(\text{OP\\_BITCOIN\\_TO\\_QUANTUM\\_MONEY}||s||y,\sigma)=1$, (ii) $y<x$, where $x$ is the value of the bitcoins associated with the verification key (bitcoin address) $vk$, and (iii) as long as the miner fee is attractive enough. Under the above conditions, eventually, a block which contains that transaction would be mined, verified by all other nodes, and would become part of the longest chain. At this point, the original owner of the $x$ bitcoins does not possess any “standard” bitcoins, since they are considered spent. Instead, she holds the quantum state $(\ket{\$},s)$ with the “value” $y$. She could then spend the quantum money by sending it to other others, and it would be treated as having the same value as $y$ bitcoins. To verify the quantum money $(\ket{\psi},s,y)$, the receiver would first check that $\text{OP\\_BITCOIN\\_TO\\_QUANTUM\\_MONEY}||s||y$ indeed appears in the blockchain. This step requires storing the blockchain. Any old version that contains the signed message will do. Techniques such as Simplified Payment Verification (SPV) could be used to eliminate the need to store that information, with some downsides in terms of privacy and security [Nak08, NBF+16]. Then, the receiver would accept the quantum money if $\mathsf{verify}_{pk}(\ket{\psi},s)=1$. The receiver could in turn spend the received quantum money to other users via the same procedure. ### 7.2 Bitcoin to Quantum Money: Space Optimization The approach mentioned above has a limitation. Unlike bitcoin, which is divisible almost infinitely, quantum money is not (see Item 13 in p. 13 for more details). The most naive workaround is to divide the value of the $x$ bitcoins between several quantum money states, with values $y_{1},\ldots,y_{n}$ so that $\sum_{i\in[n]}y_{i}<x$. The disadvantage is in terms of space: each serial number is recorded on the blockchain, which causes high fees and somewhat of a scalability problem. We now present a much more efficient approach, in which the space required on the blockchain is independent of the number of quantum money states the user creates. Suppose the user wants to split the $x$ bitcoins into $2^{n}$ quantum money states, each having a value of $2^{-n}y$. The user first creates $2^{n}$ quantum money states with serial numbers $s_{1},\ldots,s_{n}$, and calculates the Merkle Hash Tree [Mer80]555We note that the original motivation of Merkle was very different than the one we use here. See [KL14, Section 5.6.2] and [NBF+16, Section 1.2] for the specific application we need. of these serial numbers. The user than signs and publishes only the root of the Merkle tree $r$, the total value $y$ (which must satisfy $y<x$ as before), and $n$. Each quantum money state now consists of three parts: the quantum part $\ket{\$}$, the serial number $s_{i}$, and a Merkle path from $s_{i}$ to the root $r$. Note that the information recorded on the block-chain is independent of $n$, and one could create very small denominations using this approach. Another advantage of this approach is that it allows a slightly weak variant of a Proof of Payment – see Item 17 in p. 17. One could even take this one step further. Miners could include a Merkle root of the Merkle tree of all existing serial numbers, at the end of every calendar year. This would require users that wish to verify quantum money to come online once every year to store the Merkle root. Every owner of quantum money would calculate the Merkle path from the serial number of the quantum money state, to that root, and append it to their quantum money state. That would allow these users to verify the quantum money state generated in previous years using only one hash, typically, e.g., 256 bits. Of course, this technique does not reduce the space requirement for quantum money which was generated in a year which has not ended yet. ## 8 Comparing our scheme to classical alternatives In this section, we discuss several trade-offs between Bitcoin and our payment system based on quantum money. The current roadmap for solving the scalability problem of Bitcoin goes through a second layer solution called the Lightning Network (LN) 666Note that there is no connection between the terms quantum lightning and the lightning network. This section discusses the trade-offs between the following three different alternatives of employing Bitcoin: (a) a standard Bitcoin transaction, (b) a LN transaction and (c) a quantum money transaction as in our payment system (we will refer to this as QM for short, from now on). Before doing so, we give a brief overview of the LN. The LN improves upon Bitcoin, and provides a way to transact off-chain using _bidirectional payment channels_ between two parties [PD16] (see also [BDW18]). The opening and closing of the channel is done on-chain (i.e., both of these operations require $1$ transaction that would appear on the Bitcoin blockchain) while updating the balance of the channel is done completely off- chain, and thus allows for an unlimited number of transactions which do not affect the limited throughput of the Bitcoin blockchain, and do not incur the long confirmation times of on-chain transactions. At any point, each of the two parties involved can “close” the channel, by submitting the most up to date state of the channel to the blockchain. Crucially, the lightning network supports routing. What this means is that the graph induced by all bi- directional payment channels allows Alice to pay Bob, without any on-chain transaction, as long there is a path in the graph from Alice to Bob. The objectives of the LN are to increase the effective transaction throughput of the Bitcoin blockchain (by having most of the transactions happen off-chain), make transactions be effective immediately, and reduce the transaction costs. Since early 2018, the LN has been active on the Bitcoin main-net. The capacity of the LN is shown in Fig. 9, and as of August 2019, it stores only $0.004\%$ of the total bitcoins in circulation, and is still considered experimental777For example, as of August 2019, the maximal capacity of a LN channel is $0.16$ bitcoin, see https://github.com/lightningnetwork/lightning- rfc/blob/master/02-peer-protocol.md#the-open_channel-message, and the requirements regarding the parameter funding_satoshis. . Figure 9: The total capacity of the lightning network, starting on January 2018, brown in BTC, blue in USD. Source: https://bitcoinvisuals.com/ln- capacity The following list provides an exhaustive comparison between the three alternative modes of operation mentioned above. It is worth noting that quantum money, used as in our payment system, in many ways resembles (digital) cash banknotes, both in terms of the advantages and the disadvantages, and is thus the closest of the three modes to ideal digital cash. Items 1-9 present the aspects in which quantum money outperform Bitcoin and the LN, while the disadvantages of quantum money are presented in Items 10-17. 1. 1. Throughput. The throughput of Bitcoin is strongly capped. On average, in the first 6 months of 2019, the Bitcoin network had 3.9 transaction per second888Source: https://www.blockchain.com/charts/n-transactions.. This throughput cannot increase dramatically without changing the Bitcoin protocol. To receive bitcoins through the LN, Alice must have an open channel, which she will eventually need to close. This requires at least two on-chain transaction, though, the number of uses is not bounded. In this regard, transforming quantum money to Bitcoin is very similar to opening a quantum channel, and transforming the quantum money back to Bitcoin is similar to closing a channel. The balance of a LN channel can be updated, but is always bounded by the transaction that opened that channel. For example, if Alice locks $10$ bitcoins in a channel with Bob, then initially she has $10$ bitcoins and Bob has zero; Alice and Bob could then update it, but Bob could never receive more than $10$ bitcoins using the channel. A user could receive and send quantum money without ever making an on-chain transaction. Quantum money has no limit on the value transferred. 2. 2. Liquidity. Suppose Alice wants to pay Bob. In Bitcoin, she could do that, assuming she has enough funds, and connection to the Bitcoin network. In the LN, this is not always the case: there needs to be a way to route money among the open channels. Sometimes no such route exists, or is inaccessible (using these channels for routing requires the cooperation of the parties along the channel). This may have an impact on the effective throughput of the LN. A quantum money state can always be sent from the receiver to the sender. 3. 3. Latency. The recommended best practice for a Bitcoin transaction is waiting for 6 confirmations, which takes 1 hour on average. Both the LN and QM need one transaction – in order to open a LN channel, or to transform bitcoin to QM. That part can be done in advance, and is only needed once, but it suffers from the same latency as a Bitcoin transaction. A LN has no inherent latency other than the delays caused by the routing. For example, a single transaction might involve several hops between distant nodes, which takes on the order of a second. QM has a slightly better performance, especially when the sender and receiver are physically close to each other – the latency in this case is only due to the quantum verification algorithm. Overall, the LN and QM have comparable latencies, which are much better than Bitcoin’s. 4. 4. Fees. Each Bitcoin transaction needs to pay a _fee_ , in order to get approved. This fee is collected by the miner, which included the transaction. The average fee per transaction in the first 6 months of 2019 was $1.41$ USD999Sources: https://www.blockchain.com/charts/n-transactions and https://www.blockchain.com/charts/transaction-fees-usd.. To encourage LN nodes to provide liquidity, the protocol uses routing fees. These fees are expected to be smaller than on-chain transaction fees, but still non-zero. No fees are needed to transact with QM. 5. 5. Dependence on mining. The Bitcoin protocol and implicitly, the LN, are based on mining (also known as Proof-of-Work [DN92]). This approach suffers from two main drawbacks: (a) As of August 2018, Bitcoin mining (also known as Proof-of- Work [DN92]) consumed slightly under 1% of the world’s electricity [Nar18]. Mining is required to secure the network. Proof-of-Stake is a competing approach with somewhat different trade-offs [KRDO17, DGKR18], with the main advantage that it does not spend so much energy. (b) The security model related to mining is convoluted, and is based on incentives, without clear assumptions nor a full security analysis. In particular, it is known that the Bitcoin protocol is _not_ incentive compatible [ES14]. In PoW and PoS, a corrupted majority can double-spend. In addition, the Bitcoin protocol does not provide finality – in principle, at any point, a roll-back (known as re- organization) could occur. Quantum money does not require PoW or PoS, and provides finality. For example, if all the bitcoins were transformed to quantum money, PoW or PoS would not be needed at all. Of course, coordinating such a change is highly non-trivial. 6. 6. Privacy. The users’ privacy is not guaranteed in Bitcoin, and certainly not by default. All the transactions that have ever occurred are publicly accessible. The common approach to provide some level of privacy is to use a new bitcoin address per transaction. Yet, there are various techniques to relate these addresses to real-world identities [RS13, RS14, MPJ+16]. There are several commercial services that deanonymize Bitcoin transactions101010E.g., https://www.chainalysis.com. They claim to have “the biggest database in the world of connections between real world entities and transactions on the blockchain”, see https://youtu.be/yNpNz-FvSYQ?t=154. . More recent works tackle this privacy issue – see [vS13, MGGR13, BCG+14, Poe16] and references therein. There is a trade-off between privacy and concurrency in the LN, see [MMK+17] and references therein. In addition, this aforementioned work explicitly leaves the question of privacy preserving routing for the LN open. Quantum money enjoys superior privacy, as only the sender and receiver take part in the transaction, and the transaction leaves no trace. The privacy of QM is analogous to that of physical banknotes. Bear in mind that banknotes have serial numbers, and these could potentially be traced, although arguably the effect on privacy is negligible. In this sense, coins provide better privacy for the users, since they are indistinguishable. _Quantum Coins_ have been formally studied: these are indistinguishable quantum states which provide an analogous level of privacy as physical coins - see [MS10, JLS18] (improving upon [TOI03]). Unfortunately, these constructions are for private quantum money (rather than public), and do not constitute a quantum lightning scheme, which is crucial for our construction, and thus cannot be used in our payment system. We leave it as an open question whether the same level of anonymity achieved using quantum coins, or the construction mentioned above, could be achieved in the context of quantum money for Bitcoin. 7. 7. Risk of Locked funds. In the LN, the parties lock funds in a channel. When both parties are cooperative, these funds can be easily unlocked, and used in the next block. In case one party is uncooperative, and refuses to close the channel, these funds are effectively locked for some period of time, typically, for a few days. Quantum money does not require users to lock any funds. 8. 8. Connectivity requirements. A Bitcoin transaction requires communication with the Bitcoin network. At the very least, the receiver needs a communication channel with at least one other Bitcoin node. A LN transaction needs even more resources, since the LN uses source routing – therefore, the sender has to be well connected to the entire network in order to find the route. A QM transaction only requires quantum communication between the sender and the receiver. 9. 9. Liveliness. For technical reasons which are outside the scope of this work, both parties participating in each channel have to be on-line occasionally, and monitor the Bitcoin network, in order to revoke a transaction, in case the other party cheats. This task can be outsourced to a _Watchtower_ , which has to be trusted to perform its job (and of course the watch-tower has to be on- line occasionally). Online availability is not required for QM, if one opts for a version of our payment system in which there is no procedure to recover the value corresponding to lost or damaged quantum banknotes. If one opts to include such a recovery mechanism, then a similar level of online availability as in the LN is required. 10. 10. Technological requirements. Transactions with quantum money require 3 main resources: long term quantum memory to store the quantum money, a universal quantum computer to verify the quantum money, and a quantum network to transmit the quantum money between users. 11. 11. Smart contracts. Bitcoin provides a scripting language which can be used to design _smart contracts_. Notable examples include _multi-sig transactions_ , in which $m$-out-of-$n$ signatures are needed to spend the money, and _atomic cross-chain swap_ which allows Alice and Bob to trade two crypto-currencies without trusting each other [NBF+16]. Both capabilities naturally extend to the LN111111The video in https://youtu.be/cBVcgzEuJ7Q demonstrates a LN atomic swap between Bitcoin and Litecoin.. Other crypto-currencies such as Ethereum have a richer programming language, which allows constructing Decentralized Applications (DApps) [Woo14, But14]. QM does not solve the consensus problem or any variant of it, and does not provide the functionality of smart contracts. 12. 12. Backup. It is pretty easy to backup a Bitcoin wallet. Typically, all that a user needs is a fairly short string called a _seed_. This seed is used to generate a Hierarchical Deterministic Wallet [Wui13]. A backup can be done once, and never needs to be updated in the lifetime of a Bitcoin wallet. LN channels are slightly harder to backup, since the protocol is stateful, and therefore, currently, backing up requires having the most up-to-date state of the channel. By definition, it is impossible to backup a quantum money state. In this work, we proposed a mechanism to _recover_ lost banknotes. In essence, a user can claim that her quantum money was lost, by depositing $d$ bitcoins. The user would receive these bitcoins after some period of time (e.g., one year). If the party that claimed that the coins were lost is dishonest, then the person holding the legitimate quantum money state can produce a certificate of this fact, and claim the $d$ bitcoins, in addition to the value of the quantum money that she originally held. To avoid theft, users that want this option available have to be on-line occasionally (in this example, at least once a year). 13. 13. Divisibility. One of the advantages of Bitcoin and the LN is that any amount, down to $10^{-8}$ bitcoin can be sent, and in principle, even smaller amounts could be used. Quantum money, on the other hand, is _not_ divisible. The user must decide, at the time of the quantum minting, what is the denomination of the quantum money, and it remains the same for the lifetime of the quantum money. 14. 14. Hidden inflation. Consider a computationally powerful adversary that attacks the Bitcoin network. Such an adversary could, for example, break the digital signature scheme and steal other people’s Bitcoins. Yet, such an adversary couldn’t “print” bitcoin from thin air, without others noticing it121212Unless there is some bug that is unrelated to cryptography. When we use quantum money, the situation is different. A powerful adversary could create new quantum money from thin air, without others being able to notice it. This might be a threat either because of invalid computational assumptions, or flaws in the implementation. This threat is not unique to quantum money. In fact, such a flaw in the implementation occurred in ZCash, a crypto-currency which is based on the ZeroCash protocol [BCG+14], see https://electriccoin.co/blog/zcash-counterfeiting-vulnerability-successfully- remediated. Inevitably, there is no definitive way to know whether that bug was exploited. 15. 15. Finality of the security parameters. One interesting feature of Bitcoin is that the level of security that is achieved can, in principle, be increased. If the level of security seems insufficient due to technological advancement, the protocol may allow users to transition to more secure schemes. This is exactly the case for the proposed post-quantum secure digital signature schemes [But13, LRS18]. It is the the responsibility (and incentive) of each individual user to transition – otherwise, her funds might be lost. The quantum money can be used too increase security in the same manner: users with QM in circulation can create new QM with the improved parameters, sign the new serial number using their the bolt-to-signature capability (and by that destroying it), and adding that signed message to the blockchain. Yet, the incentives here a slightly different: an adversary could steal the bitcoins of some user, if the security parameters are poorly chosen. In the quantum money setting, the adversary could print money from thin air – see Item 14. This makes the system, as a whole, insecure. Therefore, it is advisable to make such a transition mandatory. 16. 16. Optional Transparency. Bitcoin transactions are publicly available, and organizations or individuals that wish to, can make their accounting book completely transparent. See Bitcoin Improvement Proposal (BIP) 32 [Wui13] and Ref. [NBF+16, Chapter 4.2] for more details. Quantum money transactions leave no trace, and therefore it seems impossible to achieve this sort of transparency. 17. 17. Proof of payment. Consider the following scenario. You go to a store, and pay for an item. You hand over a valid banknote to the seller. The seller takes it to the bill checking machine, and secretly replaces your valid bill with a fake one. The seller then gives you back the fake money, and blames you for trying to fraud him. This kind of attack cannot happen in Bitcoin, if used appropriately. The seller can ask for a signed payment request, and the buyer can then verify the authenticity of that message, using the seller’s public key. After payment, the seller cannot argue that the payment was not received – the buyer can prove that the bitcoins were sent to the seller’s address, by showing the payment on the blockchain. A Bitcoin payment can be done in such a way that an honest user would have a proof of payment [AH13]. A similar functionality might be possible to achieve in the LN, though currently, as far as the authors are aware, the LN does not provide such functionality. On the other hand, QM transactions leaves no trace, and proof of payment seems harder to achieve. A possible workaround (which works for the LN as well) could be the following. Suppose Alice wants to send 10 bitcoins worth of quantum money to Bob. Instead of sending it all at a time, she could divide the payment into 100 iterations. In each iteration she would send $0.1$ bitcoin, and expect a digital signature in return approving the payment in return. If Bob fails to provide such a signature, she would abort. The worst case scenario in this case is that she would not have a proof of payment for $0.1$ bitcoins. ## 9 Conclusion In this work, we gave the first example of the use of classical smart contracts in conjunction with quantum cryptographic tools. We showed that smart contracts can be combined with quantum tools, in particular quantum lightning, to design a decentralized payment system which solves the problem of scalability of (payment) transactions. There is currently only one known secure construction of quantum lightning, which relies on a computational assumption about multi-collision resistance of certain degree-2 hash functions [Zha19]. Finding alternative constructions of quantum lightning, secure under more well-studied computational assumptions, is a very interesting open problem. Smart contracts have found several applications in classical cryptographic tasks, but their application to quantum cryptographic tasks is virtually unexplored. We hope that this work will ignite future investigations. Some candidate tasks which might potentially benefit from smart contracts are: generation of public trusted randomness, distributed delegation of quantum computation, secure multi-party quantum computation. #### Acknowledgments A.C is supported by the Simons Institute for the Theory of Computing. O.S. is supported by the Israel Science Foundation (ISF) grant No. 682/18 and 2137/19, and by the the Cyber Security Research Center at Ben-Gurion University. ## References * [Aar09] S. Aaronson. Quantum copy-protection and quantum money. In Computational Complexity, 2009. CCC’09. 24th Annual IEEE Conference on, pages 229–242. IEEE, 2009. * [AH13] G. Andresen and M. Hearn. Payment Protocol. Bitcoin Improvement Proposal (BIP) 70 https://github.com/bitcoin/bips/blob/master/bip-0070, 2013. * [BCG+14] E. Ben-Sasson, A. Chiesa, C. Garman, M. Green, I. Miers, E. Tromer, and M. Virza. Zerocash: Decentralized Anonymous Payments from Bitcoin. 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Rijmen, editors, Advances in Cryptology - EUROCRYPT 2019 - 38th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Darmstadt, Germany, May 19-23, 2019, Proceedings, Part III, volume 11478 of Lecture Notes in Computer Science, pages 408–438. Springer, 2019, arXiv: 1711.02276. ## Appendix A Appendix ### A.1 Proof of Proposition 2 ###### Proof of Proposition 2. We assume some familiarity with Zhandry’s construction (see section 6 of [Zha19] for more details). In his construction, a full bolt is a tensor product of $n$ mini-bolts. A valid mini-bolt with serial number $y$ takes the form $\ket{\Psi}^{\otimes(k+1)}$ where $\ket{\Psi}$ is a superposition of pre- images of $y$ under a certain function $H$ (this is called $f_{\mathcal{A}}$ in Zhandry’s paper, and we do not go into the details of what this function is). The serial number of the full bolt is the concatenation of the serial numbers of the mini-bolts. verify-bolt has $H$ hardcoded in its description. The computational assumption under which Zhandry’s construction is proved secure is that $H$ is $(2k+2)$-multi-collision resistant, i.e. it is hard to find $2k+2$ colliding inputs (for this particular function it is easy to find $k+1$ on the other hand). Similarly to the proof of Proposition 1, we define gen-certificate to be the QPT algorithm that measures each mini-bolt in the computational basis and outputs the concatenation of the outcomes. We define verify-certificate to be the deterministic algorithm which receives as input a serial number $(y_{1},..,y_{n})$, and the concatenation of $(z^{(i)}_{1},..,z^{(i)}_{k+1})$ for $i=1,..,n$, and checks that $H(z^{(i)}_{1})=..=H(z^{(i)}_{k+1})=y_{i}$ for all $i$. It is clear that $(I)$ holds. For property $(II)$, similarly to the proof of Proposition 1, we can construct an adversary $\mathcal{A}^{\prime}$ that breaks the $(2k+2)$-multi-collision resistance of $H$ from an adversary $\mathcal{A}$ that wins game Forge- certificate with non-negligible probability: $\mathcal{A}^{\prime}$ runs $(\textsf{gen-bolt},\textsf{verify- bolt})\leftarrow\textsf{QL.Setup}(1^{\lambda})$ and sends $(\textsf{gen- bolt},\textsf{verify-bolt},\textsf{gen-certificate},\textsf{verify- certificate})$ to $\mathcal{A}$, where the latter two are defined as above in terms of the function $H$ hardcoded in verify-bolt. $\mathcal{A}$ returns $c$ which is parsed as $(x_{1},..,x_{n})$, where each $x_{i}$ is a $(k+1)$-tuple, and $\ket{\psi}$. $\mathcal{A}^{\prime}$ then measures each of the $n$ registers of $\ket{\psi^{\prime}}$ to get $n$ $(k+1)$-tuples $(x^{\prime}_{1},..,x^{\prime}_{n})$. If there is some $i$ such that $(x_{i},x_{i}^{\prime})$ is a $(2k+2)$-collision, then $\mathcal{A}^{\prime}$ outputs this. We claim that with non-neglibile probability $\mathcal{A}^{\prime}$ outputs a $(2k+2)$-collision: in fact, since $\mathcal{A}$ wins Forge-certificate with non-negligible probability, then $\ket{\psi^{\prime}}$ must pass verify-bolt with non-negligible probability; from the analysis of Zhandry’s proof, we know that any full bolt that passes verification with non-negligible probability must be such that most mini-bolts have non-negligible weight on most pre-images (in fact they should be close to uniform superpositions over all pre-images). Indeed, in Section 6.3, p. 435, Zhandry argues: > Conditioned on acceptance, by the above arguments the resulting mini bolts > must all be far from singletons when we trace out the other bolts. This > means that if we measure the mini-bolts, the resulting superpositions will > have high min-entropy. ∎ ### A.2 Standard Security Definitions ###### Definition 14 (Digital Signature Scheme). A digital signature scheme consists of 3 PPT algorithms $\mathsf{key\textit{-}gen},\ \mathsf{sign}$ and $\mathsf{verify}$. The scheme is complete if the following holds. When a document is signed using the private key, the signature is accepted by the verification algorithm using the public key. Formally, for every $\alpha\in\\{0,1\\}^{*}$: $\Pr\left[(vk,sk)\leftarrow\mathsf{key\textit{-}gen}(\secparam);\sigma\leftarrow\mathsf{sign}_{sk}(\alpha);r\leftarrow\mathsf{verify}_{vk}\left(\alpha,\sigma)\right):r=1\right]=1$ (18) The scheme satisfies Post-Quantum Existential Unforgeability under a Chosen Message Attack (PQ-EU-CMA) if the following holds. Consider a QPT adversary with the capability of adaptively requesting documents to be signed by a signing oracle. The scheme is secure if such an adversary cannot generate a signature for any fresh document – a document which the adversary did not ask the oracle to sign. Formally, for every QPT adversary $\adv$ there exists a negligible function $\negl[]$ such that $\Pr\left[(vk,sk)\leftarrow\mathsf{key\textit{-}gen}(\secparam);(\alpha,\sigma)\leftarrow\adv^{\mathsf{sign}_{sk}}(\secparam,vk):\mathsf{verify}_{vk}(\alpha,\sigma)=1\wedge\alpha\notin Q_{\adv}^{\mathsf{sign}_{sk}}\right]\leq\negl,$ (19) where $\adv^{\mathsf{sign}_{sk}}$ is a QPT algorithm with access to the signing oracle, and $Q_{\adv}^{\mathsf{sign}_{sk}}$ is the set of queries it made to the oracle.

2024-09-04T02:54:54.976247

2002.12005

arxiv-papers

11institutetext: School of Sciences and Humanities, Nazarbayev University, Nur-Sultan, Kazakhstan 11email<EMAIL_ADDRESS> # Squashed Shifted PMI Matrix: Bridging Word Embeddings and Hyperbolic Spaces Zhenisbek Assylbekov 0000-0003-0095-9409 Alibi Jangeldin ###### Abstract We show that removing sigmoid transformation in the skip-gram with negative sampling (SGNS) objective does not harm the quality of word vectors significantly and at the same time is related to factorizing a squashed shifted PMI matrix which, in turn, can be treated as a connection probabilities matrix of a random graph. Empirically, such graph is a complex network, i.e. it has strong clustering and scale-free degree distribution, and is tightly connected with hyperbolic spaces. In short, we show the connection between static word embeddings and hyperbolic spaces through the squashed shifted PMI matrix using analytical and empirical methods. ###### Keywords: Word vectors PMI Complex networks Hyperbolic geometry ## 1 Introduction Modern word embedding models (McCann et al., 2017; Peters et al., 2018; Devlin et al., 2019) build vector representations of words in context, i.e. the same word will have different vectors when used in different contexts (sentences). Earlier models (Mikolov et al., 2013b; Pennington et al., 2014) built the so- called static embeddings: each word was represented by a single vector, regardless of the context in which it was used. Despite the fact that static word embeddings are considered obsolete today, they have several advantages compared to contextualized ones. Firstly, static embeddings are trained much faster (few hours instead of few days) and do not require large computing resources (1 consumer-level GPU instead of 8–16 non- consumer GPUs). Secondly, they have been studied theoretically in a number of works (Levy and Goldberg, 2014b; Arora et al., 2016; Hashimoto et al., 2016; Gittens et al., 2017; Tian et al., 2017; Ethayarajh et al., 2019; Allen et al., 2019; Allen and Hospedales, 2019; Assylbekov and Takhanov, 2019; Zobnin and Elistratova, 2019) but not much has been done for the contextualized embeddings (Reif et al., 2019). Thirdly, static embeddings are still an integral part of deep neural network models that produce contextualized word vectors, because embedding lookup matrices are used at the input and output (softmax) layers of such models. Therefore, we consider it necessary to further study static embeddings. With all the abundance of both theoretical and empirical studies on static vectors, they are not fully understood, as this work shows. For instance, it is generally accepted that good quality word vectors are inextricably linked with a low-rank approximation of the pointwise mutual information (PMI) matrix or the Shifted PMI (SPMI) matrix, but we show that vectors of comparable quality can also be obtained from a low-rank approximation of a Squashed SPMI matrix (Section 2). Thus, a Squashed SPMI matrix is a viable alternative to standard PMI/SPMI matrices when it comes to obtaining word vectors. At the same time, it is easy to interpret the Squashed SPMI matrix with entries in $[0,1)$ as a connection probabilities matrix for generating a random graph. Studying the properties of such a graph, we come to the conclusion that it is a so-called complex network, i.e. it has a strong clustering property and a scale-free degree distribution (Section 3). It is noteworthy that complex networks, in turn, are dual to hyperbolic spaces (Section 4) as was shown by Krioukov et al. (2010). Hyperbolic geometry has been used to train word vectors (Nickel and Kiela, 2017; Tifrea et al., 2018) and has proven its suitability — in a hyperbolic space, word vectors need lower dimensionality than in the Euclidean space. Thus, to the best of our knowledge, this is the first work that establishes simultaneously a connection between word vectors, a Squashed SPMI matrix, complex networks, and hyperbolic spaces. Figure 1 summarizes our work and serves as a guide for the reader. Squashed SPMIComplex NetworksWord EmbeddingsHyperbolic SpacesSection 2Section 3Section 4 Figure 1: Summary of our work ### Notation We let $\mathbb{R}$ denote the real numbers. Bold-faced lowercase letters ($\mathbf{x}$) denote vectors, plain-faced lowercase letters ($x$) denote scalars, $\langle\mathbf{x},\mathbf{y}\rangle$ is the Euclidean inner product, $(a_{ij})$ is a matrix with the $ij$-th entry being $a_{ij}$. ‘i.i.d.’ stands for ‘independent and identically distributed’. We use the sign $\propto$ to abbreviate ‘proportional to’, and the sign $\sim$ to abbreviate ‘distributed as’. Assuming that words have already been converted into indices, let $\mathcal{W}:=\\{1,\ldots,n\\}$ be a finite vocabulary of words. Following the setup of the widely used word2vec model (Mikolov et al., 2013b), we use two vectors per each word $i$: (1) $\mathbf{w}_{i}\in\mathbb{R}^{d}$ when $i\in\mathcal{W}$ is a center word, (2) $\mathbf{c}_{i}\in\mathbb{R}^{d}$ when $i\in\mathcal{W}$ is a context word; and we assume that $d\ll n$. In what follows we assume that our dataset consists of co-occurence pairs $(i,j)$. We say that “the words $i$ and $j$ co-occur” when they co-occur in a fixed-size window of words. The number of such pairs, i.e. the size of our dataset, is denoted by $N$. Let $\\#(i,j)$ be the number of times the words $i$ and $j$ co-occur, then $N=\sum_{i\in\mathcal{W}}\sum_{j\in\mathcal{W}}\\#(i,j)$. ## 2 Squashed SPMI and Word Vectors A well known skip-gram with negative sampling (SGNS) word embedding model of Mikolov et al. (2013b) maximizes the following objective function $\textstyle\sum_{i\in\mathcal{W}}\sum_{j\in\mathcal{W}}\\#(i,j)\left(\log\sigma(\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle)+k\cdot\mathbb{E}_{j^{\prime}\sim p}[\log\sigma(-\langle\mathbf{w}_{i},\mathbf{c}_{j^{\prime}}\rangle)]\right),$ (1) where $\sigma(x)=\frac{1}{1+e^{-x}}$ is the logistic sigmoid function, $p$ is a smoothed unigram probability distribution for words111The authors of SGNS suggest $p(i)\propto\\#(i)^{3/4}$., and $k$ is the number of negative samples to be drawn. Interestingly, training SGNS is approximately equivalent to finding a low-rank approximation of a Shifted PMI matrix (Levy and Goldberg, 2014b) in the form $\log\frac{p(i,j)}{p(i)p(j)}-\log k\approx\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle$, where the left-hand side is the $ij$-th element of the $n\times n$ shifted PMI matrix, and the right- hand side is an element of a matrix with rank $\leq d$ since $\mathbf{w}_{i},\mathbf{c}_{j}\in\mathbb{R}^{d}$. This approximation (up to a constant shift) was later re-derived by Arora et al. (2016); Assylbekov and Takhanov (2019); Allen et al. (2019); Zobnin and Elistratova (2019) under different sets of assuptions. In this section we show that constraint optimization of a slightly modified SGNS objective (1) leads to a low-rank approximation of the Squashed Shifted PMI ($\sigma$SPMI) matrix, defined as $\sigma\mathrm{SPMI}_{ij}:=\sigma(\mathrm{PMI}_{ij}-\log k)$. ###### Theorem 2.1 Assuming $0<\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle<1$, the following objective function $\mathcal{L}=\sum_{i\in\mathcal{W}}\sum_{j\in\mathcal{W}}\underbrace{\\#(i,j)\left(\log\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle+k\cdot\mathbb{E}_{j^{\prime}\sim P}[\log(1-\langle\mathbf{w}_{i},\mathbf{c}_{j^{\prime}}\rangle)]\right)}_{\ell(\mathbf{w}_{i},\mathbf{c}_{j})},$ (2) reaches its optimum at $\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle=\sigma\mathrm{SPMI}_{ij}$. ###### Proof Expanding the sum and the expected value in (2) as in Levy and Goldberg (2014b), and defining $p(i,j):=\frac{\\#(i,j)}{N}$, $p(i):=\frac{\\#(i)}{N}$, we have $\mathcal{L}=N\sum_{i\in\mathcal{W}}\sum_{j\in\mathcal{W}}p(i,j)\cdot\log\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle+p(i)\cdot p(j)\cdot k\cdot\log(1-\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle).$ (3) Thus, we can rewrite the individual objective $\ell(\mathbf{w}_{i},\mathbf{c}_{j})$ in (2) as $\ell=N\left[p(i,j)\cdot\log\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle+p(i)\cdot p(j)\cdot k\cdot\log(1-\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle)\right].$ (4) Differentiating (4) w.r.t. $\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle$ we get $\frac{\partial\ell}{\partial\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle}=N\left[\frac{p(i,j)}{\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle}-\frac{p(i)\cdot p(j)\cdot k}{1-\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle}\right].$ Setting this derivative to zero gives $\frac{p(i,j)}{p(i)p(j)}\cdot\frac{1}{k}=\frac{\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle}{1-\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle}\quad\Rightarrow\quad\log\frac{p(i,j)}{p(i)p(j)}-\log k=\log\frac{\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle}{1-\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle}\\\ \Leftrightarrow\quad\log\frac{p(i,j)}{p(i)p(j)}-\log k=\operatorname*{logit}\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle\\\ \Leftrightarrow\quad\sigma\left(\log\frac{p(i,j)}{p(i)p(j)}-\log k\right)=\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle,$ (5) where $\operatorname*{logit}(q):=\log\frac{q}{1-q}$ is the logit function which is the inverse of the logistic sigmoid function, i.e. $\sigma(\operatorname*{logit}(q))=q$. From (5) we have $\sigma\mathrm{SPMI}_{ij}=\langle\mathbf{w}_{i},\mathbf{c}_{j}\rangle$, which concludes the proof. ###### Remark 1 Since $\sigma(x)$ can be regarded as a smooth approximation of the Heaviside step function $H(x)$, defined as $H(x)=1$ if $x>0$ and $H(x)=0$ otherwise, it is tempting to consider a binarized SPMI (BSPMI) matrix $H(\mathrm{PMI}_{ij}-\log k)$ instead of $\sigma$SPMI. Being a binary matrix, BSPMI can be interpreted as an adjacency matrix of a graph, however our empirical evaluation below (Table 1) shows that such strong roughening of the $\sigma$SPMI matrix degrades the quality of the resulting word vectors. This may be due to concentration of the SPMI values near zero (Figure 5), while $\sigma(x)$ is approximated by $H(x)$ only for $x$ away enough from zero. ###### Remark 2 The objective (2) differs from the SGNS objective (1) only in that the former does not use the sigmoid function (keep in mind that $\sigma(-x)=1-\sigma(x)$). We will refer to the objective (2) as Nonsigmoid SGNS. ### Direct Matrix Factorization Optimization of the Nonsigmoid SGNS (2) is not the only way to obtain a low- rank approximation of the $\sigma$SPMI matrix. A viable alternative is factorizing the $\sigma$SPMI matrix with the singular value decomposition (SVD): $\sigma\mathrm{SPMI}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{\top}$, with orthogonal $\mathbf{U},\mathbf{V}\in\mathbb{R}^{n\times n}$ and diagonal $\mathbf{\Sigma}\in\mathbb{R}^{n\times n}$, and then zeroing out the $n-d$ smallest singular values, i.e. $\sigma\mathrm{SPMI}\approx\mathbf{U}_{1:n,1:d}\mathbf{\Sigma}_{1:d,1:d}\mathbf{V}^{\top}_{1:d,1:n},$ (6) where we use $\mathbf{A}_{a:b,c:d}$ to denote a submatrix located at the intersection of rows $a,a+1,\ldots,b$ and columns $c,c+1,\ldots,d$ of $\mathbf{A}$. By the Eckart-Young theorem (Eckart and Young, 1936), the right- hand side of (6) is the closest rank-$d$ matrix to the $\sigma$SPMI matrix in Frobenius norm. The word and context embedding matrices can be obtained from (6) by setting $\mathbf{W}^{\text{SVD}}:=\mathbf{U}_{1:n,1:d}\sqrt{\mathbf{\Sigma}_{1:d,1:d}}$, and $\mathbf{C}^{\text{SVD}}:=\sqrt{\mathbf{\Sigma}_{1:d,1:d}}\mathbf{V}^{\top}_{1:d,1:n}$. When this is done for a positive SPMI (PSPMI) matrix, defined as $\max(\mathrm{PMI}_{ij}-\log k,0)$, the resulting word embeddings are comparable in quality with those from the SGNS (Levy and Goldberg, 2014b). ### Empirical Evaluation of the $\sigma$SPMI-based Word Vectors To evaluate the quality of word vectors resulting from the Nonsigmoid SGNS objective and $\sigma$SPMI factorization, we use the well-known corpus, text8.222http://mattmahoney.net/dc/textdata.html. We ignored words that appeared less than 5 times, resulting in a vocabulary of 71,290 words. The SGNS and Nonsigmoid SGNS embeddings were trained using our custom implementation.333https://github.com/zh3nis/SGNS The SPMI matrices were extracted using the hyperwords tool of Levy et al. (2015) and the truncated SVD was performed using the scikit-learn library of Pedregosa et al. (2011). Table 1: Evaluation of word embeddings on the analogy tasks (Google and MSR) and on the similarity tasks (the rest). For word similarities evaluation metric is the Spearman’s correlation with the human ratings, while for word analogies it is the percentage of correct answers. Method | WordSim | MEN | M. Turk | Rare Words | Google | MSR ---|---|---|---|---|---|--- SGNS | .678 | .656 | .690 | .334 | .359 | .394 Nonsigm. SGNS | .649 | .649 | .695 | .299 | .330 | .330 PMI + SVD | .663 | .667 | .668 | .332 | .315 | .323 SPMI + SVD | .509 | .576 | .567 | .244 | .159 | .107 PSPMI + SVD | .638 | .672 | .658 | .298 | .246 | .207 $\sigma$SPMI + SVD | .657 | .631 | .661 | .328 | .294 | .341 BSPMI + SVD | .623 | .586 | .643 | .278 | .177 | .202 The trained embeddings were evaluated on several word similarity and word analogy tasks: WordSim (Finkelstein et al., 2002), MEN (Bruni et al., 2012), M.Turk (Radinsky et al., 2011), Rare Words (Luong et al., 2013), Google (Mikolov et al., 2013a), and MSR (Mikolov et al., 2013c). We used the Gensim tool of Řehůřek and Sojka (2010) for evaluation. For answering analogy questions ($a$ is to $b$ as $c$ is to $?$) we use the 3CosAdd method of Levy and Goldberg (2014a) and the evaluation metric for the analogy questions is the percentage of correct answers. We mention here that our goal is not to beat state of the art, but to compare SPMI-based embeddings (SGNS and SPMI+SVD) versus $\sigma$SPMI-based ones (Nonsigmoid SGNS and $\sigma$SPMI+SVD). The results of evaluation are provided in Table 1. As we can see the Nonsigmoid SGNS embeddings in general underperform the SGNS ones but not by a large margin. $\sigma$SPMI shows a competitive performance among matrix-based methods across most of the tasks. Also, Nonsigmoid SGNS and $\sigma$SPMI demonstrate comparable performance as predicted by Theorem 2.1. Although BSPMI is inferior to $\sigma$SPMI, notice that such aggressive compression as binarization still retains important information on word vectors. Figure 2: Spectral distribution of the $\sigma$SPMI-induced graphs (left and middle columns), and of scale-free random graphs with strong clustering property (right top: Goh et al. (2001), right bottom: Farkas et al. (2001)). When generating several random graphs from the same $\sigma$SPMI matrix, their eigenvalue distributions are visually indistinguishable, thus we display the results of one run per each matrix. Figure 3: Degree distributions of the $\sigma$SPMI-induced graphs. The axes are on logarithmic scales. Table 2: Clustering coefficients of the $\sigma$SPMI-induced graphs. For each corpus–window combination we generate ten graphs and report 95% confidence intervals across these ten runs. | text8 | enwik9 ---|---|--- | window $=2$ | window $=5$ | window $=2$ | window $=5$ $C$ | $.1341\pm.0006$ | $.1477\pm.0005$ | $.1638\pm.0006$ | $.1798\pm.0004$ $\bar{k}/n$ | $.0014\pm.0000$ | $.0030\pm.0000$ | $.0006\pm.0000$ | $.0012\pm.0000$ ## 3 $\sigma$SPMI and Complex Networks $\sigma$SPMI matrix has the following property: its entries $\sigma\text{SPMI}_{ij}\in[0,1)$ can be treated as connection probabilities for generating a random graph. As usually, by a graph $\mathcal{G}$ we mean a set of vertices $\mathcal{V}$ and a set of edges $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$. It is convenient to represent graph edges by its adjacency matrix $(e_{ij})$, in which $e_{ij}=1$ for $(i,j)\in\mathcal{E}$, and $e_{ij}=0$ otherwise. The graph with $\mathcal{V}:=\mathcal{W}$ and $e_{ij}\sim\text{Bernoulli}(\sigma\mathrm{SPMI}_{ij})$ will be referred to as $\sigma$SPMI-induced Graph. ### 3.1 Spectrum of the $\sigma$SPMI-induced Graph First of all, we look at the spectral properties of the $\sigma$SPMI-induced Graphs.444We define the graph spectrum as the set of eigenvalues of its adjacency matrix. For this, we extract SPMI matrices from the text8 and enwik9 datasets using the hyperwords tool of Levy et al. (2015). We use the default settings for all hyperparameters, except the word frequency threshold and context window size. We ignored words that appeared less than 100 times and 250 times in text8 and enwik9 correspondingly, resulting in vocabularies of 11,815 and 21,104 correspondingly. We additionally experiment with the context window size 5, which by default is set to 2. We generate random graphs from the $\sigma$SPMI matrices and compute their eigenvalues using the TensorFlow library (Abadi et al., 2016), and the above-mentioned threshold of 250 for enwik9 was chosen to fit the GPU memory (11GB, RTX 2080 Ti). The eigenvalue distributions are provided in Figure 2. The distributions seem to be symmetric, however, the shapes of distributions are far from resembling the Wigner semicircle law $x\mapsto\frac{1}{2\pi}\sqrt{4-x^{2}}$, which is the limiting distribution for the eigenvalues of many random symmetric matrices with i.i.d. entries (Wigner, 1955, 1958). This means that the entries of the $\sigma$SPMI-induced graph’s adjacency matrix are dependent, otherwise we would observe approximately semicircle distributions for its eigenvalues. We observe some similarity between the spectral distributions of the $\sigma$SPMI-induced graphs and of the so-called complex networks which arise in physics and network science (Figure 2). Notice that the connection between human language structure and complex networks was observed previously by Cancho and Solé (2001). A thorough review on approaching human language with complex networks was given by Cong and Liu (2014). In the following subsection we will specify precisely what we mean by a complex network. ### 3.2 Clustering and Degree Distribution of the $\sigma$SPMI-induced Graph We will use two statistical properties of a graph – degree distribution and clustering coefficient. The degree of a given vertex $i$ is the number of edges that connects it with other vertices, i.e. $\deg(i)=\sum_{j\in\mathcal{V}}e_{ij}$. The clustering coefficient measures the average fraction of pairs of neighbors of a vertex that are also neighbors of each other. The precise definition is as follows. Let us indicate by $\mathcal{G}_{i}=\\{j\in\mathcal{V}\mid e_{ij}=1\\}$ the set of nearest neighbors of a vertex $i$. By setting $l_{i}=\sum_{j\in\mathcal{V}}e_{ij}\left[\sum_{k\in\mathcal{G}_{i};\ j<k}e_{jk}\right],$ we define the local clustering coefficient as $C(i)=\frac{l_{i}}{{|\mathcal{G}_{i}|\choose 2}}$, and the clustering coefficient as the average over $\mathcal{V}$: $C=\frac{1}{n}\sum_{i\in\mathcal{V}}C(i)$. Let $\bar{k}$ be the average degree per vertex, i.e. $\bar{k}=\frac{1}{n}\sum_{j\in\mathcal{V}}e_{ij}$. For random binomial graphs, i.e. graphs with edges $e_{ij}\,\,{\stackrel{{\scriptstyle\text{iid}}}{{\sim}}}\,\,\mathrm{Bernoulli}(p)$, it is well known (Erdős and Rényi, 1960) that $C\approx\frac{\bar{k}}{n}$ and $\deg(i)\,\,\sim\,\,\mathrm{Binomial}(n-1,p)$. A complex network is a graph, for which $C\gg\frac{\bar{k}}{n}$ and $p(\deg(i)=k)\propto\frac{1}{k^{\gamma}}$, where $\gamma$ is some constant (Dorogovtsev, 2010). The latter property is referred to as scale-free (or power-law) degree distribution. We constructed $\sigma$SPMI-induced Graphs from the text8 and enwik9 datasets using context windows of sizes 2 and 5 and ignoring words that appeared less than 5 times, and computed their clustering coefficients (Table 2) as well as degree distributions (Figure 3) using the NetworKit tool (Staudt et al., 2016). NetworKit uses the algorithm of Schank and Wagner (2005) to compute the clustering coefficient. As we see, the $\sigma$SPMI-induced graphs are complex networks, and this brings us to the hyperbolic spaces. ## 4 Complex Networks and Hyperbolic Geometry Complex networks are “dual” to hyperbolic spaces as was shown by Krioukov et al. (2010). They showed that any complex network, as defined in Section 3, has an effective hyperbolic geometry underneath. Apart from this, they also showed that any hyperbolic geometry implies a complex network: they placed randomly $n$ points (nodes) into a hyperbolic disk of radius $R$, and used $p_{ij}:=\sigma\left(c[R-x_{ij}]\right)$ as connection probability for connecting nodes $i$ and $j$, where $x_{ij}$ is the hyperbolic distance between $i$ and $j$, and $c$ is a constant. An example of such random graph is shown in Figure 5. Figure 4: Rand. hyperbolic graph. Figure 5: SPMI values distr’n (top) vs $R-X$. Krioukov et al. (2010) showed that the resulting graph is a complex network. They establish connections between the clustering coefficient $C$ and the power-law exponent $\gamma$ of a complex network and the curvature of a hyperbolic space. Comparing the construction of Krioukov et al. (2010) to the way we generate a random graph from the $\sigma$SPMI matrix, and taking into account that both methods produce similar structures (complex networks), we conclude that the distribution of the SPMI values should be similar to the distribution of $R-x_{ij}$, i.e. $\text{PMI}_{ij}-\log k\sim R-x_{ij}$. To verify this claim we compare the distribution of SPMI values with the p.d.f. of a random variable $R-X$, where $X$ is a hyperbolic distance between two random points on the hyperbolic disk (the exact form of this p.d.f. is given in the Appendix 0.A). $R$ was chosen according to the formula $R=2\ln[8n/(\pi\bar{k})]$ (Krioukov et al., 2010), where $\bar{k}$ is the average degree of the $\sigma$SPMI-induced Graph. The results are shown in Figure 5. As we can see, the two distributions are indeed similar and the main difference is in the shift—distribution of $R-X$ is shifted to the left compared to the distribution of the SPMI values. This allows us reinterpreting the pointwise mutual information as the negative of hyperbolic distance (up to scaling and shifting). ## 5 Conclusion It is noteworthy that the seemingly fragmented sections of scientific knowledge can be closely interconnected. In this paper, we have established a chain of connections between word embeddings and hyperbolic geometry, and the key link in this chain is the Squashed Shifted PMI matrix. Claiming that hyperbolicity underlies word vectors is not novel (Nickel and Kiela, 2017; Tifrea et al., 2018). However, this work is the first attempt to justify the connection between hyperbolic geometry and the word embeddings. 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2024-09-04T02:54:54.997176

2002.12021

arxiv-papers

# Dataset search in biodiversity research: Do metadata in data repositories reflect scholarly information needs? Felicitas Löffler* * — 1 Valentin Wesp * — 1 Birgitta König-Ries * —1 2 3 Friederike Klan * — 2 4 (1Heinz-Nixdorf Chair for Distributed Information Systems, Department of Mathematics and Computer Science, Friedrich Schiller University Jena, Jena, Germany 2Michael-Stifel-Center for Data-Driven and Simulation Science, Jena, Germany 3German Center for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Germany 4Citizen Science Group, DLR-Institute of Data Science German Aerospace Center, Jena, Germany __) ###### Abstract The increasing amount of publicly available research data provides the opportunity to link and integrate data in order to create and prove novel hypotheses, to repeat experiments or to compare recent data to data collected at a different time or place. However, recent studies have shown that retrieving relevant data for data reuse is a time-consuming task in daily research practice. In this study, we explore what hampers dataset retrieval in biodiversity research, a field that produces a large amount of heterogeneous data. We analyze the primary source in dataset search - metadata - and determine if they reflect scholarly search interests. We examine if metadata standards provide elements corresponding to search interests, we inspect if selected data repositories use metadata standards representing scholarly interests, and we determine how many fields of the metadata standards used are filled. To determine search interests in biodiversity research, we gathered 169 questions that researchers aimed to answer with the help of retrieved data, identified biological entities and grouped them into 13 categories. The categories were evaluated with nine biodiversity scholars who assigned one of the types to pre-labeled biological entities in the questions. Our findings indicate that environments, materials and chemicals, species, biological and chemical processes, locations, data parameters and data types are important search interests in biodiversity research. The comparison with existing metadata standards shows that domain-specific standards cover search interests quite well, whereas general standards do not explicitly contain elements that reflect search interests. We inspect metadata from five large data repositories. Our results confirm that metadata currently poorly reflect search interests in biodiversity research. From these findings, we derive recommendations for researchers and data repositories how to bridge the gap between search interest and metadata provided. *<EMAIL_ADDRESS> Keywords: semantic search, query expansion, biological data, Life Sciences, biodiversity. ## Introduction Scientific progress in biodiversity research, a field dealing with the diversity of life on earth - the variety of species, genetic diversity, diversity of functions, interactions and ecosystems [idiv, 2019], is increasingly achieved by the integration and analysis of heterogeneous datasets [GBIF, 2018, Culina et al., 2018]. Therefore, locating and finding proper data for synthesis is a key challenge in daily research practice. Datasets can differ in format and size. Interesting data is often scattered across various repositories focusing on different domains. In a survey conducted by the Research Data Alliance (RDA) Data Discovery Group [SiriJodha Khalsa, 22z1], 35% of the 98 participating repositories stated that they host data from Life Science and 34% indicated they cover Earth Science. All of these are potentially of interest to biodiversity researchers. However, the offered search services at public data providers do not seem to support scholars effectively. A study by Kacprzak et al. [Kacprzak et al., 2018] reports that 40% of the users, who had sent data search requests to two open data portals, said, that they could not find the data they were interested in and thus directly requested the data from the repository manager. In several studies, ecologists report on the difficulties they had when looking for suitable datasets to reuse [Parker et al., 2016] [Ramakers et al., 2018] [Culina et al., 2018]. Scholars from research projects we are involved in also complain that data discovery is a time-consuming task. They have to search in a variety of data repositories with several different search terms to find data about species, habitats, or processes. Thus, there is a high demand for new techniques and methods to better support scholars in finding relevant data. In this study, we explore what hampers data set retrieval in biodiversity research. We analyze two building blocks in retrieval systems: _information needs (user queries)_ and underlying _data_. We want to find out how large the gap is between scholarly search interests and provided data. In order to identify scholarly search interests, we analyzed _user questions_. In contrast to user queries, which are usually formulated in a few keywords, questions represent a search context, a more comprehensive information need. Characteristic terms or phrases in these textual resources can be labeled and classified to identify biological entities [Kilicoglu et al., 2018, Nentidis et al., 2017]. _Scientific data_ are not easily accessible by classical text retrieval mechanisms as they were mainly developed for unstructured textual resources. Thus, effective data retrieval heavily relies on the availability of proper _metadata_ (structured information about the data) describing available datasets in a way that enables their _Findability_ , one principle to ensure FAIR data [Wilkinson et al., 2016]. A survey conducted by the Research Data Alliance (RDA) Data Discovery Group points out that 58% of the 98 participating data repositories index all metadata and partial metadata (52%), and only 33% integrate data dictionaries or variables [SiriJodha Khalsa, 22z1]. We argue that _Findability_ at least partially depends on how well metadata reflect scholarly information needs. Therefore, we propose the following layered approach: (A) At first, we identified main entity types (categories) that are important in biodiversity research. We collected $169$ questions provided by 73 scholars of three large and very diverse biodiversity projects in Germany, namely _AquaDiva_ [AquaDiva, 2020], GFBio - The German Federation for Biological Data [GFBio, 2020] and iDiv - The German Research Center for Integrative Biodiversity Research [idiv, 2019]. Two authors of this publication labeled and grouped all noun entities into $13$ categories (entity types), which were identified in several discussion rounds. Finally, all proposed categories were evaluated with biodiversity scholars in an online survey. The scholars assigned the proposed categories to important phrases and terms in the questions (Section “A - Information Needs in the Biodiversity Domain”). (B) Most data providers use keyword-based search engines returning data sets that exactly match keywords entered by a user [SiriJodha Khalsa, 22z1]. In dataset search, the main source are metadata that contain structured entries on measurements, data parameters or species observed rather than textual descriptions. It depends on the metadata schema used how sparse or rich the description turns out to be and which facets are provided for filtering. Therefore, we inspected common metadata standards in the Life Sciences and analyzed, to which extent their metadata schemes cover the identified information categories (Section “B - Metadata Standards in the Life Sciences”). (C) There are several data repositories that take and archive scientific data for biodiversity research. According to _Nature’s_ list of recommended data repositories [Nature, 2018], repositories such as _Dryad_ [Dryad, 2019], _Zenodo_ [Zenodo, 2019a] or _Figshare_ [Figshare, 2019] are generalist repositories and can handle different types of data. Data repositories such as _Pangaea_ [Pangaea, 2019a] (environmental data) or _GBIF_ [GBIF, 2020] (taxonomic data) are domain specific and only take data of a specific format. We harvested and parsed all publicly available metadata from these repositories and analyzed, if they utilize metadata schemes with elements reflecting search interests. For _GBIF_ , we concentrated on datasets only, as individual occurrence records are not available in the metadata API. We explored how many fields of the respective schemas are actually used and filled (Section “C - Metadata Usage in Selected Data Repositories”). (D) Finally, we discuss the results and outline how to consider and address user interests in metadata (Section “D - Discussion”). In order to foster reproducibility, questions, scripts, results, and the parsed metadata are publicly available: https://github.com/fusion- jena/QuestionsMetadataBiodiv The structure of the paper is as follows: The first part “Definitions” focuses on the clarification of various terms. This is followed by sections that explain basics in Information Retrieval (“Background”) and “Related Work”. The fourth section “Objectives” gives an overview of our research idea. The following four sections contain the individual research contributions described above. Each of these sections describes the respective methodology and results. Finally, section “Conclusion” summarizes our findings. ## Definitions Since dataset retrieval is a yet largely unexplored research field [Chapman et al., 2019], few definitions exist describing what it comprises and how it can be characterized. Here, we briefly introduce an existing definition and add our own definition from the Life Sciences’ perspective. Chapman et al [Chapman et al., 2019] define a dataset as “A collection of related observations organized and formatted for a particular purpose”. They further characterize a dataset search as an application that “involves the discovery, exploration, and return of datasets to an end user.” They distinguish between two types: (a) a basic search in order to retrieve individual datasets in data portals and (b) a constructive search where scholars create a new dataset out of various input datasets in order to analyze relationships and different influences for a specific purpose. From our perspective, this definition of a dataset is a bit too restricted. All kinds of scientific data such as experimental data, observations, environmental and genome data, simulations and computations can be considered as datasets. We therefore extend the definition of Chapman et al [Chapman et al., 2019] as follows: ###### Definition 1 A dataset is a collection of scientific data including primary data and metadata organized and formatted for a particular purpose. We agree with Chapman et al.’s definition of dataset search. We use _Dataset Search_ and _Dataset Retrieval_ synonymously and define it as follows: ###### Definition 2 Dataset Retrieval comprises the search process, the ranking and return of scientific datasets. Unger et al. [Unger et al., 2014] introduced three dimensions to take into account in Question Answering namely the _User_ and _Data_ perspective as well as the _Complexity_ of a task. We argue that these dimensions can also be applied in dataset retrieval. ### User Perspective In conventional retrieval systems users’ search interests are represented as a few keywords that are sent to the system as a search query. Keywords are usually embedded in a search context that can be expressed in a full sentence or a question. In order to understand what users are looking for, a semantic analysis is needed. _Information Extraction_ is a technique from text mining that identifies main topics (also called entity types) occurring in unstructured text [Jurafsky and Martin, 2008]. Noun entities are extracted and categorized based on rules. Common, domain-independent entity types are for instance _Person_ , _Location_ , and _Time_. When it comes to specific domains, additional entity types corresponding to core user interests need to be taken into consideration. In bio-medicine, according to [Roberts et al., 2017], the main topics are data type, disease type, biological process and organism. In new research fields such as biodiversity research these main entity types still need to be identified in order to get insights into users’ information needs and to be able to later adapt systems to user requirements. ### Data Perspective From the data perspective, a dataset search can be classified into two types based on the source of data: _primary data_ and _metadata_. ###### Definition 3 Primary data are scientific raw data. They are the result of scientific experiments, observations, or simulations and vary in type, format, and size. ###### Definition 4 Metadata are structured, descriptive information of primary data and answer the W-questions: _What?_ has been measured by Whom?, When?, Where? and Why?. Metadata are created for different purposes such as search, classification, or knowledge derivation. Dataset retrieval approaches focussing on primary data as source data have to deal with different data formats such as tabular data, images, sound files, or genome data. This requires specific query languages such as QUIS [Chamanara et al., 2017] to overcome the ensuing heterogeneity and is out of scope of this paper. Here, we solely focus on dataset retrieval approaches that use metadata as input for search. A variety of metadata standards in the Life Sciences are introduced in Section “B - Metadata Standards in the Life Sciences”. ### Complexity Scholarly search interests are as heterogeneous as data are. Information needs can range from specific questions where users expect datasets to contain the complete answer to broader questions that are answered partially only by datasets. Furthermore, users construct new datasets out of various input datasets. Unger et al. [Unger et al., 2014] characterize the complexity in retrieval tasks along four dimensions: _Semantic complexity_ describes how complex, vague, and ambiguous a question is formulated and if heterogeneous data have to be retrieved. _Answer locality_ denotes if the answer is completely contained in one dataset or if parts of various datasets need to be composed or if no data can be found to answer the question. _Derivability_ describes if the answer contains explicit or implicit information. The same applies for the question. If broad or vague terms appear in the question or answer, additional sources have to be integrated to enrich both, question and/or answer. _Semantic tractability_ denotes if the natural language question can be transformed into a formal query. In this work, we do not further explore the complexity of questions. We focus on the analysis of user interests and metadata, only. ## Background This section provides background information on which parts are involved in a search process, how the system returns a result based on a user’s query and what evaluation methods and metrics exist in Information Retrieval. ### The Retrieval Process A retrieval system consists of a collection of documents (a _corpus_) and a user’s information needs that are described with a few keywords (_query_). The main aim of the retrieval process is to return a ranked list of documents that match a user’s query. The architecture of a retrieval system is depicted in Figure (1): If the document corpus is not given, an optional _Crawling Process_ has to be run beforehand to retrieve and collect documents [Baeza- Yates and Ribeiro-Neto, 2008]. The _Indexing Process_ comprises pre-processing steps such as stopword removal, stemming, and spell checks important to clean documents from unnecessary information and to analyze only those terms that truly represent the content of a document. Afterwards, the system counts word frequencies within a document and across all documents. The result is an inverted index. Similar to a book index, this is a list of terms together with the number of occurrences of each term in each document and across all documents. These statistics, generated regularly in background processes, form the basis for a fast access to the documents at search time. The actual search takes place in the _Retrieval and Ranking Process_ whenever a user sends a query to the system and results in a ranked result set being returned to the user. Based on the underlying _Retrieval Model_ , different ranking functions have been developed to produce a score for the documents with respect to the query. Top-scored documents are returned first. In larger corpora, paging functions allow a subsequent retrieval of further documents. Classical retrieval models are for instance: the _Boolean Model_ [Manning et al., 2008] where only documents are returned that exactly match a query. In this model all documents in the retrieved set are equally relevant and therefore it is not considered as a ranking algorithm. It is often used in search engines in combination with further retrieval models such as the _Vector Space Model_ [Manning et al., 2008]. Here, documents are represented by vectors that consist of term weights. The similarity of documents and queries is determined by computing the distance between the vectors. _Probabilistic Models_ [Manning et al., 2008] are based on computations of the probability of a document belonging to the relevant set. For languages where word boundaries are not given, e.g., in Eastern Asian Languages, _Language Models_[Jurafsky and Martin, 2000] have to be applied to get a mathematical representation of the documents. The system analyzes the text documents by means of character-based sliding windows (_n- grams_) to determine word boundaries and compute statistics. All these classical retrieval models are keyword-based. Thus, retrieval systems only return documents that exactly match the user query. Figure 1: The architecture of an Information Retrieval system based on [Baeza- Yates and Ribeiro-Neto, 2008]: An optional _Crawling Process_ (blue, dashed line) gathers documents or web pages. In the _Indexing Process_ (blue) the documents are pre-processed before an index can be established. The _Retrieval Process_ (orange) comprises the transformation of a user query into a format the search engine understands before the actual search and ranking takes place. Finally, users receive a ranked list of documents that match their query. ### Evaluation in Information Retrieval When setting up a retrieval system, various design decisions influencing different parts of the system have to be made. Examples of such decisions are whether to stem terms in the pre-processing phase or which terms to include in the stopword list. Numerous evaluation measures have been developed to determine the _effectiveness_ of the systems, i.e., the accuracy of the result returned by a given retrieval algorithm. For this purpose, a test collection is required that consists of three things [Manning et al., 2008]: (1) a corpus of documents, (2) representative information needs expressed as queries, (3) a set of relevance judgments provided by human judges containing assessments of the relevance of a document for given queries. If judgments are available for the entire corpus they serve as baseline (“gold standard”) and can be used to determine how many relevant documents a search system finds for a specific topic. User queries should be representative for the target domain. Queries are either obtained from query logs of a similar application or domain users are asked to provide example queries [Croft et al., 2009]. The number of example questions influences the evaluation result. TREC (Text REtrieval Conference) is a long-running, very influential annual Information Retrieval competition that considers different retrieval issues in a number of _Tracks_ , e.g., Genomics Track or Medical Track (https://trec.nist.gov/). Various TREC experiments have shown that the number of queries used for the evaluation matters more than the number of documents judged per query [Croft et al., 2009]. Therefore, TREC experiments usually consist of around 150 queries (or so-called “topics”) per track. Common evaluation metrics with respect to effectiveness are _Precision and Recall (PR)_ , _F-Measure_ and _Mean Average Precision (MAP)_ [Manning et al., 2008]. Precision denotes which fraction of the documents in the result set is relevant for a query, whereas recall describes which fraction of relevant documents was successfully retrieved. Both metrics are based on binary judgments, i.e., raters can only determine, if a document is relevant or non- relevant. The F-Measure is the harmonic mean of Precision and Recall. Precision and Recall can only be used, when a gold standard is provided containing the total number of documents in a corpus that are relevant for a query. However, in applied domains where corpora are established specifically for a particular research field, gold standards are usually not available. Therefore, with the recall being unknown, _MAP_ only requires to get ratings for the TopN-ranked documents to compute an average precision. The assumption here is that users are only interested in the first entries of a search result and usually do not navigate to the last page. The top-ranked documents get higher scores than the lower ranked ones [Croft et al., 2009]. Another metric proposed by Järvelin and Kekäläinen[Järvelin and Kekäläinen, 2002] is the _Discounted Cumulated Gain (DCG)_ , a metric that uses a Likert-scale as rating scheme and allows non-binary ratings. All entries of the scheme should be equally distributed. However, DCG does not penalize for wrong results but only increases the scores of top-ranked documents. Other evaluation criteria concentrate on the _efficiency_ (e.g., the time, memory and disk space required by the algorithm to produce the ranking [Croft et al., 2009]), _user satisfaction_ on the provided result set and _visualization_ [Hearst, 2011]. The missing aspect in evaluation approaches in Information Retrieval is the analysis of the underlying documents. The data source in classical Information Retrieval systems is unstructured text whereas Dataset Retrieval is based on structured metadata files. Hence, retrieval success depends on the metadata format used, the experience of the curator, and the willingness of the individual scholar to describe data properly and thoroughly. This structured information could be used in the search index. For instance, if researchers provide information such as taxon, length, location or experimental method in the metadata explicitly, a search application could offer a search over a specific metadata field. Thus, we argue that there is a need to analyze the given metadata and to quantify the gap between a scholar’s actual search interests and the metadata primarily used in search applications. ## Related Work This section focuses on approaches that analyze, characterize and enhance dataset search. We discuss studies identifying users’ information needs and introduce existing question corpora. In a second part, we describe approaches that aim at improving dataset search. ### User Interests In order to understand user behavior in search, query logs or question corpora are valid sources. Kacprazal et al [Kacprzak et al., 2018] provide a comprehensive log analysis of three government open data portals from the United Kingdom (UK), Canada, and Australia and one open data portal with national statistics from the UK. 2.2 million queries from logs provided by the data portals (internal queries) and 1.1 million queries issued to external web search engines (external queries) were analyzed. Two authors manually inspected a sample set of 665 questions and determined the main query topics. Most queries were assigned to Business and Economy (20$\%$ internal queries, 10$\%$ external queries) and Society (14.7$\%$ internal queries, 18$\%$ external queries). Besides query logs, Kacprazal et al [Kacprzak et al., 2018] also explicit requests by users for data via a form on the website. Here, users provided title and description which allowed the authors to perform a deeper thematic analysis on 200 manually selected data requests. It revealed that geospatial (77.5$\%$) and temporal (44$\%$) information occurred most, often together with a specific granularity (24.5$\%$), e.g., “hourly weather and solar data set” or “prescription data per hospital”. Users were also asked why they had requested data explicitly, and more than 40$\%$ indicated that they were not able to find relevant data via the provided search. In the Life Sciences, Dognan et al [Islamaj Dogan et al., 2009] inspected one month of log data with more than 58 million user queries from PubMed [PubMed, 2019], a platform providing biomedical literature. They randomly selected 10,000 queries for a semantic analysis. Seven annotators categorized the queries along 16 given categories. They distinguished between bibliographic queries (44$\%$) containing information such as journal name, author name, or article title and non-bibliographic queries with domain specific categories. The most frequent category over all questions was “Author Name” (36$\%$) followed by “Disorder” (20$\%$) comprising diseases, abnormalities, dysfunctions etc., and “Gene/ Protein” (19 $\%$). Further main topics were abbreviations (mostly from genes/ proteins) and chemicals/drugs. A large study on user needs in biodiversity research have been conducted in the GBIF community in 2009 [Faith et al., 2013, Ariño et al., 2013]. The aim was to determine what GBIF users need in terms of primary data and to identify data gaps in the current data landscape at that time. More than 700 participants from 77 countries took part in the survey. It revealed that scholars used retrieved primary data for analyzing species diversity, taxonomy, and life histories/ phenology. That mainly required “taxon names, occurrence data and descriptive data about the species” [Ariño et al., 2013]. As biodiversity is a rapidly changing research field, the authors recommend to repeat content need assessments in frequent intervals [Faith et al., 2013]. Apart from query logs, question corpora are another source for identifying search interests. Usually, questions are collected from experts of a particular research field and important terms representing main information needs are labeled with categories or so-called _entity types_. These manually generated annotations help understanding what information users are interested in and developing tools and services to either automatically extract these interests from text (Text Mining), to retrieve relevant data (Information Retrieval) or to provide an exact answer for that information need (Question Answering). In the Life Sciences, question corpora for text retrieval have been mainly established in the medical and biomedical domains. One of the largest corpora in medicine is the Consumer Health Corpus [Kilicoglu et al., 2018], a collection of email requests (67%) received by the U.S. National Library of Medicine (NLM) customer service and search query logs (33%) of MedlinePlus, a consumer-oriented NLM website for health information. The final corpus consists of 2614 questions and has been integrated into the Medical Question Answering Task at TREC 2017 LiveQA [Abacha et al., 2017]. Six trained domain experts were involved in the annotation tasks to manually label information. The experts had to indicate named entities, e.g., problem, anatomy or measurement and labeled question topics such as the cause of a disease or complications (longer term effects of a disease). A common question corpus in biomedicine is the Genomics Track at TREC conferences [Hersh and Voorhees, 2009]. The topics of the retrieval tasks are formulated as natural language questions and contain pre-labeled main categories, e.g., _What [GENES] are involved in insect segmentation?_. A further large question corpus in biomedicine is the question corpus created for the BioASQ challenge [Nentidis et al., 2017], an annual challenge for researchers working on text mining, machine learning, information retrieval, and question answering. The tasks are split into three parts: (1) the extraction of main entities and their linkage with ontological concepts (semantic annotation), (2) the translation of natural language queries into RDF triples, and (3) the retrieval of the exact answer to a natural language query. The question corpus was created and annotated by a team of $10$ experts, selected with the goal to cover different ages and complementary expertise in the fields of medicine, biology, and bioinformatics [Polychronopoulos et al., 2013]. Each expert was asked to formulate $50$ questions in English that reflect “real-life information needs”. However, the type of questions to be formulated was restricted, e.g., the experts were instructed to provide questions of certain types typically considered in question answering systems (yes/no, factoid, etc.). These restrictions are justified to a certain degree since they affect the applicability of the resulting corpus for evaluation purposes of question answering approaches. However, they have an impact on which questions are formulated and how. This will likely lead to a bias in the question corpus. Another question corpus in the biomedical domain is the benchmark developed for the 2016 bioCADDIE Dataset Retrieval Challenge [Roberts et al., 2017]. This benchmark was explicitly created for the retrieval of datasets based on metadata and includes 137 questions, 794,992 datasets gathered from different data portals in XML structure, and relevance judgments for 15 questions. Similar to the BioASQ challenge, domain experts got instructed on how to create questions. Based on templates, the question constructors formulated questions using the most desired entity types, namely data type, disease type, biological process, and organism. At present, to the best of our knowledge, there is neither a public log analysis nor a question corpus available for biodiversity research. In order to understand genuine user interests and to improve current dataset retrieval systems, unfiltered information needs are crucial. Therefore, collecting current search interests from scholars is the first step in our top-down approach presented in Section “Objectives”. ### Dataset Search A study by the RDA Data Discovery Group points out [SiriJodha Khalsa, 22z1] that most data repositories offer search applications based on metadata and utilize one of the existing and widely spread search engines for data access, e.g., _Apache Solr_ (http://lucene.apache.org/solr/) or _elasticsearch_ (https://www.elastic.co/products/elasticsearch). Large data repositories such as _GBIF_ [GBIF, 2019], _PANGAEA_ [Pangaea, 2019b] or _Zenodo_ [Zenodo, 2019b] also use _elasticsearch_ and offer public search services. _Apache Solr_ and _elasticsearch_ are both keyword-based and return datasets that exactly match a user’s entered query terms. If the desired information need is not explicitly mentioned in the metadata, the search will fail. In recent years, a variety of approaches have emerged to improve dataset search. A common approach is to annotate metadata with entities from _schema.org_ (https://schema.org). Favored by Google [Google, 2019] and the RDA Discovery Task Group [RDA, 2019], the idea is to add descriptive information to structured data such as XML or HTML in order to increase findability and interoperability. These additional attributes help search engines to better disambiguate terms occuring in text. For example, Jaguar could be a car, an animal or an operating system. By means of _schema.org_ entities, data providers can define the context explicitly. Numerous extensions for specific domains have been developed or are still in development, e.g., _bioschemas.org_ [Michel and Community, 2018] for the Life Sciences. Since Google launched its beta version of a dataset search in Fall 2018 (https://toolbox.google.com/datasetsearch), _schema.org_ entities got more and more attention. Hence, data centers such as _PANGAEA_ [Pangaea, 2019a] or _Figshare_ [Figshare, 2019] are increasingly incorporating _schema.org_ entities in their dataset search. Other approaches favor an improved metadata schema. Pfaff et al [Pfaff et al., 2017] introduce the Essential Annotation Schema for Ecology (EASE). The schema was primarily developed in workshops and intensive discussions with scholars and aims to support scientists in search tasks. The MIBBI project [Taylor et al., 1411] (now known as _BioSharing_ or _FAIRSharing_ portal - https://fairsharing.org/) also recognized that only improved metadata allow information seekers to retrieve relevant experimental data. They propose a harmonization of minimum information checklists in order to facilitate data reuse and to enhance data discovery across different domains. Checklist developers are advised to consider “’cross-domain’ integrative activities”[Taylor et al., 1411] when creating and maintaining checklists. In addition, standards are supposed to contain information on formats (syntax), vocabularies and ontologies used. The latter points to an increasing interest in semantic techniques that have emerged over the past decade. Vocabularies such as the Data Catalog Vocabulary (DCAT) [Maali and Erickson, 2014] or the Vocabulary of Interlinked Datasets (VoID) [Alexander et al., 2011] aim to describe datasets semantically in RDF [Brickley and Guha, 2014] or OWL [W3C, 2012] format based on subject, predicate, and object triples. Fully semantic approaches such as BioFED [Hasnain et al., 2017] offer a single-point-of-access to 130 SPARQL endpoints in the Life Sciences. They integrate a variety of heterogeneous biomedical ontologies and knowledge bases. Each data source is described by VoID descriptors that facilitate federated SPARQL query processing. The user interface permits simple and complex SPARQL queries and provides support in creating federated SPARQL queries. The result set contains provenance information, i.e., where the answer has been found, “the number of triples returned and the retrieval time”[Hasnain et al., 2017]. However, improvements in the user interface still remain necessary. As BioFED is mainly focused on a linked data approach, it requires all data sources to be stored in semantic formats and users to have at least basic SPARQL knowledge. In contrast, Kunze and Auer [Kunze and Auer, 2013] consider the search process in their search over RDF datasets as an exploratory task based on semantic facets. Instead of SPARQL queries or keyword-based user interfaces, they provide parameters for filtering. This allows an unambiguous search and returns relevant datasets that match the provided filter parameters. Other federated approaches outside semantic techniques attempt to align heterogeneous data sources in one search index. That allows the use of conventional search engines and keyword-based user interfaces: DataONE is a project aiming to provide access to earth and environmental data provided by multiple member repositories [Cook et al., 2012]. Participating groups can provide data in different metadata formats such as EML, DataCite or FGDC [DataONE, 2019a]. DataONE is currently working on quantifying FAIR [DataONE, 2019b]. Their findability check determines if specific metadata items such as title, abstract or publication date are present. For title and abstract, they additionally check the length and content. Based on these criteria, they evaluated their data and found out that concerning _Findability_ around 75% of the available metadata fulfilled the self-created criteria. The German Federation for Biological Data (GFBio) [Diepenbroek et al., 2014] is a national infrastructure for research data management in the green Life Sciences and provides a search over more than six million heterogeneous datasets from environmental archives and collection data centers. It was extended to a semantic search [Löffler et al., 2017] that allows a search over scientific names, common names, or other synonyms. These related terms are obtained from GFBio’s Terminology Service [Karam et al., 2016] and are added in the background to a user’s query. As described above, numerous approaches have been proposed and developed to improve dataset search. However, what is lacking is a comprehensive analysis on what exactly needs to be improved and how large the actual gap is between user requirements and given metadata. ## Objectives Current retrieval evaluation methods are basically focused on improving retrieval algorithms and ranking. Therefore, question corpora and documents are taken as given and are not questioned. However, if the underlying data do not contain the information users are looking for, the best retrieval algorithm will fail. We argue, in dataset search, metadata, the basic source for dataset applications, need to be adapted to match users’ information needs. We want to find out how large the gap in biodiversity research is between actual user needs and provided metadata and how to overcome this obstacle. Thus, the following analysis aims to explore: * • What are genuine user interests in biodiversity research? * • Do existing metadata standards reflect information needs of biodiversity scholars? * • Are metadata standards utilized by data repositories useful for data discovery? How many metadata fields are filled? * • Do common metadata fields contain useful information? We take a top-down approach starting from scholars’ search interests, then looking at metadata standards and finally inspecting the metadata provided in selected data repositories. (A) First, we generate an annotated question corpus for the biodiversity domain: We gather questions from scholars, explore the questions and identify information categories. In an online evaluation, domain experts assign these categories to terms and phrases of the questions (Section “A - Information Needs in the Biodiversity Domain”). (B) We inspect different metadata standards in the Life Sciences and compare the metadata elements to the identified search categories from (A) (Section “B - Metadata Standards in the Life Sciences”). (C) We analyze the application programming interfaces (APIs) of selected data repositories to figure out what metadata standards are used and how many elements of a metadata schema are utilized for data description (Section “C - Metadata Usage in Selected Data Repositories”). (D) We discuss how to bridge the gap between users’ search interests and metadata. We propose an approach to overcome the current obstacles in dataset search (Section “D - Discussion”). ## A - Information Needs in the Biodiversity Domain Question corpora are common sources for getting an impression what users are interested in in a particular domain. Therefore, we asked biodiversity scholars to provide questions that are specific for their research. We analyzed the questions and identified search topics that represent scholarly information needs in this domain. ### Methodology The following subsection describes the methodology in detail, divided into four paragraphs. #### Questions: We gathered questions in three large biodiversity projects, namely _CRC AquaDiva_ [AquaDiva, 2020], _GFBio_ [GFBio, 2020] and _iDiv_ [idiv, 2019]. We explicitly requested fully expressed questions to capture the keywords in their search context. These projects vary widely in their overall setting, the scientists and disciplines involved and their main research focus. Together, they provide a good and rather broad sample of current biodiversity research topics. In total, 73 scholars with various research backgrounds in biology (e.g., ecology, bio-geochemistry, zoology and botany) and related fields (e.g., hydro-geology) provided 184 questions. This number is comparable to related question corpora in Information Retrieval (e.g., bioCADDIE [Roberts et al., 2017]) which typically consist of around 100 – 150 questions. The scholars were asked to provide up to five questions from their research background. Questions varied with respect to granularity. The corpus contains specific questions, such as _List all datasets with organisms in water samples!_ or questions with a broader scope, e.g., _Does agriculture influence the groundwater?_. We published the questionnaires that were handed out in AquaDiva and iDiv as supplementary material in our repository. In the GFBio project, questions were gathered via email and from an internal search evaluation. All questions were inspected by the authors with respect to comprehensibility. We discarded questions which were not fully understandable (e.g., missing verb, misleading grammatical structures) but left clear phrases in the corpus that were not fully expressed as a question. If scholars provided several questions, they were treated individually even if terms referred to previous questions, e.g., _Do have earthworm burrows (biopores) an impact on infiltration and transport processes during rainfall events?_ and _Are the surface properties influencing those processes?_. In this case, no further adaption towards comprehensibility has been made. The questions were also not corrected with respect to grammar and spelling since changing the grammar could lead to an altered statement. We did not want to loose the original question statement. In some questions, abbreviations occurred without explanations. In these cases, we left the questions as they are and did not provide full terms, since these abbreviations can have various meanings in different biological fields. It was up to the domain experts to either look them up or to leave the term out. After the cleaning, the final corpus consists of 169 questions and is publicly available: https://github.com/fusion-jena/QuestionsMetadataBiodiv/tree/master/questions. #### Categories: Boundaries of semantic categories are domain-dependent and fuzzy. However, in search, categories support users in finding relevant information more easily and should be valid across various research backgrounds. In a first round, two authors of this work analyzed the collected questions manually. Both have a research background in computer science and strong knowledge in scientific data management, in particular for biodiversity research. The corpus was split up and each of them inspected around 50% of it and assigned broad categories independently of the other one. Afterwards, this first classification was discussed in several sessions. This resulted in 13 categories. The naming was adapted to domain-specific denotations and ontologies. Furthermore, the categories were compared to EASE [Pfaff et al., 2017], a metadata schema which was primarily developed for an improved dataset retrieval in the field of ecology. This comparison revealed that there is an overlap with EASE but that we discovered further relevant categories [Löffler et al., 2017]. The final categories are: 1. 1. ORGANISM comprises all individual life forms including plants, fungi, bacteria, animals and microorganisms. 2. 2. All species live in certain local and global ENVIRONMENTS such as habitats, ecosystems (e.g., below 4000 m, ground water, city) and 3. 3. have certain characteristics (traits, phenotypes) that are summarized with QUALITY & PHENOTYPE, e.g., length, growth rate, reproduction rate, traits. 4. 4. Biological, chemical and physical PROCESSES are re-occurring and transform materials or organisms due to chemical reactions or other influencing factors. 5. 5. EVENTS are processes that appear only once at a specific time, such as environmental disasters, e.g., Deepwater Horizon oil spill, Tree of the Year 2016. 6. 6. Chemical compounds, rocks, sand and sediments can be grouped as MATERIALS & SUBSTANCES. 7. 7. ANATOMY comprises the structure of organisms, e.g., body or plant parts, organs, cells, and genes. 8. 8. METHOD describes all operations and experiments that have to be conducted to lead to a certain result, e.g., lidar measurements, observation, remote sensing. 9. 9. Outcomes of research methods are delivered in DATA TYPE, e.g., DNA data or sequence data is the result of genome sequencing, lidar data is the result of lidar measurements (active remote sensing). 10. 10. All kinds of geographic information is summarized with LOCATION, e.g., Germany, Hainich, Atlantic Ocean, and 11. 11. temporal data including date, date times, and geological eras are described by TIME, e.g., current, over time, triassic. 12. 12. PERSON & ORGANIZATION are either projects or authors of data. 13. 13. As reflected in the search questions, scholars in biodiversity are highly interested in HUMAN INTERVENTION on landscape and environment, e.g., fishery, agriculture, and land use. For the evaluation with domain experts we added two more categories, namely OTHER and NONE. The first permits to define an own category, if none of the given ones is appropriate. NONE applies, if the term is not relevant, or if the domain expert does not know the term or if the phrase is too fuzzy and can not be classified into one category. #### Annotation: An annotation process usually has two steps: (1) the identification of terms based on annotation rules and (2) the assignment of an appropriate category in a given context. Usually, an annotator - a domain expert - who is trained in the annotation guidelines, carries out both tasks. However, we argue that training is somewhat biased and influences annotators in their classification decision. This is an obstacle in search where an intuitive feedback for category assignment is required. Hence, we split up the annotation process. Two scholars, who collected the questions and who are familiar with the guidelines conducted the identification, whereas domain experts only received short instructions and assigned categories. Our annotation guidelines, needed to identify phrases and terms (artifacts) to label, are available as supplementary material in our repository. #### Annotators and Annotation Process: Nine domain experts (8 Postdocs, 1 Project Manager) with expertise in various biological and environmental sciences participated in the classification task. All of them have experience in ecology but in addition, each of them has individual research competence in fields such as bio-geography, zoology, evolutionary biology, botany, medicine, physiology, or biochemistry. For the category assignment, all scholars received a link to an online survey with explanations of the categories (including examples) and short instructions on how to classify the artifacts. A screenshot of the survey is presented in Figure 2. The purpose of this evaluation was also explained to them (improvement of data set retrieval systems). Multi-labeling was not allowed; only one category was permitted per artifact. Should there be no proper category, they were advised to select OTHER and if possible to provide an alternative category. If they did not know a term or phrase, they could decide either to look it up or to omit it. The latter also applied,if they considered a phrase or term to be not relevant or too complicated and fuzzy. As we wanted to obtain intuitive feedback, the experts were told not to spend too much time on the classification decision but to determine categories according to their knowledge and research perspective. The annotators also had the opportunity to skip an artifact. In this case the category NONE was applied. For each question, annotators had the opportunity to provide a comment. Figure 2: Excerpt of the survey that was set up for the classification task. The annotators were told to assign only one category per given artifact. If an artifact is a compound noun, the nested entities such as adjectives or second nouns that further describe the term were provided for tagging as well. We decided to use a combination of csv files, Python scripts and _Limesurvey_ to support the annotation process. Details on this process can be found in the supplementary material in our repository. ### Results We analyzed the user responses to determine whether the identified information categories are comprehensive and representative for biodiversity research. We computed the inter-rater agreement per artifact to determine the category that best describes an artifact. #### Representativeness of the Categories In order to verify completeness we determined the fraction of artifacts assigned to the category OTHER, i.e., if the experts deemed none of the given categories as appropriate. Figure 3 depicts the frequency of information categories and how often they were selected by the domain experts. As it turned out, the category OTHER was selected by at least $1$ expert per artifact for $46\%$ of the phrases and terms and by at least $2$ experts for $24\%$. The fraction of phrases for which at least $3$ experts selected the category OTHER was $12\%$. If at least two domain experts agree that there is no proper category for a given phrase, it is a strong indicator for a missing category or a misinterpretation. This is the case for $24\%$ out of all annotated artifacts. Hence, the coverage of the identified information categories is still high. Figure 3: The frequency of the categories and how often they were assigned to given phrases and terms, with and without QUALITY correction. However, there might be various reasons why none of the given categories fit: (1) The phrase or term to be annotated was unknown to the annotator such as _shed precipitation_. (2) Frequently, phrases that refer to data attributes (e.g., _soil moisture_ , _oxygen uptake rate_ or _amount of rain_) and which were supposed to be covered by the category QUALITY, were classified as OTHER. As alternative category, the annotators proposed “Parameter” or “Variable”. When adding these ratings to the QUALITY category, the results for the OTHER category decreased to $37\%$/$13\%$/$4\%$. That strongly indicates that renaming the QUALITY category or adding synonyms would increase comprehensibility significantly. (3) The category OTHER was often chosen for terms used in questions with a broader scope in order to express expected results. However, since this is often vague, scholars tend to use generic terms such as _signal_ , _pattern_ , _properties_ , _structure_ , _distribution_ , _driver_ or _diversity_. Hence, further discussions in the biodiversity research community are needed to define and classify these terms. In addition, we wanted to know if there are categories that were not or rarely used by the annotators. This would indicate a low relevance for biodiversity research. As depicted in Figure 3, the categories ENVIRONMENT, ORGANISM, MATERIAL & SUBSTANCES, QUALITY, PROCESS, LOCATION and DATA TYPE have been selected most frequently (assigned to more than $15\%$ of the phrases). Information related to these categories seems to be essential for biodiversity research. Although there were categories that were rarely chosen (PERSON & ORGANISATION and TIME), there was no category that was not used at all. #### Consensus of the Categories In statistics, the consensus describes how much homogeneity exists in ratings among domain experts. We determined the inter-rater agreement and inter-rater reliability using Fleiss’ Kappa ($\kappa$ statistics) [Fleiss, 1971] and GWET’s AC [Gwet, 2008]. In general, the inter-rater reliability computes the observed agreement among raters “and then adjusts the result by determining how much agreement could be expected from random chance”[Quarfoot and Levine, 2016]. $\kappa$ values vary between $-1$ and $+1$, where values less than $0$ denote poorer than chance agreement and values greater than $0$ denote better than chance agreement. As suggested by Landis and Koch [Landis and Koch, 1977], $\kappa$ values below $0.4$ indicate fair agreement beyond chance, values between $0.4$ and $0.6$ moderate agreement, values between $0.6$ and $0.8$ substantial agreement and values higher than $0.80$ indicate almost perfect agreement. However, $\kappa$ statistics can lead to a paradox: When the distribution of the raters’ scores is unbalanced, the correction for the chance agreement can result in negative $\kappa$ values even if the observed agreement is very high [Quarfoot and Levine, 2016]. Since this is the opposite of what is expected, a new and more robust statistic has emerged, the GWET’s AC [Gwet, 2008]. GWET’s AC considers the response categories in the agreement by chance and the values can range from $0$ to $1$. With a Fleiss’ Kappa of $0.48$ and GWET’s AC of $0.51$ the agreement of the annotators over all categories was moderate. Considering the QUALITY correction, the values increase slightly to $0.49$ for Fleiss’ Kappa and $0.52$ to GWET’s AC. Figure 4a reveals a more detailed picture. It shows the Fleiss’ Kappa for the individual information categories with QUALITY correction. The agreement among the experts was excellent for the categories TIME and ORGANISM and intermediate to good for the categories PERSON & ORGANIZATION, LOCATION, PROCESS, MATERIALS & SUBSTANCES and ENVIRONMENT. The experts’ agreement for the categories EVENT, HUMAN INTERVENTION, ANATOMY, DATA TYPE, METHOD and QUALITY was fair. This lack of agreement can either point to a different understanding of the categories or might indicate that the categorization of the phrase itself was difficult since some phrases, in particular longer ones with nested entities, were fuzzy and difficult to classify in one category. In the latter case, the annotators were advised not to choose a category for that phrase. Our results show, that for $5\%$ of the phrases at least $2$ annotators did not provide a category. The fraction of phrases where $3$ or more annotators did not choose a category was below $2\%$. This points out that annotators in fact interpreted the categories with poor agreement differently. This correlates with our results regarding the category QUALITY. For the categories EVENT, HUMAN INTERVENTION, ANATOMY, DATA TYPE, METHOD there is no such evidence. This should be discussed and reconsidered with biodiversity experts. (a) Fleiss’ Kappa values per category with and without QUALITY correction. (b) Fleiss’ Kappa values per category for artifacts with one and two terms (with QUALITY correction). Figure 4: Fleiss’ Kappa values for the individual information categories. #### Comparison of short and long artifacts We also analyzed the influence of longer artifacts on the result. Table 1 presents the $\kappa$ statistic and $GWET^{\prime}sAC$ for artifacts with one term, two terms, three and more terms including the quality correction. As assumed, the longer an artifact is, the more difficult it is to assign an unambiguous category. Table 1: Annotator’s agreement with QUALITY correction overall and for one term, two terms, three terms and more per artifact | Overall | One Term | Two Terms | $>=$ Three Terms ---|---|---|---|--- $Fleiss^{\prime}Kappa$ | 0.49 | 0.54 | 0.50 | 0.33 $GWET^{\prime}sAC$ | 0.52 | 0.57 | 0.53 | 0.37 Figure 4b depicts a more detailed picture on the individual categories for artifacts with one and two terms. Since artifacts with three and more terms resulted in coefficients with less than $0.4$, we left them out in this analysis. One-term artifacts got an excellent agreement ($>0.8$) for the categories ORGANISM, TIME and LOCATION and a moderate agreement for ENVIRONMENT, MATERIAL, PROCESS and DATA TYPE. It strikes that PERSON results in a negative value with a poor agreement. Since full person names usually contain two terms, there were no artifacts with one term that could be assigned to PERSON & ORGANIZATION. However, looking at the results for two terms per artifact, the PERSON category reaches an excellent agreement as well as ORGANISM. Surprisingly, PROCESS ($0.76$) got a substantial agreement for two terms pointing out that biological and chemical processes are obviously mainly defined by two terms. The same effect, a larger agreement for two terms than one term, can also be observed for the categories EVENT and HUMAN INTERVENTION. DATA TYPE got a moderate agreement for one and two terms. #### Summary All $13$ provided categories were used by the annotators to label the artifacts in the questions. However, what stands out is the high number of the category OTHER in the frequency analysis. For 45% out of 592 annotations, at least one domain expert did not assign one of the given categories but selected OTHER. That points to missing interests that are not represented by the given classes. In terms of consensus, seven information categories got a moderate agreement ($>$ 0.4) and five out of these seven were also mentioned very often ($>15\%$), namely ENVIRONMENT (e.g., habitats, climate zone, soil, weather conditions), MATERIAL (e.g., chemicals, geological information), ORGANISM (species, taxonomy), PROCESS (biological and chemical processes) and LOCATION (coordinates, altitude, geographic description) (Figure 5). We conclude that these classes are important search interests for biodiversity research. In comparison to the outcome of the content assessment analysis conducted in the GBIF community [Ariño et al., 2013] in 2009, the assumption that user interests change over time has been confirmed. Species are still an important category scholars are interested in, however, further important topics for the acquisition and description of ecosystem services are emerging. Figure 5: Frequency of category mentions and inter-rater agreement with QUALITY correction. We are aware that this result is not complete and leaves room for improvement. Some category names were misleading and confused the annotators. That is reflected in fair and bad agreement for some categories such as QUALITY (data parameters measured) or DATA TYPE (nature or genre of the primary data). Here, it should be discussed in the research community how they could be further considered in search, e.g., re-naming or merging of categories. Since the backgrounds of the annotators were quite diverse and no training took place, we did not expect completeness and perfect agreement. We wanted to get a real, genuine, and unbiased first picture of biodiversity scholars’ comprehension when looking for scientific data. In biology and biodiversity research, scholars use a specialized language with diverse content and imprecise and inconsistent naming [Thessen et al., 2012, Ananiadou et al., 2004]. Hence, labeling and extracting biological entities remain a challenge. Therefore, our thresholds for agreement ($>$ 0.4) and frequency ($>15\%$) are not as high as in similar studies in bio-medicine. Concerning the shortened methodology for the evaluation, our assumptions have been confirmed. It saved a lot time that only a few people did the identification of artifacts to be labeled and that domain experts assigned categories, only. On average, domain experts spent between two and three hours for labeling 169 questions. We conclude that our shortened annotation approach is fine for opening up new domains and getting insights in what scholars are interested in. If the aim is to achieve higher agreement per annotation, we recommend training sessions and trial rounds. However, it should be considered that in this case the unbiased feedback gets lost. For further reuse of the annotated question corpus, our analysis script also produces an XML file with all questions and annotations above a certain agreement threshold that can be set as a parameter. By default, all annotations per question with an agreement above $0.6$ will be returned. ## B - Metadata Standards in the Life Sciences In this section, we describe a selection of existing metadata standards and investigate whether their elements reflect the identified information categories. ### Methodology Metadata describe scientific primary data such as experiments, tabular data, images, sound and acoustic files in a structured format, e.g., XML or JSON. Metadata possess a _schema_ stored typically in an XSD file outlining which elements and attributes exist and which of them are mandatory and/or repeatable. Many schemes employ vocabularies or ontologies to ensure that the metadata use the same names for concepts. In order to become a metadata standard, a schema needs to be formally adopted by a standards’ organization such as the International Organization for Standardization, https://www.iso.org. There are a variety of metadata standards for different research fields. Table 2 presents a list of 13 metadata standards used in data repositories for the Life Sciences. All metadata standards were obtained from _re3data_ [re3data, 2018]. We filtered for “Life Sciences” and retrieved a list of 25 standards. The categories _Other_ and _Repository-Developed Metadata Schema_ have been left out. The _MIBBI_ standard is outdated and has been integrated into _ISA- Tab_ , so we left it out, too. All other standards that were used in at least 5 repositories have been selected. We compared them along the focused _Domain_ , _Number of Elements_ in the standard and _Mandatory Fields_ (Table 3). The standards are ranked by the number of data repositories supporting them. The number of elements and required fields were either stated on the standard’s website or they were obtained from the schema. We also examined, if there is support for the Semantic Web, namely, if the standard supports RDF or OWL formats. According to the FAIR principles [Wilkinson et al., 2016], community standards, semantic formats, and ontologies ensure interoperability and data reuse. The last two columns denote whether the standard is still maintained and provide some examples of data repositories that support the respective standard. Table 2: Metadata Schemes in the Life Sciences obtained from _re3data_ [re3data, 2018] Dublin Core (http://dublincore.org/documents/dces/) | DDI (https://www.ddialliance.org/) ---|--- Dublin Core is a widely used generic metadata standard offering basic fields for the description of research data. | The DDI (Document, Discover, Interoperate) standard addresses metadata from questionnaires and surveys in the social, behavioral, economic and health sciences. Data Cite (https://schema.datacite.org) | ISO19115 (https://www.iso.org/standard/53798.html) Data Cite relates to generic research data and comprises a set of mandatory, recommended and optional properties. | The ISO19115 metadata standard includes the identification, the extent, the quality, the spatial and temporal schema, spatial reference, and distribution of digital geographic data. DIF (https://gcmd.nasa.gov/DocumentBuilder/defaultDif10/guide) | FDGC/CSDGM (https://www.fgdc.gov/metadata/csdgm-standard) The Directory Interchange Format (DIF) is the US-predecessor of ISO 19115 and focuses on the description of geospatial metadata. | The Federal Geographic Data Committee Content Standard for Digital Geospatial Metadata is a legacy national standard for geospatial data developed in the United States. FGDC now encourages its research community to use the international ISO standards. EML (https://knb.ecoinformatics.org/) | Darwin Core (https://dwc.tdwg.org/) The Ecological Metadata Language (EML) is a series of XML document types that can be used in a modular and extensible manner to document ecological data. | The Darwin Core standard provides metadata fields for sharing biodiversity data. It is primarily based on taxa, their occurrence in nature and related information. RDF Data Cube (https://www.w3.org/TR/vocab-data-cube/) | ISA - Tab (https://isa-specs.readthedocs.io) The RDF Data Cube vocabulary aims to describe statistical data. The model is compatible with the cube model that underlies the Statistical Data and Metadata eXchange standard (SDMX, https://sdmx.org/), an ISO standard for exchanging and sharing statistical data and metadata among organizations. | The ISA specification is not a standard but a metadata framework that addresses the description and management of biological experiments. It comprises three core entities to capture experimental metadata: Investigation (the project context), Study (a unit of research) and Assay (analytical measurements). ABCD (https://github.com/tdwg/abcd) | CF (http://cfconventions.org/) The ABCD (Access to Biological Collection Data) metadata standards aims to share biological collection data. It offers a variety of metadata fields to describe specimen and observations, and it is compatible with numerous existing standards. | The Conventions for Climate and Forecast Metadata (CF) comprise geophysical quantities to describe climate and forecast data. DCAT (https://www.w3.org/TR/vocab-dcat/) | The Data Catalog Vocabulary (DCAT) facilitates interoperability between data catalogs in the web and allows dataset search across sites. | Table 3: Comparison of metadata standards and specifications used in data repositories for Life Sciences. The number in brackets denotes the number of repositories supporting the standard. Standard Name | Domain | Elements | Mandatory Elements | Semantic Support | Maintenance | Examples ---|---|---|---|---|---|--- $DublinCore(142)$ | general | 15 | No | Yes (RDFS) | Yes | Pangaea, Dryad, GBIF, Zenodo, Figshare $DDI(74)$ | questionnaires and surveys in the social, behavioral, economic, and health sciences | 1154 | 7 | No | Yes | Dataverse $DataCite(60)$ | general research data | 19 (57) | 5 | No | Yes | Pangaea, Zenodo, Figshare, Radar $ISO19115(36)$ | geospatial data | N/A | 7 | No | Yes | Pangaea, NSF Arctic Data Center, coastMap $FDGC/CSDGM(34)$ | geographic information | 342 | 74 | No | No (1998, last update: 2002) | Dataverse, NSF Arctic Data Center $EML(21)$ | ecological data | N/A | N/A | No | Yes | GBIF, GFBio, SNSB, Senckenberg, WORMS, NSF Arctic Data Center $DarwinCore(21)$ | biodiversity data | 184 | No | Yes (RDF) | Yes | GFBio, GBIF, VerNET, Atlas of Living Australia, WORMS $RDFDataCube(18)$ | statistical data | 36 | N/A | Yes | Yes | Dryad (only RDF with DublinCore) $ISA-Tab(9)$ | biological experiments | N/A | Yes (11 blocks) | Yes | Yes | Data Inra, GigaDB $DIF(7)$ | geospatial metadata | 34(219) | 8 | No | Yes | Pangaea, Australian Antarctic Data Center, Marine Environmental Data Section $CF(7)$ | climate and forecast | 4798—54—70 (lines in the standard table) | No | No | Yes | WORMS, NSF Arctic Data Center, coastMap $ABCD(7)$ | biological collection data | 1418 | 20 | No | Yes (ABCD 3.0) | GBIF, BioCase Network $DCAT(6)$ | data catalogs, data sets | 16 | N/A | Yes | Yes | Data.gov.au, European Data Portal N/A denotes that the information was not available --- The standard supported by most repositories is _Dublin Core_ , a general metadata standard based on 15 fields, such as contributor, coverage, creator, date, description, format, and identifier. In addition, data repositories utilize further domain-specific standards with richer vocabulary and structure such as _ISO19115_ for geospatial data or _EML_ for ecological data. The _RDF Data Cube Vocabulary_ is not used by any of the data centers. We suppose, the abbreviation _RDF DC_ might lead to a misunderstanding (_DublinCore_ instead of _RDF Data Cube_). All standards provide elements that can be described along the questions: Who? What? Where? When? Why? and How?. In particular, contact person, collection or publication date and location are considered with one or several metadata fields in all standards. In order to describe the main scope of the primary data, all standards offer numerous metadata fields but differ in their granularity. While simple ones such as _Dublin Core_ only offer fields such as title, description, format, and type, standards with more elements such as _EML_ or _ABCD_ even offer fields for scientific names, methods and data attributes measured. _EML_ even allows scholars to define the purpose of the study making it the only standard that supports the _Why_ question. Data reuse and citation also play an important role. As it is demanded by the Joint Declaration of Data Citation Principles [M., 2014] and practical guidelines for data repositories [Fenner et al., 2019], all standards provide several elements for digital identifiers, license information and citation. In addition, some standards provide elements for data quality checks. For instance, _ISO19115_ offers a container for data quality including lineage information and _EML_ supports quality checks with the _qualityControl_ element. Surprisingly, 52 repositories stated to use own- developed metadata schemes. That indicates that a variety of data repositories is not satisfied with the existing metadata landscape and therefore started developing their own schema. For our further analysis, we selected 12 out of the 13 standards shown in Table 3. Since _DDI_ is a standard that was mainly developed for questionnaires and surveys, we decided not to use it. ### Results In our second analysis, we compared the information categories with elements of the metadata schemes to figure out, if search interests can be explicitly described with metadata elements. Our results are presented in Table 4. For the sake of completeness, we explored all $13$ categories from the previous analysis but marked the ones with an asterisk that had a fair agreement ($<$ 0.4). The categories are sorted by frequency from left to right. The red color denotes that no element is available in the standard to express the category, orange indicates that only a general field could be used to describe the category and a light-orange cell implies that one or more elements are available in the standard for this search interest. Table 4: Comparison of metadata standards and information categories. The categories are sorted by frequency, the asterisk denotes the categories with an agreement less than 0.4 | Environment | Quality* | Material | Organism | Process | Location | Data Type* | Method* | Anatomy* | Human Intervention* | Event* | Time | Person ---|---|---|---|---|---|---|---|---|---|---|---|---|--- $DublinCore$ | | | | | | | | | | | | | $DataCite$ | | | | | | | | | | | | | $ISO19115$ | | | | | | | | | | | | | $FDGC/CSDGM$ | | | | | | | | | | | | | $EML$ | | | | | | | | | | | | | $DarwinCore$ | | | | | | | | | | | | | $RDFDataCube$ | | | | | | | | | | | | | $ISA-Tab$ | | | | | | | | | | | | | $DIF$ | | | | | | | | | | | | | $CF$ | | | | | | | | | | | | | $ABCD$ | | | | | | | | | | | | | $DCAT$ | | | | | | | | | | | | | Table key $\blacksquare$ | Not provided | $\blacksquare$ | Unspecific (generic element) | $\blacksquare$ | Available (one or more elements) ---|---|---|---|---|--- There is no schema that covers all categories. Since the interests are obtained from scholars with various and heterogeneous research backgrounds, this was also not to be expected. Some standards such as _ABCD_ or _DarwinCore_ are discipline-specific and therefore, mainly provide elements that support the respective domain (e.g., collection data). Apart from HUMAN INTERVENTION, all categories are covered by different metadata schemes. In particular, _ISA-Tab_ followed by _ABCD_ , _DarwinCore_ and _EML_ are frameworks and metadata schemes with elements that cover most of the search interests of biodiversity researchers. _EML_ provides numerous fields to describe ecological data including elements for environmental information (studyAreaDescription), species (taxonomicCoverage) and research methods used (methods). However, important search preferences such as materials (including chemicals) and biological and chemical processes are only explicitly supported by _ISA-Tab_. Widely used general standards such as _DublinCore_ or _DataCite_ offer at least a general field (dc:subject, subject) that could be used to describe the identified search categories. In _DublinCore_ , at least one metadata field each is provided to describe geographic information, e.g., where the data have been collected (LOCATION), the type of the data (DATA TYPE), the creator and contributor (PERSON & ORGANIZATION) and when it was collected or published (TIME). However, one field is often not enough to distinguish if the provided field is for instance a collection date or publication date, or if the creator of the dataset is also the same person that collected the data. In contrast, _DataCite_ provides individual fields for publication year and the date field can be used with dateType="Collected" to specify a collection date. The metadata field contributor can also be extended with a type to indicate whether the contact details belong to the data collector or the project leader. Bounding box elements are also provided to enter geographic coordinates (LOCATION). The question that still remains to be answered is whether these detailed metadata standards are actually used by data repositories. ## C - Metadata Usage in Selected Data Repositories In the following analysis, we examine what metadata standards are used in selected data repositories and how many schema elements are actually filled. In a second part, we explore, if descriptive fields of selected files contain data that might be relevant for information seekers. ### Methodology Scholarly publishers increasingly demand scientific data to be submitted along with publications. Since publishers usually do not host the data on their own, they ask scholars to upload the data at one of the repositories for their research domain. According to _Nature’s_ list of recommended data repositories [Nature, 2018], we selected five archives for our further analysis: three generalist ones (_Dryad_ , _Figshare_ and _Zenodo_) and two domain-specific ones (_PANGAEA_ \- environmental data, _GBIF_ \- taxonomic data). In the biodiversity projects we are involved in, scholars also often mention these repositories as the ones they mainly use. #### OAI-PMH Harvesting The Open Archives Initiative Protocol for Metadata Harvesting (OAI-PMH) is a client/server architecture primarily developed for providing and consuming metadata. Data repositories are required to expose their metadata in $DublinCore$ metadata format and may also support other metadata formats. Metadata consumers, e.g., other institutions or data portals, harvest that data via the provided services on the OAI-PMH server in order to integrate or reuse it in their services. The OAI-PMH protocol comprises a set of six services that are accessible via HTTP. Requests for metadata can be based on a date stamp range or can be restricted to named sets defined by the provider. We parsed all available metadata from _Figshare_ , _Dryad_ , _GBIF_ , _PANGAEA_ and _Zenodo_ in May 2019 via their respective _OAI-PMH_ interfaces. _GBIF_ only offers the metadata of their datasets in the _OAI-PMH_ interface. The individual occurrence records, which are provided in _Darwin Core_ metadata schema [Gaiji et al., 2013] and belong to a dataset, are available in the search, only. Hence, we only analyzed the metadata of the datasets. Our script parses the metadata fields of all public records per metadata schema for each of the selected data repositories (Table 5). Apart from the metadata standards introduced in the previous section, a few more standards appear in this list. _OAI-DC_ is an abbreviation for $DublinCore$, a mandatory standard in OAI-PMH interfaces. _QCD_ means _qualified DublinCore_ and denotes an extended $DublinCore$ extending or refining the $15$ core elements. _ORE_ (The Open Archives Initiative Object Reuse and Exchange (OAI-ORE)) is a standard for exchanging aggregations of web resources. It can be used together with other semantic standards such as RDF to group individual web resources. We also considered _Pan-MD_ , a metadata schema developed by _PANGAEA_. It extends $DublinCore$ with more fine-grained geographic information such as bounding boxes or adds information on data collection. The latter can range from projects, parameters, methods, and sensors to taxonomy or habitats. Table 5: Metadata schemes offered by selected data repositories in their OAI-PMH interfaces. Dryad | GBIF | PANGAEA | Zenodo | Figshare ---|---|---|---|--- METS | EML | DATACITE3 | DATACITE | CERIF OAI-DC | OAI-DC | DIF | DATACITE3 | METS ORE | | ISO19139 | DATACITE4 | OAI-DATACITE RDF | | ISO19139.IODP | MARCXML | OAI-DC | | OAI-DC | MARC21 | QDC | | PAN-MD | OAI-DATACITE | RDF | | | OAI-DATACITE3 | | | | OAI-DC | _MARC21_ , _MARCXML_ and _METS_ are metadata standards that are mainly used for bibliographic data in digital libraries. Hence, we left them out of our further explorations. We also did not consider the _Common European Research Information Format (CERIF)_ and _ORE_ as they are not focused on describing primary data but research entities and their relationships and grouping web resources, respectively. However, we decided to permit all available repository-developed schemes for Life Sciences such as _Pan-MD_ in order to get an impression how repositories extend metadata descriptions. Per metadata file we inspected which elements of the metadata standards are used, and we saved their presence (1) or non-presence (0). The result is a csv file per metadata schema that contains dataset IDs and metadata elements used. All generated files are stored in separate folders per repository and metadata format. Each request to a repository returns an XML body that includes several metadata files as records. Each record is separated in two sections, a header and a metadata section. The header section comprises general information such as example ID of the record and a date stamp. The metadata section contains elements of the metadata schema, e.g., the name of the contributors, abstract and publication year. Unused metadata fields are not included in the response. We saved a boolean value encoding whether a metadata field was used or not. The source code and a documentation on how to use it is available in our $GitHub$ repository. Table 6: The date stamps used for each metadata standard and their descriptions obtained from the standard’s website. Format | Element | URL | Description ---|---|---|--- _OAI-DC/METS/QDC RDF(Dryad)_ | dc:date | http://www.dublincore.org/specifications/dublin-core/dces/ | “A point or period of time associated with an event in the lifecycle of the resource.” _EML_ | pubDate | https://knb.ecoinformatics.org/external//emlparser/docs/eml-2.1.1/eml-resource.html#pubDate | “The ’pubDate’ field represents the date that the resource was published.” _DATACITE3/4 OAI-DATACITE/3_ | publicationYear | https://support.datacite.org/docs/schema-40 | “The year when the data was or will be made publicly available.” _DIF_ | DIF_Creation- _Date | https://gcmd.gsfc.nasa.gov/DocumentBuilder/defaultDif10/guide/metadata_dates.html | “ refers to the date the data was created” _ISO19139/ISO19139.iodp_ | gco:DateTime | https://geo-ide.noaa.gov/wiki/index.php?title=ISO_Dates | (CI-DataTypeCode=publication), publication Date _PAN-MD_ | md:dateTime | http://ws.pangaea.de/schemas/pangaea/MetaData.xsd | publication date (contact to data repository) _RDF(Figshare)_ | vivo:datePublished | https://codemeta.github.io/terms/ | ”Date of first broadcast/publication” For our further consideration, we wanted to obtain a publication date of each downloaded dataset to inspect how many datasets have been published over the years in which format per data repository. Unfortunately, a publication date is not provided in all metadata schemes. Therefore, we looked up each date related field in the schema and used the one that is (based on the description) the closest to a publication date. Table 6 depicts all date stamps utilized and their descriptions. If the respective date stamp was not found in a dataset or was empty, we left the dataset out in the following analysis. #### Content Analysis: General, descriptive metadata fields such as ‘title’, ‘description’ or ‘abstract’, and ‘subject’ might contain relevant data that are interesting for information seekers. Using conventional retrieval techniques, this data is only accessible in a full text search and if the entered query terms exactly match a term in the dataset. Hence, we aim to explore what information is available in general, descriptive metadata fields. In a first step, we downloaded descriptive metadata fields, namely, dc:title, dc:description and dc:subject in _OAI-DC_ format from all repositories in October and November 2019. Parallel to the download, we collected the keywords used in the subject field and counted their presence in a separate csv file. In order to further inspect the content with Natural Language Processing (NLP) tools, we selected a subset of representative datasets. We limited the amount to 10,000 datasets per repository as the processing of textual resources is time-consuming and resource-intensive. A variety of applications have been developed to determine Named Entities (NE) such as geographic locations, persons and dates. Thessen et. al [Thessen et al., 2012] explored the suitability of existing NLP applications for biodiversity research. Their outcome reveals that current text mining systems, which were mainly developed for the biomedical domain, are able to discover biological entities such as species, genes, proteins and enzymes. Further relevant entity types such as habitats, data parameters or processes are currently not supported by existing taggers. Thus, we concentrated on the extraction of entity types that (a) correspond to the identified search interests and for which (b) text mining pipelines are available. We used the text mining framework GATE [Cunningham et al., 2011] and its ANNIE pipeline [Cunningham et al., 2002] as well as the OrganismTagger [Naderi et al., 2011a] to extract geographic locations, persons, organizations and organisms. ### Results The overall statistics are presented in Table 7. At first, we inspected the fields concerning a publication date in a valid format. We could not use all harvested datasets as for some metadata files publication dates were not available. _Dryad_ had a large number of datasets with a status “Item is not available”, which we left out, too. The number in brackets denotes the amount of datasets we used for the following considerations. What stands out is that most repositories provide general standards, only _Pangaea_ and _GBIF_ utilize discipline-specific metadata schemes. $Dryad$ and $Figshare$ already provide metadata in semantic formats such as RDF. In addition, $Figshare$ offers _Qualified Dublin Core (QDC)_ , an extended _Dublin Core_ that allows the description of relations to other data sources. #### Timelines Based on the given publication dates, we computed timelines (Figure 6) for the introduction of the various standards over time per data repository. The code and all charts are available in the repository. As _Dryad_ provides several dc:date elements in the metadata, we used the first available date entry as publication date for the timeline chart. Per repository, the timelines for the different metadata formats are almost identical. Obviously, when introducing a new metadata format, publication dates were adopted from existing metadata formats. Only _Figshare_ uses new date stamps when a new metadata format is provided. For instance, _Figshare_ ’s timeline shows that QDC and RDF were launched in 2015. The result for RDF was too large to process it together with the other metadata formats. Hence, we produced the timeline for RDF separately. The timelines across all repositories reveal a steadily increasing number of datasets being published at _GBIF_ , _Dryad_ , _Zenodo_ and _Figshare_. For _PANGAEA_ , the timeline points to a constant number of published datasets of around 10,000 datasets a year apart from an initial release phase between 2003 and 2007. Figure 6: Timelines for all repositories presenting the number of datasets per metadata schema offered. _Figshare’s_ timeline for RDF was computed separately as the data are too large to process it together with the other metadata formats. Table 7: Total number of datasets parsed per data repository and metadata schema. The numbers in brackets denote the number of datasets used for the analysis. All datasets were harvested and parsed in May 2019. Metadata Schema | Dryad | PANGAEA | GBIF | Zenodo | Figshare ---|---|---|---|---|--- _OAI-DC_ | 186951 (142329) | 383899 (383899) | 44718 (42444) | 255000 (255000) | 3128798 (3128798) _QDC_ | | | | | 1718059 (1718059) _RDF_ | 186955 (142989) | | | | 3157347 (3157347) _DATACITE_ | | | | 1268155 (1268155) | _DATACITE3_ | | 383906 (383906) | | 1268232 (1268232) | _OAI-DATACITE_ | | | | 1266522 (1266522) | 3134958 (3134958) _OAI-DATACITE3_ | | | | 1268679 (1268679) | _DATACITE4_ | | | | 1268262 (1268262) | _EML_ | | | 44718 (42444) | | _DIF_ | | 383899 (383899) | | | _ISO19139_ | | 383899 (383899) | | | _ISO19139.iodp_ | | 383899 (383899) | | | _PAN-MD_ | | 383899 (383899) | | | #### Metadata Field Usage Figure 7 presents how many metadata elements of the best matching standard were filled. The individual results per data archive are available in our repository as supplementary material. _Dryad_ used 9 out of 15 available metadata fields from OAI-DC very often (¿ 80%) including important fields such as dc:title, dc:description and dc:subject. dc:publisher and dc:contributor were provided in less that 20%. For _GBIF_ , the EML standard does not provide a fixed number of core elements. Hence, we analyzed the 129 available fields. Most of them (89 elements) were not filled, e.g., fields describing taxonomic information. Data about author, title and description were provided in more than 80%. The general field eml:keyword was used in around 20%. Out of 124 used fields in _PANGAEA’s_ Pan-MD format, 43 fields were filled in more than 80% of the harvested metadata files including information on the author, project name, coordinates, data parameters and used devices. Fields that were less filled are supplementary fields, for instance for citation, e.g., volume, pages. For _Zenodo_ , all required fields in DataCite (identifier, creator, title, publisher, publication year) were always filled. In addition, title, rights and descriptions as well as resource type were also provided in more than 99% of the analyzed metadata files. However, in only 45% of the metadata files, keywords (subject) were present. _Figshare_ used only 12 out of 17 available fields of QDC, but these fields were always filled. Figure 7: Metadata field usage in all data repositories evaluated. #### Category - Field - Match Per data repository and metadata format, we computed charts that visualize which field was filled at what percentage rate and if they correspond to the categories introduced in Section “A - Information Needs in the Biodiversity Domain”. Table 8 presents a summary of all data repositories and their best matching standard. The individual results per repository and the concrete field-to-category mapping are available in our repository. Temporal expressions (TIME) and information about author and/or creator (PERSON) were mostly provided in all repositories. Apart from $PANGAEA$, repositories mainly provided the publication date and only partially added information about when the data was collected. Information about data type and formats was also contained in all metadata files apart from _GBIF_. The identified search categories were partially covered by two repositories. $GBIF$ with EML reflects most of the categories, but fields that correspond to ENVIRONMENT, ORGANISM, DATA TYPE and METHOD were rarely filled. Metadata files in $PANGAEA^{\prime}s$ repository-developed standard Pan-MD always contained information on data parameters (QUALITY) and geographic locations. In most cases, research methods and devices used were also given. _Dryad_ provided geographic information (LOCATION) in its dc:coverage field in at least 60%. Table 8: Comparison of data repositories and their best matching standard with the information categories. The categories are sorted by frequency. The asterisk denotes the categories with an agreement less than 0.4 | Environment | Quality* | Material | Organism | Process | Location | Data Type* | Method* | Anatomy* | Human Intervention* | Event* | Time | Person ---|---|---|---|---|---|---|---|---|---|---|---|---|--- $GBIF$ ($EML$) | (3%) | | | (11%) | | (35%) | (8%) | (18%) | | | | (publication Date - 100%, collection Date - 10%) | ($>$90%) $Dryad$ ($OAI-DC$) | | | | | | (60%) | | | | | | (publication Date) | (80%) $PANGAEA$ ($Pan-MD$) | | ($>$90%) | | | | (100%) | (100%) | (Devices used - 90%, research methods - 65%) | | | | (publication Date - 100%, collection Date - 80%) | (100%) $Zenodo$ ($OAI-Datacite$) | | | | | | | (100%) | | | | | (publication Date) | (100%) $Figshare$ ($QDC$) | | | | | | | (100%) | | | | | (publication Date) | (100%) Table key $\blacksquare$ | Unspecific (generic element) | $\blacksquare$ | Available (one or more elements) ---|---|---|--- Amount in brackets denotes the percentage the element is filled. #### Content Analysis: Table 9 presents the Top5 keywords in the metadata field dc:subject for all repositories sorted by their frequencies. The full keyword lists are available in our repository. For _GBIF_ datasets, in 81% an empty dc:subject field was returned. _Zenodo’s_ metadata provided keywords in 52% of the inspected cases. None of the repositories seem to consider upper and lower cases. For several terms, different spellings resulted in separate entries. Numerous keywords in _PANGAEA_ and _Dryad_ reveal that both repositories host marine data. _PANGAEA_ ’s list mainly contains data parameters measured and used devices. In contrast, _Dryad’s_ list indicates that terrestrial data are also provided. For instance, the lower ranked terms contain entries such as Insects (1296) (insects (180)) or pollination (471) (Pollination (170)). Geographic information , e.g., California (9817), also occured in _Dryad’s_ dc:subject field. _Zenodo’s_ and _Figshare’s_ keyword lists contain numerous terms related to collection data. We checked the term ‘Biodiversity’ in both repositories in their search interfaces on their websites. It turned out that the _Meise Botanic Garden_ (https://www.plantentuinmeise.be) provided large collection data in _Zenodo_. Hence, each occurrence record counted as a search hit and got the label ‘Biodiversity’. We also discovered that _Figshare_ harvests _Zenodo_ data which also resulted in high numbers for _Figshare_ and the keyword ‘Biodiversity’ (219022). Table 9: Top5 keywords and their frequencies in the metadata field dc:subject. | Pangaea | GBIF | Dryad | Zenodo | Figshare ---|---|---|---|---|--- | water (201102) | Occurrence (6510), occurrence (46) | Temperature (16652), temperature (15916) | Taxonomy (459877), taxonomy (105) | Medicine (1057684), medicine (240) | DEPTH (198349), Depth(71916) | Specimen (3046), specimen (22) | Integrated Ocean Observing System (16373) | Biodiversity (458336), biodiversity (8593) | Biochemistry (1015906), biochemistry (92) | Spectral irradiance (175373) | Observation (2425), observation (24) | IOOS (16373) | Herbarium (270110), herbarium (91) | Biological Sciences not elsewhere classified (983829) | DATE/TIME (128917) | Checklist (589), checklist (43) | Oceanographic Sensor Data (15015) | Terrestrial (269900), terrestrial (177) | Chemical Sciences not elsewhere classified (842865) | Temperature (118522), temperature (50) | Plantas (368), plantas (42) | continental shelf (15015) | Animalia (205242), animalia (261) | Biotechnology (792223), biotechnology (23978) empty dc:subject | 0 | 38296 | 15436 | 705730 | 0 In a second analysis, we investigated which kinds of entity occur in descriptive metadata fields. As the processing of textual resources with NLP tools is time-consuming and resource-intensive, we selected a subset of datasets. We limited the amount to 10,000 datasets per repository. Table 10 presents the filter strategies. For _PANGAEA_ and _GBIF_ , we randomly selected 10,000 datasets as they are domain-specific repositories for which all data are potentially relevant for biodiversity research. For _Dryad_ , the filter consists of a group of relevant keywords, and for _Zenodo_ and _Figshare_ we used the keyword ‘Biodiversity’. Due to the large amount of collection data with the keyword ‘Biodiversity’, we are aware that this filter strategy might have led to a certain bias in the selected data. Table 10: Filter strategies used per data repository to select 10,000 datasets. The number in brackets denotes the total number of available datasets in OAI-DC format at the time of download (October/November 2019). | Pangaea | GBIF | Dryad | Zenodo | Figshare ---|---|---|---|---|--- filter strategy | 10000 randomly selected (388254) | 10000 randomly selected GBIF (46954) | 10000 randomly with keywords: biodiversity, climate change, ecology, insects, species richness, invasive species, herbivory, pollination, endangered species, ecosystem functioning, birds (149672) | 10000 randomly with keyword: Biodiversity (1467958) | 10000 randomly with keyword: Biodiversity (3602808) Per data repository, we processed the selected 10,000 files with two open- source taggers of the text mining framework GATE [Cunningham et al., 2011]. Named Entities such as Person, Organization and Location were obtained with the ANNIE pipeline [Cunningham et al., 2002], and Organisms were obtained from the OrganismTagger [Naderi et al., 2011a]. The results are presented in Table 11. Unfortunately, the OrganismTagger pipeline aborted for _PANGAEA_ and _Zenodo_ , but in around 12% _GBIF_ files, 36% _Dryad_ files and 85% _Figshare_ files ‘Organism’ annotations were created. Probably, the number of ‘Organism’ annotations in _Figshare_ files is that high due to the mentioned bias towards collection data. The number of ‘Organism’ annotations in _GBIF_ files is low since datasets mostly describe the overall study and do not contain concrete species names but rather broader taxonomic terms such as ‘Family’ or ‘Order’. A large number of ‘Location’ annotations were extracted for files from _PANGAEA_ (91%) and _Figshare_ ( 100%). ‘Person’ and ‘Organization’ annotations are largely presented in _PANGAEA_ ( 51%) and _GBIF_ files ( 74%). The text mining pipelines were originally developed and evaluated with text corpora and not sparse datasets. Hence, the results might contain wrong (false positive) annotations. However, the results indicate that NLP tools can support the identification of biological entities. That could be an approach for generalist repositories to additionally enrich metadata. All scripts and the final results are available in our repository. Table 11: NLP analysis: Number of datasets with Named Entities (out of 10,000 processed files in a reduced OAI-DC schema) per repository. Each file contains a subset of the original metadata, namely, dc:title, dc:description, dc:subject and dc:date. | Pangaea | GBIF | Dryad | Zenodo | Figshare ---|---|---|---|---|--- Organism | N/A (pipeline aborted) | 1183 | 3603 | N/A (pipeline aborted) | 8542 Location | 9111 | 5642 | 3530 | 4641 | 9978 Person & Organization | 5048 | 7355 | 657 | 192 | 1645 ## D - Discussion In this study, we explored what hampers dataset retrieval in biodiversity research. The following section summarizes our findings and outlines a proposal on how to bridge the gap between search interests in biodiversity and given metadata. We also highlight challenges that are not fully resolved yet. ### Research Contributions #### Scholarly Search Interests in Biodiversity Research In order to understand what biodiversity scholars are interested in, we gathered 169 questions, identified biological entities and classified the entities in 13 information categories. In the subsequent evaluation with domain experts, five categories were verified and can be considered as important information needs in biodiversity research. That includes information about habitats, ecosystems, vegetation (ENVIRONMENT), chemical compounds, sediments and rocks (MATERIAL), species (ORGANISM), biological and chemical processes (PROCESS). Further categories being mentioned very often are information about data parameters (QUALITY) and the nature or type of data resources (DATA TYPE). Usually, the latter is an outcome of a certain research method. However, the naming should be discussed in the research community as the comprehensibility of these categories were fair, only. #### Comparison of Metadata Standards and User Interests We selected 13 metadata standards used in the Life Sciences from _re3data_ , and we analyzed whether the elements of the metadata schemes reflect the identified information categories. Elements of general standards cover the categories to some extent, only. LOCATION and DATA TYPE are the sole information that can be explicitly described with metadata fields of general standards such $DublinCore$ or $DataCite$. Further elements are focused on information less relevant for search such as data creator, contributor (PERSON), collection or publication data (TIME), and license information. All this information is important for data reuse and data citation and needs to be part of the metadata. However, if the dataset is not findable, it can not be cited. As a general standard, $DataCite$ provides many more fields and attributes to describe personal data, time and geographic information. Therefore, it should be provided in addition to $DublinCore$. There are numerous discipline-specific standards that describe search interests quite well. For instance, $EML$, $DarwinCore$ and $ABCD$ provide elements to describe environmental information, species, methods, and data parameters. _ISA-Tab_ , a framework for genome data and biological experiments covers all important search categories. The only drawback is that it takes time for scholars and/or data curators to fill in all these fields. In some standards such as $ABCD$ more than 1000 elements are available. With our work, we aim to provide insights on what scholars are actually interested in when looking for scientific data. We believe that our results could serve as a good start for discussions in the respective research communities to define core elements of discipline-specific standards that are not only focused on data citation but also broaden the view on search interests. #### Metadata Analysis of Selected Data Repositories We selected 5 repositories from _Nature’s_ list of recommended data archives and analyzed the metadata provided in their OAI-PMH interfaces. We wanted to know what metadata standards are used in common data repositories in the Life Sciences and how many elements of the standard are actually filled. We figured that generalist repositories such as $Dryad$, $Zenodo$ and $Figshare$ tend to use only general standards such as $DublinCore$ and $DataCite$. Even when using simple standards the repositories did not fully use all provided elements. Furthermore, the ones utilized are not always filled. That hampers successful data retrieval. Most repositories seem to be aware of that problem and enhance metadata with numerous keywords in generic fields such as dc:subject. Discipline-specific repositories, e.g., $GBIF$ and $PANGAEA$ are more likely to provide domain-related standards such as $EML$ or _Pan-MD_. That supports an improved filtering in search, however, it does not guarantee that the fields are always filled. In _GBIF’s_ case, we are aware that we could not provide a full picture as we did not analyze the occurrence records. Here, only a deeper analysis of the provided fields in the search index would deliver more answers. However, that would require technical staff support as the access to search indices is limited. ### Suggestions to Bridge the Gap In this subsection, we outline approaches to overcome the current obstacles in dataset search applications based on our findings from the preceding sections. Table 12 presents checklists for data repositories and scholars that in the following are discussed in detail. Table 12: Recommendations for data repositories and scholars to create rich metadata For Data Repositories | For Scholars ---|--- 1.) Keep metadata diversity by offering domain-specific standards. | 1.) If available, select a domain-specific repository where possible (as recommended by _Nature_). 2.) Use metadata standards that include the information categories identified in Section A - Information Needs to cover potential search interests and make your data findable. | 2.) Check if appropriate, discipline-specific metadata standards are offered to describe your data. 3.) Extend existing standards where necessary, preferably get in touch with metadata standard consortia. | 3.) Fill in all applicable metadata fields and use appropriate terms (if possible from controlled vocabularies). 4.) Fill in all metadata fields. If possible use controlled vocabularies to describe the data, preferably Linked Open Data. | 4.) If your data is available in search, check if you can find it with various search terms. 5.) Enrich your metadata with entities from _schema.org_ or _bioschemas.org_. | 5.) Contact the data repository if you notice issues in your data description or presentation. 6.) In addition to explicit information, attempt to extract implicit information from metadata fields that contain longer textual resources, e.g., title, description and abstract. | #### For Data Repositories Adherence to the FAIR principles, long-term data preservation and the creation of citable and reusable data are main targets of all data repositories. Therefore, a strong focus of data archives is on generating unique identifiers, linking the metadata to their primary data and publications. Less considered is the perspective of dataset seekers. Hence, we propose the following improvements to enhance dataset retrieval. _Keep metadata diversity_ : Scientific data are very heterogeneous. This diversity can not be reflected in one generic metadata standard. Thus, it is highly recommended to use different domain-specific standards considering the requirements from various research disciplines. _Use proper metadata fields_ : If search interests are explicitly mentioned in metadata, conventional search techniques are able to retrieve relevant datasets. Providing possible search terms in generic keyword fields supports dataset retrieval in a full-text search but does not allow proper category- based facet creation. Therefore, using proper metadata fields covering potential search interests greatly enhances dataset retrieval and filtering. In addition, metadata need to have a unique identifier and should answer the W-questions including information on how the data can be re-used. That comprises information on data owner, contact details and citation information. _Extend standards_ : Metadata standards are developed and adopted from large organizations or research communities for a specific purpose or research fields. They also discuss extensions of new fields or changes of existing elements. If the given fields are not sufficient for particular requirements, the preferred way is to get in touch with the standard organization and to propose new fields or attributes. However, since these processes usually take a long time, it is sometimes unavoidable to extend a schema or to develop a new schema. In these cases, it would be a good scientific practice to give feedback to the standard organization why and how a schema has been changed or extended. That might influence the further development of standards and would counteract the creation of numerous repository-developed schemes. _Use controlled vocabularies_ : The questions that still remain and that have not been considered so far are how metadata fields are filled - by the data submitter, the data repository or by the system - and whether a controlled vocabulary is used for the keywords and the other metadata elements. When describing scientific data it is highly recommended to use controlled vocabularies or terminologies, in particular for important fields in search. If possible, Linked Open Data [Heath and Bizer, 2011] vocabularies should be utilized to better link datasets, publications, authors and other resources. That supports data transparency, data findability and finally data reuse. In the Life Sciences, there are a variety of terminology providers. We provide a list of ontology and semantic service providers in our repository. _Utilize schema.org_ : Driven by _Google_ and the Research Data Alliance (RDA) Data Discovery Group, the enrichment of HTML with _schema.org_ (https://schema.org) entities became very popular in recent years. The enrichment helps to identify unique identifiers, persons, locations or time information in the HTML file. That supports external search engines or data providers to crawl the landing pages of search applications provided by the data repositories per dataset. As the current schema.org entities do not fully reflect scientific search interests, more attention should be paid to initiatives such as _bioschemas.org_ (https://bioschemas.org/) that aims to expand schema.org on biological entities such as species and genes. That confirms and complements our recommendations for explicit metadata fields tailored to search interests. At the time of writing this paper, _bioschemas.org_ is still in draft mode. However, in the future, efforts like this will improve dataset retrieval significantly. _Extract implicit information_ : Apart from short information such as contact details, data type or location, metadata usually contain longer textual resources such as title, description and abstract. Most of them contain useful information for search and mention species observed or describe environments where data has been gathered. These resources could be used to extract implicit information and to automatically identify further relevant data. #### For Scholars Documenting scientific data is a disliked task that also takes time. Therefore, scholars attempt to minimize the effort on describing their data and are pleased when data repositories offer not too many fields to fill in for data submission. However, scholars are responsible to properly document their data so that other researchers are able to find and reuse it. Hence, each scholar should carefully and thoroughly describe the produced research data. Based on our findings, we summarize what should be considered when submitting scientific data to a data repository. _Prefer domain-specific repositories:_ As generalist repositories tend to offer only general metadata standards for data description, preference should be given to domain-specific data archives. This is also recommend by highly influential journals such as _Nature_ [Nature, 2018]. Another advantage is that repositories that are familiar with the research domain might give more qualitative feedback on the submitted data descriptions. _Use domain-specific metadata standards:_ Even when selecting a domain- specific data repository, it does not guarantee that archives use proper metadata standards. Scholars are advised to know at least a few appropriate standards for their research field and to ask the repository if one of these standards are supported if not stated anywhere. _Fill in all relevant fields with controlled vocabularies:_ All relevant metadata fields should be filled in. That enhances the chance that datasets are retrieved. When describing the data, scholars should attempt to use controlled vocabularies. As this is a new procedure in data submission, it is currently not supported by all data repositories. However, if it is available, it is recommended to use the terminologies given and not to describe the data with one’s own words. _Search for your data:_ Once the data is available in the repositories’ search application, scholars are advised to check if they can find their data with various search terms. They should also review whether the data are accessible and all displayed information are correct. It is also recommended to repeat this checking from time to time as repositories might update or extend data presentations and/or metadata schemes used. _Get in touch with the repository:_ If scholars notice anything concerning their data, they should contact the archive. The staff at the repositories are probably grateful if attentive scholars give feedback on their submitted data or detect issues that hampers dataset retrieval. ### Challenges As stated in our summary of the question analysis, the outcomes in Section “A - Information Needs in the Biodiversity Domain” are not a complete picture of search interests but only serve as a start for discussions with biodiversity researchers to further identify possible search categories. Controlled vocabularies can only be used if appropriate terminologies exist. This is not the case for all topics. While there are numerous vocabularies for species, to the best of our knowledge, there is no vocabulary that allow the description of research methods and results. Scientific data types are also less considered in existing terminologies. Another challenge lies in the automatic identification of relevant search topics in metadata. The text mining community has already developed various taggers and pipelines to extract organisms [Naderi et al., 2011b], chemistry items [Cunningham et al., 2011] or genes [McDonald and Pereira, 2005] from text. These annotations can support automatic facet or category creation. However, for important categories such as habitats, data parameters, biological and chemical processes or research methods, taggers are still missing. In order to increase semantic linkage of datasets and other resources such as publications, authors and locations, it would be a great benefit if the annotations also contain URIs to resources in controlled vocabularies. Then, dataset retrieval could be expanded on semantically related terms such as synonyms or more specific or broader terms. An important point, however, are not standards, systems or vocabularies, but scholars themselves. Scholars need to be aware that thorough data descriptions are part of a good scientific practice. In order to preserve all kind of scientific data, independently of whether it has been used in publications or not, proper metadata in appropriate schemes are the key to successful dataset retrieval and thus, to data citation and data reuse. Data Repositories could offer data curation services to support scholars in describing research data and to encourage them to describe their data thoroughly. We are aware that it would require high efforts to introduce more domain-specific metadata schemes at generalist repositories; however, it would enhance dataset retrieval. Computer science research can contribute to improvements for dataset search by developing methods and software tools that facilitate standard-compliant metadata provision ideally at the time of data collection, thus ensuring metadata standards to be actually used by data providers. ## Conclusion Scholarly search interests are as diverse as data are and can range from specific information needs such as searches for soil samples collected in a certain environment to broader research questions inspecting relationships among species. Our findings reveal that these search interests are not entirely reflected in existing metadata. One problem are general standards that are simple and mainly contain information that support data citation. Actual search interests can only be represented if keywords and suitable search terms are provided in general, non-specific fields that are provided in most standards, e.g., dc:subject in _DublinCore_. Most data repositories utilize these fields to enrich metadata with suitable search terms. However, if search interests are not explicitly given, facet creation, e.g., filtering over species or habitats, is more difficult. Full-text searches only return data if query terms match given keywords. On the other hand, even when scholars submit their data to a domain-specific repository that uses discipline-specific metadata standards, it does not guarantee that all search- relevant fields will be filled. Data findability, one of the four FAIR principles [Wilkinson et al., 2016], at least partially relies on rich metadata descriptions reflecting scholarly information needs. If the information scholars are interested in is not available in metadata, the primary data can not be retrieved, reused and cited. In order to close this gap, we propose checklists for data archives and scholars to overcome the current obstacles. We also highlight remaining challenges. 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2024-09-04T02:54:55.023333

2002.12059

arxiv-papers

# Quantum-heat fluctuation relations in $3$-level systems under projective measurements G. Giachetti<EMAIL_ADDRESS>SISSA, Via Bonomea 265, I-34136 Trieste, Italy INFN, Sezione di Trieste, I-34151 Trieste, Italy S. Gherardini <EMAIL_ADDRESS>SISSA, Via Bonomea 265, I-34136 Trieste, Italy Department of Physics and Astronomy & LENS, University of Florence, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy A. Trombettoni<EMAIL_ADDRESS>Department of Physics, University of Trieste, Strada Costiera 11, I-34151 Trieste, Italy CNR-IOM DEMOCRITOS Simulation Center and SISSA, Via Bonomea 265, I-34136 Trieste, Italy S. Ruffo<EMAIL_ADDRESS>SISSA, Via Bonomea 265, I-34136 Trieste, Italy INFN, Sezione di Trieste, I-34151 Trieste, Italy Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy ###### Abstract We study the statistics of energy fluctuations in a three-level quantum system subject to a sequence of projective quantum measurements. We check that, as expected, the quantum Jarzynski equality holds provided that the initial state is thermal. The latter condition is trivially satisfied for two-level systems, while this is generally no longer true for $N$-level systems, with $N>2$. Focusing on three-level systems, we discuss the occurrence of a unique energy scale factor $\beta_{\rm eff}$ that formally plays the role of an effective inverse temperature in the Jarzynski equality. To this aim, we introduce a suitable parametrization of the initial state in terms of a thermal and a non- thermal component. We determine the value of $\beta_{\rm eff}$ for a large number of measurements and study its dependence on the initial state. Our predictions could be checked experimentally in quantum optics. ## I Introduction Fluctuation theorems relate fluctuations of thermodynamic quantities of a given system to equilibrium properties evaluated at the steady state Esposito2009 ; Campisi2011 ; SeifertRPP2012 ; DeffnerBook2019 . This statement finds fulfillment in the Jarzynski equality, whose validity has been extensively theoretically and experimentally discussed in the last two decades for both classical and quantum systems JarzynskiPRL1997 ; CrooksPRE1999 ; CollinNature2005 ; ToyabeNatPhys2010 ; Kafri2012 ; Albash13PRE88 ; Rastegin13JSTAT13 ; Sagawa2014 ; AnNatPhys2015 ; BatalhaoPRL2014 ; CerisolaNatComm2017 ; BartolottaPRX2018 ; Hernandez2019 . The evaluation of the relevant work originated by using a coherent modulation of the system Hamiltonian has been the subject of intense investigation TalknerPRE2007 ; CampisiPRL2009 ; MazzolaPRL2013 ; AllahverdyanPRE2014 ; TalknerPRE2016 ; JaramilloPRE2017 ; DengENTROPY2017 . A special focus was devoted to the study of heat and entropy production, obeying of the second law of thermodynamics, the interaction with one or more external bodies, and/or the inclusion of an observer JarzynskiPRL2004 ; Campisi15NJP17 ; Campisi17NJP19 ; BatalhaoPRL ; Gherardini_entropy ; ManzanoPRX ; Irreversibility_chapter ; SantosnpjQI2019 ; KwonPRX2019 ; RodriguesPRL2019 . In this respect, the energy variation and the emission/absorption of heat induced—and sometimes enhanced—by the application of a sequence of quantum measurements was studied Campisi2010PRL ; CampisiPRE2011 ; Yi2013 ; WatanabePRE2014 ; HekkingPRL2013 ; AlonsoPRL2016 ; GherardiniPRE2018 ; Hernandez2019 . In such a case, the term quantum heat has been used Elouard2016 ; GherardiniPRE2018 ; we will employ it as well in the following to refer to the fact that the fluctuations of energy exchange are induced by quantum projective measurements performed during the time evolution of the system. As recently discussed in Ref. Hernandez2019 , the information about the fluctuations of energy exchanges between a quantum system and an external environment may be enclosed in an energy scaling parameter that only depends on the initial and the asymptotic (for long times) quantum states resulting from the system dynamics. In Ref. GherardiniPRE2018 , the effect of stochastic fluctuations on the distribution of the energy exchanged by a quantum two-level system with an external environment under sequences of quantum measurements was characterized and the corresponding quantum-heat probability density function was derived. It has been shown that, when a stochastic protocol of measurements is applied, the quantum Jarzynski equality is obeyed. In this way, the quantum-heat transfer was characterized for two-level systems subject to projective measurements. Two-level systems have the property that a density matrix in the energy basis (as the one obtained after a measurement of the Hamiltonian operator TalknerPRE2007 ) can be always written as a thermal state and, therefore, the Jarzynski equality has a $1$ on its right-hand side. Therefore, a natural issue to be investigated is the study of quantum-heat fluctuation relations for $N$-level systems, e.g., $N=3$, where this property of the initial state of a two-points measurement scheme of being thermal is no longer valid. So, it would be desirable, particularly for the case of a large number of quantum measurements, to study the properties of the characteristic function of the quantum heat when initial states cannot be written as thermal states. With the goal of characterizing the effects of having arbitrary initial conditions, in this paper, we study quantum systems described by finite dimensional Hilbert spaces, focusing on the case of three-level systems. We observe that finite-level quantum systems may present peculiar features with respect to continuum systems. As shown in Ref. JaramilloPRE2017 , even when the quantum Jarzynski equality holds and the average of the exponential of the work equals the free energy difference, the variance of the energy difference may diverge for continuum systems, an exception being provided by finite-level quantum systems. In this paper we analyze, using numerical simulations, (i) the distribution of the quantum heat originated by a three-level system under a sequence of $M$ projective measurements in the limit of a large $M$, and (ii) the behavior of an energy parameter $\beta_{\rm eff}$, such that the Jarzynski equality has $1$ on its right-hand side, always in the limit of a large $M$. We also discuss the dependence of $\beta_{\rm eff}$ on the initial state, before the application of the sequence of measurements. ## II The Protocol Let us consider a quantum system described by a finite dimensional Hilbert space. We denote with $H$ the time-independent Hamiltonian of the system that admits the spectral decomposition $H=\sum^{N}_{k=1}E_{k}\lvert E_{k}\rangle\\!\langle E_{k}\rvert\ ,$ (1) where $N$ is the dimension of the Hilbert space. We assume that no degeneration occurs in the eigenstates of $H$. At time $t=0^{-}$, just before the first measurement of $H$ is performed, the system is supposed to be in an arbitrary quantum state described by the density matrix $\rho_{0}$ s.t. $[H,\rho_{0}]=0$. This allows us to write $\rho_{0}=\sum^{N}_{k=1}c_{k}\lvert E_{k}\rangle\\!\langle E_{k}\rvert\ ,$ (2) where $1\geq c_{k}\geq 0\ \forall k=1,\dots,N$ and $\sum^{N}_{k}c_{k}=1$. Then, we assume that the fluctuations of the energy variations, induced by a given transformation of the state of the system, are evaluated by means of the so-called two-point measurement (TPM) scheme TalknerPRE2007 . According to this scheme, a quantum projective measurement of the system hamiltonian is performed both at the initial and the final times of the transformation. This hypothesis justifies the initialization of the system in a mixed state, as given in Equation (2). By performing a first projective energy measurement, at time $t=0^{+}$, the system is in one of the states $\rho_{n}=\lvert E_{n}\rangle\\!\langle E_{n}\rvert$ with probability $p_{n}=\langle E_{n}\rvert\rho_{0}\lvert E_{n}\rangle$, while the system energy is $E_{n}$. Afterwards, we suppose that the system $S$ is subject to a number $M$ of consecutive projective measurements of the generic observable $O=\sum^{N}_{k=1}\Omega_{k}\lvert\Omega_{k}\rangle\\!\langle\Omega_{k}\rvert\ ,$ (3) where $\Omega_{k}$ and $\lvert\Omega_{k}\rangle$ denote, respectively, the outcomes and the eigenstates of $O$. According to the postulates of quantum mechanics, the state of the system after one of these projective measurements is given by one of the projectors $\lvert\Omega_{n}\rangle\\!\langle\Omega_{n}\rvert$. Between two consecutive measurements, the system evolves with the unitary dynamics generated by $H$, i.e., $U(\tau_{i})=e^{-iH\tau_{i}}$, where $\hbar$ has been set to unity and the waiting time $\tau_{i}$ is the time difference between the $(i-1)^{\text{th}}$ and the $i^{\text{th}}$ measurement of $O$. In general, the waiting times $\tau_{i}$ can be random variables, and the sequence $(\tau_{1},\ldots,\tau_{M})$ is distributed according to the joint probability density function $p(\tau_{1},\ldots,\tau_{M})$. The last (i.e, the $M^{\text{th}}$) measurement of $O$ is immediately followed by a second projective measurement of the energy, as prescribed by the TPM scheme. By denoting with $E_{m}$ the outcome resulting from the second energy measurement of the scheme, the final state of the system is $\rho_{m}=\lvert E_{m}\rangle\\!\langle E_{m}\rvert$ and the quantum heat $Q$ exchanged during the transformation is thus given by $Q=E_{m}-E_{n}\ .$ (4) As $Q$ is a random variable, one can define the characteristic function $G(\epsilon)\equiv\left\langle e^{-\epsilon Q}\right\rangle=\sum_{m,n}p_{m|n}p_{n}e^{-\epsilon(E_{m}-E_{n})}\ ,$ (5) where $p_{m|n}$ denotes the probability of obtaining $E_{m}$ at the end of the protocol conditioned to have measured $E_{n}$ at the first energy measurement of the TPM scheme. If the initial state is thermal, i.e., $\rho_{0}=e^{-\beta H}/Z$, then one recovers the Jarzynski equality stating that $G(\beta)=\left\langle e^{-\beta Q}\right\rangle=1\ .$ (6) Let us notice that $G(\epsilon)$ is a convex function such that $G(0)=1$ and $G(\pm\infty)\rightarrow+\infty$, as discussed in Ref. Hernandez2019 . Hence, as long as $\frac{\partial G}{\partial\epsilon}(0)\neq 0$, one can unambiguously introduce the parameter $\beta_{\rm eff}\neq 0$ defined by the relation $G(\beta_{\rm eff})=1\,,$ (7) which formally plays the role of an effective inverse temperature. Focusing on three-level systems, in the following, we will study the characteristic function $G(\epsilon)$ and the properties of such a parameter $\beta_{\rm eff}$. For comparison, we first pause in the next subsection to discuss what happens for two-level systems. ### II.1 Intermezzo on Two-Level Quantum Systems We pause here to remind the reader of the results for two-level systems. The state of any two-level system, diagonal on the Hamiltonian basis, is a thermal state for some value of $\beta$. Of course, if the state is thermal, the value of $\beta_{\rm eff}$ trivially coincides with $\beta$. In particular, in Ref. GherardiniPRE2018 , the energy exchanged between a two-level quantum system and a measurement apparatus was analyzed, with the assumption that the repeated interaction with the measurement device can be reliably modeled by a sequence of projective measurements occurring instantly and at random times. Numerically, it has been observed that, as compared with the case of measurements occurring at fixed times, the two-level system exchanges more energy in the presence of randomness when the average time between consecutive measurements is sufficiently small in comparison with the inverse resonance frequency. However, the quantum-heat Jarzynski equality, related to the equilibrium properties of the transformation applied to the system, is still obeyed, as well as when the waiting times between consecutive measurements are randomly distributed and for each random realization. These results are theoretically supported by the fact that, in the analyzed case, the dynamical evolution of the quantum system is unital Rastegin13JSTAT13 ; Sagawa2014 . A discussion on the values of the parameter $\beta_{\rm eff}$, extracted from experimental data for nitrogen-vacancy (NV) centers in diamonds subject to projective measurements in a regime where an effective two-level approximation is valid was recently presented in Ref. Hernandez2019 . In Figure 1, we plot the quantum-heat characteristic function $\langle e^{-\beta Q}\rangle$ as a function of the parameter $c_{1}$ that appears in the decomposition of the initial state $\rho_{0}$ with respect to the energy eigenstates $|E_{1}\rangle$ and $|E_{2}\rangle$ Figure 1: Quantum-heat characteristic function $\langle e^{-\beta Q}\rangle$ for a two-level quantum system as a function of $c_{1}$ in Equation (8) for three values of $|a|^{2}$, which characterizes the initial state. The function is obtained from numerical simulations performed for a system with a Hamiltonian $H=J(\lvert 0\rangle\\!\langle 1\rvert+\lvert 1\rangle\\!\langle 0\rvert)$ subject to a sequence of $M=5$ projective measurements. The latter are separated by a fixed waiting time $\tau=0.5$, averaged over $2000$ realizations, with $E_{1,2}=\pm 1$ and $\beta=3/2$ (notice that the same values of $|a|^{2}$ are used in Fig. 1 of GherardiniPRE2018 and in the corresponding caption the value of $\beta$ should read $\beta=3/2$). Units are used with $\hbar=1$ and $J=1$. $\rho_{0}=c_{1}\lvert E_{1}\rangle\\!\langle E_{1}\rvert+c_{2}\lvert E_{2}\rangle\\!\langle E_{2}\rvert\ ,$ (8) where $c_{2}=1-c_{1}$. The function is plotted for three values of the parameter $|a|^{2}$, used to parametrize the eigenstates $\\{|\Omega_{1}\rangle,|\Omega_{2}\rangle\\}$ of $O$ as a function of the energy eigenstates of the system, i.e., $|\Omega_{1}\rangle=a\lvert E_{1}\rangle-b\lvert E_{2}\rangle\,\,\,\,\,\text{and}\,\,\,\,\,|\Omega_{2}\rangle=b\lvert E_{1}\rangle+a\lvert E_{2}\rangle\ ,$ (9) with $|a|^{2}+|b|^{2}=1$ and $a^{\ast}b=ab^{\ast}$. As a result, one can observe that $\langle e^{-\beta Q}\rangle=1$ for the value of $c_{1}$ ensuring that $\rho_{0}=e^{-\beta H}/Z$. Further details can be found in Ref. GherardiniPRE2018 . The $N>2$ case is trickier: Since, in general, the initial state is no longer thermal, it is not trivial to determine the value of $\beta_{\rm eff}$, and its dependence on the initial condition is interesting to investigate. In this regard, in the following, we will numerically address the $N=3$ case by providing results on the asymptotic behavior of the system in the limit $M\gg 1$. For the sake of simplicity, from here on, we assume that the values of the waiting times $\tau_{i}$ are fixed for any $i=1,\ldots,M$. However, our findings turn out to be the same for any choice of the marginal probability distribution functions $p(\tau_{i})$, up to “pathological” cases such as $p(\tau)=\delta(\tau)$. So, having random waiting times does not significantly alter the scenario emerging from the presented results. ## III Parametrization of the Initial State In this paragraph, we introduce a parametrization of the initial state $\rho_{0}$ for the $N=3$ case, which will be useful for a thermodynamic analysis of the system. As previously recounted, for a two-level quantum system in a mixed state, as given by Equation (2), it is always formally possible to define a temperature. In particular, one can implicitly find an effective inverse temperature, making $\rho_{0}$ a thermal state, by solving the following equation for $\beta\in\mathbb{R}$ $e^{-\beta(E_{2}-E_{1})}=\frac{c_{2}}{c_{1}}\ .$ (10) In the $N=3$ case, however, we have two independent parameters in (2) (the third parameter indeed enforces the condition $\text{Tr}[\rho_{0}]=1$), and thus, in general, it is no longer possible to formally define a single temperature for the state. Here, we propose the following parametrization of $c_{1},c_{2},c_{3}$ that generalizes the one of Equation (10). We denote as _partial_ effective temperatures the three parameters $b_{1},b_{2},b_{3}$, defined through the ratios of $c_{1}$, $c_{2}$, and $c_{3}$ $\frac{c_{2}}{c_{1}}=e^{-b_{1}(E_{2}-E_{1})},\hskip 28.45274pt\frac{c_{3}}{c_{2}}=e^{-b_{2}(E_{3}-E_{2})},\hskip 28.45274pt\frac{c_{1}}{c_{3}}=e^{-b_{3}(E_{1}-E_{3})}\ ,$ (11) such that, for a thermal state, $b_{k}=\beta$, $\forall k=1,2,3$. The three parameters are not independent, as they are constrained by the relation $\frac{c_{2}}{c_{1}}\frac{c_{3}}{c_{2}}\frac{c_{1}}{c_{3}}=1\ ,$ (12) which gives in turn the following equality $b_{1}(E_{2}-E_{1})+b_{2}(E_{3}-E_{2})+b_{3}(E_{1}-E_{3})=0\ .$ (13) By introducing $\Delta_{1}=E_{2}-E_{1}$, $\Delta_{2}=E_{3}-E_{2}$, and $\Delta_{3}=E_{1}-E_{3}$, Equation (13) can be written as $\sum^{3}_{k=1}b_{k}\Delta_{k}=0\ ,$ (14) where by definition $\sum^{3}_{k=1}\Delta_{k}=0\ .$ (15) Thus, as expected, the thermal state is a solution of the condition (14) for any choice of $E_{1},E_{2},E_{3}$. This has also a geometric interpretation. In the space of the $\Delta_{k}$, $k=1,2,3$, Equation (14) becomes an orthogonality condition between the $\Delta_{k}$ and the $b_{k}$ vectors, while (15) defines a plane that is orthogonal to the vector $(1,1,1)$. When $b_{k}$ is proportional to $(1,1,1)$, the orthogonality condition is automatically satisfied and one finds a thermal state. This suggests that, in general, one can conveniently parametrize $b_{k}$ in terms of both the components that are orthogonal and parallel to the plane $\sum^{3}_{k=1}\Delta_{k}=0$. Such terms have the physical meaning of the thermal and non-thermal components of the initial state. Formally, this means that we can parametrize each $b_{k}$ through the fictitious inverse temperature $\beta$ and a deviation $\alpha$, i.e., $(b_{1},b_{2},b_{3})=\beta(1,1,1)+\frac{\alpha}{v}(\Delta_{3}-\Delta_{2},\,\Delta_{1}-\Delta_{3},\,\Delta_{2}-\Delta_{1})\ ,$ (16) where $v$ acts as a normalization constant $v^{2}=3\left(\Delta_{1}^{2}+\Delta_{2}^{2}+\Delta_{3}^{2}\right)\ .$ (17) Hence, taking into account the normalization constraint, the coefficients $c_{k}$ are given by $c_{1}=\frac{1}{1+e^{-b_{1}\Delta_{1}}+e^{b_{3}\Delta_{3}}},\hskip 28.45274ptc_{2}=\frac{1}{1+e^{-b_{2}\Delta_{2}}+e^{b_{1}\Delta_{1}}},\hskip 28.45274ptc_{3}=\frac{1}{1+e^{-b_{3}\Delta_{3}}+e^{b_{2}\Delta_{2}}},$ (18) or, in terms of the parameters $\alpha$ and $\beta$, $c_{1}=\frac{1}{\tilde{Z}}\exp\left[-\beta E_{1}+\frac{\alpha}{v}(E_{2}-E_{3})^{2}\right],\hskip 11.38092ptc_{2}=\frac{1}{\tilde{Z}}\exp\left[-\beta E_{2}+\frac{\alpha}{v}(E_{3}-E_{1})^{2}\right],\hskip 11.38092ptc_{3}=\frac{1}{\tilde{Z}}\exp\left[-\beta E_{3}+\frac{\alpha}{v}(E_{1}-E_{2})^{2}\right],$ (19) where $\tilde{Z}$ is a pseudo-partition function ensuring the normalization of the $c_{k}$’s $\tilde{Z}=\tilde{Z}(\alpha,\beta)\equiv e^{-\beta E_{1}+\frac{\alpha}{v}(E_{2}-E_{3})^{2}}+e^{-\beta E_{2}+\frac{\alpha}{v}(E_{3}-E_{1})^{2}}+e^{-\beta E_{3}+\frac{\alpha}{v}(E_{1}-E_{2})^{2}}\ .$ (20) Let us provide some physical intuition about the parameters $\alpha$ and $\beta$: For $\alpha=0$, we recover a thermal state, whereby $c_{1}>c_{2}>c_{3}$ if $\beta>0$, or vice versa if $\beta<0$. On the other hand, the non-thermal component $\alpha$ can be used to obtain a non-monotonic behavior of the coefficients $c_{k}$. For example, for $\beta=0$, since $(E_{3}-E_{1})^{2}$ is greater than both $(E_{3}-E_{2})^{2}$ and $(E_{1}-E_{2})^{2}$, one finds that $c_{2}>(<)\,c_{1},c_{3}$ if $\alpha>(<)\,0$. As a final remark, it is worth noting that we can reduce the dimension of the space of the parameters. In particular, without loss of generality, one can choose the zero of the energy by taking $E_{2}=0$ (and then $E_{3}>0$, $E_{1}<0$), or we can reduce our analysis to the cases with $\beta>0$. As a matter of fact, the parametrization is left unchanged by the transformation $\\{\beta\rightarrow-\beta,\,E_{k}\rightarrow-E_{k}\\}$, with the result that the case of $\beta<0$ can be explored by simply considering $\beta>0$ in the fictitious system with $E^{\prime}_{k}=-E_{k}$ (here, the choice of $E_{2}=0$ guarantees that this second case can be simply obtained by substituting $E_{1}$ with $E_{3}$). ## IV Large $M$ Limit Here, we numerically investigate the behavior of a three-level system subject to a sequence of $M$ projective quantum measurements with a large $M$ (asymptotic limit) and where $\tau$ is not infinitesimal. From here on, we adopt the language of spin-$1$ systems, and we thus identify $O$ with $S_{z}$. In the asymptotic limit, the behavior of the system is expected not to depend on the choice of the evolution Hamiltonian, with the exception that at least one of the eigenstates of $S_{z}$ is also an energy eigenstate. In such a case, indeed, if the energy outcome corresponding to the common eigenstate is obtained by the first measurement of the TPM scheme, then the evolution is trivially deterministic, as the system is locked in the measured eigenstate. Choosing a generic observable (with no eigenstates in common with $H$), numerical simulations (cf. Figure 2) suggest that our protocol leads the system to the completely uniform state. The latter can be interpreted as a canonical state with $\beta=0$ (notice that this result holds in the situation we are analyzing, with a finite dimensional Hilbert space). The system evolves with Hamiltonian $H=\omega_{1}S_{z}+\omega_{2}S_{x}$, where the energy units are chosen such that $\omega_{1}=1$ and $\omega_{2}=\frac{1}{2}$. It is initialized in the state $\rho_{0}$ with $\\{c_{1}=0.8,c_{2}=0.01,c_{3}=0.19\\}$, corresponding to $\alpha\approx-2,32$ and $\beta\approx 1,96$, and we performed $M=20$ projective measurements of the observable $O=S_{z}$ separated by the time $\tau=1$. Figure 2: Histogram of the initial (dashed) and final (solid) energy outcomes for the TPM scheme described in the text performed over $3\cdot 10^{5}$ realizations. While the initial state is non-uniform, the final state is practically uniform over the three energy levels. This numerical finding allows us to derive an analytic expression of the quantum-heat characteristic function. In this regard, as the final state is independent of the initial one for a large $M$, the joint probability of obtaining $E_{n}$ and $E_{m}$, after the first and the second energy measurement respectively, is equal to $p_{mn}=\frac{1}{3}c_{m}\ .$ (21) Hence, $G(\epsilon)=\left\langle e^{-\epsilon Q}\right\rangle=\frac{1}{3}\sum^{3}_{m,n=1}c_{m}e^{-\epsilon(E_{m}-E_{n})}=\frac{1}{3}\sum_{n=1}^{3}e^{-\epsilon E_{n}}\sum_{m=1}^{3}c_{m}e^{\epsilon E_{m}}\ .$ (22) As a consequence, $G$ can be expressed in terms of the partition function $Z(\beta)$ and of the pseudo-partition function introduced in Equation $\eqref{pseudoZ}$, i.e. $G(\epsilon;\alpha,\beta)=\frac{Z(\epsilon)}{Z(0)}\frac{\tilde{Z}(\alpha,\beta-\epsilon)}{\tilde{Z}(\alpha,\beta)}\ .$ (23) Regardless of the choice of the system parameters, the already known results are straightforwardly recovered, i.e., $G(0)=1$ and $G(\beta)=1$ for $\alpha=0$ (initial thermal state). In Figure 3, our analytical expression for $G(\varepsilon)$ is compared with its numerical estimate for two different Hamiltonians, showing a very good agreement. We remark that the distribution of $\rho$ after the second energy measurement could also be obtained by simply imposing the maximization of the von Neumann entropy. This is reasonable, since the measurement device is macroscopic and can provide any amount of energy. As a final remark, notice that, in the numerical findings of Figure 3, the statistics of the quantum-heat fluctuations originated by the system respect the same ergodic hypothesis that is satisfied whenever a sequence of quantum measurements is performed on a quantum system Gherardini2016NJP ; Gherardini2017QSc ; PiacentiniNatPhys2017 . In particular, in Figure 3, one can observe that the analytical expression of $G(\epsilon)$ for a large $M$ (i.e., in the asymptotic regime obtained by indefinitely increasing the time duration of the implemented protocol) practically coincides with the numerical results obtained by simulating a sequence with a finite number of measurements ($M=20$) but over a quite large number ($3\cdot 10^{5}$) of realizations. This evidence is quite important, because it means that the quantum-heat statistics is homogeneous and fully take into account even phenomena occurring with very small probability in a single realization of the protocol. (a) (b) Figure 3: Comparison of the analytic expression (22) of the asymptotic (large-$M$) quantum-heat characteristic function $G(\epsilon)$ (blue solid lines) with the numerical results averaged over $3\cdot 10^{5}$ realizations (red dots). The initial state is the same as in Figure 2 and, again, $O=S_{z}$. In panel (a), the Hamiltonian is the same as in Figure 2, while in panel (b) the Hamiltonian is $H=\omega_{1}S^{2}_{z}+\omega_{2}S_{x}$, with $\omega_{1}=2\omega_{2}=1$. ## V Estimates of $\beta_{\rm eff}$ In this section, we study the behavior of $\beta_{\rm eff}$, i.e., the nontrivial solution of $G(\beta_{\rm eff})=1$, as a function of the initial state (parametrized by $\alpha$ and $\beta$) and of the energy levels of the system. Let us first notice that, by starting from Equation (22), obtaining an analytical expression for $\beta_{\rm eff}$ in the general case appears to be a very non-trivial task. Thus, in Figure 4a, we numerically compute $\beta_{\rm eff}$ as a function of $\alpha$ (the non-thermal component of $\rho_{0}$) for different values of $\beta$. Three representative cases for the energy levels are taken, i.e., $\\{E_{1}=-1,\,E_{2}=0,\,E_{3}=3\\}$, $\\{E_{1}=-1,\,E_{2}=0,\,E_{3}=1\\}$, and $\\{E_{1}=-3,\,E_{2}=0,\,E_{3}=1\\}$, respectively. This choice allows us to deal both with the cases $E_{3}-E_{2}>E_{2}-E_{1}$ and $E_{3}-E_{2}<E_{2}-E_{1}$. The choice of the energy unit is such that the smallest energy gap between $E_{3}-E_{2}$ and $E_{2}-E_{1}$ is set to one. As stated above, we consider $\beta>0$; the corresponding negative values of the inverse temperature are obtained by taking $E^{\prime}_{k}=-E_{k}$ with $\beta^{\prime}=-\beta$. As expected, for $\alpha=0$, we have $\beta_{\rm eff}=\beta$, regardless of the values of the $E_{k}$’s. In the next two subsections, we continue discussing in detail the findings of Figure 4a, presenting the asymptotic behaviors for large positive and negative values of $\alpha$. (a) Figure 4: Behavior of $\beta_{\rm eff}$ as a function of $\alpha$ for different values of $\beta\in[0,2.5]$. We have chosen: (a) $\\{E_{1}=-1,\,E_{2}=0,\,E_{3}=3\\}$, (b) $\\{E_{1}=-1,\,E_{2}=0,\,E_{3}=1\\}$, and (c) $\\{E_{1}=-3,\,E_{2}=0,\,E_{3}=1\\}$, respectively. . . (b) (c) ### V.1 Asymptotic Behavior for a Large Positive $\alpha$ From Figure 4a, one can deduce that, for large positive values of $\alpha$ (corresponding to having as the initial density operator the pure state $\rho_{0}=\lvert E_{2}\rangle\\!\langle E_{2}\rvert$), $\beta_{\rm eff}\rightarrow\bar{\beta}_{\rm eff}$, which only depends on $E_{1}$ and $E_{3}$. This asymptotic value $\bar{\beta}_{\rm eff}$ is positive if $E_{3}-E_{2}>E_{2}-E_{1}$, negative if $E_{3}-E_{2}<E_{2}-E_{1}$, and zero when $E_{3}-E_{2}=E_{2}-E_{1}$. To better explain the plots in Figure 4a, let us consider the analytic expression of $G(\epsilon)$. In this regard, for a large $\alpha$ and finite $\beta$, we can write $\tilde{Z}(\alpha,\beta)\approx e^{-\beta E_{2}+\frac{\alpha}{v}(E_{3}-E_{1})^{2}}\ ,$ (24) so that, by using Equation (23), the condition $G(\beta_{\rm eff})=1$ reads as $e^{-\bar{\beta}_{\rm eff}(E_{1}-E_{2})}+e^{-\bar{\beta}_{\rm eff}(E_{3}-E_{2})}=2$, or, setting $E_{2}=0$, $e^{-\bar{\beta}_{\rm eff}E_{1}}+e^{-\bar{\beta}_{\rm eff}E_{3}}=2\ .$ (25) Notice that, if $E_{3}=-E_{1}$ the only solution of Equation (25) is $\bar{\beta}_{\rm eff}=0$, while a positive solution appears for $E_{3}>-E_{1}$ and a negative one if $E_{3}<-E_{1}$, thus confirming what was observed in the numerical simulations. Moreover, by replacing $E_{1}\rightarrow-E_{3}$ and $E_{3}\rightarrow-E_{3}$, the value of $\bar{\beta}_{\rm eff}$ changes its sign. Now, without loss of generality, let us fix the energy unit so that $E_{1}=-1$. The behavior of $\bar{\beta}_{\rm eff}$ as a function of $E_{3}$ is shown in Figure 5. We observe a monotonically increasing behavior of $\bar{\beta}_{\rm eff}$ up to a constant value for $E_{3}\gg|E_{1}|=1$. Once again, this value can be analytically computed from Equation (25), which, for a large value of $E_{3}$, gives $\bar{\beta}_{\rm eff}=\ln{2}$. Putting together all of the above considerations and restoring the energy scales, the limits of $\bar{\beta}_{\rm eff}$ are the following $-\frac{\ln{2}}{E_{3}-E_{2}}<\bar{\beta}_{\rm eff}<\frac{\ln{2}}{E_{2}-E_{1}}\ .$ (26) The lower and the upper bounds of $\bar{\beta}_{\rm eff}$ are also shown in Figure 5, in which $E_{1}=-1$ and $E_{2}=0$. ### V.2 Asymptotic Behavior for a Large Negative $\alpha$ From Figure 4a, one can also conclude that, for large negative values of $\alpha$, the behavior of $\beta_{\rm eff}$ is linear with $\alpha$ $\beta_{\rm eff}\approx r\alpha\ ,$ (27) with $r>0$ if $E_{3}-E_{2}>E_{2}-E_{1}$, $r=0$ when $E_{3}-E_{2}=E_{2}-E_{1}$, and $r$ is negative otherwise. This divergence is easily understood: In fact, the limit $\alpha\rightarrow-\infty$ (for finite $\beta$) corresponds to the initial state $\rho_{0}=\lvert E_{1}\rangle\\!\langle E_{1}\rvert$ when $E_{3}-E_{2}<E_{2}-E_{1}$ or $\rho_{0}=\lvert E_{3}\rangle\\!\langle E_{3}\rvert$ if $E_{3}-E_{2}>E_{2}-E_{1}$. On the other hand, those states (thermal states with $\beta_{\rm eff}=\beta=\pm\infty$) are also reached in the limits $\beta\rightarrow\pm\infty$ with a finite $\alpha$. This simple argument does not imply the linear divergence of $\beta_{\rm eff}$ as in Equation (27), nor does it provide insights about the value of $r$, which, however, can be derived from Equation (23). Although the calculation makes a distinction on the sign of $r$ depending on whether $E_{3}-E_{2}$ is greater or smaller than $E_{2}-E_{1}$, the result is independent of this detail. In particular, by considering the case $E_{3}-E_{2}>E_{2}-E_{1}$ ($r>0$) and taking into account the divergence of $\beta_{\rm eff}=r\alpha$, we find in the $\alpha\rightarrow-\infty$ regime that the characteristic function $G(\beta_{\rm eff})$ has the following form: $G(\beta_{\rm eff})=\frac{1}{3}+\text{const}\times e^{-\alpha|\Delta_{3}|\left[r-\frac{(\Delta_{1}-\Delta_{2})}{v}\right]}\ .$ (28) (a) (b) Figure 5: (a) Behavior of the asymptotic value $\bar{\beta}_{\rm eff}$ for a large positive $\alpha$ ($\alpha=20$) as a function of $E_{3}$ (solid blue line) with $E_{1}=-1$ and $E_{2}=0$. We compare the curve with its limiting (lower and upper) values, defined in Equation (26) (dash-dotted red lines). (b) Behavior of the asymptotic slope $r$, rescaled for $v$, for a large negative $\alpha$ ($\alpha=-20$) as a function of $E_{3}$. In both cases, $E_{2}=0$ and $E_{1}=-1$. Hence, in order to ensure that $G(\beta_{\rm eff})\neq\frac{1}{3}$ in the limit $\alpha\rightarrow-\infty$, the following relation has to be satisfied $r=\frac{E_{1}+E_{3}-2E_{2}}{v}\ .$ (29) The numerical estimate of $rv$ as a function of $E_{3}$ is shown in Figure 5. The numerical results confirm the linear dependence of $\beta_{\rm eff}$ as a function of $\alpha$. ### V.3 Limits of the Adopted Parametrization (a) (b) (c) Figure 6: Behavior of $\beta_{\rm eff}$ as a function of $q$, which parametrizes the initial state $\rho_{0}$ as in Equation (30), in each of the three cases (a) $E_{3}-E_{2}>E_{2}-E_{1}$, (b) $E_{3}-E_{2}=E_{2}-E_{1}$, and (c) $E_{3}-E_{2}<E_{2}-E_{1}$. The parametrization introduced in Section III is singular in correspondence of the initial states $\rho_{0}$ with one or more coefficients $c_{k}$ equal to zero. In this regard, as remarked above, initial pure states can be easily obtained in the limits $\beta\rightarrow\pm\infty$, $\alpha$ finite (corresponding to $\rho_{0}=\lvert E_{1}\rangle\\!\langle E_{1}\rvert$ and $\rho_{0}=\lvert E_{3}\rangle\\!\langle E_{3}\rvert$, respectively) and $\alpha\rightarrow+\infty$, $\beta$ finite (that provides $\rho_{0}=\lvert E_{2}\rangle\\!\langle E_{2}\rvert$). Instead, initial states with only a coefficient $c_{k}$ equal to zero, namely $\rho_{0}=q\lvert E_{1}\rangle\\!\langle E_{1}\rvert+(1-q)\lvert E_{2}\rangle\\!\langle E_{2}\rvert\ ,\hskip 14.22636pt\rho_{0}=q\lvert E_{1}\rangle\\!\langle E_{1}\rvert+(1-q)\lvert E_{3}\rangle\\!\langle E_{3}\rvert\ ,\hskip 14.22636pt\rho_{0}=q\lvert E_{2}\rangle\\!\langle E_{2}\rvert+(1-q)\lvert E_{3}\rangle\\!\langle E_{3}\rvert\ ,$ (30) with $q\in[0,1]$, cannot be easily written in terms of $\alpha$ and $\beta$. In fact, the expressions in Equation (30) correspond to the limit in which $\alpha\rightarrow-\infty$ with $\beta=a\alpha+b$ for suitable $a$, $b$. This result can be obtained, e.g., for the first of the states in Equation (30), considering the state $c_{1}=q(1-e^{-Y})$, $c_{2}=e^{-Y}$, and $c_{3}=(1-q)(1-e^{-Y})$ in the limit $Y\rightarrow+\infty$. Solving for $\alpha$ and $\beta$, we have $\alpha=-\frac{v}{\Delta_{1}\Delta_{2}}Y+O(1)\ ,\hskip 28.45274pt\beta=-\frac{r}{3}\frac{v}{\Delta_{1}\Delta_{2}}Y+O(1)\ ,$ (31) so that $a=r/3$, while the $q$ dependence is encoded in the next-to-leading term. For this reason, the parametrization in terms of $q\in[0,1]$ turns out to be the most convenient in the case of singularity. In Figure 6, the numerical estimates of $\beta_{\rm eff}$ as a function of $q$ are shown for the three cases in Equation (30), respectively for $E_{3}-E_{2}$ greater, equal to, and smaller than $E_{2}-E_{1}$. The symmetries $E_{1}\rightarrow- E_{3}$, $E_{3}\rightarrow-E_{1}$, $q\rightarrow 1-q$ and $\beta_{\rm eff}\rightarrow-\beta_{\rm eff}$, due to our choice of parametrization, can be observed. ## VI Conclusions In this paper, we studied the quantum-heat distribution originating from a three-level quantum system subject to a sequence of projective quantum measurements. As figure of merit, we analyze the characteristic function $G(\epsilon)=\langle e^{-\epsilon Q}\rangle$ of the quantum heat $Q$ by using the formalism of stochastic thermodynamics. In this regard, it is worth recalling that, as the system Hamiltonian $H$ is time-independent, the fluctuations of the energy variation during the protocol can be effectively referred of as quantum heat. As shown in Ref. Hernandez2019 , the fluctuation relation describing all the statistical moments of $Q$ is simply given by the equality $G(\beta_{\rm eff})=1$, where the energy-scaling parameter $\beta_{\rm eff}$ can be considered as an effective inverse temperature. The analytic expression of $\beta_{\rm eff}$ has been determined only for specific cases, as, for example, two-level quantum systems. Here, a three-level quantum system was considered and, in order to gain information on the value of $\beta_{\rm eff}$, we performed specific numerical simulations. In doing this, we introduced a convenient parametrization of the initial state $\rho_{0}$, such that its population values can be expressed as a function of the reference inverse temperature $\beta$ and the parameter $\alpha$, identifying the deviation of $\rho_{0}$ from the thermal state. Then, the behavior of the system when $M$, the number of projective measurements, is large is numerically analyzed. The condition of a large $M$ leads to an asymptotic regime whereby the final state of the system tends to a completely uniform state, stationary with respect to the energy basis. This means that such a state can be equivalently described by an effective thermal state with zero inverse temperature. In this regime, the value of the energy scaling $\epsilon$ allowing for the equality $G(\epsilon)=1$ (i.e., $\beta_{\rm eff}$) is evaluated. As a consequence, $\beta_{\rm eff}$, which uniquely rescales energy exchange fluctuations, implicitly encloses information on the initial state $\rho_{0}$. In other terms, once $\rho_{0}$ and the system (time-independent) Hamiltonian $H$ are fixed, $\beta_{\rm eff}$ remains unchanged by varying parameters pertaining to the measurements performed during the dynamics, e.g., the time interval between the measurements. We have also determined $\beta_{\rm eff}$ as a function of $\alpha$ and $\beta$ for large $M$. Except for few singular cases, we found that, for large negative values of $\alpha$, $\beta_{\rm eff}$ is linear with respect to $\alpha$, while it tends to become constant and independent of $\beta$ for large positive values of $\alpha$. Such conditions can be traced back to an asymptotic equilibrium regime, because any dependence from the initial state $\rho_{0}$ is lost. As a final remark, we note that, overall, the dynamics acting on the analyzed three-level system are unital Rastegin13JSTAT13 ; Sagawa2014 . As a matter of fact, this is the result of a non-trivial composition of unitary evolutions (between each couple of measurements) and projections. It would certainly also be interesting to analyze a three-level system subject to both a sequence of quantum measurements and in interaction with an external (classical or quantum) environment. In this respect, in light of the results in Refs. WoltersPRA2013 ; Hernandez2019 , the most promising platforms for this kind of experiment are NV centers in diamonds DohertyPhysRep2013 . Finally, also the analysis of general $N$-level systems, and the study of large $M$ behavior deserve further investigations. ### Acknowlegments The authors gratefully acknowledge M. Campisi, P. Cappellaro, F. Caruso, F. Cataliotti, N. Fabbri, S. Hernández-Gómez, M. Müller and F. Poggiali for useful discussions. This work was financially supported by the MISTI Global Seed Funds MIT-FVG Collaboration Grant “NV centers for the test of the Quantum Jarzynski Equality (NVQJE)”. The author (SG) also acknowledges PATHOS EU H2020 FET-OPERN grant No. 828946 and UNIFI grant Q-CODYCES. 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2002.12088

arxiv-papers

∎ 11institutetext: Miroslav Saur 22institutetext: School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100000, China 22email<EMAIL_ADDRESS>33institutetext: Fu-Sheng Yu 44institutetext: School of Nuclear Science and Technology, Lanzhou university, Lanzhou, 730000, China 44email<EMAIL_ADDRESS> # Charm $C\\!PV$: observation and prospects Miroslav Saur Fu-Sheng Yu ## 1 Introduction In physics, the symmetries and their violation always provide deep insights into the Nature. The parity ($P$) symmetry represents the system is unchanged under the space reflection. The violation of parity, firstly proposed by Lee and Yang and subsequently discovered in 1956, plays the key role in the understanding of the weak interaction which is one of the four basic forces of nature. The charge ($C$) symmetry describes a property between particles and their anti-particles. The violation of the combined charge-parity ($C\\!P$) symmetry was unexpectedly observed in kaon meson decays in 1964. The $C$ and $C\\!P$ violation ($C\\!PV$) are required to explore why there are much more matter than anti-matter in the Universe. The explanation of $C\\!PV$ was proposed by Kobayashi and Maskawa (KM) in 1973 by introducing three generations of quarks, or say six quarks, whereas only three quarks were established at the time. All the six quarks were found in the following twenty years. This theory was finally manifested by the observation of $C\\!PV$ in the bottom-quark meson system in 2001. The measured amount of $C\\!PV$ in the Standard Model (SM) of particle physics is about ten orders of magnitude smaller than required by the matter-antimatter asymmetry in the Universe. Therefore, it is important to search for new sources of $C\\!PV$ beyond the SM (BSM). The KM mechanism also predicts the existence of $C\\!PV$ in the charm-quark system which, however, had never been discovered with a lot of efforts during the past decade. The LHCb collaboration eventually observed the charm $C\\!PV$ in 2019 via measuring the difference of $C\\!P$ asymmetries of $D^{0}\rightarrow K^{+}K^{-}$ and $D^{0}\rightarrow\pi^{+}\pi^{-}$ with the result of $(1.54\pm 0.29)\times 10^{-3}$ LHCb-PAPER-2019-006 , with the significance of 5.3$\sigma$. After the establishment of $C\\!PV$ in the strange- and bottom-quark systems, the observation of charm $C\\!PV$ is a milestone of particle physics. ## 2 LHCb and recent measurement Large Hadron Collider beauty experiment (LHCb) on Large Hadron Collider (LHC) is a dedicated heavy-flavour (particles containing $c$ and $b$ quarks) experiment with a special focus on $C\\!PV$ measurements. Being a single-arm forward spectrometer with excellent vertex, interaction point and momentum resolution in combination with high efficient particle identification systems and large ${c}{\overline{c}}$ cross-section, LHCb can study charm physics, especially possible $C\\!P$ violating processes, with higher precision than previous dedicated $B$-factory experiments. In the time period from 2011 to 2018, LHCb has collected 9 fb-1 of data, roughly corresponding to the sample of decays of $10^{10}$ ${D}^{0}$ whose components are a charm quark and an anti-up quark. Charmed mesons can be produced as a direct result of proton-proton collisions (prompt production) or via weak decays of $b$-hadrons (semileptonic productions). In the case of studies using ${D}^{0}$ mesons, prompt production is in fact a strong decay ${D}^{*}(2010)^{+}\rightarrow{{D}^{0}}{{\pi}^{+}}$ and charge conjugated decay as well. Usage of this decays allows to determine exact charm charge of ${D}^{0}$ meson according to the charge of bachelor pion. Semileptonic process are then defined by the weak decay ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow{{D}^{0}}{\mu^{+}}{{\overline{\nu}}_{\mu}}X$ and charge conjugated, where $X$ stands for any allowed additional particles. In this case charm charge of ${D}^{0}$ meson is determined by the charge of muon. Recently reported observation of $C\\!PV$ in the Charm sector by LHCb is based on the new Run 2 data analysis and subsequent combination of the obtained results with the previous measurements from Run 1 LHCb-PAPER-2014-013 ; LHCb- PAPER-2015-055 . The new analysis is based on $44~{}(9)\times 10^{6}$ and $14~{}(3)\times 10^{6}$ ${D}^{0}$ $\rightarrow$ ${K}^{+}$ ${K}^{-}$ and ${D}^{0}$ $\rightarrow$ ${\pi}^{+}$ ${\pi}^{-}$ prompt (semileptonic) decays, respectively. This data set, corresponding to 6 fb-1, was recorded from 2015 to 2018 at the collision energy 13 TeV. Time dependent $C\\!P$ asymmetry of ${D}^{0}$ decays is given by $A_{CP}(f,t)\equiv\frac{\mathrm{\Gamma}(D^{0}(t)\rightarrow f)-\mathrm{\Gamma}({{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(t)\rightarrow f)}{\mathrm{\Gamma}(D^{0}(t)\rightarrow f)+\mathrm{\Gamma}({{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}(t)\rightarrow f)},$ (1) where f is a final state and $C\\!P$ eigenstate, in the case of reported analysis final state is ${K}^{+}$ ${K}^{-}$ or ${\pi}^{+}$ ${\pi}^{-}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ is the anti-particle of ${D}^{0}$. This asymmetry can be also written as the combination of direct and indirect $C\\!P$ asymmetry effect: $A_{CP}(f)\approx a_{{C\\!P}}^{\mathrm{dir}}(f)-\frac{\langle t(f)\rangle}{\tau({{D}^{0}})}A_{\Gamma}(f)$, where $\langle t(f)\rangle$ denotes the mean decay time of ${D}^{0}$ $\rightarrow~{}f$ influenced by the experimental efficiency, $a_{{C\\!P}}^{\mathrm{dir}}(f)$ is the direct $C\\!P$ asymmetry, $\tau$(${D}^{0}$) the ${D}^{0}$ lifetime and $A_{\Gamma}$ the asymmetry between the ${D}^{0}$ $\rightarrow f$ and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ $\rightarrow f$ effective decay widths. Figure 1: HFLAV fit of direct $C\\!PV$ parameter $\Delta a^{\mathrm{dir}}_{{C\\!P}}$ and indirect $C\\!PV$ parameter $a_{{C\\!P}}^{\mathrm{ind}}$ updated with the reported LHCb measurement. Reproduced from Ref. HFLAV16 . However,the $A_{CP}$ values, as defined above are not accessible directly by the experimental methods and must be extracted from the data. Directly measurable value is the difference between raw yields, $A_{\rm raw}$, of ${{D}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decays or between ${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$, respectively. $A_{\rm raw}$ can be very well approximated, up to the order $\mathcal{O}(10^{-6})$, as linear combination of physical $C\\!P$ asymmetry $A_{CP}$, detection asymmetry of ${D}^{0}$ which is equal to zero due to charge conjugated final states, mother particle production asymmetry and detection asymmetry of tagging particle. These detection and production asymmetries are cancelled by equalising kinematics between ${K}^{+}$ ${K}^{-}$ and ${\pi}^{+}$ ${\pi}^{-}$ decay modes and then taking a difference. This equalisation is done in three dimension of kinematic variables simultaneously after the removal of phase space regions with large intrinsic asymmetries due to the LHCb detector geometry. Final experimental formula is then written as following $\displaystyle\mathrm{\Delta}A_{CP}$ $\displaystyle\equiv A_{CP}(D^{0}\rightarrow{{K}^{+}}{{K}^{-}})-A_{CP}(D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}})$ $\displaystyle=A_{\rm raw}^{\rm equalised}({{K}^{+}}{{K}^{-}})-A_{\rm raw}^{\rm equalised}({{\pi}^{+}}{{\pi}^{-}}).$ (2) The difference of $C\\!P$ asymmetries in $D^{0}\rightarrow K^{+}K^{-}$ and $D^{0}\rightarrow\pi^{+}\pi^{-}$ are finally measured by LHCb as $\mathrm{\Delta}A_{CP}^{\rm prompt}=[-18.2\pm 3.2\text{\,(stat.)}\pm 0.9\text{\,(syst.)}]\times 10^{-4}$, $\mathrm{\Delta}A_{CP}^{\rm semileptonic}=[-9\pm 8\text{\,(stat.)}\pm 5\text{\,(syst.)}]\times 10^{-4}$ LHCb-PAPER-2019-006 . By combing both these results with the previous LHCb measurements with the Run I data of 3 fb-1 LHCb-PAPER-2014-013 ; LHCb- PAPER-2015-055 , it can be obtained that $\begin{split}\mathrm{\Delta}A_{CP}^{\rm combined}=(-15.4\pm 2.9)\times 10^{-4},\end{split}$ (3) where the uncertainty includes statistical and systematic contributions. This result deviates from zero $C\\!P$ asymmetry hypothesis on 5.3$\sigma$ level. This is the first observation of $C\\!P$ violation in the charm sector. With the LHCb average of $A_{\Gamma}$ HFLAV16 , the direct $C\\!P$ asymmetry can then be obtained as $\Delta a_{{C\\!P}}^{\mathrm{dir}}=(-15.7\pm 2.9)\times 10^{-4}$, which shows the sensitivity of $\mathrm{\Delta}A_{CP}$ to the direct $C\\!PV$. Finally, the combined fit of the direct and indirect $C\\!P$ asymmetries by the Heavy Flavour Averaging Groups (HFLAV) is shown in Fig. 1. The current world average result excludes the no-$C\\!PV$ hypothesis on the level of 5.44$\sigma$. ## 3 Theoretical explanations and implications In theory, $C\\!PV$ in $D^{0}\rightarrow K^{+}K^{-}$ and $\pi^{+}\pi^{-}$ results from the interference between the tree and penguin amplitudes of charm decays. It is difficult to calculate in the first-principle QCD methods due to the large non-perturbative contributions at the charm scale. Therefore, the order of magnitude of predictions on the charm $C\\!PV$ is meaningful. Before 2019, several orders of magnitude of charm $C\\!PV$ have been predicted in literatures, ranging from $10^{-4}$ to $10^{-2}$. If persisting in using the perturbative QCD, $C\\!PV$ in charm decays is naively expected as $A_{CP}\sim{\alpha_{s}(\mu_{c})\over\pi}{|V_{ub}V_{cb}^{*}|\over|V_{us}V_{cs}^{*}|}=\mathcal{O}(10^{-4})$. On the contrary, If taking the unknown non-perturbative contributions to be arbitrarily large, the charm $C\\!PV$ could be as large as $10^{-2}$, to be consistent with the experimental results in 2011 when $\mathrm{\Delta}A_{CP}$ was measured to be $(-0.82\pm 0.24)\%$ by LHCb Aaij:2011in . Due to the limit of space of this article, relevant references can be seen in FSY_implication:2019hho . The most interesting thing is that only two papers, written by Cheng and Chiang (CC) Cheng:2012xb and Li, Lü and Yu (LLY) Li:2012cfa , quantitatively predicted $\mathrm{\Delta}A_{CP}$ at the order of $10^{-3}$ before the observation. They are much smaller than the experimental measurements in 2011 and 2012, but manifested by the recent LHCb result. The comparison between the experimental measurements and the theoretical predictions by CC and LLY are shown in Fig. 2. Figure 2: Comparison between experimental measurements (in black) and theoretical predictions (in blue) on $\mathrm{\Delta}A_{CP}$ in year. Experimental results are corresponding to the world-average values for specific year as calculated by the HFLAV HFLAV16 . The theoretical approaches of CC and LLY are explained in text. The yellow band is the most recent experimental result for comparison. To predict the charm $C\\!PV$, one should reliably obtain the tree amplitudes first, to understand the dynamics at the charm scale, and then calculate the penguin amplitudes reasonably. Including all the strong-interaction effects, especially the non-perturbative contributions, the topological diagrammatic approach works well for hadronic charm-meson decays by extracting the tree amplitudes from the data of branching fractions Cheng:2012wr ; Cheng:2012xb ; Li:2012cfa . CC pointed out that the upper bound of $\mathrm{\Delta}A_{CP}$ in the SM is $-0.25\%$ Cheng:2012wr which is more than $2\sigma$ away from the experimental result at that time. Later on, they assumed that the penguin- exchange diagram is identical to the $W$-exchange diagram, $PE=E$, so that $\mathrm{\Delta}A_{CP}$ $=(-1.39\pm 0.04)\times 10^{-3}~{}\mathrm{or}~{}(-1.51\pm 0.04)\times 10^{-3}$ Cheng:2012xb , and updated with similar results in Cheng:2019ggx . To give a reasonable uncertainty of the CC approach, considering the possible difference between $PE$ and $E$, we take $PE$ ranging from $E/2$ to $2E$. Under the factorization hypothesis, LLY proposed the factorization-assisted topological-amplitude approach which relates the penguin amplitudes to the tree amplitudes without any additional free parameters. Considering the uncertainties of input parameters, it is predicted that $\mathrm{\Delta}A_{CP}$ $=(-0.57\sim-1.87)\times 10^{-3}$ Li:2012cfa , which is consistent with the latest result by the LHCb measurement. After the observation of charm $C\\!PV$ by LHCb in 2019, new explanations are explored either in the SM or in the BSM. In the SM picture, the measured result of $\mathrm{\Delta}A_{CP}$ can be explained by the non-perturbative final-state-interaction contributions from the rescattering effects Grossman:2019xcj or the near-by resonant effects Soni:2019xko . Alternatively, given that the SM contribution to the charm $C\\!PV$ is very small based on the heavy-quark expansion and the perturbative QCD, the observed $\mathrm{\Delta}A_{CP}$ are explored by the BSM explanations, such as the flavour-violating $Z^{\prime}$ model, the two-Higgs-doublet model, and vector-like quark models Lenz_CPV_BSM:2019fdb ; Dery_implication:2019ysp ; Calibbi:2019bay . Due to the non-perturbative physics at the charm scale, it is difficult to use the observed CPV to reliably search for new physics. On the contrary, the study of charm CPV could be used to test and understand the non-perturbative behaviour of the SM. The additional progress in theory and also more precise experimental measurements are needed. ## 4 Impact and prospect for the future Although the combination of ${{D}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ and ${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ was expected to be one of the best probes of $C\\!PV$ in charm, many other studies are possible or even already ongoing, such as ${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{0}_{\mathrm{S}}}$ and ${{D}^{+}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$ FSY_implication:2019hho , to test and understand the dynamics of charm decay and to search for new physics beyond the SM. Due to their generally rich decays structure multibody decays offer additional interesting possibility how to look for $C\\!PV$ signatures. However, at the same time such a studies generally require more complicated analysis methods and higher recorded luminosity. Another interesting measurement is being performed to investigate a novel effect of $C\\!PV$ in charmed meson decaying into ${K}^{0}_{\mathrm{S}}$, which comes from mother decay and daughter mixing with predicted values reaching the available experimental sensitivity Yu_CPV:2017oky . The LHCb detector is currently going through the substantial upgrade which will allow to record data with 5 times higher luminosity than during the years 2015-2018. This, in combination with the new full software trigger, a crucial point for the charm physics, will allow LHCb to achieve an unprecedented precision in the heavy flavour sector $C\\!PV$ measurements. This opens a door to measure possible $C\\!PV$ effects in rare decays, e.g. radiative and semi- leptonic decays. Another dedicated heavy-flavour experiment is Belle II, which started taking data in 2019, from which contributions to $C\\!PV$ measurements are expected, especially results from the decays with neutral particles in the final states. Another substantial upgrade of the LHCb is planned for the time period after 2030, with additional ten fold increase of luminosity. The LHCb is currently expected to be the only dedicated heavy-flavour experiment taking data during that time period. Table 1 summarises current and expected future trigger-level yields and sensitivity in promptly produced ${{D}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ and ${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays. In summary, the first experimental observation of $C\\!P$ violation in the charm sector was done with an amazing sensitivity obtained by LHCb in 2019. This is a milestone in the high energy physics. The result is consistent with the theoretical predictions by CC and LLY. It is expected that more precise measurements and more theoretical studies in the near future will help us to deeply understand the dynamics at the charm scale and to explore the new physics effects. Table 1: Overview of the recorded and predicted values for promptly produced ${{D}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}$ and ${{D}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ yields during the different data-taking periods at LHCb. All values are corresponding to the trigger-level yields within the LHCb acceptance. Last column shows expected precision of the $\mathrm{\Delta}A_{CP}$ measurements with the corresponding yields. Reproduced from Ref. LHCb-PII-Physics . Sample [fb-1] | Yield | yield | $\sigma(\mathrm{\Delta}A_{CP})$ ---|---|---|--- | ${{D}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}$ | ${{D}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ | [%] Run 1-2 (9) | $52\times 10^{6}$ | $17\times 10^{6}$ | 0.03 Run 1-3 (23) | $280\times 10^{6}$ | $94\times 10^{6}$ | 0.013 Run 1-4 (50) | $1\times 10^{9}$ | $305\times 10^{6}$ | 0.01 Run 1-5 (300) | $4.9\times 10^{9}$ | $1.6\times 10^{9}$ | 0.003 ## 5 Acknowledgement This work is partially supported by the National Natural Science Foundation of China under Grants No.U1732101 and 11975112, by Gansu Natural Science Fund under grant No.18JR3RA265, by the Fundamental Research Funds for the Central Universities under Grant No. lzujbky-2019-55 and by the University of Chinese Academy of Sciences scholarship for International students. ## References * (1) LHCb collaboration, R. Aaij, C. Abellan Beteta, M. Adinolfi, et al., _Observation of $C\\!P$ violation in charm decays_, Phys. Rev. Lett. 122 (2019) 211803, arXiv:1903.08726 * (2) LHCb collaboration, R. Aaij, B. Adeva, M. Adinolfi, et al., _Measurement of $C\\!P$ asymmetry in ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays_, JHEP 07 (2014) 041, arXiv:1405.2797 * (3) LHCb collaboration, R. Aaij, C. Abellan Beteta, B. Adeva, et al., _Measurement of the difference of time-integrated $C\\!P$ asymmetries in ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays_, Phys. Rev. Lett. 116 (2016) 191601, arXiv:1602.03160 * (4) Heavy Flavor Averaging Group, Y. Amhis, Sw. Banerjee, E. Ben-Haim et al., _Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of summer 2016_, Eur. Phys. J. C77 (2017) 895, arXiv:1612.07233, updated results and plots available at https://hflav.web.cern.ch * (5) LHCb, R. Aaij, C. Abellan Beteta, B. Adeva, et al., _Evidence for CP violation in time-integrated $D^{0}\rightarrow h^{-}h^{+}$ decay rates_, Phys. Rev. Lett. 108 (2012) 111602, arXiv:1112.0938 * (6) H.-N. Li, C.-D. Lü, and F.-S. Yu, _Implications on the first observation of charm cpv at lhcb_ , arXiv:1903.10638 * (7) H.-Y. Cheng and C.-W. Chiang, _SU(3) symmetry breaking and CP violation in D $\rightarrow$ PP decays_, Phys. Rev. D86 (2012) 014014, arXiv:1205.0580 * (8) H.-n. Li, C.-D. Lu, and F.-S. Yu, _Branching ratios and direct CP asymmetries in $D\rightarrow PP$ decays_, Phys. Rev. D86 (2012) 036012, arXiv:1203.3120 * (9) H.-Y. Cheng and C.-W. Chiang, _Direct CP violation in two-body hadronic charmed meson decays_ , Phys. Rev. D 85 (2012) 034036, arXiv:1201.0785, [Erratum: Phys.Rev.D 85, 079903 (2012)] * (10) H.-Y. Cheng and C.-W. Chiang, _Revisiting cp violation in $d\rightarrow p\\!p$ and $v\\!p$ decays_, Phys. Rev. D 100 (2019) 093002, arXiv:1909.03063 * (11) Y. Grossman and S. Schacht, _The emergence of the $\Delta U=0$ rule in charm physics_, JHEP 07 (2019) 020, arXiv:1903.10952 * (12) A. Soni, _Resonance enhancement of Charm CP_ , arXiv:1905.00907 * (13) M. Chala, A. Lenz, A. V. Rusov, and J. Scholtz, _$\delta a\\_{CP}$ within the standard model and beyond_, JHEP 07 (2019) 161, arXiv:1903.10490 * (14) A. Dery and Y. Nir, _Implications of the LHCb discovery of CP violation in charm decays_ , JHEP 12 (2019) 104, arXiv:1909.11242 * (15) L. Calibbi, T. Li, Y. Li, and B. Zhu, _Simple model for large CP violation in charm decays, B-physics anomalies, muon g-2, and Dark Matter_ , arXiv:1912.02676 * (16) F.-S. Yu, D. Wang, and H.-n. Li, _$CP$ asymmetries in charm decays into neutral kaons_, Phys. Rev. Lett. 119 (2017) 181802, arXiv:1707.09297 * (17) LHCb collaboration, R. Aaij, B. Adeva, M Adinolfi, et al., _Physics case for an LHCb Upgrade II — Opportunities in flavour physics, and beyond, in the HL-LHC era_ , arXiv:1808.08865

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2002.12122

arxiv-papers

# Energy-scale cascade and correspondence between the Mott and Kondo lattice physics Danqing Hu Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Ning-Hua Tong Department of Physics, Renmin University of China, Beijing 100872, China Yi-feng Yang<EMAIL_ADDRESS>Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China ###### Abstract We propose an energy-scale correspondence between the Mott physics and the Kondo lattice physics and construct a tentative phase diagram of their correlated electrons with two characteristic energy scales $\omega^{*}$ and $\Omega$ marking the upper boundary of a low-energy regime with well-developed long-range coherence and the lower boundary of localized spectral weight in the energy space, respectively. In between, there exists a crossover region with emergent but damped quasiparticle excitations. We argue that the presence of two separate energy scales is a generic property of correlated electrons on a lattice and reflects an intrinsic two-stage process of the quasiparticle dynamics to build up the lattice coherence. For the Hubbard model, they correspond to the kink and waterfall structures on the dispersion, while for the periodic Anderson model, they are associated with the indirect and direct hybridization gaps. Our work reveals a deep connection between the Mott and Kondo lattice physics and provides a basic ingredient for the study of many- body systems. In correlated electron systems, a “kink” denotes an abrupt slope change of the dispersive band and has often been regarded as a fingerprint of the coupling between electronic quasiparticles and collective bosonic excitations. It has been intensively studied in cuprates Lanzara2001Nature ; He2001PRL ; Shen2002PMB ; Damascelli2003RMP ; Meevasana2007 ; Armitage2010RMP , where its appearance at an energy scale of around 30-90 meV has been attributed to spin fluctuations or phonons, which may potentially participate in the superconducting pairing. Different from these “bosonic kinks” which represent the characteristic energy scales of the bosonic excitations, it has been shown later that kink may also arise purely due to electronic mechanism without involving bosonic excitations Byczuk2007NatPhys . This “electronic kink” was first seen in the Hubbard model (HM) within the framework of the dynamical mean-field theory (DMFT) Nekrasov2006 and then extended to other more complicated models Macridin2007PRL ; Grete2011PRB ; Kainz2012PRB ; Greger2013PRL ; Toschi2009 ; Toschi2010 . It might explain the so-called high energy kink observed over a wide energy range from 40 to 800 meV in cuprates and some other transition-metal oxides Yang2005PRL ; Iwasawa2005 ; Yoshida2005 ; Ingle2005PRB ; Valla2007 ; Graf2007 ; Xie2007PRL ; Aizaki2012PRL . Theoretically, the “electronic kink” has been argued to mark an important energy scale of the Mott physics, namely the boundary of its Fermi liquid ground state Byczuk2007NatPhys . Roughly speaking, the kink energy $\omega^{*}$ constrains a low-energy region of well-defined Landau quasiparticles and poses a threshold for the application of the Fermi liquid theory. A higher energy scale $\Omega$ has also been proposed as the boundary separating the many-body resonant state from the high-energy Hubbard bands. $\omega^{*}$ and $\Omega$ together define a cascade of key energy scales in the HM. Their discovery has since stimulated many debates concerning their true physical origin. Some suggested that the kink should still be of bosonic origin due to emergent collective spin fluctuations Rass2009 , while some associated it with an effective Kondo energy scale Held2013PRL . A better understanding of its origin will deepen our knowledge of the many-body physics, but a consensus has yet to be reached. In particular, its fundamental importance and potential implications have not been fully recognized. Here we point out their deep connection with the general physics of quasiparticle emergence and coherence and report a close correspondence between the energy scales of the Mott and Kondo lattice physics. While the two phenomena have mostly been studied separately in different families of correlated systems based on either the HM or the periodic Anderson model (PAM), there are increasing evidences supporting an intimate underlying connection Held2000PRL ; Logan2016 . The Kondo lattice physics has been argued to be a special case of the (orbital-selective) Mott transition Medici2005 ; Pepin2007 ; Pepin2008 ; Vojta2010 , and the Mott physics in cuprates is itself a low-energy projection of the Kondo-Heisenberg model Zhang1988 . In the lately-discovered nickelate superconductors, both seem to exist and determine the properties of the ground state and low-energy excitations Zhang2020 . Our proposed correspondence is based on the simple observation that the two energy scales of the HM have the simple relationship $\omega^{*}\propto Z$ and $\Omega\propto{Z^{1/2}}$ with the renormalization factor Z Byczuk2007NatPhys ; Bulla1999PB . In comparison, the PAM also has two well-known energy scales, namely the indirect and direct hybridization gaps Coleman1984PRB ; Auerbach1986PRL ; Benlagra2011 ; Coleman2015 , which, in the mean-field approximation, are related to an effective hybridization $\tilde{V}$ through $\Delta_{\rm ind}\propto\tilde{V}^{2}$ and $\Delta_{\rm dir}\propto\tilde{V}$. This is reminiscent of $\omega^{*}$ and $\Omega$ in the HM. Since the hybridization gaps have well-defined physical meanings, it is natural to ask if this similarity can provide some insight on the physics of the HM or reveal certain generic features of both models. In this work, we will establish such a correspondence through systematic numerical studies and explore their origin and potential implications on the physics of correlated electrons. We will show that they represent two distinct groups of related energy scales and provide a good characterization of the energy boundaries separating the fully coherent and localized spectral weights with an itinerant but damped crossover region in between. This allows us to construct a tentative phase diagram in the energy space. We argue that the separation of the two energy scales reveals an instrinsic two-stage process of the quasiparticle dynamics for building up the long-range spatial coherence and represents a generic property of a typical class of correlated electron physics on a lattice represented by the HM and PAM. We note that the present work only concerns the paramagnetic Fermi liquid side of the HM phase diagram but for the whole spectrum beyond the Landau quasiparticle regime in the energy space. There is a difference between the energy domain and temperature domain. While the Fermi liquid is usually realized at low temperatures, the two energy scales should still be observed in the energy spectrum, reflecting the critical charge or spin dynamics of low-lying quasiparticle excitations. Figure 1: Illustration of different energy scales identified from the $f$ electron DOS $\rho(\omega)=-{{\rm Im}}G(\omega)/\pi$, the real part of the $f$ electron self-energy ${\rm Re}\Sigma(\omega)$ and its slope ${\rm Re}\Sigma^{\prime}(\omega)$, and the real part of the Green function ${\rm Re}G(\omega)$ for (a) the HM with $U=3$ and (b) the PAM with $U=5.6$ and $V^{2}=0.4$. $\omega^{*}$ and $\omega_{\rm m}$ are estimated from the kink in ${\rm Re}\Sigma(\omega)$ and the maximum of ${\rm Re}G(\omega)$, respectively. For the HM, $\Omega$ is defined from the maximum of ${\rm Re}\Sigma^{\prime}(\omega)$. For the PAM, $\Delta_{\rm ind}$, $\Delta_{\rm dir}$, and $\Omega$ are obtained from the dispersion in Fig. 3. The insets are the enlarged plots for the low energy scales: $\omega^{*}$, $\omega_{\rm m}$, and $\Delta_{\rm ind}$. In both models, the Hamiltonians include two parts: $H=H_{K}+H_{U}$. The potential energy has the form, $H_{U}=U\sum_{i}\left(n_{i\uparrow}-\frac{1}{2}\right)\left(n_{i\downarrow}-\frac{1}{2}\right)$, where $U$ is the onsite Coulomb interaction and $n_{i\sigma}$ is the density operator of the local orbital. There are two general ways to delocalize the $f$ electrons. In the HM, one introduces a direct hopping between neighoring lattice sites, giving rise to a kinetic energy, $H_{K}=\sum_{k\sigma}\epsilon_{k}f_{k\sigma}^{{\dagger}}f_{k\sigma}$, where $f^{\dagger}_{k\sigma}(f_{k\sigma})$ are the creation (annihilation) operator of the $f$ electrons. In the PAM, the $f$ orbitals remain local and the delocalization is achieved by hybridizing with an additional conduction band such that $H_{K}=\sum_{k\sigma}\epsilon_{k}c_{k\sigma}^{{\dagger}}c_{k\sigma}+V\sum_{i\sigma}(f_{i\sigma}^{{\dagger}}c_{i\sigma}+{\rm H.c.})$, where $c^{\dagger}_{k\sigma}$($c_{k\sigma}$) are the creation (annihilation) operator of the conduction electrons and $V$ is the bare hybridization parameter. For simplicity, we focus on the paramagnetic state in the half-filled one-band case with particle-hole symmetry and use the numerical renormalization group (NRG) Wilson1975 ; Bulla2008 as the impurity solver within the DMFT framework Metzner1989 ; Georges1992PRB ; Georges1996RMP ; Kotliar2006RMP . The local self-energy of the $f$ electrons is calculated using $\Sigma(\omega)=UF(\omega)/G(\omega)$ with $G(\omega)=\langle\langle f_{i\sigma};f_{i\sigma}^{{\dagger}}\rangle\rangle_{\omega}$ and $F(\omega)=\langle\langle f_{i\sigma}(n_{i\overline{\sigma}}-1/2);f_{i\sigma}^{{\dagger}}\rangle\rangle_{\omega}$ Bulla1998JPCM . We then solve the DMFT self-consistent equation $G(\omega)=\int d\epsilon\rho_{0}(\epsilon)/[\omega-\epsilon-\Sigma(\omega)]$ for the HM and $G(\omega)=\int d\epsilon\rho_{0}(\epsilon)/[\omega-V^{2}/(\omega-\epsilon+i\eta)-\Sigma(\omega)]$ for the PAM Zhang1993PRL ; Pruschke2000PRB , where $\rho_{0}(\epsilon)=e^{-(\epsilon/t)^{2}}/(\sqrt{\pi}t)$ is the noninteracting density of states for a hypercubic lattice with the dimensionality $d\rightarrow\infty$. We set $t=1$ as the energy unit and choose the logarithmic discretization parameter $\Lambda=2$ and a total number of stored states $N_{\rm s}=800$ for the NRG calculations. Our calculations are limited in the paramagnetic phase concerning the emergence and development of quasiparticle coherence. Magnetic instabilities are additional issues beyond the current work. Figure 2: The scaling relation of different energy scales with respect to the renormalization factor $Z$ for (a) the HM and (b) the PAM with tuning Coulomb interaction $U$. For the PAM, the hybridization strength is set to $V^{2}=0.4$. Figure 1 illustrates how the different energy scales can be identified from the $f$ electron density of states (DOS) $\rho(\omega)=-{\rm Im}G(\omega)/\pi$, the real part of the self-energy ${\rm Re}\Sigma(\omega)$ and its $\omega$-derivative ${\rm Re}\Sigma^{\prime}(\omega)$, and the real part of the Green function, ${\rm Re}G(\omega)$. For the HM, the DOS has a three-peak structure including two Hubbard peaks at higher energy and a quasiparticle peak around zero energy. The larger scale $\Omega$ is defined here from the maximum of ${\rm Re}\Sigma^{\prime}(\omega)$. In Ref. Byczuk2007NatPhys , it was defined from the minimum of ${\rm Re}\Sigma(\omega)$, which, however, only follows the proposed scaling near the Mott transition with a small $Z$. Nonetheless, both reflect the same physics and separate roughly the quasiparticle peak and the Hubbard peaks in the DOS. The lower scale $\omega^{*}$ can be defined from an additional slope change or the kink in ${\rm Re}\Sigma(\omega)$ that constrains a low-energy region for quasiparticle excitations. A different scale $\omega_{\rm m}$ may be introduced from the maximum of ${\rm Re}G(\omega)$. $\omega^{*}$ and $\omega_{\rm m}$ are of the same origin and related through $\omega^{*}\approx(\sqrt{2}-1)\omega_{\rm m}$ for the HM Byczuk2007NatPhys . A third definition may be found from the maximum ($\omega_{s}$) of the local spin susceptibility as discussed later in Fig. 4. The relationship of these scales can be established if we tune the local Coulomb interaction $U$ and plot their variations with the renormalization factor, $Z^{-1}=1-{\rm Re}\Sigma^{\prime}(0)$. The results are summarized in Fig. 2(a). We see that $\omega^{*}\propto\omega_{m}\propto Z$ and $\Omega\propto Z^{1/2}$ over a wide range of $Z$. Thus, $\omega^{*}$ and $\Omega$ represent two distinct groups of related energy scales. The different definitions originate from the crossover nature of the underlying physics. For comparison, Fig. 1(b) shows the energy scales in the PAM. The indirect hybridization gap $\Delta_{\rm ind}$ is determined by the DOS gap and coincides with the additional minimum of ${\rm Re}G(\omega)$, while the direct hybridization gap $\Delta_{\rm dir}$ cannot be clearly discerned in these quantities and is best seen in Fig. 3 from the spectral function, $A_{k}(\omega)=-\pi^{-1}{\rm Im}[\omega-\Sigma(\omega)-\frac{V^{2}}{\omega-\epsilon_{k}+i\eta}]^{-1}$. The plot implies two separated hybridization bands for the PAM. The indirect gap $\Delta_{\rm ind}$ is the energy difference between the bottom of the upper band and the top of the lower band and marks the boundary for the low-energy gap. Although the PAM is in a Kondo insulating state, the self-energy below $\Delta_{\rm ind}$ behaves as that of a Fermi liquid Pruschke2000PRB , reflecting the establishment of full coherence on the Kondo lattice Hu2019 ; Qi2020 . By contrast, the direct hybridization gap $\Delta_{\rm dir}$ measures the minimal energy difference between the two hybridization bands, below which the spectra are governed by the itinerant $f$ character. The energy scales $\omega^{*}$ and $\omega_{\rm m}$ can still be defined similarly as in the HM and contain the flat part of the hybridization bands. We obtain $\omega^{*}\propto\omega_{\rm m}\propto\Delta_{\rm ind}$. On the other hand, the maximum of ${\rm Re}\Sigma^{\prime}(\omega)$ changes slightly with varying $U$ and is no longer a meaningful quantity separating the boundary of the itinerant and localized regions. Rather, we find it is better to define the larger energy $\Omega$ from the deviation of the hybridization bands from the original conduction bands. In Fig. 3, it is seen that $\Omega$ sets roughly the outmost boundary of the itinerant $f$ electron spectral weight. Both $\Omega$ and $\Delta_{\rm dir}$ reflect the separation of the hybridized (itinerant) and unhybridized (localized) spectral regions of the $f$ electrons in the energy space. The scaling results for the PAM are summarized in Fig. 2(b). Similarly, we have $\omega^{*}\propto\omega_{\rm m}\propto\Delta_{\rm ind}\propto Z$ and $\Omega\propto\Delta_{\rm dir}\propto Z^{1/2}$ over a wide range of $Z$. The obtained scaling relations link $\omega^{*}$ and $\Omega$ to $\Delta_{\rm ind}$ and $\Delta_{\rm dir}$ and suggest that they reflect similar physics in the PAM. However, unlike $\omega^{*}$ and $\Omega$ which are in reality hard to measure, the indirect and direct hybridization gaps have unambiguous physical meaning and can be directly probed in experiment. Their scaling relations in the PAM can be derived from the pole properties of the Green function, $G_{k}(\omega)=[\omega-\Sigma(\omega)-\frac{V^{2}}{\omega-\epsilon_{k}+i\eta}]^{-1}$. If we make the lowest order approximation, ${\rm Re}\Sigma(\omega)\approx(1-Z^{-1})\omega$, the poles are determined by $Z^{-1}\omega-\frac{V^{2}}{\omega-\epsilon_{k}}=0$. The direct gap is defined as the energy differences of the two poles at $\epsilon_{k}=0$, while the indirect gap is the energy difference at the band edges ($\epsilon_{k}=\pm D$). We have $\Delta_{\rm dir}=Z^{1/2}V$ and $\Delta_{\rm ind}=ZV^{2}/D$, confirming their relationship with $Z$. The relative deviation of the hybridization bands from the conduction band may be estimated to be $\delta=(\omega-\epsilon_{k})/\omega=ZV^{2}/\omega^{2}$. We have thus $\Omega\approx Z^{1/2}V/\delta^{1/2}$ for a chosen small cutoff $\delta$, at which the $f$ electron spectral weight on the dispersion is also reduced to the order of $Z\delta$. Since all these energy scales diminish at $V=0$, their presence reflects the delocalization of the $f$ electrons on the lattice driven by the hybridization with conduction electrons. In this respect, the larger scale $\Omega$ constrains the major spectral region of delocalized $f$ electrons, while $\omega^{*}$ puts a further restriction for well-established long-range coherence. Figure 3: Different energy scales on the intensity plot of the resolved spectral functions for the HM and PAM. The background colors reflect the magnitude of the $f$ electron spectral weight $A_{\rm k}(\omega)$. The blue dashed lines denote the dispersion $Z\epsilon_{k}$ for the HM and $\epsilon_{k}$ for the PAM. The parameters are the same as in Fig. 1. For the PAM, the localized flat $f$ bands are outside the plotted energy window. Similar analyses may help to understand $\omega^{*}$ and $\Omega$ in the HM. Figure 3 also plots the dispersion $A_{k}(\omega)=-\pi^{-1}{\rm Im}[\omega-\epsilon_{k}-\Sigma(\omega)]^{-1}$ of the HM. Indeed, we see that $\omega^{*}$ marks the boundary of well-defined Landau quasiparticles, above which the dispersion has a different renormalization factor, while $\Omega$ separates the low-energy many-body states and the high energy incoherent region, beyond which the $f$ electron weights are localized. The presence of both $\omega^{*}$ and $\Omega$ is a reflection of the special cascade structure of ${\rm Re}\Sigma^{\prime}(\omega)$, which, as shown in the middle panel of Fig. 1(a), has a maximum at $\Omega$ but, with lowering energy, decreases first to a plateau (or local minimum) on the hillside before it eventually reaches the valley floor below $\omega^{*}$ and $\omega_{\rm m}$. Interestingly, $\Omega$ lies roughly at the waterfall structure connecting the quasiparticle and Hubbard bands Weber2008 ; Kohno2012 . This poses a question concerning the relation of the two properties. For this, we note that the waterfall is an abrupt change of the pole of the Green function with increasing $\epsilon_{k}$. It can only occur around the inflection point of $\omega-{\rm Re}\Sigma(\omega)$ if one wants to tentatively avoid multiple poles. This immediately implies ${\rm Re}\Sigma^{\prime\prime}(\Omega)\approx 0$ or a maximum (minimum) in ${\rm Re}\Sigma^{\prime}(\Omega)$. Thus, $\Omega$ defined here is indeed associated with the waterfall structure in the HM. We conclude that the two energy scales correspond to two features of the dispersion: $\omega^{*}$ for the kink and $\Omega$ for the waterfall, which also reflect the boundaries for the low-energy fully coherent region and the high-energy localized region. The fact that $\Omega$ connects two regions with opposite energy dependence in the self-energy implies $Z^{-1}\Omega\sim\Omega^{-1}$ or $\Omega\propto Z^{1/2}$, similar to that in the PAM. Figure 4: The imaginary part of the local spin susceptibility, showing a maximum at $\omega_{s}$ for both models. The parameters are the same as in Fig. 1. The inset plots $\omega_{s}$ as functions of $Z$ with $V^{2}=0.4$ (PAM) and varying $U$. Further insight on $\omega^{*}$ may be obtained if we consider that $\Delta_{\rm ind}$ in the PAM is related to the spin screening, $\omega^{*}\propto\Delta_{\rm ind}\propto T_{K}$ where $T_{K}$ is the lattice Kondo temperature Jarrell1995PRB ; Chen2016 . The effect of spin screening might be best seen from the local susceptibility, $\chi_{s}(\omega)=\langle\langle S_{i}^{z};S_{i}^{z}\rangle\rangle_{\omega}$ with $S_{i}^{z}=\frac{1}{2}(n_{i\uparrow}-n_{i\downarrow})$ Shimizu2000JPSJ . Figure 4 plots the imaginary part of the local spin susceptibilities for the HM and PAM using the same parameters as in Fig. 1. Both exhibit a maximum at $\omega_{s}$, which, as shown in the inset, varies linearly with $Z$ as those of $\omega^{*}$ and $\Delta_{\rm ind}$. In fact, we find $\omega_{s}\approx\omega^{*}$ in both models (Fig. 3). Away from half filling, this kink or hybridization energy scale $\omega^{*}$ still exists and retains the same relation with $\omega_{s}$ Grete2011PRB ; Kainz2012PRB . Below $\omega_{s}$ the local susceptibility is suppressed due to the spin screening. It is therefore natural to associate the kink with an effective spin screening scale as proposed in Ref. Held2013PRL . We should emphasize, however, that the presence of both $\omega^{*}$ and $\Omega$ is a property of the lattice but not in the usual single-impurity Kondo model where only the Kondo scale exists. This implies that the lattice feedback is crucial for separating $\omega^{*}$ and $\Omega$ and causing the cascade structure of ${\rm Re}\Sigma^{\prime}(\omega)$. Although DMFT does not contain explicit spatial correlations, it necessarily includes important lattice information through the self-consistent iterative procedure. Thus, all energy scales can be classified into two categories, a lower one $\omega^{*}\propto Z$ and a higher one $\Omega\propto Z^{1/2}$, which separate the $f$ electron spectral weight into different regions of distinct properties. If we start from localized $f$ electrons, we may conclude that their spin screening and consequential delocalization is an energy-dependent process marked by the two scales. While $\Omega$ sets the outmost boundary for the many-body resonant states and covers the full energy range with delocalized $f$ electron spectral weight, $\omega^{*}$ is a lower boundary for quasiparticle excitations with well-established long-range coherence. This is common for both PAM and HM and may be best seen if we construct a tentative phase diagram in Fig. 5 in the energy space based on the properties of the $f$ electron spectra. For $Z>0$, the $\omega^{*}\propto Z$ line indicates a screening scale and marks the upper boundary of a well-developed coherent region irrespective of the Fermi liquid or Kondo insulating state, while $\Omega\propto{Z^{1/2}}$ marks the lower boundary of an incoherent region with localized $f$ spectral weight. In between, there exists a crossover region with itinerant but strongly damped excitations of the $f$ character. With decreasing $Z$ to zero, the system enters a Mott insulating state for the HM or an orbital-selective Mott state for the PAM, where the many-body resonance is suppressed and all $f$ electrons turn into fully localized magnetic moments. Since $Z$ is dimensionless, the scaling relations suggest constant prefactors roughly given by $D$ or $V^{2}/D$. Major correlation effects in both properties are encapsulated in the renormalization factor $Z$ (as a function of $U/D$) over a wide parameter region. Figure 5: A tentative phase diagram of the HM and PAM in the energy space as a function of the renormalization factor $Z$, showing different regions of correlated electron spectral weights. In the shadow region, the $f$ electrons become completely localized so that the HM is in a Mott state and the PAM is in an orbital-selective Mott state. We remark again that the presence of the two energy scales represents two related but distinct features of the correlated electron physics. It is not just a matter of theory but has real implications and observable consequences in experiment. To see this, we may consider the gradual delocalization of the $f$ electrons with lowering temperature. The two energy scales then turn into two temperature scales, as observed in recent pump-probe experiment on the heavy fermion compound CeCoIn5 Qi2020 . Determinate Quantum Monte Carlo simulations (DQMC) taking into account spatial degree of freedom for the PAM have confirmed two distinct stages of hybridization dynamics, namely, the onset of precursor hybridization fluctuations and the eventual establishment of a long-range coherent (Kondo insulating) state Hu2019 . In the HM, the two energy scales are also expected to represent two succeeding stages of the (nonequilibrium) quasiparticle dynamics, namely, a Fermi liquid phase with well-established Landau quasiparticles and a higher crossover region with itinerant but damped quasiparticles (bad metal) Pruschke1995 ; Mravlje2011PRL ; Deng2013PRL . It might also be reflected in the temperature or time evolution of the doublon-holon excitations. Exact lattice simulations on the HM may help clarify this issue but are often limited by the numerical accuracy due to small lattice size. At even higher temperatures, the localized spectral weight might be thermally excited and contribute to physical properties, but whether or not (and how) the highly damped electrons can become Anderson localized requires more rigorous study Antipov2016 . Along this line of thought, we anticipate that the existence and separation of two energy scales reflect a generic two-stage development of the electronic coherence and is an intrinsic property of correlated electrons. Before the long-range coherence is eventually established, there exists a precursor stage where the electrons become partially delocalized with damped quasiparticle excitations. On the other hand, the usual Mott transition also exists in other models such as the Falicov-Kimball model, the Ising-Kondo model, and the Hatsugai-Kohmoto model, which do not seem to possess the two energy scales. A closer inspection suggests that the Mott transitions in these models are different from that in the HM. In the Falicov-Kimball model and the Ising- Kondo model, the ground state near the transition is not a Fermi liquid due to “disorder scattering” Freericks2003 ; Yang2020 , while the Hatsugai-Kohmoto model lacks the so-called dynamical spectral weight transfer which is a key feature of the HM Yeo2019 . Thus the presence of two energy scales should not be viewed as a generic property of the Mott transition. Rather, they are associated with the emergence and establishment of coherent quasiparticles on the lattice, possibly induced by the spectral weight transfer from the high- energy localized part to the low-energy itinerant part. More elaborate studies are required to establish this important distinction. To summarize, we report systematic analyses of the energy-scale cascade in the HM and the PAM and reveal a deep connection between the Mott and Kondo lattice physics. This allows us to construct a tentative phase diagram for correlated electrons based on two energy scales marking the upper boundary of the fully coherent low-energy states and the lower boundary of the incoherent regime with localized spectral weight in the energy space. For the HM, these correspond to the kink and waterfall structures on the dispersion. For the PAM, they are associated with the indirect and direct hybridization gaps. The separation of two energy scales is an intrinsic property of the lattice models and reflects a two-stage dynamical process to build up the lattice coherence of itinerant quasiparticles. Our work clarifies the origin of these energy scales and reveals a potentially basic and generic property for understanding the key of correlated electrons on a lattice. This work was supported by the National Natural Science Foundation of China (NSFC Grant No. 11974397, No. 11774401, No. 11974420), the National Key Research and Development Program of MOST of China (Grant No. 2017YFA0303103), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB33010100). ## References * (1) A. Lanzara, P. V. Bogdanov, X. J. Zhou, S. A. Kellar, D. L. Feng, E. D. Lu, T. Yoshida, H. Eisaki, A. Fujimori, K. Kishio, J.-I. Shimoyama, T. Noda, S. Uchida, Z. Hussain, and Z.-X. Shen, Evidence for ubiquitous strong electron–phonon coupling in high-temperature superconductors, Nature (London) 412, 510 (2001). * (2) H. He, Y. Sidis, P. Bourges, G. D. Gu, A. Ivanov, N. Koshizuka, B. Liang, C. T. Lin, L. P. Regnault, E. Schoenherr, and B. Keimer, Resonant Spin Excitation in an Overdoped High Temperature Superconductor, Phys. Rev. Lett. 86, 1610 (2001). * (3) Z.-X. Shen, A. Lanzara, S. Ishihara, and N. 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2024-09-04T02:54:55.061928

2002.12144

arxiv-papers

# Fair Adversarial Networks George Čevora illumr Ltd., London, United Kingdom <EMAIL_ADDRESS> ###### Abstract The influence of human judgement is ubiquitous in datasets used across the analytics industry, yet humans are known to be sub-optimal decision makers prone to various biases. Analysing biased datasets then leads to biased outcomes of the analysis. Bias by protected characteristics (e.g. race) is of particular interest as it may not only make the output of analytical process sub-optimal, but also illegal. Countering the bias by constraining the analytical outcomes to be fair is problematic because A) fairness lacks a universally accepted definition, while at the same time some definitions are mutually exclusive, and B) the use of optimisation constraints ensuring fairness is incompatible with most analytical pipelines. Both problems are solved by methods which remove bias from the data and returning an altered dataset. This approach aims to not only remove the actual bias variable (e.g. race), but also alter all proxy variables (e.g. postcode) so the bias variable is not detectable from the rest of the data. The advantage of using this approach is that the definition of fairness as a lack of detectable bias in the data (as opposed to the output of analysis) is universal and therefore solves problem (A). Furthermore, as the data is altered to remove bias the problem (B) disappears because the analytical pipelines can remain unchanged. This approach has been adopted by several technical solutions. None of them, however, seems to be satisfactory in terms of ability to remove multivariate, non-linear and non-binary biases. Therefore, in this paper I propose the concept of _Fair Adversarial Networks_ as an easy-to-implement general method for removing bias from data. This paper demonstrates that Fair Adversarial Networks achieve this aim. Unfair treatment of individuals enacted by automated decision-makers has recently attracted a large amount of attention [14, 7, 19]. This problem is, however, not limited to automated decision-making; all Data Analytics [DA] and Machine Learning [ML] has the potential to lead to biased conclusions and therefore enact discrimination. The prominence of automated decision-makers in the discussion of discrimination is mostly due to two factors: 1) machine learning models are hard to scrutinise and therefore lack the human oversight that otherwise could prevent discrimination, and 2) automated decision-makers are easy to test and therefore they can be relatively easily proven to be biased. Many conventional uses of Data Analytics can lead to same problems though - for instance the analyst may conclude that individuals from particular postcodes are less credit-worthy. If that judgement is based on biased data and those postcodes are predominantly home to individuals of a specific race, this amounts to racial discrimination despite race not being considered by the analyst at all. The methods presented in this paper aim to prevent these exact situations and at the same time provide a generic easy-to- use solution accessible to any data analyst. Discrimination is a result of past biased human judgements that make-up the datasets, or by biased human judgements which shape the composition of the dataset [1, 4]. This paper focuses on illegal discrimination; however, the findings are applicable to all types of bias, including biases that would not be generally seen as unfair. For instance in the case of dealing with HR data collected across a number of regional offices, the bias of the region might be undesirable in employee evaluation. It has been reported [17] that it is harder for African-Americans to obtain loans than for their white-American counterparts with the same repayment ability. As race is one of the protected characteristics under US law, use of such a system to perform decision-making is illegal. However, this problem does not only concern ethics or legality. In fact, there is a solid business case for fairness. Let us consider a case in which two individuals with equal repayment ability apply for a loan but one of them is rejected based on discrimination against a group they are a member of. Besides being unfair and illegal this also makes a lost profit for the lender in question. There are two main scenarios in which racism creeps into DA. The more straightforward scenario occurs whenever the objective of the DA exercise is to predict (biased) human judgement. The need to remove the past human bias in this scenario is widely acknowledged, yet often ignored. Examples of this issue come from many of the companies that use DA or ML for hiring. They may try to link a candidate’s characteristics such as CV, performance on aptitude tests, or even appearance to figure out how well the candidate will perform in the job. However, they almost never link that information with actual performance of people hired in the past. They are more likely to simply check whether the individuals with particular characteristics were hired or not, which is the decision of a potentially biased hiring manager. This approach is pragmatic - evaluating performance is not easy and can be only done for individuals who were actually hired, not for all applicants, thus severely reducing the size of the dataset. The fairer and more accurate option is to use the actual performance of the individuals hired on the job to guide future decision making. However, this data too can be biased. The reason for this is that the datasets used for this task are almost never a uniform sample from the population. For instance, women have lower chances to be hired for many roles, therefore they are less likely to be included in the dataset used to train the automated decision- maker. This results in a situation in which the dataset used is systematically different to the population, where the difference may increase over time - an ubiquitous sight in DA/ML. This approach too offers a biased view of the population. Several attempts to address this issue have been made in the past [8, 11, 18]. However, a complete lack of consensus on which of these methods should be the gold standard led to the current status quo in which data analysts or ML engineers decide themselves how to address this problem [1]. The bias-removal methods can be broadly divided into two categories: 1) methods seeking statistical parity between groups of individuals that differ on their protected characteristic; 2) methods that seek to remove information about the protected characteristics in the datasets used for ML. ### Statistical Parity Ensuring statistical parity between groups, such as equal access to opportunity, is one of the most popular bias-aware method in DA/ML [12]. The fundamental problem with this approach is the necessity to explicitly define fairness. Unfortunately there is a complete lack of consensus on this issue. In fact Berk and colleagues [2] identify at least six different notions of fairness in the context of penal systems. Many of these notions are not mutually compatible [5], therefore the bias-aware data analysts have to pick what fairness means to them. This principle is fundamentally flawed as the notion of fairness should not be subjective. Unfortunately, while the legal system requires fairness in respect to protected characteristics (e.g. US - Human Rights Act, EU - European Convention on Human Rights), it doesn’t provide sensible111The U.S. Supreme Court asserted the principle of disparate impact, while resisting any sort of mathematical definition (p. 5, [8]). Other bodies such as the Equal Employment Opportunity Commission [6] have provided some guidance, but these are not legally binding and leave a lot to be desired. guidance on what type of statistical parity is required. A further big hurdle for achieving fairness through statistical parity is the difficulty of implementing it. Whilst matching the means and standard deviations between the groups is easy222Just a simple elastic transformation of one group’s outcomes to match the outcomes of the other group. it is also a fairly weak notion of fairness [5]. Most other notions of fairness require inclusion of a parity optimiser into the DA/ML model - adding a fairness term to the loss function. This is clearly outside the area of expertise of most data analysts, as they do not tend to write their own loss functions. It also renders most standard DA functions or packages, such as scikit learn, obsolete as there usually are no reasonable options to customize loss functions. Therefore this approach causes a significant disturbance to existing DA pipelines and significantly increases the time commitment from the analysts, as well as the technical prowess required for such a role. Furthermore, it should not be forgotten that often the data used for ML does not represent the population well. Optimising for parity on training data doesn’t guarantee parity at the population level when the training sample is heavily biased. ### Removing protected characteristics An approach that avoids many of the issues of the statistical parity is to remove the information about protected characteristics from the dataset. The logic behind this issue is beautifully elegant; an agent that is not aware of the protected characteristics can not discriminate based on those characteristics. Unfortunately, the realisation of this principle often fails to deliver on its promises. An approach that is sometimes applied to remove information on protected characteristics is a simple exclusion of these labels from the dataset. While it is widely recognized [14] that this approach is highly flawed, it is still widespread across the industry (author’s anecdotal evidence). The logic behind the failure is simple, the information about the protected characteristics is partially contained in the other characteristics relevant for the decision at hand [15]. These cases include racially segregated neighbourhoods where the postcode reveals the race of an individual, or professions that have traditionally high gender imbalance which reveal gender of the individual. At the same time, home address or previous profession are highly desirable information for a whole range of data applications; therefore can not be excluded from the dataset. It is therefore apparent that this approach cannot lead to fair and accurate DA outcomes. A more sophisticated application of the same principle is manipulating the dataset in a way that removes the ability to predict the protected characteristics for the individuals in the dataset from the rest of the data. This approach has been developed over time to a great success [3, 10, 11, 18]; however, some shortcomings are still apparent: 1) the relationship between protected and other characteristics is certainly not linear like most methods assume; 2) even in cases where a variable seems to not relate to the protected characteristic, it may do so via an interaction with another variable; 3) none of the methods can address continuous characteristics (e.g. age) and even multiple categories pose significant problems [3]. Proposing a novel method that addresses these issues is the main objective of this paper. ## 1 Fair Adversarial Networks A number of methods of removing bias from datasets have been devised, however they generally fall short at removing non-linear, non-binary and/or multivariate biases. To address the problems of the current methods this paper introduces a novel concept of _Fair Adversarial Networks_ [FANs]. Like Generative Adversarial Networks [GANs] [9] this method consists of two networks with different objectives trained iteratively. FANs are a system that creates an unbiased version of a dataset that can subsequently be used with any analytical tools. There are two main components to this system 1) an _autoencoder_ [16] function $y=\rho(x,W_{A})$ that provides $y$ a reconstruction of data $x$ given autoencoder weights $W_{A}$, and 2) a _Racist Network_ that provides estimate $\hat{r}$ of the true protected characteristics $\bar{r}$ (_race_ in this example) from $y$. Figure 1: Architecture of a Fair Adversarial Network. Two networks are trained iteratively: the Racist network minimizes error on predicting race (or other protected characteristic of interest) while the autoencoder jointly minimizes its reconstruction error and the predictive capability of the Racist network. The measure of bias we wish to minimize333Please note that the the cross entropy function $D$ is only appropriate to evaluate classifiers, not regressors, where a different objective function needs to be used. is the true performance, such as the cross-entropy function $D$, of the Racist Network after full training $\bar{D}$ given the autoencoder weights at given epoch444$\bar{D}$ is identical to $D$ when optimal training schedule of the Racist Network has been concluded.. The main problem of this approach is the complexity of such a bias measure. The measure of bias is required at each epoch on which the autoencoder is trained, but it is not easy to obtain. Unlike the discriminator accuracy in GANs, the only way find the bias of encoded data is to fully train the racist network - to measure the true potential of the data to reveal the protected characteristic. Without full training, there is no guarantee that a failure of the racist network to predict the protected characteristic (detect bias) is not simply due to recent changes in the encoding that the racist network has not been able to adapt to yet. Therefore, to find the bias of the data exactly, we would have to perform lengthy full-training of the racist network on each epoch of the autoencoder training, clearly making the algorithm impractical. Furthermore, a guarantee of optimality for the network hyperparameters would be needed. We have developed a method for approximating the fully-trained performance from a single forward pass through the network $\hat{D}(\hat{r},\bar{r})$ which is the core of the autoencoder loss function specified by the equation 1. Further developments to this measure that are required include a general mechanism to stabilise the adversarial training and a number of regularisers $\mathcal{R}$ that ensure quality of the final data encoding. While these developments cannot be shared publicly as they are core to the intellectual property of illumr Ltd., standard approaches are enough to replicate debiasing process on a one-off basis given sufficient hyperparameter tuning. Formally, autoencoder minimizes loss function $\mathcal{L}_{A}(x,r,W_{A},W_{R})=MSE(y,x)+c\hat{D}(\hat{r},\bar{r})+\mathcal{R}$ (1) where $MSE(y,x)$ is the reconstruction error (Mean Square Error) of the autoencoder, $W_{R}$ the Racist Network weights, and $c$ is a constant balancing the individual terms of loss function. The Racist Network optimizes the appropriate loss function such as $\mathcal{L}_{R}(y,\bar{r},W_{R})=D(\hat{r},\bar{r}),$ (2) which is a cross entropy function of two vectors $\bar{r}$ and $\hat{r}$. As mentioned before this cost function ($D$) is different to the estimate of the performance of the Racist Network as minimized by the autoencoder ($\hat{D}$). This is because $D$ is a bad approximation of performance after full training $\bar{D}$. This system of two networks is trained in an iterative adversarial fashion similar to GANs: Algorithm 1 Fair Adversarial Networks: removing bias from data 1:while stopping criteria is not reached do 2: Update the Autoencoder weights $W_{A}$ using the loss function 3: $\mathcal{L}_{A}(x,r,W_{A},W_{R})$. 4: Obtain reconstructed data $y=\rho(x,W_{A})$ 5: for k steps do 6: Update the Racist Network weights $W_{R}$ using the loss function 7: $\mathcal{L}_{R}(y,r,W_{R},W_{A})$. 8: end for 9:end whilereturn y $\triangleright$ debiased data Given the success of this training procedure, the end result should be a dataset that is as similar to the original as possible while it should be harder/impossible to detect the undesirable protected characteristic. However, GANs are notoriously hard to train [13]. ### 1.1 Convergence Adversarial training often leads to very unstable or even run-away behaviour [13]. Here we demonstrate acceptable convergence of our algorithm on five real-world datasets. While the convergence may seem still fairly sub-optimal, we argue that it is sufficiently good for our purpose. Crucially, we implement a ratchet mechanism which always preserves the state of the network with the lowest $\mathcal{L}_{A}$, therefore run-away behaviour after a period of convergence is not particularly problematic. While, we optimize the loss functions $\mathcal{L}_{A}$ and $\mathcal{L}_{R}$, they are not of interest for our purposes. Instead what we truly aim to achieve is to bring the predictive performance of the racist network after full training $\bar{D}(\hat{r},\bar{r})$ down to random. For this task we operationalized $\bar{D}(\hat{r},\bar{r})$ as the best performance of the Racist Network on the validation data from 3 random initializations, and subsequent 10,000 epochs of training. The Racist Network used had a single hidden layer of the same width as the dataset. The random benchmark we are using is $\bar{D}(ML(\bar{r}),\bar{r})$ i.e. always picking the most likely category. The other value of importance to us is $MSE(x,y)$ as we aim to scramble the data as little as possible. Therefore these will be the focus of the analysis of convergence. Further interesting values to observe include $D(\hat{r},\bar{r})$, the loss function of the racist network, but this does not provide a good indication of $\bar{D}(\hat{r},\bar{r})$, and therefore is fundamentally unsuitable to be a part of $\mathcal{L}_{A}$. $\hat{D}(\hat{r},\bar{r})$ is also of interest which while being noisy provides good gradients for training as a part of $\mathcal{L}_{A}$. Figures 2 to 6 clearly show that the $\bar{D}(\hat{r},\bar{r})$ has decreased to random performance and also an orderly behaviour of $MSE(x,y)$ correctly approaching minimum distortion of the data. The actual impact on Data Analytical outcomes will be discussed in another paper that is currently in preparation. Figure 2: The Absenteeism at Work data includes personal information and total absenteeism time over 3 years for employees at a courier company in Brazil. We have selected age as the undesirable protected characteristic and successfully removed it. Source: UCI Machine Learning Database. Creators: Andrea Martiniano, Ricardo Pinto Ferreira, and Renato Jose Sassi. $\bar{D}$ values denote proportion of correct predictions of race, while all other values are arbitrarily scaled. Figure 3: Performance data from schools in New York. The bias removed was a variable called ’Majority Black/Hispanic’. Source: Kaggle. Creators: PASSNYC. $\bar{D}$ values denote proportion of correct predictions of race, while all other values are arbitrarily scaled. Figure 4: The Heart Disease Dataset consists of blood measurements from a set of patients with and without heart disease. The bias variable removed was ’Sex’. Source: UCI Machine Learning Database. Creators: Hungarian Institute of Cardiology, University Hospital Zurich, University Hospital Basel, V.A. Medical Center Long Beach and Cleveland Clinic Foundation. $\bar{D}$ values denote proportion of correct predictions of gender, while all other values are arbitrarily scaled. Figure 5: The COMPAS Dataset consists of profiles of criminals from Broward County, Florida. The bias variable removed was ’Race’. Source: Kaggle. Creators: ProPublica. $\bar{D}$ values denote proportion of correct predictions of race, while all other values are arbitrarily scaled. Figure 6: The Communities and Crime Dataset includes statistics on various communities i $\bar{D}$ values denote proportion of correct predictions of race, while all other values are arbitrarily scaled.n the U.S. Bias variable removed was ’Majority Black’, indicating whether the community has a majority black population. Source: UCI Machine Learning Database. Creators: Michael Redmond. ## 2 Discussion It is apparent that the outcomes of Data Analytics [DA] and Machine Learning [ML] are often perpetuating the human bias present in the datasets and therefore enacting illegal discrimination. Constraining the DA/ML outcomes to be fair is problematic as there is no universally accepted definition of fairness while at the same time many of the notions of fairness are very hard to implement, disrupting DA pipelines and putting a significant extra load on DA resources. One apparent solution is to remove bias from the data before proceeding with DA as usual; however, these methods generally, cannot account for non-linear, non-binary and/or multivariate relationships between race (or other biasing factor) and the rest of the data [3]. This paper introduced _Fair Adversarial Networks_ [FANs] as a method that compensates for these shortcomings and provides a very significant improvement in both fairness and ease of use. There is no universally accepted, or even legally binding, notion of fairness that can be used for optimisation, while at the same time many definitions of fairness are mutually exclusive. Unless a definition of fairness is provided by regulatory bodies it seems unlikely optimising for parity between groups on a fairness measure can be a useful bias-removal approach. Even if a mathematical notion of fairness becomes agreed there is no guarantee that optimizing for parity on a training set achieves parity on the population- level. Furthermore, the need to optimise for fairness introduces an extra term into the cost function of any optimisation procedure, which is not compatible with current DA tools. Data analysts are currently not required to write their own loss functions or optimisation procedures, therefore including such a requirement would damage their ability to perform their jobs. Even if the data analysts become comfortable with this requirement, the huge time overhead of this task makes it unlikely it will be performed in practise. Removing information about protected characteristics from the data is an attractive alternative. It’s philosophically very simple - without knowledge of membership to a protected group (such as race) it should be impossible to discriminate based on it - therefore it removes the subjective nature of treating bias. It can also be made very simple, a single preprocessing step can remove bias from the data while the rest of DA/ML pipeline can remain exactly the same. However many methods removing bias from data fail. Removing the column containing the protected characteristic is clearly insufficient due to the presence of proxy variables. A number of methods go beyond removing the column containing the protected characteristic and attempt to de-correlate the other characteristics from the protected ones. However, these approaches generally cannot account for non-linear, non-binary, and/or multivariate relationships between the characteristics [3]. To counter these problems this paper has introduced FANs. FANs are a version of adversarial networks with two main components 1) an _autoencoder_ that encodes a fair representation of data, and a _Racist Network_ which is the adversary predicting the protected characteristics (e.g. race) from the data, the performance of which needs to be minimised. The autoencoder’s cost function consist of reconstruction error of the transformed data, and also the performance of the racist network, both of which are to be minimised. The Racist Network simply tries to achieve the best predictive performance on the protected characteristic, using the autoencoder’s output as its input. This system produces a data representation that is most similar to the original data, but at the same time from which the protected characteristics cannot (or at least are harder) be predicted. Any analytical methods can be subsequently used with such a representation. This paper describes the principles of FANs and demonstrates on five real- world, disparate datasets that FANs can indeed achieve their goal of removing the ability to predict the protected characteristic from the data, while minimising the difference between the original data and its fair reconstruction. The problematic part of training FANs is that we aim to remove _possibility to predict_ protected characteristic from reconstructed data. Possibility to predict implies full training, not just the current state of the adversarial process. The success of our algorithm across five disparate datasets crucially relies on our approximation of full-training performance of a neural network from a single forward pass. While this approximation will remain our trade secret, it is possible to replicate our success on a one-off basis using conventional approaches and heavy parameter-tuning. This paper has demonstrated that FANs can consistently succeed at removing bias from datasets while keeping the necessary alterations of the data to the minimum. FANs are particularly valuable because they can be used as a generic and easy to use data pre-processing step, allowing all Data Analysts to account for biases in their datasets without significant overheads. ### Limitations and Future Directions It is necessary to mention that under special circumstances FANs have the potential to make things worse for discriminated groups. FANs will remove all kinds of discrimination including the positive one, which might be a desirable way of breaking-up vicious cycles of deprivation in some areas. Optimising for two metrics at the same time is a compromise. The present paper has not attempted to analyse the residual discrimination which, while statistically insignificant, is likely still present. On the other hand the statistical insignificance can be seen as the criterion for success. Either way it is apparent that FANs provide a step in the right direction in respect to increasing the fairness. Lastly, it is unclear what the correct time to stop the training of Neural Networks is, and what the right hyper-parameters are. It is certain that our neural architecture, neither hyper-parameter choice is optimal. Especially with GANs, one wishes to have interactive, optimal control of hyper-parameters throughout training to stabilize the process and ensure convergence. Therefore we are now exploring Reinforcement Learning as a method to control these factors interactively throughout training. ## References * [1] Solon Barocas and Andrew D Selbst. Big data’s disparate impact. Cal. L. Rev., 104:671, 2016. * [2] Richard Berk, Hoda Heidari, Shahin Jabbari, Michael Kearns, and Aaron Roth. Fairness in criminal justice risk assessments: the state of the art. arXiv preprint arXiv:1703.09207, 2017. * [3] Tolga Bolukbasi, Kai-Wei Chang, James Y Zou, Venkatesh Saligrama, and Adam T Kalai. Man is to computer programmer as woman is to homemaker? debiasing word embeddings. In Advances in Neural Information Processing Systems, pages 4349–4357, 2016. * [4] Aylin Caliskan, Joanna J Bryson, and Arvind Narayanan. Semantics derived automatically from language corpora contain human-like biases. Science, 356(6334):183–186, 2017. * [5] Alexandra Chouldechova. Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. Big data, 5(2):153–163, 2017. * [6] Equal Employment Opportunity Commission et al. Uniform guidelines on employee selection procedures. Fed Register, 1:216–243, 1990. * [7] Virginia Eubanks. Automating inequality: How high-tech tools profile, police, and punish the poor. St. Martin’s Press, 2018. * [8] Michael Feldman, Sorelle A Friedler, John Moeller, Carlos Scheidegger, and Suresh Venkatasubramanian. Certifying and removing disparate impact. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 259–268. ACM, 2015. * [9] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pages 2672–2680, 2014. * [10] Sara Hajian, Josep Domingo-Ferrer, and Antoni Martinez-Balleste. Discrimination prevention in data mining for intrusion and crime detection. In Computational Intelligence in Cyber Security (CICS), 2011 IEEE Symposium on, pages 47–54. IEEE, 2011. * [11] Moritz Hardt, Eric Price, Nati Srebro, et al. Equality of opportunity in supervised learning. In Advances in neural information processing systems, pages 3315–3323, 2016. * [12] Toshihiro Kamishima, Shotaro Akaho, Hideki Asoh, and Jun Sakuma. Fairness-aware classifier with prejudice remover regularizer. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pages 35–50. Springer, 2012. * [13] Naveen Kodali, Jacob Abernethy, James Hays, and Zsolt Kira. On convergence and stability of gans. arXiv preprint arXiv:1705.07215, 2017. * [14] Cathy O’Neil. Weapons of math destruction: How big data increases inequality and threatens democracy. Broadway Books, 2016. * [15] Dino Pedreshi, Salvatore Ruggieri, and Franco Turini. Discrimination-aware data mining. In Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 560–568. ACM, 2008. * [16] Salah Rifai, Pascal Vincent, Xavier Muller, Xavier Glorot, and Yoshua Bengio. Contractive auto-encoders: Explicit invariance during feature extraction. In Proceedings of the 28th International Conference on International Conference on Machine Learning, pages 833–840. Omnipress, 2011\. * [17] Margery Austin Turner. Mortgage lending discrimination: A review of existing evidence. 1999\. * [18] Rich Zemel, Yu Wu, Kevin Swersky, Toni Pitassi, and Cynthia Dwork. Learning fair representations. In International Conference on Machine Learning, pages 325–333, 2013. * [19] James Zou and Londa Schiebinger. Ai can be sexist and racist—it’s time to make it fair, 2018.

2024-09-04T02:54:55.072158

2002.12146

arxiv-papers

FSQ-12-033 $HeadURL$ $Id$ FSQ-12-033 # Measurement of single-diffractive dijet production in proton-proton collisions at $\sqrt{s}=8\TeV$ with the CMS and TOTEM experiments The CMS and TOTEM Collaborations ###### Abstract Measurements are presented of the single-diffractive dijet cross section and the diffractive cross section as a function of the proton fractional momentum loss $\xi$ and the four-momentum transfer squared $t$. Both processes $\Pp\Pp\to\Pp\PX$ and $\Pp\Pp\to\PX\Pp$, with the proton scattering to either side of the interaction point, are measured, where $\PX$ includes at least two jets; the results of the two processes are averaged. The analyses are based on data collected simultaneously with the CMS and TOTEM detectors at the LHC in proton-proton collisions at $\sqrt{s}=8\TeV$ during a dedicated run with $\beta^{\ast}=90\unit{m}$ at low instantaneous luminosity and correspond to an integrated luminosity of $37.5\nbinv$. The single-diffractive dijet cross section $\sigma^{\Pp\PX}_{\mathrm{jj}}$, in the kinematic region $\xi<0.1$, $0.03<\abs{t}<1\GeV^{2}$, with at least two jets with transverse momentum $\pt>40\GeV$, and pseudorapidity $\abs{\eta}<4.4$, is $21.7\pm 0.9\stat\,^{+3.0}_{-3.3}\syst\pm 0.9\lum\unit{nb}$. The ratio of the single- diffractive to inclusive dijet yields, normalised per unit of $\xi$, is presented as a function of $x$, the longitudinal momentum fraction of the proton carried by the struck parton. The ratio in the kinematic region defined above, for $x$ values in the range $-2.9\leq\log_{10}x\leq-1.6$, is $R=(\sigma^{\Pp\PX}_{\mathrm{jj}}/\Delta\xi)/\sigma_{\mathrm{jj}}=0.025\pm 0.001\stat\pm 0.003\syst$, where $\sigma^{\Pp\PX}_{\mathrm{jj}}$ and $\sigma_{\mathrm{jj}}$ are the single-diffractive and inclusive dijet cross sections, respectively. The results are compared with predictions from models of diffractive and nondiffractive interactions. Monte Carlo predictions based on the HERA diffractive parton distribution functions agree well with the data when corrected for the effect of soft rescattering between the spectator partons. We dedicate this paper to the memory of our colleague and friend Sasha Proskuryakov, who started this analysis but passed away before it was completed. His contribution to the study of diffractive processes at CMS is invaluable. ## 0.1 Introduction In proton-proton ($\Pp\Pp$) collisions a significant fraction of the total cross section is attributed to diffractive processes. Diffractive events are characterised by at least one of the two incoming protons emerging from the interaction intact or excited into a low-mass state, with only a small energy loss. These processes can be explained by the exchange of a virtual object, the so-called Pomeron, with the vacuum quantum numbers [1]; no hadrons are therefore produced in a large rapidity range adjacent to the scattered proton, yielding a so-called large rapidity gap (LRG). A subleading exchange of Reggeons, as opposed to a Pomeron, also contributes to diffractive scattering, especially for large values of the proton fractional momentum loss $\xi$, and is required to describe diffractive data [2, 3, 4, 5]. While Pomerons mainly consist of gluons, Reggeons are mesons composed of a quark-antiquark pair. Hard diffraction has been studied in hadron-hadron collisions at the SPS at CERN [6], the Tevatron at Fermilab [7, 8, 9, 10, 11], the CERN LHC [12, 13], and in electron-proton ($\Pe\Pp$) collisions at the HERA collider at DESY [2, 3, 4, 5, 14]. Hard diffractive processes can be described in terms of the convolution of diffractive parton distribution functions (dPDFs) and hard scattering cross sections, which can be calculated in perturbative quantum chromodynamics (pQCD). The dPDFs have been determined by the HERA experiments [2, 4, 5] by means of fits to inclusive diffractive deep inelastic scattering data. The dPDFs have been successfully applied to describe different hard diffractive processes in $\Pe\Pp$ collisions. This success is based on the factorisation theorem proven for $\Pe\Pp$ interactions at large $Q^{2}$, and on the validity of the QCD evolution equations for the dPDFs [15, 16, 17]. However, in hard diffractive hadron-hadron collisions factorisation is broken because of the presence of soft rescattering between the spectator partons. This leads to a suppression of the observed diffractive cross section in hadron-hadron collisions [18]. The suppression factor, often called the rapidity gap survival probability ($\langle S^{2}\rangle$), is $\sim$10% at the Tevatron energies [9]. Experimentally, diffractive events can be selected either by exploiting the presence of an LRG or by measuring the scattered proton. The latter method is superior since it gives a direct measurement of $t$, the squared four momentum transfer at the proton vertex, and suppresses the contribution from events in which the proton dissociates into a low-mass state. The CMS Collaboration has previously reported a measurement of diffractive dijet production at $\sqrt{s}=7\TeV$ [12] that did not include information on the scattered proton. The ATLAS Collaboration has also measured dijet production with large rapidity gaps at $\sqrt{s}=7\TeV$ [13]. This article presents a measurement of dijet production with a forward, high longitudinal momentum proton at $\sqrt{s}=8\TeV$. It corresponds to the processes $\Pp\Pp\to\Pp\PX$ or $\Pp\Pp\to\PX\Pp$, with the proton scattering to either side of the interaction and $\PX$ including at least two jets. The system $\PX$ is measured in CMS and the scattered proton in the TOTEM roman pots (RPs). This process is referred to as single-diffractive dijet production. The single-diffractive dijet production cross section is measured as a function of $\xi$ and $t$ in the kinematic region $\xi<0.1$ and $0.03<\abs{t}<1\GeV^{2}$, in events with at least two jets, each with transverse momentum $\pt>40\GeV$ and pseudorapidity $\abs{\eta}<4.4$. The ratio of the single-diffractive to inclusive dijet cross sections is measured as a function of $x$, the longitudinal momentum fraction of the proton carried by the struck parton for $x$ values in the range $-2.9\leq\log_{10}x\leq-1.6$. This is the first measurement of hard diffraction with a measured proton at the LHC. ## 0.2 The CMS and TOTEM detectors The central feature of the CMS apparatus is a superconducting solenoid of 6m internal diameter, providing a magnetic field of 3.8T. Within the superconducting solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. Forward calorimeters extend the pseudorapidity coverage provided by the barrel and endcap detectors. The forward hadron (HF) calorimeter uses steel as an absorber and quartz fibers as the sensitive material. The two HFs are located 11.2m from the interaction region, one on each end, and together they provide coverage in the range $3.0<\abs{\eta}<5.2$. Muons are measured in gas-ionisation detectors embedded in the steel flux-return yoke outside the solenoid. When combining information from the entire detector, including that from the tracker, the jet energy resolution amounts typically to 15% at 10, 8% at 100, and 4% at 1, to be compared to about 40, 12, and 5%, respectively, obtained when ECAL and HCAL alone are used. In the region $\abs{\eta}<1.74$, the HCAL cells have widths of 0.087 in pseudorapidity and 0.087 in azimuth ($\phi$). In the $\eta$-$\phi$ plane, and for $\abs{\eta}<1.48$, the HCAL cells map on to $5{\times}5$ arrays of ECAL crystals to form calorimeter towers projecting radially outwards from close to the nominal interaction point. For $\abs{\eta}>1.74$, the coverage of the towers increases progressively to a maximum of 0.174 in $\Delta\eta$ and $\Delta\phi$. Within each tower, the energy deposits in the ECAL and HCAL cells are summed to define the calorimeter tower energies, subsequently used to provide the energies and directions of hadronic jets. The reconstructed vertex with the largest value of summed charged-particle track $\pt^{2}$ is taken to be the primary interaction vertex. Tracks are clustered based on the $z$ coordinate of the track at the point of closest approach to the beamline. In the vertex fit, each track is assigned a weight between 0 and 1, which reflects the likelihood that it genuinely belongs to the vertex. The number of degrees of freedom in the fit is strongly correlated with the number of tracks arising from the interaction region. The particle-flow (PF) algorithm [19] aims to reconstruct and identify each individual particle in an event with an optimised combination of information from the various elements of the CMS detector. The energy of photons is directly obtained from the ECAL measurement, corrected for zero-suppression effects. The energy of electrons is determined from a combination of the electron momentum at the primary interaction vertex as determined by the tracker, the energy of the corresponding ECAL cluster, and the energy sum of all bremsstrahlung photons spatially compatible with originating from the electron track. The energy of muons is obtained from the curvatures of the corresponding track. The energy of charged hadrons is determined from a combination of their momentum measured in the tracker and the matching ECAL and HCAL energy deposits, corrected for zero-suppression effects and for the response function of the calorimeters to hadronic showers. Finally, the energy of neutral hadrons is obtained from the corresponding corrected ECAL and HCAL energies. Hadronic jets are clustered from these reconstructed particles using the anti- algorithm [20, 21]. The jet momentum is determined as the vectorial sum of all PF candidate momenta in the jet, and is found from simulation to be within 5 to 10% of the true momentum over the whole spectrum and detector acceptance. Jet energy corrections are derived from simulation, and are confirmed with in situ measurements of the energy balance in dijet, multijet, photon + jet, and + jet events [22]. The jet resolution in the simulation is scaled upwards by around 15% in the barrel region, 40% in the endcaps and 20% in the forward region to match the resolution in the data. Additional selection criteria are applied to each event to remove spurious jet-like features originating from isolated noise patterns in some HCAL regions [23]. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [24]. The TOTEM experiment [25, 26] is located at the LHC interaction point (IP) 5 together with the CMS experiment. The RP system is the subdetector relevant for measuring scattered protons. The RPs are movable beam pipe insertions that approach the LHC beam very closely (few) to detect protons scattered at very small angles or with small $\xi$. The proton remains inside the beam pipe and its trajectory is measured by tracking detectors installed inside the RPs. They are organised in two stations placed symmetrically around the IP; one in LHC sector 45 (positive $z$), the other in sector 56 (negative $z$). Each station is formed by two units: near (215m from the IP) and far (220m from the IP). Each unit includes three RPs: one approaching the beam from the top, one from the bottom and one horizontally. Each RP hosts a stack of 10 silicon strip sensors (pitch 66) with a strongly reduced insensitive region at the edge facing the beam (few tens of ). Five of these planes are oriented with the silicon strips at a $+45^{\circ}$ angle with respect to the bottom of the RP and the other five have the strips at a $-45^{\circ}$ angle. The beam optics relates the proton kinematics at the IP and at the RP location. A proton emerging from the interaction vertex ($x^{\ast}$, $y^{\ast}$) at horizontal and vertical angles $\theta_{x}^{\ast}$ and $\theta_{y}^{\ast}$, with a fractional momentum loss $\xi$, is transported along the outgoing beam through the LHC magnets. It arrives at the RPs at the transverse position: $\displaystyle x(z_{\mathrm{RP}})$ $\displaystyle=L_{x}(z_{\mathrm{RP}})\,\theta_{x}^{\ast}+v_{x}(z_{\mathrm{RP}})\,x^{\ast}-D_{x}(z_{\mathrm{RP}})\,\xi,$ (1) $\displaystyle y(z_{\mathrm{RP}})$ $\displaystyle=L_{y}(z_{\mathrm{RP}})\,\theta_{y}^{\ast}+v_{y}(z_{\mathrm{RP}})\,y^{\ast}-D_{y}(z_{\mathrm{RP}})\,\xi,$ relative to the beam centre. This position is determined by the optical functions, characterising the transport of protons in the beamline and controlled via the LHC magnet currents. The effective length $L_{x,y}(z)$, magnification $v_{x,y}(z)$ and horizontal dispersion $D_{x}(z)$ quantify the sensitivity of the measured proton position to the scattering angle, vertex position, and fractional momentum loss, respectively. The dispersion in the vertical plane, $D_{y}$, is nominally zero. For the present measurement, a special beam optical setup with $\beta^{\ast}=90\unit{m}$ was used, where $\beta^{\ast}$ is the value of the amplitude function of the beam at the IP. This optical setup features parallel-to-point focussing ($v_{y}\sim 0$) and large $L_{y}$, making $y$ at RP directly proportional to $\theta_{y}^{\ast}$, and an almost vanishing $L_{x}$ and $v_{x}$, implying that any horizontal displacement at the RP is approximately proportional to $\xi$. Protons can hence be measured with large detector acceptance in the vertical RPs that approach the beam from the top and bottom. To reduce the impact of imperfect knowledge of the optical setup, a calibration procedure [27] has been applied. This method uses elastic scattering events and various proton observables to determine fine corrections to the optical functions presented in Eq. (1). For the RP alignment, a three- step procedure [26] has been applied: beam-based alignment prior to the run (as for the LHC collimators) followed by two offline steps. First, track-based alignment for the relative positions among RPs, and second, alignment with elastic events for the absolute position with respect to the beam. The final uncertainties per unit (common for top and bottom RPs) are: 2(horizontal shift), 100(vertical shift), and 0.2mrad (rotation about the beam axis). The kinematic variables ($\xi$, $\theta_{x}^{\ast}$, $\theta_{y}^{\ast}$ as well as $t$) are reconstructed with the use of parametrised proton transport functions [26]. The values of the optical functions vary with $\xi$, an effect that is taken into account by the optics parametrisation. The details of the reconstruction algorithms and optics parametrisation are discussed in Refs. [26, 28]. The momentum loss reconstruction depends mostly on the horizontal dispersion, which is determined with a precision better than 10%. The scattering angle resolution depends mainly on the angular beam divergence and in the horizontal plane also on the detector resolution, whereas the momentum loss resolution depends mainly on the optics [29]. The $\xi$ resolution is about $\sigma$($\xi$) = 0.7% and the $\theta_{y}^{\ast}$ and the $\theta_{x}^{\ast}$ resolutions 2.4$\mu$rad and 25$\mu$rad, respectively. ## 0.3 Event kinematics Figure 1 shows a schematic diagram of the single-diffractive reaction $\Pp\Pp\to\PX\Pp$ with $\PX$ including two high-$\pt$ jets. Single-diffractive dijet production is characterised by the presence of a high-energy proton, which escapes undetected by the CMS detector, and the system $\PX$, which contains high-$\pt$ jets, separated from the proton by an LRG. Figure 1: Schematic diagram of single-diffractive dijet production. The exchange of a virtual object with the vacuum quantum numbers (a Pomeron) is indicated by the symbol IP. The diagram shows an example of the $\Pg\Pg\to\text{dijet}$ hard scattering process; the $\Pq\Pq$ and $\Pg\Pq$ initial states also contribute. The proton is scattered at small angles, has small fractional momentum loss $\xi=1-{\abs{\mathbf{p}_{f}}}/{\abs{\mathbf{p}_{i}}}$, and small absolute value of the 4-momentum transfer squared $t=(p_{f}-p_{i})^{2}$, where $p_{i}$ and $p_{f}$ are the four-momenta of the incoming and outgoing protons, respectively. The scattered proton does not leave the beam pipe and can only be detected by using the TOTEM RP detectors, which make a direct measurement of $t$ and $\xi$ (hereafter referred to as $\xi_{\text{\tiny TOTEM}}$). If only CMS information is used, as in Ref. [12], $\xi$ can be estimated only from the energies and longitudinal momenta of the particles measured in CMS: ${\xi}_{\text{\tiny CMS}}^{\pm}=\frac{\sum_{i}\left(E^{i}\pm p_{z}^{i}\right)}{\sqrt{s}},$ (2) where the sum is carried out with PF objects. The positive (negative) sign corresponds to the scattered proton moving towards the positive (negative) $z$ direction. In this case, $t$ cannot be measured. The combination of the limited CMS pseudorapidity coverage ($\abs{\eta}<5$) and the detector inefficiency causes $\xi_{\text{\tiny CMS}}$ to be smaller than $\xi_{\text{\tiny TOTEM}}$ in general, $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}\leq 0$. However, the limited detector resolution may cause $\xi_{\text{\tiny CMS}}$ to be larger than $\xi_{\text{\tiny TOTEM}}$. The momentum fraction of the partons initiating the hard scattering, $x^{+}$ and $x^{-}$, can be estimated from the energies and longitudinal momenta of the measured jets as: $x^{\pm}=\frac{\sum_{\text{\tiny jets}}\left(E^{\text{\tiny jet}}\pm p_{z}^{\text{\tiny jet}}\right)}{\sqrt{s}},$ (3) where the sum is carried out over the two highest transverse momentum jets in the event, and an additional third jet, if present. The positive (negative) sign corresponds to the incoming proton moving towards the positive (negative) $z$ direction. Finally, the fraction $\beta$ of the Pomeron momentum carried by the interacting parton is measured from the values of $x^{\pm}$ and $\xi_{\text{\tiny TOTEM}}$ as $\beta=x^{\pm}/\xi_{\text{\tiny TOTEM}}$. ## 0.4 Data samples The data were collected in July 2012 during a dedicated run with low probability ($\sim$6–10%) of overlapping $\Pp\Pp$ interactions in the same bunch crossing (pileup) and a nonstandard $\beta^{*}=90\unit{m}$ beam optics configuration. These data correspond to an integrated luminosity of $\lumi=37.5\nbinv$. Events are selected by trigger signals that are delivered simultaneously to the CMS and TOTEM detectors. The first level of the CMS trigger system (L1) is used. The L1 signal is propagated to the TOTEM electronics to enable the simultaneous readout of the CMS and TOTEM subdetectors. The CMS orbit-counter reset signal, delivered to the TOTEM electronics at the start of the run, assures the time synchronisation of the two experiments. The CMS and the TOTEM events are combined offline based on the LHC orbit and bunch numbers. ## 0.5 Monte Carlo simulation The simulation of nondiffractive dijet events is performed with the pythia6 (version 6.422) [30], pythia8 (version 8.153) [31], and herwig6 [32] Monte Carlo (MC) event generators. The underlying event is simulated in pythia6 with tune Z2* [33] and in pythia8 with tunes 4C [34], CUETP8M1, and CUETP8S1 [35]. Single-diffractive dijet events are simulated with the pythia8 and pomwig (version 2.0) [36] generators. Hard diffraction is simulated in pythia8 using an inclusive diffraction model, where both low- and high-mass systems are generated [37]. High-mass diffraction is simulated using a perturbative description. Pomeron parton densities are introduced and the diffractive process is modelled as a proton-Pomeron scattering at a reduced centre-of-mass energy. The default generator settings are used, including that for the proton-Pomeron total cross section. Multiparton interactions (MPI) are included within the proton-Pomeron system to provide cross sections for parton-parton interactions. In this model, the presence of secondary interactions does not lead to a suppression of the visible diffractive cross section. Additionally, pythia8 implements a model to simulate hard-diffractive events based on a direct application of dPDFs, and a dynamical description of the rapidity gap survival probability in diffractive hadron-hadron interactions [38]. In this model an event is classified as diffractive only when no MPI are generated. We refer to this implementation as the dynamic gap (DG) model. Single-diffractive dijet events using the inclusive diffraction model are simulated with pythia8, tunes 4C and CUETP8M1. The simulation of diffractive dijet events using the DG model is performed with pythia8 version 8.223 [38] with the underlying event tune CUETP8M1. These pythia8 tunes give a fair description of the charged-particle pseudorapidity and distributions in a sample with a large fraction of single-diffractive inelastic events [39, 35, 40]. The pomwig generator is based on herwig6 and implements dPDFs to simulate hard-diffractive processes. The simulation uses dPDFs from a fit to deep inelastic scattering data (H1 fit B [2]). The pomwig generator uses a next-to- leading order dPDF fit, whereas pythia8 uses a leading order dPDF fit. When using pomwig, a constant factor $\langle S^{2}\rangle=7.4\%$ is applied to account for the rapidity gap survival probability leading to the suppression of the diffractive cross section. This value is calculated from the ratio of the measured diffractive cross section and the prediction from pomwig, as described in Section 0.8.2. Both Pomeron and Reggeon exchange contributions are generated. Reggeon exchange is not simulated in pythia8. To improve the description of the data by the MC samples, correction factors are applied event-by-event as a function of $\beta$, by a reweighting procedure. The correction modifies the event distribution as a function of $\beta$ by up to 40%, and the $\log_{10}x$ and $\xi$ distributions by as much as 30% and 8%, respectively. The correction has a negligible effect on the $t$ distribution. The generated events are processed through the simulation of the CMS detector, based on [41], and reconstructed in the same manner as the data. The acceptance and resolution of the TOTEM RP detectors are parametrised as a function of the proton kinematics, as discussed below. All samples are simulated without pileup. ### 0.5.1 Roman pot detectors acceptance and resolution The proton path from the IP to the TOTEM RPs is calculated using a parametrisation of the LHC optics [27]. To obtain a realistic simulation of the scattered proton, the following procedure is used: * • Proton transport: The simulation of the RP detectors acceptance is parametrised in terms of the vertex position, the proton scattering angles at the vertex $\theta_{x}^{\ast}$ and $\theta_{y}^{\ast}$, and $\xi$. The incident beam energy spread and beam divergence are also simulated [29]. * • Reconstruction of $t$ and $\xi$: The detector-level distributions of $t$ and $\xi$ are obtained from the scattering angles $\theta_{x}^{\ast}$ and $\theta_{y}^{\ast}$, where the correlation between the $\xi$ and $\theta_{x}^{\ast}$ uncertainties is taken into account [26]. The generated values of $\theta_{x}^{\ast}$ and $\theta_{y}^{\ast}$ are spread by $25\unit{$\mu$rad}$ and $2.4\unit{$\mu$rad}$, respectively. These values include the effects of detector resolution, as well as those of the beam optics and the beam divergence. * • Proton reconstruction inefficiency: The track reconstruction in the RPs may fail for several reasons: inefficiency of the silicon sensors, interaction of the proton with the RP mechanics, or the simultaneous presence of a beam halo particle or a proton from a pileup interaction. The silicon strips of the detectors in an RP are oriented in two orthogonal directions; this allows for good rejection of inclined background tracks, but makes it very difficult to reconstruct more than one track almost parallel to the beam direction [26]. These uncorrelated inefficiencies are evaluated from elastic scattering data [29], and amount to $\sim$6%. To correct for this, an extra normalisation factor is applied, obtained separately for protons traversing the RPs on either side of the IP. ## 0.6 Event selection Dijet events are selected online by requiring at least two jets with $\pt>20\GeV$ [42]. The efficiency of this trigger selection is estimated with a sample of minimum bias events, events collected with a loose trigger intended to select inelastic collisions with as little bias as possible, and containing a leading jet with $\pt$, as reconstructed offline, of at least 40. The fraction of dijet events accepted by the trigger is calculated as a function of the subleading jet $\pt$. The efficiency is above $94\%$ for $\pt>40\GeV$. The offline selection requires at least two jets with $\pt>40\GeV$ and $\abs{\eta}<4.4$. Jets are reconstructed from PF objects with the anti- algorithm with a distance parameter $R=0.5$. The reconstructed jet energy is corrected with the procedure described in Ref. [22]. The parton momentum fractions $x^{+}$ and $x^{-}$ are reconstructed using Eq. (3) from the two highest transverse momentum jets and an additional third jet, if present. The latter is selected with $\pt>20\GeV$. In addition, the selection requires at least one reconstructed primary interaction vertex and at least one reconstructed proton track in the RP stations. The fit of the reconstructed vertex is required to have more than four degrees of freedom. Events with protons in the RP stations on both sides are rejected if their kinematics are consistent with those of elastic scattering. Elastic scattering events, which are present in the data sample because of pileup, are identified by the presence of two proton tracks in opposite directions, in a diagonal configuration: the protons traverse the two top RPs in sector 45 and the two bottom RPs in sector 56, or vice versa. The horizontal and vertical scattering angles are required to match within the measured resolutions. These requirements are similar to those described in Ref. [29]. To avoid detector edges with rapidly varying efficiency or acceptance, as well as regions dominated by secondary particles produced by aperture limitations in the beamline upstream of the RPs, proton track candidates are selected if the corresponding hit coordinates on the RP stations satisfy the following fiducial requirements: $0<x<7\mm$ and $8.4<\abs{y}<27\mm$, where $x$ and $y$ indicate the horizontal and vertical coordinates of the hit with respect to the beam. To suppress background from secondary particles and pileup in the RPs, the reconstructed proton track is selected if it is associated to one track element in both top or both bottom RPs on a given side. The kinematic requirements $0.03<\abs{t}<1.0\GeV^{2}$ and $0<\xi_{\text{\tiny TOTEM}}<0.1$ are then applied. For signal events, one expects $\xi_{\text{\tiny CMS}}$ to be smaller than $\xi_{\text{\tiny TOTEM}}$, $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}\leq 0$ (as discussed in Section 0.3). This selection is imposed to suppress the contribution of pileup and beam halo events, in which the proton is uncorrelated with the hadronic final state $\PX$ measured in the CMS detector. Roughly $6\%$ of signal events are rejected by this requirement, as estimated from a simulation of single-diffractive dijet production. Table 0.6 shows the number of events passing each selection. The number of events with the proton detected in the RPs in sector 45 (56) after all the selections is 368 (420). A difference in the yields for events with a proton in sector 45 and 56 could notably arise from different background contributions, which is discussed in Section 0.7. Both an imperfect knowledge of the optical functions, especially the horizontal dispersion, discussed in Section 0.8.1, and statistical fluctuations of the two mostly independent event samples contribute to the difference. Number of events after each selection. Selection Sector 45 Sector 56 At least 2 jets ($\pt>40\GeV$, $\abs{\eta}<4.4$) 427689 Elastic scattering veto 405112 Reconstructed proton 9530 RP and fiducial region 2137 3033 $0.03<\abs{t}<1.0\GeV^{2}$, $0<\xi_{\text{\tiny TOTEM}}<0.1$ 1393 1806 $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}\leq 0$ 368 420 ## 0.7 Background The main background is due to the overlap of a $\Pp\Pp$ collision in the CMS detector and an additional track in the RP stations, originating from either a beam halo particle or an outgoing proton from a pileup interaction. Pileup and beam halo events are not simulated, but they are present in the data. To estimate the pileup and beam halo contribution in the data, a zero bias sample consisting of events from randomly selected, nonempty LHC bunch crossings is used. Events with a proton measured in the RP stations and with any number of reconstructed vertices are selected from the zero bias data set. Such events are denoted by ZB in the following. The RP information from events in the zero bias sample is added to diffractive and nondiffractive events generated with pomwig and pythia6, respectively. The mixture of MC and ZB events simulates data events in the presence of pileup and beam halo. The pomwig sample is normalised assuming a rapidity gap survival probability factor of 7.4%, as discussed in Section 0.5. The MC and ZB event mixture is then passed through the selection procedure illustrated in Section 0.6, except for the requirement ${\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}\leq 0}$, which is not applied. Such mixed events with a proton in the RPs are considered as signal if the proton originates from the MC simulated sample, or as background if it originates from the ZB sample. If an event has a proton from both the MC sample and the ZB sample, the proton with smaller $\xi$ is chosen. However, the probability of such a combination is small and none of these events pass all the selections. Figure 2 shows the distribution of $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}$ for the data compared to the MC+ZB event mixture. The requirement $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}\leq 0$ selects signal events and rejects the kinematically forbidden region populated by the MC+ZB background events (filled histogram). The background distribution is normalised to the data in the $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}$ region from 0.048 to 0.4, which is dominated by background events. The background is estimated separately for events with a proton traversing the two top (top-top) or the two bottom (bottom-bottom) RPs on each side. The top- top and bottom-bottom distributions are similar. Figure 2 shows the sum of the two contributions. The background contribution for events with a proton detected in sector 56 (right panel of Fig. 2) is larger than that for events with a proton detected in sector 45 (left panel of Fig. 2). The remaining contamination of background in the signal region is estimated to be $15.7\%$ for events in which the proton is detected in sector 45 and $16.8\%$ for those in which the proton is detected in sector 56. Figure 3 shows the distribution of $\xi_{\text{\tiny TOTEM}}$ for the data and the MC+ZB sample, before and after the $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}\leq 0$ requirement, as well as the distribution of $t$, after the $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}\leq 0$ selection. The sum of the top-top and bottom-bottom combinations is used. The data and the MC+ZB sample are in good agreement. Figure 2: Distribution of $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}$ for events with a reconstructed proton in sector 45 (left) and sector 56 (right). The data are indicated by solid circles. The blue histogram is the mixture of pomwig or pythia6 and zero bias (ZB) data events described in the text. An event with a proton measured in the RPs contributes to the open histogram (signal) if the proton originates from the MC sample, or to the filled histogram (background) if it originates from the ZB sample. Figure 3: Distribution of $\xi_{\text{\tiny TOTEM}}$ before (upper) and after (middle) the $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}$ requirement and distribution of $t$ after the $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}$ requirement (lower) for events in which the proton is detected in sector 45 (left) and sector 56 (right). The data are indicated by solid circles. The blue histogram is the mixture of pomwig or pythia6 and zero bias (ZB) data events described in the text. An event with the proton measured in the RPs contributes to the open histogram (signal) if the proton originates from the MC sample, or to the filled histogram (background) if it originates from the ZB sample. An alternative method, used at HERA [4], takes two events randomly chosen from the data sample. First, $\xi_{\text{\tiny CMS}}$ is sampled from events that have passed the dijet selection; $\xi_{\text{\tiny TOTEM}}$ is then taken from events with $\xi_{\text{\tiny CMS}}>0.12$ that have passed the event selection described in Section 0.6, except for the ${\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}}$ requirement, to select proton tracks considered to be mostly from background. These two values are used to plot the $\xi_{\text{\tiny CMS}}-\xi_{\text{\tiny TOTEM}}$ distribution, which is normalised to the data in a region dominated by background. The remaining contamination in the signal region is $\sim$19% both for events with a proton detected in sector 45 and for those with a proton in sector 56. The ZB method is used in this analysis. Half the difference between the results of the two methods is taken as an estimate of the systematic uncertainty of the background subtraction procedure. ## 0.8 Results In this section the measurements of the differential cross sections $\rd\sigma/{\rd}t$, $\rd\sigma/\rd\xi$, and the ratio $R(x)$ of the single- diffractive ($\sigma^{\Pp\PX}_{\mathrm{jj}}(x)$) to inclusive dijet cross sections ($\sigma_{\mathrm{jj}}(x)$) are presented. The ratio $R(x)$, normalised per unit of $\xi$, is defined by: $R(x)=\frac{\sigma^{\Pp\PX}_{\mathrm{jj}}(x)/\Delta\xi}{\sigma_{\mathrm{jj}}(x)},$ (4) where $\Delta\xi=0.1$. The cross sections are calculated in the kinematic region $\xi<0.1$, $0.03<\abs{t}<1\GeV^{2}$, with at least two jets at a stable-particle level with $\pt>40\GeV$ and $\abs{\eta}<4.4$. The ratio $R(x)$ is calculated for $x$ values in the region $-2.9\leq\log_{10}x\leq-1.6$. In the following, the estimated background is subtracted from the number of single-diffractive dijet candidates following the procedure described in the previous section. The differential cross sections for dijet production in bins of $t$ and $\xi$ are evaluated as: $\displaystyle\frac{\rd\sigma^{\Pp\PX}_{\mathrm{jj}}}{{\rd}t}$ $\displaystyle=\mathcal{U}\left\\{\frac{N^{\Pp\PX}_{\mathrm{jj}}}{\mathcal{L}A_{\text{\tiny CMS-TOTEM}}{\Delta t}}\right\\},$ (5) $\displaystyle\frac{\rd\sigma^{\Pp\PX}_{\mathrm{jj}}}{\rd\xi}$ $\displaystyle=\mathcal{U}\left\\{\frac{N^{\Pp\PX}_{\mathrm{jj}}}{\mathcal{L}A_{\text{\tiny CMS-TOTEM}}{\Delta\xi}}\right\\},$ where $N^{\Pp\PX}_{\mathrm{jj}}$ is the measured number of single-diffractive dijet candidates per bin of the distribution after subtracting the estimated background; ${\Delta t}$ and ${\Delta\xi}$ are the bin widths, and $\mathcal{L}$ is the integrated luminosity. The factors $A_{\text{\tiny CMS- TOTEM}}$ indicate the acceptance of CMS and TOTEM for single-diffractive dijet events. Unfolding corrections, represented by the symbol $\mathcal{U}$ in Eq. (5), are applied to account for the finite resolution of the reconstructed variables used in the analysis. They are evaluated with pomwig, pythia8 4C and pythia8 CUETP8M1. The results presented are the average of those obtained with the different unfolding corrections. The measured cross sections are obtained by unfolding the data using the D’Agostini method with early stopping [43]. In this method the regularisation parameter is the number of iterations used, which is optimised to obtain a relative $\chi^{2}$ variation between iterations lower than 5%. The ratio $R(x)$ of the single-diffractive to inclusive dijet cross sections is evaluated as a function of $x$ as: $R(x)=\frac{\sigma^{\Pp\PX}_{\mathrm{jj}}(x)/\Delta\xi}{\sigma_{\mathrm{jj}}(x)}=\frac{\mathcal{U}\left\\{N^{\Pp\PX}_{\mathrm{jj}}/A_{\text{\tiny CMS- TOTEM}}\right\\}/\Delta\xi}{\mathcal{U}\left\\{N_{\mathrm{jj}}/A_{\text{\tiny CMS}}\right\\}},$ (6) where $N^{\Pp\PX}_{\mathrm{jj}}$ is the number of single-diffractive dijet candidates with $\xi_{\text{\tiny TOTEM}}<0.1$, and $N_{\mathrm{jj}}$ is the total number of dijet events without the requirement of a proton detected in the RPs. This number is dominated by the nondiffractive contribution. The symbol $A_{\text{\tiny CMS-TOTEM}}$ indicates the acceptance of CMS and TOTEM for single-diffractive dijet events, evaluated with pomwig, pythia8 4C and pythia8 CUETP8M1; $A_{\text{\tiny CMS}}$ is the acceptance for nondiffractive dijet production ($\pt>40\GeV$, $\abs{\eta}<4.4$), evaluated with pythia6, pythia8 4C, pythia8 CUETP8M1, pythia8 CUETP8S1, and herwig6. The acceptance includes unfolding corrections to the data with the D’Agostini method with early stopping, denoted by the symbol $\mathcal{U}$ in Eq. (6). ### 0.8.1 Systematic uncertainties The systematic uncertainties are estimated by varying the selections and modifying the analysis procedure, as discussed in this Section. Tables 0.8.2 and 0.8.3 summarise the main systematic uncertainties of the single- diffractive cross section and the ratio of the single-diffractive and inclusive dijet cross sections, respectively, presented in Sections 0.8.2 and 0.8.3. * • Trigger efficiency: The trigger efficiency is calculated as a function of the subleading jet $\pt$ using a fit to the data. The sensitivity to the trigger efficiency determination is estimated by varying the fit parameters within their uncertainties. This variation corresponds to a trigger efficiency that increases or decreases by roughly 2% at jet $\pt=40\GeV$ and less than 1% at $\pt=50\GeV$. * • Calorimeter energy scale: The reconstruction of $\xi_{\text{\tiny CMS}}$ is affected by the uncertainty in the calorimeter energy scale and is dominated by the HF contribution. This uncertainty is estimated by changing the energy of the PF candidates by $\pm 10\%$ [12, 44]. * • Jet energy scale and resolution: The energy of the reconstructed jets is varied according to the jet energy scale uncertainty following the procedure described in Ref. [22]. The systematic uncertainty in the jet energy resolution is estimated by varying the scale factors applied to the MC, as a function of pseudorapidity. The uncertainties obtained from the jet energy scale and resolution are added in quadrature. The effect of the jet energy resolution uncertainty amounts to less than 1% of the measured cross section. * • Background: Half the difference between the results of the ZB and HERA methods used to estimate the background, described in Section 0.7, is an estimate of the effect of the systematic uncertainty of the background. * • RP acceptance: The sensitivity to the size of the fiducial region for the impact position of the proton in the RPs is estimated by modifying its vertical boundaries by $200\mum$ and by reducing the horizontal requirement by 1, to $0<x<6\mm$. Half the difference of the results thus obtained and the nominal ones is used as a systematic uncertainty. The uncertainties obtained when modifying the vertical and horizontal boundaries are added in quadrature. * • Resolution: The reconstructed variables $t$ and $\xi$ are calculated by applying two methods: either directly, with a resolution function depending on each of these variables, or indirectly from the scattering angles $\theta_{x}^{\ast}$ and $\theta_{y}^{\ast}$. Half the difference between the results using the two methods is taken as a systematic uncertainty. * • Horizontal dispersion: The reconstructed $\xi$ value depends on the optical functions describing the transport of the protons from the interaction vertex to the RP stations, and specifically the horizontal dispersion. This uncertainty is calculated by scaling the value of $\xi$ by $\pm 10\%$. This value corresponds to a conservative limit of the possible horizontal dispersion variation with respect to the nominal optics. * • $t$-slope: The sensitivity to the modelling of the exponential $t$-slope is quantified by replacing its value in pomwig by that measured in the data. Half the difference between the results thus found and the nominal results is used as an estimate of the uncertainty. * • $\beta$-reweighting: Half the difference of the results with and without the reweighting as a function of $\beta$ in pomwig (as discussed in Section 0.5) is included in the systematic uncertainty. The effect amounts to less than 1% of the single-diffractive cross section and less than about 6% of the single- diffractive to inclusive dijet cross section ratio versus $x$. * • Acceptance and unfolding: Half the maximum difference between the single- diffractive cross section results found by unfolding with pomwig, pythia8 4C, and pythia8 CUETP8M1 is taken as a further component of the systematic uncertainty. Likewise for the results obtained with pythia6 Z2*, pythia8 4C, pythia8 CUETP8M1 and pythia8 CUETP8S1 for the inclusive dijet cross section. * • Unfolding regularisation: The regularisation parameter used in the unfolding, given by the number of iterations in the D’Agostini method used in this analysis, is optimised by calculating the relative $\chi^{2}$ variation between iterations. The value is chosen such that the $\chi^{2}$ variation is below 5%. The number of iterations when the relative variation of $\chi^{2}$ is below 2% is also used and half the difference with respect to the nominal is taken as a systematic uncertainty. * • Unfolding bias: A simulated sample, including all detector effects, is unfolded with a different model. The difference between the corrected results thus obtained and those at the particle level is an estimate of the bias introduced by the unfolding procedure. Half the maximum difference obtained when repeating the procedure with all generator combinations is a measure of the systematic uncertainty related to the unfolding. * • Integrated luminosity: The uncertainty in the integrated luminosity is 4%, measured using a dedicated sample collected by TOTEM during the same data taking period [29]. The total systematic uncertainty is calculated as the quadratic sum of the individual contributions. The uncertainties in the jet energy scale and horizontal dispersion are the dominant contributions overall. ### 0.8.2 Extraction of the cross section as a function of $t$ and $\xi$ Figure 4 shows the differential cross section as a function of $t$ and $\xi$, integrated over the conjugate variable. The results from events in which the proton is detected on either side of the IP are averaged. Figure 4: Differential cross section as a function of $t$ (left) and as a function of $\xi$ (right) for single-diffractive dijet production, compared to the predictions from pomwig, pythia8 4C, pythia8 CUETP8M1, and pythia8 DG. The pomwig prediction is shown with no correction for the rapidity gap survival probability ($\langle S^{2}\rangle=1$) and with a correction of $\langle S^{2}\rangle=7.4\%$. The vertical bars indicate the statistical uncertainties and the yellow band indicates the total systematic uncertainty. The average of the results for events in which the proton is detected on either side of the interaction point is shown. The ratio between the data and the pomwig prediction, when no correction for the rapidity gap survival probability is applied, is shown in the bottom. The data are compared to pomwig, pythia8 4C, pythia8 CUETP8M1, and pythia8 DG. The pomwig prediction is shown for two values of the suppression of the diffractive cross section, the rapidity gap survival probability, represented by $\langle S^{2}\rangle$. When $\langle S^{2}\rangle=1$, no correction is applied. The resulting cross sections are higher than the data by roughly an order of magnitude, in agreement with the Tevatron results [9, 10, 11]. The pomwig prediction is also shown with the correction $\langle S^{2}\rangle=7.4\%$, calculated from the ratio of the measured diffractive cross section and the MC prediction, as discussed below. After this correction, pomwig gives a good description of the data. The pomwig prediction is shown in Fig. 4 as the sum of the Pomeron ($\Pp\mathrm{I}\\!\mathrm{P}$), Reggeon ($\Pp\mathrm{I}\\!\mathrm{R}$) and Pomeron-Pomeron ($\mathrm{I}\\!\mathrm{P}\mathrm{I}\\!\mathrm{P}$) exchange contributions, while pythia8 includes only the Pomeron ($\Pp\mathrm{I}\\!\mathrm{P}$) contribution. pythia8 4C and pythia8 CUETP8M1 predict cross sections higher than the data by up to a factor of two. The pythia8 DG model shows overall a good agreement with the data. No correction is applied to the normalisation of the pythia8 samples. The pythia8 DG model is the only calculation that predicts the cross section normalisation without an additional correction. The ratio between the data and the pomwig predictions is shown in the bottom of the left and right panels of Fig. 4. No correction is applied for the rapidity gap survival probability ($\langle S^{2}\rangle=1$). Within the uncertainties, no significant dependence on $t$ and $\xi$ is observed. The value of the cross section for single-diffractive dijet production, measured in the kinematic region $\pt>40\GeV$, $\abs{\eta}<4.4$, $\xi<0.1$ and $0.03<\abs{t}<1\GeV^{2}$, is: $\sigma^{\Pp\PX}_{\mathrm{jj}}=21.7\pm 0.9\stat\,^{+3.0}_{-3.3}\syst\pm 0.9\lum\unit{nb}.$ (7) Table 0.8.2 summarises the main systematic uncertainties of the measured cross section. The cross section is calculated independently for events in which the proton scatters towards the positive and negative $z$ directions, namely the processes $\Pp\Pp\to\Pp\PX$ and $\Pp\Pp\to\PX\Pp$, and the results are averaged. They are compatible within the uncertainties. The pythia8 DG model predicts in the same kinematic region a cross section of $23.7\unit{nb}$, consistent with the measurement. Individual contributions to the systematic uncertainties in the measurement of the single-diffractive dijet production cross section in the kinematic region $\pt>40\GeV$, $\abs{\eta}<4.4$, $\xi<0.1$, and $0.03<\abs{t}<1\GeV^{2}$. The second column indicates the relative uncertainties in the integrated cross section. The third and fourth columns represent the minimum and maximum relative uncertainties in the differential cross sections in bins of $t$ and $\xi$, respectively. The minimum relative uncertainty is not shown when it is below 1%. The total uncertainty is the quadratic sum of the individual contributions. The uncertainty of the integrated luminosity is not shown. Uncertainty source Relative uncertainty $\sigma^{\Pp\PX}_{\mathrm{jj}}$ $\rd\sigma/\rd{}t$ $\rd\sigma/\rd\xi$ Trigger efficiency $\pm$2 % 1–2% $<$2.4% Calorimeter energy scale $+$1/$-$2 % $<$7% $<$7% Jet energy scale and resolution $+$9/$-$8 % 3–32% 7–16% Background $\pm$3 % 2–27% $<$8% RP acceptance $<$1 % $<$21% $<$2% Resolution $\pm$2 % 2–30% $<$8% Horizontal dispersion $+$9/$-$12 % 8–71% 8–41% $t$-slope $<$1 % $<$16% $<$1.3% $\beta$-reweighting $<$1 % $<$1% $<$1% Acceptance and unfolding $\pm$2 % 2–50% 5–12% Unfolding bias $\pm$3 % 2–50% 5–11% Unfolding regularization $<$8% $<$1% Total $+$14/$-$15 % The differential cross section as a function of $t$ is well described by an exponential function for $\abs{t}$ values up to about $0.4\GeV^{2}$. A fit is performed with the function $\rd\sigma/{\rd}t\propto\exp\left({-b\abs{t}}\right)$ for $t$ values in the range $0.03<\abs{t}<0.45\GeV^{2}$. The resulting exponential slope is: $b=6.5\pm 0.6\stat\,^{+1.0}_{-0.8}\syst\GeV^{-2},$ (8) where the systematic uncertainties include the contributions discussed in Section 0.8.1. The results for the exponential slope of the cross section calculated independently for events in which the proton scatters towards the positive and negative $z$ directions are compatible within the uncertainties. The parametrisation obtained from the fit is shown in Fig. 4. In the fit range ($0.03<\abs{t}<0.45\GeV^{2}$), the horizontal position of the data points is calculated as the value for which the parametrised function equals its average over the bin width. The data points in the larger-$\abs{t}$ region outside the fit range ($\abs{t}>0.45\GeV^{2}$) are shown at the centre of the bins. The slope measured by CDF is $b\approx 5\text{--}6\GeV^{-2}$ for $\abs{t}\lessapprox 0.5\GeV^{2}$ [10]. In the larger-$\abs{t}$ region, the CDF data exhibit a smaller slope that becomes approximately independent of $t$ for $\abs{t}\gtrapprox 2\GeV^{2}$. The present measurement of the slope is consistent with that by CDF at small-$\abs{t}$. The data do not conclusively indicate a flattening of the $t$ distribution at larger-$\abs{t}$. An estimate of the rapidity gap survival probability can be obtained from the ratio of the measured cross section in Eq. (7) and that predicted by pomwig with $\langle S^{2}\rangle=1$. Alternatively, the pythia8 hard-diffraction model can be used if the DG suppression framework is not applied. The two results are consistent. The overall suppression factor obtained with respect to the pomwig cross section is $\langle S^{2}\rangle=7.4\,^{+1.0}_{-1.1}\%$, where the statistical and systematic uncertainties are added in quadrature. A similar result is obtained when the pythia8 unsuppressed cross section is used as reference value. The H1 fit B dPDFs used in this analysis include the contribution from proton dissociation in $\Pe\Pp$ collisions. They are extracted from the process $\Pe\Pp\to\Pe\PX\mathrm{Y}$, where Y can be a proton or a low-mass excitation with $M_{\mathrm{Y}}<1.6\GeV$ [2]. The results found when the proton is detected are consistent, apart from a different overall normalisation. The ratio of the cross sections is $\sigma(M_{\mathrm{Y}}<1.6\GeV)/\sigma(M_{\mathrm{Y}}=M_{\Pp})=1.23\pm 0.03\stat\pm 0.16\syst$ [2, 3]. No dependence on $\beta$, $Q^{2}$, or $\xi$ is observed. To account for the different normalisation, the ratio is used to correct $\langle S^{2}\rangle$; this yields $\langle S^{2}\rangle=\left(9\pm 2\right)\%$ when the pomwig cross section is taken as the reference value. A similar result is obtained with pythia8. ### 0.8.3 Extraction of the ratio of the single-diffractive to inclusive dijet yields Figure 5 shows the ratio $R(x)$ in the kinematic region $\pt>40\GeV$, $\abs{\eta}<4.4$, $\xi<0.1$, $0.03<\abs{t}<1\GeV^{2}$ and $-2.9\leq\log_{10}x\leq-1.6$. The average of the results for events in which the proton is detected on either side of the IP is shown. The yellow band represents the total systematic uncertainty (cf. Section 0.8.1). The data are compared to the ratio of the single-diffractive and nondiffractive dijet cross sections from different models. The single-diffractive contribution is simulated with pomwig, pythia8 4C, pythia8 CUETP8M1, and pythia8 DG. The nondiffractive contribution is simulated with pythia6 and herwig6 if pomwig is used for the diffractive contribution. When using pythia8 the diffractive and nondiffractive contributions are simulated with the same underlying event tune. When no correction for the rapidity gap survival probability is applied ($\langle S^{2}\rangle=1$), pomwig gives a ratio higher by roughly an order of magnitude, consistent with the results discussed in Section 0.8.2. The suppression seen in the data with respect to the simulation is not substantially different when using pythia6 or herwig6 for the nondiffractive contribution. pomwig with a correction of $\langle S^{2}\rangle=7.4\%$ gives overall a good description of the data when pythia6 is used for the nondiffractive contribution. When herwig6 is used for the nondiffractive contribution the agreement is worse, especially in the lower- and higher-$x$ regions. The agreement for pythia8 4C is fair in the intermediate $x$ region, but worse at low- and high-$x$. The agreement is worse for pythia8 CUETP8M1, with values of the ratio higher than those in the data by up to a factor of two. The pythia8 DG predictions agree well with the data overall, though the agreement is worse in the lowest-$x$ bin. No correction is applied to the pythia8 normalisation. In the lowest-$x$ bin, the ratio in the data is below the predictions. The observed discrepancy is not significant for the predictions that agree well overall with the data elsewhere, taking into account the systematic and statistical uncertainties. The measured value of the ratio, normalised per unit of $\xi$, in the full kinematic region defined above is: $R=\left(\sigma^{\Pp\PX}_{\mathrm{jj}}/\Delta\xi\right)/\sigma_{\mathrm{jj}}=0.025\pm 0.001\stat\pm 0.003\syst.$ (9) Table 0.8.3 summarises the main contributions to the systematic uncertainty of the ratio. The uncertainty of the jet energy scale is considerably smaller than in the case of the single-diffractive cross section. Individual contributions to the systematic uncertainty in the measurement of the single-diffractive to inclusive dijet yields ratio in the kinematic region $\pt>40\GeV$, $\abs{\eta}<4.4$, $\xi<0.1$, $0.03<\abs{t}<1\GeV^{2}$, and $-2.9\leq\log_{10}x\leq-1.6$. The second and third columns represent the relative uncertainties in the ratio in the full kinematic region and in bins of $\log_{10}x$, respectively. The minimum relative uncertainty is not shown when it is below 1%. The total uncertainty is the quadratic sum of the individual contributions. Uncertainty source Relative uncertainty $R$ $R(x)$ Trigger efficiency Negligible 2–3% Calorimeter energy scale $+$1/$-$2 % $<$7% Jet energy scale and resolution $\pm$2 % 1–10% Background $\pm$1 % 1–17% RP acceptance $<$1 % $<$4% Resolution $\pm$2 % $<$4% Horizontal dispersion $+$9/$-$11 % 11–23% $t$-slope $<$1 % $<$3% $\beta$-reweighting $\pm$1 % $<$6% Acceptance and unfolding $\pm$2 % 3–11% Unfolding bias $\pm$3 % 3–14% Unfolding regularization $<$11% Total $+$10/$-$13 % Figure 5: Ratio per unit of $\xi$ of the single-diffractive and inclusive dijet cross sections in the region given by $\xi<0.1$ and $0.03<\abs{t}<1\GeV^{2}$, compared to the predictions from the different models for the ratio between the single-diffractive and nondiffractive cross sections. The pomwig prediction is shown with no correction for the rapidity gap survival probability ($\langle S^{2}\rangle=1$) (left) and with a correction of $\langle S^{2}\rangle=7.4\%$ (right). The vertical bars indicate the statistical uncertainties and the yellow band indicates the total systematic uncertainty. The average of the results for events in which the proton is detected on either side of the interaction point is shown. The ratio between the data and the pomwig prediction using pythia6 or herwig6 as the nondiffractive contribution, when no correction for the rapidity gap survival probability is applied, is shown in the bottom of the left panel. Figure 6 shows the comparison between the results of Fig. 5 and those from CDF [10]. The CDF results are shown for jets with $Q^{2}$ of roughly $100\GeV^{2}$ and pseudorapidity $\abs{\eta}<2.5$, with $0.03<\xi<0.09$. In this case $Q^{2}$ is defined, per event, as the mean transverse energy of the two leading jets squared. CDF measures the ratio for $Q^{2}$ values up to $10^{4}\GeV^{2}$. A relatively small dependence on $Q^{2}$ is observed. The present data are lower than the CDF results. A decrease of the ratio of diffractive to inclusive cross sections with centre-of-mass energy has also been observed by CDF by comparing data at 630 and 1800 [11]. Figure 6: Ratio per unit of $\xi$ of the single-diffractive and inclusive dijet cross sections in the kinematic region given by $\xi<0.1$ and $0.03<\abs{t}<1\GeV^{2}$. The vertical bars indicate the statistical uncertainties and the yellow band indicates the total systematic uncertainty. The red squares represent the results obtained by CDF at $\sqrt{s}=1.96\TeV$ for jets with $Q^{2}\approx 100\GeV^{2}$ and $\abs{\eta}<2.5$, with $0.03<\xi<0.09$. ## 0.9 Summary The differential cross section for single-diffractive dijet production in proton-proton ($\Pp\Pp$) collisions at $\sqrt{s}=8\TeV$ has been measured as a function of the proton fractional momentum loss $\xi$ and the squared four momentum transfer $t$, using the CMS and TOTEM detectors. The data, corresponding to an integrated luminosity of $37.5\nbinv$, were collected using a nonstandard optics configuration with $\beta^{*}=90\unit{m}$. The processes considered are $\Pp\Pp\to\Pp\PX$ or $\Pp\Pp\to\PX\Pp$, with $\PX$ including a system of two jets, in the kinematic region $\xi<0.1$ and $0.03<\abs{t}<1.0\GeV^{2}$. The two jets have transverse momentum $\pt>40\GeV$ and pseudorapidity $\abs{\eta}<4.4$. The integrated cross section in this kinematic region is $\sigma^{\Pp\PX}_{\mathrm{jj}}=21.7\pm 0.9\stat\,^{+3.0}_{-3.3}\syst\pm 0.9\lum\unit{nb}$; it is the average of the cross sections when the proton scatters to either side of the interaction point. The exponential slope of the cross section as a function of $t$ is $b=6.5\pm 0.6\stat\,^{+1.0}_{-0.8}\syst\GeV^{-2}$. This is the first measurement of hard diffraction with a measured proton at the LHC. The data are compared with the predictions of different models. After applying a normalisation shift ascribed to the rapidity gap survival probability, pomwig agrees well with the data. The pythia8 dynamic gap model describes the data well, both in shape and normalisation. In this model the effects of the rapidity gap survival probability are simulated within the framework of multiparton interactions. The pythia8 dynamic gap model is the only calculation that predicts the cross section normalisation without an additional correction. The ratios of the measured single-diffractive cross section to those predicted by pomwig and pythia8 give estimates of the rapidity gap survival probability. After accounting for the correction of the dPDF normalisation due to proton dissociation, the value of $\langle S^{2}\rangle$ is $\left(9\pm 2\right)\%$ when using pomwig as the reference cross section value, with a similar result when pythia8 is used. The ratio of the single-diffractive to inclusive dijet cross section has been measured as a function of the parton momentum fraction $x$. The ratio is lower than that observed at CDF at a smaller centre-of-mass energy. In the region $\pt>40\GeV$, $\abs{\eta}<4.4$, $\xi<0.1$, $0.03<\abs{t}<1.0\GeV^{2}$, and $-2.9\leq\log_{10}x\leq-1.6$, the ratio, normalised per unit $\xi$, is $R=(\sigma^{\Pp\PX}_{\mathrm{jj}}/\Delta\xi)/\sigma_{\mathrm{jj}}=0.025\pm 0.001\stat\pm 0.003\syst$. ###### Acknowledgements. We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS and TOTEM institutes for their contributions to the success of the common CMS-TOTEM effort. We gratefully acknowledge work of the beam optics development team at CERN for the design and the successful commissioning of the high $\beta^{*}$ optics and thank the LHC machine coordinators for scheduling dedicated fills. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS and TOTEM detectors provided by the following funding agencies: BMBWF and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, FAPERGS, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); SENESCYT (Ecuador); MoER, ERC IUT, PUT and ERDF (Estonia); Academy of Finland, Finnish Academy of Science and Letters (The Vilho Yrjö and Kalle Väisälä Fund), MEC, Magnus Ehrnrooth Foundation, HIP, and Waldemar von Frenckell Foundation (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); the Circles of Knowledge Club, NKFIA (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); MES (Latvia); LAS (Lithuania); MOE and UM (Malaysia); BUAP, CINVESTAV, CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MOS (Montenegro); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS, RFBR, and NRC KI (Russia); MESTD (Serbia); SEIDI, CPAN, PCTI, and FEDER (Spain); MOSTR (Sri Lanka); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR, and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU (Ukraine); STFC (United Kingdom); DOE and NSF (USA). Individuals have received support from the Marie- Curie programme and the European Research Council and Horizon 2020 Grant, contract Nos. 675440, 752730, and 765710 (European Union); the Leventis Foundation; the A.P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the F.R.S.-FNRS and FWO (Belgium) under the “Excellence of Science – EOS” – be.h project n. 30820817; the Beijing Municipal Science & Technology Commission, No. Z191100007219010; the Ministry of Education, Youth and Sports (MEYS) and MSMT CR of the Czech Republic; the Nylands nation vid Helsingfors universitet (Finland); the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306; the Lendület (“Momentum”) Programme and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, the New National Excellence Program ÚNKP, the NKFIA research grants 123842, 123959, 124845, 124850, 125105, 128713, 128786, 129058, K 133046, and EFOP-3.6.1- 16-2016-00001 (Hungary); the Council of Science and Industrial Research, India; the HOMING PLUS programme of the Foundation for Polish Science, cofinanced from European Union, Regional Development Fund, the Mobility Plus programme of the Ministry of Science and Higher Education, including Grant No. MNiSW DIR/WK/2018/13, the National Science Center (Poland), contracts Harmonia 2014/14/M/ST2/00428, Opus 2014/13/B/ST2/02543, 2014/15/B/ST2/03998, and 2015/19/B/ST2/02861, Sonata-bis 2012/07/E/ST2/01406; the National Priorities Research Program by Qatar National Research Fund; the Ministry of Science and Education, grant no. 14.W03.31.0026 (Russia); the Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia María de Maeztu, grant MDM-2015-0509 and the Programa Severo Ochoa del Principado de Asturias; the Thalis and Aristeia programmes cofinanced by EU-ESF and the Greek NSRF; the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University and the Chulalongkorn Academic into Its 2nd Century Project Advancement Project (Thailand); the Kavli Foundation; the Nvidia Corporation; the SuperMicro Corporation; the Welch Foundation, contract C-1845; and the Weston Havens Foundation (USA). ## References * [1] P. 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Suarez Gonzalez Universiteit Antwerpen, Antwerpen, Belgium E.A. De Wolf, D. Di Croce, X. Janssen, J. Lauwers, A. Lelek, M. Pieters, H. Van Haevermaet, P. Van Mechelen, N. Van Remortel Vrije Universiteit Brussel, Brussel, Belgium S. Abu Zeid, F. Blekman, J. D’Hondt, J. De Clercq, K. Deroover, G. Flouris, D. Lontkovskyi, S. Lowette, I. Marchesini, S. Moortgat, L. Moreels, Q. Python, K. Skovpen, S. Tavernier, W. Van Doninck, P. Van Mulders, I. Van Parijs Université Libre de Bruxelles, Bruxelles, Belgium D. Beghin, B. Bilin, H. Brun, B. Clerbaux, G. De Lentdecker, H. Delannoy, B. Dorney, G. Fasanella, L. Favart, A. Grebenyuk, A.K. Kalsi, T. Lenzi, J. Luetic, N. Postiau, E. Starling, L. Thomas, C. Vander Velde, P. Vanlaer, D. Vannerom, Q. Wang Ghent University, Ghent, Belgium T. Cornelis, D. Dobur, A. Fagot, M. Gul, I. Khvastunov2, D. Poyraz, C. Roskas, D. Trocino, M. Tytgat, W. Verbeke, B. Vermassen, M. Vit, N. Zaganidis Université Catholique de Louvain, Louvain-la-Neuve, Belgium H. Bakhshiansohi, O. Bondu, G. Bruno, C. Caputo, P. David, C. Delaere, M. Delcourt, A. Giammanco, G. Krintiras, V. Lemaitre, A. Magitteri, K. Piotrzkowski, A. Saggio, M. Vidal Marono, P. Vischia, J. Zobec Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil F.L. Alves, G.A. Alves, G. Correia Silva, C. Hensel, A. Moraes, M.E. Pol, P. Rebello Teles Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil E. Belchior Batista Das Chagas, W. Carvalho, J. Chinellato3, E. Coelho, E.M. Da Costa, G.G. Da Silveira4, D. De Jesus Damiao, C. De Oliveira Martins, S. Fonseca De Souza, L.M. Huertas Guativa, H. Malbouisson, D. Matos Figueiredo, M. Melo De Almeida, C. Mora Herrera, L. Mundim, H. Nogima, W.L. Prado Da Silva, L.J. Sanchez Rosas, A. Santoro, A. Sznajder, M. Thiel, E.J. Tonelli Manganote3, F. Torres Da Silva De Araujo, A. Vilela Pereira Universidade Estadual Paulista a, Universidade Federal do ABC b, São Paulo, Brazil S. Ahujaa, C.A. Bernardesa, L. Calligarisa, T.R. Fernandez Perez Tomeia, E.M. Gregoresb, P.G. Mercadanteb, S.F. Novaesa, SandraS. Padulaa Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria A. Aleksandrov, R. Hadjiiska, P. Iaydjiev, A. Marinov, M. Misheva, M. Rodozov, M. Shopova, G. Sultanov University of Sofia, Sofia, Bulgaria A. Dimitrov, L. Litov, B. Pavlov, P. Petkov Beihang University, Beijing, China W. Fang5, X. Gao5, L. Yuan Department of Physics, Tsinghua University, Beijing, China Y. Wang Institute of High Energy Physics, Beijing, China M. Ahmad, J.G. Bian, G.M. Chen, H.S. Chen, M. Chen, Y. Chen, C.H. Jiang, D. Leggat, H. Liao, Z. Liu, S.M. Shaheen6, A. Spiezia, J. Tao, E. Yazgan, H. Zhang, S. Zhang6, J. Zhao State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing, China Y. Ban, G. Chen, A. Levin, J. Li, L. Li, Q. Li, Y. Mao, S.J. Qian, D. Wang Universidad de Los Andes, Bogota, Colombia C. Avila, A. Cabrera, C.A. Carrillo Montoya, L.F. Chaparro Sierra, C. Florez, C.F. González Hernández, M.A. Segura Delgado University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Split, Croatia B. Courbon, N. Godinovic, D. Lelas, I. Puljak, T. Sculac University of Split, Faculty of Science, Split, Croatia Z. Antunovic, M. Kovac Institute Rudjer Boskovic, Zagreb, Croatia V. Brigljevic, D. Ferencek, K. Kadija, B. Mesic, M. Roguljic, A. Starodumov7, T. Susa University of Cyprus, Nicosia, Cyprus M.W. Ather, A. Attikis, M. Kolosova, G. Mavromanolakis, J. Mousa, C. Nicolaou, F. Ptochos, P.A. Razis, H. Rykaczewski Charles University, Prague, Czech Republic M. Finger8, M. Finger Jr.8 Escuela Politecnica Nacional, Quito, Ecuador E. Ayala Universidad San Francisco de Quito, Quito, Ecuador E. Carrera Jarrin Academy of Scientific Research and Technology of the Arab Republic of Egypt, Egyptian Network of High Energy Physics, Cairo, Egypt A. Ellithi Kamel9, M.A. Mahmoud10,11, E. Salama11,12 National Institute of Chemical Physics and Biophysics, Tallinn, Estonia S. Bhowmik, A. Carvalho Antunes De Oliveira, R.K. Dewanjee, K. Ehataht, M. Kadastik, M. Raidal, C. Veelken Department of Physics, University of Helsinki, Helsinki, Finland P. Eerola, H. Kirschenmann, J. Pekkanen, M. Voutilainen Helsinki Institute of Physics, Helsinki, Finland J. Havukainen, J.K. Heikkilä, T. Järvinen, V. Karimäki, R. Kinnunen, T. Lampén, K. Lassila-Perini, S. Laurila, S. Lehti, T. Lindén, P. Luukka, T. Mäenpää, H. Siikonen, E. Tuominen, J. Tuominiemi Lappeenranta University of Technology, Lappeenranta, Finland T. Tuuva IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France M. Besancon, F. Couderc, M. Dejardin, D. Denegri, J.L. Faure, F. Ferri, S. Ganjour, A. Givernaud, P. Gras, G. Hamel de Monchenault, P. Jarry, C. Leloup, E. Locci, J. Malcles, G. Negro, J. Rander, A. Rosowsky, M.Ö. Sahin, M. Titov Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut Polytechnique de Paris A. Abdulsalam13, C. Amendola, I. Antropov, F. Beaudette, P. Busson, C. Charlot, R. Granier de Cassagnac, I. Kucher, A. Lobanov, J. Martin Blanco, C. Martin Perez, M. Nguyen, C. Ochando, G. Ortona, P. Paganini, J. Rembser, R. Salerno, J.B. Sauvan, Y. Sirois, A.G. Stahl Leiton, A. Zabi, A. Zghiche Université de Strasbourg, CNRS, IPHC UMR 7178, Strasbourg, France J.-L. Agram14, J. Andrea, D. Bloch, G. Bourgatte, J.-M. Brom, E.C. Chabert, V. Cherepanov, C. Collard, E. Conte14, J.-C. Fontaine14, D. Gelé, U. Goerlach, M. Jansová, A.-C. Le Bihan, N. Tonon, P. Van Hove Centre de Calcul de l’Institut National de Physique Nucleaire et de Physique des Particules, CNRS/IN2P3, Villeurbanne, France S. Gadrat Université de Lyon, Université Claude Bernard Lyon 1, CNRS-IN2P3, Institut de Physique Nucléaire de Lyon, Villeurbanne, France S. Beauceron, C. Bernet, G. Boudoul, N. Chanon, R. Chierici, D. Contardo, P. Depasse, H. El Mamouni, J. Fay, L. Finco, S. Gascon, M. Gouzevitch, G. Grenier, B. Ille, F. Lagarde, I.B. Laktineh, H. Lattaud, M. Lethuillier, L. Mirabito, S. Perries, A. Popov15, V. Sordini, G. Touquet, M. Vander Donckt, S. Viret Georgian Technical University, Tbilisi, Georgia T. Toriashvili16 Tbilisi State University, Tbilisi, Georgia Z. Tsamalaidze8 RWTH Aachen University, I. Physikalisches Institut, Aachen, Germany C. Autermann, L. Feld, M.K. Kiesel, K. Klein, M. Lipinski, M. Preuten, M.P. Rauch, C. Schomakers, J. Schulz, M. Teroerde, B. Wittmer RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany A. Albert, M. Erdmann, S. Erdweg, T. Esch, R. Fischer, S. Ghosh, A. Güth, T. Hebbeker, C. Heidemann, K. Hoepfner, H. Keller, L. Mastrolorenzo, M. Merschmeyer, A. Meyer, P. Millet, S. Mukherjee, T. Pook, M. Radziej, H. Reithler, M. Rieger, A. Schmidt, D. Teyssier, S. Thüer RWTH Aachen University, III. Physikalisches Institut B, Aachen, Germany G. Flügge, O. Hlushchenko, T. Kress, T. Müller, A. Nehrkorn, A. Nowack, C. Pistone, O. Pooth, D. Roy, H. Sert, A. Stahl17 Deutsches Elektronen- Synchrotron, Hamburg, Germany M. Aldaya Martin, T. Arndt, C. Asawatangtrakuldee, I. Babounikau, K. Beernaert, O. Behnke, U. Behrens, A. Bermúdez Martínez, D. Bertsche, A.A. Bin Anuar, K. Borras18, V. Botta, A. Campbell, P. Connor, C. Contreras-Campana, V. Danilov, A. De Wit, M.M. Defranchis, C. Diez Pardos, D. Domínguez Damiani, G. Eckerlin, T. Eichhorn, A. Elwood, E. Eren, E. Gallo19, A. Geiser, J.M. Grados Luyando, A. Grohsjean, M. Guthoff, M. Haranko, A. Harb, H. Jung, M. Kasemann, J. Keaveney, C. Kleinwort, J. Knolle, D. Krücker, W. Lange, T. Lenz, J. Leonard, K. Lipka, W. Lohmann20, R. Mankel, I.-A. Melzer-Pellmann, A.B. Meyer, M. Meyer, M. Missiroli, G. Mittag, J. Mnich, V. Myronenko, S.K. Pflitsch, D. Pitzl, A. Raspereza, A. Saibel, M. Savitskyi, P. Saxena, P. Schütze, C. Schwanenberger, R. Shevchenko, A. Singh, H. Tholen, O. Turkot, A. Vagnerini, M. Van De Klundert, G.P. Van Onsem, R. Walsh, Y. Wen, K. Wichmann, C. Wissing, O. Zenaiev University of Hamburg, Hamburg, Germany R. Aggleton, S. Bein, L. Benato, A. Benecke, T. Dreyer, A. Ebrahimi, E. Garutti, D. Gonzalez, P. Gunnellini, J. Haller, A. Hinzmann, A. Karavdina, G. Kasieczka, R. Klanner, R. Kogler, N. Kovalchuk, S. Kurz, V. Kutzner, J. Lange, D. Marconi, J. Multhaup, M. Niedziela, C.E.N. Niemeyer, D. Nowatschin, A. Perieanu, A. Reimers, O. Rieger, C. Scharf, P. Schleper, S. Schumann, J. Schwandt, J. Sonneveld, H. Stadie, G. Steinbrück, F.M. Stober, M. Stöver, B. Vormwald, I. Zoi Karlsruher Institut fuer Technologie, Karlsruhe, Germany M. Akbiyik, C. Barth, M. Baselga, S. Baur, E. Butz, R. Caspart, T. Chwalek, F. Colombo, W. De Boer, A. Dierlamm, K. El Morabit, N. Faltermann, B. Freund, M. Giffels, M.A. Harrendorf, F. Hartmann17, S.M. Heindl, U. Husemann, I. Katkov15, S. Kudella, S. Mitra, M.U. Mozer, Th. Müller, M. Musich, M. Plagge, G. Quast, K. Rabbertz, M. Schröder, I. Shvetsov, H.J. Simonis, R. Ulrich, S. Wayand, M. Weber, T. Weiler, C. Wöhrmann, R. Wolf Institute of Nuclear and Particle Physics (INPP), NCSR Demokritos, Aghia Paraskevi, Greece G. Anagnostou, G. Daskalakis, T. Geralis, A. Kyriakis, D. Loukas, G. Paspalaki National and Kapodistrian University of Athens, Athens, Greece A. Agapitos, G. Karathanasis, P. Kontaxakis, A. Panagiotou, I. Papavergou, N. Saoulidou, K. Vellidis National Technical University of Athens, Athens, Greece K. Kousouris, I. Papakrivopoulos, G. Tsipolitis University of Ioánnina, Ioánnina, Greece I. Evangelou, C. Foudas, P. Gianneios, P. Katsoulis, P. Kokkas, S. Mallios, N. Manthos, I. Papadopoulos, E. Paradas, J. Strologas, F.A. Triantis, D. Tsitsonis MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös Loránd University, Budapest, Hungary M. Bartók21, M. Csanad, N. Filipovic, P. Major, M.I. Nagy, G. Pasztor, O. Surányi, G.I. Veres Wigner Research Centre for Physics, Budapest, Hungary G. Bencze, C. Hajdu, D. Horvath22, Á. Hunyadi, F. Sikler, T.Á. Vámi, V. Veszpremi, G. Vesztergombi${}^{\textrm{\textdagger}}$ Institute of Nuclear Research ATOMKI, Debrecen, Hungary N. Beni, S. Czellar, J. Karancsi21, A. Makovec, J. Molnar, Z. Szillasi Institute of Physics, University of Debrecen, Debrecen, Hungary P. Raics, Z.L. Trocsanyi, B. Ujvari Indian Institute of Science (IISc), Bangalore, India S. Choudhury, J.R. Komaragiri, P.C. Tiwari National Institute of Science Education and Research, HBNI, Bhubaneswar, India S. Bahinipati24, C. Kar, P. Mal, K. Mandal, A. Nayak25, S. Roy Chowdhury, D.K. Sahoo24, S.K. Swain Panjab University, Chandigarh, India S. Bansal, S.B. Beri, V. Bhatnagar, S. Chauhan, R. Chawla, N. Dhingra, R. Gupta, A. Kaur, M. Kaur, S. Kaur, P. Kumari, M. Lohan, M. Meena, A. Mehta, K. Sandeep, S. Sharma, J.B. Singh, A.K. Virdi, G. Walia University of Delhi, Delhi, India A. Bhardwaj, B.C. Choudhary, R.B. Garg, M. Gola, S. Keshri, Ashok Kumar, S. Malhotra, M. Naimuddin, P. Priyanka, K. Ranjan, Aashaq Shah, R. Sharma Saha Institute of Nuclear Physics, HBNI, Kolkata, India R. Bhardwaj26, M. Bharti26, R. Bhattacharya, S. Bhattacharya, U. Bhawandeep26, D. Bhowmik, S. Dey, S. Dutt26, S. Dutta, S. Ghosh, M. Maity27, K. Mondal, S. Nandan, A. Purohit, P.K. Rout, A. Roy, G. Saha, S. Sarkar, T. Sarkar27, M. Sharan, B. Singh26, S. Thakur26 Indian Institute of Technology Madras, Madras, India P.K. Behera, A. Muhammad Bhabha Atomic Research Centre, Mumbai, India R. Chudasama, D. Dutta, V. Jha, V. Kumar, D.K. Mishra, P.K. Netrakanti, L.M. Pant, P. Shukla, P. Suggisetti Tata Institute of Fundamental Research-A, Mumbai, India T. Aziz, M.A. Bhat, S. Dugad, G.B. Mohanty, N. Sur, RavindraKumar Verma Tata Institute of Fundamental Research-B, Mumbai, India S. Banerjee, S. Bhattacharya, S. Chatterjee, P. Das, M. Guchait, Sa. Jain, S. Karmakar, S. Kumar, G. Majumder, K. Mazumdar, N. Sahoo Indian Institute of Science Education and Research (IISER), Pune, India S. Chauhan, S. Dube, V. Hegde, A. Kapoor, K. Kothekar, S. Pandey, A. Rane, A. Rastogi, S. Sharma Institute for Research in Fundamental Sciences (IPM), Tehran, Iran S. Chenarani28, E. Eskandari Tadavani, S.M. Etesami28, M. Khakzad, M. Mohammadi Najafabadi, M. Naseri, F. Rezaei Hosseinabadi, B. Safarzadeh29, M. Zeinali University College Dublin, Dublin, Ireland M. Felcini, M. Grunewald INFN Sezione di Bari a, Università di Bari b, Politecnico di Bari c, Bari, Italy M. Abbresciaa,b, C. Calabriaa,b, A. Colaleoa, D. Creanzaa,c, L. Cristellaa,b, N. De Filippisa,c, M. De Palmaa,b, A. Di Florioa,b, F. Erricoa,b, L. Fiorea, A. Gelmia,b, G. Iasellia,c, M. Incea,b, S. Lezkia,b, G. Maggia,c, M. Maggia, G. Minielloa,b, S. Mya,b, S. Nuzzoa,b, A. Pompilia,b, G. Pugliesea,c, R. Radognaa, A. Ranieria, G. Selvaggia,b, A. Sharmaa, L. Silvestrisa, R. Vendittia, P. Verwilligena INFN Sezione di Bologna a, Università di Bologna b, Bologna, Italy G. Abbiendia, C. Battilanaa,b, D. Bonacorsia,b, L. Borgonovia,b, S. Braibant- Giacomellia,b, R. Campaninia,b, P. Capiluppia,b, A. Castroa,b, F.R. Cavalloa, S.S. Chhibraa,b, G. Codispotia,b, M. Cuffiania,b, G.M. Dallavallea, F. Fabbria, A. Fanfania,b, E. Fontanesi, P. Giacomellia, C. Grandia, L. Guiduccia,b, F. Iemmia,b, S. Lo Meoa,30, S. Marcellinia, G. Masettia, A. Montanaria, F.L. Navarriaa,b, A. Perrottaa, F. Primaveraa,b, A.M. Rossia,b, T. Rovellia,b, G.P. Sirolia,b, N. Tosia INFN Sezione di Catania a, Università di Catania b, Catania, Italy S. Albergoa,b, A. Di Mattiaa, R. Potenzaa,b, A. Tricomia,b, C. Tuvea,b INFN Sezione di Firenze a, Università di Firenze b, Firenze, Italy G. Barbaglia, K. Chatterjeea,b, V. Ciullia,b, C. Civininia, R. D’Alessandroa,b, E. Focardia,b, G. Latino, P. Lenzia,b, M. Meschinia, S. Paolettia, L. Russoa,31, G. Sguazzonia, D. Stroma, L. Viliania INFN Laboratori Nazionali di Frascati, Frascati, Italy L. Benussi, S. Bianco, F. Fabbri, D. Piccolo INFN Sezione di Genova a, Università di Genova b, Genova, Italy F. Ferroa, R. Mulargiaa,b, E. Robuttia, S. Tosia,b INFN Sezione di Milano- Bicocca a, Università di Milano-Bicocca b, Milano, Italy A. Benagliaa, A. Beschib, F. Brivioa,b, V. Cirioloa,b,17, S. Di Guidaa,b,17, M.E. Dinardoa,b, S. Fiorendia,b, S. Gennaia, A. Ghezzia,b, P. Govonia,b, M. Malbertia,b, S. Malvezzia, D. Menascea, F. Monti, L. Moronia, M. Paganonia,b, D. Pedrinia, S. Ragazzia,b, T. Tabarelli de Fatisa,b, D. Zuoloa,b INFN Sezione di Napoli a, Università di Napoli ’Federico II’ b, Napoli, Italy, Università della Basilicata c, Potenza, Italy, Università G. Marconi d, Roma, Italy S. Buontempoa, N. Cavalloa,c, A. De Iorioa,b, A. Di Crescenzoa,b, F. Fabozzia,c, F. Fiengaa, G. Galatia, A.O.M. Iorioa,b, L. Listaa, S. Meolaa,d,17, P. Paoluccia,17, C. Sciaccaa,b, E. Voevodinaa,b INFN Sezione di Padova a, Università di Padova b, Padova, Italy, Università di Trento c, Trento, Italy P. Azzia, N. Bacchettaa, D. Biselloa,b, A. Bolettia,b, A. Bragagnolo, R. Carlina,b, P. Checchiaa, M. Dall’Ossoa,b, P. De Castro Manzanoa, T. Dorigoa, U. Dossellia, F. Gasparinia,b, U. Gasparinia,b, A. Gozzelinoa, S.Y. Hoh, S. Lacapraraa, P. Lujan, M. Margonia,b, A.T. Meneguzzoa,b, J. Pazzinia,b, M. Presillab, P. Ronchesea,b, R. Rossina,b, F. Simonettoa,b, A. Tiko, E. Torassaa, M. Tosia,b, M. Zanettia,b, P. Zottoa,b, G. Zumerlea,b INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy A. Braghieria, A. Magnania, P. Montagnaa,b, S.P. Rattia,b, V. Rea, M. Ressegottia,b, C. Riccardia,b, P. Salvinia, I. Vaia,b, P. Vituloa,b INFN Sezione di Perugia a, Università di Perugia b, Perugia, Italy M. Biasinia,b, G.M. Bileia, C. Cecchia,b, D. Ciangottinia,b, L. Fanòa,b, P. Laricciaa,b, R. Leonardia,b, E. Manonia, G. Mantovania,b, V. Mariania,b, M. Menichellia, A. Rossia,b, A. Santocchiaa,b, D. Spigaa INFN Sezione di Pisa a, Università di Pisa b, Scuola Normale Superiore di Pisa c, Pisa, Italy K. Androsova, P. Azzurria, G. Bagliesia, L. Bianchinia, T. Boccalia, L. Borrello, R. Castaldia, M.A. Cioccia,b, R. Dell’Orsoa, G. Fedia, F. Fioria,c, L. Gianninia,c, A. Giassia, M.T. Grippoa, F. Ligabuea,c, E. Mancaa,c, G. Mandorlia,c, A. Messineoa,b, F. Pallaa, A. Rizzia,b, G. Rolandi32, P. Spagnoloa, R. Tenchinia, G. Tonellia,b, A. Venturia, P.G. Verdinia INFN Sezione di Roma a, Sapienza Università di Roma b, Rome, Italy L. Baronea,b, F. Cavallaria, M. Cipriania,b, D. Del Rea,b, E. Di Marcoa,b, M. Diemoza, S. Gellia,b, E. Longoa,b, B. Marzocchia,b, P. Meridiania, G. Organtinia,b, F. Pandolfia, R. Paramattia,b, F. Preiatoa,b, S. Rahatloua,b, C. Rovellia, F. Santanastasioa,b INFN Sezione di Torino a, Università di Torino b, Torino, Italy, Università del Piemonte Orientale c, Novara, Italy N. Amapanea,b, R. Arcidiaconoa,c, S. Argiroa,b, M. Arneodoa,c, N. Bartosika, R. Bellana,b, C. Biinoa, A. Cappatia,b, N. Cartigliaa, F. Cennaa,b, S. Comettia, M. Costaa,b, R. Covarellia,b, N. Demariaa, B. Kiania,b, C. Mariottia, S. Masellia, E. Migliorea,b, V. Monacoa,b, E. Monteila,b, M. Montenoa, M.M. Obertinoa,b, L. Pachera,b, N. Pastronea, M. Pelliccionia, G.L. Pinna Angionia,b, A. Romeroa,b, M. Ruspaa,c, R. Sacchia,b, R. Salvaticoa,b, K. Shchelinaa,b, V. Solaa, A. Solanoa,b, D. Soldia,b, A. Staianoa INFN Sezione di Trieste a, Università di Trieste b, Trieste, Italy S. Belfortea, V. Candelisea,b, M. Casarsaa, F. Cossuttia, A. Da Rolda,b, G. Della Riccaa,b, F. Vazzolera,b, A. Zanettia Kyungpook National University, Daegu, Korea D.H. Kim, G.N. Kim, M.S. Kim, J. Lee, S. Lee, S.W. Lee, C.S. Moon, Y.D. Oh, S.I. Pak, S. Sekmen, D.C. Son, Y.C. Yang Chonnam National University, Institute for Universe and Elementary Particles, Kwangju, Korea H. Kim, D.H. Moon, G. Oh Hanyang University, Seoul, Korea B. Francois, J. Goh33, T.J. Kim Korea University, Seoul, Korea S. Cho, S. Choi, Y. Go, D. Gyun, S. Ha, B. Hong, Y. Jo, K. Lee, K.S. Lee, S. Lee, J. Lim, S.K. Park, Y. Roh Sejong University, Seoul, Korea H.S. Kim Seoul National University, Seoul, Korea J. Almond, J. Kim, J.S. Kim, H. Lee, K. Lee, K. Nam, S.B. Oh, B.C. Radburn- Smith, S.h. Seo, U.K. Yang, H.D. Yoo, G.B. Yu University of Seoul, Seoul, Korea D. Jeon, H. Kim, J.H. Kim, J.S.H. Lee, I.C. Park Sungkyunkwan University, Suwon, Korea Y. Choi, C. Hwang, J. Lee, I. Yu Riga Technical University, Riga, Latvia V. Veckalns34 Vilnius University, Vilnius, Lithuania V. Dudenas, A. Juodagalvis, J. Vaitkus National Centre for Particle Physics, Universiti Malaya, Kuala Lumpur, Malaysia Z.A. Ibrahim, M.A.B. Md Ali35, F. Mohamad Idris36, W.A.T. Wan Abdullah, M.N. Yusli, Z. Zolkapli Universidad de Sonora (UNISON), Hermosillo, Mexico J.F. Benitez, A. Castaneda Hernandez, J.A. Murillo Quijada Centro de Investigacion y de Estudios Avanzados del IPN, Mexico City, Mexico H. Castilla-Valdez, E. De La Cruz-Burelo, M.C. Duran-Osuna, I. Heredia-De La Cruz37, R. Lopez-Fernandez, J. Mejia Guisao, R.I. Rabadan-Trejo, M. Ramirez- Garcia, G. Ramirez-Sanchez, R. Reyes-Almanza, A. Sanchez-Hernandez Universidad Iberoamericana, Mexico City, Mexico S. Carrillo Moreno, C. Oropeza Barrera, F. Vazquez Valencia Benemerita Universidad Autonoma de Puebla, Puebla, Mexico J. Eysermans, I. Pedraza, H.A. Salazar Ibarguen, C. Uribe Estrada Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico A. Morelos Pineda University of Auckland, Auckland, New Zealand D. Krofcheck University of Canterbury, Christchurch, New Zealand S. Bheesette, P.H. Butler National Centre for Physics, Quaid-I-Azam University, Islamabad, Pakistan A. Ahmad, M. Ahmad, M.I. Asghar, Q. Hassan, H.R. Hoorani, W.A. Khan, M.A. Shah, M. Shoaib, M. Waqas National Centre for Nuclear Research, Swierk, Poland H. Bialkowska, M. Bluj, B. Boimska, T. Frueboes, M. Górski, M. Kazana, M. Szleper, P. Traczyk, P. Zalewski Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland K. Bunkowski, A. Byszuk38, K. Doroba, A. Kalinowski, M. Konecki, J. Krolikowski, M. Misiura, M. Olszewski, A. Pyskir, M. Walczak Laboratório de Instrumentação e Física Experimental de Partículas, Lisboa, Portugal M. Araujo, P. Bargassa, C. Beirão Da Cruz E Silva, A. Di Francesco, P. Faccioli, B. Galinhas, M. Gallinaro, J. Hollar, N. Leonardo, J. Seixas, G. Strong, O. Toldaiev, J. Varela Joint Institute for Nuclear Research, Dubna, Russia S. Afanasiev, P. Bunin, M. Gavrilenko, I. Golutvin, I. Gorbunov, A. Kamenev, V. Karjavine, A. Lanev, A. Malakhov, V. Matveev39,40, P. Moisenz, V. Palichik, V. Perelygin, S. Shmatov, S. Shulha, N. Skatchkov, V. Smirnov, N. Voytishin, A. Zarubin Petersburg Nuclear Physics Institute, Gatchina (St. Petersburg), Russia V. Golovtsov, Y. Ivanov, V. Kim41, E. Kuznetsova42, P. Levchenko, V. Murzin, V. Oreshkin, I. Smirnov, D. Sosnov, V. Sulimov, L. Uvarov, S. Vavilov, A. Vorobyev Institute for Nuclear Research, Moscow, Russia Yu. Andreev, A. Dermenev, S. Gninenko, N. Golubev, A. Karneyeu, M. Kirsanov, N. Krasnikov, A. Pashenkov, A. Shabanov, D. Tlisov, A. Toropin Institute for Theoretical and Experimental Physics named by A.I. Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia V. Epshteyn, V. Gavrilov, N. Lychkovskaya, V. Popov, I. Pozdnyakov, G. Safronov, A. Spiridonov, A. Stepennov, V. Stolin, M. Toms, E. Vlasov, A. Zhokin Moscow Institute of Physics and Technology, Moscow, Russia T. Aushev P.N. Lebedev Physical Institute, Moscow, Russia V. Andreev, M. Azarkin, I. Dremin40, M. Kirakosyan, A. Terkulov Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia A. Belyaev, E. Boos, A. Ershov, A. Gribushin, L. Khein, V. Klyukhin, O. Kodolova, I. Lokhtin, O. Lukina, S. Obraztsov, S. Petrushanko, V. Savrin, A. Snigirev Novosibirsk State University (NSU), Novosibirsk, Russia A. Barnyakov43, V. Blinov43, T. Dimova43, L. Kardapoltsev43, Y. Skovpen43 Institute for High Energy Physics of National Research Centre ‘Kurchatov Institute’, Protvino, Russia I. Azhgirey, I. Bayshev, S. Bitioukov, V. Kachanov, A. Kalinin, D. Konstantinov, P. Mandrik, V. Petrov, R. Ryutin, S. Slabospitskii, A. Sobol, S. Troshin, N. Tyurin, A. Uzunian, A. Volkov National Research Tomsk Polytechnic University, Tomsk, Russia A. Babaev, S. Baidali, V. Okhotnikov University of Belgrade: Faculty of Physics and VINCA Institute of Nuclear Sciences P. Adzic44, P. Cirkovic, D. Devetak, M. Dordevic, P. Milenovic45, J. Milosevic Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain J. Alcaraz Maestre, A. Álvarez Fernández, I. Bachiller, M. Barrio Luna, J.A. Brochero Cifuentes, M. Cerrada, N. Colino, B. De La Cruz, A. Delgado Peris, C. Fernandez Bedoya, J.P. Fernández Ramos, J. Flix, M.C. Fouz, O. Gonzalez Lopez, S. Goy Lopez, J.M. Hernandez, M.I. Josa, D. Moran, A. Pérez-Calero Yzquierdo, J. Puerta Pelayo, I. Redondo, L. Romero, S. Sánchez Navas, M.S. Soares, A. Triossi Universidad Autónoma de Madrid, Madrid, Spain C. Albajar, J.F. de Trocóniz Universidad de Oviedo, Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias (ICTEA), Oviedo, Spain J. Cuevas, C. Erice, J. Fernandez Menendez, S. Folgueras, I. Gonzalez Caballero, J.R. González Fernández, E. Palencia Cortezon, V. Rodríguez Bouza, S. Sanchez Cruz, J.M. Vizan Garcia Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, Santander, Spain I.J. Cabrillo, A. Calderon, B. Chazin Quero, J. Duarte Campderros, M. Fernandez, P.J. Fernández Manteca, A. García Alonso, J. Garcia-Ferrero, G. Gomez, A. Lopez Virto, J. Marco, C. Martinez Rivero, P. Martinez Ruiz del Arbol, F. Matorras, J. Piedra Gomez, C. Prieels, T. Rodrigo, A. Ruiz-Jimeno, L. Scodellaro, N. Trevisani, I. Vila, R. Vilar Cortabitarte University of Ruhuna, Department of Physics, Matara, Sri Lanka N. Wickramage CERN, European Organization for Nuclear Research, Geneva, Switzerland D. Abbaneo, B. Akgun, E. Auffray, G. Auzinger, P. Baillon, A.H. Ball, D. Barney, J. Bendavid, M. Bianco, A. Bocci, C. Botta, E. Brondolin, T. Camporesi, M. Cepeda, G. Cerminara, E. Chapon, Y. Chen, G. Cucciati, D. d’Enterria, A. Dabrowski, N. Daci, V. Daponte, A. David, A. De Roeck, N. Deelen, M. Dobson, M. Dünser, N. Dupont, A. Elliott-Peisert, F. Fallavollita46, D. Fasanella, G. Franzoni, J. Fulcher, W. Funk, D. Gigi, A. Gilbert, K. Gill, F. Glege, M. Gruchala, M. Guilbaud, D. Gulhan, J. Hegeman, C. Heidegger, V. Innocente, G.M. Innocenti, A. Jafari, P. Janot, O. Karacheban20, J. Kieseler, A. Kornmayer, M. Krammer1, C. Lange, P. Lecoq, C. Lourenço, L. Malgeri, M. Mannelli, A. Massironi, F. Meijers, J.A. Merlin, S. Mersi, E. Meschi, F. Moortgat, M. Mulders, J. Ngadiuba, S. Nourbakhsh, S. Orfanelli, L. Orsini, F. Pantaleo17, L. Pape, E. Perez, M. Peruzzi, A. Petrilli, G. Petrucciani, A. Pfeiffer, M. Pierini, F.M. Pitters, D. Rabady, A. Racz, T. Reis, M. Rovere, H. Sakulin, C. Schäfer, C. Schwick, M. Selvaggi, A. Sharma, P. Silva, P. Sphicas47, A. Stakia, J. Steggemann, D. Treille, A. Tsirou, A. Vartak, M. Verzetti, W.D. Zeuner Paul Scherrer Institut, Villigen, Switzerland L. Caminada48, K. Deiters, W. Erdmann, R. Horisberger, Q. Ingram, H.C. Kaestli, D. Kotlinski, U. Langenegger, T. Rohe, S.A. Wiederkehr ETH Zurich - Institute for Particle Physics and Astrophysics (IPA), Zurich, Switzerland M. Backhaus, L. Bäni, P. Berger, N. Chernyavskaya, G. Dissertori, M. Dittmar, M. Donegà, C. Dorfer, T.A. Gómez Espinosa, C. Grab, D. Hits, T. Klijnsma, W. Lustermann, R.A. Manzoni, M. Marionneau, M.T. Meinhard, F. Micheli, P. Musella, F. Nessi-Tedaldi, F. Pauss, G. Perrin, L. Perrozzi, S. Pigazzini, M. Reichmann, C. Reissel, D. Ruini, D.A. Sanz Becerra, M. Schönenberger, L. Shchutska, V.R. Tavolaro, K. Theofilatos, M.L. Vesterbacka Olsson, R. Wallny, D.H. Zhu Universität Zürich, Zurich, Switzerland T.K. Aarrestad, C. Amsler49, D. Brzhechko, M.F. Canelli, A. De Cosa, R. Del Burgo, S. Donato, C. Galloni, T. Hreus, B. Kilminster, S. Leontsinis, I. Neutelings, G. Rauco, P. Robmann, D. Salerno, K. Schweiger, C. Seitz, Y. Takahashi, S. Wertz, A. Zucchetta National Central University, Chung-Li, Taiwan T.H. Doan, R. Khurana, C.M. Kuo, W. Lin, A. Pozdnyakov, S.S. Yu National Taiwan University (NTU), Taipei, Taiwan P. Chang, Y. Chao, K.F. Chen, P.H. Chen, W.-S. Hou, Y.F. Liu, R.-S. Lu, E. Paganis, A. Psallidas, A. Steen Chulalongkorn University, Faculty of Science, Department of Physics, Bangkok, Thailand B. Asavapibhop, N. Srimanobhas, N. Suwonjandee Çukurova University, Physics Department, Science and Art Faculty, Adana, Turkey A. Bat, F. Boran, S. Cerci50, S. Damarseckin, Z.S. Demiroglu, F. Dolek, C. Dozen, I. Dumanoglu, E. Eskut, G. Gokbulut, Y. Guler, E. Gurpinar, I. Hos51, C. Isik, E.E. Kangal52, O. Kara, A. Kayis Topaksu, U. Kiminsu, M. Oglakci, G. Onengut, K. Ozdemir53, A. Polatoz, D. Sunar Cerci50, U.G. Tok, S. Turkcapar, I.S. Zorbakir, C. Zorbilmez Middle East Technical University, Physics Department, Ankara, Turkey B. Isildak54, G. Karapinar55, M. Yalvac, M. Zeyrek Bogazici University, Istanbul, Turkey I.O. Atakisi, E. Gülmez, M. Kaya56, O. Kaya57, S. Ozkorucuklu58, S. Tekten, E.A. Yetkin59 Istanbul Technical University, Istanbul, Turkey M.N. Agaras, A. Cakir, K. Cankocak, Y. Komurcu, S. Sen60 Institute for Scintillation Materials of National Academy of Science of Ukraine, Kharkov, Ukraine B. Grynyov National Scientific Center, Kharkov Institute of Physics and Technology, Kharkov, Ukraine L. Levchuk University of Bristol, Bristol, United Kingdom F. Ball, J.J. Brooke, D. Burns, E. Clement, D. Cussans, O. Davignon, H. Flacher, J. Goldstein, G.P. Heath, H.F. Heath, L. Kreczko, D.M. Newbold61, S. Paramesvaran, B. Penning, T. Sakuma, D. Smith, V.J. Smith, J. Taylor, A. Titterton Rutherford Appleton Laboratory, Didcot, United Kingdom K.W. Bell, A. Belyaev62, C. Brew, R.M. Brown, D. Cieri, D.J.A. Cockerill, J.A. Coughlan, K. Harder, S. Harper, J. Linacre, K. Manolopoulos, E. Olaiya, D. Petyt, T. Schuh, C.H. Shepherd-Themistocleous, A. Thea, I.R. Tomalin, T. Williams, W.J. Womersley Imperial College, London, United Kingdom R. Bainbridge, P. Bloch, J. Borg, S. Breeze, O. Buchmuller, A. Bundock, D. Colling, P. Dauncey, G. Davies, M. Della Negra, R. Di Maria, P. Everaerts, G. Hall, G. Iles, T. James, M. Komm, C. Laner, L. Lyons, A.-M. Magnan, S. Malik, A. Martelli, J. Nash63, A. Nikitenko7, V. Palladino, M. Pesaresi, D.M. Raymond, A. Richards, A. Rose, E. Scott, C. Seez, A. Shtipliyski, G. Singh, M. Stoye, T. Strebler, S. Summers, A. Tapper, K. Uchida, T. Virdee17, N. Wardle, D. Winterbottom, J. Wright, S.C. Zenz Brunel University, Uxbridge, United Kingdom J.E. Cole, P.R. Hobson, A. Khan, P. Kyberd, C.K. Mackay, A. Morton, I.D. Reid, L. Teodorescu, S. Zahid Baylor University, Waco, USA K. Call, J. Dittmann, K. Hatakeyama, H. Liu, C. Madrid, B. McMaster, N. Pastika, C. Smith Catholic University of America, Washington, DC, USA R. Bartek, A. Dominguez The University of Alabama, Tuscaloosa, USA A. Buccilli, S.I. Cooper, C. Henderson, P. Rumerio, C. West Boston University, Boston, USA D. Arcaro, T. Bose, D. Gastler, S. Girgis, D. Pinna, C. Richardson, J. Rohlf, L. Sulak, D. Zou Brown University, Providence, USA G. Benelli, B. Burkle, X. Coubez, D. Cutts, M. Hadley, J. Hakala, U. Heintz, J.M. Hogan64, K.H.M. Kwok, E. Laird, G. Landsberg, J. Lee, Z. Mao, M. Narain, S. Sagir65, R. Syarif, E. Usai, D. Yu University of California, Davis, Davis, USA R. Band, C. Brainerd, R. Breedon, D. Burns, M. Calderon De La Barca Sanchez, M. Chertok, J. Conway, R. Conway, P.T. Cox, R. Erbacher, C. Flores, G. Funk, W. Ko, O. Kukral, R. Lander, M. Mulhearn, D. Pellett, J. Pilot, S. Shalhout, M. Shi, D. Stolp, D. Taylor, K. Tos, M. Tripathi, Z. Wang, F. Zhang University of California, Los Angeles, USA M. Bachtis, C. Bravo, R. Cousins, A. Dasgupta, S. Erhan, A. Florent, J. Hauser, M. Ignatenko, N. Mccoll, S. Regnard, D. Saltzberg, C. Schnaible, V. Valuev University of California, Riverside, Riverside, USA E. Bouvier, K. Burt, R. Clare, J.W. Gary, S.M.A. Ghiasi Shirazi, G. Hanson, G. Karapostoli, E. Kennedy, F. Lacroix, O.R. Long, M. Olmedo Negrete, M.I. Paneva, W. Si, L. Wang, H. Wei, S. Wimpenny, B.R. Yates University of California, San Diego, La Jolla, USA J.G. Branson, P. Chang, S. Cittolin, M. Derdzinski, R. Gerosa, D. Gilbert, B. Hashemi, A. Holzner, D. Klein, G. Kole, V. Krutelyov, J. Letts, M. Masciovecchio, S. May, D. Olivito, S. Padhi, M. Pieri, V. Sharma, M. Tadel, J. Wood, F. Würthwein, A. Yagil, G. Zevi Della Porta University of California, Santa Barbara - Department of Physics, Santa Barbara, USA N. Amin, R. Bhandari, C. Campagnari, M. Citron, V. Dutta, M. Franco Sevilla, L. Gouskos, R. Heller, J. Incandela, H. Mei, A. Ovcharova, H. Qu, J. Richman, D. Stuart, I. Suarez, S. Wang, J. Yoo California Institute of Technology, Pasadena, USA D. Anderson, A. Bornheim, J.M. Lawhorn, N. Lu, H.B. Newman, T.Q. Nguyen, J. Pata, M. Spiropulu, J.R. Vlimant, R. Wilkinson, S. Xie, Z. Zhang, R.Y. Zhu Carnegie Mellon University, Pittsburgh, USA M.B. Andrews, T. Ferguson, T. Mudholkar, M. Paulini, M. Sun, I. Vorobiev, M. Weinberg University of Colorado Boulder, Boulder, USA J.P. Cumalat, W.T. Ford, F. Jensen, A. Johnson, E. MacDonald, T. Mulholland, R. Patel, A. Perloff, K. Stenson, K.A. Ulmer, S.R. Wagner Cornell University, Ithaca, USA J. Alexander, J. Chaves, Y. Cheng, J. Chu, A. Datta, K. Mcdermott, N. Mirman, J.R. Patterson, D. Quach, A. Rinkevicius, A. Ryd, L. Skinnari, L. Soffi, S.M. Tan, Z. Tao, J. Thom, J. Tucker, P. Wittich, M. Zientek Fermi National Accelerator Laboratory, Batavia, USA S. Abdullin, M. Albrow, M. Alyari, G. Apollinari, A. Apresyan, A. Apyan, S. Banerjee, L.A.T. Bauerdick, A. Beretvas, J. Berryhill, P.C. Bhat, K. Burkett, J.N. Butler, A. Canepa, G.B. Cerati, H.W.K. Cheung, F. Chlebana, M. Cremonesi, J. Duarte, V.D. Elvira, J. Freeman, Z. Gecse, E. Gottschalk, L. Gray, D. Green, S. Grünendahl, O. Gutsche, J. Hanlon, R.M. Harris, S. Hasegawa, J. Hirschauer, Z. Hu, B. Jayatilaka, S. Jindariani, M. Johnson, U. Joshi, B. Klima, M.J. Kortelainen, B. Kreis, S. Lammel, D. Lincoln, R. Lipton, M. Liu, T. Liu, J. Lykken, K. Maeshima, J.M. Marraffino, D. Mason, P. McBride, P. Merkel, S. Mrenna, S. Nahn, V. O’Dell, K. Pedro, C. Pena, O. Prokofyev, G. Rakness, F. Ravera, A. Reinsvold, L. Ristori, A. Savoy-Navarro66, B. Schneider, E. Sexton-Kennedy, A. Soha, W.J. Spalding, L. Spiegel, S. Stoynev, J. Strait, N. Strobbe, L. Taylor, S. Tkaczyk, N.V. Tran, L. Uplegger, E.W. Vaandering, C. Vernieri, M. Verzocchi, R. Vidal, M. Wang, H.A. Weber University of Florida, Gainesville, USA D. Acosta, P. Avery, P. Bortignon, D. Bourilkov, A. Brinkerhoff, L. Cadamuro, A. Carnes, D. Curry, R.D. Field, S.V. Gleyzer, B.M. Joshi, J. Konigsberg, A. Korytov, K.H. Lo, P. Ma, K. Matchev, N. Menendez, G. Mitselmakher, D. Rosenzweig, K. Shi, D. Sperka, J. Wang, S. Wang, X. Zuo Florida International University, Miami, USA Y.R. Joshi, S. Linn Florida State University, Tallahassee, USA A. Ackert, T. Adams, A. Askew, S. Hagopian, V. Hagopian, K.F. Johnson, T. Kolberg, G. Martinez, T. Perry, H. Prosper, A. Saha, C. Schiber, R. Yohay Florida Institute of Technology, Melbourne, USA M.M. Baarmand, V. Bhopatkar, S. Colafranceschi, M. Hohlmann, D. Noonan, M. Rahmani, T. Roy, M. Saunders, F. Yumiceva University of Illinois at Chicago (UIC), Chicago, USA M.R. Adams, L. Apanasevich, D. Berry, R.R. Betts, R. Cavanaugh, X. Chen, S. Dittmer, O. Evdokimov, C.E. Gerber, D.A. Hangal, D.J. Hofman, K. Jung, J. Kamin, C. Mills, M.B. Tonjes, N. Varelas, H. Wang, X. Wang, Z. Wu, J. Zhang The University of Iowa, Iowa City, USA M. Alhusseini, B. Bilki67, W. Clarida, K. Dilsiz68, S. Durgut, R.P. Gandrajula, M. Haytmyradov, V. Khristenko, J.-P. Merlo, A. Mestvirishvili, A. Moeller, J. Nachtman, H. Ogul69, Y. Onel, F. Ozok70, A. Penzo, C. Snyder, E. Tiras, J. Wetzel Johns Hopkins University, Baltimore, USA B. Blumenfeld, A. Cocoros, N. Eminizer, D. Fehling, L. Feng, A.V. Gritsan, W.T. Hung, P. Maksimovic, J. Roskes, U. Sarica, M. Swartz, M. Xiao The University of Kansas, Lawrence, USA A. Al-bataineh, P. Baringer, A. Bean, S. Boren, J. Bowen, A. Bylinkin, J. Castle, S. Khalil, A. Kropivnitskaya, D. Majumder, W. Mcbrayer, M. Murray, C. Rogan, S. Sanders, E. Schmitz, J.D. Tapia Takaki, Q. Wang Kansas State University, Manhattan, USA S. Duric, A. Ivanov, K. Kaadze, D. Kim, Y. Maravin, D.R. Mendis, T. Mitchell, A. Modak, A. Mohammadi Lawrence Livermore National Laboratory, Livermore, USA F. Rebassoo, D. Wright University of Maryland, College Park, USA A. Baden, O. Baron, A. Belloni, S.C. Eno, Y. Feng, C. Ferraioli, N.J. Hadley, S. Jabeen, G.Y. Jeng, R.G. Kellogg, J. Kunkle, A.C. Mignerey, S. Nabili, F. Ricci-Tam, M. Seidel, Y.H. Shin, A. Skuja, S.C. Tonwar, K. Wong Massachusetts Institute of Technology, Cambridge, USA D. Abercrombie, B. Allen, V. Azzolini, A. Baty, R. Bi, S. Brandt, W. Busza, I.A. Cali, M. D’Alfonso, Z. Demiragli, G. Gomez Ceballos, M. Goncharov, P. Harris, D. Hsu, M. Hu, Y. Iiyama, M. Klute, D. Kovalskyi, Y.-J. Lee, P.D. Luckey, B. Maier, A.C. Marini, C. Mcginn, C. Mironov, S. Narayanan, X. Niu, C. Paus, D. Rankin, C. Roland, G. Roland, Z. Shi, G.S.F. Stephans, K. Sumorok, K. Tatar, D. Velicanu, J. Wang, T.W. Wang, B. Wyslouch University of Minnesota, Minneapolis, USA A.C. Benvenuti${}^{\textrm{\textdagger}}$, R.M. Chatterjee, A. Evans, P. Hansen, J. Hiltbrand, Sh. Jain, S. Kalafut, M. Krohn, Y. Kubota, Z. Lesko, J. Mans, R. Rusack, M.A. Wadud University of Mississippi, Oxford, USA J.G. Acosta, S. Oliveros University of Nebraska-Lincoln, Lincoln, USA E. Avdeeva, K. Bloom, D.R. Claes, C. Fangmeier, F. Golf, R. Gonzalez Suarez, R. Kamalieddin, I. Kravchenko, J. Monroy, J.E. Siado, G.R. Snow, B. Stieger State University of New York at Buffalo, Buffalo, USA A. Godshalk, C. Harrington, I. Iashvili, A. Kharchilava, C. Mclean, D. Nguyen, A. Parker, S. Rappoccio, B. Roozbahani Northeastern University, Boston, USA G. Alverson, E. Barberis, C. Freer, Y. Haddad, A. Hortiangtham, G. Madigan, D.M. Morse, T. Orimoto, A. Tishelman-charny, T. Wamorkar, B. Wang, A. Wisecarver, D. Wood Northwestern University, Evanston, USA S. Bhattacharya, J. Bueghly, O. Charaf, T. Gunter, K.A. Hahn, N. Odell, M.H. Schmitt, K. Sung, M. Trovato, M. Velasco University of Notre Dame, Notre Dame, USA R. Bucci, N. Dev, R. Goldouzian, M. Hildreth, K. Hurtado Anampa, C. Jessop, D.J. Karmgard, K. Lannon, W. Li, N. Loukas, N. Marinelli, F. Meng, C. Mueller, Y. Musienko39, M. Planer, R. Ruchti, P. Siddireddy, G. Smith, S. Taroni, M. Wayne, A. Wightman, M. Wolf, A. Woodard The Ohio State University, Columbus, USA J. Alimena, L. Antonelli, B. Bylsma, L.S. Durkin, S. Flowers, B. Francis, C. Hill, W. Ji, T.Y. Ling, W. Luo, B.L. Winer Princeton University, Princeton, USA S. Cooperstein, P. Elmer, J. Hardenbrook, N. Haubrich, S. Higginbotham, A. Kalogeropoulos, S. Kwan, D. Lange, M.T. Lucchini, J. Luo, D. Marlow, K. Mei, I. Ojalvo, J. Olsen, C. Palmer, P. Piroué, J. Salfeld-Nebgen, D. Stickland, C. Tully University of Puerto Rico, Mayaguez, USA S. Malik, S. Norberg Purdue University, West Lafayette, USA A. Barker, V.E. Barnes, S. Das, L. Gutay, M. Jones, A.W. Jung, A. Khatiwada, B. Mahakud, D.H. Miller, N. Neumeister, C.C. Peng, S. Piperov, H. Qiu, J.F. Schulte, J. Sun, F. Wang, R. Xiao, W. Xie Purdue University Northwest, Hammond, USA T. Cheng, J. Dolen, N. Parashar Rice University, Houston, USA Z. Chen, K.M. Ecklund, S. Freed, F.J.M. Geurts, M. Kilpatrick, Arun Kumar, W. Li, B.P. Padley, R. Redjimi, J. Roberts, J. Rorie, W. Shi, Z. Tu, A. Zhang University of Rochester, Rochester, USA A. Bodek, P. de Barbaro, R. Demina, Y.t. Duh, J.L. Dulemba, C. Fallon, T. Ferbel, M. Galanti, A. Garcia-Bellido, J. Han, O. Hindrichs, A. Khukhunaishvili, E. Ranken, P. Tan, R. Taus The Rockefeller University, New York, USA R. Ciesielski, K. Goulianos Rutgers, The State University of New Jersey, Piscataway, USA B. Chiarito, J.P. Chou, Y. Gershtein, E. Halkiadakis, A. Hart, M. Heindl, E. Hughes, S. Kaplan, R. Kunnawalkam Elayavalli, S. Kyriacou, I. Laflotte, A. Lath, R. Montalvo, K. Nash, M. Osherson, H. Saka, S. Salur, S. Schnetzer, D. Sheffield, S. Somalwar, R. Stone, S. Thomas, P. Thomassen University of Tennessee, Knoxville, USA H. Acharya, A.G. Delannoy, J. Heideman, G. Riley, S. Spanier Texas A&M University, College Station, USA O. Bouhali71, A. Celik, M. Dalchenko, M. De Mattia, A. Delgado, S. Dildick, R. Eusebi, J. Gilmore, T. Huang, T. Kamon72, S. Luo, D. Marley, R. Mueller, D. Overton, L. Perniè, D. Rathjens, A. Safonov Texas Tech University, Lubbock, USA N. Akchurin, J. Damgov, F. De Guio, P.R. Dudero, S. Kunori, K. Lamichhane, S.W. Lee, T. Mengke, S. Muthumuni, T. Peltola, S. Undleeb, I. Volobouev, Z. Wang, A. Whitbeck Vanderbilt University, Nashville, USA S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K. Padeken, F. Romeo, P. Sheldon, S. Tuo, J. Velkovska, M. Verweij, Q. Xu University of Virginia, Charlottesville, USA M.W. Arenton, P. Barria, B. Cox, R. Hirosky, M. Joyce, A. Ledovskoy, H. Li, C. Neu, T. Sinthuprasith, Y. Wang, E. Wolfe, F. Xia Wayne State University, Detroit, USA R. Harr, P.E. Karchin, N. Poudyal, J. Sturdy, P. Thapa, S. Zaleski University of Wisconsin - Madison, Madison, WI, USA J. Buchanan, C. Caillol, D. Carlsmith, S. Dasu, I. De Bruyn, L. Dodd, B. Gomber73, M. Grothe, M. Herndon, A. Hervé, U. Hussain, P. Klabbers, A. Lanaro, K. Long, R. Loveless, T. Ruggles, A. Savin, V. Sharma, N. Smith, W.H. Smith, N. Woods †: Deceased 1: Also at Vienna University of Technology, Vienna, Austria 2: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France 3: Also at Universidade Estadual de Campinas, Campinas, Brazil 4: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil 5: Also at Université Libre de Bruxelles, Bruxelles, Belgium 6: Also at University of Chinese Academy of Sciences, Beijing, China 7: Also at Institute for Theoretical and Experimental Physics named by A.I. Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia 8: Also at Joint Institute for Nuclear Research, Dubna, Russia 9: Now at Cairo University, Cairo, Egypt 10: Also at Fayoum University, El-Fayoum, Egypt 11: Now at British University in Egypt, Cairo, Egypt 12: Now at Ain Shams University, Cairo, Egypt 13: Also at Department of Physics, King Abdulaziz University, Jeddah, Saudi Arabia 14: Also at Université de Haute Alsace, Mulhouse, France 15: Also at Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia 16: Also at Tbilisi State University, Tbilisi, Georgia 17: Also at CERN, European Organization for Nuclear Research, Geneva, Switzerland 18: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany 19: Also at University of Hamburg, Hamburg, Germany 20: Also at Brandenburg University of Technology, Cottbus, Germany 21: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary, Debrecen, Hungary 22: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary 23: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös Loránd University, Budapest, Hungary, Budapest, Hungary 24: Also at IIT Bhubaneswar, Bhubaneswar, India, Bhubaneswar, India 25: Also at Institute of Physics, Bhubaneswar, India 26: Also at Shoolini University, Solan, India 27: Also at University of Visva-Bharati, Santiniketan, India 28: Also at Isfahan University of Technology, Isfahan, Iran 29: Also at Plasma Physics Research Center, Science and Research Branch, Islamic Azad University, Tehran, Iran 30: Also at Italian National Agency for New Technologies, Energy and Sustainable Economic Development, Bologna, Italy 31: Also at Università degli Studi di Siena, Siena, Italy 32: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy 33: Also at Kyung Hee University, Department of Physics, Seoul, Korea 34: Also at Riga Technical University, Riga, Latvia, Riga, Latvia 35: Also at International Islamic University of Malaysia, Kuala Lumpur, Malaysia 36: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia 37: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico 38: Also at Warsaw University of Technology, Institute of Electronic Systems, Warsaw, Poland 39: Also at Institute for Nuclear Research, Moscow, Russia 40: Now at National Research Nuclear University ’Moscow Engineering Physics Institute’ (MEPhI), Moscow, Russia 41: Also at St. Petersburg State Polytechnical University, St. Petersburg, Russia 42: Also at University of Florida, Gainesville, USA 43: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia 44: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia 45: Also at University of Belgrade: Faculty of Physics and VINCA Institute of Nuclear Sciences, Belgrade, Serbia 46: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy, Pavia, Italy 47: Also at National and Kapodistrian University of Athens, Athens, Greece 48: Also at Universität Zürich, Zurich, Switzerland 49: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria, Vienna, Austria 50: Also at Adiyaman University, Adiyaman, Turkey 51: Also at Istanbul Aydin University, Application and Research Center for Advanced Studies (App. & Res. Cent. for Advanced Studies), Istanbul, Turkey 52: Also at Mersin University, Mersin, Turkey 53: Also at Piri Reis University, Istanbul, Turkey 54: Also at Ozyegin University, Istanbul, Turkey 55: Also at Izmir Institute of Technology, Izmir, Turkey 56: Also at Marmara University, Istanbul, Turkey 57: Also at Kafkas University, Kars, Turkey 58: Also at Istanbul University, Istanbul, Turkey 59: Also at Istanbul Bilgi University, Istanbul, Turkey 60: Also at Hacettepe University, Ankara, Turkey 61: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom 62: Also at School of Physics and Astronomy, University of Southampton, Southampton, United Kingdom 63: Also at Monash University, Faculty of Science, Clayton, Australia 64: Also at Bethel University, St. Paul, Minneapolis, USA, St. Paul, USA 65: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey 66: Also at Purdue University, West Lafayette, USA 67: Also at Beykent University, Istanbul, Turkey, Istanbul, Turkey 68: Also at Bingol University, Bingol, Turkey 69: Also at Sinop University, Sinop, Turkey 70: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey 71: Also at Texas A&M University at Qatar, Doha, Qatar 72: Also at Kyungpook National University, Daegu, Korea, Daegu, Korea 73: Also at University of Hyderabad, Hyderabad, India ## .11 The TOTEM Collaboration G. Antcheva, P. Aspell9, I. Atanassova, V. Avati7,9, J. Baechler9, C. Baldenegro Barrera11, V. Berardi4a,4b, M. Berretti2a, V. Borchsh8, E. Bossini9,6b, U. Bottigli6b, M. Bozzo5a,5b, H. Burkhardt9, F. S. Cafagna4a, M. G. Catanesi4a, M. Csanád3a,b, T. Csörgő3a,3b, M. Deile9, F. De Leonardis4c,4a, M. Doubek1c, D. Druzhkin8,9, K. Eggert10, V. Eremind, A. Fiergolski9, L. Forthomme2a,2b, F. Garcia2a, V. Georgiev1a, S. Giani9, L. Grzanka7, J. Hammerbauer1a, T. Isidori11, V. Ivanchenko8, M. Janda1c, A. Karev9, J. Kašpar1b,9, B. Kaynake, J. Kopal9, V. Kundrát1b, S. Lami6a, R. Linhart1a, C. Lindsey11, M. V. Lokajíček1b,†, L. Losurdo6b, F. Lucas Rodríguez9, M. Macrí5a, M. Malawski7, N. Minafra11, S. Minutoli5a, T. Naaranoja2a,2b, F. Nemes9,3a, H. Niewiadomski10, T. Novák3b, E. Oliveri9, F. Oljemark2a,2b, M. Oriunnof, K. Österberg2a,2b, P. Palazzi9, V. Passaro4c,4a, Z. Peroutka1a, J. Procházka1b, M. Quinto4a,4b, E. Radermacher9, E. Radicioni4a, F. Ravotti9, C. Royon11, G. Ruggiero9, H. Saarikko2a,2b, V.D. Samoylenkoc, A. Scribano6a, J. Siroky1a, J. Smajek9, W. Snoeys9, R. Stefanovitch9, J. Sziklai3a, C. Taylor10, E. Tcherniaev8, N. Turini6b, O. Urban1a, V. Vacek1c, O. Vavroch1a, J. Welti2a,2b, J. Williams11, J. Zich1a, K. Zielinski7 1aUniversity of West Bohemia, Pilsen, Czech Republic. 1bInstitute of Physics of the Academy of Sciences of the Czech Republic, Prague, Czech Republic. 1cCzech Technical University, Prague, Czech Republic. 2aHelsinki Institute of Physics, University of Helsinki, Helsinki, Finland. 2bDepartment of Physics, University of Helsinki, Helsinki, Finland. 3aWigner Research Centre for Physics, RMKI, Budapest, Hungary. 3bEKU KRC, Gyöngyös, Hungary. 4aINFN Sezione di Bari, Bari, Italy. 4bDipartimento Interateneo di Fisica di Bari, Bari, Italy. 4cDipartimento di Ingegneria Elettrica e dell’Informazione — Politecnico di Bari, Bari, Italy. 5aINFN Sezione di Genova, Genova, Italy. 5bUniversità degli Studi di Genova, Italy. 6aINFN Sezione di Pisa, Pisa, Italy. 6bUniversità degli Studi di Siena and Gruppo Collegato INFN di Siena, Siena, Italy. 7AGH University of Science and Technology, Krakow, Poland. 8Tomsk State University, Tomsk, Russia. 9CERN, Geneva, Switzerland. 10Case Western Reserve University, Dept. of Physics, Cleveland, OH, USA. 11The University of Kansas, Lawrence, USA. ††a INRNE-BAS, Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria.††b Department of Atomic Physics, ELTE University, Budapest, Hungary.††c NRC ‘Kurchatov Institute’–IHEP, Protvino, Russia.††d Ioffe Physical - Technical Institute of Russian Academy of Sciences, St. Petersburg, Russian Federation.††e Istanbul University, Istanbul, Turkey.††f SLAC National Accelerator Laboratory, Stanford CA, USA.††† Deceased.

2024-09-04T02:54:55.091623

2002.12151

arxiv-papers

# Vortex: OpenCL Compatible RISC-V GPGPU Fares Elsabbagh Georgia Tech <EMAIL_ADDRESS>Blaise Tine Georgia Tech <EMAIL_ADDRESS>Priyadarshini Roshan Georgia Tech <EMAIL_ADDRESS>Ethan Lyons Georgia Tech <EMAIL_ADDRESS>Euna Kim Georgia Tech <EMAIL_ADDRESS>Da Eun Shim Georgia Tech <EMAIL_ADDRESS>Lingjun Zhu Georgia Tech <EMAIL_ADDRESS>Sung Kyu Lim Georgia Tech <EMAIL_ADDRESS>Hyesoon Kim Georgia Tech <EMAIL_ADDRESS> ###### Abstract The current challenges in technology scaling are pushing the semiconductor industry towards hardware specialization, creating a proliferation of heterogeneous systems-on-chip, delivering orders of magnitude performance and power benefits compared to traditional general-purpose architectures. This transition is getting a significant boost with the advent of RISC-V with its unique modular and extensible ISA, allowing a wide range of low-cost processor designs for various target applications. In addition, OpenCL is currently the most widely adopted programming framework for heterogeneous platforms available on mainstream CPUs, GPUs, as well as FPGAs and custom DSP. In this work, we present Vortex, a RISC-V General-Purpose GPU that supports OpenCL. Vortex implements a SIMT architecture with a minimal ISA extension to RISC-V that enables the execution of OpenCL programs. We also extended OpenCL runtime framework to use the new ISA. We evaluate this design using 15nm technology. We also show the performance and energy numbers of running them with a subset of benchmarks from the Rodinia Benchmark suite. ###### Index Terms: GPGPU, OpenCL, Vector processors ## I Introduction The emergence of data parallel architectures and general purpose graphics processing units (GPGPUs) have enabled new opportunities to address the power limitations and scalability of multi-core processors, allowing new ways to exploit the abundant data parallelism present in emerging big-data parallel applications such as machine learning and graph analytics. GPGPUs in particular, with their Single Instruction Multiple-Thread (SIMT) execution model, heavily leverage data-parallel multi-threading to maximize throughput at relatively low energy cost, leading the current race for energy efficiency (Green500 [12]) and applications support with their accelerator-centric parallel programming model (CUDA [19] and OpenCL [17]). The advent of RISC-V [21, 20, 2], open-source and free instruction set architecture (ISA), provides a new level of freedom in designing hardware architectures at lower cost, leveraging its rich eco-system of open-source software and tools. With RISC-V, computer architects have designed several innovative processors and cores such as BOOM v1 and BOOM v2 [4] out-of-order cores, as well as system-on-chip (SoC) platforms for a wide range of applications. For instance, Gautschi et al. [9] have extended RISC-V to digital signal processing (DSP) for scalable Internet-of-things (IoT) devices. Moreover, vector processors [22] [15] [3] and processors integrated with vector accelerators [16] [10] have been designed and fabricated based on RISC-V. In spite of the advantages of the preceding works, not enough attention has been devoted to building an open-source general-purpose GPU (GPGPU) system based on RISC-V. Although a couple of recent work have been proposed for massively parallel computations on FPGA using RISC-V, (GRVI Phalanx) [11], (Simty) [6], none of them have implemented the full-stack by extending the RISC-V ISA, synthesizing the microarchitecture, and implementing the software stack to execute OpenCL programs. We believe that such an implementation is in fact necessary to achieve the level of usability and customizability in massively parallel platforms. In this paper, we propose an ISA RISC-V extension for GPGPU programs and microarchitecture. We also extend a software stack to support OpenCL. This paper makes the following key contributions: * • We propose a highly configurable SIMT-based General Purpose GPU architecture targeting the RISC-V ISA and synthesized the design using a Synopsys library with our RTL design. * • We show that the minimal set of five instructions on top of RV32IM (RISC-V 32 bit integer and multiply extensions) enables SIMT execution. * • We describe the necessary changes in the software stack that enable the execution of OpenCL programs on Vortex. We demonstrate the portability by running a subset of Rodinia benchmarks [5]. Figure 1: Vortex System Overview ## II Background ### II-A Open-Source OpenCL Implementations POCL [13] implements a flexible compilation backend based on LLVM, allowing it to support a wider range of device targets including general purpose processors (e.g. x86, ARM, Mips), General Purpose GPU (e.g. Nvidia), and TCE- based processors [14] and custom accelerators. The custom accelerator support provides an efficient solution for enabling OpenCL applications to use hardware devices with specialized fixed-function hardware (e.g. SPMV, GEMM). POCL is comprised of two main components: A back-end compilation engine, and a front-end OpenCL runtime API. The POCL runtime implements the device-independent common interface where the target implementation of each device plugs into POCL to specialize their operations. At runtime, POCL invokes the back-end compiler with the provided OpenCL kernel source. POCL supports target-specific execution models including SIMT, MIMD, SIMD, and VLIW. On platforms supporting MIMD and SIMD execution models such as CPUs, the POCL compiler attempts to pack as many OpenCL work- items to the same vector instruction, then the POCL runtime will distribute the remaining work-items among the active hardware threads on the device with provided synchronization. On platforms supporting SIMT execution model such as GPUs, the POCL compiler delegates the distribution of the work-items to the hardware to spread the execution among its hardware threads, relying on the device to also handle the necessary synchronization. On platforms supporting VLIW execution models such as TCE-based accelerators, the POCL compiler attempts to ”unroll” the parallel regions in the kernel code such that the operations of several independent work-items can be statically scheduled to the multiple function units of the target device. ## III The OpenCL Software Stack Figure 2: Vortex Runtime Library ⬇ 1vx_intrinsic.s 2vx_split: 3 .word 0x0005206b # split a0 4 ret 5vx_join: 6 .word 0x0000306b #join 7 ret 8 9kernel.cl 10#define __if(cond) split(cond); \ 11 if(cond) 12 13#define __endif join(); 14void opencl_kernel(){ 15 int id = vx_getTid(); 16 __if(id<4) { 17 // Path A 18 } else { 19 // Path B 20 } __endif 21} Figure 3: This figure shows the control divergent __if __endif macro definitions and how they could be used to enable control divergence in OpenCL kernels. Currently, this process is done manually for each kernel. ### III-A Vortex Native Runtime The Vortex software stack implements a native runtime library for developing applications that will run on Vortex and take advantage of the new RISC-V ISA extension. Figure 2 illustrates the Vortex runtime layer, which is comprised of three main components: 1) Low-level intrinsic library exposing the new ISA interface, 2) A support library that implements NewLib stub functions [7], 3) a native runtime API for launching POCL kernels. #### III-A1 Instrinsic Library To enable Vortex runtime kernel to utilize the new instructions without modifying the existing compilers, we implemented an intrinsic layer that implements the new ISA. Figure 2 shows the functions and ISA supported by the intrinsic library. We leverage RISC-V’s ABI which guarantees function arguments being passed through the argument registers and return values begin passed through a0 register. Thus, these intrinsic functions have only two assembly instructions: 1) The encoded 32-bit hex representation of the instruction that uses the argument registers as source registers, and 2) a return instruction that returns back to the C++ program. An example of these intrinsic functions is illustrated in Figure 3. In addition, to handle control divergence, which is frequent in OpenCL kernels, we implement __if and __endif macros shown in Figure 3 to handle the insertion of these intrinsic functions with minimal changes to the code. These changes are currently done manually for the OpenCL kernels. This approach achieves the required functionality without restricting the platform or requiring any modifications to the RISC-V compilers. #### III-A2 Newlib Stubs Library Vortex software stack uses the NewLib [7] library to enable programs to use the C/C++ standard library without the need to support an operating system. NewLib defines a minimal set of stub functions that client applications need to implement to handle necessary system calls such as file I/O, allocation, time, process, etc.. #### III-A3 Vortex Native API The Vortex native API implements some general purpose utility routines for applications to use. One of such routines is pocl_spawn() which allows programs to schedule POCL kernel execution on Vortex. pocl_spawn() is responsible for mapping work groups requested by POCL to the hardware: 1) It uses the intrinsic layer to find out the available hardware resources, 2) Uses the requested work group dimension and numbers to divide the work equally among the hardware resources, 3) For each OpenCL dimension, it assigns a range of IDs to each available warp in a global structure, 4) It uses the intrinsic layer to spawn the warps and activate threads, and finally 5) Each warp will loop through the assigned IDs, executing the kernel every time with a new OpenCL global_id. Figure 4 shows an example. In the original OpenCL code, the kernel is called once with global/local sizes as arguments. POCL wraps the kernel with three loops and sequentially calls with the logic that converts x,y,z to global ids. For a Vortex version, warps and threads are spawned, then each thread is assigned a different work-group to execute the kernel. POCL provides the feature to map the correct wid, which was a part of the baseline POCL implementation to support various hardware such as vector architecture. Figure 4: SIMT Kernel invocation modification for Vortex in POCL ### III-B POCL Runtime We modified the POCL runtime, adding a new device target to its common device interface to support Vortex. The new device target is essentially a variant of the POCL basic CPU target with the support for pthreads and other OS dependencies removed to target the NewLib interface. We also modified the single-threaded logic for executing work-items to use Vortex’s pocl_spawn runtime API. ### III-C Barrier Support Synchronizations within work-group in OpenCL is supported by barriers. POCL’s back-end compiler splits the control-flow graph (CFG) of the kernel around the barrier and splits the kernel into two sections that will be executed by all local work-groups sequentially. ## IV Vortex Parallel Hardware Architecture Figure 5: Vortex Microarchitecture. Figure 6: This figure shows the Warp Scheduler under different scenarios. In the actual microarchitecture implementation, the instruction is only known the next cycle, however it’s displayed in the same cycle in this figure for simplicity. ### IV-A SIMT Hardware Primitives SIMT, Single Instruction-Multiple Threads, execution model takes advantage of the fact that in most parallel applications, the same code is repeatedly executed but with different data. Thus, it provides the concept of Warps [19], which is a group of threads that share the same PC and follows the same execution path with minimal divergence. Each thread in a warp has a private set of general purpose registers, and the width of the ALU is matched with the number of threads. However, the fetching, decoding, and issuing of instructions is shared within the same warp which reduces execution cycles. However, in some cases, the threads in the same warp will not agree on the direction of branches. In such cases, the hardware must provide a thread mask to predicate instructions for each thread, and an IPDOM stack, to ensure all threads execute correctly, which are explained in Section IV-C. ### IV-B Warp Scheduler The warp scheduler is in the fetch stage which decides what to fetch from I-cache as shown in Figure 5. It has two components: 1) A set of warp masks to choose the warp to schedule next, and 2) a warp table that includes private information for each warp. There are 4 thread masks that the scheduler uses: 1) an active warps mask, one bit indicating whether a warp is active or not, 2) a stalled warp mask, which indicates which warps should not be scheduled temporarily ( e.g., waiting for a memory request), 3) a barrier warps stalled mask, which indicates warps that have been stalled because of a barrier instruction, and 4) a visible warps mask to support hierarchical scheduling policy [18]. Each cycle, the scheduler selects one warp from the visible warp mask and invalidates that warp. When visible warp mask is zero, the active mask is refilled by checking which warps are currently active and not stalled. An example of the warp scheduler is shown in Figure 6(a). This figure shows the normal execution; The first cycle warp zero executes an instruction, then in the second cycle warp zero is invalidated in the visible warp mask, and warp one is scheduled. In third cycle, because there are no more warps to be scheduled, the scheduler uses the active warps to refill the visible mask, and schedules warp zero for execution. Figure 6(b) shows how the warp scheduler handles a stalled warp. In the first cycle the scheduler schedules warp zero. In the second cycle, the scheduler schedules warp one. At the same cycle, because the decode stage identified that warp zero’s instruction requires a change of state, it stalls warp zero. In the third cycle, because warp zero is stalled, the scheduler only sets warp one to visible warp mask and schedules warp one again. When warp zero updates its thread mask, the bit in stalled mask will be set to 0 to allow scheduling. Figure 6(c) shows an example of spawning warps. When warp zero executes a wspawn instruction (Table I) , which activates warps and manipulates the active warps mask by setting warps two and three to be active. When it’s time to refill the visible mask, because it no longer has any warps to schedule, it includes warps two and three. Warps will stay in the Active Mask until they set their thread mask’s value to zero, or warp zero utilizes wspawn to deactivate these warps. ### IV-C Threads Masks and IPDOM Stack To support the thread concepts provided in Section IV-A, a thread mask register and an IPDOM stack have been added to the hardware similar to other SIMT architectures [8]. The thread mask register acts like a predicate for each thread, controlling which threads are active. If the bit in the thread mask for a specific thread is zero, no modifications would be made to that thread’s register file and no changes to the cache would be made based on that thread. The IPDOM stack, illustrated in Figure 5 is used to handle control divergence and is controlled by the split and join instructions. These instructions utilize the IPDOM stack to enable divergence as shown in Figure 3. When a split instruction is executed by a warp, the predicate value for each thread is evaluated. If there is only one thread active, or all threads agree on a direction, the split acts like a nop instruction and does not change the state of the warp. When there is more than one active thread that contradicts on the value of the predicate, three microarchitecture events occur: 1) The current thread mask is pushed into the IPDOM stack as a fall-through entry, 2) The active threads that evaluate the predicate as false are pushed into the stack with PC+4 (i.e., size of instruction) of the split instruction, and 3) The current thread mask is updated to reflect the active threads that evaluate the predicate to be true. When a join instruction is executed, an entry is popped out of the stack which causes one of two scenarios: 1) If the entry is not a fall-through entry, the PC is set to the entry’s PC and the thread mask is updated to the value of the entry’s mask, which enables the threads evaluating the predicate as false to follow their own execution path, and 2) If the entry is a fall-through entry, the PC continues executing to PC+4 and the thread mask is updated to the entry’s mask, which is the case when both paths of the control divergence have been executed. ### IV-D Warp Barriers Warp barriers are important in SIMT execution, as it provides synchronization between warps. Barriers are provided in the hardware to support global synchronization between work-groups. Each barrier has a private entry in barrier table, shown in Figure 5, with the following information: 1) Whether that barrier is currently valid, 2) the number of warps left that need to execute the barrier instruction with that entry’s ID for the barrier to be released, and 3) a mask of the warps that are currently stalled by that barrier. However, Figure 5 only shows the per core barriers. There is also another table on multi-core configurations that allows for global barriers between all the cores. The MSB of the barrier ID indicates whether the instruction uses local barriers or global barriers. When a barrier instruction is executed, the microarchitecture checks the number of warps executed with the same barrier ID. If the number of warps is not equal to one, the warp is stalled until that number is reached and the release mask is manipulated to include that warp. Once the same number of warps have been executed, the release mask is used to release all the warps stalled by the corresponding barrier ID. The same method works for both local and global barriers; however, global barrier tables have a release mask per each core. TABLE I: Proposed SIMT ISA extension. Instructions | Description ---|--- wspawn %numW, %PC | Spawn W new warps at PC tmc %numT | Change the thread mask to activate threads split %pred | Control flow divergence join | Control flow reconvergence bar %barID, %numW | Hardware Warps Barrier (a) GDS Layout (b) Power density distribution across Vortex Figure 7: GDS layouts for our Vortex with 8 warp, 4 thread configuration (4KB register file). The design was synthesized for 300Mhz and produced a total power output of 46.8mW. 4KB 2 ways 4 banks-data cache, 8KB with 4 banks-shared memory, 1Kb 2 way cache, one bank-I cache. ## V Evaluation This section will evaluate both the RTL Verilog model for Vortex and the software stack. ### V-A Micro-architecture Design space explorations In Vortex design, we can increase the data-level parallelism either by increasing the number of threads or by increasing the number of warps. Increasing the number of threads is similar to increasing the SIMD width and involves the following changes to the hardware: 1) Increasing the GPR memory width for reads and writes, 2) Increasing the number of ALUs to match the number of threads, 3) increasing the register width for every pipeline stage after the GPR read stage, 4) increasing the arbitration logic required in both the cache and the shared memory to detect bank conflicts and handle cache misses, and 5) increasing the number of IPDOM enteries. Whereas, increasing the number of warps does not require increasing the number of ALUs because the ALUs are multiplexed by a higher number of warps. Increasing the number of warps involves the following changes to the hardware: 1) increasing the logic for the warp scheduler, 2) increasing the number of GPR tables, 3) increasing the number of IPDOM stacks, 4) increasing the number of register scoreboards, and 5) increasing the size of the warp table. It’s important to note that the cost of increasing the number of warps is dependant on the number of threads in that warp; thus increasing warps for bigger thread configurations becomes more expensive. This is because the size of each GPR table, IPDOM stack, and warp table are dependant on the number of threads. Figure 8 shows the increases in the area and power as we increase the number of threads and warps. The number is normalized to 1 warp and 1 thread support. All the data includes 1Kb 2 way instruction cache, 4 Kb 2 way 4 banks data cache, and an 8kb 4 banks shared memory module. Figure 8: Synthesized results for power, area and cell counts for different number of warps and threads ### V-B Benchmarks All the benchmarks used for evaluations were taken from the Rodinia [5], a popular GPGPU benchmark suite.111The benchmarks that are not evaluated in this paper is due to the lack of support from LLVM RISC-V. ### V-C simX Simulator Because the benchmarks used in Rodinia Benchmark Suite have large data-sets that took Modelsim a long time to simulate, we used simX, a C++ cycle-level in-house simulator for Vortex with a cycle accuracy within 6% of the actual Verilog model. Please note that power and area numbers are synthesized from the RTL. ### V-D Performance Evaluations Figure 9: Performance of a subset of Rodinia benchmark suite (# of warps x # of threads Figure 9 shows the normalized execution time of the benchmarks normalized to the 2warps x 2threads configuration. As we predict, most of the time, as we increase the number of threads (i.e., increasing the SIMD width.), the performance is improved, but not too much from increasing the number of warps. Some benchmarks get benefits from increasing the number of warps such as bfs, but in most of the cases increasing the number of warps is not translated into performance benefit. The main reason is that to reduce the simulation time, we warmed up caches and reduced the data set size, thereby the cache hit rate in the evaluated benchmarks was high. Increasing the number of warps is typically useful to hide long latency operations such as cache misses by increasing TLP and MLP; Thus, the benchmark that benefited the most from the high warp count is BFS which is an irregular benchmark. As we increase the number of threads and warps, the power consumption increases but they do not necessarily produce more performance. Hence, the most power efficient design points vary depending on the benchmark. Figure 10 shows a power efficiency metric (similar to performance per watt) which is normalized to the 2 warps x 2 threads configuration. The results show that for many benchmarks, the most power efficient design is the one with fewer number of warps and 32 threads except for the BFS benchmark. As we discussed earlier since BFS benchmark gets the best performance from the 32 warps x 32 threads configuration, it also shows the most power efficient design point. Figure 10: Power efficiency (# of warps x # of threads (Power efficiency and Energy) ### V-E Placement and layout We synthesized our RTL using a 15-nm educational library. Using Innovus, we also performed Place and route (PnR). Figure 7 shows the GDS layout and the power density map of our Vortex processor. From the power density map, we observe that the power is well distributed among the cell area. In addition, we observe the memory including the GPR, data cache, instruction icache and the shared memory have a higher power consumption. ## VI Related Work ARA [3] is a RISC-V Vector Processor that implements a variable-length single- instruction multiple-data execution model where vector instructions are streamed into vector lanes and their execution is time-multiplexed over the shared execution units to increase energy efficiency. Ara design is based on the open-source RISC-V Vector ISA Extension proposal [1] taking advantage of its vector-length agnostic ISA and its relaxed architectural vector registers. Maximizing the utilization of the vector processors can be challenging, specifically when dealing with data dependent control flow. That is where SIMT architectures like Vortex present an advantage with their flexible scalar- threads that can diverge independently. HWACHA [15] is a RISC-V scalar processor with a vector accelerator that implements a SIMT-like architecture where vector arithmetic instructions are expanded into micro-ops and scheduled on separate processing threads on the accelerator. An advantage that Hwacha has over pure SIMT processors like Vortex is its ability to overlap the execution of scalar instructions on the scalar processor which increases hardware complexity for hazard management. Simty [6] processor implements a specialized RISC-V architecture that supports SIMT execution similar to Vortex, but with different control flow divergence handling. In their work, only microarchitecture was implemented as a proof of concept and there was no software stack, and none of GPGPU applications were executed with the architecture. ## VII Conclusions In this paper we proposed Vortex that supports an extended version of RISC-V for GPGPU applications. We also modified OpenCL software stack (POCL) to run various OpenCL kernels and demonstrated that. We plan to release the Vortex RTL and POCL modifications to the public.222currently the github is private for blind review. We believe that an Open Source version of RISC-V GPGPU will enrich the RISC-V echo systems and accelerate other researchers that study GPGPUs in wider topics since the entire software stack is also based on Open Source implementations. ## References * [1] K. Asanovic, _RISC-V Vector Extension_. [Online]. Available: https://github.com/riscv/riscv-v-spec/blob/master/v-spec.adoc * [2] K. Asanović _et al._ , “Instruction sets should be free: The case for risc-v,” _EECS Department, University of California, Berkeley, Tech. Rep. UCB/EECS-2014-146_ , 2014. * [3] M. A. Cavalcante _et al._ , “Ara: A 1 GHz+ scalable and energy-efficient RISC-V vector processor with multi-precision floating point support in 22 nm FD-SOI,” _CoRR_ , vol. abs/1906.00478, 2019. * [4] C. Celio _et al._ , “Boomv2: an open-source out-of-order risc-v core,” in _First Workshop on Computer Architecture Research with RISC-V (CARRV)_ , 2017\. * [5] S. Che _et al._ , “Rodinia: A benchmark suite for heterogeneous computing,” ser. IISWC ’09. IEEE Computer Society, 2009. * [6] S. Collange, “Simty: generalized simt execution on risc-v,” in _First Workshop on Computer Architecture Research with RISC-V (CARRV 2017)_ , 2017, p. 6. * [7] J. J. Corinna Vinschen, “Newlib,” http://sourceware. org/newlib, 2001. * [8] W. W. L. Fung _et al._ , “Dynamic warp formation and scheduling for efficient gpu control flow,” ser. MICRO 40. IEEE Computer Society, 2007, pp. 407–420. * [9] M. Gautschi _et al._ , “Near-threshold risc-v core with dsp extensions for scalable iot endpoint devices,” _IEEE Transactions on Very Large Scale Integration (VLSI) Systems_ , vol. 25, no. 10, pp. 2700–2713, 2017. * [10] G. Gobieski _et al._ , “Manic: A vector-dataflow architecture for ultra-low-power embedded systems,” ser. MICRO ’52. ACM, 2019, pp. 670–684. * [11] J. Gray, “Grvi phalanx: A massively parallel risc-v fpga accelerator accelerator,” in _2016 IEEE 24th Annual International Symposium on Field-Programmable Custom Computing Machines (FCCM)_. IEEE, 2016, pp. 17–20. * [12] Green500, “Green500 list - june 2019,” 2019. [Online]. Available: https://www.top500.org/lists/2019/06/ * [13] P. Jaaskelainen _et al._ , “Pocl: Portable computing language,” _International Journal of Parallel Programming_ , pp. 752–785, 2015. * [14] P. O. Jäskeläinen _et al._ , “Opencl-based design methodology for application-specific processors,” in _2010 International Conference on Embedded Computer Systems: Architectures, Modeling and Simulation_ , July 2010, pp. 223–230. * [15] Y. Lee _et al._ , “A 45nm 1.3ghz 16.7 double-precision gflops/w risc-v processor with vector accelerators,” in _ESSCIRC 2014 - 40th European Solid State Circuits Conference (ESSCIRC)_ , Sep. 2014, pp. 199–202. * [16] Y. Lee _et al._ , “A 45nm 1.3 ghz 16.7 double-precision gflops/w risc-v processor with vector accelerators,” in _ESSCIRC 2014-40th European Solid State Circuits Conference (ESSCIRC)_. IEEE, 2014, pp. 199–202. * [17] A. Munshi, “The opencl specification,” in _2009 IEEE Hot Chips 21 Symposium (HCS)_ , Aug 2009, pp. 1–314. * [18] V. Narasiman _et al._ , “Improving gpu performance via large warps and two-level warp scheduling,” ser. MICRO-44. New York, NY, USA: ACM, 2011, pp. 308–317. * [19] NVIDIA, “Cuda binary utilities,” NVIDIA Application Note, 2014. * [20] A. Waterman _et al._ , “The risc-v instruction set manual. volume 1: User-level isa, version 2.0,” EECS Department, UC Berkeley, Tech. Rep., 2014\. * [21] A. Waterman _et al._ , “The risc-v instruction set manual, volume i: Base user-level isa,” _EECS Department, UC Berkeley, Tech. Rep. UCB/EECS-2011-62_ , vol. 116, 2011. * [22] B. Zimmer _et al._ , “A risc-v vector processor with tightly-integrated switched-capacitor dc-dc converters in 28nm fdsoi,” in _2015 Symposium on VLSI Circuits (VLSI Circuits)_. IEEE, 2015, pp. C316–C317.

2024-09-04T02:54:55.100310

2002.12164

arxiv-papers

# Performance Analysis of Semi-supervised Learning in the Small-data Regime using VAEs Varun Mannam Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 <EMAIL_ADDRESS> &Arman Kazemi Department of Computer Science and Engineering University of Notre Dame Notre Dame, IN 46556 <EMAIL_ADDRESS> Varun Mannam is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, 46556 USA e-mail<EMAIL_ADDRESS> ###### Abstract Extracting large amounts of data from biological samples is not feasible due to radiation issues, and image processing in the small-data regime is one of the critical challenges when working with a limited amount of data. In this work, we applied an existing algorithm named Variational Auto Encoder (VAE) that pre-trains a latent space representation of the data that captures the features in a lower-dimension for the small-data regime input. The latent space representation will be fine-tuned, and its weights will be fixed. The latent space will be used as a segment of the neural network that can be used for classification. Here we will present the performance analysis of the VAE algorithm with various latent space sizes in the semi-supervised learning using the CIFAR-10 dataset. _K_ eywords VAE, CIFAR-10, Small Data ## 1 Introduction Artificial neural networks (ANNs), specifically Convolutional Neural Networks (CNNS), have become popular due to their success in image classification, feature extraction and object recognition and detection [1] in the recent years. CNNs leverage the huge amount of labeled data available to train networks that outperform humans in image recognition tasks. However, in the small-data regime, the accuracy of trained networks using a limited number of labeled samples is low [2]. This is a typical case when working with biological samples where exposure to radiation (in order to capture an image) is detrimental to the well-being of the sample. More images can be derived from the initial data by some augmentation methods, but it is unhelpful due to lack of labeled images. To address this problem, there exists a framework called “Auto Encoder” (AE [3]) that uses all the input data, labeled and unlabeled, to train a low- dimensional embedding. AE is a neural network that takes unlabeled images as the input and regards the input itself as the label. As illustrated in Figure 1, AE is comprised of two parts: the encoder and the decoder. The encoder part tries to embed the features in a latent space that can extract the features of the original image and the decoder tries to restore the image to the original image. The process called “pre-training” trains the weights for both encoder and decoder parts. Once trained, the encoder part of the AE will be a representation of all the labeled and unlabeled data. This increases the amount of usable information from all of the images. Traditional semi-supervised models consists of pretraining using Restricted Boltzmann machine (RBM)[4] or Gaussian-Restricted Boltzmann machine (G-RBM)[4]. RBM is an energy based model that is represented using an undirected graph containing a layer of observable variables and a single layer of latent variables (similar to hidden units in a multi-layer perceptron) [1, 5]. This energy based model was first introduced in 1980’s [6] and have been implemented using diverse datasets including image [4] and medical data [7]. Hinton and Salakhutdinov [8] showed that RBMs can be stacked and trained in a greedy manner. This deep learning model, Deep belief networks(DBN), that utilizes RBM as the learning model has been implemented on various unsupervised and supervised learning problems. Later, Bengio et. al. [9] showed that the pre-trained undirected graphical model in semi-supervised setting performs well with deep architectures. However, the challenge working with RBM is that it is constructed using sigmoid functions as the activation function between the input and the hidden layer. As we are aware that the major drawback of using sigmoid activation function is the vanishing gradient problem. Hence, in this work, we pre-train the model using “Variational Auto Encoder” (VAE [3]). After pre-training the encoder is able to observe similar images to the training images and extract the valuable features from it in a lower dimension than the initial image. Now it is possible to couple the encoder with a small neural network and train that network for classification tasks. The present work is similar to reinforcement learning where the model is trained with one dataset and uses the feature extraction part of that model to train another model for a different dataset. It is important to mention that the encoder weights are fixed and can’t be changed. It is only the small neural network that will be trained. The input to this network is the small set of labeled data. This is called “fine-tuning”. In this project we will implement VAE that tries to capture not only the compressed representation of the images, but also the parameters of a probability distribution representing the data. We will examine the effect of different size of the latent spaces and how it affects the accuracy of the model. Later, we will analyze the performance of the semi-supervised model with the optimum latent space. ## 2 Background In this section we enumerate the basic details about AE and VAE. ### 2.1 Auto Encoder An autoencoder is a type of ANN used to learn efficient data encoding in an unsupervised manner. The aim of an autoencoder is to learn a representation of a set of data, typically for dimensionality reduction, by training the network to ignore signal noise. Along with the reduction side, a reconstructing side is learned, where the autoencoder tries to generate from the reduced encoding a representation as close as possible to its original input. An autoencoder always consists of two parts, the encoder and the decoder, which can be defined as transitions $\phi$ and $\psi$ such that [3]: $\phi:X\rightarrow F$ (1) $\psi:F\rightarrow Y\newline $ (2) $\phi,\psi=argmin_{\phi,\psi}||X-(\psi o\phi)(X)^{2}||$ (3) where the given input is $X$ and the predicted output is $Y$. If the feature space $F$ has lower dimensionality than the input space $X$, then the feature vector $\phi(x)$ can be regarded as a compressed representation of the input $X$. Figure 1: Autoencoder block diagram ### 2.2 VAE To extend the idea used in the AE, there is a variation called VAE which uses the “KL-divergence” between the predicted probability distribution function and actual posterior distribution in the latent space whereas traditional AEs try to find the accurate mapping functions in the encoder (between input and latent space) as well as in the decoder (between the latent space and output). Using VAE, we generate a large dataset by adding the noise in the latent space which is similar to input data augmentation (adding noise to images to increase the number of examples in the input dataset). VAE uses a variational approach for latent representation learning, which results in an additional loss component. It assumes that the data is generated by a directed graphical model $p(\textbf{X}|\textbf{Z})$ and that the encoder is learning an approximation $q_{\phi}(\textbf{Z}|\textbf{X})$ of the posterior distribution $p_{\theta}(\textbf{X}|\textbf{Z})$ where $\phi$ and $\theta$ denote the parameters of the encoder (recognition model) and decoder (generative model) respectively. We can write the conditional or posterior distribution $p(\bm{z}|\bm{x})=\frac{p(\bm{z},\bm{x})}{p(\bm{x})}=\frac{p(\bm{x}|\bm{z})p(\bm{z})}{p(\bm{x})}$ (4) The denominator of above equation is the marginal distribution of the observations and is calculated by marginalizing out the latent variables from the joint distribution, i.e. $p(x)=\int_{z}p(z,x)dz$ (5) In many cases of interest this integral is not available in closed form or is intractable (requires exponential time to compute). Hence, we consider variational approximation as follows: consider a tractable distribution q(z). The goal is to find the best approximation, e.g., the one that satisfies the following optimization problem: $Minimize:D_{KL}[q_{\phi}(\bm{z}|\bm{x})||p_{\theta}(\bm{z}|\bm{x})]$ (6) Therefore, the objective of the variational autoencoder in this case has the following form: $\mathcal{L}(\phi,\theta,\mathbf{x})=D_{\mathrm{KL}}\left(q_{\phi}(\mathbf{z}|\mathbf{x})\|p_{\theta}(\mathbf{z})\right)-\mathbb{E}_{q_{\phi}(\mathbf{z}|\mathbf{x})}\left(\log p_{\theta}(\mathbf{x}|\mathbf{z})\right)$ (7) where $D_{KL}$ stands for the KL-divergence. In the VAE, the principle is to minimize the loss between input and the restored image along with the loss generated by the latent space to represent the features in the input images. ## 3 Methodology Figure 2: Network architecture: pre-training with VAE(first row) In the second row, parameter $\theta$ is underlined and bold to indicate that these parameters are freezed when the fine-tuning the network Consider a surrogate model $y=f(x,\theta)$ which is trained using limited simulation data $\mathcal{D}=\\{x^{i},y^{i}\\}_{i=1}^{N},\\{x^{j}\\}_{j=N+1}^{D}$. Where the input data, $x^{i}\in\mathbb{R}^{d_{x}\times H\times W}$ is the input from CIFAR-10 dataset. Here H and W are the height and width respectively and $d_{x}$ is the number of dimensions for the input x at one location. $x^{j}$ is the additional data utilized for pretraining the model. $y^{i}\in\mathbb{R}^{1}$ is the classified result. $\theta$ is the model parameter and N is the total number of training data utilized during fine- tuning and D is the total number of data utilized for pre-training. In semi- supervised model, we pre-train the model with the input data $\mathbb{R}^{d_{x}\times H\times W}$ and then perform image classification problem $\mathbb{R}^{d_{x}\times H\times W}\rightarrow\mathbb{R}^{1}$. For both the pre-training and fine tuning, we used stochastic gradient descent with Adam optimizer to update the network weights and biases. The simulation was performed using Pytorch machine learning package in Python. ### 3.1 VAE pre-training We implemented dense-net [10],[11] version of VAE for the pre-training part. Dense-net contains the encoder and decoder blocks along with the dense-layer that has the simple and complex features. ### 3.2 VAE fine-tuning We implemented a simple fully connected layer to classify the input images on the CIFAR-10 [12] data. This is due to the expectation that the latent space is smaller (either 4 $\times$ 4 or 8 $\times$ 8) (image size is 32 $\times$ 32). If the number of channels are large at the latent space, we will add more fully connected layers for classification of images. ## 4 Data We perform the simulations on the CIFAR-10 dataset with ten image classes with three input channels $(C=3)$ of size 32 $\times$ 32 (W $\times$ H). CIFAR-10 dataset has 50000 training images and 10000 test images. ## 5 Results In this section we enumerate the results obtained using CIFAR-10 data. We consider the following latent dimensions: 6400, 10,000 and 14,400. In order to evaluate the performance of the model for above three latent space, we consider the distribution estimated for the values at various pixel location. ### 5.1 Pre-training For the results presented in this section, we have a dataset with 50,000 $\\{x^{k}\\}_{k=1}^{50000}$ examples for pre-training and the test set consist of 10000, $\\{x^{k}\\}_{k=1}^{10000}$ examples. Adam optimizer was used for training 100 epochs, with learning rate of 1e-4 and a plateau scheduler on the test RMSE. Batch size is always smaller than the number of training data. In this work, a batch size of 16 for pre-training was used. Weight decay was set to 1e-3 for pre-training. We consider Equation: 7 loss function to evaluate the trained model on test data and also to monitor the convergence. Figure 3: Error v/s epoch for 6400 latent space (top), Error v/s epoch for 10000 latent space (middle) and Error v/s epoch for 14400 latent space (bottom) From figure 1, we observe that the solution is converged after 50 epochs and most importantly the loss for the three latent spaces is similar. Figure 4: Distribution estimate for the values at various location of the square domain for 6400, 10000 and 14400 latent space From figure 4, we observe that even when the latent size is small (Batch $\times$ 100 (channels) $\times$ 8 $\times$ 8) and (Batch $\times$ 100 (channels) $\times$ 10 $\times$ 10) the reconstructed density estimate is close to actual input data. The PDF with the latent size 10000 is closer to the actual input and also 6400 & 14400 latent space. Since, all the latent spaces yield the similar outputs,we fine-tune and compare the classification accuracy in the next section. ### 5.2 Fine-tuning In this section we freeze the parameters (weights and bias) used in the pre- training stage and fine tune the parameters (weights and bias) in the classification network. For this problem, we consider small data from the given CIFAR-10 dataset and use fully connected layers to perform classification. Cross entropy loss function is commonly used for all classification problems, we implemented cross entropy to measures the performance of a classification mode. Figure 5: Fine tuning results with three different latent space models From figure 5, we observe the latent space 10000 yields better accuracy than other two latent spaces (6400 and 14400). The smaller the latent space the lower the test accuracy, this is due to insufficient features to classify the data. Also, for the large latent space, the test accuracy is low, this is due to model complexity [1]. ## 6 Conclusions and Future work The present document outlines the development of surrogate model for semi- supervised problem. In this work, we have implemented VAE as a pre-training model and a feed forward deep learning model for the classification. The results obtained for differently sized latent spaces are presented. It was observed that there is a slight improvement in the test accuracy when the latent space is 10000 in comparison with latent space of 6400 and 14400. For future work, a Bayesian approach can be explored. Due to a limited amount of data, it is necessary to model appropriate surrogate, since it is important to quantify the epistemic uncertainty induced by limited data [13], [14] and hence, a Bayesian probabilistic approach is a natural way of addressing this challenge. ## References * [1] Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep learning. MIT press, 2016. * [2] Nitesh V Chawla et al. Learning from labeled and unlabeled data: An empirical study across techniques and domains. Journal of Artificial Intelligence Research, 23:331–366, 2005. * [3] Autoencoder. Autoencoder — Wikipedia, the free encyclopedia, 2019. * [4] Geoffrey E Hinton and Ruslan R Salakhutdinov. Reducing the dimensionality of data with neural networks. science, 313(5786):504–507, 2006. * [5] Kevin P Murphy. Machine learning: a probabilistic perspective. MIT press, 2012. * [6] Paul Smolensky. Information processing in dynamical systems: Foundations of harmony theory. Technical report, Colorado Univ at Boulder Dept of Computer Science, 1986\. * [7] Tu Dinh Nguyen, Truyen Tran, Dinh Phung, and Svetha Venkatesh. Latent patient profile modelling and applications with mixed-variate restricted boltzmann machine. In Pacific-Asia conference on knowledge discovery and data mining, pages 123–135. Springer, 2013. * [8] G Hinton and R Salakhutdinov. An efficient learning procedure for deep boltzmann machines. Neural Computation, 24(8):1967–2006, 2012. * [9] Yoshua Bengio, Pascal Lamblin, Dan Popovici, and Hugo Larochelle. Greedy layer-wise training of deep networks. In Advances in neural information processing systems, pages 153–160, 2007. * [10] Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4700–4708, 2017. * [11] Yide Zhang, Yinhao Zhu, Evan Nichols, Qingfei Wang, Siyuan Zhang, Cody Smith, and Scott Howard. A poisson-gaussian denoising dataset with real fluorescence microscopy images. arXiv preprint arXiv:1812.10366, 2018. * [12] Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, Citeseer, 2009. * [13] Rohit K Tripathy and Ilias Bilionis. Deep uq: Learning deep neural network surrogate models for high dimensional uncertainty quantification. Journal of Computational Physics, 375:565–588, 2018. * [14] Yinhao Zhu and Nicholas Zabaras. Bayesian deep convolutional encoder–decoder networks for surrogate modeling and uncertainty quantification. Journal of Computational Physics, 366:415–447, 2018.

2024-09-04T02:54:55.109051

2002.12172

arxiv-papers

11institutetext: Nordita, KTH Royal Institute of Technology and Stockholm University, 10691 Stockholm, Sweden 11email<EMAIL_ADDRESS>22institutetext: Section for Meteorology and Oceanography, Department of Geosciences, University of Oslo, P.O. Box 1022 Blindern, 0315, Oslo, Norway # Coagulation of inertial particles in supersonic turbulence Xiang-Yu Li 1122 Lars Mattsson 11 Coagulation driven by supersonic turbulence is primarily an astrophysical problem because coagulation processes on Earth are normally associated with incompressible fluid flows at low Mach numbers, while dust aggregation in the interstellar medium (ISM) for instance is an example of the opposite regime. We study coagulation of inertial particles in compressible turbulence using high-resolution direct and shock-capturing numerical simulations with a wide range of Mach numbers from nearly incompressible to moderately supersonic. The particle dynamics is simulated by representative particles and the effects on the size distribution and coagulation rate due to increasing Mach number is explored. We show that the time evolution of particle size distribution mainly depends on the compressibility (Mach number). We find that the average coagulation kernel $\langle C_{ij}\rangle$ scales linearly with the average Mach number $\mathcal{M}_{\rm rms}$ multiplied by the combined size of the colliding particles, that is, $\langle C_{ij}\rangle\sim\langle(a_{i}+a_{j})^{3}\rangle\,\mathcal{M}_{\rm rms}\tau_{\eta}^{-1}$, which is qualitatively consistent with expectations from analytical estimates. A quantitative correction $\langle C_{ij}\rangle\sim\langle(a_{i}+a_{j})^{3}\rangle(v_{\rm p,rms}/c_{\rm s})\tau_{\eta}^{-1}$ is proposed and can serve as a benchmark for future studies. We argue that the coagulation rate $\langle R_{c}\rangle$ is also enhanced by compressibility-induced compaction of particles. ###### Key Words.: Coagulation, inertial particles, turbulence, compressibility, shock waves ## 1 Introduction The kinetics of inertial particles that are (finite-size particles that are massive enough to have significant inertia) in turbulence has drawn much attentions for decades. It has been driven by wide applications in astrophysics, atmospheric sciences, and engineering. The preferential concentration and fractal clustering of inertial particles in (nearly) incompressible turbulence has been simulated extensively (see Maxey, 1987; Squires & Eaton, 1991; Eaton & Fessler, 1994; Bec, 2003, 2005; Bec et al., 2007b, a; Bhatnagar et al., 2018; Yavuz et al., 2018). In combination with the theory of coagulation of particles, this has an important application in planet-formation theory (see e.g. Pan et al., 2011; Birnstiel et al., 2016; Johansen et al., 2012; Johansen & Lambrechts, 2017). However, proto-planetary discs are dominated by low Mach-number turbulence, which is not the case in many other astrophysical environments. One example is the cold phase of the interstellar medium (ISM), where turbulence is highly compressible with Mach numbers of order 10 and thus is dominated by shock waves. Only a few studies of inertial particles in high Mach-number turbulence can be found in the literature (e.g. Hopkins & Lee, 2016; Mattsson et al., 2019, 2019), and direct numerical simulations of turbulence-driven coagulation of inertial particles have not been performed so far. Exploring the effects of compressibility (high Mach numbers) on coagulation is therefore an important branch of research that is now becoming possible through the rapid development of computing power. From an astrophysical perspective, cosmic dust grains, which are made of mineral or carbonaceous material and are a perfect example of inertial particles, are ubiquitous throughout the universe. Rapid interstellar dust growth by accretion of molecules is thought to be necessary to compensate for various dust destruction processes (see e.g. Mattsson, 2011; Valiante et al., 2011; Rowlands et al., 2014). Grains may also grow by aggregation or coagulation (which does not increase the dust mass), however, when the growth by accretion has produced large enough grains for coagulation to become efficient, that is, once the ‘coagulation bottleneck’ has been passed (Mattsson, 2016). How efficient the coagulation is in molecular clouds (MCs) is not fully understood, although models and simulations have suggested that turbulence is the key to high growth efficiency (Elmegreen & Scalo, 2004; Hirashita & Yan, 2009; Hirashita, 2010; Pan et al., 2014a, b; Pan & Padoan, 2015; Hopkins & Lee, 2016; Mattsson et al., 2019, 2019) and observations indicate the presence of very large grains (which can be tens of $\mu$m across) in the cores of MCs (Hirashita et al., 2014). We aim to study the role that the Mach number plays for the coagulation of inertial particles in compressible turbulence. Specifically, we strive to study how coagulation of particles depends on Mach number in regions where the particles cluster and form filaments. The purpose is not to target any specific astrophysical context, but to explore the Mach-number dependence to the extent that this is computationally feasible. There are three main challenges. First, the dynamics of inertial particles in compressible turbulence is poorly understood. Second, the coagulation process is a non- equilibrium process, as the particle size distribution evolves with time. Third, the coagulation timescale and the characteristic timescale of the turbulence are very different in dilute systems, such as those typically studied in astrophysics. In classical kinetic theory, the collision kernel $C_{ij}$ is a function of the relative velocity $\Delta\bm{v}_{ij}$ of two particles $i$ and $j$, which is difficult to calculate analytically, except in some special cases. For the same reason, it is difficult to calculate the coagulation rate using analytical models. More exactly, the classical Smoluchowski (1916) problem has only three known exact solutions (Aldous, 1999), and numerical solution of the coagulation equation is only feasible if treated as a local or ‘zero-dimensional’ problem. The main objective of the present work is to offer a way to quantify and possibly parametrise the effects of turbulent gas dynamics and hydrodynamic drag on the coagulation rate in such a way that it can be included for instance in traditional models of galactic chemical evolution (including dust), which are based on average physical quantities (Mattsson, 2016). A major problem when simulating the dust growth in the ISM is that the system is large scale and dilute. The coagulation rate is extremely low in such a system, which leads to very different timescales for the turbulent gas dynamics and coagulation. ## 2 Turbulence and kinetic drag In this section, equations governing compressible flow and particle dynamics of inertial particles (e.g. dust grains) are presented. The Pencil Code with HDF5 IO (Brandenburg et al., 2020; Bourdin, 2020) is used to solve these equations. Since the carrier fluid in our study is isothermal, its turbulence described by Eq. (1) is scale free, that is, the box size $L$, the mean mass density $\langle\rho\rangle$, and the sound speed $c_{\rm s}$ are the unit length, unit density, and unit velocity, respectively. These quantities can thus be scaled freely. However, the inclusion of coagulation process means our simulation is no longer scale free. This requires a careful treatment of initial conditions and scaling of units, which is discussed in more detail in Section 3.4. ### 2.1 Momentum equation of the carrier flow The motion of the gas flow is governed by the Navier-Stokes equation, ${\partial{\bm{u}}\over\partial t}+\bm{u}\cdot{\bm{\nabla}}\bm{u}={\bm{f}}-\rho^{-1}{\bm{\nabla}}p+\rho^{-1}\bm{F}_{\rm visc},$ (1) where ${\bm{f}}$ is a forcing function (Brandenburg, 2001), $p$ is the gas pressure, and $\rho$ is the fluid or gas density obeying the continuity equation, ${\partial\rho\over\partial t}+{\bm{\nabla}}\cdot(\rho\bm{u})=0.$ (2) For the case of direct numerical simulation with a constant kinetic viscosity of the gas flow, the viscosity term $\bm{F}_{\rm visc}$ equals the physical viscosity term $\bm{F}_{\rm visc}^{\nu}$ given by $\bm{F}_{\rm visc}^{\nu}=\rho\nu\left({\bm{\nabla}}^{2}\bm{u}+\frac{1}{3}\nabla\nabla\cdot\bm{u}+2\bm{{\sf S}}\cdot\nabla\ln\rho\right),$ (3) where $\bm{{\sf S}}={1\over 2}\left[{\bm{\nabla}}\bm{u}+\left({\bm{\nabla}}\bm{u}\right)^{T}\right]-{1\over 3}\left({\bm{\nabla}}\cdot\bm{u}\right){\sf I}$ is the rate-of-strain tensor (${\sf I}$ is the unit tensor) resulting from the shear effect and the deformation due to compression. For the case with shock-capturing viscosity, the viscosity term becomes $\bm{F}_{\rm visc}=\bm{F}_{\rm visc}^{\nu}+\rho\zeta_{\rm shock}\nabla\nabla\cdot\bm{u}+\left(\nabla\cdot\bm{u}\right)\nabla\left(\rho\zeta_{\rm shock}\right).$ (4) The shock viscosity $\zeta_{\rm shock}$ is given by $\zeta_{\rm shock}=c_{\rm shock}\langle\rm{max}[(-{\bm{\nabla}}\cdot\bm{u})_{+}]\rangle(\min(\delta x,\delta y,\delta z))^{2},$ (5) where $c_{\rm shock}$ is a constant defining the strength of the shock viscosity (Haugen et al., 2004). The length of the lattice is given by $\delta x$, $\delta y$, and $\delta z$, respectively. $\bm{F}_{\rm visc}^{\rm shock}$ is used in simulations with high Mach number, where we strive to use the highest spatial resolution to capture the shocks. Nevertheless, it is necessary to introduce this term to handle the strongest shocks. Two dimensionless parameters characterise compressible turbulence: the Reynolds number Re and the root-mean-square (rms) Mach number $\mathcal{M}_{\rm rms}$. Re is defined as ${\rm Re}\equiv u_{\rm rms}L_{\rm inj}/\nu,$ (6) where $u_{\rm rms}$ is the rms turbulent velocity and $L_{\rm inj}$ is the energy injection length scale. The compressibility of the flow is characterised by $\mathcal{M}_{\rm rms}$, which is defined as ${\cal{M}_{\rm rms}}=u_{\rm rms}/c_{\rm s},$ (7) where $c_{\rm s}$ is the sound speed. The sound speed is kept constant because the compressible flow to be investigated here is assumed to be isothermal such that $c_{\rm s}^{2}=\gamma p/\rho$, where $\gamma=c_{\rm P}/c_{\rm v}=1$ with the specific heats $c_{\rm P}$ and $c_{\rm V}$ at constant pressure and constant volume, respectively. Another quantity is the mean energy dissipation rate $\langle\bar{\epsilon}\rangle$, which measures how vigorous the small eddies are in turbulence. It can be calculated from the trace of $\bm{{\sf S}}_{ij}$ as $\langle\bar{\epsilon}\rangle=2\nu\,\textstyle{\overline{{\rm Tr\,}{\sf S_{ij}}{\sf S_{ji}}}}$. $\langle\bar{\epsilon}\rangle$ determines the smallest scales of the turbulence, for example, the Kolmogorov length scale is defined as $\eta=(\nu^{3}/{\langle\bar{\epsilon}\rangle})^{1/4}$ and the timescale is defined as $\tau_{\eta}=(\nu/\langle\bar{\epsilon}\rangle)^{1/2}$. Becausee the Saffman- Turner collision rate is proportional to $\tau_{\eta}^{-1}$ (Saffman & Turner, 1956), $\langle\bar{\epsilon}\rangle$ indirectly determines the coagulation rate of particles in an incompressible flow. Coagulation occurs at the small scales of turbulence, and the strength of the small eddies determines the particle velocity (Li et al., 2018). Therefore it is worth investigating whether and how $\langle\bar{\epsilon}\rangle$ affects the coagulation rate in compressible turbulence as well. We show that it does not affect the coagulation rate in the compressible case. The stochastic solenoidal forcing $\bm{f}$ is given by $\bm{f}(\bm{x},t)=\mbox{Re}\\{N\bm{f}_{\bm{k}(t)}\exp[i\bm{k}(t)\cdot\bm{x}+i\phi(t)]\\},$ (8) where $\bm{k}(t)$ is the wave space, $\bm{x}$ is position, and $\phi(t)$ ($|\phi|<\pi$) is a random phase. The normalization factor is given by $N=f_{0}c_{\rm s}(kc_{\rm s}/\Delta t)^{1/2}$, where $f_{0}$ is a non- dimensional factor, $k=|\bm{k}|$, and $\Delta t$ is the integration time step (Brandenburg & Dobler, 2002). We chose a completely non-helical forcing, that is, $\bm{f}_{\bm{k}}=\left(\bm{k}\times\bm{e}\right)/\sqrt{\bm{k}^{2}-(\bm{k}\cdot\bm{e})^{2}},$ (9) where $\bm{e}$ is the unit vector. To achieve different $\cal{M}_{\rm rms}$ with fixed ${\rm Re}$ and $\langle\bar{\epsilon}\rangle$ in the simulations, we need to change $u_{\rm rms}$, $\nu$, and the simulation box $L$ simultaneously according to Eq. (6) and Eq. (7), and we also considered $\langle\bar{\epsilon}\rangle\sim\frac{u_{\rm rms}^{3}}{L_{\rm inj}}.$ (10) Since $u_{\rm rms}$ is essentially determined by the amplitude of forcing $f_{0}$, we changed $f_{0}$ in the simulation as well. ### 2.2 Particle dynamics The trajectories of inertial particles is determined by $\frac{d\bm{x}_{i}}{dt}=\bm{v}_{i}$ (11) and $\frac{d\bm{v}_{i}}{dt}=\frac{1}{\tau_{i}}(\bm{u}-\bm{v}_{i})\,,$ (12) where $\tau_{i}=\sqrt{\pi\over 8}{\rho_{\rm mat}\over\rho}{a\over c_{\rm s}}\left(1+{9\pi\over 128}{|\bm{u}-\bm{v}_{i}|^{2}\over c_{\rm s}^{2}}\right)^{-1/2},$ (13) is the stopping time, that is, the kinetic-drag timescale. In the equation above, $a$ is the radius of a particle, $\rho_{\rm mat}$ is the material density of particles, and $\rho$ is the mass density of the gas. We assumed that particles are well described in the Epstein limit because the mean-free- path $\lambda$ is large and particles are small in most astrophysical contexts (large Knudsen number, Kn $=\lambda/a\gg 1$ Armitage, 2010). The stopping time at low relative Mach number ($\mathcal{W}=|\bm{u}-\bm{v}_{i}|/c_{\rm s}\ll 1$) is $\tau_{i}(\mathcal{W}\ll 1)=\sqrt{\pi\over 8}{\rho_{\rm mat}\over\rho}{a\over c_{\rm s}}.$ (14) The term in parentheses of Eq. (13) is a correction for high $\mathcal{W}$. Eq. (13) is essentially a quadratic interpolation between the two expressions for the limits $\mathcal{W}\ll 1$ and $\mathcal{W}\gg 1$ derived from the exact expression for $\tau_{i}$ (see Schaaf, 1963; Kwok, 1975; Draine & Salpeter, 1979). To characterize the inertia of particles, we define a ‘grain-size parameter’ as $\alpha={\rho_{\rm mat}\over\langle\rho\rangle}{a\over L},$ (15) which is the parametrisation used by Hopkins & Lee (2016). Because the total mass of a simulation box of size $L$, as well as the mass of a grain of a given radius $a$, is constant, the quantity $\alpha$ is solely determined by $a$ regardless of the characteristics of the simulated flow. In general, the inertia of particles is characterised by the Stokes number ${\rm St}=\tau_{i}/\tau_{\eta}$. The disadvantage of ${\rm St}$ as the size parameter for inertial particles in a highly compressible carrier fluid is that a fluid flow with ${\rm Re}\gg 1$ cannot be regarded as a Stokes flow. If $\mathcal{M}_{\rm rms}\gg 1$ as well, ${\rm St}$ is not even well defined as an average quantity in a finite simulation domain. The parameter $\alpha$ is therefore a better dimensionless measure of grain size than the average Stokes number $\langle{\rm St}\rangle$ for a supersonic compressible flow. Moreover, $\langle{\rm St}\rangle$ is not only a function of the size, but also a function of the mean energy dissipation rate $\langle\bar{\epsilon}\rangle$, which complicates the picture even further. ### 2.3 Averages In the following we frequently refer to the mean or average quantities of three different types. For each of them we use a different notation. First, we use the bracket notation $\langle Q\rangle$ for volume averages, taken over the whole simulation box unless stated otherwise. Second, we use the over-bar notion $\bar{Q}$ for straight time-averaged quantities. Third, we use the tilde notation $\tilde{Q}$ for ensemble averages, that is, averages defined by the first moment of the distribution function of the particles. The rms value of a fluctuating physical quantity has been mentioned in section 1. In terms of the above notion, rms values always refer to $Q_{\rm rms}\equiv\sqrt{\langle Q^{2}\rangle}$. ## 3 Coagulation ### 3.1 Numerical treatment of coagulation The most physically consistent way to model coagulation is to track each individual Lagrangian particle and to measure the collisions among them when they overlap in space, which is computationally challenging because the coagulation timescale of inertial particles is often much shorter than the Kolmogorov timescale. We also used $10^{7}$ representative particles, which means solving a large $N$-body problem. Because of the aforementioned computational load, a super-particle approach is often used to study the coagulation of dust grains (Zsom & Dullemond, 2008; Johansen et al., 2012; Li et al., 2017). Instead of tracking each individual particles, super-particles consisting of several identical particles are followed. Within each super- particle, all the particles have the same velocity $\bm{v}_{i}$ and size $a$. The super-particle approach is a Monte Carlo approach, which treats coagulation of dust grains in a stochastic manner (Bird, 1978, 1981; Jorgensen et al., 1983). Each super-particle is assigned a virtual volume that is the same as the volume of the lattice, therefore a number density $n_{j}$. When two super-particles $i$ and $j$ reside in the same grid cell, the probability of coagulation is $p_{\rm c}=\tau_{\rm c}^{-1}\Delta t$, where $\tau_{\rm c}$ is the coagulation time and $\Delta t$ is the integration time step. A coagulation event occurs when $p_{\rm c}>\eta_{c}$, where $\eta_{c}$ is a random number. The coagulation timescale $\tau_{\rm c}$ is defined as $\tau_{\rm c}^{-1}=\sigma_{\rm c}n_{j}\,|{\bm{w}}_{ij}|\,E_{\rm c},$ (16) where $\sigma_{\rm c}=\pi(a_{i}+a_{j})^{2}$ and ${\bm{w}}_{ij}$ are the geometric coagulation cross section and the absolute velocity difference between two particles with radii $a_{i}$ and $a_{j}$, respectively, and $E_{\rm c}$ is the coagulation efficiency (Klett & Davis, 1973). For simplicity, we set $E_{\rm c}$ to unity. This means that all particles coalesce upon collision, that is, bouncing and fragmentation are neglected. This treatment may overestimate the collision rate but does not affect the dynamics of particles. Therefore the $\mathcal{M}$ dependence should not be affected. Compared with the super-particle algorithm that is widely used in planet formation (Zsom & Dullemond, 2008; Johansen et al., 2012), our algorithm provides better collision statistics (Li et al., 2017). We refer to Li et al. (2017) for a detailed comparison of the super-particle algorithm used in Johansen et al. (2012) and Li et al. (2017, 2018, 2020). ### 3.2 Timescale differences Before we describe the basic theory of coagulation of particles in a turbulent carrier fluid, it is important that we consider the different timescales that are involved in this complex and composite problem. In Eq. (16), we introduced $\tau_{\rm c}$. The other important timescale in a model of coagulation of particles in turbulence is the flow timescale of the carrier fluid, in this case, the large-eddy turnover time $\tau_{L}=L/u_{\rm rms}$, where $u_{\rm rms}$ is the rms flow velocity. Clearly, $\tau_{L}$ depends on the scaling of the simulation. When we compare the two timescales, we find that ${\tau_{L}\over\langle{\tau_{\rm c}\rangle}}\sim N_{\rm p}{a^{2}\over L^{2}}{\langle{|\bm{w}_{ij}|}\rangle\over u_{\rm rms}},$ (17) where $N_{\rm p}$ is the total number of particles in the simulated volume. We note that $\langle|{\bm{w}}_{ij}|\rangle/u_{\rm rms}\ll 1$ as long as the particles do not decouple completely from the carrier flow. In order to avoid slowing down the simulation too much, we aim for $\tau_{L}/\langle\tau_{\rm c}\rangle\sim 1$. From this we may conclude that $N_{\rm p}\gg(L/a)^{2}$, which implies that if we have an upper bound of $N_{\rm p}$ for computational reasons, we cannot simulate tiny particles in a large volume. The ratio $\tau_{L}/\langle{\tau_{\rm c}\rangle}$ shows how difficult it can be to simulate the coagulation in astrophysical contexts, in particular when the details of coagulation of inertial particles are simulated in a carrier fluid representing well-resolved compressible turbulence. In addition to the two timescales discussed above, we must also consider the stopping time $\tau_{i}$ of the particles because we study inertial particles. For $\alpha\lesssim 0.1$, $\tau_{i}$ is typically smaller than $\tau_{L}$. Hence, the competing timescales would rather be $\tau_{\rm c}$ and $\tau_{i}$, which suggests that the ratio $\tau_{i}/\tau_{\rm c}$ should be of order unity to avoid slowing down the simulation compared to the case of non-interacting particles. By the same assumptions as above ($k_{\rm f}\approx 3$ and $E_{\rm c}\sim 1$), we can show that ${\tau_{i}\over\langle{\tau_{\rm c}\rangle}}\sim{\langle|\bm{w}_{ij}|\rangle\over c_{\rm s}}{\rho_{\rm p}\over\rho}\,a_{i}^{3},$ (18) where $\rho_{\rm p}{\equiv\rho_{\rm mat}\,n_{i}}$ is the mass density of particles (not to be confused with the bulk material density $\rho_{\rm mat}$). In many astrophysical contexts (in particular, cold environments) $\langle|\bm{w}_{ij}|\rangle/c_{\rm s}\sim 1$, which then suggests we must have $\rho_{\rm p}/\rho\sim 1$. This is always inconsistent with cosmic dust abundances, however, whether in stars, interstellar clouds, or even proto- planetary discs. In the cold ISM, $\rho_{\rm p}/\rho\sim 10^{-2}$ and $\langle|\bm{w}_{ij}|\rangle/c_{\rm s}\sim 1$, which implies that $\tau_{i}/\tau_{\rm c}\ll 1$ and thus the time step of a simulation of coagulation in such an environment is limited by $\tau_{i}$. In practice, this means that it will be difficult (or even impossible) to target coagulation in cold molecular clouds in the ISM without highly specialised numerical methods. The goal of the present study is primarily to investigate how coagulation of inertial particles depends on $\mathcal{M}_{\rm rms}$ and not to simulate coagulation in a realistic and dilute astrophysical environment. We note, however, that any result on coagulation of particles in compressible turbulence is primarily of importance for astrophysics, for instance, the processing of dust grains in the ISM and various types of circumstellar environments. Therefore we tried to make the simulation system as dilute, while still ensuring statistical convergence and computational feasibility. ### 3.3 Theory of coagulation of inertial particles in turbulence Coagulation, as described by the Smoluchowski (1916) equation, is determined by the coagulation rate $R_{ij}$ between two grains species (sizes) $i$ and $j$ and the associated rates of depletion. In general, we have $R_{ij}={1\over 2}n_{i}\,n_{j}\,C_{ij}$, where $n_{i}$, $n_{j}$ are the number densities of the grains $i$ and $j$, and $C_{ij}$ is the collision kernel. Turbulence has been proposed to have a profound effect on $C_{ij}$, and we focus this theory section on what happens to $C_{ij}$. Assuming the distribution of particle pairs can be separated into distinct spatial and velocity distributions, we have (Sundaram & Collins, 1997) $\langle C_{ij}\rangle=\pi(a_{i}+a_{j})^{2}\,g(r,a_{i},a_{j})\int\bm{w}_{ij}\,P(\bm{w}_{ij},a_{i},a_{j})\,d\bm{w}_{ij},$ (19) where $g$ is the radial distribution function (RDF) and $P$ is the probability density distribution of relative velocities $\bm{w}_{ij}$. Below, we review the basic theory of coagulation in the tracer particle limit and large-inertial limit. Since small-clustering is negligible for both small- and large-inertial particles, we implicitly assumed that $g(r)=1$. Moreover, in case of Maxwellian velocities, that is, $P(\bm{w}_{ij},a_{i},a_{j})$ follows a Maxwellian distribution, the integral part of Eq. (19) becomes $\langle\bm{w}_{ij}\rangle=\sqrt{8\pi/3\,\langle\bm{w}_{ij}\cdot\bm{w}_{ij}\rangle}$. Thus, the collision kernel in Eq. (19) reduces to $\langle C_{ij}\rangle=\sqrt{8\pi/3}\,(a_{i}+a_{j})^{2}\sqrt{\langle\bm{w}_{ij}\cdot\bm{w}_{ij}\rangle},$ (20) which is the form assumed in the two following subsections. #### 3.3.1 Tracer-particle limit In the low-inertial limit, also known as the Saffman-Turner limit or the tracer-particle limit (Saffman & Turner, 1956), $\langle\bm{w}_{ij}^{2}\rangle$ is a simple function of $a_{i}$ and $a_{j}$. In case of a mono-dispersed grain population ($a=a_{i}=a_{j}$) suspended in a turbulent low Mach-number medium, we may use the Saffman & Turner (1956) assumption, $\langle\bm{w}_{ij}^{2}\rangle={1\over 5}(a/\tau_{\eta})^{2}$, where $\tau_{\eta}=\sqrt{\nu/\langle\bar{\epsilon}\rangle}$ is the Kolmogorov timescale. This is a reasonable approximation if $\mathcal{M}_{\rm rms}$ is not too large. The Saffman & Turner (1956) theory relies on $\bm{w}_{ij}$ having a Gaussian distribution (Maxwellian velocities) and the final expression for $\langle C_{ij}\rangle$ becomes $\langle C_{i}\rangle=\sqrt{8\pi\over 15}\frac{\left(2a_{i}\right)^{3}}{\tau_{\eta}}.$ (21) For the multi-dispersed case, we replace $2a_{i}$ by $(a_{i}+a_{j})$ in Eq. (21). At first sight, compressibility does not seem to play any role at all, given the collision kernel $\langle C_{ij}\rangle$ above. However, it can affect the number density of particles $n_{i}$, and therefore, $R_{ij}$. In the tracer particle limit, the spatial distribution of particles is statistically the same as for gas. Simulations have shown that the gas density of isothermal hydrodynamic turbulence exhibits a lognormal distribution (e.g. Federrath et al., 2009; Hopkins & Lee, 2016; Mattsson et al., 2019) with a standard deviation of $\sigma_{\rho}^{2}$ that is empirically related to ${\cal M}_{\rm rms}$ (see e.g. Passot & Vázquez-Semadeni, 1998; Price et al., 2011; Federrath et al., 2010). Consequentially, $n_{i}$ depends on ${\cal M}_{\rm rms}$, and so does $R_{ij}$. #### 3.3.2 Large-inertia limit In the opposite limit, the large-inertia limit, particles should behave according to kinetic theory. As shown by Abrahamson (1975), we have in this limit that particles are randomly positioned and follow a Maxwellian velocity distribution. In this case, we may conclude that $\langle\bm{w}_{ij}^{2}\rangle=\langle\bm{v}_{i}^{2}\rangle+\langle\bm{v}_{j}^{2}\rangle$ because the particles are then statistically independent and thus they have a covariance that is identically zero. As in the tracer-particle limit, the expression for a mono-disperesed population becomes $\langle C_{i}\rangle=a^{2}\left({16\pi\over 3}\langle\bm{v}_{i}^{2}\rangle\right)^{1/2}.$ (22) Previous theoretical work on inertial particles in turbulent flows (e.g. Abrahamson, 1975; Hopkins & Lee, 2016; Mattsson et al., 2019; Pan & Padoan, 2013; Wang et al., 2000) have shown that the rms velocity $v_{\rm rms}$ of the particles is a function of their size. More precisely, we can parametrise in terms of grain size such that $v_{\rm rms}^{2}(a_{i})/u_{\rm rms}^{2}\approx a_{0}/a_{i}$ for $a/a_{0}\gg 1$, where $a_{0}$ is a scaling radius that can be estimated from the stopping time $\tau_{i}$ and the integral timescale $\tau_{L}$ (Hedvall & Mattsson, 2019; Mattsson & Hedvall, 2021). Assuming Maxwellian velocity distributions again, we have that $\langle\bm{w}_{ij}\cdot\bm{w}_{ij}\rangle=v_{\rm rms}^{2}(a_{i})+v_{\rm rms}^{2}(a_{j})$. Thus, the mean collision kernel for the multi-dispersed case can be expressed as $\langle C_{ij}\rangle\approx\sqrt{8\pi\over 3}\,(a_{i}+a_{j})^{2}\,u_{\rm rms}\,\left({a_{0}\over a_{i}}+{a_{0}\over a_{j}}\right)^{1/2}.$ (23) Here, we may note that $\langle C_{ij}\rangle$ is proportional to $u_{\rm rms}$, implying that the collision rate should scale with $\mathcal{M}_{\rm rms}$. For particles of equal size, that is, $a_{i}=a_{j}$, $\langle C_{ij}\rangle$ reduces to the form given above in Eq. (22), which also means that $\langle C_{i}\rangle\sim a_{i}^{5/2}$. In the absence of external body forces other than kinetic drag, $\langle{\bm{v}}_{i}^{2}\rangle\to 0$ for very large inertia particles. Thus, $\langle C_{ij}\rangle\to 0$, eventually. To summarise, we note that for the tracer-particle limit (small-inertia particles, St $\ll 1$), density variations in high-${\cal M}_{\rm rms}$ turbulence, and the locally elevated number densities ($n_{i}$) of dust particles that follows consequently, can have a significant impact on the average collision rate $\langle R_{ij}\rangle$, although not on $\langle C_{ij}\rangle$. For particles with large inertia, beyond the effect of compressibility on $\langle R_{ij}\rangle$, caustics in the particle phase may also contribute to $\langle C_{ij}\rangle$ (see Appendix A for details of the discussion). ### 3.4 Initial conditions Super-particles were initially distributed randomly in the simulation box and mono-dispersed in size ($\alpha_{0}=10^{-4}$). As discussed in section 3.1, each super-particle was assigned a virtual volume $(\delta x)^{3}$, where $\delta x$ is the lateral size of the lattice. With an initial number density of dust grains $n_{0}$, the total number of dust grains in the computational domain is given by $n_{0}L^{3}=N_{\rm s}(n_{j}\delta x^{3}),$ (24) where $N_{\rm s}$ is initial total number of super-particles. Since $(L/\delta x)^{3}=N_{\rm grid}$ with $N_{\rm grid}$ the number of grid cells, Eq. (24) can be rewritten as $n_{0}N_{\rm grid}=N_{\rm s}n_{j},$ (25) where $n_{j}$ is the number density within each super-particle at $t=0$. The number of physical particles in each super-particle $N_{\rm p/s}$ is determined by $N_{\rm p/s}=N_{\rm p}/N_{\rm s}=\frac{L^{3}}{N_{\rm grid}}n_{j},$ (26) which means that $N_{\rm p/s}$ is uniquely determined by $L^{3}$ when $N_{\rm s}$, $N_{\rm grid}$, and $n_{j}$ are fixed. To avoid running out of memory while executing the simulation, we must limit the number of super-particles to $N_{\rm s}\sim 10^{7}$, which leads to a required resolution $N_{\rm grid}=512^{3}$. The value of $n_{0}$ must be chosen for computational feasibility, and we also kept the number of particles within each super-particle to a minimum to avoid averaging out too much of the turbulence effects (as explained in Section 3.1). We can take the physical parameters of dust grains in the ISM as an example of how difficult it is to simulate a dilute system, even on a current modern supercomputer. According to Eq. (16) and Eq. (24), the collision frequency is proportional to $n_{0}$. With $n_{0}=3.33\times 10^{-7}\,\rm{cm}^{-3}$, $a_{0}=10^{-6}\,\rm{cm}$, $|\bm{v}_{i}-\bm{v}_{j}|\approx 10^{5}\,\rm{cm\,s^{-1}}$, and $E_{c}\approx 1$, the initial collision frequency is $\tau_{c}^{-1}\approx 10^{-9}\,\rm{s}^{-1}$. The simulation time step must match the corresponding physical coagulation timescale for the particles, which is far beyond the computational power at our disposal in case of a small number of particles within each super-particle. ### 3.5 Diagnostics The coagulation process is sensitive to the large-particle tail of the particle size distribution $f(a,t)$ because particle coagulation is strongly dependent on the total cross section. The tails of $f(a,t)$ can be characterised using the normalised moments of radius $a$ (Li et al., 2017), $a_{\zeta}=\left({M_{\zeta}}/{M_{0}}\right)^{1/\zeta},$ (27) where $M_{\zeta}(t)=\int_{0}^{\infty}f(a,t)\,a^{\zeta}\,{\rm d}a,$ (28) is the $\zeta$th moment of $a$. We adopted $\zeta=24$ to characterise the large-particle tail. To follow the overall evolution of $f(a,t)$, we also considered the mean radius, defined by the first-order normalised moment, $\tilde{a}=a_{1}=M_{1}/M_{0}$. The relative standard deviation of $f(a,t)$ can be defined as $\sigma_{a}/\tilde{a}=(a_{2}^{2}-a_{1}^{2})^{1/2}/a_{1},$ (29) where $\sigma_{a}=(a_{2}^{2}-a_{1}^{2})^{1/2}$ is the standard deviation of $a$. ## 4 Results Table 1: Parameter values used in different simulation runs. The sound speed is $c_{s}=1.0\,[L/T]$ in all simulations except for Run F, where $c_{s}=0.5\,[L/T]$. Run | $f_{0}$ | $L_{x}$ | $N_{\rm grid}$ | $N_{\rm s}$ | $\alpha_{\rm ini}$ | $\nu$ | $c_{\rm shock}$ | ${\cal M}_{\rm rms}$ | ${\rm Re}$ | $\langle\bar{\epsilon}\rangle$ | $\eta$ | $\rm{St}^{L}(t=0)$ | $\rm{St}^{\eta}(t=0)$ ---|---|---|---|---|---|---|---|---|---|---|---|---|--- A | $1.0$ | $2\pi$ | $512^{3}$ | $15624960$ | 0.016 | $2.5\times 10^{-3}$ | 2.0 | 1.56 | 200 | 0.80 | 0.012 | 0.047 | 1.071 C | $0.73$ | $7.47$ | $512^{3}$ | $15624960$ | 0.013 | $2.5\times 10^{-3}$ | 2.0 | 1.27 | 194 | 0.42 | 0.014 | 0.032 | 0.769 D | $0.02$ | $0.05$ | $256^{3}$ | $1953120$ | 2.000 | $1\times 10^{-5}$ | – | 0.066 | 83 | 0.76 | 0.0002 | 0.200 | 2.778 E | $1.20$ | $0.16$ | $256^{3}$ | $1953120$ | 0.625 | $5\times 10^{-5}$ | 2.0 | 0.55 | 90 | 1.13 | 0.0007 | 0.667 | 10.000 F | $3.50$ | $\pi$ | $512^{3}$ | $15624960$ | 0.032 | $2.5\times 10^{-3}$ | 8.0 | 2.58 | 82 | 1.03 | 0.01 | 0.167 | 2.600 G | $1.00$ | 1.60 | $256^{3}$ | $1953120$ | 0.032 | $1\times 10^{-3}$ | 2.0 | 0.99 | 81 | 0.80 | 0.006 | 0.118 | 1.500 H | $1.00$ | 0.60 | $256^{3}$ | $1953120$ | 0.167 | $2.5\times 10^{-4}$ | 2.0 | 0.75 | 92 | 0.79 | 0.002 | 0.240 | 3.000 I | $1.00$ | 0.80 | $256^{3}$ | $1953120$ | 0.125 | $1\times 10^{-3}$ | 2.0 | 0.74 | 30 | 0.79 | 0.006 | 0.176 | 1.500 To investigate how coagulation depends on the Mach number $\mathcal{M}_{\rm rms}$, we performed simulations for different $\mathcal{M}_{\rm rms}$ ranging from 0.07 to 2.58 while keeping ${\rm Re}$ and $\langle\bar{\epsilon}\rangle$ fixed (see the details of the simulation setup in Table 1). As shown in Fig. 1, the power spectra follow the classical Kolmogorov $-5/3$ law. Because the Reynolds number that can be achieved in DNS studies is much lower than the one in the ISM, large-scale separation of turbulence is not observed. Figure 1: Power spectra for simulations of different ${\cal M}_{\rm rms}$: ${\cal M}_{\rm rms}=0.07$ (solid dotted black curve), $0.55$ (dashed cyan curve), $0.99$ (dotted blue curve), and $2.58$ (red curve). The dashed black curve shows the Kolmogorov $-5/3$ law. Next, we inspected the time evolution of the dust size distribution $f(a,t)$. As shown in Fig. 2, the tail of $f(a,t)$ widens with increasing ${\cal M}_{\rm rms}$. The broadening of $f(a,t)$ is slowest for the nearly incompressible flow with ${\cal M}_{\rm rms}=0.07$ (solid dotted black curve). A transition is observed when the flow pattern changes from subsonic (${\cal M}_{\rm rms}\sim 0.5$) to transonic or supersonic (${\cal M}_{\rm rms}\gtrsim 1$)111A turbulent flow may be categorised in the following way according to Mach number: subsonic range (${\cal M}_{\rm rms}<0.8$), transonic range ($0.8<{\cal M}_{\rm rms}<1.3$), and the supersonic range ($1.3<{\cal M}_{\rm rms}<5.0$)., where the broadening and the extension of the tail of $f(a,t)$ become prominent. This is further evidenced by the simulations with ${\cal M}_{\rm rms}=0.75$ (dash dotted magenta curve) and ${\cal M}_{\rm rms}=0.99$ (dotted blue curve), in the intermediate transonic regime. The supersonic case with ${\cal M}_{\rm rms}=2.58$ displays a significant broadening of the tail. Figure 2: Time evolution of $f(a,t)$ for simulations in Fig. 1 and for an additional simulation run H in Table 1. Fig. 3 shows the time evolution of the mean radius $\tilde{a}$ normalised by the initial size of particles. It is obvious that $\tilde{a}/a_{\rm ini}$ increases with increasing ${\cal M}_{\rm rms}$. Although this does not say much about tail effects, $\tilde{a}$ is a good measurement of the mean evolution of $f(a,t)$. Figure 3: Time evolution of $\tilde{a}/a_{\rm ini}$ for simulations in Fig. 2. According to Eq. (19), the coagulation rate depends on the total cross section of the two colliding particles. Therefore growth by coagulation is sensitive to the large tail of $f(a,t)$. As discussed in Section 3.5, the tail of $f(a,t)$ can be characterised by $a_{24}$, that is, the 24th normalised moment. Fig. 4(a) shows that the rate of increase of $a_{24}$ increases with ${\cal M}_{\rm rms}$. The corresponding relative dispersion of $f(a,t)$, $\sigma_{a}/\tilde{a}$, is shown in Fig. 4(b), which exhibits the same ${\cal M}_{\rm rms}$ dependence as $a_{24}$. However, the form of $\sigma_{a}/\tilde{a}$ as a function of $a_{24}/a_{\rm ini}$ is essentially independent of ${\cal M}_{\rm rms}$ , as shown by the inset in Fig. 4(b). Figure 4: Time evolution of (a) the $a_{24}$ and (b) $\sigma_{a}/\tilde{a}$ for simulations in Fig. 2. The inset shares the same y-axis as the main plot in panel (b). The dashed magenta curve in the inset shows ${\rm ln}(a_{24}/a_{\rm ini})$. As mentioned in section 3.3, the mean collision kernel $\langle C_{ij}\rangle$ depends on ${\cal M}_{\rm rms}$. Fig. 5(a) shows the collision kernel $\langle C_{ij}\rangle$ normalised according to $(a_{i}+a_{j})_{\rm ini}^{3}$, that is, the initial particle size. The Saffman-Turner model should not apply to coagulation of inertial particles in compressible turbulence as it assumes that particles act as passive tracers and are advected by the turbulent motion of the carrier. In spite of this, $\langle C_{ij}\rangle$ appears to scale with particle size as $a^{3}$, which is shown in Fig. 5(b), where $\langle C_{ij}\rangle$ is normalised to $\mathcal{M}_{\rm rms}\,(a_{i}+a_{j})^{3}$. The reason for this is not obvious. We recall, however, that we consider turbulence in highly compressible flows, and more importantly, that the trajectories of inertial particles tend to deviate from the flow. This leads to higher particle densities by compaction and clustering in the convergence zones in between vortex tubes (Maxey, 1987). Moreover, depending on the particle masses, it may also lead to the formation of caustics, which are the singularities in phase-space of suspended inertial particles (Falkovich et al., 2002; Wilkinson & Mehlig, 2005). This will lead to large velocity differences between colliding particles and thus to large $\langle C_{ij}\rangle$. The net result is a rather complex coagulation process, where $\langle C_{ij}\rangle$ varies strongly from one location to the next, which is further discussed below. According to Fig. 5(a), $\langle C_{ij}\rangle$ exhibits a clear increase from subsonic to supersonic turbulence (cf. ${\cal M}_{\rm rms}=0.55$, cyan curve in Fig. 5(a) and ${\cal M}_{\rm rms}=2.58$, red curve in Fig. 5(a)). As we argued in section 3.3, $\langle C_{ij}\rangle$ is proportional to ${\cal M}_{\rm rms}$ under the assumption of Maxwellian velocity distributions. Fig. 5 (b) shows $\langle C_{ij}\rangle$ normalised also by ${\cal M}_{\rm rms}$. We note that a linear scaling seems to be applicable from the subsonic regime to the supersonic regime. This means that the simple analytical theory is reasonable. The cyan curve deviates from other curves because the initial Stokes number for this simulation is about 10, as listed in Table 1. The Stokes number dependence of $\langle C_{ij}\rangle$ can be further confirmed by Fig. 5(c). As $\langle C_{ij}\rangle$ is determined by the relative velocity of colliding pairs, we normalised $\langle C_{ij}\rangle$ by $v_{\rm p,rms}/c_{\rm s}$ , as shown in Fig. 5(c). It is obvious that $\langle C_{ij}\rangle$ scales linearly with $v_{\rm p,rms}/c_{\rm s}$ up to the supersonic regime. Fig. 6 shows the distribution of the magnitudes of the particle velocities, which indeed is very similar to a Maxwellian velocity distribution. It also shows that particle velocities become higher with increasing ${\cal M}_{\rm rms}$. Especially the tail of the particle velocity distribution becomes more populated. This indicates that stronger shocks accelerate inertial particles more and may therefore increase the coagulation rate. Figure 5: Measured collision kernel $\langle C\rangle$ normalised to $(a_{i}+a_{j})^{3}$. Panel (a) shows the case of constant (initial) $a_{i}$, while panel (b) shows the case where $(a_{i}+a_{j})^{3}$ evolves and where ${\cal M}_{\rm rms}$ is also included in the normalisation. The simulations are the same as in Fig. 2. Panel (c) shows the same normalisation as panel (b), but with ${\cal M}_{\rm p,rms}$. Figure 6: Particle velocity distribution at $105\tau_{L}$. The dashed line is a Maxwellian fit of the particle velocity of run G. According to Eq. (19), the collision kernel is determined by the relative velocity $\bm{w}_{ij}$ and the relative separation $\Delta\bm{r}$ of two colliding particles. The former scenario is known as caustics (Wilkinson et al., 2006) and the latter as clustering (Gustavsson & Mehlig, 2016), as discussed above. Our simulations involve coagulation, which leads to evolution of $\alpha(a,t)$. This makes it difficult to analyse clustering and caustics based on these simulations. Below we try to understand the ${\cal M}_{\rm rms}$-dependence based on the spatial distribution and velocity statistics of the particles. As shown in Fig. 7, the spatial distribution of particles exhibits different behaviours in the three $\alpha$ ranges we considered. When $\alpha<0.1$, particles tend to be trapped in regions where high gas density occurs. This is consistent with the findings of Hopkins & Lee (2016) and Mattsson et al. (2019), even though coagulation was not considered in their studies. When $0.1\leq\alpha\leq 0.3$, particles still accumulate in the high-density regions, but are also spread out in regions with low gas density. This dispersion is expected as $\tau_{i}$ increases. Finally, when $\alpha>0.3$, particles more or less decouple from the flow, demonstrating essentially a random-walk behaviour. When we compare with ${\cal M}(\bm{x},t)$ instead of $\ln{\rho}$, we see that particles accumulate in regions with low local ${\cal M}(\bm{x},t)$, as shown in Fig. 8. That is, low ${\cal M}(\bm{x},t)$ corresponds to high $\ln{\rho}(\bm{x},t)$. The physical picture is the following. Strong shocks generated in these local supersonic regions push particles to low ${\cal M}(\bm{x},t)$ regions, which is then how particle densities increase due to compression of the gas. This compaction of particles is different from the fractal clustering of inertial particles, which mainly occurs as a result of accumulation of particles in the convergence zones between vortices. Statistically, the spatial distribution of particles can be characterised by $g(r)$, which contributes to the mean collision rate as expressed in Eq. (19). However, $g(r)$ is only useful as a diagnostic for a mono-dispersed particle distribution or fixed size bins (Pan et al., 2011). Therefore we only show the spatial distributions of particles and do not go into details about the quantitative statistics. Figure 7: Spatial distribution of inertial particles and the gas density at 80 time units in a slab with thickness $\eta$ for run F. Figure 8: Same as Fig. 7, but with ${\cal M}({\bm{x}},t)$ as the contour map. As the collision kernel $\langle C_{ij}\rangle$ is also determined by $|{\bm{w}}_{ij}|$, we also examined the magnitude of particle velocities for different ranges of $\alpha$. Fig. 9 shows the PDF of $|\bm{v}_{p}|/u_{\rm rms}$ for $\alpha<0.1$, $0.1\leq\alpha\leq 0.3$, and $\alpha>0.3$, respectively. It is evident that the velocity magnitude of particles that are coupled to the flow is higher than that of particles that are decoupled from the flow. Thus, the ${\cal M}_{\rm rms}$ dependence of $\langle C_{ij}\rangle$ could very well be due to enhanced caustics and compression-induced concentration with increasing ${\cal M}_{\rm rms}$. Figure 9: Corresponding PDF for Fig. 7. In incompressible turbulence, the collision kernel depends on $\tau_{\eta}$, which is determined by $\langle\bar{\epsilon}\rangle$, while it is insensitive to ${\rm Re}$ (Grabowski & Wang, 2013; Li et al., 2018). We examined the $\langle\bar{\epsilon}\rangle$ and ${\rm Re}$ dependences of $a_{24}$ and $\sigma_{a}$ in compressible turbulence. As shown in Fig. 10, $a_{24}$ and $\sigma_{a}/\tilde{a}$ have only a weak dependence on $\langle\bar{\epsilon}\rangle$ in the supersonic regime (e.g. simulations A and C have similar ${\cal M}_{\rm rms}$ but differ by a factor of two in $\langle\bar{\epsilon}\rangle$). By inspection of Fig. 10, changing ${\rm Re}$ (run H and I) does not obviously affect $a_{24}$ and $\sigma_{a}/\tilde{a}$ in the transonic regime, which may seem to be consistent with the simulation results for incompressible turbulence. Figure 10: Time evolution of (a) $a_{24}$ and (b) dispersion of $f(a,t)$ for different $\langle\bar{\epsilon}\rangle$ and ${\rm Re}$ with the same ${\cal M}_{\rm rms}$ (see runs A, C, H, and I in Table 1). ## 5 Discussion We have observed that as ${\cal M}_{\rm rms}$ increases, the tail of the distribution increases. This poses mainly two questions: (1) how the compression-induced density variation (simply compaction) affects $\langle R_{ij}\rangle$, and (2) how the velocity dispersion of particles due to shocks affects $\langle C_{ij}\rangle$. According to Eq. (19), the coagulation rate $\langle R_{ij}\rangle$ is determined by $g(a,r)$ and $|\bm{w}_{ij}|,$ as discussed in section 4. Particles tend to stay in regions where the gas density is high due to shock- induced compaction. With larger ${\cal M}_{\rm rms}$, the gas density fluctuations become stronger. This leads to somewhat higher concentrations of particles, especially for particles with $\alpha\leq 0.3$. It is important to note that particles accumulate in low ${\cal M}\left(\bm{x},t\right)$ regions, where the gas density is high because particles are pushed to lower ${\cal M}\left(\bm{x},t\right)$ regions by the shocks. Supersonic flows cover a wide range of ${\cal M}\left(\bm{x},t\right)$, which results in stronger density variations of particles. Local concentrations of particles might lead to higher $\langle R_{ij}\rangle$. Therefore the enhanced $\langle R_{ij}\rangle$ with increasing ${\cal M}_{\rm rms}$ is indeed due to the change in the flow structure. In addition to the compressibility, fractal clustering due to inertia effect (large St) of particles might also enhance $\langle C_{ij}\rangle$. Higher ${\cal M}_{\rm rms}$ results in stronger shocks and thus higher particle velocities, which also leads to larger $\langle C_{ij}\rangle$. In particular, for supersonic flows, the local fluctuations of ${\cal M}\left(\bm{x},t\right)$ are strong. This leads to significant local dispersion in the particle velocities and a consequent enhancement of $\langle C_{ij}\rangle$. The coagulation rate time series almost collapse on top of each other when normalised by $v_{\rm p,rms}/c_{\rm s}$ up to the supersonic regime. This indicates that the simple description of the collision kernel, Eq. (23), applies up to the supersonic regime. As particles grow larger in the simulation, they decouple from the flow. Statistically, this decoupling can be roughly described by the difference between $u_{\rm rms}$ and $v_{\rm p,rms}$. This probably causes these curves to collapse on each other when $\langle C_{ij}\rangle$ is normalised by $v_{\rm p,rms}/c_{\rm s}$. We therefore propose that $\langle C_{ij}\rangle$ is proportional to $v_{\rm p,rms}/c_{\rm s}$ instead of ${\cal M}_{\rm rms}$ in Eq. (23). Since the inertial range is determined by ${\rm Re}$, we also examined how $\langle C_{ij}\rangle$ depends on ${\rm Re}$. As shown in Fig. 10, the $\rm{Re}$ dependence is weak because $\tau_{i}\ll\tau_{L}$. Pan & Padoan (2014) have suggested that the collision kernel is independent of $\rm{Re}$ in the subsonic regime. We show here that this also appears to apply to the transonic regime and likely also the supersonic regime. As we discussed in section 2.1, for incompressible turbulence, $\langle C_{ij}\rangle$ is determined by $\langle\bar{\epsilon}\rangle$ through $\tau_{\eta}$. Fig. 10 shows that the $\langle\bar{\epsilon}\rangle$ dependence observed in incompressible flows vanishes in compressible flows, however, which is quite expected. We demonstrated that the coagulation rate of inertial particles is mainly affected by ${\cal M}_{\rm rms}$, essentially the level of compression of the flow. We conclude that ${\cal M}_{\rm rms}$ is the main parameter determining $\langle C_{ij}\rangle$ in the trans- and supersonic regimes. The pioneering work of Saffman & Turner (1956) suggested that $\langle C_{i}\rangle\sim\tau_{\eta}^{-1}$ for specified sizes of particles. Because $\tau_{\eta}=(\nu/\langle\bar{\epsilon}\rangle)^{1/2}$, $\langle C_{i}\rangle\sim\langle\bar{\epsilon}\rangle^{1/2}$, which has been confirmed in many studies of incompressible turbulence. More importantly, all simulation works have found that $\langle C_{i}\rangle$ is independent of Re (see Grabowski & Wang, 2013; Li et al., 2018, and the references therein). This is contradictory, however, because $\langle\bar{\epsilon}\rangle$ is a parameter that is determined by Re. We argue that this is because of the forcing term in the N-S equation, which is inevitable in turbulence simulations. This forcing term invokes a third dimensional parameter $\langle\bar{\epsilon}\rangle$ that determines $\langle C_{i}\rangle$ in incompressible turbulence. For compressible turbulence, however, our study showed that $\langle C_{i}\rangle$ is determined by ${\cal M}_{\rm rms}$ alone and independent of $\langle\bar{\epsilon}\rangle$. We can then avoid the contradiction described above. However, this contradiction is a more fundamental problem that requires a solution, but this is beyond the scope of our study. In short, compressible turbulence implies that $\langle\bar{\epsilon}\rangle$ is not only determined by Re, but also by ${\cal M}_{\rm rms}$. Even though $\langle\bar{\epsilon}\rangle$ does not affect $\langle C_{i}\rangle$ directly, it affects the Stokes number ${\rm St}=\tau_{\rm i}/\tau_{\eta}$. Therefore the $\langle C_{i}\rangle$ time series collapse on top of each other when they are normalised by $\tau_{\eta}$ in Fig. 5(c). Andersson et al. (2007) proposed that in 2D compressible flow with a Gaussian random velocity, fractal particle (inertialess) clustering can lead to a higher coagulation rate. In addition to the aforementioned assumptions, the collision rate suggested in Andersson et al. (2007) involves the fractal dimension $D_{2}$, which is difficult to measure in our case because we have an evolving size distribution of particles. A direct comparison between our simulation and the theory of Andersson et al. (2007) is therefore not feasible. Nevertheless, we do observe fractal clustering in our simulation, which could indeed elevate the coagulation rate because Fig. 6 shows that high ${\cal M}_{\rm rms}$ flow results in higher particle velocities. Because we have a wide range of Stokes numbers, the fractal clustering (Falkovich et al., 2001) could also be enhanced. ## 6 Summary and conclusion Coagulation of inertial particles in compressible turbulence was investigated by direct and shock-capturing numerical simulations. Particle interaction was tracked dynamically in a Lagrangian manner, and the consequential coagulation events were counted at each simulation time step. We specifically explored the Mach-number dependence of the coagulation rate and the effects on the widening of the particle-size distribution. To our knowledge, this is the first time that this has been done. We showed that the coagulation rate is determined by Mach number ${\cal M}_{\rm rms}$ in compressible turbulence. This is fundamentally different from the incompressible case, where the coagulation rate is mainly determined by $\langle\bar{\epsilon}\rangle$ through the Kolmogorov timescale. The dispersion or variance of $f(a,t)$, $\sigma_{a}$, increases with increasing ${\cal M}_{\rm rms}$. We also note that $\sigma_{a}$ is a simple and more or less universal function of the size of the largest particles, measured by $a_{24}$, and is apparently independent of ${\cal M}_{\rm rms}$. All effects on coagulation increase progressively with ${\cal M}_{\rm rms}$, which shows the importance of compressibility for coagulation processes. Taken at face value, our simulations appear to suggest that existing theories of the ${\cal M}_{\rm rms}$ dependence of $\langle C_{ij}\rangle$, which imply an underlying linear scaling with ${\cal M}_{\rm rms}$ , are correct to first order, but we cannot draw any firm conclusions at this point. For this we will need more simulations with a wider range of ${\cal M}_{\rm rms}$ values. We note that $\langle C_{ij}\rangle$ scales as $\langle C_{ij}\rangle\sim(a_{i}+a_{j})^{3}\,\mathcal{M}_{\rm rms}/\tau_{\eta}$. When the collision kernel $\langle C_{ij}\rangle$ is normalised by $v_{\rm p,rms}/c_{\rm s}$, the curves collapse on top of each other. We therefore suggest that $\langle C_{ij}\rangle$ is proportional to $v_{\rm p,rms}/c_{\rm s}$, rather than $\mathcal{M}_{\rm rms}$. This finding may serve as a benchmark for future studies of coagulation of dust grains in highly compressible turbulence. We propose two mechanisms that might be behind the ${\cal M}_{\rm rms}$ dependence of the broadening of $f(a,t)$ even though it is still not fully understood due to the non-equilibrium nature of the coagulation process in compressible turbulence. The first mechanism is the compaction-induced concentration of particles. Supersonic flow exhibits stronger fluctuations of local ${\cal M}(\bm{x},t)$. The consequent vigorous shocks compact small particles (e.g. $\alpha<0.3$) into low-${\cal M}(\bm{x},t)$ regions. This leads to high densities of particles ($n_{i}$) and then potentially to a higher coagulation rate $\langle R_{ij}\rangle$. The second mechanism is larger dispersion of particle velocities caused by stronger shocks. Again, stronger local fluctuations ${\cal M}(\bm{x},t)$ lead to a larger dispersion of particle velocities, which increases the coagulation rate. Simulating the coagulation problem in compressible and supersonic turbulence, we achieved ${\cal M}_{\rm rms}=2.58$, but with a non-astrophysical scaling. This is smaller than the ${\cal M}_{\rm rms}\geq 10$ observed in cold clouds. To explore whether a saturation limit of the ${\cal M}_{\rm rms}$ dependence of the coagulation rate exists, a direct numerical simulation coupled with coagulation would have to reach at least ${\cal M}_{\rm rms}\sim 10$. We also note that the simulated systems in our study have flow timescales (turn-over times) that are of the same order as the coagulation timescale, that is, $\tau_{\rm c}/\tau_{L}<1$, which is computationally convenient, but very different from dust in the ISM, for example, where $\tau_{c}/\tau_{L}\gg 1$. Nonetheless, our study provides a benchmark for simulations of dust-grain growth by coagulation in the ISM and other dilute astrophysical environments. Reaching really high ${\cal M}_{\rm rms}$, and astrophysical scales in general is currently being explored. Fragmentation is also omitted in this study, which may overestimate the coagulation rate. Adding fragmentation is a topic for future work. ## Acknowledgement Xiang-Yu Li wishes to thank Axel Brandenburg, Nils Haugen, and Anders Johansen for illuminating discussions about the simulation code used in this study, the Pencil Code. Lars Mattsson wishes to thank the Swedish Research Council (Vetenskapsrdet, grant no. 2015-04505) for financial support. We thank the referee for very helpful suggestions to improve the manuscript. Our simulations were performed using resources provided by the Swedish National Infrastructure for Computing (SNIC) at the Royal Institute of Technology in Stockholm, Linköping University in Linköping, and Chalmers Centre for Computational Science and Engineering (C3SE) in Gothenburg. The Pencil Code is freely available on https://github.com/pencil-code/. 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(19) appears to be a natural expression for the collision kernel between particle pairs. However, it is not a physically complete and always representative model (Wilkinson et al., 2006). Two mechanisms contribute to the collision kernel $C_{ij}$ due to particle inertia: clustering and caustics. The former is characterised by $g(r)\propto r^{D_{2}-d}$ at a fixed Stokes number, where $g$ is the radial distribution function, $d$ is the spatial dimension, and $D_{2}$ is the correlation dimension (Reade & Collins, 2000; Procacia et al., 1983). The latter is the effect of singularities in the particle phase space at non-zero values of the Stokes number (Falkovich et al., 2002; Wilkinson et al., 2006; Gustavsson & Mehlig, 2014). It appears when phase-space manifolds fold over. In the fold region, the velocity field at a given point in space becomes multi-valued, allowing for large velocity differences between nearby particles, which results in a temporarily increased particle-interaction rate and more efficient coagulation (Gustavsson & Mehlig, 2014). As indicated above, the relative velocity between two colliding pairs obeys a power law $|\langle w_{ij}\rangle|\propto(r_{i}+r_{j})^{d-D_{2}}$. Thus, the product of $g(r)$ and $|\langle w_{ij}\rangle|$ is independent of $(r_{i}+r_{j})$, that is, caustics and clustering cancel each other out in this formulation. Therefore Wilkinson et al. (2006) proposed that $C_{ij}$ is a superposition of clustering and caustics, which was confirmed by numerical simulations (Voßkuhle et al., 2014). In section 3.3 of our study we ignored the effects of caustics mainly because caustics in high $\mathcal{M}_{\rm rms}$ compressible turbulence are associated with shock interaction and density variance in the carrier fluid, in which case the resultant increase in particle number density is a far greater effect than the caustics.

2024-09-04T02:54:55.127369

2002.12216

arxiv-papers

# Sum-of-squares decompositions for a family of noncontextuality inequalities and self-testing of quantum devices Debashis Saha Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland Rafael Santos Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland Remigiusz Augusiak Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland ###### Abstract Violation of a noncontextuality inequality or the phenomenon referred to ‘quantum contextuality’ is a fundamental feature of quantum theory. In this article, we derive a novel family of noncontextuality inequalities along with their sum-of-squares decompositions in the simplest (odd-cycle) sequential- measurement scenario capable to demonstrate Kochen-Specker contextuality. The sum-of-squares decompositions allow us to obtain the maximal quantum violation of these inequalities and a set of algebraic relations necessarily satisfied by any state and measurements achieving it. With their help, we prove that our inequalities can be used for self-testing of three-dimensional quantum state and measurements. Remarkably, the presented self-testing results rely on weaker assumptions than the ones considered in Kochen-Specker contextuality. To realize genuine quantum technologies such as cryptographic systems, quantum simulators or quantum computing devices, the back-end user should be ensured that the quantum devices work as specified by the provider. Methods to certify that a quantum device operates in a nonclassical way are therefore needed. The most compelling one, developed in the cryptographic context, is self-testing [MY04]. It exploits nonlocality, i.e., the existence of quantum correlations that cannot be reproduced by the local-realist models, and provides the complete form of device-independent 111With the requirement of the spatial separation between measurements on subsystems, and without any assumption on the internal features of the devices. characterization of quantum devices only from the statistical data the devices generate. Thus, it is being extensively studied in recent years [YVB+14, BP15, CGS17]. However, since self-testing, as defined in Ref. [MY04], stands on nonlocality [Bel64] (or, in other words, quantum correlations that violate local-realist inequalities), it is restricted to preparations of composite quantum systems and local measurements on them. Therefore, it poses a fundamental question: presuming the minimum features of the devices how to characterize $(i)$ quantum systems of prime dimension that are not capable of exhibiting nonlocal correlations, and $(ii)$ quantum systems without entanglement or spatial separation between subsystems? A possible way to address such instances is to employ quantum contextuality (Kochen-Specker contextuality), a generalization of nonlocal correlations obtained from the statistics of commuting measurements that are performed on a single quantum system [KS75, Cab08, CSW14, KCBbuS08]. Indeed, the recent study [BRV+19b, IMOK20, BRV+19a] provides self-testing statements based on contextual correlations (or correlations that violate noncontextuality inequality). Since quantum contextual correlations are essential in many aspects of quantum computation [HWVE14, Rau13] and communication [GHH+14, SHP19], self-testing statements are crucial for certifying quantum technology [BRV+19a]. Apart from that, it is, nonetheless, fundamentally interesting to seek the maximum information one can infer about the quantum devices only from the observed statistics in a contextuality experiment. In the context of nonlocality, sum-of-squares (SOS) decomposition of quantum operators associated with local-realist inequalities has been the key mathematical tool in recent years to obtain optimal quantum values and self- testing properties of quantum devices [BP15, ŠASA16, SAT+17, KŠT+19, SSKA19, ASTA19, Kan19, CMMN19]. Whether this line of study, albeit, restricted to nonlocal correlations, can further be extended to contextuality scenario is of great interest from the perspective of unified approach to non-classical correlations [CSW14, AC18]. In this work, we consider Klyachko-Can-Binicioğlu-Shumovsky (KCBS) scenario which comprises of one preparation and $n$ (where $n\geqslant 5$ is odd) number of measurements [KCBbuS08, AQB+13, LSW11]. This is the simplest scenario capable to exhibit contextual correlations using a three-dimensional quantum system and five binary outcome measurements. It also has several implications in quantum foundation and quantum information [GBC+14, GHH+14, SBA17, Cab13, KanCK14, SR17, XSS+16]. We first introduce a modified version of KCBS expression for $n=5$ involving correlation between the outcomes of two sequential measurements, along with an SOS decomposition of the respective quantum operator. We describe our methodology to obtain SOS and simultaneously, generalize for $n$-cycle KCBS scenario where $n=2^{m}+1,m\in\mathbbm{N}$. Interestingly, the SOS decomposition holds even without the idealizations that the measurement effects are projectors and satisfy cyclic orthogonality conditions. By virtue of this decomposition, we obtain the maximum quantum value of our modified $n$-cycle expression and a set of algebraic relations involving any quantum state and measurements that yield those maximum values. By solving those relations, we show the existence of a three-dimensional vector-space invariant under the algebra of measurement operators. Subsequently, we prove the uniqueness of the projected three- dimensional measurements and state up to unitary equivalence, that is, self- testing property of the quantum devices. The presented self-testing relies on the premise that the Kraus operator corresponding to one of the outcomes outcome is uniquely determined by the respective measurement effect, and it does not assume any other idealizations of the measurements or the dimension of the preparation. ## 1 Preliminaries We begin by illustrating our scenario and specifying the assumptions. Sequential-measurement set-up. Each run of the experimental observation comprises of preparation of a physical system followed by two measurements in a sequence using one non-demolishing measurement device as depicted in Fig. 1. The measurement device has $n$ (odd) different settings, each of which yields $\pm 1$ outcome. Let’s denote the first and second measurement settings by $\mathcal{A}_{i}$ and $\mathcal{A}_{j}$ where $i,j\in\\{1,\dots,n\\}$. The settings are chosen such that $j=i\pm 1$, where from now on the subscript $i$ is taken modulo $n$, that is, $\mathcal{A}_{i\pm n}=\mathcal{A}_{i}$. We first make the following assumption. ###### Assumption 1. The measurement device only returns the actual post-measurement state. This assumption is necessary, otherwise, any quantum statistics can be reproduced by classical systems. By repeating this experiment many times we can obtain joint probabilities $p(a_{i},a_{i\pm 1}|\mathcal{A}_{i},\mathcal{A}_{i\pm 1})$ of two measurements and single probabilities $p(a_{i}|\mathcal{A}_{i})$ of the first measurement, and consequently, their correlation functions, $\displaystyle\langle\mathcal{A}_{i}\mathcal{A}_{i\pm 1}\rangle$ $\displaystyle=$ $\displaystyle\sum_{a_{i},a_{i\pm 1}}a_{i}a_{i\pm 1}p(a_{i},a_{i\pm 1}|\mathcal{A}_{i},\mathcal{A}_{i\pm 1}),$ $\displaystyle\langle\mathcal{A}_{i}\rangle$ $\displaystyle=$ $\displaystyle\sum_{a_{i}}a_{i}p(a_{i}|\mathcal{A}_{i}),$ (1) where the measurement outcomes are denoted as $a_{i}=\pm 1$. Figure 1: Sequential-measurement set-up. The simplest contextuality scenario comprises of one preparation $\mathcal{P}$ and one measurement device with settings $\mathcal{A}_{i}$ each of them returns $\pm 1$ outcome. In quantum theory a two-outcome measurement $\mathcal{A}_{i}$ is represented by a pair of operators $K_{i},K^{\prime}_{i}$ acting on some finite- dimensional Hilbert space $\mathcal{H}$, such that $F_{i}\equiv K^{\dagger}_{i}K_{i}\leqslant\mathbbm{1}$ and $(K^{\prime}_{i})^{\dagger}K^{\prime}_{i}=\mathbbm{1}-F_{i}$. It is often convenient to represent such a binary measurement with the aid of the following operator $A_{i}=2F_{i}-\mathbbm{1}$ (2) that acts on $\mathcal{H}$. The preparation is represented by a quantum state that, without loss of generality, can be considered pure; we denote it by $|\psi\rangle$. Kochen-Specker contextuality [CSW14] pertains to the following assumptions: $(i)$ the measurements are projective, that is, $K_{i},K^{\prime}_{i}$ are projectors, and $(ii)$ the projectors satisfy certain orthogonality relations, particularly in this scenario, $F_{i}F_{i\pm 1}=0$ for all $i$, implying $[A_{i},A_{i\pm 1}]=0$. Such prerequisites about the measurement device are difficult to justify in practice. Since we aim to characterize the quantum devices from their minimal features, we do not make these assumptions. Instead, we consider a single assumption as given below, which is even weaker than the requirement of projectivity. We will see later that projectivity and orthogonality relations between measurement effects will be derived facts from the maximal violation of our inequality. ###### Assumption 2. The measurements are realized in a particular way such that $K_{i}$ are Hermitian or, equivalently, $K_{i}=\sqrt{F_{i}}$, for all $i$. In other words, whenever the measurement outcome is ‘+’ after measuring $A_{i}$ (2) the post-measurement state reduces according to the rule of Lüders instrument, that is, $|\psi\rangle\rightarrow\frac{\sqrt{F_{i}}|\psi\rangle}{\sqrt{\langle\psi|F_{i}|\psi\rangle}}.$ Although, a general expression of $K_{i}=U\sqrt{F_{i}}$ for some unitary $U$, such rule for post-measurement evolution appears naturally for unsharp measurements [Bus86] and other physical scenarios [BS98, PK98]. A general linear expression that can be considered to test nonclassicality (or noncontextuality in the usual scenario) in this set-up is given by, $\mathcal{B}=\sum_{i}c_{i}(\langle\mathcal{A}_{i}\mathcal{A}_{i+1}\rangle+\langle\mathcal{A}_{i+1}\mathcal{A}_{i}\rangle)+\sum_{i}d_{i}\langle\mathcal{A}_{i}\rangle.$ (3) The optimal quantum value of the above expression is defined as $\eta^{Q}=\sup_{|\psi\rangle,A_{i}}\langle\psi|B|\psi\rangle,$ (4) where $B=\sum_{i}c_{i}\\{A_{i},A_{i+1}\\}+\sum_{i}d_{i}A_{i}$ is the quantum operator associated with the expression $\mathcal{B}$ and $A_{i}$ are of the form (2). Notice that in the usual scenario, due to commutativity relations, $\\{A_{i},A_{i+1}\\}$ can be replaced by $2A_{i}A_{i+1}$. The maximal classical value $\eta^{C}$ (or noncontextual value in the usual scenario 222 Since any noncontextual value assignment pertains to projective measurements with certain orthogonality conditions, here we refer to $\eta^{C}$ as to the classical value for the relaxed scenario. Indeed, under Assumption 1 the optimal value in classical theory or any other theory where measurement does not affect the system is given by Eq. (5). In the ideal scenario $\eta^{C}$ reduces to the noncontextual value ) is defined as $\eta^{C}=\max_{a_{i}\in\\{1,-1\\}}\left\\{2\sum_{i}c_{i}a_{i}a_{i+1}+\sum_{i}d_{i}a_{i}\right\\}.$ (5) KCBS inequality. The well known $n$-cycle KCBS noncontextuality inequality [AQB+13] is of the form $\mathcal{B}_{\mathrm{KCBS}}:=-\sum^{n}_{i=1}\langle\mathcal{A}_{i}\mathcal{A}_{i+1}\rangle\leqslant\eta^{C}=n-2.$ (6) The maximal quantum violation of this inequality is $\eta^{Q}=\frac{3\cos{(\pi/n)}-1}{1+\cos{(\pi/n)}}n$ (7) and it is achieved by the following quantum state $|\widehat{\psi}\rangle=|0\rangle\equiv(1,0,0)^{T},$ (8) and observables $\widehat{A}_{i}=2|\widehat{v}_{i}\rangle\\!\langle\widehat{v}_{i}|-\mathbbm{1},$ (9) where $|\widehat{v}_{i}\rangle$ are three-dimensional real vectors defined as $|\widehat{v}_{i}\rangle=(\cos{\theta},\sin{\theta}\sin{\phi_{i}},\sin{\theta}\cos{\phi_{i}})^{T}$ (10) where $\theta$ is defined as $\cos\theta=\sqrt{1/(1+2\alpha)}$, where $\alpha=\frac{1}{2}\sec\left(\frac{\pi}{n}\right)$ (11) and $\phi_{i}=\frac{n-1}{n}\pi i.$ (12) Note that $\alpha$ and $\phi_{i}$ are functions of $n$, which for the sake of simplification is not explicitly specified in their notation. Let us also remark that $|\widehat{\psi}\rangle\in\mathbbm{C}^{3}$ and $\widehat{A}_{i}$ acting on $\mathbbm{C}^{3}$ denote a particular example of quantum realizations achieving the maximal quantum value of the KCBS inequality (6). The self-testing properties of the above-mentioned state and measurements based on the violation of KCBS inequality are shown in [BRV+19b]. The proof is based on the optimization method of semidefinite programming under the usual assumptions of contextuality. Sum-of-squares decomposition. Let us finally discuss the concept of sum-of- squares decompositions. Consider a quantum operator $B$ corresponding to some noncontextuality expression $\mathcal{B}$ like the one in (4). Now, if for any choice of quantum measurements $A_{i}$ and some $\eta\in\mathbbm{R}$ one can decompose the shifted operator $\eta\mathbbm{1}-B$ as $\eta\mathbbm{1}-B=\sum_{k}E^{\dagger}_{k}E_{k},$ (13) the maximal quantum value of $\mathcal{B}$ is upper bounded by $\eta$, i.e., $\langle\psi|B|\psi\rangle\leqslant\eta$ for any quantum state $|\psi\rangle$. We call (13) a sum-of-squares decomposition associated to $B$. Typically $E_{k}$ are constructed from the measurement operators $A_{i}$. The bound $\eta$ is realized by a state and a set of measurements if and only if the following algebraic relation holds true for all $k$, $\quad E_{k}|\psi\rangle=0.$ (14) Our self-testing proofs heavily rely on the above relations. Let us remark that Ref. [LSW11] provides an SOS decomposition for the conventional KCBS operator under the assumptions that the measurements $A_{i}$ are projective and satisfy $[A_{i},A_{i\pm 1}]=0$. In what follows we derive an alternative noncontextuality inequality together with the corresponding SOS decomposition of the form (13) which does not require making these assumptions. Furthermore, our SOS is designed in such a way that the algebraic relations (14) it implies can be used for self-testing. ## 2 Modified KCBS inequality with sum-of-squares decomposition We are now ready to present our results. For pedagogical purposes we begin with the simplest case of $n=5$ and consider the following modified KCBS expression $\mathcal{B}=-\frac{1}{2}\sum^{5}_{i=1}(\langle\mathcal{A}_{i}\mathcal{A}_{i+1}\rangle+\langle\mathcal{A}_{i+1}\mathcal{A}_{i}\rangle)-\alpha^{2}\sum^{5}_{i=1}\langle\mathcal{A}_{i}\rangle,$ (15) where $\alpha$ is given in (11) with $n=5$. Following (5) it is not difficult to find the maximal classical value of $B$ is $\eta^{C}=3+\alpha^{2}$. ###### Result 1 (Modified KCBS inequality with SOS). The maximal quantum value of $\mathcal{B}$ given in Eq. (15) with $\alpha=(1/2)\sec(\pi/n)$ is $\eta^{Q}=3(1+\alpha^{2})$. ###### Proof. To prove this statement we present the SOS decomposition for the modified KCBS operator $B=-\frac{1}{2}\sum_{i}\\{A_{i},A_{i+1}\\}-\alpha^{2}\sum_{i}A_{i}.$ (16) Let us first define the following Hermitian operators for $i=1,\dots,5$, $\displaystyle M_{i,1}$ $\displaystyle=$ $\displaystyle-\frac{1}{\alpha^{3}}(A_{i}+\alpha A_{i-1}+\alpha A_{i+1}),$ $\displaystyle M_{i,2}$ $\displaystyle=$ $\displaystyle-\frac{1}{\alpha^{4}}(-\alpha A_{i}+A_{i-2}+A_{i+2}),$ (17) and observe that they satisfy the following relations $-\frac{\alpha^{5}}{5}\sum_{i}\left(2M_{i,1}+\alpha^{3}M_{i,2}\right)=\alpha^{2}\sum_{i}A_{i},$ (18) and $\displaystyle\frac{\alpha^{5}}{5}\sum_{i}\left(M^{2}_{i,1}+\frac{\alpha^{3}}{2}M^{2}_{i,2}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{i}\\{A_{i},A_{i+1}\\}$ (19) $\displaystyle+\frac{1}{2\alpha}\sum_{i}A^{2}_{i},$ where we have used the identity $\alpha^{2}+\alpha=1$ for $\alpha$ given in Eq. (11) with $n=5$. With the aid of these relations it is straightforward to verify that $\displaystyle\frac{\alpha^{5}}{5}\sum_{i}\\!\left(\mathbbm{1}-M_{i,1}\right)^{2}\\!+\\!\frac{\alpha^{8}}{10}\sum_{i}\\!\left(\mathbbm{1}-M_{i,2}\right)^{2}\\!+\\!\frac{1}{2\alpha}\sum_{i}\\!\left(\mathbbm{1}-A^{2}_{i}\right)$ $\displaystyle=\left(\alpha^{5}+\frac{\alpha^{8}}{2}+\frac{5}{2\alpha}\right)\mathbbm{1}-\frac{\alpha^{5}}{5}\sum_{i}\left(2M_{i,1}+\alpha^{3}M_{i,2}\right)$ $\displaystyle\hskip 14.22636pt+\frac{\alpha^{5}}{5}\sum_{i}\left(M^{2}_{i,1}+\frac{\alpha^{3}}{2}M^{2}_{i,2}\right)-\frac{1}{2\alpha}\sum_{i}A^{2}_{i}$ $\displaystyle=3(1+\alpha^{2})\mathbbm{1}-B,$ (20) where $B$ is given in Eq. (16). Thus, using the fact that $A^{2}_{i}\leqslant\mathbbm{1}$, the above equation constitutes a SOS decomposition (13) of the modified KCBS operator in which $E_{k}=\sqrt{\frac{\alpha^{5}}{5}}(\mathbbm{1}-M_{k,1})$ (21) for $k=1,\ldots,5$; $E_{k}=\sqrt{\frac{\alpha^{8}}{10}}(\mathbbm{1}-M_{k-5,2})$ (22) for $k=6,\ldots,10$; $E_{k}=\sqrt{\frac{1}{2\alpha}\left(\mathbbm{1}-A^{2}_{k-10}\right)}$ (23) for $k=11,\dots,15$; and $3+3\alpha^{2}=4.146$ is the quantum bound of $B$. We can validate that the state and measurements in dimension three (8)-(9) responsible for optimal value of KCBS inequality achieve this bound. ∎ Inspired by the above $n=5$ case, let us now derive our modified KCBS expression for more measurements. Our aim is to obtain a general expression for which the sum-of-squares decomposition can easily be constructed as the one in Eq. (2) and later directly used for self-testing. To reach this goal, let us consider $n$ two-outcome quantum measurements represented by operators $A_{i}$ (2) acting on some Hilbert space of unknown but finite dimension. Let us then consider the expression (13) in which the operators $E_{k}$ are of the form $\mathbbm{1}-M_{k}$ with some positive multiplicative factors, where $M_{k}$ are constructed from $A_{i}$. Notice that for such a choice, Eq. (14) implies that $M_{k}$ must be stabilizing operators of the state $|\psi\rangle$ maximally violating our modified KCBS expression, that is, $M_{k}|\psi\rangle=|\psi\rangle$. Now, to design the explicit form of $M_{k}$ we can use the optimal quantum realization (8)-(9) of the $n$-cycle KCBS inequality (6), which gives us (see Appendix A for details of the derivation) $M_{i,k}=\bar{\alpha}\left[\left(1-2\beta_{k}\right)A_{i}+\beta_{k}(A_{i+k}+A_{i-k})\right],$ (24) where $i=1,\dots,n$ and $k=1,\dots,(n-1)/2$, whereas the coefficients $\beta_{k}$ and $\bar{\alpha}$ are given by $\beta_{k}=\frac{1}{2(1-\cos{\phi_{k}})}$ (25) and $\bar{\alpha}=\frac{1+2\alpha}{1-2\alpha},$ (26) where $\alpha$, $\phi_{k}$ are defined in Eqs. (11) and (12), respectively. Let us remark that $M_{i,k},\bar{\alpha},\beta_{i}$ are all functions of $n$ which for the sake of simplification is not specified explicitly. Moreover, the operators $M_{i,k}$ defined in (24) act on unknown Hilbert space $\mathcal{H}$ of finite dimension. We now go back to the SOS decomposition (13) which is deemed to be of the form $\sum_{i,k}c_{k}\left[\mathbbm{1}-M_{i,k}\right]^{2}+d\sum_{i}\left[\mathbbm{1}-A^{2}_{i}\right]$ (27) with some non-negative parameters $c_{k},d$ to be determined. By plugging the expression of $M_{i,k}$ (24) into it and after some rearrangement of indices, we obtain $\displaystyle\sum_{i,k}c_{k}\left[\mathbbm{1}-M_{i,k}\right]^{2}+d\sum_{i}\left[\mathbbm{1}-A^{2}_{i}\right]$ $\displaystyle=$ $\displaystyle\left(\sum_{k}c_{k}+d\right)n\mathbbm{1}-\left(2\bar{\alpha}{\sum}\limits_{k}c_{k}\right)\sum_{i}A_{i}$ (28) $\displaystyle+\left[\bar{\alpha}^{2}\sum_{k}c_{k}\left(1+6\beta_{k}^{2}-4\beta_{k}\right)-d\right]\sum_{i}A^{2}_{i}$ $\displaystyle+\bar{\alpha}^{2}\sum_{i}\left[2c_{1}\beta_{1}\left(1-2\beta_{1}\right)+c_{\frac{n-1}{2}}\beta^{2}_{\frac{n-1}{2}}\right]\\{A_{i},A_{i+1}\\}$ $\displaystyle+\bar{\alpha}^{2}\sum_{i}\sum^{(n-3)/2}_{k=2}\left[2c_{k}\beta_{k}\left(1-2\beta_{k}\right)+c_{f\left(\frac{k}{2}\right)}\beta^{2}_{f\left(\frac{k}{2}\right)}\right]\\{A_{i},A_{i+k}\\},$ where $\displaystyle f\left(\frac{k}{2}\right)=\begin{cases}k/2,&\text{ if $k$ is even}\\\ (n-k)/2,&\text{ if $k$ is odd}.\end{cases}$ (29) We want to choose the coefficient $c_{k}$ so that they are non-negative and all the anti-commutators $\\{A_{i},A_{i+k}\\}$ vanish except for $k=\pm 1$. For that purpose we consider $n=2^{m}+1$ for $m\in\mathbbm{N}\setminus\\{1\\}$. First we take $c_{k}=0$ whenever $k\neq 2^{x}$, where $x=0,\dots,m-1$. It follows from (28) that our requirement is fulfilled if the following set of equations is satisfied $2c_{2^{x}}\beta_{2^{x}}\left(1-2\beta_{2^{x}}\right)+c_{2^{x-1}}\beta_{2^{x-1}}^{2}=0$ (30) for $x=1,\dots,m-1$. The above equation (30) implies for all $x=1,\dots,m-1$ $\displaystyle\frac{c_{2^{x}}}{c_{1}}$ $\displaystyle=$ $\displaystyle\frac{1}{2^{x}}\prod^{x}_{j=1}\frac{\beta_{2^{j-1}}^{2}}{\beta_{2^{j}}\left(2\beta_{2^{j}}-1\right)}$ (31) $\displaystyle=$ $\displaystyle\left(\frac{\beta_{1}}{2^{x}\beta_{2^{x}}}\right)^{2}\prod^{x}_{j=1}\sec(\phi_{2^{j}}).$ Since $\sec(\phi_{2^{j}})$ is positive for all $j$ 333Note that $\cos{\phi_{2^{j}}}=\cos{(\pi 2^{j}/n)}$ and $0<\pi 2^{j}/n<\pi/2,\forall j=1,2,\dots,m-1$., $c_{2^{x}}/c_{1}$ is also positive. Now, to provide a plausible solution of $c_{2^{x}}$, it suffices to choose a positive $c_{1}$. Due to (30) the remaining anti-commutators in (28) are $\\{A_{i},A_{i+1}\\}$ with a factor $\bar{\alpha}^{2}\left[2c_{1}\beta_{1}\left(1-2\beta_{1}\right)+c_{2^{m-1}}\beta_{2^{m-1}}^{2}\right].$ (32) For simplicity we choose this factor to be 1/2 which implies that $c_{1}$ is such that $4c_{1}\beta_{1}\left(1-2\beta_{1}\right)+2c_{2^{m-1}}\beta_{2^{m-1}}^{2}=\frac{1}{\bar{\alpha}^{2}}.$ (33) After substituting $c_{2^{m-1}}$ from Eq. (31), the above gives $c_{1}=\frac{2^{2m-3}}{\bar{\alpha}^{2}}\frac{1}{2^{2m-1}\beta_{1}\left(1-2\beta_{1}\right)+\beta_{1}^{2}\ \prod\limits^{m-1}_{j=1}\sec(\phi_{2^{j}})}.$ (34) One can readily verify that $c_{1}$ is positive. Further, we choose $d=\bar{\alpha}^{2}\ {\sum}\limits_{k}c_{k}\left(1+6\beta_{k}^{2}-4\beta_{k}\right)$ (35) so that all the $A^{2}_{i}$ terms vanish and $d$ is positive. Finally, due to (30), (33) and (35), Eq. (28) reads as, $\sum_{i,k}c_{k}\left[\mathbbm{1}-M_{i,k}\right]^{2}+d\sum_{i}\left[\mathbbm{1}-A^{2}_{i}\right]=\eta_{n}\mathbbm{1}-B_{n},$ (36) where $B_{n}=-\frac{1}{2}\sum_{i}\\{A_{i},A_{i+1}\\}-\gamma\sum_{i}A_{i}\ ,$ (37) $\gamma=-2\bar{\alpha}\sum_{k}c_{k}\ ,$ (38) and $\displaystyle\eta_{n}=n\bar{\alpha}^{2}\sum_{k}c_{k}\left(\frac{1}{\bar{\alpha}^{2}}+1+6\beta_{k}^{2}-4\beta_{k}\right),$ (39) and $c_{k},d,M_{i,k}$ are defined in (31), (34), (35) and (24). From Eq. (25) we know that $\bar{\alpha}$ is a negative quantity and hence $\gamma$ is positive. Thus, our modified $n$-cycle KCBS inequality is $\mathcal{B}_{n}:=-\frac{1}{2}\sum_{i}(\langle\mathcal{A}_{i}\mathcal{A}_{i+1}\rangle+\langle\mathcal{A}_{i+1}\mathcal{A}_{i}\rangle)-\gamma\sum_{i}\langle\mathcal{A}_{i}\rangle\leqslant\eta_{n}^{C}$ (40) whose quantum bound is $\eta_{n}$ (39) and the classical value $\eta^{C}_{n}$ is provided in Result 3. It follows from the construction of the SOS (36) that the qutrit quantum state and measurements defined in Eqs. (8)-(12) satisfy the stabilizing relations $M_{i,k}|\psi\rangle=|\psi\rangle$ and $A_{i}^{2}|\psi\rangle=|\psi\rangle$, implying the bound $\eta_{n}$ is tight, or, in other words, the maximal quantum value of (40) equals $\eta_{n}$. To put the above mathematical analysis in a nutshell, the expression of the noncontextuality inequality (40) is derived such that it meets a SOS decomposition (13) of certain form. This leads us to the following result. ###### Result 2 (Modified $n$-cycle expression with SOS). The maximum quantum value of modified $n$-cycle noncontextuality expression (40) with a SOS decomposition (36) is $\eta_{n}$ (39) (where $n=2^{m}+1,m\in\mathbbm{N}\setminus\\{1\\})$. Let us finally prove the classical bound of our new noncontextuality expression. ###### Result 3 (Maximal classical value). The classical value of $\mathcal{B}_{n}$ in Eq. (40) is given by $n+\gamma-2$. ###### Proof. The classical value can be obtained by assigning $\pm 1$ values to the observables appearing in (40), that is, $\eta_{n}^{C}=\max_{a_{i}\in\\{1,-1\\}}\left\\{-\sum^{n}_{i=1}a_{i}a_{i+1}-\gamma\sum^{n}_{i=1}a_{i}\right\\},$ (41) where $\gamma$ is positive. Let us say in the optimal assignment there are $k$ number of $a_{i}$ which are $-1$. We first assume $k>n/2$. When there are $k$ number of $-1$, and $n-k$ number of $+1$, the minimum value of $\sum_{i}a_{i}a_{i+1}=4k-3n$, and the quantity $\sum_{i}a_{i}=n-2k$. Substituting these values in (41) we see $\eta_{n}^{C}=\left(3-\gamma\right)n-\left(4-2\gamma\right)k.$ (42) Therefore, the optimal value of $\eta_{n}^{C}$ is obtained for the minimum value of $k$, that is, for $k=(n+1)/2$. This implies the right-hand-side of (42) is $n+\gamma-2$. Similarly, if $k<n/2$, then we have $(n-k)>n/2$, and following a similar argument we can obtain the same bound. ∎ ## 3 Self-testing of quantum devices An exact self-testing statement provides us the certification of quantum devices, given that we observe an optimal violation of a noncontextuality inequality. However, the observed statistics are unchanged in the presence of auxiliary degrees of freedom (or auxiliary systems) and a global unitary. Therefore, self-testing in the context of state-dependent quantum contextual correlation [BRV+19b, IMOK20] infers unique state and measurements up to these equivalences. More formally, self-testing of preparation $|\bar{\psi}\rangle\in\mathbbm{C}^{d}$ and a set of measurements $\\{\bar{A}_{i}\\}^{n}_{i=1}$ acting on $\mathbbm{C}^{d}$ are defined as follows: if a set of observables $\\{A_{i}\\}^{n}_{i=1}$ acting on unknown finite-dimensional Hilbert space $\mathcal{H}$ and a state $|\psi\rangle\in\mathcal{H}$ maximally violate a noncontextuality inequality, then there exists an isometry $\Phi:\mathcal{H}\mapsto\mathbbm{C}^{d}$ such that 1. 1. $\Phi|\psi\rangle=|\bar{\psi}\rangle$ , 2. 2. $\Phi A_{i}\Phi^{\dagger}=\bar{A}_{i}$ for all $i=1,\ldots,n$. To obtain self-testing only from the reduced Assumptions mentioned in section 1, we consider a modified version of the expression $\mathcal{B}_{n}$ $\eqref{Bn}$ of the following form $\tilde{\mathcal{B}}_{n}:=\mathcal{B}_{n}-\sum_{i}[p(++|\mathcal{A}_{i+1},\mathcal{A}_{i})+p(++|\mathcal{A}_{i-1},\mathcal{A}_{i})].$ (43) Since the additional term is non-positive, the classical and quantum bounds of $\tilde{\mathcal{B}}_{n}$ are the same as for $\mathcal{B}_{n}$. Moreover, it follows from (36) that the SOS decomposition of $\tilde{B}_{n}$ is $\displaystyle\eta_{n}\mathbbm{1}-\tilde{B}_{n}$ $\displaystyle=$ $\displaystyle\sum_{i,k}c_{k}\left[\mathbbm{1}-M_{i,k}\right]^{2}+d\sum_{i}\left[\mathbbm{1}-A^{2}_{i}\right]$ (44) $\displaystyle+\sum_{i}(K_{i}K_{i+1})^{\dagger}(K_{i}K_{i+1})$ $\displaystyle+\sum_{i}(K_{i}K_{i-1})^{\dagger}(K_{i}K_{i-1}),$ where $\displaystyle\tilde{B}_{n}=B_{n}-\sum_{i}(K_{i}K_{i+1})^{\dagger}(K_{i}K_{i+1})$ $\displaystyle-\sum_{i}(K_{i}K_{i-1})^{\dagger}(K_{i}K_{i-1}),$ (45) and $\eta_{n}$ is again the optimal quantum value of $\tilde{B}_{n}$ 444 Note that ${K_{i\pm 1}|\psi\rangle/\langle\psi|K_{i\pm 1}^{\dagger}K_{i\pm 1}|\psi\rangle}$ is the post-measurement quantum state after $A_{i\pm 1}$ is measured and the ‘+’ outcome is obtained. Consequently, ${p(++|\mathcal{A}_{i\pm 1},\mathcal{A}_{i})=\langle\psi|(K_{i}K_{i\pm 1})^{\dagger}(K_{i}K_{i\pm 1})|\psi\rangle}$ . Let us now show that our inequality (43) can be used to make a self-testing statement, according to the above definition, for the state and observables (8)-(9) maximally violating it. ###### Result 4 (Self-testing). Under the Assumptions 1 and 2 stated in Sec. 1, if a quantum state $|\psi\rangle\in\mathcal{H}$ and a set of $n$ (where $n=2^{m}+1,m\in\mathbbm{N}\setminus\\{1\\})$ measurements $A_{i}$ acting on $\mathcal{H}$ violate the inequality (43) maximally, then there exists a projection $P:\mathcal{H}\to\mathbbm{C}^{3}$ and a unitary $U$ acting on $\mathbbm{C}^{3}$ such that $\displaystyle U(PA_{i}P^{\dagger})U^{\dagger}=2|\widehat{v}_{i}\rangle\\!\langle\widehat{v}_{i}|-\mathbbm{1}_{3},$ $\displaystyle\quad U(P|\psi\rangle)=(1,0,0)^{T},$ (46) where $|\widehat{v}_{i}\rangle$ are defined in (10). ###### Proof. Taking the expectation value of the state $|\psi\rangle$ on both side of the SOS decomposition (44) of $\mathcal{B}$, we obtain by virtue of (14) that for any $i$ and $k$, $M_{i,k}|\psi\rangle=|\psi\rangle.$ (47) In the particular $k=1$ case this condition when combined with the explicit form of $M_{i,1}$ given in Eq. (24) together with the fact that $\beta_{1}=\alpha/(1+2\alpha)$, leads to the following relations for all $i=1,\dots,n$, $(A_{i}+\alpha A_{i+1}+\alpha A_{i-1})|\psi\rangle=(1-2\alpha)|\psi\rangle.$ (48) Similarly, from the second term of the SOS decomposition (44) we get that for all $i=1,\ldots,n$, $A^{2}_{i}|\psi\rangle=|\psi\rangle.$ (49) Additionally, due to the last two terms of SOS decomposition (44) we know $K_{i}K_{i\pm 1}|\psi\rangle=0$ for all $i$, which using Assumption 2 read $\sqrt{F_{i}}\sqrt{F_{i\pm 1}}|\psi\rangle=0$. Since, $\sqrt{F_{i}}$ and $F_{i}$ have the same support, this conditions further imply the following relations for all $i=1,\dots,n,$ $\displaystyle F_{i}F_{i\pm 1}|\psi\rangle=0.$ (50) Given the relations (48), (49) and (50), the next Theorem provides the proof for the self-testing statement. ∎ The self-testing property implies our modified inequality (43) are non-trivial since any classical value assignment is not equivalent to the realization given in (4). ###### Theorem. If a set of Hermitian operators $\\{A_{i}\\}^{n}_{i=1}$ (where $n$ is odd) of the form (2) acting on arbitrary finite-dimensional Hilbert space $\mathcal{H}$ and a unit vector $|\psi\rangle\in\mathcal{H}$ satisfy the relations (48),(49) and (50), then there exists a projection operator $P:\mathcal{H}\to\mathbbm{C}^{3}$ and a unitary $U$ acting on $\mathbbm{C}^{3}$ such that (4) holds true. ###### Proof. We prove this theorem in two steps. Step 1. In the first step, we deduce the effective dimensionality of the observables $A_{i}$ and the state $|\psi\rangle$. Let us define a vector space $V=\text{Span}\\{|\psi\rangle,A_{1}|\psi\rangle,A_{3}|\psi\rangle\\}.$ Due to Lemma 1 (stated in Appendix B), it suffices to consider the observables $A_{i}$ and the state $|\psi\rangle$ restricted to $V$. In other words, Lemma 1 points out that the Hilbert space $\mathcal{H}$ can be decomposed as $V\oplus V^{\bot}$ and all the operators $A_{i}$ have the following block structure $A_{i}=\left(\begin{array}[]{@{}c|c@{}}\tilde{A}_{i}&\mathbb{O}\\\ \hline\cr\mathbb{O}&A^{\prime}_{i}\end{array}\right),$ (51) wherein $\tilde{A}_{i},A^{\prime}_{i}$ are acting on $V,V^{\bot}$ respectively. This allows us to define $\displaystyle\tilde{A}_{i}=PA_{i}P^{\dagger}=2\tilde{F}_{i}-\mathbbm{1},$ $\displaystyle|\tilde{\psi}\rangle=P|\psi\rangle,$ (52) where $P$ is the projection operator from $\mathcal{H}$ to $V$, $\tilde{F}_{i}=PF_{i}P^{\dagger}\geqslant 0$ and $\mathbbm{1}$ is the identity operator acting on $V$. It follows from Eq. (2) and Eqs. (50), (48), (49) that the projected measurements $\tilde{F}_{i}$ and the state $|\tilde{\psi}\rangle$ satisfy the following sets of relations for all $i=1,\dots,n$, $\displaystyle\quad\tilde{F}_{i}\tilde{F}_{i\pm 1}|\tilde{\psi}\rangle=0,$ (53) $\displaystyle\left(\tilde{F}_{i}+\alpha\tilde{F}_{i-1}+\alpha\tilde{F}_{i+1}\right)|\tilde{\psi}\rangle=|\tilde{\psi}\rangle,$ (54) $\displaystyle\tilde{F}^{2}_{i}|\tilde{\psi}\rangle=\tilde{F}_{i}|\tilde{\psi}\rangle.$ (55) Step 2. In the second step, we characterize the observables $\tilde{A}_{i}$. With the help of Lemma 2 given in Appendix B, we first show that all observables $\tilde{A}_{i}$ are of the form $\tilde{A}_{i}=2|v_{i}\rangle\\!\langle v_{i}|-\mathbbm{1}$ (56) for some normalized vectors $|v_{i}\rangle\in\mathbbm{C}^{3}$ such that $\langle v_{i}|v_{i\pm 1}\rangle=0$. The remaining part is the characterization of $|v_{i}\rangle$. By plugging Eq. (56) into Eq. (54) we obtain that for all $i$, $(|v_{i}\rangle\\!\langle v_{i}|+\alpha|v_{i-1}\rangle\\!\langle v_{i-1}|+\alpha|v_{i+1}\rangle\\!\langle v_{i+1}|)|\tilde{\psi}\rangle=|\tilde{\psi}\rangle.$ (57) We use the fact that $|v_{i}\rangle,|v_{i\pm 1}\rangle$ are orthogonal and multiply $\langle v_{i-1}|$ and $\langle v_{i+1}|$ with Eq. (57), which lead us to the following equations $\alpha\langle v_{i-1}|v_{i+1}\rangle\langle v_{i+1}|\tilde{\psi}\rangle=(1-\alpha)\langle v_{i-1}|\tilde{\psi}\rangle$ (58) and $\alpha\langle v_{i+1}|v_{i-1}\rangle\langle v_{i-1}|\tilde{\psi}\rangle=(1-\alpha)\langle v_{i+1}|\tilde{\psi}\rangle$ (59) for all $i$. By substituting the term $\langle v_{i-1}|\tilde{\psi}\rangle$ from the first equation to the second one, we arrive at the following conditions $\forall i,\quad|\langle v_{i-1}|v_{i+1}\rangle|=\frac{1-\alpha}{\alpha}.$ (60) Note that, here we use the fact that $\langle v_{i+1}|\tilde{\psi}\rangle\neq 0$ 555If ${\langle v_{j+1}|\tilde{\psi}\rangle}=0$ for some $j$, then (58) implies ${\langle v_{j-1}|\tilde{\psi}\rangle}$ is also 0, and further (57) implies ${|v_{j}\rangle\\!\langle v_{j}|\tilde{\psi}\rangle=|\tilde{\psi}\rangle}$. Substituting these in (57) taking $i=j+1$, we arrive at a relation ${|v_{j+2}\rangle\\!\langle v_{j+2}|\tilde{\psi}\rangle=(1-\alpha)/\alpha|\tilde{\psi}\rangle}$ which cannot be true for any finite $n$ since ${|v_{j+2}\rangle\\!\langle v_{j+2}|}$ has eigenvalues 1,0.. Considering the absolute value of both side of (59) and using (60) we obtain another set of conditions $\forall i,\quad|\langle\tilde{\psi}|v_{i-1}\rangle|=|\langle\tilde{\psi}|v_{i+1}\rangle|.$ (61) And since $n$ is odd, as a consequence of the above equation, $\forall i,j,\quad|\langle\tilde{\psi}|v_{i}\rangle|=|\langle\tilde{\psi}|v_{j}\rangle|.$ (62) Let us try to see what is the most general form of $|v_{i}\rangle$ compatible with the above conditions. First let us exploit the fact that observed probabilities do not change if we rotate the state and measurements by a unitary operation. We thus choose it so that $U|\tilde{\psi}\rangle=(1,0,0)^{T}\equiv|0\rangle$. We also notice that any unitary of the following form $\left(\begin{array}[]{cc}1&0\\\ 0&U^{\prime}\end{array}\right)$ (63) with $U^{\prime}$ being any $2\times 2$ unitary does not change $|0\rangle$. Later we will use this freedom. Due to the fact that we are characterizing projectors $|v_{i}\rangle\\!\langle v_{i}|$ rather than the vectors themselves, we can always assume the first element of the vector is positive, that is, $|v_{i}\rangle$ has the form, $|v_{i}\rangle=\left(\cos\theta_{i},e^{\mathbbm{i}a_{i}}\sin\theta_{i}\sin\phi_{i},e^{\mathbbm{i}b_{i}}\sin\theta_{i}\cos\phi_{i}\right)^{T}.$ (64) The condition (62) implies that all $\cos\theta_{i}$ are equal and therefore let us denote $\theta_{i}=\theta$. Plugging these forms of $|v_{i}\rangle$ and $|\tilde{\psi}\rangle=|0\rangle$ into Eq. (57), the first element of the vector equation leads to $\cos\theta=\frac{1}{\sqrt{1+2\alpha}}.$ (65) Next, we use the freedom given by (63) to bring one of the vectors, say $|v_{n}\rangle$, to $(\cos\theta,0,\sin\theta)^{T}$ by taking $\sin\phi_{n}=0,\quad e^{\mathbbm{i}b_{n}}=1.$ (66) Then, due to the condition $\langle v_{1}|v_{n}\rangle=\langle v_{n-1}|v_{n}\rangle=0$ we infer $e^{\mathbbm{i}b_{1}},e^{\mathbbm{i}b_{n-1}}$ are real and without loss of generality we can take $e^{\mathbbm{i}b_{1}}=e^{\mathbbm{i}b_{n-1}}=1$ (67) by absorbing the sign in $\cos\phi_{1},\cos\phi_{n-1}$. Further, we can get rid one of the phases in $|v_{1}\rangle$, that is, $e^{\mathbbm{i}a_{1}}=1,$ (68) and take $\sin(\phi_{1})$ to be non-negative by applying another unitary of the form (63), $U^{\prime}=\mathrm{diag}[\pm\exp(-\mathbbm{i}a_{1}),1]$ (69) that does not change the simplified form of $|v_{n}\rangle$. Equating the second and third element of the vector equation (57) for $|\tilde{\psi}\rangle=|0\rangle$, we obtain the relations $e^{\mathbbm{i}a_{i}}\sin\phi_{i}+\alpha e^{\mathbbm{i}a_{i-1}}\sin\phi_{i-1}+\alpha e^{\mathbbm{i}a_{i+1}}\sin\phi_{i+1}=0,$ (70) and $e^{\mathbbm{i}b_{i}}\cos\phi_{i}+\alpha e^{\mathbbm{i}b_{i-1}}\cos\phi_{i-1}+\alpha e^{\mathbbm{i}b_{i+1}}\cos\phi_{i+1}=0.$ (71) With the aid of (66) and (68), Eq. (70) for $i=n$ points out $\sin(\phi_{1})=-e^{\mathbbm{i}a_{n-1}}\sin(\phi_{n-1})$ which allows us to consider $e^{\mathbbm{i}a_{n-1}}=1$. Taking $i=1$ in Eqs. (70) and (71) and replacing the values of $\sin\phi_{n},\cos\phi_{n},e^{\mathbbm{i}a_{1}},e^{\mathbbm{i}b_{1}},e^{\mathbbm{i}b_{n}}$ we obtain, $\displaystyle\sin\phi_{1}+\alpha e^{\mathbbm{i}a_{2}}\sin\phi_{2}=0,$ (72) $\displaystyle\cos\phi_{1}+\alpha+\alpha e^{\mathbbm{i}b_{2}}\cos\phi_{2}=0.$ (73) Thus, $e^{\mathbbm{i}a_{2}},e^{\mathbbm{i}b_{2}}$ are real and can be taken to be 1. Note, here we use the fact that $\sin{\phi_{1}}\neq 0$ 666If $\sin{\phi_{1}}=0$, then $\cos{\phi_{1}}=\pm 1$ and consequently ${\langle v_{n}|v_{1}\rangle=\cos{(\theta\mp\theta)}}$ which contradicts the relation ${\langle v_{n}|v_{1}\rangle=0}$. Analogously, if we suppose $\cos{\phi_{2}}=0$, then $\cos{\phi_{1}}+\alpha=0$ and $\sin\phi_{2}=\pm 1$. Now, the first equation holds only if $2\alpha^{2}=1$.. Similarly, by taking $i=2,\dots,n-2$ we conclude for all $i$ $e^{\mathbbm{i}a_{i}}=e^{\mathbbm{i}b_{i}}=1.$ (74) On the other hand, the condition $\langle v_{i}|v_{i+1}\rangle=0$ implies, $\displaystyle\phi_{i+1}-\phi_{i}$ $\displaystyle=$ $\displaystyle\cos^{-1}\left(-\frac{\cos^{2}\theta}{\sin^{2}\theta}\right)$ (75) $\displaystyle=$ $\displaystyle\frac{(n-1)\pi}{n}.$ Finally, considering $i=n$ in the above Eq. (75) and using $\sin\phi_{n}=0$ we deduce $\phi_{1}=(n-1)\pi/n$. We discard the possibility $\phi_{1}=-(n-1)\pi/n$ since $\sin\phi_{1}$ is taken to be non-negative. Thus, the equations (65), (74), and (75) together with $\phi_{1}$ establish that the unknown vectors $|v_{i}\rangle$ in (64) are unitarily equivalent to $|\widehat{v}_{i}\rangle$. This completes the proof. ∎ ## 4 Conclusion Kochen-Specker contextuality captures the intrinsic nature of quantum theory that essentially departs from classicality. It also offers a generalization of quantum correlations beyond nonlocality to a larger class of quantum systems and minimizes the demands to test non-classicality. Therefore, it is a fundamental problem to understand what is the maximal information about the underlying quantum system that can be inferred from the correlations observed in a contextuality experiment, and whether this information can be used for certification of quantum devices from minimal assumptions of their internal functioning. In this work, we derive self-testing statements for $n$-cycle scenario using weaker assumptions than those made in previous approaches based on Kochen- Specker contextuality [CSW14, BRV+19b, IMOK20, BRV+19a]. In particular, we do not assume orthogonality relations between measurement effects and projectivity of the measurements. Instead, we consider general two-outcome measurements which nevertheless obey a single assumption (Assumption 2) that the Kraus operators representing one of the outcomes are uniquely determined by their corresponding measurement effects. While ideal measurements are the prerequisite for noncontextuality, our self-testing statement holds beyond such idealizations inferring nonclassicality from the correlations obtained in sequential measurements. Moreover, we take a different approach, that is, we use the sum-of-squares ’technique’ that has successfully been used in the Bell scenario to derive maximal quantum violation of certain Bell inequalities as well as in making self-testing statements [BP15, ŠASA16, SAT+17, KŠT+19, SSKA19, CMMN19, Kan19, ASTA19], but has never been explored for self-testing in the contextuality scenario. We further remark that self-testing from quantum contextuality is not fully device-independent as far as its original definition is concerned, while, its experimental test does not require space-like separation. The assumption 1 is critical to verify for practical purposes, however, in future studies, one may try to overcome it by restricting the computational power or the memory of the measurement device. Nonetheless, it is way more powerful than the usual process of tomography. It is also distinct from the self-testing approach in prepare-and-measure scenario [TKV+18, FK19] since no restriction on the dimensionality of the preparation is imposed here. Although the SOS decompositions hold for a certain number of measurements, a suitable adaptation of our approach in future studies may lead to SOS decompositions for an arbitrary odd number of measurements. Another direction for further study is to explore whether our approach can be applied to states and measurements of higher dimension than three and whether our self-testing statements can be made robust to experimental imperfections. 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Let us assume that these operators are in the following form $\widehat{M}_{i,k}=a\widehat{A}_{i}+b\widehat{A}_{i+k}+b^{\prime}\widehat{A}_{i-k},$ (76) where the coefficients $a$, $b$ and $b^{\prime}$ are to be determined as a solution to the equation $(a\widehat{A}_{i}+b\widehat{A}_{i+k}+b^{\prime}\widehat{A}_{i-k})|\widehat{\psi}\rangle=|\widehat{\psi}\rangle,$ (77) and $|\widehat{\psi}\rangle,\widehat{A}_{i}$ are given in Eqs. (8)-(9). To solve the above we first notice the following relation, $\widehat{A}_{i}|\widehat{\psi}\rangle=(\cos{2\theta},\sin{2\theta}\sin{\phi_{i}},\sin{2\theta}\cos{\phi_{i}})^{T},$ (78) which when substituted into Eq. (77) leads one to a system of equations $\begin{bmatrix}a(1+\frac{b}{a}+\frac{b^{\prime}}{a})\cos{2\theta}\\\\[4.30554pt] a\sin{2\theta}\big{(}\sin{\phi_{i}}+\frac{b}{a}\sin{\phi_{i+k}}+\frac{b^{\prime}}{a}\sin{\phi_{i-k}}\big{)}\\\\[4.30554pt] a\sin{2\theta}\big{(}\cos{\phi_{i}}+\frac{b}{a}\cos{\phi_{i+k}}+\frac{b^{\prime}}{a}\cos{\phi_{i-k}}\big{)}\\\ \end{bmatrix}=\begin{bmatrix}1\\\ 0\\\ 0\\\ \end{bmatrix}.$ (79) Assuming that $a\neq 0$ and taking into account that $\sin{2\theta}\neq 0$, the last two equations in the above system can be rewritten as $\displaystyle\begin{bmatrix}\sin{\phi_{i}}&\sin{\phi_{i+k}}&\sin{\phi_{i-k}}\\\\[4.30554pt] \cos{\phi_{i}}&\cos{\phi_{i+k}}&\cos{\phi_{i-k}}\\\ \end{bmatrix}\begin{bmatrix}1\\\ b/a\\\ b^{\prime}/a\end{bmatrix}=\begin{bmatrix}0\\\ 0\end{bmatrix}.$ (80) After multiplying the above equation from left by $\begin{bmatrix}\sin{\phi_{i}}&\cos{\phi_{i}}\\\ \cos{\phi_{i}}&-\sin{\phi_{i}}\\\ \end{bmatrix}$ (81) and using the fact $\phi_{i+k}-\phi_{i}=\phi_{k}$, Eq. (80) simplifies to, $\displaystyle\begin{bmatrix}1&\cos{\phi_{k}}&\cos{\phi_{k}}\\\ 0&\sin{\phi_{k}}&-\sin{\phi_{k}}\\\ \end{bmatrix}\begin{bmatrix}1\\\ b/a\\\ b^{\prime}/a\end{bmatrix}=\begin{bmatrix}0\\\ 0\end{bmatrix}.$ (82) In this way we remark that the dependence of $i$ in (80) disappears and the system of equations (82) imply $\frac{b}{a}=\frac{b^{\prime}}{a}=-\frac{1}{2}\sec{\phi_{k}}.$ (83) Substitution of above in the first vector equality of (79) leads to $a=\frac{1}{(1-\sec{\phi_{k}})(2\cos^{2}{\theta}-1)},$ (84) and thus, we obtain a unique solution of $a,b,b^{\prime}$. Finally, substituting $a,b,b^{\prime}$ into Eq. (77) we can conveniently state $\widehat{M}_{i,k}$ operators in the following way $\displaystyle\widehat{M}_{i,k}:=\left(\frac{1+2\alpha}{1-2\alpha}\right)\left[(1-2\beta_{k})\widehat{A}_{i}+\beta_{k}(\widehat{A}_{i+k}+\widehat{A}_{i-k})\right],$ where $\beta_{k}=\frac{1}{2(1-\cos{\phi_{k}})},\qquad\alpha=\frac{1}{2}\sec\left(\frac{\pi}{n}\right).$ (86) ## Appendix B Lemma 1-2 In this appendix, we provide two Lemmas that are used in the proof of the Theorem. ###### Lemma 1. If a set of Hermitian operators $\\{A_{i}\\}^{n}_{i=1}$ (where $n$ is odd) of the form (2) and a vector $|\psi\rangle$ satisfy the relations (50), (48) and (49), then the vector space $V=\mathrm{span}\\{|\psi\rangle,A_{1}|\psi\rangle,A_{3}|\psi\rangle\\}$ (87) is invariant under the algebra generated by $A_{i}$. ###### Proof. To prove this statement it suffices to show that $A_{i}|\psi\rangle$ for all $i=1,\ldots,n$ as well as all $A_{i}A_{j}|\psi\rangle$ with $i\neq j$ can be expressed as linear combinations of the basis vectors $|\psi\rangle$, $A_{1}|\psi\rangle$ and $A_{3}|\psi\rangle$. Let us begin by noting that Eq. (48) for $i=2$ gives us directly such a linear combination for $A_{2}|\psi\rangle$ and so $A_{2}|\psi\rangle\in V$. Then, the fact that $A_{i}|\psi\rangle\in V$ for $i=4,\ldots,n$ follows from Eq. (48); it is enough to rewrite the latter as $A_{i}|\psi\rangle=\frac{1-2\alpha}{\alpha}|\psi\rangle-\frac{1}{\alpha}A_{i-1}|\psi\rangle- A_{i-2}|\psi\rangle.$ (88) Let us now move on to showing that $A_{i}A_{j}|\psi\rangle\in V$ for all $i\neq j$. To this end, we first observe that using (50) we obtain $\displaystyle A_{i}A_{i\pm 1}|\psi\rangle$ $\displaystyle=$ $\displaystyle(2F_{i}-\mathbbm{1})(2F_{i\pm 1}-\mathbbm{1})|\psi\rangle$ (89) $\displaystyle=$ $\displaystyle-(A_{i}+A_{i\pm 1}+\mathbbm{1})|\psi\rangle,$ which due to the fact that $A_{i}|\psi\rangle\in V$, allows us to conclude that for all $i$, $A_{i}A_{i\pm 1}|\psi\rangle\in V$. Let us then consider the vectors $A_{i}A_{j}|\psi\rangle$ for pairs $i,j$ such that $|i-j|=2$. Using Eq. (49) and the fact $[A_{i},A_{i\pm 1}]|\psi\rangle=0$ which is a consequence of Eq. (50), we get $\displaystyle A_{i}A_{i\pm 2}|\psi\rangle$ $\displaystyle=$ $\displaystyle A_{i}A_{i\pm 2}(A_{i\pm 1})^{2}|\psi\rangle$ (90) $\displaystyle=$ $\displaystyle(A_{i}A_{i\pm 1})(A_{i\pm 1}A_{i\pm 2})|\psi\rangle.$ Since we have already shown $A_{i}A_{i\pm 1}|\psi\rangle\in V$, the above equation implies $A_{i}A_{i\pm 2}|\psi\rangle\in V$. Given that $A_{i}A_{j}|\psi\rangle\in V$ for $|i-j|=1$ and $|i-j|=2$ we can then prove, applying the same argument as above, that $A_{i}A_{j}|\psi\rangle$ belong to $V$ for any pair $i,j$ such that $|i-j|=3$. In fact, following this approach recursively we can prove that $A_{i}A_{j}|\psi\rangle\in V$ for $i,j$ such that $|i-j|=k$ with $k=3,\ldots,n-1$, which completes the proof. ∎ Let us remark that the subspace $V$ is in fact spanned by any triple of the vectors $|\psi\rangle$, $A_{i}|\psi\rangle$ and $A_{j}|\psi\rangle$ with $i\neq j$. This is a consequence of the fact that, as proven above, any vector $A_{i}|\psi\rangle$ is a linear combination of $|\psi\rangle$, $A_{1}|\psi\rangle$ and $A_{3}|\psi\rangle$. ###### Lemma 2. If a set of positive operators $\\{\tilde{F}_{i}\\}^{n}_{i=1}$ acting on $\mathbbm{C}^{3}$ and a vector $|\tilde{\psi}\rangle$ satisfy the relations (53), (54) and (55), then for each $i$ there exists a normalized vector $|v_{i}\rangle\in\mathbbm{C}^{3}$ such that $\tilde{F}_{i}=|v_{i}\rangle\\!\langle v_{i}|$ and, moreover, $\langle v_{i}|v_{i\pm 1}\rangle=0$. ###### Proof. Let us begin by showing that $\tilde{F}_{i}|\tilde{\psi}\rangle\neq 0$ for any $i$. Assume to this end that there exist $j$ such that $\tilde{F}_{j}|\tilde{\psi}\rangle=0$. Using then Eq. (54) for $i=j-1$ we arrive at $(\tilde{F}_{j-1}+\alpha\tilde{F}_{j-2})|\tilde{\psi}\rangle=|\tilde{\psi}\rangle.$ (91) After applying $\tilde{F}_{j-2}$ to both sides of this equation and using Eq. (53), we obtain $\alpha\tilde{F}^{2}_{j-2}|\tilde{\psi}\rangle=\tilde{F}_{j-2}|\tilde{\psi}\rangle$ which is consistent with Eq. (55) if and only if $\tilde{F}_{j-2}|\tilde{\psi}\rangle=0$. Therefore, due to Eq. (91) we have $\tilde{F}_{j-1}|\tilde{\psi}\rangle=|\tilde{\psi}\rangle$. Again, substituting these relations in (54) taking $i=j$, we arrive at $\tilde{F}_{j+1}|\tilde{\psi}\rangle=[(1-\alpha)/\alpha]|\tilde{\psi}\rangle$ which contradicts Eq. (55). Let us now show that all the operators $\tilde{F}_{i}$ are of rank one. We first prove that none of them can be of rank three. Assume for this purpose that $\mathrm{rank}(\tilde{F}_{j})=3$ for some $j$. Then, the condition (55) gives $\tilde{F}_{j}|\tilde{\psi}\rangle=|\tilde{\psi}\rangle$. This, after taking into account that $\tilde{F}_{j+1}\tilde{F}_{j}|\tilde{\psi}\rangle=0$ implies $\tilde{F}_{j+1}|\tilde{\psi}\rangle=0$ which contradicts the fact $\tilde{F}_{i}|\tilde{\psi}\rangle\neq 0$ for all $i$, as shown before. Let us then prove that none of $\tilde{F}_{i}$ can be of rank two. To this end, assume that there is $j$ such that $\mathrm{rank}(\tilde{F}_{j})=2$ and consider the eigendecomposition of $\tilde{F}_{j}$, $\tilde{F}_{j}=\lambda|1\rangle\\!\langle 1|+\lambda^{\prime}|2\rangle\\!\langle 2|,$ (92) where $|1\rangle,|2\rangle,|3\rangle$ are the eigenvectors, forming an orthonormal basis in $\mathbbm{C}^{3}$, whereas $\lambda,\lambda^{\prime}\in(0,1]$ are the eigenvalues. Subsequently, $|\tilde{\psi}\rangle$ can be expressed as $|\tilde{\psi}\rangle=x_{1}|1\rangle+x_{2}|2\rangle+x_{3}|3\rangle$ (93) for some $x_{1},x_{2},x_{3}\in\mathbbm{C}$. It follows from Eq. (55) that $\lambda=1\quad\text{ or }\quad x_{1}=0,$ (94) and $\lambda^{\prime}=1\quad\text{ or }\quad x_{2}=0.$ (95) Note that $x_{1}=x_{2}=0$ is not possible since it requires $F_{j}|\tilde{\psi}\rangle=0$. Similarly, $x_{3}\neq 0$ as otherwise $F_{j}|\tilde{\psi}\rangle=|\tilde{\psi}\rangle$ which implies $F_{j\pm 1}|\tilde{\psi}\rangle=0$. Now, employing the fact that $\tilde{F}_{j}$ is supported on $\mathrm{span}\\{|1\rangle,|2\rangle\\}$, it follows from the condition $\tilde{F}_{j}\tilde{F}_{j\pm 1}|\tilde{\psi}\rangle=0$ that $\tilde{F}_{j\pm 1}|\tilde{\psi}\rangle=q_{3,\pm}|3\rangle$ for some $q_{3,\pm}\in\mathbbm{C}$. By combining this with (55) we find that $\tilde{F}_{j\pm 1}|3\rangle=|3\rangle,$ (96) that is, $|3\rangle$ is the eigenvector of $\tilde{F}_{j\pm 1}$ with eigenvalue one, which, due to the fact that $\tilde{F}_{j\pm 1}\leqslant\mathbbm{1}$, implies that $\tilde{F}_{j\pm 1}$ decompose as $\tilde{F}_{j\pm 1}=\tilde{F}_{j\pm 1}^{\prime}+|3\rangle\\!\langle 3|$ (97) with $\tilde{F}_{j\pm 1}^{\prime}$ being positive matrices supported on $\mathrm{span}\\{|1\rangle,|2\rangle\\}$. By finally plugging Eqs. (92) and (97) into Eq. (54) and projecting the obtained equation onto $|3\rangle$ we see that $2\alpha=1$, which is not satisfied for any $n$. As a result all the operators $\tilde{F}_{i}$ are of rank one and therefore they can be expressed as $\tilde{F}_{i}=\lambda_{i}|v_{i}\rangle\\!\langle v_{i}|$ (98) for some $\lambda_{i}\in(0,1]$ and $|v_{i}\rangle\in\mathbbm{C}^{3}$. The condition (55) holds only if $\lambda_{i}=1$. Furthermore, since $F_{i}|\tilde{\psi}\rangle\neq 0$, Eq. (53) implies $\langle v_{i}|v_{i\pm 1}\rangle=0$. This completes the proof. ∎

2024-09-04T02:54:55.176078

2002.12229

arxiv-papers

# Woodbury Transformations for Deep Generative Flows You Lu Department of Computer Science Virginia Tech Blacksburg, VA <EMAIL_ADDRESS> &Bert Huang Department of Computer Science Tufts University Medford, MA <EMAIL_ADDRESS> ###### Abstract Normalizing flows are deep generative models that allow efficient likelihood calculation and sampling. The core requirement for this advantage is that they are constructed using functions that can be efficiently inverted and for which the determinant of the function’s Jacobian can be efficiently computed. Researchers have introduced various such flow operations, but few of these allow rich interactions among variables without incurring significant computational costs. In this paper, we introduce _Woodbury transformations_ , which achieve efficient invertibility via the Woodbury matrix identity and efficient determinant calculation via Sylvester’s determinant identity. In contrast with other operations used in state-of-the-art normalizing flows, Woodbury transformations enable (1) high-dimensional interactions, (2) efficient sampling, and (3) efficient likelihood evaluation. Other similar operations, such as 1x1 convolutions, emerging convolutions, or periodic convolutions allow at most two of these three advantages. In our experiments on multiple image datasets, we find that Woodbury transformations allow learning of higher-likelihood models than other flow architectures while still enjoying their efficiency advantages. ## 1 Introduction Deep generative models are powerful tools for modeling complex distributions and have been applied to many tasks such as synthetic data generation [27, 39], domain adaption [40], and structured prediction [33]. Examples of these models include autoregressive models [13, 28], variational autoencoders [21, 31], generative adversarial networks [11], and normalizing flows [6, 30, 7, 22]. Normalizing flows are special because of two advantages: They allow efficient and exact computation of log-likelihood and sampling. Flow-based models are composed of a series of invertible functions, which are specifically designed so that their inverse and determinant of the Jacobian are easy to compute. However, to preserve this computational efficiency, these functions usually cannot sufficiently encode dependencies among dimensions of a variable. For example, affine coupling layers [6] split a variable to two parts and require the second part to only depend on the first. But they ignore the dependencies among dimensions in the second part. To address this problem, Dinh et al. [6, 7] introduced a fixed permutation operation that reverses the ordering of the channels of pixel variables. Kingma and Dhariwal [22] introduced a 1$\times$1 convolution, which are a generalized permutation layer, that uses a weight matrix to model the interactions among dimensions along the channel axis. Their experiments demonstrate the importance of capturing dependencies among dimensions. Relatedly, Hoogeboom et al. [15] proposed emerging convolution operations, and Hoogeboom et al. [15] and Finz et al. [9] proposed periodic convolution. These two convolution layers have $d\times d$ kernels that can model dependencies along the spatial axes in addition to the channel axis. However, the increase in representational power comes at a cost: These convolution operations do not scale well to high-dimensional variables. The emerging convolution is a combination of two autoregressive convolutions [10, 23], whose inverse is not parallelizable. To compute the inverse or determinant of the Jacobian, the periodic convolution requires transforming the input and the convolution kernel to Fourier space. This transformation is computationally costly. In this paper, we develop _Woodbury transformations_ for generative flows. Our method is also a generalized permutation layer and uses spatial and channel transformations to model dependencies among dimensions along spatial and channel axes. We use the Woodbury matrix identity [37] and Sylvester’s determinant identity [35] to compute the inverse and Jacobian determinant, respectively, so that both the training and sampling time complexities are linear to the input variable’s size. We also develop a memory-efficient variant of the Woodbury transformation, which has the same advantage as the full transformation but uses significantly reduced memory when the variable is high-dimensional. In our experiments, we found that Woodbury transformations enable model quality comparable to many state-of-the-art flow architectures while maintaining significant efficiency advantages. ## 2 Deep Generative Flows In this section, we briefly introduce the deep generative flows. More background knowledge can be found in the appendix. A normalizing flow [30] is composed of a series of invertible functions $\mathbf{f}=\mathbf{f}_{1}\circ\mathbf{f}_{2}\circ...\circ\mathbf{f}_{K}$, which transform $\mathbf{x}$ to a latent code $\mathbf{z}$ drawn from a simple distribution. Therefore, with the _change of variables_ formula, we can rewrite the log-likelihood $\log p_{\theta}(\mathbf{x})$ to be $\log p_{\theta}(\mathbf{x})=\log p_{Z}(\mathbf{z})+\sum_{i=1}^{K}\log\left|\det\left(\frac{\partial\mathbf{f}_{i}}{\partial\mathbf{r}_{i-1}}\right)\right|,$ (1) where $\mathbf{r}_{i}=\mathbf{f}_{i}(\mathbf{r}_{i-1})$, $\mathbf{r}_{0}=\mathbf{x}$, and $\mathbf{r}_{K}=\mathbf{z}$. Flow-based generative models [6, 7, 22] are developed on the theory of normalizing flows. Each transformation function used in the models is a specifically designed neural network that has a tractable Jacobian determinant and inverse. We can sample from a trained flow $\mathbf{f}$ by computing $\mathbf{z}\sim p_{Z}(\mathbf{z}),\quad\mathbf{x}=\mathbf{f}^{-1}(\mathbf{z})$. There have been many operations, i.e., layers, proposed in recent years for generative flows. In this section, we discuss some commonly used ones, and more related works will be discussed in Section 4. Actnorm layers [22] perform per-channel affine transformations of the activations using scale and bias parameters to improve training stability and performance. The actnorm is formally expressed as $\mathbf{y}_{:,i,j}=\mathbf{s}\odot\mathbf{x}_{:,i,j}+\mathbf{b}$, where both the input $\mathbf{x}$ and the output $\mathbf{y}$ are $c\times h\times w$ tensors, $c$ is the channel dimension, and $h\times w$ are spatial dimensions. The parameters $\mathbf{s}$ and $\mathbf{b}$ are $c\times 1$ vectors. Affine coupling layers [6, 7] split the input $\mathbf{x}$ into two parts, $\mathbf{x}_{a},\mathbf{x}_{b}$. And then fix $\mathbf{x}_{a}$ and force $\mathbf{x}_{b}$ to only relate to $\mathbf{x}_{a}$, so that the Jacobian is a triangular matrix. Formally, we compute $\displaystyle\mathbf{x}_{a},\mathbf{x}_{b}=\text{split}(\mathbf{x}),\quad\quad\quad\quad~{}~{}\mathbf{y}_{a}=\mathbf{x}_{a},$ $\displaystyle\mathbf{y}_{b}=\mathbf{s}(\mathbf{x}_{a})\odot\mathbf{x}_{b}+\mathbf{b}(\mathbf{x}_{a}),\quad\mathbf{y}=\text{concat}(\mathbf{y}_{a},\mathbf{y}_{b}),$ where $\mathbf{s}$ and $\mathbf{b}$ are two neural networks with $\mathbf{x}_{a}$ as input. The split and the concat split and concatenate the variables along the channel axis. Usually, $s$ is restricted to be positive. An additive coupling layer is a special case when $\mathbf{s}=\mathbf{1}$. Actnorm layers only rescale the dimensions of $\mathbf{x}$, and affine coupling layers only relate $\mathbf{x}_{b}$ to $\mathbf{x}_{a}$ but omit dependencies among different dimensions of $\mathbf{x}_{b}$. Thus, we need other layers to capture local dependencies among dimensions. Invertible convolutional layers [22, 15, 9] are generalized permutation layers that can capture correlations among dimensions. The 1$\times$1 convolution [22] is $\mathbf{y}_{:,i,j}=\mathbf{Mx}_{:,i,j}$, where $\mathbf{M}$ is a $c\times c$ matrix. The Jacobian of a 1$\times$1 convolution is a block diagonal matrix, so that its log-determinant is $hw\log|\det(\mathbf{M})|$. Note that the 1$\times$1 convolution only operates along the channel axis and ignores the dependencies along the spatial axes. (a) Flow step (b) Multi-scale arch. Figure 1: Overview of architecture of generative flows. We can design the flow step by selecting a suitable convolutional layer and a coupling layer based on the task. Glow [22] uses 1$\times$1 convolutions and affine coupling. Emerging convolutions [15] combine two autoregressive convolutions [10, 23]. Each autoregressive convolution masks out some weights to force an autoregressive structure, so that the Jacobian is a triangular matrix and computing its determinant is efficient. One problem of emerging convolution is the computation of inverse is non-parallelizable, so that is inefficent for high-dimensional variables. Periodic convolutions [15, 9] transform the input and kernel to the Fourier domain using discrete Fourier transformations, so the convolution function is an element-wise matrix product with a block-diagonal Jacobian. The computational cost of periodic convolutions is $\mathcal{O}(chw\log(hw)+c^{3}hw)$. Thus, when the input is high-dimensional, both training and sampling are expensive. Multi-scale architectures [7] compose flow layers to generate rich models, using _split layers_ to factor out variables and _squeeze layers_ to shuffle dimensions, resulting in an architecture with $K$ flow steps and $L$ levels. See Fig. 1. ## 3 Woodbury Transformations In this section, we introduce Woodbury transformations as an efficient means to model high-dimensional correlations. ### 3.1 Channel and Spatial Transformations Suppose we reshape the input $\mathbf{x}$ to be a $c\times n$ matrix, where $n=hw$. Then the 1$\times$1 convolution can be reinterpreted as a matrix transformation $\mathbf{y}=\mathbf{W}^{(c)}\mathbf{x},$ (2) where $\mathbf{y}$ is also a $c\times n$ matrix, and $\mathbf{W}^{(c)}$ is a $c\times c$ matrix. For consistency, we will call this a channel transformation. For each column $\mathbf{x}_{:,i}$, the correlations among channels are modeled by $\mathbf{W}^{(c)}$. However, the correlation between any two rows $\mathbf{x}_{:,i}$ and $\mathbf{x}_{:,j}$ is not captured. Inspired by Eq. 2, we use a spatial transformation to model interactions among dimensions along the spatial axis $\mathbf{y}=\mathbf{xW}^{(s)},$ (3) where $\mathbf{W}^{(s)}$ is an $n\times n$ matrix that models the correlations of each row $\mathbf{x}_{i,:}$. Combining Equation 2 and Equation 3, we have $\displaystyle\mathbf{x}_{c}=\mathbf{W}^{(c)}\mathbf{x},~{}~{}~{}~{}~{}~{}~{}~{}\mathbf{y}=\mathbf{x}_{c}\mathbf{W}^{(s)}.$ (4) For each dimension of output $\mathbf{y}_{i,j}$, we have $\mathbf{y}_{i,j}=\sum_{v=1}^{c}\left(\sum_{u=1}^{n}\mathbf{W}^{(c)}_{i,u}\cdot\mathbf{x}_{u,v}\right)\cdot\mathbf{W}^{(s)}_{v,j}$. Therefore, the spatial and channel transformations together can model the correlation between any pair of dimensions. However, in this preliminary form, directly using Eq. 4 is inefficient for large $c$ or $n$. First, we would have to store two large matrices $\mathbf{W}^{c}$ and $\mathbf{W}^{s}$, so the space cost is $\mathcal{O}(c^{2}+n^{2})$. Second, the computational cost of Eq. 4 is $\mathcal{O}(c^{2}n+n^{2}c)$—quadratic in the input size. Third, the computational cost of the Jacobian determinant is $\mathcal{O}(c^{3}+n^{3})$, which is far too expensive in practice. ### 3.2 Woodbury Transformations We solve the three scalability problems by using a low-rank factorization. Specifically, we define $\displaystyle\mathbf{W}^{(c)}=\mathbf{I}^{(c)}+\mathbf{U}^{(c)}\mathbf{V}^{(c)},~{}~{}~{}~{}~{}~{}~{}~{}\mathbf{W}^{(s)}=\mathbf{I}^{(s)}+\mathbf{U}^{(s)}\mathbf{V}^{(s)},$ where $\mathbf{I}^{(c)}$ and $\mathbf{I}^{(s)}$ are $c$\- and $n$-dimensional identity matrices, respectively. The matrices $\mathbf{U}^{c}$, $\mathbf{V}^{c}$, $\mathbf{U}^{s}$, and $\mathbf{V}^{s}$ are of size $c\times d_{c}$, $d_{c}\times c$, $n\times d_{s}$, and $d_{c}\times n$, respectively, where $d_{c}$ and $d_{s}$ are constant latent dimensions of these four matrices. Therefore, we can rewrite Equation 4 as $\displaystyle\mathbf{x}_{c}=(\mathbf{I}^{(c)}+\mathbf{U}^{(c)}\mathbf{V}^{(c)})\mathbf{x},~{}~{}~{}~{}~{}~{}~{}~{}\mathbf{y}=\mathbf{x}_{c}(\mathbf{I}^{(s)}+\mathbf{U}^{(s)}\mathbf{V}^{(s)}).$ (5) We call Eq. 5 the Woodbury transformation because the Woodbury matrix identity [37] and Sylvester’s determinant identity [35] allow efficient computation of its inverse and Jacobian determinant. Woodbury matrix identity.111A more general version replaces $\mathbf{I}^{(n)}$ and $\mathbf{I}^{(k)}$ with arbitrary invertible $n\times n$ and $k\times k$ matrices. But this simplified version is sufficient for our tasks. Let $\mathbf{I}^{(n)}$ and $\mathbf{I}^{(k)}$ be $n$\- and $k$-dimensional identity matrices, respectively. Let $\mathbf{U}$ and $\mathbf{V}$ be $n\times k$ and $k\times n$ matrices, respectively. If $\mathbf{I}^{(k)}+\mathbf{VU}$ is invertible, then $(\mathbf{I}^{(n)}+\mathbf{UV})^{-1}=\mathbf{I}^{(n)}-\mathbf{U}(\mathbf{I}^{k}+\mathbf{VU})^{-1}\mathbf{V}$. Sylvester’s determinant identity. Let $\mathbf{I}^{(n)}$ and $\mathbf{I}^{(k)}$ be $n$\- and $k$-dimensional identity matrices, respectively. Let $\mathbf{U}$ and $\mathbf{V}$ be $n\times k$ and $k\times n$ matrices, respectively. Then, $\det(\mathbf{I}^{(n)}+\mathbf{UV})=\det(\mathbf{I}^{(k)}+\mathbf{VU})$. Based on these two identities, we can efficiently compute the inverse and Jacobian determinant $\displaystyle\mathbf{x}_{c}=\mathbf{y}(\mathbf{I}^{(s)}-\mathbf{U}^{(s)}(\mathbf{I}^{(d_{s})}+\mathbf{V}^{(s)}\mathbf{U}^{(s)})^{-1}\mathbf{V}^{(s)}),$ $\displaystyle\mathbf{x}=(\mathbf{I}^{(c)}-\mathbf{U}^{(c)}(\mathbf{I}^{(d_{c})}+\mathbf{V}^{(c)}\mathbf{U}^{(c)})^{-1}\mathbf{V}^{(c)})\mathbf{x}_{c},$ (6) and $\displaystyle\log\left|\det\left(\frac{\partial\mathbf{y}}{\partial\mathbf{x}}\right)\right|$ $\displaystyle=n\log\left|\det(\mathbf{I}^{(d_{c})}+\mathbf{V}^{(c)}\mathbf{U}^{(c)})\right|+c\log\left|\det(\mathbf{I}^{(d_{s})}+\mathbf{V}^{(s)}\mathbf{U}^{(s)})\right|,$ (7) where $\mathbf{I}^{(d_{c})}$ and $\mathbf{I}^{(d_{s})}$ are $d_{c}$\- and $d_{s}$-dimensional identity matrices, respectively. A Woodbury transformation is also a generalized permutation layer. We can directly replace an invertible convolution in Figure 1 with a Woodbury transformation. In contrast with 1$\times$1 convolutions, Woodbury transformations are able to model correlations along both channel and spatial axes. We illustrate this in Figure 2. To implement Woodbury transformations, we need to store four weight matrices, i.e., $\mathbf{U}^{(c)},\mathbf{U}^{(s)},\mathbf{V}^{(c)}$, and $\mathbf{V}^{(s)}$. To simplify our analysis, let $d_{c}\leq d$ and $d_{s}\leq d$, where $d$ is a constant. This setting is also consistent with our experiments. The size of $\mathbf{U}^{(c)}$ and $\mathbf{V}^{(c)}$ is $\mathcal{O}(dc)$, and the size of $\mathbf{U}^{(c)}$ and $\mathbf{V}^{(c)}$ is $\mathcal{O}(dn)$. The space complexity is $\mathcal{O}(d(c+n))$. For training and likelihood computation, the main computational bottleneck is computing $\mathbf{y}$ and the Jacobian determinant. To compute $\mathbf{y}$ with Equation 4, we need to first compute the channel transformation and then compute the spatial transformation. The computational complexity is $\mathcal{O}(dcn)$. To compute the determinant with Equation 7, we need to first compute the matrix product of $\mathbf{V}$ and $\mathbf{U}$, and then compute the determinant. The computational complexity is $\mathcal{O}(d^{2}(c+n)+d^{3})$. For sampling, we need to compute the inverse transformations, i.e., Equation 6. With the Woodbury identity, we actually only need to compute the inverses of $\mathbf{I}^{(d_{s})}+\mathbf{V}^{(s)}\mathbf{U}^{(s)}$ and $\mathbf{I}^{(d_{c})}+\mathbf{V}^{(c)}\mathbf{U}^{(c)}$, which are computed with time complexity $\mathcal{O}(d^{3})$. To implement the inverse transformations, we can compute the matrix chain multiplication, so we can avoid computing the product of two large matrices twice, yielding cost $\mathcal{O}(c^{2}+n^{2})$. For example, for the inverse spatial transformation, we can compute it as $\mathbf{x}_{c}=\mathbf{y}-((\mathbf{yU}^{(s)})(\mathbf{I}^{(d_{s})}+\mathbf{V}^{(s)}\mathbf{U}^{(s)})^{-1})\mathbf{V}^{(s)}$, so that its complexity is $\mathcal{O}(d^{3}+cd^{2}+cnd)$. The total computational complexity of Equation 6 is $\mathcal{O}(dcn+d^{2}(n+c)+d^{3})$. In practice, we found that for a high-dimensional input, a relatively small $d$ is enough to obtain good performance, e.g., the input is $256\times 256\times 3$ images, and $d=16$. In this situation, $nc\geq d^{3}$. Therefore, we can omit $d$ and approximately see the spatial complexity as $\mathcal{O}(c+n)$, and the forward or inverse transformation as $\mathcal{O}(nc)$. They are all linear to the input size. We do not restrict $\mathbf{U}$ and $\mathbf{V}$ to force $\mathbf{W}$ to be invertible. Based on analysis by Hoogeboom et al. [15], the training maximizes the log-likelihood, which implicitly pushes $\det(\mathbf{I}+\mathbf{VU})$ away from $0$. Therefore, it is not necessary to explicitly force invertibility. In our experiments, the Woodbury transformations are as robust as other invertible convolution layers. (a) 1$\times$1 convolution (b) Woodbury (c) ME-Woodbury Figure 2: Visualization of three transformations. The 1$\times$1 convolution only operates along the channel axis. The Woodbury transformation operates along both the channel and spatial axes, modeling the dependencies of one channel directly via one transformation. The ME-Woodbury transformation operates along three axes. It uses two transformations to model spatial dependencies. ### 3.3 Memory-Efficient Variant In Eq. 4, one potential challenge arises from the sizes of $\mathbf{U}^{(s)}$ and $\mathbf{V}^{((s))}$, which are linear in $n$. The challenge is that $n$ may be large in some practical problems, e.g., high-resolution images. We develop a memory-efficient variant of Woodbury transformations, i.e., ME- Woodbury, to solve this problem. The ME version can effectively reduce space complexity from $\mathcal{O}(d(c+hw))$ to $\mathcal{O}(d(c+h+w))$. The difference between ME-Woodbury transformations and Woodbury transformations is that the ME form cannot directly model spatial correlations. As shown in Figure 2(c), it uses two transformations, for height and width, together to model the spatial correlations. Therefore, for a specific channel $k$, when two dimensions $\mathbf{x}_{k,i,j}$ and $\mathbf{x}_{k,u,v}$ are in two different heights, and widths, their interaction will be modeled indirectly. In our experiments, we found that this limitation only slightly impacts ME-Woodbury’s performance. More details on ME-Woodbury transformations are in the appendix. ## 4 Related Work Rezende and Mohamed [30] developed planar flows for variational inference $\mathbf{z}_{t+1}=\mathbf{z}_{t}+\mathbf{u}\delta(\mathbf{w}^{T}\mathbf{z}_{t}+b)$, where $\mathbf{z}$, $\mathbf{w}$, and $\mathbf{u}$ are $d$-dimensional vectors, $\delta()$ is an activation function, and $b$ is a scalar. Berg et al. [3] generalized these to Sylvester flows $\mathbf{z}_{t+1}=\mathbf{z}_{t}+\mathbf{QR}\delta(\tilde{\mathbf{R}}\mathbf{Q}^{T}\mathbf{z}_{t}+\mathbf{r})$, where $\mathbf{R}$ and $\tilde{\mathbf{R}}$ are upper triangular matrices, $\mathbf{Q}$ is composed of a set of orthonormal vectors, and $\mathbf{r}$ is a $d$-dimensional vector. The resulting Jacobian determinant can be efficiently computed via Sylvester’s identity, just as our methods do. However, Woodbury transformations have key differences from Sylvester flows. First, Berg et al. only analyze their models on vectors. The inputs to our layers are matrices, so our method operates on high-dimensional input, e.g., images. Second, though Sylvester flows are inverse functions, computing their inverse is difficult. One possible way is to apply iterative methods [2, 34, 5] to compute the inverse. But this research direction is unexplored. Our layers can be inverted efficiently with the Woodbury identity. Third, our layers do not restrict the transformation matrices to be triangular or orthogonal. In fact, Woodbury transformations can be seen as another generalized variant of planar flows on matrices, with $\delta(\mathbf{x})=\mathbf{x}$, and whose inverse is tractable. Roughly speaking, Woodbury transformations can also be viewed as applying the planar flows sequentially to each row of the input matrix. After this work was completed and submitted, we learned that the TensorFlow software [1] also uses the Woodbury identity in their affine bijector. Normalizing flows have also been used for variational inference, density estimation, and generative modeling. Autoregressive flows [23, 29, 17, 25] restrict each variable to depend on those that precede it in a sequence, forcing a triangular Jacobian. Non-linear coupling layers replace the affine transformation function. Specifically, spline flows [26, 8] use spline interpolation, and Flow++ [14] uses a mixture cumulative distribution function to define these functions. Flow++ also uses variational dequantization to prevent model collapse. Many works [22, 15, 9, 18] develop invertible convolutional flows to model interactions among dimensions. MintNet [34] is a flexible architecture composed of multiple masked invertible layers. I-ResNet [2, 5] uses discriminative deep network architecture as the flow. These two models require iterative methods to compute the inverse. Discrete flows [36, 16] and latent flows [41] can be applied to discrete data such as text. Continuous-time flows [4, 12] have been developed based on the theory of ordinary differential equations. ## 5 Experiments In this section, we compare the performance of Woodbury transformations against other modern flow architectures, measuring running time, bit per- dimension ($\log_{2}$-likelihood), and sample quality. Figure 3: Running time comparison. Sampling with emerging convolutions is slow, since their inverses are not parallelizable. Periodic convolutions are costly for larger inputs. Both 1$\times$1 convolutions and Woodbury transformations are efficient in training and sampling. Running Time We follow Finz et al. [9] and compare the per-sample running time of Woodbury transformations to other generalized permutations: 1$\times$1 [22], emerging [15], and periodic convolutions [15, 9]. We test the training time and sampling time. In training, we compute (1) forward propagation, i.e., $\mathbf{y}=\mathbf{f}(\mathbf{x})$, of a given function $\mathbf{f}()$, (2) the Jacobian determinant, i.e., $\det\left(\left|\frac{\partial\mathbf{y}}{\partial\mathbf{x}}\right|\right)$, and (3) the gradient of parameters. For sampling, we compute the inverse of transformation $\mathbf{x}=\mathbf{f}^{-1}(\mathbf{y})$. For emerging and periodic convolutions, we use $3\times 3$ kernels. For Woodbury transformations, we fix the latent dimension $d=16$. For fair comparison, we implement all methods in Pytorch and run them on an Nvidia Titan V GPU. We follow Hoogeboom et al. [15] and implement the emerging convolution inverse in Cython, and we compute it on a 4 Ghz CPU (the GPU version is slower than the Cython version). We first fix the spatial size to be $64\times 64$ and vary the channel number. We then fix the channel number to be $96$ and vary the spatial size. The results are shown in Figure 3. For training, the emerging convolution is the fastest. This is because its Jacobian is a triangular matrix, so computing its determinant is much more efficient than other methods. The Woodbury transformation and ME-Woodbury are slightly slower than the 1x1 convolution, since they contain more transformations. Emerging convolutions, Woodbury transformations, and 1x1 convolutions only slightly increase with input size, rather than increasing with $\mathcal{O}(c^{3})$. This invariance to input size is likely because of how the GPU parallelizes computation. The periodic convolution is efficient only when the input size is small. When the size is large, it becomes slow, e.g., when the input size is $96\times 64\times 64$, it is around $30$ times slower than Woodbury transformations. In our experiments, we found that the Fourier transformation requires a large amount of memory. According to Finz et al. [9], the Fourier step may be the bottleneck that impacts periodic convolution’s scalability. A more efficient implementation of Fourier transformation, e.g., [18], may improve its running time. For sampling, both 1$\times$1 convolutions and Woodbury transformations are efficient. The 1$\times$1 convolution is the fastest, and the Woodbury transformations are only slightly slower. Neither is sensitive to the change of input size. Emerging convolutions and periodic convolutions are much slower than Woodbury transformations, and their running time increases with the input size. When the input size is $96\times 128\times 128$, they are around $100$ to $200$ times slower than Woodbury transformations. This difference is because emerging convolutions cannot make use of parallelization, and periodic transformations require conversion to Fourier form. Based on these results, we can conclude that both emerging convolution and periodic convolution do not scale well to high-dimensional inputs. In contrast, Woodbury transformations are efficient in both training and sampling. Quantitative Evaluation We compare Woodbury transformations with state-of-the- art flow models, measuring bit per-dimension (bpd). We train with the CIFAR-10 [24] and ImageNet [32] datasets. We compare with three generalized permutation methods—1$\times$1 convolution, emerging convolution, and periodic convolution—and two coupling layers—neural spline coupling [8] and MaCow [25]. We use Glow (Fig. 1, [22]) as the basic flow architecture. For each method, we replace the corresponding layer. For example, to construct a flow with Woodbury transformations, we replace the 1$\times$1 convolution with a Woodbury transformation, i.e., Eq. 4. For all generalized permutation methods, we use affine coupling. For each of the coupling layer baselines, we substitute it for the affine coupling. We tune the parameters of neural spline coupling and MaCow so that their sizes are close to affine coupling. We follow Hoogeboom et al. [15] and test the performance of small models. For $32\times 32$ images, we set the number of levels to $L=3$ and the number of steps per- level to $K=8$. For $64\times 64$ images, we use $L=4$ and $K=16$. More details are in the appendix. Table 1: Quantitative evaluation results. | Quantitative measure (bpd) | Model sizes (# parameters) ---|---|--- | CIFAR-10 | ImageNet | ImageNet | 32x32 images | 64x64 images | 32x32 | 32x32 | 64x64 | | 1$\times$1 convolution | 3.51 | 4.32 | 3.94 | 11.02M | 37.04M Emerging | 3.48 | 4.26 | 3.91 | 11.43M | 40.37M Periodic | 3.49 | 4.28 | 3.92 | 11.21M | 38.61M Neural spline | 3.50 | 4.24 | 3.95 | 10.91M | 38.31M MaCow | 3.48 | 4.34 | 4.15 | 11.43M | 37.83M ME-Woodbury | 3.48 | 4.22 | 3.91 | 11.02M | 36.98M Woodbury | 3.47 | 4.20 | 3.87 | 11.10M | 37.60M Woodbury-Glow Iteration 50,000 Woodbury-Glow Iteration 100,000 Woodbury-Glow Iteration 600,000 Glow Iteration 50,000 Glow Iteration 100,000 Glow Iteration 600,000 Figure 4: Random samples $64\times 64$ drawn from models trained on CelebA with temperature $0.7$. The test-set likelihoods are listed in Table 1 left. Our scores are worse than those reported by Kingma and Dhariwal [22], Hoogeboom et al. [15] because we use smaller models and train each model on a single GPU. Based on the scores, 1$\times$1 convolutions perform the worst. Emerging convolutions and periodic convolutions score better than the 1$\times$1 convolutions, since they are more flexible and can model the dependencies along the spatial axes. Neural spline coupling works well on $32\times 32$ images, but do slightly worse than 1$\times$1 convolution on $64\times 64$ images. MaCow does not work well on ImageNet. This trend demonstrates the importance of permutation layers. They can model the interactions among dimensions and shuffle them, which coupling layers cannot do. Without a good permutation layer, a better coupling layer still cannot always improve the performance. The Woodbury transformation models perform the best, likely because they can model the interactions between the target dimension and all other dimensions, while the invertible convolutions only model the interactions between target dimension its neighbors. ME-Woodbury performs only slightly worse than the full version, showing that its restrictions provide a useful tradeoff between model quality and efficiency. We list model sizes in Table 1 (right). Despite modeling rich interactions, Woodbury transformations are not the largest. With $32\times 32$ images, ME- Woodbury and 1$\times$1 convolution are the same size. When the image size is $64\times 64$, ME-Woodbury is the smallest. This is because we use the multi- scale architecture, i.e., Fig. 1, to combine layers. The squeeze layer doubles the input variable’s channels at each level, so larger $L$ suggests larger $c$. The space complexities of invertible convolutions are $\mathcal{O}(c^{2})$, while the space complexity of ME-Woodbury is linear to $c$. When $c$ is large, the weight matrices of invertible convolutions are larger than the weight matrices of ME-Woodbury. Table 2: Evaluation of different $d$ (bpd). | Woodbury | ME-Woodbury ---|---|--- $d=2$ | 3.54 | 3.53 $d=4$ | 3.51 | 3.51 $d=8$ | 3.48 | 3.48 $d=16$ | 3.47 | 3.48 $d=32$ | 3.47 | 3.48 Latent Dimension Evaluation We test the impact of latent dimension $d$ on the performance of Woodbury-Glow. We train our models on CIFAR-10, and use bpd as metric. We vary $d$ within $\\{2,4,8,16,32\\}$. The results are in Table 2. When $d<8$, the model performance will be impacted. When $d>16$, increasing $d$ will not improve the bpd. This is probably because when $d$ is too small, the latent features cannot represent the input variables well, and when $d$ is too big, the models become hard to train. When $8\leq d\leq 16$, the Woodbury transformations are powerful enough to model the interactions among dimensions. We also test two values of $d$, i.e., $16,32$, of Woodbury-Glow on ImageNet $64\times 64$. The bpds of both $d$ are $3.87$, which are consistent with our conclusion. Figure 5: Learning curves on CelebA-HQ 64x64. The NLL of Woodbury Glow decreases faster than Glow. Sample Quality Comparisons We train Glow and Woodbury-Glow on the CelebA-HQ dataset [19]. We use $5$-bit images and set the size of images to be $64\times 64$, $128\times 128$, and $256\times 256$. Due to our limited computing resources, we use relatively small models in our experiments. We follow Kingma and Dhariwal [22] and choose a temperature parameter to encourage higher quality samples. Detailed parameter settings are in the appendix. We compare samples from Glow and Woodbury-Glow during three phases of training, displayed in Fig. 4. The samples show a clear trend where Woodbury-Glow more quickly learns to generate reasonable face shapes. After 100,000 iterations, it can already generate reasonable samples, while Glow’s samples are heavily distorted. Woodbury-Glow samples are consistently smoother and more realistic than samples from Glow in all phases of training. The samples demonstrate Woodbury transformations’ advantages. The learning curves in Figure 5 also show that the NLL of Woodbury Glow decreases faster, which is consistent to the sample comparisons. In the appendix, we show analogous comparisons using higher resolution versions of CelebA data, which also exhibit the trend of Woodbury-Glow generating more realistic images than Glow at the same training iterations. ## 6 Conclusion In this paper, we develop Woodbury transformations, which use the Woodbury matrix identity to compute the inverse transformations and Sylvester’s determinant identity to compute Jacobian determinants. Our method has the same advantages as invertible $d\times d$ convolutions that can capture correlations among all dimensions. In contrast to the invertible $d\times d$ convolutions, our method is parallelizable and the computational complexity of our methods are linear to the input size, so that it is still efficient in computation when the input is high-dimensional. One potential limitation is that Woodbury transformations do not have parameter sharing scheme as in convolutional layers, so one potential future research is to develop partially Woodbury transformations that can share parameters. We test our models on multiple image datasets and they outperform state-of-the-art methods. ## Broader Impact This paper presents fundamental research on increasing the expressiveness of deep probabilistic models. Its impact is therefore linked to the various applications of such models. By enriching the class of complex deep models for which we can train with exact likelihood, we may enable a wide variety of applications that can benefit from modeling of uncertainty. However, a potential danger of this research is that deep generative models have been recently applied to synthesize realistic images and text, which can be used for misinformation campaigns. ## Acknowledgments We thank NVIDIA’s GPU Grant Program and Amazon’s AWS Cloud Credits for Research program for their support. The work was completed while both authors were affiliated with the Virginia Tech Department of Computer Science. Bert Huang was partially supported by an Amazon Research Award and a grant from the U.S. Department of Transportation Safe-D Program for work on separate projects not directly related to this paper. ## References * Abadi et al. 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We optimize $\theta$ by minimizing the negative log- likelihood $\mathcal{L}(\mathcal{D})=\sum_{i=1}^{D}-\log p_{\theta}(\mathbf{x}_{i}).$ (8) For some settings, variable $\tilde{\mathbf{x}}$ is discrete, e.g., image pixel values are often integers. In these cases, we dequantize $\tilde{\mathbf{x}}$ by adding continuous noise $\bm{\mu}$ to it, resulting in a continuous variable $\mathbf{x}=\tilde{\mathbf{x}}+\bm{\mu}$. As shown by Ho et al. [14], the log-likelihood of $\tilde{\textbf{x}}$ is lower-bounded by the log-likelihood of $\mathbf{x}$. Normalizing flows enable computation of $p_{\theta}(\mathbf{x})$, even though it is usually intractable for many other model families. A normalizing flow [30] is composed of a series of invertible functions $\mathbf{f}=\mathbf{f}_{1}\circ\mathbf{f}_{2}\circ...\circ\mathbf{f}_{K}$, which transform $\mathbf{x}$ to a latent code $\mathbf{z}$ drawn from a simple distribution. Therefore, with the _change of variables_ formula, we can rewrite $\log p_{\theta}(\mathbf{x})$ to be $\log p_{\theta}(\mathbf{x})=\log p_{Z}(\mathbf{z})+\sum_{i=1}^{K}\log\left|\det\left(\frac{\partial\mathbf{f}_{i}}{\partial\mathbf{r}_{i-1}}\right)\right|,$ (9) where $\mathbf{r}_{i}=\mathbf{f}_{i}(\mathbf{r}_{i-1})$, $\mathbf{r}_{0}=\mathbf{x}$, and $\mathbf{r}_{K}=\mathbf{z}$. ### A.2 Invertible $d\times d$ Convolutions Emerging convolutions [15] combine two autoregressive convolutions [10, 23]. Formally, $\displaystyle\mathbf{M}^{\prime}_{1}=\mathbf{M}_{1}\odot\mathbf{A}_{1},~{}~{}~{}~{}~{}~{}~{}~{}\mathbf{M}^{\prime}_{2}=\mathbf{M}_{2}\odot\mathbf{A}_{2},~{}~{}~{}~{}~{}~{}~{}~{}\mathbf{y}=\mathbf{M}^{\prime}_{2}\star(\mathbf{M}^{\prime}_{1}\star\mathbf{x}),$ where $\mathbf{M}_{1},\mathbf{M}_{2}$ are convolutional kernels whose size is $c\times c\times d\times d$, and $\mathbf{A}_{1},\mathbf{A}_{2}$ are binary masks. The symbol $\star$ represents the convolution operator.222In practice, a convolutional layer is usually implemented as an aggregation of cross- correlations. We follow Hoogeboom et al. [15] and omit this detail. An emerging convolutional layer has the same receptive fields as standard convolutional layers, which can capture correlations between a target pixel and its neighbor pixels. However, like other autoregressive convolutions, computing the inverse of an emerging convolution requires sequentially traversing each dimension of input, so its computation is not parallelizable and is a computational bottleneck when the input is high-dimensional. Periodic convolutions [15, 9] use discrete Fourier transformations to transform both the input and the kernel to Fourier domain. A periodic convolution is computed as $\mathbf{y}_{u,:,:}=\sum_{v}\mathcal{F}^{-1}(\mathcal{F}(\mathbf{M}^{(p)}_{u,v,:,:})\odot\mathcal{F}(\mathbf{x}_{v,:,:})),$ where $\mathcal{F}$ is a discrete Fourier transformation, and $\mathbf{M}^{(p)}$ is the convolution kernel whose size is $c\times c\times d\times d$. The computational complexity of periodic convolutions is $\mathcal{O}(c^{2}hw\log(hw)+c^{3}hw)$. In our experiments, we found that the Fourier transformation requires a large amount of memory. These two problems impact the efficiency of both training and sampling when the input is high- dimensional. ## Appendix B Memory-Efficient Woodbury Transformations Memory-Efficient Woodbury transformations can effectively reduce the space complexity. The main idea is to perform spatial transformations along the height and width axes separately, i.e., a height transformation and a width transformation. The transformations are: $\displaystyle\mathbf{x}_{c}$ $\displaystyle=$ $\displaystyle(\mathbf{I}^{(c)}+\mathbf{U}^{(c)}\mathbf{V}^{(c)})\mathbf{x},$ $\displaystyle\mathbf{x}_{w}$ $\displaystyle=$ $\displaystyle\text{reshape}(\mathbf{x}_{c},(ch,w)),$ $\displaystyle\mathbf{x}_{w}$ $\displaystyle=$ $\displaystyle\mathbf{x}_{c}(\mathbf{I}^{(w)}+\mathbf{U}^{(w)}\mathbf{V}^{(w)}),$ $\displaystyle\mathbf{x}_{h}$ $\displaystyle=$ $\displaystyle\text{reshape}(\mathbf{x}_{w},(cw,h)),$ $\displaystyle\mathbf{y}$ $\displaystyle=$ $\displaystyle\mathbf{x}_{h}(\mathbf{I}^{(h)}+\mathbf{U}^{(h)}\mathbf{V}^{(h)}),$ $\displaystyle\mathbf{y}$ $\displaystyle=$ $\displaystyle\text{reshape}(\mathbf{y},(c,hw)),$ (10) where $\text{reshape}(\mathbf{x},(n,m))$ reshapes $\mathbf{x}$ to be an $n\times m$ matrix. Matrices $\mathbf{I}^{(w)}$ and $\mathbf{I}^{(h)}$ are $w$\- and $h$-dimensional identity matrices, respectively. Matrices $\mathbf{U}^{(w)},\mathbf{V}^{(w)},\mathbf{U}^{(h)}$, and $\mathbf{V}^{(h)}$ are $w\times d_{w}$, $d_{w}\times w$, $w\times d_{w}$, and $d_{w}\times w$ matrices, respectively, where $d_{w}$ and $d_{h}$ are constant latent dimensions. Using the Woodbury matrix identity and the Sylvester’s determinant identity, we can compute the inverse and Jacobian determinant: $\displaystyle\mathbf{y}$ $\displaystyle=$ $\displaystyle\text{reshape}(\mathbf{y},(cw,h)),$ $\displaystyle\mathbf{x}_{h}$ $\displaystyle=$ $\displaystyle\mathbf{y}(\mathbf{I}^{(h)}-\mathbf{U}^{(h)}(\mathbf{I}^{(d_{h})}+\mathbf{V}^{(h)}\mathbf{U}^{(h)})^{-1}\mathbf{V}^{(h)}),$ $\displaystyle\mathbf{x}_{w}$ $\displaystyle=$ $\displaystyle\text{reshape}(\mathbf{x}_{h},(ch,w)),$ $\displaystyle\mathbf{x}_{w}$ $\displaystyle=$ $\displaystyle\mathbf{x}_{w}(\mathbf{I}^{(w)}-\mathbf{U}^{(w)}(\mathbf{I}^{(d_{w})}+\mathbf{V}^{(w)}\mathbf{U}^{(w)})^{-1}\mathbf{V}^{(w)}),$ $\displaystyle\mathbf{x}_{c}$ $\displaystyle=$ $\displaystyle\text{reshape}(\mathbf{x}_{w},(c,hw)),$ $\displaystyle\mathbf{x}$ $\displaystyle=$ $\displaystyle(\mathbf{I}^{(c)}-\mathbf{U}^{(c)}(\mathbf{I}^{(d_{c})}+\mathbf{V}^{(c)}\mathbf{U}^{(c)})^{-1}\mathbf{V}^{(c)})\mathbf{x}_{c},$ (11) $\displaystyle\log\left|\det(\frac{\partial\mathbf{y}}{\partial\mathbf{x}})\right|$ $\displaystyle=hw\log\left|\det(\mathbf{I}^{(d_{c})}+\mathbf{V}^{(c)}\mathbf{U}^{(c)})\right|+ch\log\left|\det(\mathbf{I}^{(d_{w})}+\mathbf{V}^{(w)}\mathbf{U}^{(w)})\right|$ (12) $\displaystyle+cw\log\left|\det\left(\mathbf{I}^{(d_{h})}+\mathbf{V}^{(h)}\mathbf{U}^{(h)}\right)\right|,$ where $\mathbf{I}^{(d_{w})}$ and $\mathbf{I}^{(d_{h})}$ are $d_{w}$\- and $d_{h}$-dimensional identity matrices, respectively. The Jacobian of the $\text{reshape}()$ is an identity matrix, so its log-determinant is $0$. We call Equation 10 the memory-efficient Woodbury transformation because it reduces space complexity from $\mathcal{O}(c+hw)$ to $\mathcal{O}(c+h+w)$. This method is effective when $h$ and $w$ are large. To analyze its complexity, we let all latent dimensions be less than $d$ as before. The complexity of forward transformation is $\mathcal{O}(dchw)$; the complexity of computing the determinant is $\mathcal{O}(d(c+h+w)+d^{3})$; and the complexity of computing the inverse is $\mathcal{O}(dchw+d^{2}(c+ch+cw)+d^{3})$. The same as Woodbury transformations, when the input is high dimensional, we can omit $d$. Therefore, the computational complexities of the memory-efficient Woodbury transformation are also linear with the input size. We list the complexities of different methods in Table 3. We can see that the computational complexities of Woodbury transformations are comparable to other methods, and maybe smaller when the input is high-dimensional, i.e., the $c,h,w$ are big. Table 3: Comparisons of computational complexities. Method | Forward | Backward ---|---|--- 1x1 convolution | $\mathcal{O}(c^{2}hw+c^{3})$ | $\mathcal{O}(c^{2}hw)$ Periodic conolution | $\mathcal{O}(chw\log(hw)+c^{3}hw)$ | $\mathcal{O}(chw\log(hw)+c^{2}hw)$ Emerging convolution | $\mathcal{O}(c^{2}hw)$ | $\mathcal{O}(c^{2}hw)$ ME-Woodbury transformation | $\mathcal{O}(dchw)$ | $\mathcal{O}(dchw)$ Woodbury transformation | $\mathcal{O}(dchw)$ | $\mathcal{O}(dchw)$ ## Appendix C Parameter Settings In this section, we present additional details about our experiments to aid reproducibility. ### C.1 Experiments of Quantitative Evaluation In the experiments of qualitative evalution, we compare Woodbury transformations with $3$ permutation layer baselines, i.e., 1x1 convolution, emerging convolution, and periodic coupling, and $2$ coupling layer baselines, i.e., neural spline coupling, and MaCow. For all generalized permutation methods, we use affine coupling, which is composed of $3$ convolutional layers, and the $2$ latent layers have $512$ channels. For the neural spline coupling, we set the number of spline bins to $4$. The spline parameters are generated by a neural network, which is also composed of convolutional layers. For $32\times 32$ images, we set the number of channels to $256$, and for $64\times 64$ images, we set it to $224$. Ma et al. [25] used steps containing a MaCow unit, i.e., $4$ autoregressive convolution coupling layers, and a full Glow step. For fair comparison, we directly use the MaCow unit to replace the affine coupling. For $32\times 32$ images, we set the convolution channel to $384$, and for $64\times 64$ images, we set it to $296$. We run each method to fixed number of iterations and test it every $10,000$ iterations. The bpds reported in our main paper are the best bpds obtained by each method. The bpds are single-run results. This is because each run of the experiment requires 3 to 5 days, and running each model multiple times is a major cost. We found in our experiments that for the same model and parameter settings, the bpds’ standard deviation of multiple runs are very small, i.e., around $0.003$, so single run results are sufficient for comparing bpd. ### C.2 Hyper-parameter Settings We use Adam [20] to tune the learning rates, with $\alpha=0.001$, $\beta_{1}=0.9$, and $\beta_{2}=0.999$. We use uniform dequantization. The sizes of models we use, and mini-batch sizes for training in our experiments are listed in Table 4. Table 4: Model sizes and mini-batch sizes. Dataset | Mini-batch size | Levels(L) | Steps(K) | Coupling channels ---|---|---|---|--- CIFAR-10 32x32 | 64 | 3 | 8 | 512 ImageNet 32x32 | 64 | 3 | 8 | 512 ImageNet 64x64 | 32 | 4 | 16 | 512 LSUN Church 96x96 | 16 | 5 | 16 | 256 CelebA-HQ 64x64 | 8 | 4 | 16 | 512 CelebA-HQ 128x128 | 4 | 5 | 24 | 256 CelebA-HQ 256x256 | 4 | 6 | 16 | 256 ### C.3 Latent Dimension Settings In all our experiments, we set the latent dimensions of Woodbury transformations, and ME-Woodbury transformations as in Table 5. Table 5: Latent dimensions of Woodbury transformations and ME-Woodbury transformations. The numbers in the brackets represent the latent dimension used in that level. For example, the $d_{c}:\\{8,8,16\\}$, represents that the settings of $d_{c}$ at the three levels are $8$, $8$, and $16$. Dataset | Woodbury | ME-Woodbury ---|---|--- CIFAR-10 32x32 | $d_{c}:\\{8,8,16\\}$ | $d_{c}:\\{8,8,16\\}$ | $d_{s}:\\{16,16,8\\}$ | $d_{h}:\\{16,16,8\\}$ | | $d_{w}:\\{16,16,8\\}$ ImageNet 32x32 | $d_{c}:\\{8,8,16\\}$ | $d_{c}:\\{8,8,16\\}$ | $d_{s}:\\{16,16,8\\}$ | $d_{h}:\\{16,16,8\\}$ | | $d_{w}:\\{16,16,8\\}$ ImageNet 64x64 | $d_{c}:\\{8,8,16,16\\}$ | $d_{c}:\\{8,8,16,16\\}$ | $d_{s}:\\{16,16,8,8\\}$ | $d_{h}:\\{16,16,8,8\\}$ | | $d_{w}:\\{16,16,8,8\\}$ LSUN Church 96x96 | $d_{c}:\\{8,8,16,16,16\\}$ | — | $d_{s}:\\{16,16,16,8,8\\}$ | CelebA-HQ 64x64 | $d_{c}:\\{8,8,16,16\\}$ | — | $d_{s}:\\{16,16,8,8\\}$ | CelebA-HQ 128x128 | $d_{c}:\\{8,8,16,16,16\\}$ | — | $d_{s}:\\{16,16,16,8,8\\}$ | CelebA-HQ 256x256 | $d_{c}:\\{8,8,16,16,16,16\\}$ | — | $d_{s}:\\{16,16,16,16,8,8\\}$ | ## Appendix D Sample Quality Comparisons We compare the samples generated by Woodbury-Glow and Glow models trained on the CelebA-HQ dataset. We follow Kingma and Dhariwal [22] and randomly hold out 3,000 images as a test set. We use $5$-bits images. We use $64\times 64$, $128\times 128$, $256\times 256$ images. Due to the our limited computing resources, we use relatively small models. The model sizes and other settings are listed in Table 4 and Table 5. We generate samples from the models during different phases of training and display them in Figure 6, and Figure 7 (The results of $64\times 64$ images are shown in the main paper). For the $128\times 128$ images, both Glow and Woodbury-Glow generate distorted images at iteration 100,000, but Woodbury-Glow seems to improve in later stages, stabilizing the shapes of faces and structure of facial features. Glow, continues generating faces with distorted overall shapes as training continues. For the $256\times 256$ images, neither model ever trains sufficiently to generate highly realistic faces, but Woodbury-Glow makes significantly more progress in these 300,000 iterations than Glow. Glow’s samples at 300,000 are still mostly random swirls with an occasional recognizable face, while almost all of Woodbury-Glow’s samples look like faces, though distorted. Due to limits on our computational resources, we stopped the higher resolution experiments at 300,000 iterations (rather than running to 600,000 iterations as we did for the $64\times 64$ experiments in the main paper). With a larger model and longer training time, it seems Woodbury-Glow would reach higher sample quality much faster than Glow. The likelihoods of test set under the trained model are listed in Table 3. For the $64\times 64$ and $128\times 128$ images, Woodbury-Glow scores higher likelihood than Glow. For the $256\times 256$ images, their likelihoods are almost identical, and are better than the score reported in [22]. This may be due to three possible reasons: (1) We use affine coupling rather than additive coupling, which is a non-volume preserving layer and may improve the likelihoods; (2) Since the test set is randomly collected, it is different from the one used in [22]; And (3) The model used in [22] is very large, so it may be somewhat over-fitting. Surprisingly, the clear difference in sample quality is not reflected by the likelihoods. This discrepancy may be because we use $5$-bit images, and the images are all faces, so the dataset is less complicated than other datasets such as ImageNet. Moreover, even though Glow cannot generate reasonable $256\times 256$ samples, the colors of these samples already match the colors of real images well, so these strange samples may non-intuitively be equivalently likely as the face-like samples from Woodbury-Glow. Table 6: Bit per-dimension results on CelebA-HQ Size of images | Glow | Woodbury-Glow ---|---|--- $64\times 64$ | 1.27 | 1.23 $128\times 128$ | 1.09 | 1.04 $256\times 256$ | 0.93 | 0.93 Woodbury-Glow Iteration 100,000 Woodbury-Glow Iteration 200,000 Woodbury-Glow Iteration 300,000 Glow Iteration 100,000 Glow Iteration 200,000 Glow Iteration 300,000 Figure 6: Random samples of $128\times 128$ images drawn with temperature $0.7$ from a model trained on CelebA data. Woodbury-Glow Iteration 150,000 Woodbury-Glow Iteration 220,000 Woodbury-Glow Iteration 300,000 Glow Iteration 150,000 Glow Iteration 220,000 Glow Iteration 300,000 Figure 7: Random samples of $256\times 256$ images drawn with temperature $0.7$ from a model trained on CelebA data. ## Appendix E Additional Samples In this section, we include additional samples from Woodbury-Glow models trained on our various datasets. These samples complement our quantitative analysis. We train our models on CIFAR-10 [24], ImageNet [32], the LSUN church dataset [38], and the CelebA-HQ dataset [19]. Specifically, for ImageNet, we use $32\times 32$ and $64\times 64$ images. For the LSUN dataset, we use the same approach as Kingma and Dhariwal [22] to resize the images to be $96\times 96$. For the CelebA-HQ dataset, we use $64\times 64$, $128\times 128$, and $256\times 256$ images. For LSUN and CelebA-HQ datasets, we use $5$-bit images. The parameter settings of our models are in Table 4 and Table 5. The samples are in Figures 8, 9, 10, 11, 12, 13, and 14. Figure 8: CIFAR-10 $32\times 32$ Woodbury-Glow samples. Figure 9: ImageNet $32\times 32$ Woodbury-Glow samples. Figure 10: ImageNet $64\times 64$ Woodbury-Glow samples. Figure 11: LSUN church $96\times 96$ Woodbury-Glow samples (temperature $0.875$). Figure 12: CelebA-HQ $64\times 64$ Woodbury-Glow samples (temperature $0.7$). Figure 13: CelebA-HQ $128\times 128$ Woodbury-Glow samples (temperature $0.5$). Figure 14: Selected CelebA-HQ $256\times 256$ Woodbury-Glow samples (temperature $0.5$).

2024-09-04T02:54:55.187903

2002.12248

arxiv-papers

††thanks: These two authors contributed equally††thanks: These two authors contributed equally # Non-trivial quantum magnetotransport oscillations in pure and robust topological $\alpha$-Sn films Ivan Madarevic Quantum Solid State Physics, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium Niels Claessens Quantum Solid State Physics, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium IMEC, Kapeldreef 75, 3001 Leuven, Belgium Aleksandr Seliverstov Quantum Solid State Physics, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium Chris Van Haesendonck Quantum Solid State Physics, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium Margriet J. Van Bael Quantum Solid State Physics, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium ###### Abstract We report experimental evidence of topological Dirac fermion charge carriers in pure and robust $\alpha$-Sn films grown on InSb substrates. This evidence was acquired using standard macroscopic four-point contact resistance measurements, conducted on uncapped films with a significantly reduced bulk mobility. We analyzed and compared electrical characteristics of the constituting components of the $\alpha$-Sn/InSb sample, and propose a three- band drift velocity model accordingly. A surface band, with low carrier density and high mobility, is identified as the origin of the observed Shubnikov – de Haas oscillations. The analysis of these quantum oscillations results in a non-trivial value of the phase shift $\gamma=0$, characteristic for topologically protected Dirac fermions. For the same uncapped samples we estimate the momentum relaxation time $\tau\approx 300\ \mathrm{fs}$, which is significantly larger in comparison with the previous reports on grown $\alpha$-Sn films. ††preprint: APS/123-QED The alpha phase of Sn ($\alpha$-Sn) is a diamond structured crystal which is a zero band gap semimetal. Due to its suitable band structure, $\alpha$-Sn was predicted as a material capable of acquiring several topological phases [1, 2]. Crystal strain can bring about the topologically protected states in $\alpha$-Sn, more specifically: in-plane tensile strain turns it into a 3D topological insulator (3DTI), while in-plane compressive strain converts it into a topological Dirac semimetal (DSM). Topologically protected bands of $\alpha$-Sn crystalline thin films were characterized by experiments during the last decade. This was particularly done using angle-resolved photoemission spectroscopy (ARPES), which revealed a Dirac cone near the Fermi level in the surface electronic structure of Te and/or Bi doped $\alpha$-Sn thin films, compressively strained on InSb substrates [3, 4, 5, 6, 7]. Interestingly, the most recent (soft X-ray) ARPES investigations have suggested the existence of a topologically protected bulk band (DSM phase) in compressively strained $\alpha$-Sn films [8, 5, 9]. These reports confirm that this material is an excellent platform to explore several topological phases [1, 2], making it very interesting for possible spintronics [10, 11] and quantum computing applications [12, 13, 14]. Moreover, as a very well structurally defined non- toxic elemental material with a robust topological phase, $\alpha$-Sn could be more favorable for applications when compared to prototypical DSM binary compounds Na3Bi [15, 16] and Cd3As2 [17, 18, 19]. There are recent reports indicating topological magnetotransport in $\alpha$-Sn. In the case of $\alpha$-Sn on InSb there is, until now, one report [7] suggesting topologically protected carriers based on magnetotransport measurements. However, these experiments were conducted on a relatively complicated system (Bi-doped $\alpha$-Sn$\mid$InSb$\mid$GaAs), in which the Dirac point was positioned below the Fermi level. There is another report on the $\alpha$-Sn magnetotransport properties in the case of a thin film grown on CdTe [20]. The use of insulating CdTe should allow more straightforward detection of the $\alpha$-Sn topological bands. However, both Cd and Te bond with Sn (by metallic and covalent bonding respectively), complicating the morphology of the films, disturbing its epitaxial growth and the introduction of strain [21, 20]. At this moment, no unambiguous experimental evidence has been presented for topological Dirac charge carriers in completely pure (no dopants) $\alpha$-Sn films. Very recently we presented [22] a straightforward method to grow pure, compressively strained $\alpha$-Sn films on InSb(100) substrates. The structure and morphology of these films were fully characterized and the presence of a Dirac cone was confirmed by ARPES. Moreover, these films demonstrated an excellent surface quality and a notable robustness against ambient conditions. In this letter we report prominent accessibility of the topologically protected Dirac charge carriers in pure and uncapped 30 nm thick $\alpha$-Sn films compressively strained on InSb(100) [22]. This was made possible due to the decreased mobility of the trivial bulk charge carriers in the grown films – most likely a consequence of increased grain-boundary barrier due to the small grain size. We present evidence for the existence of the Dirac charge carriers, as an absolutely intrinsic property of the topological $\alpha$-Sn films. This evidence is based on our analysis of the Shubnikov – de Haas (SdH) oscillations originating from a surface band. The Dirac charge carriers exhibit a significantly enhanced relaxation time compared to previous reports [7, 11]. For the unambiguous interpretation of electronic transport experiments on composite systems (i.e. film/substrate) it is generally required to analyze the transport properties of the constituting components. Therefore, along with the transport measurements on the 30 nm $\alpha$-Sn$\mid$InSb(100) system, we conducted the same measurements on a non-covered InSb(100) substrate, prepared according to the procedure presented in our previous report [22]. The magnetotransport experiments were conducted at a temperature of 1.5 K, with silver paste contacts in a four-point geometry, at constant current (between 50 and 100 $\mu$A), and magnetic field ($B$) up to 8 T. Figure 1 depicts the results of the conducted Hall effect measurements, revealing a clear qualitative difference between the off-diagonal magnetoresistance ($R_{xy}$) of the InSb(100) substrate and the grown $\alpha$-Sn$\mid$InSb(100) sample. While the substrate’s off-diagonal magnetoresistance shows a linear behavior ($n$-type), the grown sample clearly displays a non-linear Hall effect, indicating the presence of multiple types of carriers. At low fields ($<$ 2 T), the substrate is shunting the $\alpha$-Sn film, and the behavior of $R_{xy}$ is slightly non-linear. From the results of our detailed study of the transport properties of the substrate, there are some indications that the substrate preparation (Ar+ plasma etching and thermal annealing [22]) leaves it electronically heterogenous in depth, which may cause such non-linear behavior. However, it is beyond the scope of this letter to discuss the cause of the non-linear behavior at low fields. Since it can be attributed to the substrate, for which the linear $n$-type behavior becomes dominant at higher fields, we omit the low field region from further analysis. Figure 1: Measured off-diagonal magnetoresistance tensor element ($R_{xy}$) of a 30 nm thick film of $\alpha$-Sn(100) on InSb(100), and a non-covered InSb(100) substrate. Measurements were done using the Van der Pauw technique [23]. The inset shows the four-point contact (1 – 4) configuration on the surface of the sample. Figure 2: Fit based on the three-band drift velocity model (orange) for the high-field off-diagonal magnetoresistance above 2 T (black) of a 30 nm thick film of $\alpha$-Sn on InSb(100). The method to identify different contributions leading to the observed Hall effect is to fit a multi-band drift velocity model to the measured $R_{xy}$ data. $R_{xy}$ can, in general, be expressed using the following equations: $R_{xy}=\ \frac{{\sigma}_{xy}}{{{\sigma}_{xy}}^{2}+{{\sigma}_{xx}}^{2}}~{},$ (1) with the conductance matrix components of the different contributions ($i$) to $\sigma$ given by: ${\sigma}^{i}_{xx}=\ \frac{n^{i}_{s}\cdot e\cdot{\mu}^{i}}{1+{\left({\mu}^{i}\cdot B\right)}^{2}}~{},$ (2) ${\sigma}^{i}_{xy}=\ \pm\frac{n^{i}_{s}\cdot e\cdot{\left({\mu}^{i}\right)}^{2}\cdot B}{1+{\left({\mu}^{i}\cdot B\right)}^{2}}~{},$ (3) under the following constraint for the measured sheet resistance at zero field: $R_{s}=\ \left(\sum^{k}_{1}{n^{i}_{s}\cdot}e\cdot{\mu}^{i}\right)^{-1}~{}.{}$ (4) Fitting the measured $R_{xy}$ behavior of the $\alpha$-Sn sample to a two-band ($k=2$) drift velocity model results in one $p$-type band ($i=p$) and one $n$-type band ($i=n$), with sheet carrier concentrations $n^{p}_{s}=3.2(1)\cdot{10}^{14}\ \mathrm{cm}^{-2}$, $n^{n}_{s}=1.65(4)\cdot{10}^{13}\ \mathrm{cm}^{-2}$, and mobilities ${\mu}^{p}=0.184(2)\cdot 10^{3}\ \mathrm{cm^{2}V^{-1}s^{-1}}$, ${\mu}^{n}=1.4(8)\cdot{10}^{4}\ \mathrm{cm^{2}V^{-1}s^{-1}}$. In this case, the reduced chi-square value (${\chi}^{2}$) for the fit equals 0.78, at a $R_{xy}$ relative error of 5 %. On the other hand, repeating the procedure for a three-band ($k=3$) drift velocity model (Fig. 2) results in two $p$-type bands ($i=p1,~{}p2$) and one $n$-type band ($i=n$), with sheet carrier concentrations $n^{p1}_{s}=3.2(1)\cdot{10}^{13}\ \mathrm{cm}^{-2}$, $n^{p2}_{s}=1.3(2)\cdot{10}^{12}\ \mathrm{cm}^{-2}$, $n^{n}_{s}=1.65(4)\cdot{10}^{13}\ \mathrm{cm}^{-2}$, and mobilities ${\mu}^{p1}=0.177(2)\cdot 10^{3}\ \mathrm{cm^{2}V^{-1}s^{-1}}$, ${\mu}^{p2}=3.47(12)\cdot 10^{3}\ \mathrm{cm^{2}V^{-1}s^{-1}}$, ${\mu}^{n}=1,4(8)\cdot{10}^{4}\ \mathrm{cm^{2}V^{-1}s^{-1}}$, with ${\chi}^{2}=0.53$ at the same relative error. Considering these values, it is tempting to choose the latter model over the first one. Indeed, knowing the film/substrate thickness [22], the bands with the large number of carriers ($n$ and $p1$) are typical bulk bands expected for InSb [24] and $\alpha$-Sn grown on InSb [25], which exhibits a reduced mobility compared to the bulk single crystal case [26]. At the same time, the band with the high mobility and low carrier density ($p2$) may be identify as a possible topological surface band of $\alpha$-Sn [27]. Although the second model fits the data better, a nicer fit on its own does not prove the correctness of the model. To exclude one of the two models, we extracted the SdH oscillations from the diagonal magnetoresistance $R_{xx}$ measurement of a grown $\alpha$-Sn sample (Fig. 3). The experiments were again conducted at a temperature of 1.5 K and with magnetic field (up to 8 T) applied perpendicularly to the surface of the samples ($\vec{B}\perp(100)$), with macroscopic silver paste contacts arranged in a linear four-point geometry. Figure 4 (inlay) depicts magnetoresistance oscillations ($\Delta R$) of a grown $\alpha$-Sn sample, extracted after removing the background which was obtained from a $3^{\mathrm{rd}}$ order polynomial fit. The maximum of the fast Fourier transform (FFT) of $\Delta R$ vs. $1/B$ provides an estimate of the characteristic frequency $f=14(6)\ \mathrm{T}$ of the oscillations. A more accurate value of 13.5(8) T was extracted from the Landau index plot (Fig. 4), as shown later. Figure 3: Measured diagonal magnetoresistance tensor element ($R_{xx}$) of a 30 nm thick film of $\alpha$-Sn(100) on InSb(100) and a non-covered InSb(100) substrate, with the magnetic field perpendicular to the film/substrate surface. The inset shows the four-point contact configuration on the surface of the sample. Assuming the oscillations originate from a bulk band, the estimated value for $f$ would correspond to a carrier density of $n=(4\pi f/\mathrm{\Phi_{0}})^{{3}/{2}}/{3{\pi}^{2}}=2.8(2)\cdot{10}^{17}\ \mathrm{cm}^{-3}$, where ${\mathrm{\Phi}}_{0}=h/e$ is the elementary flux quantum, with Planck’s constant $h$ and the electron charge $e$. If we convert this to the hypothetical sheet carrier densities for such bulk bands of the $\alpha$-Sn film and InSb substrate, by multiplying with their thicknesses [22], this would give $8.4(7)\cdot{10}^{11}\ \mathrm{cm}^{-2}$ and $1.75(16)\cdot~{}{10}^{16}\ \mathrm{cm}^{-2}$ respectively. From these sheet carrier densities, which differ from the bulk bands extracted using the Hall effect measurements ($p1$ and $n1$, respectively), it appears that neither the two- nor three-band model would be compatible with the observed SdH oscillations if the oscillations would originate from a bulk band. Figure 4: Landau index plot of the magnetoresistance oscillations after removal of the background ($\vec{B}\perp(100)$). The Landau index for the minima and maxima of the observed SdH oscillations is plotted against the inverse of the field strength (red dots) at which they are observed. The fit of Eq. (6) (black line) through the data points is shown together with the low field limit ($A_{2}=0$) (red dashed line). The inlay shows magnetoresistance oscillations in the $\vec{B}\perp(100)$ (blue line) and the $\vec{B}\parallel(100)$ orientation (black dashed line). If we instead assume that the experimentally observed quantum oscillations arise from a surface band, the sheet carrier density corresponding to the observed SdH frequency is given by $n_{s}=2\cdot n=2\cdot(2f/{\mathrm{\Phi}}_{0})={1.30(7)\cdot{10}^{12}\ \mathrm{cm}^{-2}}$. This carrier density value is in very good agreement with the second $p$-type band ($p2$) in the above fitted three-band drift velocity model, but in contradiction with the two-band model. Note as well that the SdH oscillations are only observable when their cyclotron frequency is larger than the inverse of the relaxation time [28] – equivalent to $\mu>{1}/{B}\approx 3.33\cdot 10^{3}\ \mathrm{cm^{2}V^{-1}s^{-1}}$ for oscillations starting at $\sim 3$ T. This condition is satisfied for the $p2$ band in the three-band drift velocity model, but it is not met for the two-band model. Moreover, the reduced mobility of the trivial bulk carriers (increased grain boundary scattering) in this model (${\mu}^{p1}$) is consistent with the fact that the grown $\alpha$-Sn films have a granular morphology, with grain size around 10 – 20 nm [22]. Therefore, it can be concluded that the two-band drift velocity model fails, while the three-band model explains our observations. The second $p$-type band of the three-band model ($p2$) can thus be identified as a surface band of the $\alpha$-Sn(100) film. The first $p$-type band ($p1$) we assign to the bulk $\alpha$-Sn and the $n$-type band to the bulk of the InSb substrate. Using Eq. (4) we can now estimate the sheet resistance contribution of the bands separately: $R^{~{}p1}_{s}=1.10(4)\ \mathrm{k\Omega/\Box}$ ($\alpha$-Sn bulk band), $R^{~{}p2}_{s}=1.4(2)\ \mathrm{k\Omega/\Box}$ ($\alpha$-Sn surface band) and $R^{~{}n}_{s}=27(15)\ \mathrm{\Omega/\Box}$ (InSb bulk band). We will now investigate the possible Dirac nature of the carriers in the second $p$ band ($i=p2$). The phase offset $\gamma$ of the SdH oscillations contains the information on the dispersion of the corresponding charge carriers [29]. The SdH oscillations can be expressed as: $\Delta R\propto{\mathrm{cos}\left[2\pi\cdot\left(\frac{f}{B}-\gamma\right)\right]}~{},$ (5) where $\gamma$ can be related to Berry’s phase acquired by the carriers within a cyclotron orbit [30]. When the zero-field electronic dispersion would be followed by the carriers, then $\gamma=0$ (non-trivial Berry’s phase of $\pi$) for Dirac fermions, while $\gamma=1/2$ (trivial Berry’s phase of 0) for “normal” fermions [31]. The intercept of a linear fit of the Landau level index N with respect to $1/B$ then gives the $\gamma$ value. However, for finite field strengths, as pointed out by Wright and McKenzie, the fermions can no longer be expected to follow the zero-field dispersion, and a deviation from a perfect quantization, where $\gamma=\gamma(B)$, is to be expected [31]. For that reason, a more accurate low-field fit procedure was proposed to extract the zero-field phase offset: $N=\frac{f}{B}+A_{1}+A_{2}\cdot B~{},{}$ (6) where $f$ (characteristic frequency), $A_{1}$ and $A_{2}$ are the fit- parameters. The zero-field limit of the Eq. (6) is equivalent with $\gamma(0)=A_{1}$. Therefore, the parameter $A_{1}$ is the quantized offset related to the zero-field Berry’s phase. Figure 4 presents the fit of Eq. (6) to the Landau indices of the observed SdH oscillations minima and maxima extracted from the magnetoresistance measurement with the magnetic field applied perpendicular to the sample surface plane ($\vec{B}\perp\mathrm{(100)}$). Here, due to the dominance of bulk contributions to the conductance [32], the minima of the SdH oscillations correspond to integer Landau level indices ($N$), while the maxima correspond to half-integer Landau level indices ($N+1/2$). The resulting fit-parameters are $A_{1}=0.02(15)$ and $A_{2}=0.00(6)$, and $f=13.5(8)\ \mathrm{T}$. Contrary to the near-zero $A_{1}$ value for the $\alpha$-Sn samples, the fit of the SdH oscillations extracted from the magnetoresistance measurement on a non-covered InSb substrate gave $A_{1}=0.52(13)$, $A_{2}=-0.04(2)$ and $f=39.4(8)\ \mathrm{T}$. The characteristic frequencies were determined by a linear fit of the Landau index plot (Fig. 4) and held fixed as the linear fit yields more accurate results on the frequency compared to the estimation of the FFT maximum of $\Delta R$, which is the usual procedure in order to reduce the number of fit parameters. As the analysis in the case of $\alpha$-Sn gives $A_{1}\approx 0$, it can be concluded that the second $p$-type band ($i=p2$) indeed has a topological Dirac fermion nature. Based on this conclusion, we can now estimate the value of the characteristic Fermi wavevector for this $p2$ band as $k_{F}=\sqrt{{4\pi}{f/\Phi_{0}}}=0.203(6)\ \mathrm{nm}^{-1}$. Assuming a linear energy dispersion [33] and using the previously extracted Fermi velocity value [22], we then estimate the Fermi energy at $E_{F}=\hslash k_{F}v_{F}=67(2)\ \mathrm{meV}$. These values, estimated from the transport measurements, are in agreement with the ones extracted from our recent ARPES and scanning tunneling spectroscopy (STS) study ($k_{F}^{~{}ARPES}\sim 0.18\ \mathrm{nm}^{-1}$, $E_{F}^{~{}ARPES}\sim 60\ \mathrm{meV}$ and $E_{F}^{~{}STS}\sim 70\ \mathrm{meV}$ below the Dirac point) [22]. From the calculated $R^{~{}p2}_{s}$ value, using the resistance expression for non-degenerate 2D bands with a linear dispersion, i.e., $R^{p2}_{s}=(4\hbar^{2}\pi)/(e^{2}E_{F}\tau)$ [34], we estimate the momentum relaxation time $\tau=300(40)\ \mathrm{fs}$. This value is about five times larger than the one estimated for $\alpha$-Sn films with an AlOx capping layer [7], and almost two orders of magnitude larger compared with $\alpha$-Sn films covered with Ag [11], indicating that there are fewer parallel momentum relaxation channels in our samples. In the case of a 2D gas of Dirac fermions existing on top (and at the bottom) of the grown $\alpha$-Sn film, one expects the SdH oscillations to vanish for the $\vec{B}\parallel\mathrm{(100)}$ alignment. However, for our samples this is not the case. Surprisingly, in the same type of grown $\alpha$-Sn samples, SdH oscillations were also observed (inlay of Fig. 4), when the magnetic field is applied parallel to the film surface (equivalent to $\vec{B}\perp\mathrm{(010)}$). These SdH oscillations have a characteristic frequency $f=10.1(15)\ \mathrm{T}$, which is quite similar to the $\vec{B}\perp\mathrm{(100)}$ case, and which cannot be attributed to InSb or $\alpha$-Sn bulk bands. The phase offset analysis of these oscillations gave a value $A_{1}=0.1(3)$ close to the non-trivial Berry’s phase of $\pi$. From the topographic details of our samples [22] we know that the surface of the film is not atomically flat, but in fact consists of crystalline grains with non- zero roughness. Therefore, a significant portion of the surface area is never in parallel alignment with the applied magnetic field. All of the above presented results do strongly support the existence of topological Dirac fermion charge carriers, but our discussion so far has not yet specified which type of Dirac material, 3DTI or DSM, our $\alpha$-Sn films belong to. In the case of the 3DTI $\alpha$-Sn, a surface phase with a clear topological Dirac signature should arise in the electrical transport experiments [1, 2]. The detected $p2$ surface band appears to agree with the 3DTI picture. However, this phase is predicted to emerge for tensile strained $\alpha$-Sn, while our samples are compressively strained [22]. On the other hand, the most recent ARPES studies confirmed that for in-plane compressive strain (which is the case for our samples) $\alpha$-Sn is a DSM [8, 9]. This portrays $\alpha$-Sn as a material which is topologically non-trivial regardless of the type of strain. For in-plane compressively strained film, the $\alpha$-Sn DSM phase should host two Dirac nodes, projected to the (100) plane as a single Dirac cone [1]. Topological surface states and Fermi arcs (between multiple Dirac points) can form as a consequence of strain, and/or the applied magnetic field [2]. However, magnetotransport experiments, aside of the detected topological Dirac-like 2D surface band, do not show the existence of the (DSM) Dirac bulk charge carriers in our samples (e.g. through an additional frequency in the SdH oscillations). For our experiments the lack of sensitivity to the topology of the bulk is due to the low mobility of the film bulk charge carriers and the obvious low resistance of the InSb substrate (resulting in pronounced shunting of the $\alpha$-Sn film). For the above stated reasons, based on our transport results it is not possible to unambiguously resolve the type of topological phase (3DTI or DSM) responsible for the Dirac cone observed in our $\alpha$-Sn films [22]. We conclude that we obtained evidence for the existence of topological Dirac fermion charge carriers in uncapped pure and robust $\alpha$-Sn(100) films. A $p$-type surface band with low carrier density and high mobility has been identified to be the origin of the observed SdH oscillations. To analyze these quantum oscillations, Landau index plots were fitted, revealing a non-trivial value of the phase shift, characteristic for topologically protected Dirac fermions. 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2024-09-04T02:54:55.198550

2002.12257

arxiv-papers

Further author information: (Send correspondence to Hector Santos-Villalobos) Hector Santos-Villalobos: E-mail<EMAIL_ADDRESS>Telephone: 1 865 574 0215 David S. Bolme: E-mail<EMAIL_ADDRESS>Telephone: 1 865 576 0300 Notice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe- public-access-plan). # The Mertens Unrolled Network (MU-Net): A High Dynamic Range Fusion Neural Network for Through the Windshield Driver Recognition Max Ruby Purdue University, 610 Purdue Mall, West Lafayette, USA David S. Bolme Oak Ridge National Laboratory, 1 Bethel Valley Rd, Oak Ridge, USA Joel Brogan Oak Ridge National Laboratory, 1 Bethel Valley Rd, Oak Ridge, USA David Cornett III Oak Ridge National Laboratory, 1 Bethel Valley Rd, Oak Ridge, USA Baldemar Delgado Texas A&M University-Kingsville, 700 University Blvd, Kingsville, USA Gavin Jager University of Michigan, 500 S. State St, Ann Arbor, USA Christi Johnson Oak Ridge National Laboratory, 1 Bethel Valley Rd, Oak Ridge, USA Jose Martinez-Mendoza Texas A&M University- Kingsville, 700 University Blvd, Kingsville, USA Hector Santos-Villalobos Oak Ridge National Laboratory, 1 Bethel Valley Rd, Oak Ridge, USA Nisha Srinivas Oak Ridge National Laboratory, 1 Bethel Valley Rd, Oak Ridge, USA ###### Abstract Face recognition of vehicle occupants through windshields in unconstrained environments poses a number of unique challenges ranging from glare, poor illumination, driver pose and motion blur. In this paper, we further develop the hardware and software components of a custom vehicle imaging system to better overcome these challenges. After the build out of a physical prototype system that performs High Dynamic Range (HDR) imaging, we collect a small dataset of through-windshield image captures of known drivers. We then re- formulate the classical Mertens-Kautz-Van Reeth HDR fusion algorithm as a pre- initialized neural network, which we name the Mertens Unrolled Network (MU- Net), for the purpose of fine-tuning the HDR output of through-windshield images. Reconstructed faces from this novel HDR method are then evaluated and compared against other traditional and experimental HDR methods in a pre- trained state-of-the-art (SOTA) facial recognition pipeline, verifying the efficacy of our approach. ###### keywords: MU-Net, HDR, Facial Recognition, Through Windshield ## 1 INTRODUCTION Face recognition of vehicle occupants in unconstrained environments, particularly through the windshield, poses a number of challenges. Previous research has shown that artifacts introduced by a windshield can greatly impact a camera’s ability to image within the vehicle interior [1]. Additionally, images captured in this scenario are typically from long distances and moderate speeds, which reduces the amount of light available to the sensor. Low-intensity ultraviolet or near-infrared (NIR) illumination has become increasingly ineffective in alleviating these issues as tinted windshields that block these wavelengths gain popularity. Likewise, increasing the exposure time cannot fix the problems associated with that since the vehicle is moving, and motion blur artifacts caused by increasing exposure time significantly degrades face recognition quality. Furthermore, windshields are reflective, which produces several unique challenges. First, it further reduces the light available to the sensors. Second, the windshield will reflect light from other sources into the sensor. If that light propagates directly from a light source, this will often cause obstructive glare (Figure 1). Even if that light is reflected off of an object onto the windshield, the object will appear as an unwanted artifact overlaid with the desired image of the driver. Current solutions for this problem include flashing visible lights at the driver, such as the system devised by Gatekeeper Security [2]. This is highly undesirable, as it both distracts the driver from safe vehicle operation and has the potential to cause obstructive glare due to the windshield. To provide the best possible input to a deep learning algorithm, a custom multi-camera imaging system was developed to specifically mitigate these hurdles while remaining non-intrusive [3]. The system is modular in design, where each unit is composed of both an imaging system and an associated computational system, seen in Figures 2, 3, and 4. The raw images captured by the system can subsequently be processed by any HDR method to ultimately provide the processed input to facial recognition software. Figure 1: A typical example of an HDR burst captured from the Through the Windshield System, including heavy glaring artifacts. This paper extends the previous work in two main ways: First, we collect and annotate a new dataset for the purpose of training HDR reconstruction models fine-tuned for our specific imaging hardware. Second, we propose an novel network architecture for HDR reconstruction by using an unrolling technique to model the Mertens-Kautz-Van Reeth HDR algorithm [4] as a neural network - we call this network the Mertens-Unrolled Network, or MU-Net. Finally, to analyze the end-to-end system, SOTA facial detection and recognition algorithms are used to evaluate the system’s image capture and HDR reconstruction quality in its overall goal to detect and identify faces through the windshield of moving vehicles. We also compare our efficacy of MU-Net against other classical and deelply-learned HDR methods from previous works in the same recognition pipeline. ## 2 Motivation The applications of this system are numerous. For example, the ability to recognize drivers through the windshield has the potential to speed up security checkpoints in restricted areas. These can also be permanently deployed along roads to locate criminals and terrorists. Moreover, a more agile version of this system can be used to greatly accelerate the flow of traffic through temporary checkpoints which have been set up to catch identified criminals who have recently committed a high-profile crime or escaped from prison. Alternatively, it can be used to help locate a child who is the focus of an amber alert, which can possibly return them to safety faster. However, contrast within the vehicle is a function of the ambient light, the shadows from the vehicle, and the tint of the windshield. We should expect that these conditions may change quickly; by the addition or removal of cloud cover, or by a passing driver with their lights on. If we have a single camera with fixed optics, then we cannot expect the image to have good contrast most of the time. Unfortunately, automatic camera gain is difficult given these situations. This is why HDR fusion techniques for our camera system are so valuable - we expect that if we take pictures from multiple cameras with varying optics, we should be able to get a high-quality image by combining them. ## 3 Related Work There is a notable body of work in the field of HDR fusion, a review of which can be found in [5]. Note the algorithms given by Mertens et. al. [4] and Debevec et. al. [6] usually perform adequate HDR fusion under normal circumstances. However, when dealing with through-windshield environments, these methods are not designed to account for the unfortunately common cases of obstructive glare, which obscures the face within one or more images in the HDR burst. Most external driver recognition research has been performed in the commercial sector [7] and public sector [8]. Beyond an initial case study at Oak Ridge National Laboratory [3], very little open research has been published on the topic. While little work has been done in the field of driver recognition through the windshield , there has been significant related work two related fields: Vehicle occupancy detection, and glass artifact suppression. In vehicle occupancy detection [9, 10, 11, 12], algorithms aim to detect and accurately count the number of occupants in an automobile, usually for the purpose of automatically enforcing HOV and carpool lane rules. In glass artifact suppression, algorithms look to effectively remove artifacts such as reflection, refraction, and glare, that are introduced by glass into real- world images [13, 14, 15, 16]. While these works are useful within their own context, they are solutions designed around separate problems and therefore sub-optimal to solve the through-windshield recognition task, due to less- stringent time or accuracy constraints on their output Convolutional Neural Networks (CNNs) are the the current SOTA approaches for a large number of image processing tasks. Some recent successes of the neural networks can be attributed to their ability learn and optimize the behavior previously established deterministic algorithms, as is pointed out in [17]. Moreover, the technique of taking an algorithm and turning it into a neural network - for example, through “deconstructing” or “Unrolling” - has been shown to effectively incorporate domain-specific priors into a given model[18, 19]. This technique can provide a straightforward improvement to a problem’s solution, as long as the solution in question is already acceptably effective and easily posed as a network architecture. If we have designed and initialize a network with parameters to accommodate a target algorithm, we can ensure that the original algorithm can not only be mimicked by the neural network, but further improve upon it through fine-tuning [20]. We should expect an algorithm re-formulated as a neural network to perform no worse than the original. There is a growing body of work focused around performing HDR reconstruction using a CNN. Many of them attempt to do this using a single shot [21, 22] or by simply having a extreme number of weights [21, 23, 24, 22]. For real-time, robust driver recognition, both of these are unacceptable. In the former case, we expect that a single shot will have a high probability of either misfiring or inaccurately capturing the face, due to the potential for obstructive glare. In the latter case, an extremely large model cannot be run locally, within an embedded system on-device, and expected to perform with adequate throughput for the frequency and volume of highway traffic. Previous work attempts to overcome these issues using a GAN approach [3], which this work improves upon. While our proposed Deep HDR Pipeline, the MU-Net, is designed and trained to work specifically with the our custom face imaging system, it’s architectural design mimicking the general Merten’s HDR algorithm inherently improves the method’s generalizability across all domains, not just through- windshield faces. ## 4 Computational Imaging System Overview Our custom imaging system was developed to withstand harsh environments as well as produce quick results for biometric assessment in field deployment. The overall system design is modular, where each computational unit added can perform stand-alone, or input its data into a central analysis system. A single computational imaging unit is comprised of two primary components: weatherproof camera enclosure and a ruggedized computer. The camera enclosure is shown in Figure 2 and the associated computing system can be seen in Figure 3. The camera enclosure houses an array of three Basler GigE cameras ($2048\times 1536$ pixel matrix, 3.45 $\mu$m pixel size), mounted on a 3D-printed bracket and arranged horizontally. The cameras are linked to the associated computer unit via Power over Ethernet (PoE) connections. They are aligned to have a common focal point at 10 meters. Gain and exposure time is programmatically adjusted at the time of data acquisition. The cameras are equipped with filters as follows: First, all cameras are equipped with a 50 mm focal length and 34 mm aperture lens. Next, all cameras are equipped with a Midwest Optical Systems UVIR filter to sample only visible light. Third, Cameras #1 and #2 are equipped with a Midwest Optical Systems 0.9 and 0.3 ND filter, respectively. Last, cameras #1, #2, and #3 are equipped with Meadowlark Optics linear polarizers (extinction ratio greater than 10,000,000:1) placed at angles of $80^{\circ}$, $90^{\circ}$, and $100^{\circ}$, respectively. The ruggedized computer is also housed in a 3D-printed, weatherize enclosure. The computer has 32GB of memory, Intel Core i7 CPU with 8 processing cores and a NVIDIA GTX 1050 Ti GPU. The computer was chosen for its PoE ports used to interface with the camera unit as well as its GPU so that it can handle the computational burden of the implemented deep learning algorithms. To trigger the imaging system an Optex through beam photoelectric sensor was used. To ensure frame synchronization across multiple systems the same trigger signal was distributed to all systems. Once triggered, an imaging unit captures 20 frame-sets (20 x 3 cameras = 60 images per frame-set). The system software is also modular by design, and leverages the Google Remote Procedure Call (gRPC) library. The gRPC library allows for the use of independent servers so that each task of the image acquistion and processing pipeline can be managed independently. For the all experiments mentioned in this paper, two modular units were used - one on either side of the driving lane that test subjects would drive through. This configuration was used to account for driver head pose, ensuring that a sufficiently frontal face shot could be captured for detection and identification tasks. The layout of the units can be observed in Figure 4. Figure 2: Computational imaging unit - weatherproof camera array. Figure 3: Computational imaging unit - weatherproof ruggedized computer. Figure 4: Two- unit camera setup for dataset collection at security portal. ## 5 Overview of Image Pipeline Before images are acquired, the cameras are calibrated to give an initial estimate for coarse image registration; After this, we use FaRO’s [25] implementation of RetinaFace face detection on the three images. Using these detection locations, a secondary registration is performed (in the case that a face was not detected in a given image, this step is skipped). We crop the images to 600 $\times$ 600 pixels and perform fine registration using a modified form FFT-based image registration [26]; We modify the algorithm to multiply the images using a Kaiser window with $\beta=4$ as to “prefer” smaller shifts (we expect the fine registration to be, in our case, usually between $30$ and $150$ pixels in any given direction, as we have already been coarsely registered). We utilize this method of registration to achieve accurate correspondence and alignment without the need to detect facial landmark points, which are often noisy or incorrect within our dataset. It should be noted that this method deviates from the previous work, [3], which uses Hamming windows to perform alignment instead. The images are then cropped to $256\times 256$ pixels and then fed into the HDR fusion neural network outlined in Figure 5. This produces a final face tile which can then be fed to a biometric template extractor. ## 6 MU-Net: Mertens Unrolled Network Motivated by the simplicity of the Mertens HDR algorithm, we first wanted to show that the algorithm itself could be reformulated within a CNN architecture. Therefore, MU-Net was conceived by applying an “Unrolling”, first used in [27], to the Mertens algorithm. For completeness, a rough sketch of the Mertens HDR algorithm follows. First, for each pixel, the contrast and well-exposedness of each pixel is computed, by the formulas $\begin{split}C_{i,j,k}=|R_{i-1,j,k}+R_{i+1,j,k}+R_{i,j-1,k}\\\ +R_{i,j+1,k}-4R_{i,j,k}|\end{split}$ (1) and $X_{i,j,k}=\exp\left(-\frac{1}{2}\left(\frac{R_{i,j,k}-0.5}{\sigma}\right)^{2}\right)$ (2) where $R_{i,j,k}\in[0,1]$ is the $(i,j)$th pixel of the $k$th image, and $\sigma$ is some standard deviation (in the standard implementation $\sigma=0.2$.) Concretely, $X$ is a measure of image exposure, while $C$ is a measure of contrast. We note that for a color image, there is also a measure of saturation; this network can be extended to include this saturation measure, but in our greyscale implementation it is not necessary. After this, the $C$ and $X$ measurements are multiplied to give a measure of “quality” of each pixel of each image. This quality metric is then normalized, and then the images are mixed using a Laplacian Blending scheme, as in [28]. We determine that the way to proceed is to develop a network composed of three blocks, following the scheme above. First, a Contrast Block, or “C-Block”, which is initialized to compute the contrast as above. Second, a Well-Exposure Block, or “X-Block,”, which is initialized to compute the Well-Exposure as above. Third, a Pyramid Blending Block, or “P-Block,”, which is initialized to perform Laplacian Pyramid Blending. The output of the C-Block and X-Block is multiplied together, a convolutional layer is applied, and the resulting “Image Quality” is fed into the P-Block along with the original images. A sketch of the neural network, as well as the blocks, are in Figure 5. Note that only 3 “steps” of the P-Block are shown in Figure 5, for brevity. In reality, our network contains 8 (that is, the network should be naturally extended so that the output of the upper-right and lower-right 2D convolutional layers are $1\times 1$ pixel each, as each convolutional layer has a stride of 2) Figure 5: An overview of the MU-Net architecture. From left to right: i) The Contrast Block (C-Block), ii) The Exposure Block (X-Block), iii) the Concatenation Block (P-Block). Note that P-Block shows only 3 unrolled loops for brevity. Each convolutional and upsampling layer has a stride of 2 with 3 output channels, unless denoted by a superscript. Each conv layer utilizes tanh activation unless otherwise denoted with a ReLU block. * - contains an ELU, Negative, Exp, and Tanh activation. Unless denoted with a RelU block, each convolutional layer in Figure 5 is followed by a tanh activation. The C-Block’s first convolutional layer has 6 output channels, while its second has 3 output channels, using $7\times 7$ kernels. Each layer in the X-Block has 3 channels with $3\times 3$ kernels. The ReLU layer that has been starred within the X-Block has an ELU, Negative, Exp, and Tanh activation. The output of the last Conv2D layer represents an exponent, whose value we want to remain negative so that our network’s output never “explodes”. The ELU and Negative force that condition on that parameter. The Conv2D layer between the P-Block has $3$ output channels and a $3\times 3$ kernel. Every Conv2D layer in the P-block has a $7\times 7$ kernel and a tanh activation function. All of them have 3 output channels, with the exception of the ones which are superscripted. As we developed this generator by unrolling the Laplacian Blending scheme in the Mertens algorithm, we call it the Mertens Unrolled Network, or “MU-Net.” Note that the architecture is rather light-weight: our current architecture has under $45,000$ parameters. A significant advantage to this approach is that we avoid treating our neural network like a “black box.” Specifically, we expect that our C-Block measures something similar to the contrast, our X-Block measures something similar to the Well-Exposedness, and that our P-Block performs something similar to Laplacian Pyramid Blending. This means that it should be possible to debug the MU-Net, and we can identify when and where our training has gone badly. ### 6.1 MU-Net Discriminator: The NU-Net We use a Feature Pyramid Network (FPN) architecture to design our discriminator [29]. The FPN provides a multi-scale framework for learning a feature to detect real and fake images at different frequency scales. If we take the maximum element in each channel (that is, if we maxpool down to $1\times 1$ pixel), this should serve as an indicator of how much each feature exists within the input image. This pooled result is then feed that a classifier layer with a single dimension output. We decided to downsample more aggressively than in the MU-Net to reduce the number of parameters required; it was found that it was easy for the architecture to overtrain if it was given more parameters, so we opt to use a smaller network. Moreover, we feed the contrast well-exposure maps from the C-Block and X-Block into the discriminator, as these are pertinent features to generate correctly when producing a fused HDR image. Figure 6: NU-Net (Discriminator) Architecture. The architecture for NU-Net is shown in Figure 6. The upper half of the diagram is the same as in the MU-Net, utilizing the C- and X-Block. The leftmost Conv2D layers all have a $7\times 7$ kernel, except for the last, which has a $4\times 4$ kernel, due to reduced resolution. They have 4,8,16, and 32 output channels, from top to bottom, respectively. The upsampling Conv2D layers have 16 output channels and a $7\times 7$ kernel, except the first, which has a $3\times 3$ kernel due to reduced resolution. The rightmost Conv2D layers all have $8$ output channels, with $5\times 5$ kernels. As we developed this discriminator by doing something other than unrolling, we call it the Not Unrolled Network, or “NU-Net.” Both the MU-Net and NU-Net were built in Keras, using the Tensorflow backend. ### 6.2 Training We were able to take a number of photos at the security gate of a restricted access vehicle portal. From this set of photos, we were able to capture $1957$ triples of images which were of faces, as well as $2371$ images of faces which were of sufficiently high quality for training the discriminator. Of this, we used $1440$ of the triples and face tiles for training, $160$ for validation, and the rest as a testing set. We determined that the use of Batch Normalization layers to overcome the exploding gradients problem was not a good solution; naive approaches to Batch Normalization eliminate the advantages of our initialization, due to our initialization passing values of nonzero mean and nonunit variance between layers. We opted to perform $L1$ normalization on the layers, with $\lambda=0.001$, along with gradient clipping, with our clipnorm value being $0.1$. Moreover, we determined that pretraining the adversary for $10$ epochs would improve the initial training, as our generator had a “head-start” due to its good initialization. We used ADAM to optimize, with an initial learning rate of $0.005$, a learning rate decay of $0.000001$, and a value of epsilon equal to $0.001$; we set all other parameters to the defaults. During training, we noticed that the output of our GAN began to degenerate once the adversary began to learn. It seems that this is due to the generator learning a “cheap trick” that fools the adversary, but does not produce realistic looking images. This was solved by taking the training result before degeneration; we save output images after each epoch, and take the result that looks qualitatively best. Although we trained for $100$ epochs, we determined that the result of training after the $71$st epoch provided the best aesthetic quality. ## 7 Experimental Evaluation and Results This section details our effort to evaluate and analyze the effectiveness of the MU-Net HDR algorithm for the purpose of through-windshield facial recognition. Figure 7: ROC performance of MU-Net compared to other HDR Methods. ### 7.1 The Driver Dataset To evaluate MU-Net on the Image pipeline from Section 5, we capture a new dataset of known drivers, disjoint from the training data captured in Section 6.2. The dataset was captured at a facility security portal using live driving conditions over the course of two days. For this experiment there were 22 participants enrolled as gallery subjects for the recognition set. These participants entered the facility through the security portal under normal driving conditions. The images captured during this experiment represented the scene illumination changes that occur through sunrise to sundown. There were 471 vehicle triggered events that produced 56,520 raw images. For each of these events that involved an enrolled participant, each of the three raw image frame-sets were fused via each of the evaluated HDR methods to be used as probes by the face identification systems. ### 7.2 Baseline Methods Classical HDR fusion can be performed in a number of ways, usually combining images taken at different radiance ranges and fuses them into a single radiance image. This method allows the fused image to display a range of radiance levels that could not be captured by a single image. For our application, HDR imaging provides a means to overcome the artifacts introduced into face images from windshield glare and reflection. A brief summary of all methods we will compare to our novel approach is laid out below: * • Matlab HDR[30] is the standard Matlab implementation for HDR fusion. It is used as a standard comparison against the other methods. * • Mertens-Kautz-Van HDR (Mertens)[4] is based on the Mertens algorithm utilizes contrast and exposure estimates to fuse the three raw frames for detection and recognition. * • Debevec HDR[6] is based on Debevec radiance mapping with Durand tone mapping. This method is also a physics-based implementation of HDR. * • PeceptionGAN[3] is a GAN network trained specifically for visual perception and was the initial design for the system used in [3]. Figure 8: Sampled images from cameras (top), input to HDR (middle), Mertens HDR reconstruction (bottom left), and MU-Net HDR reconstruction (bottom right). To analyze the efficacy of MU-Net, we performed a verification study on the dataset introduced in Section 7.1, comparing performance against the 5 baselines described in Section 7.2. We also include the baseline Single Camera, which uses a randomly-selected, non-processed image from each HDR burst. For each method, we detect faces using RetinaFace [31] and subsequently extract feature templates using Additive Angular Margin Loss for Deep Face Recognition (ArcFace)[32]. Because our driver dataset is relatively small and was not used for network training, we report results over the entire set of subjects. Figure 7 Shows the final results of all 6 methods. The MU-Net provides the best performance out of all those evaluated, improving upon the PerceptionGAN by almost $7\%$. As can be seen by the AUC’s of most other methods, this problem is inherently difficult to solve, and classical HDR methods do not provide high enough quality reconstructions for accurate face recognition. Figure 8 shows that classical methods like Debevec and Merten’s are not robust towards artifacts introduced by windshield glare and reflection, which significantly impairs verification performance. Quantitative and Qualitative analysis of the MU-Net pipeline revealed two key features of the system: First, MU-Net is better able to discriminate areas of poor quality, including glare and reflection artifacts, than the original Mertens or Debevec HDR methods, as seen in Figure 8 columns 3,4, and 5. Second, MU-Net provides output reconstructions that are more stable than that of PerceptionGan. Because the underlying architecture of MU-Net was designed and initialized from a state that already performed classical HDR, the network overfits our data less, resulting in fewer GAN-like artifacts (see Figure 8 columns 6 and 7). We see in general that MU-Net consistently outputs a more stable image free from windshield artifacts than its competing baselines. While MU-Net improves upon previous methods for through-windshield recognition, there is still significant work to be done to achieve a highly reliable system. For instance, MU-Net can suffer when running in low-light conditions. This highlights the need for more uniform data collection, ideally over the course of an entire day. Moreover, the MU-Net was trained on faces - it does not appear to work properly on images that are not faces. ## 8 Conclusion and Discussion The MU-Net architecture described in this paper represents a significant step forward in the use of neural network architectures for HDR fusion for face recognition through the windshield. Using this method we are able to obtain higher quality images than classical fusion methods while maintaining a relatively low computational overhead. While other works have performed HDR using deep learning, our method is unique in that it is designed on top of the original Merten’s HDR algorithm, and has a relatively low number of parameters in contrast to other comparative algorithms. The MU-Net shows that we are able to successfully incorporate exposure and contrast quality priors into our deep learning architecture, allowing for faster training, more stable fusion results that introduce significantly fewer GAN-related generative artifacts. While the MU-Net is a major improvement over previous methods, significant work must be done to further enhance facial recognition in the adverse conditions of through-windshield imaging. We believe it would be beneficial to train the network stacked end-to-end with a face recognition algorithm, eliminating the need for training with an ad-hoc discriminator, and optimizing the HDR fusion expressly for the task of facial recognition. Efforts are currently being made to expand the custom dataset of through the windshield images. This expansion would include more enrolled subjects, a more expansive training set, and a wider array of weather and illumination conditions. Additionally, the custom imaging system is being redesigned with better optics, utilizing polarized filter arrays to mitigate glare for different angles of incidence. ###### Acknowledgements. This research was supported in part by an appointment to the Oak Ridge National Laboratory Post-Bachelor’s Research Associate Program, sponsored by the U.S. Department of Energy and administered by the Oak Ridge Institute for Science and Education, the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists (WDTS) under the Science Undergraduate Laboratory Internships Program (SULI), and the Texas A&M University-Kingsville, Office of University Programs, Science and Technology Directorate, Department of Homeland Security Grant Award # 2012-ST -062-000054. ## References * [1] Bhise, V. and Sethumadhavan, S., “Effect of windshield veiling glare on driver visibility,” Transportation Research Record 2056(1), 1–8 (2008). * [2] Harmon, D., “Facial recognition and registration plate reading,” (2017). * [3] Cornett, D., Yen, A., Nayola, G., Montez, D., Johnson, C. R., Baird, S. T., Santos-Villalobos, H., and Bolme, D. 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A., “Convolutional neural networks for inverse problems in imaging: A review,” IEEE Signal Processing Magazine 34, 85–95 (2017). * [18] Adler, J. and Öktem, O., “Learned primal-dual reconstruction,” (2017). * [19] Putzky, P. and Welling, M., “Recurrent inference machines for solving inverse problems,” (2017). * [20] Rao, Y. and Ni, J., “A deep learning approach to detection of splicing and copy-move forgeries in images,” in [2016 IEEE International Workshop on Information Forensics and Security (WIFS) ], 1–6, IEEE (2016). * [21] Huo, Y. and Zhu, X., “High dynamic range image forensics using cnn,” CoRR abs/1902.10938 (2019). * [22] Marnerides, D., Bashford-Rogers, T., Hatchett, J., and Debattista, K., “Expandnet: A deep convolutional neural network for high dynamic range expansion from low dynamic range content,” CoRR abs/1803.02266 (2018). * [23] WANG, J., Wang, W., XU, G., and Liu, H., “End-to-end exposure fusion using convolutional neural network,” IEICE Transactions on Information and Systems E101.D, 560–563 (02 2018). * [24] Eilertsen, G., Kronander, J., Denes, G., Mantiuk, R., and Unger, J., “Hdr image reconstruction from a single exposure using deep cnns,” ACM Transactions on Graphics (TOG) 36(6) (2017). * [25] Bolme, D. S., III, D. C. C., and Srinivas, N., “FaRO: FAce Recognition from Oak ridge.” https://github.com/ORNL/faro (2019). * [26] Reddy, B. S. and Chatterji, B. N., “An fft-based technique for translation, rotation, and scale-invariant image registration,” IEEE Transactions on Image Processing 5, 1266–1271 (Aug 1996). * [27] Gregor, K. and LeCun, Y., “Learning fast approximations of sparse coding,” in [Proceedings of the 27th International Conference on International Conference on Machine Learning ], ICML’10, 399–406, Omnipress, USA (2010). * [28] ADELSON, H., “Laplacian pyramid as a compact image code peter j. burt,” (07 2019). * [29] Lin, T., Dollár, P., Girshick, R. B., He, K., Hariharan, B., and Belongie, S. 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2024-09-04T02:54:55.212797

2002.12271

arxiv-papers

empty # Deep Reinforcement Learning Based Intelligent Reflecting Surface for Secure Wireless Communications Helin Yang, , Zehui Xiong, , Jun Zhao, , Dusit Niyato, , Liang Xiao, , and Qingqing Wu This research is supported by the National Research Foundation (NRF), Singapore, under Singapore Energy Market Authority (EMA), Energy Resilience, NRF2017EWT-EP003-041, Singapore NRF2015-NRF-ISF001-2277, Singapore NRF National Satellite of Excellence, Design Science and Technology for Secure Critical Infrastructure NSoE DeST-SCI2019-0007, A*STAR-NTU-SUTD Joint Research Grant on Artificial Intelligence for the Future of Manufacturing RGANS1906, Wallenberg AI, Autonomous Systems and Software Program and Nanyang Technological University (WASP/NTU) under grant M4082187 (4080), Singapore Ministry of Education (MOE) Tier 1 (RG16/20), and NTU-WeBank JRI (NWJ-2020-004), Alibaba Group through Alibaba Innovative Research (AIR) Program, Alibaba-NTU Singapore Joint Research Institute (JRI), Nanyang Technological University (NTU) Startup Grant, Singapore Ministry of Education Academic Research Fund Tier 1 RG128/18, Tier 1 RG115/19, Tier 1 RT07/19, Tier 1 RT01/19, and Tier 2 MOE2019-T2-1-176, NTU-WASP Joint Project, Singapore National Research Foundation under its Strategic Capability Research Centres Funding Initiative: Strategic Centre for Research in Privacy-Preserving Technologies & Systems, Energy Research Institute @NTU , Singapore NRF National Satellite of Excellence, Design Science and Technology for Secure Critical Infrastructure NSoE DeST-SCI2019-0012, AI Singapore 100 Experiments (100E) programme, NTU Project for Large Vertical Take-Off & Landing Research Platform, and the Natural Science Foundation of China under Grant 61971366. A part of this paper was accepted in IEEE Global Communications Conference, December 2020, Taipei, Taiwan [40]. _(Corresponding author: Zehui Xiong.)_ H. Yang, Z. Xiong, J. Zhao, and D. Niyato are with the School of Computer Science and Engineering, Nanyang Technological University, Singapore 639798 (e-mail: <EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>[email protected]). L. Xiao is with the Department of Information and Communication Engineering, Xiamen University, Xiamen 361005, China (e-mail: [email protected]).Q. Wu is with State Key Laboratory of Internet of Things for Smart City, University of Macau, Macau, 999078 China. (email:[email protected]). ###### Abstract In this paper, we study an intelligent reflecting surface (IRS)-aided wireless secure communication system, where an IRS is deployed to adjust its reflecting elements to secure the communication of multiple legitimate users in the presence of multiple eavesdroppers. Aiming to improve the system secrecy rate, a design problem for jointly optimizing the base station (BS)’s beamforming and the IRS’s reflecting beamforming is formulated considering different quality of service (QoS) requirements and time-varying channel conditions. As the system is highly dynamic and complex, and it is challenging to address the non-convex optimization problem, a novel deep reinforcement learning (DRL)-based secure beamforming approach is firstly proposed to achieve the optimal beamforming policy against eavesdroppers in dynamic environments. Furthermore, post-decision state (PDS) and prioritized experience replay (PER) schemes are utilized to enhance the learning efficiency and secrecy performance. Specifically, a modified PDS scheme is presented to trace the channel dynamic and adjust the beamforming policy against channel uncertainty accordingly. Simulation results demonstrate that the proposed deep PDS-PER learning based secure beamforming approach can significantly improve the system secrecy rate and QoS satisfaction probability in IRS-aided secure communication systems. _Index Terms_ —Secure communication, intelligent reflecting surface, beamforming, secrecy rate, deep reinforcement learning. ## I Introduction Physical layer security (PLS) has attracted increasing attention as an alternative of cryptography-based techniques for wireless communications [1], where PLS exploits the wireless channel characteristics by using signal processing designs and channel coding to support secure communication services without relying on a shared secret key [1], [2]. So far, a variety of approaches have been reported to improve PLS in wireless communication systems, e.g., cooperative relaying strategies [3], [4], artificial noise- assisted beamforming [5], [6], and cooperative jamming [7], [8]. However, employing a large number of active antennas and relays in PLS systems incurs an excessive hardware cost and the system complexity. Moreover, cooperative jamming and transmitting artificial noise require extra transmit power for security guarantees. To tackle these shortcomings of the existing approaches [3]-[8], a new paradigm, called intelligent reflecting surface (IRS) [9]-[13], has been proposed as a promising technique to achieve high spectrum efficiency and energy efficiency, and enhance secrecy rate in the fifth generation (5G) and beyond wireless communication systems. In particular, IRS is a uniform planar array which is comprised of a number of low-cost passive reflecting elements, where each of elements adaptively adjusts its reflection amplitude and/or phase to control the strength and direction of the electromagnetic wave, hence IRS is capable of enhancing and/or weakening the reflected signals at different users [9]. As a result, the reflected signal by IRS can increase the received signal at legitimate users while suppressing the signal at the eavesdroppers [9]-[13]. Hence, from the PLS perspective, some innovative studies have been recently devoted to performance optimization for IRS-aided secure communications [14]-[25]. ### I-A Related Works Initial studies on IRS-aided secure communication systems have reported in [14]-[17], where a simple system model with only a single-antenna legitimate user and a single-antenna eavesdropper was considered in these works. The authors in [14] and [15] applied the alternative optimization (AO) algorithm to jointly optimize the transmit beamforming vector at the base station (BS) and the phase elements at the IRS for the maximization of the secrecy rate, but they did not extend their models to multi-user IRS-assisted secure communication systems. To minimize the transmit power at the BS subject to the secrecy rate constraint, the authors in [18] utilized AO solution and semidefinite programming (SDP) relaxation to address the optimization problem with the objective to jointly optimize the power allocation and the IRS reflecting beamforming. In addition, Feng $et~{}al$. [19] also studied the secure transmission framework with an IRS to minimize the system transmit power in cases of rank-one and full-rank BS-IRS links, and derived a closed- form expression of beamforming matrix. Different from these studies [14]-[19] which considered only a single eavesdropper, secure communication systems comprising multiple eavesdroppers were investigated in [20]-[22]. Chen $et~{}al$. [20] presented a minimum-secrecy-rate maximization design to provide secure communication services for multiple legitimate users while keeping them secret from multiple eavesdroppers in an IRS-aided multi-user multiple-input single-output (MISO) system, but the simplification of the optimization problem may cause a performance loss. The authors in [23] and [24] studied an IRS-aided multiple-input multiple-output (MIMO) channel, where a multi-antenna BS transmits data stream to a multi-antenna legitimate user in the presence of an eavesdropper configured with multiple antennas, and a suboptimal secrecy rate maximization approach was presented to optimize the beamforming policy. In addition to the use of AO or SDP in the system performance optimization, the minorization-maximization (MM) algorithm was recently utilized to optimize the joint transmit beamforming at the BS and phase shift coefficient at the IRS [16], [23]. Moreover, the authors in [22] and [25] employed the artificial noise-aided beamforming for IRS-aided MISO secure communication systems to improve the system secrecy rate, and an AO based solution was applied to jointly optimize the BS’s beamforming, artificial noise interference vector and IRS’s reflecting beamforming with the goal to maximize the secrecy rate. All these existing studies [14]-[20], [22]-[25] assumed that perfect channel state information (CSI) of legitimate users or eavesdroppers is available at the BS, which is not a practical assumption. The reason is that acquiring perfect CSI at the BS is challenging since the corresponding CSI may be outdated when the channel is time-varying due to the transmission delay, processing delay, and high mobility of users. Hence, Yu $et~{}al$. [21] investigated an optimization problem with considering the impacts of outdated CSI of the eavesdropping channels in an IRS-aided secure communication system, and a robust algorithm was proposed to address the optimization problem in the presence of multiple eavesdroppers. The above mentioned studies [14]-[25] mainly applied the traditional optimization techniques e.g., AO, SDP or MM algorithms to jointly optimize the BS’s beamforming and the IRS’s reflecting beamforming in IRS-aided secure communication systems, which are less efficient for large-scale systems. Inspired by the recent advances of artificial intelligence (AI), several works attempted to utilize AI algorithms to optimize IRS’s reflecting beamforming [26]-[29]. Deep learning (DL) was exploited to search the optimal IRS reflection matrices that maximize the achievable system rate in an IRS-aided communication system, and the simulation demonstrated that DL significantly outperforms conventional algorithms. Moreover, the authors in [28] and [29] proposed deep reinforcement learning (DRL) based approach to address the non- convex optimization problem, and the phase shifts at the IRS are optimized effectively. However, the works [26]-[29] merely considered to maximize the system achievable rate of a single user without considering the scenario of multiple users, secure communication and imperfect CSI in their models. The authors in [30] and [31] applied reinforcement learning (RL) to achieve smart beamforming at the BS against an eavesdropper in complex environments, but the IRS-aided secure communication system needs to optimize the IRS’s reflect beamforming in addition to the BS’s transmit beamforming. To the best of our knowledge, RL or DRL has not been explored yet in prior works to optimize both the BS’s transmit beamforming and the IRS’s reflect beamforming in dynamic IRS-aided secure communication systems, under the condition of multiple eavesdroppers and imperfect CSI, which thus motivates this work. ### I-B Contributions In this paper, we investigate an IRS-aided secure communication system with the objective to maximize the system secrecy rate of multiple legitimate users in the presence of multiple eavesdroppers under realistic time-varying channels, while guaranteeing quality of service (QoS) requirements of legitimate users. A novel DRL-based secure beamforming approach is firstly proposed to jointly optimize the beamforming matrix at the BS and the reflecting beamforming matrix (reflection phases) at the IRS in dynamic environments. The major contributions of this paper are summarized as follows: * • The physical secure communication based on IRS with multiple eavesdroppers is investigated under the condition of time-varying channel coefficients in this paper. In addition, we formulate a joint BS’s transmit beamforming and IRS’s reflect beamforming optimization problem with the goal of maximizing the system secrecy rate while considering the QoS requirements of legitimate users. * • An RL-based intelligent beamforming framework is presented to achieve the optimal BS’s beamforming and the IRS’s reflecting beamforming, where the central controller intelligently optimizes the beamforming policy by using a Markov decision process (MDP) according to the instantaneous observations from dynamic environment. Specifically, a QoS-aware reward function is constructed by covering both the secrecy rate and users’ QoS requirements into the learning process. * • A DRL-based secure beamforming approach is proposed to improve the learning efficiency and secrecy performance by fully exploiting the information of complex structure of the beamforming policy domain, where a modified post- decision state (PDS) learning is presented to trace the channel dynamic against channel uncertainty, and prioritized experience replay (PER) is applied to enhance the learning efficiency. * • Extensive simulation results are provided to demonstrate the effectiveness of the proposed deep PDS-PER leaning based secure beamforming approach in terms of improving the secrecy rate and the QoS satisfaction probability, compared with other existing approaches. For instance, the proposed learning approach achieves the secrecy rate and QoS satisfaction level improvements of 17.21% and 8.67%, compared with the approach [14] in time-varying channel condition. The rest of this paper is organized as follows. Section II presents the system model and problem formulation. The optimization problem is formulated as an RL problem in Section III. Section IV proposes a deep PDS-PER based secure beamforming approach. Section V provides simulation results and Section VI concludes the paper. Notations: In this paper, vectors and matrices are represented by Boldface lowercase and uppercase letters, respectively. ${\rm{Tr}}(\cdot)$, ${(\cdot)^{*}}$ and ${(\cdot)^{H}}$ denote the trace, the conjugate and the conjugate transpose operations, respectively. $|\cdot|$ and $||\cdot||$ stand for the absolute value of a scalar and the Euclidean norm of a vector or matrix, respectively. $\mathbb{E}[\cdot]$ denotes the expectation operation. ${\mathbb{C}^{M\times N}}$ represents the space of complex-valued matrices. ## II System Model and Problem Formulation ### II-A System Model Figure 1: IRS-aided secure communication under multiple eavesdroppers. We consider an IRS-aided secure communication system, as shown in Fig. 1, where the BS is equipped with $N$ antennas to serve $K$ single-antenna legitimate mobile users (MUs) in the presence of $M$ single-antenna eavesdroppers. An IRS with $L$ reflecting elements is deployed in the system to assist secure wireless communications from the BS to the MUs. The IRS is equipped with a controller to coordinate with the BS. For the ease of practical implementation, the maximal reflection without power loss at the IRS is considered since the reflecting elements are designed to maximize the reflected desired signal power to the MUs [13]-[23]. In addition, unauthorized eavesdroppers aim to eavesdrop any of the data streams of the MUs. Hence, the use of reflecting beamforming at IRS is also investigated to improve the achievable secrecy rate at the MUs while suppressing the wiretapped data rate at the eavesdroppers. In addition, we explicitly state that the eavesdroppers cannot collide [5], [6], [18]-[21]. Let $\mathcal{K}=\\{1,2,\ldots,K\\}$, $\mathcal{M}=\\{1,2,\ldots,M\\}$ and $\mathcal{L}=\\{1,2,\ldots,L\\}$ denote the MU set, the eavesdropper set and the IRS reflecting element set, respectively. Let ${\bf{H}_{{\rm{br}}}}\in{\mathbb{C}^{L\times N}}$, ${\bf{h}}_{{\rm{bu,}}k}^{H}\in{\mathbb{C}^{1\times N}}$, ${\bf{h}}_{{\rm{ru,}}k}^{H}\in{\mathbb{C}^{1\times L}}$, ${\bf{h}}_{{\rm{be,}}m}^{H}\in{\mathbb{C}^{1\times N}}$, and ${\bf{h}}_{{\rm{re,}}m}^{H}\in{\mathbb{C}^{1\times L}}$ denote the channel coefficients from the BS to the IRS, from the BS to the $k$-th MU, from the IRS to the $k$-th MU, from the BS to the $m$-th eavesdropper, and from the IRS to the $m$-th eavesdropper, respectively. All the above mentioned channel coefficients in the system are assumed to be small-scale fading with path loss which follows the Rayleigh fading model [11]-[14], [21]. Let ${\bf{\Psi}}={\rm{diag}}({\chi_{1}}{e^{j{\theta_{1}}}},{\chi_{2}}{e^{j{\theta_{2}}}},\ldots,{\chi_{L}}{e^{j{\theta_{L}}}})$ denote the reflection coefficient matrix associated with effective phase shifts at the IRS, where ${\chi_{l}}\in[0,1]$ and ${\theta_{l}}\in[0,2\pi]$ denote the amplitude reflection factor and the phase shift coefficient on the combined transmitted signal, respectively. As each phase shift is desired to be designed to achieve full reflection, we consider that ${\chi_{l}}=1$, $\forall l\in\mathcal{L}$ in the sequel of the paper. At the BS side, the beamforming vector for the $k$-th MU is denoted as ${{\bf{v}}_{k}}\in{\mathbb{C}^{N\times 1}}$, which is the continuous linear precoding [11]-[16], [23]. Thus, the transmitted signal for all MUs at the BS is written as ${\bf{x}}=\sum\nolimits_{k=1}^{K}{{{\bf{v}}_{k}}{s_{k}}}$, where ${s_{k}}$ is the transmitted symbol for the $k$-th MU which can be modelled as independent and identically distributed (i.i.d.) random variables with zero mean and unit variance [11]-[16], [23], and ${s_{k}}\sim\mathcal{CN}(0,1)$. The total transmit power at the BS is subject to the maximum power constraint: $\begin{split}\mathbb{E}[||{\bf{x}}{||^{2}}]={\rm{Tr(}}{\bf{V}}{\bf{V}}^{H}{\rm{)}}\leq{P_{\max}}\end{split}$ (1) where ${\bf{V}}\buildrel\Delta\over{=}[{{\bf{v}}_{1}},{{\bf{v}}_{2}},\ldots,{{\bf{v}}_{K}}]\in{\mathbb{C}^{M\times K}}$, and ${P_{\max}}$ is the maximum transmit power at the BS. When the BS transmits a secret message to the $k$-th MU, the MU will receive the signal from the BS and the reflected signal from the IRS. Accordingly, the received signal at MU $k$ can be given by $\begin{split}\begin{array}[]{l}{y_{k}}=\underbrace{\left({{\bf{h}}_{{\rm{ru,}}k}^{H}{\bf{\Psi}}{{\bf{H}}_{{\rm{br}}}}+{\bf{h}}_{{\rm{bu,}}k}^{H}}\right){{\bf{v}}_{k}}{s_{k}}}_{{\rm{desired}}\;{\rm{signal}}}\\\ \;\;\;\;+\underbrace{\sum\limits_{i\in{\mathcal{K}},i\neq k}{\left({{\bf{h}}_{{\rm{ru,}}k}^{H}{\bf{\Psi}}{{\bf{H}}_{{\rm{br}}}}+{\bf{h}}_{{\rm{bu,}}k}^{H}}\right){{\bf{v}}_{i}}{s_{i}}}}_{{\rm{inter- userinterference}}}+{n_{k}}\end{array}\end{split}$ (2) where ${n_{k}}$ denotes the additive complex Gaussian noise (AWGN) with the with zero mean and variance $\delta_{k}^{2}$ at the $k$-th MU. In (2), we observe that in addition to the received desired signal, each MU also suffers inter-user interference (IUI) in the system. In addition, the received signal at eavesdropper $m$ is expressed by $\begin{split}{y_{m}}=\left({{\bf{h}}_{{\rm{re,}}m}^{H}{\bf{\Psi}}{{\bf{H}}_{{\rm{br}}}}+{\bf{h}}_{{\rm{be,}}m}^{H}}\right)\sum\limits_{k\in{\mathcal{K}}}{{{\bf{v}}_{k}}{s_{k}}}+{n_{m}}\end{split}$ (3) where ${n_{m}}$ is the AWGN of eavesdropper $m$ with the variance $\delta_{m}^{2}$ . In practical systems, it is not easy for the BS and the IRS to obtain perfect CSI [9], [21]. This is due to the fact that both the transmission delay and processing delay exist, as well as the mobility of the users. Therefore, CSI is outdated at the time when the BS and the IRS transmit the data stream to MUs [21]. Once this outdated CSI is employed for beamforming, it will lead to a negative effect on the demodulation at the MUs, thereby leading to substantial performance loss [21]. Therefore, it is necessary to consider outdated CSI in the IRS-aided secure communication system. Let ${T_{{\rm{delay}}}}$ denote the delay between the outdated CSI and the real-time CSI. In other words, when the BS receives the pilot sequences sent from the MUs at the time slot $t$, it will complete the channel estimation process and begin to transmit data stream to the MUs at the time slot $t+{T_{{\rm{delay}}}}$. Hence, the relation between the outdated channel vector ${\bf{h}}(t)$ and the real-time channel vector ${\bf{h}}(t+{T_{{\rm{delay}}}})$ can be expressed by $\begin{split}{\bf{h}}(t+{T_{{\rm{delay}}}})=\rho{\bf{h}}(t)+\sqrt{1-{\rho^{2}}}\hat{\bf{h}}(t+{T_{{\rm{delay}}}}).\end{split}$ (4) In (4), $\hat{\bf{h}}(t+{T_{{\rm{delay}}}})$ is independent identically distributed with ${\bf{h}}(t)$ and ${\bf{h}}(t+{T_{{\rm{delay}}}})$, and it is with zero-mean and unit-variance complex Gaussian entries. $\rho$ is the autocorrelation function (outdated CSI coefficient) of the channel gain ${\bf{h}}(t)$ and $0\leq\rho\leq 1$, which is given by $\begin{split}\rho={J_{0}}(2{\pi_{{\rm{pi}}}}{f_{D}}{T_{{\rm{delay}}}})\end{split}$ (5) where ${J_{0}}(\cdot)$ is the zeroth-order Bessel function of the first kind, ${f_{D}}$ is the Doppler spread which is generally a function of the velocity ($\upsilon$) of the transceivers, the carrier frequency (${f_{c}}$) and the speed of light $(c)$, i.e., ${f_{D}}=\upsilon{f_{c}}/c$. Note that $\rho=1$ indicates the outdated CSI effect is eliminated, whereas $\rho=0$ represents no CSI. As the outdated CSI introduces the channel uncertainty in practical dynamic systems, the actual channel coefficients can be rewritten as $\begin{split}\begin{array}[]{l}{{\bf{h}}_{{\rm{bu}},k}}={{{\bf{\tilde{h}}}}_{{\rm{bu}},k}}+\Delta{{\bf{h}}_{{\rm{bu}},k}},\;\forall k\in\mathcal{K},\\\ {{\bf{h}}_{{\rm{ru}},k}}={{{\bf{\tilde{h}}}}_{{\rm{ru}},k}}+\Delta{{\bf{h}}_{{\rm{ru}},k}},\;\forall k\in\mathcal{K},\\\ {{\bf{h}}_{{\rm{be}},m}}={{{\bf{\tilde{h}}}}_{{\rm{be}},m}}+\Delta{{\bf{h}}_{{\rm{be}},m}},\;\forall m\in\mathcal{M},\\\ {{\bf{h}}_{{\rm{re}},m}}={{{\bf{\tilde{h}}}}_{{\rm{re}},m}}+\Delta{{\bf{h}}_{{\rm{re}},m}},\;\forall m\in\mathcal{M},\end{array}\end{split}$ (6) where ${{\bf{\tilde{h}}}_{{\rm{bu}},k}}$, ${{\bf{\tilde{h}}}_{{\rm{ru}},k}}$, ${{\bf{\tilde{h}}}_{{\rm{be}},m}}$ and ${{\bf{\tilde{h}}}_{{\rm{re}},m}}$ denote the estimated channel vectors; $\Delta{{\bf{h}}_{{\rm{bu}},k}}$, $\Delta{{\bf{h}}_{{\rm{ru}},k}}$, $\Delta{{\bf{h}}_{{\rm{be}},m}}$ and $\Delta{{\bf{h}}_{{\rm{re}},m}}$ are the corresponding channel error vectors. In the paper, generally, the channel error vectors of each MU and each eavesdropper can be bounded with respect to the Euclidean norm by using norm- bounded error model, i.e., $\begin{split}\begin{array}[]{l}||\Delta{{\bf{h}}_{{\rm{bu}}}}|{|^{2}}\leq{({\varsigma_{{\rm{bu}}}})^{2}},\;\;||\Delta{{\bf{h}}_{{\rm{ru}}}}|{|^{2}}\leq{({\varsigma_{{\rm{ru}}}})^{2}},\\\ ||\Delta{{\bf{h}}_{{\rm{be}}}}|{|^{2}}\leq{({\varsigma_{{\rm{be}}}})^{2}},\;\;||\Delta{{\bf{h}}_{{\rm{re}}}}|{|^{2}}\leq{({\varsigma_{{\rm{re}}}})^{2}},\end{array}\end{split}$ (7) where ${\varsigma_{{\rm{bu}}}}$, ${\varsigma_{{\rm{ru}}}}$, ${\varsigma_{{\rm{be}}}}$, and ${\varsigma_{{\rm{re}}}}$ refer to the radii of the deterministically bounded error regions. Under the channel uncertainty model, the achievable rate of the $k$-th MU is given by $\begin{split}R_{k}^{\rm{u}}={\log_{2}}\left({1+\frac{{{{\left|{({\bf{h}}_{{\rm{ru,}}k}^{H}{\bf{\Psi}}{{\bf{H}}_{{\rm{br}}}}+{\bf{h}}_{{\rm{bu,}}k}^{H}){{\bf{v}}_{k}}}\right|}^{2}}}}{{{{|{\sum\limits_{i\in{\mathcal{K}},i\neq k}{({\bf{h}}_{{\rm{ru,}}k}^{H}{\bf{\Psi}}{{\bf{H}}_{{\rm{br}}}}+{\bf{h}}_{{\rm{bu,}}k}^{H}){{\bf{v}}_{i}}}}|}^{2}}+\delta_{k}^{2}}}}\right).\end{split}$ (8) If the $m$-th eavesdropper attempts to eavesdrop the signal of the $k$-th MU, its achievable rate can be expressed by $\begin{split}\begin{array}[]{l}R_{m,k}^{\rm{e}}=\\\ {\log_{2}}\left({1+\frac{{{{\left|{({\bf{h}}_{{\rm{re,}}m}^{H}{\bf{\Psi}}{{\bf{H}}_{{\rm{br}}}}+{\bf{h}}_{{\rm{be,}}m}^{H}){{\bf{v}}_{k}}}\right|}^{2}}}}{{{{|{\sum\limits_{i\in{\mathcal{K}},i\neq k}{({\bf{h}}_{{\rm{re,}}m}^{H}{\bf{\Psi}}{{\bf{H}}_{{\rm{br}}}}+{\bf{h}}_{{\rm{be,}}m}^{H}){{\bf{v}}_{i}}}}|}^{2}}+\delta_{m}^{2}}}}\right).\end{array}\end{split}$ (9) Since each eavesdropper can eavesdrop any of the $K$ MUs’ signal, according to [14]-[25], the achievable individual secrecy rate from the BS to the $k$-th MU can be expressed by $\begin{split}R_{k}^{\sec}={\left[{R_{k}^{\rm{u}}-\mathop{\max}\limits_{\forall m}R_{m,k}^{\rm{e}}}\right]^{+}}\end{split}$ (10) where ${[z]^{+}}=\max(0,z)$. ### II-B Problem Formulation Our objective is to jointly optimize the robust BS’s transmit beamforming matrix ${\bf{V}}$ and the robust IRS’s reflecting beamforming matrix ${\bf{\Psi}}$ from the system beamforming codebook $\mathcal{F}$ to maximize the worst-case secrecy rate with the worst-case secrecy rate and data rate constraints, the total BS transmit power constraint and the IRS reflecting unit constraint. As such, the optimization problem is formulated as $\begin{split}\begin{array}[]{l}\mathop{\max}\limits_{{\bf{V}},{\bf{\Psi}}}\mathop{\min}\limits_{\\{\Delta{\bf{h}}\\}}\sum\limits_{k\in\mathcal{K}}{R_{k}^{\sec}}\\\ s.t.\;\;({\rm{a}}):\;R_{k}^{\sec}\geq R_{k}^{\sec,\min},\;\forall k\in\mathcal{K},\\\ \;\;\;\;\;\;\;({\rm{b}}):\;\mathop{\min}\limits_{\scriptstyle||\Delta{{\bf{h}}_{{\rm{bu}}}}|{|^{2}}\leq{({\varsigma_{{\rm{bu}}}})^{2}},\hfill\atop\scriptstyle||\Delta{{\bf{h}}_{{\rm{ru}}}}|{|^{2}}\leq{({\varsigma_{{\rm{ru}}}})^{2}}\hfill}(R_{k}^{\rm{u}})\geq R_{k}^{\min},\;\forall k\in\mathcal{K},\\\ \;\;\;\;\;\;\;({\rm{c}}):\;{\rm{Tr}}\left({{\bf{V}}{{\bf{V}}^{H}}}\right)\leq{P_{\max}},\\\ \;\;\;\;\;\;\;({\rm{d}}):\;|\chi{e^{j{\theta_{l}}}}|=1,\;0\leq{\theta_{l}}\leq 2\pi,\;\forall l\in\mathcal{L},\end{array}\end{split}$ (11) where $R_{k}^{\sec{\rm{,min}}}$ is the target secrecy rate of the $k$-th MU, and $R_{k}^{{\rm{min}}}$ denotes its target data rate. The constraints in (11a) and (11b) are imposed to satisfy the worst-case secrecy rate and data rate requirements, respectively. The constraint in (11c) is set to satisfy the BS’s maximum power constraint. The constraint in (11d) is the constraint of the IRS reflecting elements. Obviously, it is challenging to obtain an optimal solution to the optimization (11), since the objective function in (11) is non-concave with respect to either ${\bf{V}}$ or ${\bf{\Psi}}$, and the coupling of the optimization variables (${\bf{V}}$ and ${\bf{\Psi}}$) and the unit-norm constraints in (11d) are non-convex. In addition, we would consider the robust beamforming design to maximize the worst-case achievable secrecy rate of the system while guaranteeing the worst-case constraints. ## III Problem Transformation Based on RL The optimization problem given in (11) is difficult to address as it is a non- convex problem. In addition, in realistic IRS-aided secure communication systems, the capabilities of MUs, the channel quality, and the service applications will change dynamically. Moreover, the problem in (11) is just a single time slot optimization problem, which may converge to a suboptimal solution and obtain the greedy-search like performance due to the ignorance of the historical system state and the long term benefit. Hence, it is generally infeasible to apply the traditional optimization techniques (AO, SDP, and MM) to achieve an effective secure beamforming policy in uncertain dynamic environments. Model-free RL is a dynamic programming tool which can be adopted to solve the decision-making problem by learning the optimal solution in dynamic environments [32]. Hence, we model the secure beamforming optimization problem as an RL problem. In RL, the IRS-aided secure communication system is treated as an environment, the central controller at the BS is regarded as a learning agent. The key elements of RL are defined as follows. State space: Let ${\mathcal{S}}$ denote the system state space. The current system state $s\in{\mathcal{S}}$ includes the channel information of all users, the secrecy rate, the transmission data rate of the last time slot and the QoS satisfaction level, which is defined as $\begin{split}\begin{array}[]{l}s=\left\\{{{{\\{{{\bf{h}}_{k}}{\rm{\\}}}}_{k\in{\mathcal{K}}}},{{\\{{{\bf{h}}_{m}}{\rm{\\}}}}_{m\in{\mathcal{M}}}},{{{\rm{\\{}}R_{k}^{\sec}{\rm{\\}}}}_{k\in{\mathcal{K}}}},{{\\{{R_{k}}\\}}_{k\in{\mathcal{K}}}},}\right.\\\ \;\;\;\;\;\;\left.{{{\\{{\rm{Qo}}{{\rm{S}}_{k}}\\}}_{k\in{\mathcal{K}}}}}\right\\}\end{array}\end{split}$ (12) where ${{\bf{h}}_{k}}$ and ${{\bf{h}}_{m}}$ are the channel coefficients of the $k$-th MU and $m$-th eavesdropper, respectively. ${\rm{Qo}}{{\rm{S}}_{k}}$ is the feedback QoS satisfaction level of the $k$-th MU, where the QoS satisfaction level consists of both the minimum secrecy rate satisfaction level in (11a) and the minimum data rate satisfaction level in (11b). Other parameters in (12) are already defined in Section II. Action space: Let ${\mathcal{A}}$ denote the system action space. According to the observed system state $s$, the central controller chooses the beamforming vector ${\\{{{\bf{v}}_{k}}\\}_{k\in{\mathcal{K}}}}$ at the BS and the IRS reflecting beamforming coefficient (phase shift) ${\\{{\theta_{l}}\\}_{l\in{\mathcal{L}}}}$ at the IRS. Hence, the action $a\in{\mathcal{A}}$ can be defined by $\begin{split}a=\left\\{{{{\\{{{\bf{v}}_{k}}\\}}_{k\in{\mathcal{K}}}},{{\\{{\theta_{l}}\\}}_{l\in{\mathcal{L}}}}}\right\\}.\end{split}$ (13) Transition probability: Let ${\mathcal{T}}(s^{\prime}|s,a)$ represent the transition probability, which is the probability of transitioning to a new state $s^{\prime}\in{\mathcal{S}}$, given the action $a$ executed in the sate $s$. Reward function: In RL, the reward acts as a signal to evaluate how good the secure beamforming policy is when the agent executes an action at a current state. The system performance will be enhanced when the reward function at each learning step correlates with the desired objective. Thus, it is important to design an efficient reward function to improve the MUs’ QoS satisfaction levels. In this paper, the reward function represents the optimization objective, and our objective is to maximize the system secrecy rate of all MUs while guaranteeing their QoS requirements. Thus, the presented QoS-aware reward function is expressed as $\begin{split}r=\underbrace{\sum\limits_{k\in{\mathcal{K}}}{R_{k}^{\sec}}}_{{\rm{part}}\;{\rm{1}}}-\underbrace{\sum\limits_{k\in{\mathcal{K}}}{{\mu_{1}}p_{k}^{\sec}}}_{{\rm{part}}\;2}-\underbrace{\sum\limits_{k\in{\mathcal{K}}}{{\mu_{2}}p_{k}^{\rm{u}}}}_{{\rm{part}}\;3}\end{split}$ (14) where $\begin{split}p_{k}^{\sec}=\left\\{\begin{array}[]{l}1,\;{\rm{if}}\;R_{k}^{\sec}<R_{k}^{\sec{\rm{,min}}},\forall k\in{\mathcal{K}},\\\ 0,\;{\rm{otherwise}}{\rm{,}}\end{array}\right.\end{split}$ (15) $\begin{split}p_{k}^{\rm{u}}=\left\\{\begin{array}[]{l}1,\;{\rm{if}}\;{R_{k}}<R_{k}^{{\rm{min}}},\forall k\in{\mathcal{K}},\\\ 0,\;{\rm{otherwise}}{\rm{.}}\end{array}\right.\end{split}$ (16) In (14), the part 1 represents the immediate utility (system secrecy rate), the part 2 and the part 3 are the cost functions which are defined as the unsatisfied secrecy rate requirement and the unsatisfied minimum rate requirement, respectively. The coefficients ${\mu_{1}}$ and ${\mu_{2}}$ are the positive constants of the part 2 and the part 3 in (14) , respectively, and they are used to balance the utility and cost [33]-[35]. The goals of (15) and (16) are to impose the QoS satisfaction levels of both the secrecy rate and the minimum data rate requirements, respectively. If the QoS requirement is satisfied in the current time slot, then $p_{k}^{\sec}={\rm{0}}$ or $p_{k}^{\rm{u}}={\rm{0}}$, indicating that there is no punishment of the reward function due to the successful QoS guarantees. The goal of the learning agent is to search for an optimal policy ${\pi^{*}}$ ($\pi$ is a mapping from states in ${\mathcal{S}}$ to the probabilities of choosing an action in ${\mathcal{A}}$: $\pi(s):{\mathcal{S}}\to{\mathcal{A}}$) that maximizes the long-term expected discounted reward, and the cumulative discounted reward function can be defined as $\begin{split}{U_{t}}=\sum\limits_{\tau=0}^{\infty}{{\gamma_{\tau}}{r_{t+\tau+1}}}\end{split}$ (17) where $\gamma\in(0,1]$ denotes the discount factor. Under a certain policy $\pi$, the state-action function of the agent with a state-action pair ($s$, $a$) is given by $\begin{split}{Q^{\pi}}({s_{t}},{a_{t}})={\mathbb{E}_{\pi}}\left[{{U_{t}}|{s_{t}}=s,{a_{t}}=a}\right].\end{split}$ (18) The conventional Q-Learning algorithm can be adopted to learn the optimal policy. The key objective of Q-Learning is to update Q-table by using the Bellman’s equation as follows: $\begin{split}\begin{array}[]{l}{Q^{\pi}}({s_{t}},{a_{t}})={\mathbb{E}_{\pi}}\left[{{r_{t}}+\gamma\sum\limits_{{s_{t+1}}\in{\mathcal{S}}}{{\mathcal{T}}({s_{t+1}}|{s_{t}},{a_{t}})}}\right.\\\ \;\;\;\;\;\;\;\;\;\;\left.{\sum\limits_{{a_{t+1}}\in{\mathcal{A}}}{\pi({s_{t+1}},{a_{t+1}}){Q^{\pi}}({s_{t+1}},{a_{t+1}})}}\right]\end{array}\end{split}$ (19) The optimal action-value function in (17) is equivalent to the Bellman optimality equation, which is expressed by $\begin{split}{Q^{*}}({s_{t}},{a_{t}})={r_{t}}+\gamma\mathop{\max}\limits_{{a_{t+1}}}{Q^{*}}({s_{t+1}},{a_{t+1}})\end{split}$ (20) and the state-value function is achieved as follows: $\begin{split}V({s_{t}})=\mathop{\max}\limits_{{a_{t}}\in{\mathcal{A}}}Q({s_{t}},{a_{t}}).\end{split}$ (21) In addition, the Q-value is updated as follows: $\begin{split}\begin{array}[]{l}{Q_{t+1}}({s_{t}},{a_{t}})=(1-{\alpha_{t}}){Q_{t}}({s_{t}},{a_{t}})+{\alpha_{t}}\left({{r_{t}}+\gamma{V_{t}}({s_{t{\rm{+1}}}})}\right)\end{array}\end{split}$ (22) where ${\alpha_{t}}\in(0,1]$ is the learning rate. Q-Learning generally constructs a lookup Q-table $Q(s,a)$, and the agent selects actions based on the greedy policy for each learning step [32]. In the $\varepsilon-$greedy policy, the agent chooses the action with the maximum Q-table value with probability ${\rm{1}}-\varepsilon$, whereas a random action is picked with probability $\varepsilon$ to avoid achieving stuck at non-optimal policies [32]. Once the optimal Q-function ${Q^{*}}(s,a)$ is achieved, the optimal policy is determined by $\begin{split}{\pi^{*}}(s,a)=\arg\mathop{\max}\limits_{a\in{\mathcal{A}}}{Q^{*}}(s,a).\end{split}$ (23) ## IV Deep PDS-PER Learning Based Secure Beamforming The secure beamforming policy discussed in Section III can be numerically achieved by using Q-Learning, policy gradient, and deep Q-Network (DQN) algorithms [32]. However, Q-Learning is not an efficient learning algorithm because it cannot deal with continuous state space and it has slow learning convergence speed. The policy gradient algorithm has the ability to handle continuous state-action spaces, but it may converge to a suboptimal solution. In addition, it is intractable for Q-learning and policy gradient algorithms to solve the optimization problem under high-dimensional input state space. Although DQN performs well in policy learning under high-dimensional state space, its non-linear Q-function estimator may lead to unstable learning process. Considering the fact that the IRS-aided secure communication system has high- dimensional and high-dynamical characteristics according to the system state that is defined in (12) and uncertain CSI that is shown in (4) and (6), we propose a deep PDS-PER learning based secure beamforming approach, as shown in Fig. 2, where PDS-learning and PER mechanisms are utilized to enable the learning agent to learn and adapt faster in dynamic environments. In detail, the agent utilizes the observed state (i.e, CSI, previous secrecy rate, QoS satisfaction level), the feedback reward from environment as well as the historical experience from the replay buffer to train its learning model. After that, the agent employs the trained model to make decision (beamforming matrices ${\bf{V}}$ and ${\bf{\Psi}}$) based on its learned policy. The procedures of the proposed learning based secure beamforming are provided in the following subsections. Figure 2: Deep PDS-PER learning based beamforming for IRS-aided secure communications. Note that the policy optimization (in terms of the BS’s beamforming matrix ${\bf{V}}$ and the RIS’s reflecting beamforming matrix ${\bf{\Psi}}$) in the IRS-aided secure communication system can be performed at the BS and that the optimized reflecting beamforming matrix can be transferred in an offline manner to the IRS by the controller to adjust the corresponding reflecting elements accordingly. ### IV-A Proposed Deep PDS-PER Learning As discussed in Section II, CSI is unlikely to be known accurately due to the transmission delay, processing delay, and mobility of users. At the same time, beamforming with outdated CSI will decrease the secrecy capacity, and therefore, a fast optimization solution needs to be designed to reduce processing delay. PDS-learning as a well-known algorithm has been used to improve the learning speed by exploiting extra partial information (e.g., the previous location information and the mobility velocity of MUs or eavesdroppers that affect the channel coefficients) and search for an optimized policy in dynamic environments [33]-[35]. Motivated by this, we devise a modified deep PDS-learning to trace the environment dynamic characteristics, and then adjust the transmit beamforming at the BS and the reflecting elements at the IRS accordingly, which can speed up the learning efficiency in dynamic environments. PDS-learning can be defined as an immediate system state ${\tilde{s}_{t}}\in{\mathcal{S}}$ happens after executing an action ${a_{t}}$ at the current state ${s_{t}}$ and before the next time state ${s_{t+1}}$ . In detail, the PDS-learning agent takes an action ${a_{t}}$ at state ${s_{t}}$, and then will receive known reward ${r^{\rm{k}}}({s_{t}},{a_{t}})$ from the environment before transitioning the current state ${s_{t}}$ to the PDS state ${\tilde{s}_{t}}$ with a known transition probability ${{\mathcal{T}}^{\rm{k}}}({\tilde{s}_{t}}|{s_{t}},{a_{t}})$. After that, the PDS state further transform to the next state ${s_{t+1}}$ with an unknown transition probability ${{\mathcal{T}}^{\rm{u}}}({s_{t+1}}|{\tilde{s}_{t}},{a_{t}})$ and an unknown reward ${r^{\rm{u}}}({s_{t}},{a_{t}})$, which corresponds to the wireless CSI dynamics. In PDS-learning, ${s_{t+1}}$ is independent of ${s_{t}}$ given the PDS state ${\tilde{s}_{t}}$, and the reward $r({s_{t}},{a_{t}})$ is decomposed into the sum of ${r^{\rm{k}}}({s_{t}},{a_{t}})$ and ${r^{\rm{u}}}({s_{t}},{a_{t}})$ at ${\tilde{s}_{t}}$ and ${s_{t+1}}$, respectively. Mathematically, the state transition probability in PDS-learning from ${s_{t}}$ to ${s_{t+1}}$ admits $\begin{split}{\mathcal{T}}({s_{t+1}}|{s_{t}},{a_{t}})=\sum\nolimits_{{{\tilde{s}}_{t}}}{{{\mathcal{T}}^{\rm{u}}}({s_{t+1}}|{{\tilde{s}}_{t}},{a_{t}}){{\mathcal{T}}^{\rm{k}}}({{\tilde{s}}_{t}}|{s_{t}},{a_{t}})}.\end{split}$ (24) Moreover, it can be verified that the reward of the current state-action pair $({s_{t}},{a_{t}})$ is expressed by $\begin{split}r({s_{t}},{a_{t}})={r^{\rm{k}}}({s_{t}},{a_{t}})+\sum\nolimits_{{{\tilde{s}}_{t}}}{{{\mathcal{T}}^{\rm{k}}}({{\tilde{s}}_{t}}|{s_{t}},{a_{t}}){r^{\rm{u}}}({{\tilde{s}}_{t}},{a_{t}})}.\end{split}$ (25) At the time slot $t$, the PDS action-value function $\tilde{Q}({\tilde{s}_{t}},{a_{t}})$ of the current PDS state-action pair $({\tilde{s}_{t}},{a_{t}})$ is defined as $\begin{split}\tilde{Q}({\tilde{s}_{t}},{a_{t}})={r^{\rm{u}}}({\tilde{s}_{t}},{a_{t}})+\gamma\sum\limits_{{s_{t+1}}}{{{\mathcal{T}}^{\rm{u}}}({s_{t+1}}|{{\tilde{s}}_{t}},{a_{t}})V({s_{t+1}})}.\end{split}$ (26) By employing the extra information (the known transition probability ${{\mathcal{T}}^{\rm{k}}}({\tilde{s}_{t}}|{s_{t}},{a_{t}})$ and known reward ${r^{\rm{k}}}({s_{t}},{a_{t}})$), the Q-function $\hat{Q}({s_{t}},{a_{t}})$ in PDS-learning can be further expanded under all state-action pairs $(s,a)$, which is expressed by $\begin{split}\hat{Q}({s_{t}},{a_{t}})={r^{\rm{k}}}({s_{t}},{a_{t}})+\sum\nolimits_{{{\tilde{s}}_{t}}}{{{\mathcal{T}}^{\rm{k}}}({{\tilde{s}}_{t}}|{s_{t}},{a_{t}})\tilde{Q}({{\tilde{s}}_{t}},{a_{t}})}.\end{split}$ (27) The state-value function in PDS-learning is defined by $\begin{split}{\hat{V}_{t}}({s_{t}})=\sum\nolimits_{{s_{t+1}}}{{{\mathcal{T}}^{\rm{k}}}({s_{t+1}}|{s_{t}},{a_{t}})\tilde{V}({s_{t+1}})}\end{split}$ (28) where ${\tilde{V}_{t}}({s_{t+1}})=\mathop{\max}\limits_{{a_{t}}\in{\mathcal{A}}}{\tilde{Q}_{t}}({\tilde{s}_{t+1}},{a_{t}})$. At each time slot, the PDS action-value function $\tilde{Q}({\tilde{s}_{t}},{a_{t}})$ is updated by $\begin{split}\begin{array}[]{l}{{\tilde{Q}}_{{}_{t+1}}}({{\tilde{s}}_{t}},{a_{t}})=(1-{\alpha_{t}}){{\tilde{Q}}_{t}}({{\tilde{s}}_{t}},{a_{t}})\\\ \;\;\;\;\;\;\;\;\;\;\;\;\;+{\alpha_{t}}\left({{r^{\rm{u}}}({{\tilde{s}}_{t}},{a_{t}})+\gamma{{\hat{V}}_{t}}({s_{t+1}})}\right)\end{array}\end{split}$ (29) After updating ${\tilde{Q}_{{}_{t+1}}}({\tilde{s}_{t}},{a_{t}})$, the action- value function ${\hat{Q}_{{}_{t+1}}}({s_{t}},{a_{t}})$ can be updated by plugging ${\tilde{Q}_{{}_{t+1}}}({\tilde{s}_{t}},{a_{t}})$ into (27). After presenting in the above modified PDS-learning, a deep PDS learning algorithm is presented. In the presented learning algorithm, the traditional DQN is adopted to estimatete the action-value Q-function $Q(s,a)$ by using $Q(s,a;{\bm{\theta}})$, where ${\bm{\theta}}$ denote the DNN parameter. The objective of DQN is to minimize the following loss function at each time slot $\begin{split}\begin{array}[]{l}{\mathcal{L}}({{\bm{\theta}}_{t}})=\left[{{{\\{{{\hat{V}}_{t}}({s_{t}};{{\bm{\theta}}_{t}})-\hat{Q}({s_{t}},{a_{t}};{{\bm{\theta}}_{t}})\\}}^{2}}}\right]=\left[{\\{r({s_{t}},{a_{t}})}\right.\\\ \;\;\;\;\;\;\;\left.{+\gamma\mathop{\max}\limits_{{a_{t+1}}\in{\mathcal{A}}}{{\hat{Q}}_{t}}({s_{t+1}},{a_{t+1}};{{\bm{\theta}}_{t}})-\hat{Q}({s_{t}},{a_{t}};{{\bm{\theta}}_{t}}){\\}^{2}}}\right]\end{array}\end{split}$ (30) where ${\hat{V}_{t}}({s_{t}};{{\bm{\theta}}_{t}})=r({s_{t}},{a_{t}})+\gamma\mathop{\max}\limits_{{a_{t+1}}\in{\mathcal{A}}}{\hat{Q}_{t}}({s_{t+1}},{a_{t+1}};{{\bm{\theta}}_{t}})$ is the target value. The error between ${\hat{V}_{t}}({s_{t}};{{\bm{\theta}}_{t}})$ and the estimated value $\hat{Q}({s_{t}},{a_{t}};{{\bm{\theta}}_{t}})$ is usually called temporal- difference (TD) error, which is expressed by $\begin{split}{\delta_{t}}={\hat{V}_{t}}({s_{t}};{{\bm{\theta}}_{t}})-\hat{Q}({s_{t}},{a_{t}};{{\bm{\theta}}_{t}}).\end{split}$ (31) The DNN parameter ${\bm{\theta}}$ is achieved by taking the partial differentiation of the objective function (30) with respect to ${\bm{\theta}}$, which is given by $\begin{split}{{\bm{\theta}}_{t+1}}={{\bm{\theta}}_{t}}+\beta\nabla{\mathcal{L}}({{\bm{\theta}}_{t}}).\end{split}$ (32) where $\beta$ is the learning rate of ${\bm{\theta}}$, and $\nabla(\cdot)$ denotes the first-order partial derivative. Accordingly, the policy ${\hat{\pi}_{t}}(s)$ of the modified deep PDS-learning algorithm is given by $\begin{split}{\hat{\pi}_{t}}(s)=\arg\mathop{\max}\limits_{{a_{t}}\in{\mathcal{A}}}\hat{Q}({s_{t}},{a_{t}};{{\bm{\theta}}_{t}}).\end{split}$ (33) Although DQN is capable of performing well in policy learning with continuous and high-dimensional state space, DNN may learn ineffectively and cause divergence owing to the nonstationary targets and correlations between samples. Experience replay is utilized to avoid the divergence of the RL algorithm. However, classical DQN uniformly samples each transition ${e_{t}}=\langle{s_{t}},{a_{t}},{r_{t}},{\tilde{s}_{t}},{s_{t+1}}\rangle$ from the experience replay, which may lead to an uncertain or negative effect on learning a better policy. The reason is that different transitions (experience information) in the replay buffer have different importance for the learning policy, and sampling every transition equally may unavoidably result in inefficient usage of meaningful transitions. Therefore, a prioritized experience replay (PER) scheme has been presented to address this issue and enhance the sampling efficiency [36], [37], where the priority of transition is determined by the values of TD error. In PER, a transition with higher absolute TD error has higher priority in the sense that is has more aggressive correction for the action-value function. In the deep PDS-PER learning algorithm, similar to classical DQN, the agent collects and stores each experience ${e_{t}}=\langle{s_{t}},{a_{t}},{r_{t}},{\tilde{s}_{t}},{s_{t+1}}\rangle$ into its experience replay buffer, and DNN updates the parameter by sampling a mini-batch of tuples from the replay buffer. So far, PER was adopted only for DRL and Q-learning, and has never been employed with the PDS-learning algorithm to learn the dynamic information. In this paper, we further extend this PER scheme to enable prioritized experience replay in the proposed deep PDS-PER learning framework, in order to improve the learning convergence rate. The probability of sampling transition $i$ (experience $i$) based on the absolute TD-error is defined by $\begin{split}p(i)={{|\delta(i){|^{{\eta_{1}}}}}\mathord{\left/{\vphantom{{|\delta(i){|^{{\eta_{1}}}}}{\sum\nolimits_{j^{\prime}}{|\delta(j^{\prime}){|^{{\eta_{1}}}}}}}}\right.\kern-1.2pt}{\sum\nolimits_{j^{\prime}}{|\delta(j^{\prime}){|^{{\eta_{1}}}}}}}\end{split}$ (34) where the exponent ${\eta_{1}}$ weights how much prioritization is used, with ${\eta_{1}}=0$ corresponding to being uniform sampling. The transition with higher $p(i)$ will be more likely to be replayed from the replay buffer, which is associated with very successful attempts by preventing the DNN from being over-fitting. With the help of PER, the proposed deep PDS-PER learning algorithm tends to replay valuable experience and hence learns more effectively to find the best policy. It is worth noting that experiences with high absolute TD-error are more frequently replayed, which alters the visitation frequency of some experiences and hence causes the training process of the DNN prone to diverge. To address this problem, importance-sampling (IS) weights are adopted in the calculation of weight changes $\begin{split}W(i)={\left({D\cdot p(i)}\right)^{-{\eta_{2}}}}\end{split}$ (35) where $D$ is the size of the experience replay buffer, and the parameter ${\eta_{2}}$ is used to adjust the amount of correction used. Accordingly, by using the PER scheme into the deep PDS-PER learning, the DNN loss function (30) and the corresponding parameters are rewritten respectively as follows: $\begin{split}{\mathcal{L}}\left({{{\bm{\theta}}_{t}}}\right)=\frac{1}{H}\sum\limits_{i=1}^{H}{\left({{W_{i}}{{\mathcal{L}}_{i}}({{\bm{\theta}}_{t}})}\right)}\end{split}$ (36) $\begin{split}{{\bm{\theta}}_{t+1}}={{\bm{\theta}}_{t}}+\beta{\delta_{t}}{\nabla_{\bm{\theta}}}{\mathcal{L}}({{\bm{\theta}}_{t}}))\end{split}$ (37) _Theorem 1:_ The presented deep PDS-PER learning can converge to the optimal $\hat{Q}({s_{t}},{a_{t}})$ of the MDP with probability 1 when the learning rate sequence ${\alpha_{t}}$ meets the following conditions ${\alpha_{t}}\in[0,1)$, $\sum\nolimits_{t=0}^{\infty}{{\alpha_{t}}}=\infty$ and $\sum\nolimits_{t=0}^{\infty}{\alpha_{t}^{2}}<\infty$, where the aforementioned requirements have been appeared in most of the RL algorithms [32] and they are not specific to the proposed deep PDS-PER learning algorithm [32]. _Proof:_ If each action can be executed with an infinite number of learning steps at each system state, or in other words, the learning policy is greedy with the infinite explorations, the Q-function $\hat{Q}({s_{t}},{a_{t}})$ in PDS-learning and its corresponding policy strategy $\pi(s)$ will converge to the optimal points, respectively, with probability of 1 [33]-[35]. The existing references [34] and [35] have provided the proof. ### IV-B Secure Beamforming Based on Proposed Deep PDS-PER Learning Similar to most DRL algorithms, our proposed deep PDS-PER learning based secure beamforming approach consists of two stages, i.e., the training stage and implementation stage. The training process of the proposed approach is shown in Algorithm 1. A central controller at the BS is responsible for collecting environment information and making decision for secure beamforming. In the training stage, similar to RL-based policy control, the controller initializes network parameters and observes the current system state including CSI of all users, the previous predicted secrecy rate and the transmission data rate. Then, the state vector is input into DQN to train the learning model. The $\varepsilon$-greedy scheme is leveraged to balance both the exploration and exploitation, i.e., the action with the maximum reward is selected probability 1- $\varepsilon$ according to the current information (exploitation, which is known knowledge), while a random action is chosen with probability $\varepsilon$ based on the unknown knowledge (i.e., keep trying new actions, hoping it brings even higher reward (exploration, which is unknown knowledge) ). After executing the selected action, the agent receives a reward from the environment and observes the sate transition from ${s_{t}}$ to PDS state ${\tilde{s}_{t}}$ and then to the next state ${s_{t+1}}$. Then, PDS-learning is used to update the PDS action-value function $\tilde{Q}({\tilde{s}_{t}},{a_{t}};{{\bm{\theta}}_{t}})$ and Q-function $\hat{Q}({s_{t}},{a_{t}};{{\bm{\theta}}_{t}})$, before collecting and storing the transition tuple (also called experience) ${e_{t}}=\langle{s_{t}},{a_{t}},{r_{t}},{\tilde{s}_{t}},{s_{t+1}}\rangle$ into the experience replay memory buffer ${\mathcal{D}}$, which includes the current system state, selected action, instantaneous reward and PDS state along with the next state. The experience in the replay buffer is selected by the PER scheme to generate mini-batches and they are used to train DQN. In detail, the priority of each transition $p(i)$ is calculated by using (34) and then get its IS weight $W(i)$ in (35), where the priorities ensure that high- TD-value ($\delta(i)$) transitions are replayed more frequently. The weight $W(i)$ is integrated into deep PDS learning to update both the loss function ${\mathcal{L}}({\bf{\theta}})$ and DNN parameter ${\bm{\theta}}$. Once DQN converges, the deep PDS-PER learning model is achieved. After adequate training in Algorithm 1, the learning model is loaded for the implementation stage. During the implementation stage, the controller uses the trained learning model to output its selected action $a$ by going through the DNN parameter ${\bm{\theta}}$, with the observed state $s$ from the IRS-aided secure communication system. Specifically, it chooses an action $a$, with the maximum value based on the trained deep PDS-PER learning model. Afterwards, the environment feeds back an instantaneous reward and a new system state to the agent. Finally, the beamforming matrix ${{\bf{V}}^{*}}$ at the BS and the phase shift matrix ${{\bf{\Psi}}^{*}}$ (reflecting beamforming) at the IRS are achieved according to the selected action. We would like to point out that the training stage needs a powerful computation server which can be performed offline at the BS while the implementation stage can be completed online. The trained learning model requires to be updated only when the environment (IRS-aided secure communication system) has experienced greatly changes, mainly depending on the environment dynamics and service requirements. Algorithm 1 Deep PDS-PER Learning Based Secure Beamforming 1: Input: IRS-aided secure communication simulator and QoS requirements of all MUs (e.g., minimum secrecy rate and transmission rate). 2: Initialize: DQN with initial Q-function $Q(s,a;{\bm{\theta}})$, parameters ${\bm{\theta}}$, learning rate $\alpha$ and $\beta$. 3: Initialize: experience replay buffer ${\mathcal{D}}$ with size $D$, and mini-batch size $H$. 4: for each episode =1, 2, …, ${N^{{\rm{epi}}}}$ do 5: Observe an initial system state $s$; 6: for each time step $t$=0, 1, 2, …, $T$ do 7: Select action based on the $\varepsilon$-greedy policy at current state ${s_{t}}$: choose a random action ${a_{t}}$ with probability $\varepsilon$; 8: Otherwise, ${a_{t}}=\arg\mathop{\max}\limits_{{a_{t}}\in{\mathcal{A}}}Q({s_{t}},{a_{t}};{{\bf{\theta}}_{t}})$; 9: Execute action ${a_{t}}$, receive an immediate reward ${r^{\rm{k}}}({s_{t}},{a_{t}})$ and observe the sate transition from ${s_{t}}$ to PDS state ${\tilde{s}_{t}}$ and then to the next state ${s_{t+1}}$; 10: Update the reward function $r({s_{t}},{a_{t}})$ under PDS-learning using (25); 11: Update the PDS action-value function $\tilde{Q}({\tilde{s}_{t}},{a_{t}};{{\bm{\theta}}_{t}})$ using (29); 12: Update the Q-function $\hat{Q}({s_{t}},{a_{t}};{{\bm{\theta}}_{t}})$ using (25); 13: Store PDS experience ${e_{t}}=\langle{s_{t}},{a_{t}},{r_{t}},{\tilde{s}_{t}},{s_{t+1}}\rangle$ in experience replay buffer ${\mathcal{D}}$, if ${\mathcal{D}}$ is full, remove least used experience from ${\mathcal{D}}$; 14: for $i$= 1, 2, …, $H$ do 15: Sample transition $i$ with the probability $p(i)$ using (34); 16: Calculate the absolute TD-error $|\delta(i)|$ in (31); 17: Update the corresponding IS weight ${W_{i}}$ using (35); 18: Update the priority of transition $i$ based on $|\delta(i)|$; 19: end for 20: Update the loss function ${\mathcal{L}}\left({\bm{\theta}}\right)$ and parameter ${\bm{\theta}}$ of DQN using (36) and (37), respectively; 21: end for 22: end for 23: Output: Return the deep PDS-PER learning model. ### IV-C Computational Complexity Analysis For the training stage, in DNN, let $L$, ${Z_{0}}$ and ${Z_{l}}$ denote the training layers, the size of the input layer (which is proportional to the number of states) and the number of neurons in the $l$-th layer, respectively. The computational complexity in each time step for the agent is $O({Z_{0}}{Z_{l}}+\sum\nolimits_{l=1}^{L-1}{{Z_{l}}{Z_{l+1}}})$. In the training phase, each mini-batch has ${N^{{\rm{epi}}}}$ episodes with each episode being $T$ time steps, each trained model is completed iteratively until convergence. Hence, the total computational complexity in DNN is $O\left({{N^{{\rm{epi}}}}T({Z_{0}}{Z_{l}}+\sum\nolimits_{l=1}^{L-1}{{Z_{l}}{Z_{l+1}}})}\right)$. The high computational complexity of the DNN training phase can be performed offline for a finite number of episodes at a centralized powerful unit (such as the BS). In our proposed deep PDS-PER learning algorithm, PDS-learning and PER schemes are utilized to improve the learning efficiency and enhance the convergence speed, which requires extra computational complexity. In PDS-learning leaning, since the set of PDS states is the same as the set of MDP states ${\mathcal{S}}$ [30]-[32], the computational complexity of the classical DQN algorithm and the deep PDS-learning algorithm are $O(|{\mathcal{S}}{|^{2}}\times|{\mathcal{A}}|)$ and $O(2|{\mathcal{S}}{|^{2}}\times|{\mathcal{A}}|)$, respectively. In PER, since the relay buffer size is $D$, the system requires to make both updating and sampling $O\left({{{\log}_{2}}D}\right)$ operations, so the computational complexity of the PER scheme is $O\left({{{\log}_{2}}D}\right)$. According the above analysis, the complexity of the classical DQN algorithm and the proposed deep PDS-PER learning algorithm are respectively $O\left({I{N^{{\rm{epi}}}}T({Z_{0}}{Z_{l}}+\sum\nolimits_{l=1}^{L-1}{{Z_{l}}{Z_{l+1}}})+|{\mathcal{S}}{|^{2}}\times|{\mathcal{A}}|}\right)$ and $O\left({I{N^{{\rm{epi}}}}T({Z_{0}}{Z_{l}}+\sum\nolimits_{l=1}^{L-1}{{Z_{l}}{Z_{l+1}}})+2|{\mathcal{S}}{|^{2}}\times|{\mathcal{A}}|+{{\log}_{2}}D}\right)$, indicating that the complexity of the proposed algorithm is slightly higher than the classical DQN learning algorithm. However, our proposed algorithm achieves better performance than that of the classical DQN algorithm, which will be shown in the next section. ### IV-D Implementation Details of DRL This subsection provides extensive details regarding the generation of training, validation, and testing dataset production. Generation of training: As shown in Fig. 3, $K$ single-antenna MUs and $M$ single-antenna eavesdroppers are randomly located in the $100m\times 100m$ half right-hand side rectangular of Fig. 3 (light blue area) in a two- dimensional x-y rectangular grid plane. The BS and the IRS are located at (0, 0) and (150, 100) in meter (m), respectively. The x-y grid has dimensions 100m$\times$100m with a resolution of 2.5 m, i.e., a total of 1600 points. Figure 3: Simulation setup. In the presented IRS-assisted system, the system beamforming codebook ${\mathcal{F}}$ includes the BS’s beamforming codebook ${\mathcal{{F_{{\rm{BS}}}}}}$ and the IRS’s beamforming codebook ${\mathcal{{F_{{\rm{IRS}}}}}}$. Both the BS’s beamforming matrix ${\bf{V}}$ and the IRS’s reflection beamforming matrix ${\bf{\Psi}}$ are picked from the pre-defined codebook ${\mathcal{{F_{{\rm{BS}}}}}}$ and ${\mathcal{{F_{{\rm{IRS}}}}}}$, respectively. The data points of sampled channel vector and the corresponding reward vector $\left\langle{{\bf{h}},{\bf{r}}}\right\rangle$ is added into the DNN training data set ${\mathcal{D}}$. The sampled channel, ${\bf{h}}$, is the input to DQN. All the simples are normalized by using the normalization scheme to realize a simple per-dataset scaling. After training, the selected BS’s beamforming matrix ${\bf{V}}$ and IRS’s beamforming matrix ${\bf{\Psi}}$ with the highest achievable reward are used to reflect the security communication performance. The DQN learning model is trained using empirically hyper-parameter, where DNN trained for 1000 epochs with 128 mini-batches being utilized in each epoch. In the training process, 80% and 20% of all generated data are selected as the training and validation (test) datasets, respectively. The experience replay buffer size is 32000 where the corresponding samples are randomly sampled from this number of the most recently experiences. DQN structure: The DQN model is designed as a Multi-Layer Perceptron network, which is also referred to as the feedforward Fully Connected network. Note here that Multi-Layer Perceptron network is widely used to build an advanced estimator, which fulfills the relation between the environment descriptors and the beamforming matrices (both the BS’s beamforming matrix and the IRS’s reflecting beamforming matrix). The DQN model is comprised of $L$ layers, as illustrated in Fig. 2, where the first layer is input layer, the last layer is output layer and the remaining layers are the hidden layers. The $l$-th hidden layer in the network has a stack of neurons, each of which connects all the outputs of the previous layer. Each unit operates on a single input value outputting another single value. The input of the input layer consist of the systems states, i.e., channel samples, the achievable rate and QoS satisfaction level information in the last time slot, while the output layer outputs the predicted reward values with beamforming matrices in terms of the BS’s beamforming matrix and the IRS’s reflecting beamforming matrix. The DQN construction is used for training stability. The network parameters will be provided in the next section. Training loss function: The objective of DRL model is to find the best beamforming matrices, i.e., ${\bf{V}}$ and ${\bf{\Psi}}$, from the beamforming codebook with the highest achievable reward from the environment. In this case, having the highest achievable reward estimation, the regression loss function is adopted to train the learning model, where DNN is trained to make its output, ${\bf{\hat{r}}}$, as close as possible to the desired normalized reward, ${\bf{\bar{r}}}$. Formally, the training is driven by minimizing the loss function, ${\mathcal{L}}({\bm{\theta}})$, defined as $\begin{split}{\mathcal{L}}({\bm{\theta}})=MSE\left({{\bf{\hat{r}}},{\bf{\bar{r}}}}\right),\end{split}$ (38) where ${\bm{\theta}}$ is the set of all DNN parameters, $MSE(\cdot)$ denotes the mean-squared-error between ${\bf{\hat{r}}}$ and ${\bf{\bar{r}}}$. Note that the outputs of DNN, ${\bf{\hat{r}}}$ can be acted as functions of ${\bm{\theta}}$ and the inputs of DNN are the system states shown in (12) in the paper. ## V Simulation Results and Analysis This section evaluates the performance of the IRS-aided secure communication system. The background noise power of MUs and eavesdroppers is equal to -90 dBm. We set the number of antennas at the BS is $N=4$, the number of MUs is $K=2$ and the number of eavesdroppers is $M=2$. The transmit power ${P_{\max}}$ at the BS varies between 15 dBm and 40 dBm, the number of IRS elements $L$ varies between 10 and 60, and the outdated CSI coefficient $\rho$ varies from 0.5 to 1 for different simulation settings. The minimum secrecy rate and the minimum transmission data rate are 3 bits/s/Hz and 5 bits/s/Hz, respectively. The path loss model is defined by $PL=\left({P{L_{0}}-10\varsigma\log 10(d/{d_{0}})}\right)$ dB, where $P{L_{0}}=30\;{\rm{dB}}$ is the path loss at the reference distance ${d_{0}}=1\;{\rm{m}}$ [9], [38], $\varsigma=3$ is the path loss exponent, and $d$ is the distance from the transmitter to the receiver. The learning model consists of three connected hidden layers, containing 500, 250, and 200 neurons [39], respectively. The learning rate is set to $\alpha=0.002$ and the discount factor is set to $\gamma=0.95$. The the exploration rate $\varepsilon$ is linearly annealed from 0.8 to 0.1 over the beginning 300 episodes and remains constant afterwards. The parameters ${\mu_{1}}$ and ${\mu_{2}}$ in (12) are set to ${\mu_{1}}={\mu_{2}}=2$ to balance the utility and cost [33]-[35]. Similar to the IRS-aided communication systems [9], [13] and [17], the path loss exponents from the BS to the UEs is set to 3.2, from BS to IRS is set to 2.2, and from IRS to UEs is set to 2.2. The selection of the network parameters decide the learning convergence speed and efficiency. Here, we take the network parameters, i.e., the learning rate, as an example to demonstrate the importance of the network parameters selection. Fig. 4 shows the average system reward versus training episodes under different learning rates, i.e., $\alpha=\\{0.1,0.01,0.001,0.0001\\}$. It can be observed that different learning rates have different effect on the performance of the deep reinforcement learning algorithm. Specifically, there exits oscillations in behavior for the too-large learning rate of $\alpha=0.1$, and its reward performance is much lower than that of $\alpha=0.001$. In addition, if we set the too-small learning rate of $\alpha=0.0001$, it requires the longer time to achieve the convergence. We can see that the model is able to learn the problem well with the learning rate of $\alpha=0.001$. Hence, we select a suitable learning rate, neither too large nor too small, and the value can be around 0.001 in our test. Figure 4: Average reward performance versus episodes under different learning rates. To analyze the loss function of the presented DRL shown in Section IV.D, Fig. 5 illustrates the training/validation loss value during training versus the number of epochs. We can observe that the training/validation loss value decreases significantly in the first few decades epochs and then tend to be approximately at a horizontal level after 150 epochs. Furthermore, the validation loss is only slightly higher than the training loss, which demonstrates that the DNN weights designed have the ability to provide a good fit in terms of the mapping between input samples and output samples. It is worth noting that Fig. 5 is used to investigate how well the DNN weight parameters are designed. If the validation loss is high while the training loss is low, this means that the DQN model is over-fitting and thus the regularization factors may need to be adjusted; if both the validation and training loss values are high, this shows that the DQN model is under-fitting and hence the number of neurons may require to be adjusted. Figure 5: The training and validation losses of DNN employed. In addition, simulation results are provided to evaluate the performance of the proposed deep PDS-PER learning based secure beamforming approach (denoted as deep PDS-PER beamforming) in the IRS-aided secure communication system, and compare the proposed approach with the following exiting approaches: * • The classical DQN based secure beamforming approach (denoted as DQN-based beamforming), where DNN is employed to estimate the Q-value function, when acting and choosing the secure beamforming policy corresponding to the highest Q-value. * • The existing secrecy rate maximization approach which optimizes the BS’s transmit beamforming and the IRS’s reflect beamforming by fixing other parameters as the constants by using the iterative algorithm, which is similar to the suboptimal solution [14] (denoted as Baseline 1 [14]). * • The optimal BS’s transmit beamforming approach without IRS assistance (denoted as optimal BS without IRS). Without IRS, the optimization problem (11) is transformed as $\begin{split}\begin{array}[]{l}\mathop{\max}\limits_{{\bf{V}}}\mathop{\min}\limits_{\\{\Delta{\bf{h}}\\}}\sum\limits_{k\in{\mathcal{K}}}{R_{k}^{\sec}}\\\ s.t.\;\;({\rm{a}}):\;R_{k}^{\sec}\geq R_{k}^{\sec,\min},\;\forall k\in{\mathcal{K}},\\\ \;\;\;\;\;\;\;({\rm{b}}):\;\mathop{\min}\limits_{\scriptstyle||\Delta{{\bf{h}}_{{\rm{bu}}}}|{|^{2}}\leq{({\varsigma_{{\rm{bu}}}})^{2}},\hfill\atop\scriptstyle||\Delta{{\bf{h}}_{{\rm{ru}}}}|{|^{2}}\leq{({\varsigma_{{\rm{ru}}}})^{2}}\hfill}(R_{k}^{\rm{u}})\geq R_{k}^{\min},\;\forall k\in{\mathcal{K}},\\\ \;\;\;\;\;\;\;({\rm{c}}):\;{\rm{Tr}}\left({{\bf{V}}{{\bf{V}}^{H}}}\right)\leq{P_{\max}},\\\ \;\;\;\;\;\;\;({\rm{d}}):\;|\chi{e^{j{\theta_{l}}}}|=1,\;0\leq{\theta_{l}}\leq 2\pi,\;\forall l\in{\mathcal{L}}.\end{array}\end{split}$ (39) From the optimization problem (39), they system only needs to optimize the BS’s transmit beamforming matrix . Problem (39) is non-convex due to the rate constraints, and hence we consider semidefinite programming (SDP) relaxation to solve it. After transforming problem (39), into a convex optimization problem, we can use CVX to obtain the solution [12]-[16]. Figure 6: Performance comparisons versus the maximum transmit power at the BS. Fig. 6 shows the average secrecy rate and QoS satisfaction probability (consists of the secrecy rate satisfaction probability (11a) and the minimum rate satisfaction probability (11b)) versus the maximum transmit power ${P_{\max}}$, when $L=40$ and $\rho=0.95$. As expected, both the secrecy rate and QoS satisfaction probability of all the approaches enhance monotonically with increasing ${P_{\max}}$. The reason is that when ${P_{\max}}$ increases, the received SINR at MUs improves, leading to the performance improvement. In addition, we find that our proposed learning approach outperforms the Baseline1 approach. In fact, our approach jointly optimizes the beamforming matrixes ${\bf{V}}$ and ${\bf{\Psi}}$, which can simultaneously facilitates more favorable channel propagation benefit for MUs and impair eavesdroppers, while the Baseline1 approach optimizes the beamforming matrixes in an iterative way. Moreover, our proposed approach has higher performance than DQN in terms of both secrecy rate and QoS satisfaction probability, due to its efficient learning capacity by utilizing PDS-learning and PER schemes in the dynamic environment. From Fig. 6, we also find that the three IRS assisted secure beamforming approaches provide significant higher secrecy rate and QoS satisfaction probability than the traditional system without IRS. This indicates that the IRS can effectively guarantee secure communication and QoS requirements via reflecting beamforming, where reflecting elements (IRS- induced phases) at the IRS can be adjusted to maximize the received SINR at MUs and suppress the wiretapped rate at eavesdroppers. Figure 7: Performance comparisons versus the number of IRS elements. In Fig. 7, the achievable secrecy rate and QoS satisfaction level performance of all approaches are evaluated through changing the IRS elements, i.e., from $L=10$ to 60, when ${P_{\max}}=30\;{\rm{dBm}}$ and $\rho=0.95$. For the secure beamforming approaches assisted by the IRS, their achievable secrecy rates and QoS satisfaction levels significantly increase with the number of the IRS elements. The improvement results from the fact that more IRS elements, more signal paths and signal power can be reflected by the IRS to improve the received SINR at the MUs but to decrease the received SINR at the eavesdroppers. In addition, the performance of the approach without IRS remains constant under the different numbers of the IRS elements. From Fig. 7(a), it is found that the secrecy rate of the proposed learning approach is higher than those of the Baseline 1 and DQN approaches, especially, their performance gap also obviously increases with $L$, this is because that with more reflecting elements at the IRS, the proposed deep PDS- PER learning based secure communication approach becomes more flexible for optimal phase shift (reflecting beamforming) design and hence achieves higher gains. In addition, from Fig. 7(b) compared with the Baseline 1 and DQN approaches, as the reflecting elements at the IRS increases, we observe that the proposed learning approach is the first one who attains 100% QoS satisfaction level. This superior achievements are based on the particular design of the QoS-aware reward function shown in (14) for secure communication. Figure 8: Performance comparisons versus outdated CSI coefficient $\rho$. In Fig. 8, we further analyze how the system secrecy rate and QoS satisfaction level performances are affected by the outdated CSI coefficient $\rho$ in the system, i.e., from $\rho=0.5$ to 1, when ${P_{\max}}=30\;{\rm{dBm}}$ and $L=40$. Note that as $\rho$ decreases, the CSI becomes more outdated as shown in (4) and (6), and $\rho=1$ means non-outdated CSI. It can be observed from all beamforming approaches, when CSI becomes more outdated (as $\rho$ decreases), the average secrecy rate and QoS satisfaction level decrease. The reason is that a higher value of $\rho$ indicates more accurate CSI, which will enable all the approaches to optimize secure beamforming policy to achieve higher average secrecy rate and QoS satisfaction level in the system. It can be observed that reducing $\rho$ has more effects on the performance of the other three approaches while our proposed learning approach still maintains the performance at a favorable level, indicating that the other three approaches are more sensitive to the uncertainty of CSI and the robust of the proposed learning approach. For instance, the proposed learning approach achieves the secrecy rate and QoS satisfaction level improvements of 17.21% and 8.67%, compared with the Baseline 1 approach when $\rho=0.7$. Moreover, in comparison, the proposed learning approach achieves the best performance among all approaches against channel uncertainty. The reason is that the proposed learning approach considers the time-varying channels and takes advantage of PDS-learning to effectively learn the dynamic environment. ## VI Conclusion In this work, we have investigated the joint BS’s beamforming and IRS’s reflect beamforming optimization problem under the time-varying channel conditions. 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Mobile Comput._ , vol. 12, no. 4, pp. 694-709, Apr. 2013. * [36] T. Schaul, J. Quan, I. Antonoglou, and D. Silver,“Prioritized experience replay,” in _Proc. 4th Int. Conf. Learn. Represent. (ICLR)_ , San Juan, US, May. 2016, pp. 1–21. * [37] H. Gacanin and M. Di Renzo, “Wireless 2.0: Towards an intelligent radio environment empowered by reconfigurable meta-surfaces and artificial intelligence,” 2020. [Online]. Available: https://arxiv.org/abs/2002.11040. * [38] C. W. Huang, $et~{}al.$, “Holographic MIMO surfaces for 6G wireless networks: opportunities, challenges, and trends,” to appear in _IEEE Wireless Commun._ , Apr. 2020. * [39] F. B. Mismar, B. L. Evans, and A. Alkhateeb, “Deep reinforcement learning for 5G networks: Joint beamforming, power control, and interference coordination,” _IEEE Trans. Commun._ , vol. 68, no. 3, pp. 1581-1592, Mar. 2020. * [40] H. Yang, $et~{}al.$, “Deep reinforcement learning based intelligent reflecting surface for secure wireless communications,” will be presented in _Proc. IEEE Global Commun. Conf. (GLOBECOM)_ , Dec. 2020, Taipei, Taiwan. | Helin Yang (S’15) received the B.S. and M.S. degrees in the School of Telecommunications Information Engineering from Chongqing University of Posts and Telecommunications in 2013, and 2016, and the Ph.D. degree from School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, in 2020\. His current research interests include wireless communication, visible light communication, Internet of Things and resource management. ---|--- | Zehui Xiong (S’17) is currently a researcher with Alibaba-NTU Singapore Joint Research Institute, Nanyang Technological University, Singapore. He received the Ph.D. degree at Nanyang Technological University, Singapore. He received the B.Eng degree with the highest honors at Huazhong University of Science and Technology, Wuhan, China. He is the visiting scholar with Princeton University and University of Waterloo. His research interests include network economics, wireless communications, blockchain, and edge intelligence. He has published more than 60 peer-reviewed research papers in leading journals and flagship conferences, and 3 of them are ESI Highly Cited Papers. He has won several Best Paper Awards. He is an Editor for Elsevier Computer Networks (COMNET) and Elsevier Physical Communication (PHYCOM), and an Associate Editor for IET Communications. He is the recipient of the Chinese Government Award for Outstanding Students Abroad in 2019, and NTU SCSE Outstanding PhD Thesis Runner-Up Award in 2020. ---|--- | Jun Zhao (S’10-M’15) is currently an Assistant Professor in the School of Computer Science and Engineering at Nanyang Technological University (NTU) in Singapore. He received a PhD degree in Electrical and Computer Engineering from Carnegie Mellon University (CMU) in the USA (advisors: Virgil Gligor, Osman Yagan; collaborator: Adrian Perrig), affiliating with CMU’s renowned CyLab Security & Privacy Institute, and a bachelor’s degree from Shanghai Jiao Tong University in China. Before joining NTU first as a postdoc with Xiaokui Xiao and then as a faculty member, he was a postdoc at Arizona State University as an Arizona Computing PostDoc Best Practices Fellow (advisors: Junshan Zhang, Vincent Poor). His research interests include communications, networks, security, and AI. ---|--- | Dusit Niyato (M’09-SM’15-F’17) is currently a professor in the School of Computer Science and Engineering, at Nanyang Technological University, Singapore. He received B.Eng. from King Mongkuts Institute of Technology Ladkrabang (KMITL), Thailand in 1999 and Ph.D. in Electrical and Computer Engineering from the University of Manitoba, Canada in 2008. His research interests are in the area of energy harvesting for wireless communication, Internet of Things (IoT) and sensor networks. ---|--- | Liang Xiao (M’09-SM’13) is currently a Professor in the Department of Information and Communication Engineering, Xiamen University, Xiamen, China. She has served as an associate editor of IEEE Trans. Information Forensics and Security and guest editor of IEEE Journal of Selected Topics in Signal Processing. She is the recipient of the best paper award for 2016 INFOCOM Big Security WS and 2017 ICC. She received the B.S. degree in communication engineering from Nanjing University of Posts and Telecommunications, China, in 2000, the M.S. degree in electrical engineering from Tsinghua University, China, in 2003, and the Ph.D. degree in electrical engineering from Rutgers University, NJ, in 2009. She was a visiting professor with Princeton University, Virginia Tech, and University of Maryland, College Park. ---|--- | Qingqing Wu (S’13-M’16) received the B.Eng. and the Ph.D. degrees in Electronic Engineering from South China University of Technology and Shanghai Jiao Tong University (SJTU) in 2012 and 2016, respectively. He is currently an assistant professor in the Department of Electrical and Computer Engineering at the University of Macau, China, and also affiliated with the State key laboratory of Internet of Things for Smart City. He was a Research Fellow in the Department of Electrical and Computer Engineering at National University of Singapore. His current research interest includes intelligent reflecting surface (IRS), unmanned aerial vehicle (UAV) communications, and MIMO transceiver design. He has published over 70 IEEE journal and conference papers. He was the recipient of the IEEE WCSP Best Paper Award in 2015, the Outstanding Ph.D. Thesis Funding in SJTU in 2016, the Outstanding Ph.D. Thesis Award of China Institute of Communications in 2017. He was the Exemplary Editor of IEEE Communications Letters in 2019 and the Exemplary Reviewer of several IEEE journals. He serves as an Associate Editor for IEEE Communications Letters, IEEE Open Journal of Communications Society, and IEEE Open Journal of Vehicular Technology. He is the Lead Guest Editor for IEEE Journal on Selected Areas in Communications on ”UAV Communications in 5G and Beyond Networks”, and the Guest Editor for IEEE Open Journal on Vehicular Technology on “6G Intelligent Communications” and IEEE Open Journal of Communications Society on “Reconfigurable Intelligent Surface-Based Communications for 6G Wireless Networks”. He is the workshop co-chair for IEEE ICC 2019-2021 workshop on “Integrating UAVs into 5G and Beyond”, and the workshop co-chair for IEEE GLOBECOM 2020 workshop on “Reconfigurable Intelligent Surfaces for Wireless Communication for Beyond 5G”. He serves as the Workshops and Symposia Officer of Reconfigurable Intelligent Surfaces Emerging Technology Initiative and Research Blog Officer of Aerial Communications Emerging Technology Initiative. ---|---

2024-09-04T02:54:55.227312

2002.12274

arxiv-papers

# Empirical Analysis of Indirect Internal Conversions in Cryptocurrency Exchanges Paz Grimberg, Tobias Lauinger, Damon McCoy New York University ## I Abstract Algorithmic trading is well studied in traditional financial markets. However, it has received less attention in centralized cryptocurrency exchanges. The Commodity Futures Trading Commission (CFTC) attributed the $2010$ flash crash, one of the most turbulent periods in the history of financial markets that saw the Dow Jones Industrial Average lose $9\%$ of its value within minutes, to automated order “spoofing” algorithms. In this paper, we build a set of methodologies to characterize and empirically measure different algorithmic trading strategies in Binance, a large centralized cryptocurrency exchange, using a complete data set of historical trades. We find that a sub-strategy of triangular arbitrage is widespread, where bots convert between two coins through an intermediary coin, and obtain a favorable exchange rate compared to the direct one. We measure the profitability of this strategy, characterize its risks, and outline two strategies that algorithmic trading bots use to mitigate their losses. We find that this strategy yields an exchange ratio that is $0.144\%$, or $14.4$ basis points (bps) better than the direct exchange ratio. $2.71\%$ of all trades on Binance are attributable to this strategy. ## II Introduction Cryptocurrency exchanges today handle more than $50b in daily trade volume [5]. Most of it occurs on centralized exchanges, which hold assets and settle trades on behalf of their customers. Traders on these exchanges can convert different coins at certain exchange ratios, just like traditional foreign exchange (FX) markets. The ability to convert between coins creates potential arbitrage opportunities, where traders can make a profit through a series of conversions. The case involving three conversions, coin $c_{1}$ converted to coin $c_{2}$, which is then converted to coin $c_{3}$ and back to $c_{1}$, is called triangular arbitrage if the proceeds of the conversions are greater than the initial quantity. The existence of triangular arbitrage in foreign exchange markets is well documented [14] [19] [18] [26]. The characteristics of cryptocurrency exchanges and their relationship to traditional foreign exchange markets have been studied as well [17]. However, to the best of our knowledge, triangular arbitrage has never been studied within the context of a centralized cryptocurrency exchange. In this paper, we measure arbitrage activity in Binance, a centralized cryptocurrency exchanges, by empirically exploring its complete historical trade data. Since different traders cannot be identified from trade data, we cluster sequences of consecutive trades that match in their quantities and timing and attribute them to the same trader. To the best of our knowledge, we are the firsts to employ a clustering methodology for identifying triangular arbitrage traders, based on trade-by-trade data. We find that triangular arbitrage is rarely accomplished. Participants predominantly engage in an alternative strategy, which we call indirect internal conversions, where coin $A$ is converted to coin $B$ through an intermediary coin $x$, at a favorable exchange ratio compared to directly converting $A$ to $B$. This activity accounts for $2.71$% of the total daily volume, and offers an exchange ratio that is $14.4$ bps better on average. We believe that the fee structure in cryptocurrency exchanges makes it unprofitable for participants to engage in triangular arbitrage. Instead, participants turn to indirect conversions as an efficient way to rebalance their holdings. ## III Background ### III-A Exchanges An exchange is an organized market where tradable securities, commodities, foreign exchange/cryptocurrencies (or “coins”) and derivatives are sold and bought (collectively referred to as instruments). In a centralized exchange, the deposits and assets of participants are held and settled by the exchange. In decentralized exchanges (or “DEXes”), a smart contract (a program executing on a blockchain) or other form of peer-to-peer network executes exchange functionality. In DEXes, funds cannot be stolen by the exchange operator, because their custody and exchange logic is processed and guaranteed by the smart contract. In centralized cryptocurrency exchanges, different cryptocurrencies can be exchanged to others, such as Bitcoin and Ethereum. In addition, some exchanges list ERC20 tokens, or simply “tokens,” that can also be exchanged to cryptocurrencies. Tokens are essentially smart contracts that make use of the Ethereum blockchain [13]. Participants can place orders on an exchange. An order is an instruction to buy or sell some traded instrument. These instructions can be simple or complicated, and can be sent to either a broker or directly to an exchange via direct market access. There are some standard instructions for such orders. For example, a market order is a buy or sell order to be executed immediately at the current market prices, i.e., buy at the lowest asking price or sell to the highest bidding price. Market orders are typically used when certainty of execution is a priority over the price of execution. A limit order is an order to buy an instrument at no more than a specific price, or to sell at no less than a specific price (called “or better” for either direction). This gives the trader control over the price at which the trade is executed; however, the order may never be executed (“filled”). Limit orders are typically used when the trader wishes to control price rather than certainty of execution. Each instrument traded on an exchange has an order book. The order book refers to an electronic list of buy and sell orders for a specific security or financial instrument organized by price level. An order book lists the number of shares being bid on or offered at each price point, or market depth [7]. ### III-B Arbitrage The traditional financial industry has settled on three main types of quantitative strategies that are sustainable because they provide real economic value to the market: arbitrage, market-making and market-taking [8]. In market-taking strategies, traders post both buy and sell orders in the same financial instrument, hoping to make a profit on the bid-ask spread [25]. In market-taking strategies, traders engage in longer-term trading, subject to some rules-based investment methodology. Market-taking strategies on centralized cryptocurrency exchanges have been studied in [23]. Arbitrage and its economic benefits have been well understood for quite some time and documented by academia [27] [15]. Competitive arbitrageurs on centralized exchanges have at least one of three advantages. 1. 1. Scale: participants who trade large volumes are often compensated in the form of kickbacks, rebates, and low (or zero) trading fees, which provide such participants the opportunity to make profits in cases where others with less scale cannot. 2. 2. Speed: the ability to access order book information and trade faster than others provides an opportunity to gain from mispricings. For example, triangular arbitrage on FX products is a major impetus for the Go West and Hibernia microwave telecommunications projects [9] [1], where multi-million dollar network infrastructure was developed for the purpose of shaving off milliseconds in electronic trading latency. 3. 3. Queue position: being able to enter one leg of an arbitrage trade by placing orders ahead of time, without crossing the spreads, i.e., placing orders that execute immediately without being queued in the order book, significantly reduces fees, which enables profitable completion of triangular arbitrage trades. Participants are compensated for the risk of queuing orders in the order book. Arbitrage strategies involving multiple centralized cryptocurrency exchanges, exploiting different prices of same assets, have been studied [22] [21] [24] [20]. Arbitrage strategies on decentralized exchanges are executed by bots who pay high transaction fees and optimize their network latency to front-run ordinary users’ trades [16]. At time of writing, the vast majority of cryptocurrencies trading volume (over $99\%$) is done in centralized exchanges [5], therefore in this paper, we focus on triangular arbitrage in a single centralized cryptocurrency exchange (Binance). Triangular arbitrage is an arbitrage strategy resulting from a discrepancy between three coins that occurs when their exchange rates give rise to a profitable sequence of trades, i.e., trades of the form $c_{1}\mapsto c_{2}\mapsto c_{3}\mapsto c_{1}$, where $c_{1},c_{2},c_{3}$ are coins, and the ending balance of $c_{1}$ is greater than the initial balance. We call $c_{1}$ the base coin, $c_{2}$ and $c_{3}$ intermediary coins, and the sequence a cycle [10]. To the best of our knowledge, we are the firsts to employ a clustering methodology for identifying triangular arbitrage traders. ## IV Dataset We use historical cryptocurrency trade data and order book data provided by Kaiko from Nov $9$th, $2017$ through Aug $6$th, $2019$ [6]. We conducted our measurements on Binance’s exchange data, as it is the largest centralized exchange by daily volume and has the highest number of listed symbols (the “ticker” of an exchange pair). Kaiko’s data consists of trade-by-trade data and market depth data for all cryptocurrencies and ERC20 tokens traded on Binance. There are $1,964,461,269$ trades. Figure 1 shows the monthly number of trades executed on Binance. Figure 1: Number of monthly trades, in millions, executed on Binance, based on Kaiko’s data set. Every coin in Binance is denominated by at least one of the following coins: BNB (Binance’s native token), Bitcoin, ALTS (Etheruem, Ripple, Tron) and stable coins. Stable coins are fiat-pegged cryptocurrencies designed to minimize the volatility of price fluctuations. We call these coins anchor coins, as these coins can be directly exchanged to many other coins and have historically had the greatest market value (in US dollar terms) [3]. There are $204$ different coins and $704$ different possible conversions. We denote such conversions by $c_{1}\Leftrightarrow c_{2}$, where $c_{1}$ and $c_{2}$ are two different coins. We call $c_{1}\Leftrightarrow c_{2}$, a pair. For example, if $c_{1}=\text{BTC}$ and $c_{2}$ = ETH, then the pair is denoted $\text{BTC}\Leftrightarrow\text{ETH}$. Traders can exchange their BTC to ETH (or vice versa) using this conversion. We write $c_{1}\mapsto c_{2}$ when considering the side of the $c_{1}$ holder. There are 6,534 possible cycles in Binance, i.e., sequences of the form $c_{1}\mapsto c_{2}\mapsto c_{3}\mapsto c_{1}$. In $3,900$ of them, one of $c_{2}$ and $c_{3}$ is an anchor. In $2,634$ of them, both $c_{2}$ and $c_{3}$ are anchors. There are no cases where both $c_{2}$ and $c_{3}$ are non- anchors. $c_{1}$ can be anchor or non-anchor. Cycle statistics in Binance are summarized in Table I. There are $1,982$ cycles with a non-anchor coin as the base coin. These cycles have the potential to create arbitrage gains in non-anchor coins. However, anchor coins represent over $92\%$ of the total market value of all listed coins and historically had lower volatility, compared to non-anchor coins. Therefore, we focus on cycles with anchor base coins. As future work, we could explore cycles with no-anchor base coins. However, we note that there is inherent risk in operating in non-anchor coins, due to volatility. Base coin | Number of cycles ---|--- BTC | 986 BNB | 818 ALTS | 908 Stable coins | 1,840 Other coins | 1,982 Total | 6,534 Table I: Number of cycles in Binance by base coin. BTC,BNB,ALTS and stable coins represent over $92\%$ of the total market value of all listed coins and historically had lower volatility. Therefore, we omit other cycles in this study. ### IV-A Data Fields Kaiko’s historical trade data fields are described in Table II. The granularity of the date field is in ms. Since multiple trades can execute within the same ms, we use the monotonically increasing trade ids as a way to order the trades. Field | Description ---|--- id | Incremental trade id exchange | Exchange id symbol | Pair symbol date | Time (in epoch) when the trade took place price | Trade price amount | Trade quantity (quoted in base asset) sell | | Binary. If TRUE, then the --- holder of $c_{1}$, swapping to $c_{2}$, was a liquidity taker, and $c_{1}$ is the base asset in the pair listed on the exchange Table II: Kaiko’s trade data fields. Kaiko’s historical order book data is given on a 60-second basis, that is, for every pair listed, its entire order book is given in 60 second increments of the date field. Note that the quotes’ depth are not given and need to be reconstructed separately. The data fields are described in Table III. Field | Description ---|--- symbol | Pair symbol date | Time (in epoch) when the order was placed type | Bid or ask amount | Order quantity (quoted in base asset) Table III: Kaiko’s order book data fields. ### IV-B Data Limitations Kaiko’s order book data is given on a $60$-second interval basis and Binance’s API does not provide historical order-book data [4]. Complete order book data could reveal all arbitrage opportunities, not just the ones executed with actual trades, as it would accurately paint the traders’ view of the market at any point in time. When merging Kaiko’s historical trade data with its order book, only about $5\%$ of the trade data has matching order book data available within a $100ms$ interval. (During our measurements, we used smaller time intervals than $100ms$.) For profitability measurements, about $0.5\%$ of the time, we were able to use the order book’s bid/ask directly. For the remainder, we approximated the bid/ask price with the volume-weighted average price of that time interval, based on the trade data. In addition, Kaiko does not include trade fees or historical fee schedules, so we had to reconstruct Binance’s historical fee time series. ### IV-C Binance Trading Fees An important data point for our analysis are the trading fees collected by Binance [2]. For every trade executed, Binance collects a fee from the proceeds of the trade. The fees are collected either in the currency of the proceeds or in Binance’s native ERC20 token, BNB, depending on the trader’s BNB balance. If the trader has enough BNB balance, Binance will calculate the conversion rate between the proceeds and BNB, and withdraw the fees from the BNB balance. In general, Binance rewards participants who trade large volumes by charging them a lower fee. In addition, for high-volume traders, Binance distinguishes between liquidity-taking trades, i.e., market order trades or limit order trades that were matched immediately at the best bid/ask level, and market-making trades, which are trades that were placed somewhere in the order book but were not executed immediately. Binance charges less fees for market-making trades, as they wish to encourage participants to “fill up” the order book, thereby narrowing the bid/ask spread and increasing liqudity. This is a common practice; however, in traditional financial exchanges, market- makers pay zero or even negative fees (rebates, kickbacks). We assume that arbitrageurs are operating at the lowest fee level. To track Binance’s fee schedule, we used the Wayback Machine [12], a digital archive of the World Wide Web, to view Binance’s fee web page historically. In our analysis time span, Binance’s fee web page changed $46$ times. However, the lowest fee level remained constant at $1.2$bps for makers and $2.4$bps for takers. ## V Methodology In this section, we describe our methodology for discovering triangular arbitrage trade sequences in Binance by examining the trade data. During our analysis, we discovered that a sub-strategy of triangular arbitrage, involving only the first two conversions, is widely deployed. We refined our original methodology to identify such conversions. These conversions exhibit some risks, one of them is the scenario where multiple traders compete for the same intermediary coin, potentially harming laggards’ profitability. We identify these risks and outline a methodology for clustering competing events. Lastly, we observe different strategies traders take to mitigate this risk and describe a methodology to detect it. ### V-A Discovering Triangular Arbitrage Trading Sequences To discern triangular arbitrage trading sequences, we design a methodology to identify likely arbitrage trading sequences. We look for trading sequences of the form $c_{1}\mapsto c_{2}\mapsto c_{3}\mapsto c_{1}$, where $c_{1}$ is an anchor coin (BTC, BNB, ALTS, Stable coins). Arbitrageurs start with some quantity $Q_{1}$ of $c_{1}$, exchange it to $c_{2}$ at price $p_{12}$, resulting in $Q_{2}$ units of $c_{2}$. In practice, Binance deducts the fee upon conversion, therefore $Q_{2}$ will be slightly lower than the conversion price. Next, $Q_{2}$ units of $c_{2}$ are converted to $c_{3}$, resulting in $Q_{3}$ units, minus fees. Lastly, $Q_{3}$ units of $c_{3}$ are converted back to $c_{1}$, minus fees, resulting in $Q_{1}^{\prime}$ units of $c_{1}$. If $Q_{1}^{\prime}>Q_{1}$, then the arbitrage sequence is profitable. To successfully profit from this opportunity, arbitrageurs need to execute $3$ trades, $c_{1}\mapsto c_{2}$, $c_{2}\mapsto c_{3}$ and $c_{3}\mapsto c_{1}$. First, an arbitrageur needs to calculate the initial quantity $Q_{1}$ of $c_{1}$, such that during conversions, fee payments and other trading constraints will leave no or minimal residue in the intermediary coins. Second, since Binance does not support batch trading, i.e., grouping multiple orders in a single request, arbitrageurs need to ensure correct timing of their trades. For example, if the order $c_{2}\mapsto c_{3}$ arrives before $c_{1}\mapsto c_{2}$, then it will fail as the arbitrageur does not hold $c_{2}$ yet. Furthermore, the arbitrageur competes with other arbitrageurs for these trades, so speed is an important factor. In order to identify triangular arbitrage sequences, we first need to ensure that the same quantity is flowing through different coins, ending at the base coin. Binance quotes quantities based on how the pair is listed. Different pairs have different quantity/price restrictions, namely, minimum, maximum and increment size. Since quantities and prices change in discrete steps, a small quantity might not be converted, leaving a residue. These small residues need to be accounted for when identifying trades with equal quantities. While these residues have a (small) value, in practice, they cannot be converted immediately to anchor coins, due to minimum size restrictions. As residue accumulates beyond the exchange threshold, arbitrageurs can convert it back to anchor coins. We found that less than $0.01\%$ of the time, the profitability was decided by the residue. Therefore, in our detection process we ignore the residue value. To illustrate coin exchanges, we follow an actual trade example from Binance. #### V-A1 BTC$\Leftrightarrow$ETH Exchange Example Consider the conversion between BTC and ETH. It is listed as ETH/BTC in Binance. ETH is called base asset and BTC is called the quote asset. The quantity is always given in base asset terms. This pair has a $0.001$ minimum quantity (base asset), $100,000$ maximum quantity, $0.001$ quantity increments, $0.000001$ minimum price (quote asset), $100,000$ maximum price and $10^{-6}$ price increments. At the time of writing, the last trade of ETH/BTC was executed at a price of $0.018876$ and quantity $0.153$. The “sell” flag was on, meaning that the holder of ETH was the one to initiate the trade and met the buyer at the buyer’s bid price. Assume both participants are at the cheapest fee level, currently at $2.4$bps for takers and $1.2$bps for makers. From the ETH holder’s perspective: Since ETH is the base asset and the “sell” flag is on, this means the ETH holder initiated the trade and met the buyer at the buyer’s bid price, therefore paying a liquidity taking fee of $2.4$bps. The holder exchanged $0.153$ units of his ETH at a price of $0.018876$ BTC per $1$ ETH. This means the holder ended with $0.153\cdot 0.018876\cdot 0.99976=0.002887334873$ units of BTC. Note that if $0.002887334873$ BTC are to be exchanged for some other coin, only $0.00288733$ could be converted, leaving $0.000000004873$ BTC residue. From the BTC holder’s perspective: Since BTC is the quote asset and the “sell” flag is on, this means the BTC holder provided the liquidity for the trade and his price was met by the seller, thus paying a liquidity making fee of $1.2$bps. The BTC holder exchanged $0.153\cdot 0.018876=0.002888028$ units of BTC at a price of $0.018876$ BTC per $1$ ETH, while paying $1.2$bps fee, resulting in $(0.002888028/0.018876)\cdot 0.99988=0.15298164$ ETH. From Binance’s perspective: They collect $2.4$bps of the BTC proceeds from the ETH holder, i.e., $0.00024\cdot 0.002888028\approx 6.93\cdot 10^{-7}$ units of BTC. If the ETH holder also holds BNB, then this amount is actually converted to BNB terms (using an average of recent exchange ratios between BTC and BNB), and deducted from the BNB balance. If the ETH holder does not own BNB, then $6.93\cdot 10^{-7}$ BTC are deducted from the proceeds. In addition, Binance also collect $1.2$bps of the ETH proceeds from the BTC holder, i.e., $0.153\cdot 0.00012\approx 18.36\cdot 10^{-6}$ units of ETH. If the BTC holder also holds BNB, this amount is collected in BNB terms as well. Note that if the seller/buyer did not hold BNB, the fee would have been higher. In our analysis, we assumed arbitrageurs operate at the lowest fee level. #### V-A2 Identifying Equal Quantities in Triangular Arbitrage Sequences We identify sequences of trades, $c_{1}\mapsto c_{2}\mapsto c_{3}\mapsto c_{1}$, where the same quantity is passed. The quantity is quoted in the base asset and depends on whether $c_{1}/c_{2}$ or $c_{2}/c_{1}$ is listed, whether $c_{2}/c_{3}$ or $c_{3}/c_{2}$ is listed and whether $c_{1}/c_{3}$ or $c_{3}/c_{1}$ is listed. When matching quantities between trades of two pairs, having a common coin, $c_{1}\mapsto c_{2}$ and $c_{2}\mapsto c_{3}$, we translate the quantities to the common coin’s terms, $c_{2}$, and check if they are equal, up to fees and residue resulting from the trade. In Table IV, we describe the translation process, based on different listing scenarios in Binance. Binance listing | Matching condition ---|--- $c_{2}/c_{3}$, $c_{1}/c_{2}$ | $q_{12}p_{12}-q_{23}=\text{fee + residue}$ $c_{2}/c_{3}$, $c_{2}/c_{1}$ | $q_{12}-q_{23}=\text{fee + residue}$ $c_{3}/c_{2}$, $c_{1}/c_{2}$ | $q_{12}p_{12}-q_{23}p_{23}=\text{fee + residue}$ $c_{3}/c_{2}$, $c_{2}/c_{1}$ | $q_{12}-q_{23}p_{23}=\text{fee + residue}$ Table IV: Matching equal quantities in conversions with common coin. $q_{ij}$ and $p_{ij}$ are the quantities and prices quoted in the trade data. #### V-A3 Identifying Trade Latency in Triangular Arbitrage Sequences When a triangular arbitrage opportunity presents itself, arbitrageurs compete amongst each other to be the first to execute the trade sequences. There are three trades to be executed, $c_{1}\mapsto c_{2}$, $c_{2}\mapsto c_{3}$ and $c_{3}\mapsto c_{1}$. Since Binance does not support batch trading, an arbitrageur will have to send three limit orders with prices $p_{12},p_{23},p_{31}$ and quantities $q_{12},q_{23},q_{31}$. These quantities are equal in the conversion process, in the sense explained previously and the prices match the prices quoted in the order book. To ensure that trades are received in their original order, the arbitrageur will take into account network latency and wait a small amount of time between sending consecutive orders. Analysis of the trade data suggests that the average latency between consecutive trades is $20$ ms with a standard deviation of $15$ ms. Around $95\%$ of trades with matching quantities are executed within $10$ms-$30$ms of each other. Therefore, clustering trades using longer latency does not have material impact on the number of trades identified. We find that using $\Delta t\approx 50$ms as an approximation for arbitrage latency between consecutive trades maximizes precision and minimizes recall. We found that less than $0.01\%$ of triangular arbitrage sequences contain a liquidity-making trade. We believe this behavior is caused by Binance’s fee model, which charges traders a commission even for liquidity making trades. If traders used liquidity-making orders, they would need to pay the fee in case they were filled. At that point, it is not guaranteed that an arbitrage opportunity will exist, while the fee is already paid and exposure to what is mostly an illiquid coin is already established. To further enhance our precision, we require arbitrage trade sequences to be all liquidity-taking. Furthermore, we found a surprisingly low number of triangular arbitrage trades, 1,381,928 in total, accounting for $0.24\%$ of total number of trades. However, in the course of clustering triangular arbitrage trades, we witnessed a much larger number of partial arbitrage sequences, 20,892,825, or $2.71\%$ of total trades, where traders executed the first two trades, $c_{1}\mapsto c_{2}$ and $c_{2}\mapsto c_{3}$, but did not execute the third trade, $c_{3}\mapsto c_{1}$. Executing the first two trades effectively converts $c_{1}$ to $c_{3}$ at an exchange ratio of $p_{12}\cdot p_{23}$, minus fees and residue. Interestingly, $95\%$ of the time these trades resulted in an exchange ratio that was favorable to the exchange ratio of the direct trade $c_{1}\mapsto c_{3}$. We call this trading strategy indirect internal conversions and explain below in more details what is means to have a “favorable” rate. We refine our methodology to identify indirect internal conversions and study the root cause of unprofitable instances. We elaborate on the risks associated with this trading strategy in the discussion section. ### V-B Discovering Indirect Internal Conversion Attempts We refined our original methodology for discovering triangular arbitrage sequences by relaxing the constraint for the third trade to be executed. We identify equal quantities in the first two trades in the same way as before. To determine the trade latency, we empirically explored different time constraints. $93\%$ of trades with equal quantities have a latency between $10$ms and $100$ms. Latency lower than $10$ms gives poor recall, with less than $5\%$ of total attempts. Latency greater than $100$ms only accounts for $2\%$ of all attempts. The average latency is $22$ms with a standard deviation of $15$ms. We find that, as before, $\Delta t\approx 50$ms is an approximation that likely provides both high precision and recall for identifying indirect conversion trading sequences. However, we lack ground truth to definitively evaluate this part of our methodology. #### V-B1 Determining Profitability of Internal Conversions The first two trades of a triangular arbitrage sequence, $c_{1}\mapsto c_{2}\mapsto c_{3}\mapsto c_{1}$ are $c_{1}\mapsto c_{2}$ and $c_{2}\mapsto c_{3}$. Executing these two trades gives an indirect exchange ratio between $c_{1}$ and $c_{3}$. If this exchange ratio, net of fees and residue, is greater than the exchange ratio of $c_{1}\mapsto c_{3}$, then this conversion offers a favorable exchange ratio to the direct one. We call such favorable conversions indirect internal conversions. Since the exchange ratio between $c_{1}$ and $c_{3}$ fluctuates, it is important to define the time it is taken. We wish to approximate the trader’s view of the market upon executing the internal conversion. Therefore, we take the exchange ratio at a short period of time prior to the first trade’s execution time. As discussed above, $50$ms is a good delay approximation for the trader’s view. Kaiko’s order book data is given on a $60$-second basis, so it is unlikely that this $50$ms time window falls within Kaiko’s data intervals. To approximate the order book’s best bid/ask price at the time, we take the volume weighted average price (VWAP) [11] of $c_{1}\mapsto c_{3}$, over the period of $50$ms, prior to the execution time of the first trade. These trades tell us the best bid/ask level at that time, and taking an average, weighted by volume, is a good approximation of the trader’s view of the order book. However, it is possible that within that time period, other participants posted bids and asks and then cancelled them, giving the trader a completely different view of $c_{1}\mapsto c_{3}$. In our analysis, we assume that this did not occur. If there were no trades during that period and order book data is unavailable, we cannot determine if the conversion is profitable and did not include such conversions in our profitability analysis. We found $26,603,038$ indirect conversions, which is $2.71\%$ of the total number of trades. $95\%$ of the indirect conversions resulted in a favorable exchange ratio. We hypothesized that unprofitable conversions occur when multiple traders obtain the same intermediary coin $c_{2}$, and simultaneously attempt to convert it to $c_{3}$. For example, one trader is looking to convert $c_{1}\mapsto c_{2}\mapsto c_{3}$ for a favorable exchange ratio to $c_{1}\mapsto c_{3}$, and a second trader is looking to convert $y\mapsto c_{2}\mapsto c_{3}$ at a favorable exchange ratio to $y\mapsto c_{3}$ ($y$ could be the same coin as $c_{1}$). When both traders obtain $c_{2}$, there could be a scenario where only one trader is able to convert $c_{2}$ to $c_{3}$ at the best bid/ask level. This happens when the first one to execute $c_{2}\mapsto c_{3}$ consumes the entire quantity of the best bid/ask and causes the order book to change. The laggard in this case engages in a loss mitigating strategy. One option is to convert $c_{2}$ to $c_{3}$ at a worse exchange ratio than originally intended, potentially resulting in a conversion that is worse than the direct one. Another option is to convert $c_{2}$ to several other coins, with the goal of minimizing the potential losses from unfavorable exchange ratios. We call the former strategy a full-exit strategy and the latter partial-exit strategy. To corroborate our hypothesis with the trade data, we refine our methodology to cluster competing conversions and identify loss-mitigating exit strategies. ### V-C Clustering Competing Indirect Internal Conversion Attempts Using our methodology to discover indirect conversions, we identified conversions that were initiated around the same time and had an overlapping intermediary coin. This is because competing conversions try to complete the same second trade. Therefore, for a given second trade $c_{2}\mapsto c_{3}$, we look at indirect conversions attempts of the form $x\mapsto c_{2}\mapsto c_{3}$ that started within $100$ms of each other. Every cluster of competing conversions can have winning conversions, i.e., indirect conversions that were completed at a favorable rate to the direct rate and losing conversions, conversions that completed at an unfavorable rate to the direct rate. Losing conversions can have many causes, such as mistiming or an inaccurate view of the order book. We believe that one of those reasons is that traders who successfully completed the first trade, and failed to complete the second trade, are unloading their $c_{2}$ coin at an unfavorable rate to avoid having directional exposure to $c_{2}$. #### V-C1 Identifying Loss-Mitigating Trading Strategies For losing conversions, we wanted to see if the loss was the result of a competing winning conversion that utilized all the capacity of $c_{2}\mapsto c_{3}$. Determining whether a conversion utilized all capacity is impossible without knowing the order book at that time. However, we can approximate it by observing the next trade of $c_{2}\mapsto c_{3}$. Since we have the trade id, which is monotonically increasing with the trades, we can tell whether the next trade of $c_{2}\mapsto c_{3}$ was completed at the same price or not. If it was not completed at the same price, it is likely that the previous trade used up the capacity of the previous level. This is only a heuristic, as traders can post and cancel orders at any time. Loss-mitigating conversions are losing conversions where a winning conversion in the same cluster used up the intermediary’s coin capacity. By analyzing the trade data, we found that $17.7\%$ of all losing conversions corresponded to loss-mitigating traders. We identified two sub-strategies of loss-mitigating traders. #### V-C2 Full-Exit Loss Mitigating Strategy These are conversions that converted an equal quantity $Q_{1}$ from $x\mapsto c_{2}$ and $c_{2}\mapsto c_{3}$, i.e., these are traders who converted all their $c_{2}$ into one coin. #### V-C3 Partial-Exit Loss Mitigating Strategy These are conversions that converted a quantity $Q_{1}$ from $x\mapsto c_{2}$ at price $p_{12}$, but converted a lower quantity $Q_{2}<Q_{1}$ from $c_{2}\mapsto c_{3}$, i.e., these are traders who unloaded some, but not all of their $c_{2}$ into one coin. This strategy can be attributed to traders solving the following minimization problem: Find a set of $k$ coins $\\{d_{1},d_{2},\ldots d_{k}\\}$ having exchange ratios $\\{p_{1},p_{2},\ldots p_{k}\\}$, and a set of $k$ quantities $\\{q_{1},q_{2},\ldots q_{k}\\}$, where $\sum_{j=1}^{k}q_{j}=Q_{1}$ such that the following loss function is minimized: $\displaystyle Loss=\sum_{j=1}^{k}q_{j}\left(\frac{p_{13}}{p_{12}p_{j}}\right)$ where $p_{13}$ is the direct exchange ratio of $x\mapsto c_{3}$. To detect partial exits, we iterate over all different combinations of trades from $c_{2}\mapsto c_{3}$ having quantities less than $Q_{1}$, such that they sum exactly to $Q_{1}$, up to fees and residue. We found that $85\%$ of loss-mitigating strategies are partial exits and $15\%$ are full exits. It also turns out that solving a loss minimization problems is effective, as the average partial-exit loss is $21$bps while the full-exit average loss is $25$bps. ## VI Results ### VI-A Volume We found $26,603,038$ indirect conversions, which is $2.71\%$ of total trades. We found $1,381,928$ triangular arbitrage sequences, accounting for $0.24\%$ of total trades. The time series of the number of favorable indirect conversion attempts as a percentage of number of direct conversions, on a daily basis, is shown in Figure 2. The number of favorable indirect conversion attempts, on a daily basis, in shown in Figure 3. The number of triangular arbitrage attempts, on a daily basis, in shown in Figure 4. Figure 2: Percentage of daily volume of direct conversions trades that is also done indirectly. Typically, $1\%-3\%$ of the daily volume of a direct conversion pair is also executed via indirect conversion sequences. Figure 3: Daily number of indirect conversion attempts. From October $2017$ \- October $2018$, the number of indirect conversions has been trending down and since October $2018$ has been trending up. Figure 4: Daily number of triangular arbitrage attempts. From October $2017$ \- April $2019$, the number of triangular arbitrage attempts has been trending down and since April $2019$ has been trending up, with a sharp spike in July $2019$ ### VI-B Latency We found that indirect conversion sequences have, on average, $21.43$ms of delay between the first trade and the second, with a standard deviation of $11.04$ms. $78.94\%$ have a latency of $30$ms or less. Triangular arbitrage sequences are faster than indirect conversions. This could be an indication that this space is more competitive. We found that triangular arbitrage sequences have, on average, $20.7$ms of delay between consecutive trades, with a standard deviation of $11.04$ms. $85.43\%$ have a latency of $30$ms or less. The latency statistics of triangular arbitrage sequences and indirect conversions are shown in Table V. Type | Metric | Value ---|---|--- Triangular Arbitrage | Latency Average | $20.7$ms Triangular Arbitrage | Latency Stdev | $11.04$ms Triangular Arbitrage | % Below $30$ms | $85.43\%$ Indirect Conversion | Latency Average | $21.43$ms Indirect Conversion | Latency Stdev | $13.7$ms Indirect Conversion | % Below $30$ms | $78.94\%$ Table V: Latency statistics for triangular arbitrage and indirect conversions. The triangular arbitrage strategy exhibits lower latency than indirect conversions, possibly a sign that this strategy is more competitive The latency distribution, in ms, of indirect conversions and triangular arbitrage is shown in Figure 5. Figure 5: Left: Latency distribution, in ms, of indirect conversion trade sequences. Right: Latency distribution, in ms, of triangular arbitrage trade sequences. Indirect conversion trades exhibit higher latency with a fatter tail, indicating that triangular arbitrage is more competitive. ### VI-C Profitability We calculate the equal-weighted return and the return on capital for triangular arbitrage trades and indirect conversions. The equal-weighted return is the average of returns. The return on capital is the arithmetic average of returns, weighted by quantity of base coin committed to the trade, i.e., larger trades receive higher weight. Indirect conversion attempts were profitable $93.92\%$ of the time with an equal-weighted net return of $11.8$bps. The return on capital for indirect conversions is $22\%$ greater, at $14.4$bps, which means that traders efficiently commit more coins to more profitable opportunities. Triangular arbitrage trades were profitable $94.97\%$ of the time with an equal-weighted net return of $9.3$bps. The return on capital for triangular arbitrage sequences is $5\%$ greater, at $9.8$bps, which is slightly higher than the equal-weighted return, but not significantly. It appears that triangular arbitrage traders are unable to significantly allocate more coins to more profitable opportunities. One potential explanation is that the space is more crowded, or “arb-ed out,” than the indirect conversions strategy. ### VI-D Loss-Mitigating Strategies We saw $1,617,464$ instances of indirect conversion completed at an unfavorable rate compared to the direct rate, or $6.08\%$ of the total number of indirect conversions. $17.7\%$ of such conversions were loss-mitigating trades, triggered by a competing indirect conversion that used all the intermediary coin’s capacity. When this happened, there were $1.06$ competing conversions on average. The highest number of competing conversions we saw in one cluster was $5$. In $15\%$ of such cases, the entire base quantity was unloaded into another coin, i.e., they were full exits. The return on capital for the full exit strategy was $-27.4$bps. $85\%$ of the time, the base coin quantity was unloaded to several other coins, i.e., they were partial exits. On average, the number of coins used to exit was $2.7$, and the return on capital was $-23.8$bps. Intuitively, this shows that solving a loss minimization problem mitigates losses by $3.6$bps compared to a full exit. Note, however, that a full exit could potentially be the solution to the minimization problem as well. ## VII Discussion The existence of triangular arbitrage in traditional foreign exchange markets is well documented and studied in academia. In centralized cryptocurrency exchanges such as Binance, triangular arbitrage happens at a small scale. However, we found that a different strategy is taking place, which converts two coins through an intermediary coin, at a favorable rate to the direct conversion rate. This strategy comes with a risk, however, that multiple competing conversions occur at the same time, preventing slower ones from completing their conversions. We saw that when this happens, traders engage in a loss-mitigating strategy. We believe that the rationale for this strategy stems from Binance’s fee structure, where there is a fee for market-making, thus creating frictions for the triangular arbitrage strategy. We believe arbitrageurs adapted to the fee structure by executing the indirect conversions strategy. While triangular arbitrage increases the arbitrageur’s holding in the base asset, indirect conversions simply provide a more favorable ratio to the direct one. Participants who already have a stake in the cryptocurrency ecosystem and wish to rebalance their portfolio might choose to engage in this strategy. ## VIII Conclusion We found that $0.24\%$ of daily trades on Binance can be attributed to triangular arbitrage trades. We found that a different strategy is $11$ times more prevalent, at $2.71\%$ of daily trades, which involves exchanging coins through an intermediary coin at a favorable rate to the direct exchange rate. We designed a methodology to measure such events and discovered that $93.92\%$ of the time the rate obtained from these conversions is $14.4$bps better than the direct one. When it is not, $17.7\%$ of the time traders engage in loss- mitigating strategies, and we identified two of them. ## Acknowledgements The authors would like to thank Andrew Papanicolaou and David Yermack for their feedback and suggestions to improve the quality of our manuscript. We acknowledge funding support under National Science Foundation award 1844753. ## References * [1] The $300m cable that will save traders milliseconds. https://www.telegraph.co.uk/technology/news/8753784/The-300m-cable-that-will-save-traders-milliseconds.html. * [2] Binance fee schedule. https://www.binance.com/en/fee/schedule. * [3] Binance markets page. https://www.binance.com/en/markets. * [4] Binance official API docs. https://github.com/binance-exchange/binance-official-api-docs/blob/master/rest-api.md. * [5] Coinmarketcap cryptocurrency exchanges trade volume. https://coinmarketcap.com/rankings/exchanges/. * [6] Kaiko historical trade data. https://www.kaiko.com/pages/historical-data. * [7] Order book definition. https://www.investopedia.com/terms/o/order-book.asp. * [8] Quantitative trading summary. https://blog.headlandstech.com/2017/08/03/quantitative-trading-summary. * [9] Ready, set, go west: Chicago-to-Tokyo trading is revving up. https://www.bloomberg.com/news/articles/2017-09-07/ready-set-go-west-chicago-to-tokyo-trading-is-revving-up. * [10] Triangular arbitrage definition. https://www.investopedia.com/terms/t/triangulararbitrage.asp. * [11] Volume weighted average price (VWAP) definition. https://www.investopedia.com/terms/v/vwap.asp. * [12] Wayback machine. https://archive.org/web/. * [13] What is ERC-20 and what does it mean for Ethereum. https://www.investopedia.com/news/what-erc20-and-what-does-it-mean-ethereum/. * [14] Y. 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Pawel, W. Marcin, and D. Stanislaw. Detecting correlations and triangular arbitrage opportunities in the Forex by means of multifractal detrended cross-correlations analysis. Nonlinear Dynamics, 98(3):2349–2364, Nov 2019. * [27] A. Shleifer and R. W. Vishny. The limits of arbitrage. The Journal of finance, 52(1):35–55, 1997. ## IX Appendix In this appendix, we define notations that are specific to the market microstructure of Binance and formally define triangular arbitrage and indirect conversions. Let $\mathbb{P}$ be the set of all pairs $x/y$ traded on Binance. Every pair facilitates two conversions; from $x$ to $y$ and from $y$ to $x$, denoted by $(x\mapsto y)$ and $(y\mapsto x)$, respectively. ###### Definition IX.1. (exchange ratio) Let $\psi_{ij}$ be the proceeds of converting 1 unit of coin $c_{i}$ to $c_{j}$. We call $\psi_{ij}$ the exchange ratio and it is given by $\displaystyle\psi_{ij}=\begin{cases}\frac{1}{p_{c_{j}/c_{i}}^{ask}}&\text{if $c_{j}/c_{i}\in\mathbb{P}$}\\\ p_{c_{i}/c_{j}}^{bid}&\text{if $c_{i}/c_{j}\in\mathbb{P}$}\\\ \end{cases}$ ###### Definition IX.2. (pair capacity) Denote $\eta_{ij}$ the maximum quantity $c_{i}$ that can be converted to $c_{j}$. We call $\eta_{ij}$ the capacity of $(c_{i}\mapsto c_{j})$ and it is given by $\displaystyle\eta_{ij}=\begin{cases}p_{c_{j}/c_{i}}^{ask}q_{c_{j}/c_{i}}^{ask}&\text{if $c_{j}/c_{i}\in\mathbb{P}$}\\\ q_{c_{i}/c_{j}}^{bid}&\text{if $c_{i}/c_{j}\in\mathbb{P}$}\\\ \end{cases}$ ###### Definition IX.3. (bid-ask spread) We write $\displaystyle\Delta_{ij}=\begin{cases}p_{c_{j}/c_{i}}^{ask}-p_{c_{j}/c_{i}}^{bid}&\text{if $c_{j}/c_{i}\in\mathbb{P}$}\\\ p_{c_{i}/c_{j}}^{ask}-p_{c_{i}/c_{j}}^{bid}&\text{if $c_{i}/c_{j}\in\mathbb{P}$}\\\ \end{cases}$ the bid ask spread of pair $(c_{i}\mapsto c_{j})$ ###### Definition IX.4. (trade quantity) suppose we wish to convert $q$ units of $c_{i}$ to $c_{j}$. The trade quantity passed to the exchange as a parameter is denoted $T(q)$ and is given by $\displaystyle T_{ij}(q)=\begin{cases}\frac{q}{p_{c_{j}/c_{i}}^{ask}}&\text{if $c_{j}/c_{i}\in\mathbb{P}$}\\\ q&\text{if $c_{i}/c_{j}\in\mathbb{P}$}\\\ \end{cases}$ ###### Definition IX.5. (minimum trade lot size) Let $m_{ij}$ be the minimum amount of $c_{i}$ that must be converted to $c_{j}$ in $\left(\text{$c_{i}$}\mapsto\text{$c_{j}$}\right)$, i.e., when converting $q$ units of $c_{i}$ to $c_{j}$, $m_{ij}\leq q$ ###### Definition IX.6. (high-level parameters) the $h$-th order book exchange ratio and capacity are denoted $\psi_{ij}^{h}$ and $\eta_{ij}^{h}$, respectively. ###### Definition IX.7. (minimum price increment) Denote $dx_{ij}$ the the minimum order book price increments of $(c_{i}\mapsto c_{j})$. ###### Remark. It holds that $dx_{ij}=dx_{ji}$. In practice, when converting $q$ units of $c_{i}$ to $c_{j}$, the trade quantity passed to the exchange as a parameter has to be a multiple of $dx_{ij}$, therefore only $\left\lfloor\frac{T(q)}{dx_{ij}}\right\rfloor dx_{ij}$ units are passed. Denote $\left\lfloor x\right\rfloor_{y}\mathrel{\mathop{:}}=\left\lfloor\frac{x}{y}\right\rfloor y$ the rounding operation of $x$ to the nearest multiple of $y$. ###### Definition IX.8. (residuals) Let $r_{ij}(q)$ be the quantity of $c_{i}$ left when converting $q$ units of $c_{i}$ to $c_{j}$. We write $\displaystyle r_{ij}(q)=\begin{cases}\left(T(q)-\left\lfloor T(q)\right\rfloor_{dx_{ij}}\right)p_{c_{j}/c_{i}}^{ask}&\text{if $c_{j}/c_{i}\in\mathbb{P}$}\\\ T(q)-\left\lfloor T(q)\right\rfloor_{dx_{ij}}&\text{if $c_{i}/c_{j}\in\mathbb{P}$}\\\ \end{cases}$ Therefore, converting $x$ units of $c_{i}$ yields $\psi_{ij}\left[x-r_{ij}(x)\right]$ units of $c_{j}$ and $r_{ij}(x)$ residual units of $c_{i}$ ### IX-A Cycle Arbitrage The set of pairs $C=\\{(c_{1}\mapsto c_{2}),(c_{2}\mapsto c_{3}),\ldots,(c_{n}\mapsto c_{n+1})\\}$ is called a cycle, if $c_{n+1}=c_{1}$ and $c_{j}\not\in\\{c_{1},\ldots\,c_{j-1}\\}$ for $1<j\leq n$. In the context of cycles, we use the shortened notation $\psi_{i},\eta_{i},dx_{i},r_{i}$ when referring to pairs of the form $(c_{i}\mapsto c_{i+1})$. A cycle with $n=3$ is called a triangular sequence. ###### Definition IX.9. The capacity of cycle $C=\\{(c_{1}\mapsto c_{2}),\ldots,(c_{n}\mapsto c_{n+1})\\}$ is the maximum quantity of $c_{1}$ that can be converted through its pairs. The capacity of a cycle is given by $\displaystyle Q=\min_{1\leq i\leq n}\left\\{\eta_{i}\left(\prod_{k=1}^{i-1}\psi_{k}\right)^{-1}\right\\}$ (1) The balance of $c_{i}$, denoted $q_{i}$, is given by the recurrence relation $\displaystyle q_{1}$ $\displaystyle=Q-r_{1}(Q)$ $\displaystyle q_{i+1}$ $\displaystyle=\psi_{i}\left[q_{i}-r_{i}(q_{i})\right],\qquad 1\leq i\leq n$ ###### Remark. $r_{i}\Big{(}x-r_{i}(x)\Big{)}=0$ ### IX-B Binance Fee Structure The fees paid on $(c_{i}\mapsto c_{i+1})$ are $fq_{i+1}b_{i+1}$ where $f=5\cdot 10^{-4}$ and $b_{i+1}$ is the last traded price of $(c_{i+1}\mapsto\text{BNB})$ $\displaystyle b_{i}=\begin{cases}\frac{1}{p_{\text{BNB}/c_{i}}}&\text{if $\text{BNB}/c_{i}\in\mathbb{P}$}\\\ p_{c_{i}/\text{BNB}}&\text{if $c_{i}/\text{BNB}\in\mathbb{P}$}\\\ \end{cases}$ The gain/loss in $c_{1}$ terms is given by $\displaystyle G=(q_{n+1}-q_{1})+\underbrace{\sum_{i=1}^{n}r_{i}(q_{i})\psi_{i1}}_{\text{residuals $c_{1}$ value}}-\underbrace{f\sum_{i=1}^{n}q_{i+1}b_{i+1}\psi_{B1}}_{\text{fees $c_{1}$ value}}$ (2) Note that $G$ is a function of $\\{\psi_{21},\ldots,\psi_{n1},\psi_{B1},b_{2},\ldots,b_{n+1}\\}$ as well as $\psi_{1},\ldots\psi_{n}$. ###### Definition IX.10. a cycle is called an arbitrage free cycle if $G\leq 0$ ###### Definition IX.11. a cycle is called an open cycle if $G>0$ ###### Remark. One can argue that there are additional costs for converting the residuals to $c_{1}$ and purchasing BNB tokens ahead of time to be able to pay fees, which are not accounted for in (2). However, this can be done in infrequent bulk trades. We make the assumption that small directional exposure to BNB and residuals is negligible compared to the accumulated gains over a period of time. ###### Remark 1. If we assume zero residuals, i.e., $\forall i:r_{i}=0,$ then $q_{i+1}=Q\prod_{j=1}^{i-1}\psi_{j}$ and (2) reduces to $G=Q\left(\prod_{i=1}^{n}\psi_{i}-1\right)-f\sum_{i=1}^{n}Q\left(\prod_{j=1}^{i}\psi_{j}\right)b_{i+1}\psi_{B1}$ ### IX-C Indirect Internal Conversions The set of pairs $V=\\{(c_{1}\mapsto c_{2}),(c_{2}\mapsto c_{3}),\ldots,(c_{n}\mapsto c_{n+1})\\}$ is called a conversion if $c_{j}\not\in\\{c_{1},\ldots\,c_{j-1}\\}$ for $1<j\leq n+1$. In the context of conversions, we use notation $\psi_{i},\eta_{i},dx_{i},r_{i}$ when referring to pairs of the form $(c_{i}\mapsto c_{i+1})$. When the intermediate pairs are clear, we use the shortened notation $c_{1}\stackrel{{\scriptstyle V}}{{\leadsto}}c_{n+1}$ ###### Definition IX.12. The capacity of conversion $V=\\{(c_{1}\mapsto c_{2}),\ldots,(c_{n}\mapsto c_{n+1})\\}$ is the maximum quantity of $c_{1}$ that can be converted through its pairs and is defined similarly to the capacity of a cycle. ###### Definition IX.13. (conversion proceeds) Let $c_{1}\stackrel{{\scriptstyle V}}{{\leadsto}}c_{n+1}$ be a conversion from $c_{1}$ to $c_{n+1}$ with capacity $Q$. The conversion proceeds of $q\leq Q$ units of $c_{1}$ is the quantity of $c_{n+1}$ that results from converting through the pairs of $V$, accounting for residuals and fees, i.e., $\displaystyle Pr(q,V)=q_{n+1}+\underbrace{\sum_{i=1}^{n}r_{i}(q_{i})\psi_{i(n+1)}}_{\text{residuals $c_{n+1}$ value}}-\underbrace{f\sum_{i=1}^{n}q_{i+1}b_{i+1}\psi_{B(n+1)}}_{\text{fees $c_{n+1}$ value}}$ ###### Definition IX.14. (profitable conversions) Let $x\stackrel{{\scriptstyle V_{1}}}{{\leadsto}}y$ and $x\stackrel{{\scriptstyle V_{2}}}{{\leadsto}}y$ be two conversions from $x$ to $y$, with capacities $Q_{1}$ and $Q_{2}$, respectively. Let $Q=\min\\{Q_{1},Q_{2}\\}$. We say that $V_{1}$ is a profitable conversion w.r.t. $V_{2}$ if the proceeds of conversion $V_{1}$ are greater than the proceeds of conversion $V_{2}$, i.e., $Pr(Q,V_{1})>Pr(Q,V_{2})$ ### IX-D Binance Order Types Binance supports the following order types: 1. 1. LIMIT \- an order to buy/sell a pair at a specified price and can only be executed at that price (or better). Not guaranteed to execute 2. 2. MARKET \- an order to buy/sell a pair at the best current market price, i.e., lowest ask or highest bid. 3. 3. STOP_LOSS_LIMIT \- an order to buy (sell) a pair, once its price exceeds (drops below) the specified price. In contrast to a LIMIT order, the price should be above (below) the lowest ask (highest bid). The execution price is guaranteed to be the specified price. 4. 4. STOP_LOSS \- same as STOP_LOSS_LIMIT, but when the price threshold is breached, a MARKET order is executed and the execution price is not guaranteed. 5. 5. TAKE_PROFIT_LIMIT \- equivalent to a LIMIT order 6. 6. TAKE_PROFIT \- automatically places a MARKET order when the specified price level is met 7. 7. LIMIT_MAKER \- LIMIT orders that will be rejected if they would immediately match and trade as a taker. 8. 8. ICEBERG \- an order used for large quantities as it automatically breaks down to multiple LIMIT orders with different prices. The goal is to hide the actual order quantity and prevent trend-followers from unfavorably moving the price.

2024-09-04T02:54:55.239469

2002.12276

arxiv-papers

# Stochastic dynamics of dissolving active particles Alexander Chamolly Eric Lauga<EMAIL_ADDRESS>Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom ###### Abstract The design of artificial microswimmers has generated significant research interest in recent years, for promise in applications such as nanomotors and targeted drug-delivery. However, many current designs suffer from a common problem, namely the swimmers remain in the fluid indefinitely, posing risks of clogging and damage. Inspired by recently proposed experimental designs, we investigate mathematically the dynamics of degradable active particles. We develop and compare two distinct chemical models for the decay of a swimmer, taking into account the material composition and nature of the chemical or enzymatic reaction at its surface. These include a model for dissolution without a reaction, as well as models for a reacting swimmer studied in the limit of large and small Damköhler number. A new dimensionless parameter emerges that allows the classification of colloids into ballistic and diffusive type. Using this parameter, we perform an asymptotic analysis to derive expressions for colloid lifetimes and their total mean-squared displacement from release and validate these by numerical Monte Carlo simulations of the associated Langevin dynamics. Supported by general scaling relationships, our theoretical results provide new insight into the experimental applicability of a wide range of designs for degradable active colloids. ## I Introduction In recent years, scientists from a wide variety of different fields have given considerable attention to the subject of synthetic microswimmers. This focus in research is no coincidence, as such colloids show great promise in biomedical and engineering applications Wang and Gao (2012); Wang _et al._ (2013); Nelson _et al._ (2010). The design of autonomous swimmers in particular has received significant theoretical and experimental attention Elgeti _et al._ (2015); Moran and Posner (2017). In an effort to exploit the peculiarities of the associated low-Reynolds number hydrodynamics Purcell (1977), many different propulsion mechanisms have been invented. These include self-phoretic propulsion, such as chemophoresis Michelin and Lauga (2014); Golestanian _et al._ (2007); Brady (2011); Walther and Mueller (2013) and electrophoresis Ebbens _et al._ (2014); Paxton _et al._ (2006); Moran and Posner (2011), as well as ultrasound propulsion Gallino _et al._ (2018); Mou _et al._ (2015); Wang _et al._ (2012), bubble propulsion Gibbs and Zhao (2009); Wang and Wu (2014) and magnetic propulsion Zhang _et al._ (2009); Ghosh and Fischer (2009). Despite this remarkable progress, common experimental designs still need to be improved in order to be suitable for sensitive applications, such as non- invasive medicine. Next to potential toxicity of swimmer components or their fuel Gao _et al._ (2015), the question of waste disposal remains largely open. This can be a serious problem, since artificial micron sized particles in the blood stream have the potential to cause clogging Bächer _et al._ (2017); Sauret _et al._ (2018); Fogelson and Neeves (2015) and may thus pose a significant health risk Nesbitt _et al._ (2009); Fogelson and Neeves (2015). It is therefore essential to develop designs for microswimmers that degrade after fulfilling their purpose. Very recently, novel experimental designs have begun to address these issues. Examples of such colloids include non-toxic magnesium-based bubble propelled swimmers Chen _et al._ (2018) suitable for aqueous environments, as well as other kinds of inorganic compositions driven by reactions in either acidic or alkaline environments Chen _et al._ (2016). More designs have been proposed using organic compounds that may be 3D-printed Wang _et al._ (2018) or that self-assemble into nanomotors Tu _et al._ (2017). These experimental advances raise new theoretical questions. While the dynamics of classical non-dissolving colloids have been studied extensively, the time-evolution of colloid size modifies its stochastic behaviour, and new quantities characterising its physics emerge. The purpose of this paper is therefore to provide theoretical answers to two fundamental questions. First, we examine which material and environmental parameters determine the lifetime of a dissolving spherical microswimmer. Second, we study the influence of dissolution on the stochastic behaviour of both passive and self-propelled colloids. Here, a new dimensionless quantity arises which splits microswimmers into two categories: those that are subject to sufficient amounts of thermal noise during their life time to evolve diffusively, and those that exhibit near-ballistic trajectories that may be exploited for delivery applications. We show that both scenarios may enter for realistic values of the material and environmental parameters. Knowledge of these and their scaling relations is thus essential for the application-specific engineering of degradable microswimmer designs. The structure of this paper is as follows. We begin by presenting two theoretical models for the dissolution process in §II, one suitable for designs in which the dissolution process is not driven by a reaction with a fuel in the solvent (such as dissolution by hydrogen bonding), and one for swimmers whose matrix is decomposed by means of a reaction (chemical or enzymatic). For further analysis the latter case is considered in the two limits of slow and fast reaction, the former corresponding to a fixed material flux boundary condition. In all these models we find expressions for the time dependence of the swimmer size, as well as their total lifetime in terms of the essential physical parameters. We present the necessary modification to classical Brownian motion in §III, and derive expressions for the passive mean squared displacement of not self-propelling colloids. Based on this, we next derive corresponding expressions for active motion in §IV and validate our results numerically. Finally we discuss the implications of our research on future studies in §V. ## II Dissolution models Inspired by recent experimental realisations, we propose two models for the dissolution of a spherical colloid based on different possibilities for the boundary conditions at its surface. Specifically, we distinguish between the case in which dissolution occurs through binding colloid material to fluid molecules (for example, the case of ionic dissolution in water), which we call non-reacting, and the case of dissolution through a chemical or enzymatic reaction that consumes a fuel. In the latter scenario we distinguish further between the limits of slow and fast reaction, and discuss their physical implications. As a preamble, we note that, unlike geophysical melting processes Woods (1992), enthalpy plays no role in the dissolution processes considered in our paper. This means the Stefan boundary condition does not apply and the dynamics we derived is different from e.g. the dissolution of ice crystals in water. While the general dynamics of diffusive dissolution have been considered in the geophysical literature Zhang _et al._ (1989), there has to the best of our knowledge been no study that derived the asymptotic solutions we compute below. This is likely due to the dominance of convection driven processes on relevant geophysical scales that require different modelling Kerr (1995). ### II.1 Non-reacting swimmer Figure 1: Schematic presentation of non-reacting dissolution dynamics. The matrix of the swimmer consists of a substance that dissolves by bonding to the fluid (thus acting as a solvent). Near the boundary, solute is present at a saturation concentration $c_{0}$ and subject to advective-diffusive transport in the bulk. Dissolution emerges through maintaining a normal concentration gradient at the swimmer surface. In our first model, we assume that the colloidal particle is composed of a material that dissolves in the surrounding fluid through bonding of solute colloid material to fluid molecules, as illustrated schematically in Fig. 1. We consider this an appropriate model for non-reacting dissolution processes, such as dissolution of many organic compounds as well as ionic salts in water. In order to keep the mathematics simple we make the simplifying assumption that only one species of solute is dissolved into the bulk. This allows us to define the (mass) concentration, $c(\bm{r},t)$, of solute defined as the mass of solute dissolved in a unit volume of solvent, with $c=c_{\infty}\geq 0$ far away from the colloid. Note that this differs from the definition of molar concentration common in chemistry by a factor equal to the molar mass of the solute. We make this choice in order to avoid clutter that would arise from the application of mass conservation below. In this model and the following we assume the absence of any background flow that would disturb the distribution of solute or reactant in the bulk fluid. This assumption is of course violated for self-propelled particles moving relative to a background fluid. However, we can use a scaling argument to show that this does not affect our leading-order results. Since typical propulsion velocities $U$ are expected to be on the order of a few microns per second, initial colloid radii $R_{0}$ on the scale of microns Elgeti _et al._ (2015) and for many ions in water at room temperature the solute diffusivity is approximately $D_{s}\sim 10^{-9}$ m2/s Haynes (2014), the Péclet number quantifying the relative important of advection to diffusion for the solute is $\text{Pe}_{\text{sol}}=R_{0}U/D_{s}\sim 10^{-4}-10^{-3}$. This indicates that advection of solute can be safely neglected. This remains true even when the Péclet number associated with motion of the colloid, $\text{Pe}_{\text{col}}=R_{0}U/D$ is large, since the particle is several orders of magnitude larger than a solvent molecule and therefore has a much smaller diffusivity. The same result applies to phoretic slip flows, which are typically of the same strength as the propulsion velocity. In the context of dissolution dynamics, the flows arising from propulsion can therefore be neglected in the transport processes of solute and reactant. We further assume that the swimmer has a homogeneous mass density, $\rho_{s}$, and the fluid solvent a constant density, $\rho_{l}$. In general the density of the solvent depends weakly on the amount of solute dissolved Haynes (2014). However, we will soon develop an asymptotic analysis based on the assumption that the solubility is weak and therefore can neglect this effect. Finally, we assume also that the swimmer remains spherical at all times, and that that the dissolution dynamics is independent of any self-propulsion mechanism or background flow. Both these assumptions will be justified _a posteriori_ in section §II.1.3. A brief discussion of the case of a partially dissolving swimmer is included our discussion §V. #### II.1.1 Mathematical model We consider a spherically symmetric colloid or radius $R(t)$ with initial condition $R(0)=R_{0}>0$. Near the boundary, there is chemical equilibrium between solute attached to the swimmer surface and present in the fluid. In this case the dissolution process is driven by removal (through diffusion) of solute from a boundary layer into the bulk and subsequent replenishment from the swimmer surface (Fig. 1). We model this effect by imposing the boundary condition $c(R(t),t)=c_{0}>c_{\infty},\quad t\geq 0,$ (1) where $c_{0}$ is the saturation concentration of solute in the solvent. This condition assumes that the boundary layer is negligibly thin and that the surface reaches chemical equilibrium instantaneously, which may be justified by noting that time scales of interest will be much larger than the molecular collision time, $\tau_{MC}\approx 10^{-13}\text{ s}$ Haynes (2014). The other condition we impose is the requirement that the solute is initially distributed homogeneously in the bulk, i.e. $c(r,0)=c_{\infty},\quad r>R_{0}.$ (2) Conservation of solute at the boundary gives $\displaystyle 4\pi R^{2}\rho_{s}\frac{dR}{dt}$ $\displaystyle=-\text{(solute flux into the fluid)}$ $\displaystyle=-\left(-D_{s}4\pi R^{2}\frac{\partial c}{\partial r}\bigg{\rvert}_{r=R}\right),$ (3) and therefore $\frac{dR}{dt}=\frac{D_{s}}{\rho_{s}}\frac{\partial c}{\partial r}\bigg{\rvert}_{r=R},$ (4) where $D_{s}$ is the diffusivity of solute in the solvent. Furthermore, in the case of unequal densities we also get a non-zero fluid flux at the boundary since by mass conservation there is equality $-\dot{R}\rho_{s}=(-\dot{R}+{\bm{u}}\cdot{\hat{\bm{r}}})\rho_{l}$ (5) and thus ${\bm{u}}\cdot{\hat{\bm{r}}}=\dot{R}\frac{\rho_{l}-\rho_{s}}{\rho_{l}},$ (6) where $\hat{\bm{r}}$ denotes a unit vector in the outward radial direction. For a self-propelled microscopic colloid in water the Reynolds number, defined as the ratio of colloid radius times velocity divided by kinematic viscosity, is typically on the order of $10^{-9}\ll 1$. Therefore the fluid dynamics obey the incompressible Stokes equations, $\mu\nabla^{2}\bm{u}=\nabla p,\quad\nabla\cdot\bm{u}=0,$ (7) where $\mu$ is dynamic viscosity and $p$ is the pressure field. Solving these with the boundary condition given in Eq. (6) at $r=R$ leads to the flow of a point source $\bm{u}=\dot{R}\frac{\rho_{l}-\rho_{s}}{\rho_{l}}\frac{R^{2}}{r^{2}}\hat{{\bm{r}}}.$ (8) The transport equation for $c(r,t)$ is the standard advection-diffusion equation $\displaystyle\frac{\partial c}{\partial t}+\nabla\cdot(c\bm{u})=D_{s}\nabla^{2}c.$ (9) Using the result of Eq. (8) together with incompressibility and assuming radial symmetry of the solute concentration, this becomes $\displaystyle\frac{\partial c}{\partial t}+\frac{\rho_{l}-\rho_{s}}{\rho_{l}}\frac{D_{s}}{\rho_{s}}\frac{R^{2}}{r^{2}}\frac{\partial c}{\partial r}\bigg{\rvert}_{r=R}\frac{\partial c}{\partial r}=D_{s}\left(\frac{\partial^{2}c}{\partial r^{2}}+\frac{2}{r}\frac{\partial c}{\partial r}\right),$ (10) Next we non-dimensionalise this transport equation using the scalings $c^{*}=\frac{c-c_{\infty}}{c_{0}-c_{\infty}},\,R^{*}=\frac{R}{R_{0}},\,r^{*}=\frac{r}{R_{0}},\,t^{*}=\frac{D_{s}t}{R_{0}^{2}}.$ (11) Substituting in Eq. (10) and dropping stars in what follows for notational convenience, we obtain the colloid dynamics as solution to $\frac{dR}{dt}=\alpha_{1}\frac{\partial c}{\partial r}\bigg{\rvert}_{r=R}$ (12) with $c$ solution to $\frac{\partial c}{\partial t}+\frac{R^{2}}{r^{2}}(\alpha_{1}-\beta_{1})\frac{\partial c}{\partial r}\bigg{\rvert}_{r=R}\frac{\partial c}{\partial r}=\left(\frac{\partial^{2}c}{\partial r^{2}}+\frac{2}{r}\frac{\partial c}{\partial r}\right),$ (13) with dimensionless boundary conditions $c(R(t),t)=1,\,\,t\geq 0,\quad{\rm and}\quad c(r,0)=0,\,\,r>1,$ (14) where we have defined the two dimensionless parameters $\alpha_{1}=\frac{c_{0}-c_{\infty}}{\rho_{s}},\quad\beta_{1}=\frac{c_{0}-c_{\infty}}{\rho_{l}}\cdot$ (15) We note that despite a negligibly small solute Péclet number, it was necessary to include an advective term due to volume conservation, whose relative strength is given by $(\alpha_{1}-\beta_{1})$. It is therefore independent of the Péclet number and its irrelevance at leading order will be only a consequence of the weak solubility assumption. Only when there is no density mismatch between colloid and fluid is this term identically zero. Furthermore, the swimmer radius remains constant when the solvent is saturated with solute, as may be expected intuitively. #### II.1.2 Asymptotic solution In order to make analytical progress, we make the assumptions that $\alpha_{1},\beta_{1}\ll 1,$ (16) which corresponds to a low-solubility limit for the colloid material. We can then develop an asymptotic expansion to solve for $c$ and $R$. Here we will only calculate the leading-order solution, but our setup allows for calculations to arbitrarily high orders. We proceed by a rescaling of our spatial coordinate as $x=\frac{r}{R},\quad y(x,t)=xc(x,t),$ (17) so that our system becomes $\displaystyle R^{2}\frac{\partial y}{\partial t}$ $\displaystyle+R\dot{R}y+(\alpha_{1}-\beta_{1})\left(\frac{1}{x^{2}}\frac{\partial y}{\partial x}-\frac{y}{x^{3}}\right)\left(\frac{\partial y}{\partial x}\bigg{\rvert}_{x=1}-1\right)$ $\displaystyle=\frac{\partial^{2}y}{\partial x^{2}}$ (18) and $R^{2}=1+2\alpha_{1}\left(\int_{0}^{t}\frac{\partial y}{\partial x}\bigg{\rvert}_{x=1}dt^{\prime}-t\right)$ (19) with boundary conditions $y(1,t)=1,\quad y(x,0)=0.$ (20) The solution may be written as $y(x,t;\alpha_{1},\beta_{1})=y_{0}(x,t)+\alpha_{1}y_{\alpha}(x,t)+\beta_{1}y_{\beta}(x,t)+o(\alpha_{1},\beta_{1}).$ (21) The problem for $y_{0}$ reduces to the one-dimensional heat equation with Dirichlet boundary conditions and its solution is well known to be $y_{0}(x,t)=\text{erfc}\left(\frac{x-1}{2\sqrt{t}}\right),$ (22) whence to leading order $R^{2}=1-2\alpha_{1}\left(t+2\sqrt{\frac{t}{\pi}}\right),$ (23) or, after reinserting dimensions, we obtain our desired result $R(t)=R_{0}\sqrt{1-2\alpha_{1}\left(\frac{t}{t_{s}}+\frac{2}{\sqrt{\pi}}\sqrt{\frac{t}{t_{s}}}\right)}.$ (24) where $t_{s}=R_{0}^{2}/D_{s}$ is the diffusive time scale for the solute. An illustration of this decay, along with a comparison to the reacting model is presented in Fig. 3. Denoting by $T_{d}$ the finite time at which the particle disappears, and taking into account the order of terms we neglect, we can deduce that $\begin{split}T_{d}&=\frac{t_{s}}{2\alpha_{1}}\left(1-\sqrt{\frac{8}{\pi}}\sqrt{\alpha_{1}}+\mathcal{O}(\alpha_{1},\beta_{1})\right).\end{split}$ (25) Therefore at leading order, the lifetime of the colloid scales inversely proportional with the solubility and diffusivity of its material, but quadratically with the initial colloid radius $R_{0}$. However, the correction from the next-to-leading order term remains significant for $\alpha_{1}\gtrsim 10^{-3}$ due to its slow square-root like decay. #### II.1.3 Physical interpretation The aim of this section is to provide some physical interpretation for Eq. (24). For many ions in water at room temperature, the diffusivity is approximately $D_{s}\sim 10^{-9}$ m2/s Haynes (2014). In the case of an initially micron-sized colloid this gives $t_{s}\sim 10^{-3}\,{\rm s}.$ (26) The other (previously unknown) time scale in the problem is the swimmer lifetime $T_{d}$. There is a separation of scales that is to leading order inversely proportional to $\alpha_{1}$. In the specific example of calcium carbonate with $\alpha_{1}\approx 10^{-6}$ Haynes (2014), we obtain $T_{d}\sim 10^{3}\,{\rm s}\sim 10\text{ min},$ (27) which is a conceivably desirable lifetime for a microswimmer. The separation of scales has further consequences for the decay rate. For $t\ll t_{s}$ we have $R^{2}\sim 1-4\sqrt{t\alpha_{1}^{2}/t_{s}\pi}$, while for $t\gg t_{s}$ we obtain the behaviour $R^{2}\sim 1-2\alpha_{1}t/t_{s}$. Therefore the particle size satisfies $R\sim\sqrt{1-2\alpha_{1}t/t_{s}}$ except for a short, transient period on the order of $t_{s}$. This feature may be explained physically. Initially, the discontinuity in concentration at $r=R$ causes a large concentration gradient and fast dissolution but on the (fast) scale of solute diffusion the system relaxes to equilibrium in a boundary layer of thickness $\sim\sqrt{D_{s}t_{s}}$, which is on the order of the colloid size, $R_{0}$. From this point onwards the colloid is surrounded by a cloud of solute in equilibrium and the process becomes quasi-static. At leading order, the dissolution dynamics therefore reduces to steady diffusion. This gives simultaneously justification to our assumption of sphericity, since the diffusive boundary layer smooths out any surface inhomogeneities. As an aside, we note while the dissolution process of microbubbles is driven by capillary pressures Michelin _et al._ (2018), the $R\sim\sqrt{1-t}$ behaviour also emerges in the absence of surface tension, essentially also due to the dominance of diffusive effects. Finally, we point out that $\alpha_{1}$ and $\beta_{1}$ depend only on the material chosen for the swimmer (and its abundance in the bulk fluid). Unsurprisingly, only materials that are considered insoluble on the macroscale yield appreciable microswimmer life times. Hence, together with fine tuning of the initial radius $R_{0}$, full control of the dissolution dynamics can be achieved through the microswimmer design. ### II.2 Dissolution through reaction Figure 2: Schematic presentation of the molecular dynamics near the boundary of a reacting colloid. In this example, motivated by experiments in Ref. Chen _et al._ (2016), zinc is dissolved in acid forming zinc-ions and molecular hydrogen. If $\text{Da}=0$, i.e. infinitely much H+ is present to sustain the reaction, the dissolution rate is constant. If $\text{Da}>0$, the reaction rate will depend on the amount of fuel present, but not on the amount of product. Artificial microswimmers are rarely composed solely of chemically inert materials. Indeed, autophoretic swimmers often consume a fuel in the solvent, like in the widely studied case of catalytic platinum swimmers splitting hydrogen peroxide into water and oxygen Moran and Posner (2017). A sketch of the process is illustrated in Fig. 2 in the specific case of zinc dissolving in acid as realised experimentally by Chen et. al. Chen _et al._ (2016). An analogous picture may be imagined for the case of biodegradation by enzymes. A degradable autophoretic colloid might therefore consist of a reactant that will then dissolve into the fluid. To this end, let us consider a fixed reaction-rate boundary condition. It will be important to distinguish between the concentration of fuel $c_{f}(\bm{r},t)$ and concentration of swimmer substrate $c_{s}(\bm{r},t)$. For example, in the case of zinc, the fuel concentration might be provided by hydrogen ions in acid, which relates their concentration directly to the pH-value of the solvent, while the concentration of substrate influences the dissolution rate through mass conservation. Notation-wise, we will use the subscript $f$ to refer below to the fuel and the subscript $s$ to the substrate. #### II.2.1 Mathematical model The mathematical development is similar to the non-reacting swimmer, with an important change to the boundary conditions. Indeed, unlike Eq. (4) where the concentration at the boundary was fixed, the boundary conditions for the fields $c_{s}$ and $c_{f}$ are now given by $-D_{s}{\bm{n}}\cdot\nabla c_{s}|_{R}=k_{s}c_{f},\quad- D_{f}{\bm{n}}\cdot\nabla c_{f}|_{R}=-k_{f}c_{f},$ (28) where $k_{s}$ and $k_{f}$ are the constant reaction rates for solute and fuel respectively and $D_{f}$ the diffusivity of fuel in the solution. Mass conservation for the colloidal particle leads to $\frac{dR}{dt}=-\frac{k_{s}c_{f}(R)}{\rho_{s}}.$ (29) Furthermore, we once again have conservation of fluid volume giving rise to a source flow $\bm{u}=\dot{R}(1-\rho_{s}/\rho_{l})\frac{R^{2}}{r^{2}}{\hat{\bm{r}}}.$ (30) Similar to what was done above, we assume that the Péclet numbers associated with the solute and the fuel dynamics are small, so that only volume conservation gives rise to advective flows. We can then write the advection- diffusion equation for $c_{f}$ as $\frac{\partial c_{f}}{\partial t}-(1-\rho_{s}/\rho_{l})\frac{k_{s}c_{f}(R)}{\rho_{s}}\frac{R^{2}}{r^{2}}\frac{\partial c_{f}}{\partial r}=D_{f}\left(\frac{\partial^{2}c_{f}}{\partial r^{2}}+\frac{2}{r}\frac{\partial c_{f}}{\partial r}\right).$ (31) Introducing non-dimensionalised variables as $c_{f}^{*}=\frac{c_{f}}{c_{f,\infty}},R^{*}=\frac{R}{R_{0}},r^{*}=\frac{r}{R_{0}},t^{*}=\frac{D_{f}t}{R_{0}^{2}},$ (32) where $c_{f,\infty}$ is the mass concentration of fuel in the bulk, we may substitute in Eqs. (29) and (31) and dropping stars immediately we find $\frac{\partial c_{f}}{\partial t}-\text{Da}(\alpha_{2}-\beta_{2})c_{f}(R)\frac{R^{2}}{r^{2}}\frac{\partial c}{\partial r}=\frac{\partial^{2}c}{\partial r^{2}}+\frac{2}{r}\frac{\partial c}{\partial r},$ (33) $\frac{dR}{dt}=-\text{Da}\alpha_{2}c_{f}(R),$ (34) with the boundary conditions $\displaystyle c_{f}\to 1,\quad$ $\displaystyle r\to\infty,$ (35) $\displaystyle\frac{\partial c_{f}}{\partial r}=\text{Da}c_{f},\quad$ $\displaystyle r=1,$ $\displaystyle c_{f}(r,0)=1,\quad$ $\displaystyle r>1,$ $\displaystyle R(0)=1,$ where we have defined the three dimensionless numbers $\text{Da}=\frac{R_{0}k_{f}}{D_{f}},\quad\alpha_{2}=\frac{c_{f,\infty}k_{s}}{\rho_{s}k_{f}},\quad\beta_{2}=\alpha_{2}\frac{\rho_{s}}{\rho_{l}}\cdot$ (36) Here Da is a Damköhler number for the fuel, indicating the ratio between reactive and diffusive fluxes, while $\alpha_{2}$ and $\beta_{2}$ may be interpreted as dimensionless ratios comparing the mass of fuel consumed against the mass of solute shed in the reaction. Upon rescaling our coordinates according to $x=\frac{r}{R},\quad y(x,t)=c_{f}x,$ (37) our system becomes $R^{2}\frac{\partial y}{\partial t}+R\dot{R}y-\text{Da}(\alpha_{2}-\beta_{2})R\left(\frac{1}{x^{2}}\frac{\partial y}{\partial x}-\frac{y}{x^{3}}\right)y(1,t)=\frac{\partial^{2}y}{\partial x^{2}},$ (38) and $R=1-\text{Da}\alpha_{2}\int_{0}^{t}y(1,t^{\prime})dt^{\prime},$ (39) with $y(x,0)=1,\quad\frac{\partial y}{\partial x}(1,t)=\text{Da}y(1,t).$ (40) From here, we can again proceed by means of an asymptotic expansion. #### II.2.2 Asymptotic expansion We next assume $\alpha_{2}\text{Da},\,\beta_{2}\text{Da}\ll 1$ and write the solution as a power expansion $\displaystyle y(x,t;\alpha_{2},\beta_{2};\text{Da})=$ $\displaystyle y_{0}(x,t;\text{Da})+\alpha_{2}y_{\alpha}(x,t;\text{Da})+\beta_{2}y_{\beta}(x,t;\text{Da})+{\rm h.o.t.}$ (41) The boundary condition in Eq. (40) consitutes a Robin problem and can be solved by considering the quantity $\phi=y-\text{Da}^{-1}\partial y/\partial x$ subject to Cauchy conditions Carslaw and Jaeger (1959). The solution for $y_{0}$ is $\displaystyle y_{0}(x,t;\text{Da})=$ $\displaystyle\text{erf}\left(\frac{x-1}{2\sqrt{t}}\right)$ $\displaystyle+e^{\text{Da}(x-1)+\text{Da}^{2}t}\text{erfc}\left(\frac{x-1}{2\sqrt{t}}+\text{Da}\sqrt{t}\right).$ (42) It follows that $y_{0}(1,t;\rm{Da})=e^{\text{Da}^{2}t}\text{erfc}\left(\text{Da}\sqrt{t}\right),$ (43) and hence to leading order in $\alpha_{2}$, $R(t)=1-2\alpha_{2}\sqrt{\frac{t}{\pi}}-\frac{\alpha_{2}}{\text{Da}}\left[e^{\text{Da}^{2}t}\text{erfc}\left(\text{Da}\sqrt{t}\right)-1\right].$ (44) Upon reinserting dimensions we finally arrive at $\displaystyle R(t)=R_{0}\Bigg{\\{}$ $\displaystyle 1-\alpha_{2}\frac{2}{\sqrt{\pi}}\sqrt{\frac{t}{t_{f}}}$ $\displaystyle\left.-\frac{\alpha_{2}}{\text{Da}}\left[e^{\text{Da}^{2}t/t_{f}}\text{erfc}\left(\text{Da}\sqrt{\frac{t}{t_{f}}}\right)-1\right]\right\\}.$ (45) where $t_{f}=R_{0}^{2}/D_{f}$ is the diffusive time scale for the fuel. #### II.2.3 Slow reaction limit (fixed solute flux) Inspired by a study of boundary conditions in the context of finite Péclet- number propulsion in Ref. Michelin and Lauga (2014), we may consider separately the limits $\text{Da}\to 0$ and $\text{Da}\to\infty$. Each of these limits will lead to a different model that we will consider in the remainder of this paper. For small Damköhler number, we find $\displaystyle R(t)=R_{0}\left[1-\frac{\alpha_{2}\text{Da}}{t_{f}}t+\mathcal{O}\left(\text{Da}^{2}{\left(\frac{t}{t_{f}}\right)}^{3/2}\right)\right],$ $\displaystyle\text{as Da}\to 0,\quad\frac{t}{t_{f}}\lesssim\text{Da}^{-2}.$ When $\text{Da}=0$, no reaction takes place and the radius of the colloid remains constant. At next to leading order we have linear decay, so the lifetime $T_{d}$ is $T_{d}=\frac{t_{f}}{\alpha_{2}}\text{Da}^{-1}=\frac{R_{0}\rho_{s}}{c_{f,\infty}k_{s}}\quad(\text{Da}\to 0),$ (46) which is consistent with the asymptotic expansion to this order. Thus we arrive at a model for the dissolution with a constant solute flux. We note the different scaling compared to the non-reacting model where the lifetime scaled as $T_{d}\sim R_{0}^{2}$. This is indicative of the absence of diffusion in this limit. Note that the model can be recovered from simply applying mass conservation to a flux boundary condition of the form $-D_{s}\frac{\partial c}{\partial r}\bigg{|}_{r=R(t)}=c_{f,\infty}k_{s},$ (47) which shows that the flux is equal to $c_{f,\infty}k_{s}$. #### II.2.4 Fast reaction limit Conversely, as $\text{Da}\to\infty$ (still with $\alpha_{2}\text{Da}\ll 1$), we find that $\displaystyle R(t)=R_{0}\left\\{1-\frac{2\alpha_{2}}{\sqrt{\pi}}\sqrt{\frac{t}{t_{f}}}+\mathcal{O}\left(\text{Da}^{-1}\right)\right\\},$ $\displaystyle\text{as Da}\to\infty,\quad\frac{t}{t_{f}}\gtrsim\text{Da}^{-2}.$ In this limit the reaction is infinitely fast, so the boundary condition on the fuel effectively reduces to instantaneous depletion, $c_{f}(R,t)=0$, and the dissolution rate is limited by the diffusive flux of fuel from the bulk. Correspondingly the lifetime $T_{d}$ in dimensional units is $T_{d}=\frac{\pi}{4\alpha_{2}^{2}}\frac{R_{0}^{2}}{D_{f}}\quad(\text{Da}\to\infty,\alpha_{2}\text{Da}\ll 1),$ (48) a result which is again consistent with the expansion. Apart from the introduction of reaction rates, this result is qualitatively different from the non-reacting swimmer insofar as the lifetime depends on the square of swimmer density and reactant concentration at infinity, rather than being inversely proportional to solubility. We remark that in the case of hydrogen ions, the concentration $c_{f}$ is directly related to the pH value of the solvent, which establishes an experimentally accessible relationship between the pH and swimmer dissolution dynamics. Figure 3: Comparison of the decay dynamics between the three models: decay of the dimensionless colloid radius as a function of dimensionless time. (i) Non- reacting (red solid line; $t_{d}/T_{d}=0.01$); (ii) Slow reaction (green dashed line); (iii) Fast reaction (blue dash-dotted line). In Fig. 3 we illustrate the different decay behaviour for our three models: (i) Non-reacting (red solid line, with $t_{d}/T_{d}=0.01$); (ii) Slow reaction (green dashed line); and (iii) Fast reaction (blue dash-dotted line). We note for the non-reacting model the decay rate increases with time, whereas it is constant for the slowly reacting, and decreasing for the fast reacting model. In the following two sections, we will explore the important consequences this has for the stochastic behaviour of dissolving microswimmers. ## III Passive dynamics of dissolving colloids After developing three models for the dissolution of a spherical colloid, we now ask what effect this reduction in size has on its fluctuating trajectory. As will be shown, the mean squared displacement of a stochastic self-propelled particle is given by the sum of the contributions from translational noise and active motion. This allows us to split the analysis into the case of a passive colloid with no intrinsic propulsion mechanism but with translational noise and an active colloid with rotational but no translational diffusion. We treat the former case in this section and consider the motion of self-propelled particles in §IV. ### III.1 Mathematical model The change in the dynamics of colloidal particles arises through the time dependence of the translational diffusion coefficient, which is given by the Stokes-Einstein relation Einstein (1905) $D(t)=\frac{k_{B}T}{6\pi\mu R(t)}\equiv D_{0}\frac{R_{0}}{R(t)},$ (49) where $k_{B}$ is Boltzmann’s constant, $T$ is absolute temperature and $D_{0}\equiv D(0)=k_{B}T/6\pi\mu R_{0}$. In analogy with classical Brownian motion, we consider the following overdamped Langevin equation for the position of the passive colloidal particle, $\bm{r}(t)$, $d\bm{r}=\sqrt{2D(t)}d\bm{W}.$ (50) Classically, $\bm{W}(t)$ is white noise with the properties that $\langle d\bm{W}\rangle={\bf{0}},\quad\langle dW_{i}(t)dW_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})dt,$ (51) with brackets denoting ensemble averages. The right-hand side of Eq. (50) therefore varies on two different time scales: the rate of change of $D$ and the time scale of the molecular chaos $\tau_{MC}$ that gives rise to noise. Typically, $\tau_{MC}=\mathcal{O}(10^{-13}s)$ Haynes (2014). The mathematical assumption of $\delta$-correlated noise only holds true if $\tau_{MC}$ is very small compared to the time scale of diffusion, which holds true for microscopic colloids. However, since the rate of change of $D$ diverges as the swimmer size tends to 0, this model is expected break down at the very end of the swimmer lifetime. In the case of the non-reacting model this singularity is integrable and poses no problem, whereas for the reacting model we will also include a physical discussion of the breakdown. For an active self-propelled particle at velocity ${\bf U}(t)$, the right-hand side of the Langevin equation Eq. (50) includes an additional term ${\bf U}(t)dt$, which is deterministic in the sense that it is uncorrelated with translational white noise (even if ${\bf U}(t)$ is subject to rotational noise). A straightforward integration using the properties in Eq. (51) then shows that the total mean squared displacement is given by the sum of active and passive contributions, $\langle r^{2}\rangle_{tot}=\langle r^{2}\rangle_{a}+\langle r^{2}\rangle_{p},$ (52) as claimed. The stochastic dynamics in Eq. (50) gives rise to a Fokker-Planck equation for the probability for the position of the particle, $P(\bm{r},t)$, as $\frac{\partial P}{\partial t}=D(t)\nabla^{2}P.$ (53) We can solve this by a rescaling of time, introducing $\tau(t)$ such that $\tau=\int_{0}^{t}D(s)ds=D_{0}\int_{0}^{t}\frac{R_{0}}{R(s)}ds,$ (54) which yields $\frac{\partial\tilde{P}}{\partial\tau}=\nabla^{2}\tilde{P}.$ (55) where $\tilde{P}(\bm{r},\tau)=P(\bm{r},t)$. In three spatial dimensions this equation has a well known Gaussian solution corresponding to the initial condition of a particle located at the origin, $\tilde{P}(\bm{r},\tau)=\tilde{P}(r=|\bm{r}|,\tau)=\frac{1}{(4\pi\tau)^{3/2}}\exp\left(-\frac{r^{2}}{4\tau}\right).$ (56) The first two moments are well known to be $\langle\bm{r}\rangle={\bf 0}$ and $\langle{r}^{2}\rangle=6\tau$. The total passive mean squared displacement of the particle in its lifetime, $\langle r^{2}\rangle_{p}\equiv\langle r^{2}\rangle(T_{d})$, is therefore given by the integral $\langle r^{2}\rangle_{p}=6D_{0}\int_{0}^{T_{d}}\frac{R_{0}}{R(t)}dt.$ (57) Note that since $R\leq R_{0}$, the integral has value larger than $T_{d}$. Therefore dissolution always enhances passive diffusion. All that remains to be done is to calculate the integral for each of our three models. ### III.2 Total root mean squared displacement In the following we consider the solutions to Eq. (57). Bearing in mind the order of terms we neglected in the derivation of Eq. (24), we can integrate Eq. (57) directly to obtain the following result for the non-reacting model $\langle{r}^{2}\rangle_{p}=6D_{0}\times\frac{t_{s}}{\alpha_{1}}\left(1-\sqrt{\frac{\pi}{2}}\sqrt{\alpha_{1}}+\mathcal{O}(\alpha_{1},\beta_{1})\right).$ (58) Comparing with Eq. (25) we can see that at leading order in $\alpha_{1}$, dissolution enhances the total mean squared displacement by a factor of two. Through the scaling of $t_{s}$ with $R_{0}$ we also find that $\langle{r}^{2}\rangle_{p}\sim R_{0}$. This may be tested easily in experiments without affecting the other parameters. Perhaps surprisingly, this also means that in contrast to fixed-size swimmers, the importance of passive Brownian effects increases with swimmer size, since the smaller diffusivity is overcompensated for by the longer life span. The scaling with $\alpha_{1}$ can be explained the same way, as a colloid with small $\alpha_{1}$ decays slower, lives longer and therefore travels further. For the slow reaction model we can use Eq. (II.2.3) in the integration of Eq. (57) to find $\langle r^{2}\rangle(t)=6D_{0}\times T_{d}\log\left(\frac{R_{0}}{R(t)}\right).$ (59) This expression diverges logarithmically as $t\to T_{d}$. This should not be taken as indicative of superdiffusion, but can be resolved by the breakdown of the Stokes-Einstein relation below a certain colloid size. Past experiments suggest this happens for colloids smaller than a few nanometres in diameter Li (2009). Compared to an initial colloid size on the scale of a few microns, this corresponds to 2 to 4 orders of magnitude. Since the divergence of the mean squared displacement is logarithmic, this will give a total mean squared displacement that is greater than that of a non-dissolving colloid by a factor of $\mathcal{O}(1)-\mathcal{O}(10)$. Furthermore, since $D_{0}T_{d}$ is independent of $R_{0}$ for this model, the contribution of passive Brownian motion only depends weakly on the initial colloid size. This is in contrast with the other models, and indicative of the absence of diffusion. Finally, using Eq. (II.2.4) in Eq. (57) we obtain for the fast reaction limit the result $\langle{r}^{2}\rangle(t)=6D_{0}\times 2T_{d}\left(\log\left(\frac{R_{0}}{R(t)}\right)+\frac{R(t)}{R_{0}}-1\right).$ (60) where again we have a logarithmic divergence as $t\to T_{d}$. Using previous definitions we find that as in the non-reacting model $\langle{r}^{2}\rangle_{p}\sim R_{0}$ (+ logarithmic corrections) and also that $\langle{r}^{2}\rangle_{p}\sim\alpha_{2}^{-2}$. The passive mean squared displacement therefore depends rather sensitively on the availability of fuel for the reaction. ## IV Active motion of dissolving colloids After examining the dynamics of passive particles, we now turn to the effect of dissolution on self-propelled microswimmers. For the case of active particles subject to rotational diffusion with coefficient $D_{r}$, it is well known that self-propulsion at velocity $U$ gives rise to an effective enhanced translational diffusivity Golestanian _et al._ (2007) $D_{\text{eff}}=D+\frac{U^{2}}{6D_{r}},$ (61) for times much longer than $D_{r}^{-1}$, the time scale of rotational diffusion (i.e. in the limit $tD_{r}\gg 1$). On scales much shorter than this the motion is instead ballistic, i.e. $\langle r^{2}\rangle\sim U^{2}t^{2}$. Figure 4: 2D-projections of sample trajectories for different values of $\gamma$. The colloids initially swim from left to right (see arrow) and dissolve according to the non-reacting model with the same length scale and lifetime. In this new scenario however, an additional scale is introduced through the swimmer lifetime, $T_{d}$. It is therefore vital to consider the dimensionless quantity $\gamma:=D_{r,0}T_{d},$ (62) where we define $D_{r,0}=k_{B}T/8\pi\mu R_{0}^{3}$. If $\gamma\lesssim 1$, then the particle disappears before displaying macroscopically diffusive behaviour. Conversely, if $\gamma\gtrsim 1$ we expect trajectories that are qualitatively similar to that of a classically diffusive colloid at long time scales. The qualitative role of $\gamma$ is illustrated in Fig. 4 where we observe three trajectories becoming more curly as time progresses, since diffusivity increases as the swimmer dissolves. However, only colloids with large values of $\gamma$ (here, $\gamma=10$) exist long enough for this effect to become significant, giving rise to a macroscopically ‘diffusive’ trajectory. Conversely, for small $\gamma$ (here, $\gamma=0.1$) trajectories appear macroscopically ‘ballistic’. Depending on the application, it may be desirable to design swimmers that belong to either of these two regimes. In water at room temperature we have $D_{r,0}^{-1}\approx 6(R_{0}/\mu\text{m})^{3}\text{ s}$ Haynes (2014), so depending on the initial colloid size the threshold lifetime ranges from seconds to hours. Therefore both regimes are conceivable for applications and thus relevant to study. We proceed with the development of our theoretical framework to derive expressions for the active mean squared displacement and present analytical solutions for each model both as $\gamma\to 0$ and as $\gamma\to\infty$. We then validate our theoretical results against numerical simulations of the associated Langevin dynamics. ### IV.1 Mathematical model In the rest of this section we assume that the colloid is subject to Langevin dynamics as $\displaystyle d\bm{r}$ $\displaystyle=U\bm{e}dt,$ (63) $\displaystyle d\bm{e}$ $\displaystyle=-2D_{r}(t)\bm{e}dt+\sqrt{2D_{r}(t)}\bm{\Pi}(\bm{e})\cdot d\bm{W},$ (64) to be understood in the Itô formulation of stochastic calculus. Here $U$ is the particle self-propulsion speed, $\bm{e}$ the unit vector along the direction of propul d $\Pi_{ij}=\delta_{ij}-e_{i}e_{j}$. As is the case for a wide range of phoretic swimmers Moran and Posner (2017), we assume the velocity $U$ to be independent of the swimmer size. Moreover, we set $D=0$ to isolate the effect of active diffusion, which generally exceeds that of (regularised) passive diffusion discussed previously. Since both contribute independently however, they may simply be added together if the total mean squared displacement is desired. We also neglect the details of the propulsion mechanism and possible interactions with our dissolution models. As in the classical case, the $\bm{e}$-dynamics decouple from the $\bm{r}$-dynamics. With the same assumptions regarding the separation of time scales as in the passive case, $\bm{e}(\theta,\phi)$ is therefore subject to the Fokker-Planck equation $\frac{\partial}{\partial t}P(\theta,\phi,t)=D_{r}(t)\nabla_{\text{ang}}^{2}P,$ (65) where $\nabla_{\text{ang}}^{2}$ denotes the angular part of the Laplacian operator. By introducing a rescaled time $\tau_{r}(t)$ as $\tau_{r}=\int_{0}^{t}D_{r}(s)ds=D_{r,0}\int_{0}^{t}\left(\frac{R_{0}}{R(s)}\right)^{3}ds,$ (66) this may be used to show that $\langle\bm{e}(t)\cdot\bm{e}(0)\rangle=\exp(-2\tau_{r})$. Therefore we have the following expression for the total active mean squared displacement, $\langle r^{2}\rangle_{a}=2U^{2}\int_{0}^{T_{d}}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}\exp\bigg{\\{}{-2\left[\tau_{r}(t^{\prime})-\tau_{r}(t^{\prime\prime})\right]}\bigg{\\}}.$ (67) Substituting values for our models and rescaling variables, this gives the following general expressions. $\displaystyle\langle r^{2}\rangle_{a}$ $\displaystyle=U^{2}T_{d}^{2}\int_{0}^{\infty}dx^{\prime}\int_{0}^{x^{\prime}}dx^{\prime\prime}\frac{2e^{-2\gamma(x^{\prime}-x^{\prime\prime})}}{(1+x^{\prime}/2)^{3}(1+x^{\prime\prime}/2)^{3}},$ (non-reacting) (68) $\displaystyle\langle r^{2}\rangle_{a}$ $\displaystyle=U^{2}T_{d}^{2}\int_{0}^{\infty}dx^{\prime}\int_{0}^{x^{\prime}}dx^{\prime\prime}\frac{2e^{-2\gamma(x^{\prime}-x^{\prime\prime})}}{(1+2x^{\prime})^{3/2}(1+2x^{\prime\prime})^{3/2}},$ (slow reaction) (69) $\displaystyle\langle r^{2}\rangle_{a}$ $\displaystyle=U^{2}T_{d}^{2}\int_{0}^{\infty}dx^{\prime}\int_{0}^{x^{\prime}}dx^{\prime\prime}\frac{2e^{-2\gamma(x^{\prime}-x^{\prime\prime})}}{(1+\sqrt{x^{\prime}})^{3}(1+\sqrt{x^{\prime\prime}})^{3}}.$ (fast reaction) (70) Unfortunately, while these are exact results, it is not possible to evaluate these integrals analytically for arbitrary values of $\gamma$. However, we can derive asymptotic solutions in both the diffusive and ballistic limits, as we now show. #### IV.1.1 Diffusive limit ($\gamma\to\infty$) In the diffusive limit, $\gamma\gg 1$, we can use Watson’s lemma to develop an asymptotic expansion, with details given in the Appendix. In the case of a non-reacting swimmer, we find $\langle r^{2}\rangle_{a}\sim\frac{2}{5}\frac{U^{2}T_{d}}{D_{r,0}}\left[1-\frac{5}{8\gamma}+\dots\right],\quad\gamma\to\infty\quad\text{(non- react.)}.$ (71) As expected, the behaviour is diffusive and the leading-order scaling is $\langle r^{2}\rangle_{a}\sim\frac{U^{2}\mu\rho_{s}R_{0}^{5}}{k_{B}TD_{s}(c_{0}-c_{\infty})},\quad\gamma\to\infty\quad\text{(non- reacting)}.$ (72) We notice the appearance of the $2/5$ factor in Eq. (71), indicating that the enhancement of the diffusivity through active motion is reduced dramatically, to just 40% of that of a comparable classical colloid. Furthermore, the active mean squared displacement scales as $\sim R_{0}^{5}$, making the range of the swimmer extremely sensitive to its initial size. This scaling breaks down for very large swimmers, since it is necessary that $\gamma\sim R_{0}^{-1}$ is sufficiently large for this expansion to remain valid. For the slowly reacting swimmer we find in a similar fashion that $\langle r^{2}\rangle_{a}\sim\frac{1}{4}\frac{U^{2}T_{d}}{D_{r,0}}\left[1-\frac{1}{\gamma}+\dots\right],\quad\gamma\to\infty\quad\text{(slow react.)}.$ (73) with the leading-order scaling $\langle r^{2}\rangle_{a}\sim\frac{U^{2}\mu\rho_{s}R_{0}^{4}}{k_{B}Tc_{f,\infty}k_{s}},\quad\gamma\to\infty\quad\text{(slow reaction)}.$ (74) We see that the diffusivity in Eq. (73) is reduced even further, to 25% that of a classical colloid. Finally for the fast reacting swimmer we obtain $\langle r^{2}\rangle_{a}\sim\frac{1}{10}\frac{U^{2}T_{d}}{D_{r,0}}\left[1-\frac{5}{2\gamma}+\dots\right],\quad\gamma\to\infty\quad\text{(fast react.)/},$ (75) and the leading-order scaling $\langle r^{2}\rangle_{a}\sim\frac{U^{2}\mu\rho_{s}^{2}k_{f}^{2}R_{0}^{5}}{k_{B}TD_{f}c_{f,\infty}^{2}k_{s}^{2}},\quad\gamma\to\infty\quad(\text{fast reaction}).$ (76) This third dissolution model gives the strongest reduction of the active mean squared displacement in the diffusive regime, to just 10% that of a classical colloid. The strong reduction in mean squared displacement across all three models suggests that it is impractical to rely on active diffusion to transport dissolving microswimmers. Instead designs may be aimed at exploiting the ballistic regime ($\gamma\ll 1$) or making use of external flows and geometries to direct swimmers. #### IV.1.2 Ballistic limit ($\gamma\to 0$) The asymptotic expansions in the ballistic limit are more complicated, and rely on careful splitting of the integration range to tame divergences. With all details shown in the Appendix, we obtain the following leading-order results: $\displaystyle\langle r^{2}\rangle_{a}$ $\displaystyle=U^{2}T_{d}^{2}\left(1-\frac{16}{3}\gamma+\mathcal{O}(\gamma^{3/2})\right),$ (non-reacting) (77) $\displaystyle\langle r^{2}\rangle_{a}$ $\displaystyle=U^{2}T_{d}^{2}\left(1-2\sqrt{\pi}\sqrt{\gamma}+\mathcal{O}(\gamma\log\gamma)\right),$ (slow reaction) (78) $\displaystyle\langle r^{2}\rangle_{a}$ $\displaystyle=U^{2}T_{d}^{2}\left(1-4\sqrt{2\pi}\sqrt{\gamma}+\mathcal{O}(\gamma\log\gamma)\right).$ (fast reaction) (79) Once again, we observe the same hierarchy among the three models, with the non-reacting swimmer exhibiting the smallest decrease in range compared to a classical colloid, in contrast with a fast reacting swimmer with the same lifetime $T_{d}$. Note that in this limit not only the coefficient but also the leading-order scaling varies between the models. We obtain therefore that in both the ballistic and diffusive limit there exists a hierarchy among the three models. The mean squared displacement for a given value of $\gamma$ is always largest for the non-reacting swimmer, followed by the slowly reacting and finally the fast reacting colloid. This may be explained by considering the decay behaviour in Fig. 3. Since the decay rate of the non-reacting swimmer is accelerating, it is only significantly smaller than its original size for a comparatively short proportion of its total lifetime. Since rotational diffusion is strongest for particles of small radius, this means that it is comparatively weakly affected by the enhancement in rotational diffusion. In contrast, colloids decaying according the other two models experience strong rotational diffusion for a significantly longer proportion of their lifetime, leading to less directed motion and smaller overall displacement. In Fig. 8 and 9 we illustrate this further using results from our numerical simulations. ### IV.2 Computational results #### IV.2.1 Validation of the method Figure 5: Normalised active mean squared displacement as a function $\gamma$ for the non-reacting model. The solid black line corresponds to direct numerical integration of Eq. (68), while the dashed orange line is our theoretical prediction in Eq. (71) for the large $\gamma$ limit. Each scatter point represents the mean and one standard deviation obtained from $10^{3}$ Monte-Carlo simulations of the associated Langevin equations. Inset: the small $\gamma$ behaviour, comparing Eq. (68) (solid black) with the asymptotic solution Eq. (77) (dashed orange). Figure 6: Normalised active mean squared displacement against $\gamma$ for the slow reaction limit of the reacting model. The solid black line corresponds to direct numerical integration of Eq. (69), the dashed orange lines to the theoretical predictions of Eq. (73) and Eq. (78), and the scatter points to Monte-Carlo simulations in analogy with Fig. 5. Figure 7: Normalised active mean squared displacement against $\gamma$ for the slow reaction limit of the reacting model. The solid black line corresponds to direct numerical integration of Eq. (70), the dashed orange lines to the theoretical predictions of Eq. (75) and Eq. (79), and the scatter points to Monte-Carlo simulations in analogy with Fig. 5. In order to test our theoretical approach, we perform direct numerical integrations of our integral expressions for the active mean squared displacement in Eqs. (68)-(70). We compare them with Monte-Carlo simulations of the associated Langevin dynamics to assert its validity, and subsequently with our analytical predictions for the asymptotic behaviour. The results are shown in Figs. 5, 6 and 7 for the non-reacting, slowly reacting and fast reacting models respectively. Since the large $\gamma$ limit corresponds to strong rotational diffusion and long lifetimes, the Monte Carlo simulations necessitate very small time steps and very long run times. Depending on the model, such simulations therefore become prohibitively expensive even for moderate values of $\gamma$. Since rotational diffusion is strongest for small colloids, this effect is most pronounced for the fast reacting swimmer whose rate of dissolution is decreasing since this swimmer spends the longest proportion of its lifetime in this regime. Conversely, the non-reacting swimmer is the least expensive to simulate. As can be seen in Fig. 5, we obtain excellent agreement between the Langevin dynamics and the predicted mean-squared displacement for a wide range of $\gamma$ values. In the diffusive limit ($\gamma\gg 1$), the next-to leading order asymptotics agree extremely well with the exact result down to $\gamma=\mathcal{O}(1)$ on a log-log scale. In the ballistic limit, divergences begin to appear at $\gamma=\mathcal{O}(10^{-1})$. Similar conclusions hold for the slowly reacting swimmer, as shown in Fig. 6. In the case of the fast reacting swimmer, shown in Fig. 7, the active mean squared displacement is a less smooth function of $\gamma$, leading to stronger diversion from the asymptotic expressions. #### IV.2.2 Distribution of spread Figure 8: Histograms illustrating the distribution of root mean squared displacement from the initial position for different values of $\gamma$, scaled by the ballistic length scale $L_{b}=UT_{d}$ for $\gamma=0.1$ and $\gamma=1$, and the diffusive length scale $L_{d}=L_{b}/\sqrt{\gamma}$ for $\gamma=10$. Each histogram is generated from $10^{3}$ Monte Carlo simulations. Dashed lines indicate sample means. Figure 9: Cloud scatter plot of lateral displacement, $r_{\perp}=\sqrt{x^{2}+y^{2}}$, vs. vertical displacement, $z$, of $10^{3}$ Monte Carlo simulations in the weakly ballistic regime for our three models compared to the non-dissolving case. All simulations are started at the coordinate origin (filled circle) with initial orientation vertically upwards in Cartesian coordinates $(x,y,z)$. Symbols indicate positions of the colloids at time of disappearance. The non- dissolving data points are generated by initialising a simulation with a given rotational diffusivity $D_{r}$ and terminating after a time $T$ such that $TD_{r}=\gamma$. Lengths are scaled by the ballistic length scale $UT_{d}$. From these Monte-Carlo simulations, we can deduce further information regarding the spread of particle trajectories. As predicted in §IV.1, a hierarchy between the models is revealed that applies for a wide range of values of $\gamma$, covering both the ballistic and the diffusive regime. This is illustrated in Fig. 8, where we show histograms of root-mean-square displacement distributions. For equal values of $\gamma$, the non-reacting model consistently produces the largest displacement. The distribution is strongly peaked for small $\gamma$ (ballistic), but spreads as $\gamma$ shifts to larger values. This may be attributed to the general shift towards diffusion. Contrastingly however, the distribution of the fast-reacting colloids is spread rather widely even in the ballistic regime and in fact peaked much more strongly in the diffusive regime than both the non-reacting and the slowly reacting particles, whose distribution lies between the two others. This is indicative of fast-reacting dissolution fostering diffusive behaviour independent of the parameter $\gamma$. In order to further illustrate this point, we examine the lateral spread of colloid trajectories in the weakly ballistic regime. In Fig. 9, we plot the final positions of colloids with identical initial orientations, including non-dissolving particles for comparison. A clear stratification between the models is visible with non-dissolving colloids being closely confined to a spherical cap on the one extreme, and fast reacting colloids in a near- spherical diffusive cloud close to the origin. These also exhibit the smallest absolute lateral spread, while the classical colloids are the most spread out. However, the average angular spread is similar between the models. ## V Discussion In this paper we provide two fundamental models for the dissolution and stochastic dynamics of self-propelled artificial microswimmers. Inspired by recent experimental realisations, we seek to identify the swimmer decay rates and their influence on translational and rotational diffusivity, and in turn analyse both theoretically and numerically how changes in these modify the distribution of swimmer trajectories. We identify a new dimensionless parameter, $\gamma$ defined as the product of lifetime and initial rotational diffusivity, that classifies colloids with finite lifetime into ‘ballistic’ and ‘diffusive’ types independent of the dissolution process, and study the differences between our dissolution models in three distinct limits for various values of this parameter. We find that for a given value of $\gamma$, particles dissolving in the absence of a reaction behave the most ballistic, whereas colloids reacting at high Damköhler number, defined as the ratio of fuel reactivity and diffusive replenishment, behave the most diffusively. We find that this is due to increasing and decreasing dissolution rates respectively for the different models. Furthermore we derive asymptotic expressions of their mean squared displacement for both small and large values of $\gamma$, and perform extensive Monte Carlo simulations to validate our theoretical results and derive more information about the distribution of spread. Under experimental conditions, Damköhler numbers of more than about 10 are often very difficult to realise. However, this does not really constrain the applicability of our fast-reacting model, since we only require $t_{f}/\text{Da}^{2}\ll T_{d}$ for the expansion to be valid on the scale of dissolution dynamics. Since typically $t_{f}\ll T_{d}$ anyway, we find that even Damköhler numbers of order unity are sufficient for this limit. On the other hand, this argument implies that very small Damköhler numbers are required in the slow-reaction asymptotic limit, a situation which might not be realisable experimentally. Note however that we include also the general expression of the decay for arbitrary Damköhler number in Eq. (II.2.2), for which computations similar to the ones provided in §IV.2 may be performed. Despite this, not all our models can apply to all kinds of microswimmer designs. Specifically, the non-reacting model might be at odds with phoretic self-propulsion. Therefore this model only describes colloids that propel through different mechanisms, such as magnetic swimmers. Furthermore, our statistical results only hold true for microswimmers that are fully degradable. A Janus colloid with, e.g., degradable and inert halves is not going to exhibit divergent diffusivity since the relevant length scale is bounded. Instead such a swimmer would show a decrease in velocity, which if known can be dealt with in a manner similar to our theoretical approach. In this case, however, the changing geometry of the swimmer would likely have to be solved for numerically. Another important problem that remains to be investigated is the influence of directed motion, such as chemotaxis. Breaking the isotropy of orientational dynamics prevents an analytical investigation similar to the one carried out in this paper since it relies on the result that the directional correlation of a particle decays exponentially. However, we can still address the issue directly in at least one special case. It was shown recently in Ref. Tatulea- Codrean and Lauga (2018) that artificial colloids perform chemotaxis by adjusting their trajectory by means of rotation, translation in the direction to a chemical gradient, and translation at an angle, each with a coefficient of strength that can be calculated from the surface activity and mobility of the colloid. In the case of uniform surface activity, the only coefficient that is non-zero is the one giving rise to translation in the direction of a chemical gradient. In particular, the rotational dynamics remain unaffected. In that case, the swimmer trajectories behave therefore just like we describe in our paper, plus a constant velocity displacing the colloid in the direction of the chemical gradient. Furthermore numerical work will be required to address the full interplay between chemotaxis behaviour and dissolution dynamics. Before degradable designs may be employed in real-world applications, it will be furthermore necessary to examine the effects of collective dissolution. Since our models are sensitive to the background distribution of fuel and/or solute, the influence of other nearby colloids on their dissolution will be noticeable. It is conceivable that, in analogy with bubbles Michelin _et al._ (2018), different decay patterns and complex stochastic behaviour emerges. Similar effects may also be triggered by confinement and also warrant further investigation. ###### Acknowledgements. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement 682754 to EL). ## Author contributions EL conceived the study, AC developed models and performed computations, all authors contributed to the interpretation and writing of the manuscript. ## Conflicts of interest There are no conflicts to declare. ## Appendix A Details of the asymptotics for active MSD ### A.1 Diffusive limit ($\gamma\to\infty$) The general expression for the active mean squared displacement is $\langle r^{2}\rangle_{a}=2U^{2}\int_{0}^{T_{d}}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}\exp\left\\{{-2\left[\tau_{r}(t^{\prime})-\tau_{r}(t^{\prime\prime})\right]}\right\\}.$ (80) In the case of the non-reacting swimmer we have $R\approx R_{0}\sqrt{1-t/T_{d}}$, and thus $\tau_{r}=D_{r,0}T_{d}\int_{0}^{t/T_{d}}\frac{dt^{\prime}}{(1-t^{\prime})^{3/2}}=2\gamma\left(\frac{1}{\sqrt{1-t/T_{d}}}-1\right).$ (81) We can use this to change integration variables in Eq. (80) by setting $x=\tau_{r}/\gamma$ and obtain $\langle r^{2}\rangle_{a}=2U^{2}T_{d}^{2}\int_{0}^{\infty}dx^{\prime}\int_{0}^{x^{\prime}}dx^{\prime\prime}\frac{e^{-2\gamma(x^{\prime}-x^{\prime\prime})}}{(1+x^{\prime}/2)^{3}(1+x^{\prime\prime}/2)^{3}}.$ (82) This transformation can be interpreted as mathematically equivalent to the motion of a non-dissolving colloid with constant rotational diffusivity and algebraically decaying velocity. We switch variables again to $\displaystyle y^{\prime}$ $\displaystyle=x^{\prime},$ (83) $\displaystyle y^{\prime\prime}$ $\displaystyle=x^{\prime}-x^{\prime\prime}.$ and obtain $\langle r^{2}\rangle_{a}=2U^{2}T_{d}^{2}\int_{0}^{\infty}dy^{\prime}\int_{0}^{y^{\prime}}dy^{\prime\prime}e^{-2\gamma y^{\prime\prime}}\left(1+\frac{y^{\prime}}{2}\right)^{-3}\left(1+\frac{y^{\prime}-y^{\prime\prime}}{2}\right)^{-3}.$ (84) It is then possible to write the $y^{\prime\prime}$-integral in terms of auxiliary Gamma functions. These may be expanded in the limit $\gamma\to\infty$ to give $\langle r^{2}\rangle_{a}=U^{2}T_{d}^{2}\int_{0}^{\infty}dy^{\prime}\frac{64}{\gamma(2+y^{\prime})^{6}}+\frac{96}{\gamma^{2}(2+y^{\prime})^{7}}-\frac{8e^{-2\gamma y^{\prime}}}{\gamma(2+y^{\prime})^{3}}+\mathcal{O}(\gamma^{-3}).$ (85) The first two terms can be evaluated directly, while the last one may be expanded using Watson’s lemma. We find that $\langle r^{2}\rangle_{a}=U^{2}T_{d}^{2}\left(\frac{2}{5\gamma}-\frac{1}{4\gamma^{2}}+\mathcal{O}\left(\gamma^{-3}\right)\right),$ (86) which is the same as Eq. (71). The case of a slowly reacting swimmer can be solved in a very similar fashion. This time we have $x=\frac{1}{2}\left(\frac{1}{(1-t/T_{d})^{2}}-1\right).$ (87) It follows that the active part of the mean squared displacement may be written as $\langle r^{2}\rangle_{a}=2U^{2}T_{d}^{2}\int_{0}^{\infty}dx^{\prime}\int_{0}^{x^{\prime}}dx^{\prime\prime}\frac{e^{-2\gamma(x^{\prime}-x^{\prime\prime})}}{(1+2x^{\prime})^{3/2}(1+2x^{\prime\prime})^{3/2}}.$ (88) Developing an asymptotic expansion as before we get $\displaystyle\langle r^{2}\rangle_{a}$ $\displaystyle=U^{2}T_{d}^{2}\int_{0}^{\infty}dy^{\prime}\frac{1}{\gamma(1+2y^{\prime})^{3}}+\frac{3}{2\gamma^{2}(1+2y^{\prime})^{4}}-\frac{e^{-2\gamma y^{\prime}}}{\gamma(1+2y^{\prime})^{3/2}}+\mathcal{O}(\gamma^{-3})$ $\displaystyle=U^{2}T_{d}^{2}\left(\frac{1}{4\gamma}-\frac{1}{4\gamma^{2}}+\mathcal{O}\left(\gamma^{-3}\right)\right),$ (89) which is Eq. (73). Finally, for the fast reacting swimmer we have $x=\frac{t/T_{d}}{(1-\sqrt{t/T_{d}})^{2}},$ (90) from which we can derive that $\langle r^{2}\rangle_{a}=2U^{2}T_{d}^{2}\int_{0}^{\infty}dx^{\prime}\int_{0}^{x^{\prime}}dx^{\prime\prime}\frac{e^{-2\gamma(x^{\prime}-x^{\prime\prime})}}{(1+\sqrt{x^{\prime}})^{3}(1+\sqrt{x^{\prime\prime}})^{3}}.$ (91) In this case it is easier to interchange the integrals as $\int_{0}^{\infty}dx^{\prime}\int_{0}^{x^{\prime}}dx^{\prime\prime}=\int_{0}^{\infty}dy^{\prime\prime}\int_{y^{\prime\prime}}^{\infty}dy^{\prime}$ and perform the $y^{\prime}$-integral first. The resulting expression produced by Wolfram Mathematica 11 contains 1692 terms, but may again be expanded and simplified significantly upon the application of Watson’s lemma, giving $\displaystyle\langle r^{2}\rangle_{a}$ $\displaystyle=U^{2}T_{d}^{2}\int_{0}^{\infty}dy^{\prime\prime}e^{-2\gamma y^{\prime\prime}}\left(\frac{1}{5}-y^{\prime\prime}+\mathcal{O}\left(y^{\prime\prime 3/2}\right)\right)$ (92) $\displaystyle=U^{2}T_{d}^{2}\left(\frac{1}{10\gamma}-\frac{1}{4\gamma^{2}}+\mathcal{O}\left(\gamma^{-5/2}\right)\right),$ (93) as claimed in Eq. (75). ### A.2 Ballistic limit ($\gamma\to 0$) First, the non-reacting swimmer. We have $\langle r^{2}\rangle_{a}=2U^{2}T_{d}^{2}\int_{0}^{\infty}dx\int_{0}^{x}dy\frac{e^{-2\gamma(x-y)}}{(1+x/2)^{3}(1+y/2)^{3}},$ (94) and are interested in the limit $\gamma\to 0$. We set $U^{2}T_{d}^{2}=1$ to keep the notation clean. Since the denominator decays rapidly enough at $\infty$ we can Taylor expand the exponential to pick up the two leading-order contributions to the integral. $\displaystyle\langle r^{2}\rangle_{a}$ $\displaystyle=\int_{0}^{\infty}dx\int_{0}^{x}dy\frac{2-4\gamma(x-y)+\dots}{(1+x/2)^{3}(1+y/2)^{3}}$ (95) $\displaystyle=1-\frac{16}{3}\gamma+o(\gamma),$ (96) which is Eq. (77). For the slowly reacting swimmer we have $\langle r^{2}\rangle_{a}=2\int_{0}^{\infty}dx\int_{0}^{x}dy\frac{e^{-2\gamma(x-y)}}{(1+2x)^{3/2}(1+2y)^{3/2}}.$ (97) Because of the slower decay, it is necessary to divide and conquer from the start. We set $z=x-y$ and note that $\int_{0}^{\infty}dx\int_{0}^{x}dy=\int_{0}^{\infty}dz\int_{z}^{\infty}dx$. Upon performing the inner integral we have $\langle r^{2}\rangle_{a}=\int_{0}^{\infty}dz\frac{e^{-2\gamma z}}{1+\sqrt{1+2z}+z(2+\sqrt{1+2z})}.$ (98) We define $\delta$ such that $1\ll\delta\ll\gamma^{-1}$ and split the integral into $\displaystyle I_{1}=\int_{0}^{\delta}dz\frac{e^{-2\gamma z}}{1+\sqrt{1+2z}+z(2+\sqrt{1+2z})},$ $\displaystyle I_{2}=\int_{\delta}^{\infty}dz\frac{e^{-2\gamma z}}{1+\sqrt{1+2z}+z(2+\sqrt{1+2z})}.$ (99) Upon expanding the exponential in $I_{1}$ and taking $\delta\to\infty$ we have $I_{1}=1+(2-2\log 2)\gamma+\mathcal{O}(\gamma^{2})+\text{terms depending on }\delta.$ (100) Meanwhile, we rescale $z\to\gamma z$ in $I_{2}$ and expand the denominator for small $\gamma$. $I_{2}=\int_{\gamma\delta}^{\infty}dze^{-2z}\left(\frac{\gamma^{1/2}}{\sqrt{2}z^{3/2}}-\frac{\gamma}{z^{2}}+\dots\right).$ (101) Performing the integral and taking the limit $\delta\to 0$ we arrive at $\displaystyle I_{2}=-2\sqrt{\pi}\gamma^{1/2}-2\gamma\log\gamma+\left(2-2\gamma_{e}-2\log 2\right)\gamma$ $\displaystyle+o(\gamma)+\text{terms depending on }\delta,$ (102) where $\gamma_{e}$ is the Euler-Mascheroni constant. Since $\delta$ is arbitrary, the divergent terms in both integrals must cancel. In summary, we have for the slowly reacting swimmer that $\langle r^{2}\rangle_{a}=1-2\sqrt{\pi}\gamma^{1/2}-2\gamma\log\gamma+\left(4-2\gamma_{e}-4\log 2\right)\gamma+o(\gamma),$ (103) which is Eq. (78). Finally, for the fast reacting swimmer we have $\langle r^{2}\rangle_{a}=2\int_{0}^{\infty}dx\int_{0}^{x}dy\frac{e^{-2\gamma(x-y)}}{(1+\sqrt{x})^{3}(1+\sqrt{y})^{3}}.$ (104) This time there is no closed-form expression for the inner integral, forcing us to split both integrals in two domains. We define $\delta$ as before and write $\langle r^{2}\rangle_{a}=\underbrace{\int_{0}^{\delta}dx\int_{0}^{x}dy}_{I_{1}}+\underbrace{\int_{\delta}^{\infty}dx\int_{0}^{x}dy}_{I_{2}}\frac{2e^{-2\gamma(x-y)}}{(1+\sqrt{x})^{3}(1+\sqrt{y})^{3}}.$ (105) The first part, $I_{1}$, is straightforward to do once the exponential is expanded and yields $I_{1}=1-\frac{296}{3}\gamma+\mathcal{O}(\gamma^{2})+\text{terms depending on }\delta.$ (106) To perform $I_{2}$ we write $I_{2}=\int_{\delta}^{\infty}dx\frac{2e^{-2\gamma x}}{(1+\sqrt{x})^{3}}\underbrace{\int_{0}^{x}dy\frac{e^{2\gamma y}}{(1+\sqrt{y})^{3}}}_{J(x)},$ (107) and split the range of $J(x)$ again with the goal to obtain an expansion valid for small $\gamma$. Defining $\delta_{1}$, $J_{1}$ and $J_{2}$ in a similar fashion, we find $J_{1}=1+10\gamma+\mathcal{O}(\gamma^{2})+\text{terms depending on }\delta_{1},$ (108) whereas for $J_{2}$ we have $\displaystyle J_{2}=$ $\displaystyle\frac{3e^{2\gamma x}}{x}-\frac{2e^{2\gamma x}}{\sqrt{x}}+2\sqrt{2\pi}\gamma^{1/2}\text{Erfi}\left(\sqrt{2\gamma x}\right)$ $\displaystyle-6\gamma\text{Ei}\left(2\gamma x\right)+2\gamma\log\gamma+\gamma\left(6\gamma_{e}-6+6\log 2\right)$ $\displaystyle+o(\gamma)+\text{terms depending on }\delta_{1},$ (109) where $\text{Erfi}(z)=\text{Erf}(iz)/i$ and $\text{Ei}(z)=-\int_{-z}^{\infty}e^{-t}/t\,dt$. 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2024-09-04T02:54:55.253187

2002.12303

arxiv-papers

# Electrostatic potential shape of gate defined quantum point contacts M. Geier Dahlem Center for Complex Quantum Systems and Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany J. Freudenfeld Paul-Drude-Institut für Festkörperelektronik, Leibniz-Institut im Forschungsverbund Berlin e.V., Hausvogteiplatz 5-7, 10117 Berlin, Germany J. T. Silva Paul-Drude-Institut für Festkörperelektronik, Leibniz-Institut im Forschungsverbund Berlin e.V., Hausvogteiplatz 5-7, 10117 Berlin, Germany V. Umansky Weizmann Institute of Science, Rehovot 76100, Israel D. Reuter111Present address: Department of Physics, Paderborn University, Warburger Straße 100, 33098 Paderborn Angewandte Festkörperphysik, Ruhr- Universität Bochum, Universitätsstrasse 150, 44780 Bochum, Germany A. D. Wieck Angewandte Festkörperphysik, Ruhr-Universität Bochum, Universitätsstrasse 150, 44780 Bochum, Germany P. W. Brouwer Dahlem Center for Complex Quantum Systems and Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany S. Ludwig Paul-Drude-Institut für Festkörperelektronik, Leibniz-Institut im Forschungsverbund Berlin e.V., Hausvogteiplatz 5-7, 10117 Berlin, Germany ###### Abstract Quantum point contacts (QPC) are fundamental building blocks of nanoelectronic circuits. For their emission dynamics as well as for interaction effects such as the 0.7-anomaly the details of the electrostatic potential are important, but the precise potential shapes are usually unknown. Here, we measure the one-dimensional subband spacings of various QPCs as a function of their conductance and compare our findings with models of lateral parabolic versus hard wall confinement. We find that a gate-defined QPC near pinch-off is compatible with the parabolic saddle point scenario. However, as the number of populated subbands is increased Coulomb screening flattens the potential bottom and a description in terms of a finite hard wall potential becomes more realistic. ###### pacs: ## I introduction Given the importance of quantum point contacts (QPC) as fundamental building blocks of nanoelectronic circuits and the vast amount of literature about them Landauer (1981); van Wees et al. (1988a); Wharam et al. (1988); Berggren and Pepper (2010); Micolich (2011), surprisingly little is known about the shape of their electrostatic potential as a function of gate voltages. However, knowledge of the precise confinement potential is crucial for understanding interaction effects in QPCs Rejec and Meir (2006); Koop et al. (2007); Bauer et al. (2013); Heyder et al. (2015) as well as their carrier emission dynamics Topinka et al. (2000); Freudenfeld et al. (2020), which is central for optimizing a quantum electronic circuit. The lateral confinement defines the mode structure of the one-dimensional (1D) channel while the longitudinal potential shape governs the coupling of the 1D modes into the surrounding 2DES. Populating the 1D channel with electrons by increasing the voltage applied to the split gates enhances Coulomb screening inside the constriction. As a consequence, the lateral confinement potential undergoes a transition from an unscreened approximately parabolic shape near pinch-off towards a screened potential for many occupied 1D subbands. Such a transition had been theoretically predicted Laux et al. (1988). Here, we experimentally demonstrate it using transport spectroscopy at finite source-drain voltage. Details of the confinement vary between individual devices produced by various layouts based on different methods, which include the field effect Wharam et al. (1988); van Wees et al. (1988a), etching Martin et al. (2008) or oxidation Senz et al. (2001) techniques and more Fricke et al. (2006); Rössler et al. (2010). The manifestation of 1D conductance quantization, $G=NG_{\text{Q}}$ with $G_{\text{Q}}=2e^{2}/h$ and $N=1,2,3,\dots$, at cryogenic temperatures is often seen as a quality feature of QPCs. An “optimally” designed QPC has several conductance steps that are approximately equidistant in gate voltage as the QPC is opened up starting from pinch-off at $G=0$. It is tempting to interpret the presence of equidistant conductance steps van Wees et al. (1988a); Taboryski et al. (1995); Thomas et al. (1998); Hew et al. (2008); Rössler et al. (2011); Burke et al. (2012) as a signature of a parabolic transverse confinement potential as introduced in Ref. Büttiker (1990), since such a potential has transverse modes at equally spaced energies. However, this interpretation is questionable as the distance of the conductance steps as a function of gate voltage is not one-to-one related with the energy spacing of the 1D modes Rössler et al. (2011); Micolich and Zülicke (2011). Figure 1: (a) Scanning electron microscope (SEM) picture of Ti/Au gates (light gray) on the wafer surface (dark) of QPC1 and sketch of the electric field effect device. Negative voltage $V_{\text{g}}$ applied to the gates (yellow) is used to locally deplete the 2DES (blue, where conducting, gray where depleted) 107 nm beneath the surface. (b) Pinch-off curve $G(V_{\text{g}})/G_{\text{Q}}$ of QPC1 using the source-drain voltage $V=-0.5$ mV; solid line: raw data; dots: corrected for lead resistance $R_{\text{lead}}=4.62$ k$\Omega$ which includes $4.4$ k$\Omega$ resistance of external RC filters; inset: simplified circuit diagram of the measurement. (c) Finite bias spectroscopy of QPC1, $\text{d}g/\text{d}V_{\text{g}}(V_{\text{QPC}},V_{\text{g}})$, accounting for the lead resistance (see main text). Local maxima of $\text{d}g/\text{d}V_{\text{g}}$ (white lines) indicate transitions between adjacent conductance plateaus. (d) SEM picture of a screen gate equivalent to that of QPC2. As shown in the sketch, the actual device is covered with a 130 nm thick layer of cross-linked PMMA which carries a global top gate. (e) Pinch-off curves of QPC2 corrected for a gate voltage dependent lead resistance, including a constant $4.4$ k$\Omega$ resistance of external RC filters, cf. Fig. 2: $G(V_{\text{t}})/G_{\text{Q}}$ for $V_{\text{s}}=0.5\,$V and $G(V_{\text{s}})/G_{\text{Q}}$ for $V_{\text{t}}=-3.4\,$V at $V=-0.1$ mV. (f) $\text{d}g/\text{d}V_{\text{g}}(V_{\text{QPC}},V_{\text{t}})$ of QPC2, accounting for the lead resistance. Additional lines and symbols in panels (c) and (f) are explained in the main text. We study QPCs of two different designs, but both defined using gate voltages by means of the electric field effect. In agreement with previous publications Büttiker (1990); Berggren and Pepper (2010); Burke et al. (2012); Heyder et al. (2015) our findings are consistent with a parabolic confinement potential near the pinch-off point of the QPCs. However, as the conductance of a QPC is increased, more and more carriers populate the 1D subbands and thereby arrange themselves to partially screen the electric field induced by the applied gate voltages. The resulting effective potential is then a function of the position of all charges, which also includes the usually not well-known distribution of surface states and charged bulk defects. A precise theoretical description of this screening effect requires a three-dimensional self-consistent calculation solving the classical Poisson equations together with the quantum mechanical Schrödinger equations Laux et al. (1988); Yakimenko et al. (2013); Birner et al. (2007). A self consistent Poisson-Schrödinger calculation performed for a set of fictitious boundary conditions and for the case of a standard split- gate defined QPC suggests a transition from a parabolic lateral confinement for $N=1$ towards a truncated parabola and, eventually, a hard wall confinement as $N$ is increased Laux et al. (1988). To test this scenario we measure non-linear response transport through our QPC from which we identify the energy spacings between its highest occupied 1D modes. We compare our results to the two extreme scenarios for the lateral electrostatic confinement: parabolic confinement as, e.g., in Refs. Weisz and Berggren (1989); Hew et al. (2008); Song et al. (2009); Rössler et al. (2011) and a hard-wall confinement as, e.g., in Refs. van Wees et al. (1988b); Gloos et al. (2006). Our results are inconsistent with parabolic confinement for $N\geq 4$ but are consistent with a transition from a parabolic lateral confinement at $N=1$ towards a hard wall potential as the QPCs are opened up. ## II Transport spectroscopy of quantum point contacts Our QPCs are formed using the electric field effect in a 2D electron system (2DES) embedded 107 nm beneath the surface of a (Al,Ga)As/GaAs heterostructure. The 2DESs Fermi energy and mobility measured at cryogenic temperatures are $E_{\text{F}}\simeq 10.9$ meV and $\mu_{e}\simeq 2.6\times 10^{6}\,\text{cm}^{2}/$Vs for QPC1 and similar for QPC2. We performed all measurements in a helium-3 evaporation cryostat at temperatures near $T=250$ mK. In Fig. 1(a) and (d) we present scanning electron microscope images of the two QPC samples and sketches of the gate layouts. For QPC1 shown in panel (a) we use a standard split-gate layout and define the 1D constriction of the 2DES by applying a negative voltage $V_{\text{g}}$ to both gates while the 2DES and a back gate approximately 500 $\mu$m below the surface are at ground potential. The resulting linear response pinch-off curve, $G(V_{\text{g}})/G_{\text{Q}}$, is presented in Fig. 1(b). It features clear and, for $N<6$, nearly equidistant steps of quantized conductance. To create the second QPC2, see Fig. 1 (d), we use a global top gate to globally deplete the 2DES. Only below a screen gate placed in between the top gate and the 2DES we induce a finite density of free electrons Bachsoliani et al. (2017). The screen gate shapes a narrow constriction, i.e., a QPC between 2D leads. Both, the QPC conductance and the carrier density in the leads are controlled by the combination of the voltages $V_{\text{t}}$ and $V_{\text{s}}$ applied to the top gate and screen gate, respectively. We present example pinch-off curves $G(V_{\text{t}})$ for fixed $V_{\text{s}}=0.5\,$V and $G(V_{\text{s}})$ for constant $V_{\text{t}}=-3.4\,$V in Fig. 1(e). Note, that the screen gate voltage is restricted to $V_{\text{s}}\lesssim 0.5\,$V as larger $V_{\text{s}}$ causes a leakage current from the gate into the 2DES (as expected for a Schottky barrier). All our pinch-off curves feature smooth transitions between quantized conductance plateaus. They indicate that the potential varies slowly and smoothly in current direction, reminiscent of a parabolic potential profile in the longitudinal direction, which results in reflectionless contacts between constriction and leads. Quantized conductance is a consequence of the energy quantization in a 1D channel caused by the lateral confinement of the constriction. To experimentally determine the energies of the 1D modes we need a known energy scale to compare with. For this reason we measure the differential conductance $g=\text{d}I/\text{d}V$ (e.g., using a lock-in amplifier) as a function of source-drain voltage $V$ along the pinch-off curves. In panels (c) and (f) of Fig. 1 we plot the differential transconductances $\text{d}g/\text{d}V_{\text{g}}$ ($\text{d}g/\text{d}V_{\text{t}}$) for the two QPCs as a function of the gate voltage and the bias voltage $V_{\text{QPC}}$ (defined below) dropping across the QPC. In these plots steps of the conductance $G(V_{\text{g}},V_{\text{QPC}})$ [$G(V_{\text{t}},V_{\text{QPC}})$] appear as lines of positive differential transconductance (white). Red lines are a guide for the eye, indicating resonances between the 1D modes and the chemical potentials of the source and drain leads. Along the $N$th line of positive (negative) slope counted from the bottom of the plot, the $N$th 1D subband bottom energy is equal to the chemical potential in the source (drain) lead, $\varepsilon_{N}=\mu_{\text{S}}$ ($\varepsilon_{N}=\mu_{\text{D}}$). The lines frame diamond shaped regions around $V_{\text{QPC}}=0$. Within these regions the conductance takes the quantized values $G=NG_{\text{Q}}$. Intersection points at $V_{\text{QPC}}=0$ indicate steps of the linear response pinch-off curves, i.e., $G=(N-0.5)G_{\text{Q}}$. At intersection points at finite $V_{\text{QPC}}\neq 0$ the chemical potential drop across a QPC equals the energy spacing between the corresponding 1D modes, $|\mu_{\text{S}}-\mu_{\text{D}}|=eV_{\text{QPC}}=\varepsilon_{N}-\varepsilon_{M}$. The additional curved lines of enhanced differential transconductance within the $N=1$ diamond indicate the 0.7-anomaly Rejec and Meir (2006); Koop et al. (2007); Micolich (2011); Bauer et al. (2013); Heyder et al. (2015) which is not a topic of this article. Since the source-drain voltage $V$ is applied across the QPC and its leads (which is always the case, because of the finite contact sizes even for a four-terminal measurement), the voltage drop across a QPC is $V_{\text{QPC}}=V-V_{\text{lead}}=V-R_{\text{lead}}I$, cf. sketch in Fig. 1(b). The lead resistance can be directly determined from the linear response pinch-off curves by forcing the conductance plateaus to their quantized values, $R_{\text{lead}}=V/I-(NG_{\text{Q}})^{-1}$. Our pinch-off curves in panels (b) and (e) of Fig. 1 are already corrected for the lead resistances, while for QPC1 we additionally plot the uncorrected curve, i.e., the raw data as a solid line. For completeness we present the lead resistances for all three pinch-off curves in Fig. 2. Figure 2: Resistances $R_{\text{lead}}$ of the leads to the QPCs, cf. sketch in Fig. 1(b). For the split gate design of QPC1 $R_{\text{lead}}$ is constant while it is a function of gate voltages for QPC2. From these we determine the voltage drop across the QPC, $V_{\text{QPC}}=V/(R_{\text{lead}}G+1)$, which is the x-axis in panels (c) and (f) of Fig. 1. The tapered shape of the region of plotted data is a result of correcting for the lead resistances (we measured between $-8\text{mV}\leq V\leq 8\,$mV). At the intersection points marked by red squares in panels (c) and (f) of Fig. 1 the bias $V_{\text{QPC}}$ is precisely equal to the energy spacings between subsequent subbands, $\displaystyle\delta\varepsilon(N)=$ $\displaystyle\,\varepsilon_{N+1}-\varepsilon_{N}$ $\displaystyle=$ $\displaystyle\,eV_{\text{QPC}}.$ (1) We plot $\delta\varepsilon(N)$ in Fig. 3 Figure 3: Subband spacings $\delta\epsilon(N)$ of both QPCs for the three pinch-off curves presented in Fig. 1. Lines are guides for the eyes. At the intersection of the two lines of QPC2 the gate voltages $V_{\text{s}}$ and $V_{\text{t}}$ would be identical for both measurements. Error bars reflect the uncertainties of the red lines in Figs. 1(c) and (d). for all three pinch-off curves. Related to the variations in geometry the three implementations of QPCs have different subband spacings. However, as a general feature we observe a strong decrease of $\delta\varepsilon(N)$ as the QPCs are opened and $N$ is increased. ## III Hard wall versus parabolic lateral confinement Given reflectionless contacts the conductance of a QPC is limited by its strongest lateral confinement in the center of the constriction. The measured subband spacings are uniquely related with this lateral confinement. In the following, we compare the two most common models describing the lateral confinement, namely a hard-wall versus a parabolic potential. These two models may be considered the extreme limits of a “continuum” of realistic scenarios for the transverse confinement. ### III.1 Lateral hard-wall potential Figure 4: Comparison between hard-wall (left column) and parabolic (right column) potential models of the lateral confinement. (a) Width of hard-wall potential $W(N)$. (b) Offset of hard-wall potential $\Phi_{0}(N)$. (c) Shape of hard-wall potential for $1\leq N\leq 9$, only for QPC1. (d) Curvature of parabolic potential $\omega_{y}(N)$. (e) Offset of parabolic potential $\Phi_{0}(N)$. (f) Shape of parabolic potential for $1\leq N\leq 9$, only for QPC1. Error bars in panels (a), (b), (d) and (e) are calculated by error propagation from the error of $\delta\varepsilon(N)$, cf. Fig. 3. For the lateral hard-wall potential we model the transverse confinement potential $\Phi(y)$ as $\Phi(y)=\begin{cases}\Phi_{0},&|y|\leq{W}/{2}\\\ \infty,&|y|>{W}/{2}\,,\end{cases}$ (2) where the two parameters $W$ and $\Phi_{0}$ are the width and offset of the hard-wall potential well. An offset can be caused by a partial depletion of the constriction related to the incomplete screening in a semi conductor with a small carrier density. The threshold energies for the transverse modes are $E_{n}=\frac{\pi^{2}\hbar^{2}n^{2}}{2m^{\star}W^{2}}+\Phi_{0}\,$ (3) where $m^{\star}=0.067m_{0}$ is the effective mass of the electrons in GaAs, $m_{0}$ being the free electron mass. Using Eq. (II) to relate the bias voltage at the intersection points marked by the red squares in Fig. 1(c) and (f) to the subband spacing $\delta\varepsilon(N)=E_{N+1}-E_{N}$, we calculate the widths $W(N)=\pi\hbar\sqrt{\frac{2N+1}{2m\delta\varepsilon(N)}}.$ (4) Neglecting additional screening effects from the applied bias voltage, these values of $W(N)$ apply everywhere along the (almost horizontal) lines connecting pairs of red squares, see the yellow lines for $N=2$ in Fig. 1(c) and (f). In particular, this allows us to extend our estimate of the width $W(N)$ to $V_{\text{QPC}}=0$, indicated for $N=2$ by the yellow dot in Fig. 1(c) and (f). Substituting $W$ in Eq. (3) with $W(N)$ we then find the potential offset $\Phi_{0}$ using the relation $E_{\text{F}}\simeq E_{N}+0.5\delta\epsilon(N)$, which gives $\Phi_{0}(N)\simeq E_{\text{F}}-\delta\varepsilon(N)\,\left(\frac{N^{2}}{2N+1}+\frac{1}{2}\right)\,.$ (5) The potential shift by $0.5\delta\epsilon(N)$ accounts for the difference between the $N$th subband bottom $E_{N}$ and the Fermi level $E_{\text{F}}$ in the center of each diamond at $V_{\text{QPC}}=0$, assuming symmetric coupling between the 1D constriction and both leads. (The assumption of symmetric coupling is confirmed by the fact that the lines connecting pairs of red squares in Fig. 1(c) and (f) are almost horizontal.) ### III.2 Lateral parabolic potential To model a lateral parabolic potential we use $\Phi(y)=\Phi_{0}+\frac{m\omega_{y}^{2}y^{2}}{2},$ (6) where $\omega_{y}$ and $\Phi_{0}$ are the characteristic frequency and offset of the parabolic potential well. In analogy to the analysis assuming hard-wall potentials we determine the two parameters from the measured subband spacings. At the intersection points indicated with red squares in Fig. 1(c) and (f) we find $\hbar\omega_{y}(N)=eV_{\text{QPC}}=\delta\varepsilon(N)$ (7) and in the centers of the diamonds at $V_{\text{QPC}}=0$ in addition $\Phi_{0}(N)\simeq E_{F}-N\hbar\omega_{y}\,.$ (8) ### III.3 Comparison of the two potential shapes In Fig. 4 we directly compare our results for the hard-wall potential shown in the left column and for assuming parabolic confinement plotted on the right hand side. We present the parameters $W$ and $\Phi_{0}$ as a function of the subband number $N$ for all three QPC implementations for the hard-wall potential in panels (a) and (b) and $\omega_{y}$ and $\Phi_{0}$ for the parabolic potential in panels (d) and (e). The results are qualitatively similar for the various implementations of QPCs; the variations in $W$ or $\omega_{y}$ between QPCs indicate that the lateral confinement potential of QPC2 is slightly wider compared to QPC1. In panels (c) and (f) showing the actual potentials for QPC1, for comparison we indicate the lithographic distance of 250 nm between the gates seen in the inset of Fig. 1(a). It corresponds to the white area between regions of gray background. The width of the hard-wall potential slightly exceeds the lithographic width for $N=9$. QPC1 does not show further plateaus for $N>9$. Comparing the two models a substantial difference is visible in $\Phi_{0}(N)$. While for $N=1$ the potential offset is similar for both models with $\Phi_{0}/E_{\text{F}}\simeq 0.6$, in case of the hard-wall potential it slowly decreases to $\Phi_{0}/E_{\text{F}}\simeq 0.4$ at $N=4$ and stays approximately constant at that level as the QPC is opened further. In contrast, the decrease of the offset $\Phi_{0}(N)$ of the parabolic potential with $N$ is much steeper, such that for $N\gtrsim 4$ it moves below the bottom of the conduction band in the 2D leads, indicated as a dashed line at $\Phi=0$. We are not aware of a realistic mechanism that could lead to such an over-screening of the negative voltages applied to the control gates ($V_{\text{g}}$ for QPC1 or $V_{\text{t}}$ for QPC2). ## IV Discussion and summary The main result of our simple analysis starting from the measured subband spacings $\delta\varepsilon(N)$ is, that for $N\geq 4$ we can exclude a parabolic lateral confinement potential for our QPCs. Based on a self- consistent calculation it has been suggested that the increasing population of the 1D constriction with $N$ as a QPC is opened up leads to an increased screening of the electric field originating from the charged control gates. For a gate defined QPC this process can cause a transition from a parabolic confinement for the case of little screening, i.e., $N=1$ towards a truncated potential with a flat bottom at larger $N$ where many carriers populate the constriction Laux et al. (1988). Our findings are in favor of such a scenario. The hard-wall potential presents a somewhat unrealistic extreme case of strong screening. Nevertheless, for $N\geq 4$ it seems more realistic than the other extreme, namely the parabolic potential. The true shape of the lateral confinement potential of a QPC for $N\geq 4$ likely lies between these two extremes, maybe close to a truncated parabola Laux et al. (1988); Wharam et al. (1989), i.e., a parabola with a flat bottom identical to that of a hard- wall potential but with smoothly increasing side walls of constant curvature as the case for a parabola. In summary, a parabolic saddle point potential is likely a realistic description of a QPC near pinch-off, although our measurement can also be explained with a hard-wall confinement in this regime. However, as the QPC is opened up beyond $N\simeq 4$, the parabolic lateral confinement turns out to be a bad approximation. In this regime of enhanced screening a hard-wall potential is the better approximation. ## V appendix ### V.1 Coupling between control gates and the QPC The electrostatic potential shaping the QPCs is generated and controlled via the field effect by applying voltages to nearby metal gates. The size of the plateaus of quantized conductance in the pinch-off curves as a function of gate voltage, cf. Fig. 1(b) and (e), is proportional to the capacitive coupling between the control gates and the QPC, which we approximate as a conducting 1D-channel with the carrier density $n_{\text{1D}}$. We determine the approximate capacitance per unit length between gate and QPC as $c_{\text{1D}}=e\delta n_{\text{1D}}/\delta V_{\text{gate}}\,,$ (9) where $\delta n_{\text{1D}}$ is the carrier density increase as the voltage on the control gate is increased by $\delta V_{\text{gate}}$. If we take for $\delta V_{\text{gate}}$ the voltage difference between two subsequent intersection points of the source- and drain-resonances at $V_{\text{QPC}}=0$ in Fig. 1(c) and (f), $\delta n_{\text{1D}}$ corresponds to the difference of the values of $n_{\text{1D}}$ at these points with $N$ versus $N+1$ subbands being populated. The 1D carrier density is $n_{\text{1D}}(N)=\int_{0}^{\infty}D_{\text{1D}}(E)f(E)\text{d}E\,,$ (10) where $D_{\text{1D}}=\frac{1}{\pi\hbar}\sqrt{\frac{2m^{\star}}{E}}$ is the 1D electron density of states and $f(E)$ the Fermi-Dirac distribution. Given $k_{\text{B}}T\ll E_{\text{F}}$ we approximate $f(E)=1$ for $E<E_{\text{F}}$ and $f(E)=0$ for $E>E_{\text{F}}$. Summing up all 1D modes which are actually populated for the QPC tuned to the conductance $G=NG_{\text{Q}}$ we find $\displaystyle n_{\text{1D}}(N)$ $\displaystyle=\frac{\sqrt{2m^{\star}}}{\pi\hbar}\sum_{n=1}^{N}\int_{E_{n}}^{E_{\text{F}}}\frac{1}{\sqrt{E-E_{n}}}dE$ $\displaystyle=\frac{\sqrt{8m^{\star}}}{\pi\hbar}\sum_{n=1}^{N}\sqrt{E_{\text{F}}-E_{n}}\,.$ (11) Inserting $\delta n_{\text{1D}}(N)=n_{\text{1D}}(N+1)-n_{\text{1D}}(N)$ from Eq. (V.1) in Eq. (9) we finally determine the 1D capacitance density as $c_{\text{1D}}(N)=\frac{\sqrt{8m^{\star}e^{2}}}{\pi\hbar}\,\frac{\sqrt{E_{\text{F}}-E_{N+1}}}{\delta V_{\text{gate}}(N)}\,,$ (12) where $\delta V_{\text{gate}}(N)$ is the width of the $Nth$ plateau of the pinch-off curve, cf. Fig. 1, measured between the conductance $(N+0.5)G_{\text{Q}}$ and $(N-0.5)G_{\text{Q}}$. Substituting $E_{N+1}$ with the according eigen-energy of the hard-wall potential using Eq. (3) we can now determine $c_{\text{1D}}(N)$. In Fig. 5 Figure 5: 1D carrier density $n_{\text{1D}}(N)$ assuming an infinitely long hard-wall 1D channel of width $W(N)$ and depth $E_{\text{F}}-\Phi_{0}(N)$ of QPC1 (red squares, rhs axis) and corresponding 1D capacitance density $c_{\text{1D}}(N)$ (blue triangles, lhs axis). we present the 1D capacitance density $c_{\text{1D}}(N)$, which is the slope of the also shown 1D carrier density $n_{\text{1D}}(N)$. The strong decrease of the capacitance with $N$ for $N\leq 4$ is a direct signature of the increase of the screening of the electric field of the gates with growing carrier density. In addition, the variations in capacitance as a function of $N$ explain the counter-intuitive result that the subband spacings $\delta\epsilon(N)$ strongly vary in a region of almost equal widths of the plateaus of quantized conductance of the pinch-off curve, cf. Figs. 1(b),(e) and Fig. 3. ### V.2 Width of the 1D constriction as a function of gate voltage In Fig. 4(a) we have presented the width of the hard-wall potential $W(N)$. In Fig. 6 Figure 6: Width of the hard-wall potential $W(V_{\text{g}})$ for QPC1 [same data as the $W(N)$ in Fig. 4(a)]. The slope of the red line is $\text{d}W/\text{d}V_{\text{g}}=300\,\text{nm}/$V, cf. main text. we plot $W(V_{\text{g}})$ for QPC1. Next, we compare this result with the dependence of the depletion region of a gate voltage using a different sample on the same wafer material. The sample shown in Fig. 7(a) Figure 7: (a) QPC nominally identical to QPC1 for $N=4$ coupled to a hemispherical mirror defined by a negative gate voltage $V_{\text{m}}$. (b) Conductance of the QPC as a function of $V_{\text{m}}$. (c) Fourier transform of the conductance. From the peak value, we determine the period $\delta V_{\text{m}}\simeq 150\,$mV of the oscillations in panel (b). contains a QPC nominally identical to QPC1 and a hemispherical mirror gate. The two samples have been prepared in parallel and on the same wafer. In Fig. 7(b) we present the conductance of the QPC as a function of the voltage applied to the mirror gate. The bare conductance (without mirror) is $G=4G_{\text{Q}}$. However, with the 2DES below the mirror gate depleted it is reduced roughly by a factor 2, because of enhanced back scattering through the QPC. At the same time $G(V_{\text{m}})$ oscillates with a visibility of 40 %. Both, the conductance reduction and oscillation are related to the formation of localized modes inside the hemispherical resonator. The oscillation can be interpreted in analogy to the oscillations of the standing wave in a Fabry- Pérot resonator, while here, the coherent electrons generate the standing wave. By increasing the gate voltage $V_{\text{m}}$ we decrease the area of 2DES depleted next to the mirror gate and thereby increase the length of the resonator (the distance between the QPC and the mirror). Per period of the conductance oscillation the length of the resonator is reduced by half of the Fermi wavelength $\text{d}L_{\text{res}}/\text{d}V_{\text{m}}=0.5\lambda_{\text{F}}/\delta V_{\text{m}}$ with the resonator length $L_{\text{res}}$. We determine the averaged period from the fast Fourier transform of the oscillation, cf. Fig. 7(c), and find $\delta V_{\text{m}}\simeq 150\,$mV. With $\lambda_{\text{F}}=45\,$nm we finally estimate the rate of the depletion length reduction as $\text{d}L_{\text{res}}/\text{d}V_{\text{m}}=150\,\text{nm}/$V. Changing the voltage applied to the QPC gates instead of the mirror gates results in the same conclusion while in this case the interference pattern appears on top of the QPC pinch-off curve. To estimate the depletion of the electron system between the QPC gates we have to add up the electric fields of both gates. Based on the fact that the same voltage is applied to both gates and the distance between gates is more than two times larger than the distance between each gate and the 2DES we neglect any influences that a gate has on the depletion caused by the other gate. From the slope of the red line in Fig. 6 we find the dependence of the width of the hard-wall potential $W(V_{\text{g}})$ as a function of gate voltage to be $\text{d}W/\text{d}V_{\text{g}}\simeq 300\,\text{nm}/$V, twice as large as the effect of a single mirror gate. This finding supports the applicability of the hard-wall model for QPCs with $N\geq 4$. ## VI Acknowledgments We thank Philipp Altpeter for technical support and are grateful for financial support from the DFG via Grant No. LU 819/11-1. M.G. acknowledges support by project A03 of the CRC-TR 183. 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arxiv-papers

# Generating Followup Questions for Interpretable Multi-hop Question Answering Christopher Malon NEC Laboratories America Princeton, NJ 08540 <EMAIL_ADDRESS> Bing Bai NEC Laboratories America Princeton, NJ 08540 <EMAIL_ADDRESS> ###### Abstract We propose a framework for answering open domain multi-hop questions in which partial information is read and used to generate followup questions, to finally be answered by a pretrained single-hop answer extractor. This framework makes each hop interpretable, and makes the retrieval associated with later hops as flexible and specific as for the first hop. As a first instantiation of this framework, we train a pointer-generator network to predict followup questions based on the question and partial information. This provides a novel application of a neural question generation network, which is applied to give weak ground truth single-hop followup questions based on the final answers and their supporting facts. Learning to generate followup questions that select the relevant answer spans against downstream supporting facts, while avoiding distracting premises, poses an exciting semantic challenge for text generation. We present an evaluation using the two-hop bridge questions of HotpotQA. ## 1 Introduction Multi-hop question answering tests the ability of a system to retrieve and combine multiple facts to answer a single question. HotpotQA Yang et al. (2018) introduces a task where questions are free-form text, supporting facts come from Wikipedia, and answer text and supporting facts are labeled. The questions in HotpotQA are further categorized as bridge-type questions or comparison-type questions. For comparison questions, often all necessary facts may be retrieved using terms in the question itself. For challenging bridge- type questions, it may not be possible to retrieve all the necessary facts based on the terms present in the original question alone. Rather, partial information must first be retrieved and used to formulate an additional query. Although many systems have been submitted to the HotpotQA leaderboard, surprisingly, only a few have directly addressed the challenge of followups. Systems can either be evaluated in a distractor setting, where a set of ten paragraphs containing all supporting facts is provided, or in a full wiki setting, where supporting facts must be retrieved from all of Wikipedia. The systems that compete only in the distractor setting can achieve good performance by combining and ranking the information provided, without performing followup search queries. Furthermore, even in the distractor setting, Min et al. (2019a) found that only 27% of the questions required multi-hop reasoning, because additional evidence was redundant or unnecessary or the distractors were weak. They trained a single-hop model that considered each paragraph in isolation and ranked confidences of the answers extracted from each, to obtain competitive performance. Of the nine systems with documentation submitted to the full wiki HotpotQA leaderboard as of 24 November 2019, four of them (Nie et al., 2019; Ye et al., 2019; Nishida et al., 2019; Yang et al., 2018) attempt to retrieve all relevant data with one search based on the original question, without any followups. Fang et al. (2019) retrieves second hop paragraphs simply by following hyperlinks from or to the first hop paragraphs. Qi et al. (2019), Ding et al. (2019), and Feldman and El-Yaniv (2019) form various kinds of followup queries without writing a new question to be answered. Qi et al. (2019) trains a span extractor to predict the longest common subsequence between the question plus the first hop evidence and the (unseen) second hop evidence. At inference time, these predicted spans become followup search queries. In Ding et al. (2019), a span extractor is trained using the titles of the second hop evidence. Feldman and El-Yaniv (2019) trains a neural retrieval model that uses maximum inner product with an encoding of the question plus first hop evidence to retrieve second hop evidence. Min et al. (2019b) forms not just followup queries but followup questions. They use additional specially labeled data to train a pointer network to divide the original question into substrings, and use handcrafted rules to convert these substrings into subquestions. The original question is answered by the second subquestion, which incorporates a substitution of the answer to the first subquestion. While performing followup retrievals of some sort should be essential for correctly solving the most difficult multi-hop problems, formulating a followup question whose answer becomes the answer to the original question is motivated primarily by interpretability rather than accuracy. In this paper, we pursue a trained approach to generating followup questions that is not bound by handcrafted rules, posing a new and challenging application for abstractive summarization and neural question generation technologies. Our contributions are to define the task of a followup generator module (Section 2), to propose a fully trained solution to followup generation (Section 3), and to establish an objective evaluation of followup generators (Section 5). ## 2 Problem Setting Our technique is specifically designed to address the challenge of discovering new information is needed that is not specified by the terms of the original question. At the highest level, comparison questions do not pose this challenge, because each quantity to be compared is specified by part of the original question. (They also pose different semantics than bridge questions because a comparison must be applied after retrieving answers to the subquestions.) Therefore we focus only on bridge questions in this paper. Figure 1: The architecture of our system to generate intermediate questions for answer extraction. Figure 1 shows our pipeline to answer a multi-hop bridge question. As partial information is obtained, an original question is iteratively reduced to simpler questions generated at each hop. Given an input question or subquestion, possible premises which may answer the subquestion are obtained from an information retrieval module. Each possible premise is classified against the question as irrelevant, containing a final answer, or containing intermediate information, by a three-way controller module. For premises that contain a final answer, the answer is extracted with a single-hop question answering module. For premises that contain intermediate information, a question generator produces a followup question, and the process may be repeated with respect to this new question. It is this question generator that is the focus of this paper. Various strategies may be used to manage the multiple reasoning paths that may be produced by the controller. Details are in section 5. Although our method applies to bridge questions with arbitrary number of hops, for simplicity we focus on two-hop problems and on training the followup question generator. Let $Cont$ denote the controller, $SingleHop$ denote the answer extractor, and $Followup$ denote the followup generator. Let $Q_{1}$ be a question with answer $A$ and gold supporting premises $\hat{P_{1}}$ and $\hat{P_{2}}$, and suppose that $\hat{P_{2}}$ but not $\hat{P_{1}}$ contains the answer. The task of the followup generator is to use $Q_{1}$ and $\hat{P_{1}}$ to generate a followup question $Q_{2}$ such that $\displaystyle SingleHop(Q_{2},\hat{P_{2}})$ $\displaystyle=$ $\displaystyle A$ (1) $\displaystyle Cont(Q_{2},\hat{P_{2}})$ $\displaystyle=$ $\displaystyle Final$ (2) $\displaystyle\mbox{and}\;Cont(Q_{2},P)$ $\displaystyle=$ $\displaystyle Irrel\,\mbox{for}\,P\neq\hat{P_{2}}\ldotp$ (3) Failure of any of these desiderata could harm label accuracy in the HotpotQA full wiki or distractor evaluations. Some questions labeled as bridge type in HotpotQA have a different logical structure, called “intersection” by Min et al. (2019b). Here the subquestions specify different properties that the answer entity is supposed to satisfy, and the intersection of possible answers to the subquestions is the answer to the original question. Our approach is not oriented towards this type of question, but there is no trivial way to exclude them from the dataset. One non-interpretable implementation of our pipeline would be for $Followup$ to simply output $Q_{1}$ concatenated with $P_{1}$ as the “followup question.” Then $SingleHop$ would operate on input that really does not take the form of a single question, along with $P_{2}$, to determine the final answer. Effectively, $SingleHop$ would be doing multi-hop reasoning. To ensure that $Followup$ gets credit only for forming real followup questions, we insist that $SingleHop$ is first trained as a single-hop answer extractor, by training it on SQuAD 2.0 (Rajpurkar et al., 2018), then freeze it while $Followup$ and $Cont$ are trained. ## 3 Method Ideally, we might train $Followup$ using cross entropy losses inspired by equations 1, 2, and 3 with $SingleHop$ and $Cont$ fixed, but the decoded output $Q_{2}$ is not differentiable with respect to $Followup$ parameters. Instead, we train $Followup$ with a token-based loss against a set of weakly labeled ground truth followup questions. The weakly labeled ground truth followups are obtained using a neural question generation (QG) network. Given a context $\overline{C}$ and an answer $\overline{A}$, QG is the task of finding a question $\overline{Q}=\mbox{argmax}_{Q}Prob(Q|\overline{C},\overline{A})$ (4) most likely to have produced it. We use reverse SQuAD to train the QG model of Zhao et al. (2018), which performs near the top of an extensive set of models tested by Tuan et al. (2019) and has an independent implementation available. Applied to our training set with $\overline{C}=\hat{P_{2}}$ and $\overline{A}=A$, it gives us a weak ground truth followup $\overline{Q_{2}}$. We instantiate the followup question generator, which uses $Q_{1}$ and $P_{1}$ to predict $Q_{2}$, with a pointer-generator network (See et al., 2017). This is a sequence to sequence model whose decoder repeatedly chooses between generating a word from a fixed vocabulary and copying a word from the input. Typically, pointer-generator networks are used for abstractive summarization. Although the output serves a different role here, their copy mechanism is useful in constructing a followup that uses information from the original question and premise. We train $Cont$ with cross-entropy loss for ternary classification on the ground truth triples $(Q_{1},\hat{P_{1}},Intermediate)$, $(Q_{1},\hat{P_{2}},Final)$ if $SingleHop(Q_{1},\hat{P_{2}})\cap A\neq\emptyset$, and $(Q_{1},P,Irrel)$ for all other $P$. In this way the controller learns to predict when a premise has sufficient or necessary information to answer a question. Both $Cont$ and $SingleHop$ are implemented by BERT following the code by Devlin et al. (2019). ## 4 Related Work Evaluating a followup question generator by whether its questions are answered correctly is analogous to verifying the factual accuracy of abstractive summarizations, which has been studied by many, including Falke et al. (2019), who estimate factual correctness using a natural language inference model, and find that it does not correlate with ROUGE score. Contemporaneous work by Zhang et al. (2019) uses feedback from a fact extractor in reinforcement learning to optimize the correctness of a summary, suggesting an interesting future direction for our work. A recent neural question generation model has incorporated feedback from an answer extractor into the training of a question generator, rewarding the generator for constructing questions the extractor can answer correctly (Klein and Nabi, 2019). Although the loss is not backpropagated through both the generator and extractor, the generator is penalized by token level loss against ground truth questions when the question is answered wrongly, but by zero loss when it constructs a variant that the extractor answers correctly. ## 5 Experiments To isolate the effect of our followup generator on the types of questions for which it was intended, our experiments cover the subset of questions in HotpotQA labeled with exactly two supporting facts, with the answer string occurring in exactly one of them. There are 38,806 such questions for training and 3,214 for development, which we use for testing because the structure of the official test set is not available. For a baseline we compare to a trivial followup generator that returns the original question $Q_{1}$ without any rewriting. | EM | F1 ---|---|--- Oracle setting | | Trained $Q_{2}$ | 14.7 | 19.0 $Q_{2}=Q_{1}$ | 27.6 | 34.9 $Q_{1}$ else $Q_{2}$ | 34.7 | 43.8 Full system | | One hop ($Q_{1}$ only) | 16.8 | 21.5 Two hops (trained $Q_{2}$) | 19.8 | 25.4 Table 1: Answer accuracy on filtered subset of HotpotQA development set in the distractor setting. $Q_{1}$ | Truth | $Q_{1}$ Answer | $Q_{2}$ | $Q_{2}$ Answer ---|---|---|---|--- Randall Cunningham II was a multi-sport athlete at the high school located in what Nevada city? | Summerlin | — | where is bishop gorman high school located? | Summerlin, Nevada Alexander Kerensky was defeated and destroyed by the Bolsheviks in the course of a civil war that ended when? | October 1922 | — | what was the name of the russian civil war? | The Russian Civil War Peter Marc Jacobson is best known as the co-creator of the popular sitcom ”The Nanny”, which he created and wrote with his then wife an actress born in which year ? | 1957 | 1993 | what year was fran drescher born in? | 1957 Who did the Star and Dagger bass player marry? | Sean Yseult. | Sean Yseult | what was the name of the american rock musician? | Chris Lee Table 2: Example generated followup questions $Q_{2}$, evaluated against oracle $\hat{P_{2}}$. First, we evaluate performance using an oracle controller, which forwards only $(Q_{1},\hat{P_{1}})$ to the followup generator, and only $(Q_{2},\hat{P_{2}})$ to the answer extractor. Results are shown in Table 1. Best performance is achieved using the system “$Q_{1}$ else $Q_{2}$,” which answers with $SingleHop(Q_{1},\hat{P_{2}})$ or $SingleHop(Q_{2},\hat{P_{2}})$, whichever is non-null. Thus, although many questions are really single-hop and best answered using the original question, using the followup questions when a single-hop answer cannot be found helps the F1 score by 8.9%. Table 2 shows followup generations and extracted answers in two typical successful and two typical failed cases. Next we consider the full system of Figure 1. We use the distractor paragraphs provided. We run the loop for up to two hops, collecting all answer extractions requested by the controller, stopping after the first hop where a non-null extracted answer was obtained. If multiple extractions were requested for the same problem, we take the answer in where $SingleHop$ had the highest confidence. The controller requested 2,989 followups, and sent 975 $(Q,P)$ pairs for answer extraction in hop one, and 1,180 in hop two. The performance gain shows that the followup generator often can generate questions which are good enough for the frozen single hop model to understand and extract the answer with, even when the question must be specific enough to avoid distracting premises. ## 6 Conclusion Followup queries are essential to solving the difficult cases of multi-hop QA, and real followup questions are an advance in making this process interpretable. We have shown that pointer generator networks can effectively learn to read partial information and produce a fluent, relevant question about what is not known, which is a complement to their typical role in summarizing what is known. 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2002.12393

arxiv-papers

# Cleo: Learned Cost Models for Query Optimization in Shared Clouds Tarique Siddiqui1,2, Alekh Jindal1, Shi Qiao1, Hiren Patel1, Wangchao Le1 1Microsoft 2University of Illinois, Urbana-Champaign (2020) # Learned Cost Models for Cloud Data Services Tarique Siddiqui1,2, Alekh Jindal1, Shi Qiao1, Hiren Patel1, Wangchao Le1 1Microsoft 2University of Illinois, Urbana-Champaign (2020) # Learned Cost Models for Query Optimization in the Cloud Tarique Siddiqui1,2, Alekh Jindal1, Shi Qiao1, Hiren Patel1, Wangchao Le1 1Microsoft 2University of Illinois, Urbana-Champaign (2020) # Cleo: Accurate Cost Models for Query Optimization Tarique Siddiqui1,2, Alekh Jindal1, Shi Qiao1, Hiren Patel1, Wangchao Le1 1Microsoft 2University of Illinois, Urbana-Champaign (2020) # Learned Cost Models for Optimizing Big Data Queries Tarique Siddiqui1,2, Alekh Jindal1, Shi Qiao1, Hiren Patel1, Wangchao Le1 1Microsoft 2University of Illinois, Urbana-Champaign (2020) # Cost Models for Serverless Query Processing: Learning, Retrofitting, and Our Findings Tarique Siddiqui1,2, Alekh Jindal1, Shi Qiao1, Hiren Patel1, Wangchao Le1 1Microsoft 2University of Illinois, Urbana-Champaign (2020) # CLEO: Learned Cost Models for Query Optimization in Big Data Systems Tarique Siddiqui1,2, Alekh Jindal1, Shi Qiao1, Hiren Patel1, Wangchao Le1 1Microsoft 2University of Illinois, Urbana-Champaign (2020) # CLEO: Learned Cost Models for Cloud-based Big Data Systems Tarique Siddiqui1,2, Alekh Jindal1, Shi Qiao1, Hiren Patel1, Wangchao Le1 1Microsoft 2University of Illinois, Urbana-Champaign (2020) # Cost Models for Big Data Query Processing: Learning, Retrofitting, and Our Findings Tarique Siddiqui1,2, Alekh Jindal1, Shi Qiao1, Hiren Patel1, Wangchao Le1 1Microsoft 2University of Illinois, Urbana-Champaign (2020) ###### Abstract. Query processing over big data is ubiquitous in modern clouds, where the system takes care of picking both the physical query execution plans _and_ the resources needed to run those plans, using a cost-based query optimizer. A good cost model, therefore, is akin to better resource efficiency and lower operational costs. Unfortunately, the production workloads at Microsoft show that costs are very complex to model for big data systems. In this work, we investigate two key questions: (i) can we learn accurate cost models for big data systems, and (ii) can we integrate the learned models within the query optimizer. To answer these, we make three core contributions. First, we exploit workload patterns to learn a large number of individual cost models and combine them to achieve high accuracy and coverage over a long period. Second, we propose extensions to Cascades framework to pick optimal resources, i.e, number of containers, during query planning. And third, we integrate the learned cost models within the Cascade-style query optimizer of SCOPE at Microsoft. We evaluate the resulting system, Cleo, in a production environment using both production and TPC-H workloads. Our results show that the learned cost models are $2$ to $3$ orders of magnitude more accurate, and $20\times$ more correlated with the actual runtimes, with a large majority ($70\%$) of the plan changes leading to substantial improvements in latency as well as resource usage. ††journalyear: 2020††copyright: acmcopyright††conference: Proceedings of the 2020 ACM SIGMOD International Conference on Management of Data; June 14–19, 2020; Portland, OR, USA††booktitle: Proceedings of the 2020 ACM SIGMOD International Conference on Management of Data (SIGMOD’20), June 14–19, 2020, Portland, OR, USA††price: 15.00††doi: 10.1145/3318464.3380584††isbn: 978-1-4503-6735-6/20/06 ## 1\. Introduction Figure 1. Impact of manual tuning and cardinality feedback on cost models in SCOPE There is a renewed interest in cost-based query optimization in big data systems, particularly in modern cloud data services (e.g., Athena (aws-athena, ), ADLA (adla, ), BigSQL (ibm-bigsql, ), and BigQuery (google-bigquery, )) that are responsible for picking both the query execution plans and the resources (e.g., number of containers) needed to run those plans. Accurate cost models are therefore crucial for generating efficient combination of plan and resources. Yet, the traditional wisdom from relational databases is that cost models are less important and fixing cardinalities automatically fixes the cost estimation (leis2015good, ; lohman2014query, ). The question is whether this also holds for the new breed of big data systems. To dig deeper, we analyzed one day’s worth of query logs from the big data infrastructure (SCOPE (scopeVLDB2008, ; scopeVLDBJ12, )) at Microsoft. We feed back the actual runtime cardinalities, i.e., the ideal estimates that any cardinality estimator, including learned models (stillger2001leo, ; cardLearner, ; lightweightCardModels, ; corrJoinsCardModels, ) can achieve. Figure 1 compares the ratio of cost estimates with the actual runtimes for two cost models in SCOPE: 1) a default cost model, and 2) a manually-tuned cost model that is partially available for limited workloads. The vertical dashed-line at $10^{0}$ corresponds to an ideal situation where all cost estimates are equal to the actual runtimes. Thus, the closer a curve is to the dashed line, the more accurate it is. The dotted lines in Figure 1(b) show that fixing cardinalities reduces the over-estimation, but there is still a wide gap between the estimated and the actual costs, with the Pearson correlation being as low as $0.09$. This is due to the complexity of big data systems coupled with the variance in cloud environments (cloudVariance, ), which makes cost modeling incredibly difficult. Furthermore, any improvements in cost modeling need to be consistent across workloads and over time since performance spikes are detrimental to the reliability expectations of enterprise customers. Thus, accurate cost modeling is still a challenge in SCOPE like big data systems. In this paper, we explore the following two questions: * (1) Can we learn accurate, yet robust cost models for big data systems? This is motivated by the presence of massive workloads visible in modern cloud services that can be harnessed to accurately model the runtime behavior of queries. This helps not only in dealing with the various complexities in the cloud, but also specializing or instance optimizing (marcus2019neo, ) to specific customers or workloads, which is often highly desirable. Additionally, in contrast to years of experience needed to tune traditional optimizers, learned cost models are potentially easy to update at a regular frequency. * (2) Can we effectively integrate learned cost models within the query optimizer? This stems from the observation that while some prior works have considered learning models for predicting query execution times for a given physical plan in traditional databases (ganapathi2009predicting, ; bach2002kernel, ; akdere2012learning, ; li2012robust, ), none of them have integrated learned models within a query optimizer for selecting physical plans. Moreover, in big data systems, resources (in particular the number of machines) play a significant role in cost estimation (raqo, ), making the integration even more challenging. Thus, we investigate the effects of learned cost models on query plans by _extending_ the SCOPE query optimizer in a _minimally invasive_ way for _predicting costs_ in a _resource-aware manner_. To the best of our knowledge, this is the first work to integrate learned cost models within an industry-strength query optimizer. Our key ideas are as follows. We note that the cloud workloads are quite diverse in nature, i.e., there is no representative workload to tune the query optimizer, and hence there is no single cost model that fits the entire workload, i.e., no-one-size-fits-all. Therefore, we learn a large collection of smaller-sized cost models, one for each common subexpressions that are typically abundant in production query workloads (cloudviews, ; bigsubs, ). While this approach results in specialized cost models that are very accurate, the models do not cover the entire workload: expressions that are not common across queries do not have models. The other extreme is to learn a cost model per operator, which covers the entire workload but sacrifices the accuracy with very general models. Thus, _there is an accuracy-coverage trade-off that makes cost modeling challenging_. To address this, we define the notion of cost model _robustness_ with three desired properties: (i) high accuracy, (ii) high coverage, and (iii) high retention, i.e., stable performance for a long- time period before retraining. We achieve these properties in two steps: First, we bridge the accuracy-coverage gap by learning additional mutually enhancing models that improve the coverage as well as the accuracy. Then, we learn a combined model that automatically corrects and combines the predictions from multiple individual models, providing accurate and stable predictions for a sufficiently long window (e.g., more than 10 days). We implemented our ideas in a Cloud LEarning Optimizer (Cleo) and integrated it within SCOPE. Cleo uses a feedback loop to periodically train and update the learned cost models within the Cascades-style top-down query planning (cascades95, ) in SCOPE. We extend the optimizer to invoke the learned models, instead of the default cost models, to estimate the cost of candidate operators. However, in big data systems, the cost depends heavily on the resources used (e.g., number of machines for each operator) by the optimizer (raqo, ). Therefore, we extend the Cascades framework to explore resources, and propose mechanisms to explore and derive optimal number of machines for each stage in a query plan. Moreover, instead of using handcrafted heuristics or assuming fixed resources, we leverage the learned cost models to find optimal resources as part of query planning, thereby _using learned models for producing both runtime as well as resource-optimal plans_. In summary, our key contributions are as follows. * (1) We motivate the cost estimation problem from production workloads at Microsoft, including prior attempts for manually improving the cost model (Section 2). * (2) We propose machine learning techniques to learn highly accurate cost models. Instead of building a generic cost model for the entire workload, we learn a large collection of smaller specialized models that are resource-aware and highly accurate in predicting the runtime costs (Section 3). * (3) We describe the accuracy and coverage trade-off in learned cost models, show the two extremes, and propose additional models to bridge the gap. We combine the predictions from individual models into a robust model that provides the best of both accuracy and coverage over a long period (Section 4). * (4) We describe integrating Cleo within SCOPE, including periodic training, feedback loop, model invocations during optimization, and novel extensions for finding the optimal resources for a query plan (Section 5). * (5) Finally, we present a detailed evaluation of Cleo, using both the production workloads and the TPC-H benchmark. Our results show that Cleo improves the correlation between predicted cost and actual runtimes from $0.1$ to $0.75$, the accuracy by $2$ to $3$ orders of magnitude, and the performance for 70% of the changed plans (Section 6). In Section 6.7, we further describe practical techniques to address performance regressions in our production settings. ## 2\. Motivation In this section, we give an overview of SCOPE, its workload and query optimizer, and motivate the cost modeling problem from production workloads at Microsoft. ### 2.1. Overview of SCOPE SCOPE (scopeVLDB2008, ; scopeVLDBJ12, ) is the big data system used for internal data analytics across the whole of Microsoft to analyze and improve its various products. It runs on a hyper scale infrastructure consisting of hundreds of thousands of machines, running a massive workload of hundreds of thousands of jobs per day that process exabytes of data. SCOPE exposes a job service interface where users submit their analytical queries and the system takes care of automatically provisioning resources and running queries in a distributed environment. SCOPE query processor partitions data into smaller subsets and processes them in parallel. The number of machines running in parallel (i.e., degree of parallelism) depends on the number of partitions of the input. When no specific partitioning is required by upstream operators, certain physical operators (e.g., Extract and Exchange (also called Shuffle)), decide partition counts based on data statistics and heuristics. The sequence of intermediate operators that operate over the same set of input partitions are grouped into a stage — all operators in a stage run on the same set of machines. Except for selected scenarios, Exchange operator is commonly used to re-partition data between two stages. ### 2.2. Recurring Workloads SCOPE workloads primarily consist of recurring jobs. A recurring job in SCOPE is used to provide periodic (e.g., hourly, six-hourly, daily, etc.) analytical result for a specific application functionality. Typically, a recurring job consists of a script template that accepts different input parameters similar to SQL modules. Each instance of the recurring job runs on different input data, parameters and have potentially different statements. As a result, each instance is different in terms of input/output sizes, query execution plan, total compute hour, end-to-end latency, etc. Figure 2 shows $150$ instances of an hourly recurring job that extracts facts from a production clickstream. Over these $150$ instances, we can see a big change in the total input size and the total execution time, from $69,859$ GiB to $118,625$ GiB and from $40$ mins and $50$ seconds to $2$ hours and $21$ minutes respectively. Note that a smaller portion of SCOPE workload is ad-hoc as well. Figure 3 shows our analysis from four of the production clusters. We can see that $7\%-20\%$ jobs are ad-hoc on a daily basis, with the fraction varying over different clusters and different days. However, compared to ad-hoc jobs, recurring jobs represent long term business logic with critical value, and hence the focus of several prior research works (bruno2013continuous, ; recurrJobOpt, ; recurrJobOptScope, ; redoop, ; rope, ; jockey, ; jyothi2016morpheus, ; cloudviews, ; bigsubs, ; cardLearner, ) and also the primary focus for performance improvement in this paper. Figure 2. $150$ instances of an hourly recurring job that extracts facts from a production clickstream. Figure 3. Illustrating ad-hoc jobs in SCOPE. ### 2.3. Overview of SCOPE Optimizer SCOPE uses a cost-based optimizer based on the Cascades Framework (cascades95, ) for generating the execution plan for a given query. Cascades (cascades95, ) transforms a logical plan using multiple tasks: (i) Optimize Groups, (ii) Optimize Expressions, (iii) Explore Groups, and (iv) Explore Expressions, and (v) Optimize Inputs. While the first four tasks search for candidate plans via transformation rules, our focus in this work is essentially on the Optimize Inputs tasks, where the cost of a physical operator in estimated. The cost of an operator is modeled to capture its runtime latency, estimated using a combination of data statistics and hand-crafted heuristics developed over many years. Cascades performs optimization in a top-down fashion, where physical operators higher in the plan are identified first. The exclusive (or local) costs of physical operators are computed and combined with costs of children operators to estimate the total cost. In some cases, operators can have a _required property_ (e.g., sorting, grouping) from its parent that it must satisfy, as well as can have a derived property from its children operators. In this work, we optimize _how partition counts are derived_ as it is a key factor in cost estimation for massively parallel data systems. Overall, our goal is to improve the cost estimates with minimal changes to the Cascades framework. We next analyze the accuracy of current cost models in SCOPE. ### 2.4. Cost Model Accuracy The solid red line in Figure 1 shows that the cost estimates from the default cost model range between an under-estimate of $100\times$ to an over-estimate of $1000\times$, with a Pearson correlation of just $0.04$. As mentioned in the introduction, this is because of the difficulty in modeling the highly complex big data systems. Current cost models rely on hand-crafted heuristics that combine statistics (e.g., cardinality, average row length) in complex ways to estimate each operator’s execution time. These estimates are usually way off and get worse with constantly changing workloads and systems in cloud environments. Big data systems, like SCOPE, further suffer from the widespread use of custom user code that ends up as black boxes in the cost models. One could consider improving a cost model by considering newer hardware and software configurations, such as machine SKUs, operator implementations, or workload characteristics. SCOPE team did attempt this path and put in significant efforts to improve their default cost model. This alternate cost model is available for SCOPE queries under a flag. We turned this flag on and compared the costs from the improved model with the default one. Figure 1b shows the alternate model in solid blue line. We see that the correlation improves from $0.04$ to $0.10$ and the ratio curve for the manually improved cost model shifts a bit up, i.e., it reduces the over-estimation. However, it still suffers from the wide gap between the estimated and actual costs, again indicating that cost modeling is non-trivial in these environments. Finally, as discussed in the introduction and shown as dotted lines in Figure 1(b), fixing cardinalities to the perfect values, i.e., that best that any cardinality estimator (stillger2001leo, ; cardLearner, ; lightweightCardModels, ; corrJoinsCardModels, ) could achieve, does not fill the gap between the estimated and the actual costs in SCOPE-like systems. ## 3\. Learned Cost Models In this section, we describe how we can leverage the common sub-expressions abundant in big data systems to learn a large set of smaller-sized but highly accurate cost models. We note that it is practically infeasible to learn a single global model that is equally effective for all operators. This is why even traditional query optimizers model each operator separately.A single model is prone to errors because operators can have very different performance behavior (e.g., hash group by versus merge join), and even the performance of same operator can vary drastically depending on interactions with underneath operators via pipelining, sharing of sorting and grouping properties, as well as the underlying software or hardware platform (or the cloud instance). In addition, because of the complexity, learning a single model requires a large number of features, that can be prohibitively expensive to extract and combine for every candidate operator during query optimization. ### 3.1. Specialized Cost Models As described in Section 2.2, shared cloud environments often have a large portion of recurring analytical queries (or jobs), i.e., the same business logic is applied to newer instances of the datasets that arrive at regular intervals (e.g., daily or weekly). Due to shared inputs, such recurring jobs often end up having one or more common subexpressions across them. For instance, the SCOPE query processing system at Microsoft has more than $50\%$ of jobs as recurring, with a large fraction of them appearing daily (jyothi2016morpheus, ), and as high as 60% having common subexpressions between them (cloudviews, ; bigsubs, ; cardLearner, ). Common subexpression patterns have also been reported in other production workloads, including Spark SQL queries in Microsoft’s HDInsight (sparkcruise, ), SQL queries from risk control department at Ant Financial Services Group (equitas, ), and iterative machine learning workflows (helix, ). Figure 4. Illustrating common subexpressions. Figure 4 illustrates a common subexpression, consisting of a scan followed by a filter, between two queries. We exploit these common subexpressions by learning a large number of specialized models, one for each unique operator- sub- graph template representing the subexpression. An operator-subgraph template covers the root physical operator (e.g., Filter) and all prior (descendants) operators (e.g., scan) from the root operator of the subexpression. However, parameters and inputs in operator-subgraphs _can vary over time_ , and are used as features for the model (along with other logical and physical features) as discussed in Section 3.3. The operator-subgraph templates essentially _capture the context of root operator, i.e, learn the behavior of root physical operator conditioned on the operators beneath it in the query plan_. This is helpful because of two reasons. First, the execution time of an operator depends on whether it is running in a pipelined manner, or is blocked until the completion of underneath operators. For example, the latency of a hash operator running on top of a filter operator is typically smaller compared to when running over a sort operator. Similarly, the grouping and sorting properties of operators beneath the root operator can influence the latency of root operator (cascades95, ). Second, the estimation errors (e.g., of cardinality) grow quickly as we move up the query plan, with each intermediate operator building upon the errors of children operators. The operator-subgraph models mitigates this issue partially since the intermediate operators are fixed and the cost of root operator depends only on the leaf level inputs. Moreover, when the estimation errors are systematically off by certain factors (e.g., 10x), the subgraph models can adjust the weights such that the predictions are close to actual (discussed subsequently in Section 3.4). This is similar to adjustments learned explicitly in prior cardinality estimation work (stillger2001leo, ). These adjustments generalize well since recurring jobs share similar schemas and the data distributions remain relatively stable, even as the input sizes change over time. Accurate cardinality estimations are, however, still needed in cases where simple adjustment factors do not exist (cardLearner, ). Next, we discuss the learning settings, feature selection, and our choice of learning algorithm for operator-subgraphs. In Section 5, we describe the training and integration of learned models with the query optimizer. ### 3.2. Learning Settings Target variable. Given an operator-subgraph template, we learn the _exclusive cost_ of the root operator as our target. At every intermediate operator, we predict the exclusive cost of the operator conditioned on the subgraph below it. The exclusive cost is then combined with the costs of the children subgraphs to compute the total cost of the sub-graph, similar to how default cost models combine the costs. Loss Function | Median Error ---|--- Median Absolute Error | 246% Mean Absolute Error | 62% Mean Squared Error | 36% Mean Squared-Log Error | 14% Table 1. Median error using 5-fold CV over the production workload for regression loss functions Loss function. As the loss function, we use mean-squared log error between the predicted exclusive cost ($p$) and actual exclusive latency ($a$) of the operator: $\frac{\sum_{n}(log(p+1)-log(a+1))^{2}}{n}$, here $1$ is added for mathematical convenience. Table 1 compares the average median errors using 5-fold cross validation (CV) of mean-squared log error with other commonly used regression loss functions, using elastic net as the learning model (described subsequently Section 3.4). We note that not taking the log transformation makes learning more sensitive to extremely large differences between actual and predicted costs. However, large differences often occur when the job’s running time itself is long or even due to outlier samples because of machine or network failures (typical in big data systems). We, therefore, minimize the relative error (since $log(p+1)-log(a+1)=log(\frac{p+1}{a+1})$), that reduces the penalty for large differences. Moreover, our chosen loss function helps penalize under- estimation more than over-estimation, since under estimation can lead to under allocation of resources which is typically decided based on cost estimates. Finally, log transformation implicitly ensures that the predicted costs are always positive. ### 3.3. Feature Selection It is expensive to extract and combine a large number of features every time we predict the cost of an operator — a typical query plan in big data systems can involve 10s of physical operators, each of which can have multiple possible candidates. Moreover, large feature sets require more number of training samples, while many operator-subgraph instances typically have much fewer samples. Thus, we perform an offline analysis to identify a small set of useful features. For selecting features, we start with a set of basic statistics that are frequently used for estimating costs of operators in the default cost model. These include the cardinality, the average row length, and the number of partitions. We consider three kinds of cardinalities: 1) base cardinality: the total input cardinality of the leaf operators in the subgraph, 2) input cardinality: the total input cardinality from the children operators, and 3) output cardinality: the output cardinality of the operator-subgraph. We also consider normalized inputs (ignoring dates and numbers) and parameters to the job that typically vary over time for the recurring jobs. We ignore features such as job name or cluster name, since they could induce strong bias and make the model brittle to the smallest change in names. We further combine basic features to create additional derived features to capture the behavior of operator implementations and other heuristics used in default cost models. We start a large space of possible derivations by applying (i) logarithms, square root, squares, and cubes of basic features, (ii) pairwise products among basic features and derived features listed in (i), and (iii) cardinality features divided by the number of partitions (i.e. machine). Given this set of candidate features, we use a variant of elastic net (zou2005regularization, ) model to select a subset of useful features that have at least one non-zero weight over all subgraph models. Feature | Description ---|--- Input Cardinality (I) | Total Input Cardinality from children operators Base Cardinality (B) | Total Input Cardinality at the leaf operators Output Cardinality (C) | Output Cardinality from the current operator AverageRowLength (L) | Length (in bytes) of each tuple Number of Partitions (P) | Number of partitions allocated to the operator Input (IN) | Normalized Inputs (ignored dates, numbers) Parameters (PM) | Parameters Table 2. Basic Features Category | Features ---|--- Input or Output data | $\sqrt{I}$, $\sqrt{B}$, L*I, L*B, L*log(B), L*log(I), L*log(C) Input $\times$ Output | B*C,I*C,B*log(C),I*log(C),log(I)*log(C),I*C, log(B)*log(C) Input or Output per partition | I/P, C/P, I*L/P, C*L/P, $\sqrt{I}$/P, $\sqrt{C}$/P,log(I)/P Table 3. Derived Features Table 2 and Table 3 depict the selected basic and derived features with non- zero weights. We group the derived features into three categories (i) input or output data, capturing the amount of data read, or written, (ii) the product of input and output, covering the data processing and network communication aspects, and finally (iii) per-machine input or output, capturing the partition size. Further, we analyze the influence of each feature. While the influence of each feature varies over different subgraph models, Figure 5 shows the aggregated influence over all subgraph models of each feature. Given a total of $K$ non zero features and $N$ subgraph models, with $w_{in}$ as the weight of feature $i$ in model $n$, we measure the influence of feature $i$ using normalized weight $nw_{i}$, as $nw_{i}=\frac{\sum_{N}abs(w_{in})}{\sum_{K}\sum_{N}abs(w_{kn})}$. Figure 5. Features weights (Op-Subgraph model) ### 3.4. Choice of learning model For learning costs over operator-subgraphs, we considered a number of variants of linear-regression, SVM, decision tree and their ensembles, as well as neural network models. On 5-fold cross-validation over our production workload, the following models give more accurate results compared to the default cost model: (i) Neural network. 3-layers, hidden layer size = 30, solver = adam, activation = relu, l2 regularization = 0.005, (ii) Decision tree: depth =15, (ii) Random forest number of trees = 20, depth = 5, (iii) FastTree Regression (a variant of Gradient Boosting Tree): number of trees = 20, depth = 5, and (iv) Elastic net: $\alpha$ =1.0, fit intercept=True, l1 ratio=0.5. We observe that the strict structure of subgraph template helps reduce the complexity, making the simpler models, e.g., linear- and decision tree-based regression models, perform reasonably well with the chosen set of features. A large number of operator-subgraph templates have fewer training samples, e.g., more than half of the subgraph instances have $<30$ training samples for the workload described in Section 2. In addition, because of the variance in cloud environments (e.g., workload fluctuations, machine failures, etc.), training samples can have noise both in their features (e.g., inaccurate statistics estimates) and the class labels (i.e., execution times of past queries). Together, both these factors lead to over-fitting, making complex models such as neural network as well as ensemble-based models such as gradient-boost perform worse. Elastic net (zou2005regularization, ), a $L_{1}$ and $L_{2}$ regularized linear regression model, on the other hand, is relatively less prone to overfitting. In many cases, the number of candidate features (ranging between $25$ to $30$) is as many as the number of samples, while only a select few features are usually relevant for a given subgraph. The relevant features further tend to differ across subgraph instances. Elastic net helps perform _automatic feature selection_ , by selecting a few relevant predictors for each subgraph independently. Thus, we train all subgraphs with the same set of features, and let elastic net select the relevant ones. Another advantage of elastic net model is that it is intuitive and easily interpretable, like the default cost models which are also weighted sums of a number of statistics. This is an important requirement for effective debugging and analysis of production jobs. Table 4 depicts the Pearson correlation and median error of the five machine learning models over the production workload. We see that operator-subgraphs models trained using elastic net can make sufficiently accurate cost predictions (14% median error), with a high correlation (more than $0.92$) with the actual runtime, a substantial improvement over the default cost model (median error of $258$% and Pearson correlation of $0.04$). In addition, elastic net models are fast to invoke during query optimization, and have low storage footprint that indeed makes it feasible for us to learn specialized models for each possible subgraph. ## 4\. Robustness We now discuss how we learn robust cost models. As defined in Section 1, robust cost models cover the entire workload with high accuracy for a substantial time period before requiring retraining. In contrast to prior work on robust query processing (markl2004robust, ; dutt2016plan, ; lookahead-ip, ) that either modify the plan during query execution, or execute multiple plans simultaneously, we leverage the massive cloud workloads to learn robust models offline and integrate them with the optimizer to generate one robust plan with minimum runtime overhead. In this section, we first explain the coverage and accuracy tradeoff for the operator-subgraph model. Then, we discuss the other extreme, namely an operator model, and introduce additional models to bridge the gap between the two extremes. Finally, we discuss how we combine predictions from individual models to achieve robustness. ### 4.1. Accuracy-Coverage Tradeoff The operator-subgraph model presented in Section 3 is highly specialized. As a result, this model is likely to be highly accurate. Unfortunately, the operator-subgraph model does not cover subgraphs that are not repeated in the training dataset, i.e., it has limited coverage. For example, over $1$ day of Microsoft production workloads, operator-subgraphs have learned models for only 54% of the subgraphs. Note that we create a learned model for a subgraph if it has at least $5$ occurrences over the single day worth of training data. Thus, it is difficult to predict costs for arbitrary query plans consisting of subgraphs never seen in training dataset. The other extreme: Operator model. In contrast to the operator-subgraph model, we can learn a model for each physical operator, similar to how traditional query optimizers model the cost. The operator models estimate the execution latency of a query by composing the costs of individual operators in a hierarchical manner akin to how default cost models derive query costs. As a result, operator models can predict the cost of any query in the workload, including those previously unseen in the training dataset. However, similar to traditional cost model, operator models also suffer from poor accuracy since the behavior of an operator changes based on what operations appear below it. Furthermore, the estimation errors in the features or statistics at the lower level operators of a query plan are propagated to the upper level operators that significantly degrades the final prediction accuracy. On $5$-fold cross- validation over $1$ day of Microsoft production workloads, operator models results in $42\%$ median error and $0.77$ Pearson correlation, which although better than the default cost model (258% median error and 0.04 Pearson correlation), is relatively lower compared to that of operator-subgraph models (14% median error and $0.92$ Pearson correlation). Thus, there is an accuracy and coverage tradeoff when learning cost models, with operator-subgraph and operator models being the two extremes of this tradeoff. ### 4.2. Bridging the Gap We now present additional models that fall between the two extreme models in terms of the accuracy-coverage trade-off. Operator-input model. An improvement to per-operator is to learn a model for all jobs that share similar inputs. Similar inputs also tend to have similar schema and similar data distribution even as the size of the data changes over time, thus operator models learned over similar inputs often generalize over future job invocations. In particular, we create a model for each operator and input template combination. An input template is a normalized input where we ignore the dates, numbers, and parts of names that change over time for the same recurring input, thereby allowing grouping of jobs that run on the same input schema over different sessions. Further, to partially capture the context, we featurize the intermediate subgraph by introducing two additional features: 1) the number of logical operator in the subgraph (CL) and 2) the depth of the physical operator in the sub-graph (D). This helps in distinguishing subgraph instances that are extremely different from each other. Model | Correlation | Median Error ---|---|--- Default | 0.04 | 258% Neural Network | 0.89 | 27% Decision Tree | 0.91 | 19 % Fast-Tree regression | 0.90 | 20% Random Forest | 0.89 | 32% Elastic net | 0.92 | 14% Table 4. Correlation and error w.r.t. actual runtime for the operator- subgraphs Figure 6. Feature weights (all other models) Operator-subgraphApprox model. While operator-sub- graph models exploit the overlapping and recurring nature of big data analytics, there is also a large number (about 15-20%) of subgraphs that are similar but are not exactly the same. To bridge this gap, we relax the notion of subgraph similarity, and learn one model for all subgraphs that have the same inputs and the same _approximate_ underlying subgraph. We consider two subgraphs to be approximately same if they have the same physical operator at the root, and consist of the _same frequency of each logical operator in the underneath subgraph (ignoring the ordering between operators)._ Thus, there are two relaxations: (1) we use frequency of logical operators instead of physical operators (note that this is one of the additional features in Operator-input model) and (ii) we ignore the ordering between operators. This relaxed similarity criteria allows grouping of similar subgraphs without substantially reducing the coverage. Overall, operator-subgraphApprox model is a hybrid of the operator-subgraph and operator-input models: it achieves much higher accuracy compared to operator or operator-input models, and more coverage compared to operator-subgraph model. Table 5 depicts the Pearson correlation, the median accuracy using 5-fold cross-validation, as well as the coverage of individual cost models using elastic net over production workloads. As we move from more specialized to more generalized models (i.e., operator-subgraph to operator-subgraphApprox to operator-input to operator), we see that the model accuracy decreases while the coverage over the workload increases. Figure 6 shows the feature weights for each of the intermediate models. We see that while the weight for specialized models, like the operator-subgraph model (Figure 5), are concentrated on a few features, the weights for more generalized models, like the per-operator model, are more evenly distributed. ### 4.3. The Combined Model Given multiple learned models, with varying accuracy and coverage, the strawman approach is to select a learned model in decreasing order of their accuracy over training data, starting from operator-subgraphs to operator- subgraphApprox to operator-inputs to operators. However, as discussed earlier, there are subgraph instances where more accurate models result in poor performance over test data due to over-fitting. Figure 7. Heatmap of errors over $42K$ operators from production jobs. Each point in the heatmap depicts the error of learned cost with respect to the actual runtime Model | Correlation | Median Error | Coverage ---|---|---|--- Default | 0.04 | 258% | 100% Op-Subgraph | 0.92 | 14% | 54% Op-Subgraph Approx | 0.89 | 16 % | 76% Op-Input | 0.85 | 18% | 83% Operator | 0.77 | 42% | 100% Combined | 0.84 | 19% | 100% Table 5. Performance of learned models w.r.t to actual runtimes To illustrate, Figure 7 depicts the heat-map representation of the accuracy and coverage of different individual models, over more than $42K$ operator instances from our production workloads. Each point in the heat-map represents the error of the predicted cost with respect to the actual runtime: the more green the point, the less the error. The empty region at the top of the chart depicts that the learned model does not cover those subgraph instances. We can see that the predictions are mostly accurate for operator-subgraph models, while Operator models have relatively more error (i.e, less green) than the operator-subgraph models. Operator-input have more coverage with marginally more error (less green) compared to operator-subgraphs. However, for regions b, d, and f, as marked on the left side of the figure, we notice that operator-input performs better (more blue and green) than operator-subgraph. This is because operators in those regions have much fewer training samples that results in over-fitting. Operator-input, on the other hand, has more training samples, and thus performs better. Thus, it is difficult to decide a rule or threshold that always select the best performing model for a given subgraph instance. Model | Correlation | Median Error ---|---|--- Default | 0.04 | 258% Neural Network | 0.79 | 31% Decision Tree | 0.73 | 41 % FastTree Regression | 0.84 | 19% Random Forest | 0.80 | 28% Elastic net | 0.68 | 64% Table 6. Correlation and errror w.r.t actual runtimes for the Combined Model Learning a meta-ensemble model. We introduce a meta-ensemble model that uses the predictions from specialized models as meta features, along with the following extra features: (i) cardinalities (I, B, C), (ii) cardinalities per partition (I/P, B/P, C/P), and (iii) number of partitions (P) to output a more accurate cost. Table 6 depicts the performance of different machine learning models that we use as a meta-learner on our production workloads. We see that the FastTree regression (fasttree, ) results in the most accurate predictions. FastTree regression is a variant of the gradient boosted regression trees (friedman2002stochastic, ) that uses an efficient implementation of the MART gradient boosting algorithm (mart, ). It builds a series of regression trees (estimators), with each successive tree fitting on the residual of trees that precede it. Using $5$-fold cross-validation, we find that the maximum of only $20$ regression trees with mean-squared log error as the loss function and the sub-sampling rate of $0.9$ is sufficient for optimal performance. As depicted in Figure 7, using FastTree Regression as a meta-learner has three key advantages. First, FastTree regression can effectively characterize the space where each model performs well. The regression trees recursively split the space defined by the predictions from individual models and features, creating fine-grained partitions such that prediction in each partition are highly accurate. Second, FastTree regression performs better for operator instances where individual models perform worse. For example, some of the red dots found in outlier regions b, d and f of the individual model heat-maps are missing in the combined model. This is because FastTree regression makes use of sub- sampling to build each of the regression trees in the ensemble, making it more resilient to overfitting and noise in execution times of prior queries. Similarly, for region a where subgraph predictions are missing, FastTree regression creates an extremely large number of fine-grained splits using extra features and operator predictions to give even better accuracy compared to Operator models. Finally, the combined model is naturally capable of covering all possible plans, since it uses Operator model as one of the predictors. Further, as depicted in Figure 7, the combined model is comparable to that of specialized models, and almost always has better accuracy than Operator model. The combined model is also flexible enough to incorporate additional models or features, or replace one model with another. However, on also including the default cost model, it did not result in any improvement on SCOPE. Next, we discuss how we integrate the learned models within the query optimizer. ## 5\. Optimizer Integration In this section, we describe our two-fold integration of Cleo within SCOPE. First, we discuss our end-to-end feedback loop to learn cost models and generate predictions during optimization. Then, we discuss how we extend the optimizer for supporting resource-exploration using learned models. (a) Resource-aware planning (b) Example query plan (c) Model look-ups for partition exploration Figure 8. Integrating learned cost models with the query optimizer ### 5.1. Integrating Learned Cost Models The integration of learned models with the query optimizer involves the following three components. Instrumentation and Logging. Big data systems, such as SCOPE, are already instrumented to collect logs of query plan statistics such as cardinalities, estimated costs, as well as runtime traces for analysis and debugging. For uniquely identifying each operator and the subplan, query optimizers annotate each operator with a _signature_ (bruno2013continuous, ), a 64-bit hash value that can be recursively computed in a bottom-up fashion by combining (i) the signatures of children operators, (ii) hash of current operator’s name, and (iii) hash of operator’s logical properties. We extend the optimizer to compute three more signatures, one for each individual sub-graph mode, and extract additional statistics (i.e., features) that were missing for Cleo. Since all signatures can be computed simultaneously in the same recursion, and that there are only $25-30$ features (most of which were already extracted), the additional overhead is minimal ($\leq$ 10%), as we describe in Section 6. This overhead includes both the logging and the model lookup that we discuss subsequently. Training and Feedback. Given the logs of past query runs, we learn each of the four individual elastic net-based models independently and in parallel using our SCOPE-based parallel model trainer. Experimentally, we found that a training window of two days and a training frequency of every ten days results in acceptable accuracy and coverage (Section 6). We then use the predictions from individual models on the next day queries to learn the combined FastTree regression model. Since individual models can be trained in parallel, the training time is not much, e.g., it takes less than $45$ minutes for training $25K$ models from over $50,000$ jobs using a cluster of $200$ nodes. Once trained, we serialize the models and feedback them to the optimizer. The models can be served either from a text file, using an additional compiler flag, or using a web service that is backed by a SQL database. Look-up. All models relevant for a cluster are loaded upfront by the optimizer, into a hash map with keys as signatures of models, to avoid expensive lookup calls during optimization. When loaded simultaneously, all $25K$ models together take about $600$ MB of memory, which is within an acceptable range. Finally, for cost estimation, we modify the Optimize Input phase of Cascade optimizer to invoke learned models. Figure 8(a) highlights the the key steps that we added in blue. Essentially, we replace the calls to the default cost-models with the learned model invocations (Step $10$ in Figure 8(a)) to predict the exclusive cost of an operator, which is then combined with costs of children operators, similar to how default cost models combine costs. Moreover, since there is an operator model and a combined model for every physical operator, Cleo can cover all possible query plans, and can even generate a plan unseen in training data. All the features that learned models need are available during query optimization. However, we do not reuse the partition count derived by the default cost model, rather we try to find a more optimal partition count (steps $3$, $4$, $9$) as it drives the query latency. We discuss the problem with existing partition count selection and our solution below. ### 5.2. Resource-aware Query Planning The degree of parallelism (i.e., the number of machines or containers allocated for each operator) is a key factor in determining the runtime of queries in massively parallel databases (raqo, ), which implicitly depends on the partition count. This makes partition count as an important feature in determining the cost of an operator (as noted in Figures 5–6). Unfortunately, in existing optimizers, the partition count is not explored for all operators, rather partitioning operators (e.g., Exchange for stage $2$ in Figure 8(b)) set the partition count for the entire stage based on their local statistics (yint2018bubble, ). The above operators on the same stage simply derive the partition count set by the partitioning operator. For example, in stage $2$ of Figure 8(b), Exchange sets a partition count to $2$ as it results in its smallest local cost (i.e., $15$). The operators above Exchange (i.e., Reduce and Output) derive the same partition count, resulting in a total cost of $305$ for the entire stage. However, we can see that the partition count of $16$ results in a much lower overall cost of $125$, though it’s not locally optimal for Exchange. Thus, _not optimizing the partition count for the entire stage results in a sub-optimal plan_. To address this, we explore the partition counts during query planning, by making query planning resource-aware. Figures 8(a) and 8(b) illustrate our resource-aware query planning approach. We introduce the notion of a _resource-context_ , within the optimizer-context, for tracking costs of partitions across operators in a stage. Furthermore, we add a _partition exploration_ step, where each physical operator attaches a list of learned costs for different partition counts to the resource context (step 3 in Figure 8(a)). For example, in Figure 8(b), the resource-context for stage 2 shows the learned cost for different partition counts for each of the three operators. On reaching the stage boundary, the partitioning operator Exchange performs _partition optimization_ (step $9$ in Figure 8(a)) to set its local partition count to $16$, that results in the lowest total cost of $125$ across for the stage. Thereafter, the higher level operators simply derive the selected partition count (line 8 in Figure 8(a)), like in the standard query planning, and estimate their local costs using learned models. Note that when a partition comes as required property (cascades95, ) from upstream operators, we set the partition count to the required value without any exploration (Figure 8(a) step 2). SCOPE currently does not allow varying other resources, therefore we focus only on partition counts in this work. However, the resource-aware query planning with the three new abstractions to Cascades framework, namely the resource-context, partition-exploration, and partition-optimization, is general enough to incorporate additional resources such as memory sizes, number of cores, VM instance types, and other infrastructure level decisions to jointly optimize for both plan and resources. Moreover, our proposed extensions can also be applied to other big data systems such as Spark, Flink, and Calcite that use variants of Cascades optimizers and follow a similar top- down query optimization as SCOPE. Our experiments in Section 6 show that the resource-aware query planning not only generates better plans in terms of latency, but also leads to resource savings. However, the challenge is that estimating the cost for every single partition count for each operator in the query plan can explode the search space and make query planning infeasible. We discuss our approach to address this next. ### 5.3. Efficient Resource Exploration We now discuss two techniques for efficiently exploring the partition counts in Cleo, without exploding the search space. Sampling-based approach. Instead of considering every single partition count, one option is to consider a uniform sample over the set of all possible containers for the tenant or the cluster. However, the relative change in partition is more interesting when considering its influence on the cost, e.g., a change from $1$ to $2$ partitions influences the cost more than a change from $1200$ to $1210$ partitions. Thus, we sample partition counts in a geometrically increasing sequence, where a sample $x_{i+1}$ is derived from previous sample $x_{i}$ using: $x_{i+1}=\left\lceil x_{i}+x_{i}/s\right\rceil$ with $x_{0}=1$, $x_{1}=2$. Here, $s$ is a skipping coefficient that decides the gap between successive samples. A large $s$ leads to a large number of samples and more accurate predictions, but at the cost of higher model look- ups and prediction time. Figure 9. Workload consisting of $0.5$ million jobs from $4$ different production clusters over $3$ days Analytical approach. We reuse the individual learned models to directly model the relationship between the partition count and the cost of an operator. The key insight here is that only features where partition count is present are relevant for partition exploration, while the rest of the features can be considered constants since their values are fixed during partition exploration. Thus, we can express operator cost as follows: $cost\propto\frac{(\theta_{1}*I+\theta_{2}*C+\theta_{3}*I*C)}{P}+\theta_{c}*P$ where $I$, $C$, and $P$ refer to input cardinality, cardinality and partition count respectively. During optimization, we know $I$, $C$, $I*C$, therefore: $cost\propto\frac{\theta_{P}}{P}+\theta_{c}*P$. Extending the above relationship across all operators (say $n$ in number) in a stage, the relationship can be modeled as: $cost\propto\frac{\sum_{i=1}^{n}\theta_{P_{i}}}{P}+\sum_{i=1}^{n}\theta_{C_{i}}*P$ Thus, during partition exploration, each operator calculates $\theta_{P}$ and $\theta_{C}$ and adds them to resource-context, and the partitioning operator selects the optimal partition count by optimizing the above function. There are three possible scenarios: (i) $\sum_{i=1}^{n}\theta_{P_{i}}$ is positive while $\sum_{i=1}^{n}\theta_{C_{i}}$ is negative: we can have the maximum number of partitions for the stage since there is no overhead of increasing the number of partitions, (ii) $\sum_{i=1}^{n}\theta_{P_{i}}$ is negative while $\sum_{i=1}^{n}\theta_{C_{i}}$ is positive: we set the partition count to minimum as increasing the partition count increases the cost, (iii) $\sum_{i=1}^{n}\theta_{P_{i}}$ and $\sum_{i=1}^{n}\theta_{C_{i}}$ are either both positive or both negative: we can derive the optimal partition count by differentiating the cost equation with respect to P. Overall, for $m$ physical operator, and the maximum possible partition count of $P_{max}$, the analytical model makes $5\cdot m\cdot log_{\frac{s+1}{s}}P_{max}$ cost model look-ups. Figure 8(c) shows the number of model look-ups for sampling and analytical approaches as we increase the number of physical operators from $1$ to $40$ in a plan. While the analytical model incurs a maximum of $200$ look- ups, the sampling approach can incur several thousands depending on the skipping coefficients. In section 6.5, we further analyze the accuracy of the sampling strategy with the analytical model on the production workload as we vary the sample size. Our results show that the analytical model is at least $20$ $\times$ more efficient than the sampling approach for achieving the same accuracy. Thus, we use the analytical model as our default partition exploration strategy. ## 6\. Experiments Figure 10. Illustrating workload changes over different clusters and different days. Figure 11. Cross-validation results of ML algorithms for each learned model on Cluster 4 workload | All jobs | Ad-hoc jobs ---|---|--- | Correlation | Median Error | 95%tile Error | Coverage | Correlation | Median Error | 95%tile Error | Coverage Default | 0.12 | 182% | 12512% | 100% | 0.09 | 204% | 17791% | 100% Op-Subgraph | 0.86 | 9% | 56% | 65% | 0.81 | 14% | 57% | 36% Op-Subgraph Approx | 0.85 | 12 % | 71% | 82% | 0.80 | 16 % | 79% | 64% Op-Input | 0.81 | 23% | 90% | 91% | 0.77 | 26% | 103% | 79% Operator | 0.76 | 33% | 138% | 100% | 0.73 | 42% | 186% | 100% Combined | 0.79 | 21% | 112% | 100% | 0.73 | 29% | 134% | 100% Table 7. Breakdown of accuracy and coverage of each learned model for all jobs and ad-hoc jobs separately on Cluster1. In this section, we present an evaluation of our learned optimizer Cleo. For fairness, we feed the same statistics (e.g., cardinality, average row length) to learned models that are used by the SCOPE default cost model. Our goals are five fold: (i) to compare the prediction accuracy of our learned cost models over all jobs as well as over only ad-hoc jobs across multiple clusters, (ii) to test the coverage and accuracy of learned cost models over varying test windows, (iii) to compare the Cleo cost estimates with those from CardLearner, (iv) to explore why perfect cardinality estimates are not sufficient for query optimization, (iv) to evaluate the effectiveness of sampling strategies and the analytical approach proposed in Section 5.2 in finding the optimal resource (i.e., partition count), and (v) to analyze the performance of plans produced by Cleo with those generated from the default optimizer in Cleo using both the production workloads and the TPC-H benchmark (v) to understand the training and runtime overheads when using learned cost models. Cluster | Default (all jobs) | Learned (all jobs) | Learned (ad-hoc jobs) ---|---|---|--- | Correlation | Median Accuracy | Correlation | Median Accuracy | Correlation | Median Accuracy Cluster 1 | 0.12 | 182% | 0.79 | 21% | 0.73 | 29% Cluster 2 | 0.08 | 256% | 0.77 | 33% | 0.75 | 40% Cluster 3 | 0.15 | 165% | 0.83 | 26% | 0.81 | 38% Cluster 4 | 0.05 | 153% | 0.74 | 15% | 0.72 | 26% Table 8. Pearsion Correlation and Median accuracy of default and combined learned model over all jobs and ad-hoc jobs on each cluster. Workload. As summarized in Figure 9, we consider a large workload trace from $4$ different production clusters comprising of $423$ virtual clusters, with each virtual cluster roughly representing a business unit within Microsoft, and consisting a total of $\approx 0.5$ million jobs over $3$ days, that ran with a total processing time of $\approx 6$ million hours, and use a total of $\approx 1.4$ billion containers. The workload exhibits variations in terms of the load (e.g., more than $3\times$ jobs on Cluster1 compared to Cluster4), as well as in terms of job properties such as average number of operators per job ($50s$ in Cluster1 compared to $30s$ in Cluster4) and the average total processing time of all containers per job (around $17$ hours in Cluster1 compared to around $5$ hours in Cluster4). The workload also varies across days from a $30\%$ decrease to a $20\%$ increase of different job characteristics on different clusters(Figure 10). Finally, the workload consists of a mix of both recurring and ad-hoc jobs, with about $7\%-20\%$ ad- hoc jobs on different clusters and different days. Figure 12. Accuracy results on all jobs (recurring + ad-hoc) over four different clusters Figure 13. Accuracy results on only ad-hoc jobs over four different clusters. ### 6.1. Cross Validation of ML Models We first compare default cost model with the five machine learning algorithms discussed in Section 3.4. (i) Elastic net, a regularized linear-regression model, (ii) DecisionTree Regressor, (iii) Random Forest, (iv) Gradient Boosting Tree, and (v) Multilayer Perceptron Regressor (MLP). Figure 11 (a to d) depicts the $5$-fold cross-validation results for the ML algorithms for operator-subgraph, operator-input, operator, and combined models respectively. We skip the results for operator-subgraphApprox as it has similar results to that of operator-input. We observe that all algorithms result in better accuracy compared to the default cost model for each of the models. For operator-subgraph and operator-input models, the performance of most of the algorithms (except the neural network) is highly accurate. This is because there are a large number of specialized models, highly optimized for specific sub-graph and input instances respectively. The accuracy degrades as the heterogeneity of model increases from operator-subgraph to operator-input to operator. For individual models, the performances of elastic-net and decision- tree are similar or better than complex models such as neural network and ensemble models. This is because the complex models are more prone to overfitting and noise as discussed in Section 3.4. For the combined model, the FastTree Regression does better than other models because of its ability to better characterize the space where each model performs well. The operator- subgraph and operator-input models have the highest accuracy (at between $45$% to $85$% coverage), followed by the combined model (at $100$% coverage) and then the operator model (at $100$% coverage). ### 6.2. Accuracy Next, we first compare the accuracy and correlation of learned models to that of the default cost model for each of the clusters. We use the elastic net model for individual learned model and the Fast-Tree regression for the combined model. We learn the models on day 1 and day 2, and predict on day 3 of the workload as described in Section 5.1. Table 8 shows the Pearson correlation and median accuracy of default cost model and the combined model for all jobs and only ad-hoc jobs separately for each of the clusters on day 3. We further show the breakdown of results for each of the individual models on cluster 1 jobs in Table 7. Figure 12 and Figure 13 show the CDF distribution for estimated vs actual ratio of predicted costs for each of the learned models over different clusters. All jobs. We observe that learned models result between $8\times$ to $10\times$ better accuracy and between $6\times$ to $14\times$ better correlation compared to the default cost model across the $4$ clusters. For operator-subgraph, the performance is highly accurate (9% median error and .86 correlation), but at lower coverage of 65%. This is because there are a large number of specialized models, highly optimized for specific sub-graph instances. As the coverage increases from operator-subgraphApprox to operator- input to operator, the accuracy decreases. Overall, the combined model is able to provide the best of both worlds, i.e., accuracy close to those of individual models and $100$% coverage like that of the operator model. The $50^{th}$ and the $95^{th}$ percentile errors of the combined model are about $10$$\times$ and $1000$$\times$ better than the default SCOPE cost model. These results show that it is possible to learn highly accurate cost models from the query workload. Figure 14. Coverage and accuracy with respect to default optimizer over 1 month Only ad-hoc Jobs. Interestingly, _the accuracy over ad-hoc jobs drop slightly but it is still close to those over all jobs (Table 8 and Table 7)_. This is because: (i) ad-hoc jobs can still have one or more similar subexpressions as other jobs (e.g., they might be scanning and filtering the same input before doing completely new aggregates), which helps them to leverage the subgraph models learned from other jobs. This can be seen in Table 7 — the sub-graph learned models still have substantial coverage of subexpression on ad-hoc jobs. For example, the coverage of $64\%$ for Op-Subgraph Approx model means that $64\%$ of the sub-expressions on ad-hoc jobs had matching Op-Subgraph Approx learned model. (ii) Both the operator and the combined models are learned on a per-operator basis, and have much lower error ($42\%$ and $29\%$) than the default model ($182\%$). This is because the query processor, the operator implementations, etc. still remain the same, and their behavior gets captured in the learned cost models. Thus, even if there is no matching sub- expression in ad-hoc jobs, the operator and the combined models still result in better prediction accuracy. ### 6.3. Robustness We now look at the robustness (as defined in Section 1) of learned models in terms of accuracy, coverage, and retention over a month long test window on cluster 1 workloads. Coverage over varying test window. Figure 14a depicts the coverage of different subgraph models as we vary the test window over a duration of 1 month. The coverage of per-operator and combined model is always $100\%$ since there is one model for every physical operator. The coverage of per-subgraph models, strictest among all, is about $58\%$ after $2$ days, and decreases to $37\%$ after $28$ days. Similarly, the coverage of per-subgraphApprox ranges between $75\%$ and $60\%$. The per-operator-input model, on the other hand remains stable between $78\%$ and $84\%$. Error and correlation over varying test window. Figures 14b and 14c depict the median and $95\%$ile error percentages respectively over a duration of one month. While the median error percentage of learned models improves on default cost model predictions by $3$-$15$x, the $95\%$ile error percentage is better by over three orders of magnitude. For specialized learned models, the error increases slowly over the first two weeks and then grows much faster due to decrease in the coverage. Moreover, the $95\%$ile error of the subgraph models grows worse than their median error. Similarly, in Figure 14d, we see that predicted costs from learned models are much more correlated with the actual runtimes, having high Pearson correlation (generally between $0.70$ and $0.96$), compared to default cost models that have a very small correlation (around $0.1$). A high correlation over a duration month of $1$ month shows that learned models can better discriminate between two candidate physical operators. Overall, based on these results, we believe re-training every $10$ days should be acceptable, with a median error of about $20\%$, $95\%$ error of about $200\%$, and Pearson correlation of around $0.80$. Robustness of the combined model. From Figure 14, we note that the combined model (i) has 100% coverage, (ii) matches the accuracy of best performing individual model at any time (visually illustrated in Figure 7), while having a high correlation ($>$ 0.80) with actual runtimes, and (iii) gives relatively stable performance with graceful degradation in accuracy over longer run. Together, these results show that the combined model in Cleo is indeed robust. Figure 15. Comparison with CardLearner Figure 16. Hash join operator having different weights over different sets of sub-expressions. Figure 17. Partition Exploration Accuracy vs. Efficiency ### 6.4. Impact of Cardinality Comparison with Cardlearner. Next, we compare the accuracy and the Pearson correlation of Cleo and the default cost model with CardLearner (cardLearner, ), a learning-based cardinality estimation that employs a Poisson regression model to improve the cardinality estimates, but uses the default cost model to predict the cost. For comparison, we considered jobs from a single virtual cluster from cluster 4, consisting of about $900$ jobs. While Cleo and the default cost model use the cardinality estimates from the SCOPE optimizer, we additionally consider a variant (Cleo +CardLearner) where Cleo uses the cardinality estimates from CardLearner. Overall, we observe that the median error of default cost with CardLearner (211%) is better than that of default cost model alone (236%) but much still worse than that of Cleo’s (18%) and Cleo \+ CardLearner (13%). Figure 17 depicts the CDF of estimated and actual costs ratio, where we can see that by learning costs, Cleo significantly reduces both the under-estimation as well as over-estimation, while CardLearner only marginally improves the accuracy for both default cost model and Cleo. Similarly, the Pearson correlation of CardLearner’s estimates (.01) is much worse than that of Cleo’s ($0.84$) and Cleo \+ CardLearner ($0.86$). These results are consistent with our findings from Section 2 where we show that fixing the cardinality estimates is not sufficient for filling the wide gap between the estimated and the actual costs in SCOPE-like systems. Interestingly, we also observe that the Pearson correlation of CardLearner is marginally less than that of the default cost model (0.04) inspite of better accuracy, which can happen when a model makes both over and under estimation over different instances of the same operator. Why cardinality alone is not sufficient? To understand why fixing cardinality alone is not enough, we performed the following two micro-experiments on a subset of jobs, consisting of about $200$ K subexpressions from cluster 4. 1\. The need for a large number of features. First, starting with perfect cardinality estimates (both input and output) as only features, we fitted an elastic net model to find / tune the best weights that lead to minimum cost estimation error (using the loss function described in Section 3.4). Then, we incrementally added more and more features and also combined feature with previously selected features, retraining the model with every addition. Figure 18 shows the decrease in cost model error as we cumulatively add features from left to right. We use the same notations for features as defined in Table 2 and Table 3. Figure 18. Impact on median error as we cumulatively add features from left to right. The first two features are perfect output (C) and input (I) cardinalities. We make the following observations. * (1) When using only perfect cardinalities, the median error is about $110\%$. However, when adding more features, we observe that the error drops by more than half to about $40\%$. This is because of the more complex environments and queries in big data processing that are very hard to cost with just the cardinalities (as also discussed in Section 2.3). * (2) Deriving additional statistics by transforming (e.g., sqrt, log, squares) input and output cardinalities along with other features helps in reducing the error. However, it is hard to arrive at these transformations in hand coded cost models. * (3) We also see that features such as parameter (PM), input (IN), and partitions (P) are quite important, leading to sharp drops in error. While partition count is good indicator of degree of parallelism (DOP) and hence the runtime of job; unfortunately query optimizers typically use a fixed partition count for estimating cost as discussed in Section 5.2. * (4) Finally, there is a pervasive use of unstructured data (with schema imposed at runtime) as well as custom user code (e.g., UDFs) that embed arbitrary application logic in big data applications. It is hard to come up with generic heuristics, using just the cardinality, that effectively model the runtime behavior of all possible inputs and UDFs. 2\. Varying optimal weights. We now compare the optimal weights of features of physical operators when they occur in different kinds of sub-expressions. We consider the popular hash join operator and identified two sets of sub- expressions in the same sample dataset: (i) hash join appears on top of two scan operators, and (ii) hash join appears on top of two other join operators, which in turn read from two scans. Figure 17 shows the weights of the top $10$ features of the hash join cost model for the two sets. We see that the partition count is more influential for set 2 compared to set 1. This is because there is more network transfer of data in set 2 than in set 1 because of two extra joins. Setting the partition count that leads to minimum network transfer is therefore important. On the other hand, for jobs in set 1, we notice that hash join typically sets the partition count to the same value as that of inputs, since that minimizes repartitioning. Thus, partition count is less important in set 1. To summarize, even when cardinality is present as a raw or derived feature, its relative importance is instance specific (heavily influenced by the partitioning and the number of machines) and hard to capture in a static cost model. (a) Changes in Latency (b) Changes in Total Processing Time (c) Optimization Time Overhead Figure 19. Performance comparison on production jobs with changed plans ### 6.5. Efficacy of Partition Exploration In this section, we explore the effectiveness of partition exploration strategies proposed in Section 5.2 in selecting the partition count that leads to lowest cost for learned models. We used a subset of $200$ sub-expression instances from the production workload from cluster 1, and exhaustively probe the learned models for all partition counts from $0$ to $3000$ (the maximum capacity of machines on a virtual cluster) to find the most optimal cost. Figure 17 depicts the median error in cost accuracy with respect to the optimal cost for (i) three sampling strategies: random, uniform, and geometric as we vary the number of samples of partition counts (ii) the analytical model (dotted blue line) that selects a single partition count. We note that the analytical model, although approximate, gives more accurate results compared to sampling-based approaches until the sample size of about $15$ to $20$, thereby requiring much fewer model invocations. Further, for a sample size between $4$ to $20$, we see that the geometrically increasing sampling strategy leads to more accurate results compared to uniform and random approaches. This is because it picks more samples when the values are smaller, where costs tend to change more strongly compared to higher values. During query optimization, each sample leads to five learned cost model predictions, four for individual models and one for the combined model. Thus, for a typical plan in big data system that consists of $10$ operators, the sampling approach requires $20*5*10=1000$ model invocations, while the analytical approach requires only $5*10=50$ invocations for achieving the same accuracy. This shows that the analytical approach is practically more effective when we consider both efficiency and accuracy together. Hence, Cleo uses the analytical approach as the default partition exploration strategy. ### 6.6. Performance We split the performance evaluation of Cleo into three parts: (i) the runtime performance over production workloads, (ii) a case-study on the TPC-H benchmark (poess2000new, ), explaining in detail the plan changes caused by the learned cost models, and (iii) the overheads incurred due to the learned models. For these evaluations, we deployed a new version of the SCOPE runtime with Cleo optimizer (i.e., SCOPE+Cleo) on the production clusters and compared the performance of this runtime with the default production SCOPE runtime. We used the analytical approach for partition exploration. #### 6.6.1. Production workload Since production resources are expensive, we selected a subset of the jobs, similar to prior work on big data systems (cardLearner, ), as follows. We first recompiled all the jobs from a single virtual cluster from cluster 4 with Cleo and found that $182$ (22%) out of $845$ jobs had plan changes when partition exploration was turned off, while $322$ ($39$%) had plan changes when we also applied partition exploration. For execution, we selected jobs that had at least one change in physical operator implementations, e.g, for aggregation (hash vs stream grouping), join (hash vs merge), addition or removal of grouping, sort or exchange operators. We picked $17$ such jobs and executed them with and without Cleo over the same production data, while redirecting the output to a dummy location, similar to prior work (rope, ). Figure 19(a) shows the end-to-end latency for each of the jobs, compared to their latency when using the default cost model. We see that the learned cost models improve latency in 70% (12 jobs) cases, while they degrade latency in the remaining 30% cases. Overall, the average improvement across all jobs is $15.35$%, while the cumulative latency of all jobs improves by $21.3$%. Interestingly, Cleo was able to improve the end-to-end latency for $10$ out of $12$ jobs with less degree of parallelism (i.e., the number of partitions). This is in contrary to the typical strategy of scaling out the processing by increasing the degree of parallelism, which does not always help. Instead, resource-awareness plan selection can reveal more optimal combinations of plan and resources. To understand the impact on resource consumption, Figure 19(b) shows the total processing time (CPU hours). Learned cost models reduce the Overall, we see the total processing time reducing by $32.2$% on average and $40.4\%$ cumulatively across all 17 jobs— a significant operational cost saving in big data computing environments. Thus, the learned cost models could reduce the overall costs while still improving latencies in most cases. Below, we dig deeper into the plans changes using the TPC-H workload. #### 6.6.2. TPC-H workload We generated TPC-H dataset with a scale factor of $1000$, i.e., total input of $1TB$. We ran all $22$ queries $10$ times, each time with randomly chosen different parameters, to generate the training dataset. We then trained our cost models on this workload and feedback the learned models to re-run all $22$ queries. We compare the performance with- and without the feedback as depicted in Figure 20 . For each observation, we take the average of $3$ runs. Overall, $6$ TPC-H queries had plan changes when using resource-aware cost model. Out of these, $4$ changes (Q8, Q9, Q16, Q20) improve latency as well as total processing time, 1 change improves only the latency (Q11), and 1 change (Q17) leads to performance regression in both. We next discuss the key changes observed in the plans. _1\. More optimal partitioning._ In Q8, for Part (100 partitions) $\Join_{partKey}$ Lineitem (200 partitions) and in Q9 for Part (100 partitions) $\Join_{partKey}$ (Lineitem $\Join$ Supplier) (250 partitions), the default optimizer performs the merge join over 250 partitions, thereby re- partitioning both Part and the other join input into $250$ partitions over the Partkey column. Learned cost models, on other hand, perform the merge join over $100$ partitions, which requires re-partitioning only the other join inputs and not the Part table. Furthermore, re-partitioning $200$ or $250$ partitions into $100$ partitions is cheaper, since it involves partial merge (bruno2014advanced, ), compared to re-partitioning them into $250$ partitions. In $Q16$, for the final aggregation and top-k selection, both the learned and the default optimizer repartition the $250$ partitions from the previous operator’s output on the final aggregation key. While the default optimizer re-partitions into $128$ partitions, the learned cost models pick $100$ partitions. The aggregation cost has almost negligible change due to change in partition counts. However, repartitioning from $250$ to $100$ turns out to be substantially faster than re-partitioning from $250$ to $128$ partitions. _2\. Skipping exchange (shuffe) operators._ In Q8, for Part (100 partitions) $\Join_{partKey}$ Lineitem (200 partitions), the learned cost model performs join over $100$ partitions and thus it skips the costly Exchange operator over the Part table. On the other hand, the default optimizer creates two Exchange operator for partitioning each input into $250$ partitions. _3\. More optimal physical operator:_ For both Q8 and Q20, the learned cost model performs join between Nations and Supplier using merge join instead of hash join (chosen by default optimizer). This results in an improvement of $10\%$ to $15\%$ in the end-to-end latency, and $5\%$ to $8\%$ in the total processing time (cpu hours). _4\. Regression due to partial aggregation._ For Q17, the learned cost models add local aggregation before the global one to reduce data transfer. However, this degrades the latency by $10\%$ and total processing time by $25\%$ as it does not help in reducing data. Currently, learned models do not learn from their own execution traces. We believe that doing so can potentially resolve some of the regressions. #### 6.6.3. Training and Runtime Overheads. We now describe the training and run-time overheads of Cleo. It takes less than 1 hour to analyze and learn models for a cluster running about $800$ jobs a day, and less than 4 hours for training over $50$K jobs instances at Microsoft. We use a parallel model trainer that leverages SCOPE to train and validate models independently and in parallel, which significantly speeds up the training process. For a single cluster of about $800$ jobs, Cleo learns about $23$K models which when simultaneously loaded takes about $600$ MB of memory. About $80$% of the memory is taken by the individual models while the remaining is used by the combined model. The additional memory usage is not an issue for big data environments, where an optimizer can typically have $100$s of GBs of memory. Finally, we saw between $5$-$10$% increase in the optimization time for most of the jobs when using learned cost models, which involves the overhead of signature computation of sub-graph models, invoking learned models, as well as any changes in plan exploration strategy due to learned models. Figure 19(c) depicts the overhead in the optimization time for each of the $17$ production jobs we executed. Since the optimization time is often in orders of few hundreds of milliseconds, the overhead incurred here is negligible compared to the overall compilation time (orders of seconds) of jobs in big data systems as well as the potential gains we can achieve in the end-to-end latency (orders of 10s of minutes). Figure 20. Performance change with SCOPE + CLEO for TPC-H queries (higher is better) ### 6.7. Discussion We see that it is possible to learn accurate yet robust cost models from cloud workloads. Given the complexities and variance in the modern cloud environments, the model accuracies in Cleo are not perfect. Yet, they offer two to three orders of magnitude more accuracy and improvement in correlation from less than $0.1$ to greater than $0.7$ over the current state-of-the-art. The combined meta model further helps to achieve full workload coverage without sacrificing accuracy significantly. In fact, the combined model consistently retains the accuracy over a longer duration of time. While the accuracy and robustness gains from Cleo are obvious, the latency implications are more nuanced. These are often due to various plan explorations made to work with the current cost models. For instance, SCOPE jobs tend to over- partition at the leaf levels and leverage the massive scale-out possible for improving latency of jobs. There are several ways to address performance regressions in production workloads. One option is to revisit the search and pruning strategies for plan exploration (cascades95, ) in the light of newer learned cost models. For instance, one problem we see is that current pruning strategies may sometimes skip operators without invoking their learned models. Additionally, we can also configure optimizer to not invoke learned models for specific operators or jobs. Another improvement is to optimize a query twice (each taking only few orders of milliseconds), with and without Cleo, and select the plan with the better overall latency, as predicted using learned models since they are highly accurate and correlated to the runtime latency. We can also monitor the performance of jobs in pre-production environment and isolate models that lead to performance regression (or poor latency predictions), and discard them from the feedback. This is possible since we do not learn a single global model in the first place. Furthermore, since learning cost models and feeding them back is a continuous process, regression causing models can self-correct by learning from future executions. Finally, regressions for a few queries is not really a problem for ad-hoc workloads, since majority of the queries improve their latency anyways and reducing the overall processing time (and hence operational costs) is generally more important. Finally, in this paper, we focused on the traditional use-case of a cost model for picking the physical query plan. However, several other cost model use- cases are relevant in cloud environments, where accuracy of predicted costs is crucial. Examples include performance prediction (perforator, ), allocating resources to queries (jyothi2016morpheus, ), estimating task runtimes for scheduling (boutin2014apollo, ), estimating the progress of a query especially in server-less query processors (lee2016operator, ), and running what-if analysis for physical design selection (chaudhuri2007self, ). Exploring these would be a part of future work. Examples include performance prediction (perforator, ), allocating resources to queries (jyothi2016morpheus, ), estimating task runtimes for scheduling (boutin2014apollo, ), estimating the progress of a query especially in server-less query processors (lee2016operator, ), and running what-if analysis for physical design selection (chaudhuri2007self, ). Exploring these would be a part of future work. ## 7\. Related Work Machine learning has been used for estimating the query execution time of a given physical plan in centralized databases (ganapathi2009predicting, ; akdere2012learning, ; li2012robust, ). In particular, the operator and sub- plan-level models in (akdere2012learning, ) share similarities with our operator and operator-subgraph model. However, we discovered the coverage- accuracy gap between the two models to be substantially large. To bridge this gap, we proposed additional mutually enhancing models and then combined the predictions of these individual models to achieve the best of accuracy and coverage. There are other works on query progress indicators (chaudhuri2004estimating, ; luo2004toward, ) that use the run time statistics from the currently running query to tell how much percentage of the work has been completed for the query. Our approach, in contrast, uses compile time features to make the prediction before the execution starts. Cardinality is a key input to cost estimation and several learning and feedback-driven approaches (cardLearner, ; rope, ; stillger2001leo, ). However, these works have either focused only on recurring or strict subgraphs (cardLearner, ; rope, ), or learn only the ratio between the actual and predicted cardinality (stillger2001leo, ) that can go wrong in many situations, e.g., partial coverage results in erroneous estimates due to mixing of disparate models. Most importantly, as we discuss in Section 2, fixing cardinalities alone do not always lead to accurate costs in big data systems. There are other factors such as resources (e.g., partitions) consumed, operator implementations (e.g., custom operators), and hardware characteristics (e.g., parallel distributed environment) that could determine cost. In contrast to cardinality models, Cleo introduces novel learning techniques (e.g., multiple models, coupled with an ensemble) and extensions to optimizer to robustly model cost. That said, cardinality is still an important feature (see Figure 5), and is also key to deciding partition counts, memory allocation at runtime, as well as for speculative execution in the job manager. A more detailed study on cardinality estimates in big data systems is an interesting avenue for future work. Several works find the optimal resources given a physical query execution plan (ernest, ; perforator, ; cherrypick, ). They either train a performance model or apply non-parametric Bayesian optimization techniques with a few sample runs to find the optimal resource configuration. However, the optimal execution plan may itself depend on the resources, and therefore in this work, we jointly find the optimal cost and resources. Nevertheless, the ideas from resource optimization work can be leveraged in our system to reduce the search space, especially if we consider multiple hardware types. Generating efficient combination of query plans and resources are also relevant to the new breed of serverless computing, where users are not required to provision resources explicitly and they are billed based on usage (serverless, ). For big data queries, this means that the optimizer needs to accurately estimate the cost of queries for given resources and explore different resource combinations so that users do not end up over-paying for their queries. Finally, several recent works apply machine learning techniques to improve different components of a data system (kraska2019sagedb, ; marcus2019neo, ). The most prominent being learned indexes (kraska2018case, ), which overfits a given stored data to a learned model that provides faster lookup as well as smaller storage footprint. (marcus2019neo, ) takes a more disruptive approach, where the vision is to replace the traditional query optimizers with one built using neural networks. In contrast, our focus in this paper is on improving the cost-estimation of operators in big data systems and our goal is to integrate learned models into existing query optimizers in a minimally invasive manner. ## 8\. Conclusion Accurate cost prediction is critical for resource efficiency in big data systems. At the same time, modeling query costs is incredibly hard in these systems. In this paper, we present techniques to learn cost models from the massive cloud workloads. We recognize that cloud workloads are highly heterogeneous in nature and no one model fits all. Instead, we leverage the common subexpression patterns in the workload and learn specialized models for each pattern. We further describe the accuracy and coverage trade-off of these specialized models and present additional mutual enhancing models to bridge the gap. 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2024-09-04T02:54:55.297589

2002.12399

arxiv-papers

# ConQUR: Mitigating Delusional Bias in Deep Q-learning Andy Su Jayden Ooi Tyler Lu Dale Schuurmans Craig Boutilier ###### Abstract _Delusional bias_ is a fundamental source of error in approximate Q-learning. To date, the only techniques that explicitly address delusion require comprehensive search using tabular value estimates. In this paper, we develop efficient methods to mitigate delusional bias by training Q-approximators with labels that are “consistent” with the underlying greedy policy class. We introduce a simple penalization scheme that encourages Q-labels used _across training batches_ to remain (jointly) consistent with the expressible policy class. We also propose a search framework that allows multiple Q-approximators to be generated and tracked, thus mitigating the effect of premature (implicit) policy commitments. Experimental results demonstrate that these methods can improve the performance of Q-learning in a variety of Atari games, sometimes dramatically. Machine Learning, ICML ## 1 Introduction _Q-learning_ (Watkins & Dayan, 1992; Sutton & Barto, 2018) lies at the heart of many of the recent successes of deep reinforcement learning (RL) (Mnih et al., 2015; Silver et al., 2016), with recent advancements (e.g., van Hasselt (2010); Bellemare et al. (2017); Wang et al. (2016); Hessel et al. (2017)) helping to make it among the most widely used methods in applied RL. Despite these successes, many properties of Q-learning are poorly understood, and it is challenging to successfully apply deep Q-learning in practice. Various modifications have been proposed to improve convergence or approximation error (Gordon, 1995, 1999; Szepesvári & Smart, 2004; Melo & Ribeiro, 2007; Maei et al., 2010; Munos et al., 2016); but it remains difficult to reliably attain both robustness and scalability. Recently, Lu et al. (2018) identified a source of error in Q-learning with function approximation known as _delusional bias_. This bias arises because Q-learning updates the value of state-action pairs using estimates of (sampled) successor-state values that can be _mutually inconsistent given the policy class induced by the approximator_. This can result in unbounded approximation error, divergence, policy cycling, and other undesirable behavior. To handle delusion, the authors propose a _policy-consistent backup_ operator that maintains multiple Q-value estimates organized into _information sets_. Each information set has its own backed-up Q-values and corresponding “policy commitments” responsible for inducing these values. Systematic management of these sets ensures that only _consistent_ choices of maximizing actions are used to update Q-values. All potential solutions are tracked to prevent premature convergence on specific policy commitments. Unfortunately, the proposed algorithms use tabular representations of Q-functions, so while this establishes foundations for delusional bias, the function approximator is used neither for generalization nor to manage the size of the state/action space. Consequently, this approach is not scalable to practical RL problems. In this work, we develop ConQUR (_CONsistent Q-Update Regression_), a general framework for integrating policy-consistent backups with regression-based function approximation for Q-learning and for managing the search through the space of possible regressors (i.e., information sets). With suitable search heuristics, the proposed framework provides a computationally effective means for minimizing the effects of delusional bias, while scaling to practical problems. Our main contributions are as follows. First, we define novel augmentations of Q-regression to increase the degree of policy consistency across training batches. Since testing exact consistency is expensive, we introduce an efficient _soft-consistency penalty_ that promotes consistency of labels with earlier policy commitments. Second, using information-set structure (Lu et al., 2018), we define a search space over Q-regressors to explore multiple sets of policy commitments. Third, we propose heuristics to guide the search, critical given the combinatorial nature of information sets. Finally, experimental results on the Atari suite (Bellemare et al., 2013) demonstrate that ConQUR can add (sometimes dramatic) improvements to Q-learning. These results further show that delusion does emerge in practical applications of Q-learning. We also show that straightforward consistency penalization on its own (i.e., without search) can improve both standard and double Q-learning. ## 2 Background We assume a discounted, infinite horizon _Markov decision process (MDP)_ , $\mathbf{M}=(\mathcal{S},A,P,p_{0},R,\gamma)$. The state space $\mathcal{S}$ can reflect both discrete and continuous features, but we take the action space $A$ to be finite (and practically enumerable). We consider _Q-learning_ with a function approximator $Q_{\theta}$ to learn an (approximately) optimal Q-function (Watkins, 1989; Sutton & Barto, 2018), drawn from some approximation class parameterized by $\Theta$ (e.g., the weights of a neural network). When the approximator is a deep network, we generically refer to this as _DQN_ , the method at the heart of many RL successes (Mnih et al., 2015; Silver et al., 2016). For online Q-learning, at a transition $s,a,r,s^{\prime}$, the Q-update is given by: $\theta\leftarrow\theta+\alpha\Big{(}r+\gamma\max_{a^{\prime}\in A}Q_{\theta}(s^{\prime},a^{\prime})-Q_{\theta}(s,a)\Big{)}\nabla_{\theta}Q_{\theta}(s,a).$ (1) Batch versions of Q-learning are similar, but fit a regressor repeatedly to batches of training examples (Ernst et al., 2005; Riedmiller, 2005), and are usually more data efficient and stable than online Q-learning. Batch methods use a sequence of (possibly randomized) data batches $D_{1},\ldots,D_{T}$ to produce a sequence of regressors $Q_{\theta_{1}},\ldots,Q_{\theta_{T}}=Q_{\theta}$, estimating the Q-function.111We describe our approach using batch Q-learning, but it can accommodate many variants, e.g., where the estimators generating max-actions and value estimates are different, as in double Q-learning (van Hasselt, 2010; Hasselt et al., 2016); indeed, we experiment with such variants. For each $(s,a,r,s^{\prime})\in D_{k}$, we use a prior estimator $Q_{\theta_{k-1}}$ to bootstrap the _Q-label_ $q=r+\gamma\max_{a^{\prime}}Q_{\theta_{k-1}}(s^{\prime},a^{\prime})$. We then fit $Q_{\theta_{k}}$ to this data using a regression procedure with a suitable loss function. Once trained, the (implicit) induced policy $\pi_{\theta}$ is the _greedy policy_ w.r.t. $Q_{\theta}$, i.e., $\pi_{\theta}(s)=\operatorname*{arg\\!max}_{a\in A}Q_{\theta}(s,a)$. Let $\mathcal{F}(\Theta)$ (resp., $G(\Theta)$) be the class of expressible Q-functions (resp., greedy policies). Intuitively, _delusional bias_ occurs whenever a backed-up value estimate is derived from action choices that are not (jointly) realizable in $G(\Theta)$ (Lu et al., 2018). Standard Q-updates back up values for each $(s,a)$ pair by _independently_ choosing maximizing actions at the corresponding next states $s^{\prime}$. However, such updates may be “inconsistent” under approximation: if no policy in $G(\Theta)$ can jointly express all past action choices, _backed up values may not be realizable by any expressible policy_. Lu et al. (2018) show that delusion can manifest itself with several undesirable consequences (e.g., divergence). Most critically, it can prevent Q-learning from learning the optimal representable policy in $G(\Theta)$. To address this, they propose a non-delusional _policy consistent_ Q-learning (PCQL) algorithm that provably eliminates delusion. We refer to the original paper for details, but review the main concepts.222While delusion may not arise in other RL approaches (e.g., policy iteration, policy gradient), our contribution focuses on mitigating delusion to derive maximum performance from widely used Q-learning methods. The first key concept is that of _policy consistency_. For any $S\subseteq\mathcal{S}$, an _action assignment_ $\sigma_{S}:S\rightarrow A$ associates an action $\sigma(s)$ with each $s\in S$. We say $\sigma$ is _policy consistent_ if there is a greedy policy $\pi\in G(\Theta)$ s.t. $\pi(s)=\sigma(s)$ for all $s\in S$. We sometimes equate a set $\mathit{SA}$ of state-action pairs with the implied assignment $\pi(s)=a$ for all $(s,a)\in\mathit{SA}$. If $\mathit{SA}$ contains multiple pairs with the same state $s$, but different actions $a$, it is a _multi-assignment_ (we use the term “assignment” when there is no risk of confusion). In (batch) Q-learning, each new regressor uses training labels generated by assuming maximizing actions (under the prior regressor) are taken at its successor states. Let $\sigma_{k}$ be the collection of states and corresponding maximizing actions used to generate labels for regressor $Q_{\theta_{k}}$ (assume it is policy consistent). Suppose we train $Q_{\theta_{k}}$ by bootstrapping on $Q_{\theta_{k-1}}$. Now consider a training sample $(s,a,r,s^{\prime})$. Q-learning generates label $r+\gamma\max_{a^{\prime}}Q_{\theta_{k-1}}(s^{\prime},a^{\prime})$ for input $(s,a)$. Notice, however, that taking action $a^{*}=\operatorname*{arg\\!max}_{a^{\prime}}Q_{\theta_{k}}(s^{\prime},a^{\prime})$ at $s^{\prime}$ may not be _policy consistent_ with $\sigma_{k}$. Thus Q-learning will estimate a value for $(s,a)$ assuming execution of a policy that cannot be realized given the approximator. PCQL prevents this by ensuring that any assignment used to generate labels is consistent with earlier assignments. This means Q-labels will often _not_ be generated using maximizing actions w.r.t. the prior regressor. The second key concept is that of _information sets_. One will generally not be able to use maximizing actions to generate labels, so tradeoffs can be made when deciding which actions to assign to different states. Indeed, even if it is feasible to assign a maximizing action $a$ to state $s$ early in training, say at batch $k$, since it may prevent assigning a maximizing $a^{\prime}$ to $s^{\prime}$ later, say batch $k+\ell$, we may want to use a different assignment to $s$ to give more flexibility to maximize at other states later. PCQL does not anticipate the tradeoffs—rather it maintains _multiple information sets_ , each corresponding to a different assignment to the states seen in the training data this far. Each gives rise to a _different Q-function estimate_ , resulting in multiple hypotheses. At the end of training, the best hypothesis is that with maximum expected value w.r.t. an initial state distribution. PCQL provides strong convergence guarantees, but it is a tabular algorithm: the function approximator _restricts_ the policy class, but is not used to generalize Q-values. Furthermore, its theoretical guarantees come at a cost: it uses _exact_ policy consistency tests—tractable for linear approximators, but impractical for large problems and DQN; and it maintains _all_ consistent assignments. As a result, PCQL cannot be used for large RL problems of the type tackled by DQN. ## 3 The ConQUR Framework We develop the ConQUR framework to provide a practical approach to reducing delusion in Q-learning, specifically addressing the limitations of PCQL identified above. ConQUR consists of three main components: a practical soft- constraint penalty that promotes policy consistency; a search space to structure the search over multiple regressors (information sets, action assignments); and heuristic search schemes (expansion, scoring) to find good Q-regressors. ### 3.1 Preliminaries We assume a set of training data consisting of quadruples $(s,a,r,s^{\prime})$, divided into (possibly non-disjoint) _batches_ $D_{1},\ldots,D_{T}$ for training. This perspective is quite general: online RL corresponds to $|D_{i}|=1$; offline batch training (with sufficiently exploratory data) corresponds to a single batch (i.e., $T=1$); and online or batch methods with replay are realized when the $D_{i}$ are generated by sampling some data source with replacement. For any batch $D$, let $\chi(D)=\\{s^{\prime}:(s,a,r,s^{\prime})\in D\\}$ be the set of _successor states_ of $D$. An _action assignment_ $\sigma_{D}$ for $D$ is an assignment (or multi-assignment) from $\chi(D)$ to $A$, dictating which action $\sigma_{D}(s^{\prime})$ is considered “maximum” when generating a Q-label for pair $(s,a)$; i.e., $(s,a)$ is assigned training label $r+\gamma Q(s^{\prime},\sigma(s^{\prime}))$ rather than $r+\gamma\max_{a^{\prime}\in A}Q(s^{\prime},a^{\prime})$. The set of all such assignments $\Sigma(D)=A^{\chi(D)}$ grows exponentially with $|D|$. Given a Q-function parameterization $\Theta$, we say $\sigma_{D}$ is _$\Theta$ -consistent (w.r.t. $D$)_ if there is some $\theta\in\Theta$ s.t. $\pi_{\theta}(s^{\prime})=\sigma(s^{\prime})$ for all $s^{\prime}\in\chi(D)$.333We suppress mention of $D$ when clear from context. This is simple policy consistency, but with notation that emphasizes the policy class. Let $\Sigma_{\Theta}(D)$ denote the set of all $\Theta$-consistent assignments over $D$. The union $\sigma_{1}\cup\sigma_{2}$ of two assignments (over $D_{1},D_{2}$, resp.) is defined in the usual way. ### 3.2 Consistency Penalization Enforcing strict $\Theta$-consistency as regressors $\theta_{1},\theta_{2},\ldots,\theta_{T}$ are generated is computationally challenging. Suppose the assignments $\sigma_{1},\ldots,\sigma_{k-1}$, used to generate labels for $D_{1},\ldots D_{k-1}$, are jointly $\Theta$-consistent (let $\sigma_{\leq k-1}$ denote their multi-set union). Maintaining $\Theta$-consistency when generating $\theta_{k}$ imposes two requirements. First, one must generate an assignment $\sigma_{k}$ over $D_{k}$ s.t. $\sigma_{\leq k-1}\cup\sigma_{k}$ is consistent. Even testing assignment consistency can be problematic: for linear approximators this is a linear feasibility program (Lu et al., 2018) whose constraint set grows linearly with $|D_{1}\cup\ldots\cup D_{k}|$. For DNNs, this is a complex, more expensive polynomial program. Second, the regressor $\theta_{k}$ should itself be consistent with $\sigma_{\leq k-1}\cup\sigma_{k}$. This too imposes a severe burden on regression optimization: in the linear case, it is a constrained least-squares problem (solvable, e.g., as a quadratic program); while with DNNs, it can be solved, say, using a more involved projected SGD. However, the sheer number of constraints makes this impractical. Rather than enforcing consistency, we propose a simple, computationally tractable scheme that “encourages” it: a penalty term that can be incorporated into the regression itself. Specifically, we add a penalty function to the usual squared loss to encourage updates of the Q-regressors to be consistent with the underlying information set, i.e., the prior action assignments used to generate its labels. When constructing $\theta_{k}$, let $D_{\leq k}=\cup\\{D_{j}:j\leq k\\}$, and $\sigma\in\Sigma_{\Theta}({D_{\leq k}})$ be the collective assignment used to generate labels for all prior regressors (including $\theta_{k}$ itself). The multiset of pairs $B=\\{(s^{\prime},\sigma(s^{\prime}))|s^{\prime}\in\chi(D_{\leq k})\\}$, is called a _consistency buffer_. The assignment need not be consistent (as we elaborate below), nor does regressor $\theta_{k}$ need to be consistent with $\sigma$. Instead, we use the following _soft consistency penalty_ when constructing $\theta_{k}$: $\displaystyle C_{\theta}(s^{\prime},a)$ $\displaystyle=\sum_{a^{\prime}\in A}[Q_{\theta}(s^{\prime},a^{\prime})-Q_{\theta}(s^{\prime},a)]_{+},$ (2) $\displaystyle C_{\theta}(B)$ $\displaystyle=\sum_{(s^{\prime},\sigma(s^{\prime}))\in B}C_{\theta}(s^{\prime},\sigma(s^{\prime})),$ (3) where $[x]_{+}=\max(0,x)$. This penalizes Q-values of actions at state $s$ that are larger than that of action $\sigma(s)$. Notice $\sigma$ is $\Theta$-consistent iff $\min_{\theta\in\Theta}C_{\theta}(B)=0$. We add this penalty into our regression loss for batch $D_{k}$: $L_{\theta}(D_{k},B)=\sum_{(s,a,r,s^{\prime})\in D_{k}}\Big{[}r+\gamma Q_{\theta_{k-1}}(s^{\prime},\sigma(s^{\prime}))-\\\ Q_{\theta}(s,a)\Big{]}^{2}+\lambda C_{\theta}(B).$ (4) Here $Q_{\theta_{k}}$ is the prior estimator on which labels are bootstrapped (other regressors may be used). The penalty effectively acts as a “regularizer” on the squared Bellman error, where $\lambda$ controls the degree of penalization, allowing a tradeoff between Bellman error and consistency with the assignment used to generate labels. It thus _promotes_ consistency without incurring the expense of _enforcing_ strict consistency. It is straightforward to replace the classic Q-learning update (1) with one using our consistency penalty: $\theta_{k}\leftarrow\theta_{k-1}+\sum_{(s,a,r,s^{\prime})\in D_{k}}\alpha\Big{[}r+\gamma Q_{\theta_{k-1}}(s^{\prime},\sigma(s^{\prime}))\\\ -Q_{\theta}(s,a)\Big{]}\nabla_{\theta}Q_{\theta}(s,a)+\alpha\lambda\nabla_{\theta}C_{\theta}(B)\Bigr{\rvert}_{\theta=\theta_{k-1}}.$ (5) This scheme is quite general. First, it is agnostic as to how the prior action assignments are made (e.g., standard maximization w.r.t. the prior regressor as in DQN, Double DQN (DDQN) (Hasselt et al., 2016), or other variants). It can also be used in conjunction with a search through alternate assignments (see below). Second, the consistency buffer $B$ may be populated in a variety of ways. Including all max-action choices from all past training batches promotes full consistency. However, this may be too constraining since action choices early in training are generally informed by inaccurate value estimates. $B$ may be implemented to focus only on more recent data (e.g., with a sliding recency window, weight decay, or subsampling); and the degree of recency bias may adapt during training (e.g., becoming more inclusive as training proceeds and the Q-function converges). Reducing the size of $B$ also has computational benefits. We discuss other ways of promoting consistency in Sec. 5. The proposed consistency penalty resembles the temporal-consistency loss of Pohlen et al. (2018), but our aims are very different. Their temporal consistency notion penalizes changes in a next state’s Q-estimate over all actions, whereas we discourage inconsistencies in the greedy policy induced by the Q-estimator, regardless of the actual estimated values. ### 3.3 The Search Space Figure 1: A generic search tree. Ensuring optimality requires that PCQL track _all $\Theta$-consistent assignments_. While the set of such assignments has polynomial size (Lu et al., 2018), it is impractical to track in realistic problems. As such, in ConQUR we recast information set tracking as a _search problem_ and propose several strategies for managing the search process. We begin by defining the search space and discussing its properties. We discuss search procedures in Sec. 3.4. As above, assume training data is divided into batches $D_{1},\ldots,D_{T}$ and we have some initial Q-function estimate $\theta_{0}$ (for bootstrapping $D_{1}$’s labels). The regressor $\theta_{k}$ for $D_{k}$ can, in principle, be trained with labels generated by _any assignment_ $\sigma\in\Sigma_{\Theta}(D_{k})$ of actions to its successor states $\chi(D_{k})$, not necessarily maximizing actions w.r.t. $\theta_{k-1}$. Each $\sigma$ gives rise to a different updated Q-estimator $\theta_{k}$. There are several restrictions we can place on “reasonable” $\sigma$-candidates: (i) $\sigma$ is $\Theta$-consistent; (ii) $\sigma$ is jointly $\Theta$-consistent with all $\sigma_{j}$, for $j<k$, used to construct the prior regressors on which we bootstrap $\theta_{k-1}$; (iii) $\sigma$ is not _dominated_ by any $\sigma^{\prime}\in\Sigma_{\Theta}(D_{k})$, where we say $\sigma^{\prime}$ dominates $\sigma$ if $Q_{\theta_{k-1}}(s^{\prime},\sigma^{\prime}(s^{\prime}))\geq Q_{\theta_{k-1}}(s^{\prime},\sigma(s^{\prime}))$ for all $s^{\prime}\in\chi(D)$, and this inequality is strict for at least one $s^{\prime}$. Conditions (i) and (ii) are the strict consistency requirements of PCQL. We relax these below as discussed in Sec. 3.2. Condition (iii) is inappropriate in general, since we may add additional assignments (e.g., to new data) that render all non-dominated assignments inconsistent, requiring that we revert to some dominated assignment. This gives us a generic _search space_ for finding policy-consistent, delusion-free Q-function (see Fig. 1). Each node $n^{i}_{k}$ at depth $k$ in the search tree is associated with a regressor $\theta^{i}_{k}$ defining $Q_{\theta^{i}_{k}}$ and assignment $\sigma^{i}_{k}$ that justifies the labels used to train $\theta^{i}_{k}$ ($\sigma^{i}_{k}$ can be viewed as an information set). The root $n_{0}$ is based on an initial $\theta_{0}$, and has an empty assignment $\sigma_{0}$. Nodes at level $k$ of the tree are defined as follows. For each node $n_{k-1}^{i}$ at level $k-1$—with regressor $\theta_{k-1}^{i}$ and $\Theta$-consistent assignment $\sigma_{k-1}^{i}$—we have one child $n_{k}^{j}$ for each $\sigma_{k}^{j}\in\Sigma_{\Theta}(D_{k})$ such that $\sigma_{k-1}^{i}\cup\sigma_{k}^{j}$ is $\Theta$-consistent. Node $n_{k}^{j}$’s assignment is $\sigma_{k-1}^{i}\cup\sigma_{k}^{j}$, and its regressor $\theta_{k}^{i}$ is trained using the data set: $\\{(s,a)\mapsto r+\gamma Q_{\theta_{k-1}^{i}}(s^{\prime},\sigma_{k}^{j}(s^{\prime}))\,:\,(s,a,r,s^{\prime})\in D_{k}\\}.$ The entire search space constructed in this fashion to a maximum depth of $T$. See Appendix B, Algorithm 1 for pseudocode of a simple depth-first recursive specification. The exponential branching factor in this search tree would appear to make complete search intractable; however, since we only allow $\Theta$-consistent “collective” assignments we can bound the size of the tree—it is _polynomial_ in the VC-dimension of the approximator. ###### Theorem 1. The number of nodes in the search tree is no more than $O(nm\cdot[\binom{m}{2}n]^{\operatorname*{\mathsf{VCDim}}(\mathcal{G})})$ where $\operatorname*{\mathsf{VCDim}}(\cdot)$ is the VC-dimension (Vapnik, 1998) of a set of boolean-valued functions, and $\mathcal{G}$ is the set of boolean functions defining all feasible greedy policies under $\Theta$: $\mathcal{G}=\\{g_{\theta}(s,a,a^{\prime}):={\mathbf{1}}[f_{\theta}(s,a)-f_{\theta}(s,a^{\prime})>0],\forall s,a\neq a^{\prime}~{}|~{}\theta\in\Theta\\}.$ A linear approximator with a fixed set of $d$ features induces a policy- indicator function class $\mathcal{G}$ with VC-dimension $d$, making the search tree polynomial in the size of the MDP. Similarly, a fixed ReLU DNN architecture with $W$ weights and $L$ layers has VC-dimension of size $O(WL\log W)$ again rendering the tree polynomially sized. Even with this bound, navigating the search space exhaustively is generally impractical. Instead, various search methods can be used to explore the space, with the aim of reaching a “high quality” regressor at some leaf of the tree. ### 3.4 Search Heuristics Even with the bound in Thm. 1, traversing the search space exhaustively is generally impractical. Moreover, as discussed above, enforcing consistency when generating the children of a node, and their regressors, may be intractable. Instead, various search methods can be used to explore the space, with the aim of reaching a “high quality” regressor at some (depth $T$) leaf of the tree. We outline three primary considerations in the search process: child generation, node evaluation or scoring, and the search procedure. Generating children. Given node $n^{i}_{k-1}$, there are, in principle, exponentially many action assignments, or children, $\Sigma_{\Theta}(D_{k})$ (though Thm. 1 limits this if we enforce consistency). Thus, we develop heuristics for generating a small set of children, driven by three primary factors. The first factor is a preference for generating _high-value assignments_. To accurately reflect the intent of (sampled) Bellman backups, we prefer to assign actions to state $s^{\prime}\in\chi(D_{k})$ with larger predicted Q-values i.e., a preference for $a$ over $a^{\prime}$ if ${Q}_{\theta_{k-1}^{j}}(s^{\prime},a)>{Q}_{\theta_{k-1}^{j}}(s^{\prime},a^{\prime})$. However, since the maximizing assignment may be $\Theta$-inconsistent (in isolation, jointly with the parent information set, or with future assignments), candidate children should merely have _higher probability_ of a high-value assignment. Second, we need to ensure _diversity_ of assignments among the children. Policy commitments at stage $k$ constrain the assignments at subsequent stages. In many search procedures (e.g., beam search), we avoid backtracking, so we want the stage-$k$ commitments to offer flexibility in later stages. The third factor is the degree to which we enforce consistency. There are several ways to generate high-value assignments. We focus on one natural technique: sampling action assignments using a Boltzmann distribution. Let $\sigma$ be the assignment of some node (parent) at level $k-1$ in the tree. We generate an assignment $\sigma_{k}$ for $D_{k}$ as follows. Assume some permutation $s_{1}^{\prime},\ldots,s^{\prime}_{|D_{k}|}$ of $\chi(D_{k})$. For each $s^{\prime}_{i}$ in turn, we sample $a_{i}$ with probability proportional to $e^{Q_{\theta_{k-1}}(s^{\prime}_{i},a_{i})/\tau}$. This can be done _without regard to consistency_ , in which case we use the consistency penalty when constructing the regressor $\theta_{k}$ for this child to “encourage” consistency rather than enforce it. If we want strict consistency, we can use rejection sampling without replacement to ensure $a_{i}$ is consistent with $\sigma^{j}_{k-1}\cup\sigma_{\leq i-1}$ (we can also use a subset of $\sigma^{j}_{k-1}$ as a less restrictive consistency buffer).444Notice that at least one action for state $s^{\prime}_{i}$ must be consistent with any previous (consistent) information set. The temperature parameter $\tau$ controls the degree to which we focus on maximizing assignments versus diverse, random assignments. While sampling gives some diversity, this procedure biases selection of high-value actions to states that occur early in the permutation. To ensure further diversity, we use a new random permutation for each child. Scoring children. Once the children of some expanded node are generated, we must assess the quality of each child to decide which new nodes to expand. One possiblity is to use the average Q-label (overall, or weighted using an initial distribution), Bellman error, or loss incurred by the regressor. However, care must be taken when comparing nodes at different depths of the tree. Since deeper nodes have a greater chance to accrue rewards or costs, simple calibration methods can be used. Alternatively, when a simulator is available, rollouts of the induced greedy policy can be used evaluate the node quality. However, rollouts incur considerable computational expense during training relative to the more direct scoring methods. Search Procedure. Given a method for generating and scoring children, different search procedures can be applied: best-first search, beam search, local search, etc. all fit very naturally within the ConQUR framework. Moreover, hybrid strategies are possible—one we develop below is a variant of beam search in which we generate multiple children only at certain levels of the tree, then do “deep dives” using consistency-penalized Q-regression at the intervening levels. This reduces the size of the search tree considerably and, when managed properly, adds only a constant-factor (proportional to beam size) slowdown to methods like DQN. ### 3.5 An Instantiation of the ConQUR Framework We now outline a specific instantiation of the ConQUR framework that effectively navigates the large search spaces that arise in practical RL settings. We describe a heuristic, modified beam-search strategy with backtracking and priority scoring. We outline only key features (see details in Algorithm 2, Appendix B). Our search process alternates between two phases. In an _expansion phase_ , parent nodes are expanded, generating one or more child nodes with assignments sampled from the Boltzmann distribution. For each child, we create target Q-labels, then optimize its regressor using consistency-penalized Bellman error Eq. 4, foregoing strict policy consistency. In a _dive phase_ , each parent generates _one_ child, whose action assignment is given by standard max-action selection w.r.t. the parent’s regressor. No diversity is considered but we continue to use consistency-penalized regression. From the root, the search begins with an expansion phase to create $c$ children—$c$ is the _splitting factor_. Each child inherits its parent’s consistency buffer to which we add the new assignments used for that child’s Q-labels. To limit the tree size, we track a subset of the children (the _frontier_), selected using some scoring function. We select the top $\ell$-nodes for expansion, proceed to a dive phase and iterate. We consider backtracking strategies that return to unexpanded nodes at shallower depths of the tree below. ### 3.6 Related Work Other work has considered multiple hypothesis tracking in RL. One direct approach uses ensembling, with multiple Q-approximators updated in parallel (Faußer & Schwenker, 2015; Osband et al., 2016; Anschel et al., 2017) and combined to reduce instability and variance. Population-based methods, inspired by evolutionary search, are also used. Conti et al. (2018) combine novelty search and quality diversity to improve hypothesis diversity and quality. Khadka & Tumer (2018) augment an off-policy RL method with diversified population information derived from an evolutionary algorithm. These techniques do not target a specific weaknesses of Q-learning, such as delusion. ## 4 Empirical Results We assess the performance of ConQUR using the Atari test suite (Bellemare et al., 2013). Since ConQUR directly tackles delusion, any performance improvement over Q-learning baselines strongly suggests the presence of delusional bias in the baselines in these domains. We first assess the impact of our consistency penalty in isolation (without search), treating it as a “regularizer” that promotes consistency with both DQN and DDQN. We then test our modified beam search to assess the full power of ConQUR. We do not directly compare ConQUR to policy gradient or actor-critic methods—which for some Atari games offer state-of-the-art performance (Schrittwieser et al., 2019; Kapturowski et al., 2020)—because our aim with ConQUR is to improve the performance of (widely used) Q-learning-based algorithms. ### 4.1 Consistency Penalization We first study the effects of augmenting both DQN and DDQN with soft-policy consistency in isolation. We train models using an open-source implementation of DQN and DDQN, using default hyperparameters (Guadarrama et al., 2018) . We refer to the consistency-augmented algorithms as $\mathsf{DQN}{(\lambda)}$ and $\mathsf{DDQN}{(\lambda)}$, respectively, where $\lambda$ is the penalty weight (see Eq. 4). When $\lambda=0$, these correspond to DQN and DDQN themselves. This policy-consistency augmentation is lightweight and can be applied readily to any regression-based Q-learning method. Since we do not use search (i.e., do not track multiple hypotheses), these experiments use a small consistency buffer drawn only from the _current_ data batch by sampling from the replay buffer—this prevents getting “trapped” by premature policy commitments. No diversity is used to generate action assignments—standard action maximization is used. We evaluate $\mathsf{DQN}{(\lambda)}$ and $\mathsf{DDQN}{(\lambda)}$ for $\lambda\in\\{0.25,0.5,1,1.5,2\\}$ on 19 Atari games.555These 19 games were selected arbitrarily simply to test soft-consistency in isolation. See Appendix C for details. In training, $\lambda$ is initialized at $0$ and annealed to the desired value to avoid premature commitment to poor assignments.666The annealing schedule is $\lambda_{t}=\lambda_{\textrm{final}}t/(t+2\times 10^{6})$. Without annealing, the model tends anchor on poorly informed assignments during early training, adversely impacting performance. Unsurprisingly, the best $\lambda$ tends to differ across games depending on the extent of delusional bias. Despite this, $\lambda=0.5$ works well across all games tested. Fig. 2 illustrates the effect of increasing $\lambda$ on two games. In Gravitar, it results in better performance in both $\mathsf{DQN}{}$ and $\mathsf{DDQN}{}$, while in SpaceInvaders, $\lambda=0.5$ improves both baselines, but relative performance degrades at $\lambda=2$. We also compare performance on each game for each $\lambda$ value, as well as using the best $\lambda_{\textrm{best}}$ (see Fig. 8, Table 3 in Appendix C.4). $\mathsf{DQN}{(\lambda_{\text{best}})}$ and $\mathsf{DDQN}{(\lambda_{\text{best}})}$ outperform their “potentially delusional” counterparts in all but 3 and 2 games, respectively. In 9 games, both $\mathsf{DQN}{(\lambda_{\text{best}})}$ and $\mathsf{DDQN}{(\lambda_{\text{best}})}$ beat _both_ baselines. With a fixed $\lambda=0.5$, $\mathsf{DQN}{(\lambda)}$ and $\mathsf{DDQN}{(\lambda)}$ each beat their respective baseline in 11 games. These results suggest that consistency penalization—independent of the general ConQUR model—can improve the performance of DQN and DDQN by addressing delusional bias. Moreover, promoting policy consistency appears to have a different effect on learning than double Q-learning, which addresses maximization bias. Indeed, consistency penalization, when applied to $\mathsf{DQN}{}$, achieves greater gains than $\mathsf{DDQN}{}$ in 15 games. Finally, in 9 games $\mathsf{DDQN}{(\lambda)}$ improves unaugmented $\mathsf{DQN}{(\lambda)}$. Further experiment details and results can be found in Appendix C. Figure 2: Varying penalization $\lambda$ (no search procedure). ### 4.2 Full ConQUR Figure 3: Policy value (game scores) of nodes, sorted. Figure 4: Training curves of 8 sorted nodes. Figure 5: Performance effects of varying $\lambda$. Figure 6: Improvements of ConQUR($\lambda=10$) over multi-DQN baseline on all 59 games. A frontier $F=16$ nodes was used. We test the full ConQUR framework using our modified beam search (Sec. 3.5) on the full suite of 59 Atari games. Rather than training a full Q-network using ConQUR, we leverage pre-trained networks from the Dopamine package (Castro et al., 2018),777See https://github.com/google/dopamine and use ConQUR to learn final layer weights, i.e., a new “linear approximator” w.r.t. the learned feature representation. We do this for two reasons. First, this allows us to test whether delusional bias occurs in practice. By freezing the learned representation, any improvements offered by ConQUR when learning a linear Q-function over those same features provides direct evidence that (a) delusion is present in the original trained baselines, and (b) ConQUR does in fact mitigate its impact (without relying on novel feature discovery). Second, from a practical point of view, this “linear tuning” approach offers a relatively inexpensive way to apply our methodology in practice. By bootstrapping a model trained in standard fashion and extracting performance gains with a relatively small amount of additional training (e.g., linear tuning requires many fewer training samples, as our results show), we can offset the cost of the ConQUR search process itself. We use DQN-networks with the same architecture as in Mnih et al. (2015), trained on 200M frames as our baseline. We use ConQUR to retrain only the last (fully connected) layer (freezing other layers), which can be viewed as a linear Q-approximator over the features learned by the CNN. We train Q-regressors in ConQUR using _only 4M additional frames_.888This reduces computational/memory footprint of our experiments, and suffices since we re- train a simpler approximator. Nothing in the framework _requires_ this reduced training data. We use a splitting factor of $c=4$ and frontier size 16. The dive phase is always of length nine (i.e., nine batches of data), giving an expansion phase every ten iterations. Regressors are trained using soft-policy consistency (Eq. 4), with the consistency buffer comprising _all_ prior action assignments. We run ConQUR with $\lambda\in\\{1,10\\}$ and select the best performing policy. We use larger $\lambda$ values than in Sec. 4.1 since full ConQUR maintains multiple Q-regressors and can “discard” poor performers. This allows more aggressive consistency enforcement—in the extreme, with exhaustive search and $\lambda\to\infty$, ConQUR behaves like PCQL, finding a near- optimal greedy policy. See Appendix D for further details (e.g., hyperparameters) and results. We first test two approaches to scoring nodes: (i) policy evaluation using rollouts; and (ii) scoring using the loss function (Bellman error with soft consistency). Results on a small selection of games are shown in Table 1. While rollouts, unsurprisingly, tend to induce better-performing policies, consistent-Bellman scoring is competitive. Since the latter much less computationally intense, and does not require a simulator (or otherwise sampling the environment), we use it throughout our remaining experiments. We next compare ConQUR with the value of the pre-trained DQN. We also evaluate a “multi-DQN” baseline that trains multiple DQNs independently, warm-starting from the same pre-trained DQN. It uses the same number of frontier nodes as ConQUR, and is trained identically to ConQUR, but uses direct Bellman error (no consistency penalty). This gives DQN the same advantage of multiple- hypothesis tracking as ConQUR (without its policy consistency). | Rollouts | Bellman + Consistency Penalty ---|---|--- BattleZone | 33796.30 | 32618.18 BeamRider | 9914.00 | 10341.20 Boxing | 83.34 | 83.03 Breakout | 379.21 | 393.00 MsPacman | 5947.78 | 5365.06 Seaquest | 2848.04 | 3000.78 SpaceInvader | 3442.31 | 3632.25 StarGunner | 55800.00 | 56695.35 Zaxxon | 11064.00 | 10473.08 Table 1: Results of ConQUR with 8 nodes (split 2) on 9 games: comparing loss- based node scoring with scoring using rollouts. We test on 59 games. ConQUR with frontier size 16 and expansion factor 4 and splitting factor 4 (16-4-4) with backtracking (as described in the Appendix D) results in significant improvements over the pre-trained DQN, with an average score improvement of 189%. The only games without improvement are Montezuma’s Revenge, Tennis, Freeway, Pong, PrivateEye and BankHeist. This demonstrates that, _even when simply retraining the last layer of a highly tuned DQN network_ , removing delusional bias frequently improves policy performance significantly. ConQUR exploits the reduced parameterization to obtain these gains with only 4M frames of training data. A half-dozen games have outsized improvements over pre-trained DQN, including Venture (35 times greater value), ElevatorAction (23 times), Tutankham (5 times) and Solaris (5 times).999This may be in part, but not fully, due to the sticky-action training of the pre- trained model. We found that $\lambda=10$ provided the best performance across all games. Fig. 6 shows the percentage improvement of ConQUR($\lambda=10$) over the multi-DQN baseline for all 59 games. The improvement is defined as $(s_{C}-s_{B})/|s_{B}|$ where $s_{C}$ and $s_{B}$ are the average scores (over 5 runs) of the policy generated by ConQUR and that by the multi-DQN baseline (16 nodes), respectively. Compared to this stronger baseline, ConQUR wins by a margin of at least 10% in 16 games, while 19 games see improvements of 1–10%, 16 games show little effect ($\pm 1$%) and 8 games show a decline of greater than 1%. Tables of complete results and figures of training curves (all games) appears in Appendix D.3, Table 4 and Fig. 11. Figs. 3 and 4 (smoothed, best frontier node) show node policy values and training curves, respectively, for Solaris. When examining nodes ranked by their policy value (Fig. 3), we see that nodes of any given rank generated by ConQUR dominate their by multi-DQN (baseline) counterparts: the three highest- ranked nodes exceed their baseline counterparts by 18%, 13% and 15%, respectively, while the remaining nodes show improvements of roughly 11–12%. Fig. 5 (smoothed, best frontier node) shows the effect of varying $\lambda$. In Alien, increasing $\lambda$ from 1 to 10 improves performance, but performance starts to decline for higher $\lambda$ (we tested both 100 and 1000). This is similar to patterns observed in Sec. 4.1 and represents a trade-off between emphasizing consistency and not over-committing to action assignments. In Atlantis, stronger penalization tends to degrade performance. In fact, the stronger the penalization, the worse the performance. ## 5 Concluding Remarks We have introduced ConQUR, a framework for mitigating delusional bias in various forms of Q-learning that relaxes some of the strict assumptions of exact delusion-free algorithms like PCQL to ensure scalability. Its main components are a search procedure used to maintain diverse, promising Q-regressors (and corresponding information sets); and a consistency penalty that encourages “maximizing” actions to be consistent with the approximator class. ConQUR embodies elements of both value-based and policy-based RL: it can be viewed as using partial policy constraints to bias the Q- value estimator, and as a means of using candidate value functions to bias the search through policy space. Empirically, we find that ConQUR can improve the quality of existing approximators by removing delusional bias. Moreover, the consistency penalty applied on its own, in either DQN or DDQN, can improve policy quality. There are many directions for future research. Other methods for nudging regressors to be policy-consistent include exact consistency (i.e., constrained regression), other regularization schemes that push the regressor to fall within the information set, etc. Further exploration of search, child- generation, and node-scoring strategies should be examined within ConQUR. Our (full) experiments should also be extended beyond those that warm-start from a DQN model. 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In _Proceedings of the International Conference on Machine Learning (ICML-16)_ , 2016. * Watkins (1989) Watkins, C. J. C. H. _Learning from Delayed Rewards_. PhD thesis, King’s College, Cambridge, UK, May 1989. * Watkins & Dayan (1992) Watkins, C. J. C. H. and Dayan, P. Q-learning. _Machine Learning_ , 8:279–292, 1992. ## Appendix A An Example of Delusional Bias We describe an example, taken directly from (Lu et al., 2018), to show concretely how delusional bias causes problems for Q-learning with function approximation. The MDP in Fig. 7 illustrates the phenomenon: Lu et al. (2018) use a linear approximator over a specific set of features in this MDP to show that: 1. (a) No $\pi\in G(\Theta)$ can express the optimal (unconstrained) policy (which requires taking $a_{2}$ at each state); 2. (b) The optimal _feasible_ policy in $G(\Theta)$ takes $a_{1}$ at $s_{1}$ and $a_{2}$ at $s_{4}$ (achieving a value of $0.5$). 3. (c) Online Q-learning (Eq. 1) with data generated using an $\varepsilon$-greedy behavior policy must converge to a fixed point (under a range of rewards and discounts) corresponding to a “compromise” admissible policy which takes $a_{1}$ at both $s_{1}$ and $s_{4}$ (value of $0.3$). Q-learning fails to find a reasonable fixed-point because of delusion. Consider the backups at $(s_{2},a_{2})$ and $(s_{3},a_{2})$. Suppose $\hat{\theta}$ assigns a “high” value to $(s_{3},a_{2})$, so that $Q_{\hat{\theta}}(s_{3},a_{2})>Q_{\hat{\theta}}(s_{3},a_{1})$ as required by $\pi_{\theta^{\ast}}$. They show that any such $\hat{\theta}$ also accords a “high” value to $(s_{2},a_{2})$. But $Q_{\hat{\theta}}(s_{2},a_{2})>Q_{\hat{\theta}}(s_{2},a_{1})$ is inconsistent the first requirement. As such, any update that makes the Q-value of $(s_{2},a_{2})$ higher _undercuts the justification_ for it to be higher (i.e., makes the “max” value of its successor state $(s_{3},a_{2})$ lower). This occurs not due to approximation error, but the inability of Q-learning to find the value of the optimal _representable_ policy. Figure 7: A simple MDP (Lu et al., 2018). ## Appendix B Algorithms The pseudocode of (depth-first) version of the ConQUR search framework is listed in Algorithm 1. Algorithm 1 ConQUR Search (Generic, depth-first) 0: Data sets $D_{k},D_{k+1},\ldots D_{T}$; regressor $\hat{Q}_{k-1}$; and assignment $\sigma$ over $D_{\leq k-1}=\cup_{1\leq j\leq k-1}D_{j}$ reflecting prior data; policy class $\Theta$. 1: Let $\Sigma_{\Theta,\sigma}=\\{\sigma_{k}\in\Sigma_{\Theta}(D_{j}):\sigma_{k}\cup\sigma\textrm{ is consistent}\\}$ 2: for all $\sigma_{k}^{j}\in\Sigma_{\Theta,\sigma}$ do 3: Training set $S\leftarrow\\{\\}$ 4: for all $(s,a,r,s^{\prime})\in D_{k}$ do 5: $q\leftarrow r+\gamma\hat{Q}_{k-1}(s^{\prime},\sigma_{k}^{j}(s^{\prime}))$ 6: $S\leftarrow S\cup\\{((s,a),q)\\}$ 7: end for 8: Train $\hat{Q}^{j}_{k}$ using training set $S$ 9: if $k=T$ then 10: Return $\hat{Q}^{j}_{k}$ // terminate 11: else 12: Return Search($D_{k+1},\ldots D_{T};\,\hat{Q}^{j}_{k};\,\sigma_{k}^{j}\cup\sigma;\,\Theta$) // recurse 13: end if 14: end for As discussed in Sec. 3.5, a more specific instantiation of the ConQUR algorithm is listed in Algorithm 2. Algorithm 2 Modified Beam Search Instantiation of ConQUR Algorithm 0: Search control parameters: $m$, $\ell$, $c$, $d$, $T$ 1: Maintain list of data batches $D_{1},...,D_{k}$, initialized empty 2: Maintain candidate pool $P$ of at most $m$ nodes, initialized $P=\\{n_{0}\\}$ 3: Maintain frontier list $F$ of $\ell^{c}$ nodes 4: Maintain for each node $n^{i}_{k}$ a regressor $\theta^{i}_{k}$ and an ancestor assignment $\sigma^{i}_{k}$ 5: for each search level $k\leq T$ do 6: Find top scoring node $n^{1}\in P$ 7: Use $\varepsilon$-greedy policy extracted from $Q_{\theta^{1}}$ to collect next data batch $D_{k}$ 8: if $k$ is an expansion level then 9: Select top $\ell$ scoring nodes $n^{1},...,n^{\ell}\in P$ 10: for each selected node $n^{i}$ do 11: Generate $c$ children $n^{i,1},...,n^{i,c}$ using Boltzmann sampling on $D_{k}$ with $Q_{\theta^{i}}$ 12: for each child $n^{i,j}$ do 13: Let assignment history $\sigma^{i,j}$ be $\sigma^{i}\cup\\{\textit{new assignment}\\}$ 14: Determine regressor $\theta^{i,j}$ by applying update (5) from $\theta^{i}$ 15: end for 16: Score and add child nodes to the candidate pool $P$ 17: Assign frontier nodes to set of child nodes, $F=\\{n^{i,j}\\}$ 18: if $|P|>m$ then 19: evict bottom scoring nodes, keeping top $m$ in $P$ 20: end if 21: end for 22: end if 23: if $k$ is a refinement (”dive”) level then 24: for each frontier node $n^{i,j}\in F$ do 25: Update regressor $\theta^{i,j}$ by applying update (5) to $\theta^{i,j}$ 26: end for 27: end if 28: Run $d$ ”dive” levels after each expansion level 29: end for ## Appendix C Additional Detail: Effects of Consistency Penalization ### C.1 Delusional bias in DQN and DDQN Both DQN and DDQN uses a delayed version of the $Q$-network $Q_{\theta^{-}}(s^{\prime},a^{\prime})$ for label generation, but in a different way. In DQN, $Q_{\theta^{-}}(s^{\prime},a^{\prime})$ is used for both value estimate and action assignment $\sigma_{\text{DQN}}(s^{\prime})=\operatorname*{arg\\!max}_{a^{\prime}}Q_{\theta_{k}}(s^{\prime},a^{\prime})$, whereas in DDQN, $Q_{\theta^{-}}(s^{\prime},a^{\prime})$ is used only for value estimate and the action assignment is computed from the current network $\sigma_{\text{DDQN}}(s^{\prime})=\operatorname*{arg\\!max}_{a^{\prime}}Q_{\theta_{k}}(s^{\prime},a^{\prime})$. With respect to delusional bias, action assignment of DQN is consistent for all batches after the latest network weight transfer, as $\sigma_{\text{DQN}}(s^{\prime})$ is computed from the same $Q_{\theta^{-}}(s^{\prime},a^{\prime})$ network. DDQN, on the other hand, could have very inconsistent assignments, since the action is computed from the current network that is being updated at every step. ### C.2 Training Methodology and Hyperparameters We implement consistency penalty on top of the DQN and DDQN algorithm by modifying the open-source TF-Agents library (Guadarrama et al., 2018). In particular, we modify existing DqnAgent and DdqnAgent by adding a consistency penalty term to the original TD loss. We use TF-Agents implementation of DQN training on Atari with the default hyperparameters, which are mostly the same as that used in the original DQN paper (Mnih et al., 2015). For conveniece to the reader, some important hyperparameters are listed in Table 2. The reward is clipped between $[-1,1]$ following the original DQN. Hyper-parameter | Value ---|--- Mini-batch size | 32 Replay buffer capacity | 1 million transitions Discount factor $\gamma$ | 0.99 Optimizer | RMSProp Learning rate | 0.00025 Convolution channel | $32,64,64$ Convolution filter size | $(8\times 8),(4\times 4),(3\times 3)$ Convolution stride | 4, 2, 1 Fully-connected hidden units | 512 Train exploration $\varepsilon_{\text{train}}$ | 0.01 Eval exploration $\varepsilon_{\text{eval}}$ | 0.001 Table 2: Hyperparameters for training DQN and DDQN with consistency penalty on Atari. ### C.3 Evaluation Methodology We empirically evaluate our modified DQN and DDQN agents trained with consistency penalty on 15 Atari games. Evaluation is run using the training and evaluation framework for Atari provided in TF-Agents without any modifications. ### C.4 Detailed Results Fig. 8 shows the effects of varying $\lambda$ on both DQN and DDQN. Table 3 summarizes the best penalties for each game and their corresponding scores. Fig. 9 shows the training curves of the best penalization constants. Finally, Fig. 10 shows the training curves for a fixed penalization of $\lambda=0.5$. The datapoints in each plot of the aforementioned figures are obtained by averaging over window size of 30 steps, and within each window, we take the largest policy value (and over $\approx$2–5 multiple runs). This is done to reduce visual clutter. Figure 8: DQN and DDQN training curves for different penalty constant $\lambda$. | $\mathsf{DQN}{}$ | $\lambda_{\text{best}}$ | $\mathsf{DQN}{(\lambda_{\text{best}})}$ | $\mathsf{DDQN}{}$ | $\lambda^{\prime}_{\text{best}}$ | $\mathsf{DDQN}{(\lambda^{\prime}_{\text{best}})}$ ---|---|---|---|---|---|--- Assault | 2546.56 | 1.5 | 3451.07 | 2770.26 | 1 | 2985.74 Atlantis | 995460.00 | 0.5 | 1003600.00 | 940080.00 | 1.5 | 999680.00 BattleZone | 67500.00 | 2 | 55257.14 | 47025.00 | 2 | 48947.37 BeamRider | 7124.90 | 0.5 | 7216.14 | 5926.59 | 0.5 | 6784.97 Boxing | 86.76 | 0.5 | 90.01 | 82.80 | 0.5 | 91.29 Breakout | 220.00 | 0.5 | 219.15 | 214.25 | 0.5 | 242.73 Enduro | 1206.22 | 0.5 | 1430.38 | 1160.44 | 1 | 1287.50 Gravitar | 475.00 | 1.5 | 685.76 | 462.94 | 1.5 | 679.33 JourneyEscape | -1020.59 | 0.25 | -696.47 | -794.71 | 1 | -692.35 MsPacman | 4104.59 | 2 | 4072.12 | 3859.64 | 0.5 | 4008.91 NameThisGame | 7230.71 | 1 | 9013.48 | 9618.18 | 0.5 | 10210.00 Qbert | 13270.64 | 0.5 | 14111.11 | 13388.92 | 1 | 12884.74 Seaquest | 5849.80 | 1 | 6123.72 | 12062.50 | 1 | 7969.77 SpaceInvaders | 2389.22 | 0.5 | 2707.83 | 3007.72 | 0.5 | 4080.57 StarGunner | 40393.75 | 0.5 | 55931.71 | 55957.89 | 0.5 | 60035.90 TimePilot | 4205.83 | 2 | 7612.50 | 6654.44 | 2 | 7964.10 Tutankham | 222.76 | 1 | 265.86 | 243.20 | 0.25 | 247.17 VideoPinball | 569502.19 | 0.25 | 552456.00 | 509373.50 | 0.25 | 562961.50 Zaxxon | 5533.33 | 1 | 10520.00 | 7786.00 | 0.5 | 10333.33 Table 3: Consistency penalty ablation results on best penalty constants for DQN and DDQN. Figure 9: DQN and DDQN training curves for the respective best $\lambda$ and baseline. Figure 10: DQN and DDQN training curves for $\lambda=0.5$ and the baseline. ## Appendix D Additional Detail: ConQUR Results Our results use a frontier queue of size ($F$) 16 (these are the top scoring leaf nodes which receive gradient updates and rollout evaluations during training). To generate training batches, we select the best node’s regressor according to our scoring function, from which we generate training samples (transitions) using $\varepsilon$-greedy. Results are reported in Table 4, and training curves in Fig. 11. We used Bellman error plus consistency penalty as our scoring function. During the training process, we also calibrated the scoring to account for the depth difference between the leaf nodes at the frontier versus the leaf nodes in the candidate pool. We calibrated by taking the mean of the difference between scores of the current nodes in the frontier with their parents. We scaled this difference by multiplying with a constant of 2.5. In our implementation, we initialized our Q-network with a pre-trained DQN. We start with the expansion phase. During this phase, each parent node splits into $l$ children nodes and the Q-labels are generated using action assignments from the Boltzmann sampling procedure, in order to create high quality and diversified children. We start the dive phase until the number of children generated is at least $F$. In particular, with $F=16$ configuration, we performed the expansion phase at the zero-th and first iterations, and then at every tenth iteration starting at iteration 10, then at 20, and so on until ending at iteration 90. All other iterations execute the “dive” phase. For every fifth iteration, Q-labels are generated from action assignments sampled according to the Boltzmann distribution. For all other iterations, Q-labels are generated in the same fashion as the standard Q-learning (taking the max Q-value). The generated Q-labels along with the consistency penalty are then converted into gradient updates that applies to one or more generated children nodes. ### D.1 Training Methodology and Hyperparameters Each iteration consists of 10k transitions sampled from the environment. Our entire training process has 100 iterations which consumes 1M transitions or 4M frames. We used RMSProp as the optimizer with a learning rate of $2.5\times 10^{-6}$. One training iteration has 2.5k gradient updates and we used a batch size of 32. We replace the target network with the online network every fifth iteration and reward is clipped between $[-1,1]$. We use a discount value of $\gamma=0.99$ and $\varepsilon$-greedy with $\varepsilon=0.01$ for exploration. Details of hyper-parameter settings can be found in Table 5, 6. ### D.2 Evaluation Methodology We empirically evaluate our algorithms on 59 Atari games (Bellemare et al., 2013), and followed the evaluation procedure as in Hasselt et al. (2016). We evaluate our agents on every 10-th iteration (and also the initial and first iteration) by suspending our training process. We evaluate on 500k frames, and we cap the length of the episodes for 108k frames. We used $\varepsilon$-greedy as the evaluation policy with $\varepsilon=0.001$. We evaluated our algorithm under the _no-op starts_ regime—in this setting, we insert a random number of “do-nothing” (or _no-op_) actions (up to 30) at the beginning of each episode. ### D.3 Detailed Results Fig. 11 shows training curves of ConQUR with 16 nodes under different penalization strengths $\lambda\in\\{1,10\\}$. While each game has its own optimal $\lambda$, in general, we found that $\lambda=10$ gave the best performance for most games. Each plotted step of each training curve (including the baseline) shows the best performing node’s policy value as evaluated with full rollouts. Table 4 shows the summary of the highest policy values achieved for all 59 games for ConQUR and the baseline under 16 nodes. Both the baseline and ConQUR improve overall, but ConQUR’s advantage over the baseline is amplified. These results all use a splitting factor of $c=4$. (We show results with 8 nodes and a splitting factor of 2 below.) Figure 11: Training curves on 16 nodes with up to 30 no-op starts. | ConQUR($\lambda_{\text{best}}$) (16 nodes) | Baseline (16 nodes) | Checkpoint ---|---|---|--- AirRaid | 10613.01 | 9656.21 | 6916.88 Alien | 3253.18 | 2713.05 | 2556.64 Amidar | 555.75 | 446.88 | 203.08 Assault | 2007.81 | 2019.99 | 1932.55 Asterix | 5483.41 | 4649.52 | 2373.44 Asteroids | 1249.13 | 1196.56 | 701.96 Atlantis | 958932.00 | 931416.00 | 902216.00 BankHeist | 1002.59 | 965.34 | 872.91 BattleZone | 31860.30 | 32571.80 | 26559.70 BeamRider | 9009.14 | 9052.38 | 6344.91 Berzerk | 671.95 | 664.69 | 525.06 Bowling | 38.36 | 39.79 | 25.04 Boxing | 81.37 | 81.26 | 80.89 Breakout | 372.31 | 359.17 | 286.83 Carnival | 4889.19 | 4860.66 | 4708.14 Centipede | 4025.57 | 2408.23 | 758.21 ChopperCommand | 7818.22 | 6643.07 | 2991.00 CrazyClimber | 134974.00 | 119194.00 | 63181.14 DemonAttack | 11874.80 | 11445.20 | 7564.14 DoubleDunk | -14.04 | -15.25 | -16.66 ElevatorAction | 24.67 | 28.67 | 0.00 Enduro | 879.84 | 835.11 | 556.97 FishingDerby | 16.28 | 13.22 | 6.92 Freeway | 32.65 | 32.63 | 32.52 Frostbite | 289.25 | 230.29 | 166.44 Gopher | 11959.20 | 9084.00 | 4879.02 Gravitar | 489.22 | 446.64 | 394.46 Hero | 20827.00 | 20765.70 | 20616.30 IceHockey | -3.15 | -3.55 | -8.59 Jamesbond | 710.78 | 681.05 | 624.36 JourneyEscape | 902.22 | 1437.06 | -947.18 Kangaroo | 11017.65 | 10743.10 | 10584.20 Krull | 9556.53 | 9487.49 | 3998.90 MontezumaRevenge | 0.00 | 0.00 | 0.00 MsPacman | 5444.31 | 5487.12 | 4160.50 NameThisGame | 9104.40 | 8445.43 | 5524.73 Phoenix | 5325.33 | 5430.49 | 4801.18 Pitfall | 0.00 | 0.00 | -4.00 Pong | 21.00 | 21.00 | 20.95 Pooyan | 5898.46 | 5728.05 | 4393.09 PrivateEye | 100.00 | 100.00 | 100.00 Qbert | 13812.40 | 15189.00 | 8625.88 Riverraid | 15895.10 | 15370.10 | 11364.90 RoadRunner | 50820.40 | 47481.10 | 45073.25 Robotank | 62.74 | 57.66 | 53.08 Seaquest | 3046.34 | 2691.88 | 1060.77 Skiing | -13638.80 | -14908.21 | -29897.07 Solaris | 1991.33 | 1202.89 | 285.46 SpaceInvaders | 3556.10 | 3520.96 | 2895.30 StarGunner | 55679.27 | 55176.90 | 51490.60 Tennis | 0.00 | 0.00 | 0.00 TimePilot | 6698.88 | 7327.71 | 3806.54 Tutankham | 252.51 | 220.90 | 36.07 UpNDown | 31258.84 | 34455.20 | 5956.24 Venture | 37.37 | 3.64 | 0.00 VideoPinball | 423012.59 | 383105.41 | 268476.04 WizardOfWor | 8154.73 | 4782.11 | 2012.24 YarsRevenge | 26188.24 | 26330.31 | 25000.36 Zaxxon | 11723.20 | 11589.90 | 5334.44 Table 4: Summary of scores with $\varepsilon$-greedy ($\varepsilon=0.001$) evaluation with up to 30 no-op starts. We ran ConQUR with 16 nodes and with $\lambda\in\\{1,10\\}$. We report the scores of the best $\lambda_{\text{best}}$ for each game. For most games, $\lambda_{\text{best}}$ is $10$. Hyperparameters | Description | Value ---|---|--- Dive levels $d$ to run | We run $d$ levels of diving phase after each expansion phase | 9 Boltzmann Iteration | Every module this number of iteration/level, Q-labels are generated from Boltzmann distribution in order to create diversified node. | 5 Online network target network swap frequency | Iteration (Frequency) at which the online network parameters swap with the target network | 5 Evaluation frequency | Iteration (Frequency) at which we perform rollout operation (testing with the environment). | 10 Learning Rate | Learning rate for the optimizer. | $2.5\times 10^{-6}$ Optimizer | Optimizer for training the neural network. | RMSprop Iteration training data transition size | For each iteration, we generate this number of transitions and use it as training data. | 10k Training step frequency | For each iteration, we perform (iteration training data transition size / training step frequency) number of gradient updates. | 4 Mini-batch size | Size of the mini batch data used to train the Q-network. | 32 $\varepsilon_{\textrm{train}}$ | $\varepsilon$-greedy policy for exploration during training. | 0.01 $\varepsilon_{\textrm{eval}}$ | $\varepsilon$-greedy policy for evaluating Q-regressors. | 0.001 Training calibration parameter | Calibration to adjust the difference between the nodes from the candidate pool $m$ which didn’t selected during both the expansion nor the dive phases. The calibration is performed based on the average difference between the frontier nodes and their parents. We denote this difference as $\bigtriangleup$. | $2.5\bigtriangleup$ Temperature $\tau$ | Temperature parameter for Boltzmann sampling. Adaptively multiplied or divided by a factor of 1.5 or 4 respectively. | 1 Discount factor $\gamma$ | Discount factor during the training process. | 0.99 Table 5: Common Hyperparameters for ConQUR training and evaluation. Hyperparameters | Description | Value ---|---|--- Splitting factor $c$ | Number of children created from a parent node | 4 Candidate pool size $m$ | Pool of candidate leaf nodes for selection into the dive or expansion phase | 46 Maximum frontier nodes $F$ | Maximum number of child leaf nodes for the dive phase | 16 Top nodes to expand $l$ | Select the top $l$ nodes from the candidate pool for the expansion phase. | 4 Table 6: Hyperparameters for ConQUR (16 nodes) training and evaluation. Figure 12: Improvement ConQUR($\lambda=10$) with 16 nodes achieves over the initial checkpoint Q-network. ### D.4 Additional Results: ConQUR with 8 Nodes As an additional study of ConQUR, we present results of the running our method using 8 nodes (rather than the 16 used above), and compare it to a multi-DQN baseline that also uses 8 “nodes” (i.e., 8 separate DQN runs). We use a splitting factor $c=2$ for $\textsc{ConQUR}{}.$ Table 7 shows the average scores for each game using ConQUR and the baseline with 8 nodes. Unsurprisingly, ConQUR with 8 nodes does not perform as well as ConQUR with 16 nodes; but as in the 16-node case, ConQUR outperforms the baseline when each uses 8 nodes. More importantly, the average improvement of $24.5\%$ for ConQUR with 16 nodes over the corresponding baseline _exceeds_ the $19.6\%$ improvement of ConQUR in the 8-node case. This is a strong indication that increasing the number of nodes increases the performance gap _relative_ to the corresponding multi-DQN baseline; this implies that a good search heuristic is critical to effectively navigate the search space (as compared to randomly selected nodes) with a greater number of candidate hypotheses.101010Average score improvements exclude games where the baseline score is zero. | ConQUR($\lambda_{\text{best}}$) (8 nodes) | Baseline (8 nodes) | Checkpoint ---|---|---|--- AirRaid | 10647.80 | 9050.86 | 6885.72 Alien | 3341.36 | 3207.5.05 | 2556.64 Amidar | 577.45 | 573.55 | 202.74 Assault | 1892.02 | 1976.80 | 1873.05 Asterix | 5026.24 | 4935.21 | 2373.44 Asteroids | 1194.90 | 1170.11 | 704.38 Atlantis | 949012.00 | 932668.00 | 902216.00 BankHeist | 909.61 | 924.75 | 871.91 BattleZone | 32139.90 | 30983.10 | 26558.70 BeamRider | 8613.98 | 8109.63 | 6397.49 Berzerk | 659.64 | 634.83 | 524.76 Bowling | 30.07 | 25.29 | 25.04 Boxing | 81.78 | 81.48 | 80.29 Breakout | 350.11 | 362.98 | 286.14 Carnival | 4862.30 | 4800.83 | 4708.23 Centipede | 2747.89 | 2608.78 | 757.51 ChopperCommand | 7188.25 | 6737.21 | 2641.71 CrazyClimber | 131675.00 | 122424.00 | 63181.11 DemonAttack | 11346.20 | 10947.90 | 8022.08 DoubleDunk | -13.57 | -15.35 | -16.66 ElevatorAction | 28.00 | 21.33 | 0.00 Enduro | 849.07 | 811.58 | 556.56 FishingDerby | 13.34 | 11.56 | 7.15 Freeway | 32.60 | 32.60 | 32.52 Frostbite | 296.57 | 220.81 | 165.01 Gopher | 9999.61 | 8013.34 | 4879.13 Gravitar | 475.03 | 480.64 | 394.46 Hero | 20803.60 | 20774.80 | 20598.40 IceHockey | -3.23 | -4.78 | -8.63 Jamesbond | 664.98 | 669.54 | 626.53 JourneyEscape | -462.64 | 391.44 | -947.18 Kangaroo | 10974.00 | 10733.60 | 10584.20 Krull | 9503.62 | 9538.22 | 4039.78 MontezumaRevenge | 1.46 | 0.00 | 0.00 MsPacman | 5066.17 | 5227.84 | 4160.50 NameThisGame | 9181.30 | 8410.29 | 5529.50 Phoenix | 5307.46 | 5227.84 | 4801.18 Pitfall | 0.00 | 0.00 | -4.00 Pong | 21.00 | 20.99 | 20.95 Pooyan | 5778.99 | 5184.14 | 4393.09 PrivateEye | 100.00 | 100.00 | 100.00 Qbert | 11953.40 | 13965.80 | 8625.88 Riverraid | 15614.40 | 14812.40 | 11253.30 RoadRunner | 49864.80 | 46302.20 | 45073.25 Robotank | 61.92 | 56.90 | 53.08 Seaquest | 2647.82 | 2560.61 | 1034.83 Skiing | -14058.90 | -14079.80 | -29896.80 Solaris | 1956.24 | 1182.59 | 291.70 SpaceInvaders | 3436.16 | 3292.68 | 2895.30 StarGunner | 55479.00 | 54207.30 | 51419.60 Tennis | 0.00 | 0.00 | 0.00 TimePilot | 6717.62 | 6799.19 | 3806.22 Tutankham | 242.03 | 229.23 | 36.00 UpNDown | 22544.60 | 23331.20 | 5956.21 Venture | 15.41 | 1.50 | 0.00 VideoPinball | 382110.59 | 390540.41 | 209620.0 WizardOfWor | 5750.05 | 3632.17 | 2011.77 YarsRevenge | 25631.10 | 25396.70 | 25319.20 Zaxxon | 10751.80 | 11156.20 | 5186.01 Table 7: Summary of scores with $\varepsilon$-greedy ($\varepsilon=0.001$) evaluation with up to 30 no-op starts. As a side study, we ran ConQUR with 8 nodes and with $\lambda\in\\{1,10\\}$. We report the scores of the best $\lambda_{\text{best}}$ for each game. For most games, $\lambda_{\text{best}}$ is $10$. The 8 nodes configuration follows the same as in Table 5, except that $c=2,m=38,F=8,l=2.$

2024-09-04T02:54:55.315321

2002.12426

arxiv-papers

Roohi, # Reconstructing Element-by-Element Dissipated Hysteretic Energy in Instrumented Buildings: Application to the Van Nuys Hotel Testbed Milad Roohi Postdoctoral fellow, NIST Center of Excellence for Risk-Based Community Resilience Planning, Department of Civil and Environmental Engineering, Colorado State University, Fort Collins, CO 80523, USA; formerly, Graduate Research Assistant, Department of Civil and Environmental Engineering, University of Vermont, Burlington, VT 05405 USA. E-mail: <EMAIL_ADDRESS>Corresponding author: Milad Roohi, <EMAIL_ADDRESS>Eric M. Hernandez Professor of Civil Engineering, Former Provost and Senior Vice President, University of Vermont, Burlington, VT 05405 USA. E-mail<EMAIL_ADDRESS>David Rosowsky ###### Abstract The authors propose a seismic monitoring framework for instrumented buildings that employs dissipated energy as a feature for damage detection and localization. The proposed framework employs a nonlinear model-based state observer, which combines a nonlinear finite element model of a building and global acceleration measurements to estimate the time history of seismic response at all degrees of freedom of the model. This includes displacements, element forces, and plastic deformations in all structural members. The estimated seismic response is then used to 1) estimate inter-story drifts and determine the post-earthquake re-occupancy classification of the building based on performance-based criteria, 2) compare the estimated demands with code-based capacity and reconstruct element-by-element demand-to-capacity ratios and 3) reconstruct element-level normalized energy dissipation and ductility. The outcome of this process is employed for the performance-based monitoring, damage detection, and localization in instrumented buildings. The proposed framework is validated using data from the Van Nuys hotel testbed; a seven story reinforced concrete building instrumented by the California Strong Motion Instrumentation Program (Station 24386). The nonlinear state observer of the building is implemented using a distributed plasticity finite element model and seismic response measurements during the 1992 Big Bear and 1994 Northridge earthquakes. The performance and damage assessment results are compared with the post-earthquake damage inspection reports and photographic records. The results demonstrate the accuracy and capability of the proposed framework in the context of a real instrumented building that experienced significant localized structural damage. ## 1 Introduction This paper proposes a seismic monitoring framework that employs dissipated energy as a feature for damage detection and localization in instrumented moment resisting frame building structures. The main advantages of the proposed energy-based approach are: i) the proposed feature is physically meaningful and correlates well with the level of cyclic damage experienced during strong earthquakes [Uang and Bertero (1990), Sucuoglu and Erberik (2004), Teran-Gilmore and Jirsa (2007)], ii) dissipated energy can be reconstructed from element level stress-strain fields, which can be estimated from global acceleration measurements [Stephens and Yao (1987), Roohi et al. (2019a)], and iii) it can be calibrated using experimental data [Krawinkler and Zohrei (1983), Park and Ang (1985), Sucuoglu and Erberik (2004)]. Despite the immediate appeal, the application of this feature for structural health monitoring purposes has been limited [Frizzarin et al. (2010), Hernandez and May (2012)] primarily due to the challenges associated with estimating dissipated energy under dynamic loading. The main contribution of this paper consists in reconstructing element-by-element dissipated hysteretic energy using a nonlinear model-data fusion approach. This approach deviates from the traditional approach used in structural monitoring and damage identification, which seeks changes in the structural parameters before and after an earthquake. To contextualize the proposed energy-based method, current damage detection methods are briefly reviewed. Based on the damage features selected to distinguish between undamaged and damaged states of the structure, the existing methods can be widely categorized into 1) spectral, 2) wave propagation, 3) time series, 4) demand-to-capacity ratio, 5) model updating methods. The spectral methods assume that changes in spectral parameters (mode shapes and frequencies) of a structure indicate the occurrence of structural damage; where the changes are identified from vibration measurements before/after or during strong ground motion. The main challenges associated with this approach include: i) spectral parameters can be conceptually defined if the dynamic response is governed by a linear equation of motion; however, this feature does not exist for nonlinear hysteretic structural systems, ii) damage localization is a challenging task using changes in spectral parameters; this is mainly because low-frequency modes are the only reliable modes identified from vibration data, and the sensitivity of these modes to localized damage is low, and iii) changes in spectral parameters can be due to other factors such as environmental effects, which can negatively affect the reliability of this approach to detect structural damage. The wave propagation methods process the measured vibration data to extract travel times of seismic waves propagating through the structure and use this feature to detect changes in structural stiffness and subsequently infer structural damage. The main advantages of this approach include i) damage detection using a small number of instruments and ii) high sensitivity to localized damage. However, the resolution of damage detection depends on the number of instruments. This means that only two instruments are enough to determine if the building is damaged, and additional instruments are required to improve the resolution and specify the damaged part or floor of the structure. The time series methods employ a data-driven approach to detect damage based on mathematical models extracted solely from measured vibrations. These methods require no information from structural models and only track changes in the time history response or the identified black box input-output model coefficients. Thus, it is a difficult task to correlate these features with the damage sensitive structural quantities. This drawback makes these methods less appealing for seismic monitoring purposes. The demand-to-capacity ratio methods operate by comparing element-level force demands with capacities of any pertinent failure mode or comparing inter-story drifts with qualitative performance criteria to assess the level of damage in a particular member or story. The intention of selecting this damage feature for damage detection is to make the results similar to the way in which the buildings are designed, making them more interpretable. However, this approach has the drawback that the expected capacities are estimated based on codes and laboratory experiments, which can differ considerably from the actual capacities of structural elements because of the uncertainty in the stiffness and strength of building materials. The model updating approach updates the structural model parameters to minimize the error between the model estimates and vibration measurements. The structural damage can be identified by seeking changes in the model parameters. The main drawback of the model updating approach include i) the effectiveness of this approach depends on the model class, and it is necessary to examine the robustness to modeling error, and ii) the uniqueness problem may arise in the case of structural models where free parameter space becomes too large. Therefore, it is necessary to have prior knowledge regarding elements likely to be damaged. The proposed energy-based damage feature can overcome some of the drawbacks associated with existing methods if element-by-element energy demands can be reconstructed from global response measurement. In this paper, the authors propose a three step process: (1) implement a state observer to reconstruct the dynamic response at all degrees of freedom (DoF) of the model, (2) use the reconstructed response to estimate element-by-element forces and displacements, (3) use estimated displacement, forces, and constitutive laws to estimate element-level dissipated energy. The accuracy of this approach depends mostly on the performance of the state observer in reconstructing the dynamic response. Researchers have successfully implemented various nonlinear state observers including the extended Kalman filter (EKF) [Gelb (1974)], unscented Kalman filter (UKF) [Wan and Van Der Merwe (2000)], particle filter (PF) [Doucet et al. (2000)], and nonlinear model-based observer (NMBO) [Roohi et al. (2019a)] for response reconstruction in nonlinear structural systems. The EKF, UKF, and PF are computationally intensive and require the use of rather simple state-space models, which may not be capable of capturing the complexity of nonlinear structural behavior. However, the NMBO can be implemented directly as a second-order nonlinear FE model. This capability allows the NMBO to take advantage of simulation and computation using the conventional structural analysis software for the purpose of state estimation. The primary aim of this paper is to address the reconstruction of element-by- element dissipated hysteretic energy for damage detection and localization in instrumented buildings. A seismic monitoring framework is proposed that employs the NMBO to combine a nonlinear FE model with acceleration measurements for reconstructing the complete dynamic response as the vibration measurements become available for every seismic event. Then, the estimated response is processed to 1) estimate inter-story drifts and determine the post-earthquake re-occupancy classification of the building based on performance-based criteria 2) to compare the estimated demands with code-based capacity and reconstruct element-by-element demand-to-capacity ratios and 3) reconstruct element-level dissipated energy and ductility. The outcome of this process is employed for the performance-based monitoring, damage detection, and localization of instrumented buildings. A secondary objective of this paper is to validate the application of the NMBO for reconstruction of nonlinear response in the context of instrumented buildings that experience severe structural damage during an earthquake. The NMBO has been successfully validated using case study of the NEESWood Capstone project, a fully-instrumented six-story wood-frame building tested in full- scale at the E-Defense shake table in Japan. It was demonstrated that the NMBO could estimate quantities such as drifts and internal forces from a few acceleration measurements [Roohi (2019a), Roohi et al. (2019b)]. However, in this test, nonlinearity was limited and distributed throughout the building; e.g., during the maximum credible earthquake (MCE) level test corresponding to a 2%/50-year event, the damage was limited to nonstructural elements such as gypsum wallboard, and no structural damage was reported [van de Lindt et al. (2010)]. The effectiveness of the proposed energy-based damage detection and localization method is investigated using data from Van Nuys hotel testbed; a seven story reinforced concrete (RC) building instrumented by the California Strong Motion Instrumentation (CSMIP) Program (Station 24386). The Van Nuys building was severely damaged during the 1994 Northridge earthquake, and localized damage occurred in five of the nine columns in the 4th story (between floors 4 and 5) of the south longitudinal frame. In the literature, multiple researchers have studied this building for seismic damage assessment and localization. Traditionally, the main objective has been to use acceleration measurements to identify the presence of damage and reproduce its location and intensity with respect to the visual evidence. The remainder of this paper is organized as follows. First, the nonlinear dynamic analysis of building structures is discussed, and the system and measurement models of interest are presented. This is followed by a section on dissipated energy reconstruction and nonlinear model-data fusion in instrumented buildings. Then, a section discussing the case study of Van Nuys seven-story RC building is presented. The paper ends with a section presenting the validation of the proposed damage detection and localization methodology using seismic response measurements of the case-study building. ## 2 Structural Modeling for Nonlinear Model-Data Fusion Various approaches are available in the literature for the nonlinear structural modeling and dynamic analysis of moment resisting frame building structures subjected to seismic excitations. These approaches can be classified into three categories based on their scales: 1) global modeling, 2) discrete FE modeling, and 3) continuum FE modeling [Taucer et al. (1991)]. The global modeling approach condenses the nonlinear behavior of a building at selected global DoF. One example is to assign the hysteretic lateral load- displacement and energy dissipation characteristics of every story of building to an equivalent element and assemble these elements to construct a simplified model of a building. This method displays low-resolution, which depending on the specific application might be detrimental. The discrete FE modeling approach first formulates the hysteretic behavior of elements and then, assembles interconnected frame elements to construct an FE model of a structure. Two types of element formulations are used in research and practice, including 1) a concentrated plasticity formulation and 2) a distributed plasticity formulation. The concentrated plasticity formulation lumps the nonlinear behavior in springs or plastic hinges at the end of elastic elements. The distributed plasticity formulation that concentrates the nonlinear behavior at selected integration points along the element using cross-sections that are discretized to fibers, which account for stress-strain relations of corresponding material. The continuum FE element modeling approach discretizes structural elements into micro finite elements and requires localized model parameters (constitutive and geometric nonlinearity) calibration. The analysis of such high-resolution models increases the computational complexity. Therefore, this approach can be unpractical for the model-data fusion and response reconstruction applications. Figure 1 presents a schematic of five idealized nonlinear beam-column elements developed for nonlinear modeling of moment resisting frame building structures. Figure 1: Schematic of nonlinear beam-column elements (Deierlein et al. 2010) From these formulations, the concentrated and distributed plasticity formulations have been implemented in advanced structural simulation software packages such as OpenSEES, Perform, and SAP. In recent years, the fiber-based distributed plasticity FE modeling has been the most popular approach among researchers. The main reasons are: 1) the formulation accurately simulates the coupling between axial force and bending moment and also, accounts for element shear, 2) various uniaxial material models have been developed by researchers to characterize section fibers and are available for users of advanced structural simulation software, 3) the predictions using this formulation have been validated with experimental testing, and 4) the simulation and analysis are computationally efficient and accurate, even with a relatively low number of integration points per element. This paper employs a fiber-based distributed plasticity FE modeling approach for nonlinear model-data fusion and seismic response reconstruction. ### 2.1 System and measurement models of interest The global response of building structures to seismic ground motions can be accurately described as $\mathbf{M}\ddot{q}(t)+\mathbf{C}_{\xi}\dot{q}(t)+f_{R}(q(t),\dot{q}(t),z(t))=-\mathbf{M}\mathbf{b}_{1}\ddot{u}_{g}(t)+\mathbf{b}_{2}w(t)$ (1) where the vector $q(t)\in\mathbb{R}^{n}$ contains the relative displacement (with respect to the ground) of all stories. $z(t)$ is a vector of auxiliary variables dealing with material nonlinearity and damage behavior. $n$ denotes the number of geometric DoF, $\mathbf{M}=\mathbf{M}^{T}\in\mathbb{R}^{n\times n}$ is the mass matrix, $\mathbf{C}_{\xi}=\mathbf{C}_{\xi}^{T}\in\mathbb{R}^{n\times n}$ is the damping matrix, $f_{R}(\cdot)$ is the resultant global restoring force vector. The matrix $\mathbf{b}_{1}\in\mathbb{R}^{n\times r}$ is the influence matrix of the $r$ ground acceleration time histories defined by the vector $\ddot{u}_{g}(t)\in\mathbb{R}^{r}$. The matrix $\mathbf{b}_{2}\in\mathbb{R}^{n\times p}$ defines the spatial distribution the vector $w(t)\in\mathbb{R}^{p}$, which in the context of this paper represents the process noise generated by unmeasured excitations and (or) modeling errors. This study relies only on building vibrations measured horizontally in three independent and non-intersecting directions and assumes the vector of acceleration measurements, $\ddot{y}(t)\in\mathbb{R}^{m}$, is given by $\ddot{y}(t)=-\mathbf{c}_{2}\mathbf{M}^{-1}\left[\mathbf{C}_{\xi}\dot{q}(t)+f_{R}(q(t),\dot{q}(t),z(t))-\mathbf{b_{2}}w(t)\right]+\nu(t)$ (2) where $\mathbf{c}_{2}\in\mathbb{R}^{m\times n}$ is a Boolean matrix that maps the DoFs to the measurements, and $\nu(t)\in\mathbb{R}^{m\times 1}$ is the measurement noise. ## 3 Dissipated Energy Reconstruction from Response Measurements This section presents the theoretical background necessary to calculate dissipated energy induced by material nonlinearity and proposes a nonlinear model-data fusion approach to reconstruct element-by-element dissipated energy from global response measurements of building structures. ### 3.1 Theoretical background The dissipated hysteretic energy ($E_{h}$) can be defined by a change of variables and integrating equation of motion in time for multi-DoF systems as follows $\int\dot{q}(t)^{T}\mathbf{M}\ddot{q}(t)dt+\int\dot{q}(t)^{T}\mathbf{C}_{\xi}\dot{q}(t)dt+\int\dot{q}(t)^{T}f_{R}(q(t),\dot{q}(t),z(t))dt=-\int\dot{q}(t)^{T}\mathbf{M}\mathbf{b}_{1}\ddot{u}_{g}(t)dt$ (3) Equation 3 can be represented in energy-balance notation [Uang and Bertero (1990)] as follows $E_{k}+E_{\xi}+E_{s}=E_{i}$ (4) where $E_{k}$, $E_{\xi}$, $E_{s}$ and $E_{i}$ are kinetic, viscous damping, stain and input energy, respectively. The strain energy is the sum of recoverable elastic strain energy ($E_{e}$) and irrecoverable dissipated hysteretic energy ($E_{h}$). Thus, Equation 4 can be written as $E_{k}+E_{\xi}+(E_{e}+E_{h})=E_{i}$ (5) The dissipated hysteretic energy ($E_{h}$) can be calculated using element- level stress-strain or force-displacement demand by integrating the area under hysteresis loops as follows $E_{h}=\dfrac{1}{2}\int\epsilon^{T}\sigma dV$ (6) where $\sigma$ and $\epsilon$ are stress and strain demands and $V$ is the total volume of an element. In distributed plasticity beam-column elements, where energy dissipation occurs primarily due to bending, the dissipated hysteretic energy ($E_{h}$) can be calculated by integrating the moment- curvature response along the element as follows $\displaystyle E_{h}=\int_{0}^{L}M\phi dx=\sum_{j=1}^{N_{p}}(M\phi|_{x=\xi_{j}})\omega_{j}$ (7) where $M$ and $\phi$ are moment and curvature response of elements, respectively; $N_{p}$ is number of integration points along the element; $\xi_{j}$ and $\omega_{j}$ respectiveky denote locations and associated weights of integration points. As can be seen from Equations 6 and 7, the calculation of $E_{h}$ requires element-level seismic response to be known. Therefore, there is a need to employ signal processing algorithms that can accurately reconstruct the element-level seismic response from global response measurements. The next subsection addresses this need by proposing the use of a recently developed nonlinear model-data fusion algorithm for seismic response reconstruction. ### 3.2 Nonlinear model-data fusion and seismic response reconstruction Recently, [Roohi et al. (2019a)] proposed a nonlinear state observer for nonlinear model-data fusion in second-order nonlinear hysteretic structural systems. This nonlinear state observer has appealing properties for seismic monitoring application; two most important ones include: (1) it has been formulated to be realizable as a nonlinear FE model, which allows implementing the nonlinear state observer using the conventional structural analysis software; therefore, the computational costs would reduce significantly, and (2) it uses power spectral density (PSD) representation to account for measurement noise and unmeasured excitations explicitly. This property is important as it is consistent with the representation of seismic excitation in many stochastic models. The NMBO estimate of the displacement response, ${\hat{q}}(t)$, is given by the solution of the following set of ordinary differential equations $\displaystyle\mathbf{M}\ddot{\hat{q}}(t)+(\mathbf{C}_{\xi}+\mathbf{c}_{2}^{T}\mathbf{E}\mathbf{c}_{2})\dot{\hat{q}}(t)+f_{R}(\hat{q}(t),\dot{\hat{q}}(t),z(t))=\mathbf{c}_{2}^{T}\mathbf{E}\dot{y}(t)$ (8) where $\dot{y}(t)$ is the measured velocity and $\mathbf{E}\in\mathbb{R}^{m\times m}$ is the feedback gain. It can be seen that Equation 8 is of the same form of the original nonlinear model of interest in Equation 1. A physical interpretation of the NMBO can be obtained by viewing the right-hand side of Equation 8 as a set of corrective forces applied to a modified version of the original nonlinear model of interest in the left-hand side. The modification consists in adding the damping term $c_{2}^{T}\mathbf{E}c_{2}$, where the matrix $\mathbf{E}$ is free to be selected. The diagonal terms of $\mathbf{E}$ are equivalent to grounded dampers in the measurement locations, and the off-diagonal terms (typically set to zero) are equivalent to dampers connecting the respective DoF of the measurement locations. To retain a physical interpretation, the constraints on $\mathbf{E}$ are symmetry and positive definiteness. Also, the corrective forces $\mathbf{c}_{2}^{T}\mathbf{E}\dot{y}(t)$ are proportional to the velocity measurements and added grounded dampers. The velocity measurements $\dot{y}(t)$ can be obtained by integration of acceleration measurements $\ddot{y}(t)$ in Equation 2. The integration might add long period drifts in velocity measurements, and high-pass filtering can be performed to remove these baseline shifts. To determine E, the objective function to be minimized is the trace of the estimation error covariance matrix. Since for a general nonlinear multi-variable case, a closed-form solution for the optimal matrix $\mathbf{E}$ has not been found, a numerical optimization algorithm is used. To derive the optimization objective function, Equation 8 is linearized as follows $\displaystyle\mathbf{M}\ddot{\hat{q}}(t)+(\mathbf{C}_{\xi}+\mathbf{c}_{2}^{T}\mathbf{E}\mathbf{c}_{2})\dot{\hat{q}}(t)+\mathbf{K}_{0}{\hat{q}}(t)=\mathbf{c}_{2}^{T}\mathbf{E}\dot{y}(t)$ (9) where $\mathbf{K}_{0}$ is the initial stiffness matrix. By defining the state error as $e=q-\hat{q}$, it was shown in [Hernandez (2011)] that the PSD of estimation error, $\boldsymbol{\Phi}_{ee}$, is given by $\displaystyle\boldsymbol{\Phi}_{ee}(\omega)=\mathbf{H}_{o}\mathbf{b}_{2}\boldsymbol{\Phi}_{ww}(\omega)\mathbf{b}_{2}^{T}\mathbf{H}_{o}^{*}+\mathbf{H}_{o}\mathbf{c}_{2}^{T}\mathbf{E}\boldsymbol{\Phi}_{vv}(\omega)\mathbf{E}^{T}\mathbf{c}_{2}\mathbf{H}_{o}^{*}$ (10) with $\mathbf{H}_{o}$ defined as $\displaystyle\mathbf{H}_{o}=\left(-\mathbf{M}\omega^{2}+\left(\mathbf{C}_{\xi}+\mathbf{c}_{2}^{T}\mathbf{Ec}_{2}\right)i\omega+\mathbf{K}_{0}\right)^{-1}$ (11) where the matrices $\boldsymbol{\Phi}_{ww}(\omega)$ and $\boldsymbol{\Phi}_{vv}(\omega)$ are the PSDs of the uncertain excitation on the system and measurement noise, respectively. In this paper, the uncertain input corresponds to the ground motion excitation, and the measurement noise corresponds to unmeasured excitations and (or) modeling errors. To select the optimal value of $\mathbf{E}$ matrix, the following optimization problem must be solved $\displaystyle\begin{aligned} &\operatorname*{arg\,min}_{\mathbf{E}\,\in\,\mathbb{R}^{+}}\,tr(\mathbf{P})\\\ \end{aligned}$ (12) where $\mathbf{P}$ is the covariance matrix of displacement estimation error described as $\displaystyle\begin{aligned} \mathbf{P}&=\mathbb{E}\left[[q(t)-\hat{q}(t)][q(t)-\hat{q}(t)]^{T}\right]=\int_{-\infty}^{+\infty}\boldsymbol{\Phi}_{ee}(\omega)d\omega\end{aligned}$ (13) One alternative for the optimization problem in Equation 12 can be defined if the objective is minimization of the inter-story drifts (ISD) estimation error, $\mathbf{P_{\text{ISD}}}$, given by $\displaystyle\begin{aligned} &\operatorname*{arg\,min}_{\mathbf{E}\,\in\,\mathbb{R}^{+}}\,tr(\mathbf{P_{\text{ISD}}})\end{aligned}$ (14) where $\displaystyle\begin{aligned} tr(\mathbf{P_{\text{ISD}}})=\sum_{k=1}^{n}\mathbf{P_{\text{ISD}}}_{(k,k)}=\mathbf{P}_{(1,1)}+\sum_{k=2}^{n}[\mathbf{P}_{(k,k)}+\mathbf{P}_{(k-1,k-1)}-2\mathbf{P}_{(k,k-1)}]\end{aligned}$ (15) $k$ is story number and $n$ is total number of stories. Any optimization algorithm (e.g., Matlab fminsearch) can be used to solve the optimization in Equations 12 and 14 by varying the values of the diagonal elements of the $\mathbf{E}$ matrix to determine the optimized feedback matrix. Figure 2 presents a summary of the nonlinear model-data fusion using the NMBO and Figure 3 schematically illustrates the implementation of the NMBO. Also, readers are kindly referred to [Hernandez (2011), Hernandez (2013), Hernandez et al. (2018), Roohi et al. (2019a), Roohi (2019b)] for implementation examples. Figure 2: Summary of the nonlinear model-data fusion using the NMBO Figure 3: Implementation of the proposed nonlinear model-based observer ## 4 Proposed Seismic Monitoring Framework This paper proposes a seismic monitoring framework that can be accurately employed for seismic damage detection and localization in instrumented buildings subjected to seismic ground motions. This framework employs the NMBO to combine a nonlinear structural model with acceleration measurements for reconstructing the complete seismic response. Then, the estimated response is processed to 1) estimate inter-story drifts and determine the post-earthquake re-occupancy classification of the building based on performance-based criteria 2) to compare the estimated demands with code-based capacity and reconstruct element-by-element demand-to-capacity ratios and 3) reconstruct element-level dissipated energy and ductility. The outcome of this process is employed for the performance-based monitoring, damage detection, and localization in instrumented buildings. Figure 4 depicts a summary of the proposed seismic monitoring framework. The following subsections discuss each step of the framework in more detail. Figure 4: Summary of the proposed mechanistic damage quantification and seismic monitoring framework #### 4.0.1 Performance-based assessment using Inter-story Drifts The maximum inter-story drift ($\text{ISD}_{\mathrm{max}}$) estimate at each story can be calculated using the NMBO displacement estimates as follows $\displaystyle\text{ISD}_{\mathrm{max}_{k}}=\frac{\mathrm{max}\Big{\lvert}\hat{q}_{k}(t)-\hat{q}_{k-1}(t)\Big{\rvert}}{h_{k}}$ (16) where $h_{k}$ is height of $k$-th story, and the uncertainty in ISD estimation can be calculated as follows $\displaystyle\text{ISD}_{\mathrm{max}_{k}}{\>\mathbin{\leavevmode\hbox to6.46pt{\vbox to5.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.21527pt\lower-0.21527pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}\pgfsys@setlinewidth{0.43056pt}\pgfsys@invoke{ }{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.02776pt}{0.0pt}\pgfsys@moveto{3.01387pt}{0.48222pt}\pgfsys@lineto{3.01387pt}{5.54552pt}\pgfsys@moveto{0.0pt}{3.01387pt}\pgfsys@lineto{6.02776pt}{3.01387pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\>\sigma_{{\text{ISD}_{k}}}}={\mathrm{max}\Big{\lvert}\text{ISD}_{k}{\>\mathbin{\leavevmode\hbox to6.46pt{\vbox to5.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.21527pt\lower-0.21527pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}\pgfsys@setlinewidth{0.43056pt}\pgfsys@invoke{ }{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.02776pt}{0.0pt}\pgfsys@moveto{3.01387pt}{0.48222pt}\pgfsys@lineto{3.01387pt}{5.54552pt}\pgfsys@moveto{0.0pt}{3.01387pt}\pgfsys@lineto{6.02776pt}{3.01387pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\>\sqrt{\mathbf{{P}}_{\text{ISD}_{k}}}}\Big{\rvert}}$ (17) where $\sigma_{{\text{ISD}_{k}}}$ is the uncertainty standard deviation of ISD estimation for $k$-th story. The estimated ISDs are used to perform the post- earthquake evaluation of the building based on [FEMA (2000)] performance measures, including immediate occupancy (IO), life safety (LS), and collapse prevention (CP). #### 4.0.2 Demand-to-capacity ratio reconstruction The demand-to-capacity ratio (DCR) for $i$-th element is reconstructed as follows $\displaystyle\mathrm{DCR}_{i}=\frac{\mathrm{max}|\hat{S}_{i}(t)|}{R_{i}}$ (18) where $\hat{S}_{i}(t)$ and $R_{i}$ are the seismic demand and capacity estimates of any pertinent failure mode in $i$-th structural element. #### 4.0.3 Dissipated energy reconstruction for damage detection and localization The seismic damage index (DI) is reconstructed using a Park-Ang type damage model [Park and Ang (1985)] expressed as $\displaystyle DI=DI_{\mu}+DI_{E}=\dfrac{\mu_{m}}{\mu_{u}}+\psi\dfrac{E_{h}}{E_{max}}$ (19) where $DI_{\mu}$ and $DI_{E}$ represent damage due to excessive deformation and dissipated hysteretic energy, respectively; $\mu_{m}$ is the maximum ductility experienced during the earthquake, $\mu_{u}$ is the ultimate ductility capacity under monotonic loading, $\psi$ is calibration parameter, and $E_{max}$ is the maximum hysteretic energy dissipation capacity for all relevant failure modes. ## 5 Case-study: Van Nuys Hotel Testbed The proposed methodology is validated in the remaining sections using seismic response measurements from Van Nuys hotel. The CSMIP instrumented this building as Station 24386, and the recorded data of this building are available from several earthquakes, including 1971 San Fernando, 1987 Whittier Narrows, 1992 Big Bear, and 1994 Northridge earthquakes. From these data, measurements during 1992 Big Bear and 1994 Northridge earthquakes are used in this study to demonstrate the proposed framework. Researchers have widely studied the Van Nuys building [Islam (1996), Loh and LIN (1996), Li and Jirsa (1998), Browning et al. (2000), Taghavi and Miranda (2005), Goel (2005), Bernal and Hernandez (2006), Ching et al. (2006), Naeim et al. (2006), Todorovska and Trifunac (2008), Rodríguez et al. (2010), Gičev and Trifunac (2012), Trifunac and Ebrahimian (2014), Shan et al. (2016), Pioldi et al. (2017)] and the building was selected as a testbed for research studies by researchers in PEER [Krawinkler (2005)]. ### 5.1 Description of the Van Nuys building The case-study building is a 7-story RC building located in San Fernando Valley in California. The building plan is 18.9 m $\times$ 45.7 m in the North-South and East-West directions, respectively. The total height of the building is 19.88 m, with the first story of 4.11 m tall, while the rest are 2.64 m approximately. The structure was designed in 1965 and constructed in 1966. Its vertical load transfer system consists of RC slabs supported by concrete columns and spandrel beams at the perimeter. The lateral resisting systems are made up of interior concrete column-slab frames and exterior concrete column-spandrel beam frames. The foundation consists of friction piles, and the local soil conditions are classified as alluvium. The testbed building is described in more detail in [Trifunac et al. (1999), Krawinkler (2005)]. ### 5.2 Building instrumentation The CSMIP initially instrumented the building with nine accelerometers at the 1st, 4th, and roof floors. Following the San Fernando earthquake, CSMIP replaced the recording layout by 16 remote accelerometer channels connected to a central recording system. These channels are located at 1st, 2nd, 3rd, 6th, and roof floors. Five of these sensors measure longitudinal accelerations, ten of them measure transverse accelerations, and one of them measures the vertical acceleration. Figure 5 shows the location of accelerometers. Figure 5: (left) Van Nuys hotel testbed (CSMIP Station 24386) and (Right) Location of building accelerometers on the West-East elevation and floor plans ### 5.3 Earthquake damage Since the Van Nuys building was instrumented and inspected following earthquakes that affected the structure, the history of damage suffered by this building is well-documented. These documents show that the building has experienced insignificant structural and mostly nonstructural damage before the Northridge earthquake in 1994. However, the Northridge earthquake extensively damaged the building. Post-earthquake inspection red-tagged the building and revealed that the damage was severe in the south longitudinal frame (Frame A). In Frame A, five of the nine columns in the 4th story (between floors 4 and 5) were heavily damaged due to inadequate transverse reinforcement, and shear cracks ($\geq 5cm$) and bending of longitudinal reinforcement were easily visible [Trifunac and Ivanovic (2003)]. Figure 6 demonstrate the seismic damage following the 1994 Northridge earthquake in the south and north frames. Figure 6: Schematic representation and photo records of of seismic damage following the 1994 Northridge earthquake: (top) south view of Frame A, and (bottom) south view of Frame D. (Adopted from Trifunac and Ivanovic 2003) ### 5.4 Previous damage assessment studies on Van Nuys building [Browning et al. (2000)] reported the performance assessment results of the Van Nuys building based on studies of three independent research teams. These teams used nonlinear dynamic and nonlinear static analysis to localize structural damage and concluded that the various studies were successful to varying degrees. [Naeim et al. (2006)] presented a methodology for automated post-earthquake damage assessment of instrumented buildings. The methodology was applied to the measured response from Landers, Big Bear, and Northridge earthquakes. Their findings show that the building did not suffer structural damage under the Landers and Big Bear earthquakes and indicate a high probability of extensive damage to the middle floors of the building under the Northridge earthquake. They concluded that their methodology was not able to identify the exact floor level at which the damage occurs because no sensors were installed on the floor that was damaged. As previously mentioned; [Ching et al. (2006)] performed state estimation using measured data during the Northridge earthquake combined with a time-varying linear model and then, with a simplified time-varying nonlinear degradation model derived from a nonlinear finite-element model of the building. They found that state estimation using the nonlinear degradation model shows better performance and estimates the maximum ISD to be at the 4th story. They concluded that an appropriate estimation algorithm and a suitable identification model can improve the accuracy of the state estimation. [Todorovska and Trifunac (2008)] used impulse response functions computed from the recorded seismic response during 11 earthquakes, including the Northridge earthquake. They analyzed travel times of vertically propagating waves to obtain the degree and spatial distribution of changes in stiffness and infer the presence of structural damage. Their findings showed that during the Northridge earthquake, the rigidity decreased by about 60% between the ground and 2nd stories; by about 33% between 2nd and 3rd stories, and between 3rd and 6th stories; and by about 41% between the 6th story and roof. [Rodríguez et al. (2010)] implemented their proposed method called Baseline Stiffness Method to detect and assess structural and nonstructural damage to the Van Nuys building using data from the Northridge earthquake. Their approach was able to detect damage in connections with wide cracks of 5 cm or greater. On the other hand, the method identified damage in some elements of upper stories that were not detected by visual inspection reports, and also, they could not identify some of the moderate damages with small cracks. [Shan et al. (2016)] presented a model- reference damage detection algorithm of hysteretic buildings and investigated the Van Nuys hotel using measured data from Big Bear and Northridge earthquakes. The researchers concluded that their algorithm can only detect damages of certain floors and cannot detect damages in structural components or connections of the instrumented structure. ## 6 Implementation of the Proposed Seismic Monitoring Framework ### 6.1 Nonlinear modeling of the Van Nuys hotel testbed in OpenSEES The nonlinear FE model of the building was implemented using a two-dimensional fixed-base model within the environment of OpenSEES [McKenna et al. (2000)]. This model corresponds to one of the longitudinal frames of the building (Frame A in Figure 5). In the FE model, beams and columns were modeled based on distributed plasticity modeling approach, and the force-based beam-column elements were used to accurately determine yielding and plastic deformations at the integration points along the element. Gauss-Lobatto integration approach was employed to evaluate the nonlinear response of force-based elements. Each beam and column element was discretized with four integration points, and the cross-section of each element was subdivided into fibers. The uniaxial Concrete01 material was selected to construct a Kent-Scott-Park object with a degraded linear unloading and reloading stiffness and zero tensile strength. The uniaxial Steel01 material was used to model longitudinal reinforcing steel as a bilinear model with kinematic hardening. The elasticity modulus and strain hardening parameters were assumed to be 200 GPa and 0.01, respectively. Due to insufficient transverse reinforcement in beams and columns [Jalayer et al. (2017)], an unconfined concrete model was defined to model concrete. The peak and post-peak strengths were defined at a strain of 0.002 and a compressive strain of 0.006, respectively. The corresponding strength at ultimate strain was defined as $0.05f^{\prime}_{c}$ for $f^{\prime}_{c}=34.5$ MPa and $f^{\prime}_{c}=27.6$ MPa and $0.2f^{\prime}_{c}$ for $f^{\prime}_{c}=20.7$ MPa. Based on the recommendation of [Islam (1996)], the expected yield strength of Grade 40 and Grade 60 steel were defined as 345 MPa (50 ksi) and 496 MPa (72 ksi), respectively, to account for inherent overstrength in the original material and strength gained over time. ### 6.2 Formulation of the OpenSEES-NMBO of Van Nuys building The nonlinear FE model and response measurements of the Van Nuys building was employed to implement the NMBO in OpenSEES. The following subsections present the step-by-step formulation of the OpenSEES-NMBO. #### 6.2.1 PSD selection and numerical optimization The PSD of ground motion, $\boldsymbol{\Phi}_{ww}(\omega)$, was characterized using the Kanai-Tajimi PSD given by $S(\omega)=G_{0}\frac{1+4\xi_{g}^{2}(\frac{\omega}{\omega_{g}})^{2}}{\left[1-(\frac{\omega}{\omega_{g}})^{2}\right]^{2}+4\xi_{g}^{2}(\frac{\omega}{\omega_{g}})^{2}}$ (20) and the amplitude modulating function $I(t)$ was selected as $I(t)=te^{-\alpha{t}}$ (21) The parameter were defined as $\xi_{g}=0.35$ for both earthquakes, $\omega_{g}=6\pi rad/s$ for Northridge earthquake and $\omega_{g}=2\pi rad/s$ for Big Bear earthquake. The underlying white noise spectral density $G_{0}$ for each direction of measured ground motion for each shake table test was found such that about 95% of the Fourier transform of the measured ground motion lies within two standard deviations of the average from the Fourier transforms of an ensemble of 200 realizations of the Kanai-Tajimi stochastic process. $\alpha$ was selected as 0.12. Details can be found in [Roohi et al. (2019a)]. Also, the PSD of measurement noise, $\boldsymbol{\Phi}_{vv}(\omega)$, in each measured channel was taken as zero mean white Gaussian sequences with a noise-to-signal root-mean-square (RMS) ratio of 0.02. Numerical optimization was performed using Equation 14. Table 1 presents the optimized damper values for each seismic event. Table 1: Optimized damper values in kN.s/m (kips.s/in) units Earthquake | Story 1 | Story 2 | Story 5 | Story 7 ---|---|---|---|--- Big Bear | 7283.11 (41.59) | 9357.25 (53.43) | 19299.40 (110.20) | 34808.04 (198.76) Northridge | 5209.72 (29.75) | 6592.45 (37.64) | 16612.79 (94.86) | 47217.69 (269.62) #### 6.2.2 Formulation of the OpenSEES-NMBO The OpenSEES nonlinear FE model was modified by adding grounded dampers in measurement locations and was subjected to corrective forces. Dynamic analysis was performed to estimate the complete seismic response. Figure 7 presents a schematic of the Van Nuys hotel testbed (with the location of accelerometers) along with the OpenSEES-NMBO (with corresponding added viscous dampers and corrective forces in measurement locations). Figure 7: Schematic of the Van Nuys hotel testbed with location of accelerometers (left) and the OpenSEES-NMBO with corresponding added viscous dampers and corrective forces in measurement locations ### 6.3 Seismic damage reconstruction using estimated seismic response Once the complete seismic response is estimated using the OpenSEES-NMBO, the seismic damage to the building can be quantified according to the Section 4.0.3. This subsection demonstrates the procedure in more detail. #### 6.3.1 Shear DCR reconstruction The shear DCRs were estimated based on the Equation 18. The shear demands were obtained from OpenSEES-NMBO, and the capacity of columns were calculated based on the Section 6.5.2.3.1 of [FEMA (2000)]. #### 6.3.2 Ductility demand reconstruction The seismic damage caused by excessive deformation is the first term of the Equation 19 given by: $\displaystyle DI_{\mu}=\dfrac{\mu_{m}}{\mu_{u}}$ (22) where, the $\mu_{m}$ in each structural element is expressed by the maximum estimated curvature along integration points normalized by the yield curvature given by $\displaystyle\mu_{m}=max\\{{\dfrac{\phi_{m,j}}{\phi_{y}}}\\}_{j=1:N_{p}}$ (23) here, $\phi_{m,j}$ is maximum estimated curvature in integration point $j$, $\phi_{y}$ is curvature capacity and $N_{p}$ is number of integration points along element. The curvature ductility capacity ($\mu_{u}$) is obtained by $\displaystyle\begin{aligned} &\mu_{u}=\dfrac{\phi_{u}}{\phi_{y}}\end{aligned}$ (24) where $\phi_{u}$ is the ultimate curvature capacity of the section. #### 6.3.3 Dissipated hysteretic energy reconstruction The seismic damage caused by dissipated hysteretic energy, $DI_{E}$ in Equation 19, was calculated based on flexure failure mode as follows $\displaystyle DI_{E}=\psi\dfrac{E_{h}}{E_{max}}\cong\psi\dfrac{E_{h}}{M_{y}\theta_{y}\mu_{u}}$ (25) where $M_{y}$ is yield moment and $\theta_{y}$ is yield rotation angle. The main issue with the calculation of $DI_{E}$ is the determination of $\psi$, which usually is calibrated to a number between 0.05 or 0.15. A reasonable $\psi$ value should properly account for the effect of load cycles causing structural damage. The selection of small value for $\psi$ neglects the effect of the $DI_{E}$ in the overall damage index [Williams and Sexsmith (1995)]. Since the true $\psi$ is unknown for the elements of Van Nuys building and in the scope of this paper, the objective is to localize seismic damage, the calibration parameter is set to one, and the estimated value of each term in the damage index will be first reported separately and then combined. The dissipated hysteretic energy ($E_{h}$) is estimated based on Equation 7 and the seismic response estimated using OpenSEES-NMBO. The parameter $M_{y}$ was obtained based on section analysis and the value of $\theta_{y}\mu_{u}$ calculated as follows $\displaystyle\theta_{y}\mu_{u}=\theta_{p}=(\phi_{u}-\phi_{y})l_{p}=\phi_{p}l_{p}$ (26) where $l_{p}$ is the plastic hinge length and is defined using an empirically validated relationship proposed by [Bae and Bayrak (2008)] given by $\displaystyle\dfrac{l_{p}}{h}=\left[0.3\left(\dfrac{P}{P_{o}}\right)+\left(\dfrac{A_{s}}{A_{g}}\right)-1\right]\left(\dfrac{L}{h}\right)+0.25\geq 0.25$ (27) where $h$ and $L$ represent depth and length of column; $A_{g}$ and $A_{s}$ denote gross area of concrete section and area of tension reinforcement; $f^{\prime}_{c}$ and $f_{y}$ are compressive strength of concrete and yield stress of reinforcement; and $P_{o}=0.85f^{\prime}_{c}(A_{g}-A_{s})+f_{y}A_{s}$. ## 7 Seismic Response and Damage Reconstruction Results A summary of the seismic response and damage reconstruction results is presented in this section to validate the proposed seismic monitoring framework. ### 7.1 Displacement estimation results First, we compare the displacement estimates using OpenSEES-NMBO and its uncertainty with those obtained from 1) response measurements and 2) open-loop analysis under measured ground motion at instrumented and non-instrumented stories. Figures 8 and 9 present the comparison of the displacement estimates at instrumented 1st and 7th stories and non-instrumented 3rd and 6th stories during the Big Bear earthquake and Northridge earthquake, respectively. Figure 8: Comparison of displacement estimates using OpenSEES-NMBO with estimates using open-loop analysis and actual measurements in 1st floor (top left), 3rd floor (top right), 6th floor (bottom left) and 7th floor (bottom right) during Big Bear earthquake. The Measured represents measured response, the OL represents the open-loop analysis of OpenSEES model under measured ground motion and the NMBO represents the estimated response using the OpenSEES-NMBO with sensor measurements from measured location along with $1\sigma$ estimation uncertainty bound ### 7.2 Inter-story drift estimation results Figure 10 depicts the estimated $\text{ISD}_{\mathrm{max}}$ ratios and their corresponding $1\sigma$ confidence intervals using OpenSEES-NMBO. These results are compared with estimated $\text{ISD}_{\mathrm{max}}$ using open- loop analysis and those obtained from instrumented stories. The OpenSEES-NMBO ISD estimates indicate that the building could be classified as IO following the Big Bear earthquake and as LS-CP following the Northridge earthquake. The actual performance and post-earthquake inspection reports of the buildings validate the accuracy of the performance estimates. Figure 11 gives an in- depth examination of the ISD estimates during the Northridge earthquake. The left plot in this figure shows the comparison of ISDs at 3rd, 4th, and 5th stories, and the right plot shows the comparison of relative ISDs between floors 3 and 4 and also, floors 4 and 5. Here, the relative ISD is defined as follows $\displaystyle\text{RISD}_{(k,k-1)}=\text{ISD}_{(k)}-\text{ISD}_{(k-1)}$ (28) where $\text{RISD}_{(k,k-1)}$ is relative ISD between stories $k$ and $k-1$. The estimation results show that even though the $\text{ISD}_{\mathrm{max}}$ occurs in the third story, the RISD demand between floors 4 and 5 is higher than floors 3 and 4. Figure 9: Comparison of displacement time history estimates with estimates using open-loop analysis and actual measurements in 1st floor (top left), 3rd floor (top right), 6th floor (bottom left) and 7th floor (bottom right) during Northridge earthquake. Figure 10: Comparison of $\text{ISD}_{\mathrm{max}}$ ratios obtained from response measurements with those estimated using OpenSEES-NMBO and open-loop analysis during 1992 Big Bear earthquake (left) and 1994 Northridge earthquake (right). Figure 11: Comparison of ISD (left) and RISD (right) time history estimates for stories 3 to 5 during Northridge earthquake. ### 7.3 Elemet-by-element shear demand to capacity ratio reconstruction Figure 12 shows results for shear estimated element-by-element shear DCR ratios by OpenSEES-NMBO using measured seismic response of the Van Nuys building during Big Bear (left) and Northridge (right) earthquakes. Figure 12: Estimated element-by-element shear demand to capacity ratios by OpenSEES-NMBO using measured seismic response of the Van Nuys building during 1992 Big Bear (left) and 1994 Northridge (right) earthquakes. ### 7.4 Element-by-element damage index reconstruction This section presents the seismic damage quantification results using the estimated response from OpenSEES-NMBO and the damage model presented at Section 4.0.3, which is also demonstrated in more detail in Section 6.3. Figure 13 summarizes the estimated maximum curvature ductility demands ($\mu_{m}$) in two ends of columns for each earthquake. To interpret the $\mu_{m}$ demands, one needs to consider that the expected ductility capacity of columns in this building is relatively low as the columns are non-ductile. Figure 14 presents reconstructed element-by-element normalized energy dissipation. Figure 15 presents the reconstructed element-by-element damage indices. Figure 16 schematically depicts the seismic damage suffered during the Northridge earthquake to compare the reconstructed DIs with the building’s actual performance. The shear cracks with $\text{width}\geq 5cm$ are highlights in red color and the shear cracks ($0.5cm\leq\text{width}\leq 1$) are highlights in green color. As can be seen, the element-by-element comparison of estimated DIs with post-earthquake inspection results confirms the accuracy of damage localization using the proposed mechanistic approach. Figure 13: Reconstructed maximum end curvature ductility demands in columns by implementing OpenSEES-NMBO using measured seismic response of the Van Nuys building during 1992 Big Bear (left) and 1994 Northridge (right) earthquakes. Figure 14: Reconstructed element-by-element normalized energy dissipation by OpenSEES-NMBO using measured seismic response of the Van Nuys building during 1992 Big Bear (left) and 1994 Northridge (right) earthquakes. Figure 15: Reconstructed element-by-element damage indices by OpenSEES-NMBO using measured seismic response of the Van Nuys building during 1992 Big Bear (left) and 1994 Northridge (right) earthquakes. Figure 16: Seismic damage experienced during the 1994 Northridge earthquake: (left) south view of Frame D, and (right) south view of Frame A. (Adopted from Trifunac and Ivanovic 2003) ### 7.5 Discussion on damage detection and localization results The results described in the preceding sections demonstrate that a nonlinear model-data fusion using a refined distributed plasticity FE model and a limited number of response measurements can accurately reconstruct the seismic response. Subsequently, the estimated response can be used to quantify the seismic damage based on damage sensitive response parameters and damage models. The estimated ISDs indicated that the performance-based post- earthquake re-occupancy category of the building was IO during the Big Bear earthquake and LS-CP during the Northridge earthquake. The ISD and RISD analysis during the Northridge earthquake showed that the $\text{ISD}_{\mathrm{max}}$ occurred at the 3rd story, while the maximum RISD occurs at the top of the 4th story. Also, dissipated energy and ductility reconstruction detects no structural damage during the Big Bear earthquake and severe damage during the Northridge earthquake. By combining the information from estimated ISDs, RISDs, maximum curvature ductility demands, and element- by-element damage indices during the Northridge earthquake, severe damage was localized in the columns of the 4th story (between floors 4 and 5) and also, small or moderate damage was estimated for the remaining columns. The location of severe damage in the 4th story can be explained mainly by widely spaced or absent transverse reinforcing in the beam-column joints contributed to the lower shear capacity of the story; which can be accounted for by the proposed mechanistic seismic monitoring framework through high-resolution seismic response and element-by-element damage index reconstruction. Finally, it was shown that the damage assessment results were consistent with the building’s actual performance and post-earthquake inspection reports following the Big Bear and Northridge earthquakes. Therefore, the applicability of the proposed framework is validated in the context of a real-world building that experienced severe localized damage during sequential seismic events. ## 8 Conclusions This paper proposes a seismic monitoring framework to reconstruct element-by- element dissipated hysteretic energy and perform structural damage detection and localization. The framework employs a nonlinear model-based state observer (NMBO) to combine a design level nonlinear FE model with acceleration measurements at limited stories to estimate nonlinear seismic response at all DoF of the model. The estimated response is then used to reconstruct damage- sensitive response features, including 1) inter-story drifts, 2) code-based demand to capacity ratios, and 3) normalized dissipated hysteretic energy and ductility demands. Ultimately, the estimated features are used to conduct the performance-based post-earthquake assessment, damage detection, and localization. The methodology was successfully validated using measured data from the seven- story Van Nuys hotel testbed instrumented by CSMIP (Station 24386) during 1992 Big Bear and 1994 Northridge earthquakes. The NMBO of the building was implemented using a distributed plasticity finite element model and measured data to reconstruct seismic response during each earthquake. The estimated seismic response was then used to reconstruct inter-story drifts and determine the performance-based post-earthquake re-occupation category of the building following each earthquake. The performance categories were estimated as IO and LS-CP during the Big Bear and Northridge earthquakes, respectively. Analysis during the Northridge earthquake showed that the maximum inter-story drift occurred at the 3rd story, while the maximum relative inter-story drift occurred at the top of the 4th story. Column-by-column shear demand to capacity ratios, ductility demands, and normalized dissipated hysteretic energy ratios were computed. The proposed framework correctly estimated linear behavior and no damage during the Big Bear earthquake and identified the location of major damage in the beam/column joints located at the fourth floor of the south frame during the Northridge earthquake. 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2024-09-04T02:54:55.341667

2002.12502

arxiv-papers

# Spontaneous Pulse Formation in Edge-Less Photonic Crystal Resonators Su-Peng Yu<EMAIL_ADDRESS>Time and Frequency Division, NIST, Boulder, CO 80305, USA Department of Physics, University of Colorado, Boulder, CO 80309, USA Daniel C. Cole Time and Frequency Division, NIST, Boulder, CO 80305, USA Department of Physics, University of Colorado, Boulder, CO 80309, USA Hojoong Jung Time and Frequency Division, NIST, Boulder, CO 80305, USA Department of Physics, University of Colorado, Boulder, CO 80309, USA Gregory T. Moille Microsystems and Nanotechnology Division, NIST, Gaithersburg, MD 20899, USA Joint Quantum Institute, NIST/University of Maryland, College Park, MD 20742, USA Kartik Srinivasan Microsystems and Nanotechnology Division, NIST, Gaithersburg, MD 20899, USA Joint Quantum Institute, NIST/University of Maryland, College Park, MD 20742, USA Scott B. Papp Time and Frequency Division, NIST, Boulder, CO 80305, USA Department of Physics, University of Colorado, Boulder, CO 80309, USA ###### Abstract Complex systems are a proving ground for fundamental interactions between components and their collective emergent phenomena. Through intricate design, integrated photonics offers intriguing nonlinear interactions that create new patterns of light. In particular, the canonical Kerr-nonlinear resonator becomes unstable with a sufficiently intense traveling-wave excitation, yielding instead a Turing pattern composed of a few interfering waves. These resonators also support the localized soliton pulse as a separate nonlinear stationary state. Kerr solitons are remarkably versatile for applications, but they cannot emerge from constant excitation. Here, we explore an edge-less photonic-crystal resonator (PhCR) that enables spontaneous formation of a soliton pulse in place of the Turing pattern. We design a PhCR in the regime of single-azimuthal-mode engineering to re-balance Kerr-nonlinear frequency shifts in favor of the soliton state, commensurate with how group-velocity dispersion balances nonlinearity. Our experiments establish PhCR solitons as mode-locked pulses by way of ultraprecise optical-frequency measurements, and we characterize their fundamental properties. Our work shows that sub- wavelength nanophotonic design expands the palette for nonlinear engineering of light. ††preprint: APS/123-QED Integrated nonlinear photonics is a versatile engine to generate and control electromagnetic radiation, opening new application directions and enabling fundamental studies. Second- and higher-order nonlinear susceptibilities now form the basis of many photonics technologies; a good example is harmonic- Hickstein _et al._ (2017) or difference-frequencyTadanagaa _et al._ (2006) generation that realize laser sources from the ultraviolet to the infrared. In particular, third-order, Kerr processes are ubiquitous in photonics due to intensity dependence of the refractive index, $n=n_{0}+n_{2}\,I$, where $n_{2}$ is the nonlinear index and $I$ is intensity. They enable spontaneous formation of stationary configurations of electromagnetic fields that affect conversion of a laser from one color to another. More generally, modulation instability that arises from nonlinearity governs interesting behaviors in systems ranging from quantum matter Carr and Brand (2004) to desert sand dunes Parteli _et al._ (2011). Studying nonlinear behaviors in the exquisitely controlled environment of integrated photonics– where sub-wavelength features lead to new optical behaviors –can increase understanding in a variety of physical systems. Kerr resonators– optical cavities built from an $n_{2}$ material –are an attractive system for fundamental studies and applications. We understand the formation of some pattern and pulse states of the intraresonator field $\psi$ from the Lugiato-Lefever equation (LLE) $\partial_{\tau}\psi=-(1+i\alpha)\psi-\frac{i}{2}\beta\partial_{\theta}^{2}\psi+i|\psi|^{2}\psi+F$, where $\theta$ is the resonator angular coordinate, $-\frac{i}{2}\beta\partial_{\theta}^{2}\psi$ is the group-velocity dispersion (hereafter GVD or dispersion), $|\psi|^{2}\psi$ is the nonlinearity, $F$ is a traveling-wave pump-laser field originating outside the resonator with a red detuning by $\alpha$ to lower frequency than the resonator mode; see Ref. Godey _et al._ (2014) for further details. A few states stand out amongst the diverse solution space of the LLE Godey _et al._ (2014): The constant- amplitude flat state energized by a sufficiently weak pump laser; the Turing pattern that emerges when the flat state is unstable; and the Kerr soliton that is a localized pulse coexisting with, but not emerging spontaneously from, the flat state. Indeed, microresonator soliton frequency combs Kippenberg _et al._ (2018) have been engineered to support a wide range of applications, including optical communicationMarin-Palomo _et al._ (2017); F$\ddot{u}$l$\ddot{o}$p _et al._ (2018), spectroscopy Suh _et al._ (2016), and ranging Trocha _et al._ (2018). GVD engineering via the cross-sectional waveguide dimensions offers powerful control of soliton properties Yu _et al._ (2019a). Moreover, exotic photonic states have been reported using unconventional resonator-mode engineering Lobanov _et al._ (2015); Xue _et al._ (2015). Spontaneous formation of patterns from break-up of the flat state is a critical outcome in the LLE. A pattern forms spontaneously by four-wave mixing (FWM), constrained by a balance of the Kerr frequency shift $\delta_{\mu}$ of the comb mode number $\mu$, and the phase-mismatch from dispersion $\frac{1}{2}\beta\,\mu^{2}$. We count the comb modes and the resonator modes with respect to the mode closest to the pump laser (hereafter the pump mode, $\mu=0$). Importantly, $\delta_{\mu}$ for each mode depends on the intraresonator field according to $\delta_{\mu}=g\,(2\,N-|a_{\mu}|^{2})$ Herr _et al._ (2015), where $a_{\mu}$ are the Fourier decomposition amplitude for mode $\mu$, $g$ the per-photon Kerr shift, and $N$ the total photon number. The term $g=1$ is a standard normalization of the LLE. Beginning with the flat state, all $a_{\mu^{\prime}\neq 0}=0$ and $\delta_{\mu=0}=2N-N=\frac{1}{2}\delta_{\mu^{\prime}\neq 0}$, where the modes $\mu^{\prime}$ are not pumped. The difference between self- and cross-phase modulation results in a reduced Kerr shift for the pump mode by a factor of two compared to other modes. This reduced Kerr shift enables FWM for the Turing pattern at modes $\pm\mu^{\prime}$, characterized by $\frac{1}{2}\beta|\mu^{\prime}|^{2}-\delta_{\pm\mu^{\prime}}=-\delta_{\mu=0}$. Conversely, the soliton is a collective state with many modes $\mu^{\prime}$ that reach phase-matching only at large $\alpha$ where the flat-state amplitude is insufficient to support spontaneous FWM processes. These phase- matching conditions result in the disparate generation behaviors of Turing patterns and solitons. Here, we explore a re-balancing of the LLE that causes Kerr-soliton formation from break-up of the flat state, replacing the Turing pattern. To accomplish this dramatic outcome, we design and fabricate edge-less photonic-crystal resonators (PhCR), which are Kerr-microresonators with their inner wall modified by an azimuthally uniform, nanopatterned shape oscillation. The ring geometry imposes the edge-less boundary condition on the photonic waveguide, opening the PhCR bandgap – thus controllably shifting the frequency – for one azimuthal mode. We program the shift to directly phase-match the soliton with the pump laser nearly on-resonance of the pump mode. Moreover, this shifts the Turing pattern off-resonance, precluding its formation. We have realized and explored spontaneous soliton formation in wide-ranging experiments, including observing the immediate transition from the flat state to the soliton, soliton pulse bandwidth control by dispersion engineering through the bulk ring dimensions, and ultraprecise measurements of the soliton repetition frequency. Our work draws on advances in nanophotonics and photonic-crystal devices that provide access to otherwise challenging or impossible to achieve phenomena. Take, for example, exotic refractive phenomenonKocaman _et al._ (2011), strong light-matter interactions Miura _et al._ (2014), and coupling to radiofrequency or phonon modes Fang _et al._ (2016). Moreover, photonic structures have been demonstrated to suppress Petrovich _et al._ (2008) and enhance Sharma _et al._ (2015) nonlinear effects, engineer small mode volume Hu and Weiss (2016), create sophisticated group-velocity dispersion profiles Kim _et al._ (2017); Moille _et al._ (2018), realize slow-light effects McGarvey-Lechable _et al._ (2017), and control resonator mode splittings Lu _et al._ (2014). Photonic-crystal devices are dielectric structures with sub- wavelength spatial periodicity Joannopoulos _et al._ (1997) that restrict scattering to discrete momentum values $k_{m}=k_{0}+\frac{2m\pi}{\Lambda}$ not interacting with free-space modes, where $\Lambda$ is the periodicity and $m$ is an integer. In a photonic resonator, the bandgap imposes reflective boundaries to confine light as in a Fabry-Perot cavity Yu _et al._ (2019b). In our experiments, we use the bandgap instead in an edge-less boundary condition– a complete ring without edges –to modify a select mode of the PhCR. This condition, combined with an even number of nanopattern periods, frequency-aligns the bandgap to a mode of the PhCR McGarvey-Lechable and Bianucci (2014). Figure 1: Mode structure for the Kerr resonator, showing the cold- (cross) and hot-cavity (open circle) resonances, pump laser (dashed line), and light in each mode (solid circle). The left panels show the (A) Turing pattern, and (B) DKS state in the resonator, with the DKS Kerr mismatch (dashed red arrow). (C) and (D) show the resonator with photonic crystal shift (dashed blue arrow) at the corresponding Kerr shift, with (D) in the pulse state. (E) Illustration of optical pulse formation in a photonic ring resonator. (F) Simulated peak power versus pump laser detuning for the (green) and (blue) resonators, with the analytic flat amplitude (dashed gray) for reference. The corresponding intensity profiles are shown in the right panels. Figure 1 introduces the mode-frequency structure of a ring resonator () and a PhCR (), emphasizing how modifying the pump mode affects Turing-pattern and Kerr-soliton generation. The diagrams plot the modal detuning $f_{\mu}-(f_{0}+\mu\cdot FSR)$ for each mode $\mu$, showing the cold-resonator modes that correspond to comb modes $\mu$ (crosses) and the hot-resonator modes (open circles). The cold resonances follow the integrated dispersion $D_{\text{int}}=\omega_{\mu}-\omega_{0}-D_{1}\mu=1/2\,D_{2}\,\mu^{2}+\epsilon_{\text{PhC}}\cdot\left(1-\delta(\mu)\right)$, where $\omega_{\mu}$ is the angular frequency, $D_{1}$ is the free-spectral range, $\epsilon_{\text{PhC}}$ is the frequency shift of the pump mode, and $\delta(\mu)$ is the Kronecker delta function. We additionally shift the hot resonances by the Kerr shift $\delta_{\mu}$, indicating phase accumulation from the Kerr effect. At the onset of flat-state breakup, $\delta_{\mu=0}$ is half that for all other modes. Therefore a natural phase matching exists for FWM to the mode $\mu^{\prime}$, shown in Fig. 1A where the horizontal dashed line matches the shifted $D_{\text{int}}$ curve. Hence, the Turing pattern emerges, initially composed of pump and $\pm\mu^{\prime}$ modes (blue dots). The stationary soliton state (Fig. 1B) of the ring resonator involves Kerr frequency shifts to balance dispersion across many equidistant comb modes (blue dots); the horizontal line in Fig. 1B indicates the pump laser. However, since the pump-mode Kerr shift is reduced, only large $\alpha$ balances the Kerr mismatch $\xi_{\text{Kerr}}\ =\delta_{\mu\neq 0}-\delta_{\mu=0}$ This detuning precludes spontaneous formation of the Turing pattern, but also the formation of solitons, as the low flat state amplitude is below threshold. See Supplemental Sect. IV for more details. With a PhCR , we program the frequency shift $\epsilon_{\text{PhC}}$ to alleviate the $\xi_{\text{Kerr}}$ mismatch of the soliton state. The negative shift of both the cold and hot resonator at comb mode $\mu$ are apparent in Fig. 1C, D. Under this condition, the Turing pattern no longer emerges from the flat state when the pump mode is energized, since the natural FWM phase matching is removed; see the horizontal line in Fig. 1C. Importantly, the shift $\epsilon_{\text{PhC}}$ moves the cold pump mode toward lower frequency by an amount commensurate with the mismatch $\xi_{\text{Kerr}}$, thereby compensating for the reduced Kerr shift on the pump mode, bringing it approximately onto resonance with the pump laser. Operationally, soliton formation in a PhCR proceeds with the configuration shown in Fig. 1E. We integrate a PhCR device and a coupling waveguide on a silicon chip. The frequency shift $\epsilon_{\text{PhC}}$ is controlled by the nanopatterning on the ring, while the pump laser field $F$ couples evanescently into the PhCR from a waveguide. The continuous-wave pump laser energizes the PhCR and creates a stable soliton pulse-train at the output. To verify the physical understanding presented above, we use the LLE to calculate $\psi$ during a sweep of the pump-laser frequency across the pump mode; See Supplemental Sect. III for an LLE with the mode shift. Figure 1E shows the peak intensity $|\psi|^{2}$ versus detuning for the ordinary and photonic-crystal resonators, respectively. All frequency variables including $\alpha$ and $\epsilon_{\text{PhC}}$ are in unit of half-width-half-max linewidths unless otherwise specified. Aside from changing $\epsilon_{\text{PhC}}$ from 0 to 2.8 to activate the PhCR frequency shift, both simulations are performed with the same conditions, namely $F=1.5$, $\beta=-0.17$. The ring produces the 5-lobe Turing pattern (lower panel) as the pump detuning is swept completely across resonance, corresponding to a range of $\alpha$ from $-2$ to $4$. We then introduce the PhCR case and carry out the same $\alpha$ sweep. In contrast to the case, a single pulse forms with abrupt onset. Neither Turing patterns nor chaotic states form during the sweep. Furthermore, the pulse demonstrates two distinct sections of oscillatory stages, known as ‘breather’ soliton states Kippenberg _et al._ (2018). The curious reappearance of the breather state at the end of the sweep is also contrasts with resonator soliton behavior, and we observe this in our experiments. Figure 2: (A) Electron microscope image of the PhCR, showing the unit cell geometry and electric field profiles. Fabrication steps are shown as (i) substrate, (ii) electron-beam lithography, (iii) reactive ion etching, and (iv) dicing. (B) Laser frequency sweep traces of the shifted mode and nearby modes. (C) Mode frequencies for resonators with varying $A_{\text{PhC}}$, where $\mu=0$ is the $m_{\text{Ring}}=163$ resonance mode. (D) Laser frequency sweep traces of pump transmission (red) and comb power (blue) demonstrating a spontaneous step, where the stable pulse state is shaded in gray. (E) Optical spectrum of the stable state. Figure 2 presents our PhCR devices and experimental evidence for spontaneous soliton formation, according to the principles laid out above. We create a PhCR device with the oscillatory nanopattern indicated in Fig. 2A. A unit cell of the pattern is defined by a sinusoidal shape, characterized by the pattern periodicity and peak-to-peak amplitude $A_{\text{PhC}}$. The periodicity enforces a photonic bandgap that necessarily overlaps one particular PhCR mode, denoted as the pump mode $\mu=0$, in the 1550-nm wavelength range, owing to an equal azimuthal mode number of pattern periods and optical-mode fringes. The bandgap lifts the degeneracy of counter-propagating light in the PhCR, creating modes shifted to higher and lower frequency by an amount $\epsilon_{\text{PhC}}$. Since the nanopattern is edgeless– circumferentially uniform – high resonator $Q$ is maintained. The properties of the other PhCR modes ($\mu\neq 0$) with $\epsilon_{\text{PhC}}\approx 0$, including nonlinearity and GVD, are preserved under the geometric modification. In particular, the GVD depends sensitively on the thickness and ring-waveguide width (RW) as in an resonator. We fabricate our devices from a 570-nm-thick tantalum pentoxide (Ta2O5, hereafter tantala) photonics layer Jung _et al._ (2019), which is deposited on an oxidized silicon wafer. We use electron-beam lithography to define the photonics pattern for a wafer, and we transfer it to the tantala layer by use of fluorine reactive-ion etching. A final UV lithography process defines several chips on the wafer, and we dry-etch facets in the tantala and oxide layers, and the silicon wafer. See Supplemental Sect. II for more details. In our experiments with PhCRs, we characterize $\epsilon_{\text{PhC}}$ by spectroscopy measurements. We fabricate up to $\sim 75$ PhCRs on a chip with a systematic, few-linewidth variation of $\epsilon_{\text{PhC}}$ and the waveguide-resonator coupling gap to optimize the conditions for spontaneous soliton formation. To measure $\epsilon_{\text{PhC}}$, we couple light to and from the chip with a standard lensed-fiber system. Using a 1550-nm tunable laser as input, we record the transmission at the output with a photodetector. Figure 2B presents several PhCR mode resonances in the 1550-nm band, with applied frequency offsets so the resonances coincide, that demonstrate a single mode frequency splitting. We label the non-degenerate modes as upper and lower, with the latter at a setting of $\epsilon_{\text{PhC}}$ consistent with spontaneous soliton formation. Our experiments focus on gaps for near- critical coupling, and this data indicates a loaded PhCR $Q$ of $\sim$ 400,000. By adjusting the nanopattern amplitude through our e-beam lithography, we systematically vary $\epsilon_{\text{PhC}}$; see Fig. 2C. In the range of $A_{\text{PhC}}$ used in this work, the $Q$ factors are unaffected, compared to resonators fabricated on the same wafer. With a nanopattern amplitude of only a few nm, we control $\epsilon_{\text{PhC}}$ for the $\mu=0$ mode, whereas the $\mu^{\prime}\neq 0$ modes exhibit an anomalous GVD of $D_{2}=2\pi\cdot 69.0$ MHz/mode. The results confirm our fabrication process provide the high device geometry resolution and low optical loss to build PhCRs to support the pulses. We search for spontaneous soliton formation in a PhCR with $\epsilon_{\text{PhC}}=2.2$ by sweeping the frequency of the pump laser with $\sim 36$ mW of on-chip power; Fig. 2D presents a $\sim 20$ GHz sweep range from high to low frequency that spans the upper and lower resonances. With photodetectors, we monitor both transmission through the PhCR device (red trace) and the power of generated comb modes (blue trace), which we obtain by filtering out the pump. These data show the presence of thermal bistability effects, which distort the resonances into a triangle shape, and the effects of nonlinear comb generation. In particular, we observe no comb power at the upper resonance, as the upper mode is shifted away from the $\mu^{\prime}$ modes needed for FWM. Whereas at the lower resonance we observe immediate comb formation, corresponding to the step change in comb power that agrees with our simulation in Fig. 1F. We assess that this nonlinear state on the lower resonance, indicated by the shaded range in Fig. 2D, is a dissipative Kerr soliton that spontaneously forms under certain conditions of pump power and laser detuning. Additionally, we observe a nonlinear state on the lower resonance that exhibits relatively higher comb power variance, likely a breather state as indicated theoretically in Fig. 1F. The breather state at higher detuning than the stable state suggests a modified optical state phase diagram yet to be explored. Operationally, we adjust the pump power to maximize the pump-frequency existence range of the low-noise spontaneous soliton step, and we hand adjust the laser frequency into this range. Under these conditions we record the optical spectrum (Fig. 2E) of the soliton comb, which exhibits a clear $sech^{2}(\nu)$ profile as shown by the gray line. The remainder of our paper presents measurements of such spontaneous solitons. We attribute the ease of spontaneous-soliton capture to desirable thermal behaviors of the PhCR. Conventionally in a device, capturing and sustaining a soliton is difficult as a result of rapid heating and cooling of the microresonator Stone _et al._ (2018); Brasch _et al._ (2016). Soliton initiation in the resonator under CW excitation is proceeded by Turing patterns or chaotic states, which are multiple-pulse states with high average intensity. Conversely, the desired soliton state is a single pulse with a relatively low average intensity. Hence, the root of thermal instability is the transition of nonlinear state in a microresonator. The -resonator spontaneous solitons offer two primary advantages: First, in soliton initiation, we bypass the high average intensity states and avoid their heating effects to the resonator. Second, we keep the pump laser on-resonance in the soliton state (note the drop in transmission trace in Fig. 2D as the pulse forms, indicating a more resonant condition), therefore minimizing changes to the in-resonator pump amplitude as the soliton forms. Together, these factors minimize the intensity changes in the PhCR, allowing pulses capturing by hand-tuning alone. Figure 3: Optical spectra from PhCRs with average ring width (A-C) 1.3, 1.4, and 1.5 $\mu$m. (D) The two-pulse state on the 1.5 $\mu$m device. The gray traces are fits with the form $y=A_{0}+10log(sech^{2}((x-x_{0})/BW))$. (E) Spectra versus power for the RW = 1.4 $\mu$m device. To explore the universality of spontaneous-soliton formation, we demonstrate soliton bandwidth control by tuning the GVD of the PhCR and the pump-laser power. We control the GVD directly by varying the RW from 1.3 to 1.5 $\mu$m, providing decreasing anomalous GVD that we can understand from FEM calculation of the PhCR resonator mode structure. Based on the LLE, this change should affect an increasing soliton bandwidth. We tune by hand into the soliton states on these devices and acquire their optical spectra, plotted in Fig. 3A-C. The spectrum bandwidth broadens with decreasing anomalous GVD as expected. Interestingly, we acquired a stable two-pulse state at lower detuning on the RW = 1.5 $\mu$m device, shown in Fig. 3D. The two-pulse state suggests that the parameter space of the PhCR – an interplay between dispersion and mode shift – supports more steady states beyond the single spontaneous pulse. We also varied the pump laser power for the RW = 1.4 $\mu$m device, see Fig. 3E, resulting in widening of the spectral envelope consistent with the DKS. However, unlike the conventional case where increasing pump power monotonically lengthens the soliton existence range Guo _et al._ (2017), the PhCR produces strong breather states at high power. More study is underway to fully explore this behavior. Figure 4: (A) The relative intensity noise on the comb power of a breather (blue) and quiet (green) states. The dash lines show the detector noise floor corresponding to the carrier power. (B) Bridging between the 1 THz separation between comb lines (blue) by creating sidebands (red) using electro-optic modulation. The arrow indicates the overlapping mode. (C) Electronic beatnote from the overlapping mode. The gray traces show five consecutive measurements. (D) Optical spectrum of broad-band soliton. Stationary microresonator solitons output an optical pulse-train with fixed period, which composes a low-noise, equidistant frequency comb suitable for optical-frequency measurements Briles _et al._ (2018); Drake _et al._ (2019a). Therefore, verifying the spectral-noise properties of spontaneous solitons in PhCR is of utmost importance. Figure 4 presents intensity- and frequency-noise measurements, excluding the pump laser, of a spontaneous soliton, which we generate in a device with RW = 1.4 $\mu$m, $\epsilon_{\text{PhC}}$=3.0. The relative intensity noise (RIN, Fig. 4A) of a stationary soliton and a breather soliton is below $-140$ dBc/Hz over a Fourier frequency range to 1.8 GHz. Here, the photodetected soliton power is 282 $\mu$W and the spur-free dynamic range is excellent, whereas the breather state manifests a single peak at 878 MHz and supports higher power and hence lower RIN. These measurements are currently limited by the comb power and the detector noise floor. To measure the $\sim$ 1 THz PhCR soliton repetition frequency, we apply electro-optic (EO) phase modulation to create a low-frequency heterodyne beat between two soliton comb modes Drake _et al._ (2019a); the optical spectrum trace in Fig. 4B indicates the soliton modes in blue and the EO sidebands in red. We choose the EO drive frequency such that the $\pm 17^{\text{th}}$ order sidebands (arrow in Fig. 4B) generate an optical heterodyne on a photodetector, after filtering out that pair. We identify the tone thus generated as the heterodyne, as it varies with the EO drive frequency at $34.2$ MHz/MHz in agreement with the sideband orders. We present the heterodyne spectrum in Fig. 4C, which shows the typical lineshape with $\sim 50$ kHz linewidth and $<1$ MHz fluctuations. We attribute these properties to thermal noise Drake _et al._ (2019b) and thermal drift of the microresonator. Finally, we demonstrate a PhCR device with optimized dispersion to create a spontaneous DKS with near-octave bandwidth, shown in Fig. 4D. The $F^{2}$ value for this trace is estimated to be 8.7, normalized to threshold power of $\mu=\pm 1$ modes. We anticipate these optimized devices to enable f-2f self referencing in the future. In conclusion, we have presented spontaneous and deterministic generation of Kerr solitons in edge-less PhCRs, enabled by compensating the Kerr shift mismatch between the pulse state and its pump mode. Mode-shifting by nanopatterning enables spontaneous generation, whereas we retain the capability to engineer broadband dispersion with the bulk ring geometry. The importance of the nanophotonic capabilities presented in this work is two- fold: First, the ability to controllably shift modes while maintaining the bulk dispersion profile provides a tool to explore the physics occurring in a nonlinear process. 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Ilic, Journal of Research of the National Institute of Standards and Technology 121, 464 (2016). * Bao and Yang (2014) C. Bao and C. Yang, Journal of the Optical Society of America B 31, 3074 (2014). ## Acknowledgments Funding provided by the DARPA DODOS, DRINQS, and PIPES programs. We acknowledge the Boulder Microfabrication Facility, where the devices were fabricated. We thank Travis Briles and Jeff Chiles for a careful reading of the manuscript. This work is a contribution of the U.S. Government and is not subject to copyright. Mention of specific companies or trade names is for scientific communication only, and does not constitute an endorsement by NIST. ## Supplementary materials Materials and Methods Design and Fabrication Derivation of Modified LLE Kerr Shift Calculation Pulse Formation Dynamics Figs. S1 to S2 References (37-38) Supplenmental Materials for: Spontaneous Pulse Formation in Edge-Less Photonic Crystal Resonators Su-Peng Yu1,2∗, Daniel C. Cole1,2, Hojoong Jung1,2, Gregory T. Moille3,4, Kartik Srinivasan3,4, and Scott B. Papp1,2 1Time and Frequency Division, NIST, Boulder, CO 80305, USA 2 Department of Physics, University of Colorado, Boulder, CO, 80309, USA 3 Microsystems and Nanotechnology Division, NIST, Gaithersburg, MD 20899, USA 4 Joint Quantum Institute, NIST / University of Maryland, College Park, MD 20742, USA ## I Materials and Methods Here we provide details for the optical setup used in this work, illustrated in Figure S1. The light source is a C-band tunable external-cavity diode laser (ECDL) with fiber-coupled output. The light goes through a fiber isolator and then to a set of fiber polarization controllers. A 90% fused fiber coupler is added between the laser and the polarization controller to tap the laser light for a Mach-Zehnder interferometer and a wavelength meter (wavemeter, 40 MHz resolution) for frequency measurements. We use the wavemeter to precisely measure modes frequencies within the ECDL tuning range, enabling us to characterize the dispersion of PhCRs. For comb generation experiments, the laser is amplified using an erbium-doped fiber amplifier (EDFA), with a tunable band-pass filter to suppress the amplified spontaneous emission of the EDFA for cleaner spectra. For passive measurements, the EDFA and the filter are bypassed. We send the light into the photonic chip using a lens fiber mounted on a three-axis flexure stage, controlled by manual micrometers. The damage threshold of our devices is typically above 1 W incident power. The typical coupling efficiency between fiber and chip is $\sim$25% per facet, limited by the mode mismatch between the air-clad waveguides and lens fibers. The chip is placed on a copper block for thermal contact. The output is collected with another lens fiber on translation stage. For passive measurements, we measure the outcoming power using an amplified photodetector, plotting the transmission versus frequency on an oscilloscope. During the comb generation experiments, we continuously monitor a portion of the outcoupled light with an OSA. With photodetectors that have 150 MHz bandwidth, we also monitor the pump-laser transmission of the resonator and the comb power, which we obtain by filter out the pump contribution. The comb power signal provides critical information on break up of the flat background and soliton initiation, and for monitoring the intensity-noise level of soliton states. To diagnose breather soliton oscillations and perform intensity-noise measurements, we use a high-speed photodetector (1.6 GHz bandwid) and a electronic spectrum analyzer. The comb-power channel, after filtering out the pump, is also used for the beatnote measurements. We pass the comb light through two cascaded EO phase modulators, driven far above $V_{\pi}$ to introduce multiple sidebands to span the 1 THz frequency spacing between the comb lines, shown in main text Figure 4B. We choose the EO modulation frequency to be 28.000 GHz so the $\pm$17th sidebands from adjacent comb lines will come into close vicinity. To improve the signal to noise ratio for the beatnote measurements, we amplify the EO output with a semiconductor optical amplifier and select the overlapping modes using a tunable optical filter with a 50GHz passband. Figure 1: Illustration of the optical testing setup. ## II Design and Fabrication Here we describe the design process to create a PhCR. We calculate the resonator dispersion and photonic bandgap using a finite-element method program. The dispersion calculation yields the propagation constant $k_{\text{eff}}$ for each RW, ring radius R, and frequency. The azimuthal mode order m of the PhC is then calculated by the boundary condition $k_{\text{eff}}\cdot 2\pi R=2m\pi$. The PhCR modulation is then introduced with the periodicity $2\pi R/2m$ and sinusoidal peak-to-peak amplitude $A_{PhC}$ on the interior of the ring. The sinusoidal shape is chosen as it can be fabricated reliably to very small amplitude using lithography and plasma etching. A bus waveguide approaches the smooth outer edge of the resonators. The strength of the evanescent coupling between the resonator and the bus is controlled by the gap between the two. On the edges of the chips where the bus waveguides terminate, the waveguides are inversely tapered to improve mode-matching to lens fibers. We generated the mask files using a pattern-defining script and the CNST Nanolithography Toolbox Balram _et al._ (2016). Typically, we place up to 70 PhCRs and their bus waveguides per chip in an evenly spaced array. Fine sweeps of $A_{PhC}$ and coupling gap are included to achieve the correct mode shifts and near-critical coupling. The fabrication procedure of our devices is as follows: We obtain 3-inch silicon wafers with 380 $\mu$m thickness and 3 $\mu$m thermal silicon dioxide on both sides. The tantala device layer is deposited onto the wafer to 570 nm thickness by an external supplier. For lithography, we carry out a double spin-coating of ZEP520A resist to reach a total resist thickness of 1 $\mu$m, then expose the resist in electron beam lithography (EBL) operating at 100 kV. All device patterns are defined on this EBL step. We develop the resist and transfer the pattern using plasma etching with an inductively coupled plasma etching tool, and a $CHF_{3}+CF_{4}+Ar$ chemistry. The ratio between $CHF_{3}$ and $CF_{4}$ is varied to achieve vertical sidewall, while the Ar gas was found to improve sidewall smoothness. The etch selectivity is sufficient to clear the device layer with the resist thickness used. A dicing pattern is put onto the wafer using UV lithography and the SPR-220 photoresist. We etch through the bottom thermal oxide layer using a plasma etch with $CHF_{3}+O_{2}$ chemistry. The resist is stripped using solvents, and the UV lithography step is carried out again for the deep-RIE dicing using the $C_{4}F_{8}+SF_{6}$ chemistry. We then clean the wafer of the fluoro-polymer deposited during the RIE steps using DuPont EKC265 solvent, followed by a Cyantek Nanostrip soak for final cleaning. The chips are then mechanically removed from the wafer and are ready for testing. ## III Derivation of Modified LLE Here we provide a modified LLE to accommodate the influences of the shifted pump mode. Importantly, the form of the modified LLE admits the steady-state solutions of the LLE, with a effective pump field reflecting the influence of the shifted mode. We begin with the LLE in the modal basis, $\partial_{\tau}a_{\mu}=-(1+i\alpha)a_{\mu}+\frac{i}{2}\beta\mu^{2}a_{\mu}+\delta_{\mu 0}F+\hat{\mathcal{F}}\\{i|\psi(\theta)|^{2}\psi(\theta)\\}$ (1) where $a_{\mu}$ is the field amplitude in mode $\mu$, $\beta=-\frac{2}{\kappa}D_{2}$ stands for the second-order dispersion normalized to linewidth $\kappa$, $\delta_{\mu 0}$ the Kronecker delta function $\delta_{00}=1$, zero otherwise, and $\hat{\mathcal{F}}$ the Fourier transform. We generalize the equation to arbitrary dispersion profiles by identifying $\frac{i}{2}\beta\mu^{2}=-\frac{2i}{\kappa}\cdot\frac{1}{2}D_{2}\mu^{2}$ to be $-\frac{2i}{\kappa}D_{\text{int}}(\mu)$, where $D_{\text{int}}(\mu)$ is the integrated dispersion. We implement the pump mode shift with an additional term to the total dispersion: $D_{\text{int}}^{Shifted}(\mu)=D_{\text{int}}^{Base}(\mu)+\Xi(1-\delta_{\mu 0})$ (2) where a constant shift of strength $\Xi$ is applied to all modes except zero, so that the zero of detuning $\alpha$ remains defined on the pump mode. A red- shift of the pump mode is associated with $\Xi>0$. The equation becomes: $\partial_{\tau}a_{\mu}=-(1+i\alpha)a_{\mu}+\delta_{\mu 0}F+\hat{\mathcal{F}}\\{i|\psi(\theta)|^{2}\psi(\theta)\\}\\\ -\frac{2i}{\kappa}\left(D_{\text{int}}^{Base}(\mu)+\Xi(1-\delta_{\mu 0})\right)a_{\mu}$ (3) now carry out inverse Fourier transform, under the normalization that $\hat{\mathcal{F}}^{-1}(\delta_{\mu 0})=1$, and that $\hat{\mathcal{F}}^{-1}(\delta_{\mu 0}\cdot a_{\mu})=\frac{1}{2\pi}\oint\psi(\theta)d\theta:=\bar{\psi}$, we get the pump- shifted LLE in the time domain: $\partial_{\tau}\psi(\theta)=-\left(1+i(\alpha+\epsilon)\right)\psi(\theta)-\frac{i}{2}\beta\partial_{\theta}^{2}\psi(\theta)\\\ +i|\psi(\theta)|^{2}\psi(\theta)+F+i\epsilon\bar{\psi}$ (4) where $\epsilon=\frac{2\Xi}{\kappa}$ is the normalized mode shift. We make two observations from the shifted LLE formula: First, in the case of an amplitude that is a constant in $\theta$, $\bar{\psi}=\psi$ and the shift terms cancel, indicating that the resonator responses identically to the unmodified LLE prior to pattern generation. Second, assuming a time-stationary pattern $\psi$ is formed, the $\bar{\psi}$ term is constant in the resonator, and the equation can be interpreted as an LLE with modified parameters: $\alpha^{\prime}=\alpha+\epsilon\qquad F^{\prime}=|F+i\epsilon\bar{\psi}|$ This is to say any stationary-state solutions $\psi$ of the modified LLE with parameters $F,\alpha$ also satisfies the LLE with parameters $F^{\prime},\alpha^{\prime}$, the later include Turing patterns and Kerr solitons. This equivalence enables the pump-shifted LLE to produce Kerr solitons. ## IV Kerr Shift Calculation We present an interpretation of the Kerr shift term in the modal basis. Under this interpretation, each of the resonator mode $\mu$ behaves as a nonlinear harmonic resonator, therefore giving physical meanings to the hot-resonator modes in the main text. We begin by recalling the case of a single-mode nonlinear oscillator: $\partial_{t}a=-i\omega^{\prime}a-\gamma_{0}a+Fe^{i\omega t}$ (5) where the resonance frequency $\omega^{\prime}=\omega_{0}-g|a|^{2}$ depends on the field amplitude $|a|^{2}$ through nonlinear coefficient $g$. We identify this cubic term as the Kerr term $ig|a|^{2}a$ which results from the resonance frequency change induced by the field amplitude. In the case of the LLE, we start with the Kerr term instead, and assign an inferred modal frequency for each field component $a_{\mu}$ by casting the time-evolution of the mode in the form of a harmonic oscillator: $\partial_{\tau}a_{\mu}=-i\alpha a_{\mu}+i\delta_{\mu}\cdot a_{\mu}+g_{\mu}\cdot a_{\mu}+\delta_{\mu 0}F$ (6) where the detuning $\alpha$ was chosen so the pump field $F$ is not time- dependent in the rotating frame, and $\delta_{\mu}$ and $g_{\mu}$ can depend on the in-resonator field profile. This form enables us to identify the modal frequency and gain from the instantaneous rate of change of the phase and amplitude induced by the Kerr effect. We calculate these rates by Fourier transforming the Kerr term $|\psi|^{2}\psi$ for a given field profile $\psi(\theta)$: $\partial_{\tau}a_{\mu}|_{Kerr}=\hat{\mathcal{F}}\\{i|\psi(\theta)|^{2}\psi(\theta)\\},_{\mu}$ where the subscript $\mu$ for the Fourier transform specifies the $\mu$-th component. Casting this into the harmonic oscillator form, we get: $\delta_{\mu}^{Kerr}=\mathcal{R}\textit{e}\left(\hat{\mathcal{F}}\\{|\psi(\theta)|^{2}\psi(\theta)\\},_{\mu}/a_{\mu}\right)$ (7) $g_{\mu}^{Kerr}=-\mathcal{I}\textit{m}\left(\hat{\mathcal{F}}\\{|\psi(\theta)|^{2}\psi(\theta)\\},_{\mu}/a_{\mu}\right)$ (8) where $\delta_{\mu}^{Kerr}$ and $g_{\mu}^{Kerr}$ are the modal Kerr shift and induced gain on mode $\mu$. An example for this formula is the modal frequency behavior near the Turing pattern onset threshold. In order to extract the frequency of the modes, we assume the modal fields in the resonator take the form: $\psi(\theta)=a_{0}+\eta\cdot u_{\mu^{\prime}}(\theta)$ where $a_{0}$ is the pump mode amplitude, a constant in the resonator, and $a_{\mu^{\prime}}=\eta$ an infinitesimal field amplitude in the $\mu^{\prime}$-th mode, $u_{\mu^{\prime}}(\theta)=exp(i\mu^{\prime}\theta)$ is the basis function in the $\mu^{\prime}$-th mode. To obtain the pump mode shift, we evaluate the Kerr shift term $|\psi|^{2}\psi$ to zeroth order in $\eta$. Since $a_{0}$ is a constant over $\theta$, the form trivially gives: $\delta_{0}^{Kerr}=\mathcal{R}\textit{e}(\mathcal{F}\\{|a_{0}|^{2}a_{0}\\},_{0}/a_{0})=|a_{0}|^{2}$ (9) which is just the pump mode intensity. To get the shift for the $\mu^{\prime}$-th mode, we evaluate the Kerr shift term to first order in $\eta$: $|\psi|^{2}\psi=(|a_{0}|^{2}+\eta\cdot(a_{0}u_{\mu^{\prime}}^{*}+a_{0}^{*}u_{\mu^{\prime}})+\mathcal{O}(\eta^{2}))\cdot(a_{0}+\eta u_{\mu^{\prime}})$ $=|a_{0}|^{2}a_{0}+2|a_{0}|^{2}\cdot\eta u_{\mu^{\prime}}+a_{0}^{2}\cdot\eta u_{\mu^{\prime}}^{*}+\mathcal{O}(\eta^{2})$ where $u_{\mu^{\prime}}^{*}=u_{-\mu^{\prime}}$. Fourier transforming this expression and taking the $\mu^{\prime}$-th component, only the term with $u_{\mu^{\prime}}$ is non-vanishing. We get: $\delta_{\mu^{\prime}}^{Kerr}=\mathcal{R}\textit{e}(2|a_{0}|^{2}\cdot\eta/\eta)=2|a_{0}|^{2}$ (10) which is twice the shift compared to the pump mode, in agreement to the form in Herr _et al._ (2015). We are now equipped with the theoretical tools to study the Kerr balancing for the soliton states. We generate the soliton field profile using the LLE for sets of parameters $(F,\alpha,\epsilon)$, then calculate the dispersion balancing conditions for each mode, namely $-D_{\text{int}}(\mu)+\delta_{\mu}$. We find the term equals $\alpha$ for all non-pump modes, while the gain terms equal 1 to balance loss. The balancing effect enables a stationary, time-independent waveform in the reference frame of the LLE. In the case $\epsilon=0$, the balance is not achieved for the pump mode, but the mismatch is compensated by the pump field $F$ in a manner similar to the forced harmonic oscillator. To study the Kerr mismatch in response to the shifted pump mode, we carry out LLE simulations to obtain the field profiles of the stable pulse states in a resonator for some $\epsilon>0$, and calculate the Kerr shifts for each mode. Kerr shift and $D_{\text{int}}$ plots for the cases with $\epsilon$=0 and $\epsilon$=4.2 are shown in Figure S2A and B. The blue dots show the sum of the two, balancing to the horizontal lines at the value of $\alpha$, except for the pump mode, in agreement with Ref. Bao and Yang (2014). A pronounced Kerr mismatch $\xi_{Kerr}$ is observed for the $\epsilon$=0 case, but is suppressed in the $\epsilon$=4.2 case. We carry out the calculations for intermediate values of $\epsilon$, shown in Fig. S2C. The increasing of $\epsilon$ resulted in a gradual reduction of mismatch $\xi_{Kerr}$, down to approximately one quarter of a linewidth. Figure S2C also shows the $\alpha$ ranges where the soliton state is stable for each $\epsilon$. We observe that the soliton is stable in detuning ranges in the single-stability range for the given $F$ value for the shifted-pump cases, while in the $\epsilon=0$ case the soliton is only stable in the bistability range (shaded area in Fig. S2C). This leads to the difference that the flat state is stable on the lower branch of the bistability in the $\epsilon=0$ case, versus the spontaneous generation of patterns from the flat state in the shifted-pump case. Initiating the $\epsilon=0$ case with a pulse in the single-stability range results in the mode reverting spontaneously to multiple-pulse Turing patterns. This suggests that the mode shift modifies the phase diagram Godey _et al._ (2014) to enable stable soliton states in ranges where the flat background amplitude is unstable. Figure 2: The balancing of Kerr shift (red) and dispersion (black) for (A) the and (B) soliton pulses, where the sum of the two is magnified and shown in blue. (C) Calculated Kerr mismatch for various mode shift values and pump detuning. (D) Simulated time traces and (E) intensity plots during the spontaneous pulse generation process. ## V Pulse Formation Dynamics We show the time-evolution of the spontaneous generation of a single pulse in Fig. S2D-E. Here the pulse arises spontaneously from the flat state with constant pump $F$ and detuning $\alpha$, seeded only by the vacuum fluctuation. The LLE simulation of pulse generation shows several transient states the resonator goes through to arrive at the DKS state. In this simulation, the resonator is initiated with zero amplitude, and is energized with a fixed pump field F at fixed detuning $\alpha$ for some time until the pulse state stabilizes. We identify four transient states in the pulse generation, shown in Fig. S2E, starting from its bottom panel: First, the flat amplitude energizes without producing comb power, until it is sufficiently large that the flat state becomes unstable. Unlike the conventional resonator where the FWM condition can be reached by the large mode density near the pump mode to form Turing patterns order $\mu^{\prime}$ determined by dispersion, the phase matching is prohibited by the shifted pump mode. Second, with the shifted pump mode, the PhCR instead make a one-lobe sinusoidal pattern once the flat amplitude is sufficiently high. This can be intuitively understood by drawing a quadratic curve across the three modes $\mu=0,\mu^{\prime}=\pm 1$ for the PhCR mode structure. The high positive curvature of this curve affects a local strong anomalous dispersion, causing a transient Turing pattern of order $\mu^{\prime}=1$ to form. This transient pattern breaks the $\theta$-symmetry in the resonator, seeding the resonator for a single pulse. Third, the one-lobe pattern begins to sharpen. This is because unlike a true high-anomalous-dispersion resonator where $\mu^{\prime}>\pm 1$ modes are FWM- mismatched from strong dispersion, the $\mu^{\prime}>\pm 1$ modes of the PhCR follow the base dispersion, therefore are sufficiently phase-matched and can be energized. The energizing of $\mu^{\prime}>\pm 1$ modes lead to sharpening of the peak in time domain, and the broadening of its spectrum. Finally, the pulse stabilizes as the transient components decay away. Note that at this stage the flat amplitude background is significantly lower than prior to the pulse formation, a curious result from the modified LLE changing its effective pump field $F^{\prime}$ in response to the existing pulse in the resonator. The reduced flat amplitude will no longer spontaneously generate patterns, preventing further pulse generation beyond the first pulse. This set of transient steps eventually result in deterministic placement of one broad-band pulse in the resonator, shown in the top panel of Fig. S2E. ## Disclaimer This work is a contribution of the U.S. Government and is not subject to copyright. Mention of specific companies or trade names is for scientific communication only, and does not constitute an endorsement by NIST.

2024-09-04T02:54:55.353168

2002.12508

arxiv-papers

int dom Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem. § INTRODUCTION Estimating ground energy and obtaining information on the ground state of a given quantum Hamiltonian are of immense importance in condensed matter physics, quantum chemistry, and quantum information. Classical methods suffer from the exponential growth of the size of Hilbert space, and therefore quantum computers are expected to be used to overcome this difficulty. However even for quantum computer, estimating the ground energy is a hard problem: deciding whether the smallest eigenvalue of a generic local Hamiltonian is greater than $b$ or smaller than $a$ for some $a<b$ is QMA-complete [Kitaev et al., 2002, Kempe et al., 2006, Oliveira and Terhal, 2005, Aharonov et al., 2009]. Therefore to make the problem efficiently solvable we need more assumptions. We denote the Hamiltonian we are dealing with by $H$, and consider its spectral decomposition $H=\sum_k\lambda_k\ket{\psi_k}\bra{\psi_k}$ where $\lambda_{k}\leq\lambda_{k+1}$. The key assumption is that we have an initial state $\ket{\phi_0}$ which can be efficiently prepared by an oracle $U_I$, and has some overlap with the ground state $\ket{\psi_0}$ lower bounded by $\gamma$. This is a reasonable assumption in many practical scenarios. For instance, even for strongly-correlated molecules in quantum chemistry, there is often a considerable overlap between the true ground state and the Hartree-Fock state. The latter can be trivially prepared in the molecular orbital basis, and efficiently prepared in other basis [Kivlichan et al., 2018]. For the moment we also assume the spectral gap is bounded from below: $\lambda_1-\lambda_0\geq\Delta$. With these assumptions we can already use phase estimation coupled with amplitude amplification [Brassard et al., 2002] to prepare the ground state, if we further know the ground energy to high precision. To our knowledge, the most comprehensive work on ground state preparation and ground state energy estimation was done by Ge [Ge et al., 2019], which provided detailed complexity estimates for well-known methods such as phase estimation, and proposed new methods to be discussed below. As analyzed in <cit.>, in order to prepare the ground state to fidelity[In this work, the fidelity between states $\ket{x},\ket{y}$ is defined to be $\abs{\braket{x|y}}$.] $1-\epsilon$, the runtime of the controlled-time-evolution of the Hamiltonian is $\wt{\Or}(1/(\gamma^2\Delta\epsilon))$ [In this work the notation $\wt{\Or}(f)$ means $\Or(f\poly\log(f))$ unless otherwise stated.], and the number of queries to $U_I$ is $\wt{\Or}(1/\gamma)$, assuming the spectral norm of $H$ is bounded by a constant. This is however far from optimal. Poulin and Wocjan [Poulin and Wocjan, 2009] proposed a method that, by executing the inverse of phase estimation to filter out the unwanted components in the initial state, can prepare a state whose energy is in a certain given range. A different choice of parameters yields a way to prepare the ground state to fidelity $1-\epsilon$ by running the controlled-time-evolution of the Hamiltonian with $\wt{\Or}(1/(\gamma\Delta)\log(1/\epsilon))$ runtime, and using $\wt{\Or}(1/\gamma)$ queries to $U_I$ <cit.>. A key difference between ground state preparation and Hamiltonian simulation, where significant progress has been made in recent years [Lloyd, 1996, Berry et al., 2015, Berry et al., 2015, Low and Chuang, 2017, Low and Wiebe, 2018, Low and Chuang, 2019, Childs et al., 2019], is its non-unitary nature. The recent development of linear combination of unitaries (LCU) method [Berry et al., 2015, Childs et al., 2017] provided a versatile tool to apply non-unitary operators. Using LCU, Ge proposed a new method to filter the initial state by applying a linear combination of time-evolutions of different time length [Ge et al., 2019], which achieves the same complexity, up to logarithmic factors, as the modified version of Poulin and Wocjan's method discussed above. All of the above methods prepare the ground state assuming the ground energy is known to high precision. When the ground energy is unknown, Ge proposed a method to estimate the ground energy using a search method called minimum label finding [Ge et al., 2019]. This method can estimate the ground energy to precision $h$ by running the controlled-time-evolution of the Hamiltonian for $\wt{\Or}(1/(\gamma h^{3/2}))$ [In [Ge et al., 2019], the meaning of the notation $\wt{\Or}(\cdot)$ is different from that in our work. In particular, $\wt{\Or}(\cdot)$ in [Ge et al., 2019] hides all factors that are poly-logarithmic in $1/h$, $1/\epsilon$, $1/\gamma$, and $1/\Delta$, regardless of what is inside the parentheses. We preserve their notation when citing their results since these factors do not play an important role when comparing the complexities of our methods.], and querying $U_I$ $\wt{\Or}(1/(\gamma \sqrt{h}))$ times. It is worth noting that their method requires $h=\wt{\Or}(\Delta)$, and therefore is very expensive when the gap is extremely small. When the ground energy is not known , Ge proposed a method to first estimate the ground energy and then apply the LCU approach. In recent years several hybrid quantum-classical algorithms have been developed to estimate the ground energy, or to prepare the ground state, or both. The variational quantum eigenvalue solver (VQE) [Peruzzo et al., 2014] has gained much attention recently because of its low requirement for circuit depth and its variational structure. However the exact complexity of this algorithm is not clear because it relies on a proper choice of ansatz and needs to solve a non-convex optimization problem. Other such algorithms include quantum imaginary-time evolution, quantum Lanczos [Motta et al., 2019], and quantum filter diagonalization [Parrish and McMahon, 2019, Stair et al., 2019]. Their complexities are either quasi-polynomial or unknown. The recent development of block-encoding [Berry et al., 2015] and quantum signal processing (QSP) [Low et al., 2016, Low and Chuang, 2017, Gilyén et al., 2019] enables us to apply non-unitary operators, specifically polynomials of a block-encoded matrix efficiently. It uses a minimal number of ancilla qubits, and avoids the Hamiltonian simulation. These will be the basic tools of this work, of which we give a brief introduction below. Block-encoding is a powerful tool to represent a non-unitary matrix in the quantum circuit. A matrix $A\in\CC^{N\times N}$ where $N=2^n$ can be encoded in the upper-left corner of an $(m+n)$-qubit unitary matrix if \begin{equation} \norm{A-\alpha(\bra{0^m}\otimes I) U (\ket{0^m}\otimes I)}_2\leq \epsilon. \label{eqn:block_encoding} \end{equation} In this case we say $U$ is an $(\alpha,m,\epsilon)$-block-encoding of $A$. Many matrices of practical interests can be efficiently block-encoded. In particular we will discuss the block-encoding of Hamiltonians of physical systems in Section <ref>. Using the block-encoding of a Hermitian $A$, QSP enables us to construct block-encodings for a large class of polynomial eigenvalue transformations of $A$. We pay special attention to even or odd polynomials with real coefficients, because we only apply this type of polynomial eigenvalue transformation in this work. Also for simplicity we assume the block-encoding is done without error. <cit.> enables us to perform eigenvalue transformation of $A$ for polynomials of definite parity (even or odd). Let $U$ be an $(\alpha,m,0)$-block-encoding of a Hermitian matrix $A$. Let $P\in\RR[x]$ be a degree-$\ell$ even or odd real polynomial and $\abs{P(x)}\le 1$ for any $x\in[-1,1]$. Then there exists an $(1, m+1,0)$-block-encoding $\wt{U}$ of $P(A/\alpha)$ using $\ell$ queries of $U$, $U^{\dagger}$, and $\Or((m+1)\ell)$ other primitive quantum gates. <cit.> provides a singular value transformation for any square matrix $A$ and polynomials of definite parity. When $A$ is a Hermitian matrix, the eigenvalue transformation is the same as the singular value transformation <cit.>. A related statement in the same paper is <cit.>, which describes the eigenvalue transformation of a Hermitian matrix for an arbitrary polynomial, by means of a linear combination of two polynomials of even and odd parities respectively. Constructing the quantum circuit for QSP requires computing a sequence of phase factors beforehand, and there are classical algorithms capable of doing this [Haah, 2019]. Some recent progress has been made to efficiently compute phase factors for high-degree polynomials to high precision [Chao et al., 2020, Dong et al., 2020]. In this work, unless otherwise specified, we assume the phase factors are computed without error. Using the tools introduced above, we assume the Hamiltonian $H$ is given in its $(\alpha,m,0)$-block-encoding $U_H$. This, together with $U_I$, are the two oracles we assume we are given in this work. QSP enables us to filter eigenstates using fewer qubits than LCU. In [Lin and Tong, 2020] a filtering method named optimal eigenstate filtering is introduced. It is based on an explicitly constructed optimal minimax polynomial, and achieves the same asymptotic complexity, ignoring poly-logarithmic factors, as the method by Ge when applied to the ground state preparation problem if the ground energy is known exactly. In this work we first develop a filtering method that filters out all eigenstates corresponding to eigenvalues above a certain threshold. This filtering method enables us to prepare the ground state of a Hamiltonian with spectral gap bounded away from zero when only an upper bound of the ground energy is known, unlike in the filtering methods discussed above which all require either exact value or high-precision estimate of the ground energy. Our filtering method has an exponentially improved dependence on precision compared to Kitaev's phase estimation [Kitaev, 1995] and uses fewer qubits compared to other variants of the phase estimation algorithm [Poulin and Wocjan, 2009, Ge et al., 2019]. This filtering method, applied to the initial state given in our assumption, also enables us to tell whether the ground energy is smaller than $a$ or greater than $b$ for some $b>a$, with high probability. Therefore a binary search yields a ground energy estimate with success probability arbitrarily close to one. We then combine the filtering method and ground energy estimation to prepare the ground state when no non-trivial bound for the ground energy is known. A comparison of the query complexities between the method in our work and the corresponding ones in [Ge et al., 2019], which to our best knowledge achieve state-or-the-art query complexities, are shown in Table <ref>. Preparation (bound known) Ground energy Preparation (bound unknown) 2*$U_H$ This work $\Or\left(\frac{\alpha}{\gamma\Delta}\log(\frac{1}{\epsilon})\right)$ $\wt{\Or}\left(\frac{\alpha}{\gamma h}\log(\frac{1}{\vartheta})\right)$ $\wt{\Or}\left(\frac{\alpha}{\gamma \Delta}\log(\frac{1}{\vartheta\epsilon})\right)$ Ge $\wt{\Or}\left(\frac{\alpha}{\gamma\Delta}\right)$ $\wt{\Or}\left(\frac{\alpha^{3/2}}{\gamma h^{3/2}}\right)$ $\wt{\Or}\left(\frac{\alpha^{3/2}}{\gamma \Delta^{3/2}}\right)$ 2*$U_I$ This work $\Or\left(\frac{1}{\gamma}\right)$ $\wt{\Or}\left(\frac{1}{\gamma}\log(\frac{\alpha}{h})\log(\frac{1}{\vartheta})\right)$ $\wt{\Or}\left(\frac{1}{\gamma}\log(\frac{\alpha}{\Delta})\log(\frac{1}{\vartheta})\right)$ Ge $\wt{\Or}\left(\frac{1}{\gamma}\right)$ $\wt{\Or}\left(\frac{1}{\gamma}\sqrt{\frac{\alpha}{h}}\right)$ $\wt{\Or}\left(\frac{1}{\gamma}\sqrt{\frac{\alpha}{\Delta}}\right)$ Extra This work $\Or(1)$ $\Or(\log(\frac{1}{\gamma}))$ $\Or(\log(\frac{1}{\gamma}))$ qubits Ge $\Or(\log(\frac{1}{\Delta}\log(\frac{1}{\epsilon})))$ $\Or(\log(\frac{1}{h}))$ $\Or(\log(\frac{1}{\Delta}\log(\frac{1}{\epsilon})))$ The query complexities of algorithms and number of extra qubits used in our work and the corresponding ones by Ge in [Ge et al., 2019]. $\alpha,\gamma,\Delta,\epsilon$ are the same as above and $h$ is the precision of the ground energy estimate. By extra qubits we mean the ancilla qubits that are not part of the block-encoding. In this work the ground energy estimation algorithm and the algorithm to prepare ground state without bound have success probabilities lower bounded by $1-\vartheta$, while in [Ge et al., 2019] the corresponding algorithms have constant success probabilities. The complexities for algorithms by Ge are estimated assuming Hamiltonian simulation is done as in [Low and Chuang, 2019]. The usage of the notation $\wt{\Or}$ is [Ge et al., 2019] different from that in our work, as explained in footnote <ref>. From the query complexities in Table <ref> we can see our method for ground energy estimation achieves a exponential speedup in terms of the dependence of number of queries to $U_I$ on the ground energy estimate precision $h$ and a speedup of $1/\sqrt{h}$ factor in the dependence of number of queries to $U_H$ on the precision. Moreover, Ge assumes in their work that the precision $h=\wt{\Or}(\Delta)$, while we make no such assumptions. This gives our algorithm even greater advantage when the gap is much smaller than desired precision. This becomes useful in the case of preparing a low energy state (not necessarily a ground state). Because Ge used a slightly different query assumption, access to time-evolution rather than block-encoding, when computing the complexities for methods in [Ge et al., 2019] in Table <ref> we assume the Hamiltonian simulation is done with $\Or(\alpha t)$ queries to $U_H$, and the error is negligible. This can be achieved using the Hamiltonian simulation in [Low and Chuang, 2019], and cannot be asymptotically improved because of the complexity lower bound proved in [Berry et al., 2015]. Therefore the comparison here is fair even though our work makes use of a different oracle. Also [Ge et al., 2019] assumed a scaled Hamiltonian $H$ with its spectrum contained in $[0,1]$. We do not make such an assumption, and therefore the $\alpha$ factor should be properly taken into account as is done in Table <ref>. The rest of the paper is organized as follows. In Section <ref> we use QSP to construct block-encodings of reflectors and projectors associated with eigen-subspaces. In Section <ref> we use the projectors to prepare ground state when an upper bound of the ground energy is given. In Section <ref> we introduce the ground energy estimation algorithm, a hybrid quantum-classical algorithm based on the binary search, and use it to prepare the ground state when no ground energy upper bound is known . In Section <ref> we show the dependence of our query complexities on the overlap and gap is essentially optimal by considering the unstructured search problem. We also show the dependence of our ground energy estimation algorithm on the precision is nearly optimal by considering the quantum approximate counting problem. In Section <ref> we use our methods to prepare low-energy states when the spectral lower gap is unknown, or even when the ground state is degenerate. In Section <ref> we discuss practical issues and future research directions. § BLOCK-ENCODING OF REFLECTOR AND PROJECTOR A key component in our method is a polynomial approximation of the sign function in the domain $[-1,-\delta]\cup[\delta,1]$. The error scaling of the best polynomial approximation has been studied in [Eremenko and Yuditskii, 2007], and an explicit construction of a polynomial with the same error scaling is provided in [Low and Chuang, 2017] based on the approximation of the $\mathrm{erf}$ function. We quote <cit.> here with some small modification: For all $0<\delta<1$, $0<\epsilon<1$, there exists an efficiently computable odd polynomial $S(\cdot;\delta,\epsilon)\in\RR[x]$ of degree $\ell=\Or(\frac{1}{\delta}\log(\frac{1}{\epsilon}))$, such that (1) for all $x\in [-1,1]$, $|S(x;\delta,\epsilon)|\leq 1$, and (2) for all $x\in[-1,-\delta]\cup[\delta,1]$, $|S(x;\delta,\epsilon)-\mathrm{sign}(x)|\leq \epsilon$. Compared to <cit.> we have rescaled the interval from $[-2,2]$ to $[-1,1]$, and this does not result in any substantial change. When we have the $(\alpha,m,0)$-block-encoding of a Hermitian matrix $H=\sum_{k}\lambda_k\ket{\psi_k}\bra{\psi_k}\in\CC^{N\times N}$, $N=2^n$, $\lambda_k\leq \lambda_{k+1}$, we can construct a $(\alpha+|\mu|,m+1,0)$-block-encoding of matrix $H-\mu I$ using of <cit.> for any $\mu\in\RR$. Then using QSP, by Theorem <ref>, we can obtain an $(1,m+2,0)$-block-encoding of $-S(\frac{H-\mu I}{\alpha+|\mu|};\delta,\epsilon)$ for any $\delta$ and $\epsilon$. If we assume further that $\Delta/2 \leq \min_k |\mu-\lambda_k|$, then we let $\delta=\frac{\Delta}{4\alpha}$, and by Lemma <ref> all the eigenvalues of $-S(\frac{H-\mu I}{\alpha+|\mu|};\delta,\epsilon)$ are $\epsilon$-close to either 0 or 1. Therefore $-S(\frac{H-\mu I}{\alpha+|\mu|};\delta,\epsilon)$ is $\epsilon$-close, in operator norm, to the reflector about the direct sum of eigen-subspaces corresponding to eigenvalues smaller than $\mu$: \[ R_{<\mu} = \sum_{k:\lambda_k<\mu} \ket{\psi_k}\bra{\psi_k} - \sum_{k:\lambda_k>\mu} \ket{\psi_k}\bra{\psi_k}, \] and thus the block-encoding is also an $(1,m+2,\epsilon)$-block-encoding of $R_{<\mu}$. We denote this block-encoding by $\REF(\mu,\delta,\epsilon)$. We omitted the dependence on $H$ because $H$ as well as its block-encoding is usually fixed in the rest of the paper. In the above discussion we have used QSP in a black-box manner. For concreteness, we present a single-qubit illustrative example to demonstrate how to use a block-encoded Hamiltonian to construct the reflector in Appendix <ref>. Because our goal is to prepare the ground state, we will use the projector more often than the reflector. Now we construct a block-encoding of projector using $\REF(\mu,\delta,\epsilon)$ by the following circuit \begin{equation} \label{eq:circ_proj} \Qcircuit @C=1em @R=.3em{ \lstick{\ket{0}} & \qw & \gate{\mathrm{H}} & \ctrl{1} & \gate{\mathrm{H}} & \qw\\ \lstick{\ket{0^{m+2}}} & \qw & \qw & \multigate{1}{\REF(\mu,\delta,\epsilon)} & \qw & \qw \\ \lstick{\ket{\phi}} & \qw & \qw & \ghost{\REF(\mu,\delta,\epsilon)} & \qw & \qw \end{equation} where $\mathrm{H}$ is the Hadamard gate, and we denote this circuit as $\PROJ(\mu,\delta,\epsilon)$. Note that \[ \begin{aligned} &(\bra{0^{m+3}}\otimes I)\PROJ(\mu,\delta,\epsilon)(\ket{0^{m+3}}\otimes I) \\ &= \Big(\bra{+}\bra{0^{m+2}}\otimes I\Big)\Big(\ket{0}\bra{0}\otimes I\otimes I + \ket{1}\bra{1}\otimes \REF(\mu,\delta,\epsilon)\Big)\Big(\ket{+}\ket{0^{m+2}}\otimes I\Big) \\ &= \frac{1}{2}\Big(I+(\bra{0^{m+2}}\otimes I)\REF(\mu,\delta,\epsilon)(\ket{0^{m+2}}\otimes I)\Big), \end{aligned} \] and we have \[ \begin{aligned} &\|(\bra{0^{m+3}}\otimes I)\PROJ(\mu,\delta,\epsilon)(\ket{0^{m+3}}\otimes I)-P_{<\mu}\| \\ &\leq \frac{1}{2}\|(\bra{0^{m+2}}\otimes I)\REF(\mu,\delta,\epsilon)(\ket{0^{m+2}}\otimes I)-R_{<\mu}\| \\ &\leq \frac{\epsilon}{2}. \end{aligned} \] Here $P_{<\mu}$ is the projector into the direct sum of eigen-subspaces corresponding to eigenvalues smaller than $\mu$ \[ P_{<\mu} = \sum_{k:\lambda_k<\mu} \ket{\psi_k}\bra{\psi_k} \] Therefore $\PROJ(\mu,\delta,\epsilon)$ is an $(1,m+3,\epsilon/2)$-block-encoding of $P_{<\mu}$. In fact this can still be seen as an application of linear combination of block encoding <cit.>, using the relation $P_{<\mu}=\frac{1}{2}(R_{<\mu}+I)$. We use the following lemma to summarize the results Given a Hermitian matrix $H$ with its $(\alpha,m,0)$-block-encoding $U_H$, with the guarantee that $\mu\in\RR$ is separated from the spectrum of $H$ by a gap of at least $\Delta/2$, we can construct an $(1,m+2,\epsilon)$-block-encoding of $R_{<\mu}$, and an $(1,m+3,\epsilon/2)$-block-encoding of $P_{<\mu}$, both using $\Or(\frac{\alpha}{\Delta}\log(\frac{1}{\epsilon}))$ applications of $U_H$ and $U_H^\dagger$, and $\Or(\frac{m\alpha}{\Delta}\log(\frac{1}{\epsilon}))$ other one- and two-qubit gates. We remark that for the block-encoding $\PROJ(\mu,\delta,\epsilon)$, even a failed application of it can give us potentially useful information. We have \[ \PROJ(\mu,\delta,\epsilon)\ket{0^{m+3}}\ket{\phi}=\ket{0}\ket{0^{m+2}}P_{<\mu}\ket{\phi}+\ket{1}\ket{0^{m+2}}P_{>\mu}\ket{\phi}+\frac{1}{\sqrt{2}}\ket{{-}}\ket{E}, \] where $P_{>\mu} = I- P_{<\mu}$ and $\ket{E}$ satisfies $\|\ket{E}\|\leq \epsilon$. Thus when we apply the block-encoding and measure the first two registers, the first $m+3$ qubits, we have probability at least $1-\frac{\epsilon^2}{2}$ to obtain an outcome with either 0 or 1 followed by $(m+2)$ 0's. In the former case the projection has been successful, and in the latter case we have obtained an approximation of $P_{>\mu}\ket{\phi}$. If we do not treat the output of 1 followed by $m+2$ 0's as failure then there is another interpretation of the circuit $\PROJ(\mu,\delta,\epsilon)$: this is an approximate projective measurement $\{P_{<\mu},P_{>\mu}\}$. In fact the whole circuit can be seen as phase estimation on a reflector, which needs only one ancilla qubit. § ALGORITHM WITH GROUND ENERGY BOUND With the approximate projector developed in the previous section we can readily design an algorithm to prepare the ground state. We assume we have the Hamiltonian $H$ given through its block-encoding as in the last section. If we are further given an initial state $\ket{\phi_0}$ prepared by a unitary $U_{I}$, $U_{I}\ket{0^n}=\ket{\phi_0}$, and the promises that for some known $\gamma>0$, $\mu$, and $\Delta$, we have * Lower bound for the overlap: $|\braket{\phi_0|\psi_0}|\geq \gamma$, * Bounds for the ground energy and spectral gap: $\lambda_0 \leq \mu - \Delta/2 < \mu + \Delta/2 \leq \lambda_1$. Here $\mu$ is an upper bound for the ground energy, $\Delta$ is a lower bound for the spectral gap, and $\gamma$ is a lower bound for the initial overlap. Now suppose we want to prepare the ground state to precision $\epsilon$, we can use Lemma <ref> to build a block-encoding of the projector $P_{<\mu}=\ket{\psi_0}\bra{\psi_0}$, and then apply it to $\ket{\phi_0}$ which we can prepare. This will give us something close to $\ket{\psi_0}$. We use fidelity to measure how close we can get. To achieve $1-\epsilon$ fidelity we need to use circuit $\PROJ(\mu,\Delta/4\alpha,\gamma\epsilon)$, and we denote, \[ \wt{P}_{<\mu} = (\bra{0^{m+3}}\otimes I)\PROJ(\mu,\Delta/4\alpha,\gamma\epsilon)(\ket{0^{m+3}}\otimes I) \] then the resulting fidelity will be \[ \frac{|\braket{\psi_0|\wt{P}_{<\mu}|\phi_0}|}{\|\wt{P}_{<\mu}\ket{\phi_0}\|} \geq \frac{|\braket{\psi_0|\phi_0}|-\gamma\epsilon/2}{|\braket{\psi_0|\phi_0}|+\gamma\epsilon/2} \geq 1-\frac{\gamma\epsilon}{|\braket{\psi_0|\phi_0}|}\geq 1-\epsilon. \] Here we have used \[ \|\wt{P}_{<\mu}\ket{\phi_0}\|\le \norm{P_{<\mu}\ket{\phi_0}+(\wt{P}_{<\mu}-P_{<\mu})\ket{\phi_0}}\le |\braket{\psi_0|\phi_0}|+\gamma\epsilon/2. \] This is when we have a successful application of the block-encoding. The success probability is \[ \|\wt{P}_{<\mu}\ket{\phi_0}\|^2 \geq \left(\|P_{<\mu}\ket{\phi_0}\|-\frac{\gamma\epsilon}{2}\right)^2 \geq \gamma^2\left(1-\frac{\epsilon}{2}\right)^2. \] With amplitude amplification [Brassard et al., 2002] we can boost the success probability to $\Omega(1)$ with $\Or(\frac{1}{\gamma})$ applications of $\PROJ(\mu,\Delta/4\alpha,\gamma\epsilon)$ and its inverse, as well as $\Or(\frac{m}{\gamma})$ other one- and two- qubit gates. Here we are describing the expected complexity since the procedure succeeds with some constant probability. In amplitude amplification we need to use a reflector similar to the oracle used in Grover's search algorithm [Grover, 1996]. Instead of constructing a reflector from $\PROJ(\mu,\Delta/4\alpha,\gamma\epsilon)$ we can directly use $\REF(\mu,\Delta/4\alpha,\gamma\epsilon)$ constructed in the previous section. We summarize the results in the following theorem Suppose we have Hamiltonian $H=\sum_{k}\lambda_k \ket{\psi_k}\bra{\psi_k}\in\CC^{N\times N}$, where $\lambda_k\leq \lambda_{k+1}$, given through its $(\alpha,m,0)$-block-encoding $U_H$. Also suppose we have an initial state $\ket{\phi_0}$ prepared by circuit $U_{I}$, as well as the promises <ref> and <ref>. Then the ground state $\ket{\psi_0}$ can be prepared to fidelity $1-\epsilon$ with the following costs: * Query complexity: $\Or(\frac{\alpha}{\gamma\Delta}\log(\frac{1}{\gamma\epsilon}))$ queries to $U_H$ and $\Or(\frac{1}{\gamma})$ queries to $U_{I}$, * Number of qubits: $\Or(n+m)$, * Other one- and two- qubit gates: $\Or(\frac{m\alpha}{\gamma\Delta}\log(\frac{1}{\gamma\epsilon}))$. § ALGORITHM WITHOUT GROUND ENERGY BOUND Next we consider the case when we are not given a known $\mu$ to bound the ground energy from above. All other assumptions about $H$ and its eigenvalues and eigenstates are identical to the previous sections. The basic idea is to test different values for $\mu$ and perform a binary search. This leads to a quantum-classical hybrid method that can estimate the ground energy as well as preparing the ground state to high precision. All eigenvalues must be in the interval $[-\alpha,\alpha]$, thus we first partition $[-\alpha,\alpha]$ by grid points $-\alpha=x_0<x_1<\ldots<x_{G}=\alpha$, where $x_{k+1}-x_k=h$ for all $k$. Then we attempt to locate $\lambda_0$ in a small interval between two grid points (not necessarily adjacent, but close) through a binary search. To do a binary search we need to be able to tell whether a given $x_k$ is located to the left or right of $\lambda_0$. Because of the random nature of measurement we can only do so correctly with some probability, and we want to make this probability as close to 1 as possible. This is achieved using a technique we call binary amplitude estimation. Let $U$ be a unitary that acts on two registers, the first register indicating success or failure. Let $A=\|(\bra{0}\otimes I)U(\ket{0}\ket{0})\|$ be the success amplitude. Given $\gamma_0$ and $\gamma_1$, $\Delta := \gamma_1-\gamma_0>0$, provided that $A$ is either smaller than $\gamma_0$ or greater than $\gamma_1$, we can correctly distinguish between the two cases, i.e. output $0$ for the former and $1$ for the latter, with probability $1-\delta$ using $\mathcal{O}((1/\Delta)\log(1/\delta))$ applications of (controlled-) $U$ and its inverse. The proof is essentially identical to the proof for gapped phase estimation in [Ambainis, 2012, Childs et al., 2017]. We can perform amplitude estimation up to error $\Delta/4$ with $\mathcal{O}(1/\Delta)$ applications of $U$ and $U^\dagger$. This has a success probability of $8/\pi^2$ according to Theorem 12 of [Brassard et al., 2002]. We turn the estimation result into a boolean indicating whether it is larger or smaller than $(\gamma_0+\gamma_1)/2$. The boolean is correct with probability at least $8/\pi^2$. Then we do a majority voting to boost this probability. Chernoff bound guarantees that to obtain a $1-\delta$ probability of getting the correct output we need to repeat $\mathcal{O}(\log(1/\delta))$ times. Therefore in total we need to run $U$ and $U^\dagger$ $\mathcal{O}((1/\Delta)\log(1/\delta))$ times. We then apply binary amplitude estimation to the block-encoding of the projector defined in (<ref>) $\PROJ(x_k,h/2\alpha,\epsilon')$ for some precision $\epsilon'$ to be chosen. We denote the amplitude of the “good” component after applying block-encoding by A_k=\|(\bra{0^{m+3}}\otimes I)\PROJ(x_k,h/2\alpha,\epsilon')(\ket{0^{m+3}}\ket{\phi})\|, which satisfies the following: \[ \geq \gamma - \frac{\epsilon'}{2}, & \lambda_{0}\leq x_{k-1},\\ \leq \frac{\epsilon'}{2}, & \lambda_{0}\geq x_{k+1}. \end{cases} \] We can then let \[ \epsilon'=\gamma/2, \] the two amplitudes are separated by a gap lower bounded by $\gamma/2$. Therefore we can run the binary amplitude estimation, letting $U$ in Lemma <ref> be U=\PROJ(x_k,h/2\alpha,\epsilon')(I\otimes U_I), to correctly distinguish the two cases where $\lambda_0\leq x_{k-1}$ and $\lambda_0 \geq x_{k+1}$ with probability $1-\delta$, by running $\PROJ(x_k,h/2\alpha,\epsilon')$, $U_I$, and their inverses $\mathcal{O}((1/\gamma)\log(1/\delta))$ times. The output of the binary amplitude estimation is denoted by $B_k$. We then define $\mathcal{E}$ as the event that an error occurs in the final result of binary amplitude estimation when we are computing $B_k$ for some $k$ such that $x_{k+1}<\lambda_0$ or $x_{k-1}>\lambda_0$ in our search process. All future discussion is conditional on $\mathcal{E}^c$ meaning that there is no error in binary amplitude estimation for $B_k$ when $x_{k+1}<\lambda_0$ or $x_{k-1}>\lambda_0$. This has a probability that is at least $(1-\delta)^R$ where $R$ is the number of times binary amplitude estimation is run. Conditional on $\mathcal{E}^c$, almost surely (with probability 1) $B_k=1$ when $\lambda_0\leq x_{k-1}$ and $B_k=0$ when $\lambda_0\geq x_{k+1}$. Therefore $B_k=0$ tells us $\lambda_0>x_{k-1}$ and $B_k=1$ tells us $\lambda_0<x_{k+1}$. $B_k$ and $B_{k+1}$ combined give us the information as shown in Table <ref>. $B_k$ $B_{k+1}$ Position of $\lambda_0$ 1 1 $\lambda_0<x_{k+1}$ 0 0 $\lambda_0>x_k$ 0 1 $x_{k-1}<\lambda_0<x_{k+2}$ 1 0 $x_{k}<\lambda_0<x_{k+1}$ Conditional on $\mathcal{E}^c$, $B_k$ and $B_{k+1}$ can provide us with the information as shown in the table. Binary search to locate $\lambda_0$ $L \gets 0$, $U\gets G$ $k=\lfloor (L+U)/2 \rfloor$ Run binary amplitude estimation to get $B_k$ and $B_{k+1}$. $(1,1)$ $U\gets k+1$ $(0,0)$ $L\gets k$ $(0,1)$ return $k-1$, $k+2$ $(1,0)$ return $k$, $k+1$ $L$, $U$ Using the Table <ref> we can do the binary search as outlined in Algorithm <ref>. For the $\ell$-th step in Algorithm <ref> we denote the integer variables $U$ and $L$ by $U_\ell$ and $L_\ell$. In all four outcomes for $(B_k,B_{k+1})$, if the algorithm does not terminate at this step, then the new $U_{\ell+1}-L_{\ell+1}$ will be at most $(U_\ell-L_\ell)/2+1$. Since $U_0-L_0=G$ at the very beginning, we can show inductively $U_\ell-L_\ell\leq (G-2)/2^\ell+2$. Therefore when $\ell \geq \log_2(G-2)$ we have $U_\ell-L_\ell\leq 3$. Thus the algorithm must terminate in $\lceil \log_2(G) \rceil=\mathcal{O}(\log(\alpha/h))$ steps. The output we denote by $L$ and $U$. They satisfy $x_L<\lambda_0<x_U$ and $U-L\leq 3$. If we want the whole procedure to be successful with probability at least $1-\vartheta$, then we need $\mathrm{Prob}(\mathcal{E}^c)\geq 1-\vartheta$. Since \[ \mathrm{Prob}(\mathcal{E}^c)\geq (1-\delta)^{\lceil \log_2(G) \rceil}\geq (1-\delta)^{\log_2(4\alpha/h)}, \] we only need, for small $\vartheta$, \[ \delta \leq \frac{\vartheta}{2\log_2(4\alpha/h)}. \] Algorithm <ref> enables us to locate $\lambda_0$ within an interval of length at most $3h$. In total we need to run binary amplitude estimation at most $\mathcal{O}(\log(\alpha/h))$ times. Each amplitude estimation queries $\PROJ(x_k,h/2\alpha,\epsilon')$ and $U_I$ $\Or((1/\gamma)\log(1/\delta))$ times, where $\epsilon'=\gamma/2$. Therefore the number of queries to $U_H$ and $U_I$ are respectively \[ \Or\left(\frac{\alpha}{\gamma h}\log\left(\frac{\alpha}{h}\right)\log\left(\frac{1}{\gamma}\right)\log\left(\frac{\log(\alpha/h)}{\vartheta}\right)\right), \quad \Or\left(\frac{1}{\gamma}\log\left(\frac{\alpha}{h}\right)\log\left(\frac{\log(\alpha/h)}{\vartheta}\right)\right). \] In particular, in the procedure above we did not use <ref> but only used <ref>. Therefore we do not need to assume the presence of a gap. The result can be summarized into the following theorem: Suppose we have Hamiltonian $H=\sum_{k}\lambda_k \ket{\psi_k}\bra{\psi_k}\in\CC^{N\times N}$, where $\lambda_k\leq \lambda_{k+1}$, given through its $(\alpha,m,0)$-block-encoding $U_H$. Also suppose we have an initial state $\ket{\phi_0}$ prepared by circuit $U_{I}$, as well as the promise <ref>. Then the ground energy can be estimated to precision $h$ with probability $1-\vartheta$ with the following costs: * Query complexity: $\Or\left(\frac{\alpha}{\gamma h}\log\left(\frac{\alpha}{h}\right)\log\left(\frac{1}{\gamma}\right)\log\left(\frac{\log(\alpha/h)}{\vartheta}\right)\right)$ queries to $U_H$ and $\Or\left(\frac{1}{\gamma}\log\left(\frac{\alpha}{h}\right)\log\left(\frac{\log(\alpha/h)}{\vartheta}\right)\right)$ queries to $U_{I}$, * Number of qubits: $\Or(n+m+\log(\frac{1}{\gamma}))$, * Other one- and two- qubit gates: $\Or\left(\frac{m\alpha}{\gamma h}\log\left(\frac{\alpha}{h}\right)\log\left(\frac{1}{\gamma}\right)\log\left(\frac{\log(\alpha/h)}{\vartheta}\right)\right)$. The extra $\Or(\log(1/\gamma))$ qubits needed come from amplitude estimation, which uses phase estimation. If we use Kitaev's original version of phase estimation using only a single qubit [Kitaev, 1995], we can reduce the number of extra qubits to $\Or(1)$. With Theorem <ref> we can then use Algorithm <ref> to prepare the ground state without knowing an upper bound for the ground energy beforehand, when in addition to <ref> we have a lower bound for the spectral gap: * Bound for the spectral gap: $\lambda_1-\lambda_0\geq \Delta$. We first run Algorithm <ref> to locate the ground energy in an interval $[x_L,x_U]$ of length at most $\Delta$. Then we simply apply $\PROJ((x_L+x_U)/2,\Delta/4\alpha,\gamma\epsilon)$ to $\ket{\phi_0}$. This will give us an approximate ground state with at least $1-\epsilon$ fidelity. Therefore we have the following corollary: Suppose we have Hamiltonian $H=\sum_{k}\lambda_k \ket{\psi_k}\bra{\psi_k}\in\CC^{N\times N}$, where $\lambda_k\leq \lambda_{k+1}$, given through its $(\alpha,m,0)$-block-encoding $U_H$. Also suppose we have an initial state $\ket{\phi_0}$ prepared by circuit $U_{I}$, as well as the promises <ref> and <ref>. Then the ground state can be can be prepared to fidelity $1-\epsilon$ with probability $1-\vartheta$ with the following costs: * Query complexity: $\Or\left(\frac{\alpha}{\gamma \Delta}\left(\log\left(\frac{\alpha}{\Delta}\right)\log\left(\frac{1}{\gamma}\right)\log\left(\frac{\log(\alpha/\Delta)}{\vartheta}\right)+\log\left(\frac{1}{\epsilon}\right)\right)\right)$ queries to $U_H$ and $\Or\left(\frac{1}{\gamma}\log\left(\frac{\alpha}{\Delta}\right)\log\left(\frac{\log(\alpha/\Delta)}{\vartheta}\right)\right)$ queries to $U_{I}$, * Number of qubits: $\Or(n+m+\log(\frac{1}{\gamma}))$, * Other one- and two- qubit gates: $\Or\left(\frac{m\alpha}{\gamma \Delta}\left(\log\left(\frac{\alpha}{\Delta}\right)\log\left(\frac{1}{\gamma}\right)\log\left(\frac{\log(\alpha/\Delta)}{\vartheta}\right)+\log\left(\frac{1}{\epsilon}\right)\right)\right)$. It may be sometimes desirable to ignore whether the procedure is successful or not. In this case we will see the output as a mixed state whose density matrix is \[ \rho = \mathrm{Prob}(\mathcal{E}^c)\ket{\wt{\psi}_0}\bra{\wt{\psi}_0} + \rho', \] where $\ket{\wt{\psi}_0}$ is the approximate ground state with fidelity at least $1-\epsilon$, which is produced conditional on the event $\mathcal{E}^c$, and $\Tr\rho'=\mathrm{Prob}(\mathcal{E})$. Then this mixed state will have a fidelity lower bounded by \[ \braket{\psi_0|\rho|\psi_0} \geq \mathrm{Prob}(\mathcal{E}^c) |\braket{\wt{\psi}_0|\psi_0}|^2\geq (1-\vartheta)(1-\epsilon)^2. \] If we want to achieve $\sqrt{1-\xi}$ fidelity for the mixed state, we can simply let $\vartheta=\epsilon=\xi/3$. Thus the number of queries to $U_H$ and $U_I$ are $\wt{\Or}(\frac{\alpha}{\gamma\Delta}\log(\frac{1}{\xi}))$ and $\wt{\Or}(\frac{1}{\gamma}\log(\frac{\alpha}{\Delta})\log(\frac{1}{\xi}))$ respectively. § OPTIMALITY OF THE QUERY COMPLEXITIES In this section we prove for the ground state preparation algorithms outlined in Section <ref> and Section <ref> the number of queries to $U_H$ and $U_I$ are essentially optimal. We will also show our ground energy estimation algorithm has an nearly optimal dependence on the precision. We first prove the following complexity lower bounds: Suppose we have a generic Hamiltonian $H=\sum_{k}\lambda_k \ket{\psi_k}\bra{\psi_k}\in\CC^{N\times N}$, where $\lambda_k\leq \lambda_{k+1}$, given through its $(\alpha,m,0)$-block-encoding $U_H$, and $\alpha=\Theta(1)$. Also suppose we have an initial state $\ket{\phi_0}$ prepared by circuit $U_{I}$, as well as the promises <ref> and <ref>. Then the query complexities of preparing the ground state $\ket{\psi_0}$ of $H$ to fidelity at least $\sqrt{3}/2$ satisfy * When $\Delta=\Omega(1)$, and $\gamma\rightarrow 0^+$, the number of queries to $U_H$ is $\Omega(1/\gamma)$; * When $\gamma=\Omega(1)$, and $\Delta\rightarrow 0^+$, the number of queries to $U_H$ is $\Omega(1/\Delta)$; * When $\Delta=\Omega(1)$, and $\gamma\rightarrow 0^+$, it is not possible to accomplish the above task using $\Or(1/\gamma^{1-\theta})$ queries to $U_I$ and $\Or(\mathrm{poly}(1/\gamma))$ queries to $U_H$ for any $\theta>0$. We prove all three lower bounds by applying the ground state preparation algorithm to the unstructured search problem. In the unstructured search problem we try to find a $n$-bit string $t$ marked out by the oracle \[ U_t = I - 2\ket{t}\bra{t}. \] It is proved for this problem the number of queries to $U_t$ to find $t$ with probability $1/2$ is lower bounded by $\Omega(\sqrt{N})$ where $N=2^n$ [Bennett et al., 1997]. This problem can be seen as a ground state preparation problem. We find that $\ket{t}$ is the ground state of $U_t$, which is at the same time a unitary and therefore an $(1,0,0)$-block-encoding of itself. Therefore $U_t$ serves as the $U_H$ in the theorem. The spectral gap is $2$. Also, let \[ \ket{u}=\frac{1}{\sqrt{N}}\sum_s\ket{s} \] be the uniform superposition of all $n$-strings, then we have \braket{u|t}=\frac{1}{\sqrt{N}} and $\ket{u}$ can be efficiently prepared by the Hadamard transform since $\mathrm{H}^{\otimes n}\ket{0^n}=\ket{u}$. Therefore $\mathrm{H}^{\otimes n}$ serves as the $U_I$ described in the theorem. If the ground state preparation problem can be solved with $o(1/\gamma)$ queries to $U_H$ for fixed $\Delta$ to produce an approximate ground state with fidelity at least $\sqrt{3}/2$, then from the above setup we have $\gamma=1/\sqrt{N}$, and we can first find the approximate ground state and then measure in the computational basis, obtaining $t$ with probability at least $3/4$. Therefore the unstructured search problem can be solved with $o(\sqrt{N})$ queries to the oracle $U_t$, which is impossible. Thus we have proved the first lower bound in our theorem. To prove the second lower bound we want to create a situation in which the overlap is bounded from below by a constant but the gap vanishes. We need to introduce the Grover diffusion operator \begin{equation} \label{eq:grover_diffusion} D = I_n-2\ket{u}\bra{u}. \end{equation} which can be efficiently implemented. Then we define \begin{equation} \label{eq:hamilton_comb} H(\tau) = (1-\tau)D + \tau U_t, \end{equation} and consider $H(1/2)$. Because both $\operatorname{span}(\ket{u},\ket{t})$ and its orthogonal complement are invariant subspaces of $D$ and $U_t$, and both operators become the identity operator when restricted to the orthogonal complement of $\operatorname{span}(\ket{u},\ket{t})$, we only need to look for the ground state in the 2-dimensional subspace $\operatorname{span}(\ket{u},\ket{t})$. In this subspace, relative to the basis $\{\ket{u},\ket{t}\}$, the matrix representation of $H(1/2)$ is \[ \begin{pmatrix} 0 & -\braket{u|t} \\ -\braket{t|u} & 0 \end{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. \] Therefore the ground state of $H(1/2)$ is \[ \ket{\Psi}=\frac{\ket{u}+\ket{t}}{\sqrt{2+\frac{2}{\sqrt{N}}}}. \] and therefore $\braket{\Psi|u}=\braket{\Psi|t}= 1/\sqrt{2}+\Or(1/\sqrt{N})$ for large $N$. Furthermore, the gap is $\Delta(1/2)=2/\sqrt{N}$. Therefore $\ket{t}$ can be prepared in the following way: we first prepare the ground state of $H(1/2)$, whose block-encoding is easy to construct using one application of $U_t$. The resulting approximate ground state we denote by $\ket{\wt{\Psi}}$. Then we measure $\ket{\wt{\Psi}}$ in the computational basis. If there is some non-vanishing probability of obtaining $t$ then we can boost the success probability to above $1/2$ by repeating the procedure and verifying using $U_t$. If the second lower bound in the theorem does not hold, then $\ket{\wt{\Psi}}$ can be prepared with $o(1/\Delta(1/2))=o(\sqrt{N})$ queries to the block-encoding of $H(1/2)$ and therefore the same number of queries to $U_t$. Because the angle corresponding to fidelity is the great-circle distance on the unit sphere, we have the triangle inequality (using that $|\braket{\wt{\Psi}|\Psi}|\ge \sqrt{3}/{2}$) \[ \arccos |\braket{\wt{\Psi}|t}| \leq \arccos |\braket{\Psi|t}| + \arccos |\braket{\wt{\Psi}|\Psi}|\le \frac{5\pi}{12} + \Or\left(\frac{1}{\sqrt{N}}\right). \] Therefore for large $N$ we have $|\braket{\wt{\Psi}|t}|\geq \cos(5\pi/12) + \Or(1/\sqrt{N})>1/4$. The probability of getting $t$ when performing measurement is at least $1/16$. Therefore we can boost the success probability to above $1/2$ by $\Or(1)$ repetitions and verifications. The total number of queries to $U_t$ is therefore $o(\sqrt{N})$. Again, this is impossible. Therefore we have proved the second lower bound in our theorem. For the last lower bound we need to create some trade off between the gap and the overlap. We consider preparing the ground state of the Hamiltonian $H(1/2-N^{-1/2+\delta})$, $0<\delta<1/6$, whose block-encoding can be efficiently constructed with a single application of $U_t$, as an intermediate step. It is shown in Appendix <ref> that the ground state is \begin{equation} \label{eq:interm_state_lb3} \ket{\Phi}=\ket{u} + \frac{1}{4}N^{-\delta}\ket{t} + \Or(N^{-2\delta}). \end{equation} \[ \gamma_u=|\braket{\Phi|u}| = 1+\Or(N^{-2\delta}),\quad \gamma_t=|\braket{\Phi|t}| = \frac{1}{4}N^{-\delta} + \Or(N^{-2\delta}). \] Also we show in Appendix <ref> that the gap is \begin{equation} \label{eq:gap_lb3} \Delta(1/2-N^{-1/2+\delta})=4N^{\delta-1/2}+\Or(N^{-1/2-\delta}). \end{equation} We first apply the algorithm described in Section <ref> to prepare the ground state of $H(1/2-N^{-1/2+\delta})$ to fidelity $1-N^{-2\delta}/128$. Using the overlap $\gamma_u$ and the gap in (<ref>), the approximate ground state, denoted by $\ket{\wt{\Phi}}$, can be prepared with $\Or(N^{1/2-\delta}\log(N))$ queries to the block-encoding of $H(1/2-N^{-1/2+\delta})$, and therefore the same number of queries to $U_t$. The overlap between $\ket{\wt{\Phi}}$ and $\ket{t}$ can again be bounded using the triangle inequality \[ \begin{aligned} \arccos|\braket{\wt{\Phi}|t}| &\leq \arccos|\braket{\Phi|t}| + \arccos|\braket{\wt{\Phi}|\Phi}| \\ &\leq \arccos\left(\frac{N^{-\delta}}{4}\right) + \arccos\left(1-\frac{N^{-2\delta}}{128}\right) + \Or(N^{-2\delta}) \\ &\leq \frac{\pi}{2} - \frac{N^{-\delta}}{4} + \sqrt{2\times \frac{N^{-2\delta}}{128}} + \Or(N^{-2\delta}) \\ &=\frac{\pi}{2} - \frac{N^{-\delta}}{8} + \Or(N^{-2\delta}). \end{aligned} \] Therefore we have \[ \wt{\gamma}_t = |\braket{\wt{\Phi}|t}| \geq \frac{N^{-\delta}}{8} + \Or(N^{-2\delta}). \] If the last lower bound in our theorem does not hold, we can then prepare the ground state of $U_t$ by using the initial state $\ket{\wt{\Phi}}$ only $\Or(1/\wt{\gamma}^{1-\theta}_t)$ times for some $\theta>0$, and the number of queries to $U_t$ at this step, not including the queries used for preparing $\ket{\wt{\Phi}}$, is $\Or(1/\wt{\gamma}^p_t)$ for some $p>0$. Therefore the total number of queries to $U_t$ is \[ \Or\left(\frac{N^{1/2-\delta}\log(N)}{\wt{\gamma}^{1-\theta}_t}+\frac{1}{\wt{\gamma}^p_t}\right)=\Or(N^{1/2-\delta\theta}\log(N)+N^{\delta p}). \] This complexity must be $\Omega(N^{1/2})$ according to the lower bound for unstructured search problem. Therefore we need $\delta p\geq 1/2$. However we can choose $\delta$ to be arbitrarily small, and no finite $p$ can satisfy this condition. Hence we have a contradiction. This proves the last lower bound in our theorem. When we look at the query complexities of the ground state preparation algorithms in Secs. <ref> and <ref>, we can use $\wt{\Or}$ notation to hide the logarithmic factors, and both algorithms use $\wt{\Or}(\frac{\alpha}{\gamma\Delta})$ queries to $U_H$ and $\wt{\Or}(\frac{1}{\gamma})$ queries to $U_I$ when we want to achieve some fixed fidelity. Given the lower bound in Theorem <ref> we can see the algorithm with bound for ground energy essentially achieves the optimal dependence on $\gamma$ and $\Delta$. The algorithm without bound for ground energy achieves the same complexity modulo logarithmic factors, while using less information. This fact guarantees that the dependence is also nearly optimal. We will then prove the nearly optimal dependence of our ground energy estimation algorithm on the precision $h$. We have the following theorem: Suppose we have a generic Hamiltonian $H=\sum_{k}\lambda_k \ket{\psi_k}\bra{\psi_k}\in\CC^{N\times N}$, where $\lambda_k\leq \lambda_{k+1}$, given through its $(\alpha,m,0)$-block-encoding $U_H$, and $\alpha=\Theta(1)$. Also suppose we have an initial state $\ket{\phi_0}$ prepared by circuit $U_{I}$, as well as the promise that $|\braket{\phi_0|\psi_0}|=\Omega(1)$. Then estimating the ground energy to precision $h$ requires $\Omega(1/h)$ queries to $U_H$. This time we convert the quantum approximate counting problem, which is closely related to the unstructured search problem, into an eigenvalue problem. The quantum approximate counting problem is defined in the following way. We are given a set of $n$-bit strings $S\subset\{0,1\}^n$ specified by the oracle $U_f$ satisfying \[ U_f\ket{x} = \begin{cases} -\ket{x} & \ x \in S, \\ \ket{x} & \ x \notin S, \end{cases} \] for any $x\in\{0,1\}^n$. We want to estimate the size $|S|/N$ up to relative error $\epsilon$. It has been proven that this requires $\Omega\left(\frac{1}{\epsilon}\sqrt{\frac{N}{|S|}}\right)$ queries to $U_f$ for $|S|=o(N)$ <cit.>, where $N=2^n$, for the success probability to be greater than $3/4$, and this lower bound can be achieved using amplitude estimation [Brassard et al., 2002]. We convert this problem into an eigenvalue problem of a block-encoded Hamiltonian. Let $\ket{u}$ be the uniform superposition of the computational basis and $D$ be the Grover diffusion operator defined in (<ref>). Then define the following $(n+1)$-qubit unitary ($\mathrm{H}$ is the Hadamard gate) \[ U_H = (\mathrm{H}\otimes I_n)[\ket{0}\bra{0}\otimes D-\ket{1}\bra{1}\otimes (U_fDU_f)](\mathrm{H}\otimes I_n), \] which can be implemented using two applications of controlled-$U_f$. We define \[ H = (\bra{0}\otimes I_n)U_H(\ket{0}\otimes I_n)=\frac{1}{2}(D-U_f D U_f). \] Note that here $H$ is given in its $(1,1,0)$-block-encoding $U_H$. \[ \ket{u}=a\ket{u_0} + \sqrt{1-a^2}\ket{u_1} \] where the unit vectors $\ket{u_0}$ and $\ket{u_1}$ satisfy \[ U_f\ket{u_0}=-\ket{u_0},\quad U_f\ket{u_1}=\ket{u_1}, \] then we find $a=\sqrt{|S|/N}$. We only need to estimate the value of $a$ to precision $\Or(\epsilon'\sqrt{N/|S|})$ in order to estimate $|S|/N$ to precision $\epsilon'$. We analyze the eigenvalues and eigenvectors of $H$. It can be verified that $\{\ket{u_0},\ket{u_1}\}$ span an invariant subspace of $H$, and relative to this orthonormal basis $H$ is represented by the matrix \[ \begin{pmatrix} 0 & -2a\sqrt{1-a^2} \\ -2a\sqrt{1-a^2} & 0 \end{pmatrix}. \] In the orthogonal complement of this subspace, $H$ is simply the zero matrix. Therefore $H$ has only two non-zero eigenvalues $\pm 2a\sqrt{1-a^2}$ corresponding to eigenvectors \[ \ket{\psi_{\mp}}=\frac{1}{\sqrt{2}}(\ket{u_0}\mp\ket{u_1}). \] The ground state of $H$ is therefore $\ket{\psi_+}$ with ground energy $-2a\sqrt{1-a^2}$. We can use $\ket{u}$ as the initial state, with an overlap $\braket{\psi_+|u}=\frac{1}{\sqrt{2}}(a+\sqrt{1-a^2})\geq\frac{1}{\sqrt{2}}$. We use this Hamiltonian to prove Theorem <ref>: Assume toward contradiction that there exists an algorithm that estimates the ground energy to precision $h$ using only $o(1/h)$ queries to $U_H$. Then we use this algorithm to estimate the ground energy of the block-encoded Hamiltonian constructed above, for $a=o(1)$, which means $|S|=o(N)$. Estimating $2a\sqrt{1-a^2}$ to precision $\Or(h)$ enables us to estimate $a$ to precision $\Or(h)$. Setting $h=\epsilon'\sqrt{N/|S|}$, then this algorithm can estimate $|S|/N$ to precision $\epsilon'$, with success probability at least $3/4$. Since we are interested in the relative error we set $\epsilon'=\epsilon|S|/N$. Therefore the whole procedure uses only $o(1/h)=o(\frac{1}{\epsilon}\sqrt{\frac{N}{|S|}})$ queries to $U_H$ and therefore twice the amount of queries to $U_f$. This contradicts the lower bound for the approximate counting problem in [Nayak and Wu, 1999]. Theorem <ref> can also be viewed as a consequence of the optimality of the quantum phase estimation algorithm [Bessen, 2005]. If instead of the block-encoding $U_H$ we have $e^{-i\tau H}$ as the oracle for some $\tau$ such that $|\tau|\|H\|\leq \pi$, then even when given the exact ground state of $H$, <cit.> gives a query complexity lower bound $\Omega(1/h)$ for estimating the ground energy to within additive error $h$. This provides a different proof of the above theorem, since $e^{-i\tau H}$ and the block-encoding of $H$ are interconvertible: one can efficiently implement $e^{-i\tau H}$ via Hamiltonian simulation starting from a block-encoding of $H$ [Low and Chuang, 2019], and can efficiently obtain a block-encoding of $H$ by querying $e^{-i\tau H}$ according to <cit.>. § LOW-ENERGY STATE PREPARATION It is known that estimating the spectral gap $\Delta$ is a difficult task [Ambainis, 2014, Cubitt et al., 2015, Bausch et al., 2018]. Our algorithm for finding ground energy, as discussed in Theorem <ref>, does not depend on knowing the spectral gap. However both of our algorithms for preparing the ground state in Theorem <ref> and Corollary <ref> require a lower bound of the spectral gap. We would like to point out that if we only want to produce a low-energy state $\ket{\psi}$, making $\braket{\psi|H|\psi}\leq \mu$ for some $\mu > \lambda_0$, as in [Poulin and Wocjan, 2009], then this can be done without any knowledge of the spectral gap. In fact this is even possible for when the ground state is degenerate. To do this, we need to first assume we have a normalized initial state $\ket{\phi_0}$ with non-trivial overlap with the low-energy eigen-subspaces. Quantitatively this means for some $\gamma,\delta>0$, if we expand the initial state in the eigenbasis of $H$, obtaining $\ket{\phi_0}=\sum_k \alpha_k \ket{\psi_k}$, then \sum_{k:\lambda_k \leq \mu-3\delta}|\alpha_k|^2\geq \gamma^2. Then we can use the block-encoded projection operator in (<ref>) to get \[ \ket{\psi'} = (\bra{0^{m+3}}\otimes I)\PROJ(\mu-2\delta,\delta,\epsilon')(\ket{0^{m+3}}\otimes \ket{\phi_0}), \] for some precision $\epsilon'$. Now we expand $\ket{\psi'}$ in the eigenbasis to get $\ket{\psi'}=\sum_{k}\beta_k\ket{\psi_k}$, and denote $\ket{\varphi'}=\sum_{k:\lambda_k<\mu-\delta}\beta_k\ket{\psi_k}$. We then have, because of the approximation to the sign function, \[ \|\ket{\psi'}-\ket{\varphi'}\|\leq\frac{\epsilon'}{2},\quad \braket{\varphi'|\varphi'}\geq \gamma^2(1-\frac{\epsilon'}{2})^2,\quad \braket{\varphi'|H|\varphi'}\leq (\mu-\delta)\braket{\varphi'|\varphi'}. \] From the above bounds we further get \[ \frac{\braket{\psi'|H|\psi'}}{\braket{\psi'|\psi'}} \leq \frac{\braket{\varphi'|H|\varphi'}+\|H\|\epsilon'+\|H\|\epsilon'^2/4}{\braket{\varphi'|\varphi'}-\epsilon'} \leq \frac{\mu - \delta + \frac{\alpha\epsilon'+\alpha\epsilon'^2/4}{\gamma^2(1-\epsilon'/2)^2}}{1-\frac{\epsilon'}{\gamma^2(1-\epsilon'/2)^2}}. \] Now denoting $\ket{\psi}=\ket{\psi'}/\|\ket{\psi'}\|$ we can make $\braket{\psi|H|\psi}\leq \mu$ by choosing $\epsilon'=\Or(\gamma^2\delta/\alpha)$. Therefore the total number of queries to $U_H$ required is $\Or(\frac{1}{\delta\gamma}\log(\frac{\alpha}{\delta\gamma}))$ and the number of queries to $U_I$ is $\Or(\frac{1}{\gamma})$. From this we can see that if the initial state $\ket{\phi_0}$ has a overlap with the the ground state that is at least $\gamma$, and we want to prepare a state with energy upper bounded by $\lambda_0+\delta$, the required number of queries to $U_H$ and $U_I$ are $\Or(\frac{1}{\delta\gamma}\log(\frac{\alpha}{\delta\gamma}))$ and $\Or(\frac{1}{\gamma})$ respectively. If we do not know the ground energy beforehand we can use the algorithm in Theorem <ref> to estimate it first. Note that none of these procedures assumes a spectral gap. § DISCUSSIONS In this work we proposed an algorithm to prepare the ground state of a given Hamiltonian when a ground energy upper bound is known (Theorem <ref>), an algorithm to estimate the ground energy based on binary search (Theorem <ref>), and combining these two to get an algorithm to prepare the ground state without knowing an upper bound (Corollary <ref>). By solving the unstructured search problem and the approximate counting problem through preparing the ground state, we proved that the query complexities for the tasks above cannot be substantially improved, as otherwise the complexity lower bound for the two problems would be violated. All our algorithms are based on the availability of the block-encoding of the target Hamiltonian. This is a non-trivial task but we know it can be done for many important settings. For example, Childs proposed an LCU approach to block-encode the Hamiltonian of a quantum spin system [Childs et al., 2018], in which the Hamiltonian is decomposed into a sum of Pauli matrices. In [Low and Wiebe, 2018], Low and Wiebe outlined the methods to construct block-encoding of Hubbard Hamiltonian with long-range interaction, and of quantum chemistry Hamiltonian in plane-wave basis, both using fast-fermionic Fourier transform (FFFT) [Babbush et al., 2018]. The FFFT can be replaced by a series of Givens rotations which gives lower circuit depth and better utilizes limited connectivity [Kivlichan et al., 2018, Jiang et al., 2018]. Any sparse Hamiltonian whose entries can be efficiently computed can also be block-encoded using a quantum walk operator [Berry and Childs, 2009, Berry et al., 2015, Childs et al., 2017]. We remark that the quantum circuit used in our method for ground energy estimation can be further simplified. The main obstacle to applying this method to near-term devices is the need of amplitude estimation, which requires phase estimation. It is possible to replace amplitude estimation by estimating the success probability classically. In the context of binary amplitude estimation in Lemma <ref>, we need to determine whether the success amplitude is greater than $3\gamma/4$ or smaller than $\gamma/4$. This can be turned into a classical hypothesis testing to determine whether the success probability is greater than $9\gamma^2/16$ or smaller than $\gamma^2/16$. A simple Chernoff bound argument tells us that we need $\Or(\log(1/\vartheta)/\gamma^2)$ samples to distinguish the two cases with success probability at least $1-\vartheta$, as opposed to the $\Or(\log(1/\vartheta)/\gamma)$ complexity in amplitude estimation. In this approach, the only quantum circuit we need to use is the one in (<ref>). The circuit depth is therefore only $\Or((\alpha/h)\log(1/\gamma))$. It also does not require the $\Or(\log(1/\gamma))$ qubits that are introduced as a result of using amplitude estimation. These features make it suitable for near-to-intermediate term devices. In [Lin and Tong, 2020] we proposed an eigenstate filtering method (similar in spirit to the method proposed in Section <ref>), and we combined it with quantum Zeno effect [Childs et al., 2002, Boixo et al., 2009] to solve the quantum linear system problem. The resulting algorithm utilizes the fact that the desired eigenstate along the eigenpath always corresponds to the eigenvalue 0. In the setting of quantum Zeno effect based state preparation, in which we have a series of Hamiltonians and wish to incrementally prepare the ground state of each of them, our algorithm in Theorem <ref> can be used to go from the ground state of one Hamiltonian to the next one, provided that we have a known upper bound for the ground energy. In the absence of such an upper bound, there is the possibility of using the algorithm in Corollary <ref> to solve this problem. However in this setting we only want to use the initial state once for every Hamiltonian, since preparing the initial state involves going through the ground state of all previous Hamiltonians. This presents a challenge and is a topic for our future work. It is worth pointing out that none of the Hamiltonians used in the proofs of lower bounds in Section <ref> is a local Hamiltonian, and therefore our lower bounds do not rule out the possibility that if special properties such as locality are properly taken into consideration, better complexities can be achieved. § ACKNOWLEDGEMENTS This work was partially supported by the Department of Energy under Grant No. DE-SC0017867, the Quantum Algorithm Teams Program under Grant No. DE-AC02-05CH11231, the Google Quantum Research Award (L.L.), and by the Air Force Office of Scientific Research under award number FA9550-18-1-0095 (L.L. and Y.T.). We thank András Gilyén, and the anonymous referees for helpful suggestions. 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We use $\mathrm{H}$ to denote the Hadamard gate. We consider a single-qubit illustrative example of block-encoded matrix and obtain the corresponding reflector through QSP. The matrix we consider is \[ H(a) = a \sigma_x + (1-a) I, \] for $0\leq a\leq 1$. Its block-encoding can be using the following circuit \[ \Qcircuit @C=0.8em @R=1.em { & \gate{V(a)} & \ctrlo{1} & \gate{V^\dagger(a)} & \qw \\ & \qw & \gate{\sigma_x} & \qw & \qw \\ \] \[ \begin{pmatrix} \sqrt{a} & -\sqrt{1-a} \\ \sqrt{1-a} & \sqrt{a} \end{pmatrix}. \] We denote the above circuit by $U_H(a)$. This is a $(\alpha,m,0)$-block-encoding of $H(a)$ where $\alpha=1$ and $m=1$, since we can readily check that \[ (\bra{0}\otimes I)U_H(a)(\ket{0}\otimes I) = H(a). \] The eigendecomposition of $H(a)$ is \[ H(a) = \ket{+}\bra{+} + (1-2a)\ket{-}\bra{-}, \] with eigenvalues $\lambda_{+}(a) = 1,\lambda_{-}(a) = 1-2a$. Our goal is to implement the reflector \[ R_{<0}(a) = -\sign(H(a)) = -\ket{+}\bra{+} - \sign(1-2a)\ket{-}\bra{-}. \] To do this we need an odd polynomial $S(x;\delta,\epsilon)$ introduced in Lemma <ref>. Instead of the construction done in Ref. [Low and Chuang, 2017] we use the Remez algorithm [Remes, 1934] to obtain this polynomial. We choose $\delta=0.2$ and the $L^{\infty}$ error of the residual is required to be less than $10^{-4}$, i.e. $\epsilon \leq 10^{-4}$. \[ \scalebox{0.8}{ \Qcircuit @C=0.8em @R=1.em { \lstick{\ket{0}}& \gate{\mathrm{H}} & \targ & \gate{e^{-i \varphi_{d} \sigma_z}} & \targ & \qw & \targ & \gate{e^{-i \varphi_{d-1} \sigma_z}} & \targ & \qw & \qw &\raisebox{0em}{$\cdots$}&& \qw & \qw &\targ & \gate{e^{-i \varphi_0 \sigma_z}} & \targ & \qw & \gate{\mathrm{H}}&\qw \\ \lstick{\ket{0^m}}& \qw &\ctrlo{-1} & \qw & \ctrlo{-1} & \multigate{1}{U_H(a)} & \ctrlo{-1} & \qw & \ctrlo{-1} & \multigate{1}{U^{\dag}_H(a)} &\qw &\raisebox{0em}{$\cdots$} && \multigate{1}{U_H(a)} & \qw &\ctrlo{-1} & \qw & \ctrlo{-1} & \qw& \qw & \qw \\ \lstick{\ket{\psi}}& \qw &\qw &\qw&\qw &\ghost{U_H(a)} &\qw&\qw&\qw&\ghost{U_H(a)}&\qw &\raisebox{0em}{$\cdots$} && \ghost{U_H(a)} & \qw&\qw&\qw & \qw&\qw& \qw &\qw \] The circuit implementing the polynomial eigenvalue transformation through QSP for an odd polynomial with phase factors $\{\varphi_j\}_{j=0}^d$. $\mathrm{H}$ is the Hadamard gate and $\sigma_z$ is the Pauli-$Z$ gate. Given the polynomial $S(x;\delta,\epsilon)$, using the optimization method proposed in Ref. [Dong et al., 2020], we find a polynomial $P(x)\in \CC [x]$ of odd degree $d$ such that \[ \max_{x\in[-1,1]}|\Re P(x)-S(x;\delta,\epsilon)|\leq \epsilon', \] where $P(x)$ is characterized by a sequence of phase factors $\{\varphi_j\}_{j=0}^d$ satisfying \begin{equation} \label{eq:phase_fac_P} \begin{pmatrix} P(x) & \cdot \\ \cdot & \cdot \end{pmatrix} e^{i\varphi_0 \sigma_z}\prod_{j=1}^d[R(x)e^{i\varphi_j \sigma_z}], \end{equation} \[ \begin{pmatrix} x & \sqrt{1-x^2} \\ \sqrt{1-x^2} & -x \end{pmatrix}. \] The existence of the phase factors is guaranteed by <cit.>. Ref. [Dong et al., 2020] uses quasi-Newton method to solve a least squares problem to obtain these phase factors, and we terminate the iteration only when $L^\infty$ error of the residual of the real part is smaller than $\epsilon'=10^{-4}$. The circuit in Figure <ref> with phase factors $\{\varphi_j\}_{j=0}^d$ implements the transformation $H/\alpha\mapsto \Re P(H/\alpha)\approx S(H/\alpha;\delta,\epsilon)$. The various components of this circuit are explained in detail in <cit.>. An important component of this circuit is \[ \Qcircuit @C=0.8em @R=1.em { & \targ & \gate{e^{-i \varphi \sigma_z}} & \targ & \qw \\ & \ctrlo{-1} & \qw & \ctrlo{-1} & \qw \\ \] where the first register has one qubit, the second register has $m$-qubits, and the open bullet indicates control-on-zero for multiple control qubits. This component implements the operator \[ \ket{0}\bra{0}\otimes (e^{i \varphi (2\ket{0^m}\bra{0^m}-I)}) + \ket{1}\bra{1}\otimes (e^{-i \varphi (2\ket{0^m}\bra{0^m}-I)}). \] For a detailed discussion see <cit.>. The error of implementing $R_{<0}(a)$ for $a\in[0,1]$ using QSP, with polynomial $S(x;\delta,\epsilon)$ where $\delta=0.2$ and $\epsilon$ is of the order of $10^{-4}$. The vertical axis uses logarithmic scale. Using the above circuit, Lemma <ref> guarantees that when the eigenvalues of $H(a)$ are contained in $[-1,-\delta]\cup[\delta,1]$, we will have a good approximation of $R_{<0}(a)$. However when at least one eigenvalue, which in our case can only be $\lambda_-(a)=1-2a$, is in $(-\delta,\delta)$, or in other words when $a\in (0.4,0.6)$, there is no such guarantee. We plot the operator norm error between the approximate reflector obtained through QSP and the exact reflector $R_{<0}(a)$ in Figure <ref>. It can be seen in the figure that the error is smaller than $10^{-4}$ everywhere except for $a\in(0.4,0.6)$, where the error spikes. § GAP AND OVERLAP IN THE UNSTRUCTURED SEARCH PROBLEM In this appendix we compute the spectral gap of the Hamiltonian $H(1/2-N^{-1/2+\delta})$ for $H(\tau)$ defined in (<ref>), $0<\delta<1/6$, and the overlap between its ground state and $\ket{u}$ and $\ket{t}$ defined in Section <ref>. The first thing we should realize is that we only need to care about the subspace of the Hilbert space spanned by $\ket{u}$ and $\ket{t}$. In the orthogonal complement of this subspace $H(\tau)$ is simple a multiple of identity. In this subspace, with respect to the non-orthogonal basis $\{\ket{u},\ket{t}\}$, the operator $H(1/2-N^{-1/2+\delta})$ is represented by the following matrix \begin{equation} \label{eq:ham_search} \begin{pmatrix} -2 & -(N^{-\delta}+2N^{-1/2}) \\ -(N^{-\delta}-2N^{-1/2}) & 2 \end{pmatrix}. \end{equation} Direct calculation shows the eigenvalues \[ \lambda_{\pm} = \pm N^{\delta-1/2}\sqrt{4+N^{-2\delta}-4N^{-1}} = \pm N^{\delta-1/2}(2+\frac{1}{4}N^{-2\delta}+\Or(N^{-4\delta})). \] Thus we obtain the spectral gap in (<ref>). To simplify notation we let $\wt{\lambda}=N^{1/2-\delta}\lambda_+$. We then compute the ground state. We first find an eigenvector corresponding to $\lambda_-$ \[ \begin{aligned} \ket{\chi} &= N^\delta((N^{-\delta}+2N^{-1/2})\ket{u}+(-2+\wt{\lambda})\ket{t}) \\ &=(1+2N^{\delta-1/2})\ket{u} + (\frac{1}{4}N^{-\delta}+\Or(N^{-3\delta}))\ket{t} \\ &=\ket{u}+ \frac{1}{4}N^{-\delta}\ket{t}+\Or(N^{\delta-1/2}). \end{aligned} \] We still need to normalize $\ket{\chi}$. The normalization factor is \[ \begin{aligned} \|\ket{\chi}\|& = \sqrt{(1+2N^{\delta-1/2})^2 + (\frac{1}{4}N^{-\delta}+\Or(N^{-3\delta}))^2 + \frac{2}{\sqrt{N}}(1+2N^{\delta-1/2})(\frac{1}{4}N^{-\delta}+\Or(N^{-3\delta}))} \\ &= 1 + \Or(N^{-2\delta}). \end{aligned} \] Note that the third term under the square root comes from the overlap between $\ket{u}$ and $\ket{t}$, and it does not play an important role asymptotically. Therefore normalizing we have the expression for the normalized eigenstate (<ref>).

2024-09-04T02:54:55.366377

2002.12530

arxiv-papers

# Temporal Convolutional Attention-based Network For Sequence Modeling Hongyan Hao1111Equal Contribution Yan Wang2111Equal Contribution Yudi Xia3 Furao Shen1&Jian Zhao4 1Department of Computer Science and Technology, Nanjing University, Nanjing, China 2School of Artificial Intelligence, Nanjing University, Nanjing, China 3Software Institute, Nanjing University, Nanjing, China 4School of Electronic Science and Engineering, Nanjing University, Nanjing, China {haohy, yanwang<EMAIL_ADDRESS> {frshen<EMAIL_ADDRESS> ###### Abstract With the development of feed-forward models, the default model for sequence modeling has gradually evolved to replace recurrent networks. Many powerful feed-forward models based on convolutional networks and attention mechanisms were proposed and show more potential to handle sequence modeling tasks. We wonder that is there an architecture that can not only achieve an approximate substitution of recurrent networks but also absorb the advantages of feed- forward models. So we propose an exploratory architecture referred to Temporal Convolutional Attention-based Network (TCAN) which combines temporal convolutional network and attention mechanism. TCAN includes two parts, one is Temporal Attention (TA) which captures relevant features inside the sequence, the other is Enhanced Residual (ER) which extracts the shallow layer’s important information and transfers to deep layers. We improve the state-of- the-art results of bpc/perplexity to 26.92 on word-level PTB, 1.043 on character-level PTB, and 6.66 on WikiText-2. ## 1 Introduction With the development of deep learning, researchers have gradually concluded three frequently used structures for sequence modeling, i.e., convolutional networks, recurrent networks and attention mechanisms. For the tasks of sequence learning, the ”default” solutions are recurrent networks in the early years, however, there are some defects that recurrent networks are hard to avoid. Besides, the feed-forward has evolved to handle tasks of sequence modeling. We aim to explore a better architecture to implement an approximate substitution of recurrent networks using feed-forward networks. So that it can not only absorb the advantages of feed-forward models but also make up for the shortcoming of recurrent networks. Before introducing our new architecture, we first analyze the problem of sequence modeling to conclude what characteristics an effective architecture should have. As the input of task, the sequence’s data point at time step $t$ is conditioned on prior one and arbitrary two data points can be relevant. Accordingly, A feasible model should characterize causality and learn the conditional correlation between data points. Besides, for high efficiency, it should train and test data in parallel. And we hope the size of this model can be as small as possible. The researchers have done many attempts from feed-forward models to recurrent models. In the early days of deep learning, Multi-Layer Perceptron (MLP) was regarded as a complex linear function to solve the problem of prediction. An idea found in machine learning and statistical models is that for a model, parameter sharing across different parts makes it possible to extend and generalize Goodfellow et al. (2016). However, MLP trains separate parameters for each input feature, that is to say, it can’t share parameters. Waibel et al., [1989] proposed a time-delay neural network (TDNN), which achieves parameter sharing by applying the same convolution kernel at each time step. But the range of memory is limited by the delay factor, so it can’t handle tasks with intensive data. Recurrent neural network (RNN) is different from them, it uses the same parameters to process sequential input from the beginning to the current time step. Due to that, each member of output is the function of the previous member of output, so that RNN is theoretically capable of infinite memory Bai et al. (2018); Goodfellow et al. (2016) and can process variable length input. At the same time, it satisfy causality and conditional correlation for sequence modeling, This strong expressiveness makes it be ”default” choice of many sequence modeling tasks. Although RNNs (i.e. RNN based architectures) has much strength, two shortcomings are limiting their applicability in reality. One is that in both training and evaluation, The later time steps must wait for their predecessors to complete, this inherently sequential nature precludes parallelization in training and evaluating processes Vaswani et al. (2017); The other one is that with the growth of sequence’s length, RNNs pay more attention to nearby context and they are sensitive to the order of words within the most recent sentence but ignore word order in the long-range context, it is to say that the ”infinite memory” is unnecessary Khandelwal et al. (2018). While some variants and optimization methods achieved significant improvements Merity et al. (2018), but the fundamental constraints mentioned above remain. Recently, researchers devised many alternative feed-forward models for sequence modeling, which mainly refer to temporal convolutional network-based models (TCNs) Bai et al. (2018) and attention mechanism-based models Vaswani et al. (2017). The basic TCN architecture mainly contains three modules, causal convolution, dilated convolution and residual block Bai et al. (2018). Although the output member of the $t$-th step is a function of a particular number (determined by dilation factor and kernel size) of neighboring members of input before $t$. However, TCN doesn’t learn distant position’s dependency inside the sequence and it doesn’t extract internal correlation information of input. Bai et al., [2019] propose TrellisNet, characterized by weight tying across the depth and direct injection of the input into deep layers. Besides that, it combines many optimization methods to promote performance. But they make TrellisNet bigger and slower than the original TCN. The representative model of attention mechanism based is Transformer Vaswani et al. (2017) which tactfully avoids recurrence by entirely relying on attention mechanism to draw global dependencies between input and output. Based on that, researchers designed some powerful models, including GPT-2 which combines generative pre- training on a diverse corpus of unlabeled text and discriminative fine-tuning on specific tasks. The GPT-2 has achieved state of the art performance on many sequence modeling tasks. While the refinements and variants of Transformers have more strong processing power for sequential data, they are too big to take into account all the characteristics of the sequence model introduced earlier. (a) Model overall (b) TCAN block Figure 1: Architectural overall and an inter-layer transformation diagram. (a) An overall of whole architecture including Input layer, Hidden layer and Output layer, the green square indicates TCAN; (b) Inter-layer transformation of TCAN, gray squares indicate the intermediate variable of temporal attention and convolution, yellow block indicate the enhanced residual. In this work, we propose an exploratory architecture referring to the Temporal Convolutional Attention-based Network (TCAN). On the one hand, it is inspired by TCN to utilize the dilated causal network to be an analogy to RNN’s input causality. On the other hand, it combines self-attention mechanism Vaswani et al. (2017) to extract internal correlation information and learn distant position’s dependency. As illustrated in Fig 1, our model consists of two parts. One of them is Temporal Attention (TA) which is different from regular attention mechanism in the property of internal causality, TA’s $l$-th hidden layer $t$-th time step input is a function of all previous input $x_{1:t}$, which guarantee no information leakage from future to past inside hidden layer. On the contrast, self-attention captures all time steps information in the former layer. The other part is Enhanced Residual (ER), it gets contribution weights of every time step for prediction from TA. The ER will be summed with a normal skip connection as the final residual of the current layer. In summary, we try to use a convolutional network and attention mechanism to design an approximate substitution of RNN to satisfy the conditions of solving sequence modeling. For evaluating the impact of TCAN, we conduct experiments on the Penn Treebank (PTB) Marcus et al. (1993) and the WikiText-2 (WT2) data set Merity et al. (2017). The results show that TCAN attains state-of-the-art performance by significant margins. Besides, TCAN has a smaller size than Transformer based and RNN based models. It indicates that our proposed architecture could effectively extract sequence features and could be a feasible alternative to RNN in resolving sequence modeling tasks. The code for reproducing the results is open sourced and is available at https://github.com/haohy/TCAN ## 2 Methodology In this section, we will introduce generic TCAN in detail. We first present the definition of the sequence modeling task in Section 2.1. Then we describe the processing of data from input to output. We apply commonly used encoder- decoder structure for sequence modeling. The causal dilated network meets the requirement of sequential causality. Section 2.2 will give an overall look at TCAN architecture. The next two sub-sections introduce the details of the two sub-modules, including Temporal Attention in Section 2.3 and Enhanced Residual in Section 2.4. ### 2.1 Sequence Modeling Sequence modeling is a key problem in domains spanning audio, language modeling, music processing, time series forecasting, and many others Bai et al. (2018). Suppose an input sequence as $x_{1:T}=x_{1},x_{2},\cdots,x_{T}$ with length $T$ and the target sequence $y_{1:T}=y_{1},y_{2},\cdots,y_{T}$ with length $T$, the task is to find a function $\mathbf{SeqMod}$ that satisfies the following relation: $y_{1:T}=\mathbf{SeqMod}(x_{1:T})$ two constraints should be noticed: 1) $y_{t}$ should satisfy the causal constraint, it is a function of $x_{1:t}$, The model should prevent future information $x_{t+1:T}$ leakage; 2) the length of the input $x_{1:T}$ should be the same as the length of output. In essence, the function $\mathbf{SeqMod}$ is to find the network that minimizes some expected loss between the model’s outputs and ground truth which we define it to be simply the input sequence shifted by one time step. Different from natural machine translation which utilizes an entire input sequence to predict the whole sentence, sequence modeling is an auto- regressive prediction which can only depend on past information. Actually, it is this constraint that makes recurrent networks become default choice for sequence modeling rather than feed-forward models. ### 2.2 Model Architecture From a holistic perspective, we adopt a similar structure as TCN which contains encoder and decoder like most competitive neural sequence transduction models. At the beginning of the model, the encoder maps an input sequence of symbol representations $x_{1:T}=x_{1},x_{2},\cdots,x_{T}$ to a sequence of continuous representations $S_{1:T}^{(0)}=\text{Encoder}(x_{1:T})$ whose T indicate the length of the sequence and 0 indicate the $0$-th layer, i.e. the first hidden layer’s input. Then we apply the different kernel sizes of dilated causal convolution as a hidden layer across $L$ layers. After the final hidden layer, the decoder generates an output sequence $\hat{y}_{1:T}=\hat{y}_{1},\hat{y}_{2},\cdots,\hat{y}_{T}$. At the most basic level, the intermediate variable at time step $t$ and level $l+1$ ($s_{1:T}^{(l+1)}$) is computed via four steps, illustrated in Figure 1: 1. 1. The $s_{1:T}^{(l)}$ is passed through Temporal Attention (TA): $sa_{1:T}^{(l)}=\text{TA}(s_{1:T}^{(l)})$ where $sa_{1:T}^{(l)}$ indicate an intermediate variable that contains information before time steps $t$, illustrated in Figure 1(b) and will be elaborated in Section 2.3. 2. 2. Given the $sa_{1:T}^{(l)}$, we apply causal convolution on it: $sc_{1:T}^{(l)}=\text{Conv1d}(sa_{1:T}^{(l)})$ where $sc_{1:T}^{(l)}$ indicates the output of causal convolution. The causal block ($L_{b}$) can be stacked into many layers. For keeping the same length of each layer, we add zero padding of length ($(k-1)2^{l-1}$) on the left, white blocks in Figure 1(a). In this way, the left relevant information of input will gradually accumulate to the right. 3. 3. Before feature maps being passed through the activation function to get $s_{1:T}^{(l+1)}$, we add three components $s_{1:T}^{(l)}$, $sc_{1:T}^{(l)}$ and $sr_{1:T}^{(l)}$, where $sr_{1:T}^{(l)}$ represents Enhanced Residual (ER): $sr_{1:T}^{(l)}=\text{ER}(s_{1:T}^{(l)})$ which is expressed by yellow blocks in Figure 1(b) and will be detailed described in Section 2.4 4. 4. A full TCAN network is built by stacking $L$ layers of TCAN block across depth and time, which is called dilated convolution. In addition, we use dilated convolution to enable networks to have enough receptive fields, which preserves the network computational efficiency. We set the size of dilation to increase exponentially with the depth of the network (i.e., $d=2^{l}$ for layer $l$ in the network). Figure 2: Temporal Attention Block ### 2.3 Temporal Attention Temporal Attention (TA), illustrated as Figure 2, can be described as a process that integrates the influence of previous time steps into the current time step. Our model is inspired by self-attention structure. The self- attention utilizes information of all time steps, both past and future of time step $t$. But for the sequential data, we can only handle the past information, so we refine the processing of the weight matrix to satisfy the sequential nature. At the first step, we use three linear transformations $f$, $g$ and $h$ to map $s_{1:T}^{(l)}$ to three different vectors, keys ($k_{1:T}^{(l)}=f(s_{1:T}^{(l)})$), query ($q_{1:T}^{(l)}=g(s_{1:T}^{(l)})$) and values ($v_{1:T}^{(l)}=h(s_{1:T}^{(l)})$) of dimension $d_{k}$. Then for getting the weight matrix $Wa^{(l)}$, we compute the dot products of $q_{1:T}^{(l)}$ and $k_{1:T}^{(l)}$, and divided each by $\sqrt{d_{k}}$. $W_{i,j}^{(l)}=\frac{{k_{i}^{(l)}}^{\mathrm{T}}\cdot{q_{j}^{(l)}}}{\sqrt{d_{k}}}$ where $i,j=1,2,\cdots,T$. After that, we extract the lower triangular part of $W^{(l)}$ as follows: $Wl_{i,j}^{(l)}=\begin{cases}W_{i,j}^{(l)},&\text{if i $\geq$ j}\\\ 0,&\text{if i $<$ j}\end{cases}$ where $i,j=1,2,\cdots,T$. This can shield the weight of future time steps, so as to achieve the purpose of not using future information. Finally, we apply a softmax function to normalize $Wl^{(l)}$ to get $Wa^{(l)}$, yellow blocks in Figure 2. Note that we apply softmax in the first dimension of $Wl^{(l)}$. Base on this operation, the sum of each column’s weight across one row can be larger than 1, that is to say, the total contribution of previous time step $s_{1:t}^{(l)}$ to current $s_{t}^{(l)}$ would have a bigger difference than normalization in the first dimension. It will be justified by an ablation study in Section 3.4. Given the weights, we can get the weighted output by: $sa_{t}^{(l)}=\sum_{i=0}^{t}Wa_{i}^{(l)}\cdot s_{i}^{(l)}$ where $t=1,2,\cdots,T$. $sa_{t}^{(l)}$ will be regarded as input of causal convolution, described in the $2$-th step in Section 2.2. Figure 3: Enhanced Residual Block ### 2.4 Enhanced Residual Before passing the intermediate variables through the activation function to get $s_{1:T}^{(l+1)}$, we integer three parts of information, they are identity mapping $s_{1:T}^{(l)}$, convolved vectors $sc_{1:T}^{(l)}$ and Enhanced Residual (ER) which we design to extract relatively important information and transfer them to the next layer. For a practical example of enhanced residual, in a case where you try to learn something new, if someone tells you what is the relatively important part, then you reinforce learning particular parts, it will help you grasp it faster and stronger. Enhanced residual harness the weight matrix $Wa_{1:T}^{(l)}$ got from Temporal Attention. We take the sum of the weights of each row of $Wa_{1:T}^{(l)}$ to indicate the importance level of each time step, so we can get another weight vector as follows: $M_{t}=\sum_{i=0}^{t}Wa_{i}^{(l)}$ where $M_{t}$ denotes the degree of importance of time step $t$, $t=1,2,\cdots,T$. Then after the Hadamard product of $M_{t}$ and $S^{(l)}$, we get the enhanced residual ($Sr^{(l)}$). ## 3 Experiments To evaluate the effectiveness of our TCAN architecture, we conduct experiments on language modeling task (LM) task. This task is defined in Section 2.1. The different datasets for LM have been described in the following section, they share the same architecture with different parameters setting. We mainly compare TCAN with representative models of three architectures listed in Section 3.2. Besides, we will verify the effectiveness of enhanced residual and the difference between three types of softmax in temporal attention in experiments. ### 3.1 Datasets and Setting #### Penn Treebank Marcus et al. (1993): The Penn Treebank (PTB) dataset has two forms for testing the performance of models. One is word-level PTB, which contains 888K words for training, 70K for validation and 79K for testing, with a vocabulary size of 10K. Each sentence’s end is marked with $<$eos$>$ and all the numbers were replaced with a $?$ symbol. The other is character-level PTB, which contains 5M characters for training, 396K for validation and 446K for testing, with an alphabet size of 50. Note that $<$eos$>$ is considered one character here. Compared to the former, character-level PTB is a medium size data set. These two forms of PTB datasets are highly studied in sequence modeling Bai et al. (2019)Bai et al. (2018)Krueger et al. (2017) #### WikiText-2 Merity et al. (2017): The WikiText-2 (WT2) contains lightly pre-possessed Wikipedia articles. In contrast to PTB, it retains capitalization, punctuation and numbers. WikiText-2 features a vocabulary of over 30,000 words and contains approximately 2 million words, which is over two times larger than that of the PTB dataset. Taking into account different characteristics of every dataset, we set different parameters for each dataset based on the same TCAN architecture. We use the gradient clip like TCN to compare the performance with TCAN based on the same optimizations. For narrative convenience, we denote the dimension of input embedding as $D_{embed}$, the number of stacking layers as $L$, the number of convolutional layers inside the TCAN as $L_{b}$ and the dimension of the linear transformation in Temporal Attention as $D_{attn}$. As to word- level PTB, we set the kernel size $3$. In order to make the size of our model and the comparison model as equal as possible, we set $D_{embed}=300$, $L=4$, $L_{b}=1$ and $D_{attn}=600$. As to character-level PTB. Because the number of characters in the whole dataset is limited and smaller than word-level PTB, we set $D_{embed}=100$. Correspondingly, because of stronger long-term dependencies between characters, we set the convolutional kernel to $7$. At the same time, the length of input sequence is larger, so we need more layers to capture the larger receptive field, we set $L=6$, $L_{b}=1$ and $D_{attn}=100$. As to WT2, it is a larger dataset with a twice larger size of the dictionary. so we set $D_{embed}=600$, $L=6$, $L_{b}=1$ and $D_{attn}=600$. we use the Adam optimizer Kingma and Ba (2015) with learning rate 0.0001 in all above experiments. In order to verify the availability of enhanced residual, We consider two versions of the model, one with enhanced residual (TCAN) and one without enhanced residual (TCAN-no-res). The enhanced residual doesn’t add parameters to the model, so they have equal model size. ### 3.2 Compared Methods #### RNN based : Researchers proposed many variants of RNN. Among them, LSTM becomes the most used benchmark model because of solving the problem of vanishing gradients. Coupled with some regularization and optimization methods, it gets impressive results on several benchmark datasets in language modeling. AWD-LSTM Merity et al. (2018) is a representative model, it introduces weight-dropped LSTM which uses DropConnect on hidden-to-hidden weights to be a form of recurrent regularization and NT-AvSGD which is a non-monotonically triggered (NT) variant of the averaged stochastic gradient method (AvSGD), wherein the averaging trigger is determined using a NT condition as opposed to being tuned by the user. Later, many better algorithms were produced based on the AWD-LSTM Wang et al. (2019); Gong et al. (2018). Besides that, We also compare TCAN with NAS models Zoph and Le (2017), which uses reinforcement learning to maximize or minimize the objective function of the generated architectures on a validation set to generate the model descriptions of neural networks. #### CNN based : A few notable convolutional networks have been applied to sequence modeling in recent years (e.g., the WaveNet van den Oord et al. (2016a) and PixelCNN van den Oord et al. (2016b) architectures). Among some derivative models, the best one is TrellisNet Bai et al. (2019). It combines weight tying across depth and time and input insertion, besides that, it equips many techniques, including dropout, weight normalization and auxiliary loss. Putting these techniques together, it achieves a state-of-the-art performance among various derivative models of TCN. #### Attention mechanism : In recent years, the most frequently used and effective models in industry are based on Transformer Vaswani et al. (2017) structure, which is a derivative framework of attention mechanism in essence. GPT-2 Radford et al. (2019) is considered the best model at present, and it achieves state-of-the- art results on many sequence modeling benchmark tasks. GPT-2’s goal is to design a multitask learner, and it utilizes a combination of pre-training and supervised finetuning to achieve more flexible forms of transfer. Therefore it has 1542M parameters, much bigger than other comparative models. Word-level Penn Treebank (PTB) --- Models | Size | $\text{ppl}^{l}$ Generic TCN Bai et al. (2018) | 13M | 88.68 NAS Cell Zoph and Le (2017) | 54M | 62.4 AWD-LSTM Merity et al. (2018) | 24M | 58.8 TrellisNet Bai et al. (2019) | 33M | 56.80 TrellisNet-MoS Bai et al. (2019) | 34M | 54.19 GPT-2 Radford et al. (2019) | 1542M | 35.76 TCAN-no-res | 13M | 28.10 TCAN | 13M | 26.92 Table 1: Test perplexities (ppl) on word-level language modeling with the PTB corpus. $l$ means lower is better. WikiText-2 (WT2) --- Models | Size | $\text{ppl}^{l}$ Generic TCN Bai et al. (2018) | 28.6M | 138.5 AWD-LSTM Merity et al. (2018)† | 33M | 44.3 AWD-LSTM-MoS Yang et al. (2018)† | 35M | 40.68 GPT-2 Radford et al. (2019) | 1542M | 18.34 TCAN-no-res | 33M | 6.95 TCAN | 33M | 6.66 Table 2: Test perplexities (ppl) on word-level language modeling with the WikiText-2 corpus. $\dagger$ indicates using dynamic evaluation. Character- level Penn Treebank (PTB) --- Models | Size | $\text{ppl}^{l}$ Generic TCN Bai et al. (2018) | 3.0M | 1.31 IndRNN Li et al. (2018) | 12.0M | 1.23 NAS Cell Zoph and Le (2017) | 16.3M | 1.214 AWD-LSTM Merity et al. (2018) | 13.8M | 1.175 TrellisNet-MoS Bai et al. (2019) | 13.4M | 1.158 TCAN-no-res | 4.3M | 1.060 TCAN | 4.3M | 1.043 Table 3: Test bits-per-character (bpc) on character-level language modeling with the PTB corpus. ### 3.3 Results and Analysis (a) vertical (b) horizontal (c) vertival+horizontal Figure 4: The weights in the last TA layer of TCAN When the matrix $W_{l}$ is calculated by softmax in the vertical (a), horizontal (b) and mixed softmax (c). We evaluate TCAN on word-level and character-level language modeling on Penn Treebank (PTB) and WikiText-2 datasets as mentioned in Section 3.1. The prior state of the art on these datasets are set most by GPT-2. For the word-level datasets, we use PTB and wikitext-2. The former one is a small but frequently used dataset. We also conduct some further ablation studies on it. The latter one is larger, so it reduces the risk of overfitting to a certain extent. As illustrated in Table 1 and Table 2, compared with TCAN, AWD-LSTM and TrellisNet have more parameters but poorer performance (higher perplexity). GPT-2 has better performance than others, but its model size is so big. It causes that it needs so much natural language corpus and computational resources to get the pre-trained model. Actually, the largest part of parameters is the embedding module whose size is proportional to dictionary size. So that the model size of wikitext-2 is bigger than that of word-level PTB. On PTB dataset, TCAN uses fewer parameters to achieve state-of-the-art performance. As to wikitext-2 dataset, TCAN uses the same amount of parameters as AWD-LSTM and sets a new state of the art as well. For the character-level dataset, illustrated in Table 3, we also set a new state of the art. Due to the GPT-2 was trained for word-level models, it doesn’t work for this dataset. From the comparison between AWD-LSTM with TrellisNet and TCN, we found that for this task of sequence modeling, the feed-forward model can not only outperforms RNN based model, but also has a similar or even smaller size. Furthermore, TCAN has a simpler structure and better performance than TrellisNet, which suggests that the combination of TCN structure and attention mechanism is a feasible exploration. According to the comparison between TCAN-no-res and TCAN in the chart, we found that the enhanced residual can improve the performance in some extent. As mentioned in Section 2.4, we think that enhanced residual select valuable features and strengthen the memory of important information. Note that the values shown on Table 1, 2 and 3 are not the best performance results, but set for comparison. ### 3.4 Ablation Experiments To better explain the relative importance of the several conclusions we proposed, we designed two ablation experiments on the word-level PTB dataset. one is to prove that temporal attention (TA) layer is more effective than a convolutional layer, the other is to claim that the softmax applied on the first dimension of $W_{l}$ is better than on the second dimension. ER | TA | $L_{b}$ | $L$ | Size | ppll ---|---|---|---|---|--- ✗ | ✓ | 1 | 4 | 13.2M | 28.10 ✗ | ✗ | 2 | 4 | 14.7M | 151.98 Table 4: Comparative Tests on temporal attention layer and convolutional layer. When we compare the efficiency of temporal attention (TA), we discard all TA layers and replace them with convolutional layers. To guarantee the fairness of models, we adjust hyperparameters to make the convolutional model have more parameters than TCAN. The initial TCAN has 1 temporal attention block in each of the 4 layers. For comparison, we use convolutional layer to replace TA. A TA layer has similar number of parameters with a convolutional layer. Note that TCAN didn’t use the enhanced residual module, which is in order to not interfere with comparative tests. The results are listed in Table 4. Note that two models are optimized by Adam and the learning rate is 0.0001. The perplexity of two models shows that the temporal attention layer is more effective than the convolutional layer. ER | TA | Direction | ppl ---|---|---|--- ✗ | ✓ | Horizontal | 207.16 ✗ | ✓ | Vertical | 28.10 ✗ | ✓ | Horizontal+Vertical | 30.88 Table 5: Comparison of Horizontal and Vertical softmax in the TA layer. As claimed in Section 2.3, we apply softmax on the first dimension of $W_{l}$ and get relatively good results. For intuitive description, we define softmax on the first dimension as ”vertical softmax”, second dimension as ”horizontal softmax” and mixed dimension (the average of the first dimension and second dimension) as ”vertical+horizontal softmax”. The results are shown in Table 5 and Figure 4 which is drawing based on the final temporal attention layer’s $W_{l}$. According to the perplexities in Table 5, We can intuitively see that vertical softmax is more effective than mixed softmax and the latter is more effective than horizontal softmax, so we infer that vertical softmax is more practical than horizontal softmax. For the figures, because the $W_{l}$ is a lower triangular matrix, the weights concentrate on the lower left side of figures, the weight of the $i$-th row and $j$-th column indicate the contribution of the $j$-th time step to $i$-th time step. From Figure 4(a), we can see that for the time steps from about 25 to 80, about 3-time steps in front of current data contribute the most, besides that, the first 30 or so time steps also played a role. In contrast, from the Figure 4(b), horizontal softmax makes temporal attention concentrate more on the most previous data, it is unreasonable, because for prediction, the nearest several members contribute mostKhandelwal et al. (2018). So the performance of horizontal softmax is poorer. As to mixed direction softmax, it spends too much attention on the first and most recent time steps, but not the others. This ablation experiment evidence the claim that the softmax on the first dimension of $W_{l}$ can integer more relevant features by making the final weighted features more differentiated. ## 4 Conclusion In this work, we propose an exploratory architecture TCAN for sequence modeling. The sub-module temporal attention can integer internal correlative features under the condition of satisfying sequential characteristics. The other enhanced residual utilize the weight of temporal attention to emphasize the importance of certain time step, it can improve the performance without adding parameters. Our model outperforms prior state-of-the-art models on WikiText-2, word- and character-level PTB datasets by a significant margin. ## References * Bai et al. [2018] Shaojie Bai, J. Zico Kolter, and Vladlen Koltun. An empirical evaluation of generic convolutional and recurrent networks for sequence modeling. CoRR, abs/1803.01271, 2018. * Bai et al. [2019] Shaojie Bai, J. Zico Kolter, and Vladlen Koltun. Trellis networks for sequence modeling. In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019, 2019. * Gong et al. [2018] ChengYue Gong, Di He, Xu Tan, Tao Qin, Liwei Wang, and Tie-Yan Liu. FRAGE: frequency-agnostic word representation. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, 3-8 December 2018, Montréal, Canada, pages 1341–1352, 2018. * Goodfellow et al. [2016] Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep learning. MIT press, 2016. * Khandelwal et al. [2018] Urvashi Khandelwal, He He, Peng Qi, and Dan Jurafsky. Sharp nearby, fuzzy far away: How neural language models use context. In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics, ACL 2018, Melbourne, Australia, July 15-20, 2018, Volume 1: Long Papers, pages 284–294, 2018. * Kingma and Ba [2015] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015. * Krueger et al. [2017] David Krueger, Tegan Maharaj, János Kramár, Mohammad Pezeshki, Nicolas Ballas, Nan Rosemary Ke, Anirudh Goyal, Yoshua Bengio, Aaron C. Courville, and Christopher J. Pal. Zoneout: Regularizing rnns by randomly preserving hidden activations. In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings, 2017. * Li et al. [2018] Shuai Li, Wanqing Li, Chris Cook, Ce Zhu, and Yanbo Gao. Independently recurrent neural network (indrnn): Building a longer and deeper RNN. In 2018 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2018, Salt Lake City, UT, USA, June 18-22, 2018, pages 5457–5466, 2018. * Marcus et al. [1993] Mitchell P. Marcus, Beatrice Santorini, and Mary Ann Marcinkiewicz. Building a large annotated corpus of english: The penn treebank. Computational Linguistics, 19(2):313–330, 1993. * Merity et al. [2017] Stephen Merity, Caiming Xiong, James Bradbury, and Richard Socher. Pointer sentinel mixture models. In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings, 2017. * Merity et al. [2018] Stephen Merity, Nitish Shirish Keskar, and Richard Socher. Regularizing and optimizing LSTM language models. In 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings, 2018. * Radford et al. [2019] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. OpenAI Blog, 1(8), 2019. * van den Oord et al. [2016a] Aäron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew W. Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. In The 9th ISCA Speech Synthesis Workshop, Sunnyvale, CA, USA, 13-15 September 2016, page 125, 2016. * van den Oord et al. [2016b] Aäron van den Oord, Nal Kalchbrenner, Lasse Espeholt, Koray Kavukcuoglu, Oriol Vinyals, and Alex Graves. Conditional image generation with pixelcnn decoders. In Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, December 5-10, 2016, Barcelona, Spain, pages 4790–4798, 2016. * Vaswani et al. [2017] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4-9 December 2017, Long Beach, CA, USA, pages 5998–6008, 2017. * Waibel et al. [1989] Alexander H. Waibel, Toshiyuki Hanazawa, Geoffrey E. Hinton, Kiyohiro Shikano, and Kevin J. Lang. Phoneme recognition using time-delay neural networks. IEEE Trans. Acoustics, Speech, and Signal Processing, 37(3):328–339, 1989. * Wang et al. [2019] Dilin Wang, ChengYue Gong, and Qiang Liu. Improving neural language modeling via adversarial training. In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, pages 6555–6565, 2019. * Yang et al. [2018] Zhilin Yang, Zihang Dai, Ruslan Salakhutdinov, and William W. Cohen. Breaking the softmax bottleneck: A high-rank RNN language model. In 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings, 2018. * Zoph and Le [2017] Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings, 2017.

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2002.12540

arxiv-papers

# UKARA 1.0 Challenge Track 1: Automatic Short-Answer Scoring in Bahasa Indonesia Ali Akbar Septiandri Airy Jakarta, Indonesia <EMAIL_ADDRESS> Yosef Ardhito Winatmoko Jheronimus Academy of Data Science ’s-Hertogenbosch, The Netherlands <EMAIL_ADDRESS> ###### Abstract We describe our third-place solution to the UKARA 1.0 challenge on automated essay scoring. The task consists of a binary classification problem on two datasets — answers from two different questions. We ended up using two different models for the two datasets. For task A, we applied a random forest algorithm on features extracted using unigram with latent semantic analysis (LSA). On the other hand, for task B, we only used logistic regression on TF- IDF features. Our model results in F1 score of 0.812. ## 1 Introduction Automated essay scoring is the application of computers technologies to assist human grader in evaluating the score of written answers (Dikli, 2006). The first track of UKARA 1.0 is the binary classification version of essay scoring, where participants are expected to develop a model that can distinguish right and wrong answers in free text format. The organizer published the questions, the responses with the labels, and the guideline on how to determine whether an answer is acceptable. During a period of five weeks, the training set and the development set were available and we could validate our model through the score of the development set in a leaderboard. Subsequently, the test set was released, which consists of roughly four times the size of the development set. We are required to submit predicted labels based on the model that we have developed and the winner was determined by the F1 score of the submitted prediction. Our final submission to this task consists of feature extraction, such as n-grams and TF-IDF, and classical machine learning algorithms, namely logistic regression and random forest. We did not use deep learning at all in our submission. The details of the dataset and our approach will be discussed in the following sections. ## 2 Datasets The dataset consists of two questions and the respective responses collected by the organizer of the challenge. All questions and responses are in Indonesian. The first question, from now on will be referred as task A, asked about the consequence of the climate change. Concretely, what are the potential problems faced by a climate refugee when they have to migrate to a new place. The second question, referred as task B, is based on an experiment. There were potential customers, initially wanted to buy clothes, who prefer to donate the money instead when they are presented with videos of the clothes manufacturing worker condition before paying. The respondents were required to give their opinion on why do people decided to change their mind. The statistics of the responses for both tasks are shown in Table 1. | Task A | Task B ---|---|--- #Positive Train | $191(71\%)$ | $168(55\%)$ #Negative Train | $77(29\%)$ | $137(45\%)$ Avg. #Char | $87.23$ | $97.33$ #Dev | $215$ | $244$ #Test | $855$ | $974$ Table 1: Summary statistics of the dataset. ## 3 Methodology ### 3.1 Preprocessing For the preprocessing steps, we first tokenized and lemmatized the text using bahasa Indonesia tokenizer provided by spaCy (Honnibal and Montani, 2017). We then extracted the features using bag-of-words or TF-IDF. Since the resulting matrix from this feature extraction method tends to be sparse, and to encode token relations, we applied Latent Semantic Analysis (LSA) using Singular Value Decomposition (SVD) (Deerwester et al., 1990) on the matrix. Based on our observation, we noticed that the labels of the provided training set are highly inconsistent. Some responses are clearly labelled incorrectly. For illustration, in task A we found ”untuk pindah ke daerah yang aman” (to move to a safe place) labelled as 1 (correct) while clearly it does not fit the criteria based on the guideline. The mislabeling was even more prominent in task B: ”karana dengan menyumbang kita bisa membuat produksi pakaian menjadi lebih beretika” (By donating, we can make clothes production becomes more ethical) is considered wrong while ”agar upaya untuk membuat produksi pakaian menjadi lebih beretika.” (As an effort to make clothes production becomes more ethical) is approved. To approach this problem, we decided to prepare a separate training set with manually corrected labels based on our own judgment. The correction result is shown in Table 2. Finally, as the responses contain a lot of typo and slang words, we also experimented with simple typo corrector using python difflib package and Indonesian colloquial dictionary (Salsabila et al., 2018). We tried every possible combination of preprocessing steps and whether to use altered version of the training set with a parameter optimization library described in the following subsection. | | Original --- Label | Corrected --- Label Count Task A | $0$ | $1$ | $10$ $1$ | $0$ | $4$ Task B | $0$ | $1$ | $46$ $1$ | $0$ | $13$ Table 2: Corrected labels of the training set. ### 3.2 The Winning Approach After trying several machine learning algorithms, such as k-Nearest Neighbors, Naïve Bayes, logistic regression, and random forest, we found that random forest was the best model for task A. This corroborates what was found by Fernández-Delgado et al. (2014) in their comprehensive comparisons among several machine learning algorithms on different datasets. On the other hand, logistic regression with L2 regularization was the best for task B. The machine learning library used in this study is scikit-learn (Pedregosa et al., 2011). Since the dataset is quite small, we used 5-fold cross validation on the training set to avoid overfitting. For our winning approach, we set the n_estimators parameter for random forest to 200 and keep the default values for other parameters. We also found that it is the best to keep the default parameter values for the logistic regression model. ### 3.3 Alternative Approaches In parallel, we also experimented with optimization using hyperopt111http://hyperopt.github.io/hyperopt/ library, which utilizes sequential model-based optimization (Bergstra et al., 2011). We tried to optimize the hyperparameters, including the preprocessing steps, of four different machine learning algorithms: logistic regression, random forest, gradient boosting tree, and support vector machine. We trained separate models for task A and task B. In addition, we tested a voting-based ensemble model by combining all the optimized model of each algorithm. The evaluation metric used for the optimization, including for the voting-ensemble model, is F1 score. The results of the experiments are presented in the next section along with the discussion. ## 4 Results and Discussion We found that choosing different preprocessing methods resulted in different performances in the two tasks. Therefore, we varied the use of unigram or TF- IDF, and whether we should apply SVD to the resulting matrix. On the other hand, we found that it is always better to use the lemmatizer built on top of spaCy in this task. Moreover, removing stopwords did not contribute much to the performance on the training set. Table of local CV results can be seen in Table 3 and Table 4. | Precision | Recall | F1 ---|---|---|--- 1-gram+RF | $0.845\pm 0.057$ | $0.921\pm 0.037$ | $0.881\pm 0.035$ 1-gram+logreg | $0.857\pm 0.074$ | $0.869\pm 0.067$ | $0.862\pm 0.065$ 1-gram+SVD+RF | $0.794\pm 0.025$ | $0.984\pm 0.014$ | $0.879\pm 0.014$ 1-gram+SVD+logreg | $\mathbf{0.859\pm 0.069}$ | $0.874\pm 0.047$ | $0.866\pm 0.053$ TF-IDF+RF | $0.847\pm 0.054$ | $0.942\pm 0.039$ | $\mathbf{0.891\pm 0.036}$ TF-IDF+logreg | $0.748\pm 0.019$ | $0.979\pm 0.034$ | $0.848\pm 0.022$ TF-IDF+SVD+RF | $0.772\pm 0.025$ | $\mathbf{0.990\pm 0.014}$ | $0.867\pm 0.014$ TF-IDF+SVD+logreg | $0.751\pm 0.025$ | $0.979\pm 0.034$ | $0.850\pm 0.025$ Table 3: 5-fold cross validation results from task A | Precision | Recall | F1 ---|---|---|--- 1-gram+RF | $0.731\pm 0.066$ | $0.744\pm 0.054$ | $0.736\pm 0.047$ 1-gram+logreg | $\mathbf{0.735\pm 0.046}$ | $0.727\pm 0.080$ | $0.730\pm 0.059$ 1-gram+SVD+RF | $0.686\pm 0.035$ | $0.762\pm 0.046$ | $0.721\pm 0.032$ 1-gram+SVD+logreg | $0.722\pm 0.067$ | $0.726\pm 0.089$ | $0.723\pm 0.074$ TF-IDF+RF | $0.709\pm 0.036$ | $0.750\pm 0.049$ | $0.728\pm 0.034$ TF-IDF+logreg | $0.725\pm 0.035$ | $0.810\pm 0.060$ | $\mathbf{0.764\pm 0.035}$ TF-IDF+SVD+RF | $0.637\pm 0.026$ | $\mathbf{0.834\pm 0.061}$ | $0.721\pm 0.036$ TF-IDF+SVD+logreg | $0.705\pm 0.023$ | $0.809\pm 0.063$ | $0.753\pm 0.036$ Table 4: 5-fold cross validation results from task B For this challenge, we need to optimize the F1 score. Therefore, it is clear from Table 4 that we should use the model from TF-IDF with random forest algorithm for task B. From what we can see ini Table 3, we should also go with the same method for task A. However, we decided to be more pessimistic by looking at the largest F1 score after subtracting 1 standard deviation from the mean F1. Thus, we chose 1-gram + SVD + random forest for task A. Table 5 shows the performance of our best single models compared with two alternative approaches. First, we used the optimized ensemble model trained on the original training set (Ens+Ori). Second, we also use a similar method but with the label-corrected training set (Ens+Upd). While we can see in Table 5 that the F1 score on task B is higher on the label-corrected training set and got a similar result for the development set, the test score is lower than the best single models. The test set labels were most likely as noisy as the training set, thus making the training score with modified label not representative. | Train A | Train B | Dev | Test ---|---|---|---|--- Best | $0.879$ | $0.764$ | $0.810$ | $0.812$ Ens+Ori | $0.885$ | $0.764$ | $0.799$ | $0.801$ Ens+Upd | $0.898$ | $0.831$ | $0.810$ | $0.803$ Table 5: F1 score comparison with the alternative Ensemble model. To analyze how hard it is to separate the right from the wrong answers, we reduced the dimensionality of the data into 2D using 1-gram, SVD, and t-SNE (Maaten and Hinton, 2008). Figure 1 suggests that it is harder to separate the two answers in task B. Since most of the answers are short, we also cannot see “islands” from applying t-SNE to the data. Figure 1: t-SNE visualisation A similar problem was also shown in Figure 2 and Figure 3. We can see a lot more data points in the 0.4-0.6 prediction range in Figure 3. This suggests more uncertainty in the model. We argue that this is potentially because of the incorrect labels in the original training set for task B. Figure 2: Best model prediction (random forest) with probability on Task A Figure 3: Best model prediction (logistic regression) with probability on Task B ## 5 Conclusions In this report, we describe our winning approach for UKARA 1.0 Challenge Track 1. During the competition, we experimented with single models and ensemble models to predict which answer is correct given an open-ended question. We also tried to re-label the training set and pre-process with text correction which gave a boost in our local cross validation. However, we found that single models with less pre-processing performed better in the test set. The best single model for task A is random forest with unigram+SVD and for task B is logistic regression with TF-IDF. The prediction of both model achieved an overall F1 score of 0.812, which was enough to get us the third position in the final leaderboard. ## References * Bergstra et al. (2011) James S Bergstra, Rémi Bardenet, Yoshua Bengio, and Balázs Kégl. 2011\. Algorithms for hyper-parameter optimization. In _Advances in neural information processing systems_ , pages 2546–2554. * Deerwester et al. (1990) Scott Deerwester, Susan T Dumais, George W Furnas, Thomas K Landauer, and Richard Harshman. 1990. Indexing by latent semantic analysis. _Journal of the American society for information science_ , 41(6):391–407. * Dikli (2006) Semire Dikli. 2006. An overview of automated scoring of essays. _The Journal of Technology, Learning and Assessment_ , 5(1). * Fernández-Delgado et al. (2014) Manuel Fernández-Delgado, Eva Cernadas, Senén Barro, and Dinani Amorim. 2014\. Do we need hundreds of classifiers to solve real world classification problems? _The Journal of Machine Learning Research_ , 15(1):3133–3181. * Honnibal and Montani (2017) Matthew Honnibal and Ines Montani. 2017. spaCy 2: Natural language understanding with Bloom embeddings, convolutional neural networks and incremental parsing. To appear. * Maaten and Hinton (2008) Laurens van der Maaten and Geoffrey Hinton. 2008. Visualizing data using t-sne. _Journal of machine learning research_ , 9(Nov):2579–2605. * Pedregosa et al. (2011) F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. 2011. Scikit-learn: Machine learning in Python. _Journal of Machine Learning Research_ , 12:2825–2830. * Salsabila et al. (2018) Nikmatun Aliyah Salsabila, Yosef Ardhito Winatmoko, Ali Akbar Septiandri, and Ade Jamal. 2018. Colloquial indonesian lexicon. In _2018 International Conference on Asian Language Processing (IALP)_ , pages 226–229. IEEE. ## Appendix A Supplemental Material The code for this analysis can be seen on https://github.com/aliakbars/ukara/.

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2002.12568

arxiv-papers

# Reconstruction of weak bialgebra maps and its applications Michihisa Wakui Department of Mathematics, Faculty of Engineering Science, Kansai University, Suita-shi, Osaka 564-8680, Japan E-mail address<EMAIL_ADDRESS> ###### Abstract In this note we give a precise statement and a detailed proof for reconstruction problem of weak bialgebra maps. As an application we characterize indecomposability of weak algebras in categorical setting. ## 1 Introduction The concept of weak bialgebras and weak Hopf algebras was introduced by Böhm, Nill and Szlachányi [1] as a generalization of bialgebras and Hopf algebras. In some literatures they are called quantum groupoids [13]. As shown by Etingof, Nikshych and Ostrik [5] the language of weak Hopf algebras is convenient to visualize various categorical construction, including (multi-)fusion categories. At present, a lot of concepts and results for the ordinal bialgebras and Hopf algebras are generalized or extended to weak versions. By the classical Tannaka-Krein reconstruction theorem, it is well known that any finite-dimensional bialgebra $A$ over a fixed field $\boldsymbol{k}$ is determined up to isomorphism by its comodule category $\mathbb{M}^{A}$ whose objects are of finite dimension [4, 9, 15, 19]. More precisely, let $A$ and $B$ be two finite-dimensional bialgebras over $\boldsymbol{k}$, and $F:\mathbb{M}^{A}\longrightarrow\mathbb{M}^{B}$ be a $\boldsymbol{k}$-linear monoidal functor. If $F$ is fibered, then there is a bialgebra map $\varphi:A\longrightarrow B$ such that $F=\mathbb{M}^{\varphi}$, where $\mathbb{M}^{\varphi}$ is the induced monoidal functor from $f$. This statement can be found in Majid’s book [10, Theorem 2.2] and a detailed proof is given in Franco’s lecture note [6, p.80–84]. In this note we treat the weak bialgebra version of the above result. It is noted that as is well known, the reconstruction theorem for weak bialgebras has been established by several researchers [3, 7, 11, 2] (see [20] for other variants). However, it seems that there is no statement on a reconstruction theorem for “weak bialgebra map” in any papers though it is very fundamental. This note is devoted to give a precise statement of it and prove it. In fact, although the same statement with the classical one holds, the proof is a rather complicated since the unit object in the comodule category over a weak bialgebra $A$ is not necessary to the base field $\boldsymbol{k}$. Actually, it is a subalgebra, which is called the source counital subalgebra of $A$. For that reason, the proof of the reconstruction theorem for weak bialgebra maps is accomplished by re-examining the method used in the proof of the reconstruction of coalgebra map. Recently, the author study on indecomposability of weak bialgebras, and find some interesting results and unsolved problems [21]. As an application of the reconstruction theorem of weak bialgebra maps, we derive a categorical interpretation of indecomposability of a weak bialgebra. This paper is organized as follows. In Section 2 we recall the definition and basic properties of weak bialgebras, and also the comodule structure over them. In Section 3 after we overview the proof of the reconstruction theorem of coalgebra maps, we state and prove the reconstruction theorem of bialgebra maps. In Section 4, that is the final section, we apply the theorem to characterize indecomposability of finite-dimensional weak bialgebras. Throughout this note, $\boldsymbol{k}$ denotes a field, and $\text{Vect}_{\boldsymbol{k}}^{\text{f.d.}}$ stands for the monoidal category whose objects are finite-dimensional vector spaces over $\boldsymbol{k}$ and morphisms are $\boldsymbol{k}$-linear maps between them. For a weak bialgebra or a weak Hopf algebra $H$, we denote by $\Delta_{H}$, $\varepsilon_{H}$ and $S_{H}$ the comultiplication, the counit and the antipode of $H$, respectively. When it is clear that they are for $H$, they are simply denoted by $\Delta$, $\varepsilon$ and $S$, respectively. The notation $\Delta^{(2)}$ means the composition $(\Delta\otimes\text{id})\circ\Delta=(\text{id}\otimes\Delta)\circ\Delta$. We use Sweedler’s notation such as $\Delta(x)=x_{(1)}\otimes x_{(2)}$ for $x\in H$, and for a right $H$-comodule $M$ with coaction $\rho$ we also use the notation $\rho(m)=m_{(0)}\otimes m_{(1)}$ for $m\in M$. A monoidal category is described as $\mathcal{C}=(\mathcal{C},\otimes,I,a,l,r)$, as in MacLane’s book [8], and a monoidal functor between monoidal categories $\mathcal{C}$ and $\mathcal{D}$ is described as $F=(F,\Phi^{F},\omega^{F})$, where $F$ is a covariant functor from $\mathcal{C}$ to $\mathcal{D}$, and $\Phi^{F}$ is a natural transformation obtained by collecting morphisms $\phi^{F}_{M,N}:F(M)\otimes F(N)\longrightarrow F(M\otimes N)$ for all $M,N\in\mathcal{C}$, and $\omega^{F}:F(I_{\mathcal{C}})\longrightarrow I_{\mathcal{D}}$ is a morphism, where $I_{\mathcal{C}}$ and $I_{\mathcal{D}}$ are the unit objects in $\mathcal{C}$, and $\mathcal{D}$, respectively. If $\Phi^{F}$ is a natural equivalence and $\omega^{F}$ is an isomorphism, then the monoidal functor $(F,\Phi^{F},\omega^{F})$ is called strong. As in the classical case, every weak bialgebra map $\varphi:H\longrightarrow K$ induces a covariant functor $\mathbb{M}^{\varphi}:\mathbb{M}^{H}\longrightarrow\mathbb{M}^{K}$. We note that the functor $\mathbb{M}^{\varphi}$ is not monoidal but is comonoidal. By a comonoidal functor we mean a triplet $F=(F,\bar{\Phi}^{F},\bar{\omega}^{F})$, which all arrows in $\bar{\Phi}^{F}$ and $\bar{\omega}^{F}$ are in reverse in monoidal categories, that is, $\bar{\Phi}^{F}$ is a natural transformation obtained by collecting morphisms $\bar{\phi}^{F}_{M,N}:F(M\otimes N)\longrightarrow F(M)\otimes F(N)$ for all $M,N\in\mathcal{C}$, and $\bar{\omega}^{F}:I_{\mathcal{D}}\longrightarrow F(I_{\mathcal{C}})$ is a morphism. By the same condition for a monoidal functor, the concept called strong is defined for a comonoidal functor. In some literatures terminologies “colax” or “op-monoidal” are used for “comonoidal”. For general facts on Hopf algebras, we refer the reader to Montgomery’s book [12]. ## 2 Definition of weak bialgebras and structures of their comodule categories In this section we recall the definition and basic properties of weak bialgebras, and also the comodule structure over them mainly following [1] and [13]. Let $H$ be a vector space over $\boldsymbol{k}$, and $(H,\mu,\eta)$ is an algebra and $(H,\Delta,\varepsilon)$ is a coalgebra over $\boldsymbol{k}$. The $5$-tuple $(H,\mu,\eta,\Delta,\varepsilon)$ is said to be a weak bialgebra over $\boldsymbol{k}$ if the following three conditions are satisfied. 1. (WH1) $\Delta(xy)=\Delta(x)\Delta(y)$ for all $x,y\in H$. 2. (WH2) $\Delta^{(2)}(1)=(\Delta(1)\otimes 1)(1\otimes\Delta(1))=(1\otimes\Delta(1))(\Delta(1)\otimes 1)$, where $1=\eta(1)$ is the identity element of the algebra $(H,\mu,\eta)$. 3. (WH3) For all $x,y,z\in H$ 1. (i) $\varepsilon(xyz)=\varepsilon(xy_{(1)})\varepsilon(y_{(2)}z)$. 2. (ii) $\varepsilon(xyz)=\varepsilon(xy_{(2)})\varepsilon(y_{(1)}z)$. Let $S:H\longrightarrow H$ be a $\boldsymbol{k}$-linear map. The $6$-tuple $(H,\mu,\eta,\Delta,\varepsilon,S)$ is said to be a weak Hopf algebra over $\boldsymbol{k}$ if the above three conditions and the following additional condition are satisfied. 1. (WH4) For all $x\in H$ 1. (i) $x_{(1)}S(x_{(2)})=\varepsilon(1_{(1)}x)1_{(2)}$. 2. (ii) $S(x_{(1)})x_{(2)}=1_{(1)}\varepsilon(x1_{(2)})$. 3. (iii) $S(x_{(1)})x_{(2)}S(x_{(3)})=S(x)$. The above $S$ is called the antipode of $(H,\mu,\eta,\Delta,\varepsilon)$ and $(H,\mu,\eta,\Delta,\varepsilon,S)$. We note that it is unique if it exists. For a weak bialgebra $H=(H,\mu,\eta,\Delta,\varepsilon)$, by the condition (WH4)(i),(ii), the following two $\boldsymbol{k}$-linear maps $\varepsilon_{t},\varepsilon_{s}:H\longrightarrow H$ are defined: $\displaystyle\varepsilon_{t}(x)=\varepsilon(1_{(1)}x)1_{(2)},$ (2.1) $\displaystyle\varepsilon_{s}(x)=1_{(1)}\varepsilon(x1_{(2)}).$ (2.2) The maps $\varepsilon_{t}$ and $\varepsilon_{s}$ are called the target counital map and the source counital map, respectively. These maps satisfy the following properties. ###### Lemma 2.1. 1. $(1)$ $\varepsilon_{t}^{2}=\varepsilon_{t},\ \varepsilon_{s}^{2}=\varepsilon_{s}$. 2. $(2)$ For all $x\in H$ 1. $(i)$ $((\mathrm{id}\otimes\varepsilon_{t})\circ\Delta)(x)=1_{(1)}x\otimes 1_{(2)}$. 2. $(ii)$ $((\varepsilon_{s}\otimes\mathrm{id})\circ\Delta)(x)=1_{(1)}\otimes x1_{(2)}$. In particular, $1_{(1)}\otimes\varepsilon_{t}(1_{(2)})=1_{(1)}\otimes 1_{(2)}=\varepsilon_{s}(1_{(1)})\otimes 1_{(2)}.$ (2.3) 3. $(3)$ For all $x\in H$ 1. $(i)$ $\varepsilon_{t}(x)=x\ \ \Longleftrightarrow\ \ \Delta(x)=1_{(1)}x\otimes 1_{(2)}$. 2. $(ii)$ $\varepsilon_{s}(x)=x\ \ \Longleftrightarrow\ \ \Delta(x)=1_{(1)}\otimes x1_{(2)}$. Especially, by Part $(2)$$(i)$ $\displaystyle 1_{(1)}1_{[1]}\otimes 1_{(2)}\otimes 1_{[2]}$ $\displaystyle=1_{(1)}\otimes\varepsilon_{t}(1_{(2)})\otimes 1_{(3)},$ $\displaystyle 1_{(1)}\otimes 1_{[1]}\otimes 1_{(2)}1_{[2]}$ $\displaystyle=1_{(1)}\otimes\varepsilon_{s}(1_{(2)})\otimes 1_{(3)},$ where $\Delta^{(2)}(1)=1_{(1)}\otimes 1_{(2)}\otimes 1_{(3)}$ and $\Delta(1)=1_{[1]}\otimes 1_{[2]}$. ###### Lemma 2.2. Let $H$ be a weak bialgebra over $\boldsymbol{k}$. For all $x,y\in H$ 1. $(1)$ $\varepsilon_{t}(x\varepsilon_{t}(y))=\varepsilon_{t}(xy)$, $\varepsilon_{s}(\varepsilon_{s}(x)y)=\varepsilon_{s}(xy)$. 2. $(2)$ $\varepsilon(x\varepsilon_{t}(y))=\varepsilon(xy)$, $\varepsilon(\varepsilon_{s}(x)y)=\varepsilon(xy)$. 3. $(3)$ $\varepsilon\circ\varepsilon_{t}=\varepsilon=\varepsilon\circ\varepsilon_{s}$. 4. $(4)$ $x=\varepsilon_{t}(x_{(1)})x_{(2)}=x_{(1)}\varepsilon_{s}(x_{(2)})$. 5. $(5)$ $x\varepsilon_{t}(y)=\varepsilon_{t}(x_{(1)}y)x_{(2)}$, $\varepsilon_{s}(x)y=y_{(1)}\varepsilon_{s}(xy_{(2)})$. ###### Lemma 2.3. Let $H$ be a weak bialgebra over $\boldsymbol{k}$, and set $H_{t}:=\varepsilon_{t}(H),\ H_{s}:=\varepsilon_{s}(H)$. Then 1. $(1)$ $x\varepsilon_{t}(y)=\varepsilon_{t}(xy)$ for all $x\in H_{t}$ and $y\in H$. 2. $(2)$ $\varepsilon_{s}(x)y=\varepsilon_{s}(xy)$ for all $x\in H$ and $y\in H_{s}$. 3. $(3)$ 1. $(i)$ An element of $H_{t}$ and an element of $H_{s}$ commute, and 2. $(ii)$ $H_{t}$ and $H_{s}$ are a left coideal and a right coideal subalgebras of $H$, respectively. The subalgebras $H_{t}$ and $H_{s}$ are called the target and source subalgebras of $H$, respectively. By (2.3) we have $\Delta(1)\in H_{s}\otimes H_{t}.$ (2.4) 1. $(4)$ For all $x\in H$, $z\in H_{t}$ and $y\in H_{s}$ $\displaystyle\Delta(xz)=x_{(1)}z\otimes x_{(2)},\quad$ $\displaystyle\Delta(zx)=zx_{(1)}\otimes x_{(2)},$ $\displaystyle\Delta(xy)=x_{(1)}\otimes x_{(2)}y,\quad$ $\displaystyle\Delta(yx)=x_{(1)}\otimes yx_{(2)}.$ In particular $\displaystyle xz=\varepsilon(x_{(1)}z)x_{(2)},\quad$ $\displaystyle zx=\varepsilon(zx_{(1)})x_{(2)},$ $\displaystyle xy=x_{(1)}\varepsilon(x_{(2)}y),\quad$ $\displaystyle yx=x_{(1)}\varepsilon(yx_{(2)}).$ For a weak bialgebra $H$, one can also consider two $\boldsymbol{k}$-linear maps $\varepsilon_{t}^{\prime},\varepsilon_{s}^{\prime}:H\longrightarrow H$ defined by $\displaystyle\varepsilon_{t}^{\prime}(x)$ $\displaystyle=\varepsilon(x1_{(1)})1_{(2)},$ (2.5) $\displaystyle\varepsilon_{s}^{\prime}(x)$ $\displaystyle=1_{(1)}\varepsilon(1_{(2)}x)$ (2.6) for all $x\in H$. Then, $H^{\mathrm{op}}=(H,\mu^{\mathrm{op}},\eta,\Delta,\varepsilon)$, $H^{\mathrm{cop}}=(H,\mu,\eta,\Delta^{\mathrm{cop}},\varepsilon)$, $H^{\mathrm{opcop}}=(H,\mu^{\mathrm{op}},\eta,\Delta^{\mathrm{cop}}$, $\varepsilon)$ are weak bialgebras over $\boldsymbol{k}$, where $\mu^{\mathrm{op}}$ and $\Delta^{\mathrm{cop}}$ are the opposite multiplication and comultiplication, respectively. The target and the source subalgebras of them are given by $(H^{\mathrm{op}})_{t}=H_{t},\ (H^{\mathrm{op}})_{s}=H_{s}$, $(H^{\mathrm{cop}})_{t}=H_{s},\ (H^{\mathrm{cop}})_{s}=H_{t}$, $(H^{\mathrm{opcop}})_{t}=H_{s},\ (H^{\mathrm{opcop}})_{s}=H_{t}$. The target and the source counital maps of them are given by $(\varepsilon_{H^{\mathrm{op}}})_{t}=\varepsilon_{t}^{\prime},\ (\varepsilon_{H^{\mathrm{op}}})_{s}=\varepsilon_{s}^{\prime}$, $(\varepsilon_{H^{\mathrm{cop}}})_{t}=\varepsilon_{s}^{\prime},\ (\varepsilon_{H^{\mathrm{cop}}})_{s}=\varepsilon_{t}^{\prime}$, $(\varepsilon_{H^{\mathrm{opcop}}})_{t}=\varepsilon_{s},\ (\varepsilon_{H^{\mathrm{opcop}}})_{s}=\varepsilon_{t}$. If $S$ is the antipode of $H$, then it is also of $H^{\mathrm{opcop}}$. For a weak bialgebra $H$ over $\boldsymbol{k}$, we denote by $\text{\boldmath{$\mathsf{M}$}}^{H}$ the $\boldsymbol{k}$-linear category whose objects are right $H$-comodules and morphisms are $H$-comodule maps between them. The comodule category $\text{\boldmath{$\mathsf{M}$}}^{H}$ has a monoidal structure [13, Section 4] as the following lemma. ###### Lemma 2.4 ([14, Lemma 4.2]). Let $H$ be a weak bialgebra over $\boldsymbol{k}$. 1. $(1)$ For two right $H$-comodules $(M,\rho_{M}),\ (N,\rho_{N})$, the pair $(M,\rho_{M})\circledast(N,\rho_{N}):=(M\otimes_{H_{s}}N,\rho)$ is also a right $H$-comodule, where $\rho:M\otimes_{H_{s}}N\longrightarrow(M\otimes_{H_{s}}N)\otimes H$ is a $\boldsymbol{k}$-linear map defined by $\rho(m\otimes_{H_{s}}n)=(m_{(0)}\otimes_{H_{s}}n_{(0)})\otimes m_{(1)}n_{(1)}\qquad(m\in M,\ n\in N).$ (2.7) The source algebra $H_{s}$ can be regarded as a right $H$-comodule with the coaction $\Delta_{s}:=\Delta|_{H_{s}}:H_{s}:\longrightarrow H_{s}\otimes H$, and for all right $H$-comodules $L,M,N$ there are natural isomorphisms 1. $(i)$ $H_{s}\circledast M\cong M\cong M\circledast H_{s}$ as right $H$-comodules, 2. $(ii)$ $(L\circledast M)\circledast N\cong L\circledast(M\circledast N)$ as right $H$-comodules. Here the natural isomorphisms in $(i)$ are given as follows: For a right $H$-comodule $M$ $\displaystyle l_{M}:$ $\displaystyle\ H_{s}\otimes_{H_{s}}M\longrightarrow M,\ \quad l_{M}(y\otimes_{H_{s}}m)=y\cdot m\qquad(y\in H_{s},\ m\in M),$ $\displaystyle l_{M}^{-1}:$ $\displaystyle\ M\longrightarrow H_{s}\otimes_{H_{s}}M,\ \quad l_{M}^{-1}(m)=1\otimes_{H_{s}}m\qquad\quad(m\in M),$ $\displaystyle r_{M}:$ $\displaystyle\ M\otimes_{H_{s}}H_{s}\longrightarrow M,\ \quad r_{M}(m\otimes_{H_{s}}y)=m\cdot y\qquad(m\in M,\ y\in H_{s}),$ $\displaystyle r_{M}^{-1}:$ $\displaystyle\ M\longrightarrow M\otimes_{H_{s}}H_{s},\ \quad r_{M}^{-1}(m)=m\otimes_{H_{s}}1\qquad\quad(m\in M).$ The isomorphism in $(ii)$ is induced from a usual isomorphism between vector spaces. 2. $(2)$ Let $f:(M,\rho_{M})\longrightarrow(N,\rho_{N})$, $g:(M^{\prime},\rho_{M}^{\prime})\longrightarrow(N^{\prime},\rho_{N}^{\prime})$ be right $H$-comodule maps. Then $f\otimes_{H_{s}}g:M\otimes_{H_{s}}M^{\prime}\longrightarrow N\otimes_{H_{s}}N^{\prime}$ is also a right $H$-comodule map with respect to the comodule structure given in $(1)$. We denote the map $f\otimes_{H_{s}}g$ by $f\circledast g$. By Parts $(1)$, $(2)$ the abelian category $\text{\boldmath{$\mathsf{M}$}}^{H}$ becomes a $\boldsymbol{k}$-linear monoidal category whose unit object is $(H_{s},\Delta_{s})$. ###### Lemma 2.5 ([14, Proposition 4.1]). Let $H$ be a weak bialgebra over $\boldsymbol{k}$, and $(M,\rho_{M})$ be a right $H$-comodule. For $y\in H_{s}$ and $m\in M$, the elements $y\cdot m$ and $m\cdot y$ in $M$ are defined as follows: $\displaystyle y\cdot m$ $\displaystyle:=m_{(0)}\varepsilon(ym_{(1)}),$ (2.8) $\displaystyle m\cdot y$ $\displaystyle:=m_{(0)}\varepsilon(m_{(1)}y).$ (2.9) 1. $(1)$ $M$ becomes an $(H_{s},H_{s})$-bimodule equipped with the above actions. 2. $(2)$ $M\otimes H$ becomes an $(H_{s},H_{s})$-bimodule equipped with the following actions: For $y\in H_{s},\ m\in M,\ x\in H$, $\displaystyle y\cdot(m\otimes x)$ $\displaystyle:=(1_{(1)}\cdot m)\otimes(y1_{(2)}x)=m_{(0)}\otimes\varepsilon_{t}(m_{(1)})yx,$ (2.10) $\displaystyle(m\otimes x)\cdot y$ $\displaystyle:=(m\cdot 1_{(1)})\otimes(xy1_{(2)})=m_{(0)}\otimes xy\varepsilon_{t}^{\prime}(m_{(1)}).$ (2.11) 3. $(3)$ The following equations hold for all $y\in H_{s}$ and $m\in M$: 1. $(i)$ $\rho_{M}(y\cdot m)=m_{(0)}\otimes ym_{(1)}=y\cdot\rho_{M}(m)$. 2. $(ii)$ $\rho_{M}(m\cdot y)=m_{(0)}\otimes m_{(1)}y=\rho_{M}(m)\cdot y$. 3. $(iii)$ $\varepsilon_{s}^{\prime}(m_{(1)})\cdot m_{(0)}=m=m_{(0)}\cdot\varepsilon_{s}(m_{(1)})$. In particular, $\rho_{M}:M\longrightarrow M\otimes H$ is an $(H_{s},H_{s})$-bimodule map by $(i)$ and $(ii)$. 4. $(4)$ Let $(N,\rho_{N})$ be a right $H$-comodule and $f:(M,\rho_{M})\longrightarrow(N,\rho_{N})$ be an $H$-comodule map. Then $f$ becomes an $(H_{s},H_{s})$-bimodule map with respect to the bimodule structures given by $(1)$. Let $H$ be a weak bialgebra over $\boldsymbol{k}$, and denote by ${}_{H_{s}}\text{\boldmath{$\mathsf{M}$}}_{H_{s}}$ the $\boldsymbol{k}$-linear category consisting of whose objects are $(H_{s},H_{s})$-bimodules and morphisms are $(H_{s},H_{s})$-bimodule maps between them. By Lemma 2.5 for a right $H$-comodule $(M,\rho_{M})$, the underlying vector space $M$ has an $(H_{s},H_{s})$-bimodule structure, and $\rho_{M}:M\longrightarrow M\otimes H$ becomes an $(H_{s},H_{s})$-bimodule map. Thus $(M,\rho_{M})$ can be regarded as a right $H$-comodule in ${}_{H_{s}}\text{\boldmath{$\mathsf{M}$}}_{H_{s}}$. Then any $H$-comodule map $f:M\longrightarrow N$ always an $(H_{s},H_{s})$-bimodule map. We denote by ${}_{H_{s}}\text{\boldmath{$\mathsf{M}$}}_{H_{s}}^{H}$ the $\boldsymbol{k}$-linear category whose objects are right $H$-comodules in ${}_{H_{s}}\text{\boldmath{$\mathsf{M}$}}_{H_{s}}$ and morphisms are right $H$-comodule and $(H_{s},H_{s})$-bimodule maps between them. The category ${}_{H_{s}}\text{\boldmath{$\mathsf{M}$}}_{H_{s}}^{H}$ has a $\boldsymbol{k}$-linear monoidal structure whose tensor product is given by $\otimes_{H_{s}}$. As a special case of [18, Theorem 2.2] we have: ###### Lemma 2.6. Let $H$ be a weak bialgebra over $\boldsymbol{k}$. By Lemma 2.5 for a right $H$-comodule $(M,\rho_{M})$, there is an $(H_{s},H_{s})$-bimodule structure on the underlying vector space $M$, and $\rho_{M}:M\longrightarrow M\otimes H$ becomes an $(H_{s},H_{s})$-bimodule map. Thus $(M,\rho_{M})$ can be regarded as a right $H$-comodule in ${}_{H_{s}}\text{\boldmath{$\mathsf{M}$}}_{H_{s}}$. Then any $H$-comodule map $f:M\longrightarrow N$ is always an $(H_{s},H_{s})$-bimodule map. This correspondence gives rise to a $\boldsymbol{k}$-linear monoidal equivalence $\Xi^{H}:\text{\boldmath{$\mathsf{M}$}}^{H}\longrightarrow{}_{H_{s}}\text{\boldmath{$\mathsf{M}$}}_{H_{s}}^{H}$ between $\boldsymbol{k}$-linear monoidal categories. We set $\hat{U}^{H}:=U^{H}\circ\Xi^{H}:\text{\boldmath{$\mathsf{M}$}}^{H}\longrightarrow{}_{H_{s}}\text{\boldmath{$\mathsf{M}$}}_{H_{s}},$ where $U^{H}:{}_{H_{s}}\text{\boldmath{$\mathsf{M}$}}_{H_{s}}^{H}\longrightarrow{}_{H_{s}}\text{\boldmath{$\mathsf{M}$}}_{H_{s}}$ is the forgetful monoidal functor. The composition $\hat{U}^{H}$ of monoidal functors becomes a monoidal functor, whose structure is given as follows: 1. $\bullet$ $\phi_{M,N}^{\hat{U}^{H}}=\text{id}_{M\otimes_{H_{s}}N}$ for each $M,N\in\text{\boldmath{$\mathsf{M}$}}^{H}$, 2. $\bullet$ $\omega^{\hat{U}^{H}}:H_{s}\longrightarrow\hat{U}^{H}(H_{s})=H_{s}$ is the identity map. ###### Lemma 2.7. Let $H,K$ be weak bialgebras over $\boldsymbol{k}$, and $\varphi:H\longrightarrow K$ be a weak bialgebra map, namely, an algebra map and coalgebra map. Then $\varphi(H_{s})\subset K_{s}$ and $\varphi(H_{t})\subset K_{t}$. The former inclusion induces an algebra map $\varphi_{s}:=\varphi|_{H_{s}}:H_{s}\longrightarrow K_{s}$. 1. $(1)$ For a right $H$-comodule $(M,\rho_{M})$ $\text{\boldmath{$\mathsf{M}$}}^{\varphi}(M,\rho_{M}):=(M,(\text{id}_{M}\otimes\varphi)\circ\rho_{M})$ is a right $K$-comodule, and for a right $H$-comodule map $f:(M,\rho_{M})\longrightarrow(N,\rho_{N})$ $\text{\boldmath{$\mathsf{M}$}}^{\varphi}(f):=f:\text{\boldmath{$\mathsf{M}$}}^{\varphi}(M,\rho_{M})\longrightarrow\text{\boldmath{$\mathsf{M}$}}^{\varphi}(N,\rho_{N})$ is a right $K$-comodule map. These correspondences define a covariant functor $\text{\boldmath{$\mathsf{M}$}}^{\varphi}:\text{\boldmath{$\mathsf{M}$}}^{H}\longrightarrow\text{\boldmath{$\mathsf{M}$}}^{K}$. 2. $(2)$ The functor $\text{\boldmath{$\mathsf{M}$}}^{\varphi}$ is a $\boldsymbol{k}$-linear comonoidal functor. If $\varphi_{s}$ is bijective, then the $\boldsymbol{k}$-linear comonoidal functor $\text{\boldmath{$\mathsf{M}$}}^{\varphi}$ is strong. In this case it can be regarded as a $\boldsymbol{k}$-linear monoidal functor. 3. $(3)$ The algebra map $\varphi_{s}$ induces a $\boldsymbol{k}$-linear monoidal functor ${}_{\varphi_{s}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}}:{}_{K_{s}}\text{\boldmath{$\mathsf{M}$}}_{K_{s}}\longrightarrow{}_{H_{s}}\text{\boldmath{$\mathsf{M}$}}_{H_{s}}$, and if $\varphi_{s}$ is bijective, then $\hat{U}^{K}\circ\text{\boldmath{$\mathsf{M}$}}^{\varphi}={}_{\varphi_{s}^{-1}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}^{-1}}\circ\hat{U}^{H}:\text{\boldmath{$\mathsf{M}$}}^{H}\longrightarrow{}_{K_{s}}\text{\boldmath{$\mathsf{M}$}}_{K_{s}}$ as monoidal functors. ###### Proof.. (2) For right $H$-comodules $(M,\rho_{M}),(N,\rho_{N})$ we set $\displaystyle\text{\boldmath{$\mathsf{M}$}}^{\varphi}(M,\rho_{M})\circledast\text{\boldmath{$\mathsf{M}$}}^{\varphi}(N,\rho_{N})$ $\displaystyle=(M\otimes_{K_{s}}N,\rho),$ $\displaystyle\text{\boldmath{$\mathsf{M}$}}^{\varphi}\bigl{(}(M,\rho_{M})\circledast(N,\rho_{N})\bigr{)}$ $\displaystyle=(M\otimes_{H_{s}}N,\rho^{\prime}),$ where $\rho$ and $\rho^{\prime}$ are given as follows: $\displaystyle\rho:M\otimes_{K_{s}}N\longrightarrow(M\otimes_{K_{s}}N)\otimes K,\quad$ $\displaystyle\rho(m\otimes_{K_{s}}n)=m_{(0)}\otimes_{K_{s}}n_{(0)}\otimes\varphi(m_{(1)})\varphi(n_{(1)}),$ $\displaystyle\rho^{\prime}:M\otimes_{H_{s}}N\longrightarrow(M\otimes_{H_{s}}N)\otimes K,\quad$ $\displaystyle\rho^{\prime}(m\otimes_{H_{s}}n)=m_{(0)}\otimes_{H_{s}}n_{(0)}\otimes\varphi(m_{(1)}n_{(1)}).$ Since $\varphi$ is an algebra map, the identity map $\text{id}_{M\otimes N}$ induces a surjection $\iota_{M,N}:M\otimes_{H_{s}}N\longrightarrow M\otimes_{K_{s}}N$ which is a right $K$-comodule map. The restriction $\varphi_{s}:H_{s}\longrightarrow K_{s}$ can be regarded as a right $K$-comodule map from $\text{\boldmath{$\mathsf{M}$}}^{\varphi}(H_{s},(\Delta_{H})_{s})=(H_{s},\ (\text{id}_{H_{s}}\otimes\varphi)\circ(\Delta_{H})_{s})$ to $(K_{s},(\Delta_{K})_{s})$. Thus, the triplet $(\text{\boldmath{$\mathsf{M}$}}^{\varphi},\iota,\varphi_{s})$ is a comonoidal functor from $\text{\boldmath{$\mathsf{M}$}}^{H}$ to $\text{\boldmath{$\mathsf{M}$}}^{K}$, where $\iota:=\\{\iota_{M,N}\\}_{M,N\in\text{\boldmath{$\mathsf{M}$}}^{H}}$. If $\varphi_{s}$ is bijective, then $\iota_{M,N}$ for all $M,N\in\text{\boldmath{$\mathsf{M}$}}^{H}$ is an isomorphism. Thus, $(\text{\boldmath{$\mathsf{M}$}}^{\varphi},\iota^{-1},\varphi_{s}^{-1}):\text{\boldmath{$\mathsf{M}$}}^{H}\longrightarrow\text{\boldmath{$\mathsf{M}$}}^{K}$ is a strong monoidal functor. (3) For a $(K_{s},K_{s})$-bimodule $(M,\alpha_{l},\alpha_{r})$ we set ${}_{\varphi_{s}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}}(M,\alpha_{l},\alpha_{r})=\bigl{(}M,\alpha_{l}\circ(\varphi_{s}\otimes\text{id}_{M}),\alpha_{r}\circ(\text{id}_{M}\otimes\varphi_{s})\bigr{)},$ and a $(K_{s},K_{s})$-bimodule map $f:M\longrightarrow N$ we set ${}_{\varphi_{s}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}}(f)=f.$ Then, ${}_{\varphi_{s}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}}$ is a $\boldsymbol{k}$-linear covariant functor. For $(K_{s},K_{s})$-bimodules $M=(M,\alpha_{l}^{M},\alpha_{r}^{M})$ and $N=(N,\alpha_{l}^{N},\alpha_{r}^{N})$, let $\jmath_{M,N}:{}_{\varphi_{s}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}}(M)\otimes_{H_{s}}{}_{\varphi_{s}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}}(N)\longrightarrow{}_{\varphi_{s}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}}(M\otimes_{K_{s}}N)$ be the induced $\boldsymbol{k}$-linear map from the identity map $\text{id}_{M_{\otimes}N}$. This is an $(H_{s},H_{s})$-module map which is natural with respect to $M,N$. So, we have a monoidal functor $({}_{\varphi_{s}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}},\jmath,\varphi_{s})$ by setting $\jmath=\\{\jmath_{M,N}\\}_{M,N\in{}_{K_{s}}\text{\boldmath{$\mathsf{M}$}}_{K_{s}}}$. If $\varphi_{s}$ is bijective, then one can show that $\hat{U}^{K}\circ\text{\boldmath{$\mathsf{M}$}}^{\varphi}={}_{\varphi_{s}^{-1}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}^{-1}}\circ\hat{U}^{H}$ as $\boldsymbol{k}$-linear monoidal functors. ∎ ## 3 Reconstruction of a weak bialgebra map First of all, we will recall the proof of the reconstruction theorem of coalgebra maps, that is a classical and fundamental theorem for Tannakian reconstruction theorem. For a coalgebra $C$ over $\boldsymbol{k}$ we denote by $\mathbb{M}^{C}$ the $\boldsymbol{k}$-linear category consisting of whose objects are right $C$-comodules of finite dimension and morphisms are $C$-comodule maps between them, and denote by $U^{C}$ the forgetful functor from $\mathbb{M}^{C}$ to $\mathrm{Vect}_{\boldsymbol{k}}^{\mathrm{f.d.}}$. ###### Theorem 3.1. Let $C,D$ be coalgebras over $\boldsymbol{k}$, and $F:\mathbb{M}^{C}\longrightarrow\mathbb{M}^{D}$ be a $\boldsymbol{k}$-linear covariant functor. If $U^{D}\circ F=U^{C}$, then there is a unique coalgebra map $\varphi:C\longrightarrow D$ such that $F=\mathbb{M}^{\varphi}:\mathbb{M}^{C}\longrightarrow\mathbb{M}^{D}$. Here, $\mathbb{M}^{\varphi}$ denotes the $\boldsymbol{k}$-linear functor induced from $\varphi$. ###### Proof.. The proof follows from Franco’s lecture note [6, p.81–84]. Let $(M,\rho_{M})$ be a finite-dimensional $C$-comodule. By the assumption $U^{D}\circ F=U^{C}$, one can set $F(M,\rho_{M})=(M,\rho_{M}^{F})$, where $\rho_{M}^{F}:M\longrightarrow M\otimes D$ is a right coaction of $D$. Let $P$ be a finite-dimensional subcoalgebra of $C$. It can be regarded as a right $C$-comodule by the coaction $\rho_{P}:P\xrightarrow{\ \ \Delta_{P}\ \ }P\otimes P\xrightarrow{\ \ \text{id}\otimes\iota_{P}\ \ }P\otimes C,$ where $\iota_{P}$ stands for the inclusion. Since $P$ is finite-dimensional, it gives an object of $\mathbb{M}^{C}$, and therefore we have $F(P,\rho_{P})=(P,\rho_{P}^{F})\in\mathbb{M}^{D}$. Let us consider the composition $\varphi_{P}:P\xrightarrow{\ \ \rho_{P}^{F}\ \ }P\otimes D\xrightarrow{\ \varepsilon_{P}\otimes\text{id}\ }\boldsymbol{k}\otimes D\cong D.$ Then $\varphi_{P}:P\longrightarrow D$ is a coalgebra map. This fact can be verified as follows. $\bullet$ The equation $\varepsilon_{D}\circ\varphi_{P}=\varepsilon_{P}$ comes from the following commutative diagram. $P$$P\otimes D$$\boldsymbol{k}\otimes D$$\cong$$D$$P$$\cong$$P\otimes\boldsymbol{k}$$\boldsymbol{k}\otimes\boldsymbol{k}$$\cong$$\boldsymbol{k}$$\rho_{P}^{F}$$\varepsilon_{P}\otimes\text{id}$$\varepsilon_{P}\otimes\text{id}$id$\text{id}\otimes\varepsilon_{D}$$\text{id}\otimes\varepsilon_{D}$$\varepsilon_{D}$ $\bullet$ To show the equation $\Delta_{D}\circ\varphi_{P}=(\varphi_{P}\otimes\varphi_{P})\circ\Delta_{P}$, it is enough to verify the following diagram commutes: $P$$P\otimes P$$P\otimes D\otimes P\otimes D$$P\otimes D$$P\otimes P\otimes D$$P\otimes D\otimes D$$\cong$$P\otimes D\otimes\boldsymbol{k}\otimes D$$\boldsymbol{k}\otimes D\otimes\boldsymbol{k}\otimes D$$\boldsymbol{k}\otimes D\otimes D$$P\otimes D$$\boldsymbol{k}\otimes D\cong D$$D\otimes D$(A)(B)(C)$\Delta_{P}$$\rho_{P}^{F}\otimes\rho_{P}^{F}$$\text{id}\otimes\rho_{P}^{F}$$\rho_{P}^{F}$$\Delta_{P}\otimes\text{id}$$\rho_{P}^{F}\otimes\text{id}$$\text{id}\otimes\Delta_{D}$$\rho_{P}^{F}$$\varepsilon_{P}\otimes\text{id}$$\Delta_{P}$$\rho_{P}^{F}\otimes\text{id}$$\text{id}\otimes\varepsilon_{P}\otimes\text{id}$$\varepsilon_{P}\otimes\text{id}$$\cong$$\varepsilon_{P}\otimes\text{id}\otimes\varepsilon_{P}\otimes\text{id}$$\cong$ Part (A) is commutative since $\rho_{P}^{F}$ is a right $D$-coaction, and Part (C) is commutative since the following diagram is so. $P$$P\otimes P$$P\otimes D\otimes P\otimes D$$P\otimes D$$P\otimes D$$P\otimes D$$D\otimes D$$\Delta_{P}\otimes\text{id}$$\rho_{P}^{F}\otimes\text{id}$$\cong$$\text{id}\otimes\varepsilon_{P}\otimes\text{id}$$\cong$$\rho_{P}^{F}\otimes\text{id}$$\rho_{P}^{F}\otimes\text{id}$$\cong$$\text{id}\otimes\varepsilon_{P}\otimes\text{id}$ The proof the commutativity of Part (B) is a little technical. For any $\boldsymbol{k}$-linear map $\gamma:P\longrightarrow\boldsymbol{k}$, the composition $P\xrightarrow{\ \ \Delta_{P}\ \ }P\otimes P\xrightarrow{\ \gamma\otimes\text{id}_{P}\ }P$ is a right $P$-comodule map, and hence it is also a right $C$-comodule map. Sending it by $F$ we have a right $D$-comodule map $(\gamma\otimes\text{id}_{P})\circ\Delta_{P}:P\longrightarrow P$. Thus the following diagram commutes: $P$$P\otimes P$$P$$P$$P\otimes P\otimes D$$P\otimes D$$\Delta_{P}$$\gamma\otimes\text{id}$$\Delta_{P}\otimes\text{id}$$\gamma\otimes\text{id}\otimes\text{id}$$\rho_{P}^{F}$$\rho_{P}^{F}$ Combining the above diagram with $\rho_{P}^{F}\circ(\gamma\otimes\text{id}_{P})=(\gamma\otimes\text{id}_{P\otimes D})\circ(\text{id}_{P}\otimes\rho_{P}^{F})$, we have $(\gamma\otimes\text{id}_{P\otimes D})\circ(\text{id}_{P}\otimes\rho_{P}^{F})\circ\Delta_{P}=(\gamma\otimes\text{id}_{P\otimes D})\circ(\Delta_{P}\otimes\text{id}_{P})\circ\rho_{P}^{F}.$ Since this equation holds for all $\boldsymbol{k}$-linear maps $\gamma$, we see that $(\text{id}_{P}\otimes\rho_{P}^{F})\circ\Delta_{P}=(\Delta_{P}\otimes\text{id}_{P})\circ\rho_{P}^{F}$. This implies the commutativity of Part (B). By the fundamental theorem for coalgebras, $C$ is a union of finite- dimensional subcoalgebras. Based on the fact, it can be shown that the coalgebra maps $\varphi_{P}:P\longrightarrow D$ for all finite-dimensional subcoalgebras $P$ induce a well-defined coalgebra map $\varphi:C\longrightarrow D$. In fact it is easily verified that $(\varphi_{Q})|_{P}=\varphi_{P}$ for two finite-dimensional subcoalgebras $P$ and $Q$ satisfying with $P\subset Q$. In this way, it is proved that there is a coalgebra map $\varphi:C\longrightarrow D$ such that $\varphi|_{P}=\varphi_{P}$ for all finite-dimensional subcoalgebra $P$ of $C$. The coalgebra map $\varphi$ satisfies $F=\mathbb{M}^{\varphi}$. This fact can be derived as follows. $\bullet$ Let $(M,\rho_{M})$ be a finite-dimensional right $C$-comodule. Then there is a finite-dimensional subcoalgebra $P$ of $C$ such that $\rho_{M}(M)\subset M\otimes P$. Since $(M\otimes P,\text{id}_{M}\otimes\rho_{P})$ is a right $C$-comodule, we have a right $D$-comodule $(M\otimes P,\ (\text{id}_{M}\otimes\rho_{P})^{F})$. Since $M\otimes P$ is decomposed to a direct sum of finite copies of $P$ as a right $C$-comodule and $F$ is a $\boldsymbol{k}$-linear functor satisfying $U^{C}=U^{D}\circ F$, it follows that $(\text{id}_{M}\otimes\rho_{P})^{F}=\text{id}_{M}\otimes\rho_{P}^{F}:M\otimes P\longrightarrow M\otimes P\otimes D$. Let $\rho_{M}^{\prime}:M\longrightarrow M\otimes P$ be the restriction of $\rho_{M}$. Then the map $\rho_{M}^{\prime}$ is a right $C$-comodule map from $(M,\rho_{M})$ to $(M\otimes P,\text{id}\otimes\rho_{P})$. Thus, $F(\rho_{M}^{\prime})=\rho_{M}^{\prime}:M\longrightarrow M\otimes P$ is a right $D$-comodule map from $(M,\rho_{M}^{F})$ to $(M\otimes P,(\text{id}\otimes\rho_{P})^{F})=(M\otimes P,\text{id}_{M}\otimes\rho_{P}^{F})$, and hence the following diagram commutes. $M$$M\otimes P$$M\otimes D$$M\otimes P\otimes D$$\rho_{M}^{\prime}$$\rho_{M}^{\prime}\otimes\text{id}$$\rho_{M}^{F}$$\text{id}\otimes\rho_{P}^{F}$ It follows that we have the commutative diagram: $M$$M\otimes C$$M\otimes P$$M\otimes P\otimes D$$M\otimes\boldsymbol{k}\otimes D$$M\otimes D$$M\otimes D$$\rho_{M}$$\rho_{M}^{\prime}$id$\text{id}\otimes\iota_{P}$$\text{id}\otimes\varphi_{P}$$\rho_{M}^{F}$$\text{id}\otimes\varphi$$\text{id}\otimes\rho_{P}^{F}$$\text{id}\otimes\varepsilon_{P}\otimes\text{id}$$\rho_{M}^{\prime}\otimes\text{id}$$\cong$ This implies that $\rho_{M}^{F}=(\text{id}\otimes\varphi)\circ\rho_{M}$, and hence we see that $F(M,\rho_{M})=\mathbb{M}^{\varphi}(M,\rho_{M})$. $\bullet$ Let $f:(M,\rho_{M})\longrightarrow(N,\rho_{N})$ be a right $C$-comodule map between finite-dimensional right $C$-comodules. Then $\displaystyle F(f)=f$ $\displaystyle:(M,\rho_{M}^{F})\longrightarrow(N,\rho_{N}^{F}),$ $\displaystyle\mathbb{M}^{\varphi}(f)=f$ $\displaystyle:(M,(\text{id}\otimes\varphi)\circ\rho_{M})\longrightarrow(N,(\text{id}\otimes\varphi)\circ\rho_{N}).$ As shown that $\rho_{M}^{F}=(\text{id}\otimes\varphi)\circ\rho_{M}$ and $\rho_{N}^{F}=(\text{id}\otimes\varphi)\circ\rho_{N}$, we see that $F(f)=\mathbb{M}^{\varphi}(f)$. Thus $F=\mathbb{M}^{\varphi}$ as $\boldsymbol{k}$-linear functors. Finally, we show the uniqueness of $\varphi$. Suppose that a coalgebra map $\psi:C\longrightarrow D$ satisfies $F=\mathbb{M}^{\psi}$, too. Let $P$ be a finite-dimensional subcoalgebra of $C$, and regard it as the right $C$-comodule $(P,\rho_{P})$. Since $\mathbb{M}^{\varphi}(P,\rho_{P})=\mathbb{M}^{\psi}(P,\rho_{P})$, the following diagram commutes. $P$$P\otimes C$$P\otimes C$$P\otimes D$$\rho_{P}$$\text{id}\otimes\varphi$$\rho_{P}$$\text{id}\otimes\psi$ Since the diagram $P$$P\otimes C$$P\otimes D$$\boldsymbol{k}\otimes D$$\boldsymbol{k}\otimes D$ $\cong$ $\cong$ $C$$D$$\rho_{P}$$\text{id}\otimes\varphi$$\iota_{P}$$\varepsilon_{P}\otimes\text{id}$$\varepsilon_{P}\otimes\text{id}$$\text{id}\otimes\varphi$$\varphi$ commutes, and the same commutative diagram holds for $\psi$, we have $\varphi\circ\iota_{P}=\psi\circ\iota_{P}$. This implies that $\varphi=\psi$ since $C$ can be regarded as a union of finite-dimensional subcoalgebras. ∎ ###### Remark 3.2. The coalgebra map $\varphi_{P}:P\longrightarrow D$ in the above proof is a right $D$-comodule map from $(P,\rho_{P}^{F})$ to $(D,\Delta_{D})$. It follows from the following commutative diagram. $P$$D$$P\otimes D$$D\otimes D$$P\otimes D$$\boldsymbol{k}\otimes D$$P\otimes D\otimes D$$\boldsymbol{k}\otimes D\otimes D$$\mathrm{(1)}$$\mathrm{(2)}$$\mathrm{(3)}$$\varphi_{P}$$\varepsilon_{P}\otimes\mathrm{id}$$\varepsilon_{P}\otimes\mathrm{id}$$\varphi_{P}\otimes\mathrm{id}$$\rho_{P}^{F}$$\rho_{P}^{F}$$\sim$$\sim$$\rho_{P}^{F}\otimes\mathrm{id}$$\Delta_{D}$$\mathrm{id}\otimes\Delta_{D}$$\mathrm{id}\otimes\Delta_{D}$ Here, commutativity of Parts $\mathrm{(1)}$ and $\mathrm{(2)}$ come from the definition of $\varphi_{P}$, and Part $\mathrm{(3)}$ comes from what $(P,\rho_{P}^{F})$ is a right $D$-comodule. As an application of Theorem 3.1 we have: ###### Corollary 3.3. Let $C,D$ be two coalgebras over $\boldsymbol{k}$, and $F:\mathbb{M}^{C}\longrightarrow\mathbb{M}^{D}$ be a $\boldsymbol{k}$-linear functor. If $F$ is an equivalence of $\boldsymbol{k}$-linear categories and $U^{D}\circ F=U^{C}$ is satisfied, then the coalgebra map $\varphi:C\longrightarrow D$ determined by $F=\mathbb{M}^{\varphi}:\mathbb{M}^{C}\longrightarrow\mathbb{M}^{D}$ in Theorem 3.1 is an isomorphism. The above corollary can be easily proved by taking a quasi-inverse $G$ of $F$ such as $U^{C}\circ G=U^{D}$, $G\circ F=1_{\mathbb{M}^{C}}$ and $F\circ G=1_{\mathbb{M}^{D}}$. Dualizing of Theorem 3.1 we have also the following corollary. ###### Corollary 3.4. Let $A,B$ be two finite-dimensional algebras over $\boldsymbol{k}$, and $F:{}_{B}\mathbb{M}\longrightarrow{}_{A}\mathbb{M}$ be a $\boldsymbol{k}$-linear functor. If ${}_{A}U\circ F={}_{B}U$ is satisfied for the forgetful functors ${}_{A}U,{}_{B}U$ to $\text{Vect}_{\boldsymbol{k}}^{\mathrm{f.d.}}$, then 1. $(1)$ there is a unique algebra map $\varphi:A\longrightarrow B$ such that $F={}_{\varphi}\mathbb{M}:{}_{B}\mathbb{M}\longrightarrow{}_{A}\mathbb{M}$, where ${}_{\varphi}\mathbb{M}$ stands for the $\boldsymbol{k}$-linear functor induced from $\varphi$, 2. $(2)$ if $F$ is an equivalence of $\boldsymbol{k}$-linear categories, then the algebra map $\varphi:A\longrightarrow B$ given by $(1)$ is an isomorphism. Now, we show the main theorem in the present paper: ###### Theorem 3.5. Let $A,B$ be weak bialgebras over $\boldsymbol{k}$, and $F=(F,\bar{\phi}^{F},\bar{\omega}^{F}):\mathbb{M}^{A}\longrightarrow\mathbb{M}^{B}$ be a strong $\boldsymbol{k}$-linear comonoidal functor. Suppose that $U^{B}\circ F=U^{A}$ as $\boldsymbol{k}$-linear monoidal categories. Then there is a unique weak bialgebra map $\varphi:A\longrightarrow B$ such that $F=\mathbb{M}^{\varphi}$ as $\boldsymbol{k}$-linear monoidal categories and $\bar{\omega}^{F}=\varphi|_{A_{s}}:A_{s}\longrightarrow B_{s}$ is an algebra isomorphism. Furthermore, $\hat{U}^{B}\circ F={}_{\varphi_{s}^{-1}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}^{-1}}\circ\hat{U}^{A}$ is satisfied, where $\hat{U}^{A}:\mathbb{M}^{A}\longrightarrow{}_{A_{s}}\text{\boldmath{$\mathsf{M}$}}_{A_{s}},\ \hat{U}^{B}:\mathbb{M}^{B}\longrightarrow{}_{B_{s}}\text{\boldmath{$\mathsf{M}$}}_{B_{s}}$ are forgetful functors. ###### Proof.. By Theorem 3.1 there is a unique coalgebra map $\varphi:A\longrightarrow B$ such that $F=\mathbb{M}^{\varphi}$ as $\boldsymbol{k}$-linear functors. Since $U^{B}\circ F=U^{A}$ as $\boldsymbol{k}$-linear monoidal functors, for all $M,N\in\mathbb{M}^{A}$ the composition $\displaystyle U^{B}\bigl{(}F(M)\bigr{)}\otimes U^{B}\bigl{(}F(N)\bigr{)}$ $\displaystyle\xrightarrow{\ \phi^{U^{B}}_{F(M),F(N)}\ }U^{B}\bigl{(}F(M)\otimes_{B_{s}}F(N)\bigr{)}$ $\displaystyle\qquad\qquad\xrightarrow{\ U^{B}\bigl{(}(\bar{\phi}^{F}_{M,N})^{-1}\bigr{)}\ }U^{B}\bigl{(}F(M\otimes_{A_{s}}N)\bigr{)}$ coincides with the natural projection $\phi^{U^{A}}_{M,N}:U^{A}(M)\otimes U^{A}(N)\longrightarrow U^{A}(M\otimes_{A_{s}}N)$. This implies that the diagram $M\otimes N$$M\otimes N$$M\otimes_{A_{s}}A$$M\otimes_{B_{s}}N$$\text{id}_{M\otimes N}$$\bar{\phi}^{F}_{M,N}$natural proj.natural proj. commutes. This means that the map $\bar{\phi}^{F}_{M,N}:F(M\otimes_{A_{s}}N)\longrightarrow F(M)\otimes_{B_{s}}F(N)$ is induced from the identity map $\text{id}_{M\otimes N}$. Let us show that $\bar{\omega}^{F}=\varphi|_{A_{s}}:A_{s}\longrightarrow B_{s}$ is an algebra isomorphism. Since $U^{B}\circ F=U^{A}$ as $\boldsymbol{k}$-linear monoidal functors, the composition $U^{B}\bigl{(}(\bar{\omega}^{F})^{-1}\bigr{)}\circ\omega^{U^{B}}:\boldsymbol{k}\longrightarrow(U^{B}\circ F)(A_{s})=A_{s}$ coincides with $\omega^{U^{A}}:\boldsymbol{k}\longrightarrow A_{s}$. Thus $(\bar{\omega}^{F})^{-1}\circ\omega^{U^{B}}=\omega^{U^{A}}$ as maps, and hence $\bar{\omega}^{F}(1)=1$. Since $\bar{\omega}^{F}:F(A_{s})\longrightarrow B_{s}$ is a right $B$-comodule map and $F(A_{s})=\mathbb{M}^{\varphi}(A_{s})$, the following diagram commutes. $M\otimes N$$M\otimes N$$M\otimes_{A_{s}}A$$M\otimes_{A_{s}}A$$M\otimes_{B_{s}}N$$\bar{\omega}^{F}$$\bar{\omega}^{F}\otimes\text{id}$$\Delta_{A}|_{A_{s}}$$\text{id}\otimes\varphi$$\Delta_{B}|_{B_{s}}$ Thus $\Delta_{B}\bigl{(}\bar{\omega}^{F}(y)\bigr{)}=\bar{\omega}^{F}(y_{(1)})\otimes\varphi(y_{(2)})$ for all $y\in A_{s}$. Applying $\varepsilon_{B}\otimes\text{id}$ to both sides, we have $\displaystyle\bar{\omega}^{F}(y)$ $\displaystyle=\varepsilon_{B}\bigl{(}\bar{\omega}^{F}(y_{(1)})\bigr{)}\varphi(y_{(2)})$ $\displaystyle=\varepsilon_{A}(y_{(1)})\varphi(y_{(2)})$ $\displaystyle=\varphi(y).$ This implies that $\bar{\omega}^{F}=\varphi|_{A_{s}}:A_{s}\longrightarrow B_{s}$, and hence $\varphi(1)=\bar{\omega}^{F}(1)=1$. Next we show that $\bar{\omega}^{F}$ preserves products. Since $F=(F,\bar{\phi}^{F},\bar{\omega}^{F})$ is a comonoidal functor, the following diagram commutes for all $M\in\mathbb{M}^{A}$: $F(A_{s}\otimes_{A_{s}}M)$$F(M)$$F(A_{s})\otimes_{B_{s}}F(M)$$B_{s}\otimes_{B_{s}}F(M)$$F(l_{M}^{A})$$\bar{\omega}^{F}\otimes_{B_{s}}\text{id}$$\bar{\phi}^{F}_{A_{s},M}$$l^{B}_{M}$ In particular, the following diagram commutes: $F(A_{s}\otimes_{A_{s}}A_{s})$$F(A_{s})\otimes_{B_{s}}F(A_{s})$$F(A_{s})$$B_{s}\otimes_{B_{s}}F(A_{s})$$B_{s}$$B_{s}\otimes_{B_{s}}B_{s}$$\bar{\phi}_{A_{s},A_{s}}^{F}$$l_{F(A_{s})}^{B}$$l_{B_{s}}^{B}$$F(l_{A_{s}}^{A})$$\bar{\omega}^{F}\otimes_{B_{s}}\text{id}$$\bar{\omega}^{F}$$\text{id}\otimes_{B_{s}}\bar{\omega}^{F}$ It follows that $\bar{\omega}^{F}(y_{1}y_{2})=\bar{\omega}^{F}(y_{1})\bar{\omega}^{F}(y_{2})$ for all $y_{1},y_{2}\in A_{s}$ since $\bar{\phi}_{A_{s},A_{s}}^{F}$ is induced from the identity map $\text{id}_{A_{s}\otimes A_{s}}$. Therefore, it is shown that $\bar{\omega}^{F}=\varphi|_{A_{s}}$ is an algebra isomorphism. Next let us show that $\varphi$ is a weak bialgebra map. For this it is enough to show that $\varphi$ preserves products. For two subspaces $P,P^{\prime}$ of $A$, let $PP^{\prime}$ denote the subspace of $A$ spanned by the set $\\{\ pp^{\prime}\ |\ p\in P,\ p^{\prime}\in P^{\prime}\ \\}$. Let $\mu_{P,P^{\prime}}:P\otimes P^{\prime}\longrightarrow PP^{\prime}$ be the restriction of the product $\mu_{A}$ of $A$. Then $\mu_{P,P^{\prime}}$ induces a $\boldsymbol{k}$-linear map $\bar{\mu}_{P,P^{\prime}}:P\otimes_{A_{s}}P^{\prime}\longrightarrow PP^{\prime}$ since the equation $(p\cdot y)p^{\prime}=p(y\cdot p^{\prime})$ holds for $p\in P,\ y\in A_{s},\ p^{\prime}\in P^{\prime}$. Let $a,a^{\prime}\in A$, and let $P,P^{\prime}$ be finite-dimensional subcoalgebras of $A$ such that $a\in P$, $a^{\prime}\in P^{\prime}$. Then $PP^{\prime}$ is also a finite-dimensional subcoalgebra of $A$ containing $aa^{\prime}$, and the map $\bar{\mu}_{P,P^{\prime}}$ is a right $A$-comodule map. Let $\varphi_{P},\varphi_{P^{\prime}}$ be the coalgebra maps defined in the proof of Theorem 3.1. We will show that the following diagram commutes. (3.1) Here, $\bar{\mu}_{B}$ is the induced map from the product $\mu_{B}$ of $B$. Since $\varphi_{P},\varphi_{P^{\prime}}$ are right $B$-comodule maps by Remark 3.2, the map $\varphi_{P}\otimes_{B_{s}}\varphi_{P^{\prime}}$ is well-defined. Since $F=\text{\boldmath{$\mathsf{M}$}}^{\varphi}$ and $\hat{U}^{B}\circ\text{\boldmath{$\mathsf{M}$}}^{\varphi}={}_{\varphi_{s}^{-1}}\text{\boldmath{$\mathsf{M}$}}_{\varphi_{s}^{-1}}\circ\hat{U}^{A}$ as $\boldsymbol{k}$-linear monoidal functors, one can verify that $\bar{\phi}_{P,P^{\prime}}^{F}:F(P\otimes_{A_{s}}P^{\prime})\longrightarrow F(P)\otimes_{B_{s}}F(P^{\prime})$ is a $\boldsymbol{k}$-linear map given by $\bar{\phi}_{P,P^{\prime}}^{F}(p\otimes_{A_{s}}p^{\prime})=p\otimes_{B_{s}}p^{\prime}\qquad(p\in F(P)=P,\ p^{\prime}\in F(P^{\prime})=P^{\prime}).$ (3.2) Let $(F(P)\otimes_{B_{s}}F(P^{\prime}),\ \rho_{P,P^{\prime}}^{F})$ be the tensor product in $\text{\boldmath{$\mathsf{M}$}}^{B}$ of right comodules $(F(P),\rho_{P}^{F})$ and $(F(P^{\prime}),\rho_{P^{\prime}}^{F})$. Here, $\rho_{P,P^{\prime}}^{F}$ is the right coaction given by $\rho_{P,P^{\prime}}^{F}(m\otimes_{B_{s}}n)=(m_{(0)}\otimes_{B_{s}}n_{(0)})\otimes m_{(1)}n_{(1)}\qquad(m\in F(P),\ n\in F(P^{\prime})),$ and $\rho_{P}^{F}(m)=m_{(0)}\otimes m_{(1)},\ \rho_{P^{\prime}}^{F}(n)=n_{(0)}\otimes n_{(1)}$. Since $\bar{\phi}_{P,P^{\prime}}^{F}$ is a right $B$-comodule map from $(F(P\otimes_{A_{s}}P^{\prime}),\ \rho_{P\otimes_{A_{s}}P^{\prime}}^{F})$ to $(F(P)\otimes_{B_{s}}F(P^{\prime}),\ \rho_{P,P^{\prime}}^{F})$, the following diagram commutes. $F(P\otimes_{A_{s}}P^{\prime})$$F(P\otimes_{A_{s}}P^{\prime})\otimes B$$F(P)\otimes_{B_{s}}F(P^{\prime})$$\bigl{(}F(P)\otimes_{B_{s}}F(P^{\prime})\bigr{)}\otimes B$$\rho_{P\otimes_{A_{s}}P^{\prime}}^{F}$$\rho_{P,P^{\prime}}^{F}$$\bar{\phi}_{P,P^{\prime}}^{F}$$\bar{\phi}_{P,P^{\prime}}^{F}\otimes\text{id}_{B}$ On the other hand, there is a $(B_{s},B_{s})$-bimodule map $\chi_{P,P^{\prime}}:(P\otimes B)\otimes_{B_{s}}(P^{\prime}\otimes B)\longrightarrow(P\otimes_{B_{s}}P^{\prime})\otimes B$, which is defined by $\chi_{P,P^{\prime}}\bigl{(}(p\otimes b)\otimes_{B_{s}}(p^{\prime}\otimes b^{\prime})\bigr{)}=(p\otimes_{B_{s}}p^{\prime})\otimes bb^{\prime}\qquad(p\in P,\ p^{\prime}\in P^{\prime},\ b,b^{\prime}\in B).$ We also define a $\boldsymbol{k}$-linear map $\varepsilon_{P,P^{\prime}}:P\otimes_{A_{s}}P^{\prime}\longrightarrow\boldsymbol{k}$ by $\varepsilon_{P,P^{\prime}}(p\otimes_{A_{s}}p^{\prime})=\varepsilon(p1_{(1)})\varepsilon(1_{(2)}p^{\prime})\qquad(p\in P,\ p^{\prime}\in P^{\prime}).$ Then it can be shown that the following diagram commutes. $F(P\otimes_{A_{s}}P^{\prime})$$F(P)\otimes_{B_{s}}F(P^{\prime})$$B\otimes_{B_{s}}B$$F(P)\otimes_{B_{s}}F(P^{\prime})\otimes B$$(F(P)\otimes B)\otimes_{B_{s}}(F(P^{\prime})\otimes B)$$F(P\otimes_{A_{s}}P^{\prime})\otimes B$$\boldsymbol{k}\otimes B$$F(PP^{\prime})\otimes B$$F(PP^{\prime})$$B$($\ast$)$\bar{\phi}_{P,P^{\prime}}^{F}$$\varphi_{P}\otimes_{B_{s}}\varphi_{P^{\prime}}$$\rho_{P\otimes_{A_{s}}P^{\prime}}^{F}$$\rho_{P,P^{\prime}}^{F}$$\bar{\phi}_{P,P^{\prime}}^{F}\otimes\text{id}$$\chi_{P,P^{\prime}}$$\rho_{P}^{F}\otimes_{B_{s}}\rho_{P^{\prime}}^{F}$$(F(\varepsilon_{P})\otimes\text{id})_{B_{s}}(F(\varepsilon_{P^{\prime}})\otimes\text{id})$$F(\varepsilon_{P,P^{\prime}})\otimes\text{id}$$\cong$$F(\bar{\mu}_{P,P^{\prime}})\otimes\text{id}$$F(\varepsilon_{PP^{\prime}})\otimes\text{id}$$\rho_{PP^{\prime}}^{F}$$F(\bar{\mu}_{P,P^{\prime}})$$\varphi_{PP^{\prime}}$$\bar{\mu}_{B}$ In fact, the commutativity of ($\ast$) in the above diagram comes from the following three facts: 1. (i) $\bar{\phi}^{F}_{M,N}:F(M\otimes_{A_{s}}N)\longrightarrow F(M)\otimes_{B_{s}}F(N)$ is an isomorphism, 2. (ii) the equation (3.2) holds, 3. (iii) for $p\in P,\ p^{\prime}\in P^{\prime}$, 1. $\bullet$ $p\otimes_{A_{s}}p^{\prime}=(p\cdot 1)\otimes_{A_{s}}p^{\prime}=\bigl{(}p\cdot 1_{(1)}\varepsilon_{s}(1_{(2)})\bigr{)}\otimes_{A_{s}}p^{\prime}=p1_{(1)}\otimes_{A_{s}}\varepsilon_{s}(1_{(2)})p^{\prime}$, 2. $\bullet$ $\varepsilon\bigl{(}\varepsilon_{s}(1_{(2)})p^{\prime}\bigr{)}=\varepsilon(1_{(2)}p^{\prime})$ by Lemma 2.2(2). Thus, the diagram (3.1) commutes, and therefore $\varphi$ preserves products. To complete the proof we need to verify that $F=\mathbb{M}^{\varphi}$ as $\boldsymbol{k}$-linear comonoidal functors. This is an easy consequence of the proof of Lemma 2.7(2) since $\bar{\phi}^{F}_{M,N}$ is induced from the identity map $\text{id}_{M\otimes N}$ for all $M,N\in\mathbb{M}^{A}$. ∎ By Corollary 3.3 and Theorem 3.5 we have: ###### Corollary 3.6. Let $A,B$ be weak bialgebras over $\boldsymbol{k}$, and $F=(F,\bar{\phi}^{F},\bar{\omega}^{F}):\mathbb{M}^{A}\longrightarrow\mathbb{M}^{B}$ be a strong $\boldsymbol{k}$-linear comonoidal functor satisfying $U^{B}\circ F=U^{A}$ as $\boldsymbol{k}$-linear monoidal categories. If $F$ is a $\boldsymbol{k}$-linear monoidal equivalence, then the weak bialgebra map $\varphi:A\longrightarrow B$ determined in Theorem 3.5 is an isomorphism of weak bialgebras. ## 4 Categorical aspects of indecomposable weak bialgebras Let $A=(A,\Delta_{A},\varepsilon_{A})$ and $B=(B,\Delta_{B},\varepsilon_{B})$ be two weak bialgebras over $\boldsymbol{k}$, and set $H=A\oplus B$. Then, $H$ has a weak bialgebra structure such that the algebra structure is the product and the coalgebra structure is the direct sum of $A$ and $B$. The target and source counital maps $\varepsilon_{t}$ and $\varepsilon_{s}$ are given by $\displaystyle\varepsilon_{t}(x)$ $\displaystyle=(\varepsilon_{A})_{t}(a)+(\varepsilon_{B})_{t}(b),$ $\displaystyle\varepsilon_{s}(x)$ $\displaystyle=(\varepsilon_{A})_{s}(a)+(\varepsilon_{B})_{s}(b)$ for all $x=a+b\in H,\ a\in A,\ b\in B$, where $(\varepsilon_{A})_{t},(\varepsilon_{A})_{s}$ are the target and source counital maps of $A$, and $(\varepsilon_{B})_{t},(\varepsilon_{B})_{s}$ are that of $B$, respectively. Moreover, the target and source subalgebras are given as follows: $\displaystyle H_{t}$ $\displaystyle=\varepsilon_{t}(H)=(\varepsilon_{A})_{t}(A)+(\varepsilon_{B})_{t}(B)=A_{t}+B_{t},$ $\displaystyle H_{s}$ $\displaystyle=\varepsilon_{s}(H)=(\varepsilon_{A})_{s}(A)+(\varepsilon_{B})_{s}(B)=A_{s}+B_{s}.$ We call the above weak bialgebra $H$ the direct sum of $A$ and $B$. A weak bialgebra $H$ is called indecomposable if there are no weak bialgebras $A$ and $B$ such that $H\cong A\oplus B$. Any finite-dimensional weak bialgebra can be decomposed into the finitely many direct sum of indecomposable ones. More precisely, we have: ###### Theorem 4.1. Let $H$ be a finite-dimensional weak bialgebra over $\boldsymbol{k}$. Then 1. $(1)$ there are finitely many indecomposable weak bialgebras $H_{i}\ (i=1,\ldots,k)$ such that $H=H_{1}\oplus\cdots\oplus H_{k}$. 2. $(2)$ If indecomposable weak bialgebras $H_{1},\ldots,H_{k}$ and $H_{1}^{\prime},\ldots,H_{l}^{\prime}$ satisfy $H_{1}\oplus\cdots\oplus H_{k}=H=H_{1}^{\prime}\oplus\cdots\oplus H_{l}^{\prime},$ then $k=l$, and there is a permutation $\sigma\in\mathfrak{S}_{k}$ such that $H_{i}^{\prime}=H_{\sigma(i)}$ for all $i=1,\ldots,k$. The above theorem can be proved by decomposability and uniqueness of finite- dimensional algebras into indecomposable ones (see [21] for detail). A $\boldsymbol{k}$-linear monoidal category $\mathcal{C}$ is called indecomposable if there are no $\boldsymbol{k}$-linear monoidal categories $\mathcal{C}_{1},\mathcal{C}_{2}$ such that $\mathcal{C}\simeq\mathcal{C}_{1}\times\mathcal{C}_{2}$ as $\boldsymbol{k}$-linear monoidal categories. ###### Proposition 4.2. Let $H$ be a weak bialgebra over $\boldsymbol{k}$. If $H$ is decomposable, then 1. $(1)$ the left $H$-module category ${}_{H}\text{\boldmath{$\mathsf{M}$}}$ and the finite-dimensional left $H$-module category ${}_{H}\mathbb{M}$ are decomposable. 2. $(2)$ the right $H$-comodule category $\text{\boldmath{$\mathsf{M}$}}^{H}$ and the finite-dimensional right $H$-comodule category $\mathbb{M}^{H}$ are decomposable. ###### Proof.. Suppose that $H=A\oplus B$ for some weak bialgebras $A,B$ over $\boldsymbol{k}$. (1) The left $H$-module category ${}_{H}\text{\boldmath{$\mathsf{M}$}}$ is equivalent to the Cartesian product ${}_{A}\text{\boldmath{$\mathsf{M}$}}\times{}_{B}\text{\boldmath{$\mathsf{M}$}}$ as $\boldsymbol{k}$-linear monoidal categories. An equivalence is given by the following monoidal functors, which are quasi-inverse each other: $\displaystyle(F,\phi^{F},\omega^{F}):$ $\displaystyle{}_{H}\text{\boldmath{$\mathsf{M}$}}\longrightarrow{}_{A}\text{\boldmath{$\mathsf{M}$}}\times{}_{B}\text{\boldmath{$\mathsf{M}$}},$ $\displaystyle F(X)$ $\displaystyle=(1_{A}\cdot X,\ 1_{B}\cdot X),$ $\displaystyle\phi^{F}_{X,Y}$ $\displaystyle=\mathrm{id}_{F(X)\circledast F(Y)}:F(X)\circledast F(Y)\longrightarrow F(X\circledast Y),$ $\displaystyle\omega^{F}$ $\displaystyle=\mathrm{id}_{(A_{t},B_{t})}:(A_{t},B_{t})\longrightarrow F(H_{t}),$ $\displaystyle(G,\phi^{G},\omega^{G}):$ $\displaystyle{}_{A}\text{\boldmath{$\mathsf{M}$}}\times{}_{B}\text{\boldmath{$\mathsf{M}$}}\longrightarrow{}_{H}\text{\boldmath{$\mathsf{M}$}},$ $\displaystyle G(U,V)$ $\displaystyle=U\times V,$ $\displaystyle\phi^{G}_{(U_{1},V_{1}),(U_{2},V_{2})}$ $\displaystyle=\mathrm{id}_{G(U_{1},V_{1})\circledast G(U_{2},V_{2})}:G(U_{1},V_{1})\circledast G(U_{2},V_{2})\longrightarrow G\bigl{(}(U_{1},V_{1})\circledast(U_{2},V_{2})\bigr{)},$ $\displaystyle\omega^{G}$ $\displaystyle:H_{t}\longrightarrow G(A_{t},B_{t})=A_{t}\times B_{t},\ \ \omega^{G}(z)=(1_{A}\cdot z,\ 1_{B}\cdot z)\qquad(z\in H_{t}).$ By restricting the above equivalence ${}_{H}\text{\boldmath{$\mathsf{M}$}}\simeq{}_{A}\text{\boldmath{$\mathsf{M}$}}\times{}_{B}\text{\boldmath{$\mathsf{M}$}}$ to finite dimension a $\boldsymbol{k}$-linear monoidal equivalence ${}_{H}\mathbb{M}\simeq{}_{A}\mathbb{M}\times{}_{B}\mathbb{M}$ is obtained. (2) The right $H$-comodule category $\text{\boldmath{$\mathsf{M}$}}^{H}$ is equivalent to the Cartesian product $\text{\boldmath{$\mathsf{M}$}}^{A}\times\text{\boldmath{$\mathsf{M}$}}^{B}$ as $\boldsymbol{k}$-linear monoidal categories. An equivalence is given by the following monoidal functors, which are quasi-inverse each other: $\displaystyle(F,\phi^{F},\omega^{F}):$ $\displaystyle\text{\boldmath{$\mathsf{M}$}}^{H}\longrightarrow\text{\boldmath{$\mathsf{M}$}}^{A}\times\text{\boldmath{$\mathsf{M}$}}^{B},$ $\displaystyle F\bigl{(}(X,\rho)\bigr{)}$ $\displaystyle=\Bigl{(}\bigl{(}(\varepsilon_{A}\circ\pi_{A})\cdot X,\ (\mathrm{id}\otimes\pi_{A})\circ\rho|_{(\varepsilon_{A}\circ\pi_{A})\cdot X}\bigr{)},\bigl{(}(\varepsilon_{B}\circ\pi_{B})\cdot X,\ (\mathrm{id}\otimes\pi_{B})\circ\rho|_{(\varepsilon_{B}\circ\pi_{B})\cdot X}\bigr{)}\Bigr{)}$ $\displaystyle\phi^{F}_{X,Y}$ $\displaystyle=\mathrm{id}_{F(X)\circledast F(Y)}:F(X)\circledast F(Y)\longrightarrow F(X\circledast Y),$ $\displaystyle\omega^{F}$ $\displaystyle=\mathrm{id}_{(A_{s},B_{s})}:(A_{s},B_{s})\longrightarrow F(H_{s}),$ $\displaystyle(G,\phi^{G},\omega^{G}):$ $\displaystyle\text{\boldmath{$\mathsf{M}$}}^{A}\times\text{\boldmath{$\mathsf{M}$}}^{B}\longrightarrow\text{\boldmath{$\mathsf{M}$}}^{H},$ $\displaystyle G(U,V)$ $\displaystyle=U\times V,$ $\displaystyle\phi^{G}_{(U_{1},V_{1}),(U_{2},V_{2})}$ $\displaystyle=\mathrm{id}_{G(U_{1},V_{1})\circledast G(U_{2},V_{2})}:G(U_{1},V_{1})\circledast G(U_{2},V_{2})\longrightarrow G\bigl{(}(U_{1},V_{1})\circledast(U_{2},V_{2})\bigr{)},$ $\displaystyle\omega^{G}$ $\displaystyle:H_{s}\longrightarrow G(A_{s},B_{s})=A_{s}\times B_{s},\ \ \omega^{G}(y)=\bigl{(}\pi_{A}(y),\ \pi_{B}(y)\bigr{)}\qquad(y\in H_{s}),$ where $\pi_{A}$ and $\pi_{B}$ are natural surjections associated with the direct sum decomposition $H=A\oplus B$. By restricting the above equivalence $\text{\boldmath{$\mathsf{M}$}}^{H}\simeq\text{\boldmath{$\mathsf{M}$}}^{A}\times\text{\boldmath{$\mathsf{M}$}}^{B}$ to finite dimension a $\boldsymbol{k}$-linear monoidal equivalence $\mathbb{M}^{H}\simeq\mathbb{M}^{A}\times\mathbb{M}^{B}$ is obtained. ∎ The converse of the above proposition is true. To prove it we need the following reconstruction theorem of bialgebras. ###### Theorem 4.3 (Ulbrich [19], Schauenburg [17, Theorem 5.4]). Let $\mathcal{C}$ be a $\boldsymbol{k}$-linear monoidal category, and $\omega:\mathcal{C}\longrightarrow\text{Vect}_{\boldsymbol{k}}^{\text{f.d.}}$ be a faithful and exact $\boldsymbol{k}$-linear monoidal functor. Then, there are a bialgebra $B$ and a monoidal equivalence $F:\mathcal{C}\longrightarrow\mathbb{M}^{B}$ such that $U^{B}\circ F=\omega$, where $U^{B}:\mathbb{M}^{B}\longrightarrow\text{Vect}_{\boldsymbol{k}}^{\text{f.d.}}$ is the forgetful functor. ∎ Combining Theorems 3.5 and 4.3 one can show the following. ###### Theorem 4.4. Let $H$ be a finite-dimensional weak bialgebra over $\boldsymbol{k}$. Then $H$ is indecomposable as a weak bialgebra if and only if the finite-dimensional left $H$-module category ${}_{H}\mathbb{M}$ is indecomposable as a $\boldsymbol{k}$-linear monoidal category. ###### Proof.. By the contraposition of Proposition 4.2 “if part” holds. We will show that “only if part”. Suppose that $H$ is indecomposable whereas ${}_{H}\mathbb{M}$ is not. Then, there are $\boldsymbol{k}$-linear monoidal categories $\mathcal{C}_{1},\mathcal{C}_{2}$ such that ${}_{H}\mathbb{M}\simeq\mathcal{C}_{1}\times\mathcal{C}_{2}$ as $\boldsymbol{k}$-linear monoidal categories. Let ${}_{H}U:{}_{H}\mathbb{M}\longrightarrow\text{Vect}_{\boldsymbol{k}}^{\text{f.d.}}$ be the forgetful functor, and $F:\mathcal{C}_{1}\times\mathcal{C}_{2}\longrightarrow{}_{H}\mathbb{M}$ be a $\boldsymbol{k}$-linear monoidal category equivalence. Since the two $\boldsymbol{k}$-linear monoidal functors $\displaystyle\omega_{1}:$ $\displaystyle\ \mathcal{C}_{1}\cong\mathcal{C}_{1}\times 0\xrightarrow{\ \ F\ \ }{}_{H}\mathbb{M}\xrightarrow{\ \ {}_{H}U\ \ }\text{Vect}_{\boldsymbol{k}}^{\text{f.d.}},$ $\displaystyle\omega_{2}:$ $\displaystyle\ \mathcal{C}_{2}\cong 0\times\mathcal{C}_{2}\xrightarrow{\ \ F\ \ }{}_{H}\mathbb{M}\xrightarrow{\ \ {}_{H}U\ \ }\text{Vect}_{\boldsymbol{k}}^{\text{f.d.}}$ are faithful and exact, there are bialgebras $A,B$ and $\boldsymbol{k}$-linear monoidal equivalences $G_{1}:\mathcal{C}_{1}\simeq\mathbb{M}^{A},\ G_{2}:\mathcal{C}_{2}\simeq\mathbb{M}^{B}$ such that $U^{A}\circ G_{1}=\omega_{1},\ U^{B}\circ G_{2}=\omega_{2}$ by Theorem 4.3. Then $\mathcal{C}_{1}\times\mathcal{C}_{2}\simeq\mathbb{M}^{A}\times\mathbb{M}^{B}\cong\mathbb{M}^{A\oplus B}$ as $\boldsymbol{k}$-linear monoidal categories. Thus we have $\boldsymbol{k}$-linear monoidal equivalence $G:\mathbb{M}^{H^{\ast}}\longrightarrow\mathbb{M}^{A\oplus B}$. This equivalence satisfies $U^{A\oplus B}\circ G=U^{H^{\ast}}$. So, by Corollary 3.6, there is a weak bialgebra isomorphism $g:A\oplus B\longrightarrow H^{\ast}$. Therefore, $H\cong H^{\ast\ast}\cong(A\oplus B)^{\ast}\cong A^{\ast}\oplus B^{\ast}$. This contradicts the fact that $H$ is indecomposable as a weak bialgebra. ∎ ## References * [1] G. Böhm, F. Nill and K. Szlachányi, Weak Hopf algebras I. Integral theory and $C^{\ast}$-structure, J. Algebra 221 (1999), 385–438. * [2] A. Bruguieres and A. Virelizier, Hopf monads, Adv. Math. 215 (2007), 679–733. * [3] D. Chikhladze, The Tannaka representation theorem for separable Frobenius functors, Algebr. Represent. Theor. 15 (2012), 1205–1213. * [4] P. Deligne, Catégories Tannakiènnes, In: The Grothendieck Festschrift vol.2, Progr. Math. 87, edited by P.Cartier, L.Illusie, N.M.Katz, G.Laumon, Y.Manin and K.A.Ribet, pp.111–195, Birkhäuiser, Boston, 1991. * [5] P. Etingof, D. Nikshych and V. Ostrik, On fusion categories, Ann. of Math. 162, (2005), 581–642. * [6] I.L. Franco, Topics in category theory: Hopf algebras, a lecture note, noted by D. Mehrle, at Cambridge University, 2015. http://pi.math.cornell.edu/$\sim$dmehrle/notes/partiii/hopfalg partiii notes.pdf * [7] R. Häring-Oldenburg, Reconstruction of weak quasi-Hopf algebras, J. Algebra 194, (1997), 14–35. * [8] S. MacLane, Categories for the working mathematician, Grad.Texts Math. 5, Springer, New York, 1971, Second edition, Springer-Verlag, New York, 1998. * [9] S. Maijd, Tannaka-Krein theorem for quasi-Hopf algebras and other results, in “Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990)”, Contemp. Math. 134, 219–232, Amer. Math. Soc., Providence, RI, 1992. * [10] S. Maijd, Foundations of quantum group theory, Cambridge University Press, 1995. * [11] M. B. McCrudy Graphical methods for Tannaka duality of weak bialgebras and weak Hopf algebras, Theory and Appl. Cat. 26 (2012), 233–280. * [12] S. Montgomery, Hopf algebras and their action on rings, C.B.M.S.82, American Mathematical Society, 1993. * [13] D. Nikshych, V. Turaev and L. Vainerman, Invariants of knots and $3$-manifolds from finite quantum groupoids, Top. Appl. 127 (2003), 91–123. * [14] F. Nill, Axioms for weak bialgebras, arXiv:math. 9805104v1, 1998. * [15] P. Schauenburg, Tannaka duality for arbitrary Hopf algebras, Algebra Berichte 66, Verlag Reinhard Fisher, Munich, 1992. * [16] P. Schauenburg, Weak Hopf algebras and quantum groupoids, Banach Center Publ. 61 (2003), 171–188. * [17] P. Schauenburg, Hopf bigalois extensions, Comm. Algebra 24 (1996), 3797–3825. * [18] K. Szlachányi, Adjointable monoidal functors and quantum groupoids, In: “Hopf algebras in noncommutative geometry and physics”, Lecture Notes in Pure and Appl. Math. 239, 291–307, Dekker, New York, 2005. * [19] K.-H. Ulbrich, On Hopf algebras and rigid monoidal categories, Israel J. Math. 72 (1990), 252–256. * [20] J. Vercruysse, Hopf algebras–Variant notations and reconstruction theorem, arXiv:1202.3613v1, 2012. * [21] M. Wakui, Indecomposability of weak Hopf algebras, submitted, preprint, 2020.

2024-09-04T02:54:55.396490

2002.12569

arxiv-papers

# Solutions of nonhomogeneous equations involving Hardy potentials with singularities on the boundary Huyuan Chen Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China<EMAIL_ADDRESS>, Alexander Quaas Departamento de Matemática, Universidad Técnica Federico, Santa María Casilla V-110, Avda. España 1680, Valparaíso, Chile<EMAIL_ADDRESS>and Feng Zhou Center for PDEs, School of Mathematical Sciences, East China Normal University, Shanghai Key Laboratory of PMMP, Shanghai, 200241, PR China <EMAIL_ADDRESS> ###### Abstract. In this paper, we present a new distributional identity for the solutions of elliptic equations involving Hardy potentials with singularities located on the boundary of the domain. Then we use it to obtain the boundary isolated singular solutions of nonhomogeneous problems. ###### Key words and phrases: Distributional identity, Hardy potential, Boundary isolated singularity. ###### 2010 Mathematics Subject Classification: 35B40, 35J99. ## 1\. Introduction The classical Hardy inequality is stated as following: For any smooth bounded domain $\mathcal{O}$ in $\mathbb{R}^{N}$ containing the origin, there holds (1.1) $\int_{\mathcal{O}}|\nabla u|^{2}dx\geq c_{N}\int_{\mathcal{O}}|x|^{-2}|u|^{2}dx,\quad\forall\,u\in H_{0}^{1}(\mathcal{O}),$ with the best constant $c_{N}=\frac{(N-2)^{2}}{4}$. The qualitative properties of Hardy inequality and its improved versions have been studied extensively, see for example [1, 4, 19, 21], motivated by great applications in the study of stability of solutions to semilinear elliptic and parabolic equations (cf. [5, 6, 13, 30, 31]). The isolated singular solutions of Hardy problem with absorption nonlinearity have been studied in [11, 12, 23] and the one with source nonlinearity has been done in [3, 16]. The related semilinear elliptic problem involving the inverse square potential has been studied by variational methods in [15, 14, 18] and the references therein. In a very recent work [9], we established a new distributional identity with respect to a specific weighted measure and we then classify the classical isolated singular solutions of $-\Delta u+\frac{\mu}{|x|^{2}}u=f\quad{\rm in}\quad\mathcal{O}\setminus\\{0\\},$ subject to the homogeneous Dirichlet boundary condition with $\mu\geq-c_{N}$. These results allow us to draw a complete picture of the existence, non- existence and the singularities for classical solutions for the above problems (cf. [10]). It is of interest to consider the corresponding problem involving Hardy potential with singularity on the boundary. While the sharp constant $c_{N}$ in Hardy inequality (1.1) could be replaced by $\frac{N^{2}}{4}$ when the origin is addressed on the boundary of the domain, see [20, Corollary 2.4], also [7, 8, 17]. Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ with $0\in\partial\Omega$. We study boundary isolated singular solutions of nonhomogeneous problems: (1.2) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}u=f\quad{\rm in}\;\;\Omega,\\\\[4.2679pt] \phantom{L_{\beta}}\displaystyle u=g\quad{\rm on}\;\;\partial{\Omega}\setminus\\{0\\},\end{array}\right.$ where $f\in C^{\gamma}_{loc}(\bar{\Omega}\setminus\\{0\\})$ with $\gamma\in(0,1)$, $g\in C(\partial\Omega\setminus\\{0\\})$ and $\mathcal{L}_{\beta}:=-\Delta+\frac{\beta}{|x|^{2}}$ is the Hardy operator which is singular at $0$ (with $N\geq 2$, $\beta\geq\beta_{0}:=-\frac{N^{2}}{4}$). Recall that for $\beta\geq\beta_{0}$, the problem (1.3) $\left\\{\begin{array}[]{lll}\mathcal{L}_{\beta}u=0\quad{\rm in}\;\;\mathbb{R}^{N}_{+},\\\\[2.84526pt] \phantom{\mathcal{L}_{\beta}}u=0\quad{\rm on}\;\;\partial\mathbb{R}^{N}_{+}\setminus\\{0\\}\end{array}\right.$ has two special solutions with the explicit formulas as (1.4) $\Lambda_{\beta}(x)=\left\\{\begin{array}[]{lll}x_{N}|x|^{\tau_{-}(\beta)}&{\rm if}\;\;\beta>\beta_{0},\\\\[2.84526pt] \phantom{}-x_{N}|x|^{\tau_{-}(\beta)}\ln|x|&{\rm if}\;\;\beta=\beta_{0}\end{array}\right.\quad\;\;{\rm and}\quad\lambda_{\beta}(x)=x_{N}|x|^{\tau_{+}(\beta)},$ where $x=(x^{\prime},x_{N})\in\mathbb{R}^{N}_{+}:=\mathbb{R}^{N-1}\times(0,+\infty)$, and (1.5) $\tau_{-}(\beta)=-\frac{N}{2}-\sqrt{\beta-\beta_{0}}\quad{\rm and}\quad\tau_{+}(\beta)=-\frac{N}{2}+\sqrt{\beta-\beta_{0}},$ are two roots of $\beta-\tau(\tau+N)=0$. As in [10, 9], we first find a certain distributional identity which shows that the singularity of solution $\Lambda_{\beta}$ for (1.3) is associated to a Dirac mass. Let $C^{1.1}_{0}(\mathbb{R}^{N}_{+})$ be the set of functions in $C^{1.1}(\overline{\mathbb{R}^{N}_{+}})$ vanishing on the boundary and having compact support in $\overline{\mathbb{R}^{N}_{+}}$. Then we have ###### Theorem 1.1. Let $d\gamma_{\beta}:=\lambda_{\beta}(x)dx$ and (1.6) $\mathcal{L}^{*}_{\beta}:=-\Delta-\frac{2\tau_{+}(\beta)}{|x|^{2}}\,x\cdot\nabla-\frac{2}{x_{N}}\frac{\partial}{\partial x_{N}},\quad x=(x^{\prime},x_{N})\in\mathbb{R}^{N}_{+}.$ Then there holds (1.7) $\int_{\mathbb{R}^{N}_{+}}\Lambda_{\beta}\mathcal{L}^{*}_{\beta}(\frac{\zeta}{x_{N}})\,d\gamma_{\beta}=c_{\beta}\frac{\partial\zeta}{\partial x_{N}}(0),\quad\forall\,\zeta\in C^{1.1}_{0}(\mathbb{R}^{N}_{+}),$ where (1.8) $c_{\beta}=\left\\{\begin{array}[]{lll}\sqrt{\beta-\beta_{0}}\,|\mathcal{S}^{N-1}|/N&{\rm if}\;\;\beta>\beta_{0},\\\\[4.2679pt] \phantom{}|\mathcal{S}^{N-1}|/N&{\rm if}\;\;\beta=\beta_{0},\end{array}\right.$ and $\mathcal{S}^{N-1}$ is the unit sphere of $\mathbb{R}^{N}$ and $|\mathcal{S}^{N-1}|$ denotes its $(N-1)$-dimensional Hausdorff measure. From the distributional identity (1.7), $\Lambda_{\beta}$ is called as a fundamental solution of (1.3). We remark that when $\beta=0$, $\mathcal{L}^{*}_{0}=-\Delta-\frac{2}{x_{N}}\frac{\partial}{\partial x_{N}}$, $\lambda_{\beta}(x)=x_{N}$ and (1.7) could be reduced to $\displaystyle c_{0}\frac{\partial\zeta}{\partial x_{N}}(0)=\int_{\mathbb{R}^{N}_{+}}\Lambda_{0}\mathcal{L}^{*}_{0}(\frac{\zeta}{x_{N}})\,d\gamma_{\beta}=\int_{\mathbb{R}^{N}_{+}}\Lambda_{0}(-\Delta\zeta)\,dx,\quad\forall\;\zeta\in C^{1.1}_{0}(\mathbb{R}^{N}_{+}),$ which coincides with the classical distributional identity proposed in [22]. On this classical subject, it has been vastly expanded in the works [2, 26, 27, 28, 29]. For simplicity, here and in the sequel, we always assume that $\Omega$ is a bounded $C^{2}-$ domain satisfying that (1.9) $B_{r_{0}}^{+}(0)\subset\Omega\subset B_{R_{0}}^{+}(0),$ for some $0<r_{0}<R_{0}<+\infty$ where $B_{r}^{+}(0):=B_{r}(0)\cap\mathbb{R}^{N}_{+}$. Let $d\omega_{\beta}(x):=|x|^{\tau_{+}(\beta)}d\omega(x)$, where $\omega$ is the Hausdorff measure of $\partial\Omega$. We can state our main result as follows ###### Theorem 1.2. Let $\mathcal{L}^{*}_{\beta}$ be given by (1.6), $f\in C^{\theta}_{loc}(\bar{\Omega}\setminus\\{0\\})$ with $\theta\in(0,1)$, $g\in C(\partial\Omega\setminus\\{0\\})$. $(i)$ If (1.10) $\int_{\Omega}|f|\,d\gamma_{\beta}+\int_{\partial\Omega}|g|\,d\omega_{\beta}<+\infty,$ then for any $k\in\mathbb{R}$, problem (1.2) admits a unique solution $u_{k}\in C^{2}(\Omega)\cap L^{1}(\Omega,|x|^{-1}d\gamma_{\beta})$ such that (1.11) $\int_{\Omega}u_{k}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=\int_{\Omega}\frac{f\xi}{x_{N}}\,d\gamma_{\beta}-\int_{\partial\Omega}g\frac{\partial\xi}{\partial\nu}d\omega_{\beta}+c_{\beta}k\frac{\partial\xi}{\partial x_{N}}(0),\quad\forall\,\xi\in C^{1.1}_{0}(\Omega),$ where $\nu$ is the unit outward vector on $\partial\Omega$. $(ii)$ If $f,g$ are nonnegative and (1.12) $\lim_{r\to 0^{+}}\Big{(}\int_{\Omega\setminus B_{r}(0)}f\,d\gamma_{\beta}+\int_{\partial\Omega\setminus B_{r}(0)}g\,d\omega_{\beta}\Big{)}=+\infty,$ then problem (1.2) has no nonnegative solution. When $g=0$ on $\partial\Omega$ and $f=0$ in $\Omega$, we prove in Proposition 3.2 in Section 3 that problem (1.2) admits an isolated singular solution $\Lambda^{\Omega}_{\beta}$, which has the asymptotic behavior at the origin as the fundamental function $\Lambda_{\beta}$. More precisely, we have (1.13) $\lim_{t\to 0^{+}}\sup_{z\in S^{N-1}_{+}}\Big{(}\frac{\Lambda^{\Omega}_{\beta}(tz)}{\Lambda_{\beta}(tz)}-1\Big{)}=0.$ When $g=0$ on $\partial\Omega$ and $f\in C^{\theta}_{loc}(\bar{\Omega}\setminus\\{0\\})\cap L^{1}(\Omega,d\gamma_{\beta})$, Theorem 4.1 in Section 4 shows that problem (1.2) has a solution $u_{f}$ verifying the isolated singularity (see Remark 4.2) (1.14) $\lim_{t\to 0^{+}}\inf_{z\in S^{N-1}_{+}}\frac{u_{f}(tz)}{\Lambda_{\beta}(tz)}=0,$ which is less precise than (1.13) due to the lack of estimates of Green kernel of Hardy operator with singularity on the boundary. However, when $f=0$ and $g\not=0$, it is not convenient to use (1.14) to describe the singularity of the solution $u_{g}$, so we may distinguish this by the distributional identity $\int_{\Omega}u_{g}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=-\int_{\partial\Omega}g\frac{\partial\xi}{\partial\nu}d\omega_{\beta},\quad\forall\,\xi\in C^{1.1}_{0}(\Omega),$ All in all, the solution $u_{k}$ of (1.2) can be decomposed into three components $k\Lambda^{\Omega}_{\beta}$, $u_{f}$ and $u_{g}$. The method we use to prove the existence of solutions for problem (1.2) is different from the classical method of the boundary data problem used by Gmira-Véron in [22] due to the appearance of Hardy potential. They obtained the very weak solutions by approximating the Dirac mass at boundary. Then they considered the limit of the solutions to the corresponding problem where the convergence is guaranteed by the Poisson kernel. In this paper, we prove the existence of moderate singular solution by using the function $\Lambda_{\beta}$ to construct suitable solutions of problem (1.2) with the zero Dirichlet boundary condition. While for nonzero Dirichlet boundary condition, we transform the boundary data into nonhomogeneous term. However, for $\beta>0$, that transformation can not totally solve (1.2) with the nonzero Dirichlet boundary condition, and our idea is to cut off the boundary data and approximate the solutions. The rest of the paper is organized as follows. In Section 2, we start from a comparison principle for $\mathcal{L}_{\beta}$ and show the moderate singular solution of (1.2) when $g=0$. Section 3 is devoted to prove the distributional identity (1.7) for the fundamental solution $\Lambda_{\beta}$ in $\mathbb{R}^{N}_{+}$, to consider its trace, the corresponding distributional identity in bounded smooth domain. Section 4 is to study the qualitative properties of the solutions for problem (1.2) when $g=0$ and then we give the proof of Theorem 1.2 in the case of nonzero boundary data in Section 5. In what follows, we denote by $c_{i}$ a generic positive constant in the proofs of the results. ## 2\. Preliminary ### 2.1. Comparison principle We start the analysis from a comparison principle for $\mathcal{L}_{\beta}$. Let $\eta_{0}:[0,+\infty)\to[0,\,1]$ be a decreasing $C^{\infty}$ function such that (2.1) $\eta_{0}=1\quad{\rm in}\;\;[0,1]\quad{\rm and}\quad\eta_{0}=0\quad{\rm in}\;\;[2,+\infty).$ ###### Lemma 2.1. Let $\Omega$ be a bounded open set in $\mathbb{R}^{N}_{+}$, $L:\Omega\times[0,+\infty)\to[0,+\infty)$ be a continuous function satisfying that for any $x\in\Omega$, $L(x,s_{1})\geq L(x,s_{2})\quad{\rm if}\;\;s_{1}\geq s_{2},$ then $\mathcal{L}_{\beta}+L$ with $\beta\geq\beta_{0}$ verifies the comparison principle, that is, if $u,\,v\in C^{1,1}(\Omega)\cap C(\bar{\Omega})$ verify that $\mathcal{L}_{\beta}u+L(x,u)\geq\mathcal{L}_{\beta}v+L(x,v)\quad{\rm in}\;\;\Omega\quad{\rm and}\quad u\geq v\quad{\rm on}\;\;\partial\Omega,$ then $u\geq v\;{\rm in}\;\Omega.$ Proof. Let $w=u-v$ and then $w\geq 0$ on $\partial\Omega$. Denote $w_{-}=\min\\{w,0\\}$, and we claim that $w_{-}\equiv 0$. Indeed, if $\Omega_{-}:=\\{x\in\Omega:\,w(x)<0\\}$ is not empty, then it is a bounded $C^{1,1}$ domain in $\Omega$ and $w_{-}=0$ on $\partial\Omega$. We observe that $\Omega_{-}\subset\mathbb{R}^{N}_{+}$ and then from Hardy inequality [7, (1.7)] (see also [25]), it holds that $\displaystyle 0$ $\displaystyle=$ $\displaystyle\int_{\Omega_{-}}(-\Delta w_{-}+\frac{\beta}{|x|^{2}}w_{-})w_{-}dx+\int_{\Omega_{-}}[L(x,u)-L(x,v)]w_{-}\,dx$ $\displaystyle\geq$ $\displaystyle\int_{\Omega_{-}}\left(|\nabla w_{-}|^{2}+\frac{\beta}{|x|^{2}}w_{-}^{2}\right)dx\geq c_{1}\int_{\Omega_{-}}w_{-}^{2}dx,$ then $w_{-}=0$ in $\Omega_{-}$, by the continuity of $w_{-}$, which is impossible with the definition of $\Omega_{-}$. $\Box$ ###### Lemma 2.2. Assume that $\beta\geq\beta_{0}$, $f_{1}$, $f_{2}$ are two functions in $C^{\theta}_{loc}(\Omega)$ with $\theta\in(0,1)$, $g_{1}$, $g_{2}$ are two continuous functions on $\partial\Omega\setminus\\{0\\}$, and $f_{1}\geq f_{2}\quad{\rm in}\;\;\Omega\quad{\rm and}\quad g_{1}\geq g_{2}\quad{\rm on}\;\;\partial\Omega\setminus\\{0\\}.$ Let $u_{i}$ ($i=1,2$) be the classical solutions of $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}u=f_{i}\quad{\rm in}\;\;{\Omega},\\\\[4.2679pt] \phantom{L_{\beta}}\displaystyle u=g_{i}\quad{\rm on}\;\;\partial{\Omega}\setminus\\{0\\}.\end{array}\right.$ If (2.2) $\lim_{r\to 0^{+}}\inf_{x\in\partial_{+}B_{r}(0)}[u_{1}(x)-u_{2}(x)]\Lambda_{\beta}^{-1}(x)\geq 0,$ where $\partial_{+}B_{r}(0)=\partial B_{r}(0)\cap\Omega$. Then $u_{1}\geq u_{2}\quad{\rm in}\;\;\overline{\Omega}\setminus\\{0\\}.$ Proof. Let $w=u_{2}-u_{1}$, then $w$ satisfies $\left\\{\begin{array}[]{lll}\displaystyle\qquad\mathcal{L}_{\beta}w\leq 0\quad{\rm in}\;\;{\Omega},\\\\[4.2679pt] \phantom{L_{\beta}}\displaystyle\qquad w\leq 0\quad{\rm on}\;\;\partial{\Omega}\setminus\\{0\\},\\\\[4.2679pt] \phantom{}\displaystyle\lim_{r\to 0^{+}}\sup_{x\in\partial_{+}B_{r}(0)}w(x)\Lambda_{\beta}^{-1}(x)\leq 0.\end{array}\right.$ Thus for any $\epsilon>0$, there exists $r_{\epsilon}>0$ converging to zero as $\epsilon\to 0$ such that $w\leq\epsilon\Lambda_{\beta}\quad{\rm on}\quad\partial B_{r_{\epsilon}}(0)\cap\Omega.$ We observe that $w\leq 0<\epsilon\Lambda_{\beta}\ {\rm on}\ \partial\Omega\setminus B_{r_{\epsilon}}(0),$ which implies by Lemma 2.1 that $w\leq\epsilon\Lambda_{\beta}\quad{\rm in}\quad\overline{\Omega}\setminus\\{0\\}.$ Therefore we obtain that $w\leq 0$ in $\overline{\Omega}\setminus\\{0\\}$ which ends the proof. $\Box$ For any $\varepsilon>0$, denote (2.3) $\mathcal{L}_{\beta,\varepsilon}=-\Delta+\frac{\beta}{|x|^{2}+\varepsilon}.$ We remark that $\mathcal{L}_{\beta,\varepsilon}$ is strictly elliptic operator and we have the following existence result for related nonhomogeneous problem. ###### Lemma 2.3. Assume that $\varepsilon\in(0,\,1)$, $\beta\geq\beta_{0}$, $\mathcal{L}_{\beta,\varepsilon}$ is given by (2.3) and $f\in C^{\theta}_{loc}(\Omega)\cap C(\bar{\Omega})$ with $\theta\in(0,1)$ and $g\in C(\partial\Omega)$. Then the problem (2.4) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta,\varepsilon}u=f&{\rm in}\quad\Omega,\\\\[5.69054pt] \phantom{\mathcal{L}_{\beta,\varepsilon}}u=g&{\rm on}\quad\partial\Omega\end{array}\right.$ has a unique classical solution $u_{\varepsilon}\in C^{2}(\Omega)\cap C(\bar{\Omega})$, which verifies that (2.5) $\int_{\Omega}u_{\varepsilon}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=\int_{\Omega}\frac{f\xi}{x_{N}}d\gamma_{\beta}-\int_{\partial\Omega}g\frac{\partial\xi}{\partial\nu}\,d\omega_{\beta}+\beta\varepsilon\int_{\Omega}\frac{u_{\varepsilon}\xi}{(|x|^{2}+\varepsilon)|x|^{2}x_{N}}\,d\gamma_{\beta},$ for any $\xi\in C^{1.1}_{0}(\Omega)$. Assume more that $f\geq 0$ in $\Omega$ and $g\geq 0$ on $\partial\Omega$. Then the mapping $\varepsilon\mapsto u_{\varepsilon}$ is decreasing if $\beta>0$, and is increasing if $\beta_{0}\leq\beta<0$. Proof. We first prove the existence of solution to problem (2.4). We introduce Poisson kernel $P_{\Omega}$ of $-\Delta$ in $\Omega$, and denote Poisson operator as $\mathbb{P}_{\Omega}[g](x)=\int_{\partial\Omega}P_{\Omega}(x,y)g(y)dy.$ We observe that $\mathcal{L}_{\beta,\varepsilon}\mathbb{P}_{\Omega}[g]=\frac{\beta}{|x|^{2}+\varepsilon}\mathbb{P}_{\Omega}[g]\in C^{1}(\Omega)\cap C(\bar{\Omega}).$ Then the solution of (2.4) denoted by $u_{\varepsilon}$, could be reduced to $u_{\varepsilon}=\mathbb{P}_{\Omega}[g]+u_{f}$, where $u_{f}$ is the solution of (2.6) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta,\varepsilon}u=f-\frac{\beta}{|x|^{2}+\varepsilon}\mathbb{P}_{\Omega}[g]&{\rm in}\quad\Omega,\\\\[5.69054pt] \phantom{\mathcal{L}_{\beta,\varepsilon}}u=0&{\rm on}\quad\partial\Omega.\end{array}\right.$ For $\beta\geq\beta_{0}$, a solution $u_{f}$ in $H^{1}_{0}(\Omega)$ of (2.6) could be derived by Ekeland’s variational methods as the critical point of the functional $I(u)=\int_{\Omega}|\nabla u|^{2}dx+\beta\int_{\Omega}\frac{u^{2}}{|x|^{2}+\varepsilon}dx-\int_{\Omega}\Big{(}f-\frac{\beta}{|x|^{2}+\varepsilon}\mathbb{P}_{\Omega}[g]\Big{)}udx.$ That is well-defined in $H^{1}_{0}(\Omega)$ since $\beta\in(\beta_{0},0)$. From the Hardy’s inequality in [17], we have that, for any $u\in C_{0}^{2}(\Omega)$, $\int_{\Omega}|\nabla u|^{2}dx+\beta\int_{\Omega}\frac{u^{2}}{|x|^{2}+\varepsilon}dx\geq(\beta-\beta_{0})\int_{\Omega}|\nabla u|^{2}dx,$ for $\beta=\beta_{0}$, from the improved Hardy inequality in [17], it holds $\displaystyle c_{2}\int_{\Omega}u^{2}dx$ $\displaystyle\leq$ $\displaystyle\int_{\Omega}|\nabla u|^{2}dx-|\beta_{0}|\int_{\Omega}\frac{u^{2}}{|x|^{2}}dx$ $\displaystyle<$ $\displaystyle\int_{\Omega}|\nabla u|^{2}dx-|\beta_{0}|\int_{\Omega}\frac{u^{2}}{|x|^{2}+\varepsilon}dx.$ Finally it is trivial for the case $\beta\geq 0$. By the standard regularity result (e.g. [24]), we have that $u_{f}$ is a classical solution of (2.6). Then problem (2.4) admits a classical solution and the uniqueness follows by comparison principle. Finally, we prove (2.5). Multiple $\frac{\lambda_{\beta}\xi}{x_{N}}$ with $\xi\in C^{1.1}_{0}(\Omega)$ and integrate over $\Omega$, we have that $\displaystyle\int_{\Omega}\frac{\lambda_{\beta}\xi}{x_{N}}f\,dx=\int_{\Omega}\frac{\lambda_{\beta}\xi}{x_{N}}\mathcal{L}_{\beta,\varepsilon}u_{\varepsilon}\,dx$ $\displaystyle=$ $\displaystyle\int_{\Omega}u_{\varepsilon}(-\Delta\frac{\lambda_{\beta}\xi}{x_{N}})\,dx+\int_{\partial\Omega}g\frac{\partial(|x|^{\tau_{+}(\beta)}\xi)}{\partial\nu}\,d\omega(x)+\int_{\Omega}\frac{\beta}{|x|^{2}+\varepsilon}u_{\varepsilon}\lambda_{\beta}\xi\,dx$ $\displaystyle=$ $\displaystyle\int_{\Omega}u_{\varepsilon}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}+\int_{\partial\Omega}g\frac{\partial\xi}{\partial\nu}\,d\omega_{\beta}-\beta\varepsilon\int_{\Omega}\frac{u_{\varepsilon}\xi}{(|x|^{2}+\varepsilon)|x|^{2}x_{N}}\,d\gamma_{\beta}.$ Note that if $f\geq 0$ in $\Omega$ and $g\geq 0$ on $\partial\Omega$, then $u_{\varepsilon}\geq 0$ in $\Omega$. Let $\varepsilon_{1}\geq\varepsilon_{2}$ and $u_{\varepsilon_{1}},\,u_{\varepsilon_{2}}$ be two solutions of (2.4) respectively. If $\beta\geq\beta_{0}$, we observe that $\mathcal{L}_{\beta,\varepsilon_{2}}u_{\varepsilon_{1}}\geq\mathcal{L}_{\beta,\varepsilon_{1}}u_{\varepsilon_{1}}=f,$ so $u_{\varepsilon_{1}}$ is a super solution of (2.4) with $\varepsilon=\varepsilon_{2}$ and by comparison principle, it holds $u_{\varepsilon_{1}}\geq u_{\varepsilon_{2}}\;\;{\rm in}\;\Omega.$ The proof ends. $\Box$ Now we build the distributional identity for the classical solution of nonhomogeneous problem with $g=0$ and moderate singularity at the origin, i.e. (2.7) $\lim_{r\to 0^{+}}\sup_{x\in\partial_{+}B_{r}(0)}\frac{|u(x)|}{\Lambda_{\beta}(x)}=0.$ ###### Proposition 2.4. Let $\beta\geq\beta_{0}$, $N\geq 2$, $f\in C_{loc}^{\theta}(\bar{\Omega})$ with $\theta\in(0,1)$, then (2.8) $\left\\{\begin{array}[]{lll}\displaystyle\quad\mathcal{L}_{\beta}u=f\quad{\rm in}\;\;{\Omega},\\\\[5.69054pt] \phantom{L_{\beta}}\quad\displaystyle u=0\quad{\rm on}\;\;\partial{\Omega}\setminus\\{0\\},\end{array}\right.$ subjecting to (2.7), has a unique solution $u_{\beta}$, which satisfies the distributional identity (2.9) $\int_{\Omega}u_{\beta}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=\int_{\Omega}\frac{f\xi}{x_{N}}\,d\gamma_{\beta},\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ Proof. The uniqueness follows by Lemma 2.2. Since $\mathcal{L}_{\beta}$ is a linear operator, we only have to deal with the case that $f\geq 0$ in $\Omega$. Part 1: $\beta>0$. In this case, the mapping $\varepsilon\mapsto u_{\varepsilon}$ is decreasing, where $u_{\varepsilon}>0$ is the solution of (2.4) with $g=0$. Then $u_{\beta}:=\lim_{\varepsilon\to 0^{+}}u_{\varepsilon}$ exists, and by the standard regularity theory, we have that $u_{\beta}$ is a classical solution of (2.10) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}u=f\quad{\rm in}\;\;{\Omega},\\\\[5.69054pt] \phantom{L_{\beta}}\displaystyle u=0\quad{\rm on}\;\;\partial{\Omega}.\end{array}\right.$ Part 2: $\beta\in[\beta_{0},0)$. Without loss of generality, we assume that $\Omega\subset B_{\frac{1}{2}}(0)$. Denote $V_{t,s}(x):=\left\\{\begin{array}[]{lll}\displaystyle tx_{N}|x|^{-\frac{N}{2}}-sx_{N}^{2}|x|^{\tau_{+}(\beta)}&{\rm if}\;\;\beta\in(\beta_{0},0),\\\\[5.69054pt] \phantom{}\displaystyle tx_{N}|x|^{-\frac{N}{2}}(-\ln|x|)^{\frac{1}{2}}-sx_{N}^{2}|x|^{-\frac{N}{2}}&{\rm if}\;\;\beta=\beta_{0},\end{array}\right.$ where the parameters $s,t\geq 0$. Then for $\beta\in(\beta_{0},0)$, we see that $V_{t,s}(x)>0$ for $x\in\Omega$ if $t\geq s$ and $\mathcal{L}_{\beta}V_{t,s}(x)=tc_{\beta}(-N/2)x_{N}|x|^{-\frac{N}{2}-2}+2s|x|^{\tau_{+}(\beta)}+2s\tau_{+}(\beta)x_{N}^{2}|x|^{\tau_{+}(\beta)-2},$ where $c_{\beta}(-N/2)>0$ and $\tau_{+}(\beta)<0$. Since $f$ is bounded in $\Omega$, let $s_{0}=\frac{1}{2}\sup_{x\in\Omega}\frac{|f(x)|}{|x|^{\tau_{+}(\beta)}}$ and then we fix $t_{0}\geq s_{0}$ such that $t_{0}c_{\beta}(-N/2)x_{N}|x|^{-\frac{N}{2}-2}+2s_{0}\tau_{+}(\beta)x_{N}^{2}|x|^{\tau_{+}(\beta)-2}\geq 0.$ So $V_{t_{0},s_{0}}$ is a positive supersolution of (2.8). For $\beta=\beta_{0},\,\tau_{-}(\beta)=-\frac{N}{2}$, we have that $\mathcal{L}_{\beta}V_{t,s}(x)=\frac{t}{4}x_{N}|x|^{-\frac{N}{2}-2}(-\ln|x|)^{-\frac{1}{2}}+2s|x|^{-\frac{N}{2}}-2sNx_{N}^{2}|x|^{-\frac{N}{2}-2}.$ We take $s_{0}$ as above where $\beta$ is replaced by $\beta_{0}$ and we fix $t_{0}\geq s_{0}$ such that $\frac{t_{0}}{4}x_{N}|x|^{-\frac{N}{2}-2}(-\ln|x|)^{-\frac{1}{2}}-2s_{0}Nx_{N}^{2}|x|^{-\frac{N}{2}-2}\geq 0.$ So $V_{t_{0},s_{0}}$ is also a positive supersolution of (2.8) in this case which implies, by comparison principle, that we have $u_{\varepsilon}(x)\leq V_{t_{0},s_{0}}(x),\quad\forall\,x\in\Omega.$ Proof of (2.9). We need to estimate $\displaystyle\int_{\Omega}\frac{u_{\varepsilon}\xi}{(|x|^{2}+\varepsilon)|x|^{2}x_{N}}\,d\gamma_{\beta}$ for $0<\varepsilon<\varepsilon_{0}$ for some $\varepsilon_{0}>0$ fixed. we first consider the case $\beta>0$. We observe that $\displaystyle\varepsilon\int_{\Omega\setminus B_{\sqrt{\varepsilon}}(0)}\frac{u_{\varepsilon}\xi\lambda_{\beta}(x)}{(|x|^{2}+\varepsilon)|x|^{2}x_{N}}\,dx$ $\displaystyle\leq$ $\displaystyle\varepsilon\|u_{\varepsilon_{0}}\|_{L^{\infty}(\Omega)}\|\xi/\rho\|_{L^{\infty}(\Omega)}\int_{\Omega\setminus B_{\sqrt{\varepsilon}}(0)}\frac{|x|^{\tau_{+}(\beta)-2}}{|x|^{2}+\varepsilon}\,dx$ $\displaystyle\leq$ $\displaystyle\|u_{\varepsilon_{0}}\|_{L^{\infty}(\Omega)}\|\xi/\rho\|_{L^{\infty}(\Omega)}\varepsilon^{\frac{N-2+\tau_{+}(\beta)}{2}}\int_{B_{\frac{1}{2\sqrt{\varepsilon}}}(0)\setminus B_{1}(0)}|y|^{\tau_{+}(\beta)-4}\,dy$ $\displaystyle\leq$ $\displaystyle c_{3}\|u_{\varepsilon_{0}}\|_{L^{\infty}(\Omega)}\|\xi/\rho\|_{L^{\infty}(\Omega)}(2^{-\tau_{+}(\beta)+4-N}\varepsilon+\varepsilon^{\frac{N-2+\tau_{+}(\beta)}{2}})$ $\displaystyle\to$ $\displaystyle 0\ \quad{\rm as}\quad\ \varepsilon\to 0^{+}$ and $\displaystyle\varepsilon\int_{B_{\sqrt{\varepsilon}}(0)}\frac{u_{\varepsilon}\xi\lambda_{\beta}(x)}{(|x|^{2}+\varepsilon)|x|^{2}x_{N}}\,\,dx$ $\displaystyle\leq$ $\displaystyle\|u_{\varepsilon_{0}}\|_{L^{\infty}(\Omega)}\|\xi/\rho\|_{L^{\infty}(\Omega)}\int_{B_{\sqrt{\varepsilon}}(0)}|x|^{\tau_{+}(\beta)-2}\,dx$ $\displaystyle\leq$ $\displaystyle c_{4}\|u_{\varepsilon_{0}}\|_{L^{\infty}(\Omega)}\|\xi/\rho\|_{L^{\infty}(\Omega)}\varepsilon^{\frac{N-2+\tau_{+}(\beta)}{2}}$ $\displaystyle\to$ $\displaystyle 0\ \quad{\rm as}\quad\ \varepsilon\to 0^{+},$ where $\rho(x)={\rm dist}(x,\,\partial\Omega)$ and $\frac{N-2+\tau_{+}(\beta)}{2}>0$. Therefore, passing to the limit of (2.5), we obtain (2.9). For $\beta\in(\beta_{0},\,0)$, from the increasing monotonicity and the upper bound $V_{s_{0},t_{0}}$, we have that $\lim_{\varepsilon\to 0^{+}}\int_{\Omega}u_{\varepsilon}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})d\gamma_{\beta}=\int_{\Omega}u_{\beta}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})d\gamma_{\beta}$ and $\varepsilon\int_{\Omega}\frac{\xi u_{\varepsilon}\lambda_{\beta}(x)}{(|x|^{2}+\varepsilon)|x|^{2}x_{N}}dx\leq c_{5}\varepsilon\int_{\Omega}\frac{|x|^{-N+\sqrt{\beta-\beta_{0}}}}{|x|^{2}+\varepsilon}dx.$ By directly compute, we have that $\displaystyle\varepsilon\int_{\Omega\setminus B_{\sqrt{\varepsilon}}(0)}\frac{|x|^{-N+\sqrt{\beta-\beta_{0}}}}{|x|^{2}+\varepsilon}dx$ $\displaystyle\leq$ $\displaystyle c_{6}\varepsilon^{\frac{\sqrt{\beta-\beta_{0}}}{2}}\int_{B_{\frac{1}{2\sqrt{\varepsilon}}}(0)\setminus B_{1}(0)}|y|^{-N-2+\sqrt{\beta-\beta_{0}}}\,dy$ $\displaystyle\leq$ $\displaystyle c_{7}(\varepsilon+\varepsilon^{\frac{\sqrt{\beta-\beta_{0}}}{2}})\to 0\quad{\rm as}\quad\varepsilon\to 0^{+}$ and $\displaystyle\varepsilon\int_{B_{\sqrt{\varepsilon}}(0)}\frac{|x|^{-N+\sqrt{\beta-\beta_{0}}}}{|x|^{2}+\varepsilon}dx$ $\displaystyle\leq$ $\displaystyle\int_{B_{\sqrt{\varepsilon}}(0)}|x|^{-N+\sqrt{\beta-\beta_{0}}}\,dx$ $\displaystyle\leq$ $\displaystyle c_{8}\varepsilon^{\frac{\sqrt{\beta-\beta_{0}}}{2}}\to 0\quad{\rm as}\quad\varepsilon\to 0^{+},$ As a conclusion, passing to the limit in (2.5) as $\varepsilon\to 0^{+}$, we have that $u_{\beta}$ satisfies that (2.11) $\int_{\Omega}u_{\beta}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=\int_{\Omega}\frac{f\xi}{x_{N}}\,d\gamma_{\beta},\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ Finally, we prove (2.9) with $\beta=\beta_{0}$, We claim that the mapping $\beta\mapsto u_{\beta}$ with $\beta\in(\beta_{0},0)$ is decreasing. In fact, if $\beta_{0}<\beta_{1}\leq\beta_{2}<0$, we know that $\displaystyle f=\mathcal{L}_{\beta_{1}}u_{\beta_{1}}$ $\displaystyle=$ $\displaystyle-\Delta u_{\beta_{1}}+\frac{\beta_{1}}{|x|^{2}}u_{\beta_{1}}$ $\displaystyle\leq$ $\displaystyle-\Delta u_{\beta_{1}}+\frac{\beta_{2}}{|x|^{2}}u_{\beta_{1}}=\mathcal{L}_{\beta_{2}}u_{\beta_{1}},$ by Lemma 2.2, which implies that $u_{\beta_{1}}\geq u_{\beta_{2}}.$ We know that $V_{s_{0},t_{0}}$ is a super solution of (2.8) with $\beta\in(\beta_{0},0)$. So it follows by Lemma 2.2 that $\\{u_{\beta}\\}_{\beta}$ is uniformly bounded by the upper bound $V_{s_{0},t_{0}}\in L^{1}(\Omega,\frac{1}{x_{N}}d\gamma_{\beta})$. For $\xi\in C^{1.1}_{0}(\Omega)$, we have that $|\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})|\leq c_{9}(\|\frac{\xi}{x_{N}}\|_{C^{1.1}(\Omega)}+\|\frac{\xi}{x_{N}}\|_{C^{1}(\Omega)}x_{N}^{-1}),$ where $c_{9}>0$ is independent of $\beta$. From the dominate monotonicity convergence theorem and the uniqueness of the solution, we have that $u_{\beta}\to u_{\beta_{0}}\quad{\rm a.e.\ in}\ \Omega\;{\rm as}\;\;\beta\to\beta^{+}_{0}\quad{\rm and\;in}\;\;L^{1}(\Omega,\,x_{N}^{-1}d\gamma_{\beta})$ and $u_{\beta_{0}}$ is a classical solution of (2.8) with $\beta=\beta_{0}$. Passing to the limit of (2.11) as $\beta\to\beta^{+}_{0}$ to obtain that $\int_{\Omega}u_{\beta_{0}}\mathcal{L}^{*}_{\beta_{0}}(\frac{\xi}{x_{N}})\,d\gamma_{\beta_{0}}=\int_{\Omega}\frac{f\xi}{x_{N}}d\gamma_{\beta_{0}}.$ The proof ends. $\Box$ ###### Remark 2.5. We note that when $\beta\geq 0$ and $f$ is bounded, the moderate singular solution of problem (2.8) is no longer singular, that means, it is a classical solution of (2.12) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}u=f\quad{\rm in}\;\;{\Omega},\\\\[4.2679pt] \phantom{L_{\beta}}\displaystyle u=0\quad{\rm on}\;\;\partial{\Omega}.\end{array}\right.$ Now we prove the following ###### Lemma 2.6. $(i)$ The problem (2.13) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}^{*}(\frac{u}{x_{N}})=1\quad{\rm in}\;\;\Omega,\\\\[5.69054pt] \phantom{\mathcal{L}_{\beta}^{*}-\ \,\,\,}\displaystyle u=0\quad{\rm on}\;\;\partial{\Omega}\end{array}\right.$ has a unique positive solution $w_{1}\in C^{2}(\Omega)\cap C^{0.1}_{0}(\Omega)$. $(ii)$ The problem (2.14) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}^{*}(\frac{u}{x_{N}})=\frac{1}{x_{N}}&{\rm in}\;\;\Omega,\\\\[5.69054pt] \phantom{\mathcal{L}_{\beta}^{*}--}\displaystyle u=0&{\rm on}\;\;\partial{\Omega}\end{array}\right.$ has a unique positive solution $w_{2}\in C^{2}(\Omega)\cap C^{1}_{0}(\bar{\Omega}\setminus\\{0\\})\cap C^{0.1}_{0}(\Omega)$. Proof. We first claim that problem (2.8) has a unique classical positive solution $w_{\beta}$ under the constraint (2.7) when $f(x)=\lambda_{\beta}(x)$ or $f(x)=|x|^{\tau_{+}(\beta)}$. In fact, let $f_{n}(x)=\lambda_{\beta}(x)\eta_{0}(n|x|)$, where $\eta_{0}:[0,+\infty)\to[0,\,1]$ is a decreasing $C^{\infty}$ function satisfying (2.1). Then $f_{n}\in C^{\theta}(\bar{\Omega})$ with $\theta\in(0,1),f_{n}\leq f$, and by Proposition 2.4, let $w_{n}$ be the solution of problem (2.15) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}u=f_{n}&{\rm in}\;\;{\Omega},\\\\[5.69054pt] \phantom{L_{\beta}}\displaystyle u=0&{\rm on}\;\;\partial{\Omega}\setminus\\{0\\},\end{array}\right.$ subject to (2.7). We know that the mapping: $n\to w_{n}$ is increasing by the increasing monotone of $\\{f_{n}\\}$. So we only construct a suitable upper bound for $w_{n}$ in the cases that $f(x)=\lambda_{\beta}(x)$ and $f(x)=|x|^{\tau_{+}(\beta)}$ respectively. When $f(x)=\lambda_{\beta}(x)$, let $V_{t,s}(x)=t\lambda_{\beta}(x)-sx_{N}|x|^{\tau_{+}(\beta)+2}$ for $s,t>0$. It is know that $\mathcal{L}_{\beta}V_{t,s}=-sc_{\tau_{+}(\beta)+2}\lambda_{\beta}(x),\quad x\in\mathbb{R}^{N}_{+},$ for some $c_{\tau_{+}(\beta)+2}<0$. So fix $s=-1/c_{\tau_{+}(\beta)+2}$ and then fix $t>0$ such that $V_{t,s}(x)>0,\quad\forall x\in\Omega.$ The limit of $\\{w_{n}\\}_{n}$, denoting by $w_{\beta,1}$, is a solution of (2.7) satisfying $w_{\beta,1}\leq V_{t,s}(x).$ When $f(x)=|x|^{\tau_{+}(\beta)}$, let $W_{t,s,l}(x)=t\lambda_{\beta}(x)-s(x_{N}|x|^{\tau_{+}(\beta)+2}+lx_{N}^{2}|x|^{\tau_{+}(\beta)+2}),$ where $s,t,l>0$. We observe that $\mathcal{L}_{\beta}W_{t,s,l}(x)=s[-c_{\tau_{+}(\beta)+2}\lambda_{\beta}(x)+2l|x|^{\tau_{+}(\beta)}+2l\tau_{+}(\beta)x_{N}^{2}|x|^{\tau_{+}(\beta)}],\;x\in\mathbb{R}^{N}_{+},$ with the same constant $c_{\tau_{+}(\beta)+2}<0$ as above. Then we choose $l>0$ such that $-2c_{\tau_{+}(\beta)+2}l\tau_{+}(\beta)x_{N}>0$ for $x\in\Omega$, $s=\frac{1}{2l}$ and we take $t>0$ such that $W_{t,s,l}>0$ in $\Omega$ and $\mathcal{L}_{\beta}W_{t,s,l}(x)\geq|x|^{\tau_{+}(\beta)}.$ Thus, the limit of $\\{w_{n}\\}_{n}$, denoting by $w_{\beta,2}$, is a solution of (2.7) such that $w_{\beta,2}(x)\leq W_{t,s,l}(x).$ As a conclusion, for $i=1,2$, (2.16) $w_{\beta,i}\leq t\lambda_{\beta}\quad{\rm in}\quad\Omega.$ Denote $w_{i}=w_{\beta,i}x_{N}/\lambda_{\beta},$ we observe that $\displaystyle 1=\lambda_{\beta}^{-1}\mathcal{L}_{\beta}w_{\beta,1}=\lambda_{\beta}^{-1}\mathcal{L}_{\beta}(\lambda_{\beta}w_{1}/x_{N})=\mathcal{L}_{\beta}^{*}(w_{1}/x_{N})$ and $\displaystyle 1/x_{N}=\lambda_{\beta}^{-1}\mathcal{L}_{\beta}w_{\beta,2}=\lambda_{\beta}^{-1}\mathcal{L}_{\beta}(\lambda_{\beta}w_{2}/x_{N})=\mathcal{L}_{\beta}^{*}(w_{2}/x_{N}).$ Moreover, by (2.16), it follow that $w_{i}\leq tx_{N}.$ Then we have that $w_{i}\in C^{2}(\Omega)\cap C^{0.1}_{0}(\Omega)$ for $i=1,2$. Away from the origin, Hardy’s operator is uniform elliptic, thus $u\in C^{1}_{0}(\bar{\Omega}\setminus\\{0\\})$ and then $u\in C^{2}(\Omega)\cap C^{1}_{0}(\bar{\Omega}\setminus\\{0\\})\cap C^{0.1}_{0}(\Omega).$ $\Box$ Although $C^{2}(\Omega)\cap C^{1}_{0}(\bar{\Omega}\setminus\\{0\\})\cap C^{0.1}_{0}(\Omega)$ is not suitable as test function space for problem (1.2), $w_{1},\,w_{2}$ are still valid as test functions for formula (1.11) with $k=0$ in the distributional sense. For given $f\in C^{1}(\bar{\Omega})$, a direct consequence of Lemma 2.6 can be stated as follows ###### Corollary 2.7. Assume that $f\in C^{1}(\bar{\Omega}\setminus\\{0\\})$ satisfying for some $c_{10}>0$ $|f(x)|\leq\frac{c_{10}}{x_{N}}.$ Then there exists a unique solution of $w_{f}\in C^{2}(\Omega)\cap C^{0.1}_{0}(\Omega)$ of (2.17) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}^{*}(\frac{u}{x_{N}})=f&{\rm in}\;\;\Omega,\\\\[5.69054pt] \phantom{\mathcal{L}_{\beta}^{*}-\ \,\,\,}\displaystyle u=0&{\rm on}\;\;\partial{\Omega}.\end{array}\right.$ ## 3\. Fundamental solution ### 3.1. In half space In this subsection, we give the proof of Theorem 1.1. Proof of Theorem 1.1. For any $\xi\in C^{1.1}_{0}(\mathbb{R}^{N}_{+})$, we know there exists a unique $\zeta\in C^{1.1}_{c}(\ \mathbb{R}^{N})$ such that $\xi(x)=x_{N}\zeta(x)$ for $x\in\overline{\mathbb{R}^{N}_{+}}$. Moreover, we have that $\frac{\partial\xi}{\partial x_{N}}(0)=\zeta(0)$. Take $\zeta\in C^{1.1}_{c}(\mathbb{R}^{N})$, multiplying $\lambda_{\beta}\zeta$ in (1.3) and integrating over $\mathbb{R}^{N}_{+}\setminus\overline{B_{r}(0)}$, then we have that $\displaystyle 0$ $\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{N}_{+}\setminus\overline{B_{r}(0)}}\mathcal{L}_{\beta}(\Lambda_{\beta})\lambda_{\beta}\zeta\,dx=\int_{\mathbb{R}^{N}_{+}\setminus\overline{B_{r}(0)}}\Lambda_{\beta}\mathcal{L}_{\beta}^{*}(\zeta)\,d\gamma_{\beta}$ $\displaystyle+\int_{\partial_{+}B_{r}(0)}\Big{(}-\nabla\Lambda_{\beta}\cdot\frac{x}{|x|}\lambda_{\beta}+\nabla\lambda_{\beta}\cdot\frac{x}{|x|}\Lambda_{\beta}\Big{)}\zeta\,d\omega$ $\displaystyle+\int_{\partial_{+}B_{r}(0)}\Lambda_{\beta}\lambda_{\beta}\Big{(}\nabla\zeta\cdot\frac{x}{|x|}\Big{)}\,d\omega,$ where $\partial_{+}B_{r}(0)=\partial B_{r}(0)\cap\mathbb{R}^{N}_{+}$. For $\beta\geq\beta_{0}$, we see that for $r=|x|>0$ small, $\displaystyle-\nabla\Lambda_{\beta}(x)\cdot\frac{x}{|x|}\lambda_{\beta}(x)+\nabla\lambda_{\beta}(x)\cdot\frac{x}{|x|}\Lambda_{\beta}(x)$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{lll}2\sqrt{\beta-\beta_{0}}\,x_{N}^{2}r^{-N-1}&{\rm if}\;\;\beta>\beta_{0},\\\\[4.2679pt] \phantom{}x_{N}^{2}r^{-N-1}&{\rm if}\;\;\beta=\beta_{0}\end{array}\right.$ and $|\zeta(x)-\zeta(0)|\leq c_{11}r,$ then $\displaystyle\int_{\partial_{+}B_{r}(0)}\sqrt{\beta-\beta_{0}}\,x_{N}^{2}r^{-N-1}\zeta(0)x_{N}d\omega(x)$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{lll}\sqrt{\beta-\beta_{0}}\,\displaystyle\int_{\partial_{+}B_{1}(0)}x_{N}^{2}d\omega(x)\,\zeta(0)&{\rm if}\;\;\beta>\beta_{0},\\\\[5.69054pt] \phantom{}\displaystyle\int_{\partial_{+}B_{1}(0)}x_{N}^{2}d\omega(x)\,\zeta(0)&{\rm if}\;\;\beta=\beta_{0}\end{array}\right.$ $\displaystyle=$ $\displaystyle c_{\beta}\zeta(0)$ and $\displaystyle\Big{|}\int_{\partial_{+}B_{r}(0)}\Big{(}-\nabla\Lambda_{\beta}\cdot\frac{x}{|x|}\lambda_{\beta}+\nabla\lambda_{\beta}\cdot\frac{x}{|x|}\Lambda_{\beta}\Big{)}\zeta\,d\omega- c_{\beta}\zeta(0)\Big{|}$ $\displaystyle\leq$ $\displaystyle c_{12}(\sqrt{\beta-\beta_{0}}+1)\,r\int_{\partial_{+}B_{1}(0)}x_{N}^{2}d\omega(x)$ $\displaystyle\to$ $\displaystyle 0\quad{\rm as}\quad r\to 0^{+},$ that is, $\lim_{r\to 0}\Big{(}\int_{\partial_{+}B_{r}(0)}-\nabla\Lambda_{\beta}\cdot\frac{x}{|x|}\lambda_{\beta}\zeta\,d\omega+\int_{\partial B_{r}(0)}\nabla\lambda_{\beta}\cdot\frac{x}{|x|}\Lambda_{\beta}\zeta\,d\omega\Big{)}=c_{\beta}\zeta(0).$ Moreover, we see that $\Big{|}\int_{\partial_{+}B_{r}(0)}\Lambda_{\beta}\lambda_{\beta}\Big{(}\nabla\zeta\cdot\frac{x}{|x|}\Big{)}\,d\omega\Big{|}\leq\|\zeta\|_{C^{1}}\,r\int_{\partial_{+}B_{1}(0)}x_{N}^{2}d\omega\to 0\quad{\rm as}\quad r\to 0^{+}.$ Therefore, we have that $\lim_{r\to 0^{+}}\int_{\mathbb{R}^{N}\setminus\overline{B_{r}(0)}}\Lambda_{\beta}\mathcal{L}_{\beta}^{*}(\zeta)d\gamma_{\beta}=c_{\beta}\zeta(0),$ which implies (1.7). The proof ends. $\Box$ ### 3.2. Trace of $\Lambda_{\beta}$. The following theorem shows the trace of $\Lambda_{\beta}$. ###### Theorem 3.1. Let $d\omega_{\beta}(x^{\prime})=|x^{\prime}|^{\tau_{+}(\beta)}dx^{\prime}$ for $x^{\prime}\in\mathbb{R}^{N-1}$, then for any $\zeta\in C_{c}(\mathbb{R}^{N-1})$, (3.3) $\lim_{t\to 0^{+}}\int_{\mathbb{R}^{N-1}}\Lambda_{\beta}(x^{\prime},t)\zeta(x^{\prime})d\omega_{\beta}(x^{\prime})=b_{N}\zeta(0),$ where $b_{N}=\int_{\mathbb{R}^{N-1}}(1+|y^{\prime}|^{2})^{-\frac{N}{2}}dy^{\prime}>0.$ This is to say that the trace of $\Lambda_{\beta}$ is $\delta_{0}$ in the $d\gamma_{\beta}$-distributional sense. Proof. For any $\zeta\in C_{c}(\mathbb{R}^{N-1})$, there exists $R>0$ such that supp$\,\zeta\subset B_{R}^{\prime}(0)$, here and in the sequel, denoting by $B_{R}^{\prime}(0)$ the ball in $\mathbb{R}^{N-1}$. By direct computations, we have that $\displaystyle\int_{\mathbb{R}^{N-1}}\Lambda_{\beta}(x^{\prime},t)\zeta(x^{\prime})\,d\omega_{\beta}(x^{\prime})$ $\displaystyle=$ $\displaystyle\int_{B_{R}^{\prime}(0)}\Lambda_{\beta}(x^{\prime},t)\zeta(x^{\prime})\,d\omega_{\beta}(x^{\prime})$ $\displaystyle=$ $\displaystyle\int_{B_{R/t}^{\prime}(0)}(|y^{\prime}|^{2}+1)^{\frac{\tau_{-}(\beta)}{2}}|y^{\prime}|^{\tau_{+}(\beta)}\zeta(ty^{\prime})dy^{\prime}.$ For any $\varepsilon>0$, there exists $R_{\varepsilon}>1$ such that $\displaystyle\int_{B_{R/t}^{\prime}(0)\setminus B^{\prime}_{R_{\varepsilon}}(0)}(|y^{\prime}|^{2}+1)^{\frac{\tau_{-}(\beta)}{2}}|y^{\prime}|^{\tau_{+}(\beta)}\zeta(ty^{\prime})dy^{\prime}$ $\displaystyle\leq$ $\displaystyle\|\zeta\|_{L^{\infty}(\mathbb{R}^{N-1})}\int_{\mathbb{R}^{N-1}\setminus B^{\prime}_{R_{\varepsilon}}(0)}|y^{\prime}|^{-N}dy^{\prime}$ $\displaystyle\leq$ $\displaystyle\|\zeta\|_{L^{\infty}(\mathbb{R}^{N-1})}|\mathcal{S}^{N-2}|\varepsilon,$ where $R_{\varepsilon}\leq\frac{1}{\varepsilon}$. Let $A:=\int_{B^{\prime}_{R_{\varepsilon}}(0)}(|y^{\prime}|^{2}+1)^{\frac{\tau_{-}(\beta)}{2}}|y^{\prime}|^{\tau_{+}(\beta)}\zeta(ty^{\prime})dy^{\prime}-\int_{\mathbb{R}^{N-1}}(|y^{\prime}|^{2}+1)^{\frac{\tau_{-}(\beta)}{2}}|y^{\prime}|^{\tau_{+}(\beta)}\zeta(0)dy^{\prime},$ we have that $\displaystyle|A|$ $\displaystyle\leq$ $\displaystyle\int_{B^{\prime}_{R_{\varepsilon}}(0)}(|y^{\prime}|^{2}+1)^{\frac{\tau_{-}(\beta)}{2}}|y^{\prime}|^{\tau_{+}(\beta)}\left|\zeta(ty^{\prime})-\zeta(0)\right|\,dy^{\prime}+\varepsilon|\zeta(0)||\mathcal{S}^{N-2}|$ $\displaystyle\leq$ $\displaystyle t\|\zeta\|_{C^{1}(\mathbb{R}^{N-1})}\int_{B^{\prime}_{R_{\varepsilon}}(0)}(|y^{\prime}|^{2}+1)^{\frac{\tau_{-}(\beta)}{2}}|y^{\prime}|^{\tau_{+}(\beta)}dy^{\prime}+\varepsilon|\zeta(0)||\mathcal{S}^{N-2}|$ $\displaystyle=$ $\displaystyle R_{\varepsilon}t\|\zeta\|_{C^{1}(\mathbb{R}^{N-1})}+\varepsilon|\zeta(0)||\mathcal{S}^{N-2}|$ $\displaystyle\leq$ $\displaystyle\left(\|\zeta\|_{C^{1}(\mathbb{R}^{N-1})}+|\zeta(0)||\mathcal{S}^{N-2}|\right)\varepsilon,$ if we take $t=\varepsilon^{2}$. Passing to the limit as $\varepsilon\to 0$, we derive (3.3). $\Box$ ### 3.3. Fundamental solution in bounded domain In this subsection, we do an approximation of the isolated singular solution. ###### Proposition 3.2. Let $\Omega$ be a $C^{2}$ domain verifying (1.9). Then the problem (3.4) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}u=0\quad{\rm in}\;\;{\Omega},\\\\[5.69054pt] \phantom{L_{\beta}}\displaystyle u=0\quad{\rm on}\;\;\partial{\Omega}\setminus\\{0\\},\\\\[5.69054pt] \phantom{}\displaystyle\lim_{r\to 0^{+}}\sup_{x\in B_{r}^{+}(0)}\frac{|u(x)-\Lambda_{\beta}(x)|}{\Lambda_{\beta}(x)}=0\end{array}\right.$ admits a unique solution $\Lambda^{\Omega}_{\beta}$ satisfying the following distributional identity: (3.5) $\int_{\Omega}\Lambda^{\Omega}_{\beta}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=c_{\beta}\frac{\partial\xi}{\partial x_{N}}(0),\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ Proof. Let $\eta_{r_{0}}(t)=\eta_{0}(\frac{2}{r_{0}}t)$, which satisfies that (3.6) $\eta_{r_{0}}=1\quad{\rm in}\quad[0,r_{0}/2]\quad{\rm and}\quad\eta_{r_{0}}=0\quad{\rm in}\quad[r_{0},+\infty).$ For $i=1,2$ the problem (3.7) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}w_{i}=-\nabla\eta_{r_{0}}\cdot\nabla\Lambda_{\beta}-\Lambda_{\beta}\Delta\eta_{r_{0}}\quad{\rm in}\;\;{\Omega},\\\\[5.69054pt] \phantom{L_{\beta}}\displaystyle w_{i}=0\quad{\rm on}\;\;\partial{\Omega}\setminus\\{0\\},\\\\[5.69054pt] \phantom{}\displaystyle\lim_{e\in\mathcal{S}^{N}_{+},\,t\to 0^{+}}w_{i}(te)\Lambda_{\beta}^{-1}(te)=2-i,\end{array}\right.$ admits a unique solutions $w_{1}$ and $w_{2}$ respectively. Obviously, $w_{1}=\Lambda_{\beta}\eta_{r_{0}}$ and $-\nabla\eta_{r_{0}}\cdot\nabla\Lambda_{\beta}-\Lambda_{\beta}\Delta\eta_{r_{0}}$ has compact set in $\Omega\cap(\overline{B_{r_{0}}(0)\setminus B_{\frac{r_{0}}{2}}(0)})$ and then $-\nabla\eta_{r_{0}}\cdot\nabla\Lambda_{\beta}-\Lambda_{\beta}\Delta\eta_{r_{0}}$ is smooth and bounded, it follows by the proof of Proposition 2.4 that there exist $s_{0},t_{0}>0$ such that $|w_{2}|\leq V_{s_{0},t_{0}}$. For $i=1$, following the proof of Theorem 1.1, we get then for any $\xi\in C^{1.1}_{0}(\Omega)$, (3.8) $\int_{\Omega}w_{1}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=\int_{\Omega}\Big{(}-\nabla\eta_{r_{0}}\cdot\nabla\Lambda_{\beta}-\Lambda_{\beta}\Delta\eta_{r_{0}}\Big{)}\frac{\xi}{x_{N}}\,d\gamma_{\beta}+c_{\beta}\frac{\partial\xi}{\partial x_{N}}(0).$ For $i=2$, it follows by Proposition 2.4 that for any $\xi\in C^{1.1}_{0}(\Omega)$, (3.9) $\int_{\Omega}w_{2}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})d\gamma_{\beta}=\int_{\Omega}\Big{(}-\nabla\eta_{r_{0}}\cdot\nabla\Lambda_{\beta}-\Lambda_{\beta}\Delta\eta_{r_{0}}\Big{)}\frac{\xi}{x_{N}}\,d\gamma_{\beta}.$ Let $\Lambda^{\Omega}_{\beta}=\Lambda_{\beta}\eta_{r_{0}}-w_{2},$ it follows by (3.8) and (3.9) that $\int_{\Omega}\Lambda^{\Omega}_{\beta}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=c_{\beta}\frac{\partial\xi}{\partial x_{N}}(0),\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ Finally, it’s clear that if $u_{1}$ and $u_{2}$ are two solutions of (3.4), then $w:=u_{1}-u_{2}$ satisfies $\lim_{r\to 0^{+}}\sup_{x\in B_{r}^{+}(0)}\frac{|w(x)|}{\Lambda_{\beta}(x)}=0.$ Combining with the fact that $\mathcal{L}_{\beta}w=0\;\;{\rm in}\;\;{\Omega}\quad{\rm and}\quad w=0\;\;{\rm on}\;\;\partial{\Omega}\setminus\\{0\\},$ and Lemma 2.2, we have that $w\equiv 0$. Thus the uniqueness is proved. $\Box$ ## 4\. Existence ### 4.1. Zero Dirichlet boundary Our purpose in this section is to clarify the isolated singularities of the nonhomogeneous problem (4.1) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}u=f\quad{\rm in}\;\;\Omega,\\\\[4.2679pt] \phantom{L_{\beta}}\displaystyle u=0\quad{\rm on}\;\;\partial{\Omega}\setminus\\{0\\},\end{array}\right.$ where $f\in C^{\theta}_{loc}(\bar{\Omega}\setminus\\{0\\})$ with $\theta\in(0,1)$. Recall that $\mathcal{L}^{*}_{\beta}$ is given by (1.6) and $d\gamma_{\beta}(x)=\lambda_{\beta}(x)dx$. We prove the following ###### Theorem 4.1. $(i)$ Assume that $f\in L^{1}(\Omega,\,d\gamma_{\beta})$ and $u\in L^{1}(\Omega,\frac{1}{|x|}d\gamma_{\beta})$ is a classical solution of problem (4.1), then there exists some $k\in\mathbb{R}$ such that there holds (4.2) $\int_{\Omega}u\,\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=\int_{\Omega}\frac{f\xi}{x_{N}}\,d\gamma_{\beta}+k\frac{\partial\xi}{\partial x_{N}}(0),\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ $(ii)$ Inversely, assume that $f\in L^{1}(\Omega,\,d\gamma_{\beta})$, then for any $k\in\mathbb{R}$, problem (4.1) has a unique solution $u_{k}\in L^{1}(\Omega,\frac{1}{|x|}d\gamma_{\beta})$ verifying (4.2) with such $k.$ Proof. $(i)$ Let $\tilde{\Omega}$ be the interior set of $\bar{\Omega}\cup\overline{\\{(x^{\prime},-x_{N}):(x^{\prime},x_{N})\in\Omega\\}}$ and extend $u$ (resp. $f$) by the $x_{N}$-odd extension to $\tilde{u}$ (resp. $\tilde{f}$) in $\tilde{\Omega}$, then $\mathcal{L}_{\beta}\tilde{u}=\tilde{f}$. Our aim is to see the distributional property at the origin. Denote by $L$ the operator related to $\mathcal{L}_{\beta}\tilde{u}-\tilde{f}$ in the distribution sense, i.e. (4.3) $L(\zeta)=\int_{\tilde{\Omega}}\Big{(}\tilde{u}\mathcal{L}_{\beta}^{*}(\zeta)-\tilde{f}\zeta\Big{)}|x_{N}||x|^{\tau_{+}(\beta)}\,dx,\quad\forall\zeta\in C^{\infty}_{c}(\tilde{\Omega}).$ For any $\zeta\in C^{\infty}_{c}(\tilde{\Omega}\setminus\\{0\\})$, we have that $L(\zeta)=0.$ In fact, there exists $\varepsilon>0$ such that supp$(\zeta)\subset\tilde{\Omega}\setminus B_{\varepsilon}(0)$ and then $\displaystyle 0$ $\displaystyle=$ $\displaystyle 2\int_{\Omega}\zeta(\mathcal{L}_{\beta}u-f)\,d\gamma_{\beta}=\int_{\tilde{\Omega}}\zeta(\mathcal{L}_{\beta}\tilde{u}-\tilde{f})\,d\tilde{\gamma}_{\beta}$ $\displaystyle=$ $\displaystyle-\int_{\tilde{\Omega}}\tilde{f}\zeta\,d\tilde{\gamma}_{\beta}+\int_{\Omega\setminus B_{\varepsilon}(0)}u\mathcal{L}_{\beta}^{*}\zeta d\gamma_{\beta}+\int_{\partial(\Omega\setminus B_{\varepsilon}(0))\cap(\mathbb{R}^{N-1}\times\\{0\\})}\frac{\partial u}{\partial x_{N}}\zeta d\omega_{\beta}$ $\displaystyle+\int_{(-\Omega)\setminus B_{\varepsilon}(0)}(-u)\mathcal{L}_{\beta}^{*}\zeta d\tilde{\gamma}_{\beta}+\int_{\partial(-\Omega\setminus B_{\varepsilon}(0))\cap(\mathbb{R}^{N-1}\times\\{0\\})}\frac{\partial\tilde{u}}{\partial(-x_{N})}\zeta d\omega_{\beta}$ $\displaystyle=$ $\displaystyle\int_{\tilde{\Omega}\setminus B_{\varepsilon}(0)}(\tilde{u}\mathcal{L}_{\beta}^{*}\zeta-\tilde{f}\zeta)\,d\tilde{\gamma}_{\beta}$ $\displaystyle=$ $\displaystyle\int_{\tilde{\Omega}}(\tilde{u}\mathcal{L}_{\beta}^{*}\zeta-\tilde{f}\zeta)\,d\tilde{\gamma}_{\beta},$ where $d\tilde{\gamma}_{\beta}=|\tilde{\lambda}_{\beta}(x)|dx$, $\tilde{\lambda}_{\beta}$ is the odd extension of $\lambda_{\beta}$ and $\int_{\partial(\Omega\setminus B_{\varepsilon}(0))\cap(\mathbb{R}^{N-1}\times\\{0\\})}\frac{\partial u}{\partial x_{N}}\zeta d\omega_{\beta}=-\int_{\partial(-\Omega\setminus B_{\varepsilon}(0))\cap(\mathbb{R}^{N-1}\times\\{0\\})}\frac{\partial\tilde{u}}{\partial(-x_{N})}\zeta d\omega_{\beta}.$ By Theorem XXXV in [33] (see also Theorem 6.25 in [32]), it implies that (4.4) $L=\sum_{|a|=0}^{p}k_{a}D^{a}\delta_{0},$ where $p\in\mathbb{N}$, $a=(a_{1},\cdots,a_{N})$ is a multiple index with $a_{i}\in\mathbb{N}$, $|a|=\sum_{i=1}^{N}a_{i}$ and in particular, $D^{0}\delta_{0}=\delta_{0}$. Then we have that (4.5) $L(\zeta)=\int_{\tilde{\Omega}}\Big{(}\tilde{u}\mathcal{L}_{\beta}^{*}\zeta-f\zeta\Big{)}\,d\tilde{\gamma}_{\beta}=\sum_{|a|=0}^{\infty}k_{a}D^{a}\zeta(0),\quad\ \ \forall\zeta\in C^{\infty}_{c}(\tilde{\Omega}).$ For any multiple index $a=(a_{1},\cdots,a_{N})$, let $\zeta_{a}$ be a $C^{\infty}$ function such that (4.6) ${\rm supp}(\zeta_{a})\subset\overline{B_{2}(0)}\quad{\rm and}\quad\zeta_{a}(x)=k_{a}\prod_{i=1}^{N}x_{i}^{a_{i}}\quad{\rm for}\ \ x\in B_{1}(0).$ Now we use the test function $\zeta_{\varepsilon,a}(x):=\zeta_{a}(\varepsilon^{-1}x)$ for $x\in\tilde{\Omega}$ in (4.5), we have that $\sum_{|a|\leq q}k_{a}D^{a}\zeta_{\varepsilon,a}(0)=\frac{k_{a}^{2}}{\varepsilon^{|a|}}\prod^{N}_{i=1}a_{i}!,$ where $a_{i}!=a_{i}\cdot(a_{i}-1)\cdots 1>0$ and $a_{i}!=1$ if $a_{i}=0$. Let $r>0$, we obtain that $\displaystyle\Big{|}\int_{\tilde{\Omega}}\tilde{u}\mathcal{L}_{\beta}^{*}\zeta_{\varepsilon}\,d\tilde{\gamma}_{\beta}\Big{|}$ $\displaystyle=$ $\displaystyle\Big{|}\int_{B_{2\varepsilon}(0)}\tilde{u}\mathcal{L}_{\beta}^{*}\zeta_{\varepsilon}\,d\tilde{\gamma}_{\beta}\Big{|}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\varepsilon^{2}}\Big{|}\int_{B_{2\varepsilon}(0)}\tilde{u}(x)(-\Delta)\zeta_{a}(\varepsilon^{-1}x)\,d\tilde{\gamma}_{\beta}\Big{|}$ $\displaystyle+\frac{2|\tau_{+}(\beta)|}{\varepsilon}\,\Big{|}\int_{B_{2\varepsilon}(0)}\tilde{u}(x)\frac{x}{|x|^{2}}\cdot\nabla\zeta_{a}(\varepsilon^{-1}x)\,d\tilde{\gamma}_{\beta}\Big{|}$ $\displaystyle\leq$ $\displaystyle c_{13}\left[\frac{1}{\varepsilon^{2}}\int_{B_{2\varepsilon}(0)}|\tilde{u}(x)|\,d\tilde{\gamma}_{\beta}+\frac{1}{\varepsilon}\,\int_{B_{2\varepsilon}(0)}\frac{|\tilde{u}(x)|}{|x|}\,d\tilde{\gamma}_{\beta}\right]$ $\displaystyle\leq$ $\displaystyle\frac{c_{14}}{\varepsilon}\,\int_{B_{2\varepsilon}(0)}\frac{|\tilde{u}(x)|}{|x|}\,d\tilde{\gamma}_{\beta},$ then, by the fact that $u\in L^{1}(\Omega,\frac{1}{|x|}d\gamma_{\beta})$, it follows that (4.7) $\lim_{\varepsilon\to 0^{+}}\displaystyle\int_{B_{2\varepsilon}(0)}\frac{|\tilde{u}(x)|}{|x|}\,d\tilde{\gamma}_{\beta}=0\quad{\rm and}\quad\lim_{\varepsilon\to 0^{+}}\varepsilon\Big{|}\int_{\tilde{\Omega}}\tilde{u}\mathcal{L}_{\beta}^{*}\zeta_{\varepsilon}\,d\tilde{\gamma}_{\beta}\Big{|}=0.$ For $|a|\geq 1$, we have that $k_{a}^{2}\leq c_{15}\varepsilon^{|a|-1}\Big{|}\int_{\tilde{\Omega}}\tilde{u}\mathcal{L}_{\beta}^{*}\zeta_{\varepsilon}\,d\tilde{\gamma}_{\beta}\Big{|}\to 0\quad{\rm as}\quad\varepsilon\to 0,$ then we have $k_{a}=0$ by arbitrary of $\varepsilon>0$ in (4.5) with $|a|\geq 1$, thus, (4.8) $L(\zeta)=\int_{\tilde{\Omega}}\left[\tilde{u}\mathcal{L}_{\beta}^{*}\zeta-\tilde{f}\zeta\right]\,d\tilde{\gamma}_{\beta}=k_{0}\zeta(0),\quad\ \forall\,\xi\in C^{\infty}_{c}(\tilde{\Omega}).$ For any $\zeta\in C^{1.1}_{c}(\tilde{\Omega})$, by taking a sequence $\zeta_{n}\in C^{\infty}_{c}(\tilde{\Omega})$ converging to $\zeta$, we obtain that (4.8) holds for any $\zeta\in C^{1.1}_{c}(\tilde{\Omega})$. Now we fix $\xi\in C^{1.1}_{0}(\Omega)$ with compact support in $\Omega\cup\\{(x^{\prime},0)\in\mathbb{R}^{N-1}\times\mathbb{R}:|x^{\prime}|<r_{0}\\}$, then $\xi/x_{N}\in C^{1.1}(\bar{\Omega})$ and we may do $x_{N}$-even extension of $\xi/x_{N}$ in $\tilde{\Omega}$, denoting by $\tilde{\xi}$, then $\tilde{\xi}\in C^{1.1}_{c}(\tilde{\Omega})$, by the $x_{N}$-even extension, we have that $\tilde{\xi}(0)=\frac{\partial\xi}{\partial x_{N}}(0).$ So it follows from (4.8) that (4.9) $\int_{\Omega}\Big{(}\tilde{u}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})-\tilde{f}\frac{\xi}{x_{N}}\Big{)}\,d\gamma_{\beta}=k_{0}\frac{\partial\xi}{\partial x_{N}}(0),\quad\ \forall\,\xi\in C^{\infty}_{c}(\tilde{\Omega}),$ so (4.2) holds. $(ii)$ By the linearity of $\mathcal{L}_{\beta}$, we may assume that $f\geq 0$. Let $f_{n}=f\eta_{n}$, where $\eta_{n}(r)=1-\eta_{0}(nr)$ for $r\geq 0$, where $\eta_{0}$ satisfies (2.1) and let $v_{n}$ be solution of (2.8) where $f$ is replaced by $f_{n}$. We see that $f_{n}$ is bounded and for any $\xi\in C^{1.1}_{0}(\Omega)$, (4.10) $\int_{\Omega}v_{n}\,\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=\int_{\Omega}f_{n}\frac{\xi}{x_{N}}\,d\gamma_{\beta}.$ Then taking $\xi=w_{2}$ in Lemma 2.6, we have that $v_{n}$ is uniformly bounded in $L^{1}(\Omega,\,d\gamma_{\beta})$ and in $L^{1}(\Omega,\,x_{N}^{-1}d\gamma_{\beta})$, that is, $\|v_{n}\|_{L^{1}(\Omega,\,x_{N}^{-1}d\gamma_{\beta})}\leq\|\frac{\xi}{x_{N}}\|_{L^{\infty}(\Omega)}\|f_{n}\|_{L^{1}(\Omega,d\gamma_{\beta})}\leq\|\frac{\xi}{x_{N}}\|_{L^{\infty}(\Omega)}\|f\|_{L^{1}(\Omega,d\gamma_{\beta})}.$ Moreover, $\\{v_{n}\\}$ is increasing, and then there exists $v_{f}$ such that $v_{n}\to v_{f}\quad{\rm a.e.\ in}\ \ \Omega\quad{\rm and\ in}\ \ L^{1}(\Omega,\,x_{N}^{-1}d\gamma_{\beta}).$ Then we have that $\int_{\Omega}v_{f}\mathcal{L}_{\beta}^{*}(\xi)\,d\gamma_{\beta}=\int_{\Omega}f\xi\,d\gamma_{\beta},\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ Since $f\in C^{\gamma}(\overline{\Omega}\setminus\\{0\\})$, then it follows by the standard regularity theory that $v_{f}\in C^{2}(\Omega)$. We claim that $v_{f}$ is a classical solution of (4.1). From Corollary 2.8 in [31] with $L^{*}=\mathcal{L}_{\beta}^{*}$, which is strictly elliptic in $\Omega\setminus B_{r}(0)$, we have that for $q<\frac{N}{N-1}$, (4.11) $\displaystyle\|v_{n}\lambda_{\beta}\|_{W^{1,q}(\Omega_{2r})}$ $\displaystyle\leq$ $\displaystyle c_{16}\|f\lambda_{\beta}\|_{L^{1}(\Omega\setminus B_{r}(0))}+c_{16}\|v_{n}\lambda_{\beta}\|_{L^{1}(\Omega\setminus B_{r}(0))}$ $\displaystyle\leq$ $\displaystyle c_{17}\|f\|_{L^{1}(\Omega,\,d\gamma_{\beta})},$ where $\Omega_{2r}=\\{x\in\Omega\setminus B_{2r}(0):\,\rho(x)>2r\\}.$ We see that $\displaystyle-\Delta v_{n}=-\frac{\beta}{|x|^{2}}v_{n}+f.$ For any compact set $K$ in $\Omega$, it is standard to improve the regularity $v_{n}$ $\|v_{n}\|_{C^{2,\lambda}(K)}\leq c_{18}[\|f\|_{L^{1}(\Omega,\,d\gamma_{\beta})}+\|f\|_{C^{\lambda}(K)}]$ where $c_{18}>0$ is independent of $n$. Then $v_{f}$ is a classical solution of (4.1) verifying the identity (4.12) $\int_{\Omega}v_{f}\,\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=\int_{\Omega}\frac{f\xi}{x_{N}}\,d\gamma_{\beta},\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ Together with the fact that $u_{k,f}=k\Lambda^{\Omega}_{\beta}+v_{f}$, we conclude that the function $u_{k,f}$ is a solution of (4.1), verifying the identity (4.2) by (4.12). Finally, we prove the uniqueness. In fact, let $w_{k,f}$ be a solution of (4.1) verifying the identity (4.2). $\int_{\Omega}(u_{k,f}-w_{k,f})\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=0.$ For any Borel subset $O$ of $\Omega$, Corollary 2.7 implies that problem (4.13) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}^{*}(\frac{u}{x_{N}})=\zeta_{n}&{\rm in}\;\;\Omega,\\\\[2.84526pt] \phantom{\mathcal{L}_{\mu}^{*}}\displaystyle u=0&{\rm on}\;\;\partial{\Omega},\end{array}\right.$ has a solution $\eta_{\omega,n}\in C^{2}(\Omega)\cap C^{0.1}_{0}(\Omega)$, where $\zeta_{n}:\bar{\Omega}\mapsto[0,1]$ is a $C^{1}(\bar{\Omega})$ function such that $\zeta_{n}\to\chi_{O}\;{\rm{in}}\ L^{\infty}(\Omega)\;{\rm{as}}\ n\to\infty.$ Therefore by passing to the limit as $n\to\infty$, we have that $\displaystyle\int_{O}(u_{k,f}-w_{k,f})d\gamma_{\beta}=0,$ which implies that $u_{k,f}=w_{k,f}$ a.e. in $\Omega$ and then the uniqueness holds true. $\Box$ ###### Remark 4.2. Let $u_{f}$ be the solution of (4.1) verifying the identity (4.2) with $k=0$, then $u_{f}$ satisfies the isolated singular behavior (1.14). In fact, letting $f\geq 0$, then $u_{f}\geq 0$ in $\Omega$. So if (1.14) fails, it implies by the positivity of $u_{f}$, that $\liminf_{t\to 0^{+}}\inf_{z\in S^{N-1}_{+}}\frac{u_{f}(tz)}{\Lambda_{\beta}(tz)}=l_{0}>0$ and $\tilde{u}_{f}:=u_{f}-l_{0}\Lambda_{\beta}^{\Omega}$ is a solution of (4.1). By Lemma 2.2, we have that $\tilde{u}_{f}\geq 0$ in $\Omega$, By the approximating procedure, $\tilde{u}_{f}$ verifies the identity (4.2) with $k=0$, which is impossible with the fact that $u_{f}-\tilde{u}_{f}=l_{0}\Lambda_{\beta}^{\Omega}$, which satisfies $\int_{\Omega}(u_{f}-\tilde{u}_{f})\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=l_{0}c_{\beta}\frac{\partial\xi}{\partial x_{N}}(0),\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ ### 4.2. Nonzero Dirichlet boundary Recall that $P_{\Omega}$ is Poisson’s Kernel of $-\Delta$ in $\Omega$ and $\mathbb{P}_{\Omega}[g](x)=\displaystyle\int_{\partial\Omega}P_{\Omega}(x,y)g(y)d\omega(y).$ It is known that if $g$ is continuous, $\mathbb{P}_{\Omega}[g]$ is a solution of (4.14) $\left\\{\begin{array}[]{lll}\displaystyle-\Delta u=0\quad{\rm in}\;\;\Omega,\\\\[2.84526pt] \phantom{-\Delta}\displaystyle u=g\quad{\rm on}\;\;\partial{\Omega}.\end{array}\right.$ Multiply $\frac{\xi\lambda_{\beta}}{x_{N}}$ where $\xi\in C^{1.1}_{0}(\Omega)$ and integrate over $\Omega$, then we have that $\displaystyle 0$ $\displaystyle=$ $\displaystyle\int_{\Omega}(-\Delta\mathbb{P}_{\Omega}[g])\frac{\xi\lambda_{\beta}}{x_{N}}dx$ $\displaystyle=$ $\displaystyle\int_{\partial\Omega}\mathbb{P}_{\Omega}[g]\nabla(\frac{\xi\lambda_{\beta}}{x_{N}})\cdot\nu d\omega+\int_{\Omega}\mathbb{P}_{\Omega}[g]\Big{(}-\Delta(\frac{\xi\lambda_{\beta}}{x_{N}})\Big{)}dx$ $\displaystyle=$ $\displaystyle\int_{\partial\Omega}g\frac{\partial\xi}{\partial\nu}d\omega_{\beta}+\int_{\Omega}\mathbb{P}_{\Omega}[g]\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}-\beta\int_{\Omega}\frac{\mathbb{P}_{\Omega}[g]}{|x|^{2}}\frac{\xi}{x_{N}}d\gamma_{\beta},$ that is, for any $\xi\in C^{1.1}_{0}(\Omega)$, there holds (4.15) $\int_{\Omega}\mathbb{P}_{\Omega}[g]\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=\beta\int_{\Omega}\frac{\mathbb{P}_{\Omega}[g]}{|x|^{2}}\frac{\xi}{x_{N}}d\gamma_{\beta}-\int_{\partial\Omega}g\frac{\partial\xi}{\partial\nu}d\omega_{\beta}.$ ###### Lemma 4.3. Let $\beta\in[\beta_{0},\,+\infty)\setminus\\{0\\}$, $d\tilde{\omega}_{\beta}=(1+|x|^{\tau_{+}(\beta)})d\omega(x)$ and $g\geq 0$. We have that $(i)$ If $g\in C(\partial\Omega\setminus\\{0\\})\cap L^{1}(\partial\Omega,\,d\tilde{\omega}_{\beta})$, then $\frac{1}{|\cdot\;|^{2}}\mathbb{P}_{\Omega}[g]\in L^{1}(\Omega,\,d\gamma_{\beta}).$ $(ii)$ If $g\in C(\partial\Omega\setminus\\{0\\})$ and (4.16) $\lim_{r\to 0^{+}}\int_{\partial\Omega\setminus B_{r}(0)}g\,d\tilde{\omega}_{\beta}=+\infty,$ then $\lim_{r\to 0^{+}}\int_{\Omega\setminus B_{r}(0)}\frac{1}{|x|^{2}}\mathbb{P}_{\Omega}[g](x)d\gamma_{\beta}=+\infty.$ Proof. From Proposition 2.1 in [2] that (4.17) $c_{19}\rho(x)|x-y|^{-N}\leq P_{\Omega}(x,y)\leq c_{20}\rho(x)|x-y|^{-N},\quad x\in\Omega,\ y\in\partial\Omega,$ where $\rho(x)={\rm dist}(x,\partial\Omega)$. Since $g$ is continuous in $\partial\Omega\setminus\\{0\\}$ and $\Omega$ is flat near the origin, we can only consider the integrability of $\frac{1}{|\cdot\;|^{2}}\mathbb{P}_{\Omega}[g]$ near the origin. Fix $r=r_{0}/2$, let $B^{\prime}_{r}(0)=\\{x^{\prime}\in\mathbb{R}^{N-1}:|x^{\prime}|<r\\}$ and $e_{(y^{\prime},0)}=(\frac{y^{\prime}}{|y^{\prime}|},\,0)$ for $y^{\prime}\not=0$, then $\displaystyle\int_{B_{r}^{+}(0)}\frac{1}{|x|^{2}}\mathbb{P}_{\Omega}[g]d\gamma_{\beta}$ $\displaystyle\geq$ $\displaystyle c_{21}\int_{B_{r}^{+}(0)}\int_{B^{\prime}_{r}(0)\setminus\\{0\\}}g(y^{\prime})|x-(y^{\prime},0)|^{-N}\frac{x_{N}^{2}}{|x|^{2}}|x|^{\tau_{+}(\beta)}\,dy^{\prime}dx$ $\displaystyle=$ $\displaystyle c_{22}\int_{B_{r}^{\prime}(0)\setminus\\{0\\}}g(y^{\prime})|y^{\prime}|^{\tau_{+}(\beta)}\int_{B_{r/|y^{\prime}|}^{+}(0)}|z-e_{(y^{\prime},0)}|^{-N}\frac{z_{N}^{2}}{|z|^{2}}|z|^{\tau_{+}(\beta)}dzdy^{\prime}$ and $\displaystyle\int_{B_{r}^{+}(0)}\frac{1}{|x|^{2}}\mathbb{P}_{\Omega}[g]d\gamma_{\beta}$ $\displaystyle\leq$ $\displaystyle c_{23}\int_{B_{r}^{+}(0)}\int_{B^{\prime}_{r}(0)\setminus\\{0\\}}g(y^{\prime})|x-(y^{\prime},0)|^{-N}\frac{x_{N}^{2}}{|x|^{2}}|x|^{\tau_{+}(\beta)}\,dy^{\prime}dx$ $\displaystyle=$ $\displaystyle c_{24}\int_{B_{r}^{\prime}(0)\setminus\\{0\\}}g(y^{\prime})|y^{\prime}|^{\tau_{+}(\beta)}\int_{B_{r/|y^{\prime}|}^{+}(0)}|z-e_{(y^{\prime},0)}|^{-N}\frac{z_{N}^{2}}{|z|^{2}}|z|^{\tau_{+}(\beta)}dzdy^{\prime}.$ Now we do estimates for $\int_{B_{r/|y^{\prime}|}^{+}(0)}I(z)dz:=\int_{B_{r/|y^{\prime}|}^{+}(0)}|z-e_{(y^{\prime},0)}|^{-N}\frac{z_{N}^{2}}{|z|^{2}}|z|^{\tau_{+}(\beta)}dz,$ we have $\displaystyle 0<\int_{B_{\frac{1}{2}}^{+}(0)}I(z)\,dz\leq 2^{N}\int_{B_{\frac{1}{2}}^{+}(0)}|z|^{\tau_{+}(\beta)}dz,$ $\displaystyle 0<\int_{B_{\frac{1}{2}}^{+}(e_{(y^{\prime},0)})}I(z)\,dz$ $\displaystyle\leq$ $\displaystyle 2^{|\tau_{+}(\beta)|+2}\int_{B_{\frac{1}{2}}^{+}(e_{(y^{\prime},0)})}|z-e_{(y^{\prime},0)}|^{-N}z_{N}^{2}dz$ $\displaystyle\leq$ $\displaystyle 2^{|\tau_{+}(\beta)|+2}\int_{B_{\frac{1}{2}}^{+}(0)}|z|^{2-N}dz,$ and $\displaystyle\int_{B_{r/|y^{\prime}|}^{+}(0)\setminus\left(B_{\frac{1}{2}}^{+}(0)\cup B_{\frac{1}{2}}^{+}(e_{(y^{\prime},0)})\right)}I(z)dz$ $\displaystyle\leq$ $\displaystyle c_{25}\int_{B_{r/|y^{\prime}|}^{+}(0)\setminus B_{\frac{1}{2}}^{+}(0)}|z|^{-N+\tau_{+}(\beta)}\,dz$ $\displaystyle\leq$ $\displaystyle\left\\{\begin{array}[]{lll}\displaystyle c_{26}\int_{\mathbb{R}^{N}\setminus B_{\frac{1}{2}}(0)}|z|^{-N+\tau_{+}(\beta)}\,dz&{\rm if}\quad\beta<0,\\\\[5.69054pt] \phantom{}\displaystyle c_{26}|y^{\prime}|^{-\tau_{+}(\beta)}&{\rm if}\quad\beta>0\end{array}\right.$ $\displaystyle\leq$ $\displaystyle c_{27}(1+|y^{\prime}|^{-\tau_{+}(\beta)})$ and $\displaystyle\int_{B_{r/|y^{\prime}|}^{+}(0)\setminus\left(B_{\frac{1}{2}}^{+}(0)\cup B_{\frac{1}{2}}^{+}(e_{(y^{\prime},0)})\right)}I(z)\,dz$ $\displaystyle\geq$ $\displaystyle c_{28}\int_{B_{r/|y^{\prime}|}^{+}(0)\setminus B_{\frac{1}{2}}^{+}(0)}|z|^{-N+\tau_{+}(\beta)}\,dz$ $\displaystyle\geq$ $\displaystyle c_{29}(1+|y^{\prime}|^{-\tau_{+}(\beta)}).$ Thus, we have that (4.19) $c_{30}\int_{B_{r}^{\prime}(0)\setminus\\{0\\}}g(y^{\prime})d\tilde{\omega}(y^{\prime})\leq\int_{B_{r}^{+}(0)}\frac{1}{|x|^{2}}\mathbb{P}_{\Omega}[g]d\gamma_{\beta}\leq c_{31}\int_{B_{r}^{\prime}(0)\setminus\\{0\\}}g(y^{\prime})d\tilde{\omega}(y^{\prime}),$ which, together with the fact that $\mathbb{P}_{\Omega}[g]$ is nonnegative and bounded in $\Omega\setminus B_{r}^{+}(0)$, proves Lemma 4.3. $\Box$ We remark that Lemma 4.3 provides estimates for transforming the boundary data into the nonhomogeneous term. Now we are ready to prove Theorem 1.2 part $(i)$ where we distinguish two cases $\beta\in[\beta_{0},\,0]$ and $\beta>0$. Proof of Theorem 1.2. Part $(i)$. The existence for $g\in L^{1}(\partial\Omega,\,d\tilde{\omega}_{\beta})$. Let $\bar{f}=f-\frac{\beta}{|\cdot\;|^{2}}\mathbb{P}_{\Omega}[g].$ Then it follows from Lemma 4.3 part $(i)$ that $\bar{f}\in L^{1}(\Omega,\,d\gamma_{\beta})$ and applying Theorem 4.1 part $(i)$, problem (4.1) verifying (4.2) for $k\in\mathbb{R}$ and replaced $f$ by $\bar{f}$ admits a unique solution of $u_{f}$. Denote $u_{f,g}:=u_{f}+\mathbb{P}_{\Omega}[g]$, then $\mathcal{L}_{\beta}u_{f,g}=f\quad{\rm and}\quad u_{f,g}=g\quad{\rm on}\ \ \partial\Omega\setminus\\{0\\}.$ Together with (4.2) and (4.15), we have that $u_{f,g}$ verifies (1.11) and it is the unique solution of problem (4.1) verifying (4.2) for that $k$. Case of $\beta\in[\beta_{0},\,0]$. Then $d\tilde{\omega}_{\beta}$ is equivalent to $d\omega_{\beta}$, so $L^{1}(\partial\Omega,\,d\tilde{\omega}_{\beta})=L^{1}(\partial\Omega,\,d\omega_{\beta})$ and we are done. Case of $\beta>0$. We note that $L^{1}(\partial\Omega,\,d\tilde{\omega}_{\beta})\subsetneqq L^{1}(\partial\Omega,\,d\omega_{\beta}).$ So for $g\in L^{1}(\partial\Omega,\,d\omega_{\beta})\setminus L^{1}(\partial\Omega,\,d\tilde{\omega}_{\beta})$, we may assume $g\geq 0$ by linearity of $\mathcal{L}_{\beta}$. Let (4.20) $\eta_{n}(s)=1-\eta_{0}(ns)\quad{\rm and}\quad g_{n}(x)=g(x)\eta_{n}(|x|),$ where $\eta_{0}$ is defined in (2.1). Then $\\{g_{n}\\}_{n}\subset L^{1}(\partial\Omega,\,d\tilde{\omega}_{\beta})$ is an increasing sequence of functions. For simplicity, we assume that $f=0$. Then the problem (4.21) $\left\\{\begin{array}[]{lll}\mathcal{L}_{\beta}^{*}u=0&{\rm in}\;\;\Omega,\\\\[2.84526pt] \phantom{\mathcal{L}_{\beta}}u=g_{n}&{\rm on}\;\;\partial\Omega\setminus\\{0\\}\end{array}\right.$ has a unique solution of $u_{n}$ verifying the identify (4.22) $\int_{\Omega}u_{n}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=-\int_{\partial\Omega}g_{n}\frac{\partial\xi}{\partial\nu}d\omega_{\beta},\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ Since $0\leq g_{n}\leq g$ and $g\in L^{1}(\partial\Omega,\,d\omega_{\beta})$, we may expand the text function space including $w_{1}$, $w_{2}$, which are the solutions of (2.13) and (2.14) respectively. Taking $\xi=w_{1}$ and then $w_{2}$ , we derive that $\|u_{n}\|_{L^{1}(\Omega)}\leq c_{32}\|g_{n}\|_{L^{1}(\partial\Omega,\,d\omega_{\beta})}\leq c_{33}\|g\|_{L^{1}(\partial\Omega,\,d\omega_{\beta})}$ and $\|u_{n}\|_{L^{1}(\Omega,\,x_{N}^{-1}d\gamma_{\beta})}\leq c_{34}\|g\|_{L^{1}(\partial\Omega,\,d\omega_{\beta})}.$ We notice that $u_{n}\geq 0$ and the mapping $n\mapsto u_{n}$ is increasing, then by the monotone converge theorem, we have that there exists $u$ such that $u_{n}$ converging to $u$ in $L^{1}(\Omega,\frac{1}{x_{N}}d\gamma_{\beta})$. Since $\xi\in C_{0}^{1.1}(\Omega)$, we have that $|\mathcal{L}_{\beta}^{*}(\xi/x_{N})|\leq cx_{N}^{-1}.$ Pass to the limit of (4.22), we have that $u$ verifies that (4.23) $\int_{\Omega}u\mathcal{L}_{\beta}^{*}(\xi/x_{N})\,d\gamma_{\beta}=-\int_{\partial\Omega}g\frac{\partial\xi}{\partial\nu}d\omega_{\beta},\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ From standard interior regularity, we have that $u$ is a classical solution $\left\\{\begin{array}[]{lll}\mathcal{L}_{\beta}^{*}u=0\quad{\rm in}\;\;\Omega,\\\\[2.84526pt] \phantom{\mathcal{L}_{\beta}}u=g\quad{\rm on}\;\;\partial\Omega\setminus\\{0\\},\end{array}\right.$ which ends the proof. $\Box$ ## 5\. Nonexistence In this subsection, we establish the approximation of the fundamental solution $G_{\mu}$. ###### Lemma 5.1. $(i)$ Let $\\{\delta_{n}\\}_{n}$ be a sequence of nonnegative $L^{\infty}$-functions defined in $\Omega$ such that ${\rm supp}\,\delta_{n}\subset B_{r_{n}}(0)\cap\Omega,$ where $r_{n}\to 0$ as $n\to+\infty$ and $\int_{\Omega}\delta_{n}\xi dx\to\frac{\partial\xi(0)}{\partial x_{N}}\quad{\rm as}\quad n\to+\infty,\quad\forall\xi\in C_{0}^{1}(\Omega).$ For any $n$, let $w_{n}$ be the unique solution of the problem in the $d\gamma_{\beta}$-distributional sense (5.1) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}u=\delta_{n}/\lambda_{\beta}&{\rm in}\;\;{\Omega}\setminus\\{0\\},\\\\[5.69054pt] \phantom{L_{\beta}}\displaystyle u=0&{\rm on}\;\;\partial{\Omega},\\\\[5.69054pt] \phantom{}\displaystyle\lim_{r\to 0^{+}}\sup_{x\in\partial_{+}B_{r}(0)}\frac{|u(x)|}{\Lambda_{\beta}(x)}=0.\end{array}\right.$ Then $\lim_{n\to+\infty}w_{n}(x)=\frac{1}{c_{\beta}}\Lambda_{\beta}^{\Omega}(x),\quad\forall\,x\in\Omega\setminus\\{0\\}$ and for any compact set $K\subset\Omega\setminus\\{0\\}$, (5.2) $w_{n}\to\frac{1}{c_{\beta}}\Lambda_{\beta}^{\Omega}\quad{\rm as}\quad n\to+\infty\ \ {\rm in}\ \ C^{2}(K).$ $(ii)$ Let $\\{\sigma_{n}\\}_{n}$ be a sequence of nonnegative $L^{\infty}$ functions defined on $\partial\Omega$ such that ${\rm supp}\,\sigma_{n}\subset\partial\Omega\cap B_{r_{n}}(0),$ where $r_{n}\to 0$ as $n\to+\infty$ and $\int_{\partial\Omega}\sigma_{n}\zeta d\omega(x)\to\zeta(0)\quad{\rm as}\quad n\to+\infty,\quad\forall\zeta\in C^{1}(\partial\Omega).$ For any $n$, let $v_{n}$ be the unique solution of the problem (5.3) $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\beta}u=0&{\rm in}\;\;{\Omega}\setminus\\{0\\},\\\\[5.69054pt] \phantom{}\displaystyle u=\frac{\sigma_{n}}{|\cdot|^{\tau_{+}(\beta)}}&{\rm on}\;\;\partial{\Omega}\setminus\\{0\\}\end{array}\right.$ subject to $\int_{\Omega}v_{n}\mathcal{L}_{\beta}^{*}(\xi/x_{N})\,d\gamma_{\beta}=-\int_{\partial\Omega}\sigma_{n}\frac{\partial\xi}{\partial\nu}d\omega,\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ Then $\lim_{n\to+\infty}v_{n}(x)=\frac{1}{c_{\beta}}\Lambda_{\beta}^{\Omega}(x),\quad\forall\,x\in\Omega\setminus\\{0\\}$ and for any compact set $K\subset\Omega\setminus\\{0\\}$, (5.2) holds true. Proof. From Lemma 2.3, problems (5.1) and (5.3) have unique solutions $w_{n},v_{n}\geq 0$ respectively and satisfying that (5.4) $\int_{\Omega}w_{n}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=\int_{\Omega}\delta_{n}\xi\,dx,\quad\forall\,\xi\in C^{1,1}_{0}(\Omega)$ and (5.5) $\int_{\Omega}v_{n}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=-\int_{\partial\Omega}\sigma_{n}\frac{\partial\xi}{\partial\nu}d\omega_{\beta},\quad\forall\,\xi\in C^{1,1}_{0}(\Omega).$ By taking $\xi=\xi_{0}$, the solution of (2.13), we obtain that $\|w_{n}\|_{L^{1}(\Omega,\,d\gamma_{\beta})}\leq\|\xi_{0}\|_{L^{\infty}(\Omega)}\|\delta_{n}\|_{L^{1}(\Omega)}=\|\xi_{0}\|_{L^{\infty}(\Omega)}.$ For any $r>0$, take $\xi$ with the support in $\Omega\setminus B_{r}(0)$, then $\xi\in C^{1.1}_{c}(\overline{\Omega\setminus B_{r}(0)})$, $\int_{\Omega\setminus B_{r}(0)}w_{n}\mathcal{L}_{\mu}^{*}(\xi)\,d\gamma_{\beta}=0.$ Take $\xi$ the solution of (2.17) with $f(x)=\frac{1}{|x|}$, we have that (5.6) $\int_{\Omega}w_{n}|x|^{-1}\,d\gamma_{\beta}=\int_{\Omega}\delta_{n}\xi\,dx\leq\|\xi_{0}\|_{L^{\infty}(\Omega)}$ and (5.7) $\int_{\Omega}v_{n}|x|^{-1}\,d\gamma_{\beta}=-\int_{\partial\Omega}\sigma_{n}\frac{\partial\xi}{\partial\nu}d\omega_{\beta}\leq\|\nabla\xi_{0}\|_{L^{\infty}(\Omega)}.$ So $w_{n},v_{n}$ are uniform bounded in $L^{1}(\Omega,\,|x|^{-1}d\gamma_{\beta})$. From Corollary 2.8 in [31] with $L^{*}=\mathcal{L}_{\mu}^{*}$, which is strictly elliptic in $\Omega\setminus B_{r}(0)$, we have that for $q<\frac{N}{N-1}$, $\displaystyle\|w_{n}\lambda_{\beta}\|_{W^{1,q}(\Omega_{2r})}\leq c_{35}\|\delta_{n}\|_{L^{1}(\Omega\setminus B_{r}(0))}+c_{36}\|w_{n}\|_{L^{1}(\Omega\setminus B_{r}(0),\,d\gamma_{\beta})}\leq c_{37}$ and $\displaystyle\|v_{n}\lambda_{\beta}\|_{W^{1,q}(\Omega_{2r})}$ $\displaystyle\leq$ $\displaystyle c_{38}\|\sigma_{n}\|_{L^{1}(\partial\Omega\setminus B_{r}(0))}+c_{39}\|v_{n}\|_{L^{1}(\Omega\setminus B_{r}(0),\,d\gamma_{\beta})}\leq c_{40},$ where $\Omega_{2r}=\\{x\in\Omega\setminus B_{2r}(0):\,\rho(x)>2r\\}.$ By the compact embedding $W^{1,q}(\Omega_{2r})\hookrightarrow L^{1}(\Omega_{2r}),$ up to some subsequence, there exists $w_{\infty},\,v_{\infty}\in W^{1,q}_{loc}(\Omega)\cap L^{1}(\Omega,\,d\gamma_{\beta})$ such that $w_{n}\to w_{\infty}\quad{\rm as}\quad n\to+\infty\quad{\rm a.e.\ \ in}\ \ \Omega\ \ {\rm and\ in}\quad L^{1}(\Omega,\,d\gamma_{\beta})$ and it follows by (5.4) and (5.5) that for $\xi\in C^{1.1}_{0}(\Omega)$, $\int_{\Omega}w_{\infty}\mathcal{L}_{\beta}^{*}(\xi)\,d\gamma_{\beta}=\int_{\Omega}v_{\infty}\mathcal{L}_{\beta}^{*}(\xi)\,d\gamma_{\beta}=\frac{\partial\xi}{\partial x_{N}}(0).$ Furthermore, $\int_{\Omega}(w_{\infty}-\frac{1}{c_{\beta}}G_{\beta})\mathcal{L}_{\beta}^{*}(\xi)\,d\gamma_{\beta}=0.$ From the Kato’s inequality, we deduce that $w_{\infty}=v_{\infty}=\frac{1}{c_{\beta}}\Lambda_{\beta}^{\Omega}\quad{a.e.}\;\;\Omega.$ Proof of (5.2). For any $x_{0}\in\Omega\setminus\\{0\\}$, let $r_{0}=\frac{1}{4}\\{|x_{0}|,\,\rho(x_{0})\\}$ and $\mu_{n}=w_{n}\eta,$ where $\eta(x)=\eta_{0}(\frac{|x-x_{0}|}{r_{0}})$. There exists $n_{0}>0$ such that for $n\geq n_{0}$, ${\rm supp}\mu_{n}\cap B_{r_{n}}(0)=\emptyset.$ Then $\displaystyle-\Delta\mu_{n}(x)$ $\displaystyle=$ $\displaystyle-\Delta w_{n}(x)\eta(x)-2\nabla w_{n}\cdot\nabla\eta-w_{n}\Delta\eta$ $\displaystyle=$ $\displaystyle-2\nabla w_{n}\cdot\nabla\eta-w_{n}\Delta\eta,$ where $\nabla\eta$ and $\Delta\eta$ are smooth. We observe that $w_{n}\in W^{1,q}(B_{2r_{0}}(x_{0}))$ and $-2\nabla w_{n}\cdot\nabla\eta-w_{n}\Delta\eta\in L^{q}(B_{2r_{0}}(x_{0})),$ then we have that $\|\mu_{n}\|_{W^{2,q}(B_{r_{0}}(x_{0}))}\leq c\|w_{n}\|_{L^{1}(\Omega,\,d\gamma_{\beta})},$ where $c>0$ is independent of $n$. Thus, $-2\nabla w_{n}\cdot\nabla\eta- w_{n}\Delta\eta\in W^{1,q}(B_{r_{0}}(x_{0})),$ repeat above process $N_{0}$ steps, for $N_{0}$ large enough, we deduce that $\|w_{n}\|_{C^{2,\gamma}(B_{\frac{r_{0}}{2^{N_{0}}}}(x_{0}))}\leq c\|w_{n}\|_{L^{1}(\Omega,\,d\gamma_{\beta})},$ where $\gamma\in(0,1)$ and $c>0$ is independent of $n$. As a conclusion, (5.2) follows by Arzelà-Ascola theorem and Heine-Borel theorem. The above process also holds for $v_{n}$. This ends the proof. $\Box$ Proof of Theorem 1.2. Part $(ii)$. From (1.12), one of the following two cases holds true, ${\rm case}\ 1:\lim_{r\to 0^{+}}\int_{\Omega\setminus B_{r}(0)}f\,d\gamma_{\beta}=+\infty,\;\;{\rm or\ case}\ 2:\;\;\lim_{r\to 0^{+}}\int_{\partial\Omega\setminus B_{r}(0)}g\,d\omega_{\beta}=+\infty.$ Case 1. We argue by contradiction. Assume that problem (1.2) has a nonnegative solution of $u_{f}$. Let $\\{r_{n}\\}_{n}$ be a sequence of strictly decreasing positive numbers converging to $0$. From the fact $f\in C_{loc}^{\gamma}(\overline{\Omega}\setminus\\{0\\})$, for any $r_{n}$ fixed, we have that $\lim_{r\to 0^{+}}\int_{(B_{r_{n}}(0)\setminus B_{r}(0))\cap\Omega}f(x)d\gamma_{\beta}=+\infty,$ then there exists $R_{n}\in(0,r_{n})$ such that $\int_{(B_{r_{n}}(0)\setminus B_{R_{n}}(0))\cap\Omega}fd\gamma_{\beta}=n.$ Let $\delta_{n}=\frac{1}{n}\lambda_{\beta}f\chi_{B_{r_{n}}(0)\setminus B_{R_{n}}(0)}$, then the problem $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\mu}u\cdot\lambda_{\beta}=\delta_{n}\qquad{\rm in}\quad{\Omega}\setminus\\{0\\},\\\\[2.84526pt] \phantom{L_{\mu}--}\displaystyle u=0\qquad{\rm on}\quad\partial{\Omega},\\\\[2.84526pt] \phantom{}\displaystyle\lim_{x\to 0}u(x)\Phi_{\mu}^{-1}(x)=0\end{array}\right.$ has a unique positive solution $w_{n}$ satisfying (in the usual sense) $\int_{\Omega}w_{n}\mathcal{L}_{\mu}(\lambda_{\beta}\xi)dx=\int_{\Omega}\delta_{n}\xi dx,\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ For any $\xi\in C^{1.1}_{0}(\Omega)$, we have that $\int_{\Omega}w_{n}\mathcal{L}_{\mu}^{*}(\xi)\,d\gamma_{\beta}=\int_{\Omega}\delta_{n}\xi\,dx\to\frac{\partial\xi}{\partial x_{N}}(0)\quad{\rm as}\quad n\to+\infty.$ Therefore, by Lemma 5.1 for any compact set $\mathcal{K}\subset\Omega\setminus\\{0\\}$ $\|w_{n}-\Lambda_{\beta}^{\Omega}\|_{C^{1}(\mathcal{K})}\to 0\quad{\rm as}\quad{n\to+\infty}.$ We fix a point $x_{0}\in\Omega$ and let $r_{0}=\frac{1}{2}\min\\{|x_{0}|,\,\rho(x_{0})\\}$ and $\mathcal{K}=\overline{B_{r_{0}}(x_{0})}$, then there exists $n_{0}>0$ such that for $n\geq n_{0}$, (5.8) $w_{n}\geq\frac{1}{2}G_{\mu}\quad{\rm in}\quad\mathcal{K}.$ Let $u_{n}$ be the solution (in the usual sense) of $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\mu}u\cdot\lambda_{\beta}=n\delta_{n}\quad{\rm in}\;\;{\Omega}\setminus\\{0\\},\\\\[2.84526pt] \phantom{L_{\mu}--}\displaystyle u=0\quad\ \ {\rm on}\;\;\partial{\Omega},\\\\[2.84526pt] \phantom{}\displaystyle\lim_{r\to 0^{+}}\sup_{x\in\partial_{+}B_{r}(0)}\frac{|u(x)|}{\Lambda_{\beta}(x)}=0,\end{array}\right.$ then we have that $u_{n}\geq nw_{n}\;\;{\rm in}\;\;\Omega.$ Together with (5.8), we derive that $u_{n}\geq\frac{n}{2}\Lambda_{\mu}^{\Omega}\quad{\rm in}\;\;\mathcal{K}.$ Then by comparison principle, we have that $u_{f}(x_{0})\geq u_{n}(x_{0})\to+\infty\quad{\rm as}\;\;n\to+\infty,$ which contradicts to the fact that $u_{f}$ is classical solution of (4.1). Case 2. Similarly for any $n\in\mathbb{N}$, we can take $r_{n}>R_{n}>0$ such that $r_{n}\to 0$ as $n\to+\infty$ and $\int_{(B_{r_{n}}(0)\setminus B_{R_{n}}(0))\cap\partial\Omega}gd\omega_{\beta}=n.$ Let $\sigma_{n}=\frac{1}{n}g\chi_{B_{r_{n}}(0)\setminus B_{R_{n}}(0)}$, $w_{n}$ be the solution of $\left\\{\begin{array}[]{lll}\displaystyle\mathcal{L}_{\mu}u=0\quad{\rm in}\;\;{\Omega}\setminus\\{0\\},\\\\[2.84526pt] \phantom{L_{\mu}}\displaystyle u=\sigma_{n}/|\cdot|^{\tau^{+}(\beta)}\quad\ \ {\rm on}\;\;\partial{\Omega},\end{array}\right.$ subject to $\int_{\Omega}w_{n}\mathcal{L}_{\beta}^{*}(\frac{\xi}{x_{N}})\,d\gamma_{\beta}=-\int_{\partial\Omega}\sigma_{n}\frac{\partial\xi}{\partial\nu}d\omega,\quad\forall\,\xi\in C^{1.1}_{0}(\Omega).$ Repeat the procedure in Case 1, we get a contradiction which completes the proof. $\Box$ Acknowledgements: H. Chen is supported by the Natural Science Foundation of China [Nos. 11661045, 11726614]. A. 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2024-09-04T02:54:55.410763

2002.12602

arxiv-papers

# Combining TCAD and Monte Carlo Methods to Simulate CMOS Pixel Sensors with a Small Collection Electrode using the Allpix2 Framework D. Dannheim K. Dort111Also at University of Giessen, Germany D. Hynds222Now at Nikhef, Amsterdam, Netherlands M. Munker A. Nürnberg333Now at KIT, Karlsruhe, Germany W. Snoeys S. Spannagel<EMAIL_ADDRESS>CERN, Geneva, Switzerland ###### Abstract Combining electrostatic field simulations with Monte Carlo methods enables realistic modeling of the detector response for novel monolithic silicon detectors with strongly non-linear electric fields. Both the precise field description and the inclusion of Landau fluctuations and production of secondary particles in the sensor are crucial ingredients for the understanding and reproduction of detector characteristics. In this paper, a CMOS pixel sensor with small collection electrode design, implemented in a high-resistivity epitaxial layer, is simulated by integrating a detailed electric field model from finite element TCAD into a Monte Carlo based simulation with the $\mathrm{Allpix}^{2}$ framework. The simulation results are compared to data recorded in test-beam measurements and very good agreement is found for various quantities such as cluster size, spatial resolution and efficiency. Furthermore, the observables are studied as a function of the intra-pixel incidence position to enable a detailed comparison with the detector behavior observed in data. The validation of such simulations is fundamental for modeling the detector response and for predicting the performance of future prototype designs. Moreover, visualization plots extracted from the charge carrier drift model of the framework can aid in understanding the charge propagation behavior in different regions of the sensor. ###### keywords: Simulation , Monte Carlo , Silicon Detectors , High Resistivity CMOS , TCAD , Drift-Diffusion , Geant4 ††journal: Nucl. Instr. Meth. A ###### Contents 1. 1 Introduction 2. 2 The High-Resistivity CMOS Process 3. 3 Detector Design under Investigation 4. 4 Simulation Flow 1. 4.1 Electrostatic Field Modeling with TCAD 2. 4.2 Energy Deposition with Geant4 3. 4.3 Charge Carrier Transport 4. 4.4 Digitization of Signals 5. 4.5 Data Processing and Storage 5. 5 Reconstruction and Analysis 1. 5.1 Reference tracks 2. 5.2 Clustering 3. 5.3 Reconstruction of the Cluster Position 6. 6 Systematic Uncertainties 1. 6.1 Free parameters 2. 6.2 Parameters constrained by measurements 7. 7 Validation With Test-Beam Data 1. 7.1 Cluster Charge 2. 7.2 Cluster Size 8. 8 Detector Performance 1. 8.1 Intrinsic Resolution 2. 8.2 Efficiency 9. 9 Summary & Outlook ## 1 Introduction Integrated monolithic CMOS technologies with small collection electrodes [1] are emerging technologies enabling advances in the design of next-generation high-performance silicon vertex and tracking detectors for high-energy physics. These technologies have allowed significant reductions in the material budget with respect to hybrid pixel detectors, while improving the signal-to-noise ratio and the position resolution that is achievable with CMOS sensors. However, the simulation of such devices remains challenging due to the complex field configuration in the sensor. Advanced simulation tools are required to understand and model the performance of detectors built in these technologies and to optimize the design of future prototypes. This paper presents a simulation performed with a combination of commonly used tools employed in silicon detector simulation. The $\mathrm{Allpix}^{2}$ framework [2] is used to combine TCAD-simulated electric fields with a Geant4 [3, 4, 5] simulation of the particle interaction with matter, to investigate the behavior of high-resistivity CMOS detectors and to compare the predicted performance with measurements recorded in a particle beam. This allows direct access to detector performance parameters such as spatial resolution and detection efficiency by taking into account the stochastic nature of the initial energy deposition. While many of these properties could also be investigated by advanced TCAD transient simulations, this approach is not practical owing to the high computing time for a single event and the high-statistics samples required to evaluate the effects related to the strong variation of the electric field in three dimensions. Instead, a simplified charge transport algorithm is used, taking as an input the electrostatic field map calculated by the TCAD simulation of the complex field configuration within the sensor. The algorithm takes into account effects like Landau fluctuations in the energy deposition and the production of secondary particles such as delta rays. With event simulation rates of several tens of Hertz, this allows for the generation of high-statistics samples necessary for detailed studies of the detector behavior. The paper is structured as follows. Section 2 provides a brief overview of the CMOS process under investigation, while the detector properties and the simulated setup are introduced in Section 3. The simulation is described in detail in Section 4, while Section 5 introduces the reconstruction of physical properties from the detector response. The sensitivity of the simulation to a range of parameters is examined in Section 6. The simulation is validated using data recorded in test-beam measurements in Section 7, while performance quantities are derived in Section 8 and compared with the values obtained from data. Finally, Section 9 summarizes the results and provides an outlook for future investigations of this technology. ## 2 The High-Resistivity CMOS Process Monolithic CMOS technologies incorporating the readout electronics in the sensor are attractive candidates for new detector designs to simplify the production and to benefit from a reduction of the material budget. By integrating the CMOS logic in doped wells separated from the inversely doped signal collection electrode, the size of the latter can be minimized as illustrated in Figure 1. The small collection electrode design allows the sensor capacitance to be reduced down to the order of $\mathrm{fF}$, enabling detector designs with low noise and detection thresholds, low analog power consumption, and large signal-to-noise ratio (SNR) [1]. Figure 1: Schematic cross section of a single pixel cell in the CMOS process under investigation. The elements shown are not to scale. Modified from [6]. Implemented in a standard silicon substrate, only a small depleted region evolves around the _pn_ -junction surrounding the collection electrode when applying a bias voltage between the doped wells and the backside of the sensor. The applicable bias voltage is limited to $-6\text{\,}\mathrm{V}$ by the process-specific breakdown voltage of the NMOS transistors [7]. In order to achieve a sizable depletion volume around the collection electrode, an epitaxial layer with high resistivity silicon can be used. The size of the depleted region forming in this epitaxial layer is restricted to the area around the collection electrode and, without additional modifications of the process, no full depletion of the sensor volume is achieved. In the CMOS process under investigation, the depleted region has the shape of a bubble as indicated by the white line in Figure 1, resulting in contributions to the overall detector response from both drift and diffusion of charge carriers. In addition, signal contributions are expected from charge carriers that are created in the highly _p_ -doped backside substrate and subsequently diffuse into the epitaxial layer. ## 3 Detector Design under Investigation The _Investigator_ test-chip is an analog test chip that has been developed within the ALICE ITS upgrade [8]. It has been investigated by the CLICdp collaboration to evaluate this technology in terms of sensor performance focussing on precise measurements of spatial resolution and detection efficiency [6, 9]. The digitization of signals is performed off-chip in the data acquisition system using one $65\text{\,}\mathrm{MHz}$ sampling analog- to-digital converter (ADC) per channel which records the full waveform of all detector channels, once a configurable triggering threshold has been exceeded in any of them [10]. It should be noted that the threshold values for data quoted below represent the offline analysis thresholds applied in addition to the triggering threshold of about $120\text{\,}\mathrm{e}$. The chip has a total thickness of $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The upper $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of the sensor, below the implants, consist of the epitaxially grown silicon with a resisitiviy of $1-$8\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$$ in which the depleted region forms, while the additional $75\text{\,}\mathrm{\SIUnitSymbolMicro m}$ represent the undepleted low- resistivity silicon substrate [7]. \begin{overpic}[width=216.81pt]{detector2.png} \put(25.0,5.0){Sensor} \put(35.0,10.0){\vector(2,1){15.0}} \put(90.0,5.0){PCB} \put(85.0,7.0){\vector(-4,1){23.0}} \end{overpic} Figure 2: Visualization of the simulated detector setup consisting of the CMOS sensor on a printed circuit board for support. The detector is oriented perpendicular to the beam incident from the left. The colored lines represent the primary and secondary particles propagated through the setup. While the actual detector contains several sub-matrices with $8\text{\times}8$ active pixels each, with different layouts such as altered collection electrode size, only one matrix has been simulated and is compared to data. The pixel cells of the chosen matrix have a pitch of $28\text{\times}28\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and feature the following geometrical parameters: the distance between the _p_ -wells and the collection electrode is $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and an octagonal collection electrode with a size of $2\text{\times}2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is placed in the center of the pixel cell. A bias voltage of $-6\text{\,}\mathrm{V}$ is applied to the _p_ -wells and a positive voltage of $0.8\text{\,}\mathrm{V}$ is applied to the collection electrode itself. The simulated detector is placed on a printed circuit board (PCB) as visualized in Figure 2. ## 4 Simulation Flow In the following section, the simulation of the detector in the $\mathrm{Allpix}^{2}$ framework is described. In order to avoid simulating a full beam telescope setup and performing a track reconstruction, the capabilities of the framework to record the Monte Carlo truth information about primary and secondary particles are exploited. Consequently, only a single CMOS detector and the source of ionizing particles are simulated as shown in Figure 2. The figure depicts the overlay of many events, as only a single primary particle is simulated in each event. The following sections describe the individual steps of the simulation in detail, providing information on the configuration of the respective $\mathrm{Allpix}^{2}$ modules where applicable and relevant. ### 4.1 Electrostatic Field Modeling with TCAD The electrostatic field in the epitaxial layer of the sensor is modeled using a three-dimensional TCAD simulation. The doping profile is taken from [9, 11] and resembles the technology described in Section 2, with the detector geometry introduced in Section 3. The simulation comprises a single pixel cell, and periodic boundary conditions allow the field to be replicated over the entire sensor. Figure 3: Magnitude of the electric field inside the pixel cell, simulated using TCAD. The visualization only shows the upper $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of the sensor with the epitaxial layer. The gray structures represent metal contacts used as terminals for the biasing voltages. The plane C1 indicated in gray corresponds to the cut presented in Figure 4 (color online). Figure 3 shows a visualization of the magnitude of the electric field in the three-dimensional pixel cell, with the corresponding voltages applied to the terminals via metal contacts indicated as gray structures. A low electric field is present in the _p_ -well rings as indicated by the blue region on the surface of the simulated pixel cell. The center of the _p_ -well rings is fitted with a squared opening that contains the collection electrode with a high-field region evolving around it. Figure 4: Magnitude of the electric field and field lines for a cut through the 3D TCAD simulation. The plot only depicts the upper $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of the sensor with the epitaxial layer, while the undepleted substrate region is omitted (color online). The strong inhomogeneities of the electric field in different regions of the pixel cell are best observed in a cut through the collection electrode, perpendicular to the sensor surface, as depicted in Figure 4. The high electric field strength close to the _pn_ -junction around the collection electrode decreases rapidly towards the sensor backside and the pixel corners. The white line indicates the depleted volume of the pixel cell. The electric field lines, indicated as black arrows, provide a first insight into the complexity of the field configuration in the sensor and the drift effects induced by this strong non-linearity. The low electric field regions in the pixel corners result in a slower charge carrier drift and an increased impact of diffusion, leading to an enhanced charge sharing which improves the position resolution without the need to reduce the pixel pitch. In the low- resistivity substrate, recombination of charge carriers is a relevant process owing to the higher doping concentration. The electrostatic field obtained from TCAD is converted to a regularly spaced mesh using the _Mesh Converter_ tool provided with the $\mathrm{Allpix}^{2}$ framework. This conversion speeds up the look-up of field values during the simulation by several orders of magnitude since all necessary interpolations are already performed offline prior to the simulation. A regular mesh granularity of $0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is chosen to correctly replicate the field in the high-density mesh regions of the TCAD simulation close to the implant. It has been verified that the selected granularity correctly replicates the TCAD simulation by comparing the two fields. Using an even finer granularity has not shown any significant improvement on the simulation results. Loading highly granular electrostatic fields in $\mathrm{Allpix}^{2}$ does not impact the performance of the simulation, but only the memory footprint of the program during execution. ### 4.2 Energy Deposition with Geant4 $\mathrm{Allpix}^{2}$ provides the _DepositionGeant4_ module, an interface to Geant4 [3, 4, 5] which facilitates the simulation of energy deposition in the sensor. A $120\text{\,}\mathrm{GeV}$ beam of $\mathup{{{\pi}}^{\scriptstyle{+}}}$ incident on the pixel detector is simulated, replicating the beam conditions of the test-beam measurements. The beam points along the positive _z_ -axis, perpendicular to the _xy_ -plane of the detector. The cross section of the beam is chosen to be significantly smaller than the detector surface to suppress effects stemming from the altered charge sharing behavior at the sensor edge. The energy deposited in the sensor by Geant4 is translated into charge carriers with a conversion factor of $3.64\text{\,}\mathrm{eV}$ per electron-hole pair [12]. The framework also stores the Monte Carlo truth information including secondary particles such as delta rays and their relation to the primary particles. This information can be exploited to establish a link between the incident particles and the electron-hole pairs created in the detector. The simulation is performed with the Photo-Absorption Ionization model (PAI) [13] to improve the description of energy deposition in thin sensors. This is of importance in spite of the total sensor thickness of $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$, since a majority of the charge carriers forming the final signal will originate from the $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ epitaxial layer. ⬇ [DepositionGeant4] physics_list = ”FTFP_BERT_EMY” enable_pai = true particle_type = ”Pi+” source_type = ”beam” source_energy = 120GeV source_position = 0um 0um -200um beam_size = 0.5mm beam_direction = 0 0 1 number_of_particles = 1 max_step_length = 1.0um Figure 5: Configuration section for the _DepositionGeant4_ module setting up the particle source in Geant4 for the initial energy deposition in the sensor. The module configuration used for the $\mathrm{Allpix}^{2}$ framework is provided in Listing 5. ### 4.3 Charge Carrier Transport The signal formation is simulated using a simplified model for charge carrier transport based on the notion of _collected charges_. The electron-hole pairs created by the incident particle are propagated along the electric field lines through the sensor volume using an adaptive fourth-order Runge-Kutta-Fehlberg (RKF) method [14] and a mobility parametrization which depends on the electric field vector [15]. The RKF method adapts the simulated time step depending on the position uncertainty derived from a fifth-order error estimation; the allowed range for time steps was set to $$0.5\text{\,}\mathrm{ps}$\leq\Delta t\leq$0.5\text{\,}\mathrm{ns}$$. While this model is not expected to reproduce a realistic time dependence of the signal, the final state of charge collected at the sensor implants is equivalent to the integrated induced current over the respective drift time. This approximation is valid since the Shockley-Ramo weighting field [16, 17] is negligible in most of the sensor volume owing to the small ratio between signal collection electrode size and sensor thickness. In the upper $25\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ of the sensor the charge carrier motion is a superposition of drift and diffusion, while in the lower $75\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ the charge carriers are only subject to random diffusion as the electric field is negligible. The propagation algorithm is halted after $22.5\text{\,}\mathrm{ns}$, the so- called integration time, and all charge carriers within a volume of $3\times 3\times$2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{3}$$ around each of the signal collection electrodes are attributed to the respective pixel signal. The volume has been chosen to cover the electrode implant itself as well as an additional volume accounting for the uncertainty in the final position of the transported charge carriers. The integration time has been chosen such that the simulation produces clusters with the same most probable value (MPV) for the cluster charge as obtained from data. This aims to emulate the physical process of charge carrier recombination in the silicon substrate, which might be modeled directly in future simulations as briefly discussed in Section 9. The systematic uncertainty introduced by this approach is discussed in Section 6. Charge carriers are transported in groups of five instead of individually to speed up the simulation process. The group size has been chosen such that an adequate number of transport steps is retained with the expected MPV for the signal of around $1.5\text{\,}\mathrm{k}\mathrm{e}$. It has been verified that this simplification does not affect the simulation result as further elaborated in Section 6. Figure 6: Visualization of the time evolution of the collected charge. Shown are snapshots at different times after the initial energy deposition with each line representing the drift and diffusion motion of a group of charge carriers. Only charge carrier groups which have reached the implant are drawn, all other charge carriers are omitted. The ionizing particle traverses the sensor along the $z$-axis in the center of a pixel cell, each plot represents three adjacent pixels. Figure 6 visualizes this transport model and shows the collection of charge carriers at the electrodes of the sensor. In this representation, only electrons that have reached a sensor implant within the integration time are shown. Electrons that are still in motion as well as holes are suppressed. The motion of each group of charge carriers is represented by one line and is shown at different integration times after the initial energy deposition. Here, the incident particle traversed the detector along the $z$-axis through the center of one pixel cell. After the first few hundred picoseconds, only charge carriers in the vicinity of the electrode are collected. The straight lines indicate that their motion is dominated by drift. With increasing integration time, the motion patterns of further groups of charge carriers arriving at the implant exhibit a strong contribution from diffusion as indicated by the numerous kinks in the respective paths. After about $15\text{\,}\mathrm{ns}$, lateral motion enables some charge carriers to be collected in the two adjacent pixel cells. The line graphs also allow visual distinction between the substrate and the epitaxially grown high-resistivity layer, which ends about $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the top of the sensor. A faster drift motion can be observed in the high-field region close to the backside of the epitaxial layer as straight lines; the contribution from substrate charge carriers diffusing into the epitaxial layer starts only after approximately $10\text{\,}\mathrm{ns}$. Figure 7: Three-dimensional visualization of the charge carrier motion, corresponding to the $20\text{\,}\mathrm{ns}$ snapshot shown as projection in Figure 6. In Figure 7, a three-dimensional representation of the line plot at $20\text{\,}\mathrm{ns}$ is presented. The lines end at five different points, each representing a different collection electrode. ⬇ [GenericPropagation] temperature = 293K charge_per_step = 5 timestep_min = 0.5ps timestep_max = 0.5ns integration_time = 20ns Figure 8: Configuration section for the _GenericPropagation_ module used to simulate the charge transport. The configuration provided in Listing 8 has been used for the charge carrier transport. Settings for creating line graphs of the charge carrier motion can be found in the $\mathrm{Allpix}^{2}$ user manual available from the project website [18]. ### 4.4 Digitization of Signals To simulate the response of the readout electronics, the charge carriers accumulated in the region around the signal collection electrode during the integration time are transformed into a digital signal. While the detector under investigation uses off-chip ADCs for the signal digitization as described in Section 3, the simulation aims to simulate an on-chip per-pixel threshold using the _DefaultDigitizer_ module of $\mathrm{Allpix}^{2}$. Equivalent noise values have been used where applicable, as discussed below. ⬇ [DefaultDigitizer] electronics_noise = 10e threshold = 40e threshold_smearing = 5e Figure 9: Configuration section used for the _DefaultDigitizer_ module of the simulation. An additional signal contribution, randomly drawn from a Gaussian distribution with a width of $10\text{\,}\mathrm{e}$ and a mean of $0\text{\,}\mathrm{e}$ is added to the signal to account for electronics noise present during digitization. The applied threshold is varied between $40\text{\,}\mathrm{e}$ and $700\text{\,}\mathrm{e}$, and a threshold dispersion, sampled from a Gaussian distribution with a width of $5\text{\,}\mathrm{e}$ and a mean of $0\text{\,}\mathrm{e}$, is added. For simplicity, the threshold dispersion is not a fixed offset calculated per-pixel, but randomly chosen per pixel hit. The setup of the module is summarized in Listing 9. ### 4.5 Data Processing and Storage The simulation results are stored in ROOT [19] trees using the _ROOTObjectWriter_ module. In order to speed up the process, the simulation is performed in two separate steps. In the first step, the energy deposition, charge carrier transport and summing of charge at the collection electrodes is performed. The result of this step is stored to disk. In a second step, the _ROOTObjectReader_ is used to read the information from the respective file and the final digitization step is performed. This allows to re-run this final section of the simulation on the full set of Monte Carlo events with different settings applied without the need to recompute the drift motions. A full threshold scan, performed on the same set of initial simulation events, thus only takes a couple of minutes instead of several hours required to create the initial data set. Since the threshold scan performed on the test-beam data has also been performed offline on the same data set [9], this is an adequate simplification of the simulation. The central simulation data set comprises about 2.5 million primary events which have been reprocessed for every threshold setting. In addition, several smaller data sets with different integration times have been produced in order to optimize agreement with data as discussed in Section 6. ## 5 Reconstruction and Analysis In the following, the reconstruction and analysis of the Monte Carlo events are discussed. The simulation was set up using known, independent parameters of the measurement setup, such as track resolution or charge threshold. Only the cluster charge MPV was used as direct observable provided by the detector to tune the simulation. All parameters were fixed before comparison with data for observables used to quantify the performance, such as cluster size, position resolution and efficiency. This blinded approach avoids drawing premature conclusions from the figures of merit and thus influencing the parameter optimization. Using only the MPV of the cluster charge for calibrating the simulation minimizes the correlation between simulation and data, and maximizes the prediction power of the simulation. ### 5.1 Reference tracks The Monte Carlo truth information provided by the $\mathrm{Allpix}^{2}$ framework is used as reference track information. All registered particles in the sensor are filtered and only primary particles entering the sensor from the outside, i.e. those without a parent particle, are selected for further analysis. This set of particles represents external tracks, and their position in the mid-plane of the sensor is calculated by linearly interpolating their entry and exit points registered by the framework. This position is then convolved with the track resolution at the device under test (DUT) of $2.0\text{\,}\mathrm{\SIUnitSymbolMicro m}$, in accordance with the value obtained for the beam telescope used for the acquisition of the test-beam data [20]. ### 5.2 Clustering The pixel hits registered by the simulated detector are grouped into clusters by starting with a seed pixel and adding all directly adjacent pixel hits to the cluster until no additional neighbors are found. This already allows basic properties of the simulation to be compared with data, namely cluster size as well as the shape of the cluster charge distribution. The total cluster charge is given by the sum of the individual pixel charges of the cluster. Its comparison with data allows the required integration time in the simplified simulation model to be adjusted to achieve the same integrated charge as seen in data. This procedure is described in detail in Section 6. The cluster size is defined as the total number of pixels contained in the respective cluster. It has a strong dependence on the drift and diffusion of the charge carriers in the sensors and is the primary measure for charge sharing between pixel cells. It thus allows evaluation of the performance of the simulation, e.g. how well the electric field is modeled. ### 5.3 Reconstruction of the Cluster Position For assessing the performance of the detector, a particle incidence position has to be extracted from the cluster information available. To replicate the analysis performed for the test-beam data, the charge-weighted center-of- gravity position of the cluster is corrected for non-linear charge sharing by an $\eta$ algorithm [21]. Figure 10: $\eta$-distribution along the $x$ axis, derived from simulation at a threshold of $40\text{\,}\mathrm{e}$. Since the $\eta$ distribution represents the charge sharing between two pixels only, for each cluster the two pixels with the highest charge, $Q_{1}$ and $Q_{2}$, are chosen to construct the $\eta$ variable independently in $x$ and $y$: $\displaystyle\eta_{k}=\frac{\sum_{i}k_{i}\cdot Q_{i}}{\sum_{i}Q_{i}}\qquad k=\\{x,y\\}\quad i=\\{1,2\\}$ where $k_{i}$ is the relative position between the two pixel centers. An example of the $\eta$ distribution in $x$ is depicted in Figure 10 for a pixel charge threshold of $40\text{\,}\mathrm{e}$. ## 6 Systematic Uncertainties The sensitivity of the simulation to different input parameters has been examined by varying the values within their respective uncertainties, if known, or within a reasonable range otherwise. The impact on the reconstructed observables was investigated. While some parameters exhibit little or no effect on the simulation results, others have a strong influence on the outcome. ### 6.1 Free parameters For the initial deposition of energy in the sensor, the influence of the maximum allowed step length of tracking primary and secondary particles through the sensor material has been evaluated by varying the respective value between $0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and no significant difference was observed. Since large parts of the sensor volume are undepleted, a strong impact of diffusion is expected which smears the initial position of the charge carriers. The charge carrier transport is mainly dominated by the precision of the numeric integration and its granularity. The number of charge carriers transported as group has been varied from a single charge carrier up to ten per group in order to study possible effects on the distribution at the implants. The effect on the reconstruction observables is found to be negligible. ### 6.2 Parameters constrained by measurements The behavior of the sensor has been shown to be very sensitive to the simulated physical properties of the CMOS sensor, i.e. the thickness of the epitaxial layer as well as the modeled electric field. Even small changes in the sensor design, such as a more simplistic approximation of the implant doping profiles in the TCAD design cause large changes in the resulting cluster size distributions and position resolution. It is therefore of paramount importance to model the sensor as precisely as possible and to constrain the different parameters in TCAD by additional measurements [22]. The low-field regions found in the corners of the pixel cell visible in Figures 3 and 4 are strongly influenced by these modifications, and their contribution to the detector signal changes accordingly. Figure 11: Most probable cluster charge as a function of the simulated integration time. The black horizontal line represents the value obtained from data, the hatched band corresponds to the assumed systematic uncertainty of the charge calibration. The integration time currently used to stop the transport of charge carriers is also linked to the sensor design, since it is used to emulate an effective recombination of charge carriers. Their lifetime in the different regions of the sensor is dominated by the respective doping concentration, and potentially affected by the silicon wafer production process. Since this was not modeled in detail for this simulation, the integration time was chosen such that the MPV of the cluster charge matched the value obtained from data as discussed in Section 4. The corresponding uncertainty on the charge calibration of the reference data has therefore to be taken into account as systematic uncertainty of the simulation by comparing the cluster charge MPV for different integration times to the value obtained from data as shown in Figure 11. Here, the hatched band represents an assumed uncertainty of $\pm 50\text{\,}\mathrm{e}$ on the charge calibration of data [7, 9]. This translates to an uncertainty on the integration time of $22.5^{+1.5}_{-1.3}\,\textrm{ns}$, which is propagated as systematic uncertainty to the results presented in this paper. It has been observed that the overall agreement between data and simulation seems to improve for lower integration times, which might indicate either an offset in the absolute charge calibration of data or an insufficient modeling of the signal formation processes in silicon. Also the charge threshold applied to the individual pixels has a strong impact on both the cluster size and the intrinsic resolution, with decreasing influence towards higher thresholds. At a threshold of $40\text{\,}\mathrm{e}$, a change of as little as $\pm 5\text{\,}\mathrm{e}$ visibly alters the observables. Since the absolute charge calibration and the threshold value in electrons are fully correlated, the uncertainty on the applied threshold has been taken into account by varying the two parameters simultaneously and by calculating the total uncertainty arising from the variations. A variation of the threshold dispersion and electronics noise of up to $10\text{\,}\mathrm{e}$ at a threshold of $40\text{\,}\mathrm{e}$ yielded no observable effect. The values for noise and threshold dispersion have been estimated from the evaluation of the full waveform in data [9]. The residual width and the final intrinsic resolution depend on the resolution of the reference tracks at the position of the detector under investigation. This resolution has been determined for the test-beam data used, and a variation of $\pm 0.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ around this value shifts the obtained resolution accordingly. This strong influence arises from the fact that the two values are of similar size. In summary, while the free parameters of the simulation have little to no influence on the final result when varied within a reasonable range, several parameters show a high sensitivity but are constrained by measurements. ## 7 Validation With Test-Beam Data The simulation is compared to data recorded with the _Investigator_ chip, described in Section 3, at the CERN SPS accelerator with a $120\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ $\mathup{{{\pi}}^{\scriptstyle{+}}}$ beam. A total of 25660 tracks through the region of interest have been recorded, mainly limited by the very small active area of the DUT and the dead time of the data acquisition system used. More details about the test-beam setup, data samples and the analysis of data used for comparison in this paper can be found in [6, 9]. ### 7.1 Cluster Charge Figure 12: Cluster charge distributions at a pixel threshold of $120\text{\,}\mathrm{e}$ for simulation and experiment. The distributions resemble the expected Landau-Gauss distribution. The hatched band represents the total uncertainty. The cluster charge distributions for both simulation and data at a charge threshold of $120\text{\,}\mathrm{e}$ are shown in Figure 12. The distributions are fitted with the convolution of a Gaussian and Landau function. The MPV is $1.42\text{\,}\mathrm{k}\mathrm{e}$ for both data and simulation, and the width of the Gaussian is $0.21\text{\,}\mathrm{k}\mathrm{e}$/$0.22\text{\,}\mathrm{k}\mathrm{e}$ for data/simulation, respectively. A good agreement between data and simulation is observed, as also indicated by the ratio of the two distributions displayed in the lower part of the figure. While the MPV has been tuned to match data using the integration time of the simulation as discussed in Section 4, the agreement of the shapes indicates that the energy deposition in the relevant parts of the sensor as well as the collection of charge carriers is well- modeled by the simulation. The data distribution exhibits some fluctuations owing to the low statistics of the sample. ### 7.2 Cluster Size Figure 13: Cluster size distributions for experiment and simulation at a threshold of $120\text{\,}\mathrm{e}$. The distribution of the total cluster size at a threshold of $120\text{\,}\mathrm{e}$ for simulation and experiment is presented in Figure 13. Qualitatively, the distributions are in good agreement. A possible source of the observed deviations for individual cluster sizes are uncertainties in the modeled electric field of the sensor as discussed in Section 6. Figure 14: Cluster size projected in $x$ (left) and $y$ (right) at a threshold of $120\text{\,}\mathrm{e}$ for data and simulation. The projection of the cluster size in $x$ and $y$, depicted in Figure 14, provides additional details about the charge sharing process. Data and simulation agree well, but a small difference between the distributions in $x$ and $y$ can be observed in data despite the symmetry of the pixel cell layout. It has been verified that this does not stem from a remaining misalignment in data by repeating the simulation with a sensor rotated around the $x$ axis by up to $\pm 15\text{\,}\mathrm{\SIUnitSymbolDegree}$ in an attempt to reproduce the difference. The deviation might be a result of the non-symmetric layout of the circuitry in the _Investigator_ pixel. While the p-well structure has been designed to be fully symmetric in x and y, the layout of the circuitry placed in the p-wells is not symmetric, which is a possible source of the asymmetry. Figure 15: Intra-pixel representation of the cluster size for data and simulation at a threshold of $120\text{\,}\mathrm{e}$. Shown is an array of $2\times 2$ pixel cells, with the top-right pixel displaying data, taken from [9], and the other pixels showing results from the simulation (color online). The cluster size distribution is a precise measure for charge sharing as confirmed by the intra-pixel representation of the total cluster size presented in Figure 15. For the simulation, the Monte Carlo truth information is exploited to produce a multi-pixel map indicating the mean cluster size as a function of the particle incidence position within the pixel cells. Likewise, the reference track supplied by the beam telescope is used to obtain the particle incidence position for data. To increase statistics, data events from the full active matrix are folded into a single pixel cell, which is displayed in the upper-right quarter of Figure 15. The largest clusters originate from the pixel corners since the low electric field between pixel implants results in a strong contribution from diffusion of charge carriers. Single-pixel clusters, on the other hand, are almost exclusively produced if the incident particle traverses the sensor very close to the center of the pixel cell. While the overall mean size distribution is faithfully reproduced in the simulation, minor discrepancies in the pixel corners are visible. The transition from four to three-pixel clusters represented by the yellow regions is more apparent in simulation than in data. The same holds true for the transition between two to three pixel clusters corresponding to the turquoise regions in Figure 15. Particles penetrating the sensor at the corners of a pixel cell, for example, are more likely to give rise to clusters with size four in data compared to simulation. This observation is in line with the higher number of clusters with size four in the cluster size distribution displayed in Figure 13. Moreover, the cluster size is particularly sensitive to a mis-modeling in the pixel corners as the diffusion of charge carriers to neighboring pixel cells is most likely if the incident particle enters the sensor at the corners between four cells. Most notably, small modifications in the electric field in the pixel corners are capable of inhibiting or enhancing the motion of charge carriers to neighboring cells causing deviations in cluster size by up to two units as there are two cells directly adjacent to one corner. As discussed in the previous section, the low field regions in the pixel corners are strongly influenced by the exact doping profile of the sensor. Figure 16: Mean cluster size as a function of the threshold, shown for experimental data as well as simulations with TCAD-modeled and linear electric fields. The hatched band represents the total uncertainty. The mean cluster size has been studied as a function of the applied charge threshold. Figure 16 shows the curves for data and simulation. In addition, a simulation with a linear electric field replacing the TCAD model in the epitaxial layer is plotted as a dashed line for comparison. By increasing the threshold, the mean cluster size shifts to smaller values as individual pixels fall below the charge threshold. Data and simulation match well down to very low thresholds, with a maximum deviation of about $6\text{\,}\mathrm{\char 37\relax}$ at very low thresholds, while the simulation with a linear electric field produces incompatible results. This deviation from the experimental results demonstrates the significance of a precise modeling of the electric field for this type of detector. Similar results have been obtained for the mean projected cluster sizes along the $x$ and $y$ coordinates. Figure 17: Intra-pixel representation of the mean projected cluster size as obtained from simulation. Shown are arrays of $2\times 2$ pixel cells indicating the mean projected cluster size in $x$ (left) and $y$ (right) direction as a function of their relative position within the four pixel cells (color online). Figure 17 displays $2\times 2$ pixel maps of the mean projected cluster size in $x$ and $y$ as a function of the particle incidence position at a threshold of $40\text{\,}\mathrm{e}$. Instead of the uniform bands along the respective coordinate expected for uncorrelated observables, eye-shaped structures reveal a correlation between charge sharing along the two dimensions caused by the inhomogeneous electric field and the bubble-shaped depletion region described in Section 2. The same effect is observed in data as demonstrated in [9]. With increasing threshold, charge sharing effects are suppressed and the correlation between the mean cluster size in $x$ and $y$ vanishes. ## 8 Detector Performance Using the reconstructed cluster position and the Monte Carlo truth information from the primary particle, the performance of the CMOS detector is assessed in terms of spatial resolution and hit detection efficiency. The results obtained from simulation are compared to data. ### 8.1 Intrinsic Resolution Figure 18: Residuals in $x$ direction for data and simulation at a threshold of $120\text{\,}\mathrm{e}$. The hatched band represents the uncertainty on simulation. Figure 18 shows the residual in $x$, defined as the difference between the impact point of the incident particle obtained from the Monte Carlo truth and the reconstructed cluster position. The width of the residual is obtained as the root mean square (RMS) of the distribution, evaluated for the central $99.73\text{\,}\mathrm{\char 37\relax}$ of the histogram, equivalent to $\pm 3\sigma$ of a Gaussian distribution, to match the definition used in the data analysis. This allows the width of the distribution to be quantified independently from its shape while providing a statistically robust metric. The spatial resolution is then calculated by quadratically subtracting the track resolution from the residual width, i.e. $\displaystyle\sigma=\sqrt{\textrm{RMS}_{$99.73\text{\,}\mathrm{\char 37\relax}$}^{2}-\sigma_{\textrm{track}}^{2}}.$ The statistical uncertainty on the resolution is calculated using pseudo- experiments. The number of entries in each bin of the residual distribution under consideration is smeared with a Poisson distribution with a mean equivalent to the original bin content. The width obtained from the smeared histogram is stored, and the pseudo-experiment repeated $10\,000$ times. The statistical uncertainty on the residual width is then taken as the width of the resulting distribution and is propagated to the intrinsic resolution. Using these definitions, resolutions in $x$ and $y$ of $\sigma_{x}=$3.60$\pm$0.01$\,\text{(stat)}\,^{+0.24}_{-0.13}\,\text{(syst)}\,$\mathrm{\SIUnitSymbolMicro m}$$ $\sigma_{y}=$3.57$\pm$0.01$\,\text{(stat)}\,^{+0.13}_{-0.11}\,\text{(syst)}\,$\mathrm{\SIUnitSymbolMicro m}$$ have been achieved in simulation which is well below the value of pitch/$\sqrt{12}\approx$8\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ expected without charge sharing. It compares very well with the resolutions of $3.29\pm 0.02\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and $3.42\pm 0.02\text{\,}\mathrm{\SIUnitSymbolMicro m}$ measured in data for $x$ and $y$ respectively. Figure 19: Spatial resolution in $x$ (left) and $y$ (right) direction as a function of the applied charge threshold, shown for experimental data as well as simulations with TCAD-modeled and linear electric fields. The hatched band represents the total uncertainty. The resolution has been studied as a function of the charge threshold applied, shown in Figure 19 for the $x$ and $y$ coordinates separately. With increasing threshold, the information from pixels not passing the threshold is lost, leading to a deterioration of the position resolution. The comparison of data with simulation shows a very good agreement down to a threshold of about $150\text{\,}\mathrm{e}$. The discrepancy at lower thresholds is most likely to be a consequence of non-Gaussian noise in the data recorded with the analog prototype chip as well as a result of the simplification of charge carrier lifetimes described in Section 4. The disagreement is of limited importance for practical purposes since a fully integrated sensor is likely to be operated at thresholds above $150\text{\,}\mathrm{e}$. The dashed gray line in Figure 19 again represents a simulation using a linear electric field as approximation, and the deviation from data suggests that this simplification leads to an inadequate description of the CMOS sensor response. ### 8.2 Efficiency The efficiency of the detector is defined as the number of incident primary particles that can be matched to a reconstructed cluster divided by the total number of primary particles penetrating the detector. A match between an incident particle and a reconstructed cluster is made, if the cluster is located within a radius of $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ around the impact point of the incident particle, using the same matching criterion as applied to data. The statistical uncertainty of the efficiency has been calculated by considering a counting experiment with two possible outcomes: either a matched or an unmatched primary particle track. This results in an uncertainty of $\displaystyle\sigma_{\textrm{eff}}=\sqrt{\frac{p\cdot\left(1-p\right)}{N}},$ where $p$ is the probability of a matched track while $N$ is the total number of experiments conducted. Figure 20: Efficiency obtained from simulations with TCAD-modeled electric field as a function of the impact position for charge thresholds of $40\text{\,}\mathrm{e}$ (left), $450\text{\,}\mathrm{e}$ (center) and $700\text{\,}\mathrm{e}$ (right) for a single pixel cell (color online). The efficiency obtained from simulation as a function of the particle impact position within a single pixel cell is displayed in Figure 20 for three different thresholds. For the lower threshold of $40\text{\,}\mathrm{e}$, depicted in Figure 20 (left), the simulation yields an overall efficiency of $$99.95$\,^{+0.05}_{-0.23}\,\text{(syst)}\,$\mathrm{\char 37\relax}$$. The statistical uncertainty is of the order of $1\text{\times}{10}^{-8}\text{\,}$. The remaining inefficiencies are evenly distributed throughout the pixel cell and arise from delta rays which pull the cluster center far away from the particle incidence point. With increasing threshold, inefficiencies start to develop in the pixel corners, as these are the regions with the strongest charge sharing and the largest mean cluster size. The overall hit detection efficiency at the threshold of $450\text{\,}\mathrm{e}$ shown in Figure 20 (center) decreases to about $$97.62$\,^{+0.13}_{-0.58}\,\text{(syst)}\,$\mathrm{\char 37\relax}$$. At the threshold of $700\text{\,}\mathrm{e}$, depicted in Figure 20 (right), a pronounced inefficiency is observed, extending from the pixel corners into the pixel cell and leading to an overall efficiency of $$85.96$\,^{+0.53}_{-1.02}\,\text{(syst)}\,$\mathrm{\char 37\relax}$$. Figure 21: Efficiency as a function of the charge threshold, shown for experimental data as well as simulations with TCAD-modeled and linear electric fields. While the shape of the curve is well reproduced in simulation, a constant offset from data can be observed. The hatched band represents the total uncertainty. This decrease of efficiency can best be observed as a function of the charge threshold applied, as shown in Figure 21. While the shape of the curve observed in data is reproduced well, a constant offset to the measured values can be observed. This difference can be attributed to fluctuations of the pedestal as well as inefficiencies in the data acquisition system which are not modeled in simulation. The simulation using the linear electric field approximation is found to not correctly model the behavior observed at high threshold values. ## 9 Summary & Outlook In this paper, a combined 3D TCAD and Monte Carlo simulation of a CMOS pixel sensor with small collection electrode design, implemented in a high- resistivity epitaxial layer, has been presented. The simulation combines the results of a three-dimensional electrostatic TCAD simulation with the stochastic description of energy deposition by Geant4 using the $\mathrm{Allpix}^{2}$ framework. Visualizations of the charge carrier motion in the sensor produced by the simulation framework have been found to be helpful to qualitatively understand the sensor response. The simulation results have been compared to measurements of a reference detector, recorded in a test-beam, and very good agreement has been observed after tuning the simulation to match the most probable value of the cluster charge measured in data. The simplified charge transport model implemented in $\mathrm{Allpix}^{2}$ has been shown to be sufficiently precise to replicate the detector performance figures of merit such as efficiency and intrinsic resolution measured in data. The implemented simulation setup for CMOS sensors will be used for further studies of similar detector prototypes and designs, including different sensor geometries and modified production processes aiming at a full lateral depletion of the epitaxial layer. In future versions of the $\mathrm{Allpix}^{2}$ framework, a simulation of charge carrier recombination might be implemented, calculating the lifetime from the respective doping concentration as a function of their position within the sensor. This would allow for an even more realistic description of the charge transport process and would remove the necessity of setting and tuning the integration time for underdepleted detectors. Furthermore, the simulation could be extended to the detector performance in the timing domain by simulating the charge transport taking into account induced currents using the Shockley-Ramo theorem as possible with the latest version of the $\mathrm{Allpix}^{2}$ framework. The presented combination of precise electric field modeling in TCAD and inclusion of statistical fluctuations is also interesting for the simulation of other silicon detector technologies with complex field configurations such as 3D sensors or LGADs. ## Acknowledgements This work was carried out in the framework of the CLICdp Collaboration. This project has received funding from the European Union’s Horizon 2020 Research and Innovation programme under Grant Agreement no. 654168. ## References ## References * [1] W. Snoeys, Monolithic pixel detectors for high energy physics, Nucl. Instr. Meth. A 731 (2013) 125 – 130, Proceedings of PIXEL 2012. doi:10.1016/j.nima.2013.05.073. * [2] S. Spannagel, et al., Allpix2: A modular simulation framework for silicon detectors, Nucl. Instr. Meth. A 901 (2018) 164 – 172. arXiv:1806.05813, doi:10.1016/j.nima.2018.06.020. * [3] S. Agostinelli, et al., Geant4 – a simulation toolkit, Nucl. Instr. Meth. 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A 389 (1–2) (1997) 81 – 86, new Computing Techniques in Physics Research V. doi:10.1016/S0168-9002(97)00048-X. * [20] N. Alipour Tehrani, Test-beam measurements and simulation studies of thin pixel sensors for the CLIC vertex detector, Ph.D. thesis, CERN (2017). doi:10.3929/ethz-b-000164813. * [21] E. Belau, et al., Charge collection in silicon strip detectors, Nucl. Instr. Meth. 214 (2–3) (1983) 253 – 260. doi:10.1016/0167-5087(83)90591-4. * [22] W. Snoeys, et al., A process modification for CMOS monolithic active pixel sensors for enhanced depletion, timing performance and radiation tolerance, Nucl. Instr. Meth. A 871 (2017) 90 – 96. doi:10.1016/j.nima.2017.07.046.

2024-09-04T02:54:55.424321

2002.12636

arxiv-papers

# Reinforcement Learning through Active Inference Alexander Tschantz Sackler Centre for Consciousness Science Evolutionary & Adaptive Systems Research Group University of Sussex Brighton, UK <EMAIL_ADDRESS> &Beren Millidge University of Edinburgh Edinburgh, UK <EMAIL_ADDRESS> &Anil K. Seth Sackler Centre for Consciousness Science Evolutionary & Adaptive Systems Research Group University of Sussex Brighton, UK Canadian Institute for Advanced Research &Christopher L. Buckley Evolutionary & Adaptive Systems Research Group University of Sussex Brighton, UK ###### Abstract The central tenet of reinforcement learning (RL) is that agents seek to maximize the sum of cumulative rewards. In contrast, active inference, an emerging framework within cognitive and computational neuroscience, proposes that agents act to maximize the evidence for a biased generative model. Here, we illustrate how ideas from active inference can augment traditional RL approaches by (i) furnishing an inherent balance of exploration and exploitation, and (ii) providing a more flexible conceptualization of reward. Inspired by active inference, we develop and implement a novel objective for decision making, which we term the free energy of the expected future. We demonstrate that the resulting algorithm successfully balances exploration and exploitation, simultaneously achieving robust performance on several challenging RL benchmarks with sparse, well-shaped, and no rewards. ## 1 Introduction Both biological and artificial agents must learn to make adaptive decisions in unknown environments. In the field of reinforcement learning (RL), agents aim to learn a policy that maximises the sum of expected rewards (Sutton et al., 1998). This approach has demonstrated impressive results in domains such as simulated games (Mnih et al., 2015; Silver et al., 2017), robotics (Polydoros & Nalpantidis, 2017; Nagabandi et al., 2019) and industrial applications (Meyes et al., 2017). In contrast, active inference (Friston et al., 2016; 2015; 2012; 2009) \- an emerging framework from cognitive and computational neuroscience - suggests that agents select actions in order to maximise the evidence for a model that is biased towards an agent’s preferences. This framework extends influential theories of Bayesian perception and learning (Knill & Pouget, 2004; L Griffiths et al., 2008) to incorporate probabilistic decision making, and comes equipped with a biologically plausible process theory (Friston et al., 2017a) that enjoys considerable empirical support (Friston & Kiebel, 2009). Although active inference and RL have their roots in different disciplines, both frameworks have converged upon similar solutions to the problem of learning adaptive behaviour. For instance, both frameworks highlight the importance of learning probabilistic models, performing inference and efficient planning. This leads to a natural question: can insights from active inference inform the development of novel RL algorithms? Conceptually, there are several ways in which active inference can inform and potentially enhance the field of RL. First, active inference suggests that agents embody a generative model of their preferred environment and seek to maximise the evidence for this model. In this context, rewards are cast as prior probabilities over observations, and success is measured in terms of the divergence between preferred and expected outcomes. Formulating preferences as prior probabilities enables greater flexibility when specifying an agent’s goals (Friston et al., 2012; Friston, 2019a), provides a principled (i.e. Bayesian) method for learning preferences (Sajid et al., 2019), and is consistent with recent neurophysiological data demonstrating the distributional nature of reward representations (Dabney et al., 2020). Second, reformulating reward maximisation as maximizing model evidence naturally encompasses both exploration and exploitation under a single objective, obviating the need for adding ad-hoc exploratory terms to existing objectives. Moreover, as we will show, active inference subsumes a number of established RL formalisms, indicating a potentially unified framework for adaptive decision-making under uncertainty. Translating these conceptual insights into practical benefits for RL has proven challenging. Current implementations of active inference have generally been confined to discrete state spaces and toy problems (Friston et al., 2015; 2017b; 2017c) (although see (Tschantz et al., 2019a; Millidge, 2019; Catal et al., 2019)). Therefore, it has not yet been possible to evaluate the effectiveness of active inference in challenging environments; as a result, active inference has not yet been widely taken up within the RL community. In this paper, we consider active inference in the context of decision making111A full treatment of active inference would consider inference and learning, see (Buckley et al., 2017) for an overview.. We propose and implement a novel objective function for active inference - the free energy of the expected future \- and show that this quantity provides a tractable bound on established RL objectives. We evaluate the performance of this algorithm on a selection of challenging continuous control tasks. We show strong performance on environments with sparse, well-shaped, and no rewards, demonstrating our algorithm’s ability to effectively balance exploration and exploitation. Altogether, our results indicate that active inference provides a promising complement to current RL methods. ## 2 Active Inference Both active inference and RL can be formulated in the context of a partially observed Markov decision process POMDPs (Murphy, 1982). At each time step $t$, the true state of the environment $\mathbf{s}_{t}$ evolves according to the stochastic transition dynamics $\mathbf{{\textnormal{s}}}_{t}\sim\mathrm{p}(\mathbf{s}_{t}|\mathbf{s}_{t-1},\mathbf{a}_{t-1})$, where $\mathbf{a}\in\mathbb{R}^{d_{a}}$ denotes an agent’s actions. Agents do not necessarily have access to the true state of the environment, but may instead receive observations ${\textnormal{o}}_{t}\in\mathbb{R}^{d_{o}}$, which are generated according to ${\textnormal{o}}_{t}\sim\mathrm{p}({\textnormal{o}}_{t}|\mathbf{s}_{t})$. In this case, agents must operate on beliefs ${\textnormal{s}}_{t}\in\mathbb{R}^{d_{s}}$ about the true state of the environment $\mathbf{s}_{t}$. Finally, the environment generates rewards ${\textnormal{r}}_{t}$ according to ${\textnormal{r}}_{t}\sim\mathrm{p}({\textnormal{r}}_{t}|\mathbf{s}_{t})$222We use $\mathbf{x}$ and $\mathrm{p}(\cdot)$ to denote the generative process and x and $p(\cdot)$ to denote the agent’s generative model.. The goal of RL is to learn a policy that maximises the expected sum of rewards $\mathbb{E}[\sum_{t=0}^{\infty}\gamma^{t}{\textnormal{r}}_{t}]$ (Sutton et al., 1998). In contrast, the goal of active inference is to maximise the Bayesian model evidence for an agent’s generative model $p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta)$, where $\theta\in\Theta$ denote model parameters. Crucially, active inference allows that an agent’s generative model can be biased towards favourable states of affairs (Friston, 2019b). In other words, the model assigns probability to the parts of observation space that are both likely and beneficial for an agent’s success. We use the notation $p^{\Phi}(\cdot)$ to represent an arbitrary distribution encoding the agent’s preferences. Given a generative model, agents can perform approximate Bayesian inference by encoding an arbitrary distribution $q({\textnormal{s}},\theta)$ and minimising variational free energy $\mathcal{F}=D_{\mathrm{KL}}\Big{(}q({\textnormal{s}},\theta)\|p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta)\Big{)}$. When observations o are known, $\mathcal{F}$ can be minimized through standard variational methods (Bishop, 2006; Buckley et al., 2017), causing $q({\textnormal{s}},\theta)$ to tend towards the true posterior $p({\textnormal{s}},\theta|{\textnormal{o}})$. Note that treating model parameters $\theta$ as random variables casts learning as a process of inference (Blundell et al., 2015). In the current context, agents additionally maintain beliefs over policies $\pi=\\{a_{0},...,a_{T}\\}$, which are themselves random variables. Policy selection is then implemented by identifying $q(\pi)$ that minimizes $\mathcal{F}$, thus casting policy selection as a process of approximate inference (Friston et al., 2015). While the standard free energy functional $\mathcal{F}$ is generally defined for a single time point $t$, $\pi$ refers to a temporal sequence of variables. Therefore, we augment the free energy functional $\mathcal{F}$ to encompass future variables, leading to the free energy of the expected future $\mathcal{\tilde{F}}$. This quantity measures the KL-divergence between a sequence of beliefs about future variables and an agent’s biased generative model. The goal is now to infer $q(\pi)$ in order to minimise $\mathcal{\tilde{F}}$. We demonstrate that the resulting scheme naturally encompasses both exploration and exploitation, thereby suggesting a deep relationship between inference, learning and decision making. ## 3 Free energy of the expected future Let ${\textnormal{x}}_{t:T}$ denote a sequence of variables through time, ${\textnormal{x}}_{t:T}=\\{{\textnormal{x}}_{t},...,{\textnormal{x}}_{T}\\}$. We wish to minimize the free energy of the expected future $\mathcal{\tilde{F}}$, which is defined as: $\mathcal{\tilde{F}}=D_{\mathrm{KL}}\Big{(}q({\textnormal{o}}_{0:T},{\textnormal{s}}_{0:T},\theta,\pi)\|p^{\Phi}({\textnormal{o}}_{0:T},{\textnormal{s}}_{0:T},\theta)\Big{)}$ (1) where $q({\textnormal{o}}_{t:T},{\textnormal{s}}_{t:T},\theta,\pi)$ represents an agent’s beliefs about future variables, and $p^{\Phi}({\textnormal{o}}_{t:T},{\textnormal{s}}_{t:T},\theta)$ represents an agent’s biased generative model. Note that the beliefs about future variables include beliefs about future observations, ${\textnormal{o}}_{t:T}$, which are unknown and thus treated as random variables333For readers familiar with the active inference framework, we highlight that the free energy of the expected future differs from expected free energy (Friston et al., 2015). We leave a discussion of the relative merits to future work.. In order to find $q(\pi)$ which minimizes $\mathcal{\tilde{F}}$ we note that (see Appendix C): $\displaystyle\mathcal{\tilde{F}}=0\Rightarrow D_{\mathrm{KL}}\Big{(}q(\pi)\ \|\big{(}-e^{-\mathcal{\tilde{F_{\pi}}}}\big{)}\Big{)}=0$ (2) where $\displaystyle\mathcal{\tilde{F_{\pi}}}=D_{\mathrm{KL}}\Big{(}q({\textnormal{o}}_{0:T},{\textnormal{s}}_{0:T},\theta|\pi)\ \|\ p^{\Phi}({\textnormal{o}}_{0:T},{\textnormal{s}}_{0:T},\theta)\Big{)}$ (3) Thus, the free energy of the expected future is minimized when $q(\pi)=\sigma(-\mathcal{\tilde{F_{\pi}}})$, or in other words, policies are more likely when they minimise $\mathcal{\tilde{F_{\pi}}}$. ### 3.1 Exploration & exploitation In order to provide an intuition for what minimizing $\mathcal{\tilde{F_{\pi}}}$ entails, we factorize the agent’s generative models as $p^{\Phi}({\textnormal{o}}_{0:T},{\textnormal{s}}_{0:T},\theta)=p({\textnormal{s}}_{0:T},\theta|{\textnormal{o}}_{0:T})p^{\Phi}({\textnormal{o}}_{0:T})$, implying that the model is only biased in its beliefs over observations. To retain consistency with RL nomenclature, we treat ‘rewards’ r as a separate observation modality, such that $p^{\Phi}({\textnormal{o}}_{t:T})$ specifies a distribution over preferred rewards. We describe our implementation of $p^{\Phi}({\textnormal{o}}_{t:T})$ in Appendix E. In a similar fashion, $q({\textnormal{o}}_{t:T}|{\textnormal{s}}_{t:T},\theta,\pi)$ specifies beliefs about future rewards, given a policy. Given this factorization, it is straightforward to show that $-\mathcal{\tilde{F_{\pi}}}$ decomposes into an expected information gain term and an extrinsic term (see Appendix B)444The approximation in Eq. 4 arises from the approximation $q({\textnormal{s}}_{0:T},\theta|{\textnormal{o}}_{0:T},\pi)\approx p({\textnormal{s}}_{0:T},\theta|{\textnormal{o}}_{0:T},\pi)$, which is justifiable given that $q(\cdot)$ represents a variational approximation of the true posterior (Friston et al., 2017a).: $\displaystyle-\mathcal{\tilde{F_{\pi}}}\approx$ $\displaystyle-\underbrace{\mathbb{E}_{q({\textnormal{o}}_{0:T}|\pi)}\Big{[}D_{\mathrm{KL}}\Big{(}q({\textnormal{s}}_{0:T},\theta|{\textnormal{o}}_{0:T},\pi)\|q({\textnormal{s}}_{0:T},\theta|\pi)\Big{)}\Big{]}}_{\text{Expected information gain}}$ (4) $\displaystyle+\underbrace{\mathbb{E}_{q({\textnormal{s}}_{0:T},\theta|\pi)}\Big{[}D_{\mathrm{KL}}\Big{(}q({\textnormal{o}}_{0:T}|{\textnormal{s}}_{0:T},\theta,\pi)\|p^{\Phi}({\textnormal{o}}_{t:T})\Big{)}\Big{]}}_{\text{Extrinsic term}}$ Maximizing Eq.4 has two functional consequences. First, it maximises the expected information gain, which quantifies the amount of information an agent expects to gain from executing some policy. As agents maintain beliefs about the state of the environment and model parameters, this term promotes exploration in both state and parameter space. Second, it minimizes the extrinsic term - which is the KL-divergence between an agent’s (policy-conditioned) beliefs about future observations and their preferred observations. In the current context, it measures the KL-divergence between the rewards an agent expects from a policy and the rewards an agent desires. In summary, selecting policies to minimise $\tilde{\mathcal{F}}$ invokes a natural balance between exploration and exploitation. ### 3.2 Relationship to probabilistic RL In recent years, there have been several attempts to formalize RL in terms of probabilistic inference (Levine, 2018), such as KL-control (Rawlik, 2013), control-as-inference (Kappen et al., 2012), and state-marginal matching (Lee et al., 2019). In many of these approaches, the RL objective is broadly conceptualized as minimising $D_{\mathrm{KL}}\Big{(}p({\textnormal{o}}_{0:T}|\pi)\|\ p^{\Phi}(o_{0:T})\Big{)}$555We acknowledge that not all objectives follow this exact formulation.. In Appendix D, we demonstrate that the free energy of the expected future $\mathcal{\tilde{F}}$ provides a tractable bound on this objective: $\displaystyle\mathcal{\tilde{F}}\geq D_{\mathrm{KL}}\Big{(}p({\textnormal{o}}_{t:T}|\pi)\|\ p^{\Phi}(o_{t:T})\Big{)}$ (5) These results suggest a deep homology between active inference and existing approaches to probabilistic RL. ## 4 Implementation In this section, we describe an efficient implementation of the proposed objective function in the context of model-based RL. To select actions, we optimise $q(\pi)$ at each time step, and execute the first action specified by the most likely policy. This requires (i) a method for evaluating beliefs about future variables $q({\textnormal{s}}_{t:T},{\textnormal{o}}_{t:T},\theta|\pi)$, (ii) an efficient method for evaluating $\mathcal{F}_{\pi}$, and (iii) a method for optimising $q(\pi)$ such that $q(\pi)=\sigma(-\mathcal{F}_{\pi})$ #### Evaluating beliefs about the future We factorize and evaluate the beliefs about the future as: $\displaystyle q({\textnormal{s}}_{t:T},{\textnormal{o}}_{t:T},\theta|\pi)$ $\displaystyle=q(\theta)\prod_{t=\tau}^{T}q({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau},\theta,\pi)q({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\theta,\pi)$ (6) $\displaystyle q({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau},\theta,\pi)$ $\displaystyle=\mathbb{E}_{q({\textnormal{s}}_{\tau}|\theta,\pi)}\big{[}p({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau})\big{]}$ $\displaystyle q({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\theta,\pi)$ $\displaystyle=\mathbb{E}_{q({\textnormal{s}}_{\tau-1}|\theta,\pi)}\big{[}p({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\theta,\pi)\big{]}$ where we have here factorized the generative model as $p({\textnormal{o}}_{\tau},{\textnormal{s}}_{\tau},\theta|\pi)=p({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau},\pi)p({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\theta,\pi)p(\theta)$. We describe the implementation and learning of the likelihood $p({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau},\pi)$, transition model $p({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\theta,\pi)$ and parameter prior $p(\theta)$ in Appendix E. #### Evaluating $\mathcal{\tilde{F}_{\pi}}$ Note that $-\mathcal{\tilde{F}}_{\pi}=\sum_{\tau=t}^{t+H}-\mathcal{\tilde{F}}_{\pi_{\tau}}$, where $H$ is the planning horizon. Given beliefs about future variables, the free energy of the expected future for a single time point can be efficiently computed as (see Appendix G): $\displaystyle-\mathcal{\tilde{F}}_{\pi_{\tau}}$ $\displaystyle\approx E_{q({\textnormal{s}}_{\tau},\theta|\pi)}\Big{[}D_{\mathrm{KL}}\Big{(}q({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau},\theta,\pi)\|p^{\Phi}({\textnormal{o}}_{\tau})\Big{)}\Big{]}$ (7) $\displaystyle+\underbrace{\mathbf{H}[q({\textnormal{o}}_{\tau}|\pi)]-\mathbb{E}_{q({\textnormal{s}}_{\tau}|\pi)}\Big{[}\mathbf{H}[q({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau},\pi)]\Big{]}}_{\text{State information gain}}$ $\displaystyle+\underbrace{\mathbf{H}[q({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\theta,\pi)]-\mathbb{E}_{q(\theta)}\Big{[}\mathbf{H}[q({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\pi,\theta)]\Big{]}}_{\text{Parameter information gain}}$ In the current paper, agents observe the true state of the environment ${\textnormal{s}}_{t}$, such that the only partial observability is in rewards ${\textnormal{r}}_{t}$. As as a result, the second term of equation 7 is redundant, as there is no uncertainty about states. The first (extrinsic) term can be calculated analytically (see Appendix E). We describe our approximation of the final term (parameter information gain) in Appendix G. #### Optimising the policy distribution We choose to parametrize $q(\pi)$ as a diagonal Gaussian. We use the CEM algorithm (Rubinstein, 1997) to optimise the parameters of $q(\pi)$ such that $q(\pi)\propto-\mathcal{F}_{\pi}$. While this solution will fail to capture the exact shape of $-\mathcal{F}_{\pi}$, agents need only identify the peak of the landscape to enact the optimal policy. The full algorithm for inferring $q(\pi)$ is provided in Algorithm 1. Input: Planning horizon $H$ — Optimisation iterations $I$ — Number of candidate policies $J$ — Current state ${\textnormal{s}}_{t}$ — Likelihood $p({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau})$ — Transition distribution $p({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\theta,\pi)$ — Parameter distribution $P(\theta)$ — Global prior $p^{\Phi}({\textnormal{o}}_{\tau})$ Initialize factorized belief over action sequences $q(\pi)\leftarrow\mathcal{N}(0,\mathbb{I})$. for _$\mathrm{optimisation\ iteration}\ i=1...I$_ do Sample $J$ candidate policies from $q(\pi)$ for _$\mathrm{candidate\ policy}\ j=1...J$_ do $\pi^{(j)}\sim q(\pi)$ $-\mathcal{\tilde{F}}_{\pi}^{j}=0$ for _$\tau=t...t+H$_ do $q({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\theta,\pi^{(j)})=\mathbb{E}_{q({\textnormal{s}}_{\tau-1}|\theta,\pi^{(j)})}\big{[}p({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\theta,\pi^{(j)})\big{]}$ $q({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau},\theta,\pi^{(j)})=\mathbb{E}_{q({\textnormal{s}}_{\tau}|\theta,\pi^{(j)})}\big{[}p({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau})\big{]}$ $-\mathcal{\tilde{F}}_{\pi}^{j}\leftarrow-\mathcal{\tilde{F}}_{\pi}^{j}+E_{q({\textnormal{s}}_{\tau},\theta|\pi^{(j)})}\big{[}D_{\mathrm{KL}}\big{(}q({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau},\theta,\pi^{(j)})\|p^{\Phi}({\textnormal{o}}_{\tau})\big{)}\big{]}+\mathbf{H}[q({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\theta,\pi^{(j)})]-\mathbb{E}_{q(\theta)}\big{[}\mathbf{H}[q({\textnormal{s}}_{\tau}|{\textnormal{s}}_{\tau-1},\pi^{(j)},\theta)]\big{]}$ end for end for $q(\pi)\leftarrow\mathrm{refit}(-\mathcal{\tilde{F}}_{\pi}^{j})$ end for return $q(\pi)$ Algorithm 1 Inference of $q(\pi)$ ## 5 Experiments To determine whether our algorithm successfully balances exploration and exploitation, we investigate its performance in domains with (i) well-shaped rewards, (ii) extremely sparse rewards and (iii) a complete absence of rewards. We use four tasks in total. For sparse rewards, we use the Mountain Car and Cup Catch environments, where agents only receive reward when the goal is achieved. For well-shaped rewards, we use the challenging Half Cheetah environment, using both the running and flipping tasks. For domains without reward, we use the Ant Maze environment, where there are no rewards and success is measured by the percent of the maze covered (see Appendix H for details on all environments). Figure 1: (A) Mountain Car: Average return after each episode on the sparse- reward Mountain Car task. Our algorithm achieves optimal performance in a single trial. (B) Cup Catch: Average return after each episode on the sparse- reward Cup Catch task. Here, results amongst algorithms are similar, with all agents reaching asymptotic performance in around 20 episodes. (C & D) Half Cheetah: Average return after each episode on the well-shaped Half Cheetah environment, for the running and flipping tasks, respectively. We compare our results to the average performance of SAC after 100 episodes learning, demonstrating our algorithm can perform successfully in environments which do not require directed exploration. Each line is the mean of 5 seeds and filled regions show +/- standard deviation. For environments with sparse rewards, we compare our algorithm to two baselines, (i) a reward algorithm which only selects policies based on the extrinsic term (i.e. ignores the parameter information gain), and (ii) a variance algorithm that seeks out uncertain transitions by acting to maximise the output variance of the transition model (see Appendix E). Note that the variance agent is also augmented with the extrinsic term to enable comparison. For environments with well-shaped rewards, we compare our algorithm to the maximum reward obtained by a state-of-the-art model-free RL algorithm after 100 episodes, the soft-actor-critic (SAC) Haarnoja et al. (2018), which encourages exploration by seeking to maximise the entropy of the policy distribution. Finally, for environments without rewards, we compare our algorithm to a random baseline, which conducts actions at random. The Mountain Car experiment is shown in Fig. 1A, where we plot the total reward obtained for each episode over 25 episodes, where each episode is at most 200 time steps. These results demonstrate that our algorithm rapidly explores and consistently reaches the goal, achieving optimal performance in a single trial. In contrast, the benchmark algorithms were, on average, unable to successfully explore and achieve good performance. We qualitatively confirm this result by plotting the state space coverage with and without exploration (Fig. 2B). Our algorithm performs comparably to benchmarks on the Cup Catch environment (Fig. 1B). We hypothesize that this is because, while the reward structure is technically sparse, it is simple enough to reach the goal with random actions, and thus the directed exploration afforded by our method provides little benefit. Figure 1 C&D shows that our algorithm performs substantially better than a state of the art model-free algorithm after 100 episodes on the challenging Half Cheetah tasks. Our algorithm thus demonstrates robust performance in environments with well-shaped rewards and provides considerable improvements in sample-efficiency, relative to SAC. Finally, we demonstrate that our algorithm can perform well in environments with no rewards, where the only goal is exploration. Figure 2B shows that our algorithms rate of exploration is substantially higher than that of a random baseline in the ant-maze environment, resulting in a more substantial portion of the maze being covered. This result demonstrates that the directed exploration afforded by minimising the free energy of the expected future proves beneficial in environments with no reward structure. Taken together, these results show that our proposed algorithm - which naturally balances exploration and exploitation - can successfully master challenging domains with a variety of reward structures. Figure 2: (A & B) Mountain Car state space coverage: We plot the points in state-space visited by two agents - one that minimizes the free energy of the expected future (FEEF) and one that maximises reward. The plots are from 20 episodes and show that the FEEF agent searches almost the entirety of state space, while the reward agent is confined to a region that be reached with random actions. (C) Ant Maze Coverage: We plot the percentage of the maze covered after 35 episodes, comparing the FEEF agent to an agent acting randomly. These results are the average of 4 seeds. ## 6 Discussion Despite originating from different intellectual traditions, active inference and RL both address fundamental questions about adaptive decision-making in unknown environments. Exploiting this conceptual overlap, we have applied an active inference perspective to the reward maximization objective of RL, recasting it as minimizing the divergence between desired and expected futures. We derived a novel objective that naturally balances exploration and exploitation and instantiated this objective within a model-based RL context. Our algorithm exhibits robust performance and flexibility in a variety of environments known to be challenging for RL. Moreover, we have shown that our algorithm applies to a diverse set of reward structures. Conversely, by implementing active inference using tools from RL, such as amortising inference with neural networks, deep ensembles and sophisticated algorithms for planning (CEM), we have demonstrated that active inference can scale to high dimensional tasks with continuous state and action spaces. While our results have highlighted the existing overlap between active inference and RL, we end by reiterating two aspects of active inference that may be of utility for RL. First, representing preferences as a distribution over observations allows for greater flexibility in modelling and learning non-scalar and non-monotonic reward functions. This may prove beneficial when learning naturalistic tasks in complex nonstationary environments. Second, the fact that both intrinsic and extrinsic value are complementary components of a single objective - the free energy of the expected future - may suggest new paths to tackling the exploration-exploitation dilemma. Our method also admits promising directions for future work. These include investigating the effects of different distributions over reward, extending the approach to models which are hierarchical in time and space (Friston et al., 2018; Pezzulo et al., 2018), and investigating the deep connections to alternative formulations of probabilistic control. ## Acknowledgements AT is funded by a PhD studentship from the Dr. Mortimer and Theresa Sackler Foundation and the School of Engineering and Informatics at the University of Sussex. BM is supported by an EPSRC funded PhDS Studentship. CLB is supported by BBRSC grant number BB/P022197/1. AT and AKS are grateful to the Dr. Mortimer and Theresa Sackler Foundation, which supports the Sackler Centre for Consciousness Science. 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In _Data-Efficient Machine Learning workshop_ , 2016. ## Appendix A Related work #### Active inference There is an extensive literature on active inference within discrete state- spaces, covering a wide variety of tasks, such as epistemic foraging in saccades (Parr & Friston, 2017; Friston, 2019b; Schwartenbeck et al., 2019), exploring mazes (Friston et al., 2015; Pezzulo et al., 2016; Friston et al., 2016), to playing Atari games (Cullen et al., 2018). Active inference also comes equipped with a well-developed neural process theory (Friston et al., 2017a; Parr et al., 2019) which can account for a substantial range of neural dynamics. There have also been prior attempts to scale up active inference to continuous RL tasks (Tschantz et al., 2019a; Millidge, 2019; Ueltzhöffer, 2018), which we build upon here. #### Model based RL Model based reinforcement learning has been in a recent renaissance, with implementations vastly exceeding the sample efficiency of model-free methods, while also approaching their asymptotic performance (Ha & Schmidhuber, 2018; Nagabandi et al., 2018; Chua et al., 2018a; Hafner et al., 2018). There have been recent successes on challenging domains such as Atari (Kaiser et al., 2019), and high dimensional robot locomotion (Hafner et al., 2018; 2019) and manipulation (Nagabandi et al., 2019) tasks. Key advances include variational autoencoders (Kingma & Welling, 2013) to flexibly construct latent spaces in partially observed environments, Bayesian approaches such as Bayes by backprop (Houthooft et al., 2016a), deep ensembles (Shyam et al., 2018; Chua et al., 2018a), and other variational approaches (Okada & Taniguchi, 2019; Tschiatschek et al., 2018; Yarin Gal et al., 2016), which quantify uncertainty in the dynamics models, and enable the model to learn a latent space that is useful for action (Tschantz et al., 2019b; Watter et al., 2015). Finally, progress has been aided by powerful planning algorithms capable of online planning in continuous state and action spaces (Williams et al., 2016; Rubinstein, 1997). #### Intrinsic Measures Using intrinsic measures to encourage exploration has a long history in RL (Schmidhuber, 1991; 2007; Storck et al., 1995; Oudeyer & Kaplan, 2009; Chentanez et al., 2005). Recent model-free and model based-intrinsic measures that have been proposed in the literature include policy-entropy (Rawlik, 2013; Rawlik et al., 2013; Haarnoja et al., 2018),state entropy (Lee et al., 2019), information-gain (Houthooft et al., 2016b; Okada & Taniguchi, 2019; Kim et al., 2018; Shyam et al., 2019; Teigen, 2018), prediction error (Pathak et al., 2017), divergence of ensembles (Shyam et al., 2019; Chua et al., 2018b), uncertain state bonuses (Bellemare et al., 2016; O’Donoghue et al., 2017), and empowerment (de Abril & Kanai, 2018; Leibfried et al., 2019; Mohamed & Rezende, 2015). Information gain additionally has a substantial history outside the RL framework, going back to (Lindley, 1956; Still & Precup, 2012; Sun et al., 2011). ## Appendix B Derivation for the free energy of the expected future We begin with the full free energy of the expected future and decompose this into the free energy of the expected future given policies, and the negative policy entropy: $\displaystyle\mathcal{\tilde{F}}$ $\displaystyle=\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta,\pi)}[\log q({\textnormal{o}},{\textnormal{s}},\theta,\pi)-\log p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta)]$ (8) $\displaystyle=\mathbb{E}_{q(\pi)}[\mathcal{\tilde{F}}_{\pi}]-\mathbf{H}[q(\pi)]$ We now show the free energy of the expected future given policies can be decomposed into extrinsic and information gain terms: $\displaystyle\mathcal{\tilde{F}}_{\pi}$ $\displaystyle=E_{q({\textnormal{o}},{\textnormal{s}},\theta,\pi)}[\log q({\textnormal{o}},{\textnormal{s}},\theta,\pi)-\log p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta)]$ (9) $\displaystyle=\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta|\pi)}[\log q({\textnormal{s}},\theta|\pi)+\log q({\textnormal{o}}|{\textnormal{s}},\theta,\pi)-\log p({\textnormal{s}},\theta|{\textnormal{o}})-\log p^{\Phi}({\textnormal{o}})]$ $\displaystyle\approx\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta|\pi)}[\log q({\textnormal{s}},\theta|\pi)+\log q({\textnormal{o}}|{\textnormal{s}},\theta,\pi)-\log q({\textnormal{s}},\theta|{\textnormal{o}},\theta)-\log p^{\Phi}({\textnormal{o}})]$ $\displaystyle=\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta|\pi)}[\log q({\textnormal{s}},\theta|\pi)-\log q({\textnormal{s}},\theta|{\textnormal{o}},\pi)]+\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta|\pi)}[\log q({\textnormal{o}}|{\textnormal{s}},\theta,\pi)-\log p^{\Phi}({\textnormal{o}})]$ $\displaystyle-\mathcal{\tilde{F}}_{\pi}$ $\displaystyle=\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta|\pi)}[\log q({\textnormal{s}},\theta|{\textnormal{o}},\pi)-\log q({\textnormal{s}},\theta|\pi)]+\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta|\pi)}[\log p^{\Phi}({\textnormal{o}})-\log q({\textnormal{o}}|{\textnormal{s}},\theta,\pi)]$ $\displaystyle=\underbrace{\mathbb{E}_{q({\textnormal{o}}|\pi)}\Big{[}D_{\mathrm{KL}}\Big{(}q({\textnormal{s}},\theta|{\textnormal{o}},\pi)\|q({\textnormal{s}},\theta|\pi)\Big{)}\Big{]}}_{\text{Expected Information Gain}}-\underbrace{\mathbb{E}_{q({\textnormal{s}},\theta|\pi)}\Big{[}D_{\mathrm{KL}}\Big{(}q({\textnormal{o}}|{\textnormal{s}},\theta,\pi)\|p^{\Phi}({\textnormal{o}})\Big{)}\Big{]}}_{\text{Extrinsic Value}}$ Where we have assumed that $p({\textnormal{s}},\theta|{\textnormal{o}})\approx q({\textnormal{s}},\theta|{\textnormal{o}},\pi)$. We wish to minimize $\mathcal{\tilde{F}}_{\pi}$, and thus maximize $-\mathcal{\tilde{F}}_{\pi}$. This means we wish to maximize the information gain and minimize the KL- divergence between expected and preferred observations. By noting that $q({\textnormal{s}},\theta|{\textnormal{o}},\pi)\approx q({\textnormal{s}}|{\textnormal{o}},\pi)q(\theta|{\textnormal{s}})$, we can split the expected information gain term into state and parameter information gain terms: $\displaystyle\mathbb{E}_{q({\textnormal{o}}|\pi)}\Big{[}D_{\mathrm{KL}}\Big{(}q({\textnormal{s}},\theta|{\textnormal{o}},\pi)\|q({\textnormal{s}},\theta|\pi)\Big{)}\Big{]}$ (10) $\displaystyle=\mathbb{E}_{q({\textnormal{o}}|\pi)q({\textnormal{s}},\theta|{\textnormal{o}},\pi)}\big{[}\log q({\textnormal{s}},\theta|{\textnormal{o}},\pi)-\log q({\textnormal{s}},\theta|\pi)\big{]}$ $\displaystyle=\mathbb{E}_{q({\textnormal{o}}|\pi)q({\textnormal{s}},\theta|{\textnormal{o}},\pi)}\big{[}\log q({\textnormal{s}}|{\textnormal{o}},\pi)+\log q(\theta|{\textnormal{s}})-\log q({\textnormal{s}}|\theta,\pi)-\log q(\theta)\big{]}$ $\displaystyle=\mathbb{E}_{q({\textnormal{o}}|\pi)q({\textnormal{s}},\theta|{\textnormal{o}},\pi)}\big{[}\log q({\textnormal{s}}|{\textnormal{o}},\pi)-\log q({\textnormal{s}}|\theta,\pi)]\big{]}+\mathbb{E}_{q({\textnormal{o}}|\pi)q({\textnormal{s}},\theta|{\textnormal{o}},\pi)}\big{[}\log q(\theta|{\textnormal{s}})-\log q(\theta)\big{]}$ $\displaystyle=\underbrace{\mathbb{E}_{q({\textnormal{o}}|\pi)q(\theta)}\Big{[}D_{\mathrm{KL}}\big{(}q({\textnormal{s}}|{\textnormal{o}},\pi)\|q({\textnormal{s}}|\theta)\big{)}\Big{]}}_{\text{Expected State Information Gain}}+\underbrace{\mathbb{E}_{q({\textnormal{s}}|\theta)}\Big{[}D_{\mathrm{KL}}\big{(}q(\theta|{\textnormal{s}})\|q(\theta)\big{)}\Big{]}}_{\text{Expected Parameter Information Gain}}$ ## Appendix C Derivation of the optimal policy We derive the distribution for $q(\pi)$ which minimizes $\mathcal{\tilde{F}}$: $\displaystyle\mathcal{\tilde{F}}$ $\displaystyle=D_{\mathrm{KL}}\Big{(}q({\textnormal{o}},{\textnormal{s}},\theta,\pi)\|p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta)\Big{)}$ (11) $\displaystyle=\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta,\pi)}[\log q({\textnormal{o}},{\textnormal{s}},\theta|\pi)+\log q(\pi)-\log p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta,\pi)]$ $\displaystyle=\mathbb{E}_{q(\pi)}\Big{[}\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta|\pi)}[\log q(\pi)-[\log p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta)-\log q({\textnormal{o}},{\textnormal{s}},\theta|\pi)]\Big{]}$ $\displaystyle=\mathbb{E}_{q(\pi)}\Big{[}\log q(\pi)-\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta|\pi)}[\log p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta)-\log q({\textnormal{o}},{\textnormal{s}},\theta|\pi)]\Big{]}$ $\displaystyle=\mathbb{E}_{q(\pi)}\Big{[}\log q(\pi)-\big{[}-\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta|\pi)}[\log q({\textnormal{o}},{\textnormal{s}},\theta|\pi)-\log p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta)]\big{]}\Big{]}$ $\displaystyle=\mathbb{E}_{q(\pi)}\Big{[}\log q(\pi)-\log e^{-\big{[}-\mathbb{E}_{q({\textnormal{o}},{\textnormal{s}},\theta|\pi)}[\log q({\textnormal{o}},{\textnormal{s}},\theta|\pi)-\log p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta)]\big{]}}\Big{]}$ $\displaystyle=\mathbb{E}_{q(\pi)}\Big{[}\log q(\pi)-\log e^{-D_{\mathrm{KL}}\big{(}q({\textnormal{o}},{\textnormal{s}},\theta|\pi)\|p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta)\big{)}}\Big{]}$ $\displaystyle=D_{\mathrm{KL}}\Big{(}q(\pi)\|e^{-D_{\mathrm{KL}}\big{(}q({\textnormal{o}},{\textnormal{s}},\theta|\pi)\|p^{\Phi}({\textnormal{o}},{\textnormal{s}},\theta)\big{)}}\Big{)}$ $\displaystyle=D_{\mathrm{KL}}\Big{(}q(\pi)\ \|e^{-\mathcal{\tilde{F_{\pi}}}}\Big{)}$ ## Appendix D Derivation of RL bound Here we show that the free energy of the expected future is a bound on the divergence between expected and desired observations. The proof proceeds straightforwardly by importance sampling on the approximate posterior and then applying Jensen’s inequality: $\displaystyle D_{\mathrm{KL}}\big{(}q({\textnormal{o}}_{t:T}|\pi)\|\ p^{\Phi}(o_{t:T})\big{)}$ $\displaystyle=\mathbb{E}_{q({\textnormal{o}}_{t:T}|\pi)}\big{[}\log q({\textnormal{o}}_{t:T}|\pi)-\log p^{\Phi}({\textnormal{o}})\big{]}$ (12) $\displaystyle=\mathbb{E}_{q({\textnormal{o}}_{t:T}|\pi)}\bigg{[}\log\big{(}\int dx_{1:T}\int d\theta_{1:T}\frac{q({\textnormal{o}}_{t:T},{\textnormal{s}}_{t:T},\theta_{t:T}|\pi)q({\textnormal{s}}_{t:T},\theta_{t:T}|{\textnormal{o}}_{t:T})}{p^{\Phi}({\textnormal{o}}_{t:T})q({\textnormal{s}}_{t:T},\theta_{t:T}|{\textnormal{o}}_{t:T})}\big{)}\bigg{]}$ $\displaystyle\leq\mathbb{E}_{q({\textnormal{o}}_{t:T},{\textnormal{s}}_{t:T},\theta_{t:T}|\pi)}\Big{[}\log\big{(}\frac{q({\textnormal{o}}_{t:T},{\textnormal{s}}_{t:T},\theta_{t:T}|\pi)}{p^{\Phi}({\textnormal{o}}_{t:T},{\textnormal{s}}_{t:T},\theta_{t:T})}\big{)}\Big{]}$ $\displaystyle\leq D_{\mathrm{KL}}\Big{(}q({\textnormal{o}}_{t:T},{\textnormal{s}}_{t:T},\theta|\pi)\|p^{\Phi}({\textnormal{o}}_{t:T},{\textnormal{s}}_{t:T},\theta)\Big{)}=\mathcal{\tilde{F}}$ ## Appendix E Model details In the current work, we implemented our probabilistic model using an ensemble- based approach (Chua et al., 2018a; Fort et al., 2019; Chitta et al., 2018). Here, an ensemble of point-estimate parameters $\theta=\\{\theta_{0},...,\theta_{B}\\}$ trained on different batches of the dataset $\mathcal{D}$ are maintained and treated as samples as from the posterior distribution $p(\theta|\mathcal{D})$. Besides consistency with the active inference framework, probabilistic models enable the active resolution of model uncertainty, capture both epistemic and aleatoric uncertainty, and help avoid over-fitting in low data regimes (Fort et al., 2019; Chitta et al., 2018; Chatzilygeroudis et al., 2018; Chua et al., 2018b). This design choice means that we use a trajectory sampling method when evaluating beliefs about future variables (Chua et al., 2018a), as each pass through the transition model $p({\textnormal{s}}_{t}|{\textnormal{s}}_{t-1},\theta,\pi)$ evokes $B$ samples from ${\textnormal{s}}_{t}$. #### Transition model We implement the transition model as $p({\textnormal{s}}_{t}|{\textnormal{s}}_{t-1},\theta,\pi)$ as $\mathcal{N}({\textnormal{s}}_{t};f_{\theta}({\textnormal{s}}_{t-1}),f_{\theta}({\textnormal{s}}_{t-1}))$, where $f_{\theta}(\cdot)$ are a set of function approximators $f_{\theta}(\cdot)=\\{f_{\theta_{0}}(\cdot),...,f_{\theta_{B}}(\cdot)\\}$. In the current paper, $f_{\theta_{i}}({\textnormal{s}}_{t-1})$ is a two-layer feed-forward network with 400 hidden units and swish activation function. Following previous work, we predict state deltas rather than the next states (Shyam et al., 2018). #### Reward model We implement the reward model as $p({\textnormal{o}}_{\tau}|{\textnormal{s}}_{\tau},\theta,\pi)=\mathcal{N}({\textnormal{o}}_{\tau};f_{\lambda}({\textnormal{s}}_{\tau}),\mathbf{1})$, where $f_{\lambda}({\textnormal{s}}_{\tau})$ is some arbitrary function approximator666Formally, this is an observation model, but we retain RL terminology for clarity.. In the current paper, $f_{\lambda}({\textnormal{s}}_{\tau})$ is a two layer feed forward network with 400 hidden units and ReLU activation function. Learning a reward model offers several plausible benefits outside of the active inference framework, as it abolishes the requirement that rewards can be directly calculated from observations or states (Chua et al., 2018a). #### Global prior We implement the global prior $p^{\Phi}({\textnormal{o}})$ as a Gaussian with unit variance centred around the maximum reward for the respective environment. We leave it to future work to explore the effects of more intricate priors. ## Appendix F Implementation details For all tasks, we initialize a dataset $\mathcal{D}$ with a single episode of data collected from a random agent. For each episode, we train the ensemble transition model and reward model for 100 epochs, using the negative-log likelihood loss. We found cold-starting training at each episode to lead to more consistent behaviour. We then let the agent act in the environment based on Algorithm 1, and append the collected data to the dataset $\mathcal{D}$. We list the full set of hyperparameters below: Hyperparameters --- Hidden layer size | 400 Learning rate | 0.001 Training-epochs | 100 Planning-horizon | 30 N-candidates (CEM) | 700 Top-candidates (CEM) | 70 Optimisation-iterations (CEM) | 7 ## Appendix G Expected information gain In Eq. 4, expected parameter information gain was presented in the form $\mathbb{E}_{q({\textnormal{s}}|\theta)}D_{\mathrm{KL}}\big{(}q(\theta|{\textnormal{s}})\|q(\theta)\big{)}$. While this provides a nice intuition about the effect of the information gain term on behaviour, it cannot be computed directly, due to the intractability of identifying true posteriors over parameters. We here show that, through a simple application of Bayes’ rule, it is straightforward to derive an equivalent expression for the expected information gain as the divergence between the state likelihood and marginal, given the parameters, which decomposes into an entropy of an average minus an average of entropies: $\displaystyle\mathbb{E}_{q({\textnormal{s}}|\theta)}D_{\mathrm{KL}}\big{(}q(\theta|{\textnormal{s}})\|q(\theta)\big{)}$ (13) $\displaystyle=\mathbb{E}_{q({\textnormal{s}}|\theta)q(\theta|{\textnormal{s}})}\big{[}\log q(\theta|{\textnormal{s}})-\log q(\theta)\big{]}$ $\displaystyle=\mathbb{E}_{q({\textnormal{s}},\theta)}\big{[}\log q({\textnormal{s}}|\theta)+\log q(\theta)-\log q({\textnormal{s}})-\log q(\theta)\big{]}$ $\displaystyle=\mathbb{E}_{q({\textnormal{s}},\theta)}\big{[}\log q({\textnormal{s}}|\theta)-\log q({\textnormal{s}})\big{]}$ $\displaystyle=\mathbb{E}_{q(\theta)q({\textnormal{s}}|\theta)}\big{[}\log q({\textnormal{s}}|\theta)\big{]}-\mathbb{E}_{q(\theta)q({\textnormal{s}}|\theta)}\big{[}\log\mathbb{E}_{q(\theta)}q({\textnormal{s}}|\theta)\big{]}$ $\displaystyle=-\mathbb{E}_{q(\theta)}\mathbf{H}\big{[}q({\textnormal{s}}|\theta)\big{]}+\mathbf{H}\big{[}\mathbb{E}_{q(\theta)}q({\textnormal{s}}|\theta)\big{]}$ The first term is the (negative) average of the entropies. The average over the parameters $\theta$ is achieved simply by averaging over the dynamics models in the ensemble. The entropy of the likelihoods $\mathbf{H}[p({\textnormal{s}}|\theta)]$ can be computed analytically since each network in the ensemble outputs a Gaussian distribution for which the entropy is a known analytical result. The second term is the entropy of the average $\mathbf{H}[\mathbb{E}_{p(\theta)}p({\textnormal{s}}|\theta)]$. Unfortunately, this term does not have an analytical solution. However, it can be approximated numerically using a variety of techniques for entropy estimation. In our paper, we use the nearest neighbour entropy approximation (Mirchev et al., 2018). ## Appendix H Environment details The Mountain Car environment ($\mathcal{S}\subseteq\mathbb{R}^{2}\mathcal{A}\subseteq\mathbb{R}^{1}$) requires an agent to drive up the side of a hill, where the car is underactuated requiring it first to gain momentum by driving up the opposing hill. A reward of one is generated when the agent reaches the goal, and zero otherwise. The Cup Catch environment ($\mathcal{S}\subseteq\mathbb{R}^{8}\mathcal{A}\subseteq\mathbb{R}^{2}$) requires the agent to actuate a cup and catch a ball attached to its bottom. A reward of one is generated when the agent reaches the goal, and zero otherwise. The Half Cheetah environment ($\mathcal{S}\subseteq\mathbb{R}^{1}7\mathcal{A}\subseteq\mathbb{R}^{6}$) describes a running planar biped. For the running task, a reward of $v-0.1||a||^{2}$ is received, where $v$ is the agent’s velocity, and for the flipping task, a reward of $\epsilon-0.1||a||^{2}$ is received, where $\epsilon$ is the angular velocity. The Ant Maze environment ($\mathcal{S}\subseteq\mathbb{R}^{2}9\mathcal{A}\subseteq\mathbb{R}^{8}$) involves a quadruped agent exploring a rectangular maze.

2024-09-04T02:54:55.443492

2002.12710

arxiv-papers

Causal mediation analysis with double machine learning Helmut Farbmacher+, Martin Huber*, Lukáš Lafférs++, Henrika Langen*, Martin Spindler** +Max Planck Society, Munich Center for the Economics of Aging *University of Fribourg, Dept. of Economics ++Matej Bel University, Dept. of Mathematics **University of Hamburg, Faculty of Business Sciences Abstract: This paper combines causal mediation analysis with double machine learning to control for observed confounders in a data-driven way under a selection-on-observables assumption in a high-dimensional setting. We consider the average indirect effect of a binary treatment operating through an intermediate variable (or mediator) on the causal path between the treatment and the outcome, as well as the unmediated direct effect. Estimation is based on efficient score functions, which possess a multiple robustness property w.r.t. misspecifications of the outcome, mediator, and treatment models. This property is key for selecting these models by double machine learning, which is combined with data splitting to prevent overfitting in the estimation of the effects of interest. We demonstrate that the direct and indirect effect estimators are asymptotically normal and root-$n$ consistent under specific regularity conditions and investigate the finite sample properties of the suggested methods in a simulation study when considering lasso as machine learner. We also provide an empirical application to the U.S. National Longitudinal Survey of Youth, assessing the indirect effect of health insurance coverage on general health operating via routine checkups as mediator, as well as the direct effect. We find a moderate short-term effect of health insurance coverage on general health which is, however, not mediated by routine checkups. Keywords: mediation, direct and indirect effects, causal mechanisms, double machine learning, efficient score. JEL classification: C21. Addresses for correspondence: Helmut Farbmacher, Max Planck Society, Munich Center for the Economics of Aging, Amalienstr. 33, 80799 Munich, Germany; <EMAIL_ADDRESS>Martin Huber, University of Fribourg, Department of Economics, Bd. de Pérolles 90, 1700 Fribourg, Switzerland; <EMAIL_ADDRESS>Lukáš Lafférs, Matej Bel University, Department of Mathematics, Tajovskeho 40, 97411 Banská Bystrica, Slovakia; <EMAIL_ADDRESS>Henrika Langen, University of Fribourg, Department of Economics, Bd. de Pérolles 90, 1700 Fribourg, Switzerland; <EMAIL_ADDRESS>Martin Spindler, University of Hamburg, Faculty of Business Administration, Moorweidenstr. 18, 20148 Hamburg, Germany; <EMAIL_ADDRESS>Lafférs acknowledges support provided by the Slovak Research and Development Agency under contract no. APVV-17-0329 and VEGA-1/0692/20. ## 1 Introduction Causal mediation analysis aims at decomposing the causal effect of a treatment on an outcome of interest into an indirect effect operating through a mediator (or intermediate outcome) and a direct effect comprising any causal mechanisms not operating through that mediator. Even if the treatment is random, direct and indirect effects are generally not identified by naively controlling for the mediator without accounting for its likely endogeneity, see Robins and Greenland (1992). While much of the earlier literature either neglected endogeneity issues or relied on restrictive linear models, see for instance Cochran (1957), Judd and Kenny (1981), and Baron and Kenny (1986), more recent contributions consider more general identification approaches using the potential outcome framework. Some of the numerous examples are Robins and Greenland (1992), Pearl (2001), Robins (2003), Petersen, Sinisi, and van der Laan (2006), VanderWeele (2009), Imai, Keele, and Yamamoto (2010), Hong (2010), Albert and Nelson (2011), Imai and Yamamoto (2013), Tchetgen Tchetgen and Shpitser (2012), Vansteelandt, Bekaert, and Lange (2012), and Huber (2014). Using the denomination of Pearl (2001), the literature distinguishes between natural direct and indirect effects, where mediators are set to their potential values ‘naturally’ occurring under a specific treatment assignment, and the controlled direct effect, where the mediator is set to a ‘prescribed’ value. The vast majority of identification strategies relies on selection-on- observable-type assumptions implying that the treatment and the mediator are conditionally exogenous when controlling for observed covariates. Empirical examples in economics and policy evaluation include Flores and Flores-Lagunes (2009), Heckman, Pinto, and Savelyev (2013), Keele, Tingley, and Yamamoto (2015), Conti, Heckman, and Pinto (2016), Huber (2015), Huber, Lechner, and Mellace (2017), Bellani and Bia (2018), Bijwaard and Jones (2018), and Huber, Lechner, and Strittmatter (2018). Such studies typically rely on the (implicit) assumption that the covariates to be controlled for can be unambiguously preselected by the researcher, for instance based on institutional knowledge or theoretical considerations. This assumes away uncertainty related to model selection w.r.t. covariates to be included and entails incorrect inference under the common practice of choosing and refining the choice of covariates based on their predictive power. To improve upon this practice, this paper combines causal mediation analysis based on efficient score functions, see Tchetgen Tchetgen and Shpitser (2012), with double machine learning as outlined in Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018) for a data-driven control of observed confounders to obtain valid inference under specific regularity conditions. In particular, one important condition is that the number of important confounders (that make the selection-on-observables assumptions to hold approximately) is not too large relative to the sample size. However, the set of these important confounders need not be known a priori and the set of potential confounders can be even larger than the sample size.111Different from conventional semiparametric methods, the double machine learning framework does not require the set of potential confounders to be restricted by Donsker conditions, but permits the set to be unbounded and to grow with the sample size. This is particularly useful in high dimensional data with a vast number of covariates that could potentially serve as control variables, which can render researcher-based covariate selection complicated if not infeasible. We demonstrate root-$n$ consistency and asymptotic normality of the proposed effect estimators under specific regularity conditions by verifying that the general framework of Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018) for well-behaved double machine learning is satisfied in our context. Tchetgen Tchetgen and Shpitser (2012) suggest estimating natural direct and indirect effects based on the efficient score functions of the potential outcomes, which requires plug-in estimates for the conditional mean outcome, mediator density, and treatment probability. Analogous to doubly robust estimation of average treatment effects, see Robins, Rotnitzky, and Zhao (1994) and Robins and Rotnitzky (1995), the resulting estimators are semiparametrically efficient if all models of the plug-in estimates are correctly specified and remain consistent even if one model is misspecified. Our first contribution is to show that the efficient score function of Tchetgen Tchetgen and Shpitser (2012) satisfies the so-called Neyman (1959) orthogonality discussed in Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018), which makes the estimation of direct and indirect effects rather insensitive to (local) estimation errors in the plug-in estimates. Second, we show that by an application of Bayes’ Theorem, the score function of Tchetgen Tchetgen and Shpitser (2012) can be transformed in a way that avoids estimation of the conditional mediator density and show it to be Neyman orthogonal. This appears particularly useful when the mediator is a vector of variables and/or continuous. Third, we establish the score function required for estimating the controlled direct effect along with Neyman orthgonality. Neyman orthgonality is key for the fruitful application of double machine learning, ensuring robustness in the estimation of the nuisance parameters which is crucial when applying modern machine learning methods. Random sample splitting – to estimate the parameters of the plug-in models in one part of the data, while predicting the score function and estimating the direct and indirect effects in the other part – avoids overfitting the plug-in models (e.g. by controlling for too many covariates). It increases the variance by only using part of the data for effect estimation. This is avoided by cross- fitting which consists of swapping the roles of the data parts for estimating the plug-in models and the treatment effects to ultimately average over the effects estimates in either part. When combining efficient score-based effect estimation with sample splitting, $n^{-1/2}$-convergence of treatment effect estimation can be obtained under a substantially slower convergence of $n^{-1/4}$ for the plug-in estimates, see Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018). Under specific regularity conditions, this convergence rate can attained by various machine learning algorithms including lasso regression, see Tibshirani (1996). We investigate the estimators’ finite sample behavior based on the score function of Tchetgen Tchetgen and Shpitser (2012) and the alternative score suggested in this paper when using post-lasso regression as machine learner for the plug-in estimates. Furthermore, we apply our method to data from the National Longitudinal Survey of Youth 1997 (NLSY97), where a large set of potential control variables is available. We disentangle the short-term effect of health insurance coverage on general health into an indirect effect which operates via the incidence of a routine checkup in the last year and a direct effect covering any other causal mechanisms. While we find a moderate health- improving direct effect, the indirect effect is very close to zero. We therefore do not find evidence that health insurance coverage affects general health through routine checkups in the short run. We note that basing estimation on efficient score functions is not the only framework satisfying the previously mentioned robustness w.r.t. estimation errors in plug-in parameters. This property is also satisfied by the targeted maximum likelihood estimation (TMLE) framework by van der Laan and Rubin (2006), see the discussion in Díaz (2020). TMLE relies on iteratively updating (or robustifying) an initial estimate of the parameter of interest based on regression steps that involve models for the plug-in parameters. Zheng and van der Laan (2012) have developed an estimation approach for natural direct and indirect effects using TMLE, where the plug-in parameters might by estimated by machine learners, e.g. the super learner, an ensemble method suggested by van der Laan, Polley, and Hubbard (2007). This iterative estimation approach is therefore an alternative to the double machine learning-based approach suggested in this paper, for which we demonstrate $n^{-1/2}$-consistency under specific conditions. This paper proceeds as follows. Section 2 introduces the concepts of direct and indirect effect identification in the potential outcome framework. In Section 3, we present the identifying assumptions and discuss identification based on efficient score functions. Section 4 proposes an estimation procedure based on double machine learning and shows root-$n$ consistency and asymptotic normality under specific conditions. Section 5 provides a simulation study. Section 6 presents an empirical application to data from the NLSY97. Section 7 concludes. ## 2 Definition of direct and indirect effects We aim at decomposing the average treatment effect (ATE) of a binary treatment, denoted by $D$, on an outcome of interest, $Y$, into an indirect effect operating through a discrete mediator, $M$, and a direct effect that comprises any causal mechanisms other than through $M$. We use the potential outcome framework, see for instance Rubin (1974), to define the direct and indirect effects of interest, see also Ten Have, Joffe, Lynch, Brown, Maisto, and Beck (2007) and Albert (2008) for further examples in the context of mediation. $M(d)$ denotes the potential mediator under treatment value $d$ $\in$ $\\{0,1\\}$, while $Y(d,m)$ denotes the potential outcome as a function of both the treatment and some value $m$ of the mediator $M$.222Throughout this paper, capital letters denote random variables and small letters specific values of random variables. The observed outcome and mediator correspond to the respective potential variables associated with the actual treatment assignment, i.e. $Y=D\cdot Y(1,M(1))+(1-D)\cdot Y(0,M(0))$ and $M=D\cdot M(1)+(1-D)\cdot M(0)$, implying that any other potential outcomes or mediators are a priori (i.e. without further statistical assumptions) unknown. We denote the ATE by $\Delta=E[Y(1,M(1))-Y(0,M(0))]$, which comprises both direct and indirect effects. To decompose the latter, note that the average direct effect, denoted by $\theta(d)$, equals the difference in mean potential outcomes when switching the treatment while keeping the potential mediator fixed, which blocks the causal mechanism via $M$: $\displaystyle\theta(d)$ $\displaystyle=$ $\displaystyle E[Y(1,M(d))-Y(0,M(d))],\quad d\in\\{0,1\\}.$ (1) The (average) indirect effect, $\delta(d)$, equals the difference in mean potential outcomes when switching the potential mediator values while keeping the treatment fixed to block the direct effect. $\displaystyle\delta(d)$ $\displaystyle=$ $\displaystyle E[Y(d,M(1))-Y(d,M(0))],\quad d\in\\{0,1\\}.$ (2) Robins and Greenland (1992) and Robins (2003) referred to these parameters as pure/total direct and indirect effects, Flores and Flores-Lagunes (2009) as net and mechanism average treatment effects, and Pearl (2001) as natural direct and indirect effects, which is the denomination used in the remainder of this paper. The ATE is the sum of the natural direct and indirect effects defined upon opposite treatment states $d$, which can be easily seen from adding and subtracting the counterfactual outcomes $E[Y(0,M(1))]$ and $E[Y(1,M(0))]$: $\displaystyle\Delta$ $\displaystyle=$ $\displaystyle E[Y(1,M(1))-Y(0,M(0))]$ (3) $\displaystyle=$ $\displaystyle E[Y(1,M(1))-Y(0,M(1))]+E[Y(0,M(1))-Y(0,M(0))]=\theta(1)+\delta(0)$ $\displaystyle=$ $\displaystyle E[Y(1,M(0))-Y(0,M(0))]+E[Y(1,M(1))-Y(1,M(0))]=\theta(0)+\delta(1).$ The distinction between $\theta(1)$ and $\theta(0)$ as well as $\delta(1)$ and $\delta(0)$ hints to the possibility of heterogeneous effects across treatment states $d$ due to interaction effects between $D$ and $M$. For instance, the direct effect of health insurance coverage ($D$) on general health ($Y$) might depend on whether or not a person underwent routine check-ups ($M$). We note that a different approach to dealing with the interaction effects between $D$ and $M$ is a three-way decomposition of the ATE into the pure direct effect ($\theta(0)$), the pure indirect effect ($\delta(0)$) and the mediated interaction effect, see VanderWeele (2013). The so-called controlled direct effect, denoted by $\gamma(m)$, is a further parameter that received much attention in the mediation literature. It corresponds to the difference in mean potential outcomes when switching the treatment and fixing the mediator at some value $m$: $\gamma(m)=E[Y(1,m)-Y(0,m)],\quad\quad\text{for }m\text{ in the support of }M.$ (4) In contrast to $\theta(d)$, which is conditional on the potential mediator value ‘naturally’ realized for treatment $d$ which may differ across subjects, $\gamma(m)$ is conditional on enforcing the same mediator state in the entire population. The two parameters are only equivalent in the absence of an interaction between $D$ and $M$. Whether the natural or controlled direct effect is more relevant depends on the feasibility and desirability to intervene on or prescribe the mediator, see Pearl (2001) for a discussion of the ‘descriptive’ and ‘prescriptive’ natures of natural and controlled effects. There is no indirect effect parameter matching the controlled direct effect, implying that the difference between the total effect and the controlled direct effect does in general not correspond to the indirect effect, unless there is no interaction between $D$ and $M$, see e.g. Kaufman, MacLehose, and Kaufman (2004). ## 3 Assumptions and identification Our identification strategy is based on the assumption that confounding of the treatment-outcome, treatment-mediator, and mediator-outcome relations can be controlled for by conditioning on observed covariates, denoted by $X$. The latter must not contain variables that are influenced by the treatment, such that $X$ is typically evaluated prior to treatment assignment. Figure 1 provides a graphical illustration using a directed acyclic graph, with arrows representing causal effects. Each of $D$, $M$, and $Y$ might be causally affected by distinct and statistically independent sets of unobservables not displayed in Figure 1, but none of these unobservables may jointly affect two or all three elements $(D,M,Y)$ conditional on $X$. Figure 1: Causal paths under conditional exogeneity given pre-treatment covariates Formally, the first assumption invokes conditional independence of the treatment and potential mediators or outcomes given $X$. This restriction has been referred to as conditional independence, selection on observables, or exogeneity in the treatment evaluation literature, see e.g. Imbens (2004). This rules out confounders jointly affecting the treatment on the one hand and the mediator and/or the outcome on the other hand conditional on $X$. In non- experimental data, the plausibility of this assumption critically hinges on the richness of $X$. Assumption 1 (conditional independence of the treatment): $\\{Y(d^{\prime},m),M(d)\\}\bot D|X$ for all $d^{\prime},d\in\\{0,1\\}$ and $m$ in the support of $M$, where ‘$\bot$’ denotes statistical independence. The second assumption requires the mediator to be conditionally independent of the potential outcomes given the treatment and the covariates. Assumption 2 (conditional independence of the mediator): $Y(d^{\prime},m)\bot M|D=d,X=x$ for all $d^{\prime},d\in\\{0,1\\}$ and $m,x$ in the support of $M,X$. Assumption 2 rules out confounders jointly affecting the mediator and the outcome conditional on $D$ and $X$. If $X$ is pre-treatment (as is common to avoid controlling for variables potentially affected by the treatment), this implies the absence of post-treatment confounders of the mediator-outcome relation. Such a restriction needs to be rigorously scrutinized and appears for instance less plausible if the time window between the measurement of the treatment and the mediator is large in a world of time-varying variables. The third assumption imposes common support on the conditional treatment probability across treatment states. Assumption 3 (common support): $\Pr(D=d|M=m,X=x)>0$ for all $d\in\\{0,1\\}$ and $m,x$ in the support of $M,X$. The common support assumption, also known as positivity or covariate overlap assumption, restricts the conditional probability to be or not be treated given $M,X$, henceforth referred to as propensity score, to be larger than zero. It implies the weaker condition that $\Pr(D=d|X=x)>0$ such that the treatment must not be deterministic in $X$, otherwise no comparable units in terms of $X$ are available across treatment states. By Bayes’ theorem, Assumption 3 also implies that $\Pr(M=m|D=d,X=x)>0$ if $M$ is discrete or that the conditional density of $M$ given $D,X$ is larger than zero if $M$ is continuous. Conditional on $X$, the mediator state must not be deterministic in the treatment, otherwise no comparable units in terms of the treatment are available across mediator states. Assumptions 1 to 3 are standard in the causal mediation literature, see for instance Imai, Keele, and Yamamoto (2010), Tchetgen Tchetgen and Shpitser (2012), Vansteelandt, Bekaert, and Lange (2012), and Huber (2014), or also Pearl (2001), Petersen, Sinisi, and van der Laan (2006), and Hong (2010), for closely related restrictions. Tchetgen Tchetgen and Shpitser (2012) discuss identification of the counterfactual $E[Y(d,M(1-d))]$ based on the efficient score function: $\displaystyle E[Y(d,M(1-d))]$ $\displaystyle=$ $\displaystyle E[\psi_{d}],$ $\displaystyle\textrm{ with }\psi_{d}$ $\displaystyle=$ $\displaystyle\frac{I\\{D=d\\}\cdot f(M|1-d,X)}{p_{d}(X)\cdot f(M|d,X)}\cdot[Y-\mu(d,M,X)]$ (5) $\displaystyle+\frac{I\\{D=1-d\\}}{1-p_{d}(X)}\cdot\Big{[}\mu(d,M,X)-\int_{m\in\mathcal{M}}\mu(d,m,X)\cdot f(m|1-d,X)\ dm\Big{]}$ $\displaystyle+\int_{m\in\mathcal{M}}\mu(d,m,X)\cdot f(m|1-d,X)\ dm$ where $f(M|D,X)$ denotes the conditional density of $M$ given $D$ and $X$ (if $M$ is discrete, this is a conditional probability and integrals need to be replaced by sums), $p_{d}(X)=\Pr(D=d|X)$ the probability of treatment $D=d$ given $X$, and $\mu(D,M,X)=E(Y|D,M,X)$ the conditional expectation of outcome $Y$ given $D$, $M$, and $X$. (3) satisfies a multiple robustness property in the sense that estimation remains consistent even if one out of the three models for the plug-in parameters $f(M|D,X)$, $p_{d}(X)$, and $\mu(D,M,X)$ is misspecified. To derive an alternative expression for identification, note that by Bayes’ Law, $\displaystyle\frac{f(M|1-d,X)}{p_{d}(X)\cdot f(M|d,X)}=\frac{\big{(}1-p_{d}(M,X)\big{)}\cdot f(M|X)}{1-p_{d}(X)}\cdot\frac{p_{d}(X)}{p_{d}(M,X)\cdot f(M|X)\cdot p_{d}(X)}=\frac{1-p_{d}(M,X)}{p_{d}(M,X)\cdot\big{(}1-p_{d}(X)\big{)}}$ where $f(M|X)$ is the conditional distribution of $M$ given $X$ and $p_{d}(X,M)=\Pr(D=d|X,M)$. Furthermore, $\displaystyle\int\mu(d,m,X)\cdot f(m|1-d,X)dm=E\Big{[}\mu(d,M,X)\Big{|}D=1-d,X\Big{]}.$ Therefore, an alternative and also multiply robust representation of (3) is $\displaystyle E[Y(d,M(1-d))]$ $\displaystyle=$ $\displaystyle E[\psi_{d}^{*}],$ $\displaystyle\textrm{ with }\psi_{d}^{*}$ $\displaystyle=$ $\displaystyle\frac{I\\{D=d\\}\cdot(1-p_{d}(M,X))}{p_{d}(M,X)\cdot(1-p_{d}(X))}\cdot[Y-\mu(d,M,X)]$ $\displaystyle+$ $\displaystyle\frac{I\\{D=1-d\\}}{1-p_{d}(X)}\cdot\Big{[}\mu(d,M,X)-E\Big{[}\mu(d,M,X)\Big{|}D=1-d,X\Big{]}\Big{]}$ $\displaystyle+$ $\displaystyle E\Big{[}\mu(d,M,X)\Big{|}D=1-d,X\Big{]}.$ Similarly as the approaches based on inverse probability weighting (rather than efficient scores) in Huber (2014) and Tchetgen Tchetgen (2013), (3) avoids conditional mediator densities, which appears attractive if $M$ is continuous and/or multidimensional. On the other hand, it requires the estimation of an additional parameter, namely the nested conditional mean $E[\mu(d,M,X)|D=1-d,X]$, as similarly found in Miles, Shpitser, Kanki, Meloni, and Tchetgen Tchetgen (2020), who suggest a multiply robust score function for assessing path-specific effects. Alternatively to rearranging the score function by Tchetgen Tchetgen and Shpitser (2012) as outlined above, ratios of conditional densities as for instance appearing in the first component of (3) might be treated as additional nuisance parameter and estimated directly via density-ratio estimation, see e.g. Sugiyama, Kawanabe, and Chui (2010) for density-ratio estimation in high-dimensional settings. Such methods based on directly estimating the density ratio without going through estimating the densities in numerator and denominator separately are shown in several studies to compare favorably with estimating the densities separately, see e.g. Kanamori, Suzuki, and Sugiyama (2012). Efficient score-based identification of $E[Y(d,M(d))]$ under $Y(d,m)\bot\\{D,M\\}|X=x$ (see Assumption 1) has been established in the literature on doubly robust ATE estimation, see for instance Robins, Rotnitzky, and Zhao (1994) and Hahn (1998): $\displaystyle E[Y(d,M(d))]=E[\alpha_{d}]\textrm{ with }\alpha_{d}=\frac{I\\{D=d\\}\cdot[Y-\mu(d,X)]}{p_{d}(X)}+\mu(d,X)$ (7) where $\mu(D,X)=E(Y|D,M(D),X)=E(Y|D,X)$ is the conditional expectation of outcome $Y$ given $D$ and $X$. For identifying the controlled direct effect, we now assume that $M$ is discrete (while this need not be the case in the context of natural direct and indirect effects) such that for all $m$ in the support of $M$, it must hold that $\Pr(M=m)>0$. As Assumptions 1 and 2 imply $Y(d,m)\bot\\{D,M\\}|X=x$, doubly robust identification of the potential outcome $E[Y(d,m)]$, which is required for the controlled direct effect, follows from replacing $I\\{D=d\\}$ and $p_{d}(X)$ in (7) by $I\\{D=d,M=m\\}=I\\{M=m\\}\cdot I\\{D=d\\}$ and $\Pr(D=d,M=m|X)=f(m|d,X)\cdot p_{d}(X)$: $\displaystyle E[Y(d,m)]=E[\psi_{dm}]\textrm{ with }\psi_{dm}=\frac{I\\{D=d\\}\cdot I\\{M=m\\}\cdot[Y-\mu(d,m,X)]}{f(m|d,X)\cdot p_{d}(X)}+\mu(d,m,X).$ (8) ## 4 Estimation of the counterfactual with K-fold Cross-Fitting We subsequently propose an estimation strategy for the counterfactual $E[Y(d,M(1-d))]$ with $d\in\\{0,1\\}$ based on the efficient score function by Tchetgen Tchetgen and Shpitser (2012) provided in (3) and show its root-$n$ consistency under specific regularity conditions. To this end, let $\mathcal{W}=\\{W_{i}|1\leq i\leq N\\}$ with $W_{i}=(Y_{i},M_{i},D_{i},X_{i})$ for $i=1,\ldots,n$ denote the set of observations in an i.i.d. sample of size $n$. $\eta$ denotes the plug-in (or nuisance) parameters, i.e. the conditional mean outcome, mediator density and treatment probability. Their respective estimates are referred to by $\hat{\eta}=\\{\hat{\mu}(D,M,X),\hat{f}(M|D,X),\hat{p_{d}}(X)\\}$ and the true nuisance parameters by $\eta_{0}=\\{\mu_{0}(D,M,X),f_{0}(M|D,X),p_{d0}(X)\\}$. Finally, $\Psi_{d0}=E[Y(d,M(1-d))]$ denotes the true counterfactual. We suggest estimating $\Psi_{d0}$ using the following algorithm that combines orthogonal score estimation with sample splitting and is root-$n$ consistent under conditions outlined further below. Algorithm 1: Estimation of $E[Y(d,M(1-d))]$ based on equation (3) 1. 1. Split $\mathcal{W}$ in $K$ subsamples. For each subsample $k$, let $n_{k}$ denote its size, $\mathcal{W}_{k}$ the set of observations in the sample and $\mathcal{W}_{k}^{C}$ the complement set of all observations not in $k$. 2. 2. For each $k$, use $\mathcal{W}_{k}^{C}$ to estimate the model parameters of $p_{d}(X)$, $f(M|D,X)$, and $\mu(D,M,X)$ in order to predict these models in $\mathcal{W}_{k}$, where the predictions are denoted by $\hat{p_{d}}^{k}(X)$, $\hat{f}^{k}(M|D,X)$, and $\hat{\mu}^{k}(D,M,X)$. 3. 3. For each $k$, obtain an estimate of the efficient score function (see $\psi_{d}$ in (3)) for each observation $i$ in $\mathcal{W}_{k}$, denoted by $\hat{\psi}_{d,i}^{k}$ : $\displaystyle\hat{\psi}_{d,i}^{k}$ $\displaystyle=$ $\displaystyle\frac{I\\{D_{i}=d\\}\cdot\hat{f}^{k}(M_{i}|1-d,X_{i})}{\hat{p}_{d}^{k}(X_{i})\cdot\hat{f}^{k}(M_{i}|d,X_{i})}\cdot[Y_{i}-\hat{\mu}^{k}(d,M_{i},X_{i})]$ (9) $\displaystyle+\frac{I\\{D_{i}=1-d\\}}{1-\hat{p}_{d}^{k}(X_{i})}\cdot\Big{[}\hat{\mu}^{k}(d,M_{i},X_{i})-\int_{m\in\mathcal{M}}\hat{\mu}^{k}(d,m,X_{i})\cdot\hat{f}^{k}(m|1-d,X_{i})dm\Big{]}$ $\displaystyle+\int_{m\in\mathcal{M}}\hat{\mu}^{k}(d,m,X_{i})\cdot\hat{f}^{k}(m|1-d,X_{i})dm.$ 4. 4. Average the estimated scores $\hat{\psi}_{d,i}^{k}$ over all observations across all $K$ subsamples to obtain an estimate of $\Psi_{d0}=E[Y(d,M(1-d))]$ in the total sample, denoted by $\hat{\Psi}_{d}=1/n\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}\hat{\psi}_{d,i}^{k}$. Algorithm 1 can be adapted to estimate the counterfactuals required for the controlled direct effect, see (8). To this end, denote by $\Psi_{dm0}=E[Y(d,m)]$ the true counterfactual of interest, which is estimated by replacing $\psi_{d}$ and $\Psi_{d}$ by $\psi_{dm}$ and $\Psi_{dm0}$, respectively, everywhere in Algorithm 1. In order to achieve root-$n$ consistency for counterfactual estimation, we make specific assumptions about the prediction qualities of the machine learners for our plug-in estimates of the nuisance parameters. Closely following Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018), we to this end introduce some further notation. Let $(\delta_{n})_{n=1}^{\infty}$ and $(\Delta_{n})_{n=1}^{\infty}$ denote sequences of positive constants with $\lim_{n\rightarrow\infty}\delta_{n}=0$ and $\lim_{n\rightarrow\infty}\Delta_{n}=0.$ Also, let $c,\epsilon,C,\underline{f},\overline{f}$ and $q$ be positive constants such that $q>2,$ and let $K\geq 2$ be a fixed integer. Furthermore, for any random vector $Z=(Z_{1},...,Z_{l})$, let $\left\|Z\right\|_{q}=\max_{1\leq j\leq l}\left\|Z_{l}\right\|_{q},$ where $\left\|Z_{l}\right\|_{q}=\left(E\left[\left|Z_{l}\right|^{q}\right]\right)^{\frac{1}{q}}$. For the sake of easing notation, we assume that $n/K$ is an integer. For brevity, we omit the dependence of probability $\Pr_{P}(\cdot),$ expectation $E_{P}(\cdot),$ and norm $\left\|\cdot\right\|_{P,q}$ on the probability measure $P$. Assumption 4 (regularity conditions and quality of plug-in parameter estimates): For all probability laws $P\in\mathcal{P}$, where $\mathcal{P}$ is the set of all possible probability laws, the following conditions hold for the random vector $(Y,D,M,X)$ for $d\in\\{0,1\\}$: 1. (a) $\left\|Y\right\|_{q}\leq C$ and $\left\|E[Y^{2}|d,M,X]\right\|_{\infty}\leq C^{2},$ 2. (b) $\Pr(\epsilon\leq p_{d0}(X)\leq 1-\epsilon)=1,$ 3. (c) $\Pr(\underline{f}\leq f(M|D,X)\leq\overline{f})=1,$ 4. (d) $\left\|Y-\mu_{0}(d,M,X)\right\|_{2}=E\Big{[}\left(Y-\mu_{0}(d,M,X))\right)^{2}\Big{]}^{\frac{1}{2}}\geq c$ 5. (e) Given a random subset $I$ of $[n]$ of size $n_{k}=n/K,$ the nuisance parameter estimator $\hat{\eta}_{0}=\hat{\eta}_{0}(\mathcal{W}_{k}^{C})$ satisfies the following conditions. With $P$-probability no less than $1-\Delta_{n}:$ $\displaystyle\left\|\hat{\eta}_{0}-\eta_{0}\right\|_{q}$ $\displaystyle\leq$ $\displaystyle C,$ $\displaystyle\left\|\hat{\eta}_{0}-\eta_{0}\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n},$ $\displaystyle\left\|\hat{p}_{d0}(X)-1/2\right\|_{\infty}$ $\displaystyle\leq$ $\displaystyle 1/2-\epsilon,$ $\displaystyle\left\|\hat{f}_{0}(M|D,X)-(\underline{f}+\overline{f})/2\right\|_{\infty}$ $\displaystyle\leq$ $\displaystyle(\overline{f}-\underline{f})/2,$ $\displaystyle\left\|\hat{\mu}_{0}(D,M,X)-\mu_{0}(D,M,X)\right\|_{2}\times\left\|\hat{p}_{d0}(X)-p_{d0}(X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2},$ $\displaystyle\left\|\hat{\mu}_{0}(D,M,X)-\mu_{0}(D,M,X)\right\|_{2}\times\left\|\hat{f}_{0}(M|1-D,X)-f_{0}(M|1-D,X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2}.$ For demonstrating root-$n$ consistency of the proposed estimation strategy for the counterfactual, we heavily draw from Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018). We show that our estimation strategy satisfies the requirements for their double machine learning framework by first verifying the satisfaction of a specific moment condition as well as linearity and Neyman orthogonality of the score (see Appendix B.1.1). Then, as e.g. $\psi_{d}(W,\eta,\Psi_{d0})$ is smooth in $(\eta,\Psi_{d0})$, the plug-in estimators must converge with rate $n^{-1/4}$ in order to achieve $n^{-1/2}$-convergence for the estimation of $\hat{\Psi}_{d}$. This convergence rate of $n^{-1/4}$ is achievable for many commonly used machine learners such as lasso, random forest, boosting and neural nets. The rates for $L_{2}$-boosting were, for instance, derived in Luo and Spindler (2016). Theorem 1 Under Assumptions 1-4, it holds for estimating $E[Y(d,M(1-d))]$, $E[Y(d,m)]$ based on Algorithm 1: $\sqrt{n}\Big{(}\hat{\Psi}_{d}-\Psi_{d0}\Big{)}\rightarrow N(0,\sigma^{2}_{\psi_{d}})$, where $\sigma^{2}_{\psi_{d}}=E[(\psi_{d}-\Psi_{d0})^{2}]$. $\sqrt{n}\Big{(}\hat{\Psi}_{dm}-\Psi_{dm0}\Big{)}\rightarrow N(0,\sigma^{2}_{\psi_{dm}})$, where $\sigma^{2}_{\psi_{d}}=E[(\psi_{d}-\Psi_{dm0})^{2}]$. The proof is provided in Appendix B.1. Analogous results follow for the estimation of $\Lambda=E[Y(d,M(d))]$ when replacing $\hat{\psi}_{d}$ in the algorithm above by an estimate of score function $\alpha_{d}$ from (7), $\displaystyle\hat{\alpha}_{d}=\frac{I\\{D=d\\}\cdot(Y_{i}-\hat{\mu}^{k}(d,X_{i}))}{\hat{p_{d}}^{k}(X_{i})}+\hat{\mu}^{k}(d,X_{i}),$ (10) where $\hat{\mu}^{k}(d,x)$ is an estimate of $\mu(d,x)$. This approach has been discussed in literature on ATE estimation based on double machine learning, see for instance Belloni, Chernozhukov, Fernández-Val, and Hansen (2017) and Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018). Denoting by $\hat{\Lambda}$ the estimate of $\Lambda$, it follows under Assumptions 1-4 that $\sqrt{n}\Big{(}\hat{\Lambda}_{d}-\Lambda_{d}\Big{)}\rightarrow N(0,\sigma^{2}_{\alpha_{d}})$, where $\sigma^{2}_{\alpha_{d}}=E[(\alpha_{d}-\Lambda_{d})^{2}]$. Therefore, root-$n$-consistent estimates of the total as well as the direct and indirect effects are obtained as difference of the estimated potential outcomes, which we denote by $\hat{\Delta}$, $\hat{\theta}(d)$, and $\hat{\delta}(d)$. That is, $\hat{\Delta}=\hat{\Lambda}_{1}-\hat{\Lambda}_{0}$, $\hat{\theta}(1)=\hat{\Lambda}_{1}-\hat{\Psi}_{0}$, $\hat{\theta}(0)=\hat{\Psi}_{1}-\hat{\Lambda}_{0}$, $\hat{\delta}(1)=\hat{\Lambda}_{1}-\hat{\Psi}_{1}$, and $\hat{\delta}(0)=\hat{\Psi}_{0}-\hat{\Lambda}_{0}$. Naturally, the asymptotic variance of any effect is obtained based on the variance of the difference in the score functions of the potential outcomes required for the respective effect. For instance, the asymptotic variance of $\hat{\theta}(1)$ is given by $Var(\hat{\theta}(1))=Var(\alpha_{1}-\psi_{0})/n=(\sigma^{2}_{\alpha_{1}}+\sigma^{2}_{\psi_{0}}-2Cov(\alpha_{1},\psi_{0}))/n$. Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018) show that under Assumptions 1-4, $\hat{\sigma}^{2}_{\psi_{d}}$ can be estimated as: $\displaystyle\hat{\sigma}^{2}_{\psi_{d}}=1/K\sum_{k=1}^{K}\Big{[}1/n_{k}\sum_{i=1}^{n_{k}}\psi_{d}(W_{i},\hat{\eta}_{0}^{k},\hat{\Psi}_{d})^{2}\Big{]}$ (11) The asymptotic variance of $\alpha_{d}$ can be estimated accordingly, with $\psi_{d}$ and $\hat{\Psi}_{d0}$ substituted by $\alpha_{d}$ and $\hat{\Lambda}_{d0}$. We subsequently discuss estimation based on the score function $\psi_{d}^{*}$ in expression (3). We note that in this case, one needs to estimate a nested nuisance parameter $E\Big{[}\mu(d,M,X)\Big{|}D=1-d,X\Big{]}$. To avoid overfitting, the models for $\mu(d,M,X)$ and $E\Big{[}\mu(d,M,X)\Big{|}D=1-d,X\Big{]}$ are estimated in different subsamples. The plug-in estimates for the conditional mean outcome, mediator density and treatment probability are referred to by $\hat{\eta}^{*}=\\{\hat{\mu}(D,M,X),\hat{\omega}(D,M,X),\hat{p}(M,X),\hat{p}_{d}(X)\\}$ and the true nuisance parameters by $\eta_{0}^{*}=\\{\mu_{0}(D,M,X),\omega(D,M,X),p_{d0}(M,X),p_{d0}(X)\\}$. Algorithm 2: Estimation of $E[Y(d,M(1-d))]$ based on equation (3) 1. 1. Split $\mathcal{W}$ in $K$ subsamples. For each subsample $k$, let $n_{k}$ denote its size, $\mathcal{W}_{k}$ the set of observations in the sample and $\mathcal{W}_{k}^{C}$ the complement set of all observations not in $k$. 2. 2. For each $k$, use $\mathcal{W}_{k}^{C}$ to estimate the model parameters of $p_{d}(X)$ and $p_{d}(M,X)$. Split $\mathcal{W}_{k}^{C}$ into 2 nonoverlapping subsamples and estimate the model parameters of the conditional mean $\mu(d,M,X)$ and the nested conditional mean $E\Big{[}\mu(d,M,X)\Big{|}D=1-d,X\Big{]}$ in the distinct subsamples. Predict the nuisance parameters in $\mathcal{W}_{k}$, where the predictions are denoted by $\hat{p_{d}}^{k}(X)$, $\hat{p}_{d}^{k}(M,X)$, $\hat{\mu}^{k}(D,M,X)$ and $\hat{E}\Big{[}\mu(d,M,X)\ \Big{|}D=1-d,X\Big{]}^{k}$. 3. 3. For each $k$, obtain an estimate of the efficient score function (see $\psi_{d}^{*}$ in (3)) for each observation $i$ in $\mathcal{W}_{k}$, denoted by $\hat{\psi}_{d,i}^{k}$ : $\displaystyle\hat{\psi}_{d,i}^{*k}$ $\displaystyle=$ $\displaystyle\frac{I\\{D_{i}=d\\}\big{(}1-\hat{p}_{d}^{k}(M_{i},X_{i})\big{)}}{\hat{p}_{d}^{k}(M_{i},X_{i})\,\big{(}1-\hat{p}_{d}^{k}(X_{i})\big{)}}\cdot[Y-\hat{\mu}^{k}(d,M_{i},X_{i})]$ (12) $\displaystyle+\frac{I\\{D_{i}=1-d\\}}{1-\hat{p}_{d}^{k}(X_{i})}\cdot\Big{[}\hat{\mu}^{k}(d,M_{i},X_{i})-\hat{E}\Big{[}\hat{\mu}^{k}(d,M_{i},X_{i})\Big{|}D_{i}=1-d,X_{i}\Big{]}\Big{]}$ $\displaystyle+\hat{E}\Big{[}\hat{\mu}^{k}(d,M_{i},X_{i})\Big{|}D_{i}=1-d,X_{i}\Big{]}.$ 4. 4. Average the estimated scores $\hat{\psi}_{d,i}^{*k}$ over all observations across all $K$ subsamples to obtain an estimate of $\Psi_{d0}=E[Y(d,M(1-d))]$ in the total sample, denoted by $\hat{\Psi}_{d}^{*}=1/n\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}\hat{\psi}_{d,i}^{*k}$. Also this approach can be shown to be root-$n$-consistent under specific regularity conditions outlined below. Assumption 5 (regularity conditions and quality of plug-in parameter estimates): For all probability laws $P\in\mathcal{P}$ the following conditions hold for the random vector $(Y,D,M,X)$ for all $d\in\\{0,1\\}$: 1. (a) $\left\|Y\right\|_{q}\leq C$ and $\left\|E[Y^{2}|d,M,X]\right\|_{\infty}\leq C^{2},$ 2. (b) $\Pr(\epsilon\leq p_{d0}(X)\leq 1-\epsilon)=1,$ 3. (c) $\Pr(\epsilon\leq p_{d0}(M,X)\leq 1-\epsilon)=1,$ 4. (d) $\left\|Y-\mu_{0}(d,M,X)\right\|_{2}=E\Big{[}\left(Y-\mu_{0}(d,M,X))\right)^{2}\Big{]}^{\frac{1}{2}}\geq c$ 5. (e) Given a random subset $I$ of $[n]$ of size $n_{k}=n/K,$ the nuisance parameter estimator $\hat{\eta}^{*}_{0}=\hat{\eta}^{*}_{0}(\mathcal{W}_{k}^{C})$ satisfies the following conditions. With $P$-probability no less than $1-\Delta_{n}:$ $\displaystyle\left\|\hat{\eta}^{*}_{0}-\eta^{*}_{0}\right\|_{q}$ $\displaystyle\leq$ $\displaystyle C,$ $\displaystyle\left\|\hat{\eta}^{*}_{0}-\eta^{*}_{0}\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n},$ $\displaystyle\left\|\hat{p}_{d0}(X)-1/2\right\|_{\infty}$ $\displaystyle\leq$ $\displaystyle 1/2-\epsilon,$ $\displaystyle\left\|\hat{p}_{d0}(M,X)-1/2\right\|_{\infty}$ $\displaystyle\leq$ $\displaystyle 1/2-\epsilon,$ $\displaystyle\left\|\hat{\mu}_{0}(D,M,X)-\mu_{0}(D,M,X)\right\|_{2}\times\left\|\hat{p}_{d0}(X)-p_{d0}(X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2},$ $\displaystyle\left\|\hat{\mu}_{0}(D,M,X)-\mu_{0}(D,M,X)\right\|_{2}\times\left\|\hat{p}_{d0}(M,X)-p_{d0}(M,X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2},$ $\displaystyle\left\|\hat{\omega}_{0}(D,M,X)-\omega_{0}(D,M,X)\right\|_{2}\times\left\|\hat{p}_{d0}(X)-p_{d0}(X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2}.$ Theorem 2 Under Assumptions 1-3 and 5, it holds for estimating $E[Y(d,M(1-d))]$ based on Algorithm 2: $\sqrt{n}\Big{(}\hat{\Psi}_{d}^{*}-\Psi_{d0}^{*}\Big{)}\rightarrow N(0,\sigma^{2}_{\psi_{d}^{*}})$, where $\sigma^{2}_{\psi_{d}^{*}}=E[(\psi_{d}^{*}-\Psi_{d0}^{*})^{2}]$. The proof is provided in Appendix B.2. ## 5 Simulation study This section provides a simulation study to investigate the finite sample behaviour of the proposed methods based on the following data generating process: $\displaystyle Y$ $\displaystyle=$ $\displaystyle 0.5D+0.5M+0.5DM+X^{\prime}\beta+U,$ $\displaystyle M$ $\displaystyle=$ $\displaystyle I\\{0.5D+X^{\prime}\beta+V>0\\},\quad D=I\\{X^{\prime}\beta+W>0\\},$ $\displaystyle X$ $\displaystyle\sim$ $\displaystyle N(0,\Sigma),\quad U,V,W\sim N(0,1)\textrm{ independently of each other and $X$}.$ Outcome $Y$ is a function of the observed variables $D,M,X$, including an interaction between the mediator and the treatment, and an unobserved term $U$. The binary mediator $M$ is a function of $D,X$ and the unobservable $V$, while the binary treatment $D$ is determined by $X$ and the unobservable $W$. $X$ is a vector of covariates of dimension $p$, which is drawn from a multivariate normal distribution with zero mean and covariance matrix $\Sigma$. The latter is defined based on setting the covariance of the $i$th and $j$th covariate in $X$ to $\Sigma_{ij}=0.5^{|i-j|}$.333The results presented below are hardly affected when setting $\Sigma$ to the identity matrix (zero correlation across $X$). Coefficients $\beta$ gauge the impact of $X$ on $Y$, $M$, and $D$, respectively, and thus, the strength of confounding. $U,V,W$ are random and standard normally distributed scalar unobservables. We consider two sample sizes of $n=1000,4000$ and run $1000$ simulations per data generating process. We investigate the performance of effect estimation based on (i) Theorem 1 using the identification result in expression (3) derived by Tchetgen Tchetgen and Shpitser (2012) as well as (ii) Theorem 2 using the modified score function in expression (3) which avoids conditional mediator densities. The nuisance parameters are estimated by post-lasso regression based on the ‘hdm’ package by Spindler, Chernozhukov, and Hansen (2016) for the statistical software ‘R’ with its default options, using logit specifications for $p_{d}(X)$, $p_{d}(M,X)$, and $f(M|D,X)$ and linear specifications for $\mu(D,M,X)$ and $E[\mu(d,M,X)|D=1-d,X]$. The estimation of direct and indirect effects is based on 3-fold cross-fitting. For all methods investigated, we drop observations whose (products of) estimated conditional probabilities in the denominator of any potential outcome expression are close to zero, namely smaller than a trimming threshold of $0.05$ (or 5%). Our estimation procedure is available in the ‘causalweight’ package for ‘R’ by Bodory and Huber (2018). In our first simulation design, we set $p=200$ and the $i$th element in the coefficient vector $\beta$ to $0.3/i^{2}$ for $i=1,...,p$, meaning a quadratic decay of covariate importance in terms of confounding. This specification implies that the $R^{2}$ of $X$ when predicting $Y$ amounts to 0.22 in large samples, while the Nagelkerke (1991) pseudo-$R^{2}$ of $X$ when predicting $D$ and $M$ by probit models amounts to 0.10 and 0.13, respectively. The left panel of Table 1 reports the results for either sample size. For $n=1000$, double machine learning based on Theorem 2 on average exhibits a slightly lower absolute bias (‘abias’) and standard deviation (‘sd’) than estimation based on Theorem 1. The behavior of both approaches improves when increasing sample size to $n=4000$, as the absolute bias is very close to zero for any effect estimate and standard deviation is roughly cut by half. Under the larger sample size, differences in terms of root mean squared error (‘rmse’) between estimation based on Theorems 1 and 2 are very close to zero. By and large, the results suggest that the estimators converge to the true effects at root-$n$ rate. In our second simulation, confounding is increased by setting $\beta$ to $0.5/i^{2}$ for $i=1,...,p$. This specification implies that the $R^{2}$ of $X$ when predicting $Y$ amounts to 0.42, while the Nagelkerke (1991) pseudo-$R^{2}$ of $X$ when predicting $D$ and $M$ amounts to 0.23 and 0.28, respectively. The results are displayed in the right panel of Table 1. Again, estimation based on Theorem 2 slightly dominates in terms of having a smaller absolute bias and standard deviation, in particular for $n=1000$. However, in other settings, the two methods might compare differently in terms of finite sample performance. Both methods based on Theorems 1 and 2, respectively, appear to converge to the true effects at root-$n$ rate, and differences in terms of root mean squared errors are minor for $n=4000$. Table 1: Simulation results for effect estimates ($p=200$) | Coefficients given by $0.3/i^{2}$ for $i=1,...,p$ | Coefficients given by $0.5/i^{2}$ for $i=1,...,p$ ---|---|--- | abias | sd | rmse | abias | sd | rmse | true | abias | sd | rmse | abias | sd | rmse | true | $n$=1000 | $n$=4000 | | $n$=1000 | $n$=4000 | | Double machine learning based on Theorem 1 $\hat{\Delta}$ | 0.01 | 0.08 | 0.08 | 0.00 | 0.04 | 0.04 | 1.02 | 0.02 | 0.09 | 0.09 | 0.02 | 0.04 | 0.05 | 1.00 $\hat{\theta}(1)$ | 0.00 | 0.09 | 0.09 | 0.00 | 0.04 | 0.04 | 0.84 | 0.01 | 0.09 | 0.09 | 0.01 | 0.04 | 0.04 | 0.83 $\hat{\theta}(0)$ | 0.01 | 0.08 | 0.08 | 0.00 | 0.04 | 0.04 | 0.75 | 0.02 | 0.08 | 0.09 | 0.01 | 0.04 | 0.04 | 0.75 $\hat{\delta}(1)$ | 0.00 | 0.06 | 0.06 | 0.00 | 0.03 | 0.03 | 0.27 | 0.00 | 0.06 | 0.06 | 0.00 | 0.03 | 0.03 | 0.25 $\hat{\delta}(0)$ | 0.01 | 0.06 | 0.06 | 0.00 | 0.02 | 0.02 | 0.18 | 0.01 | 0.06 | 0.06 | 0.00 | 0.02 | 0.02 | 0.17 trimmed | 17.24 | 19.19 | | 80.25 | 237.50 | | Double machine learning based on Theorem 2 $\hat{\Delta}$ | 0.00 | 0.08 | 0.08 | 0.00 | 0.04 | 0.04 | 1.02 | 0.01 | 0.09 | 0.09 | 0.01 | 0.04 | 0.04 | 1.00 $\hat{\theta}(1)$ | 0.00 | 0.08 | 0.08 | 0.00 | 0.04 | 0.04 | 0.84 | 0.00 | 0.08 | 0.08 | 0.00 | 0.04 | 0.04 | 0.83 $\hat{\theta}(0)$ | 0.00 | 0.08 | 0.08 | 0.00 | 0.04 | 0.04 | 0.75 | 0.00 | 0.08 | 0.08 | 0.00 | 0.04 | 0.04 | 0.75 $\hat{\delta}(1)$ | 0.00 | 0.06 | 0.06 | 0.00 | 0.03 | 0.03 | 0.27 | 0.00 | 0.06 | 0.06 | 0.00 | 0.03 | 0.03 | 0.25 $\hat{\delta}(0)$ | 0.00 | 0.04 | 0.04 | 0.00 | 0.02 | 0.02 | 0.18 | 0.00 | 0.05 | 0.05 | 0.00 | 0.02 | 0.02 | 0.17 trimmed | 1.20 | 0.11 | | 16.76 | 25.45 | Note: ‘abias’, ‘sd’, and ‘rmse’ denote the absolute bias, standard deviation and root mean squared error of the respective effect estimate. ‘true’ provides the true effect. ‘trimmed’ is the average number of trimmed observations per simulation. The propensity score-based trimming threshold is set to $0.05$. Appendix A reports the simulation results (namely the absolute bias, standard deviation, and root mean squared error) for the standard errors obtained by an asymptotic approximation based on the estimated variance of the score functions. The results suggest that the asymptotic standard errors decently estimate the actual standard deviation of the point estimators. ## 6 Application In this section, we apply our method to data from the National Longitudinal Survey of Youth 1997 (NLSY97), a survey following a U.S. nationally representative sample of 8,984 individuals born in the years 1980-84. Since 1997, the participants have been interviewed on a wide range of demographic, socioeconomic, and health-related topics in a one- to two-year circle. We investigate the causal effect of health insurance coverage ($D$) on general health ($Y$) and decompose it into an indirect pathway via the incidence of a regular medical checkup ($M$) and a direct effect entailing any other causal mechanisms. Whether or not an individual undergoes routine checkups appears to be an interesting mediator, as it is likely to be affected by health insurance coverage and may itself have an impact on the individual’s health, because checkups can help identifying medical conditions before they get serious to prevent them from affecting a person’s general health state. The effect of health insurance coverage on self-reported health has been investigated in different countries with no compulsory medical insurance and no publicly provided universal health coverage, see for example Simon, Soni, and Cawley (2017), Sommers, Maylone, Blendon, Orav, and Epstein (2017), Baicker, Taubman, Allen, Bernstein, Gruber, Newhouse, Schneider, Wright, Zaslavsky, and Finkelstein (2013), Yörük (2016) and Cardella and Depew (2014) for the U.S. and King, Gakidou, Imai, Lakin, Moore, Nall, Ravishankar, Vargas, Tellez-Rojo, Avila, et al. (2009) for Mexico). Most of these studies find a significant positive effect of insurance coverage on self-reported health. The impact of insurance coverage on the utilization of preventive care measures, particularly routine checkups like cancer, diabetes and cardiovascular screenings, is also extensively covered in public health literature. Most studies find that health insurance coverage increases the odds of attending routine checkups. While some contributions include selected demographic, socioeconomic and health-related control variables to account for the endogeneity of health insurance status (see e.g. Faulkner and Schauffler (1997), Press (2014), Burstin, Swartz, O’Neil, Orav, and Brennan (1998), Fowler-Brown, Corbie-Smith, Garrett, and Lurie (2007)), others exploit natural experiments: Simon, Soni, and Cawley (2017) estimate a difference-in- differences model comparing states which did and did not expand Medicaid to low-income adults in 2005, while Baicker, Taubman, Allen, Bernstein, Gruber, Newhouse, Schneider, Wright, Zaslavsky, and Finkelstein (2013) exploit that the state of Oregon expanded Medicaid based on lottery drawings from a waiting list. The results of both studies suggest that the Medicaid expansions increased use of certain forms of preventive care. In a study on Mexican adults, Pagán, Puig, and Soldo (2007) use self-employment and commission pay as instruments for insurance coverage and also find a more frequent use of some types of preventive care by individuals with health insurance coverage. While the bulk of studies investigating checkups focus on one particular type of screening (rather than general health checkups), see Maciosek, Coffield, Flottemesch, Edwards, and Solberg (2010) for a literature review, several experimental contributions also assess general health checkups. For instance, Rasmussen, Thomsen, Kilsmark, Hvenegaard, Engberg, Lauritzen, and Sogaard (2007) conduct an experiment with individuals aged 30 to 49 in Denmark by randomly offering a set of health screenings, including advice on healthy living and find a significant positive effect on life expectation. In a study on Japan’s elderly population, Nakanishi, Tatara, and Fujiwara (1996) find a significantly negative correlation between the rate of attendance at health check-ups and hospital admission rates. Despite the effects of health insurance coverage and routine checkups being extensively covered in the public health literature, the indirect effect of insurance on general health operating via routine checkups as mediator has to the best of our knowledge not yet been investigated. A further distinction to most previous studies is that we consider comparably young individuals with an average age below 30. For this population, the relative importance of different health screenings might differ from that for other age groups. We also point out that our application focuses on short-term health effects. We consider a binary indicator for health insurance coverage, equal to one if an individual reports to have any kind of health insurance when interviewed in 2006 and zero otherwise. The outcome, self-reported general health, is obtained from 2008 interview and measured with an ordinal variable, taking on the values ‘excellent’, ‘very good’, ‘good’, ‘fair’ and ‘poor’. In the 2007 interview, participants were asked whether they have gone for routine checkups since the 2006 interview. This information serves as binary mediator, measured post-treatment but pre-outcome. To ensure that the control variables ($X$) are not influenced by the treatment, they come from the pre-treatment 2005 and earlier interview rounds. They cover demographic characteristics, family background and quality of the home environment during youth, education and training, labor market status, income and work experience, marital status and fertility, household characteristics, received monetary transfers, attitudes and expectations, state of physical and mental health as well as health-related behavior regarding e.g. nutrition and physical activity. For some variables, we only consider measurements from 2005 or from the initial interview round covering demographics and family related topics. For other variables we include measurements from both the indiviuals’ youth and 2005 in order to capture their social, emotional and physical development. Treatment and mediator state in the pre-treatment period (2005) are also considered as potential control variables. Item non-response in control variables is dealt with by including missing dummies for each control variable and setting the respective missing values to zero. In total, we end up with a set of 770 control variables, 601 of which are dummy variables (incl. 252 dummies for missing values). After excluding 1923 observations with either mediator or treatment status missing, we remain with 7,061 observations. Table 2 presents some descriptive statistics for a selection of control variables. It shows that the group of individuals with and without health insurance coverage differ substantially. There are significant differences with respect to most of the control variables listed in the table. Females are significantly more likely to have health insurance coverage. Education and household income also show a significant positive correlation with health insurance coverage while the number of household members for example is negatively correlated with insurance coverage. Regarding the mediator, we find a similar pattern as for the treatment. With respect to many of the considered variables, the group of individuals who went for medical checkup differs substantially from those who did not. Further, we see that the correlation between many control variables and the treatment appear to have the same sign as that with the mediator. Table 2: Descriptive Statistics | overall | $D=1$ | $D=0$ | diff | p-val | $M=1$ | $M=0$ | diff | p-val ---|---|---|---|---|---|---|---|---|--- $n$ | 7,061 | 2,335 | 4,726 | | | 3,612 | 3,449 | | Female | 0.5 | 0.55 | 0.42 | 0.13 | 0 | 0.66 | 0.34 | 0.32 | 0 Age | 28.51 | 28.47 | 28.59 | -0.12 | 0 | 28.46 | 28.55 | -0.09 | 0.01 Ethnicity | | | | | | | | | Black | 0.27 | 0.25 | 0.3 | -0.05 | 0 | 0.32 | 0.21 | 0.11 | 0 Hispanic | 0.21 | 0.19 | 0.26 | -0.07 | 0 | 0.21 | 0.21 | 0 | 0.76 Mixed | 0.01 | 0.01 | 0.01 | 0 | 0.34 | 0.01 | 0.01 | 0 | 0.42 White or Other | 0.51 | 0.55 | 0.43 | 0.12 | 0 | 0.46 | 0.56 | -0.1 | 0 Relationship/Marriage | | | | | | | | | Not Cohabiting | 0.62 | 0.61 | 0.65 | -0.03 | 0.01 | 0.61 | 0.64 | -0.03 | 0.01 Cohabiting | 0.17 | 0.16 | 0.18 | -0.03 | 0.01 | 0.16 | 0.17 | 0 | 0.72 Married | 0.18 | 0.21 | 0.14 | 0.07 | 0 | 0.2 | 0.17 | 0.03 | 0 Separated/ Widowed | 0.02 | 0.02 | 0.03 | -0.01 | 0.03 | 0.02 | 0.02 | 0 | 0.62 Missing | 0 | 0 | 0 | 0 | 0.5 | 0 | 0 | 0 | 0.79 Urban | 1.75 | 1.75 | 1.73 | 0.02 | 0.03 | 1.76 | 1.73 | 0.03 | 0.01 Missing | 0.08 | 0.08 | 0.09 | -0.01 | 0.16 | 0.08 | 0.09 | -0.01 | 0.14 HH Income 444The HH income variable is the sum of several variables measuring HH income components (different sources and receivers). These variables are capped but only a total of 11 observations are in critical cap categories | 43,406 | 48,388 | 33,322 | 15,066 | 0 | 44,217 | 42,556 | 1,661 | 0.24 Missing | 0.21 | 0.19 | 0.24 | -0.05 | 0 | 0.2 | 0.22 | -0.01 | 0.2 HH Size | 3.09 | 3.06 | 3.15 | -0.1 | 0.04 | 3.13 | 3.05 | 0.08 | 0.07 Missing | 0.06 | 0.05 | 0.07 | -0.02 | 0.01 | 0.05 | 0.07 | -0.01 | 0.03 HH Members under 18 | 0.69 | 0.65 | 0.76 | -0.11 | 0 | 0.77 | 0.6 | 0.17 | 0 Missing | 0.06 | 0.06 | 0.07 | -0.02 | 0.01 | 0.06 | 0.07 | -0.01 | 0.04 Biological Children | 0.49 | 0.47 | 0.54 | -0.07 | 0 | 0.56 | 0.43 | 0.13 | 0 Highest Grade | 12.17 | 12.65 | 11.21 | 1.44 | 0 | 12.41 | 11.93 | 0.48 | 0 Missing | 0.06 | 0.06 | 0.07 | -0.02 | 0 | 0.06 | 0.07 | -0.01 | 0.04 Employment | | | | | | | | | Employed | 0.71 | 0.73 | 0.68 | 0.05 | 0 | 0.7 | 0.72 | -0.02 | 0.05 Unemployed | 0.05 | 0.04 | 0.08 | -0.03 | 0 | 0.05 | 0.06 | -0.01 | 0.17 Out of Labor Force | 0.21 | 0.19 | 0.24 | -0.04 | 0 | 0.21 | 0.2 | 0.01 | 0.4 Military | 0.02 | 0.03 | 0.01 | 0.03 | 0 | 0.03 | 0.01 | 0.02 | 0 Missing | 0 | 0 | 0 | 0 | 0.13 | 0 | 0 | 0 | 0.45 Working Hours (per week) | 24.83 | 25.47 | 23.53 | 1.94 | 0 | 24.44 | 25.24 | -0.81 | 0.09 Missing | 0.06 | 0.05 | 0.07 | -0.02 | 0.01 | 0.05 | 0.07 | -0.01 | 0.04 Weight (pounds) | 157 | 157 | 157 | -1 | 0.64 | 154 | 160 | -6 | 0 Missing | 0.08 | 0.08 | 0.1 | -0.02 | 0.01 | 0.08 | 0.09 | -0.01 | 0.09 Height (feet) | 5.12 | 5.18 | 5.01 | 0.17 | 0 | 5.08 | 5.17 | -0.09 | 0.02 Missing | 0.09 | 0.08 | 0.11 | -0.04 | 0 | 0.08 | 0.09 | -0.01 | 0.13 Days 5+ drinks (per month) | 1.64 | 1.56 | 1.8 | -0.25 | 0.02 | 1.24 | 2.06 | -0.82 | 0 Missing | 0.09 | 0.08 | 0.1 | -0.03 | 0 | 0.08 | 0.09 | -0.02 | 0.01 Days of Exercise (per week) | 2.39 | 2.41 | 2.36 | 0.05 | 0.42 | 2.32 | 2.46 | -0.14 | 0.01 Missing | 0.05 | 0.05 | 0.06 | -0.01 | 0.03 | 0.04 | 0.06 | -0.01 | 0.03 Depressed/ Down | | | | | | | | | Never | 0.31 | 0.32 | 0.29 | 0.02 | 0.05 | 0.29 | 0.32 | -0.03 | 0.01 Sometimes | 0.51 | 0.52 | 0.48 | 0.03 | 0.01 | 0.52 | 0.49 | 0.02 | 0.06 Mostly | 0.09 | 0.09 | 0.1 | -0.01 | 0.14 | 0.1 | 0.08 | 0.02 | 0 Always | 0.02 | 0.02 | 0.03 | -0.01 | 0 | 0.02 | 0.02 | 0 | 0.68 Missing | 0.08 | 0.07 | 0.1 | -0.03 | 0 | 0.07 | 0.09 | -0.01 | 0.02 Note: ‘overall’, ‘$D=1$’, ‘$D=0$’, ‘$M=1$’, ‘$M=0$’ report the mean of the respective vaiable in the total sample, among treated, among non-treated, among mediated, and among non-mediated, respectively. ‘diff’ and ‘p-val’ provide the mean difference (across treatment or mediator states) and the p-value of a two-sample t-test, respectively. In order to assess the direct and indirect effect of health insurance coverage on general health, we consider estimation based on Theorem 1 and expression (3) derived by Tchetgen Tchetgen and Shpitser (2012) well as (ii) Theorem 2 and expression (3). We estimate the nuisance parameters and treatment effects in the same way as outlined in Section 5 (i.e. post-lasso regression for modeling the nuisance parameters and 3-fold cross fitting for effect estimation) after augmenting the set of covariates with 3101 interaction and higher order terms. The trimming threshold for discarding observations with too extreme propensity scores is set to $0.02$ (2%), such that 893 and 136 observations are dropped when basing estimation on Theorems 1 and 2, respectively. Table 3 provides the estimated effects along with the standard error (‘se’) and p-value (‘p-val’) and also provides the estimated mean potential outcome under non-treatment for comparison (‘$\hat{E}[Y(0,M(0))]$’). The ATEs of health insurance coverage on general health in the year 2008 (columns 2 and 8), estimated based on Theorems 1 or 2, are statistically significant at the 5% and 10% levels, respectively. As the outcome is measured on an ordinal scale ranging from ‘excellent’ to ‘poor’, the negative ATEs suggest a short- term health improving effect of health coverage. The direct effects under treatment (columns 3 and 9) under non-treatment (columns 4 and 10) are very similar to the ATEs and statistically significant (at least) at the 10% level in 3 out ouf 4 cases. In contrast, the indirect effects under treatment (columns 5 and 11) and non-treatment (columns 6 and 12) are generally close to zero and not statistically significant at the 10% level in 3 out of 4 cases. Thus, health insurance coverage does not seem to importantly affect general health of young adults in the U.S. through routine checkups in the short run, but rather through other mechanisms. Table 3: Total, direct, and indirect effects on general health in 2008 | Estimations based on Theorem 1 | Estimations based on Theorem 2 ---|---|--- | $\hat{\Delta}$ | $\hat{\theta}(1)$ | $\hat{\theta}(0)$ | $\hat{\delta}(1)$ | $\hat{\delta}(0)$ | $\hat{E}[Y(0,M(0))]$ | $\hat{\Delta}$ | $\hat{\theta}(1)$ | $\hat{\theta}(0)$ | $\hat{\delta}(1)$ | $\hat{\delta}(0)$ | $\hat{E}[Y(0,M(0))]$ effect | -0.06 | -0.06 | -0.06 | -0.00 | 0.00 | 2.34 | -0.05 | -0.07 | -0.05 | 0.00 | 0.02 | 2.29 se | 0.03 | 0.04 | 0.03 | 0.01 | 0.02 | 0.02 | 0.03 | 0.03 | 0.03 | 0.01 | 0.01 | 0.03 p-val | 0.04 | 0.18 | 0.04 | 0.87 | 0.99 | 0.00 | 0.10 | 0.03 | 0.10 | 0.89 | 0.07 | 0.00 Note: ‘effect’, ‘se’, and ‘p-val’ report the respective effect estimate, standard error and p-value. Lasso regression is used for the estimation of nuisance parameters. The propensity score-based trimming threshold is set to $0.02$. ## 7 Conclusion In this paper, we combined causal mediation analysis with double machine learning under selection-on-observables assumptions which avoids adhoc pre- selection of control variables. Thus, this approach appears particularly fruitful in high-dimensional data with many potential control variables. We proposed estimators for natural direct and indirect effects as well as the controlled direct effect exploiting efficient score functions, sample splitting, and machine learning-based plug-in estimates for conditional outcome means, mediator densities, and/or treatment propensity scores. We demonstrated the root-$n$ consistency and asymptotic normality of the effect estimators under specific regularity conditions. Furthermore, we investigated the finite sample behavior of the proposed estimators in a simulation study and found the performance to be decent in samples with several thousand observations. 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Appendices ## A Simulation results for standard errors Table A.1: Simulation results for standard errors ($p=200$) | Coefficients given by $0.3/i^{2}$ for $i=1,...,p$ | Coefficients given by $0.5/i^{2}$ for $i=1,...,p$ ---|---|--- | abias | sd | rmse | true | abias | sd | rmse | true | abias | sd | rmse | true | abias | sd | rmse | true | $n$=1000 | $n$=4000 | $n$=1000 | $n$=4000 | Double machine learning based on Theorem 1 $se(\hat{\Delta})$ | 0.00 | 0.00 | 0.01 | 0.08 | 0.00 | 0.00 | 0.00 | 0.04 | 0.00 | 0.01 | 0.01 | 0.09 | 0.00 | 0.00 | 0.00 | 0.04 $se(\hat{\theta}(1))$ | 0.02 | 0.01 | 0.02 | 0.09 | 0.01 | 0.00 | 0.01 | 0.04 | 0.02 | 0.01 | 0.02 | 0.09 | 0.01 | 0.00 | 0.01 | 0.04 $se(\hat{\theta}(0))$ | 0.01 | 0.00 | 0.01 | 0.08 | 0.00 | 0.00 | 0.00 | 0.04 | 0.01 | 0.01 | 0.01 | 0.08 | 0.01 | 0.00 | 0.01 | 0.04 $se(\hat{\delta}(1))$ | 0.00 | 0.00 | 0.00 | 0.06 | 0.00 | 0.00 | 0.00 | 0.03 | 0.00 | 0.01 | 0.01 | 0.06 | 0.00 | 0.00 | 0.00 | 0.03 $se(\hat{\delta}(0))$ | 0.01 | 0.01 | 0.01 | 0.06 | 0.00 | 0.00 | 0.00 | 0.02 | 0.00 | 0.01 | 0.01 | 0.06 | 0.00 | 0.00 | 0.00 | 0.02 | Double machine learning based on Theorem 2 $se(\hat{\Delta})$ | 0.00 | 0.00 | 0.01 | 0.08 | 0.00 | 0.00 | 0.00 | 0.04 | 0.00 | 0.01 | 0.01 | 0.09 | 0.00 | 0.00 | 0.00 | 0.04 $se(\hat{\theta}(1))$ | 0.00 | 0.00 | 0.01 | 0.08 | 0.00 | 0.00 | 0.00 | 0.04 | 0.00 | 0.01 | 0.01 | 0.08 | 0.00 | 0.00 | 0.00 | 0.04 $se(\hat{\theta}(0))$ | 0.00 | 0.01 | 0.01 | 0.08 | 0.00 | 0.00 | 0.00 | 0.04 | 0.01 | 0.01 | 0.01 | 0.08 | 0.00 | 0.00 | 0.00 | 0.04 $se(\hat{\delta}(1))$ | 0.00 | 0.01 | 0.01 | 0.06 | 0.00 | 0.00 | 0.00 | 0.03 | 0.00 | 0.01 | 0.01 | 0.06 | 0.00 | 0.00 | 0.00 | 0.03 $se(\hat{\delta}(0))$ | 0.00 | 0.00 | 0.00 | 0.04 | 0.00 | 0.00 | 0.00 | 0.02 | 0.00 | 0.01 | 0.01 | 0.05 | 0.00 | 0.00 | 0.00 | 0.02 Note: ‘abias’, ‘sd’, and ‘rmse’ denote the absolute bias, standard deviation and root mean squared error of the respective standard error (‘se’). ‘true’ provides the true standard deviation. ## B Proofs For the proofs of Theorems 1 and 2, it suffices verifying the conditions of Assumptions 3.1 and 3.2 underlying Theorem 3.1 and 3.2 in Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018). ### B.1 Proof of Theorem 1 We first show that Assumptions 3.1 and 3.2 in Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018) are satisfied for $\Psi_{d0}=E[Y(d,M(1-d))]$ based on (3). Then, we show that Assumption 3.1 holds for $\Psi_{dm0}=E[Y(d,m)]$ based on (8), but omit the proof of the validity of Assumption 3.2, as it follows in a very similar manner as for $\Psi_{d0}$. All bounds hold uniformly over all probability laws $P\in\mathcal{P}$, where $\mathcal{P}$ is the set of all possible probability laws, and we omit $P$ for brevity. Let $\eta=(\mu(D,M,X),f(M|D,X),p_{d}(X))$ be the vector of nuisance parameters. Also, let $\mathcal{T}_{n}$ be the set of all $\eta=(\mu,f,p_{d})$ in a neighbourhood of $\eta_{0}$ that is shrinking with increasing $n,$ consisting of $P$-square integrable functions $\mu$, $f$, and $p_{d}$ such that $\displaystyle\left\|\eta-\eta_{0}\right\|_{q}$ $\displaystyle\leq$ $\displaystyle C,$ (B.1) $\displaystyle\left\|\eta-\eta_{0}\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n},$ $\displaystyle\left\|p_{d}(X)-1/2\right\|_{\infty}$ $\displaystyle\leq$ $\displaystyle 1/2-\epsilon,$ $\displaystyle\left\|f(M|D,X)-(\underline{f}+\overline{f})/2\right\|_{\infty}$ $\displaystyle\leq$ $\displaystyle(\overline{f}-\underline{f})/2,$ $\displaystyle\left\|\mu(D,M,X)-\mu_{0}(D,M,X)\right\|_{2}\times\left\|p_{d}(X)-p_{d0}(X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2},$ $\displaystyle\left\|\mu(D,M,X)-\mu_{0}(D,M,X)\right\|_{2}\times\left\|f(M|1-D,X)-f_{0}(M|1-D,X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2}.$ We furthermore replace the sequence $(\delta_{n})_{n\geq 1}$ by $(\delta_{n}^{\prime})_{n\geq 1},$ where $\delta_{n}^{\prime}=C_{\epsilon}\max(\delta_{n},n^{-1/2}),$ where $C_{\epsilon}$ is sufficiently large constant that only depends on $C$ and $\epsilon.$ Let $R\equiv\overline{f}/\underline{f}$ stands for the maximal ratio of densities $f(m|d,X).$ #### B.1.1 Counterfactual $E[Y(d,M(1-d))]$ The score function for the counterfactual $\Psi_{d0}=E[Y(d,M(1-d))]$ proposed by Tchetgen Tchetgen and Shpitser (2012) is given by the following expression, with $W=(Y,M,D,X)$: $\displaystyle\psi_{d}(W,\eta,\Psi_{d0})$ $\displaystyle=$ $\displaystyle\frac{I\\{D=d\\}\cdot f(M|1-d,X)}{p_{d}(X)\cdot f(M|d,X)}\cdot[Y-\mu(d,M,X)]$ $\displaystyle+\frac{I\\{D=1-d\\}}{1-p_{d}(X)}\cdot\Big{[}\mu(d,M,X)-\overbrace{\int_{m\in\mathcal{M}}\mu(d,m,X)\cdot f(m|1-d,X)dm}^{=:\nu(1-d,X)}\Big{]}$ $\displaystyle+\underbrace{\int_{m\in\mathcal{M}}\mu(d,m,X)\cdot f(m|1-d,X)dm}_{=:\nu(1-d,X)}\ \ -\ \ \Psi_{d0}.$ Assumption 3.1: Moment Condition, Linear scores and Neyman orthogonality Assumption 3.1(a) Moment Condition: The moment condition $E\Big{[}\psi_{d}(W,\eta_{0},\Psi_{d0})\Big{]}=0$ is satisfied: $\displaystyle E\Big{[}\psi_{d}(W,\eta_{0},\Psi_{d0})\Big{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}\overbrace{E\Bigg{[}\frac{I\\{D=d\\}\cdot f_{0}(M|1-d,X)}{p_{d0}(X)\cdot f_{0}(M|d,X)}\cdot[Y-\mu_{0}(d,M,X)]\Bigg{|}X\Bigg{]}}^{=E[E[Y-\mu_{0}(d,M,X)|D=d,M,X]|D=1-d,X]=0}\Bigg{]}$ $\displaystyle+\ E\Bigg{[}\overbrace{E\Bigg{[}\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}\cdot[\mu_{0}(d,M,X)-\nu_{0}(1-d,X)]\Bigg{|}X\Bigg{]}}^{=E[\mu_{0}(d,M,X)-\nu_{0}(1-d,X)|D=1-d,X]=0}\Bigg{]}$ $\displaystyle+\ E[\nu_{0}(1-d,X)]\ \ -\ \ \Psi_{d0}$ $\displaystyle=$ $\displaystyle\Psi_{d0}\ \ -\ \ \Psi_{d0}\ \ =0,$ where the first equality follows from the law of iterated expectations. To better see this result, note that $\displaystyle E\Bigg{[}\frac{I\\{D=d\\}\cdot f_{0}(M|1-d,X)}{p_{d0}(X)\cdot f_{0}(M|d,X)}\cdot[Y-\mu_{0}(d,M,X)]\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}\frac{I\\{D=d\\}\cdot(1-p_{d0}(M,X))}{p_{d0}(M,X)\cdot(1-p_{d0}(X))}\cdot[Y-\mu_{0}(d,M,X)]\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}E\Bigg{[}\frac{I\\{D=d\\}}{p_{d0}(M,X)}\cdot[Y-\mu_{0}(d,M,X)]\Bigg{|}M,X\Bigg{]}\cdot\frac{(1-p_{d0}(M,X))}{(1-p_{d0}(X))}\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}E[Y-\mu_{0}(d,M,X)|D=d,M,X]\cdot\frac{(1-p_{d0}(M,X))}{(1-p_{d0}(X))}\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E[E[Y-\mu_{0}(d,M,X)|D=d,M,X]|D=1-d,X]$ $\displaystyle=$ $\displaystyle E[\mu_{0}(d,M,X)-\mu_{0}(d,M,X)|D=1-d,X]=0,$ where the first equality follows from Bayes’ Law, the second from the law of iterated expectations, the third from basic probability theory, and the fourth from Bayes’ Law. Furthermore, $\displaystyle E\Bigg{[}\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}\cdot[\mu_{0}(d,M,X)-\nu_{0}(1-d,X)]\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}E\Bigg{[}\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}\cdot[\mu_{0}(d,M,X)-\nu_{0}(1-d,X)]\Big{|}M,X\Bigg{]}\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}[\mu_{0}(d,M,X)-\nu_{0}(1-d,X)]\cdot\frac{1-p_{d0}(M,X)}{1-p_{d0}(X)}\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E[\mu_{0}(d,M,X)-\nu_{0}(1-d,X)|D=1-d,X]=E[\mu_{0}(d,M,X)|D=1-d,X]-\nu_{0}(1-d,X)$ $\displaystyle=$ $\displaystyle\nu_{0}(1-d,X)-\nu_{0}(1-d,X)=0,$ where the first equality follows from the law of iterated expectations and the third from Bayes’ Law. Assumption 3.1(b) Linearity: The score $\psi_{d}(W,\eta_{0},\Psi_{d0})$ is linear in $\Psi_{d0}$ as it can be written as: $\psi_{d}(W,\eta_{0},\Psi_{d0})=\psi_{d}^{a}(W,\eta_{0})\cdot\Psi_{d0}+\psi_{d}^{b}(W,\eta_{0})$ with $\psi_{d}^{a}(W,\eta_{0})=-1$ and $\displaystyle\psi_{d}^{b}(W,\eta_{0})$ $\displaystyle=$ $\displaystyle\frac{I\\{D=d\\}\cdot f_{0}(M|1-d,X)}{p_{d0}(X)\cdot f_{0}(M|d,X)}[Y-\mu_{0}(d,M,X)]$ $\displaystyle+$ $\displaystyle\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}\Big{[}\mu_{0}(d,M,X)-\nu_{0}(1-d,X)\Big{]}+\nu_{0}(1-d,X).$ Assumption 3.1(c) Continuity: The expression for the second Gateaux derivative of a map $\eta\mapsto E\Big{[}\psi_{d}(W,\hat{\eta},\Psi_{d0})\Big{]}$, given in (3), is continuous. Assumption 3.1(d) Neyman Orthogonality: For any $\eta\in\mathcal{T}_{n}$, the Gateaux derivative in the direction $\eta-\eta_{0}=(\mu(d,M,X)-\mu_{0}(d,M,X),f(M|D,X)-f_{0}(M|D,X),p_{d}(X)-p_{d0}(X))$ is given by: $\displaystyle\partial E\big{[}\psi_{d}(W,\eta,\Psi_{d})\big{]}\big{[}\eta-\eta_{0}\big{]}$ $\displaystyle=\leavevmode\resizebox{507.33624pt}{}{$E\Bigg{[}\frac{\big{[}f(M|1-d,X)-f_{0}(M|1-d,X)\big{]}\cdot f_{0}(M|d,X)-\big{[}f(M|d,X)-f_{0}(M|d,X)\big{]}\cdot f_{0}(M|1-d,X)}{f_{0}^{2}(M|d,X)}\cdot\overbrace{\frac{I\\{D=d\\}}{p_{d0}(X)}\cdot\Big{(}Y-\mu_{0}(d,M,X)\Big{)}}^{E[\cdot|X]=E[Y-\mu_{0}(d,M,X)|D=d,X]=0}\Bigg{]}$}$ $\displaystyle\underbrace{-\ E\Bigg{[}\underbrace{\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}}_{E[\cdot|X]=1}\cdot\partial E[\nu_{0}(1-d,X)][f(M|1-d,X)-f_{0}(M|1-d,X)]\Bigg{]}+\partial E[\nu_{0}(1-d,X)][f(M|1-d,X)-f_{0}(M|1-d,X)]\Bigg{]}}_{=0}$ $\displaystyle-\ E\Bigg{[}\underbrace{\frac{I\\{D=d\\}\cdot f_{0}(M|1-d,X)}{p_{d0}(X)\cdot f_{0}(M|d,X)}\cdot\Big{(}Y-\mu_{0}(d,M,X)\Big{)}}_{E[\cdot|X]=E[E[Y-\mu_{0}(d,M,X)|D=d,M,X]|D=1-d,X]=0}\cdot\frac{p_{d}(X)-p_{d0}(X)}{p_{d0}(X)}\Bigg{]}$ $\displaystyle+\ E\Bigg{[}\underbrace{\frac{I\\{D=1-d\\}}{(1-p_{d0}(X))}\cdot\Big{(}\mu_{0}(d,M,X)-\nu_{0}(1-d,X)\Big{)}}_{E[\cdot|X]=E[\mu_{0}(d,M,X)-\nu_{0}(1-d,X)|D=1-d,X]=0}\cdot\frac{p_{d}(X)-p_{d0}(X)}{(1-p_{d0}(X))}\Bigg{]}$ $\displaystyle-\ \underbrace{E\Bigg{[}\frac{I\\{D=d\\}\cdot f_{0}(M|1-d,X)}{p_{d0}(X)\cdot f_{0}(M|d,X)}\cdot\Big{[}\mu(d,M,X)-\mu_{0}(d,M,X)\Big{]}\Bigg{]}}_{E[\cdot]=E[E[\cdot|M,X]]=E\Big{[}\frac{p_{d0}(M,X)\cdot f_{0}(M|1-d,X)}{p_{d0}(X)\cdot f_{0}(M|d,X)}\cdot[\mu(d,M,X)-\mu_{0}(d,M,X)]\Big{]}}$ ($*$) $\displaystyle+\ \underbrace{E\Bigg{[}\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}\cdot\Big{[}\mu(d,M,X)-\mu_{0}(d,M,X)\Big{]}\Bigg{]}}_{E[\cdot]=E[E[\cdot|M,X]]=E\Big{[}\frac{1-p_{d0}(M,X)}{1-p_{d0}(X)}\cdot[\mu(d,M,X)-\mu_{0}(d,M,X)]\Big{]}}$ ($**$) $\displaystyle\underbrace{-\ E\Bigg{[}\underbrace{\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}\cdot\partial E[\nu_{0}(1-d,X)][\mu(d,M,X)-\mu_{0}(d,M,X)]}_{E[\cdot|X]=\frac{1-p_{d0}(X)}{1-p_{d0}(X)}\cdot\partial E[\nu_{0}(1-d,X)][\mu(d,M,X)-\mu_{0}(d,M,X)]}\Bigg{]}+\partial E[\nu_{0}(1-d,X)][\mu(d,M,X)-\mu_{0}(d,M,X)]}_{=0},$ where terms $(*)$ and $(**)$ cancel out by Bayes’ Law, $\frac{p_{d0}(M,X)\cdot f_{0}(M|1-d,X)}{p_{d0}(X)\cdot f_{0}(M|d,X)}=\frac{p_{d0}(M,X)\cdot(1-p_{d0}(M,X))}{p_{d0}(M,X)\cdot(1-p_{d0}(X))}=\frac{1-p_{d0}(M,X)}{1-p_{d0}(X)}$. Thus, it follows that: $\displaystyle\partial E\big{[}\psi_{d}(W,\eta,\Psi_{d})\big{]}\big{[}\eta-\eta_{0}\big{]}=0$ proving that the score function is orthogonal. Assumption 3.1(e) Singular values of $E[\psi_{d}^{a}(W;\eta_{0})]$ are bounded: This holds trivially, because $\psi_{d}^{a}(W,\eta_{0})=-1.$ Assumption 3.2: Score regularity and quality of nuisance parameter estimators Assumption 3.2(a) This assumption follows directly from the regularity conditions (Assumption 4) and the definition of $\mathcal{T}_{n}$ given in (B.1). Assumption 3.2(b) Bounds for $m_{n}$: We have $\displaystyle\left\|\mu_{0}(D,M,X)\right\|_{q}$ $\displaystyle=$ $\displaystyle\left(E\left[\left|\mu_{0}(D,M,X)\right|^{q}\right]\right)^{\frac{1}{q}}=\left(\sum_{d\in\\{0,1\\}}E\left[\left|\mu_{0}(d,M,X)\right|^{q}Pr(D=d|M,X)\right]\right)^{\frac{1}{q}}$ $\displaystyle\geq$ $\displaystyle\epsilon^{1/q}\left(\sum_{d\in\\{0,1\\}}E\left[\left|\mu_{0}(d,M,X)\right|^{q}\right]\right)^{\frac{1}{q}}$ $\displaystyle\geq$ $\displaystyle\epsilon^{1/q}\left(\max_{d\in\\{0,1\\}}E\left[\left|\mu_{0}(d,M,X)\right|^{q}\right]\right)^{\frac{1}{q}}$ $\displaystyle=$ $\displaystyle\epsilon^{1/q}\max_{d\in\\{0,1\\}}\left(E\left[\left|\mu_{0}(d,M,X)\right|^{q}\right]\right)^{\frac{1}{q}}=\epsilon^{1/q}\max_{d\in\\{0,1\\}}\left\|\mu_{0}(d,M,X)\right\|_{q}.$ The first equality follows from definition, the second from the law of total probability, the first inequality from $Pr(D=d|M,X)\geq\epsilon.$ Using the same line of arguments we get that $\left\|f_{0}(M|D,X)\right\|_{q}\geq\epsilon^{1/q}\max_{d\in\\{0,1\\}}\left\|f_{0}(M|d,X)\right\|_{q}$ Also, by Jensen’s inequality $\left\|\mu_{0}(D,M,X)\right\|_{q}\leq\left\|Y\right\|_{q}$, such that for any $d\in\\{0,1\\}$: $\displaystyle\left\|\mu_{0}(d,M,X)\right\|_{q}$ $\displaystyle\leq$ $\displaystyle C/\epsilon^{1/q},$ (B.2) $\displaystyle\left\|f_{0}(M|d,X)\right\|_{q}$ $\displaystyle\leq$ $\displaystyle C/\epsilon^{1/q},$ because of $\left\|Y\right\|_{q}\leq C$ by Assumption 4(a). Similarly, for any $\eta\in\mathcal{T}_{n}$ we obtain: $\displaystyle\left\|\mu(d,M,X)-\mu_{0}(d,M,X)\right\|_{q}$ $\displaystyle\leq$ $\displaystyle C/\epsilon^{1/q},$ $\displaystyle\left\|f(M|d,X)-f_{0}(M|d,X)\right\|_{q}$ $\displaystyle\leq$ $\displaystyle C/\epsilon^{1/q},$ due to the definition of $\mathcal{T}_{n}$ given in (B.1). Also, $\displaystyle\left\|\nu_{0}(1-d,X)\right\|_{q}$ $\displaystyle=$ $\displaystyle\left(E\left[\left|\nu_{0}(1-d,X)\right|^{q}\right]\right)^{\frac{1}{q}}=\left(E\left[\left|\int_{m\in\mathcal{M}}\mu_{0}(d,m,X)\cdot f_{0}(m|1-d,X)dm\right|^{q}\right]\right)^{\frac{1}{q}}$ $\displaystyle\leq$ $\displaystyle\left(E\left[\int_{m\in\mathcal{M}}\left|\mu_{0}(d,m,X)\right|^{q}\cdot f_{0}(m|1-d,X)dm\right]\right)^{\frac{1}{q}}$ $\displaystyle=$ $\displaystyle\left(E\left[\int_{m\in\mathcal{M}}\left|\mu_{0}(d,m,X)\right|^{q}\cdot f(m|d,X)\cdot\frac{f_{0}(m|1-d,X)}{f(m|d,X)}dm\right]\right)^{\frac{1}{q}}$ $\displaystyle\leq$ $\displaystyle R^{1/q}\left(E\left[\int_{m\in\mathcal{M}}\left|\mu_{0}(d,m,X)\right|^{q}\cdot f_{0}(m|d,X)dm\right]\right)^{\frac{1}{q}}$ $\displaystyle=$ $\displaystyle R^{1/q}\left\|\mu_{0}(d,M,X)\right\|_{q}\leq\epsilon^{1/q}R^{1/q}\left\|\mu_{0}(D,M,X)\right\|_{q}$ where we make use of the definition of $\nu_{0}$, Jensen’s inequality, and the boundedness of the ratio of densities. We therefore obtain $\left\|\nu_{0}(1-d,X)\right\|_{q}\leq C/(\epsilon^{1/q}R^{1/q})$ by inequality (B.2). This permits bounding the following quantities: $\displaystyle\left\|\mu(d,M,X)\right\|_{q}$ $\displaystyle\leq$ $\displaystyle\left\|\mu(d,M,X)-\mu_{0}(d,M,X)\right\|_{q}+\left\|\mu_{0}(d,M,X)\right\|_{q}\leq 2C/\epsilon^{1/q},$ (B.4) $\displaystyle\left\|\nu(1-d,X)\right\|_{q}$ $\displaystyle\leq$ $\displaystyle\left\|\nu(1-d,X)-\nu_{0}(1-d,X)\right\|_{q}+\left\|\nu_{0}(1-d,X)\right\|_{q}\leq 2C/(\epsilon^{1/q}R^{1/q}),$ $\displaystyle|\Psi_{d0}|$ $\displaystyle=$ $\displaystyle|E[\nu_{0}(1-d,X)]|\leq E\Big{[}\left|\nu_{0}(1-d,X)\right|^{1}\Big{]}^{\frac{1}{1}}=\left\|\nu_{0}(1-d,X)\right\|_{1}$ $\displaystyle\leq$ $\displaystyle\left\|\nu_{0}(1-d,X)\right\|_{2}\leq\left\|Y_{2}\right\|_{2}/(\epsilon^{1/2}R^{1/2})\overbrace{\leq}^{q>2}\left\|Y_{2}\right\|_{q}/(\epsilon^{1/2}R^{1/2})\leq C/(\epsilon^{1/2}R^{1/2}),$ using the triangular inequality, Jensen’s inequality, and properties of statistical $l_{q}$ norms. Finally, rearranging $\psi_{d}(W,\eta,\Psi_{d0})$ $\displaystyle\psi_{d}(W,\eta,\Psi_{d0})$ $\displaystyle=$ $\displaystyle\underbrace{\frac{I\\{D=d\\}\cdot f_{0}(M|1-d,X)}{p_{d}(X)\cdot f_{0}(M|d,X)}\cdot Y}_{=I_{1}}$ $\displaystyle+$ $\displaystyle\underbrace{\bigg{(}\frac{I\\{D=1-d\\}}{1-p_{d}(X)}-\frac{I\\{D=d\\}\cdot f_{0}(M|1-d,X)}{p_{d}(X)\cdot f_{0}(M|d,X)}\bigg{)}\cdot\mu(d,M,X)}_{=I_{2}}$ $\displaystyle+$ $\displaystyle\underbrace{\bigg{(}1-\frac{I\\{D=1-d\\}}{1-p_{d}(X)}\bigg{)}\nu(1-d,X)}_{=I_{3}}-\Psi_{d0},$ provides $\displaystyle\left\|\psi_{d}(W,\eta,\Psi_{d0})\right\|_{q}$ $\displaystyle\leq$ $\displaystyle\left\|I_{1}\right\|_{q}+\left\|I_{2}\right\|_{q}+\left\|I_{3}\right\|_{q}+\left\|\Psi_{d0}\right\|_{q}$ $\displaystyle\leq$ $\displaystyle\frac{R}{\epsilon}\left\|Y\right\|_{q}+\frac{1+R}{\epsilon}\left\|\mu(d,M,X)\right\|_{q}+$ $\displaystyle+$ $\displaystyle\frac{1-\epsilon}{\epsilon}\left\|\nu(1-d,X)\right\|_{q}+|\Psi_{d0}|$ $\displaystyle\leq$ $\displaystyle C\left(\frac{R}{\epsilon}+\frac{2}{\epsilon^{1+1/q}}\left(1+R+\frac{1-\epsilon}{R^{1/q}}\right)+\frac{1}{\epsilon^{1/2}R^{1/2}}\right),$ making use of the triangular inequality and inequalities (B.4). This provides the upper bound on $m_{n}$ in Assumption 3.2(b). Bound for $m^{\prime}_{n}$: We note that $\Big{(}E[|\psi_{d}^{a}(W,\eta)|^{q}]\Big{)}^{1/q}=1,$ which provides the upper bound on $m^{\prime}_{n}$ in Assumption 3.2(b). Assumption 3.2(c) Bound for $r_{n}$: For any $\eta=(\mu,f,p_{d})$ we have $\Big{|}E\Big{(}\psi_{d}^{a}(W,\eta)-\psi_{d}^{a}(W,\eta_{0})\Big{)}\Big{|}=|1-1|=0\leq\delta^{\prime}_{n},$ providing the bound on $r_{n}$ in Assumption 3.2(c). Bound for $r^{\prime}_{n}$: Using the triangular inequality $\displaystyle\left\|\psi_{d}(W,\eta,\Psi_{0d})-\psi_{d}(W,\eta_{0},\Psi_{0d})\right\|_{2}\leq\left\|I\\{D=d\\}\cdot Y\cdot\left(\frac{f(M|1-d,X)}{p_{d}(X)f(M|d,X)}-\frac{f_{0}(M|1-d,X)}{p_{d0}(X)f_{0}(M|d,X)}\right)\right\|_{2}$ $\displaystyle+$ $\displaystyle\left\|I\\{D=d\\}\cdot\left(\frac{\mu(d,M,X)f(M|1-d,X)}{p_{d}(X)f(M|d,X)}-\frac{\mu_{0}(d,M,X)f_{0}(M|1-d,X)}{p_{d0}(X)f_{0}(M|d,X)}\right)\right\|_{2}$ $\displaystyle+$ $\displaystyle\left\|I\\{D=1-d\\}\cdot\left(\frac{\mu(d,M,X)}{1-p_{d}(X)}-\frac{\mu_{0}(d,M,X)}{1-p_{d0}(X)}\right)\right\|_{2}+\left\|I\\{D=1-d\\}\cdot\left(\frac{\nu(1-d,X)}{1-p_{d}(X)}-\frac{\nu_{0}(1-d,X)}{1-p_{d0}(X)}\right)\right\|_{2}$ $\displaystyle+$ $\displaystyle\left\|\nu(1-d,X)-\nu_{0}(1-d,X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}\left(\frac{C\cdot R^{2}}{\epsilon^{2}}+\frac{C\cdot R^{2}}{\epsilon^{2}}\left(\frac{1}{\epsilon^{1/2}}+\frac{C}{\epsilon^{1/2}}\right)+\frac{1}{\epsilon^{2}}\left(\frac{1}{\epsilon^{1/2}}+\frac{C}{\epsilon^{1/2}}\right)+\frac{1}{\epsilon^{2}}\left(\frac{1}{\epsilon^{1/2}R^{1/2}}+\frac{C}{R^{1/2}}\right)+\frac{1}{\epsilon^{1/2}R^{1/2}}\right)\leq\delta^{\prime}_{n},$ as long as $C_{\epsilon}$ in the definition of $\delta_{n}^{\prime}$ is sufficiently large. This gives the bound on $r^{\prime}_{n}$ in Assumption 3.2(c). In order to show the second to the last inequalities, we provide bounds for the terms below, where we made use of the facts that $\left\|\mu(d,M,X)-\mu_{0}(d,M,X)\right\|_{2}\leq\delta_{n}/\epsilon^{1/2},$ and $\left\|\nu(1-d,X)-\nu_{0}(1-d,X)\right\|_{2}\leq\delta_{n}/(\epsilon^{1/2}R^{1/2})$ using similar steps as in Assumption 3.1(b) of Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018). For the first term: $\displaystyle\left\|I\\{D=d\\}\cdot Y\cdot\left(\frac{f(M|1-d,X)}{p_{d}(X)f(M|d,X)}-\frac{f_{0}(M|1-d,X)}{p_{d0}(X)f_{0}(M|d,X)}\right)\right\|_{2}\leq C\cdot\left\|\frac{f(M|1-d,X)}{p_{d}(X)f(M|d,X)}-\frac{f_{0}(M|1-d,X)}{p_{d0}(X)f_{0}(M|d,X)}\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\frac{C}{\epsilon^{2}\underline{f}^{2}}\left\|f(M|1-d,X)f_{0}(M|1-d,X)p_{d0}(X)-f(M|1-d,X)f_{0}(M|1-d,X)p_{d}(X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\frac{C\cdot\overline{f}^{2}}{\epsilon^{2}\underline{f}^{2}}\left\|p_{d0}(X)-p_{d}(X)\right\|_{2}\leq\delta_{n}\frac{C\cdot R^{2}}{\epsilon^{2}},$ where we use $\left\|E[Y^{2}|d,M,X]\right\|_{\infty}\leq C^{2}$ (see our Assumption 4(a)) in the first inequality. For the second term: $\displaystyle\left\|I\\{D=d\\}\cdot\left(\frac{\mu(d,M,X)f(M|1-d,X)}{p_{d}(X)f(M|d,X)}-\frac{\mu_{0}(d,M,X)f_{0}(M|1-d,X)}{p_{d0}(X)f_{0}(M|d,X)}\right)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\left\|\frac{\mu(d,M,X)f(M|1-d,X)}{p_{d}(X)f(M|d,X)}-\frac{\mu_{0}(d,M,X)f_{0}(M|1-d,X)}{p_{d0}(X)f_{0}(M|d,X)}\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\frac{C}{\epsilon^{2}\underline{f}^{2}}\left\|\mu(d,M,X)f(M|1-d,X)f_{0}(M|1-d,X)p_{d0}(X)-\mu_{0}(d,M,X)f(M|1-d,X)f_{0}(M|1-d,X)p_{d}(X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\frac{C\overline{f}^{2}}{\epsilon^{2}\underline{f}^{2}}\left\|\mu(d,M,X)p_{d0}(X)-\mu_{0}(d,M,X)p_{d}(X)\right\|_{2}$ $\displaystyle=$ $\displaystyle\frac{C\cdot R^{2}}{\epsilon^{2}}\left\|\mu(d,M,X)p_{d0}(X)-\mu_{0}(d,M,X)p_{d}(X)+\mu_{0}(d,M,X)p_{d0}(X)-\mu_{0}(d,M,X)p_{d0}(X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\frac{C\cdot R^{2}}{\epsilon^{2}}\big{(}\left\|p_{d0}(X)(\mu(d,M,X)-\mu_{0}(d,M,X))\right\|_{2}+\left\|\mu_{0}(d,M,X)(p_{d0}(X)-p_{d}(X))\right\|_{2}\big{)}$ $\displaystyle\leq$ $\displaystyle\frac{C\cdot R^{2}}{\epsilon^{2}}\big{(}\left\|\mu(d,M,X)-\mu_{0}(d,M,X)\right\|_{2}+C\left\|(p_{d0}(X)-p_{d}(X))\right\|_{2}\big{)}$ $\displaystyle\leq$ $\displaystyle\frac{C\cdot R^{2}}{\epsilon^{2}}\left(\frac{\delta_{n}}{\epsilon^{1/2}}+C\delta_{n}\right)=\delta_{n}\frac{C\cdot R^{2}}{\epsilon^{2}}\left(\frac{1}{\epsilon^{1/2}}+C\right)$ where the fifth inequality follows from $E[Y^{2}|D=d,M,X]\geq(E[Y|D=d,M,X])^{2}=\mu^{2}_{0}(d,M,X)$ by conditional Jensen’s inequality and therefore $\left\|\mu_{0}(d,M,X)\right\|_{\infty}\leq C^{2}.$ For the third term: $\displaystyle\left\|I\\{D=1-d\\}\cdot\left(\frac{\mu(d,M,X)}{1-p_{d}(X)}-\frac{\mu_{0}(d,M,X)}{1-p_{d0}(X)}\right)\right\|_{2}\leq\left\|\frac{\mu(d,M,X)}{1-p_{d}(X)}-\frac{\mu_{0}(d,M,X)}{1-p_{d0}(X)}\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\epsilon^{2}}\left\|\mu(d,M,X)p_{1-d,0}-\mu_{0}(d,M,X)p_{1-d}\right\|_{2}$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon^{2}}\left\|\mu(d,M,X)p_{1-d,0}-\mu_{0}(d,M,X)p_{1-d}+\mu_{0}(d,M,X)p_{1-d,0}-\mu_{0}(d,M,X)p_{1-d,0}\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\epsilon^{2}}\big{(}\left\|p_{1-d,0}(\mu(d,M,X)-\mu_{0}(d,M,X))\right\|_{2}+\left\|\mu_{0}(d,M,X)(p_{1-d,0}-p_{1-d})\right\|_{2}\big{)}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\epsilon^{2}}\big{(}\left\|\mu(d,M,X)-\mu_{0}(d,M,X)\right\|_{2}+C\left\|p_{1-d,0}-p_{1-d}\right\|_{2}\big{)}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\epsilon^{2}}\left(\frac{\delta_{n}}{\epsilon^{1/2}}+C\delta_{n}\right)=\delta_{n}\frac{1}{\epsilon^{2}}\left(\frac{1}{\epsilon^{1/2}}+C\right).$ For the fourth term: $\displaystyle\left\|I\\{D=1-d\\}\cdot\left(\frac{\nu(1-d,X)}{1-p_{d}(X)}-\frac{\nu_{0}(1-d,X)}{1-p_{d0}(X)}\right)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\epsilon^{2}}\big{(}\left\|p_{1-d,0}(\nu(1-d,X)-\nu_{0}(1-d,X))\right\|_{2}+\left\|\nu_{0}(1-d,X)(p_{1-d,0}-p_{1-d})\right\|_{2}\big{)}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\epsilon^{2}}\big{(}\left\|\nu(1-d,X)-\nu_{0}(1-d,X)\right\|_{2}+\frac{C}{R^{1/2}}\left\|p_{1-d,0}-p_{1-d}\right\|_{2}\big{)}$ $\displaystyle\leq$ $\displaystyle\frac{1}{\epsilon^{2}}\left(\frac{\delta_{n}}{\epsilon^{1/2}R^{1/2}}+\frac{C}{R^{1/2}}\delta_{n}\right)=\delta_{n}\frac{1}{\epsilon^{2}}\left(\frac{1}{\epsilon^{1/2}R^{1/2}}+\frac{C}{R^{1/2}}\right),$ where we used Jensen’s inequality similarly to B.1.1 in order to get $E[\nu^{2}_{0}(1-d,X)]\leq R\cdot E[\mu_{0}^{2}(d,M,X)]$ and hence$\left\|\nu_{0}(1-d,X)\right\|_{\infty}\leq C^{2}/R$. Bound on $\lambda^{\prime}_{n}$: Consider $f(r):=E[\psi(W,\eta_{0}+r(\eta-\eta_{0}),\Psi_{d0})]$ We subsequently omit arguments for the sake of brevity and use $p_{d}=p_{d}(X),f_{d}=f_{d}(M|d,X),\mu=\mu(d,M,X),\nu=\nu(1-d,X)$ and similarly $p_{d0},f_{0d},\mu_{0},\nu_{0}.$ For any $r\in(0,1):$ $\displaystyle\frac{\partial^{2}f(r)}{\partial r^{2}}$ $\displaystyle=$ $\displaystyle E\Bigg{[}2\cdot I\\{D=1-d\\}\frac{(\mu-\mu_{0})(p_{d}-p_{d0})}{\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{2}}\Bigg{]}+E\Bigg{[}2\cdot I\\{D=1-d\\}\frac{(\nu-\nu_{0})(p_{d}-p_{d0})}{\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{2}}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=d\\}\frac{(f_{d}-f_{d0})(f_{1-d}-f_{1-d,0})\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{d0}+r(p_{d}-p_{d0})\right)\left(f_{d0}+r(f_{d}-f_{d0})\right)^{2}}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=d\\}\frac{(p_{d}-p_{d0})(f_{1-d}-f_{1-d,0})\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{d0}+r(p_{d}-p_{d0})\right)^{2}\left(f_{d0}+r(f_{d}-f_{d0})\right)}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=d\\}\frac{(f_{d}-f_{d0})\left(f_{1-d,0}+r(f_{1-d}-f_{1-d,0})\right)\left(-(\mu-\mu_{0})\right)}{\left(p_{d0}+r(p_{d}-p_{d0})\right)\left(f_{d0}+r(f_{d}-f_{d0})\right)^{2}}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=d\\}\frac{(p_{d}-p_{d0})\left(f_{1-d,0}+r(f_{1-d}-f_{1-d,0})\right)\left(-(\mu-\mu_{0})\right)}{\left(p_{d0}+r(p_{d}-p_{d0})\right)^{2}\left(f_{d0}+r(f_{d}-f_{d0})\right)}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}(-2)\cdot I\\{D=d\\}\frac{\left(f_{1-d}-f_{1-d,0}\right)(\mu-\mu_{0})}{\left(p_{d0}+r(p_{d}-p_{d0})\right)\left(f_{d0}+r(f_{d}-f_{d0})\right)}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=d\\}\frac{(f_{d}-f_{d0})^{2}\left(f_{1-d,0}+r(f_{1-d}-f_{1-d,0})\right)\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{d0}+r(p_{d}-p_{d0})\right)\left(f_{d0}+r(f_{d}-f_{d0})\right)^{3}}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=d\\}\frac{(f_{d}-f_{d0})(p_{d}-p_{d0})\left(f_{1-d,0}+r(f_{1-d}-f_{1-d,0})\right)\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{d0}+r(p_{d}-p_{d0})\right)^{2}\left(f_{d0}+r(f_{d}-f_{d0})\right)^{2}}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=d\\}\frac{(p_{d}-p_{d0})^{2}\left(f_{1-d,0}+r(f_{1-d}-f_{1-d,0})\right)\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{d0}+r(p_{d}-p_{d0})\right)^{3}\left(f_{d0}+r(f_{d}-f_{d0})\right)}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=1-d\\}\frac{\left(\mu_{0}-\nu_{0}\right)(p_{d}-p_{d0})^{2}}{\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{3}}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=1-d\\}\frac{\left(r(\mu-\mu_{0})-r(\nu-\nu_{0})\right)(p_{d}-p_{d0})^{2}}{\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{3}}\Bigg{]}$ Note that the following inequalities can be shown to hold using similar steps as in Assumption 3.1(b) of Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018): $\displaystyle\left\|\mu-\mu_{0}\right\|_{2}$ $\displaystyle=$ $\displaystyle\left\|\mu(d,M,X)-\mu_{0}(d,M,X)\right\|_{2}\leq\left\|\mu(D,M,X)-\mu_{0}(D,M,X)\right\|_{2}/\epsilon^{1/2}\leq\delta_{n}/\epsilon^{1/2},$ $\displaystyle\left\|\nu-\nu_{0}\right\|_{2}$ $\displaystyle=$ $\displaystyle\left\|\nu(1-d,X)-\nu_{0}(1-d,X)\right\|_{2}\leq\left\|\mu(D,M,X)-\mu_{0}(D,M,X)\right\|_{2}/\epsilon^{1/2}R^{1/2}\leq\delta_{n}/(\epsilon^{1/2}R^{1/2}),$ These inequalities together with our Assumption 4 imply $\displaystyle E[Y-\mu_{0}(d,M,X)|D=d,M,X]$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle|p_{d}-p_{d0}|$ $\displaystyle\leq$ $\displaystyle 2,$ $\displaystyle\left\|\mu_{0}\right\|_{q}\leq\left\|Y\right\|_{q}/\epsilon^{1/q}$ $\displaystyle\leq$ $\displaystyle C/\epsilon^{1/q}$ $\displaystyle\left\|\mu-\mu_{0}\right\|_{2}\times\left\|p_{d}-p_{d0}\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2}/\epsilon^{1/2},$ $\displaystyle\left\|\mu-\mu_{0}\right\|_{2}\times\left\|f_{1-d}-f_{1-d,0}\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2}/(\epsilon R^{1/2}),$ for all $d\in\\{1,0\\}$ and consequently $\left\|\nu-\nu_{0}\right\|_{2}\times\left\|p_{d}-p_{d0}\right\|_{2}\leq\delta_{n}n^{-1/2}/(\epsilon^{1/2}R^{1/2}).$ Putting everything together, we get that for some value $C_{\epsilon}^{\prime\prime}$ that only depends on $C$ and $\epsilon$ $\left|\frac{\partial^{2}f(r)}{\partial r^{2}}\right|\leq C_{\epsilon}^{\prime\prime}\delta_{n}n^{-1/2}\leq\delta_{n}^{\prime}n^{-1/2}.$ This gives the upper bound on $\lambda^{\prime}_{n}$ in Assumption 3.2(c) as long as $C_{\epsilon}$ in the definition of $\delta^{\prime}_{n}$ satisfies $C_{\epsilon}\geq C_{\epsilon}^{\prime\prime}$. In order to verify that this inequality holds we consider all the terms in $\frac{\partial^{2}f(r)}{\partial r^{2}}$ separately. For the first term we obtain $\displaystyle\left|E\Bigg{[}2\cdot I\\{D=1-d\\}\frac{(\mu-\mu_{0})(p_{d}-p_{d0})}{\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{2}}\Bigg{]}\right|\leq\frac{2}{\epsilon^{3}}\left|E\Bigg{[}(\mu-\mu_{0})(p_{d}-p_{d0})\Bigg{]}\right|\leq\frac{2}{\epsilon^{3}}\frac{\delta_{n}}{\epsilon^{1/2}R^{1/2}}n^{-1/2},$ where we made use of the fact that $1\geq p_{d0}+r(p_{d}-p_{d0})=(1-r)p_{d0}+rp_{d}\geq(1-r)\epsilon+r\epsilon=\epsilon,$ $\underline{f}\leq f_{d0}+r(f_{d}-f_{d0})\leq\overline{f}$, and Holder’s inequality. For the third term we obtain $\displaystyle\left|E\Bigg{[}2\cdot I\\{D=d\\}\frac{(f_{d}-f_{d0})(f_{1-d}-f_{1-d,0})\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{d0}+r(p_{d}-p_{d0})\right)\left(f_{d0}+r(f_{d}-f_{d0})\right)^{2}}\Bigg{]}\right|$ $\displaystyle\leq$ $\displaystyle\frac{2}{\epsilon\underline{f}^{2}}(\overline{f}-\underline{f})^{2}\left|E\Bigg{[}I\\{D=d\\}\left(Y-\mu_{0}\right)\Bigg{]}\right|+\frac{2}{\epsilon\underline{f}^{2}}\left|E\Bigg{[}I\\{D=d\\}(f_{d}-f_{d0})(f_{1-d}-f_{1-d,0})r(\mu-\mu_{0})\Bigg{]}\right|$ $\displaystyle\leq$ $\displaystyle\frac{2}{\epsilon\underline{f}^{2}}(\overline{f}-\underline{f})\left|E\Bigg{[}1\cdot(f_{1-d}-f_{1-d,0})(\mu-\mu_{0})\Bigg{]}\right|\leq\frac{2}{\epsilon\underline{f}^{2}}(\overline{f}-\underline{f})\frac{\delta_{n}}{\epsilon^{1/2}}n^{-1/2}.$ And for the second to the last terms, we obtain $\displaystyle\left|E\Bigg{[}2\cdot I\\{D=1-d\\}\frac{\left(\mu_{0}-\nu_{0}\right)(p_{d}-p_{d0})^{2}}{\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{3}}\Bigg{]}\right|$ $\displaystyle=$ $\displaystyle\left|E\Bigg{[}\overbrace{I\\{D=1-d\\}\frac{\left(\mu_{0}-\nu_{0}\right)}{p_{1-d,0}}}^{E[E[\cdot|M,X]|X]=0}\cdot\frac{p_{1-d,0}(p_{d}-p_{d0})^{2}}{\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{3}}\Bigg{]}\right|=0.$ All the remaining terms are bounded similarly. Assumption 3.2(d) Finally, we consider $\displaystyle E\Big{[}(\psi_{d}(W,\eta,\Psi_{d0}))^{2}\Big{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}\Bigg{(}\underbrace{\frac{I\\{D=d\\}\cdot f_{0}(M|1-d,X)}{p_{d}(X)\cdot f_{0}(M|d,X)}\cdot\left(Y-\mu_{0}(d,M,X)\right)}_{=I_{1}}$ $\displaystyle+$ $\displaystyle\underbrace{\bigg{(}\frac{I\\{D=1-d\\}}{1-p_{d}(X)}\bigg{)}\cdot\left(\mu_{0}(d,M,X)-\nu_{0}(1-d,X)\right)}_{=I_{2}}+\underbrace{\nu_{0}(1-d,X)-\Psi_{d0}}_{=I_{3}}\Bigg{)}^{2}\Bigg{]}$ $\displaystyle=$ $\displaystyle E[I_{1}^{2}+I_{2}^{2}+I_{3}^{2}]\geq E[I^{2}_{1}]$ $\displaystyle=$ $\displaystyle E\Bigg{[}\Bigg{(}\frac{I\\{D=d\\}\cdot f_{0}(M|1-d,X)}{p_{d}(X)\cdot f_{0}(M|d,X)}\Bigg{)}^{2}\left(Y-\mu_{0}(d,M,X)\right)^{2}\Bigg{]}$ $\displaystyle\geq$ $\displaystyle\frac{\underline{f}^{2}}{(1-\epsilon)\overline{f}^{2}}E\Bigg{[}\left(Y-\mu_{0}(d,M,X)\right)^{2}\Bigg{]}\geq\frac{c^{2}}{(1-\epsilon)R^{2}}>0,$ where the second equality follows from $\displaystyle E\Big{[}I_{1}\cdot I_{2}\Big{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}\overbrace{\frac{I\\{D=d\\}\cdot f_{0}(M|1-d,X)}{p_{d}(X)\cdot f_{0}(M|d,X)}\frac{I\\{D=1-d\\}}{1-p_{d}(X)}}^{I\\{D=d\\}\cdot I\\{D=1-d\\}=0}\cdot\left(Y-\mu_{0}(d,M,X)\right)\cdot\left(\mu_{0}(d,M,X)-\nu_{0}(1-d,X)\right)\Bigg{]},$ $\displaystyle E\Big{[}I_{2}\cdot I_{3}\Big{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}\overbrace{\frac{I\\{D=1-d\\}}{1-p_{d}(X)}\cdot\left(\mu_{0}(d,M,X)-\nu_{0}(1-d,X)\right)}^{E[\cdot|X]=0}\cdot(\nu_{0}(1-d,X)-\Psi_{d0})\Bigg{]},$ $\displaystyle E\Big{[}I_{1}\cdot I_{3}\Big{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}\overbrace{\frac{I\\{D=d\\}\cdot f_{0}(M|1-d,X)}{p_{d}(X)\cdot f_{0}(M|d,X)}\cdot\left(Y-\mu_{0}(d,M,X)\right)}^{E[\cdot|X]=0}\cdot(\nu_{0}(1-d,X)-\Psi_{d0})\Bigg{]}.$ #### B.1.2 Counterfactual $E[Y(d,m)]$ The score for the estimation of $E[Y(d,m)]$ based on (8) is given by: $\displaystyle E\big{[}\psi_{dm}(W,\eta,\Psi_{dm0})\big{]}=\ $ $\displaystyle E\Bigg{[}\frac{I\\{D=d\\}\cdot I\\{M=m\\}\cdot[Y-\mu(d,m,X)]}{f(m|d,X)\cdot p_{d}(X)}+\mu(d,m,X)-\Psi_{dm0}\Bigg{]}.$ Assumption 3.1: Moment Condition, Linear scores and Neyman orthogonality Assumption 3.1(a) Moment condition: The moment condition $E\Big{[}\psi_{dm}(W,\eta_{0},\Psi_{dm0})\Big{]}=0$ is satisfied: $\displaystyle E\Big{[}\psi_{dm}(W,\eta_{0},\Psi_{dm0})\Big{]}=$ $\displaystyle\ E\Bigg{[}\frac{I\\{D=d\\}\cdot I\\{M=m\\}\cdot[Y-\mu_{0}(d,m,X)]}{f_{0}(m|d,X)\cdot p_{d0}(X)}+\mu_{0}(d,m,X)-\Psi_{dm0}\Bigg{]}$ $\displaystyle=\ $ $\displaystyle E\Bigg{[}\overbrace{E\Bigg{[}\frac{I\\{D=d\\}\cdot I\\{M=m\\}\cdot[Y-\mu_{0}(d,m,X)]}{f_{0}(m|d,X)\cdot p_{d0}(X)}\Bigg{|}X\Bigg{]}}^{=E[Y-\mu_{0}(d,m,X)|d,m,X]=0}\Bigg{]}+E\Big{[}\mu_{0}(d,m,X)\Big{]}-\Psi_{dm0}$ $\displaystyle=\ $ $\displaystyle\Psi_{dm0}-\Psi_{dm0}=0.$ Assumption 3.1(b) Linearity: The score $\psi_{dm}(W,\eta_{0},\Psi_{dm0})$ is linear in $\Psi_{dm0}$ as it can be written as: $\psi_{dm}(W,\eta_{0},\Psi_{dm0})=\psi_{d}^{a}(W,\eta_{0})\cdot\Psi_{dm0}+\psi_{d}^{b}(W,\eta_{0})$ with $\psi_{d}^{a}(W,\eta_{0})=-1$ and $\displaystyle\psi_{d}^{b}(W,\eta_{0})=\frac{I\\{D=d\\}\cdot I\\{M=m\\}\cdot[Y-\mu(d,m,X)]}{f(m|d,X)\cdot p_{d}(X)}+\mu(d,m,X)$ Assumption 3.1(c) Continuity: The expression for the second Gateaux derivative of a map $\eta\mapsto E\Big{[}\psi_{dm}(W,\eta,\Psi_{dm0})\Big{]}$, is continuous. Assumption 3.1(d) Neyman orthogonality: The Gateaux derivative in the direction $\eta-\eta_{0}=(\mu(d,M,X)-\mu_{0}(d,M,X),f(M|D,X)-f_{0}(M|D,X),p_{d}(X)-p_{d0}(X))$ is given by: $\displaystyle\partial E$ $\displaystyle\big{[}\psi_{dm}(W,\eta,\Psi_{dm})\big{]}\big{[}\eta-\eta_{0}\big{]}$ $\displaystyle=$ $\displaystyle\overbrace{-E\Bigg{[}\underbrace{\frac{I\\{D=d\\}\cdot I\\{M=m\\}}{f_{0}(m|d,X)\cdot p_{d0}(X)}}_{E[\cdot|X]=\frac{\Pr(D=d,M=m|X)}{\Pr(D=d,M=m|X)}=1}\cdot\Big{[}\mu(d,m,X)-\mu_{0}(d,m,X)\Big{]}\Bigg{]}+E\Big{[}\mu(d,m,X)-\mu_{0}(d,m,X)\Big{]}}^{=0}$ $\displaystyle-E\Bigg{[}\frac{\overbrace{I\\{D=d\\}\cdot I\\{M=m\\}\cdot[Y-\mu_{0}(d,m,X)]}^{E[\cdot|X]=E[Y-\mu_{0}(d,m,X)|d,m,X]=0}}{f_{0}(m|d,X)\cdot p_{d0}(X)}\cdot\frac{f(m|d,X)-f_{0}(m|d,X)}{f_{0}(m|d,X)}\Bigg{]}$ $\displaystyle-E\Bigg{[}\frac{\overbrace{I\\{D=d\\}\cdot I\\{M=m\\}\cdot[Y-\mu_{0}(d,m,X)]}^{E[\cdot|X]=E[Y-\mu_{0}(d,m,X)|d,m,X]=0}}{f_{0}(m|d,X)\cdot p_{d0}(X)}\cdot\frac{p_{d}(X)-p_{d0}(X)}{p_{d0}(X)}\Bigg{]}.$ Thus, it follows that: $\displaystyle\partial E\big{[}\psi_{dm}(W,\eta,\Psi_{dm})\big{]}\big{[}\eta-\eta_{0}\big{]}=0$ proving that the score function is orthogonal. Assumption 3.1(e) Singular values of $E[\psi^{a}_{d}(W;\eta_{0})]$ are bounded: This holds trivially, because $\psi^{a}_{d}(W;\eta_{0})=-1.$ Assumption 3.2: Score regularity and quality of nuisance parameter estimators This proof is omitted for the sake of brevity. It follows along similar lines as the proof for $Y(d,M(1-d))$ presented in subsection B.1.1. This concludes the proof of Theorem 1. $\hfill\square$ ### B.2 Proof of Theorem 2 The alternative score for the counterfactual based on (3) is given by: $\displaystyle\psi_{d}^{*}(W,\eta^{*},\Psi_{d0})$ $\displaystyle=$ $\displaystyle E\Bigg{[}\frac{I\\{D=d\\}\cdot(1-p_{d}(M,X))}{p_{d}(M,X)\cdot(1-p_{d}(X))}\cdot\Big{[}Y-\mu(d,M,X)\Big{]}$ $\displaystyle+$ $\displaystyle\frac{I\\{D=1-d\\}}{1-p_{d}(X)}\cdot\Bigg{[}\mu(d,M,X)-\overbrace{E\Big{[}\mu(d,M,X)\Big{|}D=1-d,X\Big{]}}^{=:\omega(1-d,X)}\Bigg{]}$ $\displaystyle+$ $\displaystyle\overbrace{E\Big{[}\mu(d,M,X)\Big{|}D=1-d,X\Big{]}}^{=:\omega(1-d,X)}\Bigg{]}-\Psi_{d0}$ with $\eta^{*}=(\mu(D,M,X),\omega(D,X),p_{d}(M,X),p_{d}(X))$. Let $\mathcal{T}^{*}_{n}$ be the set of all $\eta^{*}$ consisting of $P$-square integrable functions $\mu(D,M,X),\omega(D,X),p_{d}(M,X)$, and $p_{d}(X)$ such that $\displaystyle\left\|\eta^{*}-\eta^{*}_{0}\right\|_{q}$ $\displaystyle\leq$ $\displaystyle C,$ (B.5) $\displaystyle\left\|\eta^{*}-\eta^{*}_{0}\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n},$ $\displaystyle\left\|p_{d}(X)-1/2\right\|_{\infty}$ $\displaystyle\leq$ $\displaystyle 1/2-\epsilon,$ $\displaystyle\left\|p_{d}(M,X)-1/2\right\|_{\infty}$ $\displaystyle\leq$ $\displaystyle 1/2-\epsilon,$ $\displaystyle\left\|\mu(D,M,X)-\mu_{0}(D,M,X)\right\|_{2}\times\left\|p_{d}(X)-p_{d0}(X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2},$ $\displaystyle\left\|\mu(D,M,X)-\mu_{0}(D,M,X)\right\|_{2}\times\left\|p_{d}(M,X)-p_{d0}(M,X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2},$ $\displaystyle\left\|\omega(D,M,X)-\omega_{0}(D,M,X)\right\|_{2}\times\left\|p_{d}(X)-p_{d0}(X)\right\|_{2}$ $\displaystyle\leq$ $\displaystyle\delta_{n}n^{-1/2}.$ We replace the sequence $(\delta_{n})_{n\geq 1}$ by $(\delta_{n}^{\prime})_{n\geq 1},$ where $\delta_{n}^{\prime}=C_{\epsilon}\max(\delta_{n},n^{-1/2}),$ where $C_{\epsilon}$ is a sufficiently large value that only depends on $C$ and $\epsilon.$ Assumption 3.1: Moment Condition, Linear scores and Neyman orthogonality Assumption 3.1(a) Moment condition: The moment condition $E\Big{[}\psi_{d}^{*}(W,\eta_{0}^{*},\Psi_{d0})\Big{]}=0$ is satisfied: $\displaystyle E\Big{[}\psi_{d}^{*}(W,\eta_{0}^{*},\Psi_{d0})\Big{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}\overbrace{E\Bigg{[}\frac{I\\{D=d\\}\cdot(1-p_{d0}(M,X))}{{p_{d0}(M,X)\cdot(1-p_{d0}(X))}}\cdot[Y-\mu_{0}(d,M,X)]\Bigg{|}X\Bigg{]}}^{=E[E[Y-\mu_{0}(d,M,X)|D=d,M,X]|D=1-d,X]=0}\Bigg{]}$ $\displaystyle+\ E\Bigg{[}\overbrace{E\Bigg{[}\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}\cdot[\mu_{0}(d,M,X)-\omega_{0}(1-d,X)]\Bigg{|}X\Bigg{]}}^{=E[\mu_{0}(d,M,X)-\omega_{0}(1-d,X)|D=1-d,X]=0}\Bigg{]}$ $\displaystyle+\ E[\omega_{0}(1-d,X)]\ \ -\ \ \Psi_{d0}$ $\displaystyle=$ $\displaystyle\Psi_{d0}\ \ -\ \ \Psi_{d0}\ \ =0.$ To better see this result, note that $\displaystyle E\Bigg{[}\frac{I\\{D=d\\}\cdot(1-p_{d0}(M,X))}{p_{d0}(M,X)\cdot(1-p_{d0}(X))}\cdot[Y-\mu_{0}(d,M,X)]\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}E\Bigg{[}\frac{I\\{D=d\\}}{p_{d0}(M,X)}\cdot[Y-\mu_{0}(d,M,X)]\Bigg{|}M,X\Bigg{]}\cdot\frac{(1-p_{d0}(M,X))}{(1-p_{d0}(X))}\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}E[Y-\mu_{0}(d,M,X)|D=d,M,X]\cdot\frac{(1-p_{d0}(M,X))}{(1-p_{d0}(X))}\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E[E[Y-\mu_{0}(d,M,X)|D=d,M,X]|D=1-d,X]$ $\displaystyle=$ $\displaystyle E[\mu_{0}(d,M,X)-\mu_{0}(d,M,X)|D=1-d,X]=0,$ where the first equality follows from the law of iterated expectations, the second from basic probability theory, and the third from Bayes’ Law. Furthermore, $\displaystyle E\Bigg{[}\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}\cdot[\mu_{0}(d,M,X)-\omega_{0}(1-d,X)]\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}E\Bigg{[}\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}\cdot[\mu_{0}(d,M,X)-\omega_{0}(1-d,X)]\Big{|}M,X\Bigg{]}\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}[\mu_{0}(d,M,X)-\omega_{0}(1-d,X)]\cdot\frac{1-p_{d0}(M,X)}{1-p_{d0}(X)}\Bigg{|}X\Bigg{]}$ $\displaystyle=$ $\displaystyle E[\mu_{0}(d,M,X)-\omega_{0}(1-d,X)|D=1-d,X]=E[\mu_{0}(d,M,X)|D=1-d,X]-\omega_{0}(1-d,X)$ $\displaystyle=$ $\displaystyle\omega_{0}(1-d,X)-\omega_{0}(1-d,X)=0,$ where the first equality follows from the law of iterated expectations and the third from Bayes’ Law. Assumption 3.1(b) Linearity: The score $\psi_{d}^{*}(W,\eta^{*}_{0},\Psi_{d0})$ is linear in $\Psi_{d0}$ as it can be written as: $\psi_{d}^{*}(W,\eta^{*}_{0},\Psi_{d0})=\psi_{d}^{a}(W,\Psi_{d0})\cdot\Psi_{d0}+\psi_{d}^{b}(W,\eta^{*}_{0})$ with $\psi_{d}^{a}(W,\eta^{*}_{0})=-1$ and $\displaystyle\psi_{d}^{b}(W,\eta^{*}_{0})$ $\displaystyle=$ $\displaystyle\frac{I\\{D=d\\}\cdot(1-p_{d0}(M,X))}{p_{d0}(M,X)\cdot(1-p_{d0}(X))}\cdot\Big{[}Y-\mu_{0}(d,M,X)\Big{]}$ $\displaystyle+$ $\displaystyle\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}\cdot\Big{[}\mu_{0}(d,M,X)-\omega_{0}(1-d,X)\Big{]}+\omega_{0}(1-d,X)$ Assumption 3.1(c) Continuity: The expression for the second Gateaux derivative of a map $\eta^{*}\mapsto E\Big{[}\psi^{*}_{d}(W,\eta^{*},\Psi_{d0})\Big{]}$ is continuous. Assumption 3.1(d) Neyman orthogonality: The Gateaux derivative in the direction $\eta^{*}-\eta^{*}_{0}=(\mu_{d}(d,M,X)-\mu_{0}(d,M,X),\omega(1-d,X)-\omega_{0}(1-d,X),p_{d}(M.X)-p_{d0}(M,X),p_{d}(X)-p_{d0}(X))$ is given by: $\displaystyle\partial E$ $\displaystyle\big{[}\psi_{d}^{*}(W,\eta^{*},\Psi_{d})\big{]}\big{[}\eta^{*}-\eta^{*}_{0}\big{]}$ $\displaystyle=$ $\displaystyle E\Bigg{[}\frac{-[p_{d}(M,X)-p_{d0}(M,X)]}{p_{d0}(M,X)^{2}}\cdot\overbrace{\frac{I\\{D=d\\}}{1-p_{d0}(X)}\cdot\big{(}Y-\mu(d,M,X)\big{)}}^{E[\cdot|X]=E[Y-\mu(d,M,X)|D=d,X]\cdot\frac{p_{d0}(X)}{1-p_{d0}(X)}=0}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}\overbrace{\frac{I\\{D=d\\}\cdot(1-p_{d0}(M,X))}{p_{d0}(M,X)\cdot(1-p_{d0}(X))}\cdot\big{(}Y-\mu_{0}(d,M,X)\big{)}}^{E[\cdot|X]=E[E[Y-\mu_{0}(d,M,X)|D=d,M,X]|D=1-d,X]=0}\cdot\frac{p_{d}(X)-p_{d0}(X)}{(1-p_{d0}(X))}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}\underbrace{\frac{I\\{D=1-d\\}}{(1-p_{d0}(X))}\cdot\big{(}\mu_{0}(d,M,X)-\omega_{0}(1-d,X)\big{)}}_{E[\cdot|X]=E[\mu_{0}(d,M,X)-\omega_{0}(1-d,X)|D=1-d,X]=0}\cdot\frac{p_{d}(X)-p_{d0}(X)}{(1-p_{d0}(X))}\Bigg{]}$ $\displaystyle\underbrace{-E\Bigg{[}\underbrace{\frac{I\\{D=d\\}}{p_{d0}(M,X)}}_{E[\cdot|M,X]=1}\cdot\frac{(1-p_{d0}(M,X))}{(1-p_{d0}(X))}\cdot\Big{[}\mu(d,M,X)-\mu_{0}(d,M,X)\Big{]}\Bigg{]}+E\Bigg{[}\underbrace{\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}}_{E[\cdot|M,X]=\frac{1-p_{d0}(M,X)}{1-p_{d0}(X)}}\cdot\Big{[}\mu(d,M,X)-\mu_{0}(d,M,X)\Big{]}\Bigg{]}}_{=0}$ $\displaystyle\underbrace{-E\Bigg{[}\underbrace{\frac{I\\{D=1-d\\}}{1-p_{d0}(X)}}_{E[\cdot|X]=1}\cdot\Big{[}\omega(1-d,X)-\omega_{0}(1-d,X)\Big{]}+\Big{[}\omega(1-d,X)-\omega_{0}(1-d,X)\Big{]}\Bigg{]}}_{=0}.$ Thus, it follows that: $\displaystyle\partial E\big{[}\psi_{d}^{*}(W,\eta^{*},\Psi_{d0})\big{]}\big{[}\eta^{*}-\eta^{*}_{0}\big{]}=0$ proving that the score function is orthogonal. Assumption 3.1(e) Singular values of $E[\psi^{a}_{d}(W;\eta^{*}_{0})]$ are bounded: This holds trivially, because $\psi^{a}_{d}(W;\eta^{*}_{0})=-1.$ Assumption 3.2: Score regularity and quality of nuisance parameter estimators Bounds for $m_{n},m^{\prime}_{n},r_{n},r^{\prime}_{n}$ are omitted for the sake of brevity, because their derivations follow similarly as in the proof for $Y(d,M(1-d))$ in subsection B.1.1. However, the proof differs in establishing the bound on $\lambda^{\prime}_{n}$ in 3.2(c) of Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018), as it is based on the regularity conditions in Assumption 5 that include $p_{d}(M,X)$ and $\omega(1-d,X)$. Bound for $\lambda^{\prime}_{n}$: Consider $f(r):=E[\psi(W,\eta^{*}_{0}+r(\eta^{*}-\eta^{*}_{0}),\Psi_{d0})]$ We subsequently omit arguments for the sake of brevity and use $\mu=\mu(d,M,X),\omega=\omega(1-d,X),p_{d}=p_{d}(X),p_{dm}=p_{d}(M,X)$ and similarly $\mu_{0},\omega_{0},p_{d0},p_{dm0}.$ For any $r\in(0,1):$ $\displaystyle\frac{\partial^{2}f(r)}{\partial r^{2}}$ $\displaystyle=$ $\displaystyle E\Bigg{[}(-2)\cdot I\\{D=d\\}\frac{(p_{dm}-p_{dm0})(p_{d}-p_{d0})\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{dm0}+r(p_{dm}-p_{dm0})\right)\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{2}}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=d\\}\frac{(p_{dm}-p_{dm0})^{2}\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{dm0}+r(p_{dm}-p_{dm0})\right)^{2}\left(1-p_{d0}+r(p_{d0}-p_{d})\right)}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=d\\}\frac{\left(1-p_{dm0}+r(p_{dm0}-p_{dm})\right)(p_{d}-p_{d0})(\mu-\mu_{0})}{\left(p_{dm0}+r(p_{dm}-p_{dm0})\right)\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{2}}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}(-2)\cdot I\\{D=d\\}\frac{\left(1-p_{dm0}+r(p_{dm0}-p_{dm})\right)(p_{dm}-p_{dm0})\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{dm0}+r(p_{dm}-p_{dm0})\right)^{2}\left(1-p_{d0}+r(p_{d0}-p_{d})\right)}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}(-2)\cdot I\\{D=d\\}\frac{(p_{d}-p_{d0})^{2}\left(1-p_{dm0}+r(p_{dm0}-p_{dm})\right)\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{dm0}+r(p_{dm}-p_{dm0})\right)\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{3}}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}(-2)\cdot I\\{D=d\\}\frac{(p_{dm}-p_{dm0})(p_{d}-p_{d0})\left(1-p_{dm0}+r(p_{dm0}-p_{dm})\right)\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{dm0}+r(p_{dm}-p_{dm0})\right)^{2}\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{2}}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}(-2)\cdot I\\{D=d\\}\frac{(p_{dm}-p_{dm0})^{2}\left(1-p_{dm0}+r(p_{dm0}-p_{dm})\right)\left(Y-\mu_{0}-r(\mu-\mu_{0})\right)}{\left(p_{dm0}+r(p_{dm}-p_{dm0})\right)^{3}\left(1-p_{d0}+r(p_{d0}-p_{d})\right)}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=d\\}\frac{(p_{dm}-p_{dm0})(\mu-\mu_{0})}{\left(p_{dm0}+r(p_{dm}-p_{dm0})\right)\left(1-p_{d0}+r(p_{d0}-p_{d})\right)}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=1-d\\}\frac{(p_{d}-p_{d0})^{2}\left(\mu_{0}-\omega_{0}\right)}{\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{3}}\Bigg{]}+E\Bigg{[}2\cdot I\\{D=1-d\\}\frac{(p_{d}-p_{d0})^{2}r\left((\mu-\mu_{0})-(\omega-\omega_{0})\right)}{\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{3}}\Bigg{]}$ $\displaystyle+$ $\displaystyle E\Bigg{[}2\cdot I\\{D=1-d\\}\frac{(p_{d}-p_{d0})\left(\mu-\mu_{0}\right)}{\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{2}}\Bigg{]}+E\Bigg{[}(-2)\cdot I\\{D=1-d\\}\frac{(p_{d}-p_{d0})\left(\omega-\omega_{0}\right)}{\left(1-p_{d0}+r(p_{d0}-p_{d})\right)^{2}}\Bigg{]}$ Bounding these twelve terms proceeds similarly as in subsection B.1.1. In order to bound the eighth term, we make use of the sixth inequality in B.5. Similarly, for bounding the tenth and the twelfth terms we make use of the last inequality in B.5. Thus, we get that for some $C_{\epsilon}^{\prime\prime}$ that only depends on $C$ and $\epsilon$ $\left|\frac{\partial^{2}f(r)}{\partial r^{2}}\right|\leq C_{\epsilon}^{\prime\prime}\delta_{n}n^{-1/2}\leq\delta_{n}^{\prime}n^{-1/2}.$ This provides the upper bound on $\lambda^{\prime}_{n}$ in Assumption 3.2(c) of Chernozhukov, Chetverikov, Demirer, Duflo, Hansen, Newey, and Robins (2018) as long as $C_{\epsilon}\geq C_{\epsilon}^{\prime\prime}$. This concludes the proof of Theorem 2. $\hfill\square$

2024-09-04T02:54:55.466938

2002.12722

arxiv-papers

# Large deviation properties of the empirical measure of a metastable small noise diffusion Paul Dupuis and Guo-Jhen Wu Division of Applied Mathematics, Brown University, Providence, USA. Research supported in part by the National Science Foundation (DMS-1904992) and the AFOSR (FA-9550-18-1-0214).Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden. Research supported in part by the AFOSR (FA-9550-18-1-0214<EMAIL_ADDRESS>(Corresponding author). ###### Abstract The aim of this paper is to develop tractable large deviation approximations for the empirical measure of a small noise diffusion. The starting point is the Freidlin-Wentzell theory, which shows how to approximate via a large deviation principle the invariant distribution of such a diffusion. The rate function of the invariant measure is formulated in terms of quasipotentials, quantities that measure the difficulty of a transition from the neighborhood of one metastable set to another. The theory provides an intuitive and useful approximation for the invariant measure, and along the way many useful related results (e.g., transition rates between metastable states) are also developed. With the specific goal of design of Monte Carlo schemes in mind, we prove large deviation limits for integrals with respect to the empirical measure, where the process is considered over a time interval whose length grows as the noise decreases to zero. In particular, we show how the first and second moments of these integrals can be expressed in terms of quasipotentials. When the dynamics of the process depend on parameters, these approximations can be used for algorithm design, and applications of this sort will appear elsewhere. The use of a small noise limit is well motivated, since in this limit good sampling of the state space becomes most challenging. The proof exploits a regenerative structure, and a number of new techniques are needed to turn large deviation estimates over a regenerative cycle into estimates for the empirical measure and its moments. Keywords: Large deviations, Freidlin-Wentzell theory, small noise diffusion, empirical measure, quasipotential, Monte Carlo method ## 1 Introduction Among the many interesting results proved by Freidlin and Wentzell in the 70’s and 80’s concerning small random perturbations of dynamical systems, one of particular note is the large deviation principle for the invariant measure of such a system. Consider the small noise diffusion $dX_{t}^{\varepsilon}=b(X_{t}^{\varepsilon})dt+\sqrt{\varepsilon}\sigma(X_{t}^{\varepsilon})dW_{t},\quad X_{0}^{\varepsilon}=x,$ where $X_{t}^{\varepsilon}\in\mathbb{R}^{d}$, $b:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$, $\sigma:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\times\mathbb{R}^{k}$ (the $d\times k$ matrices) and $W_{t}\in\mathbb{R}^{k}$ is a standard Brownian motion. Under mild regularity conditions on $b$ and $\sigma$, one has that for any $T\in(0,\infty)$ the processes $\\{X_{\cdot}^{\varepsilon}\\}_{\varepsilon>0}$ satisfy a large deviation principle on $C([0,T]:\mathbb{R}^{d})$ with rate function $I_{T}(\phi)\doteq\int_{0}^{T}\sup_{\alpha\in\mathbb{R}^{d}}\left[\langle\dot{\phi}_{t},\alpha\rangle-\left\langle b(\phi_{t}),\alpha\right\rangle-\frac{1}{2}\left\|\sigma(\phi_{t})\alpha\right\|^{2}\right]dt$ when $\phi$ is absolutely continuous and $\phi(0)=x$, and $I_{T}(\phi)=\infty$ otherwise. If $\sigma(x)\sigma(x)^{\prime}>0$ (in the sense of symmetric square matrices) for all $x\in\mathbb{R}^{d}$, then one can evaluate the supremum and find $I_{T}(\phi)=\int_{0}^{T}\frac{1}{2}\left\langle\dot{\phi}_{t}-b(\phi_{t}),\left[\sigma(\phi_{t})\sigma(\phi_{t})^{\prime}\right]^{-1}(\dot{\phi}_{t}-b(\phi_{t}))\right\rangle dt.$ (1.1) To simplify the discussion we will assume this non-degeneracy condition. It is also assumed by Freidlin and Wentzell in [12], but can be weakened. Define the quasipotential $V(x,y)$ for $x,y\in\mathbb{R}^{d}$ by $V(x,y)\doteq\inf\left\\{I_{T}(\phi):\phi(0)=x,\phi(T)=y,T<\infty\right\\}.$ Suppose that $\\{X^{\varepsilon}\\}$ is ergodic on a compact manifold $M\subset\mathbb{R}^{d}$ with invariant measure $\mu^{\varepsilon}\in\mathcal{P}(M)$. Then under a number of additional assumptions, including assumptions on the structure of the dynamical system $\dot{X}_{t}^{0}=b(X_{t}^{0})$, Freidlin and Wentzell [12, Chapter 6] show how to construct a function $J:M\rightarrow[0,\infty]$ in terms of $V$, such that $J$ is the large deviation rate function for $\\{\mu^{\varepsilon}\\}_{\varepsilon>0}$: $J$ has compact level sets, and $\liminf_{\varepsilon\rightarrow 0}\varepsilon\log\mu^{\varepsilon}(G)\geq-\inf_{y\in G}J(y)\text{ for open }G\subset M,$ $\limsup_{\varepsilon\rightarrow 0}\varepsilon\log\mu^{\varepsilon}(F)\leq-\inf_{y\in F}J(y)\text{ for closed }F\subset M.$ This gives a very useful approximation to $\mu^{\varepsilon}$, and along the way many interesting related results (e.g., transition rates between metastable states) are also developed. The aim of this paper is to develop large deviation type estimates for a quantity that is closely related to $\mu^{\varepsilon}$, which is the empirical measure over an interval $[0,T^{\varepsilon}]$. This is defined by $\rho^{\varepsilon}(A)\doteq\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}1_{A}(X_{s}^{\varepsilon})ds$ (1.2) for $A\in\mathcal{B}(M)$. For reasons that will be made precise later on, we will assume $T^{\varepsilon}\rightarrow\infty$ as $\varepsilon\rightarrow 0$, and typically $T^{\varepsilon}$ will grow exponentially in the form $e^{c/\varepsilon}$ for some $c>0$. There is of course a large deviation theory for the empirical measure when $\varepsilon>0$ is held fixed and the length of the time interval tends to infinity (see e.g., [7, 8]). However, it can be hard to extract information from the corresponding rate function. Our interest in proving large deviations estimates when $\varepsilon\rightarrow 0$ and $T^{\varepsilon}\rightarrow\infty$ is in the hope that one will find it easier to extract information in this double limit, analogous to the simplified approximation to $\mu^{\varepsilon}$ just mentioned. These results will be applied in [10] to analyze and optimize a Monte Carlo method known as infinite swapping [9, 15] when the noise is small. Small noise models are common in applications, and are also the setting in which Monte Carlo methods can have the greatest difficulty. We expect that the general set of results will be useful for other purposes as well. We note that while developed in the context of small noise diffusions, the collection of results due to Freidlin and Wentzell that are discussed in [12] also hold for other classes of processes, such as scaled stochastic networks, when appropriate conditions are assumed and the finite time sample path large deviation results are available (see, e.g., [19]). We expect that such generalizations are possible for the results we prove as well. The outline of the paper is as follows. In Section 2 we explain our motivation and the relevance for studying the particular quantities that are the topic of the paper. In Section 3 we provide definitions and assumptions that are used throughout the paper, and Section 4 states the main asymptotic results as well as a related conjecture. Examples that illustrate the results are given in Section 5. In Section 6 we introduce an important tool for our analysis — the regenerative structure, and with this concept, we decompose the original asymptotic problem into two sub-problems that require very different forms of analysis. These two types of asymptotic problems are then analyzed separately in Sections 7 and Section 8. In Section 9 we combine the partial asymptotic results from Section 7 and Section 8 to prove the main large deviation type results that were stated in Section 4. Section 10 gives the proof of a key theorem from Section 8, which asserts an approximately exponential distribution for return times that arise in the decomposition based on regenerative structure, as well as a tail bound needed for some integrability arguments. The last section of the paper, Section 11, presents the proof of an upper bound for the rate of decay of the variance per unit time in the context of a special case, thereby showing for the case that the lower bounds of Section 4 are in a sense tight. To focus on the main discussion, proofs of some lemmas are collected in an Appendix. ###### Remark 1.1. There are certain time-scaling parameters that play key roles throughout this paper. For the reader’s convenience, we record here where they are first described: $h_{1}$ and $w$ are defined in (4.1) and (4.2); $c$ is introduced and its relation to $h_{1}$ and $w$ are given in Theorem 4.3; $m$ is introduced at the beginning of Subsection 6.2. ## 2 Quantities of Interest The quantities we are interested in are the higher order moments, and in particular second moments, of an integral of a risk-sensitive functional with respect to the empirical measure $\rho^{\varepsilon}$ defined in (1.2). To be more precise, the integral is of the form $\int_{M}e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)\rho^{\varepsilon}\left(dx\right)$ (2.1) for some nice (e.g., bounded and continuous) function $f:M\rightarrow\mathbb{R}$ and a closed set $A\in\mathcal{B}(M)$. Note that this integral can also be expressed as $\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt.$ (2.2) In order to understand the large deviation behavior of moments of such an integral, we must identify the correct scaling to extract meaningful information. Moreover, as will be shown, there is an important difference between centered moments and ordinary (non-centered) moments. By the use of the regenerative structure of $\\{X_{t}^{\varepsilon}\\}_{t\geq 0}$, we can decompose (2.2) [equivalently (2.1)] into the sum of a random number of independent and identically distributed (iid) random variables, plus a residual term which here we will ignore. To simplify the notation, we temporarily drop the $\varepsilon$, and without being precise about how the regenerative structure is introduced, let $Y_{j}$ denote the integral of $e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)$ over a regenerative cycle. (The specific regenerative structure we use will be identified later on.) Thus we consider a sequence $\\{Y_{j}\\}_{j\in\mathbb{N}}$ of iid random variables with finite second moments, and want to compare the scaling properties of, for example, the second moment and the second centered moment of $\frac{1}{n}\sum_{j=1}^{n}Y_{j}$. When used for the small noise system, both $n$ and moments of $Y_{i}$ will scale exponentially in $1/\varepsilon$, and $n$ will be random, but for now we assume $n$ is deterministic. The second moment is $\displaystyle E\left(\frac{1}{n}\sum_{k=1}^{n}Y_{k}\right)^{2}=\frac{1}{n^{2}}\sum_{k=1}^{n}E\left(Y_{k}\right)^{2}+\frac{1}{n^{2}}\sum_{i,j:i\neq j}E\left(Y_{i}Y_{j}\right)=\left(EY_{1}\right)^{2}+\frac{1}{n}\mathrm{Var}\left(Y_{1}\right),$ and the second centered moment is $E\left(\frac{1}{n}\sum_{k=1}^{n}\left(Y_{k}-EY_{1}\right)\right)^{2}=\mathrm{Var}\left(\frac{1}{n}\sum_{k=1}^{n}Y_{k}\right)=\frac{1}{n}\mathrm{Var}\left(Y_{1}\right).$ When analyzing the performance of the Monte Carlo schemes one is concerned of course with both bias and variance, but in situations where we would like to apply the results of this paper one assumes $T^{\varepsilon}$ is large enough that the bias term is unimportant, so that all we are concerned with is the variance. However some care will be needed to determine a suitable measure of quality of the algorithm, since as noted $Y_{i}$ could scale exponentially with in $1/\varepsilon$ with a negative coefficient (exponentially small), while $n$ will be exponentially large. In the analysis of unbiased accelerated Monte Carlo methods for small noise systems over bounded time intervals (e.g., to estimate escape probabilities), it is standard to use the second moment, which is often easier to analyze, in lieu of the variance [3, Chapter VI], [4, Chapter 14]. This situation corresponds to $n=1$. The alternative criterion is more convenient since by Jensen’s inequality one can easily establish a best possible rate of decay of the second moment, and estimators are deemed efficient if they possess the optimal rate of decay [3, 4]. However with $n$ exponentially large this is no longer true. Using the previous calculations, we see that the second moment of $\frac{1}{n}\sum_{j=1}^{n}Y_{j}$ can be completely dominated by $\left(EY_{1}\right)^{2}$, and therefore using this quantity to compare algorithms may be misleading, since our true concern is the variance of $\frac{1}{n}\sum_{j=1}^{n}Y_{j}$. This observation suggests that our study of moments of the empirical measure we should consider only centered moments, and in particular quantities like $T^{\varepsilon}\mathrm{Var}\left(\int_{M}e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)\rho^{\varepsilon}\left(dx\right)\right)=T^{\varepsilon}\mathrm{Var}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right),$ which is the variance per unit time. For Monte Carlo one wants to minimize the variance per unit time, and to make the problem more tractable we instead try to maximize the decay rate of the variance per unit time. Assuming the limit exists, this is defined by $\lim_{\varepsilon\rightarrow 0}-\varepsilon\log\left[T^{\varepsilon}\mathrm{Var}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)\right]$ and so we are especially interested in lower bounds on this decay rate. Thus our goal is to develop methods that allow the approximation of at least first and second moments of (2.2). In fact, the methods we introduce can be developed further to obtain large deviation estimates of higher moments if that were needed or desired. ## 3 Setting of the Problem, Assumptions and Definitions The process model we would like to consider is an $\mathbb{R}^{d}$-valued solution to an Itô stochastic differential equation (SDE), where the drift so strongly returns the process to some compact set that events involving exit of the process from some larger compact set are so rare that they can effectively be ignored when analyzing the empirical measure. However, to simplify the analysis we follow the convention of [12, Chapter 6], and work with a small noise diffusion that takes values in a compact and connected manifold $M\subset\mathbb{R}^{d}$ of dimension $r$ and with smooth boundary. The precise regularity assumptions for $M$ are given on [12, page 135]. With this convention in mind, we consider a family of diffusion processes $\\{X^{\varepsilon}\\}_{\varepsilon\in(0,\infty)},X^{\varepsilon}\in C([0,\infty):M)$, that satisfy the following condition. ###### Condition 3.1. Consider continuous $b:M\rightarrow\mathbb{R}^{d}$ and $\sigma:M\rightarrow\mathbb{R}^{d}\times\mathbb{R}^{d}$ (the $d\times d$ matrices), and assume that $\sigma$ is uniformly nondegenerate, in that there is $c>0$ such that for any $x$ and any $v$ in the tangent space of $M$ at $x$, $\langle v,\sigma(x)\sigma(x)^{\prime}v\rangle\geq c\langle v,v\rangle$. For absolutely continuous $\phi\in C([0,T]:M)$ define $I_{T}(\phi)$ by (1.1), where the inverse $\left[\sigma(x)\sigma(x)^{\prime}\right]^{-1}$ is relative to the tangent space of $M$ at $x$. Let $I_{T}(\phi)=\infty$ for all other $\phi\in C([0,T]:M)$. Then we assume that for each $T<\infty$, $\\{X^{\varepsilon}_{t}\\}_{0\leq t\leq T}$ satisfies the large deviation principle with rate function $I_{T}$, uniformly with respect to the initial condition [4, Definition 1.13]. We note that for such diffusion processes nondegeneracy of the diffusion matrix implies there is a unique invariant measure $\mu^{\varepsilon}\in\mathcal{P}(M)$. A discussion of weak sufficient conditions under which Condition 3.1 holds appears in [12, Section 3, Chapter 5]. ###### Remark 3.2. There are several ways one can approximate a diffusion of the sort described at the beginning of this section by a diffusion on a smooth compact manifold. One such “compactification” of the state space can be obtained by assuming that for some bounded but large enough rectangle trajectories that exit the rectangle do not affect the large deviation behavior of quantities of interest, and then extend the coefficients of the process periodically and smoothly off an even larger rectangle to all of $\mathbb{R}^{d}$ (a technique sometimes used to bound the state space for purposes of numerical approximation). One can then map $\mathbb{R}^{d}$ to a manifold that is topologically equivalent to a torus, and even arrange that the metric structure on the part of the manifold corresponding to the smaller rectangle coincides with a Euclidean metric. Define the quasipotential $V(x,y):M\times M\rightarrow[0,\infty)$ by $V(x,y)\doteq\inf\left\\{I_{T}(\phi):\phi(0)=x,\phi(T)=y,T<\infty\right\\}.$ (3.1) For a given set $A\subset M,$ define $V(x,A)\doteq\inf_{y\in A}V(x,y)$ and $V(A,y)\doteq\inf_{x\in A}V(x,y).$ ###### Remark 3.3. For any fixed $y$ and set $A,$ $V(x,y)$ and $V(x,A)$ are both continuous functions of $x$. Similarly, for any given $x$ and any set $A,$ $V(x,y)$ and $V(A,y)$ are also continuous in $y.$ ###### Definition 3.4. We say that a set $N\subset M$ is stable if for any $x\in N,y\notin N$ we have $V(x,y)>0.$ A set which is not stable is called unstable. ###### Definition 3.5. We say that $O\in M$ is an equilibrium point of the ordinary differential equation (ODE) $\dot{x}_{t}=b(x_{t})$ if $b(O)=0.$ Moreover, we say that this equilibrium point $O$ is asymptotically stable if for every neighborhood $\mathcal{E}_{1}$ of $O$ (relative to $M$) there exists a smaller neighborhood $\mathcal{E}_{2}$ such that the trajectories of system $\dot{x}_{t}=b(x_{t})$ starting in $\mathcal{E}_{2}$ converge to $O$ without leaving $\mathcal{E}_{1}$ as $t\rightarrow\infty.$ ###### Remark 3.6. An asymptotically stable equilibrium point is a stable set, but a stable set might contain no asymptotically stable equilibrium point. The following restrictions on the structure of the dynamical system in $M$ will be used. These restrictions include the assumption that the equilibrium points are a finite collection. This is a more restrictive framework than that of [12], which allows, e.g., limit cycles. In a remark at the end of this section we comment on what would be needed to extend to the general setup of [12]. ###### Condition 3.7. There exists a finite number of points $\\{O_{j}\\}_{j\in L}\subset M$ with $L\doteq\\{1,2,\ldots,l\\}$ for some $l\in\mathbb{N}$, such that $\cup_{j\in L}\\{O_{j}\\}$ coincides with the $\omega$-limit set of the ODE $\dot{x}_{t}=b(x_{t})$. Without loss of generality, we may assume that $O_{j}$ is stable if and only if $j\in L_{\rm{s}}$ where $L_{\rm{s}}\doteq\\{1,\ldots,l_{\rm{s}}\\}$ for some $l_{\rm{s}}\leq l.$ Next we give a definition from graph theory which will be used in the statement of the main results. ###### Definition 3.8. Given a subset $W\subset L=\\{1,\ldots,l\\},$ a directed graph consisting of arrows $i\rightarrow j$ $(i\in L\setminus W,j\in L,i\neq j)$ is called a $W$-graph on $L$ if it satisfies the following conditions. 1. 1. Every point $i$ $\in L\setminus W$ is the initial point of exactly one arrow. 2. 2. For any point $i$ $\in L\setminus W,$ there exists a sequence of arrows leading from $i$ to some point in $W.$ We note that we could replace the second condition by the requirement that there are no closed cycles in the graph. We denote by $G(W)$ the set of $W$-graphs; we shall use the letter $g$ to denote graphs. Moreover, if $p_{ij}$ ($i,j\in L,j\neq i$) are numbers, then $\prod_{(i\rightarrow j)\in g}p_{ij}$ will be denoted by $\pi(g).$ ###### Remark 3.9. We mostly consider the set of $\\{i\\}$-graphs, i.e., $G(\\{i\\})$ for some $i\in$ $L$, and also use $G(i)$ to denote $G(\\{i\\}).$ We occasionally consider the set of $\\{i,j\\}$-graphs, i.e., $G(\\{i,j\\})$ for some $i,j\in$ $L$ with $i\neq j.$ Again, we also use $G(i,j)$ to denote $G(\\{i,j\\}).$ ###### Definition 3.10. For all $j\in L$, define $W\left(O_{j}\right)\doteq\min_{g\in G\left(j\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right]$ (3.2) and $W\left(O_{1}\cup O_{j}\right)\doteq\min_{g\in G\left(1,j\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right].$ (3.3) ###### Remark 3.11. Heuristically, if we interpret $V\left(O_{m},O_{n}\right)$ as the “cost” of moving from $O_{m}$ to $O_{n},$ then $W\left(O_{j}\right)$ is the “least total cost” of reaching $O_{j}$ from every $O_{i}$ with $i\in L\setminus\\{j\\}.$ According to [12, Theorem 4.1, Chapter 6], one can interpret $W(O_{i})-\min_{j\in L}W(O_{j})$ as the decay rate of $\mu^{\varepsilon}(B_{\delta}(O_{i}))$, where $B_{\delta}(O_{i})$ is a small open neighborhood of $O_{i}$. ###### Definition 3.12. We use $G_{\rm{s}}\left(W\right)$ to denote the collection of all $W$-graphs on $L_{\rm{s}}=\\{1,\ldots,l_{\rm{s}}\\}$ with $W\subset L_{\rm{s}}.$ W make the following technical assumptions on the structure of the SDE. Let $B_{\delta}(K)$ denote the $\delta$-neighborhood of a set $K\subset M.$ Recall that $\mu^{\varepsilon}$ is the unique invariant probability measure of the diffusion process $\\{X^{\varepsilon}_{t}\\}_{t}.$ The existence of the limits appearing in the first part of the condition is ensured by Theorem 4.1 in [12, Chapter 6]. ###### Condition 3.13. 1. 1. There exists a unique asymptotically stable equilibrium point $O_{1}$ of the system $\dot{x}_{t}=b(x_{t})$ such that $\lim_{\delta\rightarrow 0}\lim_{\varepsilon\rightarrow 0}-\varepsilon\log\mu^{\varepsilon}(B_{\delta}(O_{1}))=0,\text{ and }\lim_{\delta\rightarrow 0}\lim_{\varepsilon\rightarrow 0}-\varepsilon\log\mu^{\varepsilon}(B_{\delta}(O_{j}))>0\mbox{ for any }j\in L\setminus\\{1\\}.$ 2. 2. All of the eigenvalues of the matrix of partial derivatives of $b$ at $O_{\ell}$ relative to $M$ have negative real parts for $\ell\in L_{\rm{s}}$. 3. 3. $b:M\rightarrow\mathbb{R}^{d}$ and $\sigma:M\rightarrow\mathbb{R}^{d}\times\mathbb{R}^{d}$ are $C^{1}$. ###### Remark 3.14. According to [12, Theorem 4.1, Chapter 6] and the first part of Condition 3.13, we know that $W(O_{j})>W(O_{1})$ for all $j\in L\setminus\\{1\\}.$ ###### Remark 3.15. We comment on the use of the various parts of the condition. Part 1 means that neighborhoods of $O_{1}$ capture more of the mass as $\varepsilon\rightarrow 0$ than neighborhoods of any other equilibrium point. It simplifies the analysis greatly, but we expect it could be weakened if desired. Parts 2 and 3 are assumed in [6], which gives an explicit exponential bound on the tail probability of the exit time from the domain of attraction. It is largely because of our reliance on the results of [6] that we must assume that equilibrium sets are points in Condition 3.7, rather than the more general compacta as considered in [12]. Both Condition 3.7 and Condition 3.13 could be weakened if the corresponding versions of the results we use from [6] were available. ###### Remark 3.16. The quantities $V(O_{i},O_{j})$ determine various key transition probabilities and time scales in the analysis of the empirical measure. The more general framework of [12], as well as the one dimensional case (i.e., $r=1$) in the present setting, require some closely related but slightly more complicated quantities. These are essentially the analogues of $V(O_{i},O_{j})$ under the assumption that trajectories used in the definition are not allowed to pass through equilibrium compacta (such as a limit cycle) when traveling from $O_{i}$ to $O_{j}$. The related quantities, which are designated using notation of the form $\tilde{V}(O_{i},O_{j})$ in [12], are needed since the probability of a direct transition from $O_{i}$ to $O_{j}$ without passing though another equilibrium structure may be zero, which means that transitions from $O_{i}$ to $O_{j}$ must be decomposed according to these intermediate transitions. To simplify the presentation we do not provide the details of the one dimensional case in our setup, but simply note that it can be handled by the introduction of these additional quantities. Consider the filtration $\\{\mathcal{F}_{t}\\}_{t\geq 0}$ defined by $\mathcal{F}_{t}\doteq\sigma(X_{s}^{\varepsilon},s\leq t)$ for any $t\geq 0.$ For any $\delta>0$ smaller than a quarter of the minimum of the distances between $O_{i}$ and $O_{j}$ for all $i\neq j$, we consider two types of stopping times with respect to the filtration $\\{\mathcal{F}_{t}\\}_{t}$. The first type are the hitting times of $\\{X^{\varepsilon}_{t}\\}_{t}$ at the $\delta$-neighborhood of all equilibrium points $\\{O_{j}\\}_{j\in L}$ after traveling a reasonable distance away from those neighborhoods. More precisely, we define stopping times by $\tau_{0}\doteq 0,$ $\sigma_{n}\doteq\inf\\{t>\tau_{n}:X_{t}^{\varepsilon}\in{\cup_{j\in L}}\partial B_{2\delta}(O_{j})\\}\text{ and }\tau_{n}\doteq\inf\\{t>\sigma_{n-1}:X_{t}^{\varepsilon}\in{\cup_{j\in L}}\partial B_{\delta}(O_{j})\\}.$ The second type of stopping times are the return times of $\\{X^{\varepsilon}_{t}\\}_{t}$ to the $\delta$-neighborhood of $O_{1}$, where as noted previously $O_{1}$ is in some sense the most important equilibrium point. The exact definitions are $\tau_{0}^{\varepsilon}\doteq 0,$ $\sigma_{n}^{\varepsilon}\doteq\inf\\{t>\tau_{n}^{\varepsilon}:X_{t}^{\varepsilon}\in{\textstyle\cup_{j\in L\setminus\\{1\\}}}\partial B_{\delta}(O_{j})\\}\text{ and }\tau_{n}^{\varepsilon}\doteq\inf\left\\{t>\sigma_{n-1}^{\varepsilon}:X_{t}^{\varepsilon}\in\partial B_{\delta}(O_{1})\right\\}.$ (3.4) We then define two embedded Markov chains $\\{Z_{n}\\}_{n\in\mathbb{N}_{0}}\doteq\\{X_{\tau_{n}}^{\varepsilon}{}\\}_{n\in\mathbb{N}_{0}}$ with state space ${\textstyle\cup_{j\in L}}\partial B_{\delta}(O_{j})$, and $\\{Z_{n}^{\varepsilon}\\}_{n\in\mathbb{N}_{0}}\doteq\\{X_{\tau_{n}^{\varepsilon}}^{\varepsilon}{}\\}_{n\in\mathbb{N}_{0}}$ with state space $\partial B_{\delta}(O_{1}).$ Let $p(x,\partial B_{\delta}(O_{j}))$ denote the one-step transition probabilities of $\\{Z_{n}\\}_{n\in\mathbb{N}_{0}}$ starting from a point $x\in{\textstyle\cup_{i\in L}}\partial B_{\delta}(O_{i}),$ namely, $p(x,\partial B_{\delta}(O_{j}))\doteq P_{x}(Z_{1}\in\partial B_{\delta}(O_{j})).$ We have the following estimates on $p(x,\partial B_{\delta}(O_{j}))$ in terms of $V$. The lemma is a consequence of [12, Lemma 2.1, Chapter 6] and the fact that under our conditions $V(O_{i},O_{j})$ and $\tilde{V}(O_{i},O_{j})$ as defined in [12] coincide. ###### Lemma 3.17. For any $\eta>0,$ there exists $\delta_{0}\in(0,1)$ and $\varepsilon_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ and $\varepsilon\in(0,\varepsilon_{0}),$ for all $x\in\partial B_{\delta}(O_{i}),$ the one-step transition probability of the Markov chain $\\{Z_{n}\\}_{n\in\mathbb{N}}$ on $\partial B_{\delta}(O_{j})$ satisfies $e^{-\frac{1}{\varepsilon}\left(V\left(O_{i},O_{j}\right)+\eta\right)}\leq p(x,\partial B_{\delta}(O_{j}))\leq e^{-\frac{1}{\varepsilon}\left(V\left(O_{i},O_{j}\right)-\eta\right)}$ for any $i,j\in L.$ ###### Remark 3.18. According to Lemma 4.6 in [16], Condition 3.1 guarantees the existence and uniqueness of invariant measures for $\\{Z_{n}\\}_{n}$ and $\\{Z_{n}^{\varepsilon}\\}_{n}.$ We use $\nu^{\varepsilon}\in\mathcal{P}(\cup_{i\in L}\partial B_{\delta}(O_{i}))$ and $\lambda^{\varepsilon}\in\mathcal{P}(\partial B_{\delta}(O_{1}))$ to denote the associated invariant measures. ## 4 Results and a Conjecture The following main results of this paper assume Conditions 3.1, 3.7 and 3.13. Although moments higher than the second moment are not considered in this paper, as noted previously one can use arguments such as those used here to identify and prove the analogous results. Recall that $\\{O_{j}\\}_{j\in L}$ are the equilibrium points and that they satisfy Condition 3.7 and Condition 3.13. In addition, $O_{j}$ is stable if and only if $j\in L_{\rm{s}}$, where $L_{\rm{s}}\doteq\\{1,\ldots,l_{\rm{s}}\\}$ for some $l_{\rm{s}}\leq l=\left|L\right|$, and $\tau^{\varepsilon}_{1}$ is the first return time to the $\delta$-neighborhood of $O_{1}$ after having first visited the $\delta$-neighborhood of any other equilibrium point. ###### Lemma 4.1. For any $\delta\in(0,1)$ smaller than a quarter of the minimum of the distances between $O_{i}$ and $O_{j}$ for all $i\neq j$, any $\varepsilon>0$ and any nonnegative measurable function $g:M\rightarrow\mathbb{R}$ $E_{\lambda^{\varepsilon}}\left(\int_{0}^{\tau_{1}^{\varepsilon}}g\left(X_{s}^{\varepsilon}\right)ds\right)=E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\cdot\int_{M}g\left(x\right)\mu^{\varepsilon}\left(dx\right),$ where $\lambda^{\varepsilon}\in\mathcal{P}(\partial B_{\delta}(O_{1}))$ is the unique invariant measure of $\\{Z_{n}^{\varepsilon}\\}_{n}=\\{X_{\tau_{n}^{\varepsilon}}^{\varepsilon}\\}_{n}$ and $\mu^{\varepsilon}\in\mathcal{P}(M)$ is the unique invariant measure of $\\{X_{t}^{\varepsilon}\\}_{t}.$ ###### Proof. We define a measure on $M$ by $\hat{\mu}^{\varepsilon}\left(B\right)\doteq E_{\lambda^{\varepsilon}}\left(\int_{0}^{\tau_{1}^{\varepsilon}}1_{B}\left(X^{\varepsilon}_{t}\right)dt\right)$ for $B\in\mathcal{B}(M),$ so that for any nonnegative measurable function $g:M\rightarrow\mathbb{R}$ $\int_{M}g\left(x\right)\hat{\mu}^{\varepsilon}\left(dx\right)=E_{\lambda^{\varepsilon}}\left(\int_{0}^{\tau_{1}^{\varepsilon}}g\left(X^{\varepsilon}_{t}\right)dt\right).$ According to the proof of Theorem 4.1 in [16], the measure given by $\hat{\mu}^{\varepsilon}\left(B\right)/\hat{\mu}^{\varepsilon}\left(M\right)$ is an invariant measure of $\\{X^{\varepsilon}_{t}\\}_{t}.$ Since we already know that $\mu^{\varepsilon}$ is the unique invariant measure of $\\{X^{\varepsilon}_{t}\\}_{t},$ this means that $\mu^{\varepsilon}(B)=\hat{\mu}^{\varepsilon}\left(B\right)/\hat{\mu}^{\varepsilon}\left(M\right)$ for any $B\in\mathcal{B}(M).$ Therefore for any nonnegative measurable function $g:M\rightarrow\mathbb{R}$ $\displaystyle E_{\lambda^{\varepsilon}}\left(\int_{0}^{\tau_{1}^{\varepsilon}}g\left(X^{\varepsilon}_{t}\right)dt\right)$ $\displaystyle=\int_{M}g\left(x\right)\mu^{\varepsilon}\left(dx\right)\cdot\hat{\mu}^{\varepsilon}\left(M\right)=\int_{M}g\left(x\right)\mu^{\varepsilon}\left(dx\right)\cdot E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}.$ ∎ Recall the definitions of $W(O_{j})$ and $W(O_{1}\cup O_{j})$ in Definition 3.10, as well as the definition of the quasipotential $V(x,y)$ in (3.1). For any $k\in L$, we define $h_{k}\doteq\min_{j\in L\setminus\\{k\\}}V(O_{k},O_{j}).$ (4.1) In addition, define $w\doteq W(O_{1})-\min_{j\in L\setminus\\{1\\}}W(O_{1}\cup O_{j}).$ (4.2) ###### Remark 4.2. The quantity $h_{k}$ is related to the time that it takes for the process to leave a neighborhood of $O_{k}$, and $W(O_{1})-W(O_{1}\cup O_{j})$ is related to the transition time from a neighborhood of $O_{j}$ to one of $O_{1}$. It turns out that our results and arguments depend on which of $h_{1}$ or $w$ is larger. Throughout the paper, the constructions used in the case when $h_{1}>w$ will be in terms of what we call a single cycle, and those for the case when $h_{1}\leq w$ in terms of a multicycle. ###### Theorem 4.3. Let $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>h_{1}\vee w$. Given $\eta>0,$ a continuous function $f:M\rightarrow\mathbb{R}$ and any compact set $A\subset M,$ there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left|E_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)-\int_{M}e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)\mu^{\varepsilon}\left(dx\right)\right|$ $\displaystyle\qquad\geq\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)+c-(h_{1}\vee w)-\eta,$ where $W(x)\doteq\min_{j\in L}[W(O_{j})+V(O_{j},x)]$. ###### Remark 4.4. Since $W(x)=\min_{j\in L}[W(O_{j})+V(O_{j},x)],$ the lower bound appearing in Theorem 4.3 is equivalent to $\min_{j\in L}\left(\inf_{x\in A}\left[f\left(x\right)+V(O_{j},x)\right]+W(O_{j})-W\left(O_{1}\right)\right)+c-(h_{1}\vee w)-\eta.$ The next result gives an upper bound on the variance per unit time, or equivalently a lower bound on its rate of decay. In the design of a Markov chain Monte Carlo method, one would maximize this rate of decay to improve the method’s performance. ###### Theorem 4.5. Let $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>h_{1}\vee w$. Given $\eta>0,$ a continuous function $f:M\rightarrow\mathbb{R}$ and any compact set $A\subset M,$ there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(T^{\varepsilon}\cdot\mathrm{Var}_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)\right)$ $\displaystyle\qquad\geq\begin{cases}\min_{j\in L}\left(R_{j}^{(1)}\wedge R_{j}^{(2)}\right)-\eta,&\text{if }h_{1}>w\\\ \min_{j\in L}\left(R_{j}^{(1)}\wedge R_{j}^{(2)}\wedge R_{j}^{(3)}\right)-\eta,&\text{otherwise}\end{cases},$ where $R_{j}^{(1)}\doteq\inf_{x\in A}\left[2f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)-W\left(O_{1}\right),$ $R_{1}^{(2)}\doteq 2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{1},x\right)\right]-h_{1},$ and for $j\in L\setminus\\{1\\}$ $\displaystyle R_{j}^{(2)}$ $\displaystyle\doteq 2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)-2W\left(O_{1}\right)+W(O_{1}\cup O_{j}),$ $R_{j}^{(3)}\doteq 2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+2W\left(O_{j}\right)-2W\left(O_{1}\right)-w.$ ###### Remark 4.6. If one mistakenly treated a single cycle case as a multicycle case in the application of Theorem 4.5, then the result is the same since with $h_{1}>w$, (4.2) implies that $R_{j}^{(3)}\geq R_{j}^{(2)}$ for any $j\in L$. ###### Remark 4.7. Although Theorems 4.3 and 4.5 as stated assume the starting distribution $\lambda^{\varepsilon}$, they can be extended to general initial distributions by using results from Section 10, which show that the process essentially forgets the initial distribution before leaving the neighborhood of $O_{1}$. ###### Remark 4.8. In this remark we interpret the use of Theorems 4.3 and 4.5 in the context of Monte Carlo, and also explain the role of the time scaling $T^{\varepsilon}$. There is a minimum amount of time that must elapse before the process can visit all stable equilibrium points often enough that good estimation of risk- sensitive integrals is possible. As is well known, this time scales exponentially in the form of $T^{\varepsilon}=e^{c/\varepsilon}$, and the issue is the selection of the constant $c>0$, which motivates the assumptions on $T^{\varepsilon}$ for the two cases. However, when designing a scheme there typically will be parameters available for selection. The growth constant in $T^{\varepsilon}$ will then depend on these parameters, which will then be chosen to (either directly or indirectly, depending on the criteria used) reduce the size of $T^{\varepsilon}$. For a compelling example we refer to [10], which shows how for a system with fixed well depths a scheme known as infinite swapping can be designed so that given any $a>0$ one can design a scheme so that an interval of length $e^{a/\varepsilon}$ suffices. Theorem 4.3 is concerned with bias, and for $T^{\varepsilon}$ as above will give a negligible contribution to the total error in comparison to the variance. Thus it is Theorem 4.5 that determines the performance of the scheme and serves as a criteria for optimization. Of particular note is that the value of $c$ does not appear in the variational problem appearing in Theorem 4.5. Theorem 4.5 gives a lower bound on the rate of decay of variance per unit time. For applications to the design of Monte Carlo schemes as in [10] there is an a priori bound on the best possible performance, and so this lower bound (which yields an upper bound on variances) is sufficient to determine if a scheme is nearly optimal. However, for other purposes an upper bound on the decay rate could be useful, and we expect the other direction holds as well. The proofs of Theorems 4.3 and 4.5 for single cycles and multicycles are almost identical with a few key differences. We focus on providing proofs in the single cycle case, and then point out the required modifications in the proofs for the multicycle case. ###### Theorem 4.9. The bound in Theorem 4.3 can be calculated using only stable equilibrium points. Specifically, 1. 1. $W(x)=\min_{j\in L_{\rm{s}}}[W(O_{j})+V(O_{j},x)]$ 2. 2. $W\left(O_{j}\right)=\min_{g\in G_{\rm{s}}\left(j\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right]$ 3. 3. $W(O_{1}\cup O_{j})=\min_{g\in G_{\rm{s}}\left(1,j\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right]$ 4. 4. $\min_{j\in L}(\inf_{x\in A}[f(x)+V(O_{j},x)]+W(O_{j}))=\min_{j\in L_{\rm{s}}}(\inf_{x\in A}[f(x)+V(O_{j},x)]+W(O_{j}))$. ###### Remark 4.10. Theorem 4.9 says that the bound appearing in Theorem 4.3 depends on the set of indices of only stable equilibrium points. This is not surprising, since in [12, Chapter 6], it has been shown that the logarithmic asymptotics of the invariant measure of a Markov process in this framework can be characterized in terms of graphs on the set of indices of just stable equilibrium points. It is natural to ask if the same property holds for the lower bound appearing in Theorem 4.5. Notice that part 4 of Theorem 4.9 implies $\min_{j\in L}R_{j}^{(1)}=\min_{j\in L_{\rm{s}}}R_{j}^{(1)}$, so if one can prove (possibly under extra conditions, for example, by considering a double-well model as in Section 11) that $\min_{j\in L}R_{j}^{(2)}=\min_{j\in L_{\rm{s}}}R_{j}^{(2)}$, then these two equations assert the property we want for the single cycle case, namely, $\min_{j\in L}(R_{j}^{(1)}\wedge R_{j}^{(2)})=\min_{j\in L_{\rm{s}}}(R_{j}^{(1)}\wedge R_{j}^{(2)}).$ An analogous comment applies for the multicycle case. ###### Conjecture 4.11. Let $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>h_{1}\vee w$. Let $f$ be continuous and suppose that $A$ is the closure of its interior. Then for any $\eta>0,$ there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(T^{\varepsilon}\cdot\mathrm{Var}_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)\right)$ $\displaystyle\qquad\leq\begin{cases}\min_{j\in L}\left(R_{j}^{(1)}\wedge R_{j}^{(2)}\right)+\eta,&\text{if }h_{1}>w\\\ \min_{j\in L}\left(R_{j}^{(1)}\wedge R_{j}^{(2)}\wedge R_{j}^{(3)}\right)+\eta,&\text{otherwise}\end{cases}.$ In Section 11 we outline the proof of Conjecture 4.11 for a special case. ## 5 Examples ###### Example 5.1. We first consider the situation depicted in Figure 1. Values of $W(O_{j})$ are given in the figure. If one interprets the figure as a potential with minimum zero then the corresponding heights of the equilibrium points are given by $W(O_{j})-W(O_{1})$. We take $f=0$ and $A$ to be a small closed interval about $O_{5}$. As we will see and should be clear from the figure, this example can be analyzed using single regenerative cycles. Recall that $R_{j}^{(1)}\doteq\inf_{x\in A}[2f(x)+V(O_{j},x)]+W(O_{j})-W(O_{1})$ $R_{1}^{(2)}\doteq 2\inf_{x\in A}[f(x)+V(O_{1},x)]-h_{1}$ and for $j>1$ $R_{j}^{(2)}\doteq 2\inf_{x\in A}[f(x)+V(O_{j},x)]+\left(W(O_{j})-W(O_{1})\right)-W(O_{1})+W(O_{1}\cup O_{j})$ If one traces through the proof of Theorem 4.5 for the case of a single cycle, then one finds that the constraining bound is given in Lemma 7.23, which is in turn based on Lemma 7.9. As we will see, in the minimization problem $\min_{j\in L}(R_{j}^{(1)}\wedge R_{j}^{(2)})$ the min on $j$ turns out to be achieved at $j=5$. This is of course not surprising, since $A$ is an interval about $O_{5}$. It is then the minimum of $R_{5}^{(1)}$ and $R_{5}^{(2)}$ which determines the dominant source of the variance of the estimator. Figure 1: Single cycle example We recall that $\tau_{1}^{\varepsilon}$ is the time for a full regenerative cycle, and that $\tau_{1}$ is the time to first reach the $2\delta$ neighborhood of an equilibrium point and then reach a $\delta$ neighborhood of a (perhaps the same) equilibrium point. The quantities that are relevant in Lemma 7.9 are $\sup_{y\in\partial B_{\delta}(O_{5})}E_{y}\left(\int_{0}^{\tau_{1}}1_{A}(X_{t}^{\varepsilon})dt\right)^{2}\text{ and }E_{x}N_{5}$ for $R_{j}^{(1)}$ and $\left[\sup_{y\in\partial B_{\delta}(O_{5})}E_{y}\int_{0}^{\tau_{1}}1_{A}(X_{t}^{\varepsilon})dt\right]^{2}\text{ , }E_{x}N_{5}\text{, and essentially }\sup_{y\in\partial B_{\delta}(O_{5})}E_{y}N_{5}$ for $R_{j}^{(2)}$. Decay rates are in turn determined by (see the proof of Lemma 7.23) $0\text{ and }W(O_{1})-W(O_{5})+h_{1}$ and $0,\text{ }W(O_{1})-W(O_{5})+h_{1}\text{ and }W(O_{1})-W(O_{1}\cup O_{5}),$ respectively. Thus for this example it is only the term $W(O_{1})-W(O_{1}\cup O_{5})$ that distinguishes between the two. Since this is always greater than zero and it appears in $R_{j}^{(2)}$ in the form $-(W(O_{1})-W(O_{1}\cup O_{5}))$ it must be the case that $R_{5}^{(2)}<$ $R_{5}^{(1)}$. The numerical values for the example are $(W(O_{1}\cup O_{j}),j=2,\ldots,5)=(5,3,5,2)$ $(V(O_{j},O_{5}),j=1,\ldots,5)=(8,4,4,0,0)$ $(W(O_{j})-W(O_{1}),j=1,\ldots,5)=(0,4,2,6,3)$ $(R_{j}^{(1)},j=1,\ldots,5)=(8,8,6,6,3)$ $(R_{j}^{(2)},j=2,\ldots,5)=(12,8,6,0)$ and $R_{1}^{(2)}=16-4=12,h_{1}=4$ and $w=5-2=3$. Since $w<h_{1}$ it falls into the single cycle case. We therefore find $\min_{j}R_{j}^{(1)}\wedge R_{j}^{(2)}$ equals to $0$ and occurs with superscript $2$ and at $j=5$. For an example where the dominant contribution to the variance is through the quantities associated with $R_{j}^{(1)}$, we move the set $A$ further to the right of $O_{5}$. All other quantities are unchanged save $\sup_{y\in\partial B_{\delta}(O_{5})}E_{y}\left(\int_{0}^{\tau_{1}}1_{A}(X_{t}^{\varepsilon})dt\right)^{2}\text{ and }\left[\sup_{y\in\partial B_{\delta}(O_{5})}E_{y}\int_{0}^{\tau_{1}}1_{A}(X_{t}^{\varepsilon})dt\right]^{2},$ whose decay rates are governed (for $j=5$) by $\inf_{x\in A}[V(O_{5},x)]\text{ and }2\inf_{x\in A}[V(O_{5},x)],$ respectively. Choosing $A$ so that $\inf_{x\in A}[V(O_{5},x)]>3$, it is now the case that $R_{5}^{(1)}<$ $R_{5}^{(2)}$. ###### Example 5.2. In this example we again take $f=0$ and $A$ to be a small closed interval about $O_{3}$. Since the well at $O_{5}$ is deeper than that at $O_{1}$ we expect that multicycles will be needed, and so recall $R_{j}^{(3)}\doteq 2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+2W\left(O_{j}\right)-2W\left(O_{1}\right)-w.$ Figure 2: Multicycle example The needed values are $(W(O_{1}\cup O_{j}),j=2,\ldots,5)=(7,5,7,2)$ $(V(O_{j},O_{3}),j=1,\ldots,5)=(4,0,0,0,5)$ $(W(O_{j})-W(O_{1}),j=1,\ldots,5)=(0,4,2,6,1)$ $(R_{j}^{(1)},j=1,\ldots,5)=(4,4,2,6,6)$ $(R_{j}^{(2)},j=2,\ldots,5)=(4,0,6,6)$ $(R_{j}^{(3)},j=1,\ldots,5)=(3,3,-1,7,7)$ and $R_{1}^{(2)}=8-4=4,h_{1}=4$ and $w=7-2=5$. Since $w>h_{1}$ a single cycle cannot be used for the analysis of the variance, and we need to use multicycles. We find $\min_{j}R_{j}^{(1)}\wedge R_{j}^{(2)}\wedge R_{j}^{(3)}$ is equal to $-1$ and occurs with superscript $3$ and $j=3$. ## 6 Wald’s Identities and Regenerative Structure To prove Theorems 4.3 and 4.5, we will use the regenerative structure to analyze the system over the interval $[0,T^{\varepsilon}]$. Since the number of regenerative cycles will be random, Wald’s identities will be useful. Recall that $\tau_{n}^{\varepsilon}$ is the $n$-th return time to $\partial B_{\delta}\left(O_{1}\right)$ after having visited the neighborhood of a different equilibrium point, and $\lambda^{\varepsilon}\in\mathcal{P}(\partial B_{\delta}(O_{1}))$ is the invariant measure of the Markov process $\\{X_{\tau_{n}^{\varepsilon}}^{\varepsilon}\\}_{n\in\mathbb{N}_{0}}$ with state space $\partial B_{\delta}(O_{1}).$ If we let the process $\\{X^{\varepsilon}_{t}\\}_{t}$ start with $\lambda^{\varepsilon}$ at time $0,$ that is, assume the distribution of $X^{\varepsilon}_{0}$ is $\lambda^{\varepsilon},$ then by the strong Markov property of $\\{X^{\varepsilon}_{t}\\}_{t},$ we find that $\\{X^{\varepsilon}_{t}\\}_{t}$ is a regenerative process and the cycles $\\{\\{X^{\varepsilon}_{\tau_{n-1}^{\varepsilon}+t}:0\leq t<\tau_{n}^{\varepsilon}-\tau_{n-1}^{\varepsilon}\\},\tau_{n}^{\varepsilon}-\tau_{n-1}^{\varepsilon}\\}$ are iid objects. Moreover, $\\{\tau_{n}^{\varepsilon}\\}_{n\in\mathbb{N}_{0}}$ is a sequence of renewal times under $\lambda^{\varepsilon}.$ ### 6.1 Single cycle Define the filtration $\\{\mathcal{H}_{n}\\}_{n\in\mathbb{N}},$ where $\mathcal{H}_{n}\doteq\mathcal{F}_{\tau_{n}^{\varepsilon}}$ and $\mathcal{F}_{t}\doteq$ $\sigma(\\{X^{\varepsilon}_{s}$; $s\leq t\\})$. With respect to this filtration, in the single cycle case (i.e., when $h_{1}>w$), we consider the stopping times $N^{\varepsilon}\left(T\right)\doteq\inf\left\\{n\in\mathbb{N}:\tau_{n}^{\varepsilon}>T\right\\}.$ Note that $N^{\varepsilon}\left(T\right)-1$ is the number of complete single renewal intervals contained in $[0,T]$. With this notation, we can bound $\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt$ from above and below by $\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)-1}S_{n}^{\varepsilon}\leq\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\leq\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon},$ (6.1) where $S_{n}^{\varepsilon}\doteq\int_{\tau_{n-1}^{\varepsilon}}^{\tau_{n}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt.$ Applying Wald’s first identity shows $\displaystyle E_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon}\right)=\frac{1}{T^{\varepsilon}}E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}.$ (6.2) Therefore, the logarithmic asymptotics of $E_{\lambda^{\varepsilon}}(\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt/T^{\varepsilon})$ are determined by those of $E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)/T^{\varepsilon}$ and $E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}.$ Likewise, to understand the logarithmic asymptotics of $T^{\varepsilon}\cdot$Var${}_{\lambda^{\varepsilon}}(\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt/T^{\varepsilon}),$ it is sufficient to identify the corresponding logarithmic asymptotics of Var${}_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)/T^{\varepsilon}$, Var${}_{\lambda^{\varepsilon}}(S_{1}^{\varepsilon}),$ $E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)/T^{\varepsilon}$ and $E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}$. This can be done with the help of Wald’s second identity, since $\displaystyle T^{\varepsilon}$ $\displaystyle\cdot\mathrm{Var}_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}{\textstyle\sum_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}}S_{n}^{\varepsilon}\right)$ (6.3) $\displaystyle\leq 2T^{\varepsilon}\cdot E_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}{\textstyle\sum_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}}S_{n}^{\varepsilon}-\frac{1}{T^{\varepsilon}}N^{\varepsilon}\left(T^{\varepsilon}\right)E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}\right)^{2}$ $\displaystyle\quad+2T^{\varepsilon}\cdot E_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}N^{\varepsilon}\left(T^{\varepsilon}\right)E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}-\frac{1}{T^{\varepsilon}}E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}\right)^{2}$ $\displaystyle=2\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}\mathrm{Var}_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}+2\frac{\mathrm{Var}_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}\left(E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}\right)^{2}.$ In the next two sections we derive bounds on $E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}$, Var${}_{\lambda^{\varepsilon}}(S_{1}^{\varepsilon})$ and $E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)$, Var${}_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)$, respectively. ### 6.2 Multicycle Recall that in the case of a multicycle we have $w\geq h_{1}$. For any $m>0$ such that $h_{1}+m>w$ and for any $\varepsilon>0$, on the same probability space as $\\{\tau^{\varepsilon}_{n}\\}$, one can define a sequence of independent and geometrically distributed random variables $\\{\mathbf{M}^{\varepsilon}_{i}\\}_{i\in\mathbb{N}}$ with parameter $e^{-m/\varepsilon}$ that are independent of $\\{\tau^{\varepsilon}_{n}\\}$. We then define multicycles according to $\mathbf{K}^{\varepsilon}_{i}\doteq\sum_{j=1}^{i}\mathbf{M}^{\varepsilon}_{j},\quad\hat{\tau}^{\varepsilon}_{i}\doteq\sum_{n=\mathbf{K}^{\varepsilon}_{i-1}+1}^{\mathbf{K}^{\varepsilon}_{i}}\tau^{\varepsilon}_{n},\quad i\in\mathbb{N}.$ (6.4) Consider the stopping times $\hat{N}^{\varepsilon}\left(T\right)\doteq\inf\left\\{n\in\mathbb{N}:\hat{\tau}_{n}^{\varepsilon}>T\right\\}.$ Note that $\hat{N}^{\varepsilon}\left(T\right)-1$ is the number of complete multicycles contained in $[0,T]$. With this notation and by following the same idea as in the single cycle case, we can bound $\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt$ from above and below by $\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)-1}\hat{S}_{n}^{\varepsilon}\leq\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\leq\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)}\hat{S}_{n}^{\varepsilon},$ (6.5) where $\hat{S}_{n}^{\varepsilon}\doteq\int_{\hat{\tau}_{n-1}^{\varepsilon}}^{\hat{\tau}_{n}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt.$ Therefore, by applying Wald’s first and second identities, we know that the logarithmic asymptotics of $E_{\lambda^{\varepsilon}}(\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt/T^{\varepsilon})$ are determined by those of $E_{\lambda^{\varepsilon}}(\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right))/T^{\varepsilon}$ and $E_{\lambda^{\varepsilon}}\hat{S}_{1}^{\varepsilon}$, and the asymptotics of $T^{\varepsilon}\cdot$Var${}_{\lambda^{\varepsilon}}(\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt/T^{\varepsilon})$ by those of Var${}_{\lambda^{\varepsilon}}(\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right))/T^{\varepsilon}$, Var${}_{\lambda^{\varepsilon}}(\hat{S}_{1}^{\varepsilon})$, $E_{\lambda^{\varepsilon}}(\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right))/T^{\varepsilon}$ and $E_{\lambda^{\varepsilon}}\hat{S}_{1}^{\varepsilon}$. In particular, we have $\displaystyle E_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)}\hat{S}_{n}^{\varepsilon}\right)=\frac{1}{T^{\varepsilon}}E_{\lambda^{\varepsilon}}\left(\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)\right)E_{\lambda^{\varepsilon}}\hat{S}_{1}^{\varepsilon}.$ (6.6) and $\displaystyle T^{\varepsilon}\cdot\mathrm{Var}_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}{\sum\nolimits_{n=1}^{\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)}}\hat{S}_{n}^{\varepsilon}\right)\leq 2\frac{E_{\lambda^{\varepsilon}}(\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right))}{T^{\varepsilon}}\mathrm{Var}_{\lambda^{\varepsilon}}\hat{S}_{1}^{\varepsilon}+2\frac{\mathrm{Var}_{\lambda^{\varepsilon}}(\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right))}{T^{\varepsilon}}\left(E_{\lambda^{\varepsilon}}\hat{S}_{1}^{\varepsilon}\right)^{2}.$ (6.7) In the next two sections we derive bounds on $E_{\lambda^{\varepsilon}}\hat{S}_{1}^{\varepsilon}$, Var${}_{\lambda^{\varepsilon}}(\hat{S}_{1}^{\varepsilon})$ and $E_{\lambda^{\varepsilon}}(\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right))$, Var${}_{\lambda^{\varepsilon}}(\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right))$, respectively. ###### Remark 6.1. It should be kept in mind that $\hat{\tau}^{\varepsilon}_{n},\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)$ and $\hat{S}_{n}^{\varepsilon}$ all depend on $m$, although this dependence is not explicit in the notation. ###### Remark 6.2. In general, for any quantity in the single cycle case, we use analogous notation with a “hat” on it to represent the corresponding quantity in the multicycle version. For instance, we use $\tau^{\varepsilon}_{n}$ for a single regenerative cycle, and $\hat{\tau}^{\varepsilon}_{n}$ for a multi regenerative cycle. ## 7 Asymptotics of Moments of $S_{1}^{\varepsilon}$ and $\hat{S}_{1}^{\varepsilon}$ In this section we will first introduce the elementary theory of an irreducible finite state Markov chain $\\{Z_{n}\\}_{n\in\mathbb{N}_{0}}$ with state space $L$, and then state and prove bounds for the asymptotics of moments of $S_{1}^{\varepsilon}$ and $\hat{S}_{1}^{\varepsilon}$. For the asymptotic analysis, the following useful facts will be used repeatedly. ###### Lemma 7.1. For any nonnegative sequences $\left\\{a_{\varepsilon}\right\\}_{\varepsilon>0}$ and $\left\\{b_{\varepsilon}\right\\}_{\varepsilon>0}$, we have $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(a_{\varepsilon}b_{\varepsilon}\right)\geq\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log a_{\varepsilon}+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log b_{\varepsilon},$ (7.1) $\limsup_{\varepsilon\rightarrow 0}-\varepsilon\log\left(a_{\varepsilon}+b_{\varepsilon}\right)\leq\min\left\\{\limsup_{\varepsilon\rightarrow 0}-\varepsilon\log a_{\varepsilon},\limsup_{\varepsilon\rightarrow 0}-\varepsilon\log b_{\varepsilon}\right\\},$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(a_{\varepsilon}+b_{\varepsilon}\right)=\min\left\\{\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log a_{\varepsilon},\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log b_{\varepsilon}\right\\}.$ (7.2) ### 7.1 Markov chains and graph theory In this subsection we state some elementary theory for finite state Markov chains taken from [1, Chapter 2]. For a finite state Markov chain, the invariant measure, the mean exit time, etc., can be expressed explicitly as the ratio of certain determinants, i.e., sums of products consisting of transition probabilities, and these sums only contain terms with a plus sign. Which products should appear in the various sums can be described conveniently by means of graphs on the set of states of the chain. This method of linking graphs and quantities associated with a finite state Markov chain was introduced by Freidlin and Wentzell in [12, Chapter 6]. Consider an irreducible finite state Markov chain $\\{Z_{n}\\}_{n\in\mathbb{N}_{0}}$ with state space $L.$ For any $i,j\in L,$ let $p_{ij}$ be the one-step transition probability of $\\{Z_{n}\\}_{n}$ from state $i$ to state $j.$ Write $P_{i}(\cdot)$ and $E_{i}(\cdot)$ for probabilities and expectations of the chain started at state $i$ at time $0.$ Recall the notation $\pi(g)\doteq\prod_{(i\rightarrow j)\in g}p_{ij}$. ###### Lemma 7.2. The unique invariant measure of $\\{Z_{n}\\}_{n\in\mathbb{N}}$ can be expressed $\lambda_{i}=\frac{\sum_{g\in G\left(i\right)}\pi\left(g\right)}{\sum_{j\in L}\left(\sum_{g\in G\left(j\right)}\pi\left(g\right)\right)}.$ ###### Proof. See Lemma 3.1, Chapter 6 in [12]. ∎ To analyze the empirical measure we will need additional results, including representations for the number of visits to a state during a regenerative cycle. Write $T_{i}\doteq\inf\left\\{n\geq 0:Z_{n}=i\right\\}$ for the first hitting time of state $i,$ and write $T_{i}^{+}\doteq\inf\left\\{n\geq 1:Z_{n}=i\right\\}.$ Observe that $T_{i}^{+}=T_{i}$ unless $Z_{0}=i,$ in which case we call $T_{i}^{+}$ the first return time to state $i.$ Let $\hat{N}\doteq\inf\\{n\in\mathbb{N}_{0}:Z_{n}\in L\setminus\\{1\\}\\}$ and $N\doteq\inf\\{n\in\mathbb{N}:Z_{n}=1,n\geq\hat{N}\\}.$ $\hat{N}$ is the first time of visiting a state other than state $1$ and $N$ is the first time of visiting state $1$ after $\hat{N}.$ For any $j\in L,$ let $N_{j}$ be the number of visits (including time $0$) of state $j$ before $N,$ i.e., $N_{j}=\left|\\{n\in\mathbb{N}_{0}:n<N\text{ and }Z_{n}=j\\}\right|.$ We would like to understand $E_{1}N_{j}$ and $E_{j}N_{j}$ for any $j\in L.$ These quantities will appear later on in Subsection 7.2. The next lemma shows how they can be related to the invariant measure of $\\{Z_{n}\\}_{n}$. ###### Lemma 7.3. 1. 1. For any $j\in L\setminus\\{1\\}$ $E_{j}N_{j}=\frac{\sum_{g\in G\left(1,j\right)}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}\text{ and }E_{j}N_{j}=\lambda_{j}\left(E_{j}T_{1}+E_{1}T_{j}\right).$ 2. 2. For any $i,j\in L,$ $j\neq i$ $P_{i}\left(T_{j}<T_{i}^{+}\right)=\frac{1}{\lambda_{i}\left(E_{j}T_{i}+E_{i}T_{j}\right)}.$ 3. 3. For any $j\in L$ $E_{1}N_{j}=\frac{1}{1-p_{11}}\frac{\lambda_{j}}{\lambda_{1}}.$ ###### Proof. See Lemma 3.4 in [12, Chapter 6] for the first assertion of part 1 and see Lemma 2.7 in [1, Chapter 2] for the second assertion of part 1\. For part 2, see Corollary 2.8 in [1, Chapter 2]. For part 3, since $E_{1}N_{j}=\sum_{\ell=1}^{\infty}P_{1}\left(N_{j}\geq\ell\right),$ we need to understand $P_{1}\left(N_{j}\geq\ell\right)$, which means we need to know how to count all the ways to get $N_{j}\geq\ell$ before returning to state $1.$ We first have to move away from state $1$, so the types of sequences are of the form $\underset{i\text{ times}}{\underbrace{1,1,\ldots,1}},k_{1},k_{2},\ldots,k_{q},1$ for some $i,q\in\mathbb{N}$ and $k_{1}\neq 1,\cdots,k_{q}\neq 1$. When $j=1,$ we do not care about $k_{1},k_{2},\ldots,k_{q},$ and therefore $P_{1}\left(N_{1}\geq i\right)=p_{11}^{i-1}\text{ and }E_{1}N_{1}=\sum\nolimits_{i=1}^{\infty}P_{1}\left(N_{1}\geq i\right)=\frac{1}{1-p_{11}}.$ For $j\in L\setminus\\{1\\},$ the event $\\{N_{j}\geq\ell\\}$ requires that within $k_{1},k_{2},\ldots,k_{q},$ we 1. 1. first visit state $j$ before returning to state $1,$ which has corresponding probability $P_{1}(T_{j}<T_{1}^{+})$, 2. 2. then start from state $j$ and again visit state $j$ before returning to state $1,$ which has corresponding probability $P_{j}(T_{j}^{+}<T_{1}).$ Step 2 needs to happen at least $\ell-1$ times in a row, and after that we do not care. Thus, $\displaystyle P_{1}\left(N_{j}\geq\ell\right)$ $\displaystyle=\sum\nolimits_{i=1}^{\infty}\left(p_{11}\right)^{i-1}P_{1}\left(T_{j}<T_{1}^{+}\right)(P_{j}(T_{j}^{+}<T_{1}))^{\ell-1}$ $\displaystyle=\frac{1}{1-p_{11}}P_{1}\left(T_{j}<T_{1}^{+}\right)(P_{j}(T_{j}^{+}<T_{1}))^{\ell-1}$ and $\displaystyle\sum\nolimits_{\ell=1}^{\infty}P_{1}\left(N_{j}\geq\ell\right)$ $\displaystyle=\frac{1}{1-p_{11}}\frac{P_{1}\left(T_{j}<T_{1}^{+}\right)}{P_{j}(T_{1}<T_{j}^{+})}=\frac{1}{1-p_{11}}\frac{\lambda_{j}\left(E_{1}T_{j}+E_{j}T_{1}\right)}{\lambda_{1}\left(E_{1}T_{j}+E_{j}T_{1}\right)}$ $\displaystyle=\frac{1}{1-p_{11}}\frac{\lambda_{j}}{\lambda_{1}}.$ The third equality comes from part 2. ∎ To apply the preceding results using the machinery developed by Freidlin and Wentzell, one must have analogues that allow for small perturbations of the transition probabilities due to the fact that initial conditions are to be taken in small neighborhoods of the equilibrium points. The addition of a tilde will be used to identify the corresponding objects, such as hitting and return times. Take as given a Markov chain $\\{\tilde{Z}_{n}\\}_{n\in\mathbb{N}_{0}}$ on a state space $\mathcal{X}={\textstyle\cup_{i\in L}}\mathcal{X}_{i},$ with $\mathcal{X}_{i}\cap\mathcal{X}_{j}=\emptyset$ $(i\neq j),$ and assume there is $a\in[1,\infty)$ such that for any $i,j\in L$ and $j\neq i,$ the transition probability of the chain from $x\in\mathcal{X}_{i}$ to $\mathcal{X}_{j}$ (denoted by $p\left(x,\mathcal{X}_{j}\right)$) satisfies the inequalities $a^{-1}p_{ij}\leq p\left(x,\mathcal{X}_{j}\right)\leq ap_{ij}$ (7.3) for any $x\in\mathcal{X}_{i}$. Write $P_{x}(\cdot)$ and $E_{x}(\cdot)$ for probabilities and expectations of the chain started at $x\in\mathcal{X}$ at time $0.$ Write $\tilde{T}_{i}\doteq\inf\\{n\geq 0:\tilde{Z}_{n}\in\mathcal{X}_{i}\\}$ for the first hitting time of $\mathcal{X}_{i},$ and write $\tilde{T}_{i}^{+}\doteq\inf\\{n\geq 1:\tilde{Z}_{n}\in\mathcal{X}_{i}\\}.$ Observe that $\tilde{T}_{i}^{+}=\tilde{T}_{i}$ unless $\tilde{Z}_{0}\in\mathcal{X}_{i},$ in which case we call $\tilde{T}_{i}^{+}$ the first return time to $\mathcal{X}_{i}.$ Recall that $l=|L|$. ###### Remark 7.4. Observe that given $j\in L$ and for any $x\in\mathcal{X}_{j}$, $1-p\left(x,\mathcal{X}_{j}\right)=\textstyle\sum_{k\in L\setminus\left\\{j\right\\}}p\left(x,\mathcal{X}_{k}\right).$ Therefore, we can apply (7.3) to obtain $a^{-1}\textstyle\sum_{k\in L\setminus\left\\{j\right\\}}p_{jk}\leq 1-p\left(x,\mathcal{X}_{j}\right)\leq a\textstyle\sum_{k\in L\setminus\left\\{j\right\\}}p_{jk}.$ ###### Lemma 7.5. 1. 1. Consider distinct $i,j,k\in L$. Then for $x\in\mathcal{X}_{k},$ $a^{-4^{l-2}}P_{k}\left(T_{j}<T_{i}\right)\leq P_{x}(\tilde{T}_{j}<\tilde{T}_{i})\leq a^{4^{l-2}}P_{k}\left(T_{j}<T_{i}\right).$ 2. 2. For any $i\in L$, $j\in L\setminus\\{i\\}$ and $x\in\mathcal{X}_{i},$ $a^{-4^{l-2}-1}P_{i}\left(T_{j}<T_{i}^{+}\right)\leq P_{x}(\tilde{T}_{j}<\tilde{T}_{i}^{+})\leq a^{4^{l-2}+1}P_{i}\left(T_{j}<T_{i}^{+}\right).$ ###### Proof. For part 1, see Lemma 3.3 in [12, Chapter 6]. We only need to prove part $2.$ Note that by a first step analysis on $\\{\tilde{Z}_{n}\\}_{n\in\mathbb{N}_{0}}$, for any $i\in L$, $j\in L\setminus\\{i\\}$ and $x\in\mathcal{X}_{i},$ $\displaystyle P_{x}(\tilde{T}_{j}<\tilde{T}_{i}^{+})$ $\displaystyle=p\left(x,\mathcal{X}_{j}\right)+\sum\nolimits_{k\in L\setminus\\{i,j\\}}\int_{\mathcal{X}_{k}}P_{y}(\tilde{T}_{j}<\tilde{T}_{i})p\left(x,dy\right)$ $\displaystyle\leq ap_{ij}+\sum\nolimits_{k\in L\setminus\\{i,j\\}}\left(a^{4^{l-2}}P_{k}\left(T_{j}<T_{i}\right)\right)\left(ap_{ik}\right)$ $\displaystyle\leq a^{4^{l-2}+1}\left(p_{ij}+\sum\nolimits_{k\in L\setminus\\{i,j\\}}P_{k}\left(T_{j}<T_{i}\right)p_{ik}\right)$ $\displaystyle=a^{4^{l-2}+1}P_{i}\left(T_{j}<T_{i}^{+}\right).$ The first inequality comes from the use of (7.3) and part 1; the last equality holds since we can do a first step analysis on $\\{Z_{n}\\}_{n}.$ Similarly, we can show the lower bound. ∎ Let $\check{N}\doteq\inf\\{n\in\mathbb{N}_{0}:\tilde{Z}_{n}\in\cup_{j\in L\setminus\\{1\\}}\mathcal{X}_{j}\\}$ and $\tilde{N}\doteq\inf\\{n\in\mathbb{N}:Z_{n}\in\mathcal{X}_{1},n\geq\check{N}\\}.$ For any $j\in L,$ let $\tilde{N}_{j}$ be the number of visits (including time $0$) of state $\mathcal{X}_{j}$ before $\tilde{N},$ i.e. $\tilde{N}_{j}=|\\{n\in\mathbb{N}_{0}:n<\tilde{N}\text{ and }Z_{n}\in\mathcal{X}_{j}\\}|.$ We would like to understand $E_{x}\tilde{N}_{j}$ for any $j\in L$ and $x\in\mathcal{X}_{1}$ or $\mathcal{X}_{j}.$ ###### Lemma 7.6. For any $j\in L$ and $x\in\mathcal{X}_{1}$ $E_{x}\tilde{N}_{j}\leq\frac{a^{4^{l-1}}}{\sum_{\ell\in L\setminus\\{1\\}}p_{1\ell}}\frac{\sum_{g\in G\left(j\right)}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}.$ Moreover, for any $j\in L\setminus\\{1\\}$ $\sum_{\ell=1}^{\infty}\sup_{x\in\mathcal{X}_{j}}P_{x}\left(\tilde{N}_{j}\geq\ell\right)\leq a^{4^{l-1}}\frac{\sum_{g\in G\left(1,j\right)}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}\text{ and }\sum_{\ell=1}^{\infty}\sup_{x\in\mathcal{X}_{1}}P_{x}\left(\tilde{N}_{1}\geq\ell\right)\leq\frac{a}{\sum_{\ell\in L\setminus\\{1\\}}p_{1\ell}}.$ ###### Proof. For any $x\in\mathcal{X}_{1},$ note that for any $\ell\in\mathbb{N},$ by a conditioning argument as in the proof of Lemma 7.3 (3), we find that for $j\in L\setminus\\{1\\}$ $P_{x}(\tilde{N}_{j}\geq\ell)\leq\frac{\sup_{y\in\mathcal{X}_{1}}P_{y}(\tilde{T}_{j}<\tilde{T}_{1}^{+})}{1-\sup_{y\in\mathcal{X}_{1}}p\left(y,\mathcal{X}_{1}\right)}\left(\sup\nolimits_{y\in\mathcal{X}_{j}}P_{y}(\tilde{T}_{j}^{+}<\tilde{T}_{1})\right)^{\ell-1}$ and $P_{x}(\tilde{N}_{1}\geq\ell)\leq\left(\sup\nolimits_{y\in\mathcal{X}_{1}}p\left(y,\mathcal{X}_{1}\right)\right)^{\ell-1}.$ Thus, for any $x\in\mathcal{X}_{1}$ and for $j\in L\setminus\\{1\\}$ $\displaystyle E_{x}\tilde{N}_{j}$ $\displaystyle=\sum_{\ell=1}^{\infty}P_{x}(\tilde{N}_{j}\geq\ell)\leq\frac{\sup_{y\in\mathcal{X}_{1}}P_{y}(\tilde{T}_{j}<\tilde{T}_{1}^{+})}{1-\sup_{y\in\mathcal{X}_{1}}p\left(y,\mathcal{X}_{1}\right)}\cdot\frac{1}{1-\sup_{y\in\mathcal{X}_{j}}P_{y}(\tilde{T}_{j}^{+}<\tilde{T}_{1})}$ $\displaystyle=\frac{\sup_{y\in\mathcal{X}_{1}}P_{y}(\tilde{T}_{j}<\tilde{T}_{1}^{+})}{\left(\inf_{y\in\mathcal{X}_{j}}\left(1-p\left(y,\mathcal{X}_{1}\right)\right)\right)(\inf_{y\in\mathcal{X}_{j}}P_{y}(\tilde{T}_{1}<\tilde{T}_{j}^{+}))}$ $\displaystyle\leq a^{4^{l-1}}\frac{P_{1}(T_{j}<T_{1}^{+})}{(\sum_{\ell\in L\setminus\\{1\\}}p_{1\ell})P_{j}(T_{1}<T_{j}^{+})}$ $\displaystyle=\frac{a^{4^{l-1}}}{\sum_{\ell\in L\setminus\\{1\\}}p_{1\ell}}\frac{\lambda_{j}}{\lambda_{1}}=\frac{a^{4^{l-1}}}{\sum_{\ell\in L\setminus\\{1\\}}p_{1\ell}}\frac{\sum_{g\in G\left(j\right)}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}.$ The second inequality is from Remark 7.4 and Lemma 7.5 (2); the third equality comes from Lemma 7.3 (2); the last equality holds due to Lemma 7.2. Also $\displaystyle E_{x}\tilde{N}_{1}=\sum_{\ell=1}^{\infty}P_{x}(\tilde{N}_{1}\geq\ell)\leq\frac{1}{1-\sup_{y\in\mathcal{X}_{1}}p\left(y,\mathcal{X}_{1}\right)}=\frac{1}{\inf_{y\in\mathcal{X}_{1}}\left(1-p\left(y,\mathcal{X}_{1}\right)\right)}\leq\frac{a}{\sum_{\ell\in L\setminus\\{1\\}}p_{1\ell}}.$ The last inequality is from Remark 7.4. This completes the proof of part 1. Turning to part 2, since for any $\ell\in\mathbb{N}$ $\sup\nolimits_{x\in\mathcal{X}_{1}}P_{x}(\tilde{N}_{1}\geq\ell)\leq\left(\sup\nolimits_{y\in\mathcal{X}_{1}}p\left(y,\mathcal{X}_{1}\right)\right)^{\ell-1},$ we have $\sum_{\ell=1}^{\infty}\sup_{x\in\mathcal{X}_{1}}P_{x}(\tilde{N}_{1}\geq\ell)\leq\frac{1}{1-\sup_{y\in\mathcal{X}_{1}}p\left(y,\mathcal{X}_{1}\right)}\leq\frac{a}{\sum_{\ell\in L\setminus\\{1\\}}p_{1\ell}}.$ Furthermore, we use the conditioning argument again to find that for any $j\in L\setminus\\{1\\}$ and $\ell\in\mathbb{N}$ $\sup\nolimits_{x\in\mathcal{X}_{j}}P_{x}(\tilde{N}_{j}\geq\ell)\leq(\sup\nolimits_{y\in\mathcal{X}_{j}}P_{y}(\tilde{T}_{j}^{+}<\tilde{T}_{1}))^{\ell-1}.$ This implies that $\displaystyle\sum_{\ell=1}^{\infty}\sup\nolimits_{x\in\mathcal{X}_{j}}P_{x}(\tilde{N}_{j}\geq\ell)$ $\displaystyle\qquad\leq\sum_{\ell=1}^{\infty}(\sup\nolimits_{y\in\mathcal{X}_{j}}P_{y}(\tilde{T}_{j}^{+}<\tilde{T}_{1}))^{\ell-1}=\frac{1}{1-\sup_{y\in\mathcal{X}_{j}}P_{y}(\tilde{T}_{j}^{+}<\tilde{T}_{1})}$ $\displaystyle\qquad=\frac{1}{\inf_{y\in\mathcal{X}_{j}}P_{y}(\tilde{T}_{1}<\tilde{T}_{j}^{+})}\leq a^{4^{l-1}}\frac{1}{P_{j}(T_{1}<T_{j}^{+})}$ $\displaystyle\qquad=a^{4^{l-1}}\lambda_{j}(E_{1}T_{j}+E_{j}T_{1})=a^{4^{l-1}}\frac{\sum_{g\in G\left(1,j\right)}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}.$ We use Lemma 7.5 (2) to obtain the second inequality and Lemma 7.3, parts (2) and (1), for the penultimate and last equalities. ∎ ### 7.2 Asymptotics of moments of $S_{1}^{\varepsilon}$ Recall that $\\{X^{\varepsilon}\\}_{\varepsilon\in(0,\infty)}\subset C([0,\infty):M)$ is a sequence of stochastic processes satisfying Condition 3.1, Condition 3.7 and Condition 3.13. Moreover, recall that $S_{1}^{\varepsilon}$ is defined by $S_{1}^{\varepsilon}\doteq\int_{0}^{\tau_{1}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt.$ (7.4) As mentioned in Section 6, we are interested in the logarithmic asymptotics of $E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}$ and $E_{\lambda^{\varepsilon}}(S_{1}^{\varepsilon})^{2}.$ To find these asymptotics, the main tool we will use is Freidlin-Wentzell theory [12]. In fact, we will generalize the results of Freidlin-Wentzell to the following: For any given continuous function $f:M\rightarrow\mathbb{R}$ and any compact set $A\subset M,$ we will provide lower bounds for $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}\left(\int_{0}^{\tau_{1}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{s}^{\varepsilon}\right)}1_{A}\left(X_{s}^{\varepsilon}\right)ds\right)\right)$ (7.5) and $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}\left(\int_{0}^{\tau_{1}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{s}^{\varepsilon}\right)}1_{A}\left(X_{s}^{\varepsilon}\right)ds\right)^{2}\right).$ (7.6) As will be shown, these two bounds can be expressed in terms of the quasipotentials $V(O_{i},O_{j})$ and $V(O_{i},x).$ ###### Remark 7.7. In the Freidlin-Wentzell theory as presented in [12], they only consider bounds for $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup\nolimits_{z\in\partial B_{\delta}(O_{1})}E_{z}\tau_{1}^{\varepsilon}\right).$ Thus, their result is a special case of (7.5) with $f\equiv 0$ and $A=M$. Moreover, we generalize their result further by considering the logarithmic asymptotics of higher moment quantities such as (7.6). Before proceeding, we recall that $L=\\{1,\ldots,l\\}$ and for any $\delta>0,$ we define $\tau_{0}\doteq 0,$ $\sigma_{n}\doteq\inf\\{t>\tau_{n}:X_{t}^{\varepsilon}\in{\textstyle\bigcup\nolimits_{j\in L}}\partial B_{2\delta}(O_{j})\\}\text{ and }\tau_{n}\doteq\inf\\{t>\sigma_{n-1}:X_{t}^{\varepsilon}\in{\textstyle\bigcup\nolimits_{j\in L}}\partial B_{\delta}(O_{j})\\}.$ Moreover, $\tau_{0}^{\varepsilon}\doteq 0,$ $\sigma_{n}^{\varepsilon}\doteq\inf\\{t>\tau_{n}^{\varepsilon}:X_{t}^{\varepsilon}\in{\textstyle\bigcup\nolimits_{j\in L\setminus\\{1\\}}}\partial B_{\delta}(O_{j})\\}\text{ and }\tau_{n}^{\varepsilon}\doteq\inf\left\\{t>\sigma_{n-1}^{\varepsilon}:X_{t}^{\varepsilon}\in\partial B_{\delta}(O_{1})\right\\}.$ In addition, $\\{Z_{n}\\}_{n\in\mathbb{N}_{0}}\doteq\\{X_{\tau_{n}}^{\varepsilon}\\}_{n\in\mathbb{N}_{0}}$ is a Markov chain on ${\textstyle\bigcup\nolimits_{j\in L}}\partial B_{\delta}(O_{j})$ and $\\{Z_{n}^{\varepsilon}\\}_{n\in\mathbb{N}_{0}}$ $\doteq\\{X_{\tau_{n}^{\varepsilon}}^{\varepsilon}\\}_{n\in\mathbb{N}_{0}}$ is a Markov chain on $\partial B_{\delta}(O_{1}).$ It is essential to keep the distinction clear: when there is an $\varepsilon$ superscript the chain makes transitions between neighborhoods of distinct equilibria, while if absent such transitions are possible, but for stable equilibria there will be many more transitions between the $\delta$ and $2\delta$ neighborhoods. Following the notation of Subsection 7.1, let $\hat{N}\doteq\inf\\{n\in\mathbb{N}_{0}:Z_{n}\in{\textstyle\bigcup\nolimits_{j\in L\setminus\\{1\\}}}\partial B_{\delta}(O_{j})\\}$, $N\doteq\inf\\{n\geq\hat{N}:Z_{n}\in\partial B_{\delta}(O_{1})\\}$, and recall $\mathcal{F}_{t}\doteq\sigma(\\{X_{s}^{\varepsilon};s\leq t\\})$. Then since $\\{\tau_{n}\\}_{n\in\mathbb{N}_{0}}$ are stopping times with respect to the filtration $\\{\mathcal{F}_{t}\\}_{t\geq 0},$ $\mathcal{F}_{\tau_{n}}$ are well-defined for any $n\in\mathbb{N}_{0}$ and we use $\mathcal{G}_{n}$ to denote $\mathcal{F}_{\tau_{n}}.$ One can prove that $\hat{N}$ and $N$ are stopping times with respect to $\\{\mathcal{G}_{n}\\}_{n\in\mathbb{N}}.$ For any $j\in L,$ let $N_{j}$ be the number of visits of $\\{Z_{n}\\}_{n\in\mathbb{N}_{0}}$ to $\partial B_{\delta}(O_{j})$ (including time $0$) before $N.$ The proofs of the following two lemmas are given in the Appendix. ###### Lemma 7.8. Given $\delta>0$ sufficiently small, for any $x\in\partial B_{\delta}(O_{1})$ and any nonnegative measurable function $g$ $:M\rightarrow\mathbb{R}$, $E_{x}\left(\int_{0}^{\tau_{1}^{\varepsilon}}g\left(X_{s}^{\varepsilon}\right)ds\right)\leq\sum_{j\in L}\left[\sup_{y\in\partial B_{\delta}(O_{j})}E_{y}\left(\int_{0}^{\tau_{1}}g\left(X_{s}^{\varepsilon}\right)ds\right)\right]\cdot E_{x}N_{j}.$ ###### Lemma 7.9. Given $\delta>0$ sufficiently small, for any $x\in\partial B_{\delta}(O_{1})$ and any nonnegative measurable function $g$ $:M\rightarrow\mathbb{R}$, $\displaystyle E_{x}\left(\int_{0}^{\tau_{1}^{\varepsilon}}g\left(X_{s}^{\varepsilon}\right)ds\right)^{2}$ $\displaystyle\leq l\sum_{j\in L}\left[\sup_{y\in\partial B_{\delta}(O_{j})}E_{y}\left(\int_{0}^{\tau_{1}}g\left(X_{s}^{\varepsilon}\right)ds\right)^{2}\right]\cdot E_{x}N_{j}$ $\displaystyle\qquad+2l\sum_{j\in L}\left[\sup_{y\in\partial B_{\delta}(O_{j})}E_{y}\left(\int_{0}^{\tau_{1}}g\left(X_{s}^{\varepsilon}\right)ds\right)\right]^{2}\cdot E_{x}N_{j}$ $\displaystyle\quad\quad\qquad\cdot\sum_{k=1}^{\infty}\sup_{y\in\partial B_{\delta}(O_{j})}P_{y}\left(k\leq N_{j}\right).$ (7.7) Although as noted the proofs are given in the Appendix, these results follow in a straightforward way by decomposing the excursion away from $O_{1}$ during $[0,\tau_{1}^{\varepsilon}]$, which only stops when returning to a neighborhood of $O_{1}$, into excursions between any pair of equilibrium points, counting the number of such excursions that start near a particular equilibrium point, and using the strong Markov property. ###### Remark 7.10. Following an analogous argument as in the proof of Lemma 7.8 and Lemma 7.9, we can prove the following: Given $\delta>0$ sufficiently small, for any $x\in\partial B_{\delta}(O_{1})$ and any nonnegative measurable function $g$ $:M\rightarrow\mathbb{R}$, $E_{x}\left(\int_{\sigma_{0}^{\varepsilon}}^{\tau_{1}^{\varepsilon}}g\left(X_{s}^{\varepsilon}\right)ds\right)\leq{\textstyle\sum_{j\in L\setminus\\{1\\}}}\left[\sup_{y\in\partial B_{\delta}(O_{j})}E_{y}\left(\int_{0}^{\tau_{1}}g\left(X_{s}^{\varepsilon}\right)ds\right)\right]\cdot E_{x}N_{j}$ and $\displaystyle E_{x}\left(\int_{\sigma_{0}^{\varepsilon}}^{\tau_{1}^{\varepsilon}}g\left(X_{s}^{\varepsilon}\right)ds\right)^{2}$ $\displaystyle\leq l{\textstyle\sum_{j\in L\setminus\\{1\\}}}\left[\sup_{y\in\partial B_{\delta}(O_{j})}E_{y}\left(\int_{0}^{\tau_{1}}g\left(X_{s}^{\varepsilon}\right)ds\right)^{2}\right]\cdot E_{x}N_{j}$ $\displaystyle\quad+2l{\textstyle\sum_{j\in L\setminus\\{1\\}}}\left[\sup_{y\in\partial B_{\delta}(O_{j})}E_{y}\left(\int_{0}^{\tau_{1}}g\left(X_{s}^{\varepsilon}\right)ds\right)\right]^{2}\cdot E_{x}N_{j}$ $\displaystyle\quad\qquad\cdot{\textstyle\sum_{\ell=1}^{\infty}}\sup_{y\in\partial B_{\delta}(O_{j})}P_{y}\left(k\leq N_{j}\right).$ The main difference is that if the integration starts from $\sigma_{0}^{\varepsilon}$ (the first visiting time of ${\textstyle\bigcup\nolimits_{j\in L\setminus\\{1\\}}}\partial B_{\delta}(O_{j})$), then any summation appearing in the upper bounds should sum over all indices in $L\setminus\\{1\\}$ instead of $L.$ Owing to its frequent appearance but with varying arguments, we introduce the notation $I^{\varepsilon}(t_{1},t_{2};f,A)\doteq\int_{t_{1}}^{t_{2}}e^{-\frac{1}{\varepsilon}f(X_{s}^{\varepsilon})}1_{A}(X_{s}^{\varepsilon})ds,$ (7.8) and write $I^{\varepsilon}(t;f,A)$ if $t_{1}=0$ and $t_{2}=t$ so that, e.g., $S_{1}^{\varepsilon}=I^{\varepsilon}(\tau_{1}^{\varepsilon};f,A)$. ###### Corollary 7.11. Given any measurable set $A\subset M$, a measurable function $f:M\rightarrow\mathbb{R},$ $j\in L$ and $\delta>0,$ we have $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}I^{\varepsilon}(\tau_{1}^{\varepsilon};f,A)\right)$ $\displaystyle\quad\geq\min_{j\in L}\left\\{\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\right)+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)\right\\},$ and $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}I^{\varepsilon}(\tau_{1};f,A)^{2}\right)\geq\min_{j\in L}\left(\hat{R}_{j}^{(1)}\wedge\hat{R}_{j}^{(2)}\right),$ where $\hat{R}_{j}^{(1)}\doteq\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)^{2}\right)+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\right)$ and $\displaystyle\hat{R}_{j}^{(2)}$ $\displaystyle\doteq 2\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\right)$ $\displaystyle\qquad+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sum\nolimits_{\ell=1}^{\infty}\sup_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\ell\leq N_{j}\right)\right).$ ###### Proof. For the first part, applying Lemma 7.8 with $g(x)=e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)$ and using (7.1) and (7.2) completes the proof. For the second part, using Lemma 7.9 with $g(x)=e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)$ and using (7.1) and (7.2) again completes the proof. ∎ ###### Remark 7.12. Owing to Remark 7.10, we can modify the proof of Corollary 7.11 and show that given any set $A\subset M,$ a measurable function $f:M\rightarrow\mathbb{R},$ $j\in L$ and $\delta>0,$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}I^{\varepsilon}(\sigma_{0}^{\varepsilon},\tau_{1}^{\varepsilon};f,A)\right)$ $\displaystyle\quad\geq\min_{j\in L\setminus\\{1\\}}\left\\{\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\right)+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)\right\\}.$ Moreover, $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}I^{\varepsilon}(\sigma_{0}^{\varepsilon},\tau_{1}^{\varepsilon};f,A)^{2}\right)\geq\min_{j\in L\setminus\\{1\\}}\left(\hat{R}_{j}^{(1)}\wedge\hat{R}_{j}^{(2)}\right),$ where the definitions of $\hat{R}_{j}^{(1)}$ and $\hat{R}_{j}^{(2)}$ can be found in Corollary 7.11. We next consider lower bounds on $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)\quad\mbox{and}\quad\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)^{2}\right)$ for $j\in L$. We state some useful results before studying the lower bounds. Recall also that $\tau_{1}$ is the time to reach the $\delta$-neighborhood of any of the equilibrium points after leaving the $2\delta$-neighborhood of one of the equilibrium points. ###### Lemma 7.13. For any $\eta>0,$ there exists $\delta_{0}\in(0,1)$ and $\varepsilon_{0}\in(0,1)$, such that for all $\delta\in(0,\delta_{0})$ and $\varepsilon\in(0,\varepsilon_{0})$ $\sup_{x\in M}E_{x}\tau_{1}\leq e^{\frac{\eta}{\varepsilon}}\text{ and }\sup_{x\in M}E_{x}\left(\tau_{1}\right)^{2}\leq e^{\frac{\eta}{\varepsilon}}.$ ###### Proof. If $x$ is not in $\cup_{j\in L}B_{2\delta}(O_{j})$ then a uniform (in $x$ and small $\varepsilon$) upper bound on these expected values follows from the corollary to [12, Lemma 1.9, Chapter 6]. If $x\in\cup_{j\in L}B_{2\delta}(O_{j})$ then we must wait till the process reaches $\cup_{j\in L}\partial B_{2\delta}(O_{j})$, after which we can use the uniform bound (and the strong Markov property). Since there exists $\delta>0$ such the lower bound $P_{x}(\inf\\{t\geq 0:X_{t}^{\varepsilon}\in\cup_{j\in L}\partial B_{2\delta}(O_{j})\leq 1)\geq e^{-\eta/2\varepsilon}$ is valid for all $x\in\cup_{j\in L}B_{2\delta}(O_{j})$ and small $\varepsilon>0$, upper bounds of the desired form follow from the Markov property and standard calculations. ∎ For any compact set $A\subset M$, we use $\vartheta_{A}$ to denote the first hitting time $\vartheta_{A}\doteq\inf\left\\{t\geq 0:X_{t}^{\varepsilon}\in A\right\\}.$ Note that $\vartheta_{A}$ is a stopping time with respect to filtration $\\{\mathcal{F}_{t}\\}_{t\geq 0}.$ The following result is relatively straightforward given the just discussed bound on the distribution of $\tau_{1}$, and follows by partitioning according to $\tau_{1}\geq T$ and $\tau_{1}<T$ for large but fixed $T$. ###### Lemma 7.14. For any compact set $A\subset M,$ $j\in L$ and any $\eta>0,$ there exists $\delta_{0}\in(0,1)$ and $\varepsilon_{0}\in(0,1)$, such that for all $\varepsilon\in(0,\varepsilon_{0})$ and $\delta\in(0,\delta_{0})$ $\sup_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{A}\leq\tau_{1}\right)\leq e^{-\frac{1}{\varepsilon}\left(\inf_{x\in A}\left[V\left(O_{j},x\right)\right]-\eta\right)}.$ ###### Lemma 7.15. Given a compact set $A\subset M$, any $j\in L$ and $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}\left[\int_{0}^{\tau_{1}}1_{A}\left(X_{s}^{\varepsilon}\right)ds\right]\right)\geq\inf_{x\in A}V\left(O_{j},x\right)-\eta$ and $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}\left(\int_{0}^{\tau_{1}}1_{A}\left(X_{s}^{\varepsilon}\right)ds\right)^{2}\right)\geq\inf_{x\in A}V\left(O_{j},x\right)-\eta.$ ###### Proof. The idea of this proof follows from the proof of Theorem 4.3 in [12, Chapter 4]. Since $I^{\varepsilon}(\tau_{1};0,A)=\int_{0}^{\tau_{1}}1_{A}\left(X_{s}^{\varepsilon}\right)ds$, for any $x\in\partial B_{\delta}(O_{j}),$ $\displaystyle E_{x}I^{\varepsilon}(\tau_{1};0,A)$ $\displaystyle=E_{x}\left[I^{\varepsilon}(\tau_{1};0,A)1_{\left\\{\vartheta_{A}\leq\tau_{1}\right\\}}\right]=E_{x}\left[E_{x}\left[\left.I^{\varepsilon}(\tau_{1};0,A)\right|\mathcal{F}_{\vartheta_{A}}\right]1_{\left\\{\vartheta_{A}\leq\tau_{1}\right\\}}\right]$ $\displaystyle=E_{x}\left[(E_{X_{\vartheta_{A}}^{\varepsilon}}I^{\varepsilon}(\tau_{1};0,A))1_{\left\\{\vartheta_{A}\leq\tau_{1}\right\\}}\right]\leq\sup\nolimits_{y\in\partial A}E_{y}\tau_{1}\cdot\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{A}\leq\tau_{1}\right).$ The inequality is due to$E_{X_{\vartheta_{A}}^{\varepsilon}}I^{\varepsilon}(\tau_{1};0,A)\leq E_{X_{\vartheta_{A}}^{\varepsilon}}\tau_{1}\leq\sup_{y\in\partial A}E_{y}\tau_{1}.$ We then apply Lemma 7.13 and Lemma 7.14 to find that for the given $\eta>0,$ there exists $\delta_{0}\in(0,1)$ and $\varepsilon_{0}\in(0,1)$, such that for all $\varepsilon\in(0,\varepsilon_{0})$ and $\delta\in(0,\delta_{0}),$ $E_{x}I^{\varepsilon}(\tau_{1};0,A)\leq\sup_{y\in\partial A}E_{y}\tau_{1}\cdot\sup_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{A}\leq\tau_{1}\right)\leq e^{\frac{\eta/2}{\varepsilon}}e^{-\frac{1}{\varepsilon}\left(\inf_{y\in A}V\left(O_{j},y\right)-\eta/2\right)}.$ Thus, $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};0,A)\right)\geq\inf_{x\in A}V\left(O_{j},x\right)-\eta.$ This completes the proof of part 1. For part 2, following the same conditioning argument as for part 1 with the use of Lemma 7.13 and Lemma 7.14 gives that for the given $\eta>0,$ there exists $\delta_{0}\in(0,1)$ and $\varepsilon_{0}\in(0,1)$, such that for all $\varepsilon\in(0,\varepsilon_{0})$ and $\delta\in(0,\delta_{0}),$ $E_{x}I^{\varepsilon}(\tau_{1};0,A)^{2}\leq\sup_{y\in\partial A}E_{y}\left(\tau_{1}\right)^{2}\cdot\sup_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{A}\leq\tau_{1}\right)\leq e^{\frac{\eta/2}{\varepsilon}}e^{-\frac{1}{\varepsilon}\left(\inf_{x\in A}V\left(O_{j},x\right)-\eta/2\right)}.$ Therefore, $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};0,A)^{2}\right)\geq\inf_{x\in A}V\left(O_{j},x\right)-\eta.$ ∎ ###### Lemma 7.16. Given compact sets $A_{1},A_{2}\subset M$, $j\in L$ and $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}\left[\left(\int_{0}^{\tau_{1}}1_{A_{1}}\left(X_{s}^{\varepsilon}\right)ds\right)\left(\int_{0}^{\tau_{1}}1_{A_{2}}\left(X_{s}^{\varepsilon}\right)ds\right)\right]\right)$ $\displaystyle\qquad\geq\max\left\\{\inf_{x\in A_{1}}V\left(O_{j},x\right),\inf_{x\in A_{2}}V\left(O_{j},x\right)\right\\}-\eta.$ ###### Proof. We set $\vartheta_{A_{i}}\doteq\inf\left\\{t\geq 0:X_{t}^{\varepsilon}\in A_{i}\right\\}$ for $i=1,2.$ For any $x\in\partial B_{\delta}(O_{j}),$ using a conditioning argument as in the proof of Lemma 7.15 we obtain that for any $\eta>0,$ there exists $\delta_{0}\in(0,1)$ and $\varepsilon_{0}\in(0,1)$, such that for all $\varepsilon\in(0,\varepsilon_{0})$ and $\delta\in(0,\delta_{0}),$ $\displaystyle E_{x}\left[\left(\int_{0}^{\tau_{1}}1_{{A}_{1}}\left(X_{s}^{\varepsilon}\right)ds\right)\left(\int_{0}^{\tau_{1}}1_{{A}_{2}}\left(X_{s}^{\varepsilon}\right)ds\right)\right]$ (7.9) $\displaystyle=E_{x}\left[\left(\int_{0}^{\tau_{1}}\int_{0}^{\tau_{1}}1_{{A}_{1}}\left(X_{s}^{\varepsilon}\right)1_{{A}_{2}}\left(X_{t}^{\varepsilon}\right)dsdt\right)1_{\left\\{\vartheta_{{A}_{1}}\vee\vartheta_{{A}_{2}}\leq\tau_{1}\right\\}}\right]$ $\displaystyle=E_{x}\left[\left(E_{X_{\vartheta_{{A}_{1}}\vee\vartheta_{{A}_{2}}}^{\varepsilon}}\left[\int_{0}^{\tau_{1}}\int_{0}^{\tau_{1}}1_{{A}_{1}}\left(X_{s}^{\varepsilon}\right)1_{{A}_{2}}\left(X_{t}^{\varepsilon}\right)dsdt\right]\right)1_{\left\\{\vartheta_{{A}_{1}}\vee\vartheta_{{A}_{2}}\leq\tau_{1}\right\\}}\right]$ $\displaystyle\leq\sup\nolimits_{y\in\partial{A}_{1}\cup\partial{A}_{2}}E_{y}\left(\tau_{1}\right)^{2}\cdot\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{{A}_{1}}\leq\tau_{1},\vartheta_{{A}_{2}}\leq\tau_{1}\right)$ $\displaystyle\leq e^{\frac{\eta/2}{\varepsilon}}\cdot\min\left\\{\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{{A}_{1}}\leq\tau_{1}\right),\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{{A}_{2}}\leq\tau_{1}\right)\right\\},$ The last inequality holds since for $i=1,2$ $\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{A_{1}}\leq\tau_{1},\vartheta_{A_{2}}\leq\tau_{1}\right)\leq\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{A_{i}}\leq\tau_{1}\right)$ and owing to Lemma 7.13, for all $\varepsilon\in(0,\varepsilon_{0})$ $\sup\nolimits_{y\in\partial A_{1}}E_{y}\left(\tau_{1}\right)^{2}\leq e^{\frac{\eta/2}{\varepsilon}}\text{ and }\sup\nolimits_{y\in\partial A_{2}}E_{y}\left(\tau_{1}\right)^{2}\leq e^{\frac{\eta/2}{\varepsilon}}.$ Furthermore, for the given $\eta>0,$ by Lemma 7.14, there exists $\delta_{i}\in(0,1)$ such that for any $\delta\in(0,\delta_{i})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{A_{i}}\leq\tau_{1}\right)\right)\geq\inf_{x\in A_{i}}V\left(O_{j},x\right)-\eta/2$ for $i=1,2.$ Hence, letting $\delta_{0}=\delta_{1}\wedge\delta_{2},$ for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}E_{z}\left[\left(\int_{0}^{\tau_{1}}1_{A_{1}}\left(X_{s}^{\varepsilon}\right)ds\right)\left(\int_{0}^{\tau_{1}}1_{A_{2}}\left(X_{s}^{\varepsilon}\right)ds\right)\right]\right)$ $\displaystyle\geq\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(e^{\frac{\eta}{2\varepsilon}}\min\left\\{\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{A_{1}}\leq\tau_{1}\right),\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{A_{2}}\leq\tau_{1}\right)\right\\}\right)$ $\displaystyle\geq-\eta/2+\max\left\\{\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{A_{1}}\leq\tau_{1}\right)\right),\right.$ $\displaystyle\left.\qquad\qquad\qquad\qquad\qquad\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup\nolimits_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\vartheta_{A_{2}}\leq\tau_{1}\right)\right)\right\\}$ $\displaystyle\geq\max\left\\{\inf\nolimits_{x\in A_{1}}V\left(O_{j},x\right),\inf\nolimits_{x\in A_{2}}V\left(O_{j},x\right)\right\\}-\eta.$ The first inequality is from (7.9). ∎ ###### Remark 7.17. The next lemma considers asymptotics of the first and second moments of a certain integral that will appear in a decomposition of $S^{\varepsilon}_{1}$. It is important to note that the variational bounds for both moments have the same structure as an infimum over $x\in A$. While one might consider it possible that the variational problem for the second moment could require a pair of parameters (e.g., infimum over $x,y\in A$), the infimum is in fact achieved on the “diagonal” $x=y$. This means that the biggest contribution to the second moment is likewise due to mass along the “diagonal.” ###### Lemma 7.18. Given a compact set $A\subset M,$ a continuous function $f:M\rightarrow\mathbb{R},$ $j\in L$ and $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)\geq\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]-\eta$ and $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)^{2}\right)\geq\inf_{x\in A}\left[2f\left(x\right)+V\left(O_{j},x\right)\right]-\eta.$ ###### Proof. Since a continuous function is bounded on a compact set, there exists $m\in(0,\infty)$ such that $-m\leq f(x)\leq m$ for all $x\in A.$ For $n\in\mathbb{N}$ and $k\in\\{1,2,\ldots,n\\},$ consider the sets $A_{n,k}\doteq\left\\{x\in A:f\left(x\right)\in\left[-m+\frac{2\left(k-1\right)m}{n},-m+\frac{2km}{n}\right]\right\\}.$ Note that $A_{n,k}$ is a compact set for any $n,k.$ In addition, for any $n$ fixed, ${\textstyle\bigcup_{k=1}^{n}}A_{n,k}=A.$ With this expression, for any $x\in\partial B_{\delta}(O_{j})$ and $n\in\mathbb{N}$ $\displaystyle E_{x}I^{\varepsilon}(\tau_{1};f,A)\leq\sum\nolimits_{k=1}^{n}E_{x}I^{\varepsilon}(\tau_{1};f,{A_{n,k}})\leq\sum\nolimits_{k=1}^{n}E_{x}I^{\varepsilon}(\tau_{1};0,{A_{n,k}})e^{-\frac{1}{\varepsilon}\left(F_{n,k}-2m/n\right)}.$ The second inequality holds because by definition of $A_{n,k},$ for any $x\in A_{n,k}$, $f(x)\geq F_{n,k}-2m/n$ with $F_{n,k}\doteq\sup_{y\in A_{n,k}}f\left(y\right)$. Next we first apply (7.2) and then Lemma 7.15 with compact sets $A_{n,k}$ for $k\in\\{1,2,\ldots,n\\}$ to get $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)$ $\displaystyle\geq\min_{k\in\left\\{1,\ldots,n\right\\}}\left\\{\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup\limits_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};0,{A_{n,k}})e^{-\frac{1}{\varepsilon}\left(F_{n,k}-\frac{2m}{n}\right)}\right)\right\\}$ $\displaystyle=\min_{k\in\left\\{1,\ldots,n\right\\}}\left\\{\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};0,{A_{n,k}})\right)+F_{n,k}\right\\}-\frac{2m}{n}$ $\displaystyle\geq\min_{k\in\left\\{1,\ldots,n\right\\}}\left\\{\sup_{x\in A_{n,k}}f\left(x\right)+\inf_{x\in A_{n,k}}V\left(O_{j},x\right)\right\\}-\eta-\frac{2m}{n}.$ Finally, we know that $V\left(O_{j},x\right)$ is bounded below by $0$, and then we use the fact that for any two functions $f,g:\mathbb{R}^{d}\rightarrow\mathbb{R}$ with $g$ being bounded below (to ensure that the right hand side is well defined) and any set $A\subset\mathbb{R}^{d},$ $\inf_{x\in A}\left(f\left(x\right)+g\left(x\right)\right)\leq\sup_{x\in A}f\left(x\right)+\inf_{x\in A}g\left(x\right)$ to find that the last minimum in the previous display is greater than or equal to $\displaystyle\min_{k\in\left\\{1,\ldots,n\right\\}}\left\\{\inf_{x\in A_{n,k}}\left[f\left(x\right)+V\left(O_{j},x\right)\right]\right\\}=\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right].$ Therefore, $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)\geq\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]-\eta-\frac{2m}{n}.$ Since $n$ is arbitrary, sending $n\rightarrow\infty$ completes the proof for the first part. Turning to part 2, we follow the same argument as for part 1. For any $n\in\mathbb{N},$ we use the decomposition of $A$ into ${\textstyle\bigcup_{k=1}^{n}}A_{n,k}$ to have that for any $x\in\partial B_{\delta}(O_{j}),$ $\displaystyle E_{x}I^{\varepsilon}(\tau_{1};f,A)^{2}\leq E_{x}\left(\sum_{k=1}^{n}I^{\varepsilon}(\tau_{1};f,{A_{n,k}})\right)^{2}=\sum_{k=1}^{n}\sum_{\ell=1}^{n}E_{x}\left[I^{\varepsilon}(\tau_{1};f,{A_{n,k}})I^{\varepsilon}(\tau_{1};f,A_{n,\ell})\right].$ Recall that $F_{n,k}$ is used to denote $\sup_{y\in A_{n,k}}f\left(y\right)$. Using the definition of $A_{n,k}$ gives that for any $k,\ell\in\\{1,\ldots,n\\}$ $\displaystyle E_{x}\left[I^{\varepsilon}(\tau_{1};f,{A_{n,k}})I^{\varepsilon}(\tau_{1};f,A_{n,\ell})\right]$ $\displaystyle\leq\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}\left[I^{\varepsilon}(\tau_{1};0,{A_{n,k}})I^{\varepsilon}(\tau_{1};0,A_{n,\ell})\right]e^{-\frac{1}{\varepsilon}\left(F_{n,k}+F_{n,\ell}-\frac{4m}{n}\right)}.$ Applying (7.2) first and then Lemma 7.16 with compact sets $A_{n,k}$ and $A_{n,\ell}$ pairwise for all $k,\ell\in\\{1,2,\ldots,n\\}$ gives that $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)^{2}\right)$ $\displaystyle\geq\min_{k,\ell\in\left\\{1,\ldots,n\right\\}}\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}\left[I^{\varepsilon}(\tau_{1};f,{A_{n,k}})I^{\varepsilon}(\tau_{1};f,A_{n,\ell})\right]$ $\displaystyle\geq\min_{k,\ell\in\left\\{1,\ldots,n\right\\}}\left\\{\max\left\\{\inf_{x\in A_{n,k}}V\left(O_{j},x\right),\inf_{x\in A_{n,\ell}}V\left(O_{j},x\right)\right\\}+F_{n,k}+F_{n,\ell}\right\\}-\eta-\frac{4m}{n}$ $\displaystyle\geq\min_{k\in\left\\{1,\ldots,n\right\\}}\left\\{\sup_{x\in A_{n,k}}\left[2f\left(x\right)\right]+\inf_{x\in A_{n,k}}V\left(O_{j},x\right)\right\\}-\eta-\frac{4m}{n}$ $\displaystyle\geq\min_{k\in\left\\{1,\ldots,n\right\\}}\left\\{\inf_{x\in A_{n,k}}\left[2f\left(x\right)+V\left(O_{j},x\right)\right]\right\\}-\eta-\frac{4m}{n}$ $\displaystyle=\inf_{x\in A}\left[2f\left(x\right)+V\left(O_{j},x\right)\right]-\eta-\frac{4m}{n}.$ Sending $n\rightarrow\infty$ completes the proof for the second part. ∎ Our next interest is to find lower bounds for $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\right)\text{ and }\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sum_{\ell=1}^{\infty}\sup_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\ell\leq N_{j}\right)\right).$ We first recall that $N_{j}$ is the number of visits of the embedded Markov chain $\\{Z_{n}\\}_{n}=\\{X_{\tau_{n}}^{\varepsilon}\\}_{n}$ to $\partial B_{\delta}(O_{j})$ within one loop of regenerative cycle. Also, the definitions of $G(i)$ and $G(i,j)$ for any $i,j\in L$ with $i\neq j$ are given in Definition 3.8 and Remark 3.9. ###### Lemma 7.19. For any $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ and for any $j\in L$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup\nolimits_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\right)\geq-\min_{\ell\in L\setminus\\{1\\}}V\left(O_{1},O_{\ell}\right)+W\left(O_{j}\right)-W\left(O_{1}\right)-\eta,\text{ }$ where $W\left(O_{j}\right)\doteq\min_{g\in G\left(j\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right].$ ###### Proof. According to Lemma 3.17 we know that for any $\eta>0,$ there exist $\delta_{0}\in(0,1)$ and $\varepsilon_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ and $\varepsilon\in(0,\varepsilon_{0}),$ for all $x\in\partial B_{\delta}(O_{i}),$ the one-step transition probability of the Markov chain $\\{Z_{n}\\}_{n}$ on $\partial B_{\delta}(O_{j})$ satisfies the inequalities $e^{-\frac{1}{\varepsilon}\left(V\left(O_{i},O_{j}\right)+\eta/4^{l-1}\right)}\leq p(x,\partial B_{\delta}(O_{j}))\leq e^{-\frac{1}{\varepsilon}\left(V\left(O_{i},O_{j}\right)-\eta/4^{l-1}\right)}.$ (7.10) We can then apply Lemma 7.6 with $p_{ij}=e^{-\frac{1}{\varepsilon}V\left(O_{i},O_{j}\right)}$ and $a=e^{\frac{1}{\varepsilon}\eta/4^{l-1}}$ to obtain that $\displaystyle\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\leq\frac{e^{\frac{1}{\varepsilon}\eta}}{\sum_{\ell\in L\setminus\\{1\\}}e^{-\frac{1}{\varepsilon}V\left(O_{1},O_{\ell}\right)}}\frac{{\textstyle\sum_{g\in G\left(j\right)}}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}\leq\frac{e^{\frac{1}{\varepsilon}\eta}}{e^{-\frac{1}{\varepsilon}\min_{\ell\in L\setminus\\{1\\}}V\left(O_{1},O_{\ell}\right)}}\frac{{\textstyle\sum_{g\in G\left(j\right)}}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}.$ Thus, $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\right)\geq-\min_{\ell\in L\setminus\\{1\\}}V\left(O_{1},O_{\ell}\right)-\eta+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\frac{\sum_{g\in G\left(j\right)}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}\right).$ Hence it suffices to show that $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\frac{\sum_{g\in G\left(j\right)}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}\right)\geq W\left(O_{j}\right)-W\left(O_{1}\right).$ Observe that by definition for any $j\in L$ and $g\in G\left(j\right)$ $\displaystyle\pi\left(g\right)={\textstyle\prod_{\left(m\rightarrow n\right)\in g}}p_{mn}={\textstyle\prod_{\left(m\rightarrow n\right)\in g}}e^{-\frac{1}{\varepsilon}V\left(O_{m},O_{n}\right)}=\exp\left\\{-\frac{1}{\varepsilon}{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right\\},$ which implies that $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\frac{\sum_{g\in G\left(j\right)}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}\right)$ $\displaystyle\qquad\geq\min_{g\in G\left(j\right)}\left[\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\exp\left\\{-\frac{1}{\varepsilon}{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right\\}\right)\right]$ $\displaystyle\qquad\qquad-\min_{g\in G\left(1\right)}\left[\limsup_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\exp\left\\{-\frac{1}{\varepsilon}{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right\\}\right)\right]$ $\displaystyle\qquad=\min_{g\in G\left(j\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right]-\min_{g\in G\left(1\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right]$ $\displaystyle\qquad=W\left(O_{j}\right)-W\left(O_{1}\right).$ The inequality is from Lemma 7.1; the last equality holds due the definition of $W\left(O_{j}\right)$. ∎ Recall the definition of $W(O_{1}\cup O_{j})$ in (3.3). In the next result we obtain bounds on, for example, a quantity close to the expected number of visits to $B_{\delta}(O_{j})$ before visiting a neighborhood of $O_{1}$, after starting near $O_{j}$. ###### Lemma 7.20. For any $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sum_{\ell=1}^{\infty}\sup_{z\in\partial B_{\delta}(O_{1})}P_{z}\left(\ell\leq N_{1}\right)\right)\geq-\min_{\ell\in L\setminus\\{1\\}}V\left(O_{1},O_{\ell}\right)-\eta$ and for any $j\in L\setminus\\{1\\}$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sum_{\ell=1}^{\infty}\sup_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\ell\leq N_{j}\right)\right)\geq W(O_{1}\cup O_{j})-W\left(O_{1}\right)-\eta.$ ###### Proof. We again use that by Lemma 3.17, for any $\eta>0$ there exist $\delta_{0}\in(0,1)$ and $\varepsilon_{0}\in(0,1),$ such that (7.10) holds for any $\delta\in(0,\delta_{0})$, $\varepsilon\in(0,\varepsilon_{0})$ and all $x\in\partial B_{\delta}(O_{i}).$ Then by Lemma 7.6 with $p_{ij}=e^{-\frac{1}{\varepsilon}V\left(O_{i},O_{j}\right)}$ and $a=e^{\frac{1}{\varepsilon}\eta/4^{l-1}}$ $\sum_{\ell=1}^{\infty}\sup_{x\in\partial B_{\delta}(O_{j})}P_{x}\left(N_{1}\geq\ell\right)\leq\frac{e^{\frac{1}{\varepsilon}\eta}}{\sum_{\ell\in L\setminus\\{1\\}}e^{-\frac{1}{\varepsilon}V\left(O_{1},O_{\ell}\right)}}$ and for any $j\in L\setminus\\{1\\}$ $\sum_{\ell=1}^{\infty}\sup_{x\in\partial B_{\delta}(O_{j})}P_{x}\left(N_{j}\geq\ell\right)\leq e^{\frac{1}{\varepsilon}\eta}\frac{\sum_{g\in G\left(1,j\right)}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}.$ Thus, $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sum_{\ell=1}^{\infty}\sup_{z\in\partial B_{\delta}(O_{1})}P_{z}\left(\ell\leq N_{1}\right)\right)\geq-\limsup_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sum\nolimits_{\ell\in L\setminus\\{1\\}}e^{-\frac{1}{\varepsilon}V\left(O_{1},O_{\ell}\right)}\right)-\eta$ and $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sum_{\ell=0}^{\infty}\sup_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\ell\leq N_{j}\right)\right)\geq\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\frac{\sum_{g\in G\left(1,j\right)}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}\right)-\eta.$ Following the same argument as for the proof of Lemma 7.19, we can use Lemma 7.1 to obtain that $-\limsup_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sum\nolimits_{\ell\in L\setminus\\{1\\}}e^{-\frac{1}{\varepsilon}V\left(O_{1},O_{\ell}\right)}\right)\geq-\min_{\ell\in L\setminus\\{1\\}}V\left(O_{1},O_{\ell}\right)$ and $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\frac{\sum_{g\in G\left(1,j\right)}\pi\left(g\right)}{\sum_{g\in G\left(1\right)}\pi\left(g\right)}\right)$ $\displaystyle\qquad\geq\min_{g\in G\left(1,j\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right]-\min_{g\in G\left(1\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right].$ Recalling (3.2) and (3.3), we are done. ∎ As mentioned at the beginning of this subsection, our main goal is to provide lower bounds for $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}\left(\int_{0}^{\tau_{1}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{s}^{\varepsilon}\right)}1_{A}\left(X_{s}^{\varepsilon}\right)ds\right)\right)$ and $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}\left(\int_{0}^{\tau_{1}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{s}^{\varepsilon}\right)}1_{A}\left(X_{s}^{\varepsilon}\right)ds\right)^{2}\right)$ for a given continuous function $f:M\rightarrow\mathbb{R}$ and compact set $A\subset M.$ We now state the main results of the subsection. Recall that $h_{1}=\min_{\ell\in L\setminus\\{1\\}}V\left(O_{1},O_{\ell}\right)$, $S_{1}^{\varepsilon}\doteq\int_{0}^{\tau_{1}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{s}^{\varepsilon}\right)}1_{A}\left(X_{s}^{\varepsilon}\right)ds$ and $W\left(O_{j}\right)\doteq\min_{g\in G\left(j\right)}[\sum_{\left(m\rightarrow n\right)\in g}V\left(O_{m},O_{n}\right)]$ and the definitions (7.8). ###### Lemma 7.21. Given a compact set $A\subset M,$ a continuous function $f:M\rightarrow\mathbb{R}$ and $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}S_{1}^{\varepsilon}\right]\geq\min_{j\in L}\left\\{\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)\right\\}-W\left(O_{1}\right)-h_{1}-\eta.$ ###### Proof. Recall that by Lemma 7.18, we have shown that for the given $\eta,$ there exists $\delta_{1}\in(0,1),$ such that for any $\delta\in(0,\delta_{1})$ and $j\in L$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)\geq\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]-\frac{\eta}{2}.$ In addition, by Lemma 7.19, we know that for the same $\eta,$ there exists $\delta_{2}\in(0,1),$ such that for any $\delta\in(0,\delta_{2})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup\nolimits_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\right)\geq-\min_{\ell\in L\setminus\\{1\\}}V\left(O_{1},O_{\ell}\right)+W\left(O_{j}\right)-W\left(O_{1}\right)-{\eta}/{2}.$ Hence for any $\delta\in(0,\delta_{0})$ with $\delta_{0}=\delta_{1}\wedge\delta_{2},$ we apply Corollary 7.11 to get $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[E_{x}I^{\varepsilon}(\tau_{1}^{\varepsilon};f,A)\right]$ $\displaystyle\geq\min_{j\in L}\left\\{\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}\left(N_{j}\right)\right)\right\\}$ $\displaystyle\geq\min_{j\in L}\left\\{\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)\right\\}-W\left(O_{1}\right)-h_{1}-\eta,$ where $\tau_{1}^{\varepsilon}$ is the time for a regenerative cycle and $\tau_{1}$ is the first visit time of neighborhoods of equilibrium points after being a certain distance away from them. ∎ ###### Remark 7.22. According to Remark 7.12 and using the same argument as in Lemma 7.21, we can find that given a compact set $A\subset M,$ a continuous function $f:M\rightarrow\mathbb{R}$ and $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}I^{\varepsilon}(\sigma_{0}^{\varepsilon},\tau_{1}^{\varepsilon};f,A)\right]$ $\displaystyle\qquad\geq\min_{j\in L\setminus\\{1\\}}\left\\{\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)\right\\}-W\left(O_{1}\right)-h_{1}-\eta.$ ###### Lemma 7.23. Given a compact set $A\subset M,$ a continuous function $f:M\rightarrow\mathbb{R}$ and $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[\sup\nolimits_{z\in\partial B_{\delta}(O_{1})}E_{z}(S_{1}^{\varepsilon})^{2}\right]\geq\min_{j\in L}\left(R_{j}^{(1)}\wedge R_{j}^{(2)}\right)-h_{1}-\eta,$ where $S_{1}^{\varepsilon}\doteq\int_{0}^{\tau_{1}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{s}^{\varepsilon}\right)}1_{A}\left(X_{s}^{\varepsilon}\right)ds$ and $h_{1}=\min_{\ell\in L\setminus\\{1\\}}V\left(O_{1},O_{\ell}\right)$, and $R_{j}^{(1)}\doteq\inf_{x\in A}\left[2f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)-W\left(O_{1}\right)$ $R_{1}^{(2)}\doteq 2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{1},x\right)\right]-h_{1}$ and for $j\in L\setminus\\{1\\}$ $R_{j}^{(2)}\doteq 2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)-2W\left(O_{1}\right)+W(O_{1}\cup O_{j}).$ ###### Proof. Following a similar argument as for the proof of Lemma 7.21, given any $\eta>0,$ owing to Lemmas 7.18, 7.19 and 7.20, there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ and for any $j\in L$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)\geq\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]-\frac{\eta}{4},$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)^{2}\right)\geq\inf_{x\in A}\left[2f\left(x\right)+V\left(O_{j},x\right)\right]-\frac{\eta}{4},$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\right)\geq- h_{1}+W\left(O_{j}\right)-W\left(O_{1}\right)-\frac{\eta}{4},$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sum_{\ell=1}^{\infty}\sup_{z\in\partial B_{\delta}(O_{1})}P_{z}\left(\ell\leq N_{1}\right)\right)\geq- h_{1}-\frac{\eta}{4},$ and for any $j\in L\setminus\\{1\\}$, $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left({\textstyle\sum_{\ell=1}^{\infty}}\sup_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\ell\leq N_{j}\right)\right)\geq W(O_{1}\cup O_{j})-W\left(O_{1}\right)-\frac{\eta}{4}.$ Hence for any $\delta\in(0,\delta_{0})$ we apply Corollary 7.11 to get $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}\left(S_{1}^{\varepsilon}\right)^{2}\right)\geq\min_{j\in L}\left(\hat{R}_{j}^{(1)}\wedge\hat{R}_{j}^{(2)}\right),$ where $\displaystyle\hat{R}_{j}^{(1)}$ $\displaystyle\doteq\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)^{2}\right)+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\right)$ $\displaystyle\geq\inf_{x\in A}\left[2f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)-W\left(O_{1}\right)-h_{1}-\eta=R_{j}^{(1)}-h_{1}-\eta$ and $\displaystyle\hat{R}_{1}^{(2)}$ $\displaystyle\doteq 2\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)$ $\displaystyle\quad+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{1}\right)+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left({\textstyle\sum_{\ell=1}^{\infty}}\sup_{z\in\partial B_{\delta}(O_{1})}P_{z}\left(\ell\leq N_{1}\right)\right)$ $\displaystyle\geq 2\left(\inf_{x\in A}\left[f\left(x\right)+V\left(O_{1},x\right)\right]-\frac{\eta}{4}\right)+\left(-h_{1}-\frac{\eta}{4}\right)+\left(-h_{1}-\frac{\eta}{4}\right)$ $\displaystyle=2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{1},x\right)\right]-2h_{1}-\eta=R_{1}^{(2)}-h_{1}-\eta$ and for $j\in L\setminus\\{1\\}$ $\displaystyle\hat{R}_{j}^{(2)}$ $\displaystyle\doteq 2\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{j})}E_{z}I^{\varepsilon}(\tau_{1};f,A)\right)$ $\displaystyle\quad+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}N_{j}\right)+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left({\textstyle\sum_{\ell=1}^{\infty}}\sup_{z\in\partial B_{\delta}(O_{j})}P_{z}\left(\ell\leq N_{j}\right)\right)$ $\displaystyle\geq 2\left(\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]-\frac{\eta}{4}\right)+\left(-h_{1}+W\left(O_{j}\right)-W\left(O_{1}\right)-\frac{\eta}{4}\right)$ $\displaystyle\quad+\left(W(O_{1}\cup O_{j})-W\left(O_{1}\right)-\frac{\eta}{4}\right)$ $\displaystyle=2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)-2W\left(O_{1}\right)+W(O_{1}\cup O_{j})-h_{1}-\eta$ $\displaystyle=R_{j}^{(2)}-h_{1}-\eta.$ ∎ ### 7.3 Asymptotics of moments of $\hat{S}_{1}^{\varepsilon}$ Recall that $\hat{S}_{n}^{\varepsilon}\doteq\int_{\hat{\tau}_{n-1}^{\varepsilon}}^{\hat{\tau}_{n}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt,$ where $\hat{\tau}_{i}^{\varepsilon}$ is a multicycle defined according to (6.4) and with $\\{\mathbf{M}^{\varepsilon}_{i}\\}_{i\in\mathbb{N}}$ being a sequence of independent and geometrically distributed random variables with parameter $e^{-m/\varepsilon}$ for some $m>0$ such that $m+h_{1}>w$. Moreover, $\\{\mathbf{M}^{\varepsilon}_{i}\\}$ is also independent of $\\{\tau^{\varepsilon}_{n}\\}$. Using the independence of $\\{\mathbf{M}^{\varepsilon}_{i}\\}$ and $\\{\tau^{\varepsilon}_{n}\\}$, and the fact that $\\{\tau^{\varepsilon}_{n}\\}$ and $\\{S_{n}^{\varepsilon}\\}$ are both iid under $P_{\lambda^{\varepsilon}}$, we find that $\\{\hat{S}_{n}^{\varepsilon}\\}$ is also iid under $P_{\lambda^{\varepsilon}}$ and $\displaystyle E_{\lambda^{\varepsilon}}\hat{S}_{1}^{\varepsilon}=E_{\lambda^{\varepsilon}}\mathbf{M}^{\varepsilon}_{1}\cdot E_{\lambda^{\varepsilon}}S^{\varepsilon}_{1}$ (7.11) and $\displaystyle\mathrm{Var}_{\lambda^{\varepsilon}}\hat{S}_{1}^{\varepsilon}$ $\displaystyle=E_{\lambda^{\varepsilon}}\mathbf{M}^{\varepsilon}_{1}\cdot\mathrm{Var}_{\lambda^{\varepsilon}}(S^{\varepsilon}_{1})+\mathrm{Var}_{\lambda^{\varepsilon}}(\mathbf{M}^{\varepsilon}_{1})\cdot(E_{\lambda^{\varepsilon}}S^{\varepsilon}_{1})^{2}$ $\displaystyle\leq E_{\lambda^{\varepsilon}}\mathbf{M}^{\varepsilon}_{1}\cdot E_{\lambda^{\varepsilon}}(S^{\varepsilon}_{1})^{2}+\mathrm{Var}_{\lambda^{\varepsilon}}(\mathbf{M}^{\varepsilon}_{1})\cdot(E_{\lambda^{\varepsilon}}S^{\varepsilon}_{1})^{2}$ (7.12) On the other hand, since $\mathbf{M}^{\varepsilon}_{1}$ is geometrically distributed with parameter $e^{-m/\varepsilon}$, this gives that $\displaystyle E_{\lambda^{\varepsilon}}\mathbf{M}^{\varepsilon}_{1}=e^{\frac{m}{\varepsilon}}\text{ and }\mathrm{Var}_{\lambda^{\varepsilon}}(\mathbf{M}^{\varepsilon}_{1})=e^{\frac{2m}{\varepsilon}}(1-e^{\frac{-m}{\varepsilon}}).$ (7.13) Therefore, by combining (7.11), (7.3) and (7.13) with Lemma 7.21 and Lemma 7.23, we have the following two lemmas. ###### Lemma 7.24. Given a compact set $A\subset M,$ a continuous function $f:M\rightarrow\mathbb{R}$ and $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log E_{\lambda^{\varepsilon}}\hat{S}_{1}^{\varepsilon}$ $\displaystyle\qquad\geq\min_{j\in L}\left\\{\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)\right\\}-W\left(O_{1}\right)-(m+h_{1})-\eta.$ ###### Lemma 7.25. Given a compact set $A\subset M,$ a continuous function $f:M\rightarrow\mathbb{R}$ and $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\mathrm{Var}_{\lambda^{\varepsilon}}(\hat{S}_{1}^{\varepsilon})\geq\min_{j\in L}\left(R_{j}^{(1)}\wedge R_{j}^{(2)}\wedge R_{j}^{(3,m)}\right)-(m+h_{1})-\eta,$ where $R_{j}^{(1)}$ and $R_{j}^{(2)}$ are defined as in Lemma 7.23, and $R_{j}^{(3,m)}\doteq 2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+2W\left(O_{j}\right)-2W\left(O_{1}\right)-(m+h_{1}).$ Later on we will optimize on $m$ to obtain the largest bound from below. This will require that we consider first $m>w-h_{1}$, so that as shown in the next section $N^{\varepsilon}(T^{\varepsilon})$ can be suitably approximated in terms of a Poisson distribution, and then sending $m\downarrow w-h_{1}$. ## 8 Asymptotics of Moments of $N^{\varepsilon}(T^{\varepsilon})$ and $\hat{N}^{\varepsilon}(T^{\varepsilon})$ Recall that the number of single cycles in the time interval $[0,T^{\varepsilon}]$ plus one is defined as $N^{\varepsilon}\left(T^{\varepsilon}\right)\doteq\inf\left\\{n\in\mathbb{N}:\tau_{n}^{\varepsilon}>T^{\varepsilon}\right\\},$ where the $\tau_{n}^{\varepsilon}$ are the return times to $B_{\delta}(O_{1})$ after ever visiting one of the $\delta$-neighborhood of other equilibrium points than $O_{1}.$ In addition, $\lambda^{\varepsilon}$ is the unique invariant measure of $\\{Z_{n}^{\varepsilon}\\}_{n}=\\{X_{\tau_{n}^{\varepsilon}}^{\varepsilon}\\}_{n}.$ The number of multicycles in the time interval $[0,T^{\varepsilon}]$ plus one is defined as $\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)\doteq\inf\left\\{n\in\mathbb{N}:\hat{\tau}_{n}^{\varepsilon}>T^{\varepsilon}\right\\},$ where $\hat{\tau}^{\varepsilon}_{i}$ are defined as in (6.4). In this section, we will find the logarithmic asymptotics of the expected value and the variance of $N^{\varepsilon}\left(T^{\varepsilon}\right)$ with $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>h_{1}$ in Lemma 8.2 and Lemma 8.4 under the assumption that $h_{1}>w$ (i.e., single cycle case), and the analogous quantities for $\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)$ with $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>w$ in Lemma 8.19 and Lemma 8.21 under the assumption that $w\geq h_{1}$ (i.e., multicycle case). ###### Remark 8.1. While the proofs of these asymptotic results are quite detailed, it is essential that we obtain estimates good enough for a relatively precise comparison of the expected value and the variance of $N^{\varepsilon}\left(T^{\varepsilon}\right)$, and likewise for $\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)$. For this, the key result needed is the characterization of $N^{\varepsilon}\left(T^{\varepsilon}\right)$ (and $\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)$) as having an approximately Poisson distribution. These follow by exploiting the asymptotically exponential character of $\tau_{n}^{\varepsilon}$ (and $\hat{\tau}_{n}^{\varepsilon}$), together with some uniform integrability properties. Lemmas 8.2 and 8.4 below are proved in Section 8.3. ###### Lemma 8.2. If $h_{1}>w$ and $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>h_{1}$, then there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left|\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}-\frac{1}{E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\right|\geq c.$ ###### Corollary 8.3. If $h_{1}>w$ and $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>h_{1}$, then there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}\geq\varkappa_{\delta},$ where $\varkappa_{\delta}\doteq\min_{y\in\cup_{k\in L\setminus\\{1\\}}\partial B_{\delta}(O_{k})}V(O_{1},y)$. ###### Lemma 8.4. If $h_{1}>w$ and $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>h_{1}$, then for any $\eta>0,$ there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{\mathrm{Var}_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}\geq h_{1}-\eta.$ Before proceeding, we mention a result from [11] and define some notation which will be used in this section. Results in Section 5 and Section 10 of [11, Chapter XI] say that for any $t>0,$ the first and second moment of $N^{\varepsilon}\left(t\right)$ can be represented as $E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(t\right)\right)=\sum\nolimits_{n=0}^{\infty}P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq t\right)\text{ and }E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(t\right)\right)^{2}=\sum\nolimits_{n=0}^{\infty}\left(2n+1\right)P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq t\right).$ (8.1) Let $\Gamma^{\varepsilon}\doteq T^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}$ and $\gamma^{\varepsilon}\doteq\left(\Gamma^{\varepsilon}\right)^{-\ell}$ with some $\ell\in(0,1)$ which will be chosen later. Intuitively, $\Gamma^{\varepsilon}$ is the typical number of regenerative cycles in $[0,T^{\varepsilon}]$ since $E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}$ is the expected length of one regenerative cycle. To simplify notation, we pretend that $\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}$ and $\left(1-2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}$ are positive integers so that we can divide $E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)$ into three partial sums which are $\mathfrak{P}_{1}\doteq\sum\nolimits_{n=\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}+1}^{\infty}P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right),\text{ }\mathfrak{P}_{2}\doteq\sum\nolimits_{n=\left(1-2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}}^{\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}}\text{ }P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)\text{ }$ and $\mathfrak{P}_{3}\doteq\sum\nolimits_{n=0}^{\left(1-2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}-1}P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right).$ (8.2) Similarly, we divide $E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)^{2}$ into $\mathfrak{R}_{1}\doteq\sum_{n=\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}+1}^{\infty}\left(2n+1\right)P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right),\text{ }\mathfrak{R}_{2}\doteq\sum_{n=\left(1-2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}}^{\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}}\text{ }\left(2n+1\right)P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)\text{ }$ and $\mathfrak{R}_{3}\doteq\sum\nolimits_{n=0}^{\left(1-2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}-1}\left(2n+1\right)P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right).$ (8.3) The next step is to find upper bounds for these partial sums, and these bounds will help us to find suitable lower bounds for the logarithmic asymptotics of $E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)$ and Var${}_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)$. Before looking into the upper bound for partial sums, we establish some properties. ###### Theorem 8.5. If $h_{1}>w$, then for any $\delta>0$ sufficiently small, $\lim_{\varepsilon\rightarrow 0}\varepsilon\log E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}=\varkappa_{\delta}\text{ and }\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\overset{d}{\rightarrow}\rm{Exp}(1).$ Moreover, there exists $\varepsilon_{0}\in(0,1)$ and a constant $\tilde{c}>0$ such that $P_{\lambda^{\varepsilon}}\left(\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}>t\right)\leq e^{-\tilde{c}t}$ for any $t>0$ and any $\varepsilon\in(0,\varepsilon_{0}).$ ###### Remark 8.6. For any $\delta>0,$ $\varkappa_{\delta}\leq h_{1}.$ The proof of Theorem 8.5 will be given in Section 10. In that section, we will first prove an analogous result for the exit time (or first visiting time to other equilibrium points to be more precise), and then show how one can extend those results to the return time. The proof of the following lemma is straightforward and hence omitted. ###### Lemma 8.7. If $h_{1}>w$ and $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>h_{1}$, then for any $\eta>0,$ there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$, $h_{1}-\eta\geq\lim_{\varepsilon\rightarrow 0}-\varepsilon\log\Gamma^{\varepsilon}\geq h_{1}-c-\eta.$ ###### Lemma 8.8. Define $\mathcal{Z}_{1}^{\varepsilon}\doteq\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}.$ Then for any $\delta>0$ sufficiently small, * • there exists some $\varepsilon_{0}\in(0,1)$ such that $\sup_{\varepsilon\in(0,\varepsilon_{0})}E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{3}<\infty,$ * • there exists some $\varepsilon_{0}\in(0,1)$ such that $\inf_{\varepsilon\in(0,\varepsilon_{0})}\mathrm{Var}_{\lambda^{\varepsilon}}(\mathcal{Z}_{1}^{\varepsilon})>0$ and $E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{2}=E_{\lambda^{\varepsilon}}\left(\tau_{1}^{\varepsilon}\right)^{2}/\left(E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)^{2}\rightarrow 2$ as $\varepsilon\rightarrow 0.$ ###### Proof. For the first part, we use Theorem 8.5 to find that there exists $\varepsilon_{0}\in(0,1)$ and a constant $\tilde{c}>0$ such that $P_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}>t\right)=P_{\lambda^{\varepsilon}}\left(\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}>t\right)\leq e^{-\tilde{c}t}$ for any $t>0$ and any $\varepsilon\in(0,\varepsilon_{0}).$ Therefore, for $\varepsilon\in(0,\varepsilon_{0})$ $E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{3}=3\int_{0}^{\infty}t^{2}P_{\lambda^{\varepsilon}}(\mathcal{Z}_{1}^{\varepsilon}>t)dt\leq 3\int_{0}^{\infty}t^{2}e^{-\tilde{c}t}dt<\infty.$ For the second assertion, since $\sup_{0<\varepsilon<\varepsilon_{0}}E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{3}<\infty,$ it implies that $\\{\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{2}\\}_{0<\varepsilon<\varepsilon_{0}}$ and $\\{\mathcal{Z}_{1}^{\varepsilon}\\}_{0<\varepsilon<\varepsilon_{0}}$ are both uniformly integrable. Moreover, because $\mathcal{Z}_{1}^{\varepsilon}\overset{d}{\rightarrow}$ Exp$(1)$ as $\varepsilon\rightarrow 0$ from Theorem 8.5 and since for $X\overset{d}{=}$ Exp$(1),$ $EX=1$ and $EX^{2}=2,$ we obtain $E_{\lambda^{\varepsilon}}\left(\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)^{2}=E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{2}\rightarrow 2\text{ and }E_{\lambda^{\varepsilon}}\mathcal{Z}_{1}^{\varepsilon}\rightarrow 1.$ as $\varepsilon\rightarrow 0.$ This implies Var${}_{\lambda^{\varepsilon}}(\mathcal{Z}_{1}^{\varepsilon})\rightarrow 1$ as $\varepsilon\rightarrow 0.$ Obviously, there exists some $\varepsilon_{0}\in(0,1)$ such that $\inf\nolimits_{\varepsilon\in(0,\varepsilon_{0})}\mathrm{Var}_{\lambda^{\varepsilon}}(\mathcal{Z}_{1}^{\varepsilon})\geq 1/2>0.$ This completes the proof. ∎ ###### Remark 8.9. Throughout the rest of this section, we will use $C$ to denote a constant in $(0,\infty)$ which is independent of $\varepsilon$ but whose value may change from use to use. ### 8.1 Chernoff bound In this subsection we will provide upper bounds for $\mathfrak{P}_{1}\doteq\sum_{n=\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}+1}^{\infty}P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)\text{ and }\mathfrak{R}_{1}\doteq\sum_{n=\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}+1}^{\infty}\left(2n+1\right)P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)$ via a Chernoff bound. The following result is well known and its proof is standard. ###### Lemma 8.10 (Chernoff bound). Let $X_{1},\ldots,X_{n}$ be an iid sequence of random variables. For any $a\in\mathbb{R}$ and for any $t\in(0,\infty)$ $P\left(X_{1}+\cdots+X_{n}\leq a\right)\leq\left(Ee^{-tX_{1}}\right)^{n}e^{ta}.$ Recall that $\Gamma^{\varepsilon}\doteq T^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}$ and $\gamma^{\varepsilon}\doteq\left(\Gamma^{\varepsilon}\right)^{-\ell}$ with some $\ell\in(0,1)$ which will be chosen later. ###### Lemma 8.11. Given any $\delta>0$ and any $\ell>0,$ there exists $\varepsilon_{0}\in(0,1)$ such that for any $\varepsilon\in(0,\varepsilon_{0})$ $P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)\leq e^{-n\left(\Gamma^{\varepsilon}\right)^{-2\ell}}$ for any $n\geq\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}.$ In addition, $\mathfrak{P}_{1}\leq C\left(\Gamma^{\varepsilon}\right)^{2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}\text{ and }\mathfrak{R}_{1}\leq C\left(\Gamma^{\varepsilon}\right)^{1+2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}+C\left(\Gamma^{\varepsilon}\right)^{4\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}.$ ###### Proof. Given $\delta>0$, $\ell>0$ and $\varepsilon\in(0,1),$ we find that for $n\geq\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}$ $\displaystyle P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)$ $\displaystyle=P_{\lambda^{\varepsilon}}\left(\frac{\tau_{1}^{\varepsilon}+\left(\tau_{2}^{\varepsilon}-\tau_{1}^{\varepsilon}\right)+\cdots+\left(\tau_{n}^{\varepsilon}-\tau_{n-1}^{\varepsilon}\right)}{E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\leq\Gamma^{\varepsilon}\right)$ $\displaystyle\leq P_{\lambda^{\varepsilon}}\left(\frac{\tau_{1}^{\varepsilon}+\left(\tau_{2}^{\varepsilon}-\tau_{1}^{\varepsilon}\right)+\cdots+\left(\tau_{n}^{\varepsilon}-\tau_{n-1}^{\varepsilon}\right)}{E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\leq\frac{n}{1+2\gamma^{\varepsilon}}\right)$ $\displaystyle\leq\left(E_{\lambda^{\varepsilon}}e^{-\gamma^{\varepsilon}\mathcal{Z}_{1}^{\varepsilon}}\right)e^{\frac{n\gamma^{\varepsilon}}{1+2\gamma^{\varepsilon}}},$ where $\mathcal{Z}_{1}^{\varepsilon}=\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}.$ We use the fact that $\\{\tau_{n}^{\varepsilon}-\tau_{n-1}^{\varepsilon}\\}_{n\in\mathbb{N}}$ are iid and apply Lemma 8.10 (Chernoff bound) with $a=n/\left(1+2\gamma^{\varepsilon}\right)$ and $t=\gamma^{\varepsilon}$ for the last inequality. Therefore, in order to verify the first claim, it suffices to show that $\left(E_{\lambda^{\varepsilon}}e^{-\gamma^{\varepsilon}\mathcal{Z}_{1}^{\varepsilon}}\right)e^{\frac{\gamma^{\varepsilon}}{1+2\gamma^{\varepsilon}}}\leq e^{-\left(\gamma^{\varepsilon}\right)^{2}}=e^{-\left(\Gamma^{\varepsilon}\right)^{-2\ell}}.$ We observe that for any $x\geq 0,$ $e^{-x}\leq 1-x+x^{2}/2,$ and this gives $\displaystyle E_{\lambda^{\varepsilon}}e^{-\gamma^{\varepsilon}\mathcal{Z}_{1}^{\varepsilon}}\leq 1-E_{\lambda^{\varepsilon}}\left(\gamma^{\varepsilon}\mathcal{Z}_{1}^{\varepsilon}\right)+E_{\lambda^{\varepsilon}}\left(\gamma^{\varepsilon}\mathcal{Z}_{1}^{\varepsilon}\right)^{2}/2=1-\gamma^{\varepsilon}+\left(\gamma^{\varepsilon}\right)^{2}E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{2}/2.$ Moreover, since we can apply Lemma 8.8 to find $E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{2}\rightarrow 2$ as $\varepsilon\rightarrow 0,$ there exists $\varepsilon_{0}\in(0,1)$ such that for any $\varepsilon\in(0,\varepsilon_{0})$, $E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{2}\leq 9/4.$ Thus, for any $\varepsilon\in(0,\varepsilon_{0})$ $\left(E_{\lambda^{\varepsilon}}e^{-\gamma^{\varepsilon}\mathcal{Z}_{1}^{\varepsilon}}\right)e^{\frac{\gamma^{\varepsilon}}{1+2\gamma^{\varepsilon}}}\leq\exp\left\\{\gamma^{\varepsilon}/(1+2\gamma^{\varepsilon})+\log(1-\gamma^{\varepsilon}+(9/8)\left(\gamma^{\varepsilon}\right)^{2})\right\\}.$ Using a Taylor series expansion we find that for all $\left|x\right|<1$ $1/(1+x)=1-x+O\left(x^{2}\right)\text{ and }\log\left(1+x\right)=x-x^{2}/2+O\left(x^{3}\right),$ which gives $\displaystyle\gamma^{\varepsilon}/(1+2\gamma^{\varepsilon})+\log(1-\gamma^{\varepsilon}+(9/8)\left(\gamma^{\varepsilon}\right)^{2})$ $\displaystyle\quad=\gamma^{\varepsilon}-2\left(\gamma^{\varepsilon}\right)^{2}+[(-\gamma^{\varepsilon}+(9/8)\left(\gamma^{\varepsilon}\right)^{2}]-[-\gamma^{\varepsilon}+(9/8)\left(\gamma^{\varepsilon}\right)^{2}]^{2}/2+O((\gamma^{\varepsilon})^{3})$ $\displaystyle\quad=-(11/8)\left(\gamma^{\varepsilon}\right)^{2}+O(\left(\gamma^{\varepsilon}\right)^{3})\leq-\left(\gamma^{\varepsilon}\right)^{2},$ for all $\varepsilon\in(0,\varepsilon_{0})$. We are done for part 1. For part 2, we use the estimate from part 1 and find $\sum_{n=\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}+1}^{\infty}P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)\leq\sum_{n=\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}+1}^{\infty}e^{-n\left(\gamma^{\varepsilon}\right)^{2}}\leq\frac{e^{-\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}\left(\gamma^{\varepsilon}\right)^{2}}}{1-e^{-\left(\gamma^{\varepsilon}\right)^{2}}}.$ Since $e^{-x}\leq 1-x+x^{2}/2$ for any $x\in\mathbb{R},$ we have $1-e^{-x}\geq x-x^{2}/2\geq x-x/2=x/2$ for all $x\in(0,1),$ and thus $1/(1-e^{-x})\leq 2/x$ for all $x\in(0,1).$ As a result $\displaystyle\sum_{n=\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}+1}^{\infty}P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)\leq\frac{e^{-\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}\left(\gamma^{\varepsilon}\right)^{2}}}{1-e^{-\left(\gamma^{\varepsilon}\right)^{2}}}\leq\frac{2}{\left(\gamma^{\varepsilon}\right)^{2}}e^{-\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}\left(\gamma^{\varepsilon}\right)^{2}}\leq 2\left(\Gamma^{\varepsilon}\right)^{2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}.$ This completes the proof of part 2. Finally, for part 3, we use the fact that for $x\in(0,1),$ and for any $k\in\mathbb{N},$ $\sum\nolimits_{n=k}^{\infty}nx^{n}=kx^{k}(1-x)^{-1}+x^{k+1}(1-x)^{-2}\leq(k(1-x)^{-1}+(1-x)^{-2})x^{k}.$ Using the estimate from part 1 once again, we have $\displaystyle\sum_{n=\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}+1}^{\infty}nP_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)$ $\displaystyle\leq\sum_{n=\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}}^{\infty}ne^{-n\left(\gamma^{\varepsilon}\right)^{2}}$ $\displaystyle\leq\left(\frac{\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}}{1-e^{-\left(\gamma^{\varepsilon}\right)^{2}}}+\left(1-e^{-\left(\gamma^{\varepsilon}\right)^{2}}\right)^{-2}\right)e^{-\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}\left(\gamma^{\varepsilon}\right)^{2}}$ $\displaystyle\leq\left(4\left(\Gamma^{\varepsilon}\right)^{1+2\ell}+4\left(\Gamma^{\varepsilon}\right)^{4\ell}\right)e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}.$ We are done. ∎ ###### Remark 8.12. If $0<\ell<1/2,$ then $\mathfrak{P}_{1}$ and $\mathfrak{R}_{1}$ converge to $0$ doubly exponentially fast as $\varepsilon\rightarrow 0$ in the sense that for any $k\in(0,\infty)$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[\left(\Gamma^{\varepsilon}\right)^{k}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}\right]=\infty.$ ### 8.2 Berry-Esseen bound In this subsection we will provide upper bounds for $\mathfrak{P}_{2}\doteq\sum_{n=\left(1-2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}}^{\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}}P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)\text{ and }\mathfrak{R}_{2}\doteq\sum_{n=\left(1-2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}}^{\left(1+2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}}\left(2n+1\right)P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)$ via the Berry-Esseen bound. We first recall that $\Gamma^{\varepsilon}=T^{\varepsilon}/E_{\lambda^{\varepsilon}}^{\varepsilon}\tau_{1}^{\varepsilon}$. The following is Theorem 1 in [11, Chapter XVI.5]. ###### Theorem 8.13 (Berry-Esseen). Let $\left\\{X_{n}\right\\}_{n\in\mathbb{N}}$ be independent real-valued random variables with a common distribution such that $E\left(X_{1}\right)=0,\text{ }\sigma^{2}\doteq E\left(X_{1}\right)^{2}>0,\text{ }\rho\doteq E\left|X_{1}\right|^{3}<\infty.$ Then for all $x\in\mathbb{R}$ and $n\in\mathbb{N},$ $\left|P\left(\frac{X_{1}+\cdots+X_{n}}{\sigma\sqrt{n}}\leq x\right)-\Phi\left(x\right)\right|\leq\frac{3\rho}{\sigma^{3}\sqrt{n}},$ where $\Phi\left(\cdot\right)$ is the distribution function of $N\left(0,1\right).$ ###### Corollary 8.14. For any $\varepsilon>0,$ let $\left\\{X_{n}^{\varepsilon}\right\\}_{n\in\mathbb{N}}$ be independent real- valued random variables with a common distribution such that $E\left(X_{1}^{\varepsilon}\right)=0,\text{ }\left(\sigma^{\varepsilon}\right)^{2}\doteq E\left(X_{1}^{\varepsilon}\right)^{2}>0,\text{ }\rho^{\varepsilon}\doteq E\left|X_{1}^{\varepsilon}\right|^{3}<\infty.$ Assume that there exists $\varepsilon_{0}\in(0,1)$ such that $\hat{\rho}\text{ }\doteq\sup\nolimits_{\varepsilon\in(0,\varepsilon_{0})}\rho^{\varepsilon}<\infty\text{ and }\hat{\sigma}^{2}\doteq\inf\nolimits_{\varepsilon\in(0,\varepsilon_{0})}\left(\sigma^{\varepsilon}\right)^{2}>0.$ Then for all $x\in\mathbb{R},n\in\mathbb{N}$ and $\varepsilon\in(0,\varepsilon_{0}),$ $\left|P\left(\frac{X_{1}^{\varepsilon}+\cdots+X_{n}^{\varepsilon}}{\sigma^{\varepsilon}\sqrt{n}}\leq x\right)-\Phi\left(x\right)\right|\leq\frac{3\rho^{\varepsilon}}{\left(\sigma^{\varepsilon}\right)^{3}\sqrt{n}}\leq\frac{3\hat{\rho}}{\hat{\sigma}^{3}\sqrt{n}}.$ ###### Lemma 8.15. Given any $\delta>0$ and any $\ell>0,$ there exists $\varepsilon_{0}\in(0,1)$ such that for any $\varepsilon\in(0,\varepsilon_{0})$ and $k\in\mathbb{N}_{0}$, $0\leq k\leq 2\gamma^{\varepsilon}\Gamma^{\varepsilon}$ $P_{\lambda^{\varepsilon}}\left(\tau_{\Gamma^{\varepsilon}+k}^{\varepsilon}\leq T^{\varepsilon}\right)\leq 1-\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right)+\frac{3\hat{\rho}}{\hat{\sigma}^{3}\sqrt{\Gamma^{\varepsilon}+k}}$ and $P_{\lambda^{\varepsilon}}\left(\tau_{\Gamma^{\varepsilon}-k}^{\varepsilon}\leq T^{\varepsilon}\right)\leq\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}\right)+\frac{3\hat{\rho}}{\hat{\sigma}^{3}\sqrt{\Gamma^{\varepsilon}-k}},$ where $(\sigma^{\varepsilon})^{2}\doteq E_{\lambda^{\varepsilon}}\left(\mathfrak{X}_{1}^{\varepsilon}\right)^{2},$ $\hat{\rho}\doteq\sup_{\varepsilon\in(0,\varepsilon_{0})}E_{\lambda^{\varepsilon}}\left|\mathfrak{X}_{1}^{\varepsilon}\right|^{3}<\infty$ and $\hat{\sigma}^{2}\doteq\inf_{\varepsilon\in(0,\varepsilon_{0})}(\sigma^{\varepsilon})^{2}>0$ with $\mathfrak{X}_{1}^{\varepsilon}\doteq\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}^{\varepsilon}\tau_{1}^{\varepsilon}-1.$ ###### Proof. For any $n\in\mathbb{N},$ we define $\mathfrak{X}_{n}^{\varepsilon}\doteq\mathcal{Z}_{n}^{\varepsilon}-E_{\lambda^{\varepsilon}}^{\varepsilon}\mathcal{Z}_{1}^{\varepsilon}$ with $\mathcal{Z}_{n}^{\varepsilon}\doteq(\tau_{n}^{\varepsilon}-\tau_{n-1}^{\varepsilon})/E_{\lambda^{\varepsilon}}^{\varepsilon}\tau_{1}^{\varepsilon}.$ Obviously, $E_{\lambda^{\varepsilon}}\mathcal{Z}_{n}^{\varepsilon}=1$ and $E_{\lambda^{\varepsilon}}\mathfrak{X}_{n}^{\varepsilon}=0$ and if we apply Lemma 8.8, then we find that there exists some $\varepsilon_{0}\in(0,1)$ such that $\sup\nolimits_{\varepsilon\in(0,\varepsilon_{0})}E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{3}<\infty\text{ and }\inf\nolimits_{\varepsilon\in(0,\varepsilon_{0})}\mathrm{Var}_{\lambda^{\varepsilon}}(\mathcal{Z}_{1}^{\varepsilon})>0.$ Since $\mathcal{Z}_{1}^{\varepsilon}\geq 0$, Jensen’s inequality implies $\left(E_{\lambda^{\varepsilon}}\mathcal{Z}_{1}^{\varepsilon}\right)^{3}\leq E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{3}$, and therefore $\displaystyle\hat{\rho}\leq 4\sup\nolimits_{\varepsilon\in(0,\varepsilon_{0})}\left(E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{3}+\left(E_{\lambda^{\varepsilon}}\mathcal{Z}_{1}^{\varepsilon}\right)^{3}\right)\leq 8\sup\nolimits_{\varepsilon\in(0,\varepsilon_{0})}E_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)^{3}<\infty,$ and $\hat{\sigma}^{2}=\inf\nolimits_{\varepsilon\in(0,\varepsilon_{0})}E_{\lambda^{\varepsilon}}\left(\mathfrak{X}_{1}^{\varepsilon}\right)^{2}=\inf\nolimits_{\varepsilon\in(0,\varepsilon_{0})}\mathrm{Var}_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}\right)>0.$ Therefore we can use Corollary 8.14 with the iid sequence $\left\\{\mathfrak{X}_{n}^{\varepsilon}\right\\}_{n\in\mathbb{N}}$ to find that for any $k\in\mathbb{N}_{0}$ and $0\leq k\leq 2\gamma^{\varepsilon}\Gamma^{\varepsilon}$ $\displaystyle P_{\lambda^{\varepsilon}}\left(\tau_{\Gamma^{\varepsilon}+k}^{\varepsilon}\leq T^{\varepsilon}\right)$ $\displaystyle=P_{\lambda^{\varepsilon}}\left(\mathcal{Z}_{1}^{\varepsilon}+\cdots+\mathcal{Z}_{\Gamma^{\varepsilon}+k}^{\varepsilon}\leq\Gamma^{\varepsilon}\right)$ $\displaystyle=P_{\lambda^{\varepsilon}}\left(\frac{\mathfrak{X}_{1}^{\varepsilon}+\cdots+\mathfrak{X}_{\Gamma^{\varepsilon}+k}^{\varepsilon}}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\leq\frac{-k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right)$ $\displaystyle\leq 1-\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right)+\frac{3\hat{\rho}}{\hat{\sigma}^{3}\sqrt{\Gamma^{\varepsilon}+k}},$ and similarly $\displaystyle P_{\lambda^{\varepsilon}}\left(\tau_{\Gamma^{\varepsilon}-k}^{\varepsilon}\leq T^{\varepsilon}\right)$ $\displaystyle\leq\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}\right)+\frac{3\hat{\rho}}{\hat{\sigma}^{3}\sqrt{\Gamma^{\varepsilon}-k}}.$ ∎ ###### Lemma 8.16. Given any $\delta>0$ and any $\ell\in(0,1/2),$ there exists $\varepsilon_{0}\in(0,1)$ such that for any $\varepsilon\in(0,\varepsilon_{0})$, $\mathfrak{P}_{2}\leq C\left(\Gamma^{\varepsilon}\right)^{\frac{1}{2}-\ell}+2\left(\Gamma^{\varepsilon}\right)^{1-\ell}.$ ###### Proof. We rewrite $\mathfrak{P}_{2}$ as $\displaystyle\mathfrak{P}_{2}=\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}P_{\lambda^{\varepsilon}}\left(\tau_{\Gamma^{\varepsilon}-k}^{\varepsilon}\leq T^{\varepsilon}\right)+P_{\lambda^{\varepsilon}}\left(\tau_{\Gamma^{\varepsilon}}^{\varepsilon}\leq T^{\varepsilon}\right)+\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}P_{\lambda^{\varepsilon}}\left(\tau_{\Gamma^{\varepsilon}+k}^{\varepsilon}\leq T^{\varepsilon}\right).$ Then we use the upper bounds from Lemma 8.15 to get $\displaystyle\mathfrak{P}_{2}$ $\displaystyle\leq\sum_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left[\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}\right)+\frac{3\hat{\rho}}{\hat{\sigma}^{3}\sqrt{\Gamma^{\varepsilon}-k}}\right]+1+\sum_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left[1-\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right)+\frac{3\hat{\rho}}{\hat{\sigma}^{3}\sqrt{\Gamma^{\varepsilon}+k}}\right]$ $\displaystyle\leq\frac{24\hat{\rho}}{\hat{\sigma}^{3}}\gamma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}}+1+2\gamma^{\varepsilon}\Gamma^{\varepsilon}+\sum_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left[\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}\right)-\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right)\right].$ The sum of the first three terms is easily bounded above by $C\left(\Gamma^{\varepsilon}\right)^{\frac{1}{2}-\ell}+2\left(\Gamma^{\varepsilon}\right)^{1-\ell}$. We will show that the last term is bounded above by a constant to complete the proof. To prove this, we observe that for any $k\leq 2\gamma^{\varepsilon}\Gamma^{\varepsilon},$ we may assume $k\leq\Gamma^{\varepsilon}/2$ by taking $\varepsilon$ sufficiently small. Then we apply the Mean Value Theorem and find $\displaystyle\left|\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}\right)-\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right)\right|\leq\sup\limits_{x\in\left[\frac{\sqrt{2/3}k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}}},\frac{\sqrt{2}k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}}}\right]}\phi\left(x\right)\cdot\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}-\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right),$ where $\phi\left(x\right)\doteq e^{-\frac{x^{2}}{2}}/\sqrt{2\pi}$ and since $0\leq k\leq\Gamma^{\varepsilon}/2,$ we have $\left[\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}},\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}\right]\subset\left[\frac{\sqrt{2/3}k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}}},\frac{\sqrt{2}k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}}}\right].$ Additionally, because $\phi\left(x\right)=e^{-\frac{x^{2}}{2}}/\sqrt{2\pi}$ is a monotone decreasing function on $[0,\infty)$, we find that $x\in[(\sqrt{2/3}k)(\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}}),(\sqrt{2}k)(\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}})]\quad\mbox{implies}\quad\phi\left(x\right)\leq e^{-\frac{k^{2}}{3\left(\sigma^{\varepsilon}\right)^{2}\Gamma^{\varepsilon}}}/{\sqrt{2\pi}}.$ Also, $\sqrt{1+x}-\sqrt{1-x}\leq 2x$ for all $x\in[0,1]$ and $k\leq\Gamma^{\varepsilon}/2$ and a little algebra give ${k}/{\sqrt{\Gamma^{\varepsilon}-k}}-{k}/{\sqrt{\Gamma^{\varepsilon}+k}}\leq{4k^{2}}/{\Gamma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}}}.$ Therefore we find $\displaystyle\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left[\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}\right)-\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right)\right]$ $\displaystyle\qquad\leq\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\frac{1}{\sqrt{2\pi}}e^{-\frac{k^{2}}{3\left(\sigma^{\varepsilon}\right)^{2}\Gamma^{\varepsilon}}}\frac{4k^{2}}{\sigma^{\varepsilon}\Gamma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}}}\leq\frac{4}{\sigma^{\varepsilon}\Gamma^{\varepsilon}}\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\int_{k-1}^{k}\frac{\left(1+x\right)^{2}}{\sqrt{2\pi\Gamma^{\varepsilon}}}e^{-\frac{x^{2}}{3\left(\sigma^{\varepsilon}\right)^{2}\Gamma^{\varepsilon}}}dx$ $\displaystyle\qquad\leq\frac{4}{\Gamma^{\varepsilon}}\sqrt{\frac{3}{2}}\int_{0}^{\infty}\frac{\left(1+x\right)^{2}}{\sqrt{3\pi\left(\sigma^{\varepsilon}\right)^{2}\Gamma^{\varepsilon}}}e^{-\frac{x^{2}}{3\left(\sigma^{\varepsilon}\right)^{2}\Gamma^{\varepsilon}}}dx\leq\frac{6}{\Gamma^{\varepsilon}}E\left(1+X^{+}\right)^{2},$ where $X\sim N(0,3\left(\sigma^{\varepsilon}\right)^{2}\Gamma^{\varepsilon}/2).$ Finally, since $E\left(1+X^{+}\right)^{2}\leq 2+2E\left(X^{2}\right)=2+3\left(\sigma^{\varepsilon}\right)^{2}\Gamma^{\varepsilon},$ this implies that $\displaystyle\sum_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left[\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}\right)-\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right)\right]$ $\displaystyle\leq\frac{6}{\Gamma^{\varepsilon}}\left(2+3\left(\sigma^{\varepsilon}\right)^{2}\Gamma^{\varepsilon}\right)\leq 12+18\hat{\rho}^{2/3},$ (8.4) where the last inequality is from $\displaystyle\sup\nolimits_{\varepsilon\in(0,\varepsilon_{0})}\sigma^{\varepsilon}$ $\displaystyle=\sup\nolimits_{\varepsilon\in(0,\varepsilon_{0})}\left(E_{\lambda^{\varepsilon}}(\mathfrak{X}_{1}^{\varepsilon}\right)^{2})^{1/2}\leq\sup\nolimits_{\varepsilon\in(0,\varepsilon_{0})}(E_{\lambda^{\varepsilon}}\left|\mathfrak{X}_{1}^{\varepsilon}\right|^{3})^{1/3}=\hat{\rho}^{1/3}.$ Since according to Lemma 8.15 $\hat{\rho}^{1/3}$ is finite, we are done. ∎ ###### Lemma 8.17. Given any $\delta>0$ and any $\ell\in(0,1/2),$ there exists $\varepsilon_{0}\in(0,1)$ and a constant $C<\infty$ such that for any $\varepsilon\in(0,\varepsilon_{0})$, $\mathfrak{R}_{2}\leq 4\left(\Gamma^{\varepsilon}\right)^{2-\ell}+C\left(\Gamma^{\varepsilon}\right)^{2\left(1-\ell\right)}.$ ###### Proof. The proof of this lemma is similar to the proof of Lemma 8.16. We rewrite $\mathfrak{R}_{2}$ as $\displaystyle\mathfrak{R}_{2}$ $\displaystyle=\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left(2\Gamma^{\varepsilon}-2k+1\right)P_{\lambda^{\varepsilon}}\left(\tau_{\Gamma^{\varepsilon}-k}^{\varepsilon}\leq T^{\varepsilon}\right)+\left(2\Gamma^{\varepsilon}+1\right)P_{\lambda^{\varepsilon}}\left(\tau_{\Gamma^{\varepsilon}}^{\varepsilon}\leq T^{\varepsilon}\right)$ $\displaystyle\quad+\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left(2\Gamma^{\varepsilon}+2k+1\right)P_{\lambda^{\varepsilon}}\left(\tau_{\Gamma^{\varepsilon}+k}^{\varepsilon}\leq T^{\varepsilon}\right).$ Then we use the upper bounds from Lemma 8.15 to get $\displaystyle\mathfrak{R}_{2}\leq\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left(2\Gamma^{\varepsilon}-2k+1\right)\left[\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}\right)+\frac{3\hat{\rho}}{\hat{\sigma}^{3}\sqrt{\Gamma^{\varepsilon}-k}}\right]+\left(2\Gamma^{\varepsilon}+1\right)$ $\displaystyle\quad\qquad+\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left(2\Gamma^{\varepsilon}+2k+1\right)\left[1-\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right)+\frac{3\hat{\rho}}{\hat{\sigma}^{3}\sqrt{\Gamma^{\varepsilon}+k}}\right].$ The next thing is to pair all the terms carefully and bound these pairs separately. We start with $\displaystyle\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left(2\Gamma^{\varepsilon}-2k+1\right)\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}\right)-\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left(2\Gamma^{\varepsilon}+2k+1\right)\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right)$ $\displaystyle\quad\leq\left(2\Gamma^{\varepsilon}+1\right)\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left[\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}-k}}\right)-\Phi\left(\frac{k}{\sigma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}+k}}\right)\right]\leq C\Gamma^{\varepsilon}.$ We use (8.4) for the last inequality. The second pair is $\displaystyle\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left(2\Gamma^{\varepsilon}-2k+1\right)\frac{3\hat{\rho}}{\hat{\sigma}^{3}\sqrt{\Gamma^{\varepsilon}-k}}+\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left(2\Gamma^{\varepsilon}+2k+1\right)\frac{3\hat{\rho}}{\hat{\sigma}^{3}\sqrt{\Gamma^{\varepsilon}+k}}$ $\displaystyle\quad=\frac{6\hat{\rho}}{\hat{\sigma}^{3}}\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left(\sqrt{\Gamma^{\varepsilon}-k}+\sqrt{\Gamma^{\varepsilon}+k}\right)+\frac{3\hat{\rho}}{\hat{\sigma}^{3}}\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left(\frac{1}{\sqrt{\Gamma^{\varepsilon}-k}}+\frac{1}{\sqrt{\Gamma^{\varepsilon}+k}}\right)$ $\displaystyle\quad\leq\frac{6\hat{\rho}}{\hat{\sigma}^{3}}\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}2\sqrt{2\Gamma^{\varepsilon}}+\frac{3\hat{\rho}}{\hat{\sigma}^{3}}\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}2\leq C\gamma^{\varepsilon}\Gamma^{\varepsilon}\sqrt{\Gamma^{\varepsilon}}+C\gamma^{\varepsilon}\Gamma^{\varepsilon}\leq C\left(\Gamma^{\varepsilon}\right)^{\frac{3}{2}-\ell},$ where the first inequality holds due to $k\leq\Gamma^{\varepsilon}/2$. The third term is $\displaystyle\sum\nolimits_{k=1}^{2\gamma^{\varepsilon}\Gamma^{\varepsilon}}\left(2\Gamma^{\varepsilon}+2k+1\right)+\left(2\Gamma^{\varepsilon}+1\right)$ $\displaystyle=4\gamma^{\varepsilon}\left(\Gamma^{\varepsilon}\right)^{2}+2\gamma^{\varepsilon}\Gamma^{\varepsilon}+4\left(\gamma^{\varepsilon}\Gamma^{\varepsilon}\right)^{2}+2\gamma^{\varepsilon}\Gamma^{\varepsilon}+\left(2\Gamma^{\varepsilon}+1\right)$ $\displaystyle\leq 4\gamma^{\varepsilon}\left(\Gamma^{\varepsilon}\right)^{2}+C\left(\gamma^{\varepsilon}\Gamma^{\varepsilon}\right)^{2}=4\left(\Gamma^{\varepsilon}\right)^{2-\ell}+C\left(\Gamma^{\varepsilon}\right)^{2\left(1-\ell\right)},$ where the inequality holds since for $\ell\in(0,1/2),$ $2-2\ell\geq 1$ and this implies that $\left(2\Gamma^{\varepsilon}+1\right)\leq C\left(\gamma^{\varepsilon}\Gamma^{\varepsilon}\right)^{2}.$ Lastly, combining all the pairs and the corresponding upper bounds, we find that for any $\ell\in(0,1/2),$ $\displaystyle\mathfrak{R}_{2}$ $\displaystyle\leq[4\left(\Gamma^{\varepsilon}\right)^{2-\ell}+C\left(\Gamma^{\varepsilon}\right)^{2\left(1-\ell\right)}]+C\Gamma^{\varepsilon}+C\left(\Gamma^{\varepsilon}\right)^{\frac{3}{2}-\ell}\leq 4\left(\Gamma^{\varepsilon}\right)^{2-\ell}+C\left(\Gamma^{\varepsilon}\right)^{2\left(1-\ell\right)},$ where $C$ is a constant which depends on $\ell$ only (and in particular is independent of $\varepsilon$). ∎ ### 8.3 Asymptotics of moments of $N^{\varepsilon}(T^{\varepsilon})$ In this subsection, we prove Lemma 8.2 and Lemma 8.4. ###### Proof of Lemma 8.2. First, recall that $E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)=\sum\nolimits_{n=0}^{\infty}P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq T^{\varepsilon}\right)=\mathfrak{P}_{1}+\mathfrak{P}_{2}+\mathfrak{P}_{3},$ where the $\mathfrak{P}_{i}$ are defined in (8.2). We can simply bound $\mathfrak{P}_{3}$ from above by $\left(1-2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}$. Applying Lemma 8.11 and Lemma 8.16 for the other terms, we have for any $\ell\in(0,1/2)$ that $\displaystyle E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)$ $\displaystyle\leq C\left(\Gamma^{\varepsilon}\right)^{2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}+(C\left(\Gamma^{\varepsilon}\right)^{\frac{1}{2}-\ell}+2\left(\Gamma^{\varepsilon}\right)^{1-\ell})+\left(1-2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}$ $\displaystyle=T^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}+C\left(\Gamma^{\varepsilon}\right)^{\frac{1}{2}-\ell}+C\left(\Gamma^{\varepsilon}\right)^{2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}.$ On the other hand, from the definition of $N^{\varepsilon}\left(T^{\varepsilon}\right),$ $E_{\lambda^{\varepsilon}}\tau_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}\geq T^{\varepsilon}.$ Using Wald’s first identity, we find $E_{\lambda^{\varepsilon}}\tau_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}=E_{\lambda^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}\left(\tau_{n}^{\varepsilon}-\tau_{n-1}^{\varepsilon}\right)=E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)\cdot E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}.$ Hence $0\leq\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}-\frac{1}{E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\leq\frac{1}{T^{\varepsilon}}[C\left(\Gamma^{\varepsilon}\right)^{\frac{1}{2}-\ell}+C\left(\Gamma^{\varepsilon}\right)^{2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}].$ Therefore, $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left|\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}-\frac{1}{E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\right|\geq\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[\frac{1}{T^{\varepsilon}}\left(C\left(\Gamma^{\varepsilon}\right)^{\frac{1}{2}-\ell}+\left(\Gamma^{\varepsilon}\right)^{2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}\right)\right].$ It remains to find an appropriate lower bound for the liminf. We use (7.2), Lemma 8.7 and Remark 8.12 to find that for any $\eta>0,$ there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ and any $\ell\in(0,1/2)$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[\frac{1}{T^{\varepsilon}}\left(C\left(\Gamma^{\varepsilon}\right)^{\frac{1}{2}-\ell}+\left(\Gamma^{\varepsilon}\right)^{2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}\right)\right]$ $\displaystyle\quad\geq\liminf_{\varepsilon\rightarrow 0}\varepsilon\log T^{\varepsilon}+\min\left\\{\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\Gamma^{\varepsilon}\right)^{1/2-\ell},\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\left(\Gamma^{\varepsilon}\right)^{2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}\right)\right\\}$ $\displaystyle\quad\geq c+\min\left\\{\left(1/2-\ell\right)\left(h_{1}-c-\eta\right),\infty\right\\}=c+\left(1/2-\ell\right)\left(h_{1}-c-\eta\right).$ We complete the proof by sending $\ell$ to $1/2$. ∎ ###### Proof of Lemma 8.4. Recall that $E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)^{2}=\sum\nolimits_{n=0}^{\infty}\left(2n+1\right)P_{\lambda^{\varepsilon}}\left(\tau_{n}^{\varepsilon}\leq t\right)=\mathfrak{R}_{1}+\mathfrak{R}_{2}+\mathfrak{R}_{3}$ where the $\mathfrak{R}_{i}$ are defined in (8.3). We can bound $\mathfrak{R}_{3}$ from above by $\displaystyle\sum\nolimits_{n=0}^{\left(1-2\gamma^{\varepsilon}\right)\Gamma^{\varepsilon}-1}\left(2n+1\right)=(1-4\gamma^{\varepsilon}+4\left(\gamma^{\varepsilon}\right)^{2})\left(\Gamma^{\varepsilon}\right)^{2}.$ Applying Lemma 8.11 and Lemma 8.17, we have for any $\ell\in(0,1/2)$ that $\displaystyle E_{\lambda^{\varepsilon}}(N^{\varepsilon}\left(T^{\varepsilon}\right))^{2}$ $\displaystyle\leq C\left(\Gamma^{\varepsilon}\right)^{1+2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}+4\left(\Gamma^{\varepsilon}\right)^{2-\ell}+C\left(\Gamma^{\varepsilon}\right)^{2\left(1-\ell\right)}+(1-4\gamma^{\varepsilon}+4\left(\gamma^{\varepsilon}\right)^{2})\left(\Gamma^{\varepsilon}\right)^{2}$ $\displaystyle\leq\left(\Gamma^{\varepsilon}\right)^{2}+C\left(\Gamma^{\varepsilon}\right)^{2\left(1-\ell\right)}+C\left(\Gamma^{\varepsilon}\right)^{1+2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}.$ As in the proof of Lemma 8.2 $E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)\geq\Gamma^{\varepsilon}.$ Thus for any $\ell\in(0,1/2)$ $\displaystyle\mathrm{Var}_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)$ $\displaystyle\leq E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)^{2}-\left(\Gamma^{\varepsilon}\right)^{2}$ $\displaystyle\leq[\left(\Gamma^{\varepsilon}\right)^{2}+C\left(\Gamma^{\varepsilon}\right)^{2\left(1-\ell\right)}+C\left(\Gamma^{\varepsilon}\right)^{1+2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}]-\left(\Gamma^{\varepsilon}\right)^{2}$ $\displaystyle=C\left(\Gamma^{\varepsilon}\right)^{2\left(1-\ell\right)}+C\left(\Gamma^{\varepsilon}\right)^{1+2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}.$ Again we use (7.2), Lemma 8.7 and Remark 8.12 to find that for any $\eta>0,$ there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ and for any $\ell\in(0,1/2)$, $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{\mathrm{Var}_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}$ $\displaystyle\quad\geq\liminf_{\varepsilon\rightarrow 0}\varepsilon\log T^{\varepsilon}+\min\left\\{\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\Gamma^{\varepsilon}\right)^{2\left(1-\ell\right)},\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\left(\Gamma^{\varepsilon}\right)^{1+2\ell}e^{-\left(\Gamma^{\varepsilon}\right)^{1-2\ell}}\right)\right\\}$ $\displaystyle\quad\geq c+\min\left\\{2\left(1-\ell\right)\left(h_{1}-c-\eta\right),\infty\right\\}=2\left(1-\ell\right)(h_{1}-\eta)+\left(2\ell-1\right)c.$ We complete the proof by sending $\ell$ to $1/2$. ∎ ### 8.4 Asymptotics of moments of $\hat{N}^{\varepsilon}(T^{\varepsilon})$ The proof of the following result is given in Section 10. ###### Theorem 8.18. If $w\geq h_{1}$, then given any $m>0$ such that $m+h_{1}>w$ and for any $\delta>0$ sufficiently small, $\lim_{\varepsilon\rightarrow 0}\varepsilon\log E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}=m+\varkappa_{\delta}\text{ and }\hat{\tau}_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}\overset{d}{\rightarrow}\mathrm{Exp}(1).$ Moreover, there exists $\varepsilon_{0}\in(0,1)$ and a constant $\tilde{c}>0$ such that $P_{\lambda^{\varepsilon}}\left(\hat{\tau}_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}>t\right)\leq e^{-\tilde{c}t}$ for any $t>0$ and any $\varepsilon\in(0,\varepsilon_{0}).$ Notice that Theorem 8.18 is a multicycle version of Theorem 8.5, which is the key to the proofs of the asymptotics of moments of $N^{\varepsilon}(T^{\varepsilon})$, namely, Lemma 8.2 and Lemma 8.4. Given Theorem 8.18, the proofs of the following analogous results follow from essentially the same arguments as those of Lemma 8.2 and Lemma 8.4. ###### Lemma 8.19. Suppose that $w\geq h_{1}$, $m+h_{1}>w$, and that $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>w$. Then there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left|\frac{E_{\lambda^{\varepsilon}}(\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right))}{T^{\varepsilon}}-\frac{1}{E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}\right|\geq c.$ ###### Corollary 8.20. Suppose that $w\geq h_{1}$, $m+h_{1}>w$ and that $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>w$. Then there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{E_{\lambda^{\varepsilon}}(\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right))}{T^{\varepsilon}}\geq m+\varkappa_{\delta}.$ ###### Lemma 8.21. Suppose that $w\geq h_{1}$, $m+h_{1}>w$ and that $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>w$. Then for any $\eta>0,$ there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{\mathrm{Var}_{\lambda^{\varepsilon}}(\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right))}{T^{\varepsilon}}\geq m+h_{1}-\eta.$ ## 9 Large Deviation Type Upper Bounds In this section we collect results from the previous sections to prove the main results of the paper, Theorems 4.3 and 4.5, which give large deviation upper bounds on the bias under the empirical measure and the variance per unit time. We also give the proof of Theorem 4.9, which shows how to simplify some expressions appearing in the large deviation bounds. Before giving the proof of the first result we establish Lemma 9.1 and Lemma 9.2 for the single cycle case, and Lemma 9.3 and Lemma 9.4 for the multicycle case, which are needed in the proof of Theorems 4.3. Recall that for any $n\in\mathbb{N}$ $S_{n}^{\varepsilon}\doteq\int_{\tau_{n-1}^{\varepsilon}}^{\tau_{n}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt.$ (9.1) ###### Lemma 9.1. If $h_{1}>w$, $A\subset M$ is compact and $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>h_{1}$, then for any $\eta>0$, there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left|\frac{E_{\lambda^{\varepsilon}}N^{\varepsilon}\left(T^{\varepsilon}\right)}{T^{\varepsilon}}E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}-\int_{M}e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)\mu^{\varepsilon}\left(dx\right)\right|$ $\displaystyle\quad\geq\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)+c-h_{1}-\eta.$ ###### Proof. To begin, by Lemma 4.1 with $g\left(x\right)=e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)$, we know that for any $\delta$ sufficiently small and $\varepsilon>0,$ $\displaystyle E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}$ $\displaystyle=E_{\lambda^{\varepsilon}}\left(\int_{0}^{\tau_{1}^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{s}^{\varepsilon}\right)}1_{A}\left(X_{s}^{\varepsilon}\right)ds\right)=E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\cdot\int_{M}e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)\mu^{\varepsilon}\left(dx\right).$ This implies that $\displaystyle\left|\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}-\int_{M}e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)\mu^{\varepsilon}\left(dx\right)\right|=E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}\cdot\left|\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}-\frac{1}{E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\right|.$ Hence, by (7.1), Lemma 7.21 and Lemma 8.2, we find that given $\eta>0,$ there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left|\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}-\int_{M}e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)\mu^{\varepsilon}\left(dx\right)\right|$ $\displaystyle\quad\geq\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left|\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}-\frac{1}{E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\right|$ $\displaystyle\quad\geq\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)+c-h_{1}-\eta.$ ∎ In the application of Wald’s identity a difficulty arises in that, owing to the randomness of $N^{\varepsilon}\left(T^{\varepsilon}\right)$, $S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}$ need not have the same distribution as $S_{1}^{\varepsilon}$. Nevertheless, such minor term can be dealt with by using technique in, for example, [18, Theorem 3.16]. The proof of the following lemma can be found in the Appendix. ###### Lemma 9.2. If $h_{1}>w$, $A\subset M$ is compact and $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>h_{1}$, then for any $\eta>0$, there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{E_{\lambda^{\varepsilon}}S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}}{T^{\varepsilon}}\geq\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)+c-h_{1}-\eta.$ We have similar results for multicycles. To be specific, we have the following two lemmas. ###### Lemma 9.3. Suppose that $w\geq h_{1}$, $m+h_{1}>w$, $A\subset M$ is compact and that $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>w$. Then for any $\eta>0$, there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left|\frac{E_{\lambda^{\varepsilon}}\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)}{T^{\varepsilon}}E_{\lambda^{\varepsilon}}\hat{S}_{1}^{\varepsilon}-\int_{M}e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)\mu^{\varepsilon}\left(dx\right)\right|$ $\displaystyle\quad\geq\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)+c-(m+h_{1})-\eta.$ ###### Lemma 9.4. Suppose that $w\geq h_{1}$, $m+h_{1}>w$, $A\subset M$ is compact and that $T^{\varepsilon}=e^{\frac{1}{\varepsilon}c}$ for some $c>w$. Then for any $\eta>0$, there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{E_{\lambda^{\varepsilon}}\hat{S}_{\hat{N}^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}}{T^{\varepsilon}}\geq\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)+c-(m+h_{1})-\eta.$ ###### Proof of Theorem 4.3. If $h_{1}>w$, then recall that $\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)-1}S_{n}^{\varepsilon}\leq\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\leq\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon},$ where $S_{n}^{\varepsilon}$ is defined in (9.1). Then we apply Wald’s first identity to obtain $\displaystyle E_{\lambda^{\varepsilon}}\left(\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)-1}S_{n}^{\varepsilon}\right)$ $\displaystyle=E_{\lambda^{\varepsilon}}\left(\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon}\right)-E_{\lambda^{\varepsilon}}S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}$ $\displaystyle=E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}-E_{\lambda^{\varepsilon}}S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}.$ Thus $\displaystyle\left|E_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)-\int_{M}e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)\mu^{\varepsilon}\left(dx\right)\right|$ $\displaystyle\quad\leq\left|\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}-\int_{M}e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)\mu^{\varepsilon}\left(dx\right)\right|+\frac{E_{\lambda^{\varepsilon}}S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}}{T^{\varepsilon}}.$ Therefore, by Lemma 9.1 and Lemma 9.2 we have that for any $\eta>0$, there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$, $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left|E_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)-\int_{M}e^{-\frac{1}{\varepsilon}f\left(x\right)}1_{A}\left(x\right)\mu^{\varepsilon}\left(dx\right)\right|$ $\displaystyle\quad\geq\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)+c-h_{1}-\eta.$ The argument for $h_{1}\leq w$ is entirely analogous but uses by Lemma 9.3 and Lemma 9.4. ∎ The following lemma bounds quantities that will arise in the proof of Theorem 4.5. Its proof is given in the Appendix. ###### Lemma 9.5. Recall the definitions $R_{1}^{(2)}\doteq 2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{1},x\right)\right]-h_{1},$ and for $j\in L\setminus\\{1\\}$, $R_{j}^{(2)}\doteq 2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)-2W\left(O_{1}\right)+W\left(O_{1}\cup O_{j}\right)$ with $h_{1}\doteq\min_{\ell\in L\setminus\\{1\\}}V(O_{1},O_{\ell}).$ Then $2\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-2W\left(O_{1}\right)-h_{1}\geq\min_{j\in L}R_{j}^{(2)}.$ ###### Proof of Theorem 4.5. We begin with the observation that is for any random variables $X,Y$ and $Z$ satisfying $0\leq Y-Z\leq X\leq Y,$ $\displaystyle\mathrm{Var}\left(X\right)$ $\displaystyle=EX^{2}-\left(EX\right)^{2}\leq EY^{2}-\left(E\left(Y-Z\right)\right)^{2}$ $\displaystyle=\mathrm{Var}\left(Y\right)+2EY\cdot EZ-\left(EZ\right)^{2}\leq\mathrm{Var}\left(Y\right)+2EY\cdot EZ.$ When $h_{1}>w$, since $0\leq\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon}-\frac{1}{T^{\varepsilon}}S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}\leq\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\leq\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon},$ we have $\displaystyle\mathrm{Var}_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)$ $\displaystyle\quad\leq\mathrm{Var}_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon}\right)+2E_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon}\right)\frac{E_{\lambda^{\varepsilon}}S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}}{T^{\varepsilon}},$ and with the help of (7.2) $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\mathrm{Var}_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)T^{\varepsilon}\right)$ $\displaystyle\quad\geq\min\left\\{\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[\mathrm{Var}_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon}\right)T^{\varepsilon}\right],\right.$ $\displaystyle\quad\qquad\left.\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[E_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon}\right)\frac{E_{\lambda^{\varepsilon}}S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}}{T^{\varepsilon}}T^{\varepsilon}\right]\right\\}.$ We complete the proof in the case of single cycle by showing both terms are bounded below by $\min\nolimits_{j\in L}(R_{j}^{(1)}\wedge R_{j}^{(2)})-\eta$, where we recall $R_{j}^{(1)}\doteq\inf\nolimits_{x\in A}\left[2f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)-W\left(O_{1}\right),$ $R_{1}^{(2)}\doteq 2\inf\nolimits_{x\in A}\left[f\left(x\right)+V\left(O_{1},x\right)\right]-h_{1},$ and for $j\in L\setminus\\{1\\}$ $\displaystyle R_{j}^{(2)}$ $\displaystyle\doteq 2\inf\nolimits_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)-2W\left(O_{1}\right)+W\left(O_{1}\cup O_{j}\right).$ For the second term, we apply Wald’s first identity, Lemma 7.21, Corollary 8.3 and Lemma 9.2 to find that given $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[E_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon}\right)\frac{E_{\lambda^{\varepsilon}}S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}}{T^{\varepsilon}}T^{\varepsilon}\right]$ $\displaystyle\quad\geq\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log T^{\varepsilon}+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log E_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon}\right)+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{E_{\lambda^{\varepsilon}}S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}}{T^{\varepsilon}}$ $\displaystyle\quad\geq-c+(\inf\nolimits_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)-h_{1}-\eta/3)+\varkappa_{\delta}$ $\displaystyle\qquad+(\inf\nolimits_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)+\left(c-h_{1}\right)-\eta/3)$ $\displaystyle\quad\geq 2\inf\nolimits_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-2W\left(O_{1}\right)-h_{1}-\eta$ $\displaystyle\quad\geq\min\nolimits_{j\in L}R_{j}^{(2)}-\eta\geq\min\nolimits_{j\in L}(R_{j}^{(1)}\wedge R_{j}^{(2)})-\eta.$ The third inequality holds by choosing $\delta$ sufficiently small $h_{\delta}\geq h_{1}-\eta/3$. The fourth inequality is from Lemma 9.5. Turning to the first term, we can bound the variance by (6.3): $\displaystyle\mathrm{Var}_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon}\right)T^{\varepsilon}$ $\displaystyle\leq 2\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}\mathrm{Var}_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}+2\frac{\mathrm{Var}_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}\left(E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}\right)^{2}$ $\displaystyle\leq 2\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}E_{\lambda^{\varepsilon}}\left(S_{1}^{\varepsilon}\right)^{2}+2\frac{\mathrm{Var}_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}\left(E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}\right)^{2}.$ If we use Corollary 8.3 and Lemma 7.23, then we know that given $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}E_{\lambda^{\varepsilon}}\left(S_{1}^{\varepsilon}\right)^{2}\right]$ $\displaystyle\geq\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log E_{\lambda^{\varepsilon}}\left(S_{1}^{\varepsilon}\right)^{2}$ $\displaystyle\geq\min_{j\in L}(R_{j}^{(1)}\wedge R_{j}^{(2)})-\eta.$ In addition, we can apply Lemma 7.21 and Lemma 8.4 to show that given $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left[\frac{\mathrm{Var}_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}\left(E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}\right)^{2}\right]$ $\displaystyle\quad\geq\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{\mathrm{Var}_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(T^{\varepsilon}\right)\right)}{T^{\varepsilon}}+2\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}$ $\displaystyle\quad\geq 2\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-2W\left(O_{1}\right)-h_{1}-\eta$ $\displaystyle\quad\geq\min_{j\in L}R_{j}^{(2)}-\eta\geq\min_{j\in L}(R_{j}^{(1)}\wedge R_{j}^{(2)})-\eta.$ The second last inequality comes from Lemma 9.5. Hence, we find that given $\eta>0,$ there exists $\delta_{0}\in(0,1),$ such that for any $\delta\in(0,\delta_{0})$ $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\mathrm{Var}_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\sum\nolimits_{n=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}S_{n}^{\varepsilon}\right)T^{\varepsilon}\right)\geq\min_{j\in L}(R_{j}^{(1)}\wedge R_{j}^{(2)})-\eta,$ and we are done for the single cycle case. For multicycle case, by using a similar argument and applying Lemmas 7.24, 7.25, 8.21, 9.4 and Corollary 8.20, we find that $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\mathrm{Var}_{\lambda^{\varepsilon}}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}e^{-\frac{1}{\varepsilon}f\left(X_{t}^{\varepsilon}\right)}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)T^{\varepsilon}\right)\geq\min_{j\in L}(R_{j}^{(1)}\wedge R_{j}^{(2)}\wedge R_{j}^{(3,m)})-\eta,$ with $R_{j}^{(3,m)}\doteq 2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+2W\left(O_{j}\right)-2W\left(O_{1}\right)-(m+h_{1}).$ We complete the proof by sending $m\downarrow w-h_{1}$. ∎ ###### Proof of Theorem 4.9. Parts 1, 2 and 3 are from Theorem 4.3, Lemma 4.3 (b) and Theorem 6.1 in [12, Chapter 6], respectively. We now turn to part 4. Before giving the proof, we state a result from [12]. The result is Lemma 4.3 (c) in [12, Chapter 6], which says that for any unstable equilibrium point $O_{j},$ there exists a stable equilibrium point $O_{i}$ such that $W(O_{j})=W(O_{i})+V(O_{i},O_{j}).$ Now suppose that $\min\nolimits_{j\in L}(\inf\nolimits_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right))$ is attained at some $\ell\in L$ such that $O_{\ell}$ is unstable (i.e., $\ell\in L\setminus L_{s}$). Then since there exists a stable equilibrium point $O_{i}$ (i.e., $i\in L_{s}$) such that $W(O_{\ell})=W(O_{i})+V(O_{i},O_{\ell})$ we find $\displaystyle\min_{j\in L}\left(\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)\right)$ $\displaystyle\quad=\inf_{x\in A}\left[f\left(x\right)+V\left(O_{\ell},x\right)\right]+W\left(O_{\ell}\right)=\inf_{x\in A}\left[f\left(x\right)+V\left(O_{\ell},x\right)\right]+V(O_{i},O_{\ell})+W(O_{i})$ $\displaystyle\quad\geq\inf_{x\in A}\left[f\left(x\right)+V\left(O_{i},x\right)\right]+W(O_{i})\geq\min_{j\in L_{\rm{s}}}\left(\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)\right)$ $\displaystyle\quad\geq\min_{j\in L}\left(\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+W\left(O_{j}\right)\right).$ The first inequality is from a dynamic programming inequality. Therefore, the minimum is also attained at $i\in L_{\rm{s}}$ and $\min_{j\in L}R_{j}^{(1)}=\min_{j\in L_{\rm{s}}}R_{j}^{(1)}.$ ∎ ## 10 Exponential Return Law and Tail Behavior In this section we give the proof of Theorem 8.5, which was the key fact needed to obtain bounds on the distribution of $N^{\varepsilon}(T^{\varepsilon})$, and the related multicycle analogy. A result of this type first appears in [6], which asserts that the time needed to escape from an open subset of the domain of attraction of a stable equilibrium point that contains the equilibrium point has an asymptotically exponential distribution. [6] also proves a nonasymptotic bound on the tail of the probability of escape before a certain time that is also of exponential form. Theorem 8.5 is a more complicated statement, in that it asserts the asymptotically exponential form for the return time to the neighborhood of $O_{1}$. To prove this we build on the results of [6], and decompose the return time into times of transitions between equilibrium points. This in turn will require the proof of a number of related results, such as establishing the independence of certain estimates with respect to initial distributions. The existence of an exponentially distributed first hitting time is a central topic in the theory of quasistationary distributions. For a recent book length treatment of the topic we refer to [5]. However, so far as we can tell the types of situations we encounter are not covered by existing results, and so as noted we develop what is needed using [6] as the starting point. See Remark 3.15. For any $j\in L,$ define $\upsilon_{j}^{\varepsilon}$ as the hitting time of $\partial B_{\delta}(O_{k})$ for some $k\in L\setminus\\{j\\}$, i.e., $\upsilon_{j}^{\varepsilon}\doteq\inf\left\\{t>0:X_{t}^{\varepsilon}\in\cup_{k\in L\setminus\\{j\\}}\partial B_{\delta}(O_{k})\right\\}.$ (10.1) We will prove the following result for first hitting times of another equilibrium point, and later extend to return times. ###### Lemma 10.1. For any $j\in L_{\rm{s}}$, there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ and any distribution $\lambda^{\varepsilon}$ on $\partial B_{\delta}(O_{j}),$ $\lim_{\varepsilon\rightarrow 0}\varepsilon\log E_{\lambda^{\varepsilon}}\upsilon_{j}^{\varepsilon}=\min_{y\in\cup_{k\in L\setminus\\{j\\}}\partial B_{\delta}(O_{k})}V(O_{j},y)\text{ and }\upsilon_{j}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{j}^{\varepsilon}\overset{d}{\rightarrow}\rm{Exp}(1).$ Moreover, there exists $\varepsilon_{0}\in(0,1)$ and a constant $\tilde{c}>0$ such that $P_{\lambda^{\varepsilon}}\left(\upsilon_{j}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{j}^{\varepsilon}>t\right)\leq e^{-\tilde{c}t}$ for any $t>0$ and any $\varepsilon\in(0,\varepsilon_{0}).$ The organization of this section is as follows. The first part of Lemma 10.1 that is concerned with mean first hitting times is proved in Subsection 10.1, while the second part that is concerned with an asymptotically exponential distribution but when starting with a special distribution is proved in Subsection 10.2. The last part of the lemma, which focuses on bounds on the tail of the hitting time of another equilibrium point but when starting with a special distribution is proved in Subsection 10.3. We then extend the second and third parts of Lemma 10.1 to general initial distributions in Subsection 10.4 and Subsection 10.5. The last two subsections then extend all of Lemma 10.1 to return times for single cycles and multicycles, respectively. ### 10.1 Mean first hitting time ###### Lemma 10.2. For any $\delta>0$ sufficiently small and $x\in\partial B_{\delta}(O_{j})$ with $j\in L_{\rm{s}}$ $\lim_{\varepsilon\rightarrow 0}\varepsilon\log E_{x}\upsilon_{j}^{\varepsilon}=\min_{y\in\cup_{k\in L\setminus\\{j\\}}\partial B_{\delta}(O_{k})}V(O_{j},y).$ (10.2) ###### Proof. For the given $j\in L_{\rm{s}}$ let $D_{j}$ denote the corresponding domain of attraction. We claim there is $k\in L\setminus L_{\rm{s}}$ such that $q_{j}\doteq\inf_{y\in\partial D_{j}}V(O_{j},y)=V(O_{j},O_{k}).$ Since $V(O_{j},\cdot)$ is continuous and $\partial D_{j}$ is compact, there is a point $y^{\ast}\in\partial D_{j}$ such that $q_{j}=V(O_{j},y^{\ast})$. If $y^{\ast}\in\cup_{k\in L\setminus L_{\rm{s}}}O_{k}$ then we are done. If this is not true, then since $y^{\ast}\notin(\cup_{k\in L_{\rm{s}}}D_{k})\cup(\cup_{k\in L\setminus L_{\rm{s}}}O_{k})$, and since the solution to $\dot{\phi}=b(\phi),\phi(0)=y^{\ast}$ must converge to $\cup_{k\in L}O_{k}$ as $t\rightarrow\infty$, it must in fact converge to a point in $\cup_{k\in L\setminus L_{\rm{s}}}O_{k}$, say $O_{k}$. Since such trajectories have zero cost, by a standard argument for any $\varepsilon>0$ we can construct by concatenation a trajectory that connects $O_{j}$ to $O_{k}$ in finite time and with cost less than $q_{j}+\varepsilon$. Since $\varepsilon>0$ is arbitrary we have $q_{j}=V(O_{j},O_{k})$. There may be more than one $l\in L\setminus L_{\rm{s}}$ such that $O_{l}\in\partial D_{j}$ and $q_{j}=V(O_{j},O_{l})$, but we can assume that for some $k\in L\setminus L_{\rm{s}}$ and $\bar{y}\in\partial B_{\delta}(O_{k})$ we attain the min in (10.2). Then $\bar{q}_{j}\doteq V(O_{j},\bar{y})\leq q_{j}$, and we need to show $\lim_{\varepsilon\rightarrow 0}\varepsilon\log E_{x}\upsilon_{j}^{\varepsilon}=\bar{q}_{j}$. Given $s<\bar{q}_{j}$, let $D_{j}(s)=\\{x:V(O_{j},x)\leq s\\}$ and assume $s$ is large enough that $B_{\delta}(O_{j})\subset D_{j}(s)^{\circ}$. Then $D_{j}(s)\subset D_{j}^{\circ}$ is closed and contained in the open set $D_{j}\setminus\cup_{l\in L\setminus\\{j\\}}B_{\delta}(O_{l})$, and thus the time to reach $\partial D_{j}(s)$ is never greater than $\upsilon_{j}^{\varepsilon}$. Given $\eta>0$ we can find a set $D_{j}^{\eta}(s)$ that is contained in $D_{j}(s)$ and satisfies the conditions of [12, Theorem 4.1, Chapter 4], and also $\inf_{z\in\partial D_{j}^{\eta}(s)}V(O_{j},z)\geq s-\eta$. This theorem gives the equality in the following display: $\displaystyle\liminf_{\varepsilon\rightarrow 0}\varepsilon\log E_{x}\upsilon_{j}^{\varepsilon}\geq\liminf_{\varepsilon\rightarrow 0}\varepsilon\log E_{x}\inf\\{t\geq 0:X_{t}^{\varepsilon}\in\partial D_{j}^{\eta}(s)\\}=\inf_{z\in\partial D_{j}^{\eta}(s)}V(O_{j},z)\geq s-\eta.$ Letting $\eta\downarrow 0$ and then $s\uparrow\bar{q}_{j}$ gives $\liminf_{\varepsilon\rightarrow 0}\varepsilon\log E_{x}\upsilon_{j}^{\varepsilon}\geq\bar{q}_{j}$. For the reverse inequality we also adapt an argument from the proof of [12, Theorem 4.1, Chapter 4]. One can find $T_{1}<\infty$ such that the probability for $X_{t}^{\varepsilon}$ to reach $\cup_{l\in L}B_{\delta}(O_{l})$ by time $T_{1}$ from any $x\in M\setminus\cup_{l\in L}B_{\delta}(O_{l})$ is bounded below by $1/2$. (This follows easily from the law of large numbers and that all trajectories of the noiseless system reach $\cup_{l\in L}B_{\delta/2}(O_{l})$ in some finite time that is bounded uniformly in $x\in M\setminus\cup_{l\in L}B_{\delta}(O_{l})$.) Also, given $\eta>0$ there is $T_{2}<\infty$ and $\varepsilon_{0}>0$ such that $P_{x}\\{X_{t}^{\varepsilon}$ reaches $\cup_{k\in L\setminus\\{j\\}}\partial B_{\delta}(O_{k})$ before $T_{2}\\}\geq\exp-(\bar{q}_{j}+\eta)/\varepsilon$ for all $x\in\partial B_{\delta}(O_{j})$. It then follows from the strong Markov property that for any $x\in M\setminus\cup_{l\in L}B_{\delta}(O_{l})$ $P_{x}\\{\upsilon_{j}^{\varepsilon}\leq T_{1}+T_{2}\\}\geq e^{-\frac{1}{\varepsilon}(\bar{q}_{j}+\eta)}/2.$ Using the ordinary Markov property we have $\displaystyle E_{x}\upsilon_{j}^{\varepsilon}$ $\displaystyle\leq\sum\nolimits_{n=0}^{\infty}(n+1)(T_{1}+T_{2})P_{x}\\{n(T_{1}+T_{2})<\upsilon_{j}^{\varepsilon}\leq(n+1)(T_{1}+T_{2})\\}$ $\displaystyle=(T_{1}+T_{2})\sum\nolimits_{n=0}^{\infty}P_{x}\\{\upsilon_{j}^{\varepsilon}>n(T_{1}+T_{2})\\}$ $\displaystyle\leq(T_{1}+T_{2})\sum\nolimits_{n=0}^{\infty}\left[1-\inf_{x\in M\setminus\cup_{l\in L}B_{\delta}(O_{l})}P_{x}\\{\upsilon_{j}^{\varepsilon}\leq T_{1}+T_{2}\\}\right]^{n}$ $\displaystyle=(T_{1}+T_{2})\left(\inf_{x\in M\setminus\cup_{l\in L}B_{\delta}(O_{l})}P_{x}\\{\upsilon_{j}^{\varepsilon}\leq T_{1}+T_{2}\\}\right)^{-1}$ $\displaystyle\leq 2(T_{1}+T_{2})e^{\frac{1}{\varepsilon}(\bar{q}_{j}+\eta)}.$ Thus $\limsup_{\varepsilon\rightarrow 0}\varepsilon\log E_{x}\upsilon_{j}^{\varepsilon}\leq\bar{q}_{j}+\eta$, and letting $\eta\downarrow 0$ completes the proof. ∎ ###### Remark 10.3. By the standard Freidlin-Wentzell theory, the convergence asserted in Lemma 10.2 is uniform on $\partial B_{\delta}(O_{j})$. Therefore, we have the first part of Lemma 10.1. ### 10.2 Asymptotically exponential distribution ###### Lemma 10.4. For each $j\in L_{\rm{s}}$ there is a distribution $u^{\varepsilon}$ on $\partial B_{2\delta}(O_{j})$ such that $\upsilon_{j}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{j}^{\varepsilon}\overset{d}{\rightarrow}\rm{Exp}(1).$ ###### Proof. To simplify notation and since it plays no role, we write $j=1$ throughout the proof. We call $\partial B_{\delta}\left(O_{1}\right)$ and $\partial B_{2\delta}\left(O_{1}\right)$ the inner and outer rings of $O_{1}$. We can then decompose the hitting time as $\upsilon_{1}^{\varepsilon}=\sum\nolimits_{k=1}^{\mathcal{N}^{\varepsilon}-1}\theta_{k}^{\varepsilon}+\zeta^{\varepsilon},$ (10.3) where $\theta_{k}^{\varepsilon}$ is the $k$-th amount of time that the process travels from the outer ring to the inner ring and back without visiting $\cup_{j\in L\setminus\\{1\\}}\partial B_{\delta}(O_{j})$, $\zeta^{\varepsilon}$ is the amount of time that the process travels from the outer ring directly to $\cup_{j\in L\setminus\\{1\\}}\partial B_{\delta}(O_{j})$ without visiting the inner ring, and $\mathcal{N}^{\varepsilon}-1$ is the number of times that the process goes back and forth between the inner ring and outer ring. (It is assumed that $\delta>0$ is small enough that $B_{2\delta}\left(O_{1}\right)\subset M\setminus\cup_{j\in L\setminus\\{1\\}}B_{2\delta}(O_{j})$.) Note that $\theta_{k}^{\varepsilon}$ grows exponentially of the order $\delta$, due to the time taken to travel from the inner ring to the outer ring, and $\zeta^{\varepsilon}$ is uniformly bounded in expected value. For any set $A,$ define the first hitting time by $\tau\left(A\right)\doteq\inf\left\\{t>0:X_{t}^{\varepsilon}\in A\right\\}.$ Consider the conditional transition probability from $x\in\partial B_{2\delta}\left(O_{1}\right)$ to $y\in\partial B_{\delta}\left(O_{1}\right)$ given by $\psi_{1}^{\varepsilon}\left(dy|x\right)\doteq P\left(X_{\tau\left(\partial B_{\delta}\left(O_{1}\right)\right)}^{\varepsilon}\in dy|X_{0}^{\varepsilon}=x,\text{ }X_{t}^{\varepsilon}\notin\cup_{j\in L\setminus\\{1\\}}\partial B_{\delta}(O_{j}),t\in[0,\tau(\partial B_{\delta}\left(O_{1}\right)))]\right),$ and the transition probability from $y\in\partial B_{\delta}\left(O_{1}\right)$ to $x\in\partial B_{2\delta}\left(O_{1}\right)$ given by $\psi_{2}^{\varepsilon}\left(dx|y\right)\doteq P\left(X_{\tau\left(\partial B_{2\delta}\left(O_{1}\right)\right)}^{\varepsilon}\in dx|X_{0}^{\varepsilon}=y\right).$ (10.4) Then we can create a transition probability from $x\in\partial B_{2\delta}\left(O_{1}\right)$ to $y\in\partial B_{2\delta}\left(O_{1}\right)$ by $\psi^{\varepsilon}\left(dy|x\right)\doteq\int_{\partial B_{\delta}\left(O_{1}\right)}\psi_{2}^{\varepsilon}\left(dy|z\right)\psi_{1}^{\varepsilon}\left(dz|x\right).$ (10.5) Since $\partial B_{2\delta}\left(O_{1}\right)$ is compact and $\\{X_{t}^{\varepsilon}\\}_{t}$ is non-degenerate and Feller, there exists an invariant measure $u^{\varepsilon}\in\mathcal{P}\left(\partial B_{2\delta}\left(O_{1}\right)\right)$ with respect to the transition probability $\psi^{\varepsilon}\left(dy|x\right).$ If we start with the distribution $u^{\varepsilon}$ on $\partial B_{2\delta}\left(O_{1}\right)$, then it follows from the definition of $u^{\varepsilon}$ and the strong Markov property that the $\\{\theta_{k}^{\varepsilon}\\}_{k<\mathcal{N}^{\varepsilon}}$ are iid. Moreover, the indicators of escape (i.e., $1_{\\{\tau(\cup_{j\in L\setminus\\{1\\}}\partial B_{\delta}(O_{j}))=\tau(\cup_{j\in L}\partial B_{\delta}(O_{j}))\\}}$) are iid Bernoulli, and we write them as $Y_{k}^{\varepsilon}$ with $P_{u^{\varepsilon}}(Y_{k}^{\varepsilon}=1)=e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon},$ where $\delta>0$ is from the construction, $h_{1}^{\varepsilon}(\delta)\rightarrow h_{1}(\delta)$ as $\varepsilon\rightarrow 0$ and $h_{1}(\delta)\uparrow h_{1}$ as $\delta\downarrow 0$ with $h_{1}=\min_{j\in L\setminus\\{1\\}}V(O_{1},O_{j})$. Note that $\mathcal{N}^{\varepsilon}=\inf\left\\{k\in\mathbb{N}:Y_{k}^{\varepsilon}=1\right\\}.$ We therefore have $P_{u^{\varepsilon}}(\mathcal{N}^{\varepsilon}=k)=(1-e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon})^{k-1}e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon},$ and thus $E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}=E_{u^{\varepsilon}}\left[\sum\nolimits_{j=1}^{\mathcal{N}^{\varepsilon}-1}\theta_{j}^{\varepsilon}\right]+E_{u^{\varepsilon}}\zeta^{\varepsilon}=E_{u^{\varepsilon}}(\mathcal{N}^{\varepsilon}-1)E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}+E_{u^{\varepsilon}}\zeta^{\varepsilon},$ where the second equality comes from Wald’s identity. Using $\sum_{k=1}^{\infty}ka^{k-1}={1/(1-a)^{2}}$ for $a\in[0,1)$, we also have $\displaystyle E_{u^{\varepsilon}}\mathcal{N}^{\varepsilon}=\sum\nolimits_{k=1}^{\infty}k(1-e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon})^{k-1}e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}=e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}e^{2h_{1}^{\varepsilon}(\delta)/\varepsilon}=e^{h_{1}^{\varepsilon}(\delta)/\varepsilon},$ and therefore $E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}=e^{h_{1}^{\varepsilon}(\delta)/\varepsilon}E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}+(E_{u^{\varepsilon}}\zeta^{\varepsilon}-E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}).$ (10.6) Next consider the characteristic function of $\upsilon_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}$ $\phi^{\varepsilon}(t)=E_{u^{\varepsilon}}e^{it\upsilon_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}=\phi_{\upsilon}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}),$ where $\phi_{\upsilon}^{\varepsilon}$ is the characteristic function of $\upsilon_{1}^{\varepsilon}.$ By (10.3), we have $\displaystyle\phi_{\upsilon}^{\varepsilon}(s)$ $\displaystyle=E_{u^{\varepsilon}}e^{is\left(\sum_{k=1}^{\mathcal{N}^{\varepsilon}-1}\theta_{k}^{\varepsilon}+\zeta^{\varepsilon}\right)}=E_{u^{\varepsilon}}e^{is\zeta^{\varepsilon}}E_{u^{\varepsilon}}e^{is\left(\sum\nolimits_{k=1}^{\mathcal{N}^{\varepsilon}-1}\theta_{k}^{\varepsilon}\right)}$ $\displaystyle=\phi_{\zeta}^{\varepsilon}(s)\sum\nolimits_{k=1}^{\infty}(1-e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon})^{k-1}e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}\phi_{\theta}^{\varepsilon}(s)^{k-1}$ $\displaystyle=\phi_{\zeta}^{\varepsilon}(s)e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}(1-[(1-e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon})\phi_{\theta}^{\varepsilon}(s)])^{-1},$ where $\phi_{\theta}^{\varepsilon}$ and $\phi_{\zeta}^{\varepsilon}$ are the characteristic functions of $\theta_{1}^{\varepsilon}$ and $\zeta^{\varepsilon}$, respectively. We want to show that for any $t\in\mathbb{R}$ $\phi^{\varepsilon}(t)=\phi_{\upsilon}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon})\rightarrow 1/(1-it)\text{ as }\varepsilon\rightarrow 0.$ We first show that $\phi_{\zeta}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon})\rightarrow 1.$ By definition, $\phi_{\zeta}^{\varepsilon}\left(t/E_{u^{\varepsilon}\upsilon_{1}^{\varepsilon}}\right)=E_{u^{\varepsilon}}\cos\left(t\zeta^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)+iE_{u^{\varepsilon}}\sin\left(t\zeta^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right).$ According to [12, Lemma 1.9, Chapter 6], we know that there exist $T_{0}\in(0,\infty)$ and $\beta>0$ such that for any $T\in(T_{0},\infty)$ and for all $\varepsilon$ sufficiently small $P_{u^{\varepsilon}}\left(\zeta^{\varepsilon}>T\right)\leq e^{-\frac{1}{\varepsilon}\beta\left(T-T_{0}\right)},$ (10.7) and therefore for any bounded and continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ $\displaystyle\left|E_{u^{\varepsilon}}f\left(t\zeta^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)-f\left(0\right)\right|\leq 2\left\|f\right\|_{\infty}P_{u^{\varepsilon}}\left(\zeta^{\varepsilon}>T\right)+E_{u^{\varepsilon}}\left[\left|f\left(t\zeta^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)-f\left(0\right)\right|1_{\left\\{\zeta^{\varepsilon}\leq T\right\\}}\right].$ The first term in the last display goes to $0$ as $\varepsilon\rightarrow 0$. For any fixed $t,$ $t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow 0$. Since $f$ is continuous, the second term in the last display also converges to $0$ as $\varepsilon\rightarrow 0.$ $\phi_{\zeta}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon})\rightarrow 1$ follows by taking $f$ to be $\sin x$ and $\cos x.$ It remains to show that for any $t\in\mathbb{R}$ $e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}\left(1-\left[(1-e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon})\phi_{\theta}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon})\right]\right)^{-1}\rightarrow 1/(1-it)$ as $\varepsilon\rightarrow 0.$ Observe that $e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}\left(1-\left[(1-e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon})\phi_{\theta}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon})\right]\right)^{-1}=\left({\frac{1-\phi_{\theta}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon})}{e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}}+\phi_{\theta}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}})\right)^{-1},$ so it suffices to show that $\phi_{\theta}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon})\rightarrow 1$ and $[1-\phi_{\theta}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon})]/e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}\rightarrow- it$ as $\varepsilon\rightarrow 0.$ For the former, note that by (10.6) $0\leq E_{u^{\varepsilon}}\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)\leq\frac{tE_{u^{\varepsilon}}\theta_{1}^{\varepsilon}}{\left(e^{h^{\varepsilon}_{1}(\delta)/\varepsilon}-1\right)E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}}\rightarrow 0$ as $\varepsilon\rightarrow 0,$ and so $t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}$ converges to $0$ in distribution. Moreover, since $e^{ix}$ is bounded and continuous, we find $\phi_{\theta}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon})\rightarrow 1$. For the second part, using $x-{x^{3}}/{3!}\leq\sin x\leq x\text{ and }1-{x^{2}}/{2}\leq\cos x\leq 1$ for $x\in\mathbb{R}$ we find that $0\leq\frac{1-E_{u^{\varepsilon}}\cos\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)}{e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}}\leq\frac{E_{u^{\varepsilon}}\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)^{2}}{2e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}}$ and $\frac{E_{u^{\varepsilon}}\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)}{e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}}-\frac{E_{u^{\varepsilon}}\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)^{3}}{3!e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}}\leq\frac{E_{u^{\varepsilon}}\sin\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)}{e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}}\leq\frac{E_{u^{\varepsilon}}\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)}{e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}}.$ From our previous observation regarding the distribution of $\zeta^{\varepsilon}$ and (10.6) $\frac{E_{u^{\varepsilon}}\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)}{e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}}\rightarrow t\text{ as }\varepsilon\rightarrow 0.$ In addition, since $\theta_{1}^{\varepsilon}$ can be viewed as the time from the outer ring to the inner ring without visiting $\cup_{j\in L\setminus\\{1\\}}\partial B_{\delta}(O_{j})$ plus the time from the inner ring to the outer ring, by applying (10.7) to the former and using [6, Theorem 4 and Corollary 1] under Condition 3.13 to the later, we find that $P_{u^{\varepsilon}}\left(\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}>t\right)\leq 2e^{-t}$ (10.8) for all $t\in[0,\infty)$ and $\varepsilon$ sufficiently small. This implies that $\displaystyle E_{u^{\varepsilon}}\left(\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}\right)^{2}=2\int_{0}^{\infty}t^{2}P_{u^{\varepsilon}}\left(\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}>t\right)dt\leq 4\int_{0}^{\infty}t^{2}e^{-t}dt=8$ and similarly $E_{u^{\varepsilon}}\left(\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}\right)^{3}=3\int_{0}^{\infty}t^{3}\leq 36$. Then combined with (10.6), we have $0\leq\frac{E_{u^{\varepsilon}}\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)^{2}}{2e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}}\leq\frac{t^{2}E_{u^{\varepsilon}}\left(\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}\right)^{2}}{2e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}(e^{h_{1}^{\varepsilon}(\delta)/\varepsilon}-1)^{2}}\rightarrow 0$ and $0\leq\frac{E_{u^{\varepsilon}}\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)^{3}}{3!e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}}\leq\frac{t^{3}E_{u^{\varepsilon}}\left(\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}\right)^{3}}{3!e^{-h_{1}^{\varepsilon}(\delta)/\varepsilon}(e^{h_{1}^{\varepsilon}(\delta)/\varepsilon}-1)^{3}}\rightarrow 0.$ Therefore, we have shown that for any $t\in\mathbb{R}$ $\frac{1-\phi_{\theta}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon})}{e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}}=\frac{1-E_{u^{\varepsilon}}\cos\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)}{e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}}-i\frac{E_{u^{\varepsilon}}\sin\left(t\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)}{e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}}\rightarrow- it.$ ∎ ###### Remark 10.5. From the proof of Lemma 10.4, we actually know that $\phi_{\upsilon}^{\varepsilon}(t/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon})\rightarrow 1/(1-it)$ uniformly on any compact set in $\mathbb{R}$ as $\varepsilon\rightarrow 0$. ### 10.3 Tail probability The goal of this subsection is to prove the following. ###### Lemma 10.6. For each $j\in L_{\rm{s}}$ there is a distribution $u^{\varepsilon}$ on $\partial B_{2\delta}(O_{j})$ and $\tilde{c}>0$ such that for any $t\in[0,\infty)$, $P_{u^{\varepsilon}}(\upsilon_{j}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{j}^{\varepsilon}>t)\leq e^{-\tilde{c}t}$ (here $\upsilon_{j}^{\varepsilon}$ and $u^{\varepsilon}$ are defined as in the last subsection). ###### Proof. As in the last subsection we give the proof for the case $j=1$. To begin we note that for any $\alpha>0$ Chebyshev’s inequality implies $P_{u^{\varepsilon}}\left(\upsilon_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}>t\right)=P_{u^{\varepsilon}}(e^{\alpha\upsilon_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}>e^{\alpha t})\leq e^{-\alpha t}\cdot E_{u^{\varepsilon}}e^{\alpha\upsilon_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}.$ By picking $\alpha=\alpha^{\ast}\doteq 1/8$, it suffices to show that $E_{u^{\varepsilon}}e^{{\alpha^{\ast}\upsilon_{1}^{\varepsilon}}/{E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}}$ is bounded by a constant. We will do this by showing how the finiteness of $E_{u^{\varepsilon}}e^{{\alpha^{\ast}\upsilon_{1}^{\varepsilon}}/{E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}}$ is implied by the finiteness of $E_{u^{\varepsilon}}e^{{\alpha^{\ast}\theta_{1}^{\varepsilon}}/{E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}}$ and $E_{u^{\varepsilon}}e^{{\alpha^{\ast}\zeta^{\varepsilon}}/{E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}}.$ Using (10.8) we find that for any $\alpha>0$ $P_{u^{\varepsilon}}(e^{\alpha\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}}>t)\leq 2e^{-\frac{1}{\alpha}\log t}=2t^{-\frac{1}{\alpha}}$ for all $t\in[1,\infty)$ and $\varepsilon$ sufficiently small. Then (10.6) implies $E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\geq\left(e^{h^{\varepsilon}_{1}\left(\delta\right)/\varepsilon}-1\right)E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}$ and therefore $\displaystyle E_{u^{\varepsilon}}e^{\alpha^{\ast}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\theta_{1}^{\varepsilon}$ $\displaystyle\leq\int_{0}^{1}P_{u^{\varepsilon}}\left(\exp\left(\alpha^{\ast}\theta_{1}^{\varepsilon}/[{(e}^{h^{\varepsilon}_{1}\left(\delta\right)/\varepsilon}-1{)}E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}]\right)>t\right)dt$ $\displaystyle\quad+\int_{1}^{\infty}P_{u^{\varepsilon}}\left(\exp\left(\alpha^{\ast}\theta_{1}^{\varepsilon}/[{(e}^{h^{\varepsilon}_{1}\left(\delta\right)/\varepsilon}-1{)}E_{u^{\varepsilon}}\theta_{1}^{\varepsilon}]\right)>t\right)dt$ $\displaystyle\leq 1+2\int_{1}^{\infty}t^{-{(e}^{h^{\varepsilon}_{1}\left(\delta\right)/\varepsilon}-1{)/\alpha}^{\ast}}dt$ $\displaystyle=1+2[{(e}^{h^{\varepsilon}_{1}\left(\delta\right)/\varepsilon}-1{)/\alpha}^{\ast}-1]^{-1}=1+2\alpha^{\ast}[{e}^{h^{\varepsilon}_{1}\left(\delta\right)/\varepsilon}-\alpha^{\ast}-1]^{-1}.$ To estimate $\zeta^{\varepsilon},$ we use that by (10.7) there are $T_{0}\in(0,\infty)$ and $\beta>0$ such that for any $t\in(T_{0},\infty)$ and for all $\varepsilon$ sufficiently small $P_{u^{\varepsilon}}\left(\zeta^{\varepsilon}>t\right)\leq e^{-\frac{1}{\varepsilon}\beta\left(t-T_{0}\right)},$ so that for any $\alpha>0$ $P_{u^{\varepsilon}}\left(e^{\alpha\zeta^{\varepsilon}}>t\right)\leq e^{-\frac{1}{\varepsilon}\beta\left(\frac{1}{\alpha}\log t-T_{0}\right)}$ for any $t\geq e^{\alpha T_{0}}.$ Given $n\in\mathbb{N},$ for all sufficiently small $\varepsilon$ we have $\alpha^{\ast}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\leq 1/n$, and thus $P_{u^{\varepsilon}}\left(e^{\alpha^{\ast}\zeta^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}>t\right)\leq P_{u^{\varepsilon}}\left(e^{\zeta^{\varepsilon}/n}>t\right)\leq e^{-\frac{1}{\varepsilon}\beta\left(n\log t-T_{0}\right)}.$ Hence for any $n$ such that $e^{T_{0}/n}\leq 3/2$ and $\left(-\beta n+1\right)\log\left(3/2\right)+\beta T_{0}<0,$ and for $\varepsilon$ small enough that $\alpha^{\ast}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}\leq 1/n,$ we have $\displaystyle E_{u^{\varepsilon}}e^{\alpha^{\ast}\zeta^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}$ $\displaystyle\leq\int_{0}^{\infty}P_{u^{\varepsilon}}\left(e^{\alpha^{\ast}\zeta^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}>t\right)dt\leq 3/2+\int_{\frac{3}{2}}^{\infty}P_{u^{\varepsilon}}\left(e^{\alpha^{\ast}\zeta^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}>t\right)dt$ $\displaystyle\leq 3/2+\int_{\frac{3}{2}}^{\infty}e^{-\frac{1}{\varepsilon}\beta\left(n\log t-T_{0}\right)}dt=3/2+e^{\frac{1}{\varepsilon}\beta T_{0}}(\beta n/{\varepsilon}-1)^{-1}\left(3/2\right)^{\frac{1}{\varepsilon}\left(-\beta n+\varepsilon\right)}$ $\displaystyle=3/2+(\beta n/{\varepsilon}-1)^{-1}e^{\frac{1}{\varepsilon}\left[\left(-\beta n+\varepsilon\right)\log\left(3/2\right)+\beta T_{0}\right]}\leq 3/2+(\beta n/{\varepsilon}-1)^{-1}\leq 2.$ We have shown that for such $\alpha^{\ast},$ $E_{u^{\varepsilon}}e^{\alpha^{\ast}\zeta^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}$ and $E_{u^{\varepsilon}}e^{\alpha^{\ast}\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}$ are uniformly bounded for all $\varepsilon$ sufficiently small. Lastly, using the same calculation as used for the characteristic function $\displaystyle E_{u^{\varepsilon}}e^{\alpha^{\ast}\upsilon_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}$ $\displaystyle=E_{u^{\varepsilon}}e^{\alpha^{\ast}\zeta^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\cdot e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}\left(1-\left[(1-e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon})E_{u^{\varepsilon}}e^{\alpha^{\ast}\theta_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\right]\right)^{-1}$ $\displaystyle\leq 2e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}\left(1-\left[(1-e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon})\left(1+\frac{2\alpha^{\ast}}{{e}^{h^{\varepsilon}_{1}\left(\delta\right)/\varepsilon}-\alpha^{\ast}-1}\right)\right]\right)^{-1}$ $\displaystyle=2e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}\left(e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}-\frac{2\alpha^{\ast}}{{e}^{h^{\varepsilon}_{1}\left(\delta\right)/\varepsilon}-\alpha^{\ast}-1}+\frac{2\alpha^{\ast}}{{e}^{h^{\varepsilon}_{1}\left(\delta\right)/\varepsilon}-\alpha^{\ast}-1}e^{-h^{\varepsilon}_{1}(\delta)/\varepsilon}\right)^{-1}$ $\displaystyle=2\left(1-2\alpha^{\ast}\frac{e^{h^{\varepsilon}_{1}(\delta)/\varepsilon}-1}{{e}^{h^{\varepsilon}_{1}\left(\delta\right)/\varepsilon}-\alpha^{\ast}-1}\right)^{-1}\leq 2/(1-4\alpha^{\ast})=4.$ ∎ ### 10.4 General initial condition This subsection presents results that will allow us to extend the results in the previous two subsections to arbitrary initial distribution $\lambda^{\varepsilon}\in\mathcal{P}(\partial B_{\delta}\left(O_{1}\right))$. Under our assumptions, for any $j\in L_{\rm{s}}$ we observe that the process model $dX_{t}^{\varepsilon}=b\left(X_{t}^{\varepsilon}\right)dt+\sqrt{\varepsilon}\sigma\left(X_{t}^{\varepsilon}\right)dW_{t}$ (10.9) has the property that $b(x)=A(x-O_{j})[1+o(1)]$ and $\sigma\left(x\right)=\bar{\sigma}[1+o\left(1\right)]$, where $o(1)\rightarrow 0$ as $\left\|x-O_{j}\right\|\rightarrow 0$, $A$ is stable and $\bar{\sigma}$ is invertible. By an invertible change of variable we can arrange so that $O_{j}=0$ and $\bar{\sigma}=I$, and to simplify we assume this in the rest of the section. Since $A$ is stable there exists a positive definite and symmetric solution $M$ to the matrix equation $AM+MA^{T}=-I$ (we can in fact exhibit the solution in the form $M=\int_{0}^{\infty}e^{At}e^{A^{T}t}dt)$. To prove the ergodicity we introduce some additional notation: $U(x)\doteq\langle x,Mx\rangle$, $B_{i}\doteq\\{x:U(x)<b_{i}^{2}\\}$ and $\mathcal{S}_{i}(\varepsilon)\doteq\\{x:U(x)<a_{i}^{2}\varepsilon\\},$ for $i=1,2$, where $0<a_{1}<a_{2}$, $0<b_{0}<b_{1}<b_{2}$. If $\varepsilon_{0}=(b_{0}^{2}/a_{2}^{2})/2$ then with cl denoting closure, ${\rm cl}(\mathcal{S}_{2}(\varepsilon_{0}))\subset B_{0},$ and we will assume $\varepsilon\in(0,\varepsilon_{0})$ henceforth. For a use later on, we will also assume that $a_{1}^{2}=2\sup\nolimits_{x\in B_{2}}\mbox{tr}[\sigma(x)\sigma(x)^{T}M].$ ###### Remark 10.7. The sets $B_{1}$ and $B_{2}$ will play the roles that $B_{\delta}(O_{1})$ and $B_{2\delta}(O_{1})$ played previously in this section. Although elsewhere in this paper as well as in the reference [12] these sets are taken to be balls with respect to the Euclidean norm, in this subsection we take them to be level sets of $U(x)$. The shape of these sets and the choice of the factor of $2$ relating the radii play no role in the analysis of [12] or in our prior use in this paper. However in this subsection it is notationally convenient for the sets to be level sets of $U$, since $U$ is a Lyapunov function for the noiseless dynamics near $0$. After this subsection we will revert to the $B_{\delta}(O_{1})$ and $B_{2\delta}(O_{1})$ notation. In addition to the restrictions $a_{1}<a_{2}$ and $a_{2}^{2}\varepsilon_{0}\leq b_{0}^{2}$, we also assume that $a_{1},a_{2}$ and $\varepsilon_{0}>0$ are such that if $\phi^{x}$ is the solution to the noiseless dynamics $\dot{\phi}=b(\phi)$ with initial condition $x$, then: (i) for all $x\in\partial\mathcal{S}_{2}(\varepsilon)$, $\phi^{x}$ never crosses $\partial B_{1}$; (i) for all $x\in\partial\mathcal{S}_{1}(\varepsilon)$, $\phi^{x}$ never exits $\mathcal{S}_{2}(\varepsilon)$. The idea that will be used to establish asymptotic independence from the starting distribution is the following. We start the process on $\partial B_{1}$. With some small probability it will hit $\partial B_{2}$ before hitting $\partial\mathcal{S}_{2}(\varepsilon)$. This gives a contribution to $\psi_{2}^{\varepsilon}(dz|x)$ defined in (10.4) that will be relatively unimportant. If instead it hits $\partial\mathcal{S}_{2}(\varepsilon)$ first, then we do a Freidlin-Wentzell type analysis, and decompose the trajectory into excursions between $\partial\mathcal{S}_{2}(\varepsilon)$ and $\partial\mathcal{S}_{1}(\varepsilon)$, before a final excursion from $\partial\mathcal{S}_{2}(\varepsilon)$ to $\partial B_{2}$. To exhibit the asymptotic independence from $\varepsilon$, we introduce the scaled process $Y^{\varepsilon}_{t}=X^{\varepsilon}_{t}/\sqrt{\varepsilon}$, which solves the SDE $dY^{\varepsilon}_{t}=\frac{1}{\sqrt{\varepsilon}}b(\sqrt{\varepsilon}Y^{\varepsilon}_{t})dt+\sigma(\sqrt{\varepsilon}Y^{\varepsilon}_{t})dW_{t}.$ Let $\mathcal{\bar{S}}_{1}=\partial\mathcal{S}_{1}(1)\text{ and }\mathcal{\bar{S}}_{2}=\partial\mathcal{S}_{2}(1).$ Let $\omega^{\varepsilon}(w|x)$ denote the density of the hitting location on $\mathcal{\bar{S}}_{2}$ by the process $Y^{\varepsilon}$, given $Y^{\varepsilon}_{0}=x\in\mathcal{\bar{S}}_{1}$. The following estimate is essential. The density function can be identified with the normal derivative of a related Green’s function, which is bounded from above by the boundary gradient estimate and bounded below by using the Hopf lemma [13]. ###### Lemma 10.8. Given $\varepsilon_{0}>0$, there are $0<c_{1}<c_{2}<\infty$ such that $c_{1}\leq\omega^{\varepsilon}(w|x)\leq c_{2}$ for all $x\in\mathcal{\bar{S}}_{1}$, $w\in\mathcal{\bar{S}}_{2}$ and $\varepsilon\in(0,\varepsilon_{0})$. Next let $p^{\varepsilon}(u|w)$ denote the density of the return location for $Y^{\varepsilon}$ on $\mathcal{\bar{S}}_{2}$, conditioned on visiting $\mathcal{\bar{S}}_{1}$ before $\partial B_{2}/\sqrt{\varepsilon}$, and starting at $w\in\mathcal{\bar{S}}_{2}$. The last lemma then directly gives the following. ###### Lemma 10.9. For $\varepsilon_{0}>0$ and $c_{1},c_{2}$ as in the last lemma $c_{1}\leq p^{\varepsilon}(u|w)\leq c_{2}$ for all $u,w\in\mathcal{\bar{S}}_{2}$ and $\varepsilon\in(0,\varepsilon_{0})$. Let $r^{\varepsilon}(w)$ denote the unique stationary distribution of $p^{\varepsilon}(u|w)$, and let $p^{\varepsilon,n}(u|w)$ denote the $n$-step transition density. The preceding lemma, [14, Theorem 10.1 Chapter 3], and the existence of a uniform strictly positive lower bound on $r^{\varepsilon}(u)$ for all sufficiently small $\varepsilon>0$ imply the following. ###### Lemma 10.10. There is $K<\infty$ and $\alpha\in(0,1)$ such that for all $\varepsilon\in(0,\varepsilon_{0})$ $\sup_{w\in\mathcal{\bar{S}}_{2}}\left|p^{\varepsilon,n}(u|w)-r^{\varepsilon}(u)\right|/r^{\varepsilon}(u)\leq K\alpha^{n}.$ Let $\eta^{\varepsilon}(dx|w)$ denote the distribution of $X^{\varepsilon}$ upon first hitting $\partial B_{2}$ given that $X^{\varepsilon}$ reaches $\partial\mathcal{S}_{1}(\varepsilon)$ before $\partial B_{2}$ and starts at $w\in\partial\mathcal{S}_{2}(\varepsilon)$. ###### Lemma 10.11. There is $\kappa>0$ and $\varepsilon_{0}>0$ such that for all $\varepsilon\in(0,\varepsilon_{0})$ $\sup_{x\in\partial B_{1}}P_{x}\left\\{X^{\varepsilon}\mbox{ reaches }\partial B_{2}\mbox{ before }\mathcal{S}_{2}(\varepsilon)\right\\}\leq e^{-\kappa/\varepsilon}.$ ###### Lemma 10.12. There are $\bar{\eta}^{\varepsilon}(dz)\in$ $\mathcal{P}(\partial B_{2})$, $s^{\varepsilon}$ that tends to $0$ as $\varepsilon\rightarrow 0$ and $\varepsilon_{0}>0$, such that for all $A\in\mathcal{B}(\partial B_{2}),w\in\partial\mathcal{S}_{2}(\varepsilon)$ and $\varepsilon\in(0,\varepsilon_{0})$ $\bar{\eta}^{\varepsilon}(A)[1-s^{\varepsilon}K/(1-\alpha)]\leq\eta^{\varepsilon}(A|w)\leq\bar{\eta}^{\varepsilon}(A)[1+s^{\varepsilon}K/(1-\alpha)],$ where $K$ and $\alpha$ are from Lemma 10.10. ###### Proof of Lemma 10.11. Recall that $a_{1}^{2}=2\sup_{x\in B_{2}}$tr$[\sigma(x)\sigma(x)^{T}M]$. We then use that $AM+MA^{T}=-I$ to get that with $U(x)\doteq\left\langle x,Mx\right\rangle$, $\left\langle DU(x),b(x)\right\rangle\leq-\varepsilon a_{1}^{2}$ (10.10) for $x\in B_{2}\setminus\mathcal{S}_{2}(\varepsilon)$, and $\left\langle DU(x),b(x)\right\rangle\leq-\frac{1}{8}b_{0}^{2}$ (10.11) for $B_{2}\setminus(B_{0}/2)$. By Itô’s formula $\displaystyle dU(X^{\varepsilon}_{t})=\left\langle DU(X^{\varepsilon}_{t}),b(X^{\varepsilon}_{t})\right\rangle dt+\frac{\varepsilon}{2}\text{tr}[\sigma(X^{\varepsilon}_{t})\sigma(X^{\varepsilon}_{t})^{T}M]dt+\sqrt{\varepsilon}\left\langle DU(X^{\varepsilon}_{t}),\sigma(X^{\varepsilon}_{t})dW_{t}\right\rangle.$ (10.12) Starting at $x\in\partial B_{1}$, we are concerned with the probability $P_{x}\left\\{U(X^{\varepsilon}_{t})\text{ reaches }b_{2}^{2}\text{ before }a_{2}^{2}\varepsilon\right\\},$ where $U(x)=b_{1}^{2}$. However, according to (10.12) and (10.11), reaching $b_{2}^{2}$ before $b_{0}^{2}/4$ is a rare event, and its probability decays exponentially in the form $e^{-\kappa/\varepsilon}$ for some $\kappa>0$ and uniformly in $x\in\partial B_{1}$. Once the process reaches $B_{0}/2$, (10.12) and (10.10) imply $U(X^{\varepsilon}_{t})$ is a supermartingale as long as it is in the interval $[0,b_{0}^{2}]$, and therefore after $X^{\varepsilon}_{t}$ reaches $B_{0}/2$, the probability that $U(X^{\varepsilon}_{t})$ reaches $a_{2}^{2}\varepsilon$ before $b_{0}^{2}$ is greater than $1/2$. ∎ ###### Proof of Lemma 10.12. Consider a starting position $w\in\partial\mathcal{S}_{2}(\varepsilon)$, and recall that $\eta^{\varepsilon}(dz|w)$ denotes the hitting distribution on $\partial B_{2}$ after starting at $w$. Let $\theta_{k}^{\varepsilon}$ denote the return times to $\partial\mathcal{S}_{2}(\varepsilon)$ after visiting $\partial\mathcal{S}_{1}(\varepsilon)$, and let $q_{n}^{\varepsilon}(w)$ denote the probability that the first $k$ for which $X^{\varepsilon}$ visits $\partial B_{2}$ before visiting $\partial\mathcal{S}_{1}(\varepsilon)$ during $[\theta_{k}^{\varepsilon},\theta_{k+1}^{\varepsilon}]$ is $n$. Then by the strong Markov property and using the rescaled process $\int_{\partial B_{2}}g(z)\eta^{\varepsilon}(dz|w)=\sum\nolimits_{n=0}^{\infty}\int_{\partial B_{2}}g(z)q_{n}^{\varepsilon}(w)\int_{\partial\mathcal{S}_{2}(\varepsilon)}\eta^{\varepsilon}(dz|u)J^{\varepsilon}(u)p^{\varepsilon,n}(\sqrt{\varepsilon}u|\sqrt{\varepsilon}w)du,$ where $J^{\varepsilon}(u)$ is the Jacobian that accounts for the mapping between $\partial\mathcal{S}_{2}(\varepsilon)$ and $\partial\mathcal{S}_{2}(1)$ and is given by $u/\sqrt{\varepsilon}$. We next use that uniformly in $w\in\partial\mathcal{S}_{2}(\varepsilon)$ $p^{\varepsilon,n}(\sqrt{\varepsilon}u|\sqrt{\varepsilon}w)\leq r^{\varepsilon}(\sqrt{\varepsilon}u)[1+K\alpha^{n}]$ to get $\displaystyle\sum\nolimits_{n=0}^{\infty}\int_{\partial B_{2}}g(z)q_{n}^{\varepsilon}(w)\int_{\partial\mathcal{S}_{2}(\varepsilon)}\eta^{\varepsilon}(dz|u)J^{\varepsilon}(u)p^{\varepsilon,n}(\sqrt{\varepsilon}u|\sqrt{\varepsilon}w)du$ $\displaystyle\quad\leq\left(\sum\nolimits_{n=0}^{\infty}\int_{\partial B_{2}}g(z)q_{n}^{\varepsilon}(w)\int_{\partial\mathcal{S}_{2}(\varepsilon)}\eta^{\varepsilon}(dz|u)J^{\varepsilon}(u)r^{\varepsilon}(\sqrt{\varepsilon}u)du\right)[1+K\alpha^{n}]$ $\displaystyle\quad=\int_{\partial B_{2}}g(z)\int_{\partial\mathcal{S}_{2}(\varepsilon)}\eta^{\varepsilon}(dz|u)J^{\varepsilon}(u)r^{\varepsilon}(\sqrt{\varepsilon}u)du\left[1+K\sum\nolimits_{n=0}^{\infty}q_{n}^{\varepsilon}(w)\alpha^{n}\right].$ Now use that $K\sum_{n=0}^{\infty}\alpha^{n}=K/(1-\alpha)<\infty$ and $\sup_{w\in\partial\mathcal{S}_{2}(\varepsilon)}\sup_{n\in\mathbb{N}_{0}}q_{n}^{\varepsilon}(w)\rightarrow 0$ as $\varepsilon\rightarrow 0$ to get the upper bound with $\bar{\eta}^{\varepsilon}(dz)\doteq\int_{\partial\mathcal{S}_{2}(\varepsilon)}\eta^{\varepsilon}(dz|u)J^{\varepsilon}(u)r^{\varepsilon}(\sqrt{\varepsilon}u)du.$ When combined with the lower bound which has an analogous proof, Lemma 10.12 follows. ∎ ### 10.5 Times to reach another equilibrium point after starting with general initial distribution ###### Lemma 10.13. For each $j\in L_{\rm{s}}$, there exist $\tilde{c}>0$ and $\varepsilon_{0}\in(0,1)$ such that for any distribution $\lambda^{\varepsilon}$ on $\partial B_{\delta}(O_{j})$, $P_{\lambda^{\varepsilon}}(\upsilon_{j}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{j}^{\varepsilon}>t)\leq e^{-\tilde{c}t}$ for all $t>0$ and $\varepsilon\in(0,\varepsilon_{0}).$ ###### Proof. We give the proof for the case $j=1$. We first show for any $r\in(0,1)$ there is $\varepsilon_{0}>0$ such that for any $\varepsilon\in(0,\varepsilon_{0})$ and $\lambda^{\varepsilon},\theta^{\varepsilon}\in\mathcal{P}(\partial B_{\delta}(O_{1}))$ $E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}/E_{\theta^{\varepsilon}}\upsilon_{1}^{\varepsilon}\geq r.$ (10.13) We use that $\upsilon_{1}^{\varepsilon}$ can be decomposed into $\bar{\upsilon}_{1}^{\varepsilon}+\hat{\upsilon}_{1}^{\varepsilon}$, where $\bar{\upsilon}_{1}^{\varepsilon}$ is the first hitting time to $\partial B_{2\delta}(O_{1})$. Since by standard large deviation theory the exponential growth rate of the expected value of $\upsilon_{1}^{\varepsilon}$ is strictly greater than that of $\bar{\upsilon}_{1}^{\varepsilon}$ (uniformly in the initial distribution) $E_{\lambda^{\varepsilon}}\bar{\upsilon}_{1}^{\varepsilon}$ (respectively $E_{\theta^{\varepsilon}}\bar{\upsilon}_{1}^{\varepsilon}$) is negligible compared to $E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}$ (respectively $E_{\theta^{\varepsilon}}\upsilon_{1}^{\varepsilon}$), and so it is enough to show $E_{\lambda^{\varepsilon}}\hat{\upsilon}_{1}^{\varepsilon}/E_{\theta^{\varepsilon}}\hat{\upsilon}_{1}^{\varepsilon}\geq r.$ Owing to Lemma 10.11 (and in particular because $\kappa>0$) the contribution to either $E_{\lambda^{\varepsilon}}\hat{\upsilon}_{1}^{\varepsilon}$ or $E_{\theta^{\varepsilon}}\hat{\upsilon}_{1}^{\varepsilon}$ from trajectories that reach $\partial B_{2\delta}(O_{1})$ before $\partial\mathcal{S}_{2}(\varepsilon)$ can be neglected. Using Lemma 10.12 and the strong Markov property gives $\inf_{w_{1},w_{2}\in\partial\mathcal{S}_{2}(\varepsilon)}\frac{E_{w_{1}}\hat{\upsilon}_{1}^{\varepsilon}}{E_{w_{2}}\hat{\upsilon}_{1}^{\varepsilon}}\geq\frac{[1-s^{\varepsilon}K/(1-\alpha)]}{[1+s^{\varepsilon}K/(1-\alpha)]},$ and the lower bound follows since $s^{\varepsilon}\rightarrow 0$. We next claim that a suitable bound can be found for $P_{\lambda^{\varepsilon}}(\hat{\upsilon}_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}>t)$. Recall that $u^{\varepsilon}\in\mathcal{P}(\partial B_{2\delta}(O_{1}))$ is the stationary probability for $\psi^{\varepsilon}$ defined in (10.5). Let $\beta^{\varepsilon}$ be the probability measure on $\partial B_{\delta}(O_{1})$ obtained by integrating the transition kernel $\psi_{1}^{\varepsilon}$ with respect to $u^{\varepsilon}$, and note that integrating $\psi_{2}^{\varepsilon}$ with respect to $\beta^{\varepsilon}$ returns $u^{\varepsilon}$. Since the diffusion matrix is uniformly nondegenerate, by using well known “Gaussian type” bounds on the transition density for the process [2] there are $K\in(0,\infty)$ and $p\in(0,\infty)$ such that $P_{x}\left\\{X_{\theta}^{\varepsilon}\in A|X_{\theta}^{\varepsilon}\in\partial B_{2\delta}(O_{1})\right\\}\leq Km(A)/\varepsilon^{p}$ for all $x\in\partial B_{\delta}(O_{1})$, where $m$ is the uniform measure on $\partial B_{2\delta}(O_{1})$ and $\theta=\inf\\{t>0:X_{t}^{\varepsilon}\in\partial B_{2\delta}(O_{1})\cup\mathcal{S}_{2}(\varepsilon)\\}$. Together with Lemmas 10.11 and 10.12, this implies that for all sufficiently small $\varepsilon>0$ and any bounded measurable function $h:\partial B_{2\delta}(O_{1})\rightarrow\mathbb{R}$, $\displaystyle\int_{\partial B_{2\delta}(O_{1})}\int_{\partial B_{\delta}(O_{1})}h(y)\psi_{2}^{\varepsilon}(dy|x)\lambda^{\varepsilon}(dx)$ $\displaystyle\leq 2\int_{\partial B_{2\delta}(O_{1})}\int_{\partial B_{\delta}(O_{1})}h(y)\psi_{2}^{\varepsilon}(dy|x)\beta^{\varepsilon}(dx)$ $\displaystyle\leq 2\int_{\partial B_{2\delta}(O_{1})}h(y)u^{\varepsilon}(dy).$ Using the last display for the first inequality, (10.13) for the second, that $\bar{\upsilon}_{1}^{\varepsilon}$ is small compared with $\hat{\upsilon}_{1}^{\varepsilon}$ for the third and Lemma 10.6 for the last, there is $\varepsilon_{1}>0$ such that $\displaystyle P_{\lambda^{\varepsilon}}(\hat{\upsilon}_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}>t)$ $\displaystyle=E_{\lambda^{\varepsilon}}(P_{X_{\bar{\upsilon}_{1}^{\varepsilon}}^{\varepsilon}}(\hat{\upsilon}_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}>t))\leq 2P_{u^{\varepsilon}}(\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}>t)$ $\displaystyle\leq 2P_{u^{\varepsilon}}(\upsilon_{1}^{\varepsilon}/E_{\beta^{\varepsilon}}\upsilon_{1}^{\varepsilon}>t/2)\leq 2P_{u^{\varepsilon}}(\upsilon_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}>t/4)\leq 2e^{-\tilde{c}t/4}$ for all $\varepsilon\in(0,\varepsilon_{1})$ and $t\geq 0$. Since as noted previously $E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}\geq E_{\lambda^{\varepsilon}}\bar{\upsilon}_{1}^{\varepsilon}$ and since by [6, Theorem 4 and Corollary 1] there exists $\varepsilon_{2}\in(0,1)$ such that $P_{\lambda^{\varepsilon}}(\bar{\upsilon}_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\bar{\upsilon}_{1}^{\varepsilon}>t)\leq 2e^{-t/2}$ for any $t>0$ and $\varepsilon\in(0,\varepsilon_{2})$, we conclude that for any $t>0$$P_{\lambda^{\varepsilon}}(\bar{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}>t/2)\leq P_{\lambda^{\varepsilon}}(\bar{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\bar{\upsilon}^{\varepsilon}_{1}>t/2)\leq 2e^{-t/2}.$ The conclusion of the lemma follows from these two bounds and $\displaystyle P_{\lambda^{\varepsilon}}(\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}$ $\displaystyle>t)\leq P_{\lambda^{\varepsilon}}(\bar{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}>t/2)+P_{\lambda^{\varepsilon}}(\hat{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}>t/2).$ ∎ ###### Lemma 10.14. For any $j\in L_{\rm{s}}$ and any distribution $\lambda^{\varepsilon}$ on $\partial B_{\delta}(O_{j})$, $\upsilon_{j}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{j}^{\varepsilon}$ converges in distribution to an Exp(1) random variable under $P_{\lambda^{\varepsilon}}.$ Moreover, $E_{\lambda^{\varepsilon}}e^{it\upsilon_{j}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{j}^{\varepsilon}}\rightarrow 1/(1-it)$ uniformly on any compact set in $\mathbb{R}$. ###### Proof. We give the proof for the case $j=1$. Recall that $E_{u^{\varepsilon}}e^{it\upsilon_{1}^{\varepsilon}/E_{u^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\rightarrow 1/(1-it)$ uniformly on any compact set in $\mathbb{R}$ as $\varepsilon\rightarrow 0$ from Remark 10.5. We would like to show that $E_{\lambda^{\varepsilon}}e^{it\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\rightarrow 1/(1-it)$ uniformly on any compact set in $\mathbb{R}$. Since $\upsilon_{1}^{\varepsilon}=\bar{\upsilon}^{\varepsilon}_{1}+\hat{\upsilon}^{\varepsilon}_{1}$ with $\bar{\upsilon}^{\varepsilon}_{1}$ the first hitting time to $\partial B_{2\delta}(O_{1}),$ we know that $E_{\lambda^{\varepsilon}}\bar{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}\rightarrow 0$ and thus $E_{\lambda^{\varepsilon}}\hat{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}\rightarrow 1.$ Observe that $\displaystyle E_{\lambda^{\varepsilon}}e^{it\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}=E_{\lambda^{\varepsilon}}\left[e^{it\bar{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\cdot E_{X^{\varepsilon}\left(\bar{\upsilon}^{\varepsilon}_{1}\right)}\left(e^{it\hat{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\right)\right],$ $E_{\lambda^{\varepsilon}}\left[E_{X^{\varepsilon}\left(\bar{\upsilon}^{\varepsilon}_{1}\right)}\left(e^{it\hat{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\right)\right]\leq\frac{[1+s^{\varepsilon}K/(1-\alpha)]}{[1-s^{\varepsilon}K/(1-\alpha)]}E_{u^{\varepsilon}}e^{it\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\rightarrow 1/(1-it)$ and $E_{\lambda^{\varepsilon}}\left[E_{X^{\varepsilon}\left(\bar{\upsilon}^{\varepsilon}_{1}\right)}\left(e^{it\hat{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\right)\right]\geq\frac{[1-s^{\varepsilon}K/(1-\alpha)]}{[1+s^{\varepsilon}K/(1-\alpha)]}E_{u^{\varepsilon}}e^{it\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\rightarrow 1/(1-it).$ Since $E_{\lambda^{\varepsilon}}\bar{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}\rightarrow 0$ and $e^{ix}$ is a bounded and continuous function, a conditioning argument gives $\left|E_{\lambda^{\varepsilon}}e^{it\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}-E_{\lambda^{\varepsilon}}\left[E_{X^{\varepsilon}\left(\bar{\upsilon}^{\varepsilon}_{1}\right)}\left(e^{it\hat{\upsilon}^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\right)\right]\right|\leq E_{\lambda^{\varepsilon}}\left|e^{it\bar{\upsilon}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}-1\right|\rightarrow 0.$ We conclude that $E_{\lambda^{\varepsilon}}e^{it\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}}\rightarrow 1/(1-it)$ uniformly on any compact set in $\mathbb{R}$. ∎ ### 10.6 Return times (single cycles) In this subsection, we will extend all the three results to return times for the single cycle case (i.e. when $h_{1}>w$). ###### Lemma 10.15. There exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ and any distribution $\lambda^{\varepsilon}$ on $\partial B_{\delta}(O_{1}),$ $\lim_{\varepsilon\rightarrow 0}\varepsilon\log E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}=\min_{y\in\cup_{k\in L\setminus\\{1\\}}\partial B_{\delta}(O_{k})}V(O_{1},y).$ ###### Proof. We have $E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}=E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}+E_{\lambda^{\varepsilon}}(\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})$, and by Lemma 10.2 we know that $\lim_{\varepsilon\rightarrow 0}\varepsilon\log E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}=\min_{y\in\cup_{k\in L\setminus\\{1\\}}\partial B_{\delta}(O_{k})}V(O_{1},y).$ Moreover, observe that $W(O_{j})>W(O_{1})$ for any $j\in L\setminus\\{1\\}$ due to Remark 3.14. Note that $\upsilon_{1}^{\varepsilon}$ as defined in (10.1) coincides with $\sigma_{0}^{\varepsilon}$ defined in (3.4). We can therefore apply Remark 7.22 with $f=0$, $A=M$ and $\eta=[\min_{j\in L\setminus\\{1\\}}W(O_{j})-W(O_{1})]/3,$ we find that there exists $\delta_{1}\in(0,1)$ such that for any $\delta\in(0,\delta_{1})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}\left(\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon}\right)\right)$ $\displaystyle\geq\min_{j\in L\setminus\\{1\\}}W(O_{j})-W(O_{1})-\min_{j\in L\setminus\\{1\\}}V(O_{1},O_{j})-\eta$ $\displaystyle=-\min_{j\in L\setminus\\{1\\}}V(O_{1},O_{j})+2\eta.$ On the other hand, by continuity of $V(O_{1},\cdot),$ for this given $\eta,$ there exists $\delta_{2}\in(0,1)$ such that for any $\delta\in(0,\delta_{2})$ $\min_{y\in\cup_{k\in L\setminus\\{1\\}}\partial B_{\delta}(O_{k})}V(O_{1},y)\geq\min_{j\in L\setminus\\{1\\}}V(O_{1},O_{j})-\eta.$ Thus, for any $\delta\in(0,\delta_{0})$ with $\delta_{0}\doteq\delta_{1}\wedge\delta_{2}$ $\displaystyle\limsup_{\varepsilon\rightarrow 0}\varepsilon\log E_{\lambda^{\varepsilon}}(\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})$ $\displaystyle\leq\limsup_{\varepsilon\rightarrow 0}\varepsilon\log\left(\sup_{z\in\partial B_{\delta}(O_{1})}E_{z}(\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})\right)$ $\displaystyle\leq\min_{j\in L\setminus\\{1\\}}V(O_{1},O_{j})-2\eta\leq\min_{y\in\cup_{k\in L\setminus\\{1\\}}\partial B_{\delta}(O_{k})}V(O_{1},y)-\eta$ $\displaystyle=\lim_{\varepsilon\rightarrow 0}\varepsilon\log E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}-\eta$ and $\displaystyle\lim_{\varepsilon\rightarrow 0}\varepsilon\log E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}$ $\displaystyle=\lim_{\varepsilon\rightarrow 0}\varepsilon\log E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}=\min_{y\in\cup_{k\in L\setminus\\{1\\}}\partial B_{\delta}(O_{k})}V(O_{1},y).$ ∎ ###### Lemma 10.16. Given $\delta>0$ sufficiently small, and for any distribution $\lambda^{\varepsilon}$ on $\partial B_{\delta}(O_{1}),$ there exist $\tilde{c}>0$ and $\varepsilon_{0}\in(0,1)$ such that $P_{\lambda^{\varepsilon}}(\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}>t)\leq e^{-\tilde{c}t}$ for all $t\geq 1$ and $\varepsilon\in(0,\varepsilon_{0}).$ ###### Proof. For any $t>0,$ $P_{\lambda^{\varepsilon}}(\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}>t)\leq P_{\lambda^{\varepsilon}}(\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}>t/2)+P_{\lambda^{\varepsilon}}((\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}>t/2)$. It is easy to see that the first term has this sort of bound due to Lemma 10.13 and $E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\geq E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}.$ It suffices to show that this sort of bound holds for the second term, namely, there exists a constant $\tilde{c}>0$ such that $P_{\lambda^{\varepsilon}}\left((\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}>t\right)\leq e^{-\tilde{c}t}$ for all $t\in[0,\infty)$ and $\varepsilon$ sufficiently small. By Chebyshev’s inequality, $P_{\lambda^{\varepsilon}}\left((\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}>t\right)=P_{\lambda^{\varepsilon}}(e^{(\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}>e^{t})\leq e^{-t}E_{\lambda^{\varepsilon}}e^{(\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}},$ and it therefore suffices to prove that $E_{\lambda^{\varepsilon}}e^{(\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}$ is less than a constant for all $\varepsilon$ sufficiently small. Observe that $\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon}=\sum\nolimits_{j\in L\setminus\\{1\\}}\sum\nolimits_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k),$ where $N_{j}$ is the number of visits of $\partial B_{\delta}(O_{j})$, and $\upsilon_{j}^{\varepsilon}(k)$ is the $k$-th copy of the first hitting time to $\cup_{k\in L\setminus\\{j\\}}\partial B_{\delta}(O_{k})$ after starting from $\partial B_{\delta}(O_{j}).$ If we consider $\partial B_{\delta}(O_{j})$ as the starting location of a regenerative cycle, as was done previously in the paper for $\partial B_{\delta}(O_{1})$, then there will be a unique stationary distribution, and if the process starts with that as the initial distribution then the times $\upsilon_{j}^{\varepsilon}(k)$ are independent from each other and from the number of returns to $\partial B_{\delta}(O_{j})$ before first visiting $\partial B_{\delta}(O_{1})$. While these random times as used here do not arise from starting with such a distribution, we can use Lemma 10.12 to bound the error in terms of a multiplicative factor that is independent of $\varepsilon$ for small $\varepsilon>0$, and thereby justify treating $N_{j}$ as though it is independent of the $\upsilon_{j}^{\varepsilon}(k)$. Recalling that $l\doteq|L|$, $\displaystyle E_{\lambda^{\varepsilon}}e^{(\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}$ $\displaystyle=E_{\lambda^{\varepsilon}}\prod\nolimits_{j\in L\setminus\\{1\\}}e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}$ $\displaystyle\leq\prod\nolimits_{j\in L\setminus\\{1\\}}\left(E_{\lambda^{\varepsilon}}\left[e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)(l-1)/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\right]\right)^{1/(l-1)},$ where we use the generalized Hölder’s inequality for the last line. Thus, if we can show for each $j\in L\setminus\\{1\\}$ that $E_{\lambda^{\varepsilon}}\exp[{(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k))(l-1)/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}]$ is less than a constant for all $\varepsilon$ sufficiently small, then we are done. Such an estimate is straightforward for the case of an unstable equilibrium, i.e., for $j\in L\backslash L_{\rm{s}}$, and so we focus on the case $j\in L_{\rm{s}}\backslash\\{1\\}$. For this case, we apply Lemma 10.13 to find that there exists $\tilde{{c}}>0$ and $\varepsilon_{0}\in(0,1)$ such that for any $j\in L$ and any distribution $\tilde{\lambda}^{\varepsilon}$ on $\partial B(O_{j}),$ $P_{\tilde{\lambda}^{\varepsilon}}(\upsilon_{j}^{\varepsilon}/E_{\tilde{\lambda}^{\varepsilon}}\upsilon_{j}^{\varepsilon}>t)\leq e^{-\tilde{{c}}t}$ (10.14) for any $t>0$ and $\varepsilon\in(0,\varepsilon_{0}).$ Hence, given any $\eta>0$, there is $\bar{\varepsilon}_{0}\in(0,\varepsilon_{0})$ such that for all $\varepsilon\in(0,\bar{\varepsilon}_{0})$ and any $j\in L\setminus\\{1\\}$ $\displaystyle E_{\lambda^{\varepsilon}}\left[e^{\upsilon_{j}^{\varepsilon}(l-1)/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\right]$ $\displaystyle\leq 1+\int_{1}^{\infty}P_{\lambda^{\varepsilon}}(e^{(l-1)\upsilon_{j}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}>t)dt$ $\displaystyle\leq 1+\int_{1}^{\infty}P_{\lambda^{\varepsilon}}\left(\upsilon_{j}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{j}^{\varepsilon}>\log t\cdot E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}/((l-1)E_{\lambda^{\varepsilon}}\upsilon_{j}^{\varepsilon})\right)dt$ $\displaystyle\leq 1+\int_{1}^{\infty}t^{-\tilde{{c}}E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}/((l-1)E_{\lambda^{\varepsilon}}\upsilon_{j}^{\varepsilon})}dt$ $\displaystyle=1+\left(\tilde{{c}}E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}/((l-1)E_{\lambda^{\varepsilon}}\upsilon_{j}^{\varepsilon})-1\right)^{-1}\leq 1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)},$ where the last inequality comes from Lemma 10.2 and Lemma 10.15, and by picking the range of $\varepsilon$ small if it needs to be. By using induction and a conditioning argument, it follows that for any $\eta>0$, for any $j\in L\setminus\\{1\\}$ and for any $n\in\mathbb{N},$ $E_{\lambda^{\varepsilon}}\left[e^{\left(\sum_{k=1}^{n}\upsilon_{j}^{\varepsilon}(k)\right)(l-1)/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\right]\leq\left(1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)}\right)^{n}.$ This implies that $E_{\lambda^{\varepsilon}}\left[e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)(l-1)/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\right]\leq E_{\lambda^{\varepsilon}}\left[\left(1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)}\right)^{N_{j}}\right].$ The next thing we need to know is the distribution of $N_{j}$, i.e., $P_{\lambda^{\varepsilon}}(N_{j}=n)$ for $n\in\mathbb{N}$. Following a similar argument as in the proof of Lemma 7.3 and the proof of Lemma 7.6, for sufficiently small $\varepsilon>0$ we find $\displaystyle P_{\lambda^{\varepsilon}}(N_{j}=n)$ $\displaystyle\leq\left(1\wedge e^{-\frac{1}{\varepsilon}(W(O_{j})-W(O_{1}\cup O_{j})-h_{1}-\eta)}\right)(1-q_{j})^{n-1}q_{j},$ where $\displaystyle q_{j}\doteq\frac{\inf_{x\in\partial B_{\delta}(O_{j})}P_{x}(\tilde{{T_{1}}}<{\tilde{T}}_{j}^{+})}{1-\sup_{y\in\partial B_{\delta}(O_{j})}p(y,\partial B_{\delta}(O_{j}))}\geq e^{-\frac{1}{\varepsilon}(W(O_{1})-W(O_{1}\cup O_{j})-h_{j}+\eta)}.$ (10.15) Therefore, $\displaystyle E_{\lambda^{\varepsilon}}\left[e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)(l-1)/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\right]$ $\displaystyle\quad\leq E_{\lambda^{\varepsilon}}\left[\left(1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)}\right)^{N_{j}}\right]=\sum\nolimits_{n=1}^{\infty}\left(1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)}\right)^{n}P_{\lambda^{\varepsilon}}(N_{j}=n)$ $\displaystyle\quad\leq\sum_{n=1}^{\infty}\left(1\wedge e^{-\frac{1}{\varepsilon}(W(O_{j})-W(O_{1}\cup O_{j})-h_{1}-\eta)}\right)\left(1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)}\right)^{n}(1-q_{j})^{n-1}q_{j}$ $\displaystyle\quad=\frac{\left(1\wedge e^{-\frac{1}{\varepsilon}(W(O_{j})-W(O_{1}\cup O_{j})-h_{1}-\eta)}\right)q_{j}\left(1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)}\right)}{1-\left(1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)}\right)(1-q_{j})}$ $\displaystyle\quad\leq\frac{\left(1\wedge e^{-\frac{1}{\varepsilon}(W(O_{j})-W(O_{1}\cup O_{j})-h_{1}-\eta)}\right)\left(1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)}\right)}{-e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)}/q_{j}+1}$ $\displaystyle\quad\leq\frac{\left(1\wedge e^{-\frac{1}{\varepsilon}(W(O_{j})-W(O_{1}\cup O_{j})-h_{1}-\eta)}\right)\left(1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)}\right)}{-e^{-\frac{1}{\varepsilon}(h_{1}+W(O_{1}\cup O_{j})-W(O_{1})-2\eta)}+1}.$ The second equality holds since $h_{1}>w\geq h_{j}$ and (10.15) imply $(1-q_{j})(1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)})<1$ for all $\varepsilon$ sufficiently small; the last inequality is from (10.15). Then we use the fact that for $x\in(0,1/2)$, $1/(1-x)\leq 1+2x$ to find that $\displaystyle E_{\lambda^{\varepsilon}}\left[e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)(l-1)/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}}\right]$ $\displaystyle\leq\left(1\wedge e^{-\frac{1}{\varepsilon}(W(O_{j})-W(O_{1}\cup O_{j})-h_{1}-\eta)}\right)\left(1+e^{-\frac{1}{\varepsilon}(h_{1}-h_{j}-\eta)}\right)\left(1+2e^{-\frac{1}{\varepsilon}(h_{1}+W(O_{1}\cup O_{j})-W(O_{1})-2\eta)}\right)$ $\displaystyle\leq\left(1\wedge e^{-\frac{1}{\varepsilon}(W(O_{j})-W(O_{1}\cup O_{j})-h_{1}-\eta)}\right)\left(1+5e^{-\frac{1}{\varepsilon}(h_{1}+W(O_{1}\cup O_{j})-W(O_{1})-2\eta)}\right)$ (10.16) $\displaystyle\leq 1\cdot 6=6.$ The third inequality holds due to the fact that $W(O_{1})\geq W(O_{1}\cup O_{j})+h_{j}$ and the last inequality comes from the assumption that $h_{1}>w$ and by picking $\eta$ to be smaller than $(h_{1}-w)/2$. This completes the proof. ∎ ###### Lemma 10.17. Given $\delta>0$ sufficiently small, and for any distribution $\lambda^{\varepsilon}$ on $\partial B_{\delta}(O_{1})$, $\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}$ converges in distribution to an Exp(1) random variable under $P_{\lambda^{\varepsilon}}.$ Moreover, $E_{\lambda^{\varepsilon}}(e^{it\left(\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)})\rightarrow 1/(1-it)$ uniformly on any compact set in $\mathbb{R}$. ###### Proof. Note that $E_{\lambda^{\varepsilon}}\left(e^{it\left(\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)}\right)=E_{\lambda^{\varepsilon}}\left(e^{it\left(\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)}E_{X^{\varepsilon}(\upsilon_{1}^{\varepsilon})}\left(e^{it\left((\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)}\right)\right).$ Since $E_{\lambda^{\varepsilon}}\left(e^{it\left(\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)}\right)=E_{\lambda^{\varepsilon}}\left(e^{it\left(E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)\left(\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}\right)}\right)$ and we know that $E_{\lambda^{\varepsilon}}\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\rightarrow 1$ from the proof of Lemma 10.15, by applying Lemma 10.14 we have $E_{\lambda^{\varepsilon}}(e^{it\left(\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)})\rightarrow 1/(1-it)$ uniformly on any compact set in $\mathbb{R}$. Also $\displaystyle\left|E_{\lambda^{\varepsilon}}\left(e^{it\left(\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)}\right)-E_{\lambda^{\varepsilon}}\left(e^{it\left(\upsilon_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)}\right)\right|\leq E_{\lambda^{\varepsilon}}\left|E_{X^{\varepsilon}(\upsilon_{1}^{\varepsilon})}\left(e^{it\left((\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)}\right)-1\right|,$ where the right hand side converges to $0$ using $E_{\lambda^{\varepsilon}}(\tau_{1}^{\varepsilon}-\upsilon_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\rightarrow 0$ and the dominated convergence theorem. The convergence of $\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}$ to an Exp(1) random variable under $P_{\lambda^{\varepsilon}}$ and uniform convergence of $E_{\lambda^{\varepsilon}}(e^{it\left(\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\right)})$ to $1/(1-it)$ on compact set in $\mathbb{R}$ follows. ∎ ### 10.7 Return times (multicycles) In this subsection, we will extend all the three results to multi regenerative cycles (when $w\geq h_{1}$). Recall that the multicycle times $\hat{\tau}^{\varepsilon}_{i}$ are defined according to (6.4) where $\\{\mathbf{M}^{\varepsilon}_{i}\\}_{i\in\mathbb{N}}$ is a sequence of independent and geometrically distributed random variables with parameter $e^{-m/\varepsilon}$ for some $m>0$ such that $m+h_{1}>w$. In addition, $\\{\mathbf{M}^{\varepsilon}_{i}\\}$ is independent of $\\{\tau^{\varepsilon}_{n}\\}$. ###### Lemma 10.18. There exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ and any distribution $\lambda^{\varepsilon}$ on $\partial B_{\delta}(O_{1}),$ $\lim_{\varepsilon\rightarrow 0}\varepsilon\log E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}=m+\min_{y\in\cup_{k\in L\setminus\\{1\\}}\partial B_{\delta}(O_{k})}V(O_{1},y).$ ###### Proof. Since $\\{\mathbf{M}^{\varepsilon}_{i}\\}$ is independent of $\\{\tau^{\varepsilon}_{n}\\}$ and $E_{\lambda^{\varepsilon}}\mathbf{M}^{\varepsilon}_{i}=e^{m/\varepsilon}$, we apply Lemma 10.15 to complete the proof. ∎ ###### Lemma 10.19. Given $\delta>0,$ for any distribution $\lambda^{\varepsilon}$ on $\partial B_{\delta}(O_{1}),$ there exist $\tilde{c}>0$ and $\varepsilon_{0}\in(0,1)$ such that $P_{\lambda^{\varepsilon}}(\hat{\tau}_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}>t)\leq e^{-\tilde{c}t}$ for all $t\geq 1$ and $\varepsilon\in(0,\varepsilon_{0}).$ ###### Proof. We divide the multicycle into a sum of two terms. The first term is the sum of all the hitting times to $\cup_{j\in L\setminus\\{1\\}}\partial B_{\delta}(O_{j})$, and the second term is the sum of all residual times. That is, $\hat{\tau}_{1}^{\varepsilon}=\hat{\upsilon}_{1}^{\varepsilon}+(\hat{\tau}_{1}^{\varepsilon}-\hat{\upsilon}_{1}^{\varepsilon})$, where $\hat{\upsilon}_{1}^{\varepsilon}=\sum\nolimits_{i=1}^{\mathbf{M}^{\varepsilon}_{1}}\upsilon_{1}^{\varepsilon}(i)\text{ and }\hat{\tau}_{1}^{\varepsilon}-\hat{\upsilon}_{1}^{\varepsilon}=\sum\nolimits_{i=1}^{\mathbf{M}^{\varepsilon}_{1}}\left(\sum\nolimits_{j\in L\setminus\\{1\\}}\sum\nolimits_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(i,k)\right).$ As discussed many times, it suffices to show that there exist $\tilde{c}>0$ and $\varepsilon_{0}\in(0,1)$ such that $P_{\lambda^{\varepsilon}}\left(\hat{\upsilon}_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}>t\right)\leq e^{-\tilde{c}t}\text{ and }P_{\lambda^{\varepsilon}}\left((\hat{\tau}_{1}^{\varepsilon}-\hat{\upsilon}_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}>t\right)\leq e^{-\tilde{c}t}$ for all $t\geq 1$ and $\varepsilon\in(0,\varepsilon_{0}).$ The first bound is relatively easy since $\upsilon_{1}^{\varepsilon}(i)$ is a sum of approximate exponentials with a tail bound of the given sort, and since the sum of geometrically many independent and identically distributed exponentials is again an exponential distribution. For the second bound, we use Chebyshev’s inequality again as in the proof of Lemma 10.16 to find that it suffices to prove that $E_{\lambda^{\varepsilon}}e^{(\hat{\tau}_{1}^{\varepsilon}-\hat{\upsilon}_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}$ is less than a constant for all $\varepsilon$ sufficiently small. Now due to the independence of $\mathbf{M}^{\varepsilon}_{1}$ and $\\{\upsilon_{j}^{\varepsilon}(i,k)\\}$, we have $\displaystyle E_{\lambda^{\varepsilon}}e^{(\hat{\tau}_{1}^{\varepsilon}-\hat{\upsilon}_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}$ $\displaystyle\qquad=\sum\nolimits_{i=1}^{\infty}\left(E_{\lambda^{\varepsilon}}\left[\prod\nolimits_{j\in L\setminus\\{1\\}}e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}\right]\right)^{i}\cdot P_{\lambda^{\varepsilon}}(\mathbf{M}^{\varepsilon}_{1}=i)$ $\displaystyle\qquad=e^{-m/\varepsilon}\cdot E_{\lambda^{\varepsilon}}\left[\prod\nolimits_{j\in L\setminus\\{1\\}}e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}\right]$ $\displaystyle\qquad\qquad\cdot\sum\nolimits_{i=1}^{\infty}\left(E_{\lambda^{\varepsilon}}\left[\prod\nolimits_{j\in L\setminus\\{1\\}}e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}\right](1-e^{-m/\varepsilon})\right)^{i-1}.$ (10.17) To do a further computation, we have to at least make sure that $E_{\lambda^{\varepsilon}}\left[\prod\nolimits_{j\in L\setminus\\{1\\}}e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}\right](1-e^{-m/\varepsilon})<1.$ (10.18) To see this, we first use the generalized Hölder’s inequality to find $\displaystyle E_{\lambda^{\varepsilon}}\left[\prod\nolimits_{j\in L\setminus\\{1\\}}e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}\right]\leq\prod\nolimits_{j\in L\setminus\\{1\\}}\left(E_{\lambda^{\varepsilon}}\left[e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)(l-1)/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}\right]\right)^{1/(l-1)}.$ Moreover, since $m+h_{1}>w$ and $E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}=E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}\cdot E_{\lambda^{\varepsilon}}\mathbf{M}^{\varepsilon}_{1}=e^{m/\varepsilon}E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}$, by the same argument that gives (10.6), for any $\eta>0$ and $j\in L\setminus\\{1\\}$ $\displaystyle E_{\lambda^{\varepsilon}}\left[e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)(l-1)/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}\right]$ $\displaystyle\quad\leq\left(1\wedge e^{-\frac{1}{\varepsilon}(W(O_{j})-W(O_{1}\cup O_{j})-h_{1}-\eta)}\right)\left(1+5e^{-\frac{1}{\varepsilon}(m+h_{1}+W(O_{1}\cup O_{j})-W(O_{1})-2\eta)}\right).$ Therefore, $\displaystyle E_{\lambda^{\varepsilon}}\left[\prod\nolimits_{j\in L\setminus\\{1\\}}e^{\left(\sum_{k=1}^{N_{j}}\upsilon_{j}^{\varepsilon}(k)\right)/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}\right](1-e^{-m/\varepsilon})\leq\prod\nolimits_{j\in L\setminus\\{1\\}}s_{j}^{1/(l-1)},$ with $\displaystyle s_{j}\doteq\left(1\wedge e^{-\frac{1}{\varepsilon}(W(O_{j})-W(O_{1}\cup O_{j})-h_{1}-\eta)}\right)\left(1+5e^{-\frac{1}{\varepsilon}(m+h_{1}+W(O_{1}\cup O_{j})-W(O_{1})-2\eta)}\right)(1-e^{-m/\varepsilon}).$ Using $(a\wedge b)(c+d)\leq ac+bd$ for positive numbers $a,b,c,d$, $s_{j}\leq\left(1+5e^{-\frac{1}{\varepsilon}(m+W(O_{j})-W(O_{1})-3\eta)}\right)(1-e^{-m/\varepsilon})\leq 1-e^{-m/\varepsilon}/2,$ where we use $W(O_{j})>W(O_{1})$ for the second inequality and pick the range of $\varepsilon$ small if it needs to be. Thus (10.18) holds, and by (10.7) $E_{\lambda^{\varepsilon}}e^{(\hat{\tau}_{1}^{\varepsilon}-\hat{\upsilon}_{1}^{\varepsilon})/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}}\leq e^{-m/\varepsilon}\cdot 2\sum_{i=1}^{\infty}\left(1-e^{-m/\varepsilon}/2\right)^{i-1}=\frac{2e^{-m/\varepsilon}}{1-\left(1-e^{-m/\varepsilon}/2\right)}=4.$ We complete the proof. ∎ ###### Lemma 10.20. Given $\delta>0,$ for any distribution $\lambda^{\varepsilon}$ on $\partial B_{\delta}(O_{1})$, $\hat{\tau}_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon}$ converges in distribution to an Exp(1) random variable under $P_{\lambda^{\varepsilon}}.$ ###### Proof. Let $\mathbf{M}$ be a geometrically distributed random variables with parameter $p\in(0,1)$ and assume that it is independent of $\\{\tau^{\varepsilon}_{n}\\}$. Then $E_{\lambda^{\varepsilon}}\left(\sum\nolimits_{n=1}^{\mathbf{M}}\tau^{\varepsilon}_{n}\right)=E_{\lambda^{\varepsilon}}\tau^{\varepsilon}_{1}/p$ and $\displaystyle E_{\lambda^{\varepsilon}}e^{it\left(\left(p\sum_{n=1}^{\mathbf{M}}\tau^{\varepsilon}_{n}\right)/E_{\lambda^{\varepsilon}}\tau^{\varepsilon}_{1}\right)}=\sum_{k=1}^{\infty}\left(E_{\lambda^{\varepsilon}}e^{it\left(p\tau^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\tau^{\varepsilon}_{1}\right)}\right)^{k}(1-p)^{k-1}p=\frac{pE_{\lambda^{\varepsilon}}e^{it\left(p\tau^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\tau^{\varepsilon}_{1}\right)}}{1-(1-p)E_{\lambda^{\varepsilon}}e^{it\left(p\tau^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\tau^{\varepsilon}_{1}\right)}}.$ Given any fixed $t\in\mathbb{R}$, consider $f_{\varepsilon}(p)=\frac{pE_{\lambda^{\varepsilon}}e^{it\left(p\tau^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\tau^{\varepsilon}_{1}\right)}}{1-(1-p)E_{\lambda^{\varepsilon}}e^{it\left(p\tau^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\tau^{\varepsilon}_{1}\right)}}\text{ and }f(p)=\frac{1}{1-it}.$ According to Lemma 10.17, $E_{\lambda^{\varepsilon}}e^{it\left(\tau^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\tau^{\varepsilon}_{1}\right)}\rightarrow 1/(1-it)$ uniformly on any compact set in $\mathbb{R}$. This implies that $f_{\varepsilon}(p)=\frac{pE_{\lambda^{\varepsilon}}e^{it\left(p\tau^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\tau^{\varepsilon}_{1}\right)}}{1-(1-p)E_{\lambda^{\varepsilon}}e^{it\left(p\tau^{\varepsilon}_{1}/E_{\lambda^{\varepsilon}}\tau^{\varepsilon}_{1}\right)}}\rightarrow\frac{p/(1-itp)}{1-(1-p)/(1-itp)}=\frac{1}{1-it}=f(p)$ uniformly on $p\in(0,1)$. Therefore, if we consider $p^{\varepsilon}\doteq e^{-m/{\varepsilon}}\rightarrow 0$, it follows from the uniform (in $p$) convergence that $E_{\lambda^{\varepsilon}}e^{it(\hat{\tau}_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\hat{\tau}_{1}^{\varepsilon})}=f_{\varepsilon}(p^{\varepsilon})\rightarrow f(0)=\frac{1}{1-it}.$ We complete the proof. ∎ ## 11 Sketch of the Proof of Conjecture 4.11 for a Special Case In this section we outline the proof of the upper bound on the decay rate (giving a lower bound on the variance per unit time) that complements Theorem 4.5 for a special case. Consider $U:\mathbb{R}\rightarrow\mathbb{R}$ shown as in Figure 3. Figure 3: Asymmetric well for lower bound In particular, assume $U$ is a bounded $C^{2}$ function satisfying the following conditions: ###### Condition 11.1. * • $U$ is defined on a compact interval $D\doteq[\bar{x}_{L},\bar{x}_{R}]\subset\mathbb{R}$ and extends periodically as a $C^{2}$ function. * • $U$ has two local minima at $x_{L}$ and $x_{R}$ with values $U(x_{L})<U(x_{R})$ and $[x_{L}-\delta,x_{R}+\delta]\subset D$ for some $\delta>0$. * • $U$ has one local maximum at $0\in(x_{L},x_{R})$. * • $U(x_{L})=0,$ $U(0)=h_{L}$ and $U(x_{R})=h_{L}-h_{R}>0.$ * • $\inf_{x\in\partial D}U(x)>h_{L}.$ Consider the diffusion process $\\{X^{\varepsilon}_{t}\\}_{t\geq 0}$ satisfying the stochastic differential equation $dX^{\varepsilon}_{t}=-\nabla U\left(X^{\varepsilon}_{t}\right)dt+\sqrt{2\varepsilon}dW_{t},$ (11.1) where $W$ is a $1$-dimensional standard Wiener process. Then there are just two stable equilibrium points $O_{1}=x_{L}$ and $O_{2}=x_{R}$, and one unstable equilibrium point $O_{3}=0.$ Moreover, one easily finds that $V(O_{1},O_{2})=h_{L}$ and $V(O_{2},O_{1})=h_{R},$ and these give that $W(O_{1})=V(O_{2},O_{1})$, $W(O_{2})=V(O_{1},O_{2})$ and $W\left(O_{1}\cup O_{2}\right)=0$ (since $L_{\rm{s}}=\\{1,2\\},$ this implies that $G_{\rm{s}}(1)=\\{(2\rightarrow 1)\\}$ and $G_{\rm{s}}(2)=\\{(1\rightarrow 2)\\}$). Another observation is that $h_{1}\doteq\min_{\ell\in\mathcal{M}\setminus\\{1\\}}V\left(O_{1},O_{\ell}\right)=V\left(O_{1},O_{3}\right)=h_{L}$ in this model. If $f\equiv 0,$ then one obtains $\displaystyle R_{1}^{(1)}$ $\displaystyle\doteq\inf_{y\in A}V(O_{1},y)+W(O_{1})-W(O_{1})=\inf_{y\in A}V(O_{1},y);$ $\displaystyle R_{1}^{(2)}$ $\displaystyle\doteq 2\inf_{y\in A}V\left(O_{1},y\right)-h_{1}=2\inf_{y\in A}V\left(O_{1},y\right)-h_{L};$ $\displaystyle R_{2}^{(1)}$ $\displaystyle\doteq\inf_{y\in A}V(O_{2},y)+W(O_{2})-W(O_{1})=\inf_{y\in A}V(O_{2},y)+h_{L}-h_{R};$ $\displaystyle R_{2}^{(2)}$ $\displaystyle\doteq 2\inf_{y\in A}V\left(O_{2},y\right)+W\left(O_{2}\right)-2W\left(O_{1}\right)+0-W\left(O_{1}\cup O_{2}\right)$ $\displaystyle=2\inf_{y\in A}V\left(O_{2},y\right)+h_{L}-2h_{R}.$ Let $A\subset[0,\bar{x}_{R}]$ and assume that it contains a nonempty open interval, so that we are computing approximations to probabilities that are small under the stationary distribution (the case of bounded and continuous $f$ can be dealt with by approximation, as in the case of the upper bound on the decay rate). We first compute the bounds one would obtain from Theorem 4.5. Case I. If $x_{R}\in A,$ then $\inf_{y\in A}V(O_{1},y)=h_{L}$ and $\inf_{y\in A}V\left(O_{2},y\right)=0.$ Thus the decay rate of variance per unit time is bounded below by $\displaystyle\min_{j=1,2}\left[R_{j}^{(1)}\wedge R_{j}^{(2)}\right]=\min\left\\{h_{L},h_{L}-2h_{R}\right\\}=h_{L}-2h_{R}.$ Case II. If $A\subset[0,x_{R}-\delta]$ for some $\delta>0$ and $\delta<x_{R},$ then $\inf_{y\in A}V(O_{1},y)=h_{L}$ and $\inf_{y\in A}V\left(O_{2},y\right)>0$ (we denote it by $b\in(0,h_{R}]$). Thus the decay rate of variance per unit time is bounded below by $\displaystyle\min_{j=1,2}\left[R_{j}^{(1)}\wedge R_{j}^{(2)}\right]=\min\left\\{h_{L},h_{L}+2\left(b-h_{R}\right)\right\\}=h_{L}+2\left(b-h_{R}\right).$ Case III. If $A\subset[x_{R}+\delta,x^{\ast}]$ with $U(x^{\ast})=h_{L}$ for some $\delta>0$ and $\delta<x^{\ast}-x_{R},$ then $\inf_{y\in A}V(O_{1},y)=h_{L}+\inf_{y\in A}V\left(O_{2},y\right)$ and $\inf_{y\in A}V\left(O_{2},y\right)>0$ (we denote it by $b\in(0,h_{R}]$). Thus the decay rate of variance per unit time is bounded below by $\displaystyle\min_{j=1,2}\left[R_{j}^{(1)}\wedge R_{j}^{(2)}\right]=\min\left\\{h_{L}+b,h_{L}+2\left(b-h_{R}\right)\right\\}=h_{L}+2\left(b-h_{R}\right).$ Case IV. If $A\subset[x^{\ast}+\delta,\bar{x}_{R}]$ with $U(x^{\ast})=h_{L}$ for some $\delta>0$ and $x^{\ast}>x_{R},$ then $\inf_{y\in A}V(O_{1},y)=h_{L}+\inf_{y\in A}V\left(O_{2},y\right)$ and $\inf_{y\in A}V\left(O_{2},y\right)>0$ (we denote it by $\bar{b}>h_{R}$). Thus the decay rate of variance per unit time is bounded below by $\displaystyle\min_{j=1,2}\left[R_{j}^{(1)}\wedge R_{j}^{(2)}\right]=\min\left\\{h_{L}+\bar{b},h_{L}+\left(\bar{b}-h_{R}\right)\right\\}=h_{L}+\left(\bar{b}-h_{R}\right).$ To find an upper bound for the decay rate of variance per unit time, we recall that $\frac{1}{T^{\varepsilon}}\sum_{j=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)-1}\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\leq\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\leq\frac{1}{T^{\varepsilon}}\sum_{j=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt$ with $\tau_{j}^{\varepsilon}$ being the $j$-th regenerative cycle. In Case I, one might guess that $\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt$ (11.2) has approximately the same distribution as the exit time from the shallow well, which has been shown to asymptotically have an exponential distribution with parameter $\exp(-h_{R}/\varepsilon).$ Additionally, since the exit time from the shallower well is exponentially smaller than $\tau_{j}^{\varepsilon},$ it suggests that the random variables (11.2) can be taken as independent of $N^{\varepsilon}\left(T^{\varepsilon}\right)$ when $\varepsilon$ is small. We also know that $EN^{\varepsilon}\left(T^{\varepsilon}\right)/T^{\varepsilon}\approx 1/E\tau_{1}^{\varepsilon}\approx\exp\left(-h_{L}(\delta)/\varepsilon\right),$ where $h_{L}(\delta)\uparrow h_{L}$ as $\delta\downarrow 0$ and $\approx$ means that quantities on either side have the same exponential decay rate. Using Jensen’s inequality to find that $E[N^{\varepsilon}(T^{\varepsilon})]^{2}\geq[EN^{\varepsilon}(T^{\varepsilon})]^{2}$ and then applying Wald’s identity, we obtain $\displaystyle T^{\varepsilon}\mathrm{Var}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)$ $\displaystyle\approx\frac{1}{T^{\varepsilon}}E\left[\sum\nolimits_{j=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt- EN^{\varepsilon}\left(T^{\varepsilon}\right)E\left(\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)\right]^{2}$ $\displaystyle=\frac{1}{T^{\varepsilon}}E\left(\sum\nolimits_{j=1}^{N^{\varepsilon}\left(T^{\varepsilon}\right)}\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)^{2}-\frac{1}{T^{\varepsilon}}(E(N^{\varepsilon}(T^{\varepsilon})))^{2}\left(E\left(\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)\right)^{2}$ $\displaystyle=\frac{1}{T^{\varepsilon}}EN^{\varepsilon}\left(T^{\varepsilon}\right)E\left(\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)^{2}-\frac{1}{T^{\varepsilon}}\left[EN^{\varepsilon}\left(T^{\varepsilon}\right)\right]^{2}\left(E\left(\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)\right)^{2}$ $\displaystyle\quad+\frac{1}{T^{\varepsilon}}\left(E\left[N^{\varepsilon}\left(T^{\varepsilon}\right)\right]^{2}-EN^{\varepsilon}\left(T^{\varepsilon}\right)\right)\left(E\left(\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)\right)^{2}$ $\displaystyle\geq\frac{EN^{\varepsilon}\left(T^{\varepsilon}\right)}{T^{\varepsilon}}\mathrm{Var}\left(\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)$ (11.3) $\displaystyle\approx\exp\left(-h_{L}(\delta)/\varepsilon\right)\cdot\exp(2h_{R}/\varepsilon)=\exp(\left(2h_{R}-h_{L}(\delta)\right)/\varepsilon).$ Letting $\delta\rightarrow 0$, we see that the decay rate of variance per unit time is bounded above by $h_{L}-2h_{R}$, which is the same as lower bound found for Case I. For the other three Cases II, III and IV, the process spends only a very small fraction of the time while in the shallower well in the set $A$. In fact, using the stopping time arguments of the sort that appear in [12, Chapter 4], the event that the process enters $A$ during an excursion away from the neighborhood of $x_{R}$ can be accurately approximated (as far as large deviation behavior goes) using independent Bernoulli random variables $\\{B^{\varepsilon}_{i}\\}$ with success parameter $e^{-b/\varepsilon}$, and when this occurs the process spends an order one amount of time in $A$ before returning to the neighborhood of $x_{R}$. There is however another sequence of independent Bernoulli random variables with success parameter $e^{-h_{R}/\varepsilon}$, and the process accumulates time in $A$ only up till the time of first success of this sequence. Then $\mathrm{Var}(\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt)$ has the same logarithmic asymptotics as $\mathrm{Var}(\sum\nolimits_{i=1}^{R^{\varepsilon}}1_{\\{B^{\varepsilon}_{i}=1\\}}),$ where $R^{\varepsilon}$ is geometric with success parameter $e^{-h_{R}/\varepsilon}$ and independent of the $\\{B^{\varepsilon}_{i}\\}$. Straightforward calculation using Wald’s identity then gives the exponential rate of decay $2h_{R}-2b$ for Cases II, III and $h_{R}-\bar{b}$ for Case IV, so according to (11) we obtain $\displaystyle T^{\varepsilon}\mathrm{Var}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)\geq\frac{EN^{\varepsilon}\left(T^{\varepsilon}\right)}{T^{\varepsilon}}\mathrm{Var}\left(\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)\approx e^{\left[\left(2\left(h_{R}-b\right)-h_{L}(\delta)\right)/\varepsilon\right]}$ for Cases II and III and $\displaystyle T^{\varepsilon}\mathrm{Var}\left(\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)\geq\frac{EN^{\varepsilon}\left(T^{\varepsilon}\right)}{T^{\varepsilon}}\mathrm{Var}\left(\int_{\tau_{j-1}^{\varepsilon}}^{\tau_{j}^{\varepsilon}}1_{A}\left(X_{t}^{\varepsilon}\right)dt\right)\approx e^{\left[\left((h_{R}-\bar{b})-h_{L}(\delta)\right)/\varepsilon\right]}$ for Case IV. Letting $\delta\rightarrow 0$, this means that the decay rate of variance per unit time is bounded above by $h_{L}+2\left(b-h_{R}\right)$ for Case II and III, and by $h_{L}+(\bar{b}-h_{R})$ for Case IV which is again the same as the corresponding lower bound. Acknowledgement. We thank the referee for corrections and suggestions that improved this paper. ## References * [1] D. Aldous and J. Fill. Reversible markov chains and random walks on graphs (monograph), 2002\. Available at https://www.stat.berkeley.edu/users/aldous/RWG/book.pdf. * [2] D.G. Aronson. Bounds for the fundamental solution of a parabolic equation. Bulletin of the American Mathematical society, 73(6):890–896, 1967\. * [3] S. Asmussen and P.W. Glynn. Stochastic Simulation: Algorithms and Analysis. Applications of Mathematics. Springer Science+Business Media, LLC, 2007\. * [4] A. Budhiraja and P. Dupuis. Analysis and Approximation of Rare Events: Representations and Weak Convergence Methods. 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Applied Probability Models with Optimization Applications. Dover Publications, 1992. * [19] A. Shwartz and A. Weiss. Large Deviations for Performance Analysis: Queues, Communication and Computing. Chapman and Hall, New York, 1995. ## 12 Appendix ###### Proof of Lemma 7.8. Given a function $g$, we define the notation $I(t_{1},t_{2};g)\doteq\int_{t_{1}}^{t_{2}}g(X^{\varepsilon}_{s})ds,$ for any $0\leq t_{1}\leq t_{2}$. By definition, $\tau_{1}^{\varepsilon}=\tau_{N}$ and observe that $\displaystyle I(0,\tau_{N};g)$ $\displaystyle=\sum\nolimits_{\ell=1}^{N}I(\tau_{\ell-1},\tau_{\ell};g)=\sum\nolimits_{\ell=1}^{\infty}I(\tau_{\ell-1},\tau_{\ell};g)\cdot 1_{\left\\{\ell\leq N\right\\}}$ $\displaystyle=\sum\nolimits_{\ell=1}^{\infty}I(\tau_{\ell-1},\tau_{\ell};g)\cdot 1_{\left\\{\ell\leq\hat{N}\right\\}}+\sum\nolimits_{\ell=1}^{\infty}I(\tau_{\ell-1},\tau_{\ell};g)\cdot 1_{\left\\{\hat{N}+1\leq\ell\leq N\right\\}}$ $\displaystyle\quad+\sum\nolimits_{j\in L\setminus\left\\{1\right\\}}\sum\nolimits_{\ell=1}^{\infty}\left(I(\tau_{\ell-1},\tau_{\ell};g)\cdot 1_{\left\\{\hat{N}+1\leq\ell\leq N,Z_{\ell-1}\in\partial B_{\delta}(O_{j})\right\\}}\right).$ Since $\hat{N}$ and $N$ are stopping times with respect to the filtration $\\{\mathcal{G}_{n}\\}_{n},$ it implies that $\\{\ell\leq\hat{N}\\}=\\{\hat{N}\leq\ell-1\\}^{c}\in\mathcal{G}_{\ell-1}$ and $\\{\hat{N}+1\leq\ell\leq N,Z_{\ell-1}\in\partial B_{\delta}(O_{j})\\}\in\mathcal{G}_{\ell-1}.$ Let $\mathfrak{S}_{1}=\sum_{\ell=1}^{\infty}I(\tau_{\ell-1},\tau_{\ell};g)\cdot 1_{\left\\{\ell\leq\hat{N}\right\\}}\text{ and }\mathfrak{S}_{j}=\sum_{\ell=1}^{\infty}\left(I(\tau_{\ell-1},\tau_{\ell};g)\cdot 1_{\left\\{\hat{N}+1\leq\ell\leq N,Z_{\ell-1}\in\partial B_{\delta}(O_{j})\right\\}}\right)$ for all $j\in L\setminus\left\\{1\right\\}.$ We find $\displaystyle E_{x}\left(\mathfrak{S}_{1}\right)$ $\displaystyle=\sum\nolimits_{\ell=1}^{\infty}E_{x}\left(E_{x}\left[\left.I(\tau_{\ell-1},\tau_{\ell};g)\cdot 1_{\left\\{\ell\leq\hat{N}\right\\}}\right|\mathcal{G}_{\ell-1}\right]\right)$ $\displaystyle=\sum\nolimits_{\ell=1}^{\infty}E_{x}\left(1_{\left\\{\ell\leq\hat{N}\right\\}}E_{Z_{\ell-1}}\left[I(0,\tau_{1};g)\right]\right)$ $\displaystyle\leq\sup\nolimits_{y\in\partial B_{\delta}(O_{1})}E_{y}\left[I(0,\tau_{1};g)\right]\cdot\left(\sum\nolimits_{\ell=1}^{\infty}P_{x}(\hat{N}\geq\ell)\right).$ In addition, for $j\in L\setminus\left\\{1\right\\},$ $\displaystyle E_{x}\left(\mathfrak{S}_{j}\right)$ $\displaystyle=\sum\nolimits_{\ell=1}^{\infty}E_{x}\left(I(\tau_{\ell-1},\tau_{\ell};g)\cdot 1_{\left\\{\hat{N}+1\leq\ell\leq N,Z_{\ell-1}\in\partial B_{\delta}(O_{j})\right\\}}\right)$ $\displaystyle=\sum\nolimits_{\ell=1}^{\infty}E_{x}\left(E_{x}\left[\left.I(\tau_{\ell-1},\tau_{\ell};g)\cdot 1_{\left\\{\hat{N}+1\leq\ell\leq N,Z_{\ell-1}\in\partial B_{\delta}(O_{j})\right\\}}\right|\mathcal{G}_{\ell-1}\right]\right)$ $\displaystyle=\sum\nolimits_{\ell=1}^{\infty}E_{x}\left(1_{\left\\{\hat{N}+1\leq\ell\leq N,Z_{\ell-1}\in\partial B_{\delta}(O_{j})\right\\}}E_{Z_{\ell-1}}\left[I(0,\tau_{1};g)\right]\right)$ $\displaystyle\leq\sup\nolimits_{y\in\partial B_{\delta}(O_{j})}E_{y}\left[I(0,\tau_{1};g)\right]\cdot\left(\sum\nolimits_{\ell=1}^{\infty}E_{x}\left(1_{\left\\{\hat{N}+1\leq\ell\leq N,Z_{\ell-1}\in\partial B_{\delta}(O_{j})\right\\}}\right)\right).$ It is straightforward to see that $\hat{N}=N_{1}$. This implies that $\sum\nolimits_{\ell=1}^{\infty}P_{x}(\hat{N}\geq\ell)=E_{x}\hat{N}=E_{x}N_{1}.$ Moreover, observe that for any $j\in L\setminus\left\\{1\right\\}$ $\sum_{\ell=1}^{\infty}1_{\left\\{\hat{N}+1\leq\ell\leq N,Z_{\ell-1}\in\partial B_{\delta}(O_{j})\right\\}}=N_{j},$ which gives that $\sum_{\ell=1}^{\infty}E_{x}(1_{\\{\hat{N}+1\leq\ell\leq N,Z_{\ell-1}\in\partial B_{\delta}(O_{j})\\}})=E_{x}N_{j}.$ Hence, $\displaystyle E_{x}\left(I(0,\tau_{N};g)\right)=\sum\nolimits_{j\in L}E_{x}\left(\mathfrak{S}_{j}\right)\leq\sum\nolimits_{j\in L}\left[\sup\nolimits_{y\in\partial B_{\delta}(O_{j})}E_{y}\left(I(0,\tau_{1};g)\right)\right]\cdot E_{x}N_{j}.$ ∎ ###### Proof of Lemma 7.9. Let $l=|L|$. For any $j\in L$ and $n\in\mathbb{N},$ $\xi_{1}^{(j)}=\inf\\{k\in\mathbb{N}_{0}:$ $Z_{k}\in\partial B_{\delta}(O_{j})\\},$ $\xi_{n}^{(j)}=\inf\\{k\in\mathbb{N}:k>\xi_{n-1}^{(j)}$ and $Z_{k}\in\partial B_{\delta}(O_{j})\\}$ i.e. $\xi_{n}^{(j)}$ is the $n$-th time of hitting $\partial B_{\delta}(O_{j}).$ Moreover, we define $N^{(j)}=\inf\\{n\in\mathbb{N}:\xi_{n}^{(j)}\geq N\\},$ recalling that $N\doteq\inf\\{n\geq\hat{N}:Z_{n}\in\partial B_{\delta}(O_{1})\\}$ and $\hat{N}\doteq\inf\\{n\in\mathbb{N}:Z_{n}\in{\textstyle\cup_{j\in L\setminus\\{1\\}}}\partial B_{\delta}(O_{j})\\}.$ Since $\xi_{n}^{(j)}$ is a stopping time with respect to $\\{\mathcal{G}_{n}\\}_{n},$ we can define the filtration $\\{\mathcal{G}_{\xi_{n}^{(j)}}\\},$ and one can verify that $N^{(j)}$ is a stopping time with respect to $\\{\mathcal{G}_{\xi_{n}^{(j)}}\\}_{n}.$ As in the proof just given, for any function $g$ and for any $0\leq t_{1}\leq t_{2}$ we define $I(t_{1},t_{2};g)\doteq\int_{t_{1}}^{t_{2}}g(X^{\varepsilon}_{s})ds$. With this notation and since by definition $\tau_{1}^{\varepsilon}=\tau_{N}$, we can write $I(0,\tau_{N};g)=\sum\nolimits_{j\in L}\sum\nolimits_{\ell=1}^{\infty}I(\tau_{\xi_{\ell}^{(j)}},\tau_{\xi_{\ell}^{(j)}+1};g)\cdot 1_{\left\\{\ell\leq N^{(j)}-1\right\\}}.$ Since $(x_{1}+\cdots+x_{l})^{2}\leq l(x_{1}^{2}+\cdots+x_{l}^{2})$ for any $(x_{1},\ldots,x_{l})\in\mathbb{R}^{l}$ and $l\in\mathbb{N},$ $\displaystyle I(0,\tau_{N};g)^{2}$ $\displaystyle\leq l\sum\nolimits_{j\in L}\left(\sum\nolimits_{\ell=1}^{\infty}I(\tau_{\xi_{\ell}^{(j)}},\tau_{\xi_{\ell}^{(j)}+1};g)\cdot 1_{\left\\{\ell\leq N^{(j)}-1\right\\}}\right)^{2}.$ Now for any $j\in L$, each square term from the right can be written an addition of two sums, where the first sum is summation of $I(\tau_{\xi_{\ell}^{(j)}},\tau_{\xi_{\ell}^{(j)}+1};g)^{2}\cdot 1_{\\{\ell\leq N^{(j)}-1\\}}$ over all $\ell$, and the second sum is twice of summation of $I(\tau_{\xi_{\ell}^{(j)}},\tau_{\xi_{\ell}^{(j)}+1};g)\cdot 1_{\\{\ell\leq N^{(j)}-1\\}}I(\tau_{\xi_{k}^{(j)}},\tau_{\xi_{k}^{(j)}+1};g)\cdot 1_{\\{\ell\leq N^{(j)}-1\\}}$ over $k,\ell$ with $k<\ell$. For the expected value of the first sum, note that $\\{\ell\leq N^{(j)}-1\\}=\\{N^{(j)}\leq\ell\\}^{c}\in\mathcal{G}_{\xi_{\ell}^{(j)}},$ we have $\displaystyle\sum_{\ell=1}^{\infty}E_{x}\left[I(\tau_{\xi_{\ell}^{(j)}},\tau_{\xi_{\ell}^{(j)}+1};g)^{2}1_{\left\\{\ell\leq N^{(j)}-1\right\\}}\right]$ $\displaystyle=\sum_{\ell=1}^{\infty}E_{x}\left[1_{\left\\{\ell\leq N^{(j)}-1\right\\}}E_{x}\left[\left.I(\tau_{\xi_{\ell}^{(j)}},\tau_{\xi_{\ell}^{(j)}+1};g)^{2}\right|\mathcal{G}_{\xi_{\ell}^{(j)}}\right]\right]$ $\displaystyle\leq\sup_{y\in\partial B_{\delta}(O_{j})}E_{y}I(0,\tau_{1};g)^{2}\sum\nolimits_{\ell=1}^{\infty}P_{x}(N^{(j)}-1\geq\ell)$ $\displaystyle=\sup_{y\in\partial B_{\delta}(O_{j})}E_{y}I(0,\tau_{1};g)^{2}E_{x}(N_{j}).$ The last equality holds since $N^{(j)}-1=N_{j}$ (recall that $N_{j}$ is the number of visits of $\\{Z_{n}\\}_{n\in\mathbb{N}_{0}}$ to $\partial B_{\delta}(O_{j})$ before $N$ including the initial position) this implies that $\sum_{\ell=1}^{\infty}P_{x}(N^{(j)}-1\geq\ell)=\sum_{\ell=1}^{\infty}P_{x}(N_{j}\geq\ell)=E_{x}(N_{j}).$ Turning to the expected value of the second sum, by conditioning on $\mathcal{G}_{\xi_{\ell}^{(j)}}$ gives $\displaystyle\sum_{\ell=2}^{\infty}\sum_{k=1}^{\ell-1}E_{x}\left[I(\tau_{\xi_{\ell}^{(j)}},\tau_{\xi_{\ell}^{(j)}+1};g)\cdot 1_{\left\\{\ell\leq N^{(j)}-1\right\\}}I(\tau_{\xi_{k}^{(j)}},\tau_{\xi_{k}^{(j)}+1};g)\cdot 1_{\left\\{k\leq N^{(j)}-1\right\\}}\right]$ $\displaystyle\leq\sup_{y\in\partial B_{\delta}(O_{j})}E_{y}\left(I(0,\tau_{1};g)\right)\sum_{\ell=2}^{\infty}\sum_{k=1}^{\ell-1}E_{x}\left[1_{\left\\{\ell\leq N^{(j)}-1\right\\}}I(\tau_{\xi_{k}^{(j)}},\tau_{\xi_{k}^{(j)}+1};g)\cdot 1_{\left\\{k\leq N^{(j)}-1\right\\}}\right].$ Now since for any $k\leq\ell-1,$ i.e. $k+1\leq\ell$, $I(\tau_{\xi_{k}^{(j)}},\tau_{\xi_{k}^{(j)}+1};g)\in\mathcal{G}_{\xi_{k}^{(j)}+1}\text{ and }1_{\left\\{\ell\leq N^{(j)}-1\right\\}}\in\mathcal{G}_{\xi_{\ell}^{(j)}},$ we have $\displaystyle E_{x}\left[1_{\left\\{\ell\leq N^{(j)}-1\right\\}}I(\tau_{\xi_{k}^{(j)}},\tau_{\xi_{k}^{(j)}+1};g)\cdot 1_{\left\\{k\leq N^{(j)}-1\right\\}}\right]$ $\displaystyle=E_{x}\left[I(\tau_{\xi_{k}^{(j)}},\tau_{\xi_{k}^{(j)}+1};g)\cdot 1_{\\{\tau_{\xi_{1}^{(j)}}<N,\ldots,\tau_{\xi_{\ell}^{(j)}}<N\\}}\right]$ $\displaystyle=E_{x}\left[E_{Z_{\xi_{k+1}^{(j)}}}\left[1_{\\{\tau_{\xi_{1}^{(j)}}<N,\ldots,\tau_{\xi_{\ell-k}^{(j)}}<N\\}}\right]1_{\\{\tau_{\xi_{k+1}^{(j)}}<N\\}}I(\tau_{\xi_{k}^{(j)}},\tau_{\xi_{k}^{(j)}+1};g)\cdot 1_{\\{\tau_{\xi_{1}^{(j)}}<N,\ldots,\tau_{\xi_{k}^{(j)}}<N\\}}\right]$ $\displaystyle=E_{x}\left[E_{Z_{\xi_{k+1}^{(j)}}}\left[1_{\left\\{\ell-k\leq N^{(j)}-1\right\\}}\right]1_{\\{\tau_{\xi_{k+1}^{(j)}}<N\\}}I(\tau_{\xi_{k}^{(j)}},\tau_{\xi_{k}^{(j)}+1};g)\cdot 1_{\left\\{k\leq N^{(j)}-1\right\\}}\right]$ $\displaystyle\leq\sup\nolimits_{y\in\partial B_{\delta}(O_{j})}P_{y}(\ell-k\leq N^{(j)}-1)E_{x}\left[I(\tau_{\xi_{k}^{(j)}},\tau_{\xi_{k}^{(j)}+1};g)\cdot 1_{\left\\{k\leq N^{(j)}-1\right\\}}\right]$ $\displaystyle=\sup\nolimits_{y\in\partial B_{\delta}(O_{j})}P_{y}\left(\ell-k\leq N_{j}\right)E_{x}\left[E_{x}\left[\left.I(\tau_{\xi_{k}^{(j)}},\tau_{\xi_{k}^{(j)}+1};g)\right|\mathcal{G}_{\xi_{k}^{(j)}}\right]\cdot 1_{\left\\{k\leq N^{(j)}-1\right\\}}\right]$ $\displaystyle\leq\sup\nolimits_{y\in\partial B_{\delta}(O_{j})}E_{y}\left(I(0,\tau_{1};g)\right)\cdot\sup\nolimits_{y\in\partial B_{\delta}(O_{j})}P_{y}\left(\ell-k\leq N_{j}\right)\cdot P_{x}\left(k\leq N_{j}\right).$ This gives that the expected value of the second sum is less than or equal to $\displaystyle\left(\sup\nolimits_{y\in\partial B_{\delta}(O_{j})}E_{y}I(0,\tau_{1};g)\right)^{2}\sum\nolimits_{\ell=2}^{\infty}\sum\nolimits_{k=1}^{\ell-1}\sup\nolimits_{y\in\partial B_{\delta}(O_{j})}P_{y}\left(\ell-k\leq N_{j}\right)\cdot P_{x}\left(k\leq N_{j}\right)$ $\displaystyle\qquad=\left(\sup\nolimits_{y\in\partial B_{\delta}(O_{j})}E_{y}I(0,\tau_{1};g)\right)^{2}\sum\nolimits_{k=1}^{\infty}\sup\nolimits_{y\in\partial B_{\delta}(O_{j})}P_{y}\left(k\leq N_{j}\right)\cdot E_{x}N_{j}.$ Therefore, putting the estimates together gives $\displaystyle E_{x}I(0,\tau_{1}^{\varepsilon};g)^{2}$ $\displaystyle\leq 2l\sum_{j\in L}\left[\sup_{y\in\partial B_{\delta}(O_{j})}E_{y}I(0,\tau_{1};g)\right]^{2}\cdot E_{x}N_{j}\cdot\sum_{\ell=1}^{\infty}\sup_{y\in\partial B_{\delta}(O_{j})}P_{y}\left(\ell\leq N_{j}\right)$ $\displaystyle\quad+l\sum_{j\in L}\left[\sup_{y\in\partial B_{\delta}(O_{j})}E_{y}I(0,\tau_{1};g)^{2}\right]\cdot E_{x}N_{j}.$ ∎ ###### Proof of Lemma 9.2. The main idea of the proof comes from [18, Theorem 3.16]. Given any $\varepsilon>0,$ we define $g^{\varepsilon}\left(t\right)\doteq E_{\lambda^{\varepsilon}}S_{N^{\varepsilon}\left(t\right)}^{\varepsilon}$ for any $t\geq 0.$ Conditioning on $\tau_{1}^{\varepsilon}$ yields $g^{\varepsilon}\left(t\right)=\int_{0}^{\infty}E_{\lambda^{\varepsilon}}[S_{N^{\varepsilon}\left(t\right)}^{\varepsilon}|\tau_{1}^{\varepsilon}=x]dF^{\varepsilon}\left(x\right),$ where $F^{\varepsilon}\left(\cdot\right)$ is the distribution function of $\tau_{1}^{\varepsilon}.$ Note that $E_{\lambda^{\varepsilon}}\left[S_{N^{\varepsilon}\left(t\right)}^{\varepsilon}|\tau_{1}^{\varepsilon}=x\right]=\left\\{\begin{array}[c]{c}g^{\varepsilon}\left(t-x\right)\text{ if }x\leq t\\\ E_{\lambda^{\varepsilon}}\left[S_{1}^{\varepsilon}|\tau_{1}^{\varepsilon}=x\right]\text{ if }x>t\end{array}\right.,$ which implies $g^{\varepsilon}\left(t\right)=\int_{0}^{t}g^{\varepsilon}\left(t-x\right)dF^{\varepsilon}\left(x\right)+h^{\varepsilon}\left(t\right),$ with $h^{\varepsilon}\left(t\right)=\int_{t}^{\infty}E_{\lambda^{\varepsilon}}\left[S_{1}^{\varepsilon}|\tau_{1}^{\varepsilon}=x\right]dF^{\varepsilon}\left(x\right).$ Since $E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}=\int_{0}^{\infty}E_{\lambda^{\varepsilon}}\left[S_{1}^{\varepsilon}|\tau_{1}^{\varepsilon}=x\right]dF^{\varepsilon}\left(x\right)<\infty,$ we have $h^{\varepsilon}\left(t\right)\leq E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}$ for all $t\geq 0.$ Moreover, if we apply Hölder’s inequality first and then the conditional Jensen’s inequality, we find that for all $t\geq 0,$ $\displaystyle h^{\varepsilon}\left(t\right)$ $\displaystyle\leq\left(\int_{t}^{\infty}\left(E_{\lambda^{\varepsilon}}\left[S_{1}^{\varepsilon}|\tau_{1}^{\varepsilon}=x\right]\right)^{2}dF^{\varepsilon}\left(x\right)\right)^{\frac{1}{2}}\left(\int_{t}^{\infty}1^{2}dF^{\varepsilon}\left(x\right)\right)^{\frac{1}{2}}$ $\displaystyle\leq\left(1-F^{\varepsilon}\left(t\right)\right)^{\frac{1}{2}}\left(\int_{t}^{\infty}E_{\lambda^{\varepsilon}}[\left(S_{1}^{\varepsilon}\right)^{2}|\tau_{1}^{\varepsilon}=x]dF^{\varepsilon}\left(x\right)\right)^{\frac{1}{2}}\leq\left(1-F^{\varepsilon}\left(t\right)\right)^{\frac{1}{2}}(E_{\lambda^{\varepsilon}}\left(S_{1}^{\varepsilon}\right)^{2})^{\frac{1}{2}}.$ Given $\ell\in(0,c-h_{1})$ let $U^{\varepsilon}\doteq e^{{\ell}/{\varepsilon}}E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}$. According to Theorem 8.5, there exists $\varepsilon_{0}\in(0,1)$ and a constant $\tilde{c}>0$ such that $1-F^{\varepsilon}\left(U^{\varepsilon}\right)=P_{\lambda^{\varepsilon}}(\tau_{1}^{\varepsilon}/E_{\lambda^{\varepsilon}}\tau_{1}^{\varepsilon}>e^{{\ell}/{\varepsilon}})\leq e^{-\tilde{c}e^{{\ell}/{\varepsilon}}}$ for any $\varepsilon\in(0,\varepsilon_{0}).$ Also by Theorem 8.5, $U^{\varepsilon}<T^{\varepsilon}$ for all $\varepsilon$ small enough. Hence for any $t\geq U^{\varepsilon}$, $1-F^{\varepsilon}\left(t\right)\leq 1-F^{\varepsilon}\left(U^{\varepsilon}\right)\leq e^{-\tilde{c}e^{{\ell}/{\varepsilon}}}\text{ and }h^{\varepsilon}\left(t\right)\leq e^{-\tilde{c}e^{{\ell}/{\varepsilon}}/2}(E_{\lambda^{\varepsilon}}\left(S_{1}^{\varepsilon}\right)^{2})^{\frac{1}{2}}.$ By Proposition 3.4 in [18], we know that for any $\varepsilon>0$, for $t\in[0,\infty)$ $g^{\varepsilon}\left(t\right)=h^{\varepsilon}\left(t\right)+\int_{0}^{t}h^{\varepsilon}\left(t-x\right)da^{\varepsilon}\left(x\right),$ where $a^{\varepsilon}\left(t\right)\doteq\int_{0}^{\infty}E_{\lambda^{\varepsilon}}\left[N^{\varepsilon}\left(t\right)|\tau_{1}^{\varepsilon}=x\right]dF^{\varepsilon}\left(x\right)=E_{\lambda^{\varepsilon}}\left(N^{\varepsilon}\left(t\right)\right).$ This implies $\displaystyle\frac{E_{\lambda^{\varepsilon}}S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}}{T^{\varepsilon}}$ $\displaystyle=\frac{h^{\varepsilon}\left(T^{\varepsilon}\right)}{T^{\varepsilon}}+\frac{1}{T^{\varepsilon}}\int_{0}^{T^{\varepsilon}-U^{\varepsilon}}h^{\varepsilon}\left(T^{\varepsilon}-x\right)da^{\varepsilon}\left(x\right)+\frac{1}{T^{\varepsilon}}\int_{T^{\varepsilon}-U^{\varepsilon}}^{T^{\varepsilon}}h^{\varepsilon}\left(T^{\varepsilon}-x\right)da^{\varepsilon}\left(x\right),$ $\displaystyle\leq\frac{E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}}{T^{\varepsilon}}+e^{-\tilde{c}e^{{\ell}/{\varepsilon}}/2}(E_{\lambda^{\varepsilon}}\left(S_{1}^{\varepsilon}\right)^{2})^{\frac{1}{2}}\frac{a^{\varepsilon}\left(T^{\varepsilon}-U^{\varepsilon}\right)}{T^{\varepsilon}}+E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}\frac{a^{\varepsilon}\left(T^{\varepsilon}\right)-a^{\varepsilon}\left(T^{\varepsilon}-U^{\varepsilon}\right)}{T^{\varepsilon}},$ where we use $h^{\varepsilon}\left(t\right)\leq E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}$ to bound the first term and the third term, and $h^{\varepsilon}\left(t\right)\leq e^{-\tilde{c}e^{{\ell}/{\varepsilon}}/2}(E_{\lambda^{\varepsilon}}\left(S_{1}^{\varepsilon}\right)^{2})^{1/2}$ for any $t\geq U^{\varepsilon}$ for the second term. To calculate the decay rate of the first term, we apply Lemma 7.21 to find that for any $\eta>0$, there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}}{T^{\varepsilon}}\geq\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)+c-h_{1}-\eta.$ (12.1) For the decay rate of the second term, given any $\delta>0$ $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(e^{-\tilde{c}e^{{\ell}/{\varepsilon}}/4}(E_{\lambda^{\varepsilon}}\left(S_{1}^{\varepsilon}\right)^{2})^{\frac{1}{2}}\frac{a^{\varepsilon}\left(T^{\varepsilon}-U^{\varepsilon}\right)}{T^{\varepsilon}}\right)$ (12.2) $\displaystyle\quad=\frac{\tilde{c}}{4}\liminf_{\varepsilon\rightarrow 0}\varepsilon e^{{\ell}/{\varepsilon}}+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left((E_{\lambda^{\varepsilon}}\left(S_{1}^{\varepsilon}\right)^{2})^{\frac{1}{2}}\frac{a^{\varepsilon}\left(T^{\varepsilon}-U^{\varepsilon}\right)}{T^{\varepsilon}}\right)=\infty,$ where the last equality holds since $\ell>0$ implies $\liminf_{\varepsilon\rightarrow 0}\varepsilon e^{{\ell}/{\varepsilon}}=\infty$ and also because Lemma 7.23 and Corollary 8.3 ensure that $\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log((E_{\lambda^{\varepsilon}}\left(S_{1}^{\varepsilon}\right)^{2})^{1/2}a^{\varepsilon}(T^{\varepsilon}-U^{\varepsilon})/T^{\varepsilon})$ is bounded below by a constant. For the last term, note that for any $\varepsilon$ fixed, the renewal function $a^{\varepsilon}\left(t\right)$ is subadditive in $t$ (see for example Lemma 1.2 in [17]), so we have $a^{\varepsilon}\left(T^{\varepsilon}\right)-a^{\varepsilon}\left(T^{\varepsilon}-U^{\varepsilon}\right)\leq a^{\varepsilon}\left(U^{\varepsilon}\right).$ Thus we apply by Lemma 7.21, Corollary 8.3 and Theorem 8.5 to find that for any $\eta>0$, there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$, $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\left(E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}\frac{a^{\varepsilon}\left(T^{\varepsilon}\right)-a^{\varepsilon}\left(T^{\varepsilon}-U^{\varepsilon}\right)}{T^{\varepsilon}}\right)$ $\displaystyle\quad\geq\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log E_{\lambda^{\varepsilon}}S_{1}^{\varepsilon}+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{a^{\varepsilon}\left(U^{\varepsilon}\right)}{U^{\varepsilon}}+\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{U^{\varepsilon}}{T^{\varepsilon}}$ $\displaystyle\quad\geq\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)+\left(c-h_{1}-\ell\right)-\eta.$ (12.3) Since (12.3) holds for all $\ell>0$, by sending $\ell$ to 0, we know that (12.3) holds with $\ell=0$. Putting the bounds (12.1), (12.2) and (12.3) with $\ell=0$ together gives that for any $\eta>0$, there exists $\delta_{0}\in(0,1)$ such that for any $\delta\in(0,\delta_{0})$, $\displaystyle\liminf_{\varepsilon\rightarrow 0}-\varepsilon\log\frac{E_{\lambda^{\varepsilon}}S_{N^{\varepsilon}\left(T^{\varepsilon}\right)}^{\varepsilon}}{T^{\varepsilon}}\geq\inf_{x\in A}\left[f\left(x\right)+W\left(x\right)\right]-W\left(O_{1}\right)+c-h_{1}-\eta.$ ∎ ###### Proof of Lemma 9.5. By the definition of $W(x),$ $\displaystyle 2\inf_{x\in A}[f\left(x\right)+W\left(x\right)]-2W\left(O_{1}\right)-h_{1}$ $\displaystyle=2\inf_{x\in A}[f\left(x\right)+\min_{j\in L}\left(V(O_{j},x)+W\left(O_{j}\right)\right)]-2W\left(O_{1}\right)-h_{1}$ $\displaystyle=\min_{j\in L}\\{2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+2W\left(O_{j}\right)-2W\left(O_{1}\right)-h_{1}\\}.$ Define $Q_{j}\doteq 2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{j},x\right)\right]+2W\left(O_{j}\right)-2W\left(O_{1}\right)-h_{1}$. Then it suffices to show that $Q_{j}\geq R_{j}^{(2)}$ for all $j\in L.$ For $j=1,$ $Q_{1}=2\inf_{x\in A}\left[f\left(x\right)+V\left(O_{1},x\right)\right]-h_{1}=R_{1}^{(2)}.$ For $j\in L\setminus\\{1\\},$ $Q_{j}\geq R_{j}^{(2)}$ if and only if $W\left(O_{j}\right)-h_{1}\geq W\left(O_{1}\cup O_{j}\right).$ Recall that $W\left(O_{j}\right)=\min_{g\in G\left(j\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right]\text{ and }W\left(O_{1}\cup O_{j}\right)=\min_{g\in G\left(1,j\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right].$ Therefore, for any $\tilde{g}\in G\left(j\right)$ such that $W\left(O_{j}\right)=\sum\nolimits_{\left(m\rightarrow n\right)\in\tilde{g}}V\left(O_{m},O_{n}\right),$ if we remove the arrow starting from $1$, and assume that it goes to $i,$ then it is easy to see that $\hat{g}\doteq\tilde{g}\setminus\\{(1,i)\\}\in G(1,j)$. Since $V(O_{1},$ $O_{j})\geq h_{1},$ we find that $\displaystyle W\left(O_{j}\right)-h_{1}$ $\displaystyle=\sum\nolimits_{\left(m\rightarrow n\right)\in\tilde{g}}V\left(O_{m},O_{n}\right)-h_{1}=\sum\nolimits_{\left(m\rightarrow n\right)\in\hat{g}}V\left(O_{m},O_{n}\right)+V(O_{1},O_{j})-h_{1}$ $\displaystyle\geq\min_{g\in G\left(1,j\right)}\left[{\textstyle\sum_{\left(m\rightarrow n\right)\in g}}V\left(O_{m},O_{n}\right)\right]=W\left(O_{1}\cup O_{j}\right).$ ∎

2024-09-04T02:54:55.501964

2002.12740

arxiv-papers

, , and (2020), The Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement for Compressible Viscous Flows, IJNMF, 2020;00:1–6. Kercher et al *Andrew D. Kercher, # The Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement for Compressible Viscous Flows A. D. Kercher A. Corrigan D. A. Kessler Andrew D. Kercher Andrew Corrigan David A. Kessler Laboratories for Computational Physics and Fluid Dynamics, U.S. Naval Research Laboratory, 4555 Overlook Ave SW, Washington, DC 20375 <EMAIL_ADDRESS> (February 2020) ###### Abstract The moving discontinuous Galerkin finite element method with interface condition enforcement (MDG-ICE) is applied to the case of viscous flows. This method uses a weak formulation that separately enforces the conservation law, constitutive law, and the corresponding interface conditions in order to provide the means to detect interfaces or under-resolved flow features. To satisfy the resulting overdetermined weak formulation, the discrete domain geometry is introduced as a variable, so that the method implicitly fits a priori unknown interfaces and moves the grid to resolve sharp, but smooth, gradients, achieving a form of anisotropic curvilinear $r$-adaptivity. This approach avoids introducing low-order errors that arise using shock capturing, artificial dissipation, or limiting. The utility of this approach is demonstrated with its application to a series of test problems culminating with the compressible Navier-Stokes solution to a Mach 5 viscous bow shock for a Reynolds number of $10^{5}$ in two-dimensional space. Time accurate solutions of unsteady problems are obtained via a space-time formulation, in which the unsteady problem is formulated as a higher dimensional steady space- time problem. The method is shown to accurately resolve and transport viscous structures without relying on numerical dissipation for stabilization. ###### keywords: High-order finite elements; Discontinuous Galerkin method; Interface condition enforcement; MDG-ICE; Implicit shock fitting; Anisotropic curvilinear $r$-adaptivity; Space-time ††articletype: Research Article00footnotetext: Abbreviations: MDG-ICE, Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement Distribution A. Approved for public release: distribution unlimited. ## 1 Introduction The discontinuous Galerkin (DG) method 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, has become a popular method for simulating flow fields corresponding to a wide range of physical phenomena, from low speed incompressible flows 11, 12, 13 to chemically reacting compressible Navier-Stokes flows 14, 15, 16, due to its ability to achieve high-order accuracy on unstructured grids and its natural support for local polynomial, $p$, adaptivity. However, the solutions are known to contain oscillations in under-resolved regions of the flow, e.g., shocks, material interfaces, and boundary layers, where stabilization, usually in the form of shock capturing, or limiting, is required. These ad hoc methods often lead to inconsistent discretizations that are no longer capable of achieving high-order accuracy 17. This lack of robustness and accuracy when treating discontinuities and regions with sharp, but smooth, gradients is one of the main obstacles preventing widespread adoption of high-order methods for the simulation of complex high-speed turbulent flow fields. The Moving Discontinuous Galerkin Method with Interface Condition Enforcement (MDG-ICE) was introduced by the present authors 18, 19 as a high-order method for computing solutions to inviscid flow problems, even in the presence of discontinuous interfaces. The method accurately and stably computes flows with a priori unknown interfaces without relying on shock capturing. In order to detect a priori unknown interfaces, MDG-ICE uses a weak formulation that enforces the conservation law and its interface condition separately, while treating the discrete domain geometry as a variable. Thus, in contrast to a standard DG method, MDG-ICE has both the means to detect via interface condition enforcement and satisfy via grid movement the conservation law and its associated interface condition. Using this approach, not only does the grid move to fit interfaces, but also to better resolve smooth regions of the flow. In this work, we apply MDG-ICE to the case of viscous flows, and therefore extend the capability of MDG-ICE to move the grid to resolve otherwise under- resolved flow features such as boundary layers and viscous shocks. In addition to enforcing the conservation law and interface (Rankine-Hugoniot) condition, for the case of viscous flows, we separately enforce a constitutive law and the corresponding interface condition, which constrains the continuity of the state variable across the interface. Thus, in contrast to a standard DG method, MDG-ICE implicitly achieves a form of anisotropic curvilinear $r$-adaptivity via satisfaction of the weak formulation. We study the utility of this approach by solving problems involving linear advection-diffusion, unsteady Burgers flow, and steady compressible Navier- Stokes flow. The ability of the method to move the grid in order to resolve boundary layer profiles is studied in the context of one-dimensional linear advection-diffusion, where convergence under polynomial refinement is also considered. The problem of space-time viscous shock formation for Burgers flow is considered in order to assess the ability of the method to accurately resolve and transport viscous shocks without relying on shock capturing or limiting. Lastly, a Mach 5 compressible Navier-Stokes flow over a cylindrical blunt body in two dimensions is studied for a series of increasing Reynolds numbers to assess the ability of MDG-ICE to simultaneously resolve multiple viscous structures, i.e., a viscous shock and boundary layer, via anisotropic curvilinear $r$-adaptivity. ### 1.1 Background In prior work, MDG-ICE was shown to be a consistent discretization for discontinuous flows and is therefore capable of using high-order approximations to achieve extremely accurate solutions for problems containing discontinuous interfaces on relatively coarse grids 19. The previously presented test cases demonstrated that MDG-ICE can be used to compute both steady and unsteady flows with a priori unknown interface topology and point singularities using higher-order elements in arbitrary-dimensional spaces. For example, MDG-ICE was applied to fit intersecting oblique planar shocks in three dimensions. The ability to fit steady shocks extends to unsteady flows using a space-time formulation, cf. Lowrie et al. 20, 21, that was applied to compute the solution to a space-time inviscid Burgers shock formation problem, where a continuous temporal initial condition steepens to form a shock, while later work presented proof-of-concept results for unsteady flow in three- and four-dimensional space-time 22. More recently, MDG-ICE was applied to shocked compressible flow problems of increased complexity, including transonic flow over a smooth bump over which an attached curved shock forms for which optimal-order convergence was verified 23, 24. Earlier attempts at aligning the grid with discontinuous interfaces present in the flow field have resulted in mixed success, cf. Moretti 25 and Salas 26, 27. These earlier $explicit$ shock fitting, or tracking, approaches were capable of attaining high-order accuracy in the presence of shocks, but the general applicability of such methods is limited. A specialized discretization and implementation strategy is required at discontinuous interfaces making it difficult to handle discontinuities whose topologies are unknown a priori or whose topologies evolve in time. In contrast, MDG-ICE is an $implicit$ shock fitting method, automatically fitting a priori unknown interfaces, their interactions, and their evolving trajectory in space-time. Another promising form of implicit shock tracking, or fitting, is the optimization-based, $r$-adaptive, approach proposed independently by Zahr and Persson 28, 29, 30, which has been used to compute very accurate solutions to discontinuous inviscid flows on coarse grids without the use of artificial stabilization. This approach retains a standard discontinuous Galerkin method as the state equation, while employing an objective function to detect and fit interfaces present in the flow. Recently, Zahr et. al. 31, 32 have extended their implicit shock tracking framework with the addition of a new objective function based on an enriched test space and an improved SQP solver. Furthermore, the regularization introduced by the current authors 18, 19, 23, 24, 22 has been modified to include a factor proportional to the inverse volume of an element that accounts for variations in the element sizes of a given grid. This may be beneficial for obtaining solutions on highly nonuniform grids. In the case of viscous flows, regions with sharp, but smooth, gradients present a unique set of challenges. The resolution required to achieve high- order convergence, or at a minimum, achieve stability, is such that computations on uniformly refined grids are prohibitively expensive. Therefore, the local resolution must be selectively increased in certain regions of the flow. Identifying these regions is not always obvious and striking a balance between computational feasibility and accuracy is an equally challenging task. Traditionally, overcoming these challenges was viewed as an a priori grid design problem with solutions including anisotropic grid generation 33, 34, 35, 36, 37, 38, 39, 40, 41, 42 and boundary layer grid generation 43, 44, 45, 46, 47, 48, 49, 50, 51. A complementary approach to problem-specific grid generation is a posteriori grid adaptation 52, 53, 10, 54. This is an iterative process in which regions of interest are locally refined with the goal of reducing the discretization error. Anisotropic grid adaptation, which combines a posteriori grid adaptation with anisotropic grid generation, has been shown to successfully enhance accuracy for a range of aerodynamic applications as reviewed by Alauzet and Loseille 55. MDG-ICE seeks to achieve similar anisotropic grid adaptation as an intrinsic part of the solver, such that the region of anisotropic refinement evolves with the flow field solution, thereby avoiding grid coarsening as the viscous layer is more accurately resolved. In the case of least-squares (LS) methods, the residual is a natural indicator of the discretization error 56, 57. In particular, for the discontinuous Petrov-Galerkin (DPG) method introduced by Demkowicz and Gopalakrishnan 58, 59, 60, 61, 62, 63, 64, in which the ultra-weak formulation corresponds to the best approximation in the polynomial space, a posteriori grid adaptation, for both the grid resolution $h$ and the polynomial degree $p$, is driven by the built-in error representation function in the form of the Riesz representation of the residual 65. In addition to such a posteriori $hp$-adaptivity strategies, MDG-ICE achieves a form of in situ $r$-adaptivity 66, 67, 68, 69, 70, 71, 72 where the resolution of the flow is continuously improved through repositioning of the grid points. For a review of $r$-adaptivity the reader is referred to the work of Budd et al. 73 and the survey of Huang and Russell 74. ## 2 Moving Discontinuous Galerkin Method with Interface Condition Enforcement for Compressible Viscous Flows In this section we develop the formulation of MDG-ICE for compressible viscous flows. We assume that $\Omega\subset\mathbb{R}^{d}$ is a given domain, which may be either a spatial domain $\Omega\subset\mathbb{R}^{d=d_{x}}$ or a space- time domain $\Omega\subset\mathbb{R}^{d=d_{x}+1}$. In many cases, the space- time domain is defined in terms of a fixed spatial domain $\Omega_{x}\subset\mathbb{R}^{d_{x}}$ and time interval $T\subset\left\\{t\in\mathbb{R}:t>0\right\\}$ by $\Omega=\Omega_{x}\times T$. In the remainder of this work, we assume that $\Omega$ is partitioned by $\mathcal{T}$, consisting of disjoint sub-domains or cells $\kappa$, so that $\overline{\Omega}=\cup_{\kappa\in\mathcal{T}}\overline{\kappa}$, with interfaces $\epsilon$, composing a set $\mathcal{E}$ so that $\cup_{\epsilon\in\mathcal{E}}\epsilon=\cup_{\kappa\in\mathcal{T}}\partial\kappa$. Furthermore, we assume that each interface $\epsilon$ is oriented so that a unit normal $n:\epsilon\rightarrow\mathbb{R}^{d}$ is defined. In order to account for space-time problems, we also consider the spatial normal $n_{x}:\epsilon\rightarrow\mathbb{R}^{d_{x}}$, which is defined such that $\left(n_{x,1},\ldots n_{x,d_{x}}\right)=\left(n_{1},\ldots n_{d_{x}}\right)$. ### 2.1 Governing equations Consider a nonlinear conservation law governing the behavior of smooth, $\mathbb{R}^{m}$-valued, functions $y$, $\displaystyle\nabla\cdot\mathcal{F}\left(y,\nabla_{x}y\right)=0$ $\displaystyle\textup{ in }\Omega,$ (2.1) in terms of a given flux function, $\mathcal{F}:\mathbb{R}^{m}\times\mathbb{R}^{m\times d_{x}}\rightarrow\mathbb{R}^{m\times d}$ that depends on the flow state variable $y$ and its $d_{x}$-dimensional spatial gradient, $\nabla_{x}y=\left(\frac{\partial y}{\partial x_{1}},\ldots,\frac{\partial y}{\partial x_{d_{x}}}\right).$ (2.2) The flux function is assumed to be defined in terms of a spatial flux function $\mathcal{F}^{x}:\mathbb{R}^{m}\times\mathbb{R}^{m\times d_{x}}\rightarrow\mathbb{R}^{m\times d_{x}}$ that itself is defined in terms of a convective flux, depending on the state variable only, and the viscous, or diffusive flux, which also depends on the spatial gradient of the state variable, $\mathcal{F}^{x}\left(y,\nabla_{x}y\right)=\mathcal{F}^{c}\left(y\right)-\mathcal{F}^{v}\left(y,\nabla_{x}y\right).$ (2.3) In the case of a spatial domain, $d=d_{x}$, the flux function $\mathcal{F}$ coincides with the spatial flux, $\mathcal{F}\left(y,\nabla_{x}y\right)=\mathcal{F}^{x}\left(y,\nabla_{x}y\right),$ (2.4) so that the divergence operator in Equation (2.1) is defined as the spatial divergence operator $\nabla\cdot\mathcal{F}\left(y,\nabla_{x}y\right)=\nabla_{x}\cdot\mathcal{F}^{x}\left(y,\nabla_{x}y\right)=\frac{\partial}{\partial x_{1}}\mathcal{F}_{1}^{x}\left(y,\nabla_{x}y\right)+\ldots+\frac{\partial}{\partial x_{d_{x}}}\mathcal{F}_{d_{x}}^{x}\left(y,\nabla_{x}y\right).$ (2.5) Otherwise, in the case of a space-time domain, $d=d_{x}+1$, the space-time flux incorporates the state variable as the temporal flux component, $\mathcal{F}\left(y,\nabla_{x}y\right)=\left(\mathcal{F}_{1}^{x}\left(y,\nabla_{x}y\right),\ldots,\mathcal{F}_{d_{x}}^{x}\left(y,\nabla_{x}y\right),y\right),$ (2.6) so that the divergence operator in (2.1) is defined as the space-time divergence operator $\nabla\cdot\mathcal{F}\left(y,\nabla_{x}y\right)=\nabla_{x}\cdot\mathcal{F}^{x}\left(y,\nabla_{x}y\right)+\frac{\partial}{\partial t}y.$ (2.7) In this work, we consider conservation laws corresponding to linear advection- diffusion, space-time viscous Burgers, and compressible Navier-Stokes flow as detailed in the following sections. #### 2.1.1 Linear advection-diffusion Linear advection-diffusion involves a single-component flow state variable $y:\Omega\rightarrow\mathbb{R}^{1}$ with a linear diffusive flux, $\mathcal{F}^{v}\left(y,\nabla_{x}y\right)=\epsilon\nabla_{x}y,$ (2.8) that is independent of the state $y$, where the coefficient $\epsilon$ corresponds to mass diffusivity. The convective flux is given as $\mathcal{F}^{c}\left(y\right)=\left(v_{1}y,\ldots,v_{d_{x}}y\right),$ (2.9) where $\left(v_{1},\ldots,v_{d_{x}}\right)\in\mathbb{R}^{d_{x}}$ is a prescribed spatial velocity that in the present setting is assumed to be spatially uniform. The corresponding spatial flux is given by $\mathcal{F}^{x}\left(y,\nabla_{x}y\right)=\left(\left(v_{1}y,\ldots,v_{d_{x}}y\right)-\epsilon\nabla_{x}y\right),$ (2.10) #### 2.1.2 One-dimensional Burgers flow As in the case of linear advection-diffusion, one-dimensional Burgers flow involves a single-component flow state variable $y:\Omega\rightarrow\mathbb{R}^{1}$ with a linear viscous flux, $\mathcal{F}^{v}\left(y,\nabla_{x}y\right)=\epsilon\nabla_{x}y,$ (2.11) which is independent of the state $y$, where the coefficient, $\epsilon$, corresponds to viscosity. The convective flux is given as $\mathcal{F}^{c}\left(y\right)=\left(\frac{1}{2}y^{2}\right),$ (2.12) so that the one-dimensional spatial flux is given by $\mathcal{F}^{x}\left(y,\nabla_{x}y\right)=\left(\frac{1}{2}y^{2}-\epsilon\nabla_{x}y\right),$ (2.13) #### 2.1.3 Compressible Navier-Stokes flow For compressible Navier-Stokes flow, the state variable $y:\Omega\rightarrow\mathbb{R}^{m}$, where $m=d_{x}+2$, is given by $y=\left(\rho,\rho v_{1},\ldots,\rho v_{d_{x}},\rho E\right).$ (2.14) The $i$-th spatial component of the convective flux, $\mathcal{F}^{c}:\mathbb{R}^{m}\rightarrow\mathbb{R}^{m\times d_{x}}$, is $\mathcal{F}_{i}^{c}\left(y\right)=\left(\rho v_{i},\rho v_{i}v_{1}+p\delta_{i1},\ldots,\rho v_{i}v_{d_{x}}+p\delta_{id_{x}},\rho Hv_{i}\right),$ (2.15) where $\delta_{ij}$ is the Kronecker delta, $\rho:\Omega\rightarrow\mathbb{R}_{+}$ is density, $\left(v_{1},\ldots,v_{d_{x}}\right):\mathbb{R}^{m}\rightarrow\mathbb{R}^{d_{x}}$ is velocity, $\rho E:\Omega\rightarrow\mathbb{R}_{+}$ is stagnation energy per unit volume, and $H=\left(\rho E+p\right)/\rho$ (2.16) is stagnation enthalpy, where $H:\mathbb{R}^{m}\rightarrow\mathbb{R}_{+}$. Assuming the fluid is a perfect gas, the pressure $p:\mathbb{R}^{m}\rightarrow\mathbb{R}_{+}$ is defined as $p=\left(\gamma-1\right)\left(\rho E-\frac{1}{2}\sum_{i=1}^{d_{x}}\rho v_{i}v_{i}\right),$ (2.17) where the ratio of specific heats for air is given as $\gamma=1.4$. The $i$-th spatial component of the viscous flux is given by $\mathcal{F}_{i}^{\nu}\left(y,\nabla_{x}y\right)=\left(0,\tau_{1i},\ldots,\tau_{d_{x}i},\sum_{j=1}^{d_{x}}\tau_{ij}v_{j}-q_{i}\right),$ (2.18) where $q:\mathbb{R}^{m}\times\mathbb{R}^{m\times d_{x}}\rightarrow\mathbb{R}^{d_{x}}$ is the thermal heat flux, $\tau:\mathbb{R}^{m}\times\mathbb{R}^{m\times d_{x}}\rightarrow\mathbb{R}^{d_{x}\times d_{x}}$ is the viscous stress tensor. The $i$-th spatial component of the thermal heat flux is given by $q_{i}=-k\frac{\partial T}{\partial x_{i}},$ (2.19) where $T:\mathbb{R}^{m}\rightarrow\mathbb{R}_{+}$ is the temperature and $k$ is thermal conductivity. The temperature $T$ is defined as $T=\frac{p}{R\rho},$ (2.20) where $R=287$ is the mixed specific gas constant for air. The $i$-th spatial component of the viscous stress tensor is given by $\tau_{i}=\mu\left(\frac{\partial v_{1}}{\partial x_{i}}+\frac{\partial v_{i}}{\partial x_{1}}-\delta_{i1}\frac{2}{3}\sum_{j=1}^{d_{x}}\frac{\partial v_{j}}{\partial x_{j}},\ldots,\frac{\partial v_{d_{x}}}{\partial x_{i}}+\frac{\partial v_{i}}{\partial x_{d_{x}}}-\delta_{id_{x}}\frac{2}{3}\sum_{j=1}^{d_{x}}\frac{\partial v_{j}}{\partial x_{j}}\right),$ (2.21) where $\mu$ is the dynamic viscosity coefficient. ### 2.2 Interface conditions for viscous flow The viscous conservation laws described in the previous sections require a constraint on the continuity of the state variable, $y$, across an interface, in addition to the interface condition considered in our previous work 19, which enforced the continuity of the normal flux across an interface. In order to deduce the interface conditions governing viscous flow, we revisit the derivation of the DG formulation for viscous flow, cf. Arnold et al. 5. This discussion follows Section 6.3 of Hartmann and Leicht 10 and restricts the presentation to a viscous flux. Upon deducing the governing interface conditions, we will reintroduce the convective flux in Section 2.3.1. Here, we consider a spatial conservation law, $\displaystyle-\nabla_{x}\cdot\left(\mathcal{F}^{v}\left(y,\nabla_{x}y\right)\right)=0$ $\displaystyle\textup{ in }\kappa\qquad\forall\kappa\in\mathcal{T},$ (2.22) defined in terms of a given viscous flux function $\mathcal{F}^{v}:\mathbb{R}^{m}\times\mathbb{R}^{m\times d_{x}}\rightarrow\mathbb{R}^{m\times d_{x}}$, for piecewise smooth functions $y$ and their spatial gradients $\nabla_{x}y$. We introduce an $\mathbb{R}^{m\times d_{x}}$-valued auxiliary variable $\sigma$ and rewrite (2.22) as a first-order system of equations $\displaystyle-\nabla_{x}\cdot\sigma=0$ $\displaystyle\textup{ in }\kappa\qquad\forall\kappa\in\mathcal{T},$ (2.23) $\displaystyle\sigma-G\left(y\right)\nabla_{x}y=0$ $\displaystyle\textup{ in }\kappa\qquad\forall\kappa\in\mathcal{T}.$ (2.24) We assume here that $\mathcal{F}^{v}$ is linear with respect to its gradient argument so that $G\left(y\right)\nabla_{x}y=\mathcal{F}^{v}\left(y,\nabla_{x}y\right)=\mathcal{F}_{\nabla_{x}y}^{v}\left(y,\nabla_{x}y\right)\nabla_{x}y$ (2.25) where $G\left(y\right)\in\mathbb{R}^{m\times d_{x}\times m\times d_{x}}$ is a tensor of rank 4 that is referred to as the _homogeneity tensor_ 10. We integrate (2.23) and (2.24) against separate test functions and upon an application of integration by parts arrive at the following weak formulation : find $\left(y,\sigma\right)\in Y\times\Sigma$ such that $\displaystyle 0=$ $\displaystyle+\sum_{\kappa\in\mathcal{T}}\left(\sigma,\nabla_{x}v\right)_{\kappa}$ $\displaystyle-\sum_{\kappa\in\mathcal{T}}\left(\sigma\cdot n_{x},v\right)_{\partial\kappa}$ $\displaystyle+\sum_{\kappa\in\mathcal{T}}\left(\sigma,\tau\right)_{\kappa}$ $\displaystyle+\sum_{\kappa\in\mathcal{T}}\left(y,\nabla_{x}\cdot\left(G(y)^{\top}\tau\right)\right)_{\kappa}$ $\displaystyle-\sum_{\kappa\in\mathcal{T}}\left(y\otimes n_{x},G(y)^{\top}\tau\right)_{\partial\kappa}\qquad\forall\left(v,\tau\right)\in V_{y}\times V_{\sigma},$ (2.26) where the solution spaces $Y\times\Sigma$ and test spaces $V_{y}\times V_{\sigma}$ are broken Sobolev spaces. Since $y$ and $\sigma$ are multi-valued across element interfaces, in a DG formulation, they are substituted with single-valued functions of their traces, $\displaystyle\hat{\sigma}=$ $\displaystyle\hat{\sigma}\left(\sigma^{+},\sigma^{-}\right),$ (2.27) $\displaystyle\hat{y}=$ $\displaystyle\hat{y}\left(y^{+},y^{-}\right),$ (2.28) cf. Table 3.1 of Arnold et al. 5 for various definitions of both $\hat{\sigma}$ and $\hat{y}$. After another application of integration by parts and transposition of the homogeneity tensor, we obtain: find $\left(y,\sigma\right)\in Y\times\Sigma$ such that $\displaystyle 0=$ $\displaystyle-\sum_{\kappa\in\mathcal{T}}\left(\nabla_{x}\cdot\sigma,v\right)_{\kappa}$ $\displaystyle+\sum_{\kappa\in\mathcal{T}}\left(\left(\sigma-\hat{\sigma}\right)\cdot n_{x},v\right)_{\partial\kappa}$ $\displaystyle+\sum_{\kappa\in\mathcal{T}}\left(\sigma-G(y)\nabla_{x}y,\tau\right)_{\kappa}$ $\displaystyle-\sum_{\kappa\in\mathcal{T}}\left(G(y)\left(\left(\hat{y}-y\right)\otimes n_{x}\right),\tau\right)_{\partial\kappa}\qquad\forall\left(v,\tau\right)\in V_{y}\times V_{\sigma}.$ (2.29) Finally, the auxiliary variable $\sigma$ is substituted with $G\left(y\right)\nabla_{x}y$, the tensor-valued test function $\tau$ is substituted with $\nabla_{x}v$, so that upon a final application of integration by parts we obtain a DG primal formulation: find $y\in Y$ such that $\displaystyle 0=$ $\displaystyle\sum_{\kappa\in\mathcal{T}}\left(G(y)\nabla_{x}y,\nabla_{x}v\right)_{\kappa}$ $\displaystyle-\sum_{\kappa\in\mathcal{T}}\left(\hat{\sigma}\cdot n_{x},v\right)_{\partial\kappa}$ $\displaystyle+\sum_{\kappa\in\mathcal{T}}\left(G(y)\left(\left(\hat{y}-y\right)\otimes n_{x}\right),\nabla_{x}v\right)_{\partial\kappa}\qquad\forall v\in V_{y},$ (2.30) cf. Equation (254) and Section 6.6 in the work of Hartmann and Leicht 10. In contrast, we propose an MDG-ICE formulation that retains the auxiliary variable and instead makes a different substitution: the test functions $v$ and $\tau$ that appear in the surface integrals of (2.29) are substituted with separate test functions $w_{y}\in W_{y}$ and $w_{\sigma}\in W_{\sigma}$ from the single-valued trace spaces of $V_{y}$ and $V_{\sigma}$. Upon accumulating contributions from adjacent elements in (2.29) to each interface, we obtain: find $\left(y,\sigma\right)\in Y\times\Sigma$ $\displaystyle 0=$ $\displaystyle-\sum_{\kappa\in\mathcal{T}}\left(\nabla_{x}\cdot\sigma,v\right)_{\kappa}$ $\displaystyle+\sum_{\epsilon\in\mathcal{E}}\left(\left\llbracket n_{x}\cdot\sigma\right\rrbracket,w_{y}\right)_{\epsilon}$ $\displaystyle+\sum_{\kappa\in\mathcal{T}}\left(\sigma-G(y)\nabla_{x}y,\tau\right)_{\kappa}$ $\displaystyle-\sum_{\epsilon\in\mathcal{E}}\left(\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket y\otimes n_{x}\right\rrbracket,w_{\sigma}\right)_{\epsilon}\qquad\forall\left(v,\tau,w_{y},w_{\sigma}\right)\in V_{y}\times V_{\sigma}\times W_{y}\times W_{\sigma}.$ (2.31) We make use of the relationship $\displaystyle\left(\sigma^{+}-\hat{\sigma}\right)\cdot n_{x}^{+}+\left(\sigma^{-}-\hat{\sigma}\right)\cdot n_{x}^{-}$ $\displaystyle=$ $\displaystyle\left(\sigma^{+}-\hat{\sigma}\right)\cdot n_{x}^{+}-\left(\sigma^{-}-\hat{\sigma}\right)\cdot n_{x}^{+}$ $\displaystyle=$ $\displaystyle\left(\sigma^{+}-\sigma^{-}\right)\cdot n_{x}^{+}$ $\displaystyle=$ $\displaystyle\left\llbracket n_{x}\cdot\sigma\right\rrbracket,$ (2.32) so that contributions from $\hat{\sigma}$ vanish, and $\displaystyle G(y^{+})\left(\left(\hat{y}-y^{+}\right)\otimes n_{x}^{+}\right)+G(y^{-})\left(\left(\hat{y}-y^{-}\right)\otimes n_{x}^{-}\right)$ $\displaystyle=$ $\displaystyle G(y^{+})\left(\frac{1}{2}\left(y^{-}-y^{+}\right)\otimes n_{x}^{+}\right)+G(y^{-})\left(\frac{1}{2}\left(y^{+}-y^{-}\right)\otimes n_{x}^{-}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(G(y^{+})\left(\left(y^{-}-y^{+}\right)\otimes n_{x}^{+}\right)+G(y_{h}^{-})\left(\left(y^{+}-y^{-}\right)\otimes n_{x}^{-}\right)\right)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(G(y^{+})+G(y^{-})\right)\left(y^{-}-y^{+}\right)\otimes n_{x}^{+}$ $\displaystyle=$ $\displaystyle-\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket y\otimes n_{x}\right\rrbracket,$ (2.33) on interior interfaces (2.49), where we define $\hat{y}=\left\\{\\!\\!\left\\{y\right\\}\\!\\!\right\\}$ , a common choice among the various DG discretizations 5, 10. From (2.31), we deduce the strong form of the viscous interface conditions to be $\displaystyle\left\llbracket n_{x}\cdot\sigma\right\rrbracket=0$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E},$ (2.34) $\displaystyle\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket y\otimes n_{x}\right\rrbracket=0$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E}.$ (2.35) The first interface condition, Equation (2.34) is the jump or Rankine-Hugoniot condition 75 that ensures continuity of the normal flux at the interface and will balance with the jump in the normal convective flux in Equation (2.38). The second interface condition (2.35) corresponds to the constitutive law (2.37) and enforces a constraint on the continuity of the state variable at the interface. ### 2.3 Formulation in physical space with fixed geometry Having deduced the interface conditions that arise in the case of viscous flow, we reintroduce the convective flux and write the second order system (2.1) as a system of first-order equations, incorporating the additional interface conditions (2.34) and (2.35). #### 2.3.1 Strong formulation Consider a nonlinear conservation law, generalized constitutive law, and their corresponding interface conditions, $\displaystyle\nabla\cdot\mathcal{F}\left(y,\sigma\right)=0$ $\displaystyle\textup{ in }\kappa\qquad\forall\kappa\in\mathcal{T},$ (2.36) $\displaystyle\sigma-G(y)\nabla_{x}y=0$ $\displaystyle\textup{ in }\kappa\qquad\forall\kappa\in\mathcal{T},$ (2.37) $\displaystyle\left\llbracket n\cdot\mathcal{F}\left(y,\sigma\right)\right\rrbracket=0$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E},$ (2.38) $\displaystyle\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket y\otimes n_{x}\right\rrbracket=0$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E},$ (2.39) governing the flow state variable $y$ and auxiliary variable $\sigma$. The interface condition (2.38) corresponding to the conservation law (2.36) is the jump or Rankine-Hugoniot condition 75, which now accounts for both the convective and viscous flux, ensuring continuity of the normal flux at the interface. The interface condition (2.39) corresponding to the constitutive law (2.37) is unmodified from (2.35) by the inclusion of the convective flux. The flux $\mathcal{F}\left(y,\sigma\right)$ is defined in terms of the spatial flux $\mathcal{F}^{x}\left(y,\sigma\right)$ analogously to (2.4) or (2.6). The spatial flux $\mathcal{F}^{x}\left(y,\sigma\right)$ is defined as $\mathcal{F}^{x}\left(y,\sigma\right)=\mathcal{F}^{c}\left(y\right)-\mathcal{\tilde{F}}^{v}\left(y,\sigma\right),$ (2.40) where $\tilde{\mathcal{F}}^{v}:\mathbb{R}^{m}\times\mathbb{R}^{m\times d_{x}}\rightarrow\mathbb{R}^{m\times d_{x}}$ is the modified viscous flux defined consistently with the primal formulation of Section 2.1, $\mathcal{\tilde{F}}^{v}\left(y,G\left(y\right)\nabla_{x}y\right)=\mathcal{F}^{v}\left(y,\nabla_{x}y\right),$ (2.41) and $G\left(y\right)\in\mathbb{R}^{m\times d_{x}\times m\times d_{x}}$ is now a generalized _constitutive tensor_ that depends on the specific choice of constitutive law. One approach to defining the constitutive law is to define a _gradient formulation_ , where the constitutive tensor $G\left(y\right)$ is taken as the identity, $G\left(y\right)\nabla_{x}y=\nabla_{x}y,$ (2.42) while the viscous flux remains unmodified, $\mathcal{\tilde{F}}^{v}\left(y,\sigma\right)=\mathcal{F}^{v}\left(y,\sigma\right).$ (2.43) The gradient formulation has been used in the context of local discontinuous Galerkin 76 and hybridized discontinuous Galerkin methods 77. This formulation results in a constitutive law (2.37) and corresponding interface condition (2.39) that are linear with respect to the state variable and do not introduce a coupling between flow variable components 76. In this case, the interface condition (2.39) reduces to $\left\llbracket y\otimes n_{x}\right\rrbracket=0,$ (2.44) which implies $\left\llbracket y\right\rrbracket=0$ at spatial interfaces, an interface condition arising in the context of elliptic interface problems 78, 79 that directly enforces the continuity of the state variable. While this choice is reasonable if the solution is smooth, this approach would not be appropriate for flows that contain discontinuities in the state variable, such as problems with inviscid sub-systems, cf. Mott et al. 80. An alternative approach is to define a _flux formulation_ , as in Section (2.2), where the constitutive tensor $G\left(y\right)$ is defined to be the homogeneity tensor (2.25) so that $G\left(y\right)\nabla_{x}y=\mathcal{F}^{v}\left(y,\nabla_{x}y\right)=\mathcal{F}_{\nabla_{x}y}^{v}\left(y,\nabla_{x}y\right)\nabla_{x}y,$ (2.45) while the modified viscous flux is defined to be the auxiliary variable, $\mathcal{\tilde{F}}^{v}\left(y,\sigma\right)=\sigma,$ (2.46) recovering a standard mixed method 81. A slight modification of the flux formulation for the case of linear advection-diffusion or viscous Burgers, where $\mathcal{F}^{v}\left(y,\nabla_{x}y\right)=\epsilon\nabla_{x}y$, is obtained by setting $G\left(y\right)=\sqrt{\epsilon}$ , which recovers the formulation advocated by Broersen and Stevenson 82, 83 and later Demkowicz and Gopalakrishnan 65 in the context of Discontinuous Petrov-Galerkin (DPG) methods for singularly perturbed problems 84. A similar approach was used in the original description of the LDG method where nonlinear diffusion coefficients were considered by Cockburn and Shu 81. In the case of compressible Navier-Stokes flow, we take an approach similar to that of Chan et al. 85 with the scaling advocated by Broersen and Stevenson also incorporated. The constitutive tensor $G\left(y\right)$ is defined such that $\left(G\left(y\right)\nabla_{x}y\right)_{i}=\mu_{\infty}^{-\nicefrac{{1}}{{2}}}\left(0,\tau_{1i},\ldots,\tau_{d_{x}i},-q_{i}\right),$ (2.47) where $\mu_{\infty}$ is the freestream dynamic viscosity. The viscous flux is defined in terms of the auxiliary variable as $\mathcal{F}_{i}^{v}\left(y,\sigma\right)=\mu_{\infty}^{\nicefrac{{1}}{{2}}}\left(\sigma_{1i},\sigma_{2i},\ldots,\sigma_{d_{x}+1i},\sigma_{i+1j}v_{j}+\sigma_{mi}\right).$ (2.48) In this way, the auxiliary variable is defined, up to a factor $\mu_{\infty}^{-\nicefrac{{1}}{{2}}}$, as the viscous stress tensor, $\tau$, given by (2.21) and thermal heat flux, $q$, given by (2.19). In contrast to Chan et. al. 85 we do not strongly enforce symmetry of the viscous stress tensor, $\tau$. However, we may explore this approach in future work as it could lead to a more computationally efficient and physically accurate formulation. #### 2.3.2 Interior and boundary interfaces We assume that $\mathcal{E}$ consists of two disjoint subsets: the interior interfaces $\left\\{\epsilon_{0}\in\mathcal{E}\,\middle|\,\epsilon_{0}\cap\partial\Omega=\emptyset\right\\}$ (2.49) and exterior interfaces $\left\\{\epsilon_{\partial}\in\mathcal{E}\,\middle|\,\epsilon_{\partial}\subset\partial\Omega\right\\},$ (2.50) so that $\mathcal{E}=\mathcal{E}_{0}\cup\mathcal{E}_{\partial}$. For interior interfaces $\epsilon_{0}\in\mathcal{E}_{0}$ there exists $\kappa^{+},\kappa^{-}\in\mathcal{T}$ such that $\epsilon_{0}=\partial\kappa^{+}\cap\partial\kappa^{-}$. On interior interfaces Equations (2.38) and (2.39) are defined as $\displaystyle\left\llbracket n\cdot\mathcal{F}\left(y,\sigma\right)\right\rrbracket=n^{+}\cdot\mathcal{F}\left(y^{+},\sigma^{+}\right)+n^{-}\cdot\mathcal{F}\left(y^{-},\sigma^{-}\right)=0,$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E}_{0},$ (2.51) $\displaystyle\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket y\otimes n_{x}\right\rrbracket=\frac{1}{2}\left(G\left(y^{+}\right)+G\left(y^{-}\right)\right)\left(y^{+}\otimes n_{x}^{+}+y^{-}\otimes n_{x}^{-}\right)=0,$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E}_{0}.$ (2.52) where $n^{+},n^{-}$ denote the outward facing normal of $\kappa^{+},\kappa^{-}$ respectively, so that $n^{+}=-n^{-}$. For exterior interfaces $\displaystyle\left\llbracket n\cdot\mathcal{F}\left(y,\sigma\right)\right\rrbracket=n^{+}\cdot\mathcal{F}\left(y^{+},\sigma^{+}\right)-n^{+}\cdot\mathcal{\mathcal{F}}_{\partial}\left(y^{+},\sigma^{+}\right)=0,$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E}_{\partial},$ (2.53) $\displaystyle\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket y\otimes n_{x}\right\rrbracket=G_{\partial}\left(y^{+}\right)\left(y^{+}\otimes n_{x}^{+}-y_{\partial}\left(y^{+}\right)\otimes n_{x}^{+}\right)=0,$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E}_{\partial}.$ (2.54) Here $n^{+}\cdot\mathcal{F}_{\partial}\left(y^{+},\sigma^{+}\right)$ is the imposed normal boundary flux, $G_{\partial}\left(y^{+}\right)$ is the boundary modified constitutive tensor, and $y_{\partial}\left(y^{+}\right)$ is the boundary state, which are functions chosen depending on the type of boundary condition. Therefore, we further decompose $\mathcal{E}_{\partial}$ into disjoint subsets of inflow and outflow interfaces $\mathcal{E}_{\partial}=\mathcal{E}_{\text{in}}\cup\mathcal{E}_{\text{out}},$ so that at an outflow interface $\epsilon_{\textup{out}}$ the boundary flux is defined as the interior convective flux, and the boundary state is defined as the interior state, $\displaystyle n^{+}\cdot\mathcal{F}_{\partial}\left(y^{+},\sigma^{+}\right)=n^{+}\cdot\mathcal{F}\left(y^{+},\sigma_{\text{out}}=0\right),$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E}_{\text{out}},$ (2.55) $\displaystyle G_{\partial}\left(y^{+}\right)=G\left(y^{+}\right),$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E}_{\text{out}},$ (2.56) $\displaystyle y_{\partial}\left(y^{+}\right)=y^{+},$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E}_{\text{out}},$ (2.57) and therefore Equations (2.53) and (2.54) are satisfied trivially. At an inflow boundary $\epsilon_{\textup{in}}\in\mathcal{E}_{\text{in}}$, the normal convective boundary flux and boundary state are prescribed values independent of the interior state $y^{+}$ , while the normal viscous boundary flux is defined as the interior normal viscous flux, $\displaystyle n^{+}\cdot\mathcal{\mathcal{F}}_{\partial}\left(y^{+},\sigma^{+}\right)=n^{+}\cdot\mathcal{\mathcal{F}}_{\text{in}}^{c}-n^{+}\cdot\mathcal{\tilde{F}}^{v}\left(y^{+},\sigma^{+}\right),$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E}_{\text{in}},$ (2.58) $\displaystyle G_{\partial}\left(y^{+}\right)=G\left(y_{\partial}\left(y^{+}\right)\right),$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E}_{\text{in}},$ (2.59) $\displaystyle y_{\partial}\left(y^{+}\right)=y_{\text{in}},$ $\displaystyle\textup{ on }\epsilon\qquad\forall\epsilon\in\mathcal{E}_{\text{in}}.$ (2.60) #### 2.3.3 Weak formulation A weak formulation in physical space is obtained by integrating the conservation law (2.36), the constitutive law (2.37), and the corresponding interface conditions (2.38), (2.39) for each element and interface against separate test functions: find $\left(y,\sigma\right)\in Y\times\Sigma$ such that $\displaystyle 0=$ $\displaystyle\;\;\;\;\sum_{\kappa\in\mathcal{T}}\left(\nabla\cdot\mathcal{F}\left(y,\sigma\right)-f,v\right)_{\kappa}$ $\displaystyle+\sum_{\kappa\in\mathcal{T}}\left(\sigma-G(y)\nabla_{x}y,\tau\right)_{\kappa}$ $\displaystyle-\sum_{\epsilon\in\mathcal{E}}\left(\left\llbracket n\cdot\mathcal{F}\left(y,\sigma\right)\right\rrbracket,w_{y}\right)_{\epsilon}$ $\displaystyle-\sum_{\epsilon\in\mathcal{E}}\left(\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket y\otimes n_{x}\right\rrbracket,w_{\sigma}\right)_{\epsilon}\qquad\forall\left(v,\tau,w_{y},w_{\sigma}\right)\in V_{y}\times V_{\sigma}\times W_{y}\times W_{\sigma}.$ (2.61) The solution spaces $Y$ and $\Sigma$ are the broken Sobolev spaces, $\displaystyle Y$ $\displaystyle=$ $\displaystyle\left\\{y\in\left[L^{2}\left(\Omega\right)\right]^{m\hphantom{\times d_{x}}}\bigl{|}\forall\kappa\in\mathcal{T},\>\>\hphantom{\nabla_{x}\cdot}\left.y\right|_{\kappa}\in\left[H^{1}\left(\kappa\right)\right]^{m}\right\\},$ (2.62) $\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle\left\\{\sigma\in\left[L^{2}\left(\Omega\right)\right]^{m\times d_{x}}\bigl{|}\forall\kappa\in\mathcal{T},\left.\nabla_{x}\cdot\sigma\right|_{\kappa}\in\left[L^{2}\left(\Omega\right)\right]^{m}\right\\},$ (2.63) while the test spaces are defined as $V_{y}=\left[L^{2}\left(\Omega\right)\right]^{m}$ and $V_{\sigma}=\left[L^{2}\left(\Omega\right)\right]^{m\times d}$, with $W_{y}$ and $W_{\sigma}$ defined to be the corresponding single-valued trace spaces, cf. Carstensen et al. 63. ### 2.4 Formulation in reference space with variable geometry Analogous to our previous work 19, the grid must be treated as a variable in order to align discrete grid interfaces with flow interfaces or more generally to move the grid to resolve under-resolved flow features. Therefore, we transform the strong formulation (2.36), (2.37), (2.38), (2.39) and weak formulation (2.61) of the flow equations from physical to reference coordinates in order to facilitate differentiation with respect to geometry. #### 2.4.1 Mapping from reference space We assume that there is a continuous, invertible mapping $u:\hat{\Omega}\rightarrow\Omega,$ (2.64) from a reference domain $\hat{\Omega}\subset\mathbb{R}^{d}$ to the physical domain $\Omega\subset\mathbb{R}^{d}$. We assume that $\hat{\Omega}$ is partitioned by $\hat{\mathcal{T}}$, so that $\overline{\hat{\Omega}}=\cup_{\hat{\kappa}\in\hat{\mathcal{T}}}\overline{\hat{\kappa}}$. Also, we consider the set of interfaces $\hat{\mathcal{E}}$ consisting of disjoint interfaces $\hat{\epsilon}$, such that $\cup_{\hat{\epsilon}\in\hat{\mathcal{E}}}\hat{\epsilon}=\cup_{\hat{\kappa}\in\hat{\mathcal{T}}}\partial\hat{\kappa}$. The mapping $u$ is further assumed to be (piecewise) differentiable with derivative or Jacobian matrix denoted $\left.\nabla u\right|_{\hat{\kappa}}:\hat{\kappa}\rightarrow\mathbb{R}^{d\times d}\qquad\forall\hat{\kappa}\in\hat{\mathcal{T}}.$ (2.65) The cofactor matrix $\left.\operatorname{cof}\left(\nabla u\right)\right|_{\hat{\kappa}}:\hat{\kappa}\rightarrow\mathbb{R}^{d\times d},$ is defined for $\hat{\kappa}\in\hat{\mathcal{T}}$, $\operatorname{cof}\left(\nabla u\left(\hat{x}\right)\right)=\operatorname{det}\left(\nabla u\left(\hat{x}\right)\right)\left(\nabla u\left(\hat{x}\right)\right)^{-\top}\qquad\forall\hat{x}\in\hat{\kappa},$ (2.66) where $\left.\operatorname{det}\left(\nabla u\right)\right|_{\hat{\kappa}}:\hat{\kappa}\rightarrow\mathbb{R}$ is the determinant of the Jacobian. As detailed in our related work 86, assuming that $y$ and $v$ are functions over reference space, the weak formulation of a conservation law in physical space can be evaluated in reference space according to $\left(\nabla\cdot\mathcal{F}\left(y\circ u^{-1}\right),v\circ u^{-1}\right)_{\kappa}=\left(\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)\cdot\mathcal{F}\left(y\right),v\right)_{\hat{\kappa}}.$ (2.67) Likewise, treating $\sigma$ and $\tau$ as functions over reference space, the constitutive law can be evaluated in reference space according to $\left(\sigma\circ u^{-1}-G(y\circ u^{-1})\nabla_{x}\left(y\circ u^{-1}\right),\tau\circ u^{-1}\right)_{\kappa}=\left(\operatorname{det}\left(\nabla u\right)\sigma-G(y)\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)_{x}y,\tau\right)_{\hat{\kappa}},$ (2.68) In order to represent the spatial gradient in a space-time setting ($d=d_{x}+1$), we define $\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)_{x}$ to be the spatial components of $\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)$, so that if $\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)y:\hat{\kappa}\rightarrow\mathbb{R}^{m\times d}$ then $\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)y:\hat{\kappa}\rightarrow\mathbb{R}^{m\times d_{x}}$, while in a spatial ($d=d_{x}$) setting $\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)=\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)_{x}$. The weak formulation of each interface condition can similarly be evaluated in reference space according to $\left(\left\llbracket n\cdot\mathcal{F}\left(y\circ u^{-1}\right)\right\rrbracket,w_{y}\circ u^{-1}\right)_{\epsilon}=\left(\left\llbracket s\left(\nabla u\right)\cdot\mathcal{F}\left(y\right)\right\rrbracket,w_{y}\right)_{\hat{\epsilon}}$ (2.69) and $\left(\left(\left\\{\\!\\!\left\\{G\left(y\circ u^{-1}\right)\right\\}\\!\\!\right\\}\left\llbracket\left(y\circ u^{-1}\right)\otimes n_{x}\right\rrbracket\right),w_{\sigma}\circ u^{-1}\right)_{\epsilon}=\left(\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket y\otimes s\left(\nabla u\right)_{x}\right\rrbracket,w_{\sigma}\right)_{\hat{\epsilon}}$ (2.70) In this setting, $\hat{\epsilon}\in\hat{\mathcal{E}}$, $\epsilon=u\left(\hat{\epsilon}\right)\in\mathcal{E}$, and $n:\epsilon\rightarrow\mathbb{R}^{d}$ is the unit normal, which can be evaluated in terms of $u$ according to $n=\left(\frac{s\left(\nabla u\right)}{\left\|s\left(\nabla u\right)\right\|}\right)\circ u^{-1},$ (2.71) where $s\left(\nabla u\right):\hat{\epsilon}\rightarrow\mathbb{R}^{d}$ is defined as follows. We assume that there exists a parameterization $\theta_{\hat{\epsilon}}:\hat{D}\rightarrow\hat{\epsilon}$, mapping from points $\left(\xi_{1},\ldots,\xi_{d-1}\right)$ in parameter space $\hat{D}\subset\mathbb{R}^{d-1}$ to points $\hat{x}$ on the reference space interface, such that the reference space tangent plane basis vectors $\partial_{\xi_{1}}\theta_{\hat{\epsilon}},\ldots,\partial_{\xi_{d-1}}\theta_{\hat{\epsilon}}$ are of unit magnitude. A parameterization of $\epsilon=u\left(\hat{\epsilon}\right)$ is then given by the composition $\theta_{\epsilon}=u\circ\theta_{\hat{\epsilon}}:\hat{D}\rightarrow\epsilon$. Given $\hat{\epsilon}\in\hat{\mathcal{E}}$, the scaled normal $\left.s\left(\nabla u\right)\right|_{\hat{\epsilon}}:\hat{\epsilon}\rightarrow\mathbb{R}^{d}$ is defined for $\hat{x}\in\hat{\epsilon}$ as the scaled normal of the tangent plane of $\epsilon$ corresponding to the parameter $\theta_{\hat{\epsilon}}^{-1}\left(\hat{x}\right)$. If $d=3$ and $\xi,\eta$ denote the parametric coordinates, then $\left.s\left(\nabla u\right)\right|_{\hat{\epsilon}}=\left(\partial_{\xi}\theta_{\epsilon}\times\partial_{\eta}\theta_{\epsilon}\right)\circ\theta_{\hat{\epsilon}}^{-1}$ (2.72) where $\partial_{\xi}\theta_{\epsilon}\times\partial_{\eta}\theta_{\epsilon}$ is the cross product of the tangent plane basis vectors. A general formula for evaluating the cross product of tangent plane basis vectors is given by the following: let $\left(\boldsymbol{x}_{1},\ldots,\boldsymbol{x}_{d}\right)$ denote the coordinate directions in $\mathbb{R}^{d}$, and the parameterization be given in terms of components $\theta_{\epsilon}=\left(\theta_{\epsilon}^{1},\ldots,\theta_{\epsilon}^{d}\right),$ then $\partial_{\xi_{1}}\theta_{\epsilon}\times\cdots\times\partial_{\xi_{d-1}}\theta_{\epsilon}=\operatorname{det}\left(\begin{array}[]{ccc}\partial_{\xi_{1}}\theta_{\epsilon}^{1}&\cdots&\partial_{\xi_{1}}\theta_{\epsilon}^{d}\\\ \vdots&\ddots&\vdots\\\ \partial_{\xi_{d-1}}\theta_{\epsilon}^{1}&\cdots&\partial_{\xi_{d-1}}\theta_{\epsilon}^{d}\\\ \boldsymbol{x}_{1}&\ldots&\boldsymbol{x}_{d}\end{array}\right).$ (2.73) By the chain rule we can express $\partial_{\xi_{i}}\theta_{\epsilon}$ in terms of $\nabla u$, $\partial_{\xi_{i}}\theta_{\epsilon}\left(\xi\right)=\nabla u\left(\theta_{\hat{\epsilon}}\left(\xi\right)\right)\cdot\partial_{\xi_{i}}\theta_{\hat{\epsilon}}\left(\xi\right),$ (2.74) so that in general the physical space scaled normal as a function of $\nabla u$ is $\left.s\left(\nabla u\right)\right|_{\hat{\epsilon}}=\left(\partial_{\xi_{1}}\theta_{\epsilon}\times\cdots\times\partial_{\xi_{d-1}}\theta_{\epsilon}\right)\circ\theta_{\hat{\epsilon}}^{-1}.$ (2.75) In the present work, we have adopted the more standard convention in the definition of the generalized cross product given by Equation (2.73), which is used to define the generalized scaled normal given by Equation (2.75). This definition differs from Equation (3.33) of our previous work 19 by a factor of $\left(-1\right)^{\left(d-1\right)}$ in order to ensure that (2.73) and (2.75) are positively oriented, cf. Massey 87. #### 2.4.2 Strong and weak formulation in reference space The strong form in reference space is $\displaystyle\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)\cdot\mathcal{F}\left(y,\sigma\right)=0$ $\displaystyle\textup{ in }\hat{\kappa}\qquad\forall\hat{\kappa}\in\hat{\mathcal{T}},$ (2.76) $\displaystyle\operatorname{det}\left(\nabla u\right)\sigma-G(y)\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)_{x}y=0$ $\displaystyle\textup{ in }\hat{\kappa}\qquad\forall\hat{\kappa}\in\hat{\mathcal{T}},$ (2.77) $\displaystyle\left\llbracket s\left(\nabla u\right)\cdot\mathcal{F}\left(y,\sigma\right)\right\rrbracket=0$ $\displaystyle\textup{ on }\hat{\epsilon}\qquad\forall\hat{\epsilon}\in\hat{\mathcal{E}},$ (2.78) $\displaystyle\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket y\otimes s\left(\nabla u\right)_{x}\right\rrbracket=0$ $\displaystyle\textup{ on }\hat{\epsilon}\qquad\forall\hat{\epsilon}\in\hat{\mathcal{E}},$ (2.79) $\displaystyle b\left(u\right)-u=0$ $\displaystyle\textup{ on }\hat{\epsilon}\qquad\forall\hat{\epsilon}\in\hat{\mathcal{E}},$ (2.80) where $\nabla u$ is the Jacobian of the mapping from reference to physical space, $\operatorname{det}\left(\nabla u\right)$ is its determinant, $\operatorname{cof}\left(\nabla u\right)$ is its cofactor matrix, and $s\left(\nabla u\right)$ is the scaled normal as defined in Section 2.4.1. Equation (2.80) imposes geometric boundary conditions that constrain points to the boundary of the physical domain via a projection operator $b:U\rightarrow U$, where $U=\left[H^{1}\left(\hat{\Omega}\right)\right]^{d}$is the $\mathbb{R}^{d}$-valued Sobolev space over $\hat{\Omega}$. Examples of $b\left(u\right)$ are given in earlier work 19, 24. We assume that $Y$ and $\Sigma$, originally defined for functions over physical space, cf. (2.62) and (2.63), now consist, respectively, of functions defined in $\mathbb{R}^{m}$-valued and $\mathbb{R}^{m\times d_{x}}$-valued broken Sobolev spaces over $\hat{\mathcal{T}}$. We further assume that the test spaces $V_{y}$, $V_{\sigma},W_{y},W_{\sigma}$ now consist of functions defined over reference space. We define a provisional state operator $\tilde{e}:Y\times\Sigma\times U\rightarrow\left(V_{y}\times V_{\sigma}\times W_{y}\times W_{\sigma}\right)^{*}$ for $\left(y,\sigma,u\right)\in Y\times\Sigma\times U$, by $\displaystyle\tilde{e}\left(y,\sigma,u\right)=\left(v,\tau,w_{y},w_{\sigma}\right)\mapsto$ $\displaystyle\;\;\;\;\sum_{\hat{\kappa}\in\hat{\mathcal{T}}}\left(\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)\cdot\mathcal{F}\left(y,\sigma\right),v\right)_{\hat{\kappa}}$ $\displaystyle+\sum_{\hat{\kappa}\in\hat{\mathcal{T}}}\left(\operatorname{det}\left(\nabla u\right)\sigma-G(y)\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)_{x}y,\tau\right)_{\hat{\kappa}}$ $\displaystyle-\sum_{\hat{\epsilon}\in\hat{\mathcal{E}}}\left(\left\llbracket s\left(\nabla u\right)\cdot\mathcal{F}\left(y,\sigma\right)\right\rrbracket,w_{y}\right)_{\hat{\epsilon}}$ $\displaystyle-\sum_{\hat{\epsilon}\in\hat{\mathcal{E}}}\left(\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket y\otimes s\left(\nabla u\right)_{x}\right\rrbracket,w_{\sigma}\right)_{\hat{\epsilon}}$ (2.81) which has a Fréchet derivative defined for perturbation $\left(\delta y,\delta\sigma,\delta u\right)\in Y\times\Sigma\times U$, and test functions $\left(v,\tau,w_{y},w_{\sigma}\right)\in V_{y}\times V_{\sigma}\times W_{y}\times W_{\sigma}$, by its partial derivative with respect to the state variable $y$, $\displaystyle\tilde{e}_{y}\left(y,\sigma,u\right)\delta y=\left(v,\tau,w_{y},w_{\sigma}\right)\mapsto$ $\displaystyle\;\;\;\;\sum_{\hat{\kappa}\in\hat{\mathcal{T}}}\left(\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)\cdot\left(\mathcal{F}_{y}\left(y,\sigma\right)\delta y\right),v\right)_{\hat{\kappa}}$ $\displaystyle+\sum_{\hat{\kappa}\in\hat{\mathcal{T}}}\left(-\left(\left(G^{\prime}(y)\delta y\right)\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)_{x}\delta y+G(y)\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)_{x}\delta y\right),\tau\right)_{\hat{\kappa}}$ $\displaystyle-\sum_{\hat{\epsilon}\in\hat{\mathcal{E}}}\left(\left\llbracket s\left(\nabla u\right)\cdot\left(\mathcal{F}_{y}\left(y,\sigma\right)\delta y\right)\right\rrbracket,w_{y}\right)_{\hat{\epsilon}}$ $\displaystyle-\sum_{\hat{\epsilon}\in\hat{\mathcal{E}}}\left(\left\\{\\!\\!\left\\{G^{\prime}\left(y\right)\delta y\right\\}\\!\\!\right\\}\left\llbracket y\otimes s\left(\nabla u\right)_{x}\right\rrbracket+\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket\delta y\otimes s\left(\nabla u\right)_{x}\right\rrbracket,w_{\sigma}\right)_{\hat{\epsilon}},$ (2.82) its partial derivative with respect to the auxiliary variable $\sigma$, $\displaystyle\tilde{e}_{\sigma}\left(y,\sigma,u\right)\delta\sigma=\left(v,\tau,w_{y},w_{\sigma}\right)\mapsto$ $\displaystyle\;\;\;\;\sum_{\hat{\kappa}\in\hat{\mathcal{T}}}\left(\left(\operatorname{cof}\left(\nabla u\right)\nabla\right)\cdot\left(\mathcal{F}_{\sigma}\left(y,\sigma\right)\delta\sigma\right),v\right)_{\hat{\kappa}}$ $\displaystyle+\sum_{\hat{\kappa}\in\hat{\mathcal{T}}}\left(\operatorname{det}\left(\nabla u\right)\delta\sigma,\tau\right)_{\hat{\kappa}}$ $\displaystyle-\sum_{\hat{\epsilon}\in\hat{\mathcal{E}}}\left(\left\llbracket s\left(\nabla u\right)\cdot\left(\mathcal{F}_{\sigma}\left(y,\sigma\right)\delta\sigma\right)\right\rrbracket,w_{y}\right)_{\hat{\epsilon}},$ (2.83) and its partial derivative with respect to the geometry variable $u$, $\displaystyle\tilde{e}_{u}\left(y,\sigma,u\right)\delta u=\left(v,\tau,w_{y},w_{\sigma}\right)\mapsto$ $\displaystyle\;\;\;\;\sum_{\hat{\kappa}\in\hat{\mathcal{T}}}\left(\left(\left(\operatorname{cof}^{\prime}\left(\nabla u\right)\nabla\delta u\right)\nabla\right)\cdot\mathcal{F}\left(y,\sigma\right),v\right)_{\hat{\kappa}}$ $\displaystyle+\sum_{\hat{\kappa}\in\hat{\mathcal{T}}}\left(\left(\operatorname{det}^{\prime}\left(\nabla u\right)\nabla\delta u\right)\sigma-G(y)\left(\left(\operatorname{cof}^{\prime}\left(\nabla u\right)\nabla\delta u\right)\nabla\right)_{x}y,\tau\right)_{\hat{\kappa}}$ $\displaystyle-\sum_{\hat{\epsilon}\in\mathcal{\hat{E}}}\left(\left\llbracket\left(s^{\prime}\left(\nabla u\right)\nabla\delta u\right)\cdot\mathcal{F}\left(y,\sigma\right)\right\rrbracket,w_{y}\right)_{\hat{\epsilon}}$ $\displaystyle-\sum_{\hat{\epsilon}\in\hat{\mathcal{E}}}\left(\left\\{\\!\\!\left\\{G\left(y\right)\right\\}\\!\\!\right\\}\left\llbracket y\otimes\left(s^{\prime}\left(\nabla u\right)\nabla\delta u\right)_{x}\right\rrbracket,w_{\sigma}\right)_{\hat{\epsilon}}.$ (2.84) The state operator $e:Y\times\Sigma\times U\rightarrow\left(V_{y}\times V_{\sigma}\times W_{y}\times W_{\sigma}\right)^{*}$, which imposes geometric boundary conditions (2.80) by composing the provisional state operator (2.81) with the projection $b\left(u\right)$, is defined by $e\left(y,\sigma,u\right)=\tilde{e}\left(y,\sigma,b\left(u\right)\right),$ (2.85) with Fréchet derivative defined, for state $\left(y,\sigma,u\right)\in Y\times\Sigma\times U$ and perturbation $\left(\delta y,\delta\sigma,\delta u\right)\in Y\times\Sigma\times U$, by $e^{\prime}\left(y,\sigma,u\right)=\left(\delta y,\delta\sigma,\delta u\right)\mapsto\tilde{e}_{y}\left(y,\sigma,b\left(u\right)\right)\delta y+\tilde{e}_{\sigma}\left(y,\sigma,b\left(u\right)\right)\delta\sigma+\tilde{e}_{u}\left(y,\sigma,b\left(u\right)\right)b^{\prime}\left(u\right)\delta u.$ (2.86) The state equation in reference coordinates is $e\left(y,\sigma,u\right)=0.$ The corresponding weak formulation in reference coordinates is: find $\left(y,\sigma,u\right)\in Y\times\Sigma\times U$ such that $\left\langle e\left(y,\sigma,u\right),\left(v,\tau,w_{y},w_{\sigma}\right)\right\rangle=0\qquad\forall\left(v,\tau,w_{y},w_{\sigma}\right)\in V_{y}\times V_{\sigma}\times W_{y}\times W_{\sigma},$ (2.87) so that the solution satisfying (2.76) and (2.78) weakly and (2.80) strongly is therefore given as $\left(y,\sigma,b\left(u\right)\right)\in Y\times\Sigma\times U$. ### 2.5 Discretization We choose discrete (finite-dimensional) subspaces $Y_{h}\subset Y$, $\Sigma_{h}\subset\Sigma$, $U_{h}\subset U$, $V_{y,h}\subset V_{y}$, $V_{\sigma,h}\subset V_{\sigma}$, $W_{y,h}\subset W_{y}$, and $W_{\sigma,h}\subset W_{\sigma}$ to discretize the weak formulation (2.87), which is restricted to the discrete subspaces via the discrete state operator, $e_{h}:Y_{h}\times\Sigma_{h}\times U_{h}\rightarrow\left(V_{y,h}\times V_{\sigma,h}\times W_{y,h}\times W_{\sigma,h}\right)^{*}$ (2.88) defined such that $e_{h}\left(y,\sigma,u\right)=e\left(y,\sigma,u\right)$ for all $\left(y,\sigma,u\right)\in Y_{h}\times\Sigma_{h}\times U_{h}$ and the $h$-subscript indicates that discrete subspaces have been selected. We use standard piecewise polynomials, cf. 10, defined over reference elements. Let $\mathcal{P}_{p}$ denote the space of polynomials spanned by the monomials $\boldsymbol{x}^{\alpha}$ with multi-index $\alpha\in\mathbb{N}_{0}^{d}$ , satisfying $\sum_{i=1}^{d}\alpha_{i}\leq p$. In the case of a simplicial grid, $\displaystyle Y_{h}$ $\displaystyle=$ $\displaystyle\left\\{y\in Y\,\middle|\,\forall\hat{\kappa}\in\hat{\mathcal{T}},\left.y\right|_{\hat{\kappa}}\in\left[\mathcal{P}_{p}\right]^{m\hphantom{\times d_{x}}}\right\\},$ (2.89) $\displaystyle\Sigma_{h}$ $\displaystyle=$ $\displaystyle\left\\{\sigma\in\Sigma\,\middle|\,\forall\hat{\kappa}\in\hat{\mathcal{T}},\left.\sigma\right|_{\hat{\kappa}}\in\left[\mathcal{P}_{p}\right]^{m\times d_{x}}\right\\}.$ (2.90) The polynomial degree of the state space and flux space are in general distinct. In the present work, we choose $V_{y,h}=Y_{h}$, $V_{\sigma,h}=\Sigma_{h}$, while $W_{y,h}$, and $W_{\sigma,h}$ are chosen to be the corresponding single-valued polynomial trace spaces. While the present approach is a discrete least squares method 88 with a priori chosen test spaces, future work will investigate a least squares finite element formulation 89, 90 with optimal test spaces automatically generated using the discontinuous Petrov–Galerkin methodology of Demkowicz and Gopalakrishnan 58, 59, 65. The discrete subspace $U_{h}$ of mappings from reference space to physical space are also discretized into $\mathbb{R}^{d}$-valued piecewise polynomials, in the case of a simplicial grid $U_{h}=\left\\{u\in U\,\middle|\,\forall\hat{\kappa}\in\hat{\mathcal{T}},\left.u\right|_{\hat{\kappa}}\in\left[\mathcal{P}_{p}\right]^{d}\right\\}.$ (2.91) The case that the chosen polynomial degree of $U_{h}$ is equal to that of $Y_{h}$ is referred to as isoparametric. It is also possible to choose the polynomial degree of $U_{h}$ to be less (sub-parametric) or greater (super- parametric) than that of $Y_{h}$. ### 2.6 Solver In general, the dimensionality of the discrete solution space and discrete residual space do not match. Therefore, the weak formulation is solved iteratively using unconstrained optimization to minimize $\frac{1}{2}\left\|e_{h}\left(y,\sigma,u\right)\right\|^{2}$, by seeking a stationary point111The stationary point (2.92) and Newton’s method (2.94) were stated incorrectly in previous work, cf. 19, Equations (77) and (80)., $e_{h}^{\prime}\left(y,\sigma,u\right)^{*}e_{h}\left(y,\sigma,u\right)=0.$ (2.92) Given an initialization $\left(y,\sigma,u\right)_{0}$ the solution is repeatedly updated $\left(y,\sigma,u\right)_{i+1}=\left(y,\sigma,u\right)_{i}+\Delta\left(y,\sigma,u\right)_{i}\qquad i=0,1,2,\ldots$ (2.93) until (2.92) is satisfied to a given tolerance. One approach is to use Newton’s method, which is a second-order method with increment given by, $\Delta\left(y,\sigma,u\right)=-\left(\left(e_{h}^{\prime\prime}\left(y,\sigma,u\right)^{*}\cdot\right)e_{h}\left(y,\sigma,u\right)+e_{h}^{\prime}\left(y,\sigma,u\right)^{*}e_{h}^{\prime}\left(y,\sigma,u\right)\right)^{-1}\left(e_{h}^{\prime}\left(y,\sigma,u\right)^{*}e_{h}\left(y,\sigma,u\right)\right).$ (2.94) Alternatively, the Gauss-Newton method neglects second derivatives, yet recovers the second-order convergence rate of Newton’s method as the residual vanishes and ensures a positive semi-definite matrix, resulting in an increment given by $\Delta\left(y,\sigma,u\right)=-\left(e_{h}^{\prime}\left(y,\sigma,u\right)^{*}e_{h}^{\prime}\left(y,\sigma,u\right)\right)^{-1}\left(e_{h}^{\prime}\left(y,\sigma,u\right)^{*}e_{h}\left(y,\sigma,u\right)\right).$ (2.95) We employ a Levenberg-Marquardt method to solve (2.92), which augments the Gauss-Newton method (2.95) with a regularization term, $\Delta\left(y,\sigma,u\right)=-\left(e_{h}^{\prime}\left(y,\sigma,u\right)^{*}e_{h}^{\prime}\left(y,\sigma,u\right)+I_{\lambda}\left(y,\sigma,u\right)\right)^{-1}\left(e_{h}^{\prime}\left(y,\sigma,u\right)^{*}e_{h}\left(y,\sigma,u\right)\right),$ (2.96) where the regularization operator, $I_{\lambda}\left(y,\sigma,u\right):\left(\delta y,\delta\sigma,\delta u\right)\mapsto\left(\lambda_{y}\delta y,\lambda_{\sigma}\delta\sigma,\lambda_{u}\delta u\right),$ (2.97) ensures invertibility and therefore positive definiteness of the linear system of equations. Separate regularization coefficients $\lambda_{y},\lambda_{\sigma},\lambda_{u}\geq 0$ are defined for each solution variable. In practice, the state and auxiliary regularization coefficients can be set to zero, $\lambda_{y}=\lambda_{\sigma}=0$, while the grid regularization coefficient $\lambda_{u}>0$ must be positive in order to ensure rank sufficiency and to limit excessive grid motion. Additional symmetric positive definite operators can be incorporated into the regularization operator 24. In the present work, we incorporate a linear elastic grid regularization, which is a symmetric positive definite operator that has the effect of distributing the grid motion to neighboring elements. The linear elastic grid regularization is a variation of the Laplacian grid regularization, $\delta u\mapsto-\lambda_{\Delta u}\left(b^{\prime}\left(u\right)^{*}\Delta b^{\prime}\left(u\right)\right)\delta u$, with $\lambda_{\Delta u}\geq 0$, that we employed in previous work 24 that offers the added benefit of introducing a compressibility effect into the grid motion that we have found useful for resolving thin viscous layers. Other possible regularization strategies include the weighted elliptic regularization proposed by Zahr et al. 31, 32. The resulting linear system of equations is positive definite and symmetric. In the present work, we employ a sparse direct solver provided by Eigen 91. The grid topology may need to be modified by the solver in order to fit a priori unknown interfaces and ensure element validity while resolving sharp gradients, for which we employ standard edge refinement and edge collapse algorithms 92. In the present work, element quality is used as an indicator for local refinement. Elements that become highly anisotropic as MDG-ICE moves the grid to resolve thin viscous structures are adaptively split by refining their longest edge. In the case of nonlinear elements, if the determinant of the Jacobian at any degree of freedom is negative, we apply a control that projects the elements to a linear shape representation and locally refines the element if projecting the cell does not recover a valid grid. Often, the introduction of the additional grid topology and resolution is sufficient for the solver to recover a valid grid. The solver does not currently incorporate any other grid smoothing or optimization terms based on element quality 29, 30, 31. ## 3 Results We now apply MDG-ICE to compute steady and unsteady solutions for flows containing sharp, smooth, gradients. Solutions to unsteady problems are solved using a space-time formulation. Unless otherwise indicated, the grid is assumed to consist of isoparametric elements, see Section 2.5. ### 3.1 Linear advection-diffusion We consider steady, one-dimensional linear advection-diffusion described in Section 2.1.1, subject to the following boundary conditions $\displaystyle y\left(x=0\right)$ $\displaystyle=0,$ $\displaystyle y\left(x=1\right)$ $\displaystyle=1.$ (3.1) The exact solution is given by $y\left(x\right)=\frac{1-\exp\left(x\cdot\mathrm{Pe}\right)}{1-\exp\left(\mathrm{Pe}\right)}$ (3.2) where $\mathrm{Pe}=\frac{1}{\varepsilon}=\frac{v\ell}{\mu}$ is the Péclet number, $v$ is the characteristic velocity, $\ell$ is the characteristic length, and $\mu$ is the mass diffusivity. In this case, the solution exhibits a boundary layer like profile at $x=1$ 93. | ---|--- | Figure 3.1: Steady, linear advection-diffusion, $\mathrm{Pe}=100$. The $L^{2}$ projection of the exact solution onto a uniform grid consisting of 8 linear line cells is compared for various polynomial degrees to MDG-ICE, which automatically moved the initially uniform grid, indicated by red tick marks, in order to resolve the boundary layer profile, resulting in the adapted grid indicated with black tick marks. Figure 3.2: Steady, linear advection- diffusion, $\mathrm{Pe}=100$. The rate of convergence with respect to polynomial degree on a log-linear plot is shown, comparing the $L^{2}$ projection onto a uniform grid to MDG-ICE, which automatically moved the initially uniform grid to resolve the boundary layer profile. Reference slopes of $10.7$ and $3.5$ are shown, illustrating the increased rate of convergence achieved using MDG-ICE. Figure 3.1 shows the MDG-ICE solution to the linear advection-diffusion problem with exact solution (3.2) for $\mathrm{Pe}=100$ as well as the corresponding $L^{2}$ projection of the exact solution onto a uniform grid of 8 linear line cells. The $L^{2}$ projection minimizes the error in the $L^{2}$ norm and therefore provides an upper bound on the accuracy attainable by methods based on a static grid, e.g., DG. By moving the grid to resolve the boundary layer profile, MDG-ICE is able to achieve accurate, oscillation-free, solutions for a range of polynomial degrees. Figure 3.2 presents the corresponding convergence results with respect to polynomial degree, i.e., $p$-refinement. The rate of convergence of MDG-ICE with respect to polynomial degree is compared to the $L^{2}$ projection of the exact solution onto a uniform grid. These results confirm that MDG-ICE resolves sharp boundary layers with enhanced accuracy compared to static grid methods. Even for a $\mathcal{P}_{1}$ approximation, MDG-ICE provides nearly two orders of magnitude improved accuracy compared to the best approximation available on a uniform grid, a gap that only widens at higher polynomial degrees. The MDG-ICE error is plotted on a log-linear plot, with a reference slope of $10.7$, indicating spectral convergence. This shows that the $r$-adaptivity provided by MDG-ICE enhances the effectiveness of $p$-refinement, even in the presence of initially under-resolved flow features. This results demonstrates the enhanced accuracy of MDG-ICE for resolving initially under-resolved flow features using high-order finite element approximation in comparison to traditional static grid methods, such as DG. ### 3.2 Space-time Burgers viscous shock formation For a space-time Burgers flow, described in Section 2.1.2, a shock will form at time $t=t_{s}=0.5$ for the following initial conditions $y\left(x,t=0\right)=\frac{1}{2\pi t_{s}}\sin\left(2\pi x\right)+y_{\infty},$ (3.3) where $y_{\infty}=0.2$ is the freestream velocity. The space-time solution was initialized by extruding the temporal inflow condition, given by Equation (3.3), throughout the space-time domain. (a) Inviscid MDG-ICE$\left(\mathcal{P}_{5}/\mathcal{P}_{1}\right)$ space-time solution computed using 200 triangle elements. (b) MDG- ICE$\left(\mathcal{P}_{5}/\mathcal{P}_{1}\right)$ space-time solution for $\epsilon=10^{-3}$ computed using 200 triangle elements. (c) MDG- ICE$\left(\mathcal{P}_{5}/\mathcal{P}_{1}\right)$ space-time solution for $\epsilon=10^{-4}$ computed using 200 triangle elements. Figure 3.3: Space-time Burgers shock formation: inviscid, viscous $\left(\epsilon=10^{-3}\right)$, and viscous $\left(\epsilon=10^{-4}\right)$ solutions computed using $\mathcal{P}_{5}$ linear triangle elements without shock capturing. Instead, the viscous shock was resolved via anisotropic space-time $r$-adaptivity. The solver was initialized by extruding the initial condition at $t=0$ in time. (a) Inviscid MDG-ICE$\left(\mathcal{P}_{5}/\mathcal{P}_{1}\right)$ at $t=0$ and $t=1$. The corresponding space-time solution is shown in Figure 3.3a. (b) Viscous MDG-ICE$\left(\mathcal{P}_{5}/\mathcal{P}_{1}\right)$ with $\epsilon={1}\mathrm{e}{-3}$ at $t=0$ and $t=1$ . The corresponding space-time solution is shown in Figure 3.3b. (c) Viscous MDG- ICE$\left(\mathcal{P}_{5}/\mathcal{P}_{1}\right)$ with $\epsilon={1}\mathrm{e}{-4}$ at $t=0$ and $t=1$ . The corresponding space-time solution is shown in Figure 3.3c. Figure 3.4: Burgers shock formation one-dimensional profiles at $t=0$ and $t=1$: inviscid, viscous $\left(\epsilon=10^{-3}\right)$, and viscous $\left(\epsilon=10^{-4}\right)$ solutions computed using $\mathcal{P}_{5}$ linear triangle elements without shock capturing. The corresponding space-time solutions are shown in Figure 3.3. Figure 3.3 presents the space-time Burgers shock formation solutions for an inviscid flow, a viscous flow with $\epsilon=10^{-3}$, and a viscous flow with $\epsilon=10^{-4}$. Figure 3.4 presents the corresponding one-dimensional profiles at $t=0$ and $t=1$. The space-time solutions were initialized by extruding the inflow condition, given by Equation (3.3), at $t=0$ in time. The initial simplicial grid was generated by converting a uniform $10\times 10$ quadrilateral grid into triangles. In the inviscid case, MDG-ICE fits the point of shock formation and tracks the shock at the correct speed. In addition to the shock, the inviscid flow solution has a derivative discontinuity that MDG-ICE also detects and tracks at the correct speed of $0.2$. For the two viscous flow cases, $\epsilon=10^{-3}$ and $\epsilon=10^{-4}$, MDG-ICE accurately resolves each viscous shock as a sharp, yet smooth, profile by adjusting the grid geometry, without modifying the grid topology. This case demonstrates the inherent ability of MDG-ICE to achieve anisotropic space-time $r$-adaptivity for unsteady flow problems. ### 3.3 Mach 5 viscous bow shock (a) 392 linear triangle cells (b) 392 linear $\mathcal{P}_{2}$ triangle cells Figure 3.5: The initial linear grid and temperature field corresponding to a shock captured DG($\mathcal{P}_{2}/\mathcal{P}_{1}$) solution for the viscous Mach 5 bow shock at $\mathrm{Re}=10^{3}$. (a) 400 isoparametric $\mathcal{P}_{4}$ triangle cells (b) 400 isoparametric $\mathcal{P}_{4}$ triangle cells (c) 400 isoparametric $\mathcal{P}_{4}$ triangle cells (d) 400 isoparametric $\mathcal{P}_{4}$ triangle cells Figure 3.6: The MDG-ICE solution computed using $\mathcal{P}_{4}$ isoparametric triangle elements for the viscous Mach 5 bow shock at $\mathrm{Re}=10^{3}$. The MDG-ICE grid was initialized by projecting the linear triangle grid shown in Figure 3.5a to the closest point on the boundary of the domain. The MDG-ICE field variables were initialized by cell averaging the interpolated the DG($\mathcal{P}_{2}/\mathcal{P}_{1}$) solution shown in Figure 3.5b. The MDG-ICE flux variables were initialized to zero for consistency with the initial piecewise constant field variables. The location of the shock along the line $x=0$ was computed as $y=1.49995$ for a stand-off distance of $0.49995$. (a) The temperature sampled along $x=0$. The exact temperature at the stagnation point, $T=2.5$, is marked with the symbol $\times$. (b) The normal velocity, $v_{n}=0$, sampled along $x=0$. The exact normal velocity at the stagnation point, $v_{n}=0$, is marked with the symbol $\times$. (c) The pressure, $p$, sampled along $x=0$. The exact pressure at the stagnation point for an inviscid flow, $p\approx 23.324$, is marked with the symbol $\times$. (d) The density, $\rho$, sampled along $x=0$. The density at the stagnation point, computed using the stagnation pressure corresponding to an inviscid flow, $\rho\approx 13.061389724919298$, is marked with the symbol $\times$. (e) The normal component of the normal viscous stress tensor, $\tau_{nn}$, sampled along $x=0$. (f) The normal thermal heat flux, $q_{n}$, sampled along $x=0$. Figure 3.7: Centerline profiles of temperature and normal velocity for the viscous Mach 5 bow shock at $\mathrm{Re}=10^{3}$ computed with MDG- ICE($\mathcal{P}_{4}$) compared to ODE and MDG-ICE($\mathcal{P}_{4}$) approximations of the exact solution for the corresponding one-dimensional viscous shock. The one-dimensional MDG-ICE($\mathcal{P}_{4}$) approximation was computed using 16 isoparametric line cells. The location of the shock was computed as $y=1.49995$ for a stand-off distance of $0.49995$. (a) The pressure coefficient, $C_{p}$, sampled at each degree of freedom on the surface of the cylinder. The exact pressure coefficient at the stagnation point for an inviscid flow, $C_{p}\approx 1.8087699607027568$, is marked with the symbol $\times$. (b) The Stanton number sampled at each degree of freedom on the surface of the cylinder. The computed Stanton number at the stagnation point is marked with the symbol $\times$. Figure 3.8: Pressure coefficient and Stanton number for the viscous Mach 5 bow shock at $\mathrm{Re}=10^{3}$ computed with MDG-ICE($\mathcal{P}_{4}$) . (a) 527 isoparametric $\mathcal{P}_{4}$ triangle cells (b) 527 isoparametric $\mathcal{P}_{4}$ triangle cells (c) 527 isoparametric $\mathcal{P}_{4}$ triangle cells (d) 527 isoparametric $\mathcal{P}_{4}$ triangle cells Figure 3.9: The MDG-ICE solution computed using $\mathcal{P}_{4}$ isoparametric triangle elements for the viscous Mach 5 bow shock at $\mathrm{Re}=10^{4}$. The MDG-ICE solution was initialized from the MDG-ICE solution at $\mathrm{Re}=10^{3}$ shown in Figure 3.5a. The location of the shock along the line $x=0$ was computed as $y=1.48437$ for a stand-off distance of $0.48437$. (a) 768 isoparametric $\mathcal{P}_{4}$ triangle cells (b) 768 isoparametric $\mathcal{P}_{4}$ triangle cells (c) 768 isoparametric $\mathcal{P}_{4}$ triangle cells (d) 768 isoparametric $\mathcal{P}_{4}$ triangle cells Figure 3.10: The MDG-ICE solution computed using $\mathcal{P}_{4}$ isoparametric triangle elements for the viscous Mach 5 bow shock at $\mathrm{Re}=10^{5}$. The MDG-ICE solution was initialized from the MDG-ICE solution at $\mathrm{Re}=10^{4}$ shown in Figure 3.5a. The location of the shock along the line $x=0$ was computed as $y=1.4809125$ for a stand-off distance of $0.4809125$. --- (a) The final grid and temperature fields of the MDG- ICE$\left(\mathcal{P}_{4}\right)$ solution computed using 400 $\mathcal{P}_{4}$ isoparametric triangle elements for the viscous Mach 5 bow shock at $10^{3}$ $\mathrm{Re}$ shown in Figure 3.6a and Figure 3.6b respectively. --- (b) The final grid and temperature fields of the MDG- ICE$\left(\mathcal{P}_{4}\right)$ solution computed using 527 $\mathcal{P}_{4}$ isoparametric triangle elements for the viscous Mach 5 bow shock at $10^{4}$ $\mathrm{Re}$ shown in Figure 3.9a and Figure 3.9b. --- (c) The final grid and temperature fields of the MDG- ICE$\left(\mathcal{P}_{4}\right)$ solution computed using 768 $\mathcal{P}_{4}$ isoparametric triangle elements for the viscous Mach 5 bow shock at $10^{5}$ $\mathrm{Re}$ shown in Figure 3.10a and Figure 3.10b. Figure 3.11: The final grid and temperature fields corresponding to the MDG- ICE solution computed using $\mathcal{P}_{4}$ isoparametric triangle elements for the viscous Mach 5 bow shock at $\mathrm{Re}=10^{3}$, $\mathrm{Re}=10^{4}$, and $\mathrm{Re}=10^{5}$. Local edge refinement was used to adaptively split highly anisotropic elements within the viscous structures as they were resolved by MDG-ICE. The viscous MDG-ICE discretization is applied to a compressible Navier-Stokes flow, described in Section (2.1.3), and used to approximate the solution to a supersonic viscous flow over a cylinder in two dimensions. The solution is characterized by the Reynolds number $\mathrm{Re}$, and the freestream Mach number $\mathrm{M}_{\infty}$. The Reynolds number is defined as, $\mathrm{Re}=\frac{\rho vL}{\mu},$ (3.4) where $L$ is the characteristic length. The freestream Mach number is defined $\mathrm{M}_{\infty}=\frac{v_{\infty}}{c_{\infty}}$ $\mathrm{M}_{\infty}=\frac{v_{\infty}}{c_{\infty}},$ (3.5) where $v_{\infty}$ is the freestream velocity, $c_{\infty}=\sqrt{\nicefrac{{\gamma P_{\infty}}}{{\rho_{\infty}}}}$ is the freestream speed of sound, $P_{\infty}$ is the freestream pressure, and $\rho_{\infty}$ is the freestream density. In this work we consider Mach 5 flows at Reynolds numbers of $10^{3}$, $10^{4}$, and $10^{5}$ traveling in the $\left(0,-1\right)$ direction, i.e., from top to bottom in Figure 3.5 , Figure 3.6, Figure 3.9, and Figure 3.10. Supersonic inflow and outflow boundary conditions are applied at the ellipse and outflow planes respectively. An isothermal no-slip wall is specified at the surface of the cylinder of radius $r=1$ centered at the origin. The temperature at the isothermal wall is given as $T_{\mathrm{wall}}=2.5T_{\infty}$, where $T_{\infty}$ is the freestream temperature. Figure 3.6 presents the MDG-ICE($\mathcal{P}_{4}$) solution at $\mathrm{Re}=10^{3}$ computed on a grid of $400$ isoparametric triangle elements. The MDG-ICE($\mathcal{P}_{4}$) solution was initialized by interpolating the DG($\mathcal{P}_{2}$) field variables and the corresponding grid. Figure 3.5a shows the grid consisting of 392 linear triangle elements that was used to initialize the MDG-ICE($\mathcal{P}_{4}$) by interpolating the field variables. The high-order isoparametric boundary faces were initialized by projecting to closest point on the boundary of the domain via the boundary operator (2.80). The MDG-ICE($\mathcal{P}_{4}$) field variables were initialized by cell averaging the interpolated DG($\mathcal{P}_{2}$) solution shown in Figure 3.5b. The MDG-ICE auxiliary variable, with spatial components given by (2.47), were initialized to zero for consistency with the initial piecewise constant field variables. As the MDG-ICE solution converged, the previously uniformly distributed points were relocated in order to resolve the viscous shock. This resulted in a loss of resolution downstream, which was conveniently handled via local edge refinement where highly anisotropic elements were adaptively split as the viscous structures were resolved. Although this was unnecessary for maintaining a valid solution, i.e., field variables that are finite and grid composed of only cells with a positive determinant of the Jacobian, we found it sufficient for maintaining a reasonable grid resolution, as compared to the initial grid resolution downstream of the shock. The location of the shock along the line $x=0$, estimated as the location corresponding to the minimum normal heat flux, was computed as $y=1.49995$, giving a stand-off distance of $0.49995$. In one dimension, the viscous shock is described by a system of ordinary differential equations that can be solved numerically, cf. 94, 93 for details. We use this solution to verify that the viscous MDG-ICE formulation predicts the correct viscous shock profile when diffusive effects are prominent, i.e., at low Reynolds number. Figure 3.7 presents a comparison of an approximation of the exact solution for a one-dimensional viscous shock to the centerline profiles of the Mach 5 bow shock at $\mathrm{Re}=10^{3}$ for the following variables: temperature, $T$, normal velocity, $v_{n}$, pressure, $p$, density, $\rho$, normal component of the normal viscous stress tensor, $\tau_{nn}$, and normal heat flux, $q_{n}$, where the normal is taken to be in the streamwise direction. As expected, the one-dimensional profiles deviate from the two-dimensional bow shock centerline profiles downstream of the viscous shock. The one-dimensional solution assumes the viscous and diffusive fluxes are zero outside of the shock. This is not the case for the two-dimensional bow shock geometry where the blunt body and corresponding boundary layer produce gradients in the solution downstream of the shock. For density, in which case the diffusive flux is zero, the jump across the viscous shock is directly comparable to the one-dimensional solution. We also directly compare the exact solution to a one-dimensional viscous shock profile computed by MDG-ICE($\mathcal{P}_{4}$) using 16 isoparametric line cells. Figure 3.7d shows that MDG-ICE accurately reproduces the exact shock structure of density profile with only a few high- order anisotropic curvilinear cells. For reference, the exact and approximate values at the stagnation point are marked on the centerline plots with the symbol $\times$. An approximate value corresponding to the inviscid solution was used when the exact value for the viscous flow was unavailable, e.g., the stagnation pressure marked in Figure 3.7c. Although the analytic stagnation pressure for a inviscid flow neglects viscous effects, it is not expected to differ significantly from the value corresponding to the viscous solution for the problem considered here, as shown in Table 1 in the work of Williams et al. 95. We also report the pressure coefficient and the Stanton number, sampled at the degrees of freedom, on the cylindrical, no-slip, isothermal surface. The pressure coefficient at the surface is defined as $C_{p}=\frac{p-p_{\infty}}{\frac{1}{2}\rho_{\infty}v_{\infty}^{2}},$ (3.6) where $p_{\infty}$, $\rho_{\infty}$, and $v_{\infty}$ are the freestream pressure, density, and velocity respectively. The Stanton number at the surface is defined as $C_{h}=\frac{q_{\mathrm{n}}}{c_{p}\rho_{\infty}v_{\infty}\left(T_{t,\infty}-T_{\mathrm{wall}}\right)},$ (3.7) where $q_{n}$ is the normal heat flux, $T_{\mathrm{wall}}$ is the wall temperature, and $T_{t,\infty}$ is the freestream stagnation temperature. In Figure 3.8a the pressure coefficient on the cylindrical surface is plotted and the exact pressure coefficient at the stagnation point for an inviscid flow, $C_{p}\approx 1.8087699607027568$, is marked with the symbol $\times$. In order to compute solutions at higher Reynolds numbers, continuation of the freestream viscosity, $\mu_{\infty}$ was employed. Figure 3.9 presents the MDG-ICE($\mathcal{P}_{4}$) solution at $\mathrm{Re}=10^{4}$ computed on a grid of $527$ isoparametric triangle elements, which was initialized from the $\mathrm{Re}=10^{3}$ MDG-ICE($\mathcal{P}_{4}$) solution. Figure 3.9 presents the MDG-ICE($\mathcal{P}_{4}$) solution at $\mathrm{Re}=10^{5}$ computed on a grid of $768$ isoparametric triangle elements, which was initialized from the $\mathrm{Re}=10^{4}$ MDG-ICE($\mathcal{P}_{4}$) solution. As in the $\mathrm{Re}=10^{3}$ case, local edge refinement was used to adaptively split highly anisotropic elements within the viscous structures as they were resolved by MDG-ICE. At these higher Reynolds numbers, local refinement was necessary to maintain a valid grid. In addition to the splitting of highly anisotropic cells, local edge refinement was also applied to cells in which the determinant of the Jacobian became non-positive. Figure 3.11 compares the MDG-ICE($\mathcal{P}_{4}$) solutions directly downstream of the viscous shock at $\mathrm{Re}=10^{3}$, $\mathrm{Re}=10^{4}$, and $\mathrm{Re}=10^{5}$. By adapting the grid to the flow field, MDG-ICE is able to simultaneously resolve the thin viscous structure over a range of Reynolds numbers. As the MDG-ICE solution converges, elements within regions that contain strong gradients become highly anisotropic and warp nonlinearly to conform to both the curved shock geometry and efficiently resolve the flow around the curved blunt body. Thus, unlike a posteriori anisotropic mesh adaptation, MDG-ICE achieves high-order anisotropic curvilinear $r$-adaptivity as an intrinsic part of the solver. Furthermore, MDG-ICE automatically repositions the nodes in order to resolve the flow field over different length scales as the Reynolds number is increased from $10^{3}$ to $10^{5}$. As such, MDG-ICE overcomes another challenge associated with a posteriori anisotropic mesh adaptation which produces regions of excessive refinement on the scale of the coarse mesh cell size and therefore must rely on grid coarsening to limit the region of refinement to the more appropriate length scale corresponding to the feature under consideration. ## 4 Conclusions and future work The Moving Discontinuous Galerkin Method with Interface Condition Enforcement (MDG-ICE) has been applied to viscous flow problems, involving both linear and nonlinear viscous fluxes, where it was shown to detect and resolve previously under-resolved flow features. In the case of linear advection-diffusion, MDG- ICE adapted the grid to resolve the initially under-resolved boundary layer, thereby achieving spectral convergence and a more accurate solution than the best possible approximation on a uniform static grid, which is given by the $L^{2}$ projection of the exact solution. Unsteady flows were computed using a space-time formulation where viscous structures were automatically resolved via anisotropic space-time $r$-adaptivity. High speed compressible Navier- Stokes solutions for a viscous Mach 5 bow shock at a Reynolds numbers of $10^{3}$, $10^{4}$, and $10^{5}$ were presented. The viscous MDG-ICE formulation was shown to produce the correct viscous shock profile in one dimension for a Mach 5 flow at $\mathrm{Re}=10^{3}$. The one-dimensional viscous shock profile was compared to the centerline profile of the two- dimensional MDG-ICE solution where it was shown to accurately compute both the shock profile and boundary layer profile simultaneously using only a few high- order anisotropic curved cells within each region and thus overcoming an ongoing limitation of anisotropic mesh adaptation. Local edge refinement was used to adaptively split highly anisotropic elements within the viscous structures as they were resolved by MDG-ICE. Finally, MDG-ICE is a consistent discretization of the governing equations that does not introduce low-order errors via artificial stabilization or limiting and treats the discrete grid as a variable. It should be noted that the internal structure of a viscous shock may not be adequately described by the compressible Navier-Stokes equations due to non- equilibrium effects 96, an issue surveyed by Powers et al. 97. While in the present work MDG-ICE was shown to provide highly accurate solutions to the compressible Navier-Stokes equations, future work will apply MDG-ICE to an improved multi-scale model that incorporates physical effects more adequately described by the kinetic theory of gases 98. MDG-ICE is a promising method to apply within such a framework due to its ability to isolate regions in which enhanced physical modeling is required. In future work, we will also develop a least-squares MDG-ICE formulation with optimal test functions by applying the DPG methodology of Demkowicz and Gopalakrishnan 58, 59, 65. Using this approach we will demonstrate high-order convergence for both linear and nonlinear problems. We also plan to mitigate the need for local $h$-refinement by considering alternative methods for maintaining grid validity. We will explore adaptively increasing the order of the local polynomial approximation for cells within thin internal and boundary layers. For instance, Chan et al. 94, 85 used a combination of $h$ and $p$ refinement to resolve viscous shocks. In their adaptive strategy, $h-$refinement is used until the local grid resolution is on the order of the viscous scale, at which point $p$-refinement is used to further enhance accuracy. 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2024-09-04T02:54:55.519637

2002.12758

arxiv-papers

# Quark masses: N3LO bridge from ${\rm RI/SMOM}$ to ${\rm\overline{MS}}$ scheme Alexander Bednyakov<EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna 141980, Russia P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Leninskii pr., 5, Moscow 119991, Russia Andrey Pikelner <EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna 141980, Russia ###### Abstract We analytically compute the three-loop corrections to the relation between the renormalized quark masses defined in the minimal-subtraction (${\rm\overline{MS}}$) and the regularization-invariant symmetric momentum- subtraction (RI/SMOM) schemes. Our result is valid in the Landau gauge and can be used to reduce the uncertainty in a lattice determination of the ${\rm\overline{MS}}$ quark masses. ## I Introduction Quark masses $m_{q}$ arise in the Standard Model (SM) from Yukawa interactions of the quarks with the Higgs field. Although not being of fundamental origin, quark masses are usually treated as parameters of the SM and for many years were the only source of information on the Higgs Yukawa couplings. As a consequence, precise knowledge of $m_{q}$ is required both to test the SM and study new physics. The values of the quark masses can be determined in several ways (for a review see, e.g., Ref. Tanabashi _et al._ (2018)). Since all colored fermions but the top are confined inside hadrons, there is no unique (“physical”) definition of the corresponding mass parameters, and one is free to choose a renormalization scheme that suits better for a problem at hand. To compare the results of different determinations, it is customary to use perturbation theory (PT) and convert the obtained values to the short-distance running mass $m_{q}^{{\rm\overline{MS}}}(\mu)$ in the minimal-subtraction scheme ${\rm\overline{MS}}$, evaluated at a fixed scale $\mu$. | ---|--- Figure 1: Momentum flow of a Green function (left), and the three-point vertex with $O_{S}=\bar{\psi}\psi$ operator insertion (right) considered in the paper. SMOM kinematics corresponds to $p_{1}^{2}=p_{2}^{2}=q^{2}$, while in the “exceptional” case $p_{1}^{2}=p_{2}^{2}$, and $q^{2}=0$. One of the approaches to the quark-mass determination, especially useful in the case of light quarks, is based on lattice computations (see, e.g., Ref. Aoki _et al._ (2019)). The resulting values, in this case, are bare quark masses $m_{\rm bare}$, corresponding to a particular discretization of QCD with the lattice spacing $a$ acting as the ultraviolet cutoff. While it is, in principle, possible to directly relate $m_{\rm bare}$ to $m_{q}^{{\rm\overline{MS}}}$, it turns out to be more convenient to relate $m_{\rm bare}$ to a mass parameter $m_{q}^{\rm RI}$ defined in a regularization-independent (RI) momentum-subtraction renormalization scheme, which can be realized directly in lattice QCD. The continuum PT is used in this case to convert the finite value $m_{q}^{\rm RI}$ to $m_{q}^{{\rm\overline{MS}}}$. Among such kind of schemes, the so-called RI/SMOM Sturm _et al._ (2009), in which certain three-point Green functions with momenta $p_{1}$, $p_{2}$, and $q=p_{1}+p_{2}$ (see, Fig. 1) are normalized at _symmetric_ kinematics ($p_{1}^{2}=p_{2}^{2}=q^{2}=-\mu^{2}$) and have advantages over original RI/MOM Martinelli _et al._ (1995) scheme. The latter utilizes “exceptional” momenta configuration with $q^{2}=0$, $p_{1}^{2}=p_{2}^{2}=-\mu^{2}$ and suffers from enhanced sensitivity to nonperturbative infrared effects (see, e.g., Ref. Aoki _et al._ (2008) for details). In addition, the RI/SMOM PT series show a much better convergence behavior than that of the RI/MOM ones. Recent state-of-the-art lattice determination Lytle _et al._ (2018) of the running ${\rm\overline{MS}}$ masses of the charm ($m_{c}^{{\rm\overline{MS}}}(3~{}{\rm GeV})=0.9896(61)$ GeV) and strange ($m_{s}^{{\rm\overline{MS}}}(3~{}{\rm GeV})=0.008536(85)$ GeV) quarks in $n_{f}{}=4$ QCD heavily relies on the two-loop (next-to-next-to-leading, or NNLO) conversion factor Gorbahn and Jager (2010); Almeida and Sturm (2010) relating ${\rm\overline{MS}}$ and ${\rm SMOM}$ schemes. According to the estimates given in this reference, the uncertainty due to the missing next-to- next-to-next-to-leading (N3LO) term is comparable with other sources of uncertainties (e.g., due to continuum extrapolation or condensate effects) and contribute a significant part to the overall error budget (for details see Table VI of Ref.Lytle _et al._ (2018)). In this letter, we report on the analytical computation of the three-loop contribution, thus, providing additional precision for such an analysis. Recently, a _numerical_ evaluation of the same quantity appeared in Ref. Kniehl and Veretin (2020). Our result confirms the estimates provided therein. ## II Details of calculation To calculate the required conversion factor $C_{m}^{{\rm SMOM}}$, we consider QCD with $n_{f}{}$ flavors and define $\displaystyle m_{q}^{{\rm\overline{MS}}}=C_{m}^{{\rm SMOM}}m_{q}^{{\rm SMOM}},\quad C_{m}^{{\rm SMOM}}=\frac{Z_{m}^{{\rm SMOM}}}{Z_{m}^{{\rm\overline{MS}}}}.$ (1) The mass parameters in ${\rm\overline{MS}}$ and ${\rm SMOM}$ schemes are related to the quark bare mass $m_{\rm bare}$ via $Z_{m}^{\rm R}=\\{Z_{m}^{{\rm\overline{MS}}},Z_{m}^{{\rm SMOM}}\\}$ $\displaystyle m_{\rm bare}=Z_{m}^{\rm R}m_{q}^{\rm R}=Z_{m}^{{\rm\overline{MS}}}m_{q}^{{\rm\overline{MS}}}=Z_{m}^{{\rm SMOM}}m_{q}^{{\rm SMOM}}.$ (2) In continuum QCD the bare mass $m_{\rm bare}$ is usually defined in dimensional regularization so that each $Z_{m}^{\rm R}$ contains poles in $\varepsilon=(4-d)/2$. To determine $Z_{m}^{\rm R}$ we do not compute massive propagators but renormalize the scalar bilinear operator $O_{S}\equiv\bar{\psi}\psi$ (see Fig. 1) in massless QCD $\displaystyle\left[\bar{\psi}\psi\right]_{\rm R}=Z_{m}^{\rm R}(\bar{\psi}\psi)_{\rm bare}.$ (3) This simplified approach neglects both valence and sea quark masses, but still provides a reasonable approximation to the conversion factor $C_{m}^{{\rm SMOM}}$ in a range of renormalization scales utilized in lattice calculations (see,e.g, Ref. Lytle _et al._ (2018) for numerical studies of the two-loop corrections due to nonzero quark masses). We compute $Z_{m}^{{\rm SMOM}}$ and $Z_{m}^{{\rm\overline{MS}}}$ order-by- order in PT by considering bare three-point one-particle-irreducible vertex function $\displaystyle\left.\Lambda_{S}(p_{1},p_{2})\right|_{sym}=\left.\langle\psi(-p_{2})O_{S}(q)\bar{\psi}(-p_{1})\rangle\right|_{p_{1}^{2}=p_{2}^{2}=q^{2}=-\mu^{2}},\quad q=p_{1}+p_{2}$ (4) in ${\rm SMOM}$ kinematics. We use Landau gauge and require that $\displaystyle 1=Z_{m}^{{\rm SMOM}}\cdot Z_{\psi}^{{\rm SMOM}}\cdot\frac{1}{12}\cdot\left.{\rm tr}\left[\Lambda^{\rm bare}_{S}\right]\right|_{sym},\quad 1=Z_{\psi}^{{\rm SMOM}}\cdot\frac{1}{12p^{2}}\left.\cdot{\rm tr}\left[iS_{\rm bare}^{-1}(p)\hat{p}\right]\right|_{p^{2}=-\mu^{2}},$ (5) where both $\Lambda_{S}^{\rm bare}$ and the bare quark inverse propagator $S_{\rm bare}^{-1}$ are reexpanded in terms of ${\rm\overline{MS}}$ strong coupling $\alpha_{s}^{{\rm\overline{MS}}}=(4\pi)a_{{\rm\overline{MS}}}$ via the well-known formula $\mu^{-2\varepsilon}a_{\rm bare}=Z_{a_{{\rm\overline{MS}}}}a_{{\rm\overline{MS}}}$ available with five- loop accuracy Chetyrkin _et al._ (2017); Luthe _et al._ (2017). In Eq. (5) the quark field renormalization constants are defined as111It is worth mentioning that, e.g., in Refs. Sturm _et al._ (2009); Almeida and Sturm (2010); Lytle _et al._ (2018), different notation can be adopted for the renormalization constants,and one should make the substitutions $Z_{\psi}\to Z_{\psi}^{-1}$ and $Z_{m}\to Z^{-1}_{m}$ to compare the results. $\displaystyle\psi_{\rm bare}=\sqrt{Z_{\psi}^{\rm R}}\psi_{\rm R},\quad{\rm R}=\\{{\rm\overline{MS}},{\rm SMOM}\\}$ (6) The conditions (5) can be implemented in lattice computations, leading to a nonperturbative determination Martinelli _et al._ (1995) of $Z_{m}^{{\rm SMOM}}$. The latter converts the bare lattice mass into $m_{q}^{{\rm SMOM}}$, providing input for $m_{q}^{{\rm\overline{MS}}}$ calculation via Eq. (1). The ${\rm\overline{MS}}$ counterparts $Z_{m}^{{\rm\overline{MS}}}$, $Z_{\psi}^{{\rm\overline{MS}}}$ of the renormalization constants in Eq. (5) required to compute $C_{m}^{{\rm SMOM}}$ are obtained by subtracting only divergent terms of the corresponding Green functions. A comment is in order regarding the determination of the wave function renormalization constant $Z_{\psi}^{{\rm SMOM}}$. Due to Ward identities, the latter can also be obtained from the (non)renormalization of vector (axial) quark bilinear operators $O^{\mu}_{V}\equiv\psi\gamma^{\mu}\psi$ ($O^{\mu}_{A}\equiv\psi\gamma^{\mu}\gamma_{5}\psi$). In the continuum, Ward identifies and chiral symmetry guarantee that $Z_{V}=Z_{A}=1$, and it can be proven Sturm _et al._ (2009) that the condition on $Z_{\psi}^{{\rm SMOM}}$ given in Eq. (5) corresponds to $\displaystyle 1=Z_{\psi}^{{\rm SMOM}}\cdot\frac{1}{12q^{2}}\cdot\left.{\rm tr}\left[q_{\mu}\Lambda^{\mu,\rm bare}_{V}\hat{q}\right]\right|_{sym},\quad 1=Z_{\psi}^{{\rm SMOM}}\cdot\frac{1}{12q^{2}}\cdot\left.{\rm tr}\left[q_{\mu}\Lambda^{\mu,\rm bare}_{A}\gamma_{5}\hat{q}\right]\right|_{sym}$ (7) with $\Lambda_{V}^{\mu}$ ($\Lambda_{A}^{\mu}$) being analogs of (4) with $O_{S}$ replaced by $O_{V}^{\mu}$ ($O_{A}^{\mu}$). It is also possible to use the so-called RI/SMOM${}_{\gamma_{\mu}}$ Sturm _et al._ (2009) and require $\displaystyle 1=Z_{\psi}^{{\rm SMOM}_{\gamma_{\mu}}}\cdot\frac{1}{48}\cdot\left.{\rm tr}\left[\gamma_{\mu}\Lambda^{\mu,\rm bare}_{V}\right]\right|_{sym},\quad 1=Z_{\psi}^{{\rm SMOM}_{\gamma_{\mu}}}\cdot\frac{1}{48}\cdot\left.{\rm tr}\left[\Lambda^{\mu,\rm bare}_{A}\gamma_{5}\gamma_{\mu}\right]\right|_{sym}.$ (8) Both RI/SMOM and RI/SMOM${}_{\gamma_{\mu}}$ conditions can be implemented on lattice (see, e.g., Refs. Blum _et al._ (2016); Aoki _et al._ (2008) for details and subtleties). In Ref. Almeida and Sturm (2010) it was demonstrated that the PT series for the quark-mass conversion factor exhibits slightly better behavior in RI/SMOM than in RI/SMOM${}_{\gamma_{\mu}}$. Given this argument we carry out our calculation in RI/SMOM. Let us mention a few technical details of our calculation. We generate Feynman graphs with DIANA Tentyukov and Fleischer (2000) and take fermion and color van Ritbergen _et al._ (1999) traces according to Eq. (5). Resulting scalar integrals are reduced to the set of master integrals identified in our previous paper Bednyakov and Pikelner (2020) on $\alpha_{s}$ renormalization in the ${\rm SMOM}$ scheme. To perform reduction we make use of the FIRE6Smirnov and Chuharev (2019) package. Substituting masters integrals evaluated previously, we end up with expressions valid for a general gauge group. The number of master integrals and the necessary expansion depth in dimensional regularization parameter $\varepsilon=(4-d)/2$ are the same as in the paperBednyakov and Pikelner (2020). It is worth noting that as a cross- check of our calculation we also consider the renormalization of the pseudoscalar quark current $O_{P}=\bar{\psi}\gamma_{5}\psi$, which can also be used to extract $Z_{m}^{{\rm SMOM}}$ from lattice calculations. ## III Results and conclusion Expressing all the renormalization constants in terms of $a_{{\rm\overline{MS}}}$, from Eq. (1) we obtain the following N3LO conversion factor $\displaystyle C_{m}^{{\rm SMOM}}$ $\displaystyle=1+x_{1}a_{{\rm\overline{MS}}}+x_{2}a_{{\rm\overline{MS}}}^{2}+x_{3}a_{{\rm\overline{MS}}}^{2}$ (9) with $\displaystyle x_{1}=$ $\displaystyle{\color[rgb]{0.01171875,0.22265625,0.421875}C_{F}}\bigg{(}-4-\frac{2}{3}\pi^{2}+\psi_{1}\bigg{)}$ (10) $\displaystyle x_{2}=$ $\displaystyle{\color[rgb]{0.01171875,0.22265625,0.421875}n_{f}T_{F}C_{F}}\bigg{(}\frac{83}{6}+\frac{40}{27}\pi^{2}-\frac{20}{9}\psi_{1}\bigg{)}$ $\displaystyle+$ $\displaystyle{\color[rgb]{0.01171875,0.22265625,0.421875}C_{F}^{2}}\bigg{(}\frac{19}{8}+\frac{28}{9}\pi^{2}-\frac{14}{3}\psi_{1}+4\zeta_{3}+\frac{58}{81}\pi^{4}-\frac{52}{27}\psi_{1}\pi^{2}+\frac{13}{9}\psi_{1}^{2}-\frac{1}{36}\psi_{3}\bigg{)}$ $\displaystyle+$ $\displaystyle{\color[rgb]{0.01171875,0.22265625,0.421875}C_{A}C_{F}}\bigg{(}-\frac{1285}{24}-\frac{385}{54}\pi^{2}+\frac{385}{36}\psi_{1}+10\zeta_{3}-\frac{8}{81}\pi^{4}+\frac{8}{27}\pi^{2}\psi_{1}-\frac{2}{9}\psi_{1}^{2}\bigg{)}$ (11) $\displaystyle x_{3}=$ $\displaystyle{\color[rgb]{0.01171875,0.22265625,0.421875}n_{f}^{2}T_{F}^{2}C_{F}}\bigg{(}-\frac{7514}{243}-\frac{800}{243}\pi^{2}+\frac{400}{81}\psi_{1}-\frac{32}{9}\zeta_{3}-\frac{32}{243}\pi^{4}+\frac{4}{81}\psi_{3}\bigg{)}$ $\displaystyle+$ $\displaystyle{\color[rgb]{0.01171875,0.22265625,0.421875}n_{f}T_{F}C_{A}C_{F}}\bigg{(}\frac{95387}{243}+\frac{13172}{243}\pi^{2}-\frac{6586}{81}\psi_{1}-\frac{152}{9}\zeta_{3}+\frac{3952}{3645}\pi^{4}$ $\displaystyle-$ $\displaystyle\frac{320}{243}\psi_{1}\pi^{2}+\frac{80}{81}\psi_{1}^{2}-\frac{23}{162}\psi_{3}+\frac{320}{81}\pi^{2}\zeta_{3}+\frac{16240}{729}\zeta_{5}-\frac{160}{27}\psi_{1}\zeta_{3}+\frac{64}{81}H_{5}\bigg{)}$ $\displaystyle+$ $\displaystyle{\color[rgb]{0.01171875,0.22265625,0.421875}n_{f}T_{F}C_{F}^{2}}\bigg{(}\frac{1109}{9}-\frac{241}{81}\pi^{2}+\frac{241}{54}\psi_{1}-\frac{1384}{9}\zeta_{3}-\frac{15392}{3645}\pi^{4}+\frac{2080}{243}\psi_{1}\pi^{2}$ $\displaystyle-$ $\displaystyle\frac{520}{81}\psi_{1}^{2}+\frac{67}{162}\psi_{3}-\frac{128}{9}\pi^{2}\zeta_{3}-\frac{32480}{729}\zeta_{5}+\frac{64}{3}\psi_{1}\zeta_{3}-\frac{128}{81}H_{5}\bigg{)}$ $\displaystyle+$ $\displaystyle{\color[rgb]{0.01171875,0.22265625,0.421875}C_{F}^{3}}\bigg{(}-\frac{3227}{12}-\frac{191}{12}\pi^{2}+\frac{191}{8}\psi_{1}-58\zeta_{3}-\frac{992}{81}\pi^{4}+\frac{232}{9}\psi_{1}\pi^{2}-\frac{58}{3}\psi_{1}^{2}+\frac{37}{27}\psi_{3}$ $\displaystyle+$ $\displaystyle\frac{80}{9}\pi^{2}\zeta_{3}-\frac{32980}{81}\zeta_{5}-\frac{40}{3}\psi_{1}\zeta_{3}-\frac{112}{9}H_{5}-\frac{131776}{98415}\pi^{6}+\frac{23992}{6561}\psi_{1}\pi^{4}-\frac{394}{81}\psi_{1}^{2}\pi^{2}$ $\displaystyle+$ $\displaystyle\frac{679}{6561}\psi_{3}\pi^{2}-\frac{679}{4374}\psi_{1}\psi_{3}+\frac{197}{81}\psi_{1}^{3}+\frac{1}{135}\psi_{5}+\frac{2}{8505}H_{6}\bigg{)}$ $\displaystyle+$ $\displaystyle{\color[rgb]{0.01171875,0.22265625,0.421875}C_{A}C_{F}^{2}}\bigg{(}\frac{18781}{72}+\frac{23231}{324}\pi^{2}-\frac{23231}{216}\psi_{1}+\frac{2879}{9}\zeta_{3}+\frac{34423}{1458}\pi^{4}-\frac{11306}{243}\psi_{1}\pi^{2}$ $\displaystyle+$ $\displaystyle\frac{5653}{162}\psi_{1}^{2}-\frac{3937}{1296}\psi_{3}-\frac{178}{9}\pi^{2}\zeta_{3}+\frac{379285}{729}\zeta_{5}+\frac{89}{3}\psi_{1}\zeta_{3}+\frac{1840}{81}H_{5}-\frac{1519}{32805}\pi^{6}$ $\displaystyle+$ $\displaystyle\frac{4}{81}\psi_{1}\pi^{4}+\frac{4}{27}\psi_{1}^{2}\pi^{2}+\frac{1}{27}\psi_{3}\pi^{2}-\frac{1}{18}\psi_{1}\psi_{3}-\frac{2}{27}\psi_{1}^{3}-\frac{77}{116640}\psi_{5}\bigg{)}$ $\displaystyle+$ $\displaystyle{\color[rgb]{0.01171875,0.22265625,0.421875}C_{A}^{2}C_{F}}\bigg{(}-\frac{3360023}{3888}-\frac{243283}{1944}\pi^{2}+\frac{243283}{1296}\psi_{1}+\frac{4511}{24}\zeta_{3}-\frac{20513}{5832}\pi^{4}+\frac{5107}{972}\psi_{1}\pi^{2}$ $\displaystyle-$ $\displaystyle\frac{5107}{1296}\psi_{1}^{2}+\frac{3433}{5184}\psi_{3}+\frac{7535}{324}\pi^{2}\zeta_{3}-\frac{1140715}{5832}\zeta_{5}-\frac{7535}{216}\psi_{1}\zeta_{3}-\frac{668}{81}H_{5}+\frac{100133}{314928}\pi^{6}$ $\displaystyle-$ $\displaystyle\frac{10619}{13122}\psi_{1}\pi^{4}+\frac{325}{324}\psi_{1}^{2}\pi^{2}-\frac{461}{13122}\psi_{3}\pi^{2}+\frac{461}{8748}\psi_{1}\psi_{3}-\frac{325}{648}\psi_{1}^{3}-\frac{611}{373248}\psi_{5}-\frac{1}{17010}H_{6}\bigg{)}.$ (12) Here $\zeta_{i}$ is the Riemann zeta function, and $\psi_{m}=\psi^{(m)}(1/3)$ corresponds to the $(m+1)$th derivative of the gamma function. Additional constants of uniform transcendental weight $H_{5}$ and $H_{6}$, introduced in Ref. Bednyakov and Pikelner (2020), $H_{5}=-23.9316195698,\quad H_{6}=248215.038289$ (13) are linear combinations of real parts of harmonic polylogarithms with six-root of unity argument from the basis constructed in Ref. Kniehl _et al._ (2017). Our result reproduces the well-known analytic one-loop Sturm _et al._ (2009) and two-loop Gorbahn and Jager (2010); Almeida and Sturm (2010) expressions, together with recent numerical evaluation of Ref. Kniehl and Veretin (2020): $\displaystyle C_{m}^{{\rm SMOM}}=1$ $\displaystyle-0.6455188560a_{{\rm\overline{MS}}}-(22.60768757-4.013539470n_{f})a_{{\rm\overline{MS}}}^{2}$ $\displaystyle-(860.2874030-164.7423004n_{f}+2.184402262n_{f}^{2})a_{{\rm\overline{MS}}}^{3}.$ (14) Given this general result (III), we are ready to provide our numerical estimates of the N3LO contribution for different $n_{f}$. Expanding the matching factor in powers of $\alpha_{s}\equiv\alpha_{s}^{{\rm\overline{MS}}}$, we obtain $\displaystyle n_{f}=0:\quad$ $\displaystyle 1$ $\displaystyle-0.05136875839\alpha_{s}-0.1431648540\alpha_{s}^{2}$ $\displaystyle-0.4335248250\alpha_{s}^{3},$ (15) $\displaystyle n_{f}=1:\quad$ $\displaystyle 1$ $\displaystyle-0.05136875839\alpha_{s}-0.1177488184\alpha_{s}^{2}$ $\displaystyle-0.3516069867\alpha_{s}^{3},$ (16) $\displaystyle n_{f}=2:\quad$ $\displaystyle 1$ $\displaystyle-0.05136875839\alpha_{s}-0.09233278278\alpha_{s}^{2}$ $\displaystyle-0.2718907211\alpha_{s}^{3},$ (17) $\displaystyle n_{f}=3:\quad$ $\displaystyle 1$ $\displaystyle-0.05136875839\alpha_{s}-0.06691674717\alpha_{s}^{2}$ $\displaystyle-0.1943760281\alpha_{s}^{3},$ (18) $\displaystyle n_{f}=4:\quad$ $\displaystyle 1$ $\displaystyle-0.05136875839\alpha_{s}-0.04150071157\alpha_{s}^{2}$ $\displaystyle-0.1190629077\alpha_{s}^{3},$ (19) $\displaystyle n_{f}=5:\quad$ $\displaystyle 1$ $\displaystyle-0.05136875839\alpha_{s}-0.01608467597\alpha_{s}^{2}$ $\displaystyle-0.04595136006\alpha_{s}^{3},$ (20) $\displaystyle n_{f}=6:\quad$ $\displaystyle 1$ $\displaystyle-0.05136875839\alpha_{s}+0.009331359638\alpha_{s}^{2}$ $\displaystyle+0.02495861498\alpha_{s}^{3}.$ (21) Given the value $\alpha_{s}^{nf=4}(3\,)=0.2545$ used by HPQCD collaboration Lytle _et al._ (2018) in the determination of charm- and strange-quark masses, we evaluate the matching factor at the reference scale $\mu_{\rm ref}=3\,$ $\displaystyle Z^{{\rm\overline{MS}}/{\rm SMOM}}_{m}\equiv C_{m}^{{\rm SMOM}}$ $\displaystyle=1-\underbrace{0.0130733}_{\alpha_{s}}-\underbrace{0.00268801}_{\alpha_{s}^{2}}-\underbrace{0.00196264}_{\alpha_{s}^{3}}=0.982276,\quad n_{f}=4,~{}\mu=3~{}.$ (22) One can see that the three-loop contribution is of the same order as the two- loop correction and is of the same size as the uncertainty $0.22\%$ quoted in Ref. Lytle _et al._ (2018) and attributed to the missing N3LO term. The comparision with the result given in Ref. Lytle _et al._ (2018) also shows that the effect of the $\alpha_{s}^{3}$ term in Eq. (22) is four times larger than the two-loop contribution due to massive charm quark in the sea and becomes an order of magnitude larger if $\mu=5$ GeV is chosen. It is also worth mentioning that the authors of Ref. Kniehl and Veretin (2020) also consider vector and tensor quark bilinears. We apply the projector (8) to the expression for the vector-operator $O_{V}$ matrix element given in Ref. Kniehl and Veretin (2020), evaluate the quark wave function renormalization in RI/SMOM${}_{\gamma_{\mu}}$, and obtain the following numeric result for the corresponding matching factor: $\displaystyle C_{m}^{{\rm SMOM}_{\gamma_{\mu}}}=1-$ $\displaystyle 1.978852189a_{{\rm\overline{MS}}}-(55.03243483-6.161687618n_{f})a_{{\rm\overline{MS}}}^{2}$ $\displaystyle-$ $\displaystyle(2086.34(14)-362.560(3)n_{f}+6.7220(1)n_{f}^{2})a_{{\rm\overline{MS}}}^{3}.$ (23) While the two-loop contribution to Eq. (23) is known in analytic form Almeida and Sturm (2010), the three-loop term is new and, to our knowledge, is not presented in the literature. One can see that numerical coefficients in RI/SMOM${}_{\gamma_{\mu}}$ (23) is indeed larger than that in RI/SMOM (III), and, e.g., at our reference scale $\mu_{\rm ref}$ we have $\displaystyle C_{m}^{{\rm SMOM}_{\gamma_{\mu}}}$ $\displaystyle=1-\underbrace{0.04007663}_{\alpha_{s}}-\underbrace{0.012463065}_{\alpha_{s}^{2}}-\underbrace{0.006177}_{\alpha_{s}^{3}}=0.941283,\quad n_{f}=4,~{}\mu=3~{}.$ (24) To conclude, we analytically calculate the three-loop correction to the matching factor in RI/SMOM scheme required to extract ${\rm\overline{MS}}$ quark masses from nonperturbative lattice computationsBlum _et al._ (2016); Lytle _et al._ (2018). Our numerical evaluation confirms the estimate of $x_{3}$ given in Ref. Kniehl and Veretin (2020). In addition, we use the results of Ref. Kniehl and Veretin (2020) to evaluate the three-loop expression for the corresponding matching factor in RI/SMOM${}_{\gamma_{\mu}}$. We believe that the obtained N3LO contribution to $C_{m}^{{\rm SMOM}}$ will increase the precision of the resulting ${\rm\overline{MS}}$ quark masses and/or provide a more reliable estimate of the uncertainties due to missing high-order terms. ###### Acknowledgements. We would like to thank Christine Davies for the correspondence regarding Ref. Lytle _et al._ (2018) and clarifying comments on the sea quark contribution. The work of A.P. is supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS.” The work of A.B. is supported by the Grant of the Russian Federation Government, Agreement No. 14.W03.31.0026 from 15.02.2018. ## References * Tanabashi _et al._ (2018) M. Tanabashi _et al._ (Particle Data Group), “Review of particle physics,” Phys.Rev.D 98, 030001 (2018). * Aoki _et al._ (2019) S. Aoki _et al._ (Flavour Lattice Averaging Group), “FLAG Review 2019,” (2019), arXiv:1902.08191 [hep-lat] . * Sturm _et al._ (2009) C. Sturm, Y. Aoki, N. H. 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Vermaseren, “Five-loop renormalisation of QCD in covariant gauges,” JHEP 10, 179 (2017), [Addendum: JHEP12,006(2017)], arXiv:1709.08541 [hep-ph] . * Luthe _et al._ (2017) Thomas Luthe, Andreas Maier, Peter Marquard, and York Schroder, “The five-loop Beta function for a general gauge group and anomalous dimensions beyond Feynman gauge,” JHEP 10, 166 (2017), arXiv:1709.07718 [hep-ph] . * Blum _et al._ (2016) T. Blum _et al._ (RBC, UKQCD), “Domain wall QCD with physical quark masses,” Phys. Rev. D93, 074505 (2016), arXiv:1411.7017 [hep-lat] . * Tentyukov and Fleischer (2000) M. Tentyukov and J. Fleischer, “A Feynman diagram analyzer DIANA,” Comput. Phys. Commun. 132, 124–141 (2000), arXiv:hep-ph/9904258 [hep-ph] . * van Ritbergen _et al._ (1999) T. van Ritbergen, A. N. Schellekens, and J. A. M. Vermaseren, “Group theory factors for Feynman diagrams,” Int. J. Mod. Phys. A14, 41–96 (1999), arXiv:hep-ph/9802376 [hep-ph] . * Bednyakov and Pikelner (2020) Alexander Bednyakov and Andrey Pikelner, “Four-loop QCD MOM beta functions from the three-loop vertices at the symmetric point,” (2020), arXiv:2002.02875 [hep-ph] . * Smirnov and Chuharev (2019) A. V. Smirnov and F. S. Chuharev, “FIRE6: Feynman Integral REduction with Modular Arithmetic,” (2019), 10.1016/j.cpc.2019.106877, arXiv:1901.07808 [hep-ph] . * Kniehl _et al._ (2017) B. A. Kniehl, A. F. Pikelner, and O. L. Veretin, “Three-loop massive tadpoles and polylogarithms through weight six,” JHEP 08, 024 (2017), arXiv:1705.05136 [hep-ph] .

2024-09-04T02:54:55.536646

2002.12812

arxiv-papers

Optical Incoherence Tomography: a method to generate tomographic retinal cross-sections with non-interferometric imaging systems Pedro Mecê1,∗, Elena Gofas-Salas2, Michel Paques2,3, Kate Grieve2,3, and Serge Meimon4 1Institut Langevin, ESPCI Paris, CNRS, PSL University, 1 rue Jussieu, 75005 Paris, France 2Quinze-Vingts National Eye Hospital, 28 Rue de Charenton, Paris, 75012, France 3Institut de la Vision, Sorbonne Université, INSERM, CNRS, F-75012, Paris, France 4DOTA, ONERA, Université Paris Saclay F-91123 Palaiseau, France <EMAIL_ADDRESS> ###### Abstract Optical tomographic cross-sectional images of biological samples were made possible by interferometric imaging techniques such as Optical Coherence Tomography (OCT) [1, 2, 3]. Owing to its unprecedented view of the sample, OCT has become a gold standard, namely for human retinal imaging in the clinical environment. In this Letter, we present Optical Incoherence Tomography (OIT): a completely digital method extending the possibility to generate tomographic retinal cross-sections to non-interferometric imaging systems such as en-face AO-ophthalmoscopes [4, 5]. We demonstrate that OIT can be applied to different imaging modalities using back-scattered and multiply-scattered light including systems without inherent optical sectioning. We show that OIT can be further used to guide focus position when the user is “blind” focusing, allowing precise imaging of translucent retinal structures [6], the vascular plexuses [7] and the retinal pigment epithelium (RPE) [8] using respectively split detection, motion contrast, and autofluorescence techniques. High-resolution in-vivo imaging of the human retina can be achieved using Adaptive Optics (AO) ophthalmoscopes, such as Flood-Illumination Ophthalmoscopes (FIO) [9] and Scanning Laser Ophthalmoscopes (SLO) [10], owing to the capacity of AO to measure and correct for static and dynamic monochromatic ocular aberrations in real-time [11, 12]. Such high-resolution retinal images play an important role in early-stage retinal disease diagnosis, monitoring the progression of retinal disease and the effect of new therapeutic drugs. [10, 13]. To be able to explore the retinal volume using AO ophthalmoscopes the control of the imaging focal position becomes crucial. At present, the positioning of the image focal plane is typically done empirically by visualizing the en-face images displayed in real-time and judging if the retinal structure of interest is sharp or not. This focus guidance approach seems sufficient when using confocal AO-SLO to image hyperreflective retinal layers such as photoreceptors, vasculature, and nerve fiber layer (NFL) [10], especially due to the optical sectioning capability of such imaging systems. Nevertheless, the same cannot be said when using nonconfocal imaging modalities as AO-FIO and AO-SLO split-detection [6], multi-offset [14], motion contrast [15], and autofluorescence [8], since displayed images present a weak signal-to-noise ratio (SNR) and a low contrast, and users are mostly “blind” focusing. As a result, the acquisition of several image stacks around the retinal layer of interest, i.e. for different focal planes, and the assessment of the image quality after the acquisition, to select the best image stack, are mandatory, time-consuming steps which are not always compatible with the clinical environment. To avoid these drawbacks, a focus-guidance tool becomes essential, especially to reveal hypo-reflective or transparent structures such as cone photoreceptor inner segments (IS) [6], retinal ganglion cells [14], perfusion in microvasculature [16, 15] or those masked by neighboring structures of high reflectivity such as RPE lying beneath photoreceptors [8]. Here, we present OIT: a digital method that enables the generation of tomographic retinal cross-sections in non-interferometric AO-ophthalmoscopes. We apply OIT to different AO-ophthalmoscopes modalities: AO-FIO, confocal AO- SLO, split-detection AO-SLO, motion contrast AO-SLO. We demonstrate that most of the retinal layers commonly resolved by OCT cross-sections can also be resolved in OIT cross-sections. Finally, we use OIT to precisely guide focus positioning, enabling imaging of all retinal vascular plexuses, photoreceptor IS and RPE with ease when using respectively motion contrast, split detection, and autofluorescence techniques. The OIT procedure is composed of three main steps (see Methods Section for further details): 1) acquisition of en-face retinal images from different focal planes, forming a Z-stack; 2) filtering out the low-spatial frequency content of each image using a high-pass filter; 3) Measurement of the image sharpness through the computation of the image energy. A crucial step of the OIT procedure is filtering out the low-spatial frequency content. Indeed, since nonconfocal imaging systems do not present inherent optical sectioning, i.e. the capacity to reject out-of-focus photons, the high-pass filter enables one to take advantage of the fact that high-spatial frequency is only present for photons coming from the in-focus plane [17], creating an axial sectioning effect. To demonstrate the axial sectioning capability of the OIT and the impact of the choice of the cut-off frequency, we applied the OIT procedure to a Z-stack obtained from a USAF target using the PARIS AO-FIO. Figures 1(a,b) present the region of interest (ROI) of one en-face image and OIT cross- sections for different normalized cut-off frequencies, where two behaviors can be noticed. On the one hand, by increasing the cut-off frequency, the axial sectioning ability of OIT is enhanced. Figure 1(c) outlines this behavior by presenting the axial sectioning as a function of the normalized cut-off frequency. Not surprisingly, the sectioning ability is limited by the depth of field (DOF) at the diffraction limit, as the latter can be defined as the distance from the best focus where the high spatial frequency content starts to lose contrast, and, consequently, the image sharpness decreases [18]. On the other hand, as the cut-off frequency is increased, the OIT cross-section loses contrast and SNR (Fig. 1(d)). This latter effect happens since the contrast is gradually reduced towards zero at a point defined by the lateral resolution of the optical imaging system [18]. We can expect to obtain sufficient OIT image contrast (higher than 80%) for normalized cut-off frequencies ranging from 1% to 25%. To distinguish different retinal layers, we decided to use a high-pass filter with a normalized cut-off frequency of 20%, representing a good trade- off between the sectioning ability ($1.2\times$ DOF) and the image contrast (90%). With this cut-off frequency, for a 7-mm diameter pupil and a light source of 850 nm, we expect to achieve an axial sectioning of $25\leavevmode\nobreak\ \mu m$. As the OIT cross-section is given as a function of the focus position, one can precisely determine beforehand the position of the imaging plane to extract a sharp USAF target image. Figure 2(a,c) presents OIT tomographic retinal cross-sections acquired on a healthy subject at 7o Nasal using the AO-FIO and AO-SLO, both in bright-field modality. We compare both OIT cross-sections with an OCT image extracted at the same retinal location with Spectralis OCT (Heidelberg Engineering, Germany, Fig. 2(b)). Although OCT can achieve an enhanced axial resolution compared to OIT (i.e. not limited by DOF but by the light source bandwidth), most of the retinal layers commonly identified in OCT can be identified in both AO-SLO and AO-FIO OIT cross-sections: 1) the NFL, which gets thicker as eccentricity increases (blue line); 2) two intermediate layers, most probably corresponding to the inner plexiform layer (IPL, red line) and outer plexiform layer (OPL, yellow line); 3) inner/outer segment junction (IS/OS, green line) and RPE (orange line). The proposed retinal layer labeling can be further confirmed by looking at the en-face images acquired when positioning the imaging focal plane at each of these layers (Figs. 2(d,e)). The RPE labeling is confirmed when applying autofluorescence with AO-SLO [8] at the given focus position (Figs. 2(f-i)). Figure 2(j) presents the radial averaged power spectral density (PSD) for both bright-field and autofluorescence AO-SLO acquired simultaneously, presenting the typical spatial frequency of, respectively, the photoreceptor and RPE mosaics. We measured a photoreceptor density of $18\leavevmode\nobreak\ 000\leavevmode\nobreak\ cells/mm^{2}$ and an RPE density of $5\leavevmode\nobreak\ 000\leavevmode\nobreak\ cells/mm^{2}$, which is consistent with previous studies for the given retinal eccentricity [8, 19]. The ROI where the OIT tomographic retinal cross-sections were generated, comprising a retinal vessel, is indicated by the white dashed-rectangle in NFL en-face images in Figs. 2(d,e). Through the OIT cross-sections, it is possible to identify the vessel present in the ROI and the OCT (red arrows) and conclude on its axial position, here at the superficial vascular plexus (SVP) [20]. One main advantage of the OIT compared to OCT is its enhanced lateral resolution, enabling visualization of some retinal features that cannot be visualized in OCT, such as interconnecting capillaries, linking different capillary plexus [7] (Supplementary Video 1). Multiply scattered light (or nonconfocal) imaging modalities such as dark- field, offset aperture and split detection, largely applied in AO-SLO [13], and recently introduced for AO-FIO [21, 16], provide excellent contrast for blood vessels and mural cells [15, 16] and translucent retinal structures [6, 14], all poorly or not visualized in back-scattered light imaging systems. Because the coherent detection of OCT limits the use of multiply scattered light, OCT is not able to generate split detection tomographic retinal cross- sections. On the other hand, the OIT procedure can be applied to Z-stacks acquired in multiply scattered light imaging modalities to generate split detection tomographic retinal cross-sections, revealing a different cross- sectional view of the retina. Figures 3(a,b) present a comparison between OIT cross-sections generated through bright-field confocal AO-SLO and nonconfocal split detection AO-SLO at 7o Nasal of a healthy subject. Colored arrows in OIT cross-sections indicate the focus position where en-face images, presented in Fig. 3(d,e), were acquired. White dashed rectangles indicate the ROI where OIT cross-sections were generated. Full Z-stacks can be visualized in Supplementary Video 2. Three main differences can be noticed when producing split detection OIT cross-sections compared to those generated from bright-field images. The first difference concerns the NFL layer which seems to disappear in split detection OIT, indicating, as previously stated, that NFL becomes mostly transparent in multiply scattered light modalities [15]. Secondly, retinal layers in the inner retina get brighter with split detection OIT. By producing the OIT corresponding to a Z-stack of perfusion map images (Fig. 3(c)), we can deduce that these layers correspond to vascular plexuses, of which we expect there to be four at 7o Nasal [7, 20]. The focus position was adjusted using split detection OIT to acquire perfusion map images of each of the four vascular plexuses (Fig. 3(f)), named according to [7]: radial peripapillary capillary plexus (RPCP), SVP, intermediate vascular plexus (IVP) and deep vascular plexus (DVP). Owing to the precise location of focus position, one can generate depth-color coded perfusion maps with ease, revealing the 3D organization of the vascular network (Fig. 3(i)). Finally, another interesting finding is a retinal layer, just above the IS/OS, that gets brighter with split detection OIT. By using OIT to position the focal plane at this layer we were able to precisely image photoreceptor IS (Figs. 3(g,h)) [6]. Figure 4 and Supplementary Video 3 present the same comparison and results but at 7o Temporal, where NFL is less dense and all three expected vascular plexuses [20] and the photoreceptor IS layers are visible in split-detection OIT, enabling focus-guidance and image acquisition of perfusion maps (for vascular plexuses) and the IS. Throughout this study, we showed the capacity of OIT to generate tomographic retinal cross-section for various non-interferometric imaging modalities. While this Letter discusses the application of OIT to retinal imaging, this method can be applied to other samples (e.g. cornea [22] and skin [23]) and for other high-resolution microscopic/imaging techniques that go beyond biological samples. Since the focus position is known, OIT can be used as a focus guidance tool to image a retinal layer of interest, even when the user is “blind” focusing. We demonstrate this asset by extracting perfusion maps from all vascular plexuses, photoreceptor IS and RPE images assisted by OIT using respectively motion contrast, split detection, and autofluorescence techniques. Moreover, the split detection cross-sectional view of the retina, not possible with OCT, may be a valuable tool to understand the origin of retinal features observed in multiply scattered light modalities [24]. Since OIT is a completely digital method, it can be easily implemented in any optical imaging system. Moreover, an OIT cross-section can be obtained in a relatively short time, even for AO-SLO, as the ROI of en-face images from the Z-stack is reduced in one direction (about $50\leavevmode\nobreak\ \mu m$), which can coincide with the slow galvanometer scanner axis. Thanks to its easy implementation, OIT can be used during imaging sessions to help focus guidance. One can start by acquiring a fly-through focus movie, generate the patient OIT image, then using the cross-sectional view and focus information to precisely position the imaging focal plane at the retinal layer of interest, avoiding loss of time in acquiring images from different depths and selecting the best image sequences afterward. Finally, laser photocoagulation retinal surgery can also benefit from OIT, by precisely focusing the therapeutic laser at the diseased tissue avoiding damaging neighboring healthy tissues [25]. ## Acknowledgements This work was supported by Agence Nationale de la Recherche CLOVIS3D grant (ANR-14-CE17-0011), and European Research Council HELMHOLTZ grant (#610110). The authors want to thank Laurent Mugnier, Cyril Petit, Yann Lai-Tim and Antoine Chen for fruitful discussions. ## Author contributions P.M. wrote the software for data processing and generation of OIT cross- sections, analyzed the results and drafted the manuscript. P.M., E.G. and K.G. designed the experiment. P.M. and S.M. developed the presented method. P.M., E.G., K.G. and M.P. collected data. M.P., K.G. and S.M. provided overall guidance to the project. All authors discussed the results, reviewed and edited the manuscript. ## Competing interests P.M. and S.M. are listed as inventors on a patent application (FR 1904271) related to the work presented in this manuscript. All other authors have nothing to disclose. Figure 1: a. ROI where the OIT method was applied. b. Generated OIT cross- sections for different normalized cut-off frequencies. c,d Influence of the cut-off frequency choice on, respectively, the axial sectioning capacity (given in terms of the DOF at the diffraction limit) and contrast of OIT cross-section. Figure 2: a-c Tomographic retinal cross-sections generated by, respectively, AO-SLO OIT, OCT and AO-FIO OIT for the same subject and retinal location, where the main retinal layer can be identified. d-g En-face retinal images obtained when precisely positioning the focal plane at the layers labelled in a-c, guided by OIT images. At RPE focal plane, the RPE signal is masked by highly reflective photoreceptors signal due to poor axial resolution, hence need for autofluorescence. h,i Fourier Transform of en-face zoomed images (orange dashed-square) f,g at RPE focal plane. j. PSD radial average of h,i outlining the spatial frequency of photoreceptor and RPE mosaic respectively. White-dashed rectangle: ROI where OIT cross-sections were extracted. Red arrows: vessel location. Scale bar: $50\mu m$. Figure 3: a-c Tomographic retinal cross-sections generated by, respectively, bright-field, split detection and motion contrast techniques in AO-SLO for the same subject at 7o Nasal where the NFL is dense and four vascular plexuses can be seen. d-h En-face retinal images obtained when precisely positioning the focal plane at the layers labelled in a-c, with the help of OIT method. i Composite perfusion map image, revealing the 3D organization of the retinal vascular network. White-dashed rectangle: ROI where OIT cross-sections were extracted. Scale bar: $100\mu m$. Figure 4: a-c Tomographic retinal cross-sections generated by, respectively, bright-field, split detection and motion contrast techniques in AO-SLO for the same subject at 7o Temporal where NFL is less thick and three vascular plexuses can be seen. d-h En-face retinal images obtained when precisely positioning the focal plane at the layers labelled in a-c, with the help of OIT method. i Composite perfusion map image, revealing the 3D organization of the retinal vascular network. White-dashed rectangle: ROI where OIT cross- sections were extracted. Scale bar: $100\mu m$. ## Methods ### Optical Incoherence Tomography procedure As mentioned in the Letter, the OIT procedure is composed of three steps: Z-stack acquisition, image filtering and computation of the image energy. #### Z-stack acquisition: The acquisition of en-face images from different focal planes, forming a Z-stack (or a fly-through movie), can be done in two different ways: step- wise, by manually changing the focus position and acquiring enough images for each imaging plane; or continuously, by operating a fly-through focus (continuous change of focus position) during imaging acquisition. While the OIT cross-section generated by the former will present a better SNR and contrast, as multiple images for each focus position can be averaged, the latter will be faster. Although in this Letter we only used the step-wise acquisition, the fly through focus acquisition might be sufficient depending on the retinal structure of interest or when the SNR is high, as in the case, for example, of confocal AO-SLO. #### Image filtering: After acquiring the Z-stack, each image has its low-spatial frequency content filtered out. Here, we empirically chose to use an order 2 Butterworth filter to avoid adding high-spatial frequency artifacts to images. #### Image energy computation: To obtain OIT cross-sectional images, similar to an OCT ”B-scan”, each image is divided into an overlapping grid of $n\times m$ pixel ROI, where each ROI is displaced from the previous by 1 pixel. To favor a trade-off between point- wise accuracy and smoothness, we empirically chose $m=4$ and $n=80$, which is equivalent of $3\mu m\times 60\mu m$. The image energy of each ROI is then computed as follows: $\sum_{i=1}^{m}\sum_{j=1}^{n}|\widetilde{I}(i,j)|^{2}$ (1) where $\widetilde{I}$ is the Fourier Transform of the filtered ROI of $m\times n$ dimensions. High energy values will be obtained in ROI presenting a significant amount of high-spatial frequency content. The axial sectioning capacity of OIT is limited by the DOF at the diffraction limit which can be defined, according to [18] as: $DOF=\frac{n\lambda}{NA^{2}}$ (2) Finally, to facilitate visualization of OIT cross-sections, we used bicubic interpolation in the axial direction. ### Experimental set-up The Z-stacks necessary to generate OIT cross-sections were obtained using the PARIS AO-FIO and a modified version of the MAORI (multimodal adaptive optics retinal imager) AO-SLO (Physical Sciences, Inc., Andover, MA, USA). Both systems were described in detail elsewhere [9, 8]. The Z-stack from the PARIS AO-FIO was obtained by translating the imaging camera parallel to the optical axis with a constant step of 30 $\mu m$ in the retinal plane. AO-SLO Z-stacks were obtained by adding constant defocus values to the deformable mirror (equivalent to an axial displacement of 20 $\mu m$ of the retinal plane). ### Subjects Image acquisition was performed on two healthy subjects aged 25 and 38. Research procedures followed the tenets of the Declaration of Helsinki. Informed consent was obtained from subjects after the nature and possible outcomes of the study were explained. The study was authorized by the appropriate ethics review boards (CPP and ANSM (IDRCB numbers: 2016-A00704-47 and 2019-A00942-55)). Before the acquisition, pupil dilation and accommodation paralysis were performed by introducing one drop of each Tropicamide and Phenylephrine 10%, assuring a constant pupil diameter and minimal interference of defocus dynamics due to natural accommodation during Z-stack acquisition [12, 26]. Subjects were seated in front of the system and stabilized with a chin and forehead rest and asked to fixate a target placed at an infinite focal conjugate. ### Imaging acquisition For each focal plane, a total of 100 images were acquired. When using the AO- SLO device, four sets of data were recorded simultaneously: bright-field, split-detection, and autofluorescence imaging; and the deformable mirror focal plane position. In this configuration, the AO-SLO light level was 1.65 mW. For the PARIS AO-FIO, the total power entering the eye from the illumination source and the wavefront sensor laser beacon were respectively 350 $\mu W$ and 1.8 $\mu W$. For both imaging systems, the light level was below the ocular safety limits established by the ISO standards for group 1 devices. ### Imaging processing Following the acquisition, to correct for fixational eye movements [27], strip-based registration (only for the AO-SLO images) and normalized cross- correlation registration (for both devices) were performed on each image sequence for a given focal plane. After registration, we selected 20 out of 100 images presenting the best quality (computed through the image energy [12]). Then the 20 selected images were averaged or had the temporal standard deviation computed to extract perfusion maps, providing the final Z-stack used to generate the OIT retinal cross-sections. Before performing the OIT procedure, images composing the Z-stacks were registered pairwise. ### Data availability The study data are available from the corresponding author upon request. ### Code availability The software to generate OIT cross-section is available upon request. ## References * [1] D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, et al., “Optical coherence tomography,” Science, vol. 254, no. 5035, pp. 1178–1181, 1991\. * [2] P. Mecê, J. Scholler, K. Groux, and C. Boccara, “High-resolution in-vivo human retinal imaging using full-field oct with optical stabilization of axial motion,” Biomedical Optics Express, vol. 11, no. 1, pp. 492–504, 2020. * [3] P. Mecê, K. Groux, J. Scholler, O. Thouvenin, M. Fink, K. Grieve, and C. Boccara, “Curved-full-field oct for high-resolution imaging of living human retina over a large field-of-view,” arXiv preprint arXiv:2001.06893, 2020. * [4] J. Liang, D. R. Williams, and D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” JOSA A, vol. 14, no. 11, pp. 2884–2892, 1997. * [5] A. Roorda, F. Romero-Borja, W. J. Donnelly III, H. Queener, T. J. Hebert, and M. C. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Optics express, vol. 10, no. 9, pp. 405–412, 2002. * [6] D. Scoles, Y. N. Sulai, C. S. Langlo, G. A. Fishman, C. A. Curcio, J. Carroll, and A. Dubra, “In vivo imaging of human cone photoreceptor inner segments,” Investigative ophthalmology & visual science, vol. 55, no. 7, pp. 4244–4251, 2014. * [7] J. Campbell, M. Zhang, T. Hwang, S. Bailey, D. Wilson, Y. Jia, and D. Huang, “Detailed vascular anatomy of the human retina by projection-resolved optical coherence tomography angiography,” Scientific reports, vol. 7, p. 42201, 2017. * [8] K. Grieve, E. Gofas-Salas, R. D. Ferguson, J. A. Sahel, M. Paques, and E. A. 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Meimon, “Higher adaptive optics loop rate enhances axial resolution in nonconfocal ophthalmoscopes,” Optics letters, vol. 44, no. 9, pp. 2208–2211, 2019. * [13] S. A. Burns, A. E. Elsner, K. A. Sapoznik, R. L. Warner, and T. J. Gast, “Adaptive optics imaging of the human retina,” Progress in retinal and eye research, vol. 68, pp. 1–30, 2019. * [14] E. A. Rossi, C. E. Granger, R. Sharma, Q. Yang, K. Saito, C. Schwarz, S. Walters, K. Nozato, J. Zhang, T. Kawakami, et al., “Imaging individual neurons in the retinal ganglion cell layer of the living eye,” Proceedings of the National Academy of Sciences, vol. 114, no. 3, pp. 586–591, 2017. * [15] T. Y. Chui, D. A. VanNasdale, and S. A. Burns, “The use of forward scatter to improve retinal vascular imaging with an adaptive optics scanning laser ophthalmoscope,” Biomedical optics express, vol. 3, no. 10, pp. 2537–2549, 2012. * [16] E. Gofas-Salas, P. Mecê, L. Mugnier, A. M. Bonnefois, C. Petit, K. Grieve, J. Sahel, M. Paques, and S. Meimon, “Near infrared adaptive optics flood illumination retinal angiography,” Biomedical optics express, vol. 10, no. 6, pp. 2730–2743, 2019. * [17] D. Lim, K. K. Chu, and J. Mertz, “Wide-field fluorescence sectioning with hybrid speckle and uniform-illumination microscopy,” Optics letters, vol. 33, no. 16, pp. 1819–1821, 2008. * [18] M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light. Elsevier, 2013. * [19] R. F. Cooper, M. A. Wilk, S. Tarima, and J. Carroll, “Evaluating descriptive metrics of the human cone mosaic,” Investigative ophthalmology & visual science, vol. 57, no. 7, pp. 2992–3001, 2016. * [20] C. Lavia, P. Mecê, M. Nassisi, S. Bonnin, J. Marie-Louise, A. Couturier, A. Erginay, R. Tadayoni, and A. Gaudric, “Retinal capillary plexus pattern and density from fovea to periphery measured in healthy eyes with swept-source optical coherence tomography angiography,” Scientific Reports, vol. 10, no. 1, pp. 1–11, 2020. * [21] S. Meimon, E. G. Salas, P. Mecê, K. Grieve, J. A. Sahel, and M. Paques, “Manipulation of the illumination geometry on adaptive optics (ao) flood illumination ophthalmoscope (fio) for dark field imaging of the retina,” Investigative Ophthalmology & Visual Science, vol. 59, no. 9, pp. 4641–4641, 2018. * [22] I. Jalbert, F. Stapleton, E. Papas, D. Sweeney, and M. Coroneo, “In vivo confocal microscopy of the human cornea,” British Journal of Ophthalmology, vol. 87, no. 2, pp. 225–236, 2003. * [23] M. Rajadhyaksha, A. Marghoob, A. Rossi, A. C. Halpern, and K. S. Nehal, “Reflectance confocal microscopy of skin in vivo: From bench to bedside,” Lasers in surgery and medicine, vol. 49, no. 1, pp. 7–19, 2017. * [24] A. Guevara-Torres, D. Williams, and J. Schallek, “Origin of cell contrast in offset aperture adaptive optics ophthalmoscopy,” Optics Letters, vol. 45, no. 4, pp. 840–843, 2020. * [25] P. Mecê, C. Petit, E. G. Salas, L. Mugnier, K. Grieve, C. Chabrier, J. A. Sahel, M. Paques, and S. Meimon, “What can adaptive optics do for laser photocoagulation?,” Investigative Ophthalmology & Visual Science, vol. 59, no. 9, pp. 6194–6194, 2018. * [26] P. Mecê, E. Gofas Salas, C. Petit, K. Grieve, C. Chabrier, M. Paques, and S. Meimon, “Visualizing and enhancing axial resolution in nonconfocal adaptive optics ophthalmoscopy,” Proc. SPIE, vol. 10858, 2019. * [27] P. Mecê, J. Jarosz, J.-M. Conan, C. Petit, K. Grieve, M. Paques, and S. Meimon, “Fixational eye movement: a negligible source of dynamic aberration,” Biomedical optics express, vol. 9, no. 2, pp. 717–727, 2018\. ## 1 Supplementary information #### Video 1 Interconnecting capillaries are visible in OIT cross-section generated with AO-FIO. Upper image: ROI used to compute the OIT cross-section. Lower image: Corresponding OIT cross-section. Arrows highlight the three-dimensional position of interconnecting capillaries. Blue arrow: capillaries connecting the superficial vascular plexus (SVP) and the deep vascular plexus (DVP). Red arrow: capillaries connecting the SVP and the intermediate vascular plexus (IVP). Green arrow: capillaries connecting the SVP with the radial peripapillary capillary plexus (RPCP). Retinal image acquired at 7o Nasal. #### Video 2 The fly-through focus movies for bright-field AO-SLO and split-detection AO- SLO modalities at 7o Nasal. #### Video 3 The fly-through focus movies for bright-field AO-SLO and split-detection AO- SLO modalities at 7o Temporal.

2024-09-04T02:54:55.546163

2002.12813

arxiv-papers

# On the canonical formula of C Lévi-Strauss, II Jack Morava Department of Mathematics, The Johns Hopkins University, Baltimore, Maryland 21218<EMAIL_ADDRESS> (Date: Imbolc 2020) §1 Introduction and Organization > We venture a leap: we grant ab initio that there is ‘something there’ to be > translated … > > George Steiner, quoted in [3] (p 19) 1.1 This is a sequel to, and elaboration of, earlier work [21, 22] on a possible formal model for the canonical formula ${\sf CF}:F_{x}(a):F_{y}(b)\;\simeq\;F_{x}(b):F_{a^{-1}}(y)$ of C Lévi-Strauss, which he proposed [1, 14, 15, 16], cf. [Appendix] as a tool for the structural analysis of mythological systems; see [12] (p 562) for a discussion of the hair-raising example ${\rm marriage}_{\rm solidarity}:{\rm rape}_{\rm hostility}\simeq{\rm marriage}_{\rm hostility}:{\rm dissociation}_{\rm rape}\;.$ The model proposed here is based on the study of small finite groups, which have proved useful in the classification of kinship systems, crystallography, and other fields [8, 19, 25, 27]; a short account of some of the mathematics involved is postponed till §3. For clarity, a sketch of the model, with technical details backgrounded, is displayed immediately below. However, because translation across the conceptual and cognitive gulfs separating anthropology and mathematics raises significant questions, a short discussion (§2) of models and metaphors precedes the (not actually very complicated) verification of the details of the model, in §4. 1.2 A model > The first formulation of the real situation would be to state that the > speaking subject perceives neither idea $a$ nor form $A$, but only the > relation $a/A$ …What he perceives is the relation between the two relations > $a/AHZ$ and $abc/A$, or $b/ARS$ and $blr/B$, etc. This is what we term the > LAST QUATERNION … > > F de Saussure, Writings…[26] (p 22), [16] Proposition: To elements $\\{x,a,y,b\\}$, e.g. $\\{1,i,j,k\\}$ of the group $Q$ of Lipschitz unit quaternions [3.2.1], the function $x,a\mapsto\Phi_{x}(a)=2^{-1/2}(x-a)\in 2\cdot O$ assigns elements of the binary octahedral group [3.4.2] such that the anti- automorphism $\lambda=*I\sigma^{2}$ [3.3iii] of that group transforms the (noncommutative [3.2.2]) ratio $\\{\Phi_{x}(a):\Phi_{y}(b)\\}$ into $\\{\Phi_{\lambda(x)}(\lambda(a)):\Phi_{\lambda(y)}(\lambda(b))\\}\;=\;\\{\Phi_{x}(b):\Phi_{a^{-1}}(y)\\}\;.$ Terms such as $x,a,y,b$ or $1,i,j,k$ will be referred to here as ‘values’, while functions such as $F$ and $\Phi\in 2\cdot O$ of ordered pairs of values will be called ‘valences’. [Citations such as [1.2] refer to sections of this paper.] §2 Wider Questions > It takes a while to learn that things like tenses and articles …are not > ‘understood’ in Burmese [as] something not uttered but implied; they just > aren’t there… > > AL Becker, Beyond Translation [3] ( p 8) 2.1 For the philologist Becker it is fundamental that languages (and cultures) have ‘exuberances and deficiencies’ that can make translation problematic; in the present case these issues are acute. It seems generally agreed [21] that the ${\sf CF}$ is underdetermined. One of the first things a mathematician notices (cf. the original statement, included below as an appendix) is the absence of quantifiers, which usually convey information about the circumstances under which a proposition holds; moreover, the reversal of function and term [condition 2] is mysterious. One issue - the assumption that the element $a$ has a natural or canonical dual $a^{-1}$ \- can perhaps be resolved by reading the ${\sf CF}$ as asserting, in the context $\\{x,a,y,b\\}$, the existence of $a^{-1}$: for example, as in [12]. On the other hand, an exuberance of the paraphrase above is a precise specification of the binary octahedral group as a repository for the values of our analog $\Phi$ of Lévi-Strauss’s $F$. 2.2 I make no claim about the uniqueness of the model proposed here: I hope there are better ones. Perhaps this is the place to say that my concern with the ${\sf CF}$ is not its validity, or ‘truth-value’; it is rather whether or not it can be usefully be interpreted as a formal mathematical assertion. Its interest as an empirical hypothesis, like Bohr’s model for the atom, seems well-established. Po moemu the key question is what ‘interpretation’, in this context, could even mean. 2.3 An answer may lie in the ancient opposition, older than Zeno, between continuous and discrete. An anthropologist, like William James’ infant [17] (Ch 13) is ‘assailed by eyes, ears, nose, skin and entrails at once, feels it all as one great blooming, buzzing confusion’; but ‘nowadays, fundamental psychological changes occur [to the mathematician]. Instead of sets, clouds of discrete elements, we envisage some sorts of vague spaces …mapped one to another. If you want a discrete set, then you pass to the set of connected components of spaces defined only up to homotopy’ [20] (p 1274), [17]. I believe these remarks of Manin point to an answer to this question, in terms of cognitive condensation of chaotic clouds of experience into discrete, classifiable conceptual entities, cf. [4]. §3 A short mathematical glossary This section summarizes more than enough background from the theory of groups for the purposes of this paper: 3.1.1 The objects of the category of groups (for example the integers ${\mathbb{Z}}=\\{\dots,-2,-1,0,1,2,\dots\\}$) consist of sets $G$ with an associated multiplication operation $\mu_{G}:G\times G\ni(g,h)\mapsto g\cdot h\in G$ and an identity element $1=1_{G}\in G$, subject to familiar rules of associativity which I will omit. A homomorphism (or map, or morphism) $\phi:G\to G^{\prime}$ between groups respects multiplication (i.e. $\phi(g\cdot g^{\prime})=\phi(g)\cdot\phi(g^{\prime})$ etc.); a composition of homomorphisms is again a homomorphism, thus defining a category. The set of one-to-one self-maps of a set with $n$ elements, for example, defines the symmetric group $\Sigma_{n}$. A group $A$ is commutative if $a\cdot a^{\prime}=a^{\prime}\cdot a$ for any $a,a^{\prime}\in A$; in such cases the multiplication operation is often written additively, i.e. with $a+a^{\prime}$ instead of $a\cdot a^{\prime}$, e.g. as in the additive group ${\mathbb{Z}}$ of integers. The order of a group is the (not necessarily finite) number of its elements. Example The set of isomorphisms $\alpha:G\to G$ with itself is similarly a group ${\rm Aut}(G)$ (of automorphisms of $G$). Any element $h\in G$ defines an inner automorphism $\alpha_{h}:G\ni g\mapsto hgh^{-1}\in G$ of $G$; it is a group homomorphism since $\alpha_{h}(g\cdot g^{\prime})=h(g\cdot g^{\prime})h^{-1}=hgh^{-1}\cdot hg^{\prime}h^{-1}=\alpha_{h}(g)\cdot\alpha_{h}(g^{\prime})\;,$ and $h\mapsto\alpha_{h}:G\to{\rm Aut}(G)$ is a homomorphism since $(\alpha_{h}\circ\alpha_{k})(g)=\alpha_{h}(kgk^{-1})=hkgk^{-1}h^{-1}=hk\cdot g\cdot(hk)^{-1}=\alpha_{hk}(g)\;;$ much mathematics consists of shuffling parentheses. The subgroup ${\rm In}(G)=\\{\alpha_{g}\in{\rm Aut}(G)\\}$ is normal in that $(\forall\beta\in{\rm Aut}(G))\;\alpha\in{\rm In}(G)\Rightarrow\beta\alpha\beta^{-1}\in{\rm In}(G)\;.$ In particular, if $G=A$ is commutative, then ${\rm In}(A)=\\{1\\}$ is the trivial group. If a subgroup $H$ of $G$ is normal, then the set $G/H=\\{gH\;|\;g\in G\\}$ (of ‘orbits’ of elements of $G$ under right multiplication by $H$) is again a group. 3.1.2 A composition $\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{K}$ of homomorphisms is exact if the image ${\rm im}\;\phi=\\{g\in G\>|\>\exists h\in H,\;\phi(h)=g\\}$ of $\phi$ equals the kernel $\ker\;\psi=\\{g\in G\>|\>\psi(g)=1_{K}\\}$ of $\psi$; this implies in particular that the composition $\psi\circ\phi$ is trivial (i.e. maps every element of $H$ to the identity element of $K$), but is more restrictive. A sequence $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$ of groups and homomorphisms is exact if its consecutive two-term compositions are exact; this implies that $\bullet$ $\phi$ is one-to-one (or is a monomorphism, or has trivial kernel), $\bullet$ $\psi$ is ‘onto’ (or surjective, or $K={\rm im}\;\psi$), and $\bullet$ ${\rm im}\;H$ is normal in $G$, and $\psi$ factors through an isomorphism $G/H\cong K$. 3.1.3 Such an exact sequence is said to split, if there is a homomorphism $\rho:K\to G$ inverse to $\psi$ in the sense that the composition $\psi\circ\rho$ $\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{K}$ is the identity map $K$. In that case there is a unique homomorphism ${\varepsilon}:K\to{\rm Aut}(H)$ such that (the ‘semi-direct product) $G\cong H\rtimes K$ is isomorphic to the group defined on the set product $H\times K$, with twisted multiplication $(h_{0},k_{0})\cdot(h_{1},k_{1})=(h_{0}\cdot{\varepsilon}(k_{0})h_{1},k_{0}k_{1})\;;$ such a split sequence will usually be displayed below as $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\cong H\rtimes K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\;.}$ 3.2 Some topological groups and algebras 3.2.1 The quaternion group $Q=\\{\pm 1,\pm i,\pm j,\pm k\\}$ of order eight is defined by three elements $i,j,k$ with multiplication $i^{2}=j^{2}=k^{2}=-1$ and $ij=k=-ji,\;jk=i=-kh,\;ki=j=-ik\;.$ The (noncommutative) division algebra ${\mathbb{H}}=\\{q=q_{0}+q_{1}i+q_{2}j+q_{3}k\>|\>q_{i}\in{\mathbb{R}}\\}\cong{\mathbb{R}}^{4}$ of Hamiltonian quaternions is the four-dimensional real vector space with multiplication extended from $Q$; alternately, it is the two-dimensional complex vector space ${\mathbb{H}}=\\{z_{0}+z_{1}j\>|\>z_{i}\in{\mathbb{C}}\\}\;,$ where $z_{0}=q_{0}+iq_{1},\;z_{1}=q_{2}+iq_{3}$. The quaternions thus extend the field ${\mathbb{C}}$ of complex numbers much as ${\mathbb{C}}$ extends the field ${\mathbb{R}}$ of real numbers. The quaternion conjugate $q^{*}=q_{0}-q_{1}i-q_{2}j-q_{3}k$ to $q$ has positive product $q^{*}\cdot q=q\cdot q^{*}=|q|^{2}=\sum q_{i}^{2}>0$ with $q$ if $q\neq 0$, implying the existence of a multiplicative inverse $q^{-1}=|q|^{-2}q^{*}$. This defines an isomorphism ${\mathbb{H}}^{\times}\ni q\mapsto(|q|^{-1}q,|q|)\in S^{3}\times{\mathbb{R}}^{\times}_{+}$ making the three-dimensional sphere $S^{3}$ a group under multiplication. This notation is nonstandard but convenient; note that $q^{**}=q$ and that quaternion conjugation $*:q\mapsto q^{*}$ is an anti-homomorphism, i.e. $(u\cdot v)^{*}=v^{*}\cdot u^{*}\;.$ [Similarly, ${\mathbb{R}}^{\times}\cong S^{0}\times{\mathbb{R}}^{\times}_{+},\;S^{0}=\\{\pm 1\\},$ while ${\mathbb{C}}^{\times}\cong S^{1}\times{\mathbb{R}}^{\times}_{+}$ (where $S^{n}=\\{{\bf x}\in{\mathbb{R}}^{n+1}\>|\>|{\bf x}|^{2}=1\\}$ is the $n$-dimensional sphere of radius one, e.g. the circle when $n=1$).] 3.2.2 The subalgebra (i.e. closed under addition and multiplication, but not division) of Lipschitz quaternions in ${\mathbb{H}}$ is the set of $q$ with integral coordinates ($q_{i}\in{\mathbb{Z}}$), while the subalgebra of Hurwitz quaternions consists of elements $q$ with all coordinates either integral or half-integral (i.e. such that each $q_{i}$ is half of an odd integer). Finally, the subalgebra of Lipschitz (integral) quaternions has an additional (commutative but nonassociative) Jordan algebra product $u,v\mapsto\\{u,v\\}={\textstyle{\frac{1}{2}}}(u\cdot v+v\cdot u)=\\{v,u\\}\;,$ e.g. $\\{1,1\\}=1,\;\\{i,i\\}=\\{j,j\\}=\\{k,k\\}=-1,\;\\{i,j\\}=\\{j,k\\}=\\{k,i\\}=0\;.$ This allows us to define a (non-commutative) Jordan ratio $\\{u:v\\}=\\{u,v^{*}\\}=|v|^{-2}\\{u,v^{-1}\\}$ for $u,v\in{\mathbb{H}}$, distributive $\\{u+u^{\prime}:v\\}=\\{u:v\\}+\\{u:v^{\prime}\\},\;\\{u:v+v^{\prime}\\}=\\{u:v\\}+\\{u:v^{\prime}\\}$ in both variables, satisfying $\\{u:v\\}^{*}=\\{u^{*}:v^{*}\\}\;.$ Remark ${\mathbb{H}}$ can be regarded as a subalgebra of the $2\times 2$ complex matrices $M_{2}({\mathbb{C}})$, in such a way that the quaternion norm $|q|^{2}$ equals the determinant of $q$, regarded as a matrix. This identifies the 3-sphere $S^{3}$ with a subgroup of the Lie group ${\rm Sl}_{2}({\mathbb{C}})$ of complex $2\times 2$ matrices with determinant one; as such, it is the maximal compact (special unitary) subgroup ${\rm SU}(2)$ of ${\mathbb{H}}^{\times}$. The special orthogonal group ${\rm SO}(3)\cong{\rm SU}(2)/\\{\pm 1\\}$ (of rotations in three dimensions) is a quotient of this group, and the various ‘binary’ (tetrahedral, octahedral etc.) groups lift the symmetry groups of the classical Platonic solids [12] to subgroups of the three-sphere. See [2], and its comments, for some very pretty animations of a certain ‘24-cell’ associated [10] (cf. note ar) to the octahedral and tetrahedral groups. [The noncompact group $SL_{2}({\mathbb{C}})$ is similarly a double cover of (the identity component of) the physicists’ Lorentz group.] 3.3 The subset $Q\subset{\mathbb{H}}^{\times}$ (of Lipschitz units) is a finite subgroup of ${\rm SU}(2)$. Similarly, the subset $Q\subset A_{24}\subset{\rm SU}(2)$ (of Hurwitz units) is the union of $Q$ with the set of sixteen elements of the form ${\textstyle{\frac{1}{2}}}[\pm 1\pm i\pm j\pm k]\;.$ It is known as well [5] as the binary tetrahedral group $2\cdot T$. Klein’s (commutative) ‘Vierergruppe’ $V=\\{1,I,J,K\\}$ with multiplication $I^{2}=J^{2}=K^{2}=1$ and $IJ=JI=K,\;JK=KJ=I,\;KI=IK=J$ can be regarded as the subgroup $V={\rm In}(Q)\subset{\rm Aut}(Q)$ defined by $\alpha_{i}=I,\;\alpha_{j}=J,\;\alpha_{k}=K$. Exercise: i) $I$ sends $i$ to itself, and $j,k$ to $-j,-k$; similarly $J$ sends $j$ to itself, while $i,k\mapsto-i,-k$, etc. The cyclic permutation $(abc)=(a\to b\to c\to a)\in\Sigma_{3}$ of three things defines a homomorphism $C_{3}\cong\\{1,\sigma,\sigma^{2}\\}\to{\rm Aut}(Q)$ sending $\sigma$ to $(ijk)$. ii) The map $C_{2}^{2}=C_{2}\times C_{2}\to V$ defined by $(1,0)\mapsto I,\;(0,1)\mapsto J,\;(1,1)\mapsto K$ (and $1\mapsto(0,0)$) is a (nonunique) isomorphism. iii) For example, in $V\rtimes C_{3}=A_{4}$ (i.e. the alternating subgroup of order 12 (see below) of $\Sigma_{4}\cong{\rm Aut}(Q)$) we have $(I\sigma^{2})\cdot(I\sigma^{2})=I\sigma^{2}(I)\cdot\sigma^{4}=IK\sigma=J\sigma\;.$ The anti-automorphism $\lambda=*I\sigma^{2}$ of $Q$ satisfies $\lambda(i)=k,\;\lambda(j)=-i,\;\lambda(k)=j,$ e.g. $\lambda(j)=(I\sigma^{2}(j))^{*}=(Ii)^{*}=i^{*}=-i$, cf. [21 §5]. 3.4.1 Some useful small groups order & name n cyclic $C_{n}=\\{0,\dots,n-1\\},\;n\geq 1$ $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}}/n{\mathbb{Z}}\cong C_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$ n! symmetric $\Sigma_{n},\;n\geq 1$ 4 Klein Vierergruppe $V$ $V=\\{1,I,J,K\\}\cong C_{2}\times C_{2}\;({\rm non-uniquely})$ 6 symmetric $\Sigma_{3}$ $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{3}\cong C_{3}\rtimes C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$ 8 (Lipschitz) quaternion units $Q=\\{\pm 1,\pm i,\pm j,\pm k\\}$ $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$ 12 alternating or tetrahedral ($T=A_{4}$) $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{A_{4}\cong V\rtimes C_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{4}\cong V\rtimes\Sigma_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$ 24 $\Sigma_{4}$ as above; binary tetrahedral $2\cdot T$ = Hurwitz units $A_{24}$ $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\cdot T=Q\rtimes C_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{4}=V\rtimes C_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$ 48 binary octahedral $2\cdot O$ $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\cdot O\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\cdot T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$ as well as $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\cdot O\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\;.}$ 3.4.2 The binary octahedral group $2\cdot O$, regarded as a subgroup of the unit quaternions [2, 6], is the disjoint union of $A_{24}$ with the set of twenty-four special elements $q=2^{-1/2}[q_{0}+q_{1}i+q_{2}j+q_{3}k]$ in which exactly two of $q_{0},\dots,q_{3}$ are nonzero and equal $\pm 1$, in which $\Phi$ takes its values. 3.4.3 We have ${\rm Aut}(Q)\cong\Sigma_{4}\cong{\rm Aut}(2\cdot T)$ and ${\rm Aut}(2\cdot O)\cong C_{2}\times(2\cdot T\cong A_{24})\;.$ 3.5 Some of the groups above can be presented in terms of matrices over finite Galois fields ${\mathbb{F}}$: in particular, $\Sigma_{3}\cong{\rm Sl}_{2}({\mathbb{F}}_{2})$ and $A_{24}\cong{\rm Sl}_{2}({\mathbb{F}}_{3})$. Similarly, the binary icosahedral group (which plays no role in this paper) is isomorphic to ${\rm Sl}_{2}({\mathbb{F}}_{5})$. It is worth mentioning that the group of $2\times 2$ matrices with entries from ${\mathbb{Z}}$ and determinant one is a quotient $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\textstyle{{\mathbb{B}}_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm Sl}_{2}({\mathbb{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$ of Artin’s three-strand braid group [7], which thus maps (by ${\mathbb{Z}}\to{\mathbb{Z}}/3{\mathbb{Z}}\cong{\mathbb{F}}_{3}$) to ${\rm Aut}(2\cdot O)$. [The set of braids on $n$ strands, imagined for example as displayed on a loom, define a group under ‘concatenation’: Technically, an $n$-strand braid can be defined as a smooth path in the space of configurations defined by $n$ distinct points in the plane, starting for example at time $t=0$ at the integral points $(1,0),(2,0),\dots,(n,0)$ and ending at time $t=1$ at the points $(1,1),(2,1),\dots,(n,1)$, though not necessarily in that order. Such braids can be composed by concatenation (i.e. glueing and rescaling), and define elements of $\Sigma_{n}$ (sending $k$ to $l$ if the strand starting at $(k,0)$ ends at $(l,1)$). The braid group ${\mathbb{B}}_{n}$ is the set of such things under the equivalence relation roughly described as straightening: thus for example any braid can be parsed into a composition of elementary moves, in which one strand over- or under- passes one of its nearest neighbors. For example, ${\mathbb{B}}_{3}$ can be presented as the group with two generators $a,b$ satisfying the ‘braid relation’ $aba=bab$, thus the map $t$ above sends 1 to the full twist $(aba)^{2}=(ab)^{3}$.] §4 A Calculation 4.1 Lévi-Strauss’s formula is expressed in terms of formal analogies, e.g. $F_{x}(a):F_{y}(b)$, understood roughly as a ratio, in a sense going back to Eudoxus; but noncommutative algebra distinguishes the left fraction $a^{-1}b=a\backslash b$ from the right fraction $ba^{-1}=b/a$. The noncommutative ratio [3.3.2] splits this difference. Thus if $x,a,y,b\in Q$ we have $\\{\Phi_{x}(a):\Phi_{y}(b)\\}={\textstyle{\frac{1}{2}}}\\{x-a:y-b\\}={\textstyle{\frac{1}{2}}}[(\\{x:y\\}+\\{a:b\\})-(\\{a:y\\}+\\{x:b\\})]\;,$ so for example if $x=1,\;a=i,\;y=j,\;b=k$ we have $\\{\Phi_{x}(a):\Phi_{y}(b)\\}=\\{\Phi_{1}(i):\Phi_{j}(k)\\}={\textstyle{\frac{1}{2}}}\\{1-i:j-k\\}=$ ${\textstyle{\frac{1}{2}}}[(\\{1:j\\}+\\{i:k\\})-(\\{i:j\\}+\\{1:k\\})]={\textstyle{\frac{1}{2}}}[(j+0)-(0+k)]={\textstyle{\frac{1}{2}}}(j-k)\;,$ while $\\{\Phi_{x}(b):\Phi_{a^{-1}}(y)\\}={\textstyle{\frac{1}{2}}}\\{1-k:-i-j\\}=$ ${\textstyle{\frac{1}{2}}}[(-\\{1:i\\}-\\{1:j\\})+(\\{k:i\\}+\\{k:j\\})]={\textstyle{\frac{1}{2}}}[-i-j+0]=-{\textstyle{\frac{1}{2}}}(i+j)\;.$ But now, applying the anti-automorphisms $\lambda$ of [20] (§5), we have $\lambda(j-k)=-(i+j)$ by 3.3iii above, and the proposition is verified. $\Box$ Remarks 4.2.1 The automorphism group $\Sigma_{4}$ of $Q$ preserves the set of special elements [3.4.2] of $2\cdot O$, i.e. the possible values of $\Phi$, as well as their Jordan ratios; but it differs from the (inner) automorphism group $A_{24}$ of $2\cdot O$. However, the term $x$ appears in the ${\sf CF}$ in the same place on both sides of the equation, and can be interpreted as playing the role of $1\in Q$, fixed by all automorphisms. The canonical formula appears in variant forms in the literature, but (as far as I know) they all feature a term in the same place on both sides of the equation, allowing the variants to be reconciled by a cyclic permutation in $\Sigma_{3}\subset\Sigma_{4}$, which lies in a quotient of ${\rm Aut}(2\cdot O)$. 4.2.2 The classic work of Thom on singularity theory [24] has turned out (e.g. under the influence of Arnol’d, McKay, and others) to have deep connections with the theory of Platonic symmetry groups. The binary orthogonal group, in particular, seems to be related to a certain ‘symbolic umbilic’ singularity [9, 11], a special case of Thom’s original classification. 4.2.3 As a closing remark: the model proposed here is in fact not that complicated. To a mathematician, perhaps the most interesting implication is its connection with the theory of braids, which is arguably related to the processing of recursion, and to cognitive evolution. Acknowledgements and thanis: I am deeply indebted to many people, especially John Baez, Ellen Contini-Morava, Fred Damon, John McKay , Tony Phillips, Emily Riehl, Dale Rolfsen, Lucien Scubla, and Roy Wagner, for conversations, insight, and advice, during the preparation of this paper. Its excesses and deficiencies, though, are my own responsibilty. Appendix From The structural study of myth [14], Journal of American Folklore 68 (1955): 7.30 Finally, when we have succeeded in organizing a whole series of variants in a kind of permutation group, we are in a position to formulate the law of that group. Although it is not possible at the present stage to come closer than an approximate formulation which will certainly need to be made more accurate in the future, it seems that every myth (considered as the collection of all its variants) corresponds to a formula of the following type: $F_{x}(a):F_{y}(b)\;\simeq\;F_{x}(b):F_{a^{-1}}(y)$ where, two terms being given as well as two functions of these terms, it is stated that a relation of equivalence still exists between two situations when terms and relations are inverted, under two conditions: 1. that one term be replaced by its contrary; 2. that an inversion be made between the function and the term value of the two elements. ## References * [1] Kwame Anthony Appiah, https://www.nybooks.com/articles/2020/02/13/claude-levi-strauss-key-to-all-mythologies/ * [2] John Carlos Baez, https://johncarlosbaez.wordpress.com/2019/08/29/the-binary-octahedral-group/ * [3] AL Becker, Beyond Translation, U of Michigan Press (1998) * [4] JL Borges, The analytical language of John Wilkins, in Otras Inquisiciones (1952) Buenos Aires: Sur. * [5] https://en.wikipedia.org/wiki/Binary_tetrahedral_group * [6] https://en.wikipedia.org/wiki/Binary_octahedral_group * [7] https://en.wikipedia.org/wiki/Braid_group * [8] https://en.wikipedia.org/wiki/Crystallographic_point_group * [9] https://en.wikipedia.org/wiki/Du_Val_singularity * [10] https://en.wikipedia.org/wiki/24-cell * [11] P Dechant, From the trinity $(A3,B3,H3)$ to an ADE correspondence, Proc. R. Soc A 474 (2018), no. 2220, 20180034, https://arxiv.org/abs/1812.02804 * [12] A Doja, Politics of mass rapes in ethnic conflict …, Crime, Law and Social Change (2019) 71 : 541 – 580, https://link.springer.com/article/10.1007/s10611-018-9800-0 * [13] N Epa, N Ganter, Platonic and alternating 2-groups, Higher Structures 11 122 – 146 (2017), https://arxiv.org/abs/1605.09192 * [14] C Lévi-Strauss, The structural study of myth, Journal of American Folklore 68 no 270 (1955) 428–444 * [15] https://fr.wikipedia.org/wiki/Formule_canonique_du_mythe * [16] https://ncatlab.org/nlab/show/canonical+formula+of+myth * [17] W James, Principles of Psychology [1890] * [18] A Joyal, Notes on quasicategories, http://www.math.uchicago.edu/~may/IMA/Joyal.pdf * [19] https://ncatlab.org/nlab/show/kinship * [20] Y Manin, We do not choose our profession …, AMS Notices 56 (2009) 1268 -– 1274 (p 1274) * [21] P. Maranda, The double twist: from ethnography to morphodynamics, U of Toronto Press (2001) * [22] J Morava, On the canonical formula of C Lévi-Strauss, https://arxiv.org/abs/math/0306174 * [23] —–, From Lévi-Strauss to chaos and complexity, in MS Mosko, FH Damon, On the order of chaos p 47 – 63, Berghan Books (2005) * [24] J Petitot, A morphodynamical schematization of the canonical formula for myths, in [19] 267 – 311 * [25] AV Phillips, A non-commutative marriage system in the South Pacific, http://www.math.stonybrook.edu/~tony/whatsnew/oct09/vanuatu2.html * [26] F de Saussure, Writings on General Linguistics, OUP 2006 * [27] A Weil, On the algebraic study of certain types of marriage laws, appendix (221–232) to C Lévi-Strauss, Elementary structures of kinship, Beacon (1969)

2024-09-04T02:54:55.556169

2002.12814

arxiv-papers

# Estimating the entropy of shallow circuit outputs is hard Alexandru Gheorghiu111Email<EMAIL_ADDRESS>Department of Computing, Goldsmiths, University of London Matty J. Hoban222Email<EMAIL_ADDRESS> ###### Abstract Calculating relevant entropic quantities of probability distributions and quantum states is a fundamental task in information processing. The decision problem version of estimating the Shannon entropy is the _Entropy Difference_ problem (ED): given descriptions of two circuits, determine which circuit produces more entropy in its output when acting on a uniformly random input. The analogous problem with quantum circuits (QED) is to determine which circuit produces the state with greater von Neumann entropy, when acting on a fixed input state and after tracing out part of the output. Based on plausible complexity-theoretic assumptions, both of these problems are believed to be intractable for polynomial-time quantum computation. In this paper, we investigate the hardness of these problems in the case where the circuits under consideration have logarithmic (ED${}_{\textrm{log}}$ and QED${}_{\textrm{log}}$) and constant depth (ED${}_{\textrm{O(1)}}$ and QED${}_{\textrm{O(1)}}$), respectively. We first show that, relative to an oracle, these problems cannot be as hard as their counterparts with polynomial-size circuits. Furthermore, we show that if a certain type of reduction from QED to QED${}_{\textrm{log}}$ exists, it implies that any polynomial-time quantum computation can be performed in log depth. While this suggests that having shallow circuits makes entropy estimation easier, we give indication that the problem remains intractable for polynomial-time quantum computation by proving a reduction from _Learning-With-Errors_ ($\mathrm{LWE}$) to ED${}_{\textrm{O(1)}}$. We then consider a potential application of our results to quantum gravity research via the AdS/CFT correspondence. First, we introduce HQED, a Hamiltonian version of QED, where one is given two local Hamiltonians and asked to estimate the _entanglement entropy_ difference in their ground states. We show that this problem is at least as hard as the circuit version by providing a reduction from QED to HQED. With these results, we then discuss a potential experiment that would make use of AdS/CFT in order to solve $\mathrm{LWE}$ efficiently. We conjecture that unless the AdS/CFT bulk to boundary map is exponentially complex, this experiment would violate the intractability assumption of $\mathrm{LWE}$. ## 1 Introduction The entropy of a probability distribution or a quantum state is a useful measure for characterising information content with numerous applications in information theory. Shannon and von Neumann entropies, both appear in data compression [Sch95] and asymptotic cryptography [DW05], and in the case of the von Neumann entropy, entanglement theory [BBPS96]. Furthermore, this link to entanglement theory has led to the use of the von Neumann entropy in condensed matter theory [ECP10] and quantum gravity research [RT06]. Given its importance in physics and information theory, it is natural to ask how difficult it is to estimate the entropy of a process. A natural way to formalise this question is in terms of _sample complexity_ : this looks at how many samples from an unknown probability distribution, or how many copies of an unknown quantum state, are needed to estimate its entropy. Previous work has considered various algorithms, both quantum and classical, for computing the entropy of a probability distribution [DKRB02, WY16, JVHW17, VV11], as well as entropies of quantum states [HGKM10, AISW19, SH19]. More recently, it has been shown that for multiple entropy measures, computing the relevant entropy is as hard as full state tomography [AISW19, SH19, OW15]. In other words, the sample complexity would scale with the support of the probability distribution, or the dimension of the Hilbert space for the quantum state, respectively. To provide some intuition for why entropy estimation is difficult, in Appendix A, we use the computational complexity tools of advice to give a simple proof that no algorithm exists for efficiently estimating the entropy of an a priori unknown quantum state. In particular, we show that if the entropy of a (mixed) quantum state could be estimated within additive error $\epsilon$ in time that scales polynomially in the number of qubits of the state and in $\log(1/\epsilon)$, then such an algorithm could be leveraged to solve _any_ problem. To be more precise, it would imply that polynomial- time quantum computation with quantum advice, could decide all languages, which is known to be false. Rather than considering sample complexity, an operational way of capturing the complexity of entropy estimation is to start with descriptions of two random processes and ask which process produces more entropy in its output. The natural decision problem for this task was defined by Goldreich and Vadhan [GV99] and is known as the _entropy difference_ (ED) problem: given two classical circuits $C_{1}$ and $C_{2}$, let $C_{1}(x)$ and $C_{2}(x)$ define the output distributions of the two circuits when acting on an $n$-bit string, $x$, drawn from the uniform distribution over $\\{0,1\\}^{n}$; the problem is to decide whether $C_{1}(x)$ has higher entropy than $C_{2}(x)$ or vice versa (promised that one of these is the case and promised that the entropy difference is at least $1/poly(n)$). What can one say about the computational hardness of this problem? In [GV99] it was shown that the problem is complete for the class $\sf SZK$. This class, known as _statistical zero-knowledge_ , contains all decision problems for which a computationally unbounded _prover_ can convince a polynomial-time _verifier_ that there exists a solution, when one exists (and fail to do so when a solution does not exist) without revealing anything about the solution to the verifier333This last condition, known as the zero-knowledge condition, is formally defined by saying that there exists a polynomial-sized circuit called a _simulator_ that can approximately produce transcripts of the protocol for accepting (or “yes”) instances.. Due to oracle separation results [Aar02] and from the fact that certain cryptographic tasks (such as finding collisions for a cryptographic hash function) are contained in $\sf SZK$, it is believed that $\sf SZK$ contains problems that are intractable even for polynomial-time quantum algorithms. Thus, the fact that ED is complete for $\sf SZK$ tells us that we should expect a similar intractability for the problem of distinguishing the entropies of general classical circuits. Transitioning to the quantum setting, Ben-Aroya, Schwartz and Ta-Shma defined the analogue of ED known as the _quantum entropy difference_ (QED) problem [BASTS10]. In this case, the input circuits $C_{1}$ and $C_{2}$ are polynomial-size quantum circuits acting on a fixed input state (say the state $\ket{00...0}$), and a fraction of the output qubits are traced out. The remaining qubits will generally be mixed quantum states. As in the classical case, the question is which of the two mixed states (the one produced by $C_{1}$ or the one produced by $C_{2}$) has higher entropy, subject to the same promise as for ED (that there is a $1/poly(n)$ gap in the entropy difference). Ben-Aroya et al showed that QED is complete for the class $\sf QSZK$, the quantum counterpart of $\sf SZK$ of problems admitting a _quantum statistical zero-knowledge_ proof protocol [BASTS10]. Assuming $\sf QSZK$ strictly contains $\sf SZK$, this would mean that QED is strictly harder than ED. For both ED and QED the circuits under consideration were assumed to be polynomial in the size of their inputs. A natural follow-up question is: does the hardness of these problems change if we reduce the depth of the circuits? Specifically, what happens if the circuits have depth that is logarithmic or even constant in the size of the input? Focusing specifically on the quantum case, there are number of reasons why one would be interested in answering these questions. From a complexity-theory perspective, it lets us compare and contrast $\sf QSZK$ to other classes in the interactive proofs model, such as $\sf QMA$ or $\sf QIP$. Both of these classes have natural complete problems with the inputs being circuits and in both cases the problems remain complete if the circuits are made to have logarithmic depth [Ros08, Ji17]. Furthermore, from a more practical perspective, given that current and near-term quantum computing devices are subject to noise and imperfections, it is expected that the states produced in these experiments will be the result of circuits of low depth. Estimating the entropy of these states would help in computing other quantities of interest such as the amount of entanglement [BBPS96] or the Gibbs free energy [GHR+16]. It is therefore important to know whether entropy estimation can be performed efficiently for states produced by shallow circuits. Lastly, we are motivated to answer these questions by recent connections between quantum gravity research and quantum information theory, in the form of the AdS/CFT correspondence [Mal99]. Briefly, AdS/CFT is a correspondence between a quantum gravity theory in a hyperbolic _bulk_ space-time known as _Anti-de Sitter_ (AdS) space and a _conformal field theory_ (CFT) on the _boundary_ of that space-time. The general idea is to compute physical quantities of interest in the bulk quantum gravity theory by mapping them to the boundary field theory. A surprising result to come out of this program is a correspondence between bulk geometry and boundary entanglement known as the _Ryu-Takayanagi formula_ [RT06]. It states that, to leading order, the area of a bulk surface is equal to the entropy of the reduced state on the part of the boundary that encloses that surface. Moreover, for certain families of boundary field theories, it is conjectured that the underlying states can be approximated by logarithmic depth tensor networks known as MERA (_multi-scale entanglement renormalization ansatz_) [Swi12]. Thus, characterising the complexity of entropy difference for shallow circuits could yield insight into the complexity of distinguishing different bulk geometries in quantum gravity. Motivated by all of these various aspects of entropy from shallow circuits, we thus initiate a study of entropy distinguishability for shallow circuits and discuss the potential implications of our results in physics. We show that both the classical and quantum versions of entropy difference are hard assuming that the Learning-With-Errors ($\mathrm{LWE}$) problem has no efficient algorithm, even for a quantum computer. Since $\mathrm{LWE}$ serves as a basis for various schemes of _post-quantum cryptography_ , our result implies that entropy estimation for shallow circuits is intractable unless these cryptographic schemes are unsecure. Remarkably, this result also holds for constant-depth quantum and classical circuits. In contrast, we also show that both versions of entropy distinguishability for shallow classical and quantum circuits are unlikely to be $\sf SZK$-complete and $\sf QSZK$-complete respectively. Therefore, the entropy difference problem for both classical and quantum shallow circuits occupies an interesting intermediate complexity. Finally, we consider a version of von Neumann entropy difference where Hamiltonians are given as input, not circuits, and show that this problem is at least as hard as the circuit-based version. This last result allows to relate these computational complexity results to physical systems. ### 1.1 Main results We start by defining EDδ (QEDδ) as the (quantum) entropy difference problem where the circuits under consideration have depth $\delta(n)$, for some monotonically increasing function $\delta:\mathbb{N}\to\mathbb{N}$. We will denote ED${}_{\textrm{polylog}}$ (QED${}_{\textrm{polylog}}$) and ED${}_{\textrm{O(1)}}$ (QED${}_{\textrm{O(1)}}$) to be the collection of all problems EDδ (QEDδ), with $\delta(n)$ in $O(polylog(n))$ and $O(1)$, respectively. Our first result gives an indication that these problems are unlikely to be as hard as their poly-size counterparts. To show this, we will consider $\sf SZK_{\delta}$ ($\sf QSZK_{\delta}$) to be the set of problems that reduce to EDδ (QEDδ) under polynomial-time reductions444These correspond to statistical zero-knowledge protocols in which the verifier and simulator circuits have depth $\delta(n)$.. For $\delta(n)=polylog(n)$, we denote these classes as $\sf SZK_{polylog}$ and $\sf QSZK_{polylog}$ respectively, and prove the following: ###### Theorem 1. There exists an oracle $\mathcal{O}$ such that $\mathsf{SZK^{\mathcal{O}}_{polylog}}\neq\mathsf{SZK}^{\mathcal{O}}$ and $\mathsf{QSZK^{\mathcal{O}}_{polylog}}\neq\mathsf{QSZK}^{\mathcal{O}}$. In particular, this means that $\mathsf{SZK^{\mathcal{O}}_{log}}\neq\mathsf{SZK}^{\mathcal{O}}$ and $\mathsf{QSZK^{\mathcal{O}}_{log}}\neq\mathsf{QSZK}^{\mathcal{O}}$. The oracle we use is the same as the one from the recent work of Chia, Chung and Lai [CCL19], showing the separation $\mathsf{BPP}^{\mathsf{QNC}^{\mathcal{O}}}\neq\mathsf{BQP}^{\mathcal{O}}$, where $\mathsf{QNC}$ denotes the set of problems that can be solved by quantum circuits of polylogarithmic depth. We conjecture that the oracle of Coudron and Menda [CM19], showing the same separation, could also be used. Our second result concerns a direct approach at trying to show that $\sf QSZK_{log}$$=$ $\sf QSZK$ and why we believe this is unlikely. To explain this approach, let us first discuss the class $\sf QIP$, of problems that can be decided using an interactive proof system having a quantum verifier. A problem that is complete for this class is the quantum circuit distinguishability problem (QCD), in which one is given as input two quantum circuits and asked to determine whether, when restricting to a subset of input and output qubits, the corresponding channels are close or far in diamond distance. It was shown by Rosgen that this problem remains $\sf QIP$-complete even when the circuits under consideration are log-depth [Ros08]. This is achieved by constructing log-depth circuits that check a “history state” of the original circuits, which uses a different construction to the Feynman-Kitaev history state [KSV02]. One can also show that any $\sf QIP$ protocol can be made to have a log-depth verifier, using the same history state construction. In analogy to Rosgen’s result, we can now suppose that any $\sf QSZK$ protocol can be made into a $\sf QSZK_{log}$ protocol by having the prover send history states of the computations that the verifier in the $\sf QSZK$ protocol would perform. It is clear from the $\sf QIP$ result that making the verifier have logarithmic depth in a $\sf QSZK$ protocol does not reduce the set of problems that can be decided by such a protocol. The question is whether _in addition_ to having a log-depth verifier, the transcript of such a protocol555Here we mean the transcript of the protocol for “yes” instances to the problem. can be produced by a log-depth circuit. We show that if this is possible, then since $\sf BQP$ $\subseteq$ $\sf QSZK$ it would be possible to simulate any $\sf BQP$ computation by “checking” this history state on a log-depth quantum computer. This result can be stated as follows: ###### Theorem 2. If there exists a polynomial-time reduction from a $\sf QSZK$ protocol with a log-depth verifier to a $\sf QSZK_{log}$ protocol which preserves the transcript of the $\sf QSZK$ protocol, then $\sf BQP=\sf BPP^{\sf QNC^{1}}$. Based on these results, we conjecture that estimating the output entropy of circuits having (poly)logarithmic or constant depth should be easier than for circuits of larger depth. Despite this fact, our next result shows that even for these shallow circuits, entropy difference is still intractable for polynomial-time classical and quantum algorithms, assuming the quantum intractability of the Learning-With-Errors ($\mathrm{LWE}$) problem: ###### Theorem 3. $\mathrm{LWE}$ $\leq_{P}$ ED${}_{\textrm{O(1)}}$ $\leq_{P}$ QED${}_{\textrm{O(1)}}$. $\mathrm{LWE}$, a problem defined by Regev in [Reg09], serves as the basis for recently proposed cryptographic protocols that are believed to be post-quantum secure. In other words, it is conjectured that no polynomial-time classical or quantum algorithm is able to solve this problem. Recent results have leveraged the versatility of this problem to achieve tasks such as verifiable delegation of quantum computation [Mah18, GV19], certifiable randomness generation [BCM+18], and self-testing of a single quantum device [MV20]. In all of these works, the protocols were based on the use of _Extended Trapdoor Claw-Free Functions_ (ETCF), introduced in [Mah18]. An ETCF family consists of a pair of one-way functions, $(f,g)$, that are hard to invert assuming the hardness of $\mathrm{LWE}$. Importantly, $f$ is a $1$-to-$1$ one-way function, whereas $g$ is a $2$-to-$1$ one-way function. An essential property of these functions is that it should be hard (based on $\mathrm{LWE}$) to determine which is the $1$-to-$1$ function and which is the $2$-to-$1$ function, given descriptions of the functions. This property is known as _injective invariance_. Consider what happens if we evaluate each of these functions on a string $x$ drawn uniformly at random from $\\{0,1\\}^{n}$. Given that $f$ is a $1$-to-$1$ function, the distribution $f(x)$ is still the uniform distribution over $n$-bit strings and will therefore have maximum entropy, $S(f)=n$. On the other hand, since $g$ is a $2$-to-$1$, $g(x)$ will be the uniform distribution on _half_ of the strings in $\\{0,1\\}^{n}$, thus having entropy $S(g)=n-1$. This means that given descriptions of $f$ and $g$, if one can distinguish the entropy of their outputs, one can also solve $\mathrm{LWE}$, which effectively shows that $\mathrm{LWE}$ $\leq_{P}$ ED. While the above argument shows a reduction from $\mathrm{LWE}$ to ED, to obtain the result of Theorem 3 one would need an ETCF pair of functions that can be evaluated in constant depth. Such a construction is not known to exist and so we adopt a different approach towards proving the result. We start by showing that the ETCF functions defined in [Mah18] can be performed using log- depth circuits, thus showing $\mathrm{LWE}$ $\leq_{P}$ ED${}_{\textrm{log}}$. This follows from the fact that the specific ETCF functions we use involve only matrix and vector multiplications, which can be parallelized and performed in logarithmic depth666The functions also require the ability to sample from a Gaussian distribution, but this can also be done in logarithmic depth (following a preprocessing step that is done as part of the reduction) [Pei10].. To then go to constant-depth circuits, we use the result of Applebaum, Ishai and Kushilevitz of compiling log-depth one-way functions to constant-depth [AIK06]. The main idea is that, for a given function $f$, that can be evaluated in log-depth, as well as an input string $x$, it is possible to produce a _randomized encoding_ of $f(x)$ using a circuit of constant depth. A randomized encoding is an encoded version of $f(x)$ from which $f(x)$ can be efficiently (and uniquely) reconstructed. The main difficulty of the proof is to show that the constant-depth randomized encoders for $f$ and $g$ remain indistinguishable, based on $\mathrm{LWE}$. This then implies the desired result, $\mathrm{LWE}$ $\leq_{P}$ ED${}_{\textrm{O(1)}}$. Since QED${}_{\textrm{O(1)}}$ is simply the quantum generalization of ED${}_{\textrm{O(1)}}$, the reduction ED${}_{\textrm{O(1)}}$ $\leq_{P}$ QED${}_{\textrm{O(1)}}$ is immediate. Having obtained this characterisation for the hardness of entropy difference with shallow circuits, we now wish to investigate the potential application of these results to physics. To make this link, we first define a Hamiltonian version of the quantum entropy difference problem, which we denote HQED. Informally, instead of being given as input two quantum circuits and being asked to determine which produces a state of higher entropy, our input will consist of the descriptions of two local Hamiltonians $H_{1}$ and $H_{2}$. Upon tracing out a certain number of qubits from the ground states of these Hamiltonians, the problem is to decide which of the two reduced states has higher entropy777This is essentially the same as asking which of the two ground states has higher entanglement entropy, for a given bipartition of the qubits in the two states.. Unsurprisingly, it can be shown that this problem is at least at hard as the circuit version: ###### Theorem 4. QED $\leq_{P}$ HQED. The proof makes use of the Feynman-Kitaev history state construction to encode the history state of a quantum circuit in the ground state of a Hamiltonian [KSV02]. Directly using the history states of the circuits, $C_{1}$ and $C_{2}$, from an instance of QED would not necessarily yield the desired result since those states have small overlap with the output state of $C_{1}$ and $C_{2}$. To get around this issue, we make use of the padding construction from [NVY18], and pad the circuits $C_{1}$ and $C_{2}$ with polynomially-many identity gates. It can then be shown that the resulting history states for these new circuits will have overlap $1-1/poly(n)$ with the output states of $C_{1}$ and $C_{2}$. Correspondingly, up to inverse polynomial factors, the entropy difference between these states and the original ones is preserved. If we now denote as HQED${}_{\textrm{log}}$ and HQED${}_{\textrm{O(1)}}$ the analogous problems in which the ground states can be approximated by circuits of logarithmic and constant depth, respectively, it follows from the previous results that QED${}_{\textrm{log}}$ $\leq_{P}$ HQED${}_{\textrm{log}}$ and QED${}_{\textrm{O(1)}}$ $\leq_{P}$ HQED${}_{\textrm{O(1)}}$, and therefore that $\mathrm{LWE}$ $\leq_{P}$ HQED${}_{\textrm{O(1)}}$ $\leq_{P}$ HQED${}_{\textrm{log}}$. ### 1.2 Discussion and open questions In Theorem 1, as mentioned, the oracle used to separate $\mathsf{SZK_{polylog}}$ and $\mathsf{SZK}$ ($\mathsf{QSZK_{polylog}}$ and $\mathsf{QSZK}$) is taken from the work of Chia et al [CCL19]. Interestingly, Chia et al, along with Coudron and Menda [CM19], speculate that their oracles could be instantiated based on a cryptographic assumption, such as the hardness of $\mathrm{LWE}$. We further conjecture that for our specific case this should also be true. This would have the intriguing implication that the intractability of $\mathrm{LWE}$ implies ED (QED) is “strictly harder” than ED${}_{\textrm{log}}$ (QED${}_{\textrm{log}}$), while at the same time, from Theorem 3, the intractability of $\mathrm{LWE}$ also implies ED${}_{\textrm{O(1)}}$ (QED${}_{\textrm{O(1)}}$) is hard for polynomial-time quantum algorithms. A natural open problem raised by our work is to characterise more precisely the complexity classes induced by the shallow-depth versions of entropy difference ($\sf SZK_{log}$, $\sf SZK_{const}$, $\sf QSZK_{log}$, $\sf QSZK_{const}$). We show that these correspond to zero-knowledge protocols in which the verifier and simulator are bounded-depth circuits, but can one give other characterisations of these classes? We have also seen that these classes appear to be _strictly_ contained in their counterparts with general polynomial-depth circuits. Thus, if estimating the entropy of shallow circuits is easier, what is the runtime of the optimal quantum algorithm for deciding QED (or ED) for log-depth and constant-depth circuits? It is also pertinent to ask this question not just for worst-case instances of the problems, but also for the average case. $\mathrm{LWE}$ is assumed to be hard on-average, and thus we can use the reduction to argue that ED should be hard on average for a particular range of parameters. Can we say anything more general? What about the entropy of random quantum circuits? States produced by random circuits will typically be highly entangled and thus typically subsystems will have the maximum amount of entropy. Does this imply that all forms of entropy calculation for random circuits is easy? Another open problem has to do with the question of _entropy ratio_ rather than entropy difference. As mentioned, the entropy difference between two circuits can be amplified to constant or even polynomial in the input size, by simply repeating the circuits in parallel. However, this would not affect the ratio between the output entropies. Could our techniques be used to show that even the entropy ratio is hard to estimate based on $\mathrm{LWE}$? We conjecture that the answer is yes and that this could be achieved by extending the ETCF family to include functions that are $2^{m}$-to-1, with $m=poly(n+k)$. The same reductions could then be used as in the entropy difference case. In our reduction from $\mathrm{LWE}$ to ED${}_{\textrm{O(1)}}$ we used the results of Applebaum et al in[AIK06] for compiling log-depth circuits to constant depth. An appealing feature of their construction is that the resulting circuits are “fairly local”, in that they have output locality 4. In other words, each output bit depends on at most 4 input bits. They conjecture that it may also be possible to achieve output locality 3. For the purpose of constructing one-way functions, this would be optimal since it is impossible to generate one-way functions having output locality 2. We also conjecture that ED for circuits with output locality 3 is hard for quantum computers. Could the compiling technique of Applebaum et al be used to base the hardness of other natural problems in (quantum) information theory on the hardness of $\mathrm{LWE}$? The fact that the entropy difference problem is complete for $\sf SZK$ gives a connection to cryptography, but can $\mathrm{LWE}$ be used to show that other natural problems are hard for a quantum computer? Going further, is it possible to have some notion of fine-grained computational complexity based on $\mathrm{LWE}$? Let us also comment on the use of $\mathrm{LWE}$ and ETCF functions to derive Theorem 3. $\mathrm{LWE}$ and lattice problems are leading candidates in the development of post-quantum cryptographic protocols due to their conjectured intractability even for quantum algorithms. Moreover, an appealing feature of $\mathrm{LWE}$ for cryptographic purposes, is that one can reduce _worst-case_ instances of lattice problems to _average-case_ instances of $\mathrm{LWE}$ [Reg09, Pei16]. Combining this with Theorem 3, means that average-case instances (relative to a certain distribution over input circuits) of ED${}_{\textrm{O(1)}}$ will also be “at least as hard” as worst-case lattice problems. Thus, using $\mathrm{LWE}$ gives strong evidence that ED${}_{\textrm{O(1)}}$ is quantum intractable. ETCF functions then provide a very natural instance of an entropy difference problem: determine whether a given function is 1-to-1 or 2-to-1. Such functions differ by one bit in entropy, under a uniform input, but by the injective invariance property it is at least as hard as solving $\mathrm{LWE}$ to tell which is which. It is then only necessary, for our results, to show that these functions can be evaluated by circuits of constant depth. As mentioned, this can be achieved in a relatively straightforward manner by first showing that the functions can be evaluated in log depth and then using the compiling technique of Applebaum, Ishai and Kushilevitz to achieve constant depth. Given the relevance of the von Neumann entropy for quantum gravity research, it is natural to ask what impact our results have for the AdS/CFT correspondence. As previously mentioned, if states on the boundary are described by log-depth tensor networks according to MERA, then computing the entropy for these states should inform the bulk geometry contained within a boundary. If computing the entropy of the boundary is difficult even for quantum computers, then this poses a challenge to the quantum version of the Extended Church Turing thesis. That is, in some sense, quantum gravitational systems cannot be simulated efficiently by a quantum computer. Bouland, Fefferman and Vazirani have also explored this challenge to the thesis from the perspective of the wormhole growth paradox using the tools of computational complexity [BFV19]. In particular, they showed that one can prepare computationally pseudorandom states on the boundary CFT. These are efficiently preparable ensembles of quantum states that require exponential complexity to distinguish from each other. Importantly, the states can have different circuit complexities and under the AdS/CFT correspondence this would correspond to different wormhole lengths. An observer in the bulk could, in principle, determine the length of a wormhole, however the circuit complexity of the boundary states should be computationally intractable to determine. They conclude that either the map from bulk to boundary in the AdS/CFT duality is exponentially complex, or a quantum computer cannot efficiently simulate such a system. With our final result in Theorem 4 as a basis, we can make tentative connections to the AdS/CFT duality and reach a similar conclusion to the Bouland et al result. If one can have ground states of differing entanglement entropy, based on $\mathrm{LWE}$, for a CFT Hamiltonian then this will correspond to differing bulk geometries. An observer in the bulk can, in principle, efficiently differentiate between these geometries. Based on this, we can devise an experiment that allows for the efficient estimation of the ground state entanglement entropy of our Hamiltonian. We conjecture that unless the AdS/CFT bulk to boundary map is exponentially complex, this experiment would violate the intractability assumption of $\mathrm{LWE}$. We leave it open whether one can more formally describe our experiment by giving a Hamiltonian for the CFT whose grounds states encode the hard problem, instead of just positing that the CFT is prepared in one of those states. ### Acknowledgements The authors would like to thank Mehmet Burak Şahinoğlu, Thomas Vidick, Grant Salton, Hrant Gharibyan and Junyu Liu for useful comments and discussions. AG is supported by MURI Grant FA9550-18-1-0161 and the IQIM, an NSF Physics Frontiers Center (NSF Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028). MJH acknowledges the FQXi large grant The Emergence of Agents from Causal Order. ## 2 Preliminaries ### 2.1 Notation We write $\mathbb{N}$ for the set of natural numbers, $\mathbb{Z}$ for the set of integers, $\mathbb{Z}_{q}$ for the set of integers modulo $q$, $\mathbb{R}$ for the set of real numbers, $\mathcal{H}$ for a finite-dimensional Hilbert space, using indices $\mathcal{H}_{A}$, $\mathcal{H}_{B}$ to specify distinct spaces. $\mathrm{L}(\mathcal{H})$ is the set of linear operators on $\mathcal{H}$. We write $\mbox{\rm Tr}(\cdot)$ for the trace, and $\mbox{\rm Tr}_{B}:\mathrm{L}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})\to\mathrm{L}(\mathcal{H}_{A})$ for the partial trace. $\mathrm{Pos}(\mathcal{H})$ is the set of positive semidefinite operators and $\mathrm{D}(\mathcal{H})=\\{X\in\mathrm{Pos}(\mathcal{H}):\mbox{\rm Tr}(X)=1\\}$ the set of density matrices (also called states). Given $A,B\in\mathrm{L}(\mathcal{H})$, $\|A\|_{1}=\mbox{\rm Tr}\sqrt{A^{\dagger}A}$ is the Schatten $1$-norm, $TD(A,B)=\frac{1}{2}\|A-B\|_{1}$ the trace distance, $F(A,B)=\left[Tr\sqrt{\sqrt{A}B\sqrt{A}}\right]^{2}$ the fidelity, and $S(A)=-Tr(A\log A)$ the Von Neumann entropy. Probability distributions will generally be over $n$-bit binary strings and so will be functions $\mathcal{D}:\\{0,1\\}^{n}\to[0,1]$ for which $\mathcal{D}(x)\geq 0$, for all $x\in\\{0,1\\}^{n}$ and $\sum_{x\in\\{0,1\\}^{n}}\mathcal{D}(x)=1$. The Shannon entropy of a distribution is $S(\mathcal{D})=-\sum_{x\in\\{0,1\\}^{n}}\mathcal{D}(x)\log(\mathcal{D}(x)).$ (1) Here we are abusing notation since $S$ denotes both the Shannon entropy and the Von Neumann entropy and it will be clear from the context which notion of entropy is being used. We will also use $h:[0,1]\to[0,1]$ to denote the binary entropy function $h(\epsilon)=-\frac{1}{\epsilon}\log(\epsilon)-\frac{1}{1-\epsilon}\log(1-\epsilon)$. We write $\textsc{Supp}(\mathcal{D})=\\{x\;|\;\mathcal{D}(x)>0\\}$ for the support of the distribution $\mathcal{D}$. The Hellinger distance between two distributions is: $H^{2}(\mathcal{D}_{1},\mathcal{D}_{2})=1-\sum_{x}\sqrt{\mathcal{D}_{1}(x)\mathcal{D}_{2}(x)}.$ (2) Note that: $||\mathcal{D}_{1}-\mathcal{D}_{2}||_{TV}=\frac{1}{2}\sum_{x}|\mathcal{D}_{1}(x)-\mathcal{D}_{2}(x)|\leq H^{2}(\mathcal{D}_{1},\mathcal{D}_{2}),$ (3) where $||\cdot||_{TV}$ is the total variation distance. For a finite set $S$, we write $x\leftarrow_{U}S$ to mean that $x$ is drawn uniformly at random from $S$. In general, $x\leftarrow_{\chi}S$ will mean that $x$ was drawn according to the distribution $\chi:S\to[0,1]$ from $S$. We say that a function $\mu:\mathbb{N}\to\mathbb{R}$ is negligible if it goes to $0$ faster than any inverse-polynomial function, i.e. for any polynomial $p:\mathbb{N}\to\mathbb{R}$, $p(n)\mu(n)\to_{n\to\infty}0$. ### 2.2 Complexity theory In this paper we reference the standard complexity classes describing polynomial-time probabilistic and quantum computation, respectively, denoted $\sf BPP$ and $\sf BQP$. In addition, we also utilize the complexity classes for computations that can be performed by polynomial-size circuits having polylogarithmic depth, NC for classical circuits and QNC for quantum circuits, logarithmic depth $\sf NC^{1}$ and $\sf QNC^{1}$, as well as constant depth, $\sf NC^{0}$ and $\sf QNC^{0}$, respectively. For formal definitions of these classes, as well as others mentioned in this paper, we refer to the Complexity Zoo [zoo]. Here, we provide the definition of $\sf QSZK$, as it is the focus of our main results: ###### Definition 1 (Quantum Statistical Zero-Knowledge ($\sf QSZK$)). A promise problem $(L_{yes},L_{no})$ belongs to $\sf QSZK$ if there exists a uniform family of polynomial-size quantum circuits $V=\\{V_{n}\\}_{n>0}$, known as a verifier such that the following conditions are satisfied: * • Completeness. For each $x\in L_{yes}$, there exists a prover $P(x)$ exchanging polynomially-many quantum messages with $V(x)$ that makes $V(x)$ accept with probability greater than $2/3$, * • Soundness. For each $x\in L_{no}$, any prover $P(x)$ exchanging polynomially- many quantum messages with $V(x)$ will make $V(x)$ accept with probability at most $1/3$, * • Zero-Knowledge. There exists a uniform family of polynomial-size quantum circuits $S=\\{S_{n}\\}_{n>0}$, known as a simulator, as well as a negligible function $\varepsilon:\mathbb{N}\rightarrow[0,1]$ such that for all $x\in L_{yes}$, $S(x)$ produces a state that is $\varepsilon$-close in trace distance to the transcript of $V(x)\leftrightarrow P(x)$. As was shown by Ben-Aroya, Schwartz and Ta-Shma [BASTS10], the following problem is $\sf QSZK$-complete under polynomial-time reductions: ###### Definition 2 (Quantum Entropy Difference (QED) [BASTS10]). Let $C_{1}$ and $C_{2}$ be quantum circuits acting on $n+k$ qubits such that $depth(C_{1})$ is $O(log(n+k))$ and $depth(C_{2})$ is $O(log(n+k))$. Define the following $n$-qubit mixed states: $\rho_{1}=Tr_{k}(C_{1}\ket{00...0}\bra{00...0}C_{1}^{\dagger})\quad\quad\quad\quad\rho_{2}=Tr_{k}(C_{2}\ket{00...0}\bra{00...0}C_{2}^{\dagger})$ (4) Given $n$, $k$ and descriptions of $C_{1}$ and $C_{2}$ as input, decide whether: $S(\rho_{1})\geq S(\rho_{2})+1/poly(n+k)$ (5) or $S(\rho_{2})\geq S(\rho_{1})+1/poly(n+k)$ (6) promised that one of these is the case. Note that the entropy difference can always be made constant by considering _parallel_ 888Hence the new circuits have the same depths as the original ones. repeated version of $C_{1}$ and $C_{2}$. Specifically, if the entropy difference for $C_{1}$ and $C_{2}$ is $D=1/poly(n+k)$, then the difference for $C_{1}^{\otimes D}$ and $C_{2}^{\otimes D}$, acting on $(n+k)D$ qubits and when tracing out $kD$ qubits, will be $O(1)$. For this reason, throughout this paper we will consider the entropy difference problem (and all its variants) with a constant entropy difference. The classical/probabilistic analogues of the above definitions are represented by the class $\sf SZK$ and the Entropy Difference (ED) problem. In analogy to QED, ED is defined as the problem of distinguishing the output entropies of classical circuits having uniformly random inputs. It was shown in [GV99] that ED is $\sf SZK$-complete. ### 2.3 Learning-With-Errors and Extended Trapdoor Claw-Free Functions In this section we give definitions for the Learning-With-Errors ($\mathrm{LWE}$) problem, introduced by Regev [Reg09], as well as Extended Trapdoor Claw-Free Functions (ETCF). Most of this section is taken from [BCM+18]. An informal description of $\mathrm{LWE}$ is that it is the problem of solving an approximate system of linear equations. In other words, given a matrix $A\in\mathbb{Z}_{q}^{n\times m}$, as well as a vector $y=Ax+e\;(mod\;q)$, with $y\in\mathbb{Z}_{q}^{m}$, $x\in\mathbb{Z}_{q}^{n}$ and $e\leftarrow_{\chi^{m}}\mathbb{Z}^{m}$ (here $\chi$ denotes a probability distribution that is described below), the problem is to determine $x$. More formally, let us start by defining the _truncated Gaussian distribution_. For a positive real $B$ and positive integer $q$, the truncated discrete Gaussian distribution over $\mathbb{Z}_{q}$ with parameter $B$ is supported on $\\{x\in\mathbb{Z}_{q}:\,\|x\|\leq B\\}$ and has density $D_{\mathbb{Z}_{q},B}(x)\,=\,\frac{e^{\frac{-\pi\lVert x\rVert^{2}}{B^{2}}}}{\sum\limits_{x\in\mathbb{Z}_{q},\,\|x\|\leq B}e^{\frac{-\pi\lVert x\rVert^{2}}{B^{2}}}}\;.$ (7) For a positive integer $m$, the truncated discrete Gaussian distribution over $\mathbb{Z}_{q}^{m}$ with parameter $B$ is supported on $\\{x\in\mathbb{Z}_{q}^{m}:\,\|x\|\leq B\sqrt{m}\\}$ and has density $\forall x=(x_{1},\ldots,x_{m})\in\mathbb{Z}_{q}^{m}\;,\qquad D_{\mathbb{Z}_{q}^{m},B}(x)\,=\,D_{\mathbb{Z}_{q},B}(x_{1})\cdots D_{\mathbb{Z}_{q},B}(x_{m})\;.$ (8) ###### Definition 3 (Learning-With-Errors [BCM+18]). For a security parameter $\lambda$, let $n,m,q\in\mathbb{N}$ be integer functions of $\lambda$. Let $\chi=\chi(\lambda)$ be a distribution over $\mathbb{Z}$. The $LWE_{n,m,q,\chi}$ problem is to distinguish between the distributions $(\\*A,\\*A\\*s+\\*e\pmod{q})$ and $(\\*A,\\*u)$, where $\\*A$ is uniformly random in $\mathbb{Z}_{q}^{n\times m}$, $\\*s$ is a uniformly random row vector in $\mathbb{Z}_{q}^{n}$, $\\*e$ is a row vector drawn at random from the distribution $\chi^{m}$, and $\\*u$ is a uniformly random vector in $\mathbb{Z}_{q}^{m}$. Often we consider the hardness of solving $\mathrm{LWE}$ for any function $m$ such that $m$ is at most a polynomial in $n\log q$. This problem is denoted $LWE_{n,q,\chi}$. As shown in [Reg09, PRSD17], for any $\alpha>0$ such that $\sigma=\alpha q\geq 2\sqrt{n}$ the $LWE_{n,q,D_{\mathbb{Z}_{q},\sigma}}$ problem, where $D_{\mathbb{Z}_{q},\sigma}$ is the discrete Gaussian distribution, is at least as hard (under a quantum poly-time reduction) as approximating the shortest independent vector problem (SIVP) to within a factor of $\gamma=\tilde{O}({n}/\alpha)$ in _worst case_ dimension $n$ lattices. The so-called “$\mathrm{LWE}$ assumption” is that no quantum polynomial-time procedure can solve the $LWE_{n,q,\chi}$ problem with more than a negligible advantage in $\lambda$, even when given access to a quantum polynomial-size state depending on the parameters $n,m,q$ and $\chi$ of the problem. For the reduction from $\mathrm{LWE}$ to QED${}_{\textrm{log}}$ we make use of “extended trapdoor claw-free functions (ETCF),” introduced in [Mah18]. Briefly, an ETCF consists of two families of functions, denoted $\mathcal{F}$ and $\mathcal{G}$. The first family, $\mathcal{F}$, is referred to as a “trapdoor injective family” and consists of a pair of functions $(f_{k,0},f_{k,1})$ that have disjoint ranges. The second, $\mathcal{G}$, is referred to as a “noisy trapdoor claw-free family (NTCF)”, introduced in [BCM+18], and consists of a pair of functions $(g_{k,0},g_{k,1})$ that have overlapping ranges999The specific functions we consider will be functions that output probability distributions. The NTCF function pair will produce distributions with overlapping supports., essentially acting as a one-way $2$-to-$1$ function. While we do not require all the properties of ETCF functions for our results, we give the full definitions of these functions for completeness. The following definitions are taken from [Mah18, Section 4], with similar definitions in [BCM+18, Section 3]. Note that we have flipped the naming convention of these functions with respect to how they were originally defined in [Mah18, BCM+18]. In other words, for us the $\mathcal{F}$ family is the injective family and the $\mathcal{G}$ family is the NTCF one, whereas in the cited results this is reversed. This was done in order to be consistent with referenced injective functions denoted $f$ from other results. We start with the definition of the NTCF family: ###### Definition 4 (NTCF Family [Mah18, BCM+18]). Let $\lambda$ be a security parameter. Let $\mathcal{X}$ and $\mathcal{Y}$ be finite sets. Let $\mathcal{K}_{\mathcal{G}}$ be a finite set of keys. A family of functions $\mathcal{G}\,=\,\big{\\{}g_{k,b}:\mathcal{X}\rightarrow\mathcal{D}_{\mathcal{Y}}\big{\\}}_{k\in\mathcal{K}_{\mathcal{G}},b\in\\{0,1\\}}$ is called a noisy trapdoor claw-free (NTCF) family if the following conditions hold: 1. 1. Efficient Function Generation. There exists an efficient probabilistic algorithm $\textrm{GEN}_{\mathcal{G}}$ which generates a key $k\in\mathcal{K}_{\mathcal{G}}$ together with a trapdoor $t_{k}$: $(k,t_{k})\leftarrow\textrm{GEN}_{\mathcal{G}}(1^{\lambda})\;.$ 2. 2. Trapdoor Injective Pair. For all keys $k\in\mathcal{K}_{\mathcal{G}}$ the following conditions hold. 1. (a) Trapdoor: For all $b\in\\{0,1\\}$ and $x\neq x^{\prime}\in\mathcal{X}$, $\textsc{Supp}(g_{k,b}(x))\cap\textsc{Supp}(g_{k,b}(x^{\prime}))=\emptyset$. Moreover, there exists an efficient deterministic algorithm $\textrm{INV}_{\mathcal{G}}$ such that for all $b\in\\{0,1\\}$, $x\in\mathcal{X}$ and $y\in\textsc{Supp}(g_{k,b}(x))$, $\textrm{INV}_{\mathcal{G}}(t_{k},b,y)=x$. 2. (b) Injective pair: There exists a perfect matching $\mathcal{R}_{k}\subseteq\mathcal{X}\times\mathcal{X}$ such that $g_{k,0}(x_{0})=g_{k,1}(x_{1})$ if and only if $(x_{0},x_{1})\in\mathcal{R}_{k}$. 3. 3. Efficient Range Superposition. For all keys $k\in\mathcal{K}_{\mathcal{G}}$ and $b\in\\{0,1\\}$ there exists a function $g^{\prime}_{k,b}:\mathcal{X}\mapsto\mathcal{D}_{\mathcal{Y}}$ such that 1. (a) For all $(x_{0},x_{1})\in\mathcal{R}_{k}$ and $y\in\textsc{Supp}(g^{\prime}_{k,b}(x_{b}))$, INV${}_{\mathcal{G}}(t_{k},b,y)=x_{b}$ and INV${}_{\mathcal{G}}(t_{k},b\oplus 1,y)=x_{b\oplus 1}$. 2. (b) There exists an efficient deterministic procedure CHKG that, on input $k$, $b\in\\{0,1\\}$, $x\in\mathcal{X}$ and $y\in\mathcal{Y}$, returns $1$ if $y\in\textsc{Supp}(g^{\prime}_{k,b}(x))$ and $0$ otherwise. Note that CHKG is not provided the trapdoor $t_{k}$. 3. (c) For every $k$ and $b\in\\{0,1\\}$, $\textsc{E}_{x\leftarrow_{U}\mathcal{X}}\big{[}\,H^{2}(g_{k,b}(x),\,g^{\prime}_{k,b}(x))\,\big{]}\,\leq\,\mu(\lambda)\;,$ for some negligible function $\mu(\cdot)$. Here $H^{2}$ is the Hellinger distance. Moreover, there exists an efficient procedure SAMPG that on input $k$ and $b\in\\{0,1\\}$ prepares the state $\frac{1}{\sqrt{|\mathcal{X}|}}\sum_{x\in\mathcal{X},y\in\mathcal{Y}}\sqrt{(g^{\prime}_{k,b}(x))(y)}\ket{x}\ket{y}\;.$ (9) 4. 4. Adaptive Hardcore Bit. For all keys $k\in\mathcal{K}_{\mathcal{G}}$ the following conditions hold, for some integer $w$ that is a polynomially bounded function of $\lambda$. 1. (a) For all $b\in\\{0,1\\}$ and $x\in\mathcal{X}$, there exists a set $G_{k,b,x}\subseteq\\{0,1\\}^{w}$ such that $\Pr_{d\leftarrow_{U}\\{0,1\\}^{w}}[d\notin G_{k,b,x}]$ is negligible, and moreover there exists an efficient algorithm that checks for membership in $G_{k,b,x}$ given $k,b,x$ and the trapdoor $t_{k}$. 2. (b) There is an efficiently computable injection $J:\mathcal{X}\to\\{0,1\\}^{w}$, such that $J$ can be inverted efficiently on its range, and such that the following holds. If $\displaystyle H_{k}$ $\displaystyle=$ $\displaystyle\big{\\{}(b,x_{b},d,d\cdot(J(x_{0})\oplus J(x_{1})))\,|\;b\in\\{0,1\\},\;(x_{0},x_{1})\in\mathcal{R}_{k},\;d\in G_{k,0,x_{0}}\cap G_{k,1,x_{1}}\big{\\}}\;,\text{}$ $\displaystyle\overline{H}_{k}$ $\displaystyle=$ $\displaystyle\\{(b,x_{b},d,c)\,|\;(b,x,d,c\oplus 1)\in H_{k}\big{\\}}\;,$ then for any quantum polynomial-time procedure $\mathcal{A}$ there exists a negligible function $\mu(\cdot)$ such that $\Big{|}\Pr_{(k,t_{k})\leftarrow\textrm{GEN}_{\mathcal{G}}(1^{\lambda})}[\mathcal{A}(k)\in H_{k}]-\Pr_{(k,t_{k})\leftarrow\textrm{GEN}_{\mathcal{G}}(1^{\lambda})}[\mathcal{A}(k)\in\overline{H}_{k}]\Big{|}\,\leq\,\mu(\lambda)\;.$ (10) We now define the trapdoor injective family. ###### Definition 5 (Trapdoor Injective Function Family [Mah18, BCM+18]). Let $\lambda$ be a security parameter. Let $\mathcal{X}$ and $\mathcal{Y}$ be finite sets. Let $\mathcal{K}_{\mathcal{F}}$ be a finite set of keys. A family of functions $\mathcal{F}\,=\,\big{\\{}f_{k,b}:\mathcal{X}\rightarrow\mathcal{D}_{\mathcal{Y}}\big{\\}}_{b\in\\{0,1\\},k\in\mathcal{K}_{\mathcal{F}}}$ is called a trapdoor injective family if the following conditions hold: 1. 1. Efficient Function Generation. There exists an efficient probabilistic algorithm $\textrm{GEN}_{\mathcal{F}}$ which generates a key $k\in\mathcal{K}_{\mathcal{F}}$ together with a trapdoor $t_{k}$: $(k,t_{k})\leftarrow\textrm{GEN}_{\mathcal{F}}(1^{\lambda})\;.$ 2. 2. Disjoint Trapdoor Injective Pair. For all keys $k\in\mathcal{K}_{\mathcal{F}}$, for all $b,b^{\prime}\in\\{0,1\\}$ and $x,x^{\prime}\in\mathcal{X}$, if $(b,x)\neq(b^{\prime},x^{\prime})$, $\textsc{Supp}(f_{k,b}(x))\cap\textsc{Supp}(f_{k,b^{\prime}}(x^{\prime}))=\emptyset$. Moreover, there exists an efficient deterministic algorithm $\textrm{INV}_{\mathcal{F}}$ such that for all $b\in\\{0,1\\}$, $x\in\mathcal{X}$ and $y\in\textsc{Supp}(g_{k,b}(x))$, $\textrm{INV}_{\mathcal{F}}(t_{k},y)=(b,x)$. 3. 3. Efficient Range Superposition. For all keys $k\in\mathcal{K}_{\mathcal{F}}$ and $b\in\\{0,1\\}$ 1. (a) There exists an efficient deterministic procedure CHKF that, on input $k$, $b\in\\{0,1\\}$, $x\in\mathcal{X}$ and $y\in\mathcal{Y}$, outputs $1$ if $y\in\textsc{Supp}(f_{k,b}(x))$ and $0$ otherwise. Note that CHKF is not provided the trapdoor $t_{k}$. 2. (b) There exists an efficient procedure SAMPF that on input $k$ and $b\in\\{0,1\\}$ returns the state $\frac{1}{\sqrt{|\mathcal{X}|}}\sum_{x\in\mathcal{X},y\in\mathcal{Y}}\sqrt{(f_{k,b}(x))(y)}\ket{x}\ket{y}\;.$ (11) ###### Definition 6 (Injective Invariance [Mah18]). A noisy trapdoor claw-free family $\mathcal{G}$ is injective invariant if there exists a trapdoor injective family $\mathcal{F}$ such that: 1. 1. The algorithms CHKF and SAMPF are the same as the algorithms CHKG and SAMPG. 2. 2. For all quantum polynomial-time procedures $\mathcal{A}$, there exists a negligible function $\mu(\cdot)$ such that $\Big{|}\Pr_{(k,t_{k})\leftarrow\textrm{GEN}_{\mathcal{F}}(1^{\lambda})}[\mathcal{A}(k)=0]-\Pr_{(k,t_{k})\leftarrow\textrm{GEN}_{\mathcal{G}}(1^{\lambda})}[\mathcal{A}(k)=0]\Big{|}\leq\mu(\lambda)$ (12) ###### Definition 7 (Extended Trapdoor Claw-Free Family [Mah18]). A noisy trapdoor claw-free family $\mathcal{G}$ is an extended trapdoor claw- free family if: 1. 1. It is injective invariant. 2. 2. For all $k\in\mathcal{K}_{\mathcal{G}}$ and $d\in\\{0,1\\}^{w}$, let: $\displaystyle H^{\prime}_{k,d}$ $\displaystyle=$ $\displaystyle\\{d\cdot(J(x_{0})\oplus J(x_{1}))|(x_{0},x_{1})\in\mathcal{R}_{k}\\}$ (13) For all quantum polynomial-time procedures $\mathcal{A}$, there exists a negligible function $\mu(\cdot)$ and a string $d\in\\{0,1\\}^{w}$ such that $\Big{|}\Pr_{(k,t_{k})\leftarrow\textrm{GEN}_{\mathcal{G}}(1^{\lambda})}[\mathcal{A}(k)\in H^{\prime}_{k,d}]-\frac{1}{2}\Big{|}\leq\mu(\lambda)$ (14) ### 2.4 Randomized encodings To prove the reduction from $\mathrm{LWE}$ to the constant depth version of the entropy difference problem, we make use of randomized encodings. The following definitions and results are taken from [AIK06]: ###### Definition 8 (Uniform randomized encoding [AIK06]). Let $f:\\{0,1\\}^{*}\to\\{0,1\\}^{*}$ be a polynomial-time computable function and $l(n)$ an output length function such that $|f(x)|=l(|x|)$, for every $x\in\\{0,1\\}^{*}$. We say that a function $\hat{f}:\\{0,1\\}^{*}\times\\{0,1\\}^{*}\to\\{0,1\\}^{*}$ is a $\delta(n)$-correct, $\varepsilon(n)$-private randomized encoding of $f$, if it satisfies the following properties: * • Length regularity. There exist polynomially-bounded and efficiently computable length functions $m(n)$, $s(n)$ such that for every $x\in\\{0,1\\}^{n}$ and $r\in\\{0,1\\}^{m(n)}$ we have that $|\hat{f}(x,r)|=s(n)$. * • Efficient evaluation. There exists a polynomial-time evaluation algorithm that, given $x\in\\{0,1\\}^{*}$ and $r\in\\{0,1\\}^{m(n)}$, outputs $\hat{f}(x,r)$. * • $\delta$-correctness. There exists a polynomial-time algorithm $C$, called a decoder, such that for every input $x\in\\{0,1\\}^{n}$, $\Pr_{r\leftarrow_{U}\\{0,1\\}^{m(n)}}[C(1^{n},\hat{f}(x,r))\neq f(x)]\leq\delta(n).$ (15) * • $\varepsilon$-privacy. There exists a probabilistic polynomial-time algorithm $S$, called a simulator such that for every $x\in\\{0,1\\}^{n}$ and $r\leftarrow_{U}\\{0,1\\}^{m(n)}$, $||S(1^{n},f(x))-\hat{f}(x,r)||_{TV}\leq\varepsilon(n).$ (16) Correctness (and privacy, respectively) can be _perfect_ when $\delta=0$ ($\varepsilon=0$, respectively), or _statistical_ when $\delta(n)$ ($\varepsilon(n)$, respectively) is a negligible function in $n$. Thus, a _perfect randomized encoding_ is one that has perfect correctness and perfect privacy111111In fact a perfect randomized encoding needs to satisfy two further properties called _balance_ and _stretch-preservation_ [AIK06], though we omit them here since they are not important for our results.. Similarly, a _statistical randomized encoding_ is one that has statistical correctness and statistical privacy. ###### Lemma 1 (Unique randomness [AIK06]). Suppose $\hat{f}$ is a perfect randomized encoding of $f$. Then, * (a) for any input $x$, $\hat{f}(x,\cdot)$ is injective, i.e. there are no distinct $r,r^{\prime}$, such that $\hat{f}(x,r)=\hat{f}(x,r^{\prime})$, * (b) if $f$ is a permutation then so is $\hat{f}$. ###### Theorem 5 ([AIK06]). Any function $f\in\sf NC^{1}$ admits a perfect randomized encoding in $\mathsf{NC}^{0}_{4}$. Moreover, constructing the randomized encoding of $f$ can be achieved in time polynomial in the size of the circuit that evaluates $f$. Here $\mathsf{NC}^{0}_{4}$ denotes the set of uniform constant-depth circuits having output locality 4. In other words, each output bit depends on at most 4 input bits. ## 3 Entropy difference for shallow circuits In this section we examine the hardness of the entropy difference problem for circuits whose depth scales at most (poly)logarithmically with the size of the input. We start by formally defining the entropy difference problem, in both the classical and the quantum cases, for circuits with depth scaling as $\delta(n)$, for inputs of size $n$ and where $\delta:\mathbb{N}\to\mathbb{N}$, is a monotonically increasing function: ###### Definition 9 (EDδ). Let $C_{1}$ and $C_{2}$ be reversible boolean circuits acting on $n+k$ bits such that $depth(C_{1})\leq\delta(n+k)$, $depth(C_{2})\leq\delta(n+k)$. Let $\mathcal{D}_{1}(y)=\Pr_{x\leftarrow_{U}\\{0,1\\}^{n+k}}\left[y=Tr_{k}(C_{1}(x))\right]\quad\quad\quad\mathcal{D}_{2}(y)=\Pr_{x\leftarrow_{U}\\{0,1\\}^{n+k}}\left[y=Tr_{k}(C_{2}(x))\right]$ (17) denote the output distributions of the circuits when restricted to the first $n$ output bits and with the input chosen uniformly at random. Given $n$, $k$ and descriptions of $C_{1}$ and $C_{2}$ as input, decide whether: $S(\mathcal{D}_{1})\geq S(\mathcal{D}_{2})+1\quad\quad\text{or}\quad\quad S(\mathcal{D}_{2})\geq S(\mathcal{D}_{1})+1$ (18) promised that one of these is the case121212As mentioned in Subsection 2.2, the gap in entropy difference can be $1/poly(n+k)$, but this can always be made constant by simply taking parallel repeated versions of the circuits and using the fact that entropy is additive. We restrict to the case where the entropy difference is at least $1$, unless otherwise specified.. ###### Definition 10 (QEDδ). Let $C_{1}$ and $C_{2}$ be quantum circuits acting on $n+k$ qubits such that $depth(C_{1})\leq\delta(n+k)$, $depth(C_{2})\leq\delta(n+k)$. Define the following $n$-qubit mixed states: $\rho_{1}=Tr_{k}(C_{1}\ket{00...0}\bra{00...0}C_{1}^{\dagger})\quad\quad\quad\quad\rho_{2}=Tr_{k}(C_{2}\ket{00...0}\bra{00...0}C_{2}^{\dagger})$ (19) Given $n$, $k$ and descriptions of $C_{1}$ and $C_{2}$ as input, decide whether: $S(\rho_{1})\geq S(\rho_{2})+1\quad\quad\text{or}\quad\quad S(\rho_{2})\geq S(\rho_{1})+1$ (20) promised that one of these is the case. Note that if $\delta(n)=O(\log(n))$ we obtain the definitions for ED${}_{\textrm{log}}$ and QED${}_{\textrm{log}}$, respectively, and if $\delta(n)=O(1)$, we obtain the definitions for ED${}_{\textrm{O(1)}}$ and QED${}_{\textrm{O(1)}}$, respectively131313We slightly abuse notation here since $O(\log(n))$ and $O(1)$ are sets of functions and so, correspondingly, ED${}_{\textrm{log}}$ (QED${}_{\textrm{log}}$) and ED${}_{\textrm{O(1)}}$ (QED${}_{\textrm{O(1)}}$) will also be sets of problems. However, our results are valid for any instances of the problems in these classes.. In addition, when $\delta(n)=poly(n)$, we recover the original definitions of ED and QED, respectively. As those problems are complete for $\sf SZK$ and $\sf QSZK$, we proceed to define the analogous classes $\sf SZK_{\delta}$ and $\sf QSZK_{\delta}$ of problems that poly-time reduce to EDδ and QEDδ. ###### Definition 11 ($\sf SZK_{\delta}$, $\sf QSZK_{\delta}$). We let $\sf SZK_{\delta}$ ($\sf QSZK_{\delta}$) consist of all promise problems $(L_{yes},L_{no})$ for which there exists a deterministic polynomial- time reduction to EDδ (QEDδ). Taking $\delta(n)=polylog(n)$, we now show that these classes have the following characterisation: ###### Lemma 2. $\sf SZK_{polylog}$ ($\sf QSZK_{polylog}$) is the set of promise problems that admit a (quantum) statistical zero-knowledge protocol in which the verifier circuit and the simulator circuit have depth $polylog(n)$, for inputs of size $n$. ###### Proof. To prove this result we need to show that ED${}_{\textrm{polylog}}$ (QED${}_{\textrm{polylog}}$) is contained and complete for the class of problems that admit a (quantum) statistical zero-knowledge protocol in which the verifier circuit and the simulator circuit have depth $polylog(n)$. This follows immediately from the proofs that ED (QED) is contained and complete for $\sf SZK$ ($\sf QSZK$), from [VW16], by replacing all circuits in those proofs with the corresponding circuits of depth $polylog(n)$. ∎ ### 3.1 Entropy difference is easier for shallow circuits With the above definitions, let us now address the question of whether ED${}_{\textrm{polylog}}$ (QED${}_{\textrm{polylog}}$) is $\sf SZK$-complete ($\sf QSZK$-complete). From the above lemma, we see that this is the same as asking whether $\sf SZK_{polylog}=SZK$ ($\sf QSZK_{polylog}=QSZK$). In other words, can any $\sf SZK$ ($\sf QSZK$) protocol be turned into a protocol having polylog-depth circuits for the simulator and verifier? Providing an _unconditional_ negative answer seems difficult without proving explicit circuit lower-bounds. Instead, we will give “complexity-theoretic evidence” that the answer is no. We first show that there exists an oracle, $\mathcal{O}$, such that, relative to $\mathcal{O}$, $\sf SZK_{polylog}$ $\neq$ $\sf SZK$ ($\sf QSZK_{polylog}$$\neq$ $\sf QSZK$). ###### Theorem 6. There exists an oracle $\mathcal{O}$ such that $\mathsf{SZK^{\mathcal{O}}_{polylog}}\neq\mathsf{SZK}^{\mathcal{O}}$ and $\mathsf{QSZK^{\mathcal{O}}_{polylog}}\neq\mathsf{QSZK}^{\mathcal{O}}$. ###### Proof. The proof will be primarily for the quantum case, since the classical case is analogous, though we will specify whenever there is a distinction between the two. The oracle in our proof will be identical to the one of Chia, Chung and Lai [CCL19], showing the separation $\mathsf{BPP}^{\mathsf{QNC}^{\mathcal{O}}}\neq\mathsf{BQP}^{\mathcal{O}}$. Their oracle provides access to a function $f:\\{0,1\\}^{n}\to\\{0,1\\}^{n}$ that is promised to be either $1$-to-$1$ or $2$-to-$1$. However, the oracle does not give direct query access to $f$. Instead, the oracle allows for the querying of $d+1$ functions $f_{0},f_{1}...,f_{d}:\\{0,1\\}^{n}\to\\{0,1\\}^{n}$ such that $f=f_{d}\circ...\circ f_{0}$, for some $d>0$. The functions $f_{0},f_{1},...,f_{d-1}$ are $1$-to-$1$ functions and $f_{d}$ is either a $1$-to-$1$ function or a $2$-to-$1$ function depending on whether $f$ is $1$-to-$1$ or $2$-to-$1$. This is referred to as a _$d$ -shuffling oracle_. Its purpose is to force any algorithm that attempts to query $f$ to first evaluate the $d+1$ functions. This is achieved by having the image of each function be a random subset of its co-domain. In other words, each function will be defined as $f_{i}:S_{i}^{(i)}\to S_{i+1}^{(i+1)}$, with $S_{i}^{(i)}\subseteq\\{0,1\\}^{n}$. The input domain, however, will be a set $S_{0}\subseteq\\{0,1\\}^{n}$ chosen uniformly at random from subsets of $n$-bit strings. Thus, the image of $f_{1}$ on $S_{0}$ will be $Im_{S_{0}}(f_{1})=f_{1}(S_{0})=S_{1}$ and in general $S_{i}=Im_{S_{i-1}}(f_{i})$. The problem that Chia, Chung and Lai define relative to this oracle is to determine whether the function $f:S_{0}\to S_{d+1}$ is $1$-to-$1$ or $2$-to-$1$. In the latter case, the function also has Simon’s property so that the problem (called _$d$ -shuffling Simon’s problem_, or $d$-SSP) can be solved efficiently in quantum polynomial time, thus showing containment in $\sf BQP^{\mathcal{O}}$. Using the properties of the $d$-shuffling oracle it is possible to show that no quantum circuit of depth smaller than $d$ can solve the problem, even when alternating these quantum circuits with classical circuits of polynomial depth. Thus, taking $d=O(n)$ is sufficient to show that the problem is not contained ${\sf BPP^{\mathsf{QNC}}}^{\mathcal{O}}$. For the proof of our result we also consider the $d$-SSP problem. Since we already know that the problem is in $\sf BQP^{\mathcal{O}}$ this immediately implies that the problem is also in $\sf QSZK^{\mathcal{O}}$. For the classical case, we would also need to show that the problem is contained in $\sf SZK^{\mathcal{O}}$. This follows from the fact that ED is in $\sf SZK$ and the problem of determining whether a function is $1$-to-$1$ or $2$-to-$1$ reduces to ED141414This is because $f$ evaluated on a uniformly random $n$-bit string will have maximum entropy $n$, when $f$ is $1$-to-$1$ and entropy $n-1$ when the function is $2$-to-$1$.. We therefore need to show that the problem is not contained in $\sf SZK_{polylog}^{\mathcal{O}}$ and $\sf QSZK_{polylog}^{\mathcal{O}}$. Figure 1: Circuit with oracle calls. The unitaries $U_{j}$ are assumed to be polylogarithmic depth quantum circuits. The unitaries $U_{\mathcal{O}}$ represent calls to the oracle $\mathcal{O}$. The circuit has polylogarithmic depth overall. To do this, consider a $\sf QSZK_{polylog}$ protocol in which the verifier, the prover and the simulator all have access to the oracle $\mathcal{O}$. In such a protocol, the verifier and the simulator are circuits of depth $O(polylog(n))$ that generically consist of alternating sequences of polylog- depth circuits and calls to the oracle, as shown in Figure 1. For a fixed input state that starts off uncorrelated with the oracle, let us examine the output state of such a circuit in the cases when the function is injective and when it is a $2$-to-$1$ function. We will denote the oracle as $\mathcal{O}(f)$ in the former case and as $\mathcal{O}(g)$ in the latter151515The interpretation of this notation is that the $d$-shuffling oracle is providing “shuffled” access to either $f$ or $g$.. We also denote the circuit under consideration, when given access to the oracle, as $C^{\mathcal{O}(f)}$ and $C^{\mathcal{O}(g)}$, respectively. These can be expressed as follows $C^{\mathcal{O}(f)}=U_{m}U_{\mathcal{O}(f)}U_{m-1}...U_{\mathcal{O}(f)}U_{1}\quad\quad C^{\mathcal{O}(g)}=U_{m}U_{\mathcal{O}(g)}U_{m-1}...U_{\mathcal{O}(g)}U_{1}$ (21) with $m=polylog(n)$ and where each $U_{i}$ is a circuit of depth one. We denote the input state to the circuit as161616The analysis remains unchanged if the input state is a mixed state. $\ket{\psi(0)}$. We will also write $\ket{\psi^{f}(i)}=U_{\mathcal{O}(f)}U_{i}\ket{\psi^{f}(i-1)}$ and $\ket{\psi^{g}(i)}=U_{\mathcal{O}(g)}U_{i}\ket{\psi^{g}(i-1)}$. Note that $\ket{\psi^{f}(m)}=C^{\mathcal{O}(f)}\ket{\psi(0)}\quad\quad\ket{\psi^{g}(m)}=C^{\mathcal{O}(g)}\ket{\psi(0)}$ (22) Following a similar analysis to that of [CCL19], we have that $TD(\ket{\psi^{f}(m)},\ket{\psi^{g}(m)})\leq TD(\ket{\psi^{f}(m)},U_{\mathcal{O}(f)}U_{m}\ket{\psi^{g}(m-1)})+\\\ TD(U_{\mathcal{O}(f)}U_{m}\ket{\psi^{g}(m-1)},\ket{\psi^{g}(m)})$ (23) which can be extended to $TD(\ket{\psi^{f}(m)},\ket{\psi^{g}(m)})\leq\sum_{i=1}^{m}TD(U_{\mathcal{O}(f)}U_{i}\ket{\psi^{g}(i-1)},\ket{\psi^{g}(i)})$ (24) Both of these equations follow from the triangle inequality. We next have that $TD(\ket{\psi^{f}(m)},\ket{\psi^{g}(m)})\leq m\max_{i\leq m}\;TD(U_{\mathcal{O}(f)}U_{i}\ket{\psi^{g}(i-1)},U_{\mathcal{O}(g)}U_{i}\ket{\psi^{g}(i-1)})$ (25) Finally, as was shown in [CCL19, Theorem 6.1], for $i\leq d$, $TD(U_{\mathcal{O}(f)}U_{i}\ket{\psi^{g}(i-1)},U_{\mathcal{O}(g)}U_{i}\ket{\psi^{g}(i-1)})\leq\frac{poly(n)}{2^{n}}$ (26) and since $m=polylog(n)$, $d=O(n)$, for sufficiently large $n$, $m<d$, hence $TD(\ket{\psi^{f}(m)},\ket{\psi^{g}(m)})\leq m\frac{poly(n)}{2^{n}}.$ (27) If we now consider $C^{\mathcal{O}}$ to be the simulator circuit in a $\sf QSZK_{polylog}$ ($\sf SZK_{polylog}$) protocol, we see that the simulator produces nearly identical transcripts irrespective of whether the oracle function is $1$-to-$1$ or $2$-to-$1$. This means that, for the “yes” instances (the function being $1$-to-$1$), the transcript that the verifier circuit acts on, upon its interaction with the prover, is _almost completely uncorrelated_ with the oracle itself. Stated differently, the transcript is $poly(n)/2^{n}$-close in trace distance to a transcript for a “no” instance. Thus, the interaction with the prover can provide the verifier with at most a $poly(n)/2^{n}$ advantage in deciding the problem correctly. Since the verifier circuit itself is polylogarithmic in depth, from the above analysis (and the result of [CCL19]), it follows that if the oracle function type is equiprobable to be $1$-to-$1$ or $2$-to-$1$, the resulting $\sf QSZK_{polylog}$ ($\sf SZK_{polylog}$) protocol will decide correctly with probability at most $1/2+poly(n)/2^{n}$. This concludes the proof. ∎ It should be noted that, following [CCL19], the above result extends to circuits of depth strictly less than $d$. The key insight of the proof is the fact that the shuffling oracle requires circuits of depth at least $d$ in order to obtain an output that is non-negligibly correlated with the oracle type. Intuitively, if we were to look strictly at instances of ED${}_{\textrm{polylog}}$ and QED${}_{\textrm{polylog}}$ in which the circuits under consideration can query the oracle, the number of queries is too small for there to be any noticeable difference in the output entropies. Thus, relative to the shuffling oracle, ED${}_{\textrm{polylog}}$ and QED${}_{\textrm{polylog}}$ are strictly weaker than ED and QED. Let us now consider a different argument for why it is unlikely that entropy difference with shallow circuits is as hard as with general poly-size circuits. We will focus specifically on the quantum case for circuits of logarithmic depth and show the following: ###### Theorem 7. If there exists a polynomial-time reduction from a $\sf QSZK$ protocol with a log-depth verifier to a $\sf QSZK_{log}$ protocol which preserves the transcript of the $\sf QSZK$ protocol, then $\sf BQP=\sf BPP^{\sf QNC^{1}}$. ###### Proof. It is straightforward to show that $\sf BQP\subseteq\sf QSZK$: the quantum verifier ignores the prover and can decide any language in $\sf BQP$ . However, one can also give a $\sf QSZK$ protocol for any language in $\sf BQP$ where there is non-trivial interaction between a prover and verifier. Furthermore, we show that such a protocol only requires the verifier’s circuit to be log depth. To show this, we adapt the proof by Rosgen that $\sf QIP$ only requires log depth quantum verifiers [Ros08]. Given a polynomial-time quantum circuit on $n$ qubits of the form $C=U_{m}U_{m-1}...U_{1}$, the following state $\ket{\psi_{U}}$ can be constructed by another $\sf BQP$ circuit, $\ket{\psi_{U}}=\ket{0...0}\otimes U_{1}\ket{0...0}\otimes...\otimes U_{m}U_{m-1}...U_{1}\ket{0...0}.$ (28) Notice the similarity to a Feynman-Kitaev history state except where one takes the tensor product of unitaries applied to the input, and not the superposition. The state $\ket{\psi_{U}}$ stores the final state of the circuit above in the rightmost $n$ qubits. Therefore, any language in $\sf BQP$ can be decided by measuring the relevant qubit in these rightmost $n$ qubits in $\ket{\psi_{U}}$. Now we can have a $\sf QSZK$ protocol where an honest prover gives the verifier the state $\ket{\psi_{U}}$, and thus a verifier has the ability to decide any language in $\sf BQP$ when given this state. In the case of a dishonest prover, we can use the techniques described by Rosgen, to verify that the state given by the prover is $\ket{\psi_{U}}$. For convenience of explanation, we divide the state given by the prover up into $m$ “registers” of $n$ qubits, where the first register should be in the state $\ket{0...0}$, the second register in the state $U_{1}\ket{0...0}$, and so on. The idea for verifying that the state is $\ket{\psi_{U}}$ is to pairwise compare the $j$th and $(j+1)$th registers with SWAP tests. More precisely the verification circuit will apply $U_{j+1}$ to the $j$th register and perform a SWAP test on the $j$th and $(j+1)$th registers: if the states are the same then swapping leaves the states invariant and the circuit accepts, otherwise it rejects. Therefore, after $n$ SWAP tests of this form, if all tests accept, then with high probability the state is $\ket{\psi_{U}}$. Importantly, all of the SWAP tests to compare registers can be done in log depth [Ros08], thus the verifier’s circuit is a log depth quantum circuit. In the above protocol we have outlined how the verifier can verify the state $\ket{\psi_{U}}$, but we have not shown that it satisfies the property of statistical zero-knowledge. Note that the state $\ket{\psi_{U}}$ produced by the prover (such that an input is accepted) can be generated by a polynomial time quantum circuit. Therefore, in the case of a $\sf QSZK$ protocol, the simulator could produce this state $\ket{\psi_{U}}$, and since this state is the whole transcript of the protocol, the protocol has the property of zero- knowledge. If we assume that there exists a polynomial-time reduction from a $\sf QSZK$ protocol to a $\sf QSZK_{log}$ protocol which preserves the transcript of the $\sf QSZK$ protocol, then the above $\sf QSZK$ protocol for deciding $\sf BQP$ can be turned into a $\sf QSZK_{log}$ protocol. Therefore, a simulator $S$ must be able to produce a state very close to $\ket{\psi_{U}}$ with a log- depth quantum circuit, in the case that the input $x$ is in the language $L_{yes}$. Furthermore, since $\sf BQP$ is closed under complement, we can take the complement of the language above, and have a $\sf QSZK_{log}$ protocol for this, and thus another simulator $S^{\prime}$ that produces a state close to $\ket{\psi_{U}}$ for $x\in L_{no}$. Now we have the situation where if the conditions of the theorem hold, we have two log-depth quantum circuits corresponding to the simulators $S$ and $S^{\prime}$ above that can be used to decide membership of a language in $\sf BQP$: if $x\in L_{yes}$ then $S$ will produce the state $\ket{\psi_{U}}$, but $S^{\prime}$ could produce anything; if $x\in L_{no}$ then $S^{\prime}$ produces the correct state $\ket{\psi_{U}}$. To decide which is which, a verifier can apply the SWAP tests outlined above individually on both of the states generate by $S$ and $S^{\prime}$: at least one of the two states will satisfy the tests and correctly accept or reject. We can now leverage these observations to prove the theorem. To collect the observations together, we have pointed out that if the conditions of the theorem hold, there are log-depth quantum circuits $S$ and $S^{\prime}$ that generate states $\ket{\psi_{U}}$, which can be used by a log-depth quantum circuit to decide membership in $\sf BQP$. Thus we could decide any language in $\sf BQP$ with a $\sf BPP^{\sf QNC^{1}}$ algorithm in the following way: a $\sf BPP$ machine computes the reduction from a $\sf QSZK$ protocol to a $\sf QSZK_{log}$ protocol, feeds the circuit descriptions of the log-depth simulators $S$ and $S^{\prime}$ to a $\sf QNC^{1}$ oracle, which can then produce states of the form $\ket{\psi_{U}}$, and carry out the necessary SWAP tests to verify this state. If the SWAP tests are passed for at least one of the two states produced by $S$ and $S^{\prime}$, then the accept/reject measurement is performed on it (again by the oracle), and the algorithm accepts if the oracle accepts, or rejects otherwise. ∎ It is worthwhile pointing out that this trick of deciding languages in $\sf BQP$ with a $\sf BPP^{\sf QNC^{1}}$ algorithm does not obviously generalise to other complexity classes. First, we would need that there are quantum statistical zero-knowledge protocols for the class, and use the property of closure under complementation. Naturally $\sf QSZK$ satisfies both of these properties, but an arbitrary protocol for languages in $\sf QSZK$ has multiple rounds of communication between prover and verifier. Our construction uses the fact that a $\sf QSZK$ protocol for any language in $\sf BQP$ has a single round of communication from prover to verifier, which facilitates verification of a state in log depth. It is far from obvious how to construct such a verification procedure for an arbitrary language in $\sf QSZK$. We can ask about the possibility of a classical analogue of Theorem 7. That is, if there is a reduction from a $\sf SZK$ protocol to one with a log-depth verifier and simulator, does this imply that polynomial-time classical computation can be parallelised to log depth? This would then imply something about the hardness of entropy distinguishability for probability distributions from log-depth classical circuits. However, it’s not clear how to do this since we cannot use the history state construction of Rosgen [Ros08] for classical circuits. For one thing, it is not obvious how an $\sf SZK$ protocol would work with communication only from the prover to the verifier. Indeed, it is a feature of quantum interactive proof systems that they can be parallelised to a constant number of rounds of interaction, but this parallelisation is not known to be a feature of classical interactive proof systems. ### 3.2 Hardness based on Learning-With-Errors In the previous subsection we showed that the polylogarithmic-depth version of the entropy difference problem, in both the classical and quantum case, is unlikely to be as hard as the polynomial-depth variant. This raises the question of whether the problem becomes tractable for polynomial-time classical or quantum algorithms. Here we give indication that the answer is no by proving a reduction from $\mathrm{LWE}$ to ED${}_{\textrm{log}}$. Using techniques from [AIK06], we then strengthen this result by also showing a reduction from $\mathrm{LWE}$ to ED${}_{\textrm{O(1)}}$. We begin with the log-depth case: ###### Theorem 8. $\mathrm{LWE}$ $\leq_{P}$ ED${}_{\textrm{log}}$. ###### Proof. The proof uses the ETCF functions defined in Subsection 2.3. As mentioned, an ETCF family consists of an injective function and a $2$-to-$1$ (or claw-free) function and there exists a reduction from $\mathrm{LWE}$ to the problem of distinguishing the two functions given their descriptions171717This is the injective invariance property of Definition 6.. By showing that the two functions can be evaluated using circuits of logarithmic depth, we will extend this reduction to ED${}_{\textrm{log}}$. While it has already been shown that certain cryptographic functions based on $\mathrm{LWE}$ can be performed in $\sf NC^{1}$ [BPR12], our result requires that we show this for the circuits that we construct from an ETCF function family. The ETCF functions we consider will be the same as the ones from [Mah18, BCM+18]: $f(b,x)=Ax+b\cdot u+e\;(mod\;q)\quad\quad\quad g(b,x)=Ax+b\cdot(As+e^{\prime})+e\;(mod\;q)$ (29) with $q\geq 2$ a prime integer181818It should be noted that $q$ itself is a function of $n$. In general $q$ is taken to be suprapolynomial in $n$, though in recent constructions $q$ can also be polynomial in $n$ while still preserving the hardness of the $\mathrm{LWE}$ instance [BLP+13]., $b\in\\{0,1\\}$, $x\in\mathbb{Z}_{q}^{n}$, $s\in\mathbb{Z}_{q}^{n}$, $A\in\mathbb{Z}^{n\times m}_{q}$, $u,e^{\prime}\in\mathbb{Z}_{q}^{m}$, $e\leftarrow_{D_{\mathbb{Z}_{q},B}^{m}}\mathbb{Z}^{m}$ as in Definition 3. These functions will be ETCF even when $A$, $e^{\prime}$ and $u$ are chosen at random as follows: $A\leftarrow_{U}\mathbb{Z}_{q}^{n\times m}$, $u\leftarrow_{U}\mathbb{Z}_{q}^{m}$, $e^{\prime}\leftarrow_{D_{\mathbb{Z}_{q},B^{\prime}}^{m}}\mathbb{Z}^{m}$. The values $B^{\prime}$ and $B$ are the ones from [Mah18, BCM+18] (there denoted as $B_{P}$ and $B_{V}$), and determine where the Gaussian distributions are truncated. Specifically, $B^{\prime}=\frac{q}{C_{T}\sqrt{mn\log(q)}}$, where $C_{T}$ is a fixed constant and $B$ is chosen so that $B^{\prime}/B$ is super- polynomial in $n$. Since it was already shown in [Mah18] that the above functions are ETCF, we need only show that they can be evaluated by log-depth circuits. First of all note that the functions from Equation 29 output probability distributions, whereas the circuits in ED${}_{\textrm{log}}$ need to output fixed outcomes. We will fix this by making the error, $e$, be part of the input. We cannot, however, make it directly part of the input since the input to the circuits in ED${}_{\textrm{log}}$ is distributed uniformly, whereas $e$ must be drawn from a truncated Gaussian distribution, $D^{m}_{\mathbb{Z}_{q},B}$. Instead, we will have as part of the input a string $e_{u}$ which we will turn into a Gaussian sample using a log-depth circuit denoted Gaussify. Implementing this procedure can be achieved using, for instance, the algorithm of Peikert from [Pei10]. Gaussify satisfies the property that if $e_{u}\leftarrow_{U}\mathbb{Z}_{q}^{m}$ then $e=\textsc{Gaussify}(e_{u})$ is distributed according to the truncated Gaussian distribution $D^{m}_{\mathbb{Z}_{q},B}$. The ED${}_{\textrm{log}}$ circuits we construct will therefore have the form $C_{f}(b,x,e_{u})=Ax+b\cdot u+\textsc{Gaussify}(e_{u})\;(mod\;q)$ (30) $C_{g}(b,x,e_{u})=Ax+b\cdot(As+e^{\prime})+\textsc{Gaussify}(e_{u})\;(mod\;q)$ (31) where $A$, $s$, $e^{\prime}$ and $u$ are fixed. Essentially, one is given $A$, $u$ and $As+e^{\prime}$ and one has to construct the above circuits and ensure that they have logarithmic depth. Note that the input length for each circuit is $(m+n)\log(q)+1$. Following [Mah18, BCM+18] we will assume that $m=\Omega(n\log(q))$, so that what we need to ensure is that the circuit depth is $O(\log(m\log(q)))$. The $\mathrm{LWE}$ assumption states that it should be computationally intractable to distinguish $u$ from $As+e^{\prime}$, when given $A$. However, as we will show, the above circuits will have different entropies in their outputs (when the inputs are chosen uniformly at random). Intuitively this is because one function is $1$-to-$1$ and the other is _approximately_ $2$-to-$1$ and so the two cases could be distinguished if we have the ability to solve ED${}_{\textrm{log}}$. This is the essence of the reduction. Let us take stock of all the operations performed by these circuits and why they are all in $\sf NC^{1}$: 1. 1. Addition and multiplication modulo $q$ can be performed in logarithmic depth with respect to the input size (which in this case is $O(\log(q))$) [Wal64], so that this operation requires only depth $O(\log(\log(q)))$. For vectors in $\mathbb{Z}^{m}_{q}$, component-wise addition can be performed in parallel by increasing the width by a factor of $m$. Thus, the overall depth remains $O(\log(\log(q)))$. 2. 2. The dot-product between two vectors in $\mathbb{Z}^{m}_{q}$ requires depth $O(\log(m\log(q)))$. One first computes the component-wise product of the two vectors. This is the same as component-wise addition and can be performed in $O(\log(\log(q)))$ depth. One then adds together all of the results (modulo $q$) and this can be performed in $O(\log(m\log(q)))$ depth with a divide-and- conquer strategy191919Divide the result vector into two vectors of equal length, recursively compute the sum of their components and add the results. Adding the results requires constant depth and the depth of the recursion tree is logarithmic in the length of the vectors.. 3. 3. Matrix-vector multiplication with matrices in $\mathbb{Z}_{q}^{n\times m}$ and vectors in $\mathbb{Z}_{q}^{n}$ can be performed in depth $O(\log(m\log(q)))$. Start by copying the vector $m$ times (one copy for each row in the matrix). This can be done in depth $O(\log(m\log(q)))$ with a divide-and-conquer strategy. Then perform the inner products between each row of the input matrix and a corresponding copy of the vector. The inner products can be performed in parallel and each requires $O(\log(m\log(q)))$ depth. Thus, the resulting circuit will have $O(\log(m\log(q)))$ depth. 4. 4. Gaussify can be performed in depth $O(\log(m\log(q)))$ as shown in [Pei10]. The procedure from [Pei10] requires a pre-processing step of $O(m^{3}\log^{2}(q))$ operations. This will be done as part of the polynomial- time reduction that generates the ED${}_{\textrm{log}}$ instance so that the circuits $C_{f}$ and $C_{g}$ already contain the results of this pre- processing step. The actual sampling procedure requires $O(m^{2})$ multiplications and additions modulo $q$ which can be performed in parallel requiring depth $O(\log(\log(q)))$. Collecting the results will then require depth at most $O(\log(m\log(q)))$. This shows that $C_{f},C_{g}\in\sf NC^{1}$. We now estimate the entropies $S(C_{f})$ and $S(C_{g})$ when the inputs of the circuits are chosen uniformly at random. We will consider $A$ to be a matrix of full rank202020For $\mathrm{LWE}$, the matrix $A$ is chosen uniformly at random from $\mathbb{Z}^{n\times m}_{q}$ and so will be full rank, with high probability., $n$. Given that $x\leftarrow_{U}\mathbb{Z}^{n}_{q}$ and since $A$ is full rank, we have that $Ax\leftarrow_{U}Im(A)$, where $Im(A)$ denotes the image of $A$. Note that $|Im(A)|=q^{n}$. For $C_{f}$, we can choose $u$ such that the distributions $Ax+u$ and $Ax$ have no overlap212121Similar to the choice of $A$, this will be true for most choices of $u$.. Thus, $Ax+b\cdot u$ will be uniform over $\\{0,1\\}\times Im(A)$ and have $n\cdot\log(q)+1$ bits of entropy. Lastly, we need to account for the Gaussian error. Since this term appears in both $C_{f}$ and $C_{g}$, we will simply denote its contribution as $S_{Gaussian}$. We therefore have that $S(C_{f})=n\cdot\log(q)+1+S_{Gaussian}$. For the case of $C_{g}$ the difference will be due to the term $b\cdot(As+e^{\prime})$. Note that this leads to overlap among different inputs. Specifically $C_{g}(0,x,e_{u})=C_{g}(1,x-s,e^{\prime}_{u})$, where $e^{\prime}_{u}$ is such that $\textsc{Gaussify}(e^{\prime}_{u})=\textsc{Gaussify}(e_{u})-e^{\prime}$. This condition on $e^{\prime}_{u}$ is true on all but a negligible fraction of error vectors (as a result of taking $B^{\prime}/B$ to be super-polynomial in $n$) [Mah18, BCM+18]. The circuit $C_{g}$ will therefore behave like a $2$-to-$1$ function on all but a negligible fraction of the input domain. We therefore have that $S(C_{g})=n\cdot\log(q)+S_{Gaussian}+\mu(n)$, where $\mu(n)$ is a negligible function. For sufficiently large $n$, the $\mu(n)$ term will be less than $1$ and we therefore have that the entropy difference between $C_{f}$ and $C_{g}$ is at least a positive constant, as desired222222The entropy difference can be made larger by simply repeating the circuits in parallel. However, an alternate approach presented by the use of ETCF functions is to instead consider the circuits: $\displaystyle C_{f}(b_{1},b_{2},x,e_{u})$ $\displaystyle=Ax+b_{1}\cdot u_{1}+b_{2}\cdot u_{2}+\textsc{Gaussify}(e_{u})\;(mod\;q)$ (32) $\displaystyle C_{g}(b_{1},b_{2},x,e_{u})$ $\displaystyle=Ax+b_{1}\cdot(As_{1}+e^{\prime}_{1})+b_{2}\cdot(As_{2}+e^{\prime}_{2})+\textsc{Gaussify}(e_{u})\;(mod\;q)$ (33) Here $f$ is still a $1$-to-$1$ function, however $g$ is $4$-to-$1$ so that the entropy difference for these circuits will be $2-negl(n)$. This construction can be generalised so that, for any constant $k$, the function $g$ can be made into a $2^{k}$-to-$1$ function. . To complete the proof, note that the reduction we have just described constructs circuits with different entropies starting from an ETCF family (and specifically starting from an $\mathrm{LWE}$ instance $(A,As+e^{\prime})$ and a uniformly random vector $u$). However, being able to distinguish between the output entropies of the circuits allows us to determine which of the two functions is $1$-to-$1$ and which is $2$-to-$1$. By the injective invariance property of ETCF functions (Definition 6), this is as hard as $\mathrm{LWE}$, concluding the proof. ∎ It is worth mentioning that in the above proof we essentially picked “worst- case” instances of $A$, $s$, $e^{\prime}$ and $u$. However, all of the above arguments hold with high probability when $A\leftarrow_{U}\mathbb{Z}_{q}^{n\times m}$, $s,u\leftarrow_{U}\mathbb{Z}_{q}^{n}$ and $e^{\prime}\leftarrow_{D^{m}_{\mathbb{Z}_{q},B^{\prime}}}\mathbb{Z}_{q}^{m}$ [Mah18, BCM+18]. This means that ###### Corollary 1. ED${}_{\textrm{log}}$ is hard-on-average, based on $\mathrm{LWE}$ when the input circuits have the structure given in Equations 30, 31 and are chosen at random according to $A\leftarrow_{U}\mathbb{Z}_{q}^{n\times m}$, $s,u\leftarrow_{U}\mathbb{Z}_{q}^{n}$ and $e^{\prime}\leftarrow_{D^{m}_{\mathbb{Z}_{q},B^{\prime}}}\mathbb{Z}_{q}^{m}$. In the previous proof we saw that the ETCF functions based on $\mathrm{LWE}$ can be evaluated by circuits of logarithmic depth. It seems unlikely, however, that the same functions could be evaluated by circuits of constant depth. To get around this issue, we make use of the compiling techniques from [AIK06] that can take a one-way function with log-depth circuit complexity and map it to a corresponding one-way function having constant-depth circuit complexity. This is achieved through the use of a randomized polynomial encoding, in which the value of a function is encoded in a series of points that can be computed using a constant-depth circuit. Formally, we have that, ###### Theorem 9. $\mathrm{LWE}$ $\leq_{P}$ ED${}_{\textrm{O(1)}}$. ###### Proof. As shown in Theorem 8, the circuits $C_{f}$ and $C_{g}$ constructed from the ETCF functions, can be evaluated in log depth. Using Theorem 5, from [AIK06], this means that there exist randomized encodings $\hat{C}_{f}$ and $\hat{C}_{g}$ for the two circuits, that can be computed in $\mathsf{NC}^{0}_{4}$. To prove our reduction we need to show two things: that the randomized encodings of $C_{f}$ and $C_{g}$ preserve the injective invariance property (i.e. distinguishing between the randomized encodings is as hard as $\mathrm{LWE}$); that the randomized encodings are $1$-to-$1$ and $2$-to-$1$ functions, respectively. This last condition is required so that when we evaluate $\hat{C}_{f}$ and $\hat{C}_{g}$ with uniform inputs, it is still the case that $\hat{C}_{f}$ has more entropy in its output than $\hat{C}_{g}$. Showing that injective invariance is satisfied is immediate. Suppose, for the sake of contradiction, that there exists an algorithm that has non-negligible advantage in distinguishing $\hat{C}_{f}$ and $\hat{C}_{g}$. It is easy to see that this leads to an efficient algorithm for distinguishing $C_{f}$ and $C_{g}$ with non-negligible advantage. Given instances of $C_{f}$ and $C_{g}$ we can construct the randomized encodings $\hat{C}_{f}$ and $\hat{C}_{g}$. This can be done in polynomial time, according to Theorem 5. We then use our distinguisher on the randomized encodings and this then allows us to distinguish between $C_{f}$ and $C_{g}$ which contradicts the injective invariance property. We now show that the randomized encodings are $1$-to-$1$ and $2$-to-$1$, respectively. Recall first that the randomized encodings take two arguments, $x$ and $r$. First, from Lemma 1 we know that for any fixed input $x$, the encodings are injective in the second argument. In addition, the perfect correctness property (from Definition 8) guarantees that there are simulators $S_{f}$ and $S_{g}$ such that $S_{f}(\hat{C}_{f}(x,r))=C_{f}(x)$ and $S_{g}(\hat{C}_{g}(x,r))=C_{g}(x)$, for all $x$ and $r$. This ensures that if $C_{f}$ is injective then $\hat{C}_{f}$ will also be injective. It also ensures that whenever there exists a collision in $C_{g}$, i.e. $x_{1}$, $x_{2}$ such that $C_{g}(x_{1})=C_{g}(x_{2})$, there will be a corresponding collision for $\hat{C}_{g}$, i.e. $\hat{C}_{g}(x_{1},r_{1})=\hat{C}_{g}(x_{2},r_{2})$, _for all_ $r_{1}$ and $r_{2}$. It follows that the randomized encodings $\hat{C}_{f}$ and $\hat{C}_{g}$ will have the same entropy difference as $C_{f}$ and $C_{g}$, concluding the proof. ∎ Just as with the log-depth case, we also have: ###### Corollary 2. ED${}_{\textrm{O(1)}}$ is hard-on-average, based on $\mathrm{LWE}$ when the input circuits have the structure given in Equations 30, 31 and are chosen at random according to $A\leftarrow_{U}\mathbb{Z}_{q}^{n\times m}$, $s,u\leftarrow_{U}\mathbb{Z}_{q}^{n}$ and $e^{\prime}\leftarrow_{D^{m}_{\mathbb{Z}_{q},B^{\prime}}}\mathbb{Z}_{q}^{m}$. ###### Proof. The argument is the same as for the log-depth case: for most choices of the circuit parameters (i.e. the matrix $A$, the vectors $s$ and $u$ and the error vector $e^{\prime}$), we obtain instances of the circuits that satisfy the injective invariance property. As the above proof shows, this remains true for the randomized encodings of these functions as well. ∎ In the above proofs, we didn’t explicitly make use of the fact that the circuits are reversible, though the same results hold in those cases as well (provided we trace out the ancilla required to performed the reversible gates). Since classical reversible circuits are a particular kind of quantum circuits, these results have the corollary that: ###### Corollary 3. $\mathrm{LWE}$ $\leq_{P}$ QED${}_{\textrm{O(1)}}$. ###### Proof. Follows from Theorem 9 together with the fact that ED${}_{\textrm{O(1)}}$ $\leq_{P}$ QED${}_{\textrm{O(1)}}$. ∎ ## 4 Hamiltonian quantum entropy difference In this section we consider a Hamiltonian analogue of QED which we call Hamiltonian Quantum Entropy Difference (HQED). The problem will be to estimate the _entanglement entropy_ difference between the ground states of two local Hamiltonians. Equivalently, if we trace out parts of the ground states and examine the resulting reduced states, we want to know which of the two has higher Von Neumann entropy. Formally: ###### Definition 12 (HQED). Let $H_{1}$ and $H_{2}$ be local Hamiltonians acting on $n+k$ qubits, whose ground states are $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$. Define the following $n$-qubit mixed states: $\rho_{1}=Tr_{k}(\ket{\psi_{1}}\bra{\psi_{1}})\quad\quad\quad\quad\rho_{2}=Tr_{k}(\ket{\psi_{2}}\bra{\psi_{2}})$ (34) Given $n$, $k$ and descriptions of $H_{1}$ and $H_{2}$ as input, decide whether: $S(\rho_{1})\geq S(\rho_{2})+1$ (35) or $S(\rho_{2})\geq S(\rho_{1})+1$ (36) promised that one of these is the case. For the cases where either of the two Hamiltonians has a degenerate groundspace, $\ket{\psi_{j}}$ will denote a state in the groundspace for which $S(\rho_{j})$ is minimal. We will refer to HQED${}_{\textrm{log}}$and HQED${}_{\textrm{O(1)}}$, respectively, as instances of HQED in which the input Hamiltonians have the additional promise that purifications of the states $\rho_{1}$ and $\rho_{2}$ can be approximated (to within a $1/poly(n+k)$ additive error in trace distance) by quantum circuits of logarithmic and constant depth, respectively. ###### Theorem 10. There exists a deterministic poly-time reduction from QED to HQED ($\textsc{QED}\leq_{P}\textsc{HQED}$). ###### Proof. We could start the reduction by constructing Hamiltonians $H_{1}$ and $H_{2}$ such that the respective ground states $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ are Feynman-Kitaev history states for quantum circuits $C_{1}$ and $C_{2}$ respectively, $\ket{\psi_{j}}=\frac{1}{\sqrt{T+1}}\sum_{t=0}^{T}U_{t}U_{t-1}...U_{1}\ket{00...0}\ket{t}$ (37) where $C_{j}=U_{T}U_{T-1}...U_{1}$, $j\in\\{1,2\\}$ and the states $\ket{t}$ are the _clock states_ in the Feyman-Kitaev history state construction. However, a priori this does not guarantee that determining the entropy difference between the reduced states of $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ implies determining the entropy difference between the reduced states created by $C_{1}$ and $C_{2}$ respectively. This is because the output state of $C_{j}$ constitutes only one term in the history state superposition. Furthermore, we have no information about the entropy difference between the other terms in $\ket{\psi_{1}}$ relative to their counterparts in $\ket{\psi_{2}}$. To resolve this, we will use a trick from [NVY18], where a circuit can be padded at the end with identities to give more weight to the “final term” in the Feynman-Kitaev history state. We will now explain this construction. Figure 2: Circuit $C_{j}$ padded with identities. Given the input circuit $C_{j}$ to the problem QED, first apply $N$ identity operators at the end of the circuit to each qubit, as depicted in Figure 2; we will call this the padded circuit. Clearly this does not affect the final state, but it will be useful in the reduction to HQED. We now construct the Feynman-Kitaev history state from the padded circuit, which is $\ket{\psi_{j}}=\frac{1}{\sqrt{T+N+1}}\sum_{t=0}^{T+N}U_{t}U_{t-1}...U_{1}\ket{00...0}\ket{t}$ (38) Note that all unitaries $U_{i}$ for $T+1\leq i\leq T+N$, $U_{i}=\mathbb{I}^{\otimes(n+k)}$, thus we can simplify the state $\ket{\psi_{j}}$ to be $\ket{\psi_{j}}=\frac{1}{\sqrt{T+N+1}}\left((N+1)U_{T}U_{T-1}...U_{1}\ket{00...0}\sum_{t^{\prime}=T}^{T+N}\ket{t^{\prime}}+\sum_{t=0}^{T-1}U_{t}U_{t-1}...U_{1}\ket{00...0}\ket{t}\right).$ (39) By the Feynman-Kitaev construction there is a local Hamiltonian, having $poly(n+k)$ terms, for which this is a ground state. This Hamiltonian will act on $n+k+N+T$ qubits due to the clock states $\ket{t}$, which are encoded in unary. Let us now examine the reduced states obtained by tracing out the clock register, which we denote as $\sigma_{j}$. In addition, to simplify the notation, we also denote $\ket{\phi_{j}(t)}=U_{t}U_{t-1}...U_{1}\ket{00...0}$ (with $\ket{\phi_{j}(0)}=\ket{00...0}$), so that the padded history state can be written as $\ket{\psi_{j}}=\frac{1}{\sqrt{T+N+1}}\left((N+1)\ket{\phi_{j}(T)}\sum_{t^{\prime}=T}^{T+N}\ket{t^{\prime}}+\sum_{t=0}^{T-1}\ket{\phi_{j}(t)}\ket{t}\right).$ (40) Note that $\ket{\phi_{j}(T)}$ is the output state of the circuit $C_{j}$. If we now trace out the clock register, we have $\sigma_{j}=Tr_{t}(\ket{\psi_{j}}\bra{\psi_{j}})=\frac{N+1}{T+N+1}\ket{\phi_{j}(T)}\bra{\phi_{j}(T)}+\frac{1}{T+N+1}\sum_{t=0}^{T-1}\ket{\phi_{j}(t)}\bra{\phi_{j}(t)}$ (41) Computing the fidelity between $\rho_{j}$ and $\ket{\phi_{j}(T)}$ we get $F(\sigma_{j},\ket{\phi_{j}(T)}\bra{\phi_{j}(T)})=\bra{\phi_{j}(T)}\sigma_{j}\ket{\phi_{j}(T)}=\frac{N+1}{T+N+1}+\frac{1}{T+N+1}\sum_{t=0}^{T-1}|\braket{\phi_{j}(t)}{\phi_{j}(T)}|^{2}$ (42) hence $F(\sigma_{j},\ket{\phi_{j}(T)}\bra{\phi_{j}(T)})\geq\frac{N+1}{T+N+1}=1-\frac{T}{T+N+1}.$ (43) By taking $N=poly(T)$, and given that $T=poly(n+k)$, we get that $F(\sigma_{j},\ket{\phi_{j}(T)}\bra{\phi_{j}(T)})\geq 1-\frac{1}{poly(n+k)}.$ (44) From the relationship between fidelity and trace distance this also means that $TD(\sigma_{j},\ket{\phi_{j}(T)}\bra{\phi_{j}(T)})\leq\frac{1}{poly(n+k)}.$ (45) If we now trace out the $k$ qubits from both of these states and use the fact that this is a trace non-increasing operation, we get $TD(\rho^{\prime}_{j},\rho_{j})\leq\frac{1}{poly(n+k)},$ (46) where $\rho_{j}$ is the output state of $C_{1}$ when tracing out the subsystem of $k$ qubits and $\rho^{\prime}_{j}$ is the analogous state for the Hamiltonian $H_{j}$. Next, we apply the Fannes-Audanaert inequality [Fan73, Aud07] relating trace distance and entropy, which says that if $TD(\rho^{\prime}_{j},\rho_{j})\leq\epsilon$ (47) then $|S(\rho^{\prime}_{j})-S(\rho_{j})|\leq\frac{\epsilon}{2}(n-1)+h\left(\frac{\epsilon}{2}\right),$ (48) where $h$ is the binary entropy function. Given that $\epsilon=1/poly(n+k)$, it follows that $|S(\rho^{\prime}_{j})-S(\rho_{j})|\leq\frac{1}{poly(n+k)}.$ (49) By the triangle inequality we get that if $S(\rho_{1})\geq S(\rho_{2})+O(1)$, then $S(\rho^{\prime}_{1})\geq S(\rho^{\prime}_{2})+O(1)$ and if $S(\rho_{2})\geq S(\rho_{1})+O(1)$, then $S(\rho^{\prime}_{2})\geq S(\rho^{\prime}_{1})+O(1)$. Thus, from the instance of QED $(n+k,k,C_{1},C_{2})$ we have constructed an instance of HQED $(n+k+N+T,k+N+T,H_{1},H_{2})$ that preserves the entropy difference of the original circuits (up to a $1/poly(n+k)$ error). This concludes the proof. ∎ ###### Corollary 4. QED${}_{\textrm{log}}$ $\leq_{P}$ HQED${}_{\textrm{log}}$, QED${}_{\textrm{O(1)}}$ $\leq_{P}$ HQED${}_{\textrm{O(1)}}$. ###### Proof. In the proof of Theorem 10, we constructed Hamiltonians $H_{1}$ and $H_{2}$ for which the reduced ground states, $\rho^{\prime}_{1}$ and $\rho^{\prime}_{2}$, are close in trace distance to the output states of the QED circuits $C_{1}$ and $C_{2}$. Furthermore, the padded construction used in the previous theorem does not alter the depth of the original circuits, since we are padding with identity gates. Thus, the depth of the circuits required to approximate $\rho^{\prime}_{1}$ and $\rho^{\prime}_{2}$ (to within additive error $1/poly(n+k)$ in trace distance) is upper bounded by the depth of $C_{1}$ and $C_{2}$ (as $\rho_{1}$ and $\rho_{2}$ are approximations of these states). For the cases in which these circuits are log-depth or constant- depth, respectively, we obtain instances of HQED${}_{\textrm{log}}$ and HQED${}_{\textrm{O(1)}}$, respectively. ∎ For the constant depth case, in Appendix B, we give a different construction for a local Hamiltonian for which the ground state is _exactly_ the output of a quantum circuit from QED${}_{\textrm{O(1)}}$. ### 4.1 Applications to holography Holographic duality is an idea inspired from early results of Bekenstein and Hawking showing that the entropy of a black hole is proportional to its area [Bek73, Haw71]. This connection between a quantum mechanical property, the Von Neumann entropy of a quantum state, and geometry, in the form of the black hole area, was later expanded upon through the AdS/CFT correspondence [Mal99]. Briefly, the AdS/CFT correspondence is a duality between a non-gravitational quantum field theory (the conformal field theory, or CFT) and a quantum gravitational theory that takes place in an Anti-de Sitter (AdS) space-time. The CFT is defined on the boundary of the AdS space. The purpose of the correspondence is to be able to relate physical observables in the bulk to observables on the boundary and vice versa through the so-called _AdS dictionary_ (or bulk-to-boundary and boundary-to-bulk maps). This would allow for the derivation of predictions in the quantum gravitational bulk theory purely from a non-gravitational boundary theory. Similar to the Bekenstein-Hawking result, Ryu and Takayanagi showed a correspondence between geometry and entanglement in AdS/CFT [RT06]. This is known as the _Ryu-Takayanagi_ formula and it states that, to leading order, the entropy of a state on the boundary CFT is given by the area of a minimal bulk surface that encloses that state. Since entropy is an important quantity of interest in AdS/CFT, we discuss potential implications of our result for this duality232323We note that $\sf QSZK$ has appeared before in holography in the context of the Harlow-Hayden decoding task [HH13]. Briefly, Harlow and Hayden considered the task of decoding information from the Hawking radiation of a black hole and showed that one could encode a $\sf QSZK$-complete problem in this task. The result was later improved by Aaronson who showed that decoding the information from the radiation would allow one to invert general one-way functions [Aar16]. In particular we propose a gedankenexperiment based on our results that gives evidence for certain instances of AdS/CFT having the AdS dictionary be computationally intractable to compute, unless $\mathrm{LWE}$ is tractable. A similar result was obtained by Bouland et al [BFV19], in the context of the wormhole growth paradox. Their result also uses cryptographic techniques in the form of pseudorandom quantum states. In contrast to our setting, they only require that such states exist and are computationally indistinguishable, whereas we are using the more fine-grained $\mathrm{LWE}$ assumption. Roughly speaking, the main idea here is that the Ryu-Takayanagi formula relates a quantity that we have shown is hard to compute even for shallow circuits (the entropy), to a quantity that seemingly can be efficiently computed, the area of a surface. Thus assuming that $\mathrm{LWE}$ is hard for polynomial time quantum computers, we arrive at a contradiction. A potential resolution is that the AdS dictionary does not efficiently translate from the bulk to the boundary. In the following thought experiment, we will assume that CFT states can be prepared efficiently starting from descriptions of functions $f$ and $g$, such as the ETCF functions used in Theorem 8, that are $1$-to-$1$ and $2$-to-$1$, respectively. Furthermore, it should be the case that there is a constant difference in entanglement entropy for the two types of states242424As alluded to in the proof of Theorem 8, we can in fact make the entropy difference be any constant, $k$, by taking $g$ to be a $2^{k}$-to-$1$ function (and changing $f$ appropriately, though keeping it a $1$-to-$1$ function).. To give arguments for why we think this is true, first note that, as stated in Theorem 4, we can construct local Hamiltonians for which the ground states will indeed encode instances of such functions. These ground states will have different entanglement entropy depending on which function was used. A second argument is based on the observation that certain quantum error- correcting codes serve as toy models for the AdS/CFT correspondence [PYHP15]. Specifically, as discussed in [Har17], codes that protect against erasure errors constitute such toy models. They satisfy the property that encoded information can be recovered by acting on only a fraction of the qubits in the encoded state. As an example of this (taken from [Har17]), if we let $\ket{\tilde{\psi}}_{123}$ be an encoding of $\ket{\psi}$ on three subsystems, it should be that there exists a unitary $U_{12}$ such that $\ket{\tilde{\psi}}=U_{12}[\ket{\psi}_{1}\otimes\ket{\chi}_{23}]$ (50) as well as corresponding unitaries $U_{13}$ and $U_{23}$ that act in a similar way. Here $\ket{\chi}$ is the maximally entangled state $\ket{\chi}=\sum_{i}\ket{i}\ket{i}$. As a toy model for AdS/CFT, the state $\ket{\psi}$ represents the quantum state of a bulk observer and $\ket{\tilde{\psi}}$ will be the corresponding CFT state that lives on the boundary of the AdS space. The indices label three different subsystems on the boundary and Equation 50 simply states that the bulk information (the state $\ket{\psi}$) can be recovered by acting on only part of the boundary. As shown in [Har17], these states satisfy a Ryu-Takayanagi formula (in addition to other properties that are satisfied by AdS/CFT). Specifically, the entanglement entropy of $\ket{\chi}$ corresponds to the area of a bulk surface. One could imagine considering the states $\ket{\chi_{f}}=\sum_{x}\ket{x}\ket{f(x)}\quad\quad\quad\ket{\chi_{g}}=\sum_{x}\ket{x}\ket{g(x)}$ (51) instead of $\ket{\chi}$, where $f$ and $g$ are $1$-to-$1$ and $2$-to-$1$, respectively. In this case the difference in entanglement entropy will be determined by whether the function that was used was $1$-to-$1$ or $2$-to-$1$. States such as the ones from Equation 51, or even analogous weighted superpositions of such states are efficiently preparable (according to the efficient range superposition properties from Definitions 4 and 5). Finally, note that since $f$ and $g$ themselves can be implemented by circuits of constant depth, as shown in Theorem 9, the quantum states derived from these functions (such as $\ket{\chi_{f}}$ or $\ket{\chi_{g}}$) could also be prepared by short depth quantum circuits. This would be consistent with a conjecture by Swingle that the underlying CFT states that lead to the Ryu- Takayanagi formula are well approximated by states resulting from MERA (_multi-scale entanglement renormalization ansatz_) tensor networks [Swi12]. Such MERA states essentially have log-depth quantum circuit descriptions. Let us now describe our thought experiment. Suppose Alice has a quantum computer and is given the description of a function, denoted $h$, which is promised to be either $f$ (that is a $1$-to-$1$) or $g$ (that is a $2$-to-$1$) from Equation 29. Alice is then asked whether the function she received is $1$-to-$1$ or $2$-to-$1$. By the injective invariance property (Definition 5) this is as hard to determine as solving $\mathrm{LWE}$. Suppose now that Alice uses her quantum computer to do the following: 1. 1. She first prepares a state $\ket{\psi^{h}_{CFT}}$ that is supposed to represent a CFT state whose entanglement entropy is determined by the type of function of $h$. In other words, if $h$ is $f$, we will say that the state has high entanglement entropy and if $h$ is $g$ we will say it has low entanglement entropy. As discussed above, we conjecture that there should exist an efficient procedure for preparing such a state, given the function description. 2. 2. By the AdS/CFT correspondence, $\ket{\psi^{h}_{CFT}}$ should be dual to a state $\ket{\psi^{h}_{bulk}}$ in the bulk. In this bulk space-time, under the Ryu-Takayanagi formula the area of a certain surface $\gamma_{h}$ will be equal252525Or be approximately equal. to the entanglement entropy of $\ket{\psi^{h}_{CFT}}$. Using the AdS dictionary, Alice then considers a bulk Hamiltonian $H_{bulk}$ such that the time evolution of $\ket{\psi^{h}_{bulk}}$ under $H_{bulk}$ corresponds to an observer in the bulk measuring the area of $\gamma_{h}$. If this fictional bulk observer notices that the area of $\gamma_{h}$ is above a certain threshold (corresponding to the case of high entropy), it will “reset itself” so that at the end of the evolution it returns to the state $\ket{\psi^{h}_{bulk}}$ (and so the corresponding CFT state returns to $\ket{\psi^{h}_{CFT}}$). If, on the other hand the area is below the threshold (corresponding to low entropy) it should then map itself into a state for which the dual CFT state is “as close to orthogonal to $\ket{\psi^{h}_{CFT}}$ as possible”. In other words, we would like this state to be distinguishable from $\ket{\psi^{h}_{CFT}}$. A schematic illustration of the fictional bulk observer’s two perspectives is shown in Figure 3. The time required for the observer to perform the measurement should be proportional to the area262626Note that since we don’t know the entanglement entropy of $\ket{\psi^{h}_{CFT}}$ in advance, we can take the time we evolve by $H_{bulk}$ to be the longest of the two choices, corresponding to the case of maximum entropy. of $\gamma_{h}$. 3. 3. Using the AdS dictionary, Alice computes the boundary CFT Hamiltonian $H_{CFT}$ that is dual to $H_{bulk}$. She then time-evolves her state $\ket{\psi^{h}_{CFT}}$ with $H_{CFT}$. Under the AdS/CFT correspondence the evolution of her state will be dual to the time-evolution of the bulk observer that is performing the measurement of $\gamma_{h}$. Alice is, in effect, simulating this process on her quantum computer. 4. 4. At the end of the evolution, Alice performs SWAP tests to check whether the state she is left with is $\ket{\psi^{h}_{CFT}}$. If this is the case, she concludes that the original function was $1$-to-$1$, otherwise she concludes that it is $2$-to-$1$. Figure 3: The two situations for the bulk observer. The observer should measure the area of the surface $\gamma$ to determine which situation it is in. If all the steps in the above procedure can be performed efficiently, then this experiment would violate the injective invariance property of ETCF functions, since it can efficiently distinguish between the two function types. Correspondingly, we would have an efficient quantum algorithm for solving $\mathrm{LWE}$. Since we believe that this is unlikely, we would need to determine which of the above steps is intractable. As mentioned, we conjecture that preparing the CFT states should be efficient. The time- evolution under a Hamiltonian, that Alice has to perform, should also be efficient using standard techniques from quantum simulation [BMK10, BCK15]. One step that seems potentially problematic is step 3. Here the bulk observer needs to affect his space-time so as to map Alice’s state to one of two efficiently distinguishable states. It certainly seems plausible that the bulk observer can do very different things depending on the area he measures, resulting in completely different bulk states. But it is unclear whether the resulting dual CFT states would then be distinguishable by Alice. The other possible source of intractability is the use of the AdS dictionary. If the dictionary is exponentially complex, Alice cannot determine the state that is dual to her CFT state or what her boundary Hamiltonian should be. An important observation to make here is that the entropy difference between the two cases is constant and one could argue that we should not expect bulk observers to be able to efficiently detect such small differences in geometry. Indeed, it might be more relevant to have a scenario in which the _entropy ratio_ is constant instead, since this would correspond to a noticeable change in area between the two cases (and would be more in line with the portrayal in Figure 3). As mentioned in the discussion from Section 1, we conjecture that even estimating the entropy ratio should be hard based on $\mathrm{LWE}$. Specifically, by considering extensions of the ETCF functions in which the function $g$ is taken to be $2^{m}$-to-$1$, with $m=poly(n+k)$, we would achieve a constant (or even polynomial) ratio between the entropies of the two functions. The results from the previous sections would still allow for these functions to be evaluated in constant depth, leading the desired result. A final comment we make about the above experiment is that it does not require holographic duality to be true for our own universe. 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IEEE Transactions on Information Theory, 62(6):3702–3720, June 2016\. * [zoo] Complexity Zoo. https://complexityzoo.uwaterloo.ca/Complexity_Zoo. ## Appendix A An unconditional lower bound on entropy estimation A question we can ask concerning entropy estimation is whether the entropy of a quantum state, $\rho$, on $n$-qubits can be estimated up to additive error $\epsilon$ in time (and with a number of copies of $\rho$) that scales as $poly(n,\log(1/\epsilon))$. While it has already been shown in [AISW19, SH19] that the answer is no, here we give a complexity-theoretic proof of this fact: ###### Theorem 11. There is no quantum algorithm running in time $poly(n,\log(1/\epsilon))$ for estimating the entropy of an $n$-qubit quantum state $\rho$ to within additive error $\epsilon>0$. ###### Proof. We prove this result by contradiction. Suppose that a $poly(n,\log(1/\epsilon))$ quantum algorithm for entropy estimation existed. We will argue that this implies $\mathsf{BQP/qpoly}=\mathsf{ALL}$, where $\mathsf{BQP/qpoly}$ denotes the set of languages that can be decided by a polynomial-time quantum algorithm with quantum advice, and $\mathsf{ALL}$ denotes the set of all languages [zoo]. A $\mathsf{BQP/qpoly}$ algorithm is a quantum algorithm that runs in polynomial time and that receives, in addition to its input, denoted $x$, a quantum state on $poly(|x|)$ qubits (the advice state) that depends only on the size of the input and not on the input itself. We leverage this fact, together with the ability to efficiently estimate entropy to provide an algorithm for deciding any language. For a given language $L\subseteq\\{0,1\\}^{*}$ and input length $n$, let $L_{n}$ denote the set of strings $x$ of length $n$, such that $x\in L$. We also let $\bar{L}_{n}$ denote the strings of length $n$ not contained in $L$, i.e. $\bar{L}_{n}=\\{0,1\\}^{n}\setminus L_{n}$. With this notation, we define the states $\ket{\psi_{n}}_{Yes}=\frac{1}{\sqrt{L_{n}}}\sum_{x\in L_{n}}\ket{x}\quad\quad\quad\ket{\psi_{n}}_{No}=\frac{1}{\sqrt{\bar{L}_{n}}}\sum_{x\in\bar{L}_{n}}\ket{x}$ (52) to be the equal superpositions over the “yes” instances of length $n$ and the “no” instances, respectively. Finally, we let $EQ_{x}$ be the following unitary operation acting on $n+1$ qubits (for $b\in\\{0,1\\}$): $EQ_{x}\ket{y}\ket{b}=\left\\{\begin{array}[]{ll}\ket{y}\ket{b\oplus 1}&\text{ if }x=y\\\ \ket{y}\ket{b}&\text{ otherwise }\\\ \end{array}\right.$ (53) Note that $EQ_{x}$ can be implemented with $poly(|x|)$-many gates. The $\mathsf{BQP/qpoly}$ algorithm works as follows. Setting $\epsilon=2^{-n-1}$, the quantum advice will consist of $poly(n)$ copies of $\ket{\psi_{n}}_{Yes}\ket{\psi_{n}}_{No}$. The algorithm then appends two qubits in the state $\ket{0}$ to each copy $\ket{\psi_{n}}_{Yes}\ket{\psi_{n}}_{No}$ to get $\ket{\psi_{n}}_{Yes}\ket{0}\ket{\psi_{n}}_{No}\ket{0}$. Then for the input $x$, the algorithm applies $EQ_{x}$ to both $\ket{\psi_{n}}_{Yes}\ket{0}$ and $\ket{\psi_{n}}_{No}\ket{0}$ individually, for all copies. Consider what happens if $x$ is a “yes” instance (the “no” instance case is analogous). The resulting states will be $EQ_{x}\ket{\psi_{n}}_{Yes}\ket{0}=\frac{1}{\sqrt{L_{n}}}\sum_{z\in L_{n},z\neq x}\ket{z}\ket{0}+\frac{1}{\sqrt{\bar{L}_{n}}}\ket{x}\ket{1}$ (54) $EQ_{x}\ket{\psi_{n}}_{No}\ket{0}=\ket{\psi_{n}}_{No}\ket{0}$ (55) If we trace out the first $n$ qubits and denote the resulting states as $\rho_{Y}$ and $\rho_{N}$, we can see that $S(\rho_{Y})=-\frac{L_{n}-1}{L_{n}}\log\left(\frac{L_{n}-1}{L_{n}}\right)-\frac{1}{L_{n}}\log\left(\frac{1}{L_{n}}\right)$ (56) $S(\rho_{N})=0$ (57) Since $L_{n}\leq 2^{n}$, we have that $S(\rho_{Y})\geq 2^{-n}$. But now, by assumption, having $poly(n)$-many copies of $\rho_{Y}$ and $\rho_{N}$ we can estimate the entropies of the two states to within additive error $2^{-n-1}$, thus being able to determine which of the two has non-zero entropy. The algorithm is therefore able to decide any language $L$, hence showing that $\mathsf{BQP/qpoly}=\mathsf{ALL}$. However, we know from [NY04] that $\mathsf{BQP/qpoly}\neq\mathsf{ALL}$ (since, in particular $\mathsf{EESPACE}\not\subset\mathsf{BQP/qpoly}$) and this provides the desired contradiction. ∎ ## Appendix B HQED with constant depth ground state In the reduction from Theorem 10 we used the history state construction to _approximately_ map the outputs of circuits $C_{1}$ and $C_{2}$ from an instance of QED to ground states of Hamiltonians $H_{1}$ and $H_{2}$ in HQED. Correspondingly, the entanglement entropy difference for the ground states of $H_{1}$ and $H_{2}$ differed from that of the output states of $C_{1}$ and $C_{2}$ by an additive term of $1/poly(n+k)$. Here we give an alternate reduction for the case where $C_{1}$ and $C_{2}$ are of constant depth $d$, based on a recent result of Bravyi, Gosset and Movassagh [BGM19]. This reduction has the appealing feature that the resulting Hamiltonians will have as ground states _exactly_ $C_{1}\ket{00...0}$ and $C_{2}\ket{00...0}$, rather than approximate versions of these states. This means that the entanglement entropy difference for the ground states of $H_{1}$ and $H_{2}$ will be identical to that of the states produced by $C_{1}$ and $C_{2}$. ###### Lemma 3. There exists a reduction QED${}_{\textrm{O(1)}}$ $\leq_{P}$ HQED${}_{\textrm{O(1)}}$ that exactly preserves the entropy difference. ###### Proof. Assuming, as before, that the circuits acting on $n+k$ qubits have the form $C_{j}=U_{d}U_{d-1}...U_{1}$, $j\in\\{1,2\\}$, where each $U_{i}$ is a quantum gate and $d$ is constant, we use the Hamiltonians considered in [BGM19]: $H_{j}=\sum_{i=1}^{n+k}C_{j}\ket{1}\bra{1}_{i}C_{j}^{\dagger}$ (58) where $\ket{1}\bra{1}_{i}$ acts non-trivially only on the $i$’th qubit. Since $C_{j}$ is a circuit of depth $d$, the locality of each term is $2^{d}$. Because $d$ is constant, the resulting Hamiltonian is local. As shown in [BGM19], the unique ground state of $H_{j}$ is $C_{j}\ket{00...0}$. Thus, the entanglement entropy difference for $H_{1}$ and $H_{2}$ is given by the entanglement entropy difference of $C_{1}\ket{00...0}$ and $C_{2}\ket{00...0}$, concluding the proof. ∎

2024-09-04T02:54:55.577533

2002.12836

arxiv-papers

A Fock Model and the Segal–Bargmann Transform for $\mathfrak{osp}(m,2|2n)$ A Fock Model and the Segal–Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra $\boldsymbol{\mathfrak{osp}(m,2|2n)}$ Sigiswald BARBIER, Sam CLAEREBOUT and Hendrik DE BIE S. Barbier, S. Claerebout and H. De Bie Department of Electronics and Information Systems, Faculty of Engineering and Architecture, Ghent University, Krijgslaan 281, 9000 Gent, Belgium <EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS> Received March 17, 2020, in final form August 12, 2020; Published online August 26, 2020 The minimal representation of a semisimple Lie group is a ‘small’ infinite- dimensional irreducible unitary representation. It is thought to correspond to the minimal nilpotent coadjoint orbit in Kirillov’s orbit philosophy. The Segal–Bargmann transform is an intertwining integral transformation between two different models of the minimal representation for Hermitian Lie groups of tube type. In this paper we construct a Fock model for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$. We also construct an integral transform which intertwines the Schrödinger model for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$ with this new Fock model. Segal–Bargmann transform; Fock model; Schrödinger model; minimal representations; Lie superalgebras; spherical harmonics; Bessel–Fischer product 17B10; 17B60; 22E46; 58C50 ## 1 Introduction The classical Segal–Bargmann transform $\displaystyle\operatorname{SB}f(z):=\exp\left(-\tfrac{1}{2}z^{2}\right)\int_{{\mathbb{R}}^{m}}\exp(2x\cdot z)\exp\big{(}{-}x^{2}\big{)}f(x)\mathrm{d}x$ is a unitary isomorphism from the space of square integrable functions on $\mathbb{R}^{m}$ to the Fock space of entire functions on $\mathbb{C}^{m}$ which are square integrable which respect to the weight function $\exp\big{(}{-}|z|^{2}\big{)}$. The Segal–Bargmann transform is defined in such a way that it maps the creation (resp. anihilation) operators on the Schrödinger space to coordinate multiplication (resp. differentiation) on the Fock space. This implies in particular that the harmonic oscillator becomes the much simpler Euler operator on the Fock space [7]. The Segal–Bargmann transform can also be interpreted as an intertwining operator between two models of the metaplectic representation (also known as the Segal–Shale–Weil or oscillator representation) of the metaplectic group, a double cover of the symplectic group. See [14] for more on the classical Segal–Bargmann transform and the metaplectic representation. The (even part of the) metaplectic representation is a prominent example of a minimal representation. Another extensively studied example is the minimal representation of ${\rm O}(p,q)$ [21, 22, 23, 20]. The minimal representation of a semisimple Lie group is the irreducible unitary representation that according to the orbit method should correspond to the minimal nilpotent coadjoint orbit [15]. The Segal–Bargmann transform has been generalized to this setting of minimal representations. Namely for a Hermitian Lie group of tube type, there exists an explicit integral transform which intertwines the Schrödinger and Fock model of the minimal representation [18]. Lie superalgebras and Lie supergroups are generalisations of Lie algebras and Lie groups. They were introduced to mathematically describe supersymmetry. Their representation theory is an active field of study with still a lot of open questions, for instance a description of the unitary irreducible representations. Since most ingredients of the orbit method still exist in the super setting, it is believed that the orbit method should also in the super setting be a good tool for the study of irreducible representations [19, Chapter 6.3]. For example, the irreducible unitary representations of nilpotent Lie supergroups have been classified that way [25, 26]. An ambitious aim in this light would be to construct minimal representations and the intertwining Segal–Bargmann transform for Lie supergroups. Recently a first step in that direction has already been taken. Namely the construction of a minimal representation of the orthosymplectic Lie supergroup ${\rm OSp}(p,q|2n)$ was accomplished in [6]. In the bosonic case (i.e., $n=0$) this realisation corresponds to the Schrödinger model for ${\rm O}(p,q)$ of [20]. In this article we achieve two further goals. First we construct a Fock model for the minimal representation of the Lie superalgebra $\mathfrak{osp}(m,2|2n)$. We also define an integral transform which intertwines the Schrödinger model of the minimal representation of $\mathfrak{osp}(m,2|2n)$ with this Fock model. Note that only for $q=2$ the Lie group ${\rm O}(p,q)$ is Hermitian of tube type and thus only in that case do we have a Segal–Bargmann transform. For that reason we have only constructed a Segal–Bargmann transform in the super setting for $\mathfrak{osp}(m,2|2n)$. Our main results hold for $m-2n\geq 4$. This restriction comes from [6], where key properties of the integral we use in the definition of the Segal–Bargmann transform are only proven for the case $m-2n\geq 4$. We will work in this paper always on the algebraic level. So we will work with representations of the Lie superalgebra $\mathfrak{osp}(m,2|2n)$ instead of the Lie supergroup ${\rm OSp}(m,2|2n)$, and we will act on super-vector spaces defined using superpolynomials instead of using global sections of a supermanifold. This allows us to circumvent the delicate technicalities associated with supergroups and supermanifolds. Note that in the bosonic case, the spaces we work with are dense in certain Hilbert spaces. Using standard techniques one can then integrate the representation to group level and extend to the whole Hilbert space, see for example [17, Theorem 2.30] or [18, Theorem 2.17]. These techniques no longer work/exist in the super case. There does exist an abstract way to integrate a so-called admissible $(\mathfrak{g},K)$-module to group level [1] which was for example used in [6] to integrate the minimal representation of $\mathfrak{osp}(p,q|2n)$ to ${\rm OSp}(p,q|2n)$. However, this approach is not very concrete. Explicit examples such as the one constructed in this paper could help to develop such integration tools and to find the correct definitions in the super setting. For example our representations ought to be ‘unitary’ in some sense. A definition of unitary representations does exists in the supersetting [8, Definition 2]. However, a large class of Lie superalgebras, including $\mathfrak{osp}(p,q|2n)$, do not allow for any super unitary representation in this sense [25, Theorem 6.2.1]. This highly unsatisfactory situation has inspired the search for a new or extended definition of a unitary representation [13, 27]. At the moment it is still unclear what the right definition should be, but we believe that the construction of explicit examples which ought to be ‘unitary’ could be useful for this endeavour. ### 1.1 Structure of the paper We structure the paper as follows. In Section 2 we fix notations and introduce the spaces and algebras we will use throughout the paper. In Section 3 we recall the Schrödinger model of the minimal representation of $\mathfrak{osp}(m,2|2n)$ defined in [6]. We also introduce an integral which leads to an $\mathfrak{osp}(m,2|2n)$-invariant, non-degenerate, superhermitian form. The next three sections contain the main body of this paper. In Section 4 we construct the Fock space as a quotient of the space of superpolynomials. We then define the Bessel–Fischer product, which gives us a non-degenerate, superhermitian form on our Fock space (Propositions 4.7 and 4.12). In the bosonic case $(n=0)$, this Bessel–Fischer product is equivalent to the inner product coming from an integral on the Fock space [18, Proposition 2.6]. Since we do no longer have this integral in the super setting, we construct a direct proof for the superhermitian property. This seems new even in the bosonic case. We also show that our Fock space has a reproducing kernel (Theorem 4.11). In Section 5 we endow this Fock space with an $\mathfrak{osp}(m,2|2n)$-module structure leading to a Fock model of the minimal representation of $\mathfrak{osp}(m,2|2n)$. We prove that this is an irreducible representation and obtain a very explicit description (Theorem 5.3). In particular, we have complete branching rules for the subalgebras $\mathfrak{osp}(m|2n)$ and $\mathfrak{osp}(m-1|2n)$. In Section 6, we define an integral transform which maps the space of functions used in the Schrödinger realisation to the space of functions of the Fock realisation (Definition 6.1). We show that this integral is an intertwining isomorphism which preserves the superhermitian form (Theorems 6.3 and 6.6). We also give an explicit inverse (Definition 6.7). As an application we use the Segal–Bargmann transform to define generalised Hermite functions. In Appendix A we gather some definitions and results on special functions which are used throughout the paper. We have also put the technical and lengthy proof of Theorem 6.3 in Appendix B. ## 2 Preliminaries and notations In this paper Jordan and Lie algebras will be defined over complex numbers $\mathbb{C}$ if they have a $\mathbb{C}$ in subscript, otherwise they are defined over the field of real numbers $\mathbb{R}$. Function spaces will always be defined over the complex field $\mathbb{C}$. We use the convention $\mathbb{N}=\\{0,1,2,\ldots\\}$. A super-vector space is defined as a $\mathbb{Z}_{2}$-graded vector space, i.e., $V=V_{0}\oplus V_{1}$, with $V_{0}$ and $V_{1}$ vector spaces. An element $v$ of a super-vector space $V$ is called homogeneous if it belongs to $V_{i}$, $i\in\mathbb{Z}_{2}$. We call $i$ the parity of $v$ and denote it by $|v|$. An homogeneous element $v$ is even if $|v|=0$ and odd if $|v|=1$. When we use $|v|$ in a formula, we are considering homogeneous elements, with the implicit convention that the formula has to be extended linearly for arbitrary elements. We denote the super-vector space $V$ with $V_{0}=\mathbb{R}^{m}$ and $V_{1}=\mathbb{R}^{n}$ as $\mathbb{R}^{m|n}$. We will always assume $m\geq 2$. ### 2.1 Superpolynomials Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$. ###### Definition 2.1. The space of superpolynomials over $\mathbb{K}$ is defined as $\displaystyle\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}:=\mathcal{P}\big{(}\mathbb{K}^{m}\big{)}\otimes_{\mathbb{C}}\Lambda\big{(}\mathbb{K}^{2n}\big{)},$ where $\mathcal{P}\big{(}\mathbb{K}^{m}\big{)}$ denotes the space of complex- valued polynomials over the field $\mathbb{K}$ in $m$ variables and $\Lambda\big{(}\mathbb{K}^{2n}\big{)}$ denotes the Grassmann algebra in $2n$ variables. The variables of $\mathcal{P}(\mathbb{K}^{m})$ and $\Lambda\big{(}\mathbb{K}^{2n}\big{)}$ are called even and odd variables, respectively. They satisfy the commutation relations $\displaystyle x_{i}x_{j}=(-1)^{|i||j|}x_{j}x_{i},$ where $|i|:=|x_{i}|$, $i\in\\{0,1,\ldots,m+2n-1\\}$. Let $\langle{\cdot\,,\cdot}\rangle_{\beta}$ be a supersymmetric, non- degenerate, even bilinear form on $\mathbb{K}^{m|2n}$ with basis $\\{x_{i}\\}_{i=0}^{m+2n-1}$. We denote the matrix components by $\beta_{ij}:=\langle{x_{i},x_{j}}\rangle_{\beta}$ and denote the components of the inverse matrix by $\beta^{ij}$, i.e., $\beta^{ij}$ is defined such that $\sum_{j}\beta_{ij}\beta^{jk}=\delta_{ik}$. Set $x^{j}=\sum_{i}x_{i}\beta^{ij}$. The differential operator $\partial^{i}$ is defined as the unique derivation in $\operatorname{End}\big{(}\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}\big{)}$ such that $\partial^{i}(x_{j})=\delta_{ij}$, with $\delta_{ij}$ the Kronecker delta. We also define $\partial_{j}=\sum_{i}\partial^{i}\beta_{ji}$. Then it holds that $\partial_{i}(x^{j})=\delta_{ij}$. When we are working with both real and complex polynomials at the same time, we will denote $\partial_{i}$ and $\partial^{i}$ for the real variable $x_{i}$ as $\partial_{x^{i}}$ and $\partial_{x_{i}}$, respectively. Similarly, we will denote $\partial_{i}$ and $\partial^{i}$ for the complex variable $z_{i}$ as $\partial_{z^{i}}$ and $\partial_{z_{i}}$, respectively. We will make frequent use of the following operators: $\displaystyle R^{2}:=\sum_{i,j}\beta^{ij}x_{i}x_{j},\qquad\mathbb{E}:=\sum_{i,j}\beta^{ij}x_{i}\partial_{j}\qquad\mbox{and}\qquad\Delta:=\sum_{ij}\beta^{ij}\partial_{i}\partial_{j}.$ (2.1) Here, the operator $R^{2}$ is called the square of the radial coordinate and acts through multiplication. The operators $\mathbb{E}$ and $\Delta$ are called the Euler operator and the Laplacian, respectively. We have the following lemma. ###### Lemma 2.2. The operators $R^{2}$, $\mathbb{E}$ and $\Delta$ satisfy $\displaystyle[\Delta,R^{2}]=4\mathbb{E}+2M,\qquad[\Delta,\mathbb{E}]=2\Delta,\qquad[R^{2},\mathbb{E}]=-2R^{2},$ where $M=m-2n$ is the superdimension. In particular, $\big{(}R^{2},\mathbb{E}+\frac{M}{2},-\frac{\Delta}{2}\big{)}$ forms an $\mathfrak{sl}_{\mathbb{K}}(2)$-triple. Furthermore, they commute in $\operatorname{End}\big{(}\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}\big{)}$ with the operators $L_{ij}:=x_{i}\partial_{j}-(-1)^{|i||j|}x_{j}\partial_{i}.$ ###### Proof. A straightforward calculation or see, for example, [12]. ∎ If we are working with two sets of variables we will add a variable indicator to avoid confusion. For example, we denote $\displaystyle R^{2}_{x}=\sum_{i,j}\beta^{ij}x_{i}x_{j},\qquad\mathbb{E}_{x}=\sum_{i,j}\beta^{ij}x_{i}\partial_{x^{j}},\qquad\Delta_{x}=\sum_{ij}\beta^{ij}\partial_{x^{i}}\partial_{x^{j}}$ and $L_{ij}^{x}=x_{i}\partial_{x^{j}}-(-1)^{|i||j|}x_{j}\partial_{x^{i}}$ for the real variables and $\displaystyle R^{2}_{z}=\sum_{i,j}\beta^{ij}z_{i}z_{j},\qquad\mathbb{E}_{z}=\sum_{i,j}\beta^{ij}z_{i}\partial_{z^{j}},\qquad\Delta_{z}=\sum_{ij}\beta^{ij}\partial_{z^{i}}\partial_{z^{j}}$ and $L^{z}_{ij}=z_{i}\partial_{z^{j}}-(-1)^{|i||j|}z_{j}\partial_{z^{i}}$ for the complex variables. ### 2.2 The orthosymplectic Lie superalgebra Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$. ###### Definition 2.3. The orthosymplectic Lie superalgebra $\mathfrak{osp}_{\mathbb{K}}(m|2n,\beta)$ is defined as the subalgebra of $\mathfrak{gl}_{\mathbb{K}}(m|2n)$ preserving a supersymmetric non-degenerate even bilinear form $\beta$. Thus $\mathfrak{osp}_{\mathbb{K}}(m|2n,\beta)$ is spanned by $X\in\mathfrak{gl}_{\mathbb{K}}(m|2n)$ such that $\displaystyle\langle{X(u),v}\rangle_{\beta}+(-1)^{|u||X|}\langle{u,X(v)}\rangle_{\beta}=0,$ for all $u,v\in\mathbb{K}^{m|2n}$. The orthosymplectic Lie superalgebra has a differential operator realisation on $\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}$. A basis in this realisation is given by $\displaystyle L_{ij}:=x_{i}\partial_{j}-(-1)^{|i||j|}x_{j}\partial_{i},\qquad\mbox{for }i<j,$ $\displaystyle L_{ii}:=2x_{i}\partial_{i},\qquad\mbox{for }|i|=1.$ ### 2.3 Spherical harmonics Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$. The space of homogeneous superpolynomials of degree $k$ is denoted by $\displaystyle\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}:=\big{\\{}p\in\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}\colon\mathbb{E}p=kp\big{\\}}.$ The space of spherical harmonics of degree $k$ is defined by $\displaystyle\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}:=\big{\\{}f\in\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}\colon\Delta f=0\big{\\}},$ i.e., it is the space of homogeneous polynomials of degree $k$ which are in the kernel of the Laplacian. The Fischer decomposition gives a decomposition of the space of superpolynomials using these spherical harmonics [12, Theorem 3]. ###### Proposition 2.4. If $m-2n\neq-2\mathbb{N}$, then $\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}$ decomposes as $\displaystyle\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}=\bigoplus_{k=0}^{\infty}\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\bigoplus_{k=0}^{\infty}\bigoplus_{j=0}^{\infty}R^{2j}\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}.$ In [24] the following generalisation of the Fischer decomposition was obtained that still holds for the exceptional case $M\in-2\mathbb{N}$. ###### Proposition 2.5 (generalised Fischer decomposition). The superspace $\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}$ decomposes as $\displaystyle\mathcal{P}\big{(}\mathbb{K}^{m|2n}\big{)}=\bigoplus_{k=0}^{\infty}\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\bigoplus_{k=0}^{\infty}\bigoplus_{j=0}^{\infty}\big{(}R^{2}\Delta R^{2}\big{)}^{j}\operatorname{\widetilde{\mathcal{H}}}_{k}\big{(}\mathbb{K}^{m|2n}\big{)},$ where $\displaystyle\operatorname{\widetilde{\mathcal{H}}}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\big{\\{}f\in\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}\colon\Delta R^{2}\Delta f=0\big{\\}}$ is the space of generalised spherical harmonics of degree $k$. From [9, Theorem 5.2] we obtain the following. ###### Proposition 2.6. If $M\not\in-2\mathbb{N}$, then $\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}$ is an irreducible $\mathfrak{osp}_{\mathbb{K}}(m|2n)$-module. The dimension of the spherical harmonics of degree $k$ is given in [12, Corollary 1]. ###### Proposition 2.7. The dimension of $\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}$, for $m\neq 0$ is given by $\displaystyle\dim\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\dim\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}-\dim\mathcal{P}_{k-2}\big{(}\mathbb{K}^{m|2n}\big{)},$ with $\displaystyle\dim\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\sum_{i=0}^{\min(k,2n)}\binom{2n}{i}\binom{k-i+m-1}{m-1}.$ We will also use the following formula for the dimension of the spherical harmonics of degree $k$. ###### Proposition 2.8. The dimension of $\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}$, for $m>1$ is given by $\displaystyle\dim\mathcal{H}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}=\dim\mathcal{P}_{k}\big{(}\mathbb{K}^{m-1|2n}\big{)}+\dim\mathcal{P}_{k-1}\big{(}\mathbb{K}^{m-1|2n}\big{)}.$ ###### Proof. If we prove $\displaystyle\dim\mathcal{P}_{k}\big{(}\mathbb{K}^{m|2n}\big{)}-\dim\mathcal{P}_{k-2}\big{(}\mathbb{K}^{m|2n}\big{)}-\dim\mathcal{P}_{k}\big{(}\mathbb{K}^{m-1|2n}\big{)}-\dim\mathcal{P}_{k-1}\big{(}\mathbb{K}^{m-1|2n}\big{)}=0,$ then the proposition follows from Proposition 2.7. First suppose $2n\leq k-2$, then the above equation becomes $\displaystyle\sum_{i=0}^{2n}\binom{2n}{i}\left(\binom{k-i+m-1}{m-1}-\binom{k-i+m-3}{m-1}\right.$ $\displaystyle\left.\qquad{}-\binom{k-i+m-2}{m-2}-\binom{k-i+m-3}{m-2}\right)=0,$ which is true since the recursive formula of binomial coefficients gives us that $\displaystyle\binom{k-i+m-1}{m-1}-\binom{k-i+m-3}{m-1}-\binom{k-i+m-2}{m-2}-\binom{k-i+m-3}{m-2}=0,$ for all $i\in\\{0,\ldots,2n\\}$. For $2n>k-2$ we have the following extra terms $\displaystyle\binom{2n}{k-1}\left(\binom{m}{m-1}-\binom{m-1}{m-2}-\binom{m-2}{m-2}\right)+\binom{2n}{k}\left(\binom{m-1}{m-1}-\binom{m-2}{m-2}\right),$ where we ignore the last term if $2n=k-1$. Using basic binomial properties, these terms are clearly equal to zero. ∎ For $n=0$ a more insightful reasoning as to why this formula holds is given by Proposition 5 in [11]. ### 2.4 The spin factor Jordan superalgebra $\boldsymbol{J}$ To each Hermitian Lie group of tube type corresponds an Euclidean Jordan algebra. These Jordan algebras were a crucial ingredient in the unified approach to construct minimal representations and the Segal–Bargmann transform [17, 18]. More concretely, one can associate with each Jordan algebra certain Lie algebras (the structure algebra and the TKK-algebra), and the Jordan algebra is also used in the construction of spaces on which these Lie algebras act. In this paper we will not use anything directly from Jordan theory, but introducing $\mathfrak{osp}(m,2|2n)$ via the spin factor Jordan superalgebra leads to a natural decomposition of $\mathfrak{osp}(m,2|2n)$ as well as to some interesting subalgebras that we will use. ###### Definition 2.9. A Jordan superalgebra is a supercommutative superalgebra $J$ satisfying the Jordan identity $\displaystyle(-1)^{|x||z|}[L_{x},L_{yz}]+(-1)^{|y||x|}[L_{y},L_{zx}]+(-1)^{|z||y|}[L_{z},L_{xy}]=0\qquad\text{for all }x,y,z\in J.$ Here the operator $L_{x}$ is (left) multiplication with $x$ and $[\cdot\,,\cdot]$ is the supercommutator, i.e., $[L_{x},L_{y}]:=L_{x}L_{y}-(-1)^{|x||y|}L_{y}L_{x}$. Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$. We will define the spin factor Jordan superalgebra associated with a supersymmetric, non- degenerate, even, bilinear form. Let $V_{\mathbb{K}}$ be a super-vector space over $\mathbb{K}$ with $\dim(V_{\mathbb{K}})=(m-1|2n)$ and a supersymmetric, non-degenerate, even, bilinear form $\langle\cdot\;,\cdot\rangle_{\tilde{\beta}}$ where, for $\mathbb{K}=\mathbb{R}$, the even part has signature $(m-1,0)$. Recall that we always assume $m\geq 2$. We choose a homogeneous basis $(e_{i})_{i=1}^{m+2n-1}$ of $V_{\mathbb{K}}$. For $u=\sum_{i}u^{i}e_{i}$ and $v=\sum_{i}v^{i}e_{i}$ we then have $\displaystyle\langle u,v\rangle_{\tilde{\beta}}=\sum_{i,j}u^{i}{\tilde{\beta}}_{ij}v^{j}\qquad\mbox{with}\quad{\tilde{\beta}}_{ij}:=\langle e_{i},e_{j}\rangle_{\tilde{\beta}}.$ ###### Definition 2.10. The spin factor Jordan superalgebra is defined as $J_{\mathbb{K}}:=\mathbb{K}e_{0}\oplus V_{\mathbb{K}}$ with $|e_{0}|=0$. The Jordan product is given by $\displaystyle(\lambda e_{0}+u)(\mu e_{0}+v)=(\lambda\mu+\langle u,v\rangle_{\tilde{\beta}})e_{0}+\lambda v+\mu u,$ for $u,v\in V_{\mathbb{K}}$ and $\lambda,\mu\in\mathbb{K}$. Thus $e_{0}$ is the unit of $J_{\mathbb{K}}$. We extend the homogeneous basis $(e_{i})_{i=1}^{m+2n-1}$ of $V_{\mathbb{K}}$ to a homogeneous basis $(e_{i})_{i=0}^{m+2n-1}$ of $J_{\mathbb{K}}$ and extend the bilinear form $\langle\cdot\;,\cdot\rangle_{\tilde{\beta}}$ as follows. Set $\beta_{00}=-1$, $\beta_{i0}=0=\beta_{0i}$ and $\beta_{ij}={\tilde{\beta}}_{ij}$ for $i,j\in\\{1,\ldots,m+2n-1\\}$. Then the corresponding form $\langle\cdot\;,\cdot\rangle_{\beta}$ is a supersymmetric, non-degenerate, even bilinear form on the super-vector space $J_{\mathbb{K}}$ where, for $\mathbb{K}=\mathbb{R}$, the even part has signature $(m-1,1)$. Define $(\beta^{ij})_{ij}$ as the inverse of $(\beta_{ij})_{ij}$. Let $\big{(}e^{i}\big{)}_{i}$ be the right dual basis of $(e_{i})_{i}$ with respect to the form $\langle\cdot\;,\cdot\rangle_{\beta}$, i.e., $\displaystyle\langle e_{i}\;,e^{j}\rangle_{\beta}=\delta_{ij}.$ Then $\displaystyle e^{j}=\sum_{i}e_{i}\beta^{ij}.$ In this paper we will assume that the orthosymplectic metric is standardized such that $\displaystyle\beta=(\beta_{ij})_{i,j=0}^{m+2n-1}=\left(\begin{array}[]{cc|cc}-1&&&\\\ &I_{m-1}&&\\\ \hline\cr&&&-I_{n}\\\ &&I_{n}&\end{array}\right),$ with $I_{d}$ the $d$-dimensional identity matrix. From now on the real spin factor Jordan superalgebra will always be denoted by $J$ or $J_{\mathbb{R}}$ and its complexified version by $J_{\mathbb{C}}$. ### 2.5 The TKK algebra With each Jordan (super)algebra one can associate a $3$-graded Lie (super)algebra via the TKK-construction. There exist different TKK- constructions in the literature, see [5] for an overview, but for the spin factor Jordan superalgebra $J_{\mathbb{K}}$ all constructions lead to the orthosymplectic Lie superalgebras $\mathfrak{osp}_{\mathbb{K}}(m,2|2n)$. We will quickly review the Koecher construction. First consider $\operatorname{Inn}(J_{\mathbb{K}})$, the subalgebra of $\mathfrak{gl}(J_{\mathbb{K}})$ of inner derivations. It is generated by the operators $[L_{u},L_{v}]$, $u,v\in J_{\mathbb{K}}$. If we add the left multiplication operators $L_{u}$ to the inner derivations we obtain the inner structure algebra: $\displaystyle\mathfrak{istr}(J_{\mathbb{K}}):=\\{L_{u}|u\in J_{\mathbb{K}}\\}\oplus\operatorname{Inn}(J_{\mathbb{K}})=\operatorname{span}_{\mathbb{K}}\\{L_{u},[L_{u},L_{v}]\colon u,v\in J_{\mathbb{K}}\\}.$ Let $J_{\mathbb{K}}^{+}$ and $J_{\mathbb{K}}^{-}$ be two copies of $J_{\mathbb{K}}$. As a vector space we define the TKK-algebra of $J_{\mathbb{K}}$ as $\displaystyle\operatorname{TKK}(J_{\mathbb{K}}):=J_{\mathbb{K}}^{-}\oplus\mathfrak{istr}(J_{\mathbb{K}})\oplus J_{\mathbb{K}}^{+}.$ The Lie bracket on $\operatorname{TKK}(J_{\mathbb{K}})$ is defined as follows. We interpret $\mathfrak{istr}(J_{\mathbb{K}})$ as a subalgebra of $\mathfrak{gl}(J_{\mathbb{K}})$ and for homogeneous $x,y\in J_{\mathbb{K}}^{+}$, $u,v\in J_{\mathbb{K}}^{-}$, $a,b\in J_{\mathbb{K}}$ we set $\displaystyle[x,u]=2L_{xu}+2[L_{x},L_{u}],\qquad$ $\displaystyle[x,y]=[u,v]=0,$ $\displaystyle[L_{a},x]=ax,\qquad$ $\displaystyle[L_{a},u]=-au,$ $\displaystyle[[L_{a},L_{b}],x]=[L_{a},L_{b}]x,\qquad$ $\displaystyle[[L_{a},L_{b}],u]=[L_{a},L_{b}]u.$ From [6, Proposition 3.1] we obtain the following ###### Theorem 2.11. We have $\displaystyle\mathfrak{istr}(J_{\mathbb{K}})=\mathfrak{osp}_{\mathbb{K}}(J_{\mathbb{K}})\oplus\mathbb{K}L_{e_{0}}\cong\mathfrak{osp}_{\mathbb{K}}(m-1,1)\oplus\mathbb{K}L_{e_{0}},\qquad\operatorname{TKK}(J_{\mathbb{K}})\cong\mathfrak{osp}_{\mathbb{K}}(m,2|2n),$ where the direct sum decomposition is as algebras. Using the bilinear form $\displaystyle\overline{\beta}=\left(\begin{array}[]{cccc|c}1&&&&\\\ &(\beta_{ij})_{i,j=1}^{m-1}&&&\\\ &&\beta_{00}&&\\\ &&&-1&\\\ \hline\cr&&&&(\beta_{ij})_{i,j=m}^{m+2n-1}\end{array}\right),$ we have the following explicit isomorphism of $\operatorname{TKK}(J_{\mathbb{K}})$ with the differential operator realisation of $\mathfrak{osp}_{\mathbb{K}}(m,2|2n)$: $\displaystyle e_{i}^{-}\mapsto L_{\tilde{i},(m+1)}+L_{\tilde{i},0},\qquad$ $\displaystyle e_{0}^{-}\mapsto L_{m,(m+1)}+L_{m,0},$ $\displaystyle L_{e_{i}}\mapsto L_{\tilde{i},m},\qquad$ $\displaystyle L_{e_{0}}\mapsto L_{0,(m+1)},$ $\displaystyle[L_{e_{i}},L_{e_{j}}]\mapsto L_{\tilde{i},\tilde{j}},$ $\displaystyle e_{i}^{+}\mapsto L_{\tilde{i},(m+1)}-L_{\tilde{i},0},\qquad$ $\displaystyle e_{0}^{+}\mapsto- L_{m,(m+1)}+L_{m,0}.$ Here $\tilde{i}=i$ if $|i|=0$ and $\tilde{i}=i+2$ if $|i|=1$. ## 3 The Schrödinger model of $\boldsymbol{\mathfrak{osp}(m,2|2n)}$ In the bosonic case (i.e., $n=0$), the Schrödinger model of the minimal representation of a Hermitian Lie group $G$ of tube type can be seen as a representation realized on the Hilbert space $L^{2}(C)$ where $C$ is a minimal orbit for the structure group of $G$, see [18]. In the super setting the classical definition of minimal orbits no longer works. Indeed, supermanifolds, in contrast to ordinary manifolds, are not completely determined by their points. In [6] an orbit was instead defined as the quotient supermanifold of the structure group by a stabilizer subgroup together with an embedding. Using this definition, a minimal orbit $C$ was constructed which can be characterized by $R^{2}=0$, with $R^{2}$ as introduced in (2.1). We refer to Section 4 of [6] for a detailed description of this minimal orbit. We will now recall the Schrödinger model constructed in [6]. A critical role in this construction is played by the Bessel operators. ### 3.1 The Bessel operator The Bessel operator $\operatorname{\mathcal{B}}_{\lambda}(x_{k})$ is a linear operator acting on $\mathcal{P}\big{(}\mathbb{R}^{m|2n}\big{)}$. It depends on a complex parameter $\lambda$ and an explicit expression is given by $\displaystyle\operatorname{\mathcal{B}}_{\lambda}(x_{k}):=(-\lambda+2\mathbb{E})\partial_{k}-x_{k}\Delta,$ where $\mathbb{E}$ and $\Delta$ are the Euler operator and Laplacian introduced in (2.1). From Proposition 4.9 of [6] we obtain the following. ###### Proposition 3.1. The Bessel operators map $\langle R^{2}\rangle$ into $\big{\langle}R^{2}\big{\rangle}$ if and only if $\lambda=2-M$, where $M=m-2n$ is the superdimension of $\mathbb{R}^{m|2n}$ and $\big{\langle}R^{2}\big{\rangle}$ is the ideal in $\mathcal{P}\big{(}\mathbb{R}^{m|2n}\big{)}$ generated by $R^{2}$. Therefore we will only use the Bessel operator with the parameter $2-M$ in this paper and we set $\operatorname{\mathcal{B}}(x_{i}):=\operatorname{\mathcal{B}}_{2-M}(x_{i})$. We obtain the following two properties of the Bessel operator from Proposition 4.2 in [4]. ###### Proposition 3.2 (supercommutativity). For all $i\in\\{0,\ldots,m+2n-1\\}$ we have $\displaystyle\operatorname{\mathcal{B}}(x_{i})\operatorname{\mathcal{B}}(x_{j})=(-1)^{|i||j|}\operatorname{\mathcal{B}}(x_{j})\operatorname{\mathcal{B}}(x_{i}).$ ###### Proposition 3.3 (product rule). For all $i\in\\{0,\ldots,m+2n-1\\}$ we have $\displaystyle\operatorname{\mathcal{B}}(x_{i})(\phi\psi)=\operatorname{\mathcal{B}}(x_{i})(\phi)\psi+(-1)^{|i||\phi|}\phi\operatorname{\mathcal{B}}(x_{i})(\psi)$ $\displaystyle\hphantom{\operatorname{\mathcal{B}}(x_{i})(\phi\psi)=}{}+2(-1)^{|i||\phi|}\mathbb{E}(\phi)\partial_{i}(\psi)+2\partial_{i}(\phi)\mathbb{E}(\psi)-2x_{i}\sum_{r,s}(-1)^{|\phi||r|}\beta^{rs}\partial_{r}(\phi)\partial_{s}(\psi).$ As a direct result from the product rule we have $\displaystyle[\operatorname{\mathcal{B}}(x_{i}),x_{j}]=\beta_{ij}(M-2+2\mathbb{E})-2L_{ij},$ (3.1) for all $i,j\in\\{0,\ldots,m+2n-1\\}$. In what follows we will mostly use the following slightly modified version of the Bessel operator. ###### Definition 3.4. The modified Bessel operator $\operatorname{\widetilde{\mathcal{B}}}_{\lambda}(x_{k})$ is given by $\displaystyle\operatorname{\widetilde{\mathcal{B}}}(x_{0}):=-\operatorname{\mathcal{B}}(x_{0}),\qquad\operatorname{\widetilde{\mathcal{B}}}(x_{k}):=\operatorname{\mathcal{B}}(x_{k}),$ for $k\in\\{1,\ldots,m+2n-1\\}$. ### 3.2 Radial superfunctions Let us denote the supervariables $x_{m},\ldots,x_{m+2n-1}$ by $\theta_{1},\ldots,\theta_{2n}$. We keep the notations for $x_{0},\ldots,x_{m-1}$. Define $\theta^{I}$ as $\theta_{1}^{i_{1}}\theta_{2}^{i_{2}}\cdots\theta_{n}^{i_{2n}}$ for $I=(i_{1},i_{2},\dots,i_{2n})\in\mathbb{Z}_{2}^{2n}$. Consider a function $h\colon\mathbb{R}\rightarrow\mathbb{R}$, $h\in\mathcal{C}^{2n}(\mathbb{R}\setminus\\{0\\})$ and a superfunction $f=f_{0}+\sum\limits_{I\neq 0}f_{I}\theta^{I}$, with $f_{0},f_{I}\in\mathcal{C}^{\infty}\big{(}\mathbb{R}^{m}\big{)}$ and where $f_{0}$ has non-zero values. Then a new superfunction $\displaystyle h(f):=\sum_{j=0}^{2n}\dfrac{1}{j!}\left(\sum_{I\neq 0}f_{I}\theta^{I}\right)^{j}h^{(j)}(f_{0})$ (3.2) is defined in [10, Definition 3]. Here $h^{(j)}$ denotes the $j$th derivative of $h$. Set the supervariable $\displaystyle r^{2}:=\sum_{i=1}^{m+2n-1}x^{i}x_{i},$ so $R^{2}=-x_{0}^{2}+r^{2}$. We can use equation (3.2) with $f=\big{(}x_{0}^{2}+r^{2}\big{)}/2$ and $h$ the square root to define the superfunction $|X|$ as $\displaystyle|X|:=\sqrt{\dfrac{x_{0}^{2}+r^{2}}{2}}.$ Using equation (3.2) again, but now with $f=|X|$ we define $\exp(|X|)$ and $\Lambda_{2,j}^{\mu,\nu}(|X|)$ as radial superfunctions. Here $\Lambda_{2,j}^{\mu,\nu}$ is the generalised Laguerre function introduced in Appendix A.2. ### 3.3 The $\boldsymbol{(\mathfrak{g},\mathfrak{k})}$-module $\boldsymbol{W}$ We introduce the notations $\displaystyle\mathfrak{g}:=\operatorname{TKK}(J)\cong\mathfrak{osp}(m,2|2n),$ $\displaystyle\mathfrak{k}:=\\{(x,I,-x)\colon I\in\operatorname{Inn}(J),\,x\in J\\},$ $\displaystyle\mathfrak{k}_{0}:=\mathfrak{k}\cap\mathfrak{istr}(J)=\operatorname{Inn}(J).$ ###### Theorem 3.5. The following isomorphisms $\displaystyle\mathfrak{k}_{0}\cong\mathfrak{osp}(m-1,0|2n),\qquad\mathfrak{k}\cong\mathfrak{osp}(m,0|2n)\oplus\mathbb{R},$ hold as algebras. ###### Proof. This follows from a straightforward verification by, for example, looking at the matrix realisation of $\mathfrak{g}$. ∎ In Section 5.2 of [6] the so-called Schrödinger model of $\mathfrak{g}$ was constructed for $M\geq 4$. Since this model will play an important role in this paper, we will give a short recall of its construction. Firstly, we consider a representation $\pi$ of $\mathfrak{g}$ acting on smooth superfunctions on $J^{-}$. Explicitly, $\pi$ of $\mathfrak{g}=\operatorname{TKK}(J)=J^{-}\oplus\mathfrak{istr}(J)\oplus J^{+}$ is given as follows: * • $\pi(e_{l},0,0)=-2\imath x_{l}$, * • $\pi(0,L_{e_{k}},0)=-x_{0}\partial_{k}+x_{k}\partial_{0}$, * • $\pi(0,[L_{e_{i}},L_{e_{j}}],0)=x_{i}\partial_{j}-(-1)^{|i||j|}x_{j}\partial_{i}$, * • $\pi(0,L_{e_{0}},0)=\frac{2-M}{2}-\mathbb{E}$, * • $\pi(0,0,e_{l})=-\frac{1}{2}\imath\operatorname{\widetilde{\mathcal{B}}}(x_{l})$, with $i,j,k\in\\{1,2,\ldots,m+2n-1\\}$, $l\in\\{0,1,\ldots,m+2n-1\\}$ and where $\imath$ is the imaginary unit. For $n=0$, our convention corresponds to the Schrödinger realisation given in [18]. It only differs from the Schrödinger realisation given in [6] by a change of basis. It can be shown that all the operators occuring in $\pi$ are tangential to $R^{2}$, with $R^{2}$ as in equation (2.1). So we can consider the representation obtained by quotienting out $R^{2}$. This quotient representation has an interesting subrepresentation consisting of $\mathfrak{k}$-finite vectors. It is generated by $\exp(-2|X|)$, with $\exp(-2|X|)$ the smooth radial superfunction on $J^{-}$ as defined in Section 3.2. This subrepresentation wil be our Schrödinger model. So the Schrödinger model is given as follows: $\displaystyle W:=U(\mathfrak{g})\exp(-2|X|)\mod\big{\langle}R^{2}\big{\rangle},$ where the $\mathfrak{g}$-module structure is given by the Schrödinger representation $\pi$. Theorem 5.3 of [6] gives the following decomposition of $W$. ###### Theorem 3.6 (decomposition of $W$). Assume $M\geq 4$. * $(1)$ The decomposition of $W$ as a $\mathfrak{k}$-module is given by $\displaystyle W=\bigoplus_{j=0}^{\infty}W_{j},\qquad\mbox{with}\quad W_{j}=U(\mathfrak{k})\Lambda_{2,j}^{M-3,-1}(2|X|),$ where $W_{j}$ and thus also $W$ are $\mathfrak{k}$-finite. * $(2)$ $W$ is a simple $\mathfrak{g}$-module. * $(3)$ An explicit decomposition of $W_{j}$ into irreducible $\mathfrak{k}_{0}$-modules is given by $\displaystyle W_{j}=\bigoplus_{k=0}^{j}\bigoplus_{l=0}^{1}\Lambda_{2,j-k}^{M-3+2k,2l-1}(2|X|)\big{(}\mathcal{H}_{k}\big{(}\mathbb{R}^{m-1|2n}\big{)}\otimes\mathcal{H}_{l}(\mathbb{R})\big{)}.$ Furthermore, if $m$ is even we also have the following $\mathfrak{k}$-isomorphism $\displaystyle W_{j}\cong\mathcal{H}_{j}\big{(}\mathbb{R}^{m|2n}\big{)}\otimes\mathcal{H}_{\frac{M-2}{2}+j}\big{(}\mathbb{R}^{2}\big{)}.$ ### 3.4 The integral and sesquilinear form In Section 8 of [6] an integral which restricts to $W$ was constructed. We will use a renormalized version of that integral, restricted to $x_{0}>0$. To give the integral explicitly we consider spherical coordinates in $\mathbb{R}^{m-1}$ by setting $x_{i}=\omega_{i}s$ with $\omega_{i}\in\mathbb{S}^{m-2}$, $i\in\\{1,\ldots,m-1\\}$ and $\displaystyle s^{2}:=\sum_{i=1}^{m-1}x_{i}^{2}.$ We also introduce the following notations $\displaystyle\theta^{2}:=\sum_{i,j=m}^{m+2n-1}\beta^{ij}x_{i}x_{j},\qquad 1+\eta:=\sqrt{1-\dfrac{\theta^{2}}{2s^{2}}},\qquad 1+\xi:=\sqrt{1+\dfrac{\theta^{2}}{2x_{0}^{2}}}$ and the morphism $\displaystyle\phi^{\sharp}(f):=\sum_{j=0}^{n}\dfrac{\theta^{2j}}{j!}\left(\dfrac{1}{4x_{0}}\partial_{x_{0}}-\dfrac{1}{4s}\partial_{s}\right)^{j}(f),$ for $f\in\mathcal{C}^{\infty}\big{(}\mathbb{R}^{m}\setminus\\{0\\}\big{)}\otimes\Lambda\big{(}\mathbb{R}^{2n}\big{)}$. We remark that in [6, Lemma 8.2] it is shown that $\phi^{\sharp}$ is actually an algebra isomorphism. The Berezin integral on $\Lambda\big{(}\mathbb{R}^{2n}\big{)}$ is defined as $\displaystyle\int_{B}:=\partial^{m+2n-1}\partial^{m+2n-2}\cdots\partial^{m}.$ ###### Definition 3.7. Suppose $M\geq 4$. For $f\in W$ we define the integral $\int_{W}$ by $\displaystyle\int_{W}f:=\dfrac{1}{\gamma}\int_{0}^{\infty}\int_{\mathbb{S}^{m-2}}\int_{B}\rho^{m-3}(1+\eta)^{m-3}(1+\xi)^{-1}\phi^{\sharp}(f)_{|s=x_{0}=\rho}\mathrm{d}\rho\mathrm{d}\omega,$ (3.3) where $\gamma\in\mathbb{C}$ is the renormalisation factor such that $\int_{W}\exp(-4|X|)=1$. We can show that the integral is well defined modulo $\big{\langle}R^{2}\big{\rangle}$. This follows from Section 8 of [6] together with the fact that $\gamma$ is non-zero. ###### Proposition 3.8. For $M=m-2n\geq 4$ we have $\displaystyle\gamma=\dfrac{2^{5-2M}}{n!}\left(\dfrac{3-m}{2}\right)_{n}\dfrac{\pi^{\frac{m-1}{2}}}{\Gamma{\big{(}\frac{m-1}{2}\big{)}}}\Gamma(M-2),$ where we used the Pochhammer symbol $(a)_{k}=a(a+1)(a+2)\cdots(a+k-1)$. Note that $\big{(}\frac{3-m}{2}\big{)}_{n}=0$ implies $M=m-2n\leq 1$. Therefore $\gamma$ is non-zero. ###### Proof. Let us denote the integral $\int_{W}$ not normalized by $\gamma$ as $\int_{W^{\prime}}$, i.e., $\int_{W^{\prime}}=\gamma\int_{W}$. We wish to calculate $\displaystyle\gamma=\int_{W^{\prime}}\exp(-4|X|).$ In [3, Lemma 6.3.5] a similar integral is calculated if one observes that $\widetilde{K}_{-\frac{1}{2}}(t)=\frac{\sqrt{\pi}}{2}\exp(-t)$, (equation (A.1) in Appendix A.1) where $\widetilde{K}_{\alpha}$ is the K-Bessel function introduced in Appendix A.1. Using the same calculations as in the proof of [3, Lemma 6.3.5], we then obtain $\displaystyle\gamma=\dfrac{1}{n!}\left(\dfrac{3-m}{2}\right)_{n}\dfrac{\pi^{\frac{m-3}{2}}}{\Gamma{\big{(}\frac{m-1}{2}\big{)}}}\dfrac{\Gamma\big{(}\frac{M}{2}\big{)}\Gamma\big{(}\frac{M}{2}-1\big{)}\Gamma\big{(}\frac{M-1}{2}\big{)}^{2}}{\Gamma(M-1)},$ provided we take into account that we need extra factors $2$ in certain places and that we restrict ourselves to $x_{0}>0$. Note that in [3, Lemma 6.3.5] the tilde in $\widetilde{K}_{-\frac{1}{2}}$ sometimes mistakenly disappears. If we use Legendre’s duplication formula: $\displaystyle\Gamma\left(\dfrac{z+1}{2}\right)\Gamma\left(\dfrac{z}{2}\right)=2^{1-z}\sqrt{\pi}\Gamma(z),$ on $\Gamma\big{(}\frac{M}{2}\big{)}\Gamma\big{(}\frac{M-1}{2}\big{)}$ and $\Gamma\big{(}\frac{M-1}{2}\big{)}\Gamma\big{(}\frac{M}{2}-1\big{)}$ the result follows. ∎ ###### Definition 3.9. For $f,g\in W$ we define the sesquilinear form $\langle{\cdot\,,\cdot}\rangle_{W}$ as $\displaystyle\langle{f,g}\rangle_{W}:=\int_{W}f\overline{g}.$ Theorem 8.13 and Lemma 8.14 in [6] give us the following two properties. ###### Proposition 3.10. Suppose $M=m-2n\geq 4$. The Schrödinger representation $\pi$ on $W$ is skew- supersymmetric with respect to $\langle{\cdot\,,\cdot}\rangle_{W}$, i.e., $\displaystyle\langle{\pi(X)f,g}\rangle_{W}=-(-1)^{|X||f|}\langle{f,\pi(X)g}\rangle_{W},$ for all $X\in\mathfrak{g}$ and $f,g\in W$. ###### Proposition 3.11. Suppose $M=m-2n\geq 4$. The form $\langle{\cdot\,,\cdot}\rangle_{W}$ defines a sesquilinear, non-degenerate form on $W$, which is superhermitian, i.e., $\displaystyle\langle{f,g}\rangle_{W}=(-1)^{|f||g|}\overline{\langle{g,f}\rangle}_{W},$ for all $f,g\in W$. Note that for both Theorem 8.13 and Lemma 8.14 in [6] there is an extra condition saying that $M$ must be even. However, since we are working in the exceptional case that corresponds with $q=2$ in [6], the proofs still hold without this extra condition. ## 4 The Fock space In [18, Section 2.3] an inner product on the polynomial space $\mathcal{P}(\mathbb{C}^{m})$ was introduced, namely the Bessel–Fischer inner product $\displaystyle\langle{p,q}\rangle_{\mathcal{B}}:=p\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}\bar{q}(z)\big{|}_{z=0},$ where $\bar{q}(z)=\overline{q(\bar{z})}$ is obtained by conjugating the coefficients of the polynomial $q$. In [18, Proposition 2.6] it is proven that, for polynomials, the Bessel–Fischer inner product is equal to the $L^{2}$-inner product of the Fock space. Since there is no immediate extension of this $L^{2}$-inner product to the super setting, we will use the Bessel–Fischer inner product on polynomials as the starting point to generalize the Fock space to superspace. ### 4.1 Definition and properties ###### Definition 4.1. We define the polynomial Fock space as the superspace $\displaystyle\operatorname{\mathcal{F}}:=\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}/\big{\langle}R^{2}\big{\rangle}.$ From the generalised Fischer decomposition of Proposition 2.5 it follows that $\displaystyle\operatorname{\mathcal{F}}\cong\bigoplus_{l=0}^{\infty}\operatorname{\widetilde{\mathcal{H}}}_{l}\big{(}\mathbb{C}^{m|2n}\big{)}.$ In particular, if the superdimension $M=m-2n$ is such that $M\not\in-2\mathbb{N}$, then the Fischer decomposition from Proposition 2.4 gives us $\displaystyle\operatorname{\mathcal{F}}\cong\bigoplus_{l=0}^{\infty}\mathcal{H}_{l}\big{(}\mathbb{C}^{m|2n}\big{)}.$ ###### Definition 4.2. For $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ we define the Bessel–Fischer product of $p$ and $q$ as $\displaystyle\langle{p,q}\rangle_{\mathcal{B}}:=p\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}\bar{q}(z)\big{|}_{z=0},$ where $\bar{q}(z)=\overline{q(\bar{z})}$ is obtained by conjugating the coefficients of the polynomial $q$ and $\operatorname{\widetilde{\mathcal{B}}}$ is the complex version of the modified Bessel operators introduced in Definition 3.4. Explicitly for $p=\sum_{\alpha}a_{\alpha}z^{\alpha}$ and $q=\sum_{\beta}b_{\beta}z^{\beta}$ we have $\displaystyle\langle{p,q}\rangle_{\mathcal{B}}=\sum_{\alpha,\beta}a_{\alpha}\bar{b}_{\beta}(-1)^{\alpha_{0}}\operatorname{\mathcal{B}}(z_{0})^{\alpha_{0}}\cdots\operatorname{\mathcal{B}}(z_{m+2n-1})^{\alpha_{m+2n-1}}z_{0}^{\beta_{0}}\cdots z_{m+2n-1}^{\beta_{m+2n-1}}\big{|}_{z=0}.$ Note that it is only an inner product in the bosonic case. However, in [13] a new definition of Hilbert superspaces was introduced where the preserved form is no longer an inner product, but rather a non-degenerate, sesquilinear, superhermitian form. We will prove that the Bessel–Fischer product is such a form when restricted to $\operatorname{\mathcal{F}}$ with $M-2\not\in-2\mathbb{N}$. ###### Proposition 4.3 (sesquilinearity). For $p,q,r,s\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ and $\alpha,\beta,\gamma,\delta\in\mathbb{C}$ we have $\displaystyle\langle{\alpha p+\gamma r,\beta q+\delta s}\rangle_{\mathcal{B}}=\alpha\bar{\beta}\langle{p,q}\rangle_{\mathcal{B}}+\alpha\bar{\delta}\langle{p,s}\rangle_{\mathcal{B}}+\gamma\bar{\beta}\langle{r,q}\rangle_{\mathcal{B}}+\gamma\bar{\delta}\langle{r,s}\rangle_{\mathcal{B}}.$ ###### Proof. This follows from the linearity of the Bessel operators. ∎ ###### Proposition 4.4 (orthogonality). For $p_{k}\in\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$ and $p_{l}\in\mathcal{P}_{l}\big{(}\mathbb{C}^{m|2n}\big{)}$ with $l\neq k$ we have $\langle{p_{k},p_{l}}\rangle_{\mathcal{B}}=0$. ###### Proof. This follows from the fact that Bessel operators lower the degree of polynomials by one. ∎ ###### Proposition 4.5. For $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ and $i\in\\{0,1,\ldots,m+2n-1\\}$ we have $\displaystyle\langle{z_{i}p,q}\rangle_{\mathcal{B}}=(-1)^{|i||p|}\big{\langle}p,\operatorname{\widetilde{\mathcal{B}}}(z_{i})q\big{\rangle}_{\mathcal{B}}.$ ###### Proof. This follows immediately from the definition of the Bessel–Fischer product. ∎ To prove that the Bessel–Fischer product is superhermitian we will use induction on the degree of the polynomials. We will need the following lemma in the induction step of the proof. ###### Lemma 4.6. Suppose we have proven that $\langle{p,q}\rangle_{\mathcal{B}}=(-1)^{|p||q|}\overline{\langle{q,p}\rangle}_{\mathcal{B}}$ for all $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ of degree lower than or equal to $k\in\mathbb{N}$. Then for $p,q\in\mathcal{P}_{l}\big{(}\mathbb{C}^{m|2n}\big{)}$, $l\leq k$ we have $\displaystyle\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=-(-1)^{(|i|+|j|)|p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}\qquad\text{and}\qquad\langle{L_{0i}p,q}\rangle_{\mathcal{B}}=(-1)^{|i||p|}\langle{p,L_{0i}q}\rangle_{\mathcal{B}},$ for all $i,j\in\\{1,\ldots,m+2n-1\\}$. ###### Proof. We will prove the $L_{ij}$ case explicitly. The $L_{0i}$ case is entirely analogous. First note that if we combine the given superhermitian property with Proposition 4.5, we get $\displaystyle\big{\langle}\operatorname{\widetilde{\mathcal{B}}}(z_{i})p,q\big{\rangle}_{\mathcal{B}}=(-1)^{|i||p|}\langle{p,z_{i}q}\rangle_{\mathcal{B}}.$ for all $p\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ of degree $k$ or lower and all $q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ of degree $k-1$ or lower. Assume $p,q\in\mathcal{P}_{l}\big{(}\mathbb{C}^{m|2n}\big{)}$, $l\leq k$. We obtain $\displaystyle\langle{z_{i}\partial_{j}p,q}\rangle_{\mathcal{B}}=(-1)^{|i||p|}\langle{\partial_{j}p,\operatorname{\widetilde{\mathcal{B}}}(z_{i})q}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{z_{i}\partial_{j}p,q}\rangle_{\mathcal{B}}}{}=(-1)^{|i|(|p|+|j|)}\big{(}(M-2+2(l-1))\langle{\partial_{j}p,\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\partial_{j}p,z_{i}\Delta q}\rangle_{\mathcal{B}}\big{)}$ $\displaystyle\hphantom{\langle{z_{i}\partial_{j}p,q}\rangle_{\mathcal{B}}}{}=(-1)^{|i|(|p|+|j|)}\big{(}\langle{\operatorname{\widetilde{\mathcal{B}}}(z_{j})p,\partial_{i}q}\rangle_{\mathcal{B}}+\langle{z_{j}\Delta p,\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\partial_{j}p,z_{i}\Delta q}\rangle_{\mathcal{B}}\big{)}$ $\displaystyle\hphantom{\langle{z_{i}\partial_{j}p,q}\rangle_{\mathcal{B}}}{}=(-1)^{(|i|+|j|)|p|+|i||j|}\langle{p,z_{j}\partial_{i}q}\rangle_{\mathcal{B}}+(-1)^{|i|(|p|+|j|)}\big{(}\langle{z_{j}\Delta p,\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\partial_{j}p,z_{i}\Delta q}\rangle_{\mathcal{B}}\big{)}.$ This gives $\displaystyle\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=\langle{z_{i}\partial_{j}p,q}\rangle_{\mathcal{B}}-(-1)^{|i||j|}\langle{z_{j}\partial_{i}p,q}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{L_{ij}p,q}\rangle_{\mathcal{B}}}{}=(-1)^{(|i|+|j|)|p|+|i||j|}\langle{p,z_{j}\partial_{i}q}\rangle_{\mathcal{B}}-(-1)^{(|i|+|j|)|p|}\langle{p,z_{i}\partial_{j}q}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=}{}+(-1)^{|i|(|p|+|j|)}\big{(}\langle{z_{j}\Delta p,\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\partial_{j}p,z_{i}\Delta q}\rangle_{\mathcal{B}}\big{)}$ $\displaystyle\hphantom{\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=}{}-(-1)^{|j||p|}\big{(}\langle{z_{i}\Delta p,\partial_{j}q}\rangle_{\mathcal{B}}-\langle{\partial_{i}p,z_{j}\Delta q}\rangle_{\mathcal{B}}\big{)}$ $\displaystyle\hphantom{\langle{L_{ij}p,q}\rangle_{\mathcal{B}}}{}=-(-1)^{(|i|+|j|)|p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}+(-1)^{|i|(|p|+|j|)}\big{(}\langle{z_{j}\Delta p,\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\partial_{j}p,z_{i}\Delta q}\rangle_{\mathcal{B}}\big{)}$ $\displaystyle\hphantom{\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=}{}-(-1)^{|j||p|}\big{(}\langle{z_{i}\Delta p,\partial_{j}q}\rangle_{\mathcal{B}}-\langle{\partial_{i}p,z_{j}\Delta q}\rangle_{\mathcal{B}}\big{)},$ from which the desired result follows if we prove $\displaystyle 0=(-1)^{|i|(|p|+|j|)}\langle{z_{j}\Delta p,\partial_{i}q}\rangle_{\mathcal{B}}-(-1)^{|i|(|p|+|j|)}\langle{\partial_{j}p,z_{i}\Delta q}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{0=}{}-(-1)^{|j||p|}\langle{z_{i}\Delta p,\partial_{j}q}\rangle_{\mathcal{B}}+(-1)^{|j||p|}\langle{\partial_{i}p,z_{j}\Delta q}\rangle_{\mathcal{B}}.$ (4.1) For the first term in right hand side of this equation we have $\displaystyle\langle{z_{j}\Delta p,\partial_{i}q}\rangle_{\mathcal{B}}=(-1)^{|p||j|}\big{\langle}\Delta p,\operatorname{\widetilde{\mathcal{B}}}(z_{j})\partial_{i}q\big{\rangle}_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{z_{j}\Delta p,\partial_{i}q}\rangle_{\mathcal{B}}}{}=(-1)^{|p||j|}\big{(}(M-2+2(l-1))\langle{\Delta p,\partial_{j}\partial_{i}q}\rangle_{\mathcal{B}}-\langle{\Delta p,z_{j}\partial_{i}\Delta q}\rangle_{\mathcal{B}}\big{)},$ such that, using similar calculations for the other three terms, equation (4.1) can be rewritten as $\displaystyle\langle{L_{ij}\Delta p,\Delta q}\rangle_{\mathcal{B}}=-(-1)^{(|i|+|j|)|p|}\langle{\Delta p,L_{ij}\Delta q}\rangle_{\mathcal{B}}.$ Since $\Delta p$ and $\Delta q$ are polynomials of degree lower than $l$ the lemma follows from a straightforward induction argument on $l$. ∎ ###### Proposition 4.7 (superhermitianity). The Bessel–Fischer product is superhermitian, i.e., $\displaystyle\langle{p,q}\rangle_{\mathcal{B}}=(-1)^{|p||q|}\overline{\langle{q,p}\rangle}_{\mathcal{B}},$ for $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$. ###### Proof. Because of the orthogonality we only need to prove the property for $p$, $q$ homogeneous polynomials of the same degree. Because of the sesquilinearity we may assume that $p$ and $q$ are monomials. We will use induction on the degree $k$ of the polynomials, the case $k=0$ being trivial. Suppose we have proven the theorem for all $p,q\in\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$. We now look at $\langle{z_{i}p,z_{j}q}\rangle_{\mathcal{B}}$ for arbitrary $i,j\in\\{0,1,\ldots,m+2n-1\\}$. To simplify the following calculations we will restrict ourselves to $i,j\neq 0$, the cases $i=0$ or $j=0$ being similar. Denote $c=(M-2+2k)$. Using the commutation of equation (3.1) and Proposition 4.5 we find $\displaystyle\langle{z_{i}p,z_{j}q}\rangle_{\mathcal{B}}$ $\displaystyle=(-1)^{|i||p|}\langle{p,\operatorname{\mathcal{B}}(z_{i})z_{j}q}\rangle_{\mathcal{B}}$ $\displaystyle=(-1)^{|i||p|+|i||j|}\langle{p,z_{j}\operatorname{\mathcal{B}}(z_{i})q}\rangle_{\mathcal{B}}+(-1)^{|i||p|}\langle{p,[\operatorname{\mathcal{B}}(z_{i}),z_{j}]q}\rangle_{\mathcal{B}}$ $\displaystyle=(-1)^{|i||p|+|i||j|}\langle{p,z_{j}\operatorname{\mathcal{B}}(z_{i})q}\rangle_{\mathcal{B}}+(-1)^{|i||p|}c\beta_{ij}\langle{p,q}\rangle_{\mathcal{B}}-2(-1)^{|i||p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}.$ Using the induction hypothesis together with Proposition 4.5 this becomes $\displaystyle\langle{z_{i}p,z_{j}q}\rangle_{\mathcal{B}}=(-1)^{(|i|+|j|)|p|+|i||j|}\langle{\operatorname{\mathcal{B}}(z_{j})p,\operatorname{\mathcal{B}}(z_{i})q}\rangle_{\mathcal{B}}+(-1)^{|i||p|}c\beta_{ij}\langle{p,q}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{z_{i}p,z_{j}q}\rangle_{\mathcal{B}}=}{}-2(-1)^{|i||p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}.$ Switching the roles of $z_{i}p$ and $z_{j}q$ we also obtain $\displaystyle\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}=(-1)^{(|i|+|j|)|q|+|i||j|}\langle{\operatorname{\mathcal{B}}(z_{i})q,\operatorname{\mathcal{B}}(z_{j})p}\rangle_{\mathcal{B}}+(-1)^{|j||q|}c\beta_{ji}\langle{q,p}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}=}{}-2(-1)^{|j||q|}\langle{q,L_{ji}p}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}}{}=(-1)^{|i||q|+|i||p|+|q||p|}\langle{\operatorname{\mathcal{B}}(z_{j})p,\operatorname{\mathcal{B}}(z_{i})q}\rangle_{\mathcal{B}}+(-1)^{|j||q|+|i||j|+|p||q|}c\beta_{ij}\langle{p,q}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}=}{}+2(-1)^{|i||q|+|q||p|+|i||j|}\langle{L_{ij}p,q}\rangle_{\mathcal{B}},$ where we used the induction hypothesis on all three terms of the right hand side. If we use Lemma 4.6 on the last term and multiply both sides of this equation with $(-1)^{(|p|+|i|)(|q|+|j|)}$ we get $\displaystyle(-1)^{(|p|+|i|)(|q|+|j|)}\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}=(-1)^{|i||j|+|i||p|+|j||p|}\langle{\operatorname{\mathcal{B}}(z_{j})p,\operatorname{\mathcal{B}}(z_{i})q}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{(-1)^{(|p|+|i|)(|q|+|j|)}\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}=}{}+(-1)^{|j||q|+|i||q|+|p||j|}c\beta_{ij}\langle{p,q}\rangle_{\mathcal{B}}-2(-1)^{|i||p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{(-1)^{(|p|+|i|)(|q|+|j|)}\langle{z_{j}q,z_{i}p}\rangle_{\mathcal{B}}}{}=\langle{z_{i}p,z_{j}q}\rangle_{\mathcal{B}},$ where we made use of $\beta_{ij}=0$ for $|i|\neq|j|$ in the last step. ∎ Lemma 4.6 can now be extended to the following proposition. ###### Corollary 4.8. We have $\displaystyle\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=-(-1)^{(|i|+|j|)|p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}\qquad\text{and}\qquad\langle{L_{0i}p,q}\rangle_{\mathcal{B}}=(-1)^{|i||p|}\langle{p,L_{0i}q}\rangle_{\mathcal{B}},$ for all $i,j\in\\{1,\ldots,m+2n-1\\}$ and $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$. We have the following proposition. ###### Proposition 4.9. For all $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ it holds that $\displaystyle\big{\langle}R^{2}p,q\big{\rangle}_{\mathcal{B}}=0=\big{\langle}p,R^{2}q\big{\rangle}_{\mathcal{B}}.$ Thus the Bessel–Fischer product is well-defined on $\operatorname{\mathcal{F}}$. ###### Proof. Because of the superhermitian property we only need to prove $\big{\langle}p,R^{2}q\big{\rangle}=0$ holds for all $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$. The complex version of Proposition 3.1 implies that for arbitrary $p,q\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ there exists a $q^{\prime}\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ such that $\displaystyle\big{\langle}p,R^{2}q\big{\rangle}_{\mathcal{B}}=p\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}R^{2}q(z)|_{z=0}=R^{2}q^{\prime}(z)|_{z=0}.$ Since the constant term of $R^{2}q^{\prime}(z)$ is always zero this proves the theorem. ∎ Note that the above proposition also shows that the Bessel–Fischer product is degenerate on the space $\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$. In the following subsection we look at the non-degeneracy of the Bessel–Fischer product when we restrict it to $\operatorname{\mathcal{F}}$. ### 4.2 The reproducing kernel and non-degeneracy In the bosonic case a reproducing kernel for the Fock space was constructed in Section 2.4 of [18]. We will prove the non-degeneracy of the Bessel–Fischer product on $\operatorname{\mathcal{F}}$ by first constructing a generalisation of this reproducing kernel in superspace. ###### Lemma 4.10. Suppose $M-2\not\in-2\mathbb{N}$ and $k\in\mathbb{N}$. Define the superfunction $\mathbb{K}^{k}(z,w)$ by $\displaystyle\mathbb{K}^{k}(z,w):=\dfrac{1}{4^{k}k!}\left(\dfrac{M}{2}-1\right)_{k}^{-1}(z|\overline{w})^{k},$ where we used the Pochhammer symbol $(a)_{k}=a(a+1)(a+2)\cdots(a+k-1)$ and $z|w$ is defined as $\displaystyle z|w:=2z_{0}w_{0}+2\sum_{i,j=1}^{m+2n-1}z_{i}\beta^{ij}w_{j}.$ For all $p\in\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$ we then have $\displaystyle\big{\langle}p,\mathbb{K}^{k}(z,w)\big{\rangle}_{\mathcal{B}}=p(w)\mod\big{\langle}R^{2}_{w}\big{\rangle}.$ ###### Proof. We have $\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{l})(z|w)^{k}$ $\displaystyle=(-1)^{\delta_{0l}}\left((M-2+2\mathbb{E})\partial_{l}-z_{l}\Delta\right)(z|w)^{k}$ $\displaystyle=2w_{l}k(M+2k-4)(z|w)^{k-1}-4z_{l}k(k-1)R^{2}_{w}(z|w)^{k-2}$ $\displaystyle=4w_{l}k\left(k+\dfrac{M}{2}-2\right)(z|w)^{k-1}-R^{2}_{w}4z_{l}k(k-1)(z|w)^{k-2}.$ Now suppose $p$ is a monomial, i.e., $p(z)=a\prod\limits_{l=0}^{m+2n-1}z_{l}^{\alpha_{l}}$, with $|\alpha|=k$ and $a\in\mathbb{C}$. Iterating the previous calculation and working modulo $R^{2}_{w}$ we obtain $\displaystyle\langle{p,(z|\overline{w})^{k}}\rangle_{\mathcal{B}}$ $\displaystyle=\left.p\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}(z|w)^{k}\right|_{z=0}=a\prod_{l=0}^{m+2n-1}\left.\operatorname{\widetilde{\mathcal{B}}}(z_{l})^{\alpha_{l}}(z|w)^{k}\right|_{z=0}$ $\displaystyle=4^{k}(k(k-1)\cdots 1)\left(k+\dfrac{M}{2}-2\right)\cdots\left(\dfrac{M}{2}-1\right)a\prod_{l=0}^{m+2n-1}w_{l}^{\alpha_{l}}$ $\displaystyle=4^{k}k!\left(\dfrac{M}{2}-1\right)_{k}p(w),$ which gives us the desired result. ∎ ###### Theorem 4.11 (reproducing kernel of $\operatorname{\mathcal{F}}$). Suppose $M-2\not\in-2\mathbb{N}$ and define the superfunction $\mathbb{K}(z,w)$ by $\displaystyle\mathbb{K}(z,w):=\Gamma\left(\dfrac{M}{2}-1\right)\widetilde{I}_{\frac{M}{2}-2}\big{(}\sqrt{(z|\overline{w})}\big{)}=\sum_{k=0}^{\infty}\dfrac{1}{4^{k}k!}\left(\dfrac{M}{2}-1\right)_{k}^{-1}(z|\overline{w})^{k},$ where $\tilde{I}_{\alpha}$ is the I-Bessel introduced in Appendix A.1. For all $p\in\operatorname{\mathcal{F}}$ we have $\displaystyle\langle{p,\mathbb{K}(z,w)}\rangle_{\mathcal{B}}=p(w).$ ###### Proof. By Lemma 4.10, the orthogonality property and the Fischer decomposition we find that $\displaystyle\sum_{k=0}^{\infty}\mathbb{K}^{k}(z,w)=\sum_{k=0}^{\infty}\dfrac{1}{4^{k}k!}\left(\dfrac{M}{2}-1\right)_{k}^{-1}(z|\overline{w})^{k},$ has the desired property. ∎ The non-degeneracy of the Bessel–Fischer product on $\operatorname{\mathcal{F}}$ is now an almost immediate result. ###### Proposition 4.12 (non-degenerate case). For $M-2\not\in-2\mathbb{N}$ the Bessel–Fischer product is non-degenerate on the polynomial Fock space $\operatorname{\mathcal{F}}$, i.e., if $\langle{p,q}\rangle_{\mathcal{B}}=0,$ for all $q\in\operatorname{\mathcal{F}}$, then $p=0$. ###### Proof. Suppose $p\in\operatorname{\mathcal{F}}$ is such that $\langle{p,q}\rangle_{\mathcal{B}}=0$ for all $q\in\operatorname{\mathcal{F}}$. Using the reproducing kernel we obtain $p(w)=\langle{p,\mathbb{K}(z,w)}\rangle_{\mathcal{B}}=0$. Hence $p=0$. ∎ Note that the previous proposition only works when $M-2\not\in-2\mathbb{N}$. For the $M-2\in{-}2\mathbb{N}$ case the Bessel–Fischer product will always be degenerate. ###### Proposition 4.13 (degenerate case). For $M-2\in-2\mathbb{N}$ the Bessel–Fischer product is degenerate on the polynomial Fock space $\operatorname{\mathcal{F}}$, i.e., there exists a $q\in\operatorname{\mathcal{F}}$, $q\neq 0$ such that $\langle{p,q}\rangle_{\mathcal{B}}=0,$ for all $p\in\operatorname{\mathcal{F}}$. ###### Proof. Suppose $M-2\in-2\mathbb{N}$ and $q\in\mathcal{H}_{2-\frac{M}{2}}\big{(}\mathbb{C}^{m|2n}\big{)}\subset\operatorname{\widetilde{\mathcal{H}}}_{2-\frac{M}{2}}\big{(}\mathbb{C}^{m|2n}\big{)}\subset\operatorname{\mathcal{F}}$. Then $\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{i})q=\pm((M-2+2\mathbb{E})\partial_{i}q-z_{i}\Delta q)=0,$ for all $i\in\\{0,1,\ldots,m+2n-1\\}$. For all $p\in\operatorname{\mathcal{F}}$ it follows then immediately from the definition that $\langle{p,q}\rangle_{\mathcal{B}}=0$. Since $\displaystyle\dim\mathcal{H}_{2-\frac{M}{2}}\big{(}\mathbb{C}^{m|2n}\big{)}\geq\dim\mathcal{P}_{2-\frac{M}{2}}\big{(}\mathbb{C}^{m-1|2n}\big{)}\geq 1,$ we conclude that such a $q\neq 0$ exists. ∎ ## 5 The Fock model of $\boldsymbol{\mathfrak{osp}(m,2|2n)}$ In [18] the Fock representation $\rho:=\pi_{\mathbb{C}}\circ c$ is obtained by twisting the complexification of the Schrödinger representation $\pi_{\mathbb{C}}$ with the Cayley transform $c$. This Cayley transform is an isomorphism of $\mathfrak{g}_{\mathbb{C}}$ which induces a Lie algebra isomorphism between $\mathfrak{k}_{\mathbb{C}}$ and $\mathfrak{istr}(J_{\mathbb{C}})$. We will use a similar approach in our construction of the Fock model. We start by defining a Cayley transform $c$ in our setting. ### 5.1 Definition and properties Let $\mathfrak{g}_{\mathbb{C}}$ and $\mathfrak{k}_{\mathbb{C}}$ be the complexified versions of $\mathfrak{g}$ and $\mathfrak{k}$ introduced in Section 3.3, i.e., $\displaystyle\mathfrak{g}_{\mathbb{C}}=\operatorname{TKK}(J_{\mathbb{C}})\cong\mathfrak{osp}_{\mathbb{C}}(m+2|2n),$ $\displaystyle\mathfrak{k}_{\mathbb{C}}=\\{(z,I,-z)\colon I\in\operatorname{Inn}(J_{\mathbb{C}}),z\in J_{\mathbb{C}}\\}\cong\mathfrak{osp}_{\mathbb{C}}(m|2n)\oplus\mathbb{C},$ Let $\imath$ denote the complex unit. Define the Cayley transform $c\in\operatorname{End}(\mathfrak{g}_{\mathbb{C}})$ as $\displaystyle c:=\exp\left(\frac{\imath}{2}\operatorname{ad}(e_{0}^{-})\right)\exp\big{(}\imath\operatorname{ad}\big{(}e_{0}^{+}\big{)}\big{)},$ with $e_{0}^{-}$ and $e_{0}^{+}$ the copies of $e_{0}$ in $J^{-}$ and $J^{+}$, respectively. ###### Proposition 5.1. Using the decomposition $\mathfrak{g}_{\mathbb{C}}=J^{-}_{\mathbb{C}}\oplus\mathfrak{istr}(J_{\mathbb{C}})\oplus J^{+}_{\mathbb{C}}$ we obtain the following explicit expression for the Cayley transform * • $c(a,0,0)=\left(\dfrac{a}{4},\imath L_{a},a\right)$, * • $c(0,L_{a}+I,0)=\left(\imath\dfrac{a}{4},I,-\imath a\right)$, * • $c(0,0,a)=\left(\dfrac{a}{4},-\imath L_{a},a\right)$, with $a\in J$ and $I\in\operatorname{Inn}(J_{\mathbb{C}})$. It induces a Lie superalgebra isomorphism: $\displaystyle c\colon\ \mathfrak{k}_{\mathbb{C}}\rightarrow\mathfrak{istr}(J_{\mathbb{C}}),\qquad(a,I,-a)\mapsto I+2\imath L_{a}.$ ###### Proof. Expanding $\exp\big{(}\frac{\imath}{2}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\big{)}\exp\big{(}\imath\operatorname{ad}\big{(}e_{0}^{+}\big{)}\big{)}$ we obtain $\displaystyle c=1+\dfrac{\imath}{2}\operatorname{ad}\big{(}e_{0}^{-}\big{)}-\dfrac{1}{8}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{-}\big{)}+\imath\operatorname{ad}\big{(}e_{0}^{+}\big{)}-\dfrac{1}{2}\operatorname{ad}\big{(}e_{0}^{+}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}-\dfrac{1}{2}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}$ $\displaystyle\hphantom{c=}{}-\dfrac{\imath}{4}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}-\dfrac{\imath}{8}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}$ $\displaystyle\hphantom{c=}{}+\dfrac{1}{16}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{-}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}\operatorname{ad}\big{(}e_{0}^{+}\big{)}.$ We have the following straightforward calculations $\displaystyle c(a,0,0)=a^{-}+0+0+2\imath L_{a}+a^{+}-a^{-}-\imath L_{a}+0+\dfrac{a^{-}}{4}=\left(\dfrac{a}{4},\imath L_{a},a\right),$ $\displaystyle c(0,L_{a}+I,0)=(L_{a}+I)+\imath\dfrac{a^{-}}{2}+0-\imath a^{+}+0-L_{a}+0-\imath\dfrac{a^{-}}{4}+0=\left(\imath\dfrac{a}{4},I,-\imath a\right),$ $\displaystyle c(0,0,a)=a^{+}-\imath L_{a}+\dfrac{a^{-}}{4}+0+0+0+0+0+0=\left(\dfrac{a}{4},-\imath L_{a},a\right),$ proving the theorem. ∎ We complexify the Schrödinger representation $\pi$ given in Section 3.3 to obtain a representation $\pi_{\mathbb{C}}$ of $\mathfrak{g}_{\mathbb{C}}=J^{-}_{\mathbb{C}}\oplus\mathfrak{istr}(J_{\mathbb{C}})\oplus J^{+}_{\mathbb{C}}$ acting on $\operatorname{\mathcal{F}}$. Explicitly, $\pi_{\mathbb{C}}$ is given by * • $\pi_{\mathbb{C}}(e_{l},0,0)=-2\imath z_{l}$, * • $\pi_{\mathbb{C}}(0,L_{e_{k}},0)=-z_{0}\partial_{k}+z_{k}\partial_{0}$, * • $\pi_{\mathbb{C}}(0,[L_{e_{i}},L_{e_{j}}],0)=z_{i}\partial_{j}-(-1)^{|i||j|}z_{j}\partial_{i}$, * • $\pi_{\mathbb{C}}(0,L_{e_{0}},0)=\frac{2-M}{2}-\mathbb{E}$, * • $\pi_{\mathbb{C}}(0,0,e_{l})=-\frac{1}{2}\imath\operatorname{\widetilde{\mathcal{B}}}(z_{l})$, with $i,j,k\in\\{1,2,\ldots,m+2n-1\\}$ and $l\in\\{0,1,2,\ldots,m+2n-1\\}$. Since $L_{ij}$, $\mathbb{E}$ and $\operatorname{\widetilde{\mathcal{B}}}(z_{l})$ map $\big{\langle}R^{2}\big{\rangle}$ into $\big{\langle}R^{2}\big{\rangle}$ this representation is well defined on $\operatorname{\mathcal{F}}$. As in the bosonic case, we will define the Fock representation $\rho$ as the composition of $\pi_{\mathbb{C}}$ with the Cayley transform $c$, $\displaystyle\rho:=\pi_{\mathbb{C}}\circ c.$ So $\rho$ of $\mathfrak{g}=J^{-}\oplus\mathfrak{istr}(J)\oplus J^{+}$ acting on $\operatorname{\mathcal{F}}$ is given as follows * • $\rho(e_{0},0,0)=-\dfrac{\imath}{2}\big{(}z_{0}+\operatorname{\widetilde{\mathcal{B}}}(z_{0})+M-2+2\mathbb{E}\big{)}$, * • $\rho(e_{k},0,0)=-\dfrac{\imath}{2}\big{(}z_{k}+\operatorname{\widetilde{\mathcal{B}}}(z_{k})+2(z_{0}\partial_{k}-z_{k}\partial_{0})\big{)}$, * • $\rho(0,L_{e_{0}},0)=\dfrac{1}{2}\big{(}z_{0}-\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}$, * • $\rho(0,L_{e_{k}},0)=\dfrac{1}{2}\big{(}z_{k}-\operatorname{\widetilde{\mathcal{B}}}(z_{k})\big{)}$, * • $\rho(0,[L_{e_{i}},L_{e_{j}}],0)=z_{i}\partial_{j}-(-1)^{|i||j|}z_{j}\partial_{i}$, * • $\rho(0,0,e_{k})=-\dfrac{\imath}{2}\big{(}z_{k}+\operatorname{\widetilde{\mathcal{B}}}(z_{k})-2(z_{0}\partial_{k}-z_{k}\partial_{0})\big{)}$, * • $\rho(0,0,e_{0})=-\dfrac{\imath}{2}\big{(}z_{0}+\operatorname{\widetilde{\mathcal{B}}}(z_{0})+2-M-2\mathbb{E}\big{)}$, with $i,j,k\in\\{1,2,\ldots,m+2n-1\\}$. ###### Proposition 5.2 ($\mathfrak{osp}(m,2|2n)$-invariance). The Fock representation $\rho$ on $\operatorname{\mathcal{F}}$ is skew- supersymmetric with respect to the Bessel–Fischer product, i.e., $\displaystyle\langle{\rho(X)p,q}\rangle_{\mathcal{B}}=-(-1)^{|X||p|}\langle{p,\rho(X)q}\rangle_{\mathcal{B}},$ for all $X\in\mathfrak{g}$ and $p,q\in\operatorname{\mathcal{F}}$. ###### Proof. Suppose $p,q\in\operatorname{\mathcal{F}}$. From Corollary 4.8 we obtain $\displaystyle\langle{\rho(e_{i},0,-e_{i})p,q}\rangle_{\mathcal{B}}=\langle{-2\imath L_{0i}p,q}\rangle_{\mathcal{B}}=-(-1)^{|i||p|}\langle{p,2\overline{\imath}L_{0i}q}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{\rho(e_{i},0,-e_{i})p,q}\rangle_{\mathcal{B}}}{}=-(-1)^{|i||p|}\langle{p,\rho(e_{i},0,-e_{i})q}\rangle_{\mathcal{B}}$ and $\displaystyle\langle{\rho(0,[L_{e_{i}},L_{e_{j}}],0)p,q}\rangle_{\mathcal{B}}=\langle{L_{ij}p,q}\rangle_{\mathcal{B}}=-(-1)^{(|i|+|j|)|p|}\langle{p,L_{ij}q}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{\rho(0,[L_{e_{i}},L_{e_{j}}],0)p,q}\rangle_{\mathcal{B}}}{}=-(-1)^{(|i|+|j|)|p|}\langle{p,\rho(0,[L_{e_{i}},L_{e_{j}}],0)q}\rangle_{\mathcal{B}},$ for $i,j\in\\{1,\ldots,m+2n-1\\}$. Furthermore, we have $\displaystyle\langle{\rho(e_{k},0,e_{k})p,q}\rangle_{\mathcal{B}}=\big{\langle}{-}\imath\big{(}z_{k}+\operatorname{\widetilde{\mathcal{B}}}(z_{k})\big{)}p,q\big{\rangle}_{\mathcal{B}}=-\imath\big{(}\langle{z_{k}p,q}\rangle_{\mathcal{B}}+\big{\langle}\operatorname{\widetilde{\mathcal{B}}}(z_{k})p,q\big{\rangle}_{\mathcal{B}}\big{)}$ $\displaystyle\hphantom{\langle{\rho(e_{k},0,e_{k})p,q}\rangle_{\mathcal{B}}}{}=-\imath(-1)^{|k||p|}\big{(}\big{\langle}p,\operatorname{\widetilde{\mathcal{B}}}(z_{k})q\big{\rangle}_{\mathcal{B}}+\langle{p,z_{k}q}\rangle_{\mathcal{B}}\big{)}$ $\displaystyle\hphantom{\langle{\rho(e_{k},0,e_{k})p,q}\rangle_{\mathcal{B}}}{}=-(-1)^{|k||p|}\big{\langle}p,\overline{\imath}\big{(}z_{k}+\operatorname{\widetilde{\mathcal{B}}}(z_{k})\big{)}q\big{\rangle}_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{\rho(e_{k},0,e_{k})p,q}\rangle_{\mathcal{B}}}{}=-(-1)^{|k||p|}\langle{p,\rho(e_{k},0,e_{k})q}\rangle_{\mathcal{B}}$ and similarly $\displaystyle\langle{\rho(0,L_{e_{k}},0)p,q}\rangle_{\mathcal{B}}=\left\langle\dfrac{1}{2}\big{(}z_{k}-\operatorname{\widetilde{\mathcal{B}}}(z_{k})\big{)}p,q\right\rangle_{\mathcal{B}}=-(-1)^{|k||p|}\langle{p,\rho(0,L_{e_{k}},0)q}\rangle_{\mathcal{B}},$ for $k\in\\{0,\ldots,m+2n-1\\}$. Because of Proposition 4.4 we may assume $p$ and $q$ are homogeneous polynomials the same degree. We now have $\displaystyle\langle{\rho(e_{0},0,-e_{0})p,q}\rangle_{\mathcal{B}}$ $\displaystyle=\langle{-2\imath(M-2+2\mathbb{E})p,q}\rangle_{\mathcal{B}}=-\langle{p,-2\imath(M-2+2\mathbb{E})q}\rangle_{\mathcal{B}}$ $\displaystyle=-\langle{p,\rho(e_{0},0,-e_{0})q}\rangle_{\mathcal{B}},$ which proves the theorem. ∎ ### 5.2 The $\boldsymbol{(\mathfrak{g},\mathfrak{k})}$-module $\boldsymbol{F}$ We define $\displaystyle F:=U(\mathfrak{g})1\mod\big{\langle}R^{2}\big{\rangle},$ where the $\mathfrak{g}$ module structure is given by the Fock representation $\rho$. We also introduce the notation $\displaystyle F_{k}:=\bigoplus_{l=0}^{k}z_{0}^{l}\mathcal{H}_{k-l}\big{(}\mathbb{C}^{m-1|2n}\big{)}\mod\big{\langle}R^{2}\big{\rangle}$ and reintroduce $r^{2}$ from Section 3.2 as its complexified version: $\displaystyle r^{2}:=\sum_{i=1}^{m+2n-1}z^{i}z_{i}.$ We have $R^{2}=-z_{0}^{2}+r^{2}$ and since we are working modulo $\big{\langle}R^{2}\big{\rangle}$ this implies $z_{0}^{2}=r^{2}$. In the following, we will work modulo $\big{\langle}R^{2}\big{\rangle}$ but omit $\big{\langle}R^{2}\big{\rangle}$ from our notation. For $M-1\not\in-2\mathbb{N}$ we now find $\displaystyle F_{k}$ $\displaystyle=\bigoplus_{l=0}^{k}z_{0}^{l}\mathcal{H}_{k-l}\big{(}\mathbb{C}^{m-1|2n}\big{)}=\bigoplus_{l=0}^{\big{\lfloor}\frac{k}{2}\big{\rfloor}}z_{0}^{2l}\mathcal{H}_{k-2l}\big{(}\mathbb{C}^{m-1|2n}\big{)}\oplus\bigoplus_{l=0}^{\big{\lfloor}\frac{k+1}{2}\big{\rfloor}-1}z_{0}^{2l+1}\mathcal{H}_{k-2l-1}\big{(}\mathbb{C}^{m-1|2n}\big{)}$ $\displaystyle=\bigoplus_{l=0}^{\big{\lfloor}\frac{k}{2}\big{\rfloor}}r^{2l}\mathcal{H}_{k-2l}\big{(}\mathbb{C}^{m-1|2n}\big{)}\oplus\bigoplus_{l=0}^{\big{\lfloor}\frac{k+1}{2}\big{\rfloor}-1}z_{0}r^{2l}\mathcal{H}_{k-2l-1}\big{(}\mathbb{C}^{m-1|2n}\big{)}$ $\displaystyle\cong\mathcal{P}_{k}(\mathbb{C}^{m-1|2n})\oplus z_{0}\mathcal{P}_{k-1}\big{(}\mathbb{C}^{m-1|2n}\big{)},$ where we made use of the Fischer decomposition (Theorem 2.4) in the last step. In particular, by Proposition 2.8, $\displaystyle\dim F_{k}=\dim\mathcal{P}_{k}\big{(}\mathbb{C}^{m-1|2n}\big{)}+\dim\mathcal{P}_{k-1}\big{(}\mathbb{C}^{m-1|2n}\big{)}=\dim\mathcal{H}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}.$ If $M\not\in-2\mathbb{N}$, then for $p\in F_{k}$ the Fischer decomposition on $\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$ also gives $\displaystyle p=\sum_{l=0}^{\left\lfloor\frac{k}{2}\right\rfloor}R^{2l}h_{k-2l}=h_{k}\mod\big{\langle}R^{2}\big{\rangle},$ with $h_{k-2l}\in\mathcal{H}_{k-2l}\big{(}\mathbb{C}^{m|2n}\big{)}$. This implies $F_{k}\cong\mathcal{H}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$ for $M\geq 2$. ###### Theorem 5.3 (decomposition of $F$). For $M\geq 3$, we have the following: * $(1)$ $F_{k}$ is an irreducible $\mathfrak{k}$-module. * $(2)$ $F$ is an irreducible $\mathfrak{g}$-module and its $\mathfrak{k}$-type decomposition is given by $\displaystyle F=\bigoplus_{k=0}^{\infty}F_{k}.$ * $(3)$ An explicit decomposition of $F_{k}$ into irreducible $\mathfrak{k}_{0}$-modules is given by $\displaystyle F_{k}=\bigoplus_{l=0}^{k}z_{0}^{l}\mathcal{H}_{k-l}\big{(}\mathbb{C}^{m-1|2n}\big{)}\mod\big{\langle}R^{2}\big{\rangle}.$ ###### Proof. We have the following elements of the action of $\mathfrak{g}$: $\displaystyle\rho^{+}_{0}:=\rho\left(c^{-1}\left(\frac{-e_{0}}{2},0,0\right)\right)=\imath z_{0},$ $\displaystyle\rho^{-}_{0}:=\rho\big{(}c^{-1}(0,0,-2e_{0})\big{)}=\imath\operatorname{\widetilde{\mathcal{B}}}(z_{0}),$ $\displaystyle\rho_{0i}:=\rho\left(-\frac{e_{i}}{2},0,\frac{e_{i}}{2}\right)=\imath L_{0i},$ $\displaystyle\rho_{ij}:=\rho(0,[L_{e_{i}},L_{e_{j}}],0)=L_{ij},$ with $i,j\in\\{1,\ldots,m+2n-1\\}$. The elements $\rho_{ij}$ for $i,j\in\\{1,\ldots,m+2n-1\\}$, $i\leq j$ give rise to an irreducible representation of $\mathfrak{k}_{0}$ on $\mathcal{H}_{k}\big{(}\mathbb{C}^{m-1|2n}\big{)}$ as a result of Proposition 2.6. These elements leave $r^{2}=z_{0}^{2}$ invariant and therefore also leave powers of $z_{0}$ invariant, which proves $(3)$. Again by Proposition 2.6 the elements $\rho_{ij}$ and $\rho_{0i}$ give rise to an irreducible representation of $\mathfrak{k}$ on $\mathcal{H}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}\cong F_{k}$, which proves $(1)$. For the first two elements we have $\displaystyle\rho^{+}_{0}\big{(}z_{0}^{k}\big{)}=\imath z_{0}^{k+1},$ $\displaystyle\rho^{-}_{0}\big{(}z_{0}^{k}\big{)}=\imath\operatorname{\widetilde{\mathcal{B}}}(z_{0})z_{0}^{k}=\imath k(M+2k-4)z_{0}^{k-1}-\imath k(k-1)z_{0}^{k-1}=\imath k(M+k-3)z_{0}^{k-1},$ which shows that $\rho^{+}_{0}$ allows us to go to polynomials of higher degrees while $\rho^{-}_{0}$ allows us to go the other direction for $M\geq 3$. Therefore we obtain $(2)$. ∎ The following isomorphism is a direct result of this theorem. ###### Corollary 5.4. Suppose $M\geq 3$ and let $\displaystyle\operatorname{\mathcal{F}}=\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}/\big{\langle}R^{2}\big{\rangle}$ be the polynomial Fock space defined in Definition 4.1. We have $\operatorname{\mathcal{F}}\cong F$. ## 6 The Segal–Bargmann transform In this section we construct the Segal–Bargmann transform and show that it is an isomorphism from $W$ as defined in Section 3.3 to $F$ as defined in Section 5.2. It will make use of the integral $\int_{W}$ we defined in Definition 3.7. This integral is only defined for $M\geq 4$. Therefore we will always assume $M\geq 4$ throughout this section. ### 6.1 Definition and properties Let $\widetilde{I}_{\alpha}(t)$ be the I-Bessel function as introduced in Appendix A.1. We define an entire function $\operatorname{\mathbb{I}}_{\alpha}$ on $\mathbb{C}$ by $\displaystyle\operatorname{\mathbb{I}}_{\alpha}(t):=\Gamma\left(\dfrac{M}{2}-1\right)\widetilde{I}_{\frac{M}{2}-2+\alpha}\big{(}2\sqrt{t}\big{)}=\Gamma\left(\dfrac{M}{2}-1\right)\sum_{l=0}^{\infty}\dfrac{1}{l!\Gamma\big{(}l+\frac{M}{2}-1+\alpha\big{)}}t^{l}.$ Clearly we have $\operatorname{\mathbb{I}}_{0}(0)=1$ and $\displaystyle\partial_{t}^{j}\operatorname{\mathbb{I}}_{0}(t)=\Gamma\left(\dfrac{M}{2}-1\right)\partial_{t}^{j}\left(\widetilde{I}_{\frac{M}{2}-2}\big{(}2\sqrt{t}\big{)}\right)=\Gamma\left(\dfrac{M}{2}-1\right)\widetilde{I}_{\frac{M}{2}-2+j}\big{(}2\sqrt{t}\big{)}=\operatorname{\mathbb{I}}_{j}(t).$ We are now able to state the Segal–Bargmann transform that extends the one from the bosonic case obtained in [18]. ###### Definition 6.1. For $f\in W$ the Segal–Bargmann transform is defined as $\displaystyle\operatorname{SB}f(z):=\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})f(x),$ where $x|z$ is defined as $\displaystyle x|z:=2x_{0}z_{0}+2\sum_{i,j=1}^{m+2n-1}x_{i}\beta^{ij}z_{j}$ and we view $\operatorname{\mathbb{I}}_{\alpha}(x|z)$ as a radial superfunction in the sense of equation (3.2). Note that $\operatorname{\mathbb{I}}_{0}(4(x|z))$ is the reproducing kernel $\mathbb{K}(x,z)$ of the Fock space we found in Theorem 4.11. ###### Proposition 6.2. For $M\geq 4$ the Segal–Bargmann transform $\operatorname{SB}$ is well defined. ###### Proof. We wish to prove that the integral $\displaystyle\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})f(x)$ is convergent for all $f\in W$ and that $\operatorname{SB}(R^{2})=0$. As shown in the proof of Theorem 8.13 in [6], the elements of $W$ can be decomposed into elements of the form $\displaystyle P_{k}\widetilde{K}_{-\frac{1}{2}+\alpha_{1}+\alpha_{2}}(2|X|),$ with $P_{k}$ a homogeneous polynomial of degree $k$. Here $\widetilde{K}_{\alpha}$ is the K-Bessel function introduced in Appendix A.1 interpreted as a radial superfunction as in Section 3.2. Furthermore $\alpha_{1},\alpha_{2}\in\mathbb{N}$ are subject to the relations $k\geq\alpha_{1}+2\alpha_{2}$ and $M\geq 2\alpha_{1}+2$. Also, observe that $\displaystyle|X|=\sqrt{\dfrac{x_{0}^{2}+r^{2}}{2}}=\sqrt{x_{0}^{2}+\dfrac{R^{2}}{2}}=\sqrt{x_{0}^{2}}\quad\mod\big{\langle}R^{2}\big{\rangle},$ (6.1) which is equal to $x_{0}$ within the domain of integration of $\int_{W}$. Because of all this and equation (A.1) it suffices to prove $\displaystyle\int_{W}P_{k}\operatorname{\mathbb{I}}_{0}(x|z)\widetilde{K}_{-\frac{1}{2}}(2|X|)\widetilde{K}_{-\frac{1}{2}+\alpha_{1}+\alpha_{2}}(2|X|)$ is convergent for all $k\in\mathbb{N}$ and $\alpha_{1},\alpha_{2}\in\mathbb{N}$ subject to the above mentioned relations. We will use the explicit description of $\int_{W}$ given in equation (3.3). The morphism $\phi^{\sharp}$ leaves the degree of a polynomial unchanged. Hence, we can expand $\displaystyle\big{(}\phi^{\sharp}(P_{k})\big{)}_{|s=x_{0}=\rho}=\sum_{j=0}^{k}\rho^{k-j}a_{j}(\theta)b_{j}(\omega),$ where $a_{j}(\theta)$ is a polynomial in $\mathcal{P}\big{(}\mathbb{R}^{0|2n}\big{)}$ of degree $j$ and $b_{j}(\omega)$ is a function depending on the spherical coordinates $\omega$. For $c\in\mathbb{Z}$ we obtain $\displaystyle(1+\eta)^{c}=\sum_{j=0}^{n}\dfrac{1}{j!}\left(\dfrac{-c}{2}\right)_{j}\dfrac{\theta^{2j}}{2^{j}s^{2j}},\qquad(1+\xi)^{c}=\sum_{j=0}^{n}\dfrac{(-1)^{j}}{j!}\left(\dfrac{-c}{2}\right)_{j}\dfrac{\theta^{2j}}{2^{j}x_{0}^{2j}}$ and $\displaystyle\phi^{\sharp}\big{(}\widetilde{K}_{\alpha}(c|X|)\big{)}=\widetilde{K}_{\alpha}(c|X|)=\sum_{j=0}^{n}\dfrac{(-1)^{j}c^{2j}\theta^{2j}}{j!8^{j}}\widetilde{K}_{\alpha+j}(c\rho),$ from the proof of Lemma 8.6 in [6]. Here $\eta$, $\xi$ and $\theta^{2}$ were defined in Section 3.4. We introduce the notations $\displaystyle\omega|z:=2\sum_{i=1}^{m-1}\omega_{i}z_{i}\qquad\mbox{and}\qquad\theta|z:=2\sum_{i,j=m}^{m+2n-1}x_{i}\beta^{ij}z_{j}.$ We can use equation (3.2) with $h=\operatorname{\mathbb{I}}_{0}$ and $f=x|z$ as a function in the $x$ variables to obtain $\displaystyle\operatorname{\mathbb{I}}_{0}(x|z)=\operatorname{\mathbb{I}}_{0}(2x_{0}z_{0}+s\omega|z+\theta|z)=\sum_{j=0}^{2n}\dfrac{1}{j!}(\theta|z)^{j}\operatorname{\mathbb{I}}_{j}(2x_{0}z_{0}+s\omega|z).$ Using the properties of $\phi^{\sharp}$ described in Lemma 8.3 of [6] and the expansion of $(1+\xi)$ and $(1+\eta)$, we now find $\displaystyle\phi^{\sharp}(\operatorname{\mathbb{I}}_{0}(x|z))_{|s=x_{0}=\rho}=\sum_{l_{1}=0}^{2n}\dfrac{1}{l_{1}!}(\theta|z)^{l_{1}}\operatorname{\mathbb{I}}_{l_{1}}(2\rho(1+\xi)z_{0}+\rho(1+\eta)\omega|z)_{|s=x_{0}=\rho}$ $\displaystyle\hphantom{\phi^{\sharp}(\operatorname{\mathbb{I}}_{0}(x|z))_{|s=x_{0}=\rho}}{}=\sum_{l_{1}=0}^{2n}\dfrac{1}{l_{1}!}(\theta|z)^{l_{1}}\operatorname{\mathbb{I}}_{l_{1}}\left(\rho\sum_{l_{2}=0}^{n}\dfrac{1}{l_{2}!}\left(-\dfrac{1}{2}\right)_{l_{2}}\dfrac{\theta^{2l_{2}}}{2^{l_{2}}\rho^{2l_{2}}}(2(-1)^{l_{2}}z_{0}+\omega|z)\right).$ We use equation (3.2) again, this time with $h=\operatorname{\mathbb{I}}_{l_{1}}$ and $f$ equal to the sum over $l_{2}$. Note that the $l_{2}=0$ term corresponds with $f_{0}$. We obtain $\displaystyle\phi^{\sharp}(\operatorname{\mathbb{I}}_{0}(x|z))_{|s=x_{0}=\rho}=\sum_{l_{1},l_{3}=0}^{2n}\dfrac{1}{l_{1}!l_{3}!}(\theta|z)^{l_{1}}\left(\rho\sum_{l_{2}=1}^{n}\dfrac{1}{l_{2}!}\left(-\dfrac{1}{2}\right)_{l_{2}}\dfrac{\theta^{2l_{2}}}{2^{l_{2}}\rho^{2l_{2}}}(2(-1)^{l_{2}}z_{0}+\omega|z)\right)^{l_{3}}$ $\displaystyle\hphantom{\phi^{\sharp}(\operatorname{\mathbb{I}}_{0}(x|z))_{|s=x_{0}=\rho}=}{}\times\operatorname{\mathbb{I}}_{l_{1}+l_{3}}(\rho(2z_{0}+\omega|z)).$ Combining all this, we see that $\displaystyle\dfrac{1}{\gamma}\int_{0}^{\infty}\int_{\mathbb{S}^{m-2}}\int_{B}\rho^{m-3}(1+\eta)^{m-3}(1+\xi)^{-1}$ $\displaystyle\qquad{}\times\phi^{\sharp}(P_{k}\operatorname{\mathbb{I}}_{0}(x|z)\widetilde{K}_{-\frac{1}{2}}(2|X|)\widetilde{K}_{-\frac{1}{2}+\alpha_{1}+\alpha_{2}}(|X|))_{|s=x_{0}=\rho}\mathrm{d}\rho\mathrm{d}\omega,$ converges if $\displaystyle\int_{0}^{\infty}\int_{B}\rho^{m-3}\sum_{j_{1}=0}^{k}\sum_{j_{2},j_{3},j_{4},j_{5}=0}^{n}\sum_{l_{1},l_{3}=0}^{2n}\dfrac{1}{j_{2}!}\left(\dfrac{3-m}{2}\right)_{j_{2}}\dfrac{\theta^{2j_{2}}}{2^{j_{2}}\rho^{2{j_{2}}}}\dfrac{(-1)^{j_{3}}}{j_{3}!}\left(\dfrac{1}{2}\right)_{j_{3}}\dfrac{\theta^{2{j_{3}}}}{2^{j_{3}}\rho^{2{j_{3}}}}$ $\displaystyle\qquad{}\times\rho^{k-j_{1}}a_{j_{1}}(\theta)\dfrac{1}{l_{1}!l_{3}!}(\theta|z)^{l_{1}}\left(\rho\sum_{l_{2}=1}^{n}\dfrac{1}{l_{2}!}\left(-\dfrac{1}{2}\right)_{l_{2}}\dfrac{\theta^{2l_{2}}}{2^{l_{2}}\rho^{2l_{2}}}c_{1}\right)^{l_{3}}\operatorname{\mathbb{I}}_{l_{1}+l_{3}}(c_{2}\rho)$ $\displaystyle\qquad{}\times\dfrac{(-1)^{j_{4}}\theta^{2{j_{4}}}}{{j_{4}}!2^{j_{4}}}\widetilde{K}_{-\frac{1}{2}+{j_{4}}}(2\rho)\dfrac{(-1)^{j_{5}}\theta^{2{j_{5}}}}{{j_{5}}!2^{j_{5}}}\widetilde{K}_{-\frac{1}{2}+\alpha_{1}+\alpha_{2}+j_{5}}(2\rho)\mathrm{d}\rho$ converges for all $c_{1},c_{2}\in\mathbb{C}$. This in turn converges if $\displaystyle\int_{0}^{\infty}\int_{B}\rho^{m-3+k-j_{1}+l_{3}-2l_{3}l_{2}-2j_{2}-2j_{3}}a_{j_{1}}(\theta)(\theta|z)^{l_{1}}\theta^{2j_{2}+2j_{3}+2j_{4}+2j_{5}+2l_{3}l_{2}}\operatorname{\mathbb{I}}_{l_{1}+l_{3}}(c\rho)$ $\displaystyle\qquad\times\widetilde{K}_{-\frac{1}{2}+{j_{4}}}(2\rho)\widetilde{K}_{-\frac{1}{2}+\alpha_{1}+\alpha_{2}+j_{5}}(2\rho)\mathrm{d}\rho$ converges for all $0\leq j_{1}\leq k$, $0\leq j_{2},j_{3},j_{4},j_{5}\leq n$, $0\leq l_{1},l_{3}\leq 2n$, $1\leq l_{2}\leq n$ and all $c\in\mathbb{C}$. The Berezin integral is zero unless $j_{1}+l_{1}+2j_{2}+2j_{3}+2j_{4}+2j_{5}+2l_{3}l_{2}=2n$. The integral $\displaystyle\int_{0}^{\infty}\rho^{\sigma-1}\widetilde{I}_{\beta_{1}}\big{(}\sqrt{a\rho}\big{)}\widetilde{K}_{\beta_{2}}(2\rho)\widetilde{K}_{\beta_{3}}(2\rho)\mathrm{d}\rho,$ with $\beta_{1}\geq 0$ converges if $\sigma>2\max\\{\beta_{2},0\\}+2\max\\{\beta_{3},0\\}$. This follows from the asymptotic behaviour of the Bessel functions, see Appendix A.1. Therefore we get the following condition $\displaystyle m-2+k-j_{1}+l_{3}-2l_{3}l_{2}-2j_{2}-2j_{3}$ $\displaystyle\qquad{}>2\max\left\\{-\frac{1}{2}+{j_{4}},0\right\\}+2\max\left\\{-\frac{1}{2}+\alpha_{1}+\alpha_{2}+j_{5},0\right\\},$ with $j_{1}+l_{1}+2j_{2}+2j_{3}+2j_{4}+2j_{5}+2l_{3}l_{2}=2n$. Taking into account $k\geq\alpha_{1}+2\alpha_{2}$ and $M\geq 2\alpha_{1}+2$ the condition reduces to $M>2$. We still need that $\operatorname{SB}\big{(}R^{2}\big{)}=0$, but this follows easily from $\big{(}\phi^{\sharp}\big{(}R^{2}\big{)}\big{)}_{|s=x_{0}=\rho}=\big{(}{-}x_{0}^{2}+s^{2}\big{)}_{|s=x_{0}=\rho}=0$. ∎ We can now show that $\operatorname{SB}$ intertwines the Schrödinger model with the Fock model. ###### Theorem 6.3 (intertwining property). For $M\geq 4$ the Segal–Bargmann transform intertwines the action $\pi$ on $W$ with the action $\rho$ on $F$, i.e., $\displaystyle\operatorname{SB}\circ\pi(X)=\rho(X)\circ\operatorname{SB},$ for all $X\in\mathfrak{g}$. ###### Proof. The proof of this theorem is a technical and long but rather straightforward calculation. We refer to Appendix B for more details. ∎ ###### Proposition 6.4. The Segal–Bargmann transform $\operatorname{SB}$ induces a $\mathfrak{g}$-module isomorphism between $W$ and $F$. ###### Proof. From the way we normalized the integral $\int_{W}$ and equation (6.1) it is clear that $\displaystyle\operatorname{SB}(\exp(-2|X|))(0)=\int_{W}\exp(-4|X|)=1.$ Therefore the Segal–Bargmann transform maps a non-zero element of $W$ to a non-zero element of $F$. It also intertwines the actions $\pi$ and $\rho$. Since Theorems 3.6 and 5.3 give us that $W$ and $F$ are irreducible $\mathfrak{g}$-modules, we conclude that $\operatorname{SB}$ is an isomorphism of $\mathfrak{g}$-modules. ∎ ###### Lemma 6.5. We have $\operatorname{SB}(\exp(-2|X|)(z)=1$. ###### Proof. Since $\exp(-2|X|)$ is in $W_{0}$ Proposition 6.4 implies that $\operatorname{SB}(\exp(-2|X|))(z)$ is in $F_{0}=\mathbb{C}$. Hence $\operatorname{SB}(\exp(-2|X|))(z)$ is a constant. From the way we normalized the integral $\int_{W}$ we have $\operatorname{SB}(\exp(-2|X|))(0)=1$ and therefore $\operatorname{SB}(\exp(-2|X|))(z)=1$. ∎ ###### Theorem 6.6 (unitary property). For $M\geq 4$ the Segal–Bargmann transform preserves the sesquilinear forms, i.e., $\displaystyle\langle{\operatorname{SB}f,\operatorname{SB}g}\rangle_{\mathcal{B}}=\langle{f,g}\rangle_{W},$ for all $f,g\in W$. ###### Proof. We first look at the case $f=\exp(-2|X|)$. Because of Lemma 6.5 and the superhermitian property of the Bessel–Fischer product we have $\displaystyle\langle{\operatorname{SB}(\exp(-2|X|)),\operatorname{SB}g}\rangle_{\mathcal{B}}=\langle{1,\operatorname{SB}g}\rangle_{\mathcal{B}}=\overline{\langle{\operatorname{SB}g,1}\rangle}_{\mathcal{B}}=\operatorname{SB}(\overline{g(x)})\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}1\big{|}_{z=0}$ $\displaystyle\hphantom{\langle{\operatorname{SB}(\exp(-2|X|)),\operatorname{SB}g}\rangle_{\mathcal{B}}}{}=\int_{W}\exp(-2x_{0})\big{(}\operatorname{\mathbb{I}}_{0}(x|\operatorname{\widetilde{\mathcal{B}}}(z))\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}1\big{)}\overline{g(x)}\big{|}_{z=0},$ for all $g\in W$. Here $\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}$ and $\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}$ should be considered as infinite power sums of the Bessel operator with $\displaystyle x|\operatorname{\widetilde{\mathcal{B}}}(z):=2x_{0}\operatorname{\widetilde{\mathcal{B}}}(z_{0})+2\sum_{i,j=1}^{m+2n-1}x_{i}\beta^{ij}\operatorname{\widetilde{\mathcal{B}}}(z_{j})=2\sum_{i,j=0}^{m+2n-1}x_{i}\beta^{ij}\operatorname{\mathcal{B}}(z_{j}).$ Since they act on a constant with respect to the variable $z$ we get $\displaystyle\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}1=\operatorname{\mathbb{I}}_{0}(0)\exp(0)1=1,$ which gives $\displaystyle\langle{\operatorname{SB}(\exp(-2|X|)),\operatorname{SB}g}\rangle_{\mathcal{B}}=\int_{W}\exp(-2x_{0})\overline{g(x)}=\langle{\exp(-2|X|),g}\rangle_{W},$ if we use equation (6.1). Now suppose $f,g\in W$. Since $W$ is an irreducible $\mathfrak{g}$-module (Theorem 3.6), there exists a $Y\in U(\mathfrak{g})$ such that $f=\pi(Y)\exp(-2|X|)$. Therefore we can reduce the general case to the previous case using the intertwining property (Theorem 6.3) and the fact that the sesquilinear forms are skew symmetric for $\pi$ and $\rho$ (Propositions 3.10 and 5.2): $\displaystyle\langle{\operatorname{SB}f,\operatorname{SB}g}\rangle_{\mathcal{B}}=\langle{\operatorname{SB}(\pi(Y)\exp(-2|X|)),\operatorname{SB}g}\rangle_{\mathcal{B}}=\langle{\rho(Y)\operatorname{SB}(\exp(-2|X|)),\operatorname{SB}g}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{\operatorname{SB}f,\operatorname{SB}g}\rangle_{\mathcal{B}}}{}=-\langle{\operatorname{SB}(\exp(-2|X|)),\rho(Y)\operatorname{SB}g}\rangle_{\mathcal{B}}=-\langle{\operatorname{SB}(\exp(-2|X|)),\operatorname{SB}(\pi(Y)g)}\rangle_{\mathcal{B}}$ $\displaystyle\hphantom{\langle{\operatorname{SB}f,\operatorname{SB}g}\rangle_{\mathcal{B}}}{}=-\langle{\exp(-2|X|),\pi(Y)g}\rangle_{W}=\langle{\pi(Y)\exp(-2|X|),g}\rangle_{W}=\langle{f,g}\rangle_{W},$ which proves the theorem. ∎ ### 6.2 The inverse Segal–Bargmann transform ###### Definition 6.7. For $p\in\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$ the inverse Segal–Bargmann transform is defined as $\displaystyle\operatorname{SB}^{-1}p(x):=\exp(-2|X|)\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}p(z)\big{|}_{z=0},$ with $\displaystyle x|\operatorname{\widetilde{\mathcal{B}}}(z):=2x_{0}\operatorname{\widetilde{\mathcal{B}}}(z_{0})+2\sum_{i,j=1}^{m+2n-1}x_{i}\beta^{ij}\operatorname{\widetilde{\mathcal{B}}}(z_{j})=2\sum_{i,j=0}^{m+2n-1}x_{i}\beta^{ij}\operatorname{\mathcal{B}}(z_{j}).$ Note that both $\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}$ and $\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}$ are infinite power sums of the Bessel operator. However, they are well defined operators on polynomials in $z$ since the Bessel operator lowers the degree of the polynomials and therefore the power operators become zero after a finite number of terms. Thus $\operatorname{SB}^{-1}$ is a well-defined operator on $\mathcal{P}\big{(}\mathbb{C}^{m|2n}\big{)}$. Because of Proposition 3.1 it maps $\big{\langle}R^{2}\big{\rangle}$ to zero and thus $\operatorname{SB}^{-1}$ can be restricted to $F$. Moreover, it is also well defined as the inverse of the Segal–Bargmann transform. This follows from the following proposition. ###### Proposition 6.8. The inverse Segal–Bargmann transform is well defined as the inverse of the Segal–Bargmann transform defined in Definition 6.1. ###### Proof. From Proposition 6.4, we know that $\operatorname{SB}$ has an inverse. Suppose the operator $A$ is this inverse. Using Theorem 6.6 we then have the following calculation: $\displaystyle\langle{Ap,\psi}\rangle_{W}$ $\displaystyle=\langle{\operatorname{SB}(Ap),\operatorname{SB}\psi}\rangle_{\mathcal{B}}=(-1)^{|p||\psi|}\overline{\langle{\operatorname{SB}\psi,p}\rangle}_{\mathcal{B}}=(-1)^{|p||\psi|}\overline{\operatorname{SB}\psi}\big{(}\operatorname{\widetilde{\mathcal{B}}}\big{)}p(z)\big{|}_{z=0}$ $\displaystyle=(-1)^{|p||\psi|}\left.\left(\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}\int_{W}\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp(-2x_{0})\overline{\psi(x)}p(z)\right)\right|_{z=0}$ $\displaystyle=\int_{W}\big{(}\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp(-2x_{0})p(z)\big{)}\big{|}_{z=0}\overline{\psi(x)}$ $\displaystyle=\big{\langle}\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp(-2|X|)p(z)\big{|}_{z=0},\psi\big{\rangle}_{W},$ where we used equation (6.1) in the last step. Since the sesquilinear form $\langle{\cdot\,,\cdot}\rangle_{W}$ is non-degenerate we obtain $A=\operatorname{SB}^{-1}$. ∎ We can make the inverse Segal–Bargmann transform more explicit on the space of homogeneous polynomials. To the best of our knowledge, this explicit expression is also new for the bosonic case. ###### Proposition 6.9. For $p\in\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$, $k\in\mathbb{N}$ the inverse Segal–Bargmann transform $\operatorname{SB}^{-1}$ is given by $\displaystyle\operatorname{SB}^{-1}p(x)=\exp(-2|X|)\sum_{j=0}^{k}\dfrac{(-1)^{j}}{j!(k-j)!}\dfrac{\Gamma\big{(}\frac{M}{2}-1\big{)}}{\Gamma\big{(}k-j+\frac{M}{2}-1\big{)}}\big{\langle}z_{0}^{j}(x|z)^{k-j},\overline{p}\big{\rangle}.$ ###### Proof. For $p\in\mathcal{P}_{k}\big{(}\mathbb{C}^{m|2n}\big{)}$ we have $\displaystyle\operatorname{SB}^{-1}p(x)$ $\displaystyle=\exp(-2|X|)\operatorname{\mathbb{I}}_{0}\big{(}x|\operatorname{\widetilde{\mathcal{B}}}(z)\big{)}\exp\big{(}{-}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\big{)}p(z)\big{|}_{z=0}$ $\displaystyle=\exp(-2|X|)\langle{\operatorname{\mathbb{I}}_{0}(x|z)\exp(-z_{0}),\overline{p}}\rangle_{\mathcal{B}}.$ Because of the orthogonality of the Bessel–Fischer product we only need to look at the homogeneous polynomial term of degree $k$ in the expansion of $\operatorname{\mathbb{I}}_{0}(x|z)\exp(-z_{0})$. Explicitly, we need the homogeneous term of degree $k$ in $\displaystyle\Gamma\left(\dfrac{M}{2}-1\right)\sum_{l=0}^{\infty}\dfrac{1}{l!\Gamma\big{(}l+\frac{M}{2}-1\big{)}}(x|z)^{l}\sum_{j=0}^{\infty}\dfrac{(-1)^{j}}{j!}z_{0}^{j},$ which is $\displaystyle\Gamma\left(\frac{M}{2}-1\right)\sum_{j=0}^{k}\dfrac{1}{(k-j)!\Gamma\big{(}k-j+\frac{M}{2}-1\big{)}}(x|z)^{k-j}\dfrac{(-1)^{j}}{j!}z_{0}^{j}$ as desired. ∎ ### 6.3 The generalized Hermite functions As a standard application of the Segal–Bargmann transform, we can construct generalized Hermite functions which extend the ones of the bosonic case given in [18]. ###### Definition 6.10. The generalized Hermite functions on $W$ are defined by $\displaystyle h_{\alpha}(x):=\exp(2|X|)\left(\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}\right)^{\alpha}\exp(-4|X|),\qquad\text{with}\quad\left(\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}\right)^{\alpha}:=\prod_{i}\dfrac{1}{2^{\alpha_{i}}}\operatorname{\widetilde{\mathcal{B}}}(e_{i})^{\alpha_{i}},$ for $\alpha\in\mathbb{N}^{m|2n}$. The generalized Hermite polynomials $H_{\alpha}$ are defined by the equation $\displaystyle h_{\alpha}(x)=H_{\alpha}(x)\exp(-2|X|).$ ###### Proposition 6.11 (Hermite to monomial property). We have $\displaystyle\operatorname{SB}h_{\alpha}=(2z)^{\alpha}.$ ###### Proof. We will use the fact that $\int_{W}$ is supersymmetric with respect to the Bessel operators [6, Proposition 8.9]: $\displaystyle\int_{W}(\operatorname{\mathcal{B}}(x_{k})f)g=(-1)^{|f||k|}\int_{W}f(\operatorname{\mathcal{B}}(x_{k})g).$ Combining this with equation (B.3) of Lemma B.1 we find $\displaystyle\operatorname{SB}h_{\alpha}(z)$ $\displaystyle=\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})h_{\alpha}(x)$ $\displaystyle=\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\prod_{i}\dfrac{1}{2^{\alpha_{i}}}\operatorname{\widetilde{\mathcal{B}}}(x_{i})^{\alpha_{i}}\exp(-4|X|)$ $\displaystyle=\exp(-z_{0})\int_{W}\prod_{i}\dfrac{1}{2^{\alpha_{i}}}\operatorname{\widetilde{\mathcal{B}}}(x_{i})^{\alpha_{i}}\left(\operatorname{\mathbb{I}}_{0}(x|z)\right)\exp(-4|X|)$ $\displaystyle=\exp(-z_{0})\int_{W}\prod_{i}\dfrac{1}{2^{\alpha_{i}}}(4z_{i})^{\alpha_{i}}\left(\operatorname{\mathbb{I}}_{0}(x|z)\right)\exp(-4|X|)$ $\displaystyle=\exp(-z_{0})\int_{W}(2z)^{\alpha}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-4|X|)$ $\displaystyle=(2z)^{\alpha}\operatorname{SB}(\exp(-2|X|))(z).$ Because of Lemma 6.5 the theorem follows. ∎ ## Appendix A Special functions ### A.1 Bessel functions The I-Bessel function $I_{\alpha}(t)$ (or modified Bessel function of the first kind) is defined by $\displaystyle I_{\alpha}(t):=\left(\dfrac{t}{2}\right)^{\alpha}\sum_{k=0}^{\infty}\dfrac{1}{k!\Gamma(k+\alpha+1)}\left(\dfrac{t}{2}\right)^{2k}$ and the K-Bessel function $K_{\alpha}$ (or modified Bessel function of the third kind) by $\displaystyle K_{\alpha}(t):=\dfrac{\pi}{2\sin(\pi\alpha)}(I_{-\alpha}(t)-I_{\alpha}(t)),$ for $\alpha,t\in\mathbb{C}$, see [2, Section 4.12]. In this paper we will need the following renormalisations $\displaystyle\widetilde{I}_{\alpha}(t):=\left(\dfrac{t}{2}\right)^{-\alpha}I_{\alpha}(t),\qquad\widetilde{K}_{\alpha}(t):=\left(\dfrac{t}{2}\right)^{-\alpha}K_{\alpha}(t).$ Remark that we have the following special case [2, equation (4.12.4)] $\displaystyle\widetilde{K}_{-\frac{1}{2}}(t)=\frac{\sqrt{\pi}}{2}\exp(-t).$ (A.1) The asymptotic behaviour of the I-Bessel function can be deducted from equations (4.12.7) and (4.12.8) of [2], $\displaystyle\widetilde{I}_{\alpha}(t)=\dfrac{1}{\sqrt{2\pi}}\big{(}{-}t^{2}\big{)}^{-\frac{2\alpha+1}{4}}\left(\exp\left(-i\left(\dfrac{(2\alpha+1)\pi}{4}-\sqrt{-t^{2}}\right)\right)\left(1+\mathcal{O}\left(\dfrac{1}{t}\right)\right)\right.$ $\displaystyle\left.\hphantom{\widetilde{I}_{\alpha}(t)=}{}+\exp\left(i\left(\dfrac{(2\alpha+1)\pi}{4}-\sqrt{-t^{2}}\right)\right)\left(1+\mathcal{O}\left(\dfrac{1}{t}\right)\right)\right),$ for $|t|\rightarrow+\infty$. The asymptotic behaviour of the K-Bessel function for $t\in\mathbb{R}$ is given in Appendix B.2 of [6], $\displaystyle\mbox{for }t\rightarrow 0\colon\quad\widetilde{K}_{\alpha}(t)=\begin{cases}\displaystyle\frac{\Gamma(\alpha)}{2}\left(\frac{t}{2}\right)^{-2\alpha}+o\big{(}t^{-2\alpha}\big{)}&\mbox{if }\alpha>0,\vspace{1mm}\\\ \displaystyle-\log\left(\frac{t}{2}\right)+o\left(\log\left(\frac{t}{2}\right)\right)&\mbox{if }\alpha=0,\vspace{1mm}\\\ \displaystyle\frac{\Gamma(-\alpha)}{2}+o(1)&\mbox{if }\alpha<0,\end{cases}$ $\displaystyle\mbox{for }t\rightarrow+\infty\colon\quad\widetilde{K}_{\alpha}(t)=\dfrac{\sqrt{\pi}}{2}\left(\dfrac{t}{2}\right)^{-\alpha-\frac{1}{2}}e^{-t}\left(1+\mathcal{O}\left(\dfrac{1}{t}\right)\right).$ ### A.2 Generalised Laguerre functions Consider the generating function $\displaystyle G_{2}^{\mu,\nu}(t,x):=\dfrac{1}{(1-t)^{\frac{\mu+\nu+2}{2}}}\widetilde{I}_{\frac{\mu}{2}}\left(\dfrac{tx}{1-t}\right)\widetilde{K}_{\frac{\nu}{2}}\left(\dfrac{x}{1-t}\right),$ for complex parameters $\mu$ and $\nu$. The generalised Laguerre functions $\Lambda_{2,j}^{\mu,\nu}(x)$ are defined in [16] as the coefficients in the expansion $\displaystyle G_{2}^{\mu,\nu}(t,x)=\sum_{j=0}^{\infty}\Lambda_{2,j}^{\mu,\nu}(x)t^{j}.$ ## Appendix B Proof of Theorem 6.3 To give the proof of Theorem 6.3 we first need a few technical lemmas. ###### Lemma B.1 (properties of $\operatorname{\mathbb{I}}_{0}$). For $k\in\\{1,\ldots,m+2n-1\\}$ we have $\displaystyle\partial_{z^{k}}\operatorname{\mathbb{I}}_{0}(x|z)=2x_{k}\operatorname{\mathbb{I}}_{1}(x|z),\qquad$ $\displaystyle\partial_{x^{k}}\operatorname{\mathbb{I}}_{0}(x|z)=2z_{k}\operatorname{\mathbb{I}}_{1}(x|z),$ (B.1) $\displaystyle\partial_{z^{0}}\operatorname{\mathbb{I}}_{0}(x|z)=-2x_{0}\operatorname{\mathbb{I}}_{1}(x|z),\qquad$ $\displaystyle\partial_{x^{0}}\operatorname{\mathbb{I}}_{0}(x|z)=-2z_{0}\operatorname{\mathbb{I}}_{1}(x|z),$ (B.2) $\displaystyle\mathbb{E}_{z}\operatorname{\mathbb{I}}_{0}(x|z)=(x|z)\operatorname{\mathbb{I}}_{1}(x|z),\qquad$ $\displaystyle\mathbb{E}_{x}\operatorname{\mathbb{I}}_{0}(x|z)=(x|z)\operatorname{\mathbb{I}}_{1}(x|z),$ and for $i\in\\{0,\ldots,m+2n-1\\}$ we have $\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{i})\operatorname{\mathbb{I}}_{0}(x|z)=4x_{i}\operatorname{\mathbb{I}}_{0}(x|z),\qquad\operatorname{\widetilde{\mathcal{B}}}(x_{i})\operatorname{\mathbb{I}}_{0}(x|z)=4z_{i}\operatorname{\mathbb{I}}_{0}(x|z).$ (B.3) The last equation expresses that $\operatorname{\mathbb{I}}_{0}(x|z)$ is an eigenfunction of $\operatorname{\widetilde{\mathcal{B}}}$. ###### Proof. Equation (B.1) follows from the chain rule. Equation (B.2) follows immediately from equation (B.1) and the definition of the Euler operator. Using the same calculation as in the proof of Lemma 4.10, we obtain $\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{k})(x|z)^{l}=4x_{k}l\left(l+\frac{M}{2}-2\right)(x|z)^{l-1}.$ We then find $\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{\mathbb{I}}_{0}(x|z)$ $\displaystyle=\Gamma\left(\dfrac{M}{2}-1\right)\sum_{l=0}^{\infty}\dfrac{1}{l!\,\Gamma\left(l+\frac{M}{2}-1\right)}\operatorname{\widetilde{\mathcal{B}}}(z_{k})(x|z)^{l},$ $\displaystyle=4x_{k}\Gamma\left(\dfrac{M}{2}-1\right)\sum_{l=1}^{\infty}\dfrac{l\left(l+\frac{M}{2}-2\right)}{l!\,\Gamma\left(l+\frac{M}{2}-1\right)}(x|z)^{l-1}$ $\displaystyle=4x_{k}\Gamma\left(\dfrac{M}{2}-1\right)\sum_{l-1=0}^{\infty}\dfrac{1}{(l-1)!\,\Gamma\left((l-1)+\frac{M}{2}-1\right)}(x|z)^{l-1}$ $\displaystyle=4x_{k}\operatorname{\mathbb{I}}_{0}(x|z).$ The calculations for the case $\operatorname{\widetilde{\mathcal{B}}}(x_{k})\operatorname{\mathbb{I}}_{0}(x|z)$ is analogous. ∎ ###### Lemma B.2 (properties of $\exp$). For $k\in\\{1,\ldots,m+2n-1\\}$ we have $\displaystyle\mathbb{E}_{z}\exp(-z_{0})=-z_{0}\exp(-z_{0}),\qquad$ $\displaystyle\mathbb{E}_{x}\exp(-2x_{0})=-2x_{0}\exp(-2x_{0}),$ (B.4) $\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{k})\exp(-z_{0})=z_{k}\exp(-z_{0}),\qquad$ $\displaystyle\operatorname{\widetilde{\mathcal{B}}}(x_{k})\exp(-2x_{0})=4x_{k}\exp(-2x_{0}),$ (B.5) $\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{0})\exp(-z_{0})=(2-M+z_{0})\exp(-z_{0}).\quad$ (B.6) ###### Proof. Equation (B.4) is immediate. For equation (B.6) we have $\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{0})\exp(-z_{0})$ $\displaystyle=(2-M-2\mathbb{E})\partial_{0}\exp(-z_{0})+z_{0}\Delta\exp(-z_{0})$ $\displaystyle=(2-M-2\mathbb{E})\exp(-z_{0})-z_{0}\partial_{0}\partial_{0}\exp(-z_{0})$ $\displaystyle=(2-M+2z_{0})\exp(-z_{0})-z_{0}\exp(-z_{0})$ $\displaystyle=(2-M+z_{0})\exp(-z_{0}),$ while for equation (B.5) we compute $\displaystyle\operatorname{\widetilde{\mathcal{B}}}(z_{k})\exp(-z_{0})=0-z_{k}\Delta\exp(-z_{0})=z_{k}\partial_{0}\partial_{0}\exp(-z_{0})=z_{k}\exp(-z_{0}).$ A similar calculation shows $\operatorname{\widetilde{\mathcal{B}}}(x_{k})\exp(-2x_{0})=4x_{k}\exp(-2x_{0})$. ∎ ###### Lemma B.3 (properties of $\operatorname{SB}$). We have $\displaystyle\mathbb{E}_{z}\operatorname{SB}f(z)=-z_{0}\operatorname{SB}f(z)+\int_{W}\left(x|z\right)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x),$ (B.7) $\displaystyle\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\operatorname{SB}f(z)=\dfrac{1}{2}(2-M+z_{0})\operatorname{SB}f(z)+2\operatorname{SB}(x_{0}f)(z)$ $\displaystyle\hphantom{\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\operatorname{SB}f(z)=}{}-\int_{W}(x|z)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x),$ (B.8) $\displaystyle\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{SB}f(z)=\dfrac{1}{2}z_{k}\operatorname{SB}f(z)+2\operatorname{SB}(x_{k}f)(z)$ $\displaystyle\hphantom{\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{SB}f(z)=}{}-2\int_{W}(z_{0}x_{k}+z_{k}x_{0})\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x),$ (B.9) $\displaystyle L_{0k}^{z}\operatorname{SB}f(z)=-z_{k}\operatorname{SB}f(z)+2\int_{W}\left(z_{0}x_{k}+z_{k}x_{0}\right)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$ (B.10) ###### Proof. Equation (B.7) follows from (B.2) and (B.4). To prove equation (B.8) we apply the product rule (Proposition 3.3) to $-\mathcal{B}(z_{0})(\exp(-z_{0})\operatorname{\mathbb{I}}_{0}(x|z))$ and substituting equations (B.3), (B.6), and the following easily verified identities $\displaystyle\mathbb{E}_{z}(\exp(-z_{0}))\partial_{z^{0}}(\operatorname{\mathbb{I}}_{0}(x|z))=2z_{0}x_{0}\exp(-z_{0})\operatorname{\mathbb{I}}_{1}(x|z),$ $\displaystyle\partial_{z^{0}}(\exp(-z_{0}))\mathbb{E}_{z}(\operatorname{\mathbb{I}}_{0}(x|z))=\left(x|z\right)\exp(-z_{0})\operatorname{\mathbb{I}}_{1}(x|z),$ $\displaystyle z_{0}\sum_{r,s}\beta^{rs}\partial_{z^{r}}(\exp(-z_{0}))\partial_{z^{s}}(\operatorname{\mathbb{I}}_{0}(x|z))=2z_{0}x_{0}\exp(-z_{0})\operatorname{\mathbb{I}}_{1}(x|z).$ Equation (B.9) follows in the same way using equations (B.3), (B.5), and $\displaystyle\mathbb{E}_{z}(\exp(-z_{0}))\partial_{z^{k}}(\operatorname{\mathbb{I}}_{0}(x|z))=-2z_{0}x_{k}\exp(-z_{0})\operatorname{\mathbb{I}}_{1}(x|z),$ $\displaystyle\partial_{z^{k}}(\exp(-z_{0}))\mathbb{E}_{z}(\operatorname{\mathbb{I}}_{0}(x|z))=0,$ $\displaystyle z_{k}\sum_{r,s}\beta^{rs}\partial_{z^{r}}(\exp(-z_{0}))\partial_{z^{s}}(\operatorname{\mathbb{I}}_{0}(x|z))=2z_{k}x_{0}\exp(-z_{0})\operatorname{\mathbb{I}}_{1}(x|z),$ while equation (B.10) follows immediately from $\displaystyle L_{0k}^{z}(\exp(-z_{0})\operatorname{\mathbb{I}}_{0}(x|z))=L_{0k}^{z}(\exp(-z_{0}))\operatorname{\mathbb{I}}_{0}(x|z)+\exp(-z_{0})L_{0k}^{z}(\operatorname{\mathbb{I}}_{0}(x|z))$ $\displaystyle\qquad{}=-z_{k}\exp(-z_{0})\operatorname{\mathbb{I}}_{0}(x|z)+\exp(-z_{0})(z_{0}(2x_{k}\operatorname{\mathbb{I}}_{1}(x|z))-z_{k}(-2x_{0}\operatorname{\mathbb{I}}_{1}(x|z)))$ $\displaystyle\qquad{}=-z_{k}\exp(-z_{0})\operatorname{\mathbb{I}}_{0}(x|z)+2\exp(-z_{0})(z_{0}x_{k}+z_{k}x_{0})\operatorname{\mathbb{I}}_{1}(x|z).$ This proves the lemma. ∎ We can now prove Theorem 6.3. For convenience we restate it here. ###### Theorem B.4 (intertwining property). For $M\geq 4$ the Segal–Bargmann transform intertwines the action $\pi$ on $W$ with the action $\rho$ on $F$, i.e., $\displaystyle\operatorname{SB}\circ\pi(X)=\rho(X)\circ\operatorname{SB},$ (B.11) for all $X\in\mathfrak{g}$. ###### Proof. We will use the decomposition $\mathfrak{g}=J^{-}\oplus\mathfrak{istr}(J)\oplus J^{+}$ to prove equation (B.11) case by case. Case 1: $X=(e_{0},0,0)$. We wish to prove $\displaystyle\operatorname{SB}(2x_{0}f)(z)=\dfrac{1}{2}z_{0}\operatorname{SB}f(z)+\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\operatorname{SB}f(z)+\left(\dfrac{M}{2}-1\right)\operatorname{SB}f(z)+\mathbb{E}_{z}\operatorname{SB}f(z).$ If we substitute (B.7) and (B.8) into the equation the result follows. Case 2: $X=(e_{k},0,0)$, $k\neq 0$. Substituting (B.9) and (B.10) in $\displaystyle\operatorname{SB}(2x_{k}f)(z)=\dfrac{1}{2}z_{k}\operatorname{SB}f(z)+\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{SB}f(z)+L_{0k}^{z}\operatorname{SB}f(z)$ proves equation (B.11) for $X=(e_{k},0,0)$. Case 3: $X=(0,L_{e_{0}},0)$. We wish to prove $\displaystyle\dfrac{1}{2}\operatorname{SB}((2-M)f)-\operatorname{SB}(\mathbb{E}_{x}f)(z)=\dfrac{1}{2}z_{0}\operatorname{SB}f(z)-\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\operatorname{SB}f(z).$ Using (B.8) this becomes $\displaystyle\operatorname{SB}((2-M)f)=\operatorname{SB}(\mathbb{E}_{x}f)(z)-2\operatorname{SB}(x_{0}f)(z)+\int_{W}(x|z)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$ From the proof of [6, Proposition 8.9] it follows that $\int_{W}(\mathbb{E}_{x}+M-2)f=0$ when $\int_{W}f$ is well defined. This gives $\displaystyle\operatorname{SB}((2-M)f)$ $\displaystyle=\exp(-z_{0})\int_{W}(2-M)\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})f(x)$ $\displaystyle=\exp(-z_{0})\int_{W}\mathbb{E}_{x}(\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})f(x))$ $\displaystyle=\int_{W}(x|z)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x)-2\operatorname{SB}(x_{0}f)(z)+\operatorname{SB}(\mathbb{E}_{x}f)(z),$ where we used (B.2) and (B.4) to obtain the last equality. Case 4: $X=(0,L_{e_{k}},0)$, $k\neq 0$. In a similar fashion to (B.10), we find $\displaystyle-\operatorname{SB}\big{(}L_{0k}^{x}f\big{)}(z)$ $\displaystyle=-\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})L_{0k}^{x}f(x)$ $\displaystyle=\exp(-z_{0})\int_{W}L_{0k}^{x}(\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0}))f(x)$ $\displaystyle=2\int_{W}(x_{0}z_{k}+z_{0}x_{k})\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x)-2\operatorname{SB}(x_{k}f)(z).$ Because of (B.9), this is equivalent to $\displaystyle-\operatorname{SB}\big{(}L_{0k}^{x}f\big{)}(z)=\dfrac{1}{2}z_{k}\operatorname{SB}f(z)-\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{SB}f(z),$ proving equation (B.11) for $X=(0,L_{e_{k}},0)$. Case 5: $X=(0,[L_{e_{i}},L_{e_{j}}],0)$, $i,j\neq 0$. Using Proposition 3.10, we obtain $\displaystyle\operatorname{SB}\big{(}L_{ij}^{x}f\big{)}(z)$ $\displaystyle=\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})L_{ij}^{x}f(x)$ $\displaystyle=-\exp(-z_{0})\int_{W}L_{ij}^{x}(\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0}))f(x)$ $\displaystyle=-2\exp(-z_{0})\int_{W}\big{(}x_{i}z_{j}-(-1)^{|i||j|}x_{j}z_{i}\big{)}\operatorname{\mathbb{I}}_{1}(x|z)\exp(-2x_{0})f(x)$ $\displaystyle=\exp(-z_{0})\int_{W}L_{ij}^{z}(\operatorname{\mathbb{I}}_{0}(x|z))\exp(-2x_{0})f(x)$ $\displaystyle=L_{ij}^{z}\operatorname{SB}f(z),$ thus equation (B.11) holds. Case 6: $X=(0,0,e_{0})$. Using (B.7) and (B.8), we can rewrite $\displaystyle\dfrac{1}{2}z_{0}\operatorname{SB}f(z)+\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{0})\operatorname{SB}f(z)+\left(1-\dfrac{M}{2}\right)\operatorname{SB}f(z)-\mathbb{E}_{z}\operatorname{SB}f(z)$ as $\displaystyle 2z_{0}\operatorname{SB}f(z)+2\operatorname{SB}(x_{0}f)(z)+(2-M)\operatorname{SB}f(z)-2\int_{W}(x|z)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$ In a similar fashion to (B.8), we find $\displaystyle\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{0})f\big{)}(z)=\dfrac{1}{2}\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})\operatorname{\widetilde{\mathcal{B}}}(x_{0})f(x)$ $\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{0})f\big{)}(z)}{}=\dfrac{1}{2}\exp(-z_{0})\int_{W}\operatorname{\widetilde{\mathcal{B}}}(x_{0})\left(\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})\right)f(x)$ $\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{0})f\big{)}(z)}{}=2z_{0}\operatorname{SB}f(z)+(2-M)\operatorname{SB}f(z)+2\operatorname{SB}(x_{0}f)(z)$ $\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{0})f\big{)}(z)=}{}-2\int_{W}(x|z)\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$ So we conclude that equation (B.11) holds for $X=(0,0,e_{0})$. Case 7: $X=(0,0,e_{k})$, $k\neq 0$. To show $\displaystyle\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)=\dfrac{1}{2}z_{k}\operatorname{SB}f(z)+\dfrac{1}{2}\operatorname{\widetilde{\mathcal{B}}}(z_{k})\operatorname{SB}f(z)-L_{0k}^{z}\operatorname{SB}f(z),$ we substitute (B.9) and (B.10) into the equation. So we obtain $\displaystyle\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)=2z_{k}\operatorname{SB}f(z)+2\operatorname{SB}(x_{k}f)(z)$ $\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)=}{}-4\int_{W}(z_{0}x_{k}+z_{k}x_{0})\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$ In a similar fashion to (B.9), we find $\displaystyle\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)=\dfrac{1}{2}\exp(-z_{0})\int_{W}\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0})\operatorname{\widetilde{\mathcal{B}}}(x_{k})f(x)$ $\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)}{}=\dfrac{1}{2}\exp(-z_{0})\int_{W}\operatorname{\widetilde{\mathcal{B}}}(x_{k})(\operatorname{\mathbb{I}}_{0}(x|z)\exp(-2x_{0}))f(x)$ $\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)}{}=2z_{k}\operatorname{SB}f(z)+2\operatorname{SB}(x_{k}f)(z)$ $\displaystyle\hphantom{\dfrac{1}{2}\operatorname{SB}\big{(}\operatorname{\widetilde{\mathcal{B}}}(x_{k})f\big{)}(z)=}{}-4\int_{W}(z_{0}x_{k}+z_{k}x_{0})\operatorname{\mathbb{I}}_{1}(x|z)\exp(-z_{0}-2x_{0})f(x).$ This proves the theorem. ∎ ### Acknowledgements SB is supported by a BOF Postdoctoral Fellowship from Ghent University. 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2024-09-04T02:54:55.600968

2002.12895

arxiv-papers

11institutetext: Department of Astronomy, University of Geneva, Ch. d’Ècogia 16, 1290, Versoix, Switzerland 22institutetext: APC, University of Paris, CNRS/IN2P3, CEA/IRFU, 10 rue Alice Domon et Leonie Duquet, Paris, France # An online data analysis system of INTEGRAL telescope A. Neronov 1122 V. Savchenko 11 A. Tramacere 11 M. Meharga 11 C. Ferrigno 11 S. Paltani 11 ###### Abstract Context. During more than 17 years of operation in space INTEGRAL telescope has accumulated large data set that contains records of hard X-ray and soft $\gamma$-ray astronomical sources. These data can be re-used in the context of multi-wavelength or multi-messenger studies of astronomical sources and have to be preserved on long time scales. Aims. We present a scientific validation of an interactive online INTEGRAL data analysis system for multi-wavelength studies of hard X-ray and soft $\gamma$-ray sources. Methods. The online data analysis system generates publication-quality high- level data products: sky images, spectra and light-curves in response to user queries that define analysis parameters, such as source position, time and energy interval and binning. The data products can be requested via a web browser interface or via Application Programming Interface (API) available as a Python package. The products for the IBIS/ISGRI instrument of INTEGRAL are generated using the Offline Science Analysis (OSA) software which is provided by the instrument teams and is conventionally used to analyse INTEGRAL data. The analysis workflow organized to preserve and re-use various intermediate analysis products, ensuring that frequently requested results are available without delay. The platform is implemented in a Docker cluster which allows operation of the software in a controlled virtual environment, and can be deployed in any compatible infrastructure. The scientific results produces by ODA are identical to those produced by OSA, since ODA simply provides a platform to retrieve the OSA results online, while leveraging a provenance- indexed database of pre-computed (cached) results to optimize and reuse the result. Results. We report the functionalities and performance of the online data analysis system by reproducing the benchmark INTEGRAL results on different types of sources, including bright steady and transient Galactic sources, and bright and weak variable extra-galactic sources. We compare the results obtained with the online data analysis system with previously published results on these sources. We also discuss limitations of the online analysis system. Conclusions. We consider the INTEGRAL online data analysis as a demonstrator of more general web-based “data analysis as a service” approach that provides a promising solution for preservation and maintenance of data analysis tools of astronomical telescopes on (multi)decade long time scales and facilitates combination of data in multi-wavelength and multi-messenger studies of astronomical sources. ###### Key Words.: Methods: data analysis – X-rays: general ## 1 Introduction The INTErnational Gamma-RAy Laboratory (INTEGRAL) (Winkler et al., 2003) is an ESA space astronomy mission that has collected data in orbit since 2002. It provides observations of astronomical sources in the keV-MeV energy range. IBIS, the Imager on Board the INTEGRAL Satellite (Ubertini et al., 2003), is a coded-aperture instrument that provides fine imaging (12′ FWHM) in a nearly squared partially coded field-of-view (FoV) of 30${}^{\circ}\times$30∘, source identification and spectral sensitivity to both continuum and broad lines between 15 keV and 10 MeV. Its focal plane is composed of two detectors, optimized for two different energy ranges: ISGRI from 15 to $\sim$1000 keV (Lebrun et al., 2003) and PICsIT from 400 keV to 10 MeV (Labanti et al., 2003). In the following, we will discuss only the ISGRI detector layer, and refer to it as the ISGRI instrument. The Joint European X-Ray Monitor (JEM-X Lund et al., 2003) provides images, timing, and spectral information in a lower energy range (3–25 keV). It has a narrower circular FoV $\sim 10^{\circ}$ diameter. The spectrometer SPI (Vedrenne, G. et al., 2003) provides high-resolution spectroscopy data with moderate angular resolution. The SPI instrument is surrounded by an Anti-Coincidence Shield (ACS) that reduces the level of background in SPI and simultaneously works as an all-sky Gamma-Ray Burst (GRB) monitor (Savchenko et al., 2012). The wide FoV of IBIS inherently includes data on a large number of astronomical sources, which are not necessarily the targets of specific observational proposals. This means that all source-specific observational campaigns of INTEGRAL possess large potential for serendipitous science. All INTEGRAL data become publicly available after a one-year proprietary period, so that analysis of all the sources found in the field of view is possible. Astronomical sources visible in X-ray and soft $\gamma$-ray band are variable on different time scales, from milliseconds to years and decades van der Klis (2006); Beckmann et al. (2007); Mészáros et al. (2019). In this respect, the 17-year-long data-set of INTEGRAL is unique, as it contains the information on long-term variability patterns of a large number of sources. Some sources spend long periods in quiescent states producing little emission and exhibit occasional outbursts or flares on irregular basis with a duty cycles spread over years or decades. The INTEGRAL archive contains data taken nearly continuously since min November 2002 with a duty cycle on nearly 90% -instruments are switched off at each perigee passage through the Earth radiation belts. Until March 2003, on-board setting were continuously adjusted to optimize the performance, making the scientific analysis a challenging. The general user is normally suggested to start using archive data after this date finding nearly 50 Ms of usable data in the direction of the Galactic center, the region with the highest exposure. Archive data contain, thus, information on previous history of quiescence and activity of sky sources ($>1200$ detected by the imager so far), including those which might undergo outbursts in future. The INTEGRAL data and Offline Science Analysis (OSA) software that includes data analysis pipelines for all instruments, including ISGRI (Goldwurm et al., 2003) and JEM-X (Westergaard et al., 2003) is distributed by the INTEGRAL Science Data Centre (ISDC, Courvoisier et al., 2003). The main components of OSA have been developed in the period preceding the INTEGRAL launch and are maintained by the ISDC. The architecture of OSA was optimized for computing environments that were common more than 20 years ago. INTEGRAL was initially planned to operate in space for five years and generate relatively small data sets. The only solution for data processing was via local installation of OSA on a user computer. This is not necessarily the case today for data sets spanning 17 years. Their analysis requires a significant amount of computing resources. Moreover, maintenance of legacy software on evolving operating systems poses more and more challenges. The development of high-performance computing (HPC) and cloud computing (CC) technologies and their applications to astronomical data management (Banek et al., 2019; Smith et al., 2019) over the last decades opens up a new possibility of deployment of OSA-based data analyses without the need for local installation of the software, and provides access to large pool of computing resources, thus significantly reducing the complexity of the analysis of INTEGRAL data. Such online data analysis (ODA) system for the ISGRI instrument has been recently developed at the Department of astronomy of the University of Geneva111https://www.astro.unige.ch in synergy with the INTEGRAL Science Data Centre (ISDC)222https://www.isdc.unige.ch/integral/, and is maintained jointly with the François Arago Centre (FACe) of Astroparticle and Cosmology laboratory in Paris333http://www.apc.univ-paris7.fr/FACe/home. This system entirely relies on the official OSA package, provided and validate by the instrument teams, integrated and distributed by by ISDC - all of the results are not only equivalent, they are identical. This design allows to leverage, in principle, the unrestrained power of the OSA software, preserving and maintaining the complete potential of the INTEGRAL data for the future explorers, without making assumptions about which products are likely to be useful. While this approach may require larger real-time computing resources, the platform exploits a dynamic innovative provenance-based database of precomputed products to eliminate analysis duplication and flexibly respond to the user requests, as detailed in Section 2.2 and Section 5. This means, in principle, that the ODA validation is in part redundant, since it is equivalent to OSA validation. On the other hand, this paper reveals how a selection of particular OSA cookbook threads, adopted in ODA, allows to reproduce the published results. In the following, we describe the system and characterize its performance via comparison of science results obtained using the system with previously published benchmark INTEGRAL results on a variety of Galactic and extra- galactic sources. We also demonstrate the usefulness of the system in the context of multi-wavelength studies and discuss advantages and limitations of this online data analysis implementation. Throughout the paper, we provide direct links to the code producing the figures in the text, showing the potential of this approach for the open access, reusability and reproducibility of science results. ## 2 Online data analysis interface Figure 1: Architecture of the INTEGRAL Online Data Analysis system. The general scheme of the ODA is shown in Fig. 1. The user has several possibilities to submit requests for analysis of INTEGRAL data: * • through a web browser by accessing the ODA website444https://www.astro.unige.ch/cdci/astrooda_ on his/her local computer and entering the analysis parameters into the parameter boxes, or * • directly specifying analysis parameters in a URL link to the ODA website (examples are given in the next section) or * • through an ODA Application Programming Interface (API), oda_api555https://github.com/oda-hub/oda_api, e.g., from a Jupyter Notebook, also on their local computer. The full process can be schematized as follows: 1. 1. The request can be sent using the front-end or the oda_api client. Both interfaces verify the syntactical correctness and completeness of user queries. 2. 2. Requests arrive at dispatcher and here processed by an internal abstraction process, which implements classes (interfaces) for specific instruments data products, such as spectra or light curves data, and post-processing products, such as mosaic images, light curves images, spectral fits. 3. 3. Each data product interface communicate with a specific backend, i.e. data server(s) implemented as Docker666https://hub.docker.com/r/odahub/ddosa- interface containers running OSA in service mode. The containers are currently deployed locally on the HPC resources of the University of Geneva, but they could be deployed on any other HPC or CC services. 4. 4. The request can be either synchronous or asynchronous. In the latter case, a continuous report from the backend brings information to the dispatcher regarding the process status until the product is ready. In the former case, a single report is provided. 5. 5. Data products provided by the data server upon analysis requests are stored in a data repository and are made available to the dispatcher. 6. 6. In the current version of ODA, the dispatcher also performs post-processing and visualisation of the data, using specific services, providing high-level products to be displayed on the front-end. 7. 7. Final products are available to the user either through the front-end or through the client API. The front-end displays sky images, spectra, light curves, and source catalogs in a virtual-desktop environment embedded in the web browser, providing the possibility to download the data products to the local user computer. ### 2.1 Front-end interface – Dispatcher – Data Server interactions Figure 2: Left: General parameter space of ODA front-end. Right: example of instrument-specific parameter field for ISGRI telescope. IBIS and JEM-X are coded-mask instruments that rely on a dithering pointing strategy with individual exposures called Science Windows (ScW) lasting 0.1–1 hour. Reconstruction of sky images creates a catalog of detected sources. This catalog should be used during the extraction of spectra and light curves of specific sources. Disentangling of signals from different sources in the field of view requires a sky model which lists positions of all possible sources. (see Goldwurm et al., 2003, for details). The default ODA workflow allows the user to select the data set, obtain a catalog of detected sources by reconstructing an image, and then manipulate this catalog to extract spectra and light curves. The data processing is initiated in response to the user queries defining the analysis parameters (Fig. 2). These queries include at the very minimum: * • Source name or sky coordinates * • Time interval for the data or Science Window (ScW) list The front-end is able to resolve astronomical source names by accessing two of the main astronomical databases (SIMBAD and NED). It accepts time parameters in several different conventional formats. The list of ScWs can be specified as a comma-separated list of their unique identifiers. The ScW data base is separately accessible through the w3browse interface on the web pages777https://www.isdc.unige.ch/integral/archive. Apart from these generic query parameters, the front-end allows the user to specify parameters that are specific to the INTEGRAL instruments: ISGRI, JEM-X, SPI-ACS. An example of parameter field for ISGRI is shown in the right panel of Fig. 2. For ISGRI and JEM-X, it is possible to specify: * • one of the two currently used versions of OSA: 10.2 and 11.0; * • radius of the “region of interest” within which pointing data are selected (which depends on the instrument field-of-view); * • One of the two units of the JEM-X instrument (JEM-X1 or JEM-X2); * • type of the data product (image, spectrum or light curve); * • energy range of the image and light curve. It should be noted that the spectrum is always computed in the full energy range with predefined resolution (16 channels for JEM-X, 128 channels for ISGRI; * • minimal detection significance of sources in the output catalog from imaging; * • time binning for the light curve; * • source catalog to be used in spectral and timing analyses. In a similar way, the parameters can also be specified in the API requests using the oda_api Python package. For example, OSA version is specifiable by setting a parameter osa_version=’OSA10.2’ in the API requests888See oda_api documentation at https://oda-api.readthedocs.io/en/latest/ for full details.. The front-end provides a display of the high-level data products (images, spectra, light curves, source catalogues) through a virtual desktop environment. It also provides the possibility of performing post-analysis of the data, like, e.g., fitting spectra with XSPEC spectral models, using public domain rendering packages999https://bokeh.pydata.org/en/latest/. ### 2.2 Data analysis and storage organization The ODA infrastructure is using the online archive of INTEGRAL raw data, provided by the ISDC101010http://www.isdc.unige.ch/integral/archive. The data server’s task is to provide high-level data products corresponding to the user requests received through the dispatcher. Running OSA is time consuming (about 50 CPU-hours for a spectrum in a typical short transient observation, or of the order of 2000 CPU-hours for an analysis of historic data for a typical source), so as far as possible it is desirable to keep pre-computed products for future (re-)uses. However, it is not possible to store high-level data products for all imaginable combinations of user input parameters. In ODA, the permanent archive of the raw data is complemented by an analysis cache containing high- and intermediate-level data products that are added or removed depending on the the user demands averaged over certain time periods. The cache storage is organized according to the data lineage (e.g. Ikeda & Widom, 2009), which is a specific case of data provenance (Gupta, 2009). The data lineage metadata comprises the information on the sequence of analysis steps (the analysis or workflow nodes) undertaken to produce given high- or intermediate-level data products. The ontology of the workflow nodes prescribes specific metadata associated with each step, and induces the collection of metadata of the final product. The lineage metadata of the cache storage contains all relevant properties of all stored data products, and only this information. This provides a possibility to re-use previously produced intermediate-stage data products for the processing of new data analysis requests by users. New data analysis workflows typically do not start processing of raw data from “from scratch”. Instead, they are formed from a combination of parts of already available workflows derived from specific intermediate or high-level data products stored in the cache, together with the provenance DAG (Directed Acyclic Graph) metadata. This approach provides an efficient way to speed-up data processing following user requests if those are repetitive or recursive or if the requests are nearly identical to those done by previous users with only moderate parameter modifications. Efficient reuse of parts of the OSA based data analysis workflow is enabled by the re-organisation of OSA in data analysis units expressed as Python classes, following the Declarative Data Analysis (DDA) approach inspired by the principles of functional programming. This development was driven by the needs of efficiently managing the data of INTEGRAL together with the information on the 17-year history of the telescope operations: by 2018, in the raw-data archive, there are about $10^{3}$ different types of data occupying 20 TB in some $2\times 10^{4}$ files. The re-factored OSA implementing the DDA approach (called DDOSA111111See https://github.com/volodymyrss/dda-ddosa/ for implementation details.) follows a simple logical scheme suitable for reproducibility of the analysis. Each analysis unit is a pure function of its input data, meaning that it depends only on its own explicit input. It transforms the input data into other data products. Any data are uniquely identified by a tree of connected analysis units that were used to produce it, or, equivalently, by its DAG “provenance graph”. In other words, DDOSA uses provenance as a data identifier (see Savchenko, 2020, for more details). The high-level data products associated to very large analysis chains may be eventually associated with a very large provenance graphs. An example of the provenance graph for a single ScW image high-level data product is shown in Fig. 3. Figure 3: Example of high-level provenance graph for a sky image derived from a single INTEGRAL/ISGRI ScW. The DAG provenance graph approach for data identification at different analysis levels is optimal not only for caching frequently re-used intermediate analysis step results, but also for the parallelization of the analysis. The DAG structure of DDOSA workflows implies the presence of different independent branches of analysis that can be naturally executed independently in a distributed environment. This is taken into account in the system of job scheduling. For each analysis unit, execution requests originate either from the users (via dispatcher) or other analysis units. Each request processing starts from the evaluation of the request, resulting either in the retrieval of the requested products from the storage cache or in the delegation of the request to a local or remote executor, a process which is transparent from the point of view of the request. A simple scheduling layer has been implemented following this approach. The advantage of this scheduler is the straightforward treatment of complex dependencies. ## 3 Benchmark analysis results The web-based ODA interface retains all the functionalities of OSA and could be used to obtain publication quality results of analysis of INTEGRAL observations with no difference from what an experienced user can obtain running locally OSA. In this section, we demonstrate the performance of the ODA based analysis on benchmark cases, by showing that the results obtainable with ODA are compatible with previously published results or improve on them, owing to upgraded algorithms and calibration. ### 3.1 Crab pulsar and nebula The Crab pulsar and Nebula complex is one of the brightest sources on the X-ray sky (Hester, 2008). Because of this property and its flux stability, it is often used as a “calibration” source in high-energy astronomy. INTEGRAL observations of Crab are reported in a number of publications (e.g, Mineo et al., 2006) and, in particular, in the context of the study of the long-term variability of the source emission (Wilson-Hodge et al., 2011). We verify the performance of ODA by reproducing the published results on Crab variability and extending the evolution study over a 15 year time period. It can be noted that the magnitude and details Crab of variability as observed by ISGRI is not identical to those reported by other instruments. The excess variability in in part due to the systematic uncertainties in ISGRI calibration, and in part due to the difference in the instrument energy ranges. Detailed evaluation of if OSA reconstruction of ISGRI observations as well as discussion ISGRI calibration challenges is beyond the scope of this work. Here we demonstrate that ODA reproduces the best available OSA results. ODA will naturally follow any upgrades to OSA software and calibration files in the future. Figure 4: Mosaic image of Crab region extracted from a sample of 50 randomly selected ScWs, together with the display of the catalog of detected sources. The result could be re-generated using the URL https://doi.org/10.5281/zenodo.3634648. The ODA interface is currently limited to single requests for data analysis based on no more than 50 science windows (ScWs), to limit the waiting time on the available resources (see Section 4 for details, future plans, and a work- around). If the requested time span of data extends over more than 50 ScWs, random selection of ScWs within the specified time limits is performed. This is the case for the results reported in Figs. 4, 8, and 5. The time interval of the analysis for this specific ScW subset is 2003-03-15 23:27:40.0 to 2018-03-16 00:03:15.0 UTC, spanning over more than 15 years. Pointings within 15 degrees from Crab position are selected. Our example Crab image could be accessed or re-generated by directly re- launching the analysis via an URL https://www.astro.unige.ch/cdci/astrooda_ in which the analysis parameters are specified after the sign `{?}` and separated by the `{\&}` sign as, for example: src_name=Crab&RA=83.633080&DEC=22.014500&radius=15 for the source name and/or sky coordinates and region of interest specification, &T1=2003-03-15T23:27:40.0 &T2=2018-03-16T00:03:15.0 &T_format=isot for the time interval, &E1_keV=20 &E2_keV=40 for the energy interval, &instrument=isgri &osa_version=OSA10.2 &product_type=isgri_image for the instrument, software version and data product specification (spaces are added for readability, but should be removed in the actual query). The parameter detection_threshold=7.5 will result in display of the sources detected at significance level higher than $7.5$. The analysis with specified parameters is launched automatically, as soon as the instrument parameter is defined: &instrument=isgri. The parameters that are not explicitly specified in the parameter field of the URL are fixed to their default values. The executable URL with all specified parameters for each data product could be obtained by pressing the “Share” button displayed in the data product window on ODA frontend (see Fig. 4). This example of analysis could also be launched from a python interface on the user laptop (e.g., from a shell or a Jupyter notebook) by providing parameters to the request from oda_api.api import DispatcherAPI disp=DispatcherAPI( host= ’www.astro.unige.ch/cdci/astrooda/dispatch-data’) disp.get_product( RA=83.633080, DEC=22.014500, radius=15, T1=’2003-03-15T23:27:40.0’, T2=’2018-03-16T00:03:15.0’, E1_keV=20.0, E2_keV=40.0, instrument=’isgri’, product=’isgri_image’, osa_version=’OSA10.2’) The API code for each data product can be obtained directly by pressing the “API code” button in the product window on the ODA front-end (see Fig. 4). A crucial part of the imaging analysis is the search for significantly detected sources both in individual ScWs and in the mosaic image. Setting the source detection threshold to $7.5\sigma$ (a parameter in the web form of ODA) results, in our example, in the detection of four sources displayed in the image in Fig. 4. Details of the source detection are available in the catalog display accessible through a button from the image display panel, as shown in Fig. 4. Occasionally, sources may have multiple appearances in the catalog display, because this table combines several output catalogs of the standard INTEGRAL analysis, namely results of the search of sources in the mosaic and in individual ScWs. This might be important, because some flaring sources are detectable in individual ScWs during short time periods, but are not detectable in mosaic images with longer exposure times (as it is typical for bursting and transient sources, see e.g..Mereghetti et al. 2020 for a recent example). The user is asked to carefully inspect the output catalog from the imaging step and adjust the source selection for the following spectral and timing analyses. Imaging, spectral extraction and timing routines of OSA use catalogues of sources to match the shadow patterns corresponding to these sources on the detector plane. The catalog used for imaging, spectral or timing analysis could be explicitly specified in the “User catalog” parameter window in the parameter panel. If no user catalog is specified, the default general INTEGRAL catalog is used. This is advisable for the imaging products, but sub-optimal for the extraction of spectra and light curves, which relies on fitting a sky model on the shadowgram. If this sky model is redundant, the fitting becomes more problematic, resulting in unreliable flux determinations. The user can edit the catalog entries in the display of the catalog output of the imaging step. This display also has a “Use catalog” button, which would push the edited catalog to the “User catalog” to be used at the subsequent stages of analysis. The catalog can also be defined explicitly in the form of a python “dictionary” in the URL parameter field. The correctly formatted catalogue embedded in the URL can be obtained by clicking the “Share” button next to the displayed data product. The display of results of spectral analysis for all sources listed in the catalog (user custom catalog or the output catalog from the imaging step analysis) provides a possibility to choose a spectral model for fitting the spectrum121212Based on Xspec package fitting (https://heasarc.gsfc.nasa.gov/xanadu/xspec/).. The display of the spectrum together with the fitted model also provides the details of the fit, such as the model parameter values with their uncertainties. Binning of the spectra is performed only at the plotting stage, the fit is performed on the spectrum at full resolution, which can be downloaded from the web interface. We notice that the 20-30 keV band is affected by the long-term evolution of the ISGRI response, as the ISGRI energy threshold is gradually increasing with time and low-energy events are lost. For consistency, only data above 30 keV are thus automatically fitted in the web interface, but data at lower energy are available, upon download of the FITS-format spectral file131313https://www.isdc.unige.ch/integral/download/osa/doc/10.2/osa_um_ibis.pdf. Fig. 5 shows the 30–100 keV light-curve of the source during a 17-years time span, extracted from the same set of 50 random ScWs and binned into ScW-long time bins. The figure shows the fit of the light-curve with constant and linearly changing flux models. There is a noticeable decrease of the source count rate toward later times, which becomes especially pronounced after MJD 56800 (mid-2014). Superimposed onto this instrumental trend, there is the true variability of the Crab nebula studied by Wilson-Hodge et al. (2011). Such rapid decrease of the count rate is due to the decrease of the instrument response at low energy and is not corrected in version 10.2 of OSA, because calibration algorithms were not able to correct this rapid evolution and calibration files were frozen at this moment. The correct instrument response after MJD 56800 is provided by version 11.0 of OSA with the relative calibration files141414It is foreseen that the OSA11 software release will cover the full mission at the end of 2020.. See Section 4 for details on the instrumental effects contributing to these results. Figure 5: Crab long-term (15 year time span) lightcurve extracted from a sample of 50 randomly selected ScWs in 30-100 keV band, using OSA10.2. The lightcurve is generated via Crab_spectra_lc.ipynb via oda-hub. Fig. 6 shows a comparison of the long-term variability of the Crab flux in the 30–100 and 100–300 keV ranges for 17 years of INTEGRAL operations together with the ones measured by Swift/BAT and Fermi/GBM telescopes (Wilson-Hodge et al., 2011). To produce this figure, we select random sets of 50 ScWs spanning one year time intervals and extracts ScW-by-ScW lightcurves by specifying 10 ks time steps in the ODA parameters for lightcurve time binning (this is longer than the duration of one ScW). These lightcurves are subsequently averaged into the time bins used by Wilson-Hodge et al. (2011) for comparison. For this workflow, we exploited the API access to ODA platform, as coded in the Crab_lc_longterm.ipynb Jupyter notebook, which is part of the https://github.com/oda-hub/oda_api_benchmark GitHub repository. In Sect. 3.4, we detail how to run online Jupyter notebooks as the one used to produce Fig. 6. Figure 6: Evolution of the Crab count rate over 17 years of INTEGRAL operations in 30–100 keV (top panel), 100–300 keV (bottom panel) energy ranges, compared to those observed by Swift/BAT (magenta) and Fermi/GBM (green data points) (Wilson-Hodge et al., 2011). Grey data points show the lightcurve binned in ScW by ScW time bins by the ODA. Blue blue data points show the rebinned individual ScW measurements. The reader can launch the Jupyter notebook Crab_lc_longterm.ipynb used to generate these long-term light curves via API access to ODA. From Fig. 6 one can see that the lightcurve extracted using OSA10.2 and OSA11.0 in their time intervals of validity are compatible with the lightcurves of Swift/BAT and Fermi/GBM, after they have been normalized to their average level. Intrinsic variability of the Crab nebula (at $\sim 5\%$ level) can be appreciated in the general trend of all instruments, as done by Wilson-Hodge et al. (2011). There is, however, some residual differences between the INTEGRAL and Swift/BAT or Fermi/GBM lightcurves, which can be used to estimate the systematic cross-calibration uncertainty. This amounts up to 10% in the 100-300 keV energy range. Count rate light curve have variations that are also due to the evolution of the instrument gain; these are accounted for using response files to recover the intrinsic source flux in the spectral extraction stage. The ODA interface allows us to extract also images, spectra and lightcurves of JEM-X instrument, using the same set of parameters as for ISGRI. Fig. 7 shows the 3-20 keV image of Crab extracted from 50 randomly selected ScWs with pointing directions within $5^{\circ}$ from Crab (to assure that the source is in the field-of-view of JEM-X. Figure 7: Image of sky region around Crab obtained by JEM-X1 instrument in the energy range 3-20 keV. The image can be regenerated via URL https://doi.org/10.5281/zenodo.3832080. Crab is detected with significance in excess of $10^{3}\sigma$ in the image. Two sources are detected with significance larger than 20 in the image: Crab and an X-ray binary 1A 0535+262. We use a catalog containing these two sources for the spectral extraction. Fig. 8 shows combined ISGRI $+$ JEM-X1 unfolded spectra of the spectrum of Crab, extracted from the 50 ScW data sets on year-by-year basis from 2003 to 2018. We have modeled the spectra with a broken power law with two break energies fixed at 20 and 100 keV. The former value is meant to catch possible differences between JEM-X1 and ISGRI spectral responses; the latter is a reasonable approximation for the most probable spectral shape of Crab (Jourdain & Roques, 2009). We applied a systematic factor of 2% to all spectral data, limited JEM-X1 data between 5 and 20 keV, ISGRI data between 20 and 300 keV. The number of degrees of freedom is 55 for OSA10.2 (before 2016) and 112 after 2016 (OSA11.0). Fig. 9 shows the year-by-year change of the Crab spectrum measurements: our results are roughly in-line with previous findings (e.g., Mineo et al., 2006). Formally non-acceptable fits are present in 2003 and 2005 with an anomalous power-law spectral index $\Gamma_{2}$ (20–100 keV), due to a change of low threshold in the ISGRI detector, which is currently taken into account in a sub-optimal way by the calibration files. Similarly, an anomalous low flux in 2017 for ISGRI is due to calibration issues being currently investigated by part of our team in a parallel effort. Figure 8: Display of the Crab unfolded spectra extracted during each year from 2003 to 2018 from a samples of 50 randomly selected ScWs. The fitted model (in black) is a double broken power law with breaks fixed at 20 and 100 keV. The analysis results could be re-generated through oda-hub notebook Fit-Crab- Spectra.ipynb run after Crab_spectra_lc.ipynb. Figure 9: Best-fit spectral parameters with uncertainty intervals at 68% confidence level on a single parameter and the reduced $\chi^{2}$ fit statistics of the Crab pulsar and nebula ISGRI $+$ JEM-X1 spectra averaged on a year-by-year basis; the first point refers to 2003 and the last one to 2018. Data range is from 3.5 to 20 keV for JEM-X and 20–300 keV for ISGRI. The model is a broken power law with break energies fixed at 20 and 100 keV; $\Gamma_{1-3}$ are the spectral indexed at increasing energies. We allowed spectra to assume different normalization through the flux parameter that we report for 5–20 keV (JEM-X1) and 20-100 keV (ISGRI). The resulting number of degrees of freedom is 55 before 2016 and 112 after 2016. ### 3.2 Extremely bright source: V404 Cygni Figure 10: Significance map of V404 Cygni region in 20-40 keV band. The image can be re-generated via URL https://doi.org/10.5281/zenodo.3634669. V404 Cygni is a microquasar that underwent a spectacular outburst in 2015 during which the source flux has reached 50 Crab (Sánchez-Fernández et al., 2017; Rodriguez et al., 2015). Such high flux might pose challenges for data analysis because of saturation or pileup effects that need to be properly taken into account. To validate the performance of ODA for the bright source case, we have reproduced the results of the analysis of V404 Cyg reported by Rodriguez et al. (2015). At the first stage we assess the overall evolution of the source throughout the activity period that lasted from 2015-06-20T15:50:00.0 UTC to 2015-06-25T04:05:59.0 UTC, analyzed by Rodriguez et al. (2015). Entering this time interval into the ODA interface via the the URL, we let the system select randomly 50 ScWs for the analysis with pointing directions within 10 degrees from the source direction and produce a mosaic image shown in Fig. 10. The source is the brightest one in the image detected with significance exceeding 1000$\sigma$. The next brightest source in the field is Cyg X-1. A very strong source produces “ghost” images due to the specific of the coded mask imaging technique. This could be readily seen in the example of V404 Cyg by extracting all the sources in the image detectable at significance threshold above 5$\sigma$: this would result in very large amount of “ghosts” that would appear in the resulting catalog as “NEW” sources. Figure 11: JEM-X1 (top) and ISGRI (bottom) Lightcurves of V404 Cygni during 2015 flaring period (red data points). Black lines are from Ref. (Rodriguez et al., 2015). Fig. 11 shows the JEM-X1 and ISGRI lightcurves of V 404 Cyg extracted for this period. Given the problem of detection of “ghost” sources around a very bright source, it is important to correctly define an input source catalog for the light curve production. Otherwise, the catalog that would be produced based on the imaging step of the analysis would include all the “ghost” sources in the procedure of fitting the detector image, which would lead to either wrong flux calculations or larger error estimates compared to the flux measurements including only real sources. From Fig. 11, one can notice that the source underwent several short time flares. The overall evolution of the source flux inferred from ODA is practically indistinguishable from that calculated by with that reported by Rodriguez et al. (2015). This is clear from direct comparison of the two results, shown in Fig. 11, where black color shows the lightcurve of Rodriguez et al. (2015) and red color shows the result extracted using ODA. ### 3.3 Crowded field: GX 5$-$1 Figure 12: Zoom of the significance map of the sky region around GX 5$-$1 in the 20–30 keV energy band obtained from a selection of 44 science windows belonging to the GPS and GCDE programs carried out in 2013 with a poiting offset of less than three degrees from GX 5$-$1\. This image can be obtained from the following URL. To verify the performance of ODA in the crowded field, we consider an example of GX 5$-$1, a persistent bright low-mass X-ray binary with a neutron star. It is located in the inner Galaxy region in a “crowded” field with many bright sources. In such situation, OSA has to take into account simultaneously all the bright sources while modeling superposition of the shadows of different sources on the detector. If this is not done properly, the signal of the source of interest could be contaminated by the overlapping shadows of unaccounted sources. Moreover, if bright sources are not included in the sky model for spectral or light curve extraction, their photons will be erroneously assigned to other sources or background, hampering a reliable estimate. However, neglecting weak sources is a minor concern if one is interested in bright sources, as contamination can reach at most a few percents of the contaminant’s flux in the worst cases. To provide a direct comparison with the published results by Paizis et al. (2005), we reproduced their selection of 44 science windows when the source was observed by the “GPS” and “GCDE” programs in 2013 with a maximum offest of three degrees. We made first an image in the 20–30 keV band (Fig. 12) to select sources with a minimum significance of 7 for the subsequent spectral extraction. This resulted in a catalog of 25 sources. It should be noted that, owing to OSA construction, some sources might appear multiple times in the catalog, so it is necessary to delete duplication from the catalog used as input in the following steps. We performed the spectral extraction of both JEM-X2 amd ISGRI data using the python API with the same selection of science windows. The dead-time and vignetting corrected exposures for JEM-X2 and ISGRI are 40.6, and 57.2 ks, respectively. We added 1% flux systematic to ISGRI and 3% to JEM-X2, we ignored JEM-X2 data below 3 and above 20 keV; ISGRI data below 20 and above 70 keV, owing tot he partucular spectral shape. We modelled the joint spectrum with the same spectral models of Paizis et al. (2005): the “western” model made of a compTT+bbodyrad and the “eastern” model, made of compBB+diskbb using Xspec. We have also introduce a cross-normalization factor fixed to one for ISGRI, to account for non-simultaneity of some data. We determined the 1$\sigma$ confidence ranges of parameters using a Monte Carlo Markov Chain with the Goodman-Weare algorithm (Goodman & Weare, 2010) as implemented in Xspec v. 12.11.0, and taking the 16, 50, 84% percentiles of the posteriors as lower, central, and upper values. For the chain, we used 40 walkers, a length of 26 000 and a burn-in phase length of 6000. Results are presented in Table 15: both model represent well the spectra. However, the evolution of instrument calibration and analysis algorithms has lead to significant differences in the parameter values between the 2005 work, made with OSA v4.2 and ours, performed with OSA v.10.2. This is inherent to OSA and not directly related to the interface used to extract these spectra. A python notebook with the full workflow is available at this URL. Table 1: Best-fit parameters of spectral modelling of GX 5$-$1 in analogy with Paizis et al. (2005). Western model --- $kT_{\mathrm{BB}}$ | 2.05 | $\pm$ 0.10 | keV normBB | 26 | ${}_{-8}^{+12}$ | $(\mathrm{km/d_{10}})^{2}$ $T_{0}$ | 1.06 | $\pm$ 0.03 | keV $kT_{e}$ | 5.4 | $\pm$ 0.5 | keV $\tau$ | 1.5 | ${}_{-0.2}^{+0.3}$ | norm | 1.1 | $\pm$ 0.1 | factor | 0.76 | $\pm$ 0.03 | $\chi^{2}$/d.o.f. | 18 | /19 | Eastern model $T_{\mathrm{in}}$ | 1.99 | $\pm 0.04$ | keV normdisk | 131 | ${}_{-10}^{+11}$ | $(\mathrm{km/d_{10}})^{2}$ $kT$ | 3.0 | ${}_{-0.1}^{+0.2}$ | keV $\tau$ | 0.27 | ${}_{-0.12}^{+0.08}$ | norm | 4.1 | ${}_{-1.5}^{+2.0}$ | factor | 0.74 | ${}_{-0.04}^{+0.03}$ | $\chi^{2}$/d.o.f. | 19 | /20 | Flux (5–20 keV)aa$a$Fluxes from both models are compatible within the uncertainties, so we report them only once. | 1.28 | $\pm 0.02\times 10^{-8}$ | $\mathrm{erg\,s^{-1}\,cm^{-2}}$ Flux (20–100 keV)aa$a$Fluxes from both models are compatible within the uncertainties, so we report them only once. | 3.97 | $\pm 0.06\times 10^{-10}$ | $\mathrm{erg\,s^{-1}\,cm^{-2}}$ 151515 Figure 13: Time-averaged spectrum of GX 5–1 extracted from the same 44 science windows as Paizis et al. (2005), collected in 2003 with a maximum pointing offset of 3∘ from the source. Red stars and black circles represent JEM-X2 and ISGRI data, respectively. ### 3.4 3C 279 In spite of its large bolometric luminosity and powerful jet (Hayashida et al., 2015), the active galactic nucleus (AGN) 3C 279 is a weak source for INTEGRAL. It has hard spectrum in the hard X-ray energy band. Its flux is at the level of the sensitivity limit of ISGRI and its detectability depends on the source state. One of the flaring episodes occurred in 2015 and INTEGRAL observations of 3C 279 during this episode were obtained as a Target-of- Opportunity campaign. The results of data analysis for this TOO are described by Bottacini et al. (2016). Following the same approach as for other sources, we first performed an exploration of the source behavior throughout the 15 year time span of the data. However, this source would likely be only marginally detected in any 50 ScW exposure, and no assessment of the variability pattern is possible in this way. Instead, the full dataset has to be explored to find the periods during which the sources is detected. We use the API access to ODA to work with the datasets longer than 50 ScWs in the follwoing way. As for the Crab lightcurve case in Sect. 3.1, the Jupyter notebooks for the 3C 279 analysis can be launched from the oda-hub/oda_api_benchmark GitHub repository, which is integrated with the Binder interactive notebook service161616https://mybinder.org. Launching the binder using the “launch binder” button in the oda-hub/oda_api_benchmark repository and choosing the notebooks for 3C 279 lightcurve and spectra found in examples makes it possible to generate the results described below online. At the first stage of analysis, we determine the bright sources in the source field. We generate a mosaic image of the field and use the output catalog of the mosaic analysis, adding explicitly 3C 279 to the catalog, as an input step for the timing and spectral analysis. Using the resulting source catalog, we process all sets of 50 ScWs in sequence to obtain a long-term lightcurve of the source shown in Fig. 14. From this figure one could see that the source is systematically detected throughout the entire 15 year time span. It shows moderate (if any) variability from year to year. Figure 14: Lightcurve of 3C 279 on 15 yr time span. Grey vertical lines show exposure periods of the source. The notebook 3C279_lc.ipynb for the calculation of the lightcurve could be launched using this URL. The 2015 flare of the source reported by Bottacini et al. (2016) is identifiable as the highest flux point in the lightcurve in Fig. 14. More detailed view of the lightcurve for the flaring episode discussed by Bottacini et al. (2016) is shown in Fig. 15, where we plot (red points) the ODA API lightcurve, extracted for the full range of the flaring period investigated in Bottacini et al. (2016), and the ISGRI light curve reported in Fig. 1 of Bottacini et al. (2016). The average flux of the Bottacini et al. (2016) lightucurve is compatible with our bin overlapping the same time span The average count rate is at the level of 1 ct/s, which agrees with the published value. This lightcurve can be re-generated using the same lightcurve extraction notebook as for the long-term lightcurve of the source, changing the time interval to focus on the flaring period, July 2015, and adjusting the energy range. This shows how the notebook available for on-the-fly re- deployment via the oda-hub/oda_api_benchmark web page can be re-used for refinement or re-use of the analysis for different energy ranges or different sources. Fig. 16 shows a comparison of the time-averaged spectrum of the source with the flaring state spectrum. We have used the same spectral range extraction of 20-100 keV, and the same spectral model (flux-pegged power- law model pegpwrlw171717Based on Xspec package fitting (https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node207.html).) as in Bottacini et al. (2016). The spectral fit of the flaring state spectrum shown in Fig. 16 has a very hard slope $\Gamma=0.81_{-0.11}^{+1.71}$ and a flux, in the 18-55 keV band, of $4_{-3}^{+5}\times 10^{-11}\,\mathrm{erg\,cm^{-2}s^{-1}}$ ( both reported at 90% c.l.), which are consistent with Bottacini et al. (2016), where the authors report a photon index of $\Gamma=1.08_{-0.15}^{+1.98}$ and a flux of $8_{-6}^{+10}\times 10^{-11}\,\mathrm{erg\,cm^{-2}s^{-1}}$. Figure 15: Lightcurve of 3C 279 in 20-100 keV range for the flare period reported by Bottacini et al. (2016) (red points), extracted using the ODA API, and the ISGRI lightcurve reported in fig. 1 of Bottacini et al. (2016) (blue points), corresponding to their ISGRI data time span. . The notebook 3C279_lc_flare.ipynb for re-generation of the result can be executed online at this URL, and is re-executable via Binder integration of oda-hub project. Figure 16: Comparison of time-averaged spectrum of 3C 279 (black) with the spectrum of the flaring state observed during the TOO period reported by Bottacini et al. (2016) in blue; the upper limit is at 3$\sigma$ confidence level. The respective models are represented as solid lines. The notebook 3C279_spectrum.ipynb for the calculation of the spectra can be launched using this URL. ### 3.5 NGC 2110 NGC 2110 is an example of moderately bright Seyfert galaxy, i.e. a representative of typical hard X-ray bright and persistent AGN (Peterson, 1997). AGN of this type are the most abundant extragalactic sources observed by INTEGRAL. Fig. 17 shows the significance image of the source region extracted from a set of 50 random ScWs spread over 15 years of INTEGRAL operations. One can see that NGC 2110 is the brightest source in the field. Its detection significance reaches $\simeq 17\sigma$ in the exposure $T\simeq 90$ ks. The dimmer source H 0614+091 is detected at significance level $10\sigma$, and there is no other source in the field detected with significance exceeding $5\sigma$. Figure 17: 20-40 keV image of the NGC 2110 region extracted from a set of 50 randomly selected ScWs. The image could be regenerated via the URL https://doi.org/10.5281/zenodo.3634584. Fig. 18 shows the long-term lightcurve of the source extracted from the sequence of 50 ScW datasets pointed within 10 degrees from the source, using the same notebook as for the 3C 279 analysis, part of the the oda- hub/oda_api_benchmark repository. The statistical uncertainty of the flux measurement in individual ScWs is too large and it is not possible to follow the long-term evolution of the source spectrum with ScW-by-Scw binning. Taking this into account, we have rebinned the lightcurve into wider time bins. Several variability episodes are clearly identifiable with such binning in the long-term lightcurve. Figure 18: Rebinned lightcurve of NGC 2110 extracted from the sequence of 50 ScW-long intervals using the notebook lc_longterm.ipynb. The result could be re-generated via notebook deployment on Binder at this URL. We used the same approach as for 3C 279 for the extraction of the source spectrum with long exposure time. We extracted the spectrum of NGC 2110 stacking the spectra in sequences of 50 ScW long exposures and calculating the weighted average using the same notebook as for the extraction of the 3C 279 spectrum, available and executable at the oda-hub/oda_api_benchmark project page. The resulting source spectrum with a total exposure of 2.4 Ms is shown in Fig. 19. The figure shows two separate spectra for the periods of applicability of OSA 10.2 and OSA 11.0. The physical origin of the hard X-ray emission from Seyfert galaxies is thought to be due to the inverse Compton scattering of emission from an accretion disk in a hot corona characterized by certain temperature (Lubiński et al., 2016). Suppression of the emission above 100 keV expected in this model is seen in the longer (1.8 Ms) exposure spectrum for OSA 10.2 applicability period. However, the exposure of OSA 11.0 period (0.6 Ms) is not sufficiently long to constrain a high-energy cut-off in the spectrum. In order to compare our results with those reported in (Lubiński et al., 2016), we have extracted ScWs for the same time span and angular extraction cone radius. We have a final exposure of $\simeq 1714$ ks. We stress that full reproduction of the analysis reported in Lubiński et al. (2016) is beyond the scope of our work, because we are not using soft X-ray data to constrain the parameters of the COMPPS Xspec model, used to describe inverse Compton emission. In our analysis we have used the same model as in Lubiński et al. (2016), and we have fixed all the parameters to those reported in their analysis, except for the electron temperature $kT_{e}$, the $y$-Compton parameter and the normalization. In detail, we have fixed the seed photons temperature $T_{bb}$ to a value of 10 eV, the amount of reflection $R$ to 0.63, and the geometry to the spherical case. We have estimated the $90\%$ parameters confidence range using the Xspec (v. 12.11.0) MCMC implementation, based on the Goodman-Weare algorithm (Goodman & Weare, 2010), with 20 walkers, and a Cauchy proposal distribution, running a 10000 steps chain, with a burn- in phase of 3000 steps. We have run the chain with an initial state starting from the Xspec best fit solution. We report the 0.05, 0.5, and 0.95 quantiles of the posterior distribution as lower, central, and upper values. We find a temperature $kT_{e}=3.5^{+1.3}_{-2.4}\times 10^{2}$ keV, a normalization constant $N=2.6^{+1.2}_{-1.0}\times 10^{8}$, and a value of the $y$-Compton parameter of $y=1.10^{+0.19}_{-0.25}$, compared to the values reported in Lubiński et al. (2016), of $kT_{e}=2.3\pm 0.5\times 10^{2}$ keV, $N=1.8^{+0.4}_{-0.2}\times 10^{8}$, and $y=0.94^{+0.10}_{-0.09}$, respectively. The spectrum and the frequentist best fit model are reported in the top panel of Fig. 19. As further benchmark we have compared the results from ODA to those published in Ursini et al. (2019). In this case the authors have distributed online the INTEGRAL/ISGRI spectra extracted with OSA used for their publication. We have extracted the INTEGRAL/ISGRI spectra with ODA for the same time span and spectral window as in Ursini et al. (2019). The results are shown in the lower panel of Fig. 19. Figure 19: Top: Spectrum of NGC 2110 extracted from the stacking of spectra obtained for 50 ScW sets in the time periods used in (Lubiński et al., 2016), using OSA 10.2. Bottom: Comparison of the spectra of NGC 2110 extracted from the stacking of spectra obtained for 50 ScW sets in the time periods used in (Ursini et al., 2019) and the spectra published in (Ursini et al., 2019). The spectra can be re-generated online using the notebook spectrum_longterm.ipynb and spectrum_longterm_Ursini19.ipynb. ## 4 Analysis limitations In the current ODA implementation, the dispatcher defines which scientific analysis workflows can be executed through the frontend, and enables two versions of INTEGRAL analysis software: standard OSA10.2 and OSA11.0. OSA10.2 can only be used for the analysis of the data taken before 2016. OSA11.0 can only be used for the ISGRI data since 2016 (at the time of writing), while it can be used for JEM-X during the full mission lifetime. This will change, as soon as updated ISGRI calibration files are made available. It should be remarked that the introduction of new releases of OSA will be made available. The maximum number of ScW allowed in a single request is set to 50 (corresponding to up to 50 CPU-hours of computing), to reduce load on the system through too long jobs. This might change if the back-end is deployed in a different location or more computing resources are made available. Large requests will also be available for authenticated users. As of now, it is possible to overcome this limitation by looping over successive sets of 50 ScWs as it is demonstrated in the examples of analysis in this paper. The INTEGRAL data analysis within ODA platform entirely relies on scientific data analysis with OSA, modifying only the mechanism with which individual components of OSA are invoked (principally to allow for preservation and re- use of intermediate analysis results, as explained above). It means that INTEGRAL analysis within ODA shares the limitations of OSA, and any future changes in OSA will be available in ODA. Detailed scientific validation of the long-term evolution of the INTEGRAL instruments and the current status of the data analysis software with OSA goes beyond the scope of this paper. ## 5 Comparison with other Online Analysis Platforms Several other online data analysis platforms offer similar services, some of them - for INTEGRAL instruments. HEASARC Hera exposes much of the HEASoft capabilities (with no relevant support for INTEGRAL) as a web service. This original and promising development has limitations of the service resources. Hera was, in part, an inspiration for the most superficial scheme of INTEGRAL ODA back-end. Swift/XRT online data analysis181818https://www.swift.ac.uk/user_objects/ is a particularly successful example of an online analysis and has encouraged adoption of this strategy for other instruments. XMM-Newton Newton’s RISA191919https://www.cosmos.esa.int/web/xmm-newton/xsa is conceptually similar to Swift/XRT online analysis, but is, arguably, less well known, perhaps due to different strategy of observation scheduling in XMM-Newton: while Swift is favoring a large number of short public observations with quick impact, XMM-Newton observations are often part of larger campaigns and private observations, favoring traditional offline analysis. Both Swift/XRT and XMM-Newton telescopes feature focusing mirrors, and their data analysis workflows require much smaller computing resources than INTEGRAL. Coded mask telescopes feature much larger FoV than the focusing ones - which in turn results in a larger number sources to be considered by the decoding process. In addition, their PSF is highly non-trivial and generally spread over the entire FoV. The combination of the typically long observations and dependency on a model with potentially large number of sources means that online analysis of a coded mask telescopes is resource-hungry, and has to adopt a very different approach to handling data analysis workflows and pre- computed data. INTEGRAL SPIDAI provides online data analysis for INTEGRAL/SPI. SPI data analysis requires considerable re-analysis for each new request. This is in principle similar to ODA ISGRI and JEM-X workflows, but without the comparably extensive re-use of pre-computed results. SPIDAI analysis can avoid this additional complexity since only relatively small number of sources are sufficiently bright to be observed by SPI. HEAVENS202020http://isdc.unige.ch/heavens/ follows very different approach from that of INTEGRAL ODA, SPIDAI, Swift/XRT, RISA, or Hera, and pre-computes the raw data using undisclosed customized procedures, creating a space-energy- time map of celestial flux, with fixed pre-defined resolution in each dimension. This considerably speeds-up certain kinds of analysis, but also implies that the scientific meaning of the HEAVENS output may be very different from that provided by expert-validated OSA. Furthermore, a particular pre-defined selection of space-time-energy resolutions restricts certain kinds of analysis (e.g. light curves with time bins less than about 3000 seconds - the INTEGRAL pointing duration). Finally, pre-computing of the results is costly (both in computing time and human effort) and does not necessarily privilege popular results, since it is not done in response to the user needs. As a result - the pre-computed HEAVENS results are available with a large delay. Finally, HEAVENS imposes additional severe restrictions on the output results, for example, the size of the requested image. To the contrary, ODA INTEGRAL analysis runs on-demand OSA analysis based exclusively on publicly available tools. Any improvements in OSA, officially validated by the instrument experts and the data center, are immediately adopted by ODA (by providing an additional OSA version in the parameter selection). Conversely, ODA operations expose some of the technical issues of OSA. While in the traditional offline analysis approach these issues would plague users with poorly understood error messages, an equivalent message in ODA-wrapped OSA can be directed straight to the OSA software developers, and the patch will be made available in the next OSA release (with a corresponding update to ODA). INTEGRAL ODA provides a much larger set of results than that available in pre- computed databases (such as HEAVENS). For example, it is possible to produce light-curves with small (down to 10 seconds) time bins and high-resolution spectra. More advanced users may upload custom source lists to use as sky model, to provide the most reliable INTEGRAL results, as recommended by OSA cookbook. Pre-computing the results with this granularity would not be feasible in platforms like HEAVENS. The ODA platform is built as a cloud-native solution, designed to be adaptable to any modern computing environment. It is already open-source, and can be readily developed via addition of new components. The platform consists of a range of independent components, following common standards of communication and interoperability. This means that the platform may grow beyond single deployment, for example it can be deployed as a part of a federated infrastructure of ESA DataLabs, https://datalabs.esa.int. ODA platform is strongly committed to interoperability and integration with the European Open Science Cloud (EOSC) https://www.eosc-portal.eu, and has been validated as an EOSC service212121https://marketplace.eosc- portal.eu/services/astronomical-online-data-analysis-astrooda/. ## 6 Conclusions We have presented the new approach for the INTEGRAL data analysis which uses the cloud computing technology to enable deployment of data analysis workflows for imaging, spectral and timing analysis online through a web interface in a web browser: https://www.astro.unige.ch/cdci/astrooda_ or through a dedicated API that can be used in Python notebooks executable locally or remotely from the oda-hub project Github repository. Such an approach provides an important boost for reproducibility of results extracted from INTEGRAL data and possibilities of sharing and re-use of data analysis methods. Virtualisation of the data analysis system also provides a viable solution for the long-term preservation of the data and analysis results. This paper demonstrates how reusable astronomical data analysis workflows can be shared and embedded in publications. Performance tests presented in this paper validate ODA for use in scientific publications making use of INTEGRAL data. ODA results are identical to those which are obtained with OSA with parameters choices following standard recommendations of the OSA hand books. INTEGRAL ODA provides on-demand analysis of any INTEGRAL ISGRI and JEM-X data, leveraging an almost complete set of OSA capabilities, yielding the results identical to the expert-validated publicly released OSA. 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2024-09-04T02:54:55.620014

2002.12903

arxiv-papers

# The estimation error of general first order methods Michael Celentano Department of Statistics, Stanford University Andrea Montanari11footnotemark: 1 Department of Electrical Engineering, Stanford University Yuchen Wu11footnotemark: 1 ###### Abstract Modern large-scale statistical models require to estimate thousands to millions of parameters. This is often accomplished by iterative algorithms such as gradient descent, projected gradient descent or their accelerated versions. What are the fundamental limits to these approaches? This question is well understood from an optimization viewpoint when the underlying objective is convex. Work in this area characterizes the gap to global optimality as a function of the number of iterations. However, these results have only indirect implications in terms of the gap to _statistical_ optimality. Here we consider two families of high-dimensional estimation problems: high- dimensional regression and low-rank matrix estimation, and introduce a class of ‘general first order methods’ that aim at efficiently estimating the underlying parameters. This class of algorithms is broad enough to include classical first order optimization (for convex and non-convex objectives), but also other types of algorithms. Under a random design assumption, we derive lower bounds on the estimation error that hold in the high-dimensional asymptotics in which both the number of observations and the number of parameters diverge. These lower bounds are optimal in the sense that there exist algorithms whose estimation error matches the lower bounds up to asymptotically negligible terms. We illustrate our general results through applications to sparse phase retrieval and sparse principal component analysis. ## 1 Introduction High-dimensional statistical estimation problems are often addressed by constructing a suitable data-dependent cost function $\mathcal{L}(\boldsymbol{\vartheta})$, which encodes the statistician’s knowledge of the problem. This cost is then minimized using an algorithm which scales well to large dimension. The most popular algorithms for high- dimensional statistical applications are first order methods, i.e., algorithms that query the cost $\mathcal{L}(\boldsymbol{\vartheta})$ by computing its gradient (or a subgradient) at a sequence of points $\boldsymbol{\theta}^{1}$,…$\boldsymbol{\theta}^{t}$. Examples include (projected) gradient descent, mirror descent, and accelerated gradient descent. This raises a fundamental question: _What is the minimal statistical error achieved by first order methods?_ In particular, we would like to understand in which cases these methods are significantly sub-optimal (in terms of estimation) with respect to statistically optimal but potentially intractable estimators, and what is the optimal tradeoff between number of iterations and estimation error. These questions are relatively well understood only from the point of view of convex optimization, namely if estimation is performed by minimizing a convex cost function $\mathcal{L}(\boldsymbol{\vartheta})$, see e.g. [CT07, BRT09]. The seminal work of Nemirovsy and Yudin [NY83] characterizes the minimum gap to global optimality $\mathcal{L}(\boldsymbol{\theta}^{t})-\min_{\boldsymbol{\vartheta}}\mathcal{L}(\boldsymbol{\vartheta})$, where $\boldsymbol{\theta}^{t}$ is the algorithm’s output $\boldsymbol{\theta}^{t}$ after $t$ iterations (i.e., after $t$ gradient evaluations). For instance, if $\mathcal{L}(\boldsymbol{\theta})$ is a smooth convex function, there exists a first order algorithm which achieves $\mathcal{L}(\boldsymbol{\theta}^{t})\leq\min_{\boldsymbol{\vartheta}}\mathcal{L}(\boldsymbol{\vartheta})+O(t^{-2})$. At the same time, no algorithm can be guaranteed to achieve a better convergence rate over all functions in this class. In contrast, if the cost $\mathcal{L}(\boldsymbol{\vartheta})$ is nonconvex, there cannot be general guarantees of global optimality. Substantial effort has been devoted to showing that –under suitable assumptions about the data distribution– certain nonconvex costs $\mathcal{L}(\boldsymbol{\theta})$ can be minimized efficiently, e.g. by gradient descent [KMO10, LW11, CC15]. This line of work resulted in upper bounds on the estimation error of first order methods. Unlike in the convex case, worst case lower bounds are typically overly pessimistic since non-convex optimization is NP-hard. Our work aims at developing precise average-case lower bounds for a restricted class of algorithms, which are applicable both to convex and nonconvex problems. We are particularly interested in problems that exhibit an information- computation gap: we know that the optimal statistical estimator has high accuracy, but existing upper bounds on first order methods are substantially sub-optimal (see examples below). Is this a limitation of our analysis, of the specific algorithm under consideration, or of first order algorithms in general? The main result of this paper is a tight asymptotic characterization of the minimum estimation error achieved by first order algorithms for two families of problems. This characterization can be used –in particular– to delineate information-computation gaps. Our results are novel even in the case of a convex cost function $\mathcal{L}(\boldsymbol{\vartheta})$, for two reasons. First, classical theory [Nes18] lower bounds the objective value $\mathcal{L}(\boldsymbol{\theta}^{t})-\min_{\boldsymbol{\vartheta}}\mathcal{L}(\boldsymbol{\vartheta})$ after $t$ iterations. This has only indirect implications on estimation error, e.g., $\|\boldsymbol{\theta}^{t}-\boldsymbol{\theta}_{*}\|_{2}$ (here $\boldsymbol{\theta}_{*}$ is the true value of the parameters, not the minimizer of the cost $\mathcal{L}(\boldsymbol{\vartheta})$). Second, the classical lower bounds on the objective value are worst case with respect to the function $\mathcal{L}(\boldsymbol{\vartheta})$ and do not take into account the data distribution. Concretely, we consider two families of estimation problems: High-dimensional regression. Data are i.i.d. pairs $\\{(y_{i},\boldsymbol{x}_{i})\\}_{i\leq n}$, where $y_{i}\in\mathbb{R}$ is a label and $\boldsymbol{x}_{i}\in\mathbb{R}^{p}$ is a feature vector. We assume $\boldsymbol{x}_{i}\sim{\mathsf{N}}(\boldsymbol{0},\boldsymbol{I}_{p}/n)$ and $y_{i}|\boldsymbol{x}_{i}\sim\mathbb{P}(y_{i}\in\,\cdot\,|\boldsymbol{x}_{i}^{\mathsf{T}}\boldsymbol{\theta})$ for a vector $\boldsymbol{\theta}\in\mathbb{R}^{p}$. Our objective is to estimate the coefficients $\theta_{j}$ from data $\boldsymbol{X}\in\mathbb{R}^{n\times p}$ (the matrix whose $i$-th row is vector $\boldsymbol{x}_{i}$) and $\boldsymbol{y}\in\mathbb{R}^{n}$ (the vector whose $i$-th entry is label $y_{i}$). Low-rank matrix estimation. Data consist of a matrix $\boldsymbol{X}\in\mathbb{R}^{n\times p}$ where $x_{ij}=\frac{1}{n}\boldsymbol{\lambda}_{i}^{\mathsf{T}}\boldsymbol{\theta}_{j}+z_{ij}$ with $\boldsymbol{\lambda}_{i},\boldsymbol{\theta}_{j}\in\mathbb{R}^{r}$ and $z_{ij}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1/n)$. We denote by $\boldsymbol{\lambda}\in\mathbb{R}^{n\times r}$ and $\boldsymbol{\theta}\in\mathbb{R}^{p\times r}$ the matrices whose rows are $\boldsymbol{\lambda}_{i}^{\mathsf{T}}$ and $\boldsymbol{\theta}_{j}^{\mathsf{T}}$ respectively. Our objective is to to estimate $\boldsymbol{\lambda},\boldsymbol{\theta}$ from data $\boldsymbol{X}$. In order to discuss these two examples in a unified fashion, we will introduce a dummy vector $\boldsymbol{y}$ (e.g., the all-zeros vector) as part of the data in the low-rank matrix estimation problem. Let us point out that our normalizations are somewhat different from, but completely equivalent to, the traditional ones in statistics. The first question to address is how to properly define ‘first order methods.’ A moment of thought reveals that the above discussion in terms of a cost function $\mathcal{L}(\boldsymbol{\theta})$ needs to be revised. Indeed, given either of the above statistical models, there is no simple way to construct a ‘statistically optimal’ cost function.111In particular, maximum likelihood is not statistically optimal in high dimension [BBEKY13]. Further, it is not clear that using a faster optimization algorithm for that cost will result in faster decrease of the estimation error. We follow instead a different strategy and introduce the class of _general first order methods_ (GFOM). In words, these include all algorithms that keep as state sequences of matrices $\boldsymbol{u}^{1},\dots,\boldsymbol{u}^{t}\in\mathbb{R}^{n\times r}$, and $\boldsymbol{v}^{1},\dots,\boldsymbol{v}^{t}\in\mathbb{R}^{p\times r}$, which are updated by two types of operations: row-wise application of a function, or multiplication by $\boldsymbol{X}$ or $\boldsymbol{X}^{\top}$. We will then show that standard first order methods, for common choices of the cost $\mathcal{L}(\boldsymbol{\theta})$, are in fact special examples of GFOMs. Formally, a GFOM is defined by sequences of functions $F^{(1)}_{t},G^{(2)}_{t}:\mathbb{R}^{r(t+1)+1}\to\mathbb{R}^{r}$, $F^{(2)}_{t},G^{(1)}_{t}:\mathbb{R}^{r(t+1)}\to\mathbb{R}^{r}$, with the $F$’s indexed by $t\geq 0$ and the $G$’s indexed by $t\geq 0$. In the high- dimensional regression problem, we set $r=1$. The algorithm produces two sequences of matrices (vectors for $r=1$) $(\boldsymbol{u}^{t})_{t\geq 1}$, $\boldsymbol{u}^{t}\in\mathbb{R}^{n\times r}$, and $(\boldsymbol{v}^{t})_{t\geq 1}$, $\boldsymbol{v}^{t}\in\mathbb{R}^{p\times r}$, $\displaystyle\boldsymbol{v}^{t+1}$ $\displaystyle=\boldsymbol{X}^{\top}F^{(1)}_{t}(\boldsymbol{u}^{1},\dots,\boldsymbol{u}^{t};\boldsymbol{y},\boldsymbol{u})+F^{(2)}_{t}(\boldsymbol{v}^{1},\dots,\boldsymbol{v}^{t};\boldsymbol{v})$ (1a) $\displaystyle\boldsymbol{u}^{t}$ $\displaystyle=\boldsymbol{X}G^{(1)}_{t}(\boldsymbol{v}^{1},\dots,\boldsymbol{v}^{t};\boldsymbol{v})+G^{(2)}_{t}(\boldsymbol{u}^{1},\dots,\boldsymbol{u}^{t-1};\boldsymbol{y},\boldsymbol{u})\,,$ (1b) where it is understood that each function is applied row-wise. For instance $\displaystyle F^{(1)}_{t}(\boldsymbol{u}^{1},\dots,\boldsymbol{u}^{t};\boldsymbol{u})=(F^{(1)}_{t}(\boldsymbol{u}_{i}^{1},\dots,\boldsymbol{u}_{i}^{t};\boldsymbol{u}_{i}))_{i\leq n}\in\mathbb{R}^{n\times r}\,,$ where $(\boldsymbol{u}_{i}^{s})^{{\mathsf{T}}}$ is the $i^{\text{th}}$ row of $\boldsymbol{u}^{s}$. Here $\boldsymbol{u},\boldsymbol{v}$ are either deterministic or random and independent of everything else. In particular, the iteration is initialized with $\boldsymbol{v}^{1}=\boldsymbol{X}^{\mathsf{T}}F_{0}^{(1)}(\boldsymbol{y},\boldsymbol{u})+F_{0}^{(2)}(\boldsymbol{v})$. The unknown matrices (or vectors) $\boldsymbol{\theta}$ and $\boldsymbol{\lambda}$ are estimated after $t_{*}$ iterations by $\hat{\boldsymbol{\theta}}=G_{*}(\boldsymbol{v}^{1},\cdots,\boldsymbol{v}^{t_{*}};\boldsymbol{v})$ and $\hat{\boldsymbol{\lambda}}=F_{*}(\boldsymbol{u}^{1},\ldots,\boldsymbol{u}^{t_{*}};\boldsymbol{y},\boldsymbol{u})$, where the latter only applies in the low-rank matrix estimation problem. Let us point out that the update also depend on additional information encoded in the two vectors $\boldsymbol{u}\in\mathbb{R}^{n}$, $\boldsymbol{v}\in\mathbb{R}^{p}$. This enables us to model side information provided to the statistician (e.g., an ‘initialization’ correlated with the true signal) or auxiliary randomness. We study the regime in which $n,p\to\infty$ with $n/p\to\delta\in(0,\infty)$ and $r$ is fixed. We assume the number of iterations $t_{*}$ is fixed, or potentially $t_{*}\to\infty$ after $n\to\infty$. In other words, we are interested in linear-time or nearly linear-time algorithms (complexity being measured relative to the input size $np$). As mentioned above, our main result is a general lower bound on the minimum estimation error that is achieved by any GFOM in this regime. The paper is organized as follows: Section 2 illustrates the setting introduced above in two examples; Section 3 contains the statement of our general lower bounds; Section 4 applies these lower bounds to the two examples; Section 5 presents an outline of the proof, deferring technical details to appendices. ## 2 Two examples ### Example $\\#1$: M-estimation in high-dimensional regression and phase retrieval Consider the high-dimensional regression problem. Regularized M-estimators minimize a cost $\displaystyle\mathcal{L}_{n}(\boldsymbol{\vartheta}):=\sum_{i=1}^{n}\ell(y_{i};\langle\boldsymbol{x}_{i},\boldsymbol{\vartheta}\rangle)+\Omega_{n}(\boldsymbol{\vartheta})=\hat{\ell}_{n}(\boldsymbol{y},\boldsymbol{X}\boldsymbol{\vartheta})+\Omega_{n}(\boldsymbol{\vartheta})\,,$ (2) Here $\ell:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is a loss function, $\hat{\ell}_{n}(\boldsymbol{y},\hat{\boldsymbol{y}}):=\sum_{i=1}^{n}\ell(y_{i},\hat{y}_{i})$ is its empirical average, and $\Omega_{n}:\mathbb{R}^{p}\to\mathbb{R}$ is a regularizer. It is often the case that $\ell$ is smooth and $\Omega_{n}$ is separable, i.e., $\Omega_{n}(\boldsymbol{\vartheta})=\sum_{i=1}^{p}\Omega_{1}(\vartheta_{i})$. We will assume this to be the case in our discussion. The prototypical first order method is proximal gradient [PB13]: $\displaystyle\boldsymbol{\theta}^{t+1}=\textsf{Prox}_{\gamma_{t}\Omega_{1}}\big{(}\boldsymbol{\theta}^{t}-\gamma_{t}\nabla_{\boldsymbol{\vartheta}}\hat{\ell}_{n}(\boldsymbol{y},\boldsymbol{X}\boldsymbol{\theta}^{t})\big{)}\,,$ $\displaystyle\textsf{Prox}_{\gamma\Omega_{1}}(y):=\arg\min_{\theta\in\mathbb{R}}\left\\{\frac{1}{2}(y-\theta)^{2}+\gamma\Omega_{1}(\theta)\right\\}\,.$ Here $(\gamma_{t})_{t\geq 0}$ is a sequence of step sizes and $\textsf{Prox}_{\gamma\Omega_{1}}$ acts on a vector coordinate-wise. Notice that $\displaystyle\nabla_{\boldsymbol{\vartheta}}\hat{\ell}_{n}(\boldsymbol{y},\boldsymbol{X}\boldsymbol{\theta}^{t})=\boldsymbol{X}^{{\mathsf{T}}}s(\boldsymbol{y},\boldsymbol{X}\boldsymbol{\theta}^{t})\,,\;\;\;\;s(\boldsymbol{h},\hat{\boldsymbol{y}})_{i}\equiv\frac{\partial\ell}{\partial\hat{y}_{i}}(y_{,}\hat{y}_{i})\,.$ (3) Therefore proximal gradient –for the cost function (2)– is an example of a GFOM. Similarly, mirror descent with a separable Bregman divergence and accelerated proximal gradient methods are easily shown to fit in the same framework. Among the countless applications of regularized M-estimation, we will focus on the sparse phase retrieval problem. We want to reconstruct a sparse signal $\boldsymbol{\theta}\in\mathbb{R}^{p}$ but only have noisy measurements of the modulus $|\langle\boldsymbol{\theta},\boldsymbol{x}_{i}\rangle|$; that is, we lose the ‘phase’ of these projections. (We will consider for simplicity the case of a real-valued signal, but the generalization of our results to the complex case should be immediate.) As a concrete model, we will assume that number of non-zero entries of $\boldsymbol{\theta}$ is $\|\boldsymbol{\theta}\|_{0}\leq s_{0}$. From an information-theoretic viewpoint, it is known that $\boldsymbol{\theta}$ can be reconstructed accurately as soon as the number of measurements satisfies $n\geq Cs_{0}\log(p/s_{0})$, with $C$ a sufficiently large constant [LV13]. Several groups have investigated practical reconstruction algorithms by exploiting either semidefinite programming relaxations [LV13] or first order methods [SR14, CLS15, CLM16]. A standard approach would be to apply a proximal gradient algorithm to the cost function (2) with $\Omega_{n}(\boldsymbol{\vartheta})=\lambda\|\boldsymbol{\vartheta}\|_{1}$. However, all existing global convergence guarantees for these methods require $n\geq Cs_{0}^{2}\log p$. Is the dependence on $s_{0}^{2}$ due to a fundamental computational barrier or an artifact of the theoretical analysis? Recently [Sol19] presented partial evidence towards the possibility of ‘breaking’ this barrier, by proving that a first order method can accurately reconstruct the signal for $n\geq Cs_{0}\log(p/s_{0})$, if it is initialized close enough to the true signal $\boldsymbol{\theta}$. ### Example $\\#2$: Sparse PCA In a simple model for sparse principal component analysis (PCA), we observe a matrix $\boldsymbol{X}=\frac{1}{n}\boldsymbol{\lambda}\boldsymbol{\theta}^{{\mathsf{T}}}+\boldsymbol{Z}\in\mathbb{R}^{n\times p}$, where $\boldsymbol{\lambda}\in\mathbb{R}^{n}$ has entries $(\lambda_{i})_{i\leq n}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1)$, $\boldsymbol{\theta}\in\mathbb{R}^{p}$ is a sparse vector with $s_{0}\ll p$ non-zero entries, and $\boldsymbol{Z}$ is a noise matrix with entries $(z_{ij})_{i\leq n,j\leq p}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1/n)$. Given data $\boldsymbol{X}$, we would like to reconstruct the signal $\boldsymbol{\theta}$. From an information-theoretic viewpoint, it is known that accurate reconstruction of $\boldsymbol{\theta}$ is possible if $n\geq Cs_{0}\log(p/s_{0})$, with $C$ a sufficiently large constant [AW08]. A number of polynomial time algorithms have been studied, ranging from simple thresholding algorithms [JL09, DM16] to sophisticated convex relaxations [AW08, MW15]. Among other approaches, one natural idea is to modify the power iteration algorithm of standard PCA by computing $\displaystyle\boldsymbol{\theta}^{t+1}=c_{t}\,\boldsymbol{X}^{{\mathsf{T}}}\boldsymbol{X}\eta(\boldsymbol{\theta}^{t};\gamma_{t})\,.$ (4) Here $(c_{t})_{t\geq 0}$ is a deterministic normalization, and $\eta(\;\cdot\;;\gamma)$ is a thresholding function at level $\gamma$, e.g., soft thresholding $\eta(x;\gamma)=\operatorname{sign}(x)(|x|-\gamma)_{+}$. It is immediate to see that this algorithm is a GFOM. More elaborate versions of non-linear power iteration were developed, for example, by [JNRS10, Ma13], and are typically equivalent to suitable GFOMs. Despite these efforts, no algorithm is known to succeed unless $n\geq Cs_{0}^{2}$. Is this a fundamental barrier or a limitation of present algorithms or analysis? Evidence towards intractability was provided by [BR13, BBH18] via reduction from the planted clique problem. Our analysis provides new evidence towards the same conclusion. ## 3 Main results In this section we state formally our general results about high-dimensional regression and low-rank matrix estimation. The next section will apply these general results to concrete instances. Throughout we make the following assumptions: * A1. The functions $F^{(1)}_{t},G^{(2)}_{t},F_{*}:\mathbb{R}^{r(t+1)+1}\to\mathbb{R}$, $F^{(2)}_{t},G^{(1)}_{t},G_{*}:\mathbb{R}^{r(t+1)}\to\mathbb{R}$, are Lipschitz continuous, with the $F$’s indexed by $t\geq 0$ and the $G$’s indexed by $t\geq 0$. * A2. The covariates matrix $\boldsymbol{X}$ (for high-dimensional regression) or the noise matrix $\boldsymbol{Z}$ (for low-rank estimation) have entries $x_{ij}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1/n)$, $z_{ij}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1/n)$. Also, we denote by $\mathscrsfs{P}_{q}(\mathbb{R}^{k})$ the set of probability distributions with finite $q$-th moment on $\mathbb{R}^{k}$ and $\mathscrsfs{P}_{\mathrm{c}}(\mathbb{R}^{k})$ those with compact support. We say a function $f:\mathbb{R}^{k}\rightarrow\mathbb{R}$ is _pseudo-Lipschitz of order 2_ if there exists constant $C$ such that $|f(\boldsymbol{x})-f(\boldsymbol{x}^{\prime})|\leq C(1+\|\boldsymbol{x}\|+\|\boldsymbol{x}^{\prime}\|)\|\boldsymbol{x}-\boldsymbol{x}^{\prime}\|$ for all $\boldsymbol{x},\boldsymbol{x}^{\prime}\in\mathbb{R}^{k}$. We call a function $\ell:(\mathbb{R}^{k})^{2}\rightarrow\mathbb{R}$ a _quadratically- bounded loss_ if it is non-negative and pseudo-Lipschitz of order 2 and there exists $C>0$ such that for all $\boldsymbol{x},\boldsymbol{x}^{\prime},\boldsymbol{d}\in\mathbb{R}^{k}$ we have $|\ell(\boldsymbol{x},\boldsymbol{d})-\ell(\boldsymbol{x}^{\prime},\boldsymbol{d})|\leq C(1+\sqrt{\ell(\boldsymbol{x},\boldsymbol{d})}+\sqrt{\ell(\boldsymbol{x}^{\prime},\boldsymbol{d})})\|\boldsymbol{x}-\boldsymbol{x}^{\prime}\|$. ### 3.1 High-dimensional regression We make the following additional assumptions for the regression problem: * R1. We sample $\\{(w_{i},u_{i})\\}_{i\leq n}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{W,U}$, $\\{(\theta_{i},v_{i})\\}_{i\leq p}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\Theta,V}$ for $\mu_{\Theta,V},\mu_{W,U}\in\mathscrsfs{P}_{2}(\mathbb{R}^{2})$. * R2. There exists a measurable function $h:\mathbb{R}^{2}\rightarrow\mathbb{R}$ such that $y_{i}=h(\boldsymbol{x}_{i}^{{\mathsf{T}}}\boldsymbol{\theta},w_{i})$. Moreover, there exists constant $C$ such that $|h(x,w)|\leq C(1+|x|+|w|)$ for all $x,w$. Notice that the description in terms of a probability kernel $\mathbb{P}(y_{i}\in\,\cdot\,|\boldsymbol{x}_{i}^{{\mathsf{T}}}\boldsymbol{\theta})$ is equivalent to the one in terms of a ‘noisy’ function $y_{i}=h(\boldsymbol{x}_{i}^{{\mathsf{T}}}\boldsymbol{\theta},w_{i})$ in most cases of interest. Our lower bound is defined in terms of a one-dimensional recursion. Let $(\Theta,V)\sim\mu_{\Theta,V}$. Let $\textsf{mmse}_{\Theta,V}(\tau^{2})$ be the minimum mean square error for estimation of $\Theta$ given observations $V$ and $\Theta+\tau G$ where $G\sim{\mathsf{N}}(0,1)$ independent of $\Theta$. Set $\tau_{\Theta}^{2}=\mathbb{E}[\Theta^{2}]$ and $\tau_{0}^{2}=\infty$, and define recursively $\begin{gathered}\tilde{\tau}_{s}^{2}=\frac{1}{\delta}\,\textsf{mmse}_{\Theta,V}(\tau_{s}^{2}),\;\;\;\;\;\;\sigma_{s}^{2}=\frac{1}{\delta}(\tau_{\Theta}^{2}-\textsf{mmse}_{\Theta,V}(\tau_{s}^{2}))\,,\\\ \frac{1}{\tau_{s+1}^{2}}=\frac{1}{\tilde{\tau}_{s}^{2}}\mathbb{E}\left[\mathbb{E}[G_{1}|Y,G_{0},U]^{2}\right],\\\ \end{gathered}$ (5) where $Y=h(\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},W)$ and the expectation is with respect to $G_{0},G_{1}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1)$ and $(W,U)\sim\mu_{W,U}$ independent. ###### Theorem 1. Under assumptions A1, A2, R1, R2 in the high-dimensional regression model and under the asymptotics $n,p\rightarrow\infty$, $n/p\rightarrow\delta\in(0,\infty)$, let $\hat{\boldsymbol{\theta}}^{t}$ be output of any GFOM after $t$ iterations ($2t-1$ matrix-vector multiplications). Then $\displaystyle\lim_{n\rightarrow\infty}\frac{1}{p}\|\hat{\boldsymbol{\theta}}^{t}-\boldsymbol{\theta}\|_{2}^{2}\geq\textsf{mmse}_{\Theta,V}(\tau_{t}^{2})\,.$ More generally, for any quadratically-bounded loss $\ell:\mathbb{R}^{2}\rightarrow\mathbb{R}_{\geq 0}$, $\displaystyle\lim_{n\rightarrow\infty}\frac{1}{p}\sum_{j=1}^{p}\ell(\theta_{j},\hat{\theta}_{j}^{t})\geq\inf_{\hat{\theta}(\,\cdot\,)}\mathbb{E}\big{\\{}\ell(\Theta,\hat{\theta}(\Theta+\tau_{t}G,V))\big{\\}}\,,$ (6) where $(\Theta,V)\sim\mu_{\Theta,V}$ independent of $G\sim{\mathsf{N}}(0,1)$, and the infimum on the right-hand side is over measurable functions $\hat{\theta}:\mathbb{R}^{2}\to\mathbb{R}$. The limits are in probability and to a constant, and they are guaranteed to exist. For all $\epsilon>0$, there exist GFOMs which satisfy these bounds to within tolerance $\epsilon$. ### 3.2 Low-rank matrix estimation We make the following additional assumption: * M1. We sample $\\{(\boldsymbol{\lambda}_{i},\boldsymbol{u}_{i})\\}_{i\leq n}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\boldsymbol{\Lambda},\boldsymbol{U}}$ and $\\{(\boldsymbol{\theta}_{j},\boldsymbol{v}_{j})\\}_{j\leq p}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\boldsymbol{\Theta},\boldsymbol{V}}$ for $\mu_{\boldsymbol{\Lambda},\boldsymbol{U}},\mu_{\boldsymbol{\Theta},\boldsymbol{V}}\in\mathscrsfs{P}_{2}(\mathbb{R}^{2r})$. Again, our lower bound is defined in terms of recursion, which this time is defined over positive semidefinite matrices $\boldsymbol{Q}_{t},\hat{\boldsymbol{Q}}_{t}\in\mathbb{R}^{r\times r}$, $\boldsymbol{Q}_{t},\hat{\boldsymbol{Q}}_{t}\succeq\boldsymbol{0}$. Set $\hat{\boldsymbol{Q}}_{0}=\boldsymbol{0}$, and define recursively $\displaystyle\boldsymbol{Q}_{t+1}={\boldsymbol{V}}_{\boldsymbol{\Lambda},\boldsymbol{U}}(\hat{\boldsymbol{Q}}_{t})\,,\;\;\;\;\;\;\;\;\;\;\;\hat{\boldsymbol{Q}}_{t}=\frac{1}{\delta}{\boldsymbol{V}}_{\boldsymbol{\Theta},\boldsymbol{V}}(\boldsymbol{Q}_{t})\,,$ (7) where we define the second moment of the conditional expectation ${\boldsymbol{V}}_{\boldsymbol{\Theta},\boldsymbol{V}}:\mathbb{R}^{r\times r}\to\mathbb{R}^{r\times r}$ by $\displaystyle{\boldsymbol{V}}_{\boldsymbol{\Theta},\boldsymbol{V}}(\boldsymbol{Q}):=\mathbb{E}\Big{\\{}\mathbb{E}[\boldsymbol{\Theta}|\boldsymbol{Q}^{1/2}\boldsymbol{\Theta}+\boldsymbol{G}=\boldsymbol{Y};\boldsymbol{V}]\mathbb{E}[\boldsymbol{\Theta}|\boldsymbol{Q}^{1/2}\boldsymbol{\Theta}+\boldsymbol{G}=\boldsymbol{Y};\boldsymbol{V}]^{{\mathsf{T}}}\Big{\\}},$ and analogously for ${\boldsymbol{V}}_{\boldsymbol{\Lambda},\boldsymbol{U}}(\hat{\boldsymbol{Q}})$. Here the expectation is with respect to $(\boldsymbol{\Theta},\boldsymbol{V})\sim\mu_{\boldsymbol{\Theta},\boldsymbol{V}}$ and an independent Gaussian vector $\boldsymbol{G}\sim{\mathsf{N}}(\boldsymbol{0},{\boldsymbol{I}}_{r})$. Notice in particular that $\mathbb{E}\\{\boldsymbol{\Theta}\boldsymbol{\Theta}^{{\mathsf{T}}}\\}-{\boldsymbol{V}}_{\boldsymbol{\Theta},\boldsymbol{V}}(\boldsymbol{Q})$ is the vector minimum mean square error when $\boldsymbol{\Theta}$ is observed in Gaussian noise with covariance $\boldsymbol{Q}^{-1}$. For $r=1$, Eq. (7) is a simple scalar recursion. ###### Theorem 2. Under assumptions A1, A2, M1 in the low-rank matrix estimation model and under the under the asymptotics $n,p\rightarrow\infty,n/p\rightarrow\delta\in(0,\infty)$, let $\hat{\boldsymbol{\theta}}^{t}$ be output of any GFOM after $t$ iterations ($2t-1$ matrix-vector multiplications). Then $\displaystyle\lim_{n\rightarrow\infty}\frac{1}{p}\|\hat{\boldsymbol{\theta}}^{t}-\boldsymbol{\theta}\|_{\mathsf{F}}^{2}\geq\mathbb{E}\\{\|\boldsymbol{\Theta}\|^{2}\\}-\operatorname{Tr}{\boldsymbol{V}}_{\boldsymbol{\Theta},\boldsymbol{V}}(\boldsymbol{Q}_{t})\,.$ More generally, for any quadratically-bounded loss $\ell:\mathbb{R}^{2r}\rightarrow\mathbb{R}_{\geq 0}$, $\displaystyle\lim_{n\rightarrow\infty}\frac{1}{p}\sum_{j=1}^{p}\ell(\boldsymbol{\theta}_{j},\hat{\boldsymbol{\theta}}_{j}^{t})\geq\inf_{\hat{\boldsymbol{\theta}}(\,\cdot\,)}\mathbb{E}\big{\\{}\ell(\boldsymbol{\Theta},\hat{\boldsymbol{\theta}}(\boldsymbol{Q}_{t}^{1/2}\boldsymbol{\Theta}+\boldsymbol{G},\boldsymbol{V}))\big{\\}}\,,$ (8) where the infimum on the right-hand side is over functions $\hat{\boldsymbol{\theta}}:\mathbb{R}^{r}\to\mathbb{R}^{r}$. The limits are in probability and to a constant, and they are guaranteed to exist. As above, for all $\epsilon>0$ there exist GFOMs which satisfy these bounds to within tolerance $\epsilon$. ### 3.3 Discussion Our motivations are similar to the ones for statistical query (SQ) lower bounds [FGR+17, FGV17]: we want to provide estimation lower bounds under a restricted computational model, that are sensitive to the data distribution. However the scope of our approach is significantly different from SQ algorithms: the latter can query data distributions and compute approximate expectations with respect to that distribution. In contrast, our algorithms work with a fixed sample (the data matrix $\boldsymbol{X}$ and responses $\boldsymbol{y}$), which is queried multiple times. These queries can be thought as weighted averages of _both rows and columns_ of $\boldsymbol{X}$ and, as such, cannot be simulated by the SQ oracle. For instance, the proximal gradient method or the nonlinear power iteration of Section 2 cannot be framed as a SQ algorithms. The lower bounds of Theorems 1 and 2 are satisfied with equality by a specific first order method that is an approximate message passing (AMP) algorithm, with Bayes updates. This can be regarded as a version of belief propagation (BP) for densely connected graphs [KF09], or an iterative implementation of the TAP equations from spin glass theory [MPV87]. Our proof builds on the asymptotically exact analysis of AMP algorithms developed in [Bol14, BM11, JM18, BMN19]. However we need to overcome three technical obstacles: $(1)$ Show that any GFOM can be reduced (in a suitable sense) to a certain AMP algorithms, whose behavior can be exactly tracked. $(2)$ Show that Bayes-AMP is optimal among all AMP algorithms. We achieve this goal by considering an estimation problem on trees and showing that, in a suitable large degree limit, it has the same asymptotic behavior as AMP on the complete graph. On trees it is immediate to see that BP is the optimal local algorithm. $(3)$ We need to prove that the asymptotic behavior of BP for trees of large degree is equivalent to the one of Bayes-AMP on the original problem. This amounts to proving a Gaussian approximation theorem for BP. While similar results were obtained in the past for discrete models [Sly09, MX16], the current setting is technically more challenging because the underlying variables $\theta_{i}$ are continuous and unbounded. While the line of argument above is –in hindsight– very natural, the conclusion is broadly useful. For instance, [AFUZ19] study a class of of message passing algorithms inspired to replica symmetry breaking and survey propagation [MPZ02], and observe that they do not perform better than Bayes AMP. These algorithms are within the scope of our Theorem 2, which implies that indeed they cannot outperform Bayes AMP, for any constant number of iterations. Finally, a sequence of recent papers characterize the asymptotics of the Bayes-optimal estimation error in the two models described above [LM19, BKM+19]. It was conjectured that, in this context, no polynomial-time algorithm can outperform Bayes AMP, provided these algorithms have access to an arbitrarily small amount of side information.222Concretely, side information can take the form $\boldsymbol{v}=\eta\boldsymbol{\theta}+\boldsymbol{g}$ for $\eta>0$ arbitrarily small, $\boldsymbol{g}\sim{\mathsf{N}}(0,{\boldsymbol{I}}_{p})$ Theorems 1 and 2 establish this result within the restricted class of GFOMs. ## 4 Applying the general lower bounds In our two examples, we will refer to the sets $B^{p}_{0}(k)\subset\mathbb{R}^{p}$ of $k$-sparse vectors and $B^{p}_{2}(R)\subset\mathbb{R}^{p}$ of vectors with $\ell_{2}$-norm bounded by $R$. ### Example $\\#1$: Sparse phase retrieval For the reader’s convenience, we follow the standard normalization in phase retrieval, whereby the ‘sensing vectors’ (i.e. the rows of the design matrix) have norm concentrated around one. In other words, we observe $y_{i}\sim p(\,\cdot\,|\tilde{\boldsymbol{x}}_{i}^{{\mathsf{T}}}\boldsymbol{\theta}){\mathrm{d}}y$, where $\tilde{\boldsymbol{x}}_{i}\sim{\mathsf{N}}(0,{\boldsymbol{I}}_{p}/p)$. In order to model the phase retrieval problem, we assume that the conditional density $p(\,\cdot\,|\,\cdot\,\,)$ satisfies the symmetry condition $p(y|x)=p(y|-x)$. In words: we only observe a noisy version of the absolute value $|\langle\tilde{\boldsymbol{x}}_{i},\boldsymbol{\theta}\rangle|$. An important role is played by the following critical value of the number of observations per dimension $\displaystyle\delta_{\mbox{\tiny{sp}}}:=\left(\int_{\mathbb{R}}\frac{\mathbb{E}_{G}[p(y|G)(G^{2}-1)]}{\mathbb{E}_{G}[p(y|G)]}\,{\mathrm{d}}y\right)^{-1}\,.$ (9) Here expectation is with respect to $G\sim{\mathsf{N}}(0,1)$. It was proved in [MM19] that, if $\|\boldsymbol{\theta}\|_{2}=\sqrt{p}$ and $n>(\delta_{\mbox{\tiny{sp}}}+\eta)p$, for some $\eta$ bounded away from zero, then there exists a simple spectral estimator $\hat{\boldsymbol{\theta}}_{\mbox{\tiny{sp}}}$ that achieves weak recovery, i.e., a positive correlation with the true signal. Namely, $\frac{|\langle\hat{\boldsymbol{\theta}}_{\mbox{\tiny{sp}}},\boldsymbol{\theta}\rangle|}{\|\hat{\boldsymbol{\theta}}_{\mbox{\tiny{sp}}}\|_{2}\|\boldsymbol{\theta}\|_{2}}$ is bounded away from zero as $p,n\to\infty$. In the case of a dense signal $\boldsymbol{\theta}$ and observation model $y_{i}=|\tilde{\boldsymbol{x}}_{i}^{{\mathsf{T}}}\boldsymbol{\theta}|+w_{i},\,w_{i}\sim{\mathsf{N}}(0,\sigma^{2})$, the oversampling ratio $\delta_{\mbox{\tiny{sp}}}$ is known to be information- theoretically optimal: for $n<(\delta_{\mbox{\tiny{sp}}}-\eta)p$ no estimator can achieve a correlation that is bounded away from $0$ [MM19]. On the other hand, if $\boldsymbol{\theta}$ has at most $p\varepsilon$ nonzero entries, it is information-theoretically possible to reconstruct it from $\delta>C\varepsilon\log(1/\varepsilon)$ phaseless measurements per dimension [LV13]. Our next result implies that no GFOM can achieve reconstruction from $O(\varepsilon\log(1/\varepsilon))$ measurements per dimension, unless it is initialized close enough to the true signal. In order to model the additional information provided by the initialization we assume to be given $\displaystyle\overline{\boldsymbol{v}}=\sqrt{\alpha}\,\boldsymbol{\theta}/\|\boldsymbol{\theta}\|_{2}+\sqrt{1-\alpha}\tilde{\boldsymbol{g}},\;\;\;\;\;\;(\tilde{g}_{i})_{i\leq p}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1/p),.$ (10) Notice that with this normalization $\|\overline{\boldsymbol{v}}\|_{2}$ concentrates tightly around $1$, and $\sqrt{\alpha}$ can be interpreted as the cosine of the angle between $\boldsymbol{\theta}$ and $\overline{\boldsymbol{v}}$. ###### Corollary 1. Consider the phase retrieval model, for a sequence of deterministic signals $\boldsymbol{\theta}\in\mathbb{R}^{p}$, and let $\mathscrsfs{T}(\varepsilon,R):=B^{p}_{0}(p\varepsilon)\cap B^{p}_{2}(R)$. Assume the noise kernel $p(\,\cdot\,|x)$ to satisfy the conditions of Theorem 1 and to be be twice differentiable with respect to $x$. Then, for any _$\delta <\delta_{\mbox{\tiny{sp}}}$_, there exists $\alpha_{*}=\alpha_{*}(\delta,\varepsilon)>0$ and $C_{*}=C_{*}(\delta,\varepsilon)$ such that, if $\alpha\leq\alpha_{*}$, then $\displaystyle\sup_{t\geq 0}\lim_{n,p\to\infty}\inf_{\boldsymbol{\theta}\in\mathscrsfs{T}(\varepsilon,\sqrt{p})}\mathbb{E}\frac{\langle\boldsymbol{\theta},\hat{\boldsymbol{\theta}}^{t}\rangle}{\|\boldsymbol{\theta}\|_{2}\|\hat{\boldsymbol{\theta}}^{t}\|_{2}}\leq C_{*}\sqrt{\alpha}\,.$ (11) The same conclusion holds if $\boldsymbol{\theta}$ is drawn randomly with i.i.d. entries $\theta_{i}\sim\mu_{\theta}:=(1-\varepsilon)\delta_{0}+(\varepsilon/2)(\delta_{\mu}+\delta_{-\mu})$, $\mu=1/\sqrt{\varepsilon}$. ### Example $\\#2$: Sparse PCA For ease of interpretation, we assume the observation model $\tilde{\boldsymbol{X}}=\boldsymbol{\lambda}\overline{\boldsymbol{\theta}}^{{\mathsf{T}}}+\tilde{\boldsymbol{Z}}$, where $(\tilde{z}_{ij})_{i\leq n,j\leq p}\sim{\mathsf{N}}(0,1)$ and $(\lambda_{i})_{i\leq n}\sim{\mathsf{N}}(0,1)$. Equivalently, conditional on $\overline{\boldsymbol{\theta}}$, the rows of $\tilde{\boldsymbol{X}}$ are i.i.d. samples $\tilde{\boldsymbol{x}}_{i}\sim{\mathsf{N}}(0,\boldsymbol{\Sigma})$, $\boldsymbol{\Sigma}={\boldsymbol{I}}_{p}+\overline{\boldsymbol{\theta}}\overline{\boldsymbol{\theta}}^{{\mathsf{T}}}$. We also assume to have access to an initialization $\overline{\boldsymbol{v}}$ correlated with $\overline{\boldsymbol{\theta}}$, as per Eq. (10). In order to apply Theorem 2, we choose a specific distribution for the spike. Defining $\boldsymbol{\theta}=\overline{\boldsymbol{\theta}}\sqrt{p}$, we assume that the entries of $\boldsymbol{\theta}$ follow a three-points sparse distribution $(\theta_{i})_{i\leq p}\sim\mu_{\theta}:=(1-\varepsilon)\delta_{0}+(\varepsilon/2)(\delta_{+\mu}+\delta_{-\mu})$. The next lemma specializes Theorem 2. ###### Lemma 1. Assume the sparse PCA model with the distribution of $\overline{\boldsymbol{\theta}}$ given above. Define $(q_{t})_{t\geq 0}$ by $\displaystyle q_{t+1}$ $\displaystyle=\frac{V_{\pm}(q_{t}+\tilde{\alpha})}{1+V_{\pm}(q_{t}+\tilde{\alpha})}\,,\;\;\;\;\;\;q_{0}=0\,,$ (12) $\displaystyle V_{\pm}(q)$ $\displaystyle:=e^{-\delta q\mu^{2}}\mu^{2}\varepsilon^{2}\mathbb{E}\left\\{\frac{\sinh(\mu\sqrt{\delta q}G)^{2}}{1-\varepsilon+\varepsilon e^{-\delta q\mu^{2}/2}\cosh(\mu\sqrt{\delta q}G)}\right\\}\,,$ (13) where $\tilde{\alpha}=\alpha/(\mu^{2}\varepsilon(1-\alpha))$. Then, for any GFOM $\displaystyle\lim_{n,p\to\infty}\frac{\langle\overline{\boldsymbol{\theta}},\hat{\boldsymbol{\theta}}^{t}\rangle}{\|\overline{\boldsymbol{\theta}}\|_{2}\|\hat{\boldsymbol{\theta}}^{t}\|_{2}}\leq\sqrt{\frac{V_{\pm}(q_{t}+\tilde{\alpha})}{\mu^{2}\varepsilon}}\,.$ (14) The bound in the last lemma holds for random vectors $\overline{\boldsymbol{\theta}}$ with i.i.d. entries from the three-points distribution. As a consequence, it implies a minimax bound for non-random vectors $\overline{\boldsymbol{\theta}}$ with given $\ell_{2}$-norm and sparsity. We state this bound in the corollary below. In order to develop explicit expressions, we analyze the recursion of Eqs. (12), (13). ###### Corollary 2. Assume the sparse PCA model, for $\overline{\boldsymbol{\theta}}\in\mathbb{R}^{p}$ a deterministic vector and $\boldsymbol{\lambda}$, $\tilde{\boldsymbol{Z}}$ random, and consider the parameter space $\mathscrsfs{T}(\varepsilon,R):=B^{p}_{0}(p\varepsilon)\cap B^{p}_{2}(R)$. 1. $(a)$ If $R^{2}<1/\sqrt{\delta}$, then there exists $\alpha_{*}=\alpha_{*}(R,\delta,\varepsilon),C_{*}=C_{*}(R,\delta,\varepsilon)$ such that, for $\alpha<\alpha_{*}$, and any GFOM $\displaystyle\sup_{t\geq 0}\lim_{n,p\to\infty}\inf_{\overline{\boldsymbol{\theta}}\in\mathscrsfs{T}(\varepsilon,R)}\mathbb{E}\frac{\langle\overline{\boldsymbol{\theta}},\hat{\boldsymbol{\theta}}^{t}\rangle}{\|\overline{\boldsymbol{\theta}}\|_{2}\|\hat{\boldsymbol{\theta}}^{t}\|_{2}}\leq C_{*}\sqrt{\alpha}\,.$ (15) 2. $(b)$ If $R^{2}<\sqrt{(1-\varepsilon)/4\delta}$, then the above statement holds with $\alpha_{*}=\left(\frac{\varepsilon}{4\delta}\wedge\frac{1}{2}\right)$, $C_{*}=3/R^{2}$. In words, the last corollary implies that for $R^{2}\delta<1$, no estimator achieves a non-vanishing correlation with the true signal $\overline{\boldsymbol{\theta}}$, unless sufficient side information about $\overline{\boldsymbol{\theta}}$ is available. Notice that for $R^{2}\delta=1$ is the threshold above which the principal eigenvector of the empirical covariance $\tilde{\boldsymbol{X}}^{{\mathsf{T}}}\tilde{\boldsymbol{X}}/n$ becomes correlated with $\overline{\boldsymbol{\theta}}$. Hence, our result implies that, simple PCA fails, then every GFOM will fail. Viceversa, if simple PCA succeed, then it can be implemented via a GFOM, provided arbitrarily weak side information if available. Indeed, assume side information $\boldsymbol{v}=\eta\boldsymbol{\theta}+\boldsymbol{g}$, with $\boldsymbol{g}\sim{\mathsf{N}}(0,{\boldsymbol{I}}_{p})$, and an $\eta$ arbitrarily small constant. Then the power method initialized at $\boldsymbol{v}$ converges to an estimate that has correlation with $\boldsymbol{\theta}$ bounded away from zero in $O(\log(1/\eta))$ iterations. ## 5 Proof of main results In this section, we prove Theorems 1 and 2 under stronger assumptions than in their statements. In the high-dimensional regression model, these assumptions are as follows. * R3. Given $\mu_{\Theta,V}\in\mathscrsfs{P}_{\mathrm{c}}(\mathbb{R}^{2})$ and $\mu_{\boldsymbol{W},U}\in\mathscrsfs{P}_{4}(\mathbb{R}^{k}\times\mathbb{R})$ for some $k\geq 1$, we sample $\\{(\theta_{i},v_{i})\\}_{i\leq p}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\Theta,V}$, $\\{(\boldsymbol{w}_{i},u_{i})\\}_{i\leq n}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\boldsymbol{W},U}$. * R4. There exists Lipschitz function $h:\mathbb{R}\times\mathbb{R}^{k}\rightarrow\mathbb{R}$ such that $y_{i}=h(\boldsymbol{x}_{i}^{{\mathsf{T}}}\boldsymbol{\theta},\boldsymbol{w}_{i})$. Measure $\mu_{\boldsymbol{W},U}$ has regular conditional probability distribution $\mu_{\boldsymbol{W}|U}(u,\cdot)$ such that, for all fixed $x,u$, the distribution of $h(x,\boldsymbol{W})$ when $\boldsymbol{W}\sim\mu_{\boldsymbol{W}|u}(u,\cdot)$ has positive and bounded density $p(y|x,u)$ with respect Lebesgue measure. Further, $\partial_{x}^{k}\log p(y|x,u)$ for $1\leq k\leq 5$ exists and is bounded. In the low-rank matrix estimation model, this assumption is as follows. * M2. Given $\mu_{\boldsymbol{\Lambda},\boldsymbol{U}},\mu_{\boldsymbol{\Theta},\boldsymbol{V}}\in\mathscrsfs{P}_{\mathrm{c}}(\mathbb{R}^{2r})$, we sample $\\{(\boldsymbol{\lambda}_{i},\boldsymbol{u}_{i})\\}_{i\leq n}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\boldsymbol{\Lambda},\boldsymbol{U}}$, $\\{(\boldsymbol{\theta}_{j},\boldsymbol{v}_{j})\\}_{j\leq p}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\boldsymbol{\Theta},\boldsymbol{V}}$. In Appendix E, we show that Theorem 1 (resp. Theorem 2) under assumptions R3 and R4 (resp. M2) implies the theorem under the weaker assumptions R1 and R2 (resp. M1). ### 5.1 Reduction of GFOMs to approximate message passing algorithms Approximate message passing (AMP) algorithms are a special class of GFOMs that admit an asymptotic characterization called _state evolution_ [BM11]. We show that, in both models we consider, any GFOM is equivalent to an AMP algorithm after a change of variables. An AMP algorithm is defined by sequences of Lipschitz functions $(f_{t}:\mathbb{R}^{r(t+1)+1}\rightarrow\mathbb{R}^{r})_{t\geq 0}$, $(g_{t}:\mathbb{R}^{r(t+1)}\rightarrow\mathbb{R}^{r})_{t\geq 1}$. It generates sequences $(\boldsymbol{a}^{t})_{t\geq 1}$, $(\boldsymbol{b}^{t})_{t\geq 1}$ of matrices in $\mathbb{R}^{p\times r}$ and $\mathbb{R}^{n\times r}$, respectively, according to $\begin{split}\boldsymbol{a}^{t+1}=\boldsymbol{X}^{\mathsf{T}}f_{t}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{t};\boldsymbol{y},\boldsymbol{u})-\sum_{s=1}^{t}g_{s}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{s};\boldsymbol{v})\boldsymbol{\xi}_{t,s}^{\mathsf{T}},\\\ \boldsymbol{b}^{t}=\boldsymbol{X}g_{t}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{t};\boldsymbol{v})-\sum_{s=0}^{t-1}f_{s}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{s};\boldsymbol{y},\boldsymbol{u})\boldsymbol{\zeta}_{t,s}^{\mathsf{T}},\end{split}$ (16) with initialization $\boldsymbol{a}^{1}=\boldsymbol{X}^{\mathsf{T}}f_{0}(\boldsymbol{y},\boldsymbol{u})$. Here $(\boldsymbol{\xi}_{t,s})_{1\leq s\leq t}$, $(\boldsymbol{\zeta}_{t,s})_{0\leq s<t}$ are deterministic $r\times r$ matrices. The we refer to the recursion (16) as to an AMP algorithm if only if the matrices $(\boldsymbol{\xi}_{t,s})_{1\leq s\leq t}$, $(\boldsymbol{\zeta}_{t,s})_{0\leq s<t}$ are determined by the functions $(f_{t})_{t\geq 0}$, $(g_{t})_{t\geq 1}$ in a specific way, which depends on the model under consideration, and we describe in Appendix B. For this special choice of the matrices $(\boldsymbol{\xi}_{t,s})_{1\leq s\leq t}$, $(\boldsymbol{\zeta}_{t,s})_{0\leq s<t}$, the iterates $\boldsymbol{a}^{t},\boldsymbol{b}^{t}$ are asymptotically Gaussian, with a covariance that can be determined via the state evolution recursion. The next lemma, proved in Appendix B, makes this precise and describes the state evolution of the resulting AMP algorithm. ###### Lemma 1. Under assumptions A1, A2, R3, R4 (for high-dimensional regression) or assumptions A1, A2, M2 (for low-rank matrix estimation), there exist Lipschitz functions $(f_{t})_{t\geq 0},(g_{t})_{t\geq 1}$ as above and $(\varphi_{t}:\mathbb{R}^{r(t+1)}\rightarrow\mathbb{R})_{t\geq 1},(\phi_{t}:\mathbb{R}^{r(t+1)+1}\rightarrow\mathbb{R})_{t\geq 1}$, such that the following holds. Let $(\boldsymbol{\xi}_{t,s})_{1\leq s\leq t},(\boldsymbol{\zeta}_{t,s})_{0\leq s<t}$ be $r\times r$ matrices determined by the general AMP prescription (see Appendix B), and define $\\{\boldsymbol{a}^{s},\boldsymbol{b}^{s}\\}_{s\geq 0}$ via the AMP algorithm (16). Then we have $\displaystyle\boldsymbol{v}^{t}=\varphi_{t}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{t};\boldsymbol{v}),\quad t\geq 1,$ $\displaystyle\boldsymbol{u}^{t}=\phi_{t}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{t};\boldsymbol{y},\boldsymbol{u}),\quad t\geq 1.$ Further, state evolution determines two collections of of $r\times r$ matrices $(\boldsymbol{T}_{s,t})_{s,t\geq 1},(\boldsymbol{\alpha}_{t})_{t\geq 1}$ such that for all pseudo-Lipschitz functions $\psi:\mathbb{R}^{r(t+2)}\rightarrow\mathbb{R}$ of order 2, $\frac{1}{p}\sum_{j=1}^{p}\psi(\boldsymbol{a}^{1}_{j},\ldots,\boldsymbol{a}^{t}_{j},\boldsymbol{v}_{j},\boldsymbol{\theta}_{j})\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\mathbb{E}[\psi(\boldsymbol{\alpha}_{1}\boldsymbol{\Theta}+\boldsymbol{Z}^{1},\ldots,\boldsymbol{\alpha}_{t}\boldsymbol{\Theta}+\boldsymbol{Z}^{t},\boldsymbol{V},\boldsymbol{\Theta})],\\\ $ (17) where $(\boldsymbol{\Theta},\boldsymbol{V})\sim\mu_{\boldsymbol{\Theta},\boldsymbol{V}}$ independent of $(\boldsymbol{Z}^{1},\ldots,\boldsymbol{Z}^{t})\sim\mathsf{N}(\boldsymbol{0},\boldsymbol{T}_{[1:t]})$. Here $\boldsymbol{T}_{[1:t]}\in\mathbb{R}^{tr\times tr}$ is a positive semi- definite block matrix with block $(s,s^{\prime})$ given by $\boldsymbol{T}_{s,s^{\prime}}$.333We emphasize that the construction of all relevant functions and matrices depend on the model. We describe these constructions and prove Lemma 1 in Appendix B. Lemma 1 implies that the estimator $\hat{\boldsymbol{\theta}}^{t}$ in Theorem 1 and 2 can alternatively be viewed as a Lipschitz function $g_{*}:\mathbb{R}^{r(t+1)}\rightarrow\mathbb{R}^{r}$ of the AMP iterates $(\boldsymbol{a}^{s})_{s\leq t}$ and side information $\boldsymbol{v}$, applied row-wise. Thus, $\ell(\boldsymbol{\theta}_{j},\hat{\boldsymbol{\theta}}_{j}^{t})$ can be viewed as a pseudo-Lipschitz function of order 2 applied to $(\boldsymbol{a}_{j}^{s})_{s\leq t},\boldsymbol{v}_{j},\boldsymbol{\theta}_{j}$; namely, $\ell(\boldsymbol{\theta}_{j},g_{*}((\boldsymbol{a}_{j}^{s})_{s\leq t},\boldsymbol{v}_{j}))$. Then, Lemma 1 implies that the limits in Theorems 1 and 2 exist and have lower bound $\inf R_{\ell}(g_{*},(\boldsymbol{\alpha}_{s}),(\boldsymbol{T}_{s,s^{\prime}})):=\inf\mathbb{E}[\ell(\boldsymbol{\Theta},g_{*}(\boldsymbol{\alpha}_{1}\boldsymbol{\Theta}+\boldsymbol{Z}^{1},\ldots,\boldsymbol{\alpha}_{t}\boldsymbol{\Theta}+\boldsymbol{Z}^{t},\boldsymbol{V}))],$ (18) where the infimum is taken over Lipschitz functions $g_{*}$ and matrices $(\boldsymbol{\alpha}_{s}),(\boldsymbol{T}_{s,s^{\prime}})$ generated by the state evolution of _some_ AMP algorithm. This lower bound is characterized in the following sections. ### 5.2 Models and message passing on the computation tree We introduce two statistical models on trees and a collection of algorithms which correspond, in a sense we make precise, to the high-dimensional regression and low-rank matrix estimation models, and AMP algorithms. We derive lower bounds on the estimation error in these models using information- theoretic, rather than algorithmic, techniques. We then transfer these to lower bounds on (18). The models are defined using an infinite connected tree $\mathcal{T}=(\mathcal{V},\mathcal{F},\mathcal{E})$ consisting of infinite collections of variable nodes $\mathcal{V}$, factor nodes $\mathcal{F}$, and edges $\mathcal{E}$. Factor nodes have degree $p$ and have only variables nodes as neighbors, and variable nodes have degree $n$ and have only factor nodes as neighbors. These properties define the tree uniquely up to isomorphism. We denote the set of neighbors of a variable $v$ by $\partial v$, and similarly define $\partial f$. We call $\mathcal{T}$ the _computation tree_. The statistical models are joint distributions over random variables associated to the nodes and edges of the computation tree. High-dimensional regression on the computation tree. The random variables $\\{(\theta_{v},v_{v})\\}_{v\in\mathcal{V}}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\Theta,V}$, $\\{(\boldsymbol{w}_{f},u_{f})\\}_{f\in\mathcal{F}}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\boldsymbol{W},U}$, and $\\{x_{fv}\\}_{(f,v)\in\mathcal{E}}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1/n)$ are generated independently. We assume $\mu_{\Theta,V}$, $\mu_{\boldsymbol{W},U}$ are as in assumption R3. We define $y_{f}=h(\sum_{v\in\partial f}x_{fv}\theta_{v},\boldsymbol{w}_{f})$ for $h$ as in assumption R4. For each $v\in\mathcal{V}$, our objective is to estimate the coefficient $\theta_{v}$ from data $(y_{f},u_{f})_{f\in\mathcal{F}}$, $(v_{v})_{v\in\mathcal{V}}$, and $(x_{fv})_{(f,v)\in\mathcal{E}}$. Low-rank matrix estimation on the computation tree. The random variables $\\{(\boldsymbol{\theta}_{v},\boldsymbol{v}_{v})\\}_{v\in\mathcal{V}}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\boldsymbol{\Theta},\boldsymbol{V}}$, $\\{(\boldsymbol{\lambda}_{f},\boldsymbol{u}_{f})\\}_{f\in\mathcal{F}}$, and $\\{z_{fv}\\}_{(f,v)\in\mathcal{E}}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1/n)$ are generated independently. We assume $\mu_{\boldsymbol{\Lambda},\boldsymbol{U}}$, $\mu_{\boldsymbol{\Theta},\boldsymbol{V}}$ are as in assumption M2. For each $v\in\mathcal{V}$, our objective is to estimate $\boldsymbol{\theta}_{v}$ from data $(x_{fv})_{(f,v)\in\mathcal{E}}$, $(\boldsymbol{v}_{v})_{v\in\mathcal{V}}$, and $(\boldsymbol{u}_{f})_{f\in\mathcal{F}}$. When ambiguity will result, we will refer to the models of Section 3 as high- dimensional regression and low-rank matrix estimation _on the graph._ 444This terminology is motivated by viewing the models of Section 3 as equivalent to the tree-based models except that they are defined with respect to a finite complete bipartite graph between factor and variable nodes. As on the graph, we introduce dummy variables $(y_{f})_{f\in\mathcal{F}}$ in the low-rank matrix estimation problem on the computation tree. To estimate $\boldsymbol{\theta}_{v}$, we introduce the class of _message passing algorithms_. A message passing algorithm is defined by sequences of Lipschitz functions $(f_{t}:\mathbb{R}^{r(t+1)+1}\rightarrow\mathbb{R}^{r})_{t\geq 0}$, $(g_{t}:\mathbb{R}^{r(t+1)}\rightarrow\mathbb{R}^{r})_{t\geq 1}$. For each edge $(f,v)\in\mathcal{E}$, it generates sequences $(\boldsymbol{a}_{v\rightarrow f}^{t})_{t\geq 1}$, $(\boldsymbol{q}_{v\rightarrow f}^{t})_{t\geq 1}$, $(\boldsymbol{b}_{f\rightarrow v}^{t})_{t\geq 1}$, and $(\boldsymbol{r}_{f\rightarrow v}^{t})_{t\geq 0}$ of vectors in $\mathbb{R}^{r}$, called _messages_ , according to $\begin{gathered}\boldsymbol{a}_{v\rightarrow f}^{t+1}=\sum_{f^{\prime}\in\partial v\setminus f}x_{f^{\prime}v}\boldsymbol{r}_{f^{\prime}\rightarrow v}^{t},\qquad\boldsymbol{r}_{f\rightarrow v}^{t}=f_{t}(\boldsymbol{b}_{f\rightarrow v}^{1},\ldots,\boldsymbol{b}_{f\rightarrow v}^{t};y_{f},\boldsymbol{u}_{f}),\\\ \boldsymbol{b}_{f\rightarrow v}^{t}=\sum_{v^{\prime}\in\partial f\setminus v}x_{fv^{\prime}}\boldsymbol{q}_{v^{\prime}\rightarrow f}^{t},\qquad\boldsymbol{q}_{v\rightarrow f}^{t}=g_{t}(\boldsymbol{a}_{v\rightarrow f}^{1},\ldots,\boldsymbol{a}_{v\rightarrow f}^{t};\boldsymbol{v}_{v}),\end{gathered}$ (19) with initialization $\boldsymbol{r}_{f\rightarrow v}^{0}=f_{0}(y_{f},\boldsymbol{u}_{f})$ and $\boldsymbol{a}_{v\rightarrow f}^{1}=\sum_{f^{\prime}\in\partial v\setminus f}x_{f^{\prime}v}\boldsymbol{r}_{f^{\prime}\rightarrow v}^{0}$. We also define for every variable and factor node the vectors $\boldsymbol{a}_{v}^{t+1}=\sum_{f\in\partial v}x_{fv}\boldsymbol{r}_{f\rightarrow v}^{t},\qquad\boldsymbol{b}_{f}^{t}=\sum_{v\in\partial f}x_{fv}\boldsymbol{q}_{v\rightarrow f}^{t}.$ (20) These are called _beliefs_. The vector $\boldsymbol{\theta}_{v}$ is estimated after $t$ iterations by $\hat{\boldsymbol{\theta}}_{v}^{t}=g_{*}(\boldsymbol{a}_{v}^{1},\ldots,\boldsymbol{a}_{v}^{t};\boldsymbol{v}_{v})$. Message passing algorithms on the computation tree correspond to AMP algorithms on the graph in the sense that their iterates are asymptotically characterized by the same state evolution. ###### Lemma 2. In both the high-dimensional regression and low-rank matrix estimation problems on the tree, the following is true. For any Lipschitz functions $(f_{t})_{t\geq 0}$, $(g_{t})_{t\geq 1}$, there exist collections of $r\times r$ matrices $(\boldsymbol{T}_{s,t})_{s,t\geq 1},(\boldsymbol{\alpha}_{t})_{t\geq 1}$ such that for any node $v$ chosen independently of the randomness on the model, fixed $t\geq 1$, and under the asymptotics $n,p\rightarrow\infty$, $n/p\rightarrow\delta\in(0,\infty)$, the message passing algorithm (19) generates beliefs at $v$ satisfying $\displaystyle(\boldsymbol{a}^{1}_{v},\ldots,\boldsymbol{a}^{t}_{v},\boldsymbol{v}_{v},\boldsymbol{\theta}_{v})\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}(\boldsymbol{\alpha}_{1}\boldsymbol{\Theta}+\boldsymbol{Z}^{1},\ldots,\boldsymbol{\alpha}_{t}\boldsymbol{\Theta}+\boldsymbol{Z}^{t},\boldsymbol{V},\boldsymbol{\Theta}),$ where $(\boldsymbol{\Theta},\boldsymbol{V})\sim\mu_{\boldsymbol{\Theta},\boldsymbol{V}}$ independent of $(\boldsymbol{Z}^{1},\ldots,\boldsymbol{Z}^{t})\sim\mathsf{N}(\boldsymbol{0},\boldsymbol{T}_{[1:t]})$, and $\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}$ denotes convergence in the Wasserstein metric of order 2 (see Appendix A). Moreover, the matrices $(\boldsymbol{T}_{s,t})_{s,t\geq 1},(\boldsymbol{\alpha}_{t})_{t\geq 1}$ agree with those in Lemma 1 when the functions $(f_{t})_{t\geq 0}$, $(g_{t})_{t\geq 1}$ also agree. We prove Lemma 2 in Appendix C. Lemma 2 and the properties of convergence in the Wasserstein metric of order 2 (see Lemma 2, Appendix A) imply that for any message passing estimator $\hat{\boldsymbol{\theta}}_{v}^{t}$ and loss $\ell$, the risk $\mathbb{E}[\ell(\boldsymbol{\theta}_{v},\hat{\boldsymbol{\theta}}_{v}^{t})]=\mathbb{E}[\ell(\boldsymbol{\theta}_{v},g_{*}(\boldsymbol{a}_{v}^{1},\ldots,\boldsymbol{a}_{v}^{t};\boldsymbol{v}_{v})]$ converges to $R_{\ell}(g_{*},(\boldsymbol{\alpha}_{s}),(\boldsymbol{T}_{s,s^{\prime}}))$, in agreement with the asymptotic error of the corresponding AMP estimator on the graph. On the computation tree, we may lower bound this limiting risk by information- theoretic techniques, as we now explain. By induction, the estimate $\hat{\boldsymbol{\theta}}_{v}^{t}$ is a function only of observations corresponding to edges and nodes in the ball of radius $2t-1$ centered at $v$ on the computation tree. We denote the observations in this local neighborhood by $\mathcal{T}_{v,2t-1}$. We lower bound the risk of $\hat{\boldsymbol{\theta}}_{v}^{t}$ by the optimal risk of any measurable estimator, possibly intractable, which depends only on $\mathcal{T}_{v,2t-1}$; we call this the _local Bayes risk_. The following lemma characterizes the local Bayes risk. ###### Lemma 3. Consider a quadratically-bounded loss $\ell:\mathbb{R}^{2r}\rightarrow\mathbb{R}_{\geq 0}$. In the high-dimensional regression (resp. low-rank matrix estimation) model on the computation tree and under the asymptotics $n,p\rightarrow\infty$, $n/p\rightarrow\delta\in(0,\infty)$, $\liminf_{n\rightarrow\infty}\inf_{\hat{\boldsymbol{\theta}}(\cdot)}\mathbb{E}[\ell(\boldsymbol{\theta}_{v},\hat{\boldsymbol{\theta}}(\mathcal{T}_{v,2t-1}))]\geq R^{*},$ where the infimum is over all measurable functions of $\mathcal{T}_{v,2t-1}$, and $R^{*}$ is equal to the right-hand side of Eq. (6) (resp. Eq. (8)). We prove Lemma 3 in Appendix D. Combining Lemma 3 with the preceding discussion, we conclude that $R_{\ell}(g_{*},(\boldsymbol{\alpha}_{s}),(\boldsymbol{T}_{s,s^{\prime}}))\geq R_{*}$ for all Lipschitz functions $g_{*}$ and matrices $(\boldsymbol{\alpha}_{s}),(\boldsymbol{T}_{s,s^{\prime}})$ generated by the state evolution of some message passing or, equivalently, by some AMP algorithm. The bounds (6) and (8) now follow. 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It is well known that $W_{2}(\mu,\mu^{\prime})$ is a metric on $\mathscrsfs{P}_{2}(\mathbb{R}^{k})$ [Vil10, pg. 94]. When a sequence of probability distributions $\mu_{n}$ converges to $\mu$ in the Wasserstein metric of order 2, we write $\mu_{n}\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}\mu$. We also write $\boldsymbol{A}_{n}\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}\boldsymbol{A}$ when $\boldsymbol{A}_{n}\sim\mu_{n}$, $\boldsymbol{A}\sim\mu$ for such a sequence. ###### Lemma 1. If $f:\mathbb{R}^{r}\rightarrow\mathbb{R}$ and $g:\mathbb{R}^{r}\rightarrow\mathbb{R}$ are pseudo-Lipschitz of order $k_{1}$ and $k_{2}$, respectively, then their product is pseudo-Lipschitz of order $k_{1}+k_{2}$. ###### Lemma 2. If a sequence of random vectors $\boldsymbol{X}_{n}\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}\boldsymbol{X}$, then for any pseudo-Lipschitz function $f$ of order $2$ we have $\mathbb{E}[f(\boldsymbol{X}_{n})]\rightarrow\mathbb{E}[f(\boldsymbol{X})]$. ###### Lemma 3. Consider a sequence of random variables $(A_{n},\boldsymbol{B}_{n})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(A,\boldsymbol{B})$ with values in $\mathbb{R}\times\mathbb{R}^{k}$ such that $(A_{n},\boldsymbol{B}_{n})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(A,\boldsymbol{B})$ and $A_{n}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}A$ for all $n$. Then, for any bounded measurable function $f:\mathbb{R}\times\mathbb{R}^{k}\rightarrow\mathbb{R}$ for which $\boldsymbol{b}\mapsto f(a,\boldsymbol{b})$ is continuous for all $a$, we have $\mathbb{E}[f(A_{n},\boldsymbol{B}_{n})]\rightarrow\mathbb{E}[f(A,\boldsymbol{B})]$. Further, for any function $\phi:\mathbb{R}\times\mathbb{R}^{k}\rightarrow\mathbb{R}^{k^{\prime}}$ (possibly unbounded) which is continuous in all but the first coordinate, we have $\phi(A_{n},\boldsymbol{B}_{n})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}\phi(A,\boldsymbol{B})$. ###### Proof of Lemma 3. Without loss of generality, $f$ takes values in $[0,1]$. First we show that for any set $S\times I$ where $S\subset\mathbb{R}$ is measurable and $I\subset\mathbb{R}^{k}$ is a rectangle whose boundary has probability 0 under $\boldsymbol{B}$ that $\mu_{A_{n},\boldsymbol{B}_{n}}(S\times I)\rightarrow\mu_{A,\boldsymbol{B}}(S\times I)\,.$ (21) First, we show this is true for $S=K$ a closed set. Fix $\epsilon>0$. Let $\phi_{K}^{\epsilon}:\mathbb{R}\rightarrow[0,1]$ be a continuous function which is 1 on $K$ and 0 for all points separted from $K$ by distance $\epsilon$. Similarly define $\phi_{I}^{\epsilon}:\mathbb{R}^{k}\rightarrow\mathbb{R}$. Then $\displaystyle\mathbb{E}[\phi_{K}^{\epsilon}(A_{n})\phi_{I}^{\epsilon}(\boldsymbol{B}_{n})]\geq\mu_{A_{n},\boldsymbol{B}_{n}}(K\times I)\geq\mathbb{E}[\phi_{K}^{\epsilon}(A_{n})\phi_{I}^{\epsilon}(\boldsymbol{B}_{n})]-\epsilon-\mu_{\boldsymbol{B}_{n}}(\mathsf{spt}(\phi_{I}^{\epsilon})\setminus I)\,.$ Because the boundary of $I$ has measure 0 under $\mu_{\boldsymbol{B}}$, we have $\lim_{\epsilon\rightarrow 0}\limsup_{n\rightarrow\infty}\mu_{\boldsymbol{B}_{n}}(\mathsf{spt}(\phi_{I}^{\epsilon})\setminus I)=0$. Also, $\lim_{\epsilon\rightarrow 0}\lim_{n\rightarrow\infty}\mathbb{E}[\phi_{K}^{\epsilon}(A_{n})\phi_{I}^{\epsilon}(\boldsymbol{B}_{n})]=\lim_{\epsilon\rightarrow 0}=\mathbb{E}[\phi_{K}^{\epsilon}(A)\phi_{I}^{\epsilon}(\boldsymbol{B})]=\mu_{A,\boldsymbol{B}}(K\times I)$. Thus, taking $\epsilon\rightarrow\infty$ after $n\rightarrow\infty$, the previous display gives $\mu_{A_{n},\boldsymbol{B}_{n}}(K\times I)\rightarrow\mu_{A,\boldsymbol{B}}(K\times I)$. For $S=G$ an open set, we can show $\mu_{A_{n},\boldsymbol{B}_{n}}(G\times I)\rightarrow\mu_{A,\boldsymbol{B}}(G\times I)$ by a similar argument: take instead $\phi_{K}^{\epsilon}$ to be 0 outside of $G$ and $1$ for all points in $G$ separated from the boundary by at least $\epsilon$, and likewise for $\phi_{I}^{\epsilon}$. By Theorem 12.3 of [Bil12], we can construct $K\subset S\subset G$ such that $K$ is closed and $G$ is open, and $\mu_{A}(K)>\mu_{A}(S)-\epsilon$, $\mu_{A}(G)<\mu_{A}(S)+\epsilon$. The previous paragraph implies that $\displaystyle\mu_{A,\boldsymbol{B}}(S\times I)-\epsilon$ $\displaystyle\leq\mu_{A,\boldsymbol{B}}(K\times I)\leq\liminf_{n\rightarrow\infty}\mu_{A_{n},\boldsymbol{B}_{n}}(S\times I)$ $\displaystyle\leq\limsup_{n\rightarrow\infty}\mu_{A_{n},\boldsymbol{B}_{n}}(S\times I)\leq\mu_{A,\boldsymbol{B}}(G\times I)\leq\mu_{A,\boldsymbol{B}}(S\times I)+\epsilon\,.$ Taking $\epsilon\rightarrow 0$, we conclude (21). We now show (21) implies the lemma. Fix $\epsilon>0$. Let $M$ be such that $\mathbb{P}(\boldsymbol{B}_{n}\in[-M,M]^{k})>1-\epsilon$ for all $n$ and $\mathbb{P}(\boldsymbol{B}\in[-M,M]^{k})>1-\epsilon$, which we may do by tightness. For each $a$, let $\delta(a,\epsilon)=\sup\\{0<\Delta\leq M\mid\|\boldsymbol{b}-\boldsymbol{b}^{\prime}\|_{\infty}<\Delta\Rightarrow|f(a,\boldsymbol{b})-f(a,\boldsymbol{b}^{\prime})|<\epsilon\\}$. Because continuous functions are uniformly continuous on compact sets, the supremum is over a non-empty, bounded set. Thus, $\delta(a,\epsilon)$ is positive and bounded above by $M$ for all $a$. Further, $\delta(a,\epsilon)$ is measurable and non-decreasing in $\epsilon$. Pick $\delta_{*}$ such that $\mathbb{P}(\delta(A,\epsilon)<\delta_{*})<\epsilon$, which we may do because $\delta(a,\epsilon)$ is positive for all $a$. We can partition $[-M,M]^{k}$ into rectangles with side-widths smaller than $\delta_{*}$ such that the probability that $\boldsymbol{B}$ lies on the boundary of one of the partitioning rectangles is 0. Define $f_{-}(a,\boldsymbol{b}):=\sum_{\iota}\mathbf{1}\\{\boldsymbol{b}\in I_{\iota}\\}\inf_{\boldsymbol{b}^{\prime}\in I_{\iota}}f(a,\boldsymbol{b}^{\prime})$ and $f_{+}(a,\boldsymbol{b}):=\sum_{\iota}\mathbf{1}\\{\boldsymbol{b}\in I_{\iota}\\}\sup_{\boldsymbol{b}^{\prime}\in I_{\iota}}f(a,\boldsymbol{b}^{\prime})$, and note that on $\\{\delta(a,\epsilon)<\delta^{*}\\}\times[-M,M]^{k}$, we have $f_{-}(a,\boldsymbol{b})\leq f(a,\boldsymbol{b})\leq f_{+}(a,\boldsymbol{b})$ and $|f(a,\boldsymbol{b})-f_{-}(a,\boldsymbol{b})|<\epsilon$ and $|f(a,\boldsymbol{b})-f_{+}(a,\boldsymbol{b})|<\epsilon$. Thus, by the boundedness of $f$ and the high-probability bound on $\\{\delta(a,\epsilon)<\delta^{*}\\}\times[-M,M]^{k}$ $\begin{gathered}\mathbb{E}[f_{-}(A_{n},\boldsymbol{B}_{n})]-2\epsilon<\mathbb{E}[f(A_{n},\boldsymbol{B}_{n})]<\mathbb{E}[f_{+}(A_{n},\boldsymbol{B}_{n})]+2\epsilon\,,\\\ \mathbb{E}[f_{-}(A,\boldsymbol{B})]-2\epsilon<\mathbb{E}[f(A,\boldsymbol{B})]<\mathbb{E}[f_{+}(A,\boldsymbol{B})]+2\epsilon\,.\end{gathered}$ (22) We show that $\mathbb{E}[f_{-}(A_{n},\boldsymbol{B}_{n})]\rightarrow\mathbb{E}[f_{-}(A,\boldsymbol{B})]$. Fix $\xi>0$. Take $0=x_{0}\leq\ldots\leq x_{N}=1$ such that $x_{j+1}-x_{j}<\xi$ for all $j$. Let $S_{j\iota}=\\{a\mid\inf_{\boldsymbol{b}^{\prime}\in I_{\iota}}f(a,\boldsymbol{b}^{\prime})\in[x_{j},x_{j+1})\\}$. Then $\sum_{\iota,j}x_{j}\mathbf{1}\\{a\in S_{j\iota},\boldsymbol{b}\in I_{\iota}\\}+\xi\geq f_{-}(a,\boldsymbol{b})\geq\sum_{\iota,j}x_{j}\mathbf{1}\\{a\in S_{j\iota},\boldsymbol{b}\in I_{\iota}\\}\,.$ By (21), we conclude $\mathbb{E}[\sum_{\iota,j}x_{j}\mathbf{1}\\{A_{n}\in S_{j\iota},\boldsymbol{B}_{n}\in I_{\iota}\\}]\rightarrow\mathbb{E}[\sum_{\iota,j}x_{j}\mathbf{1}\\{A\in S_{j\iota},\boldsymbol{B}\in I_{\iota}\\}]$. Combined with the previous display and taking $\xi\rightarrow 0$, we conclude that $\mathbb{E}[f_{-}(A_{n},\boldsymbol{B}_{n})]\rightarrow\mathbb{E}[f_{-}(A,\boldsymbol{B})]$. Similarly, we may argue that $\mathbb{E}[f_{+}(A_{n},\boldsymbol{B}_{n})]\rightarrow\mathbb{E}[f_{+}(A,\boldsymbol{B})]$. The first statment in the lemma now follows from taking $\epsilon\rightarrow 0$ after $n\rightarrow\infty$ in (22). The second statement in the lemma follows by observing that for any bounded continuous function $f:\mathbb{R}^{k^{\prime}}\rightarrow\mathbb{R}$, we have that $f\circ\phi$ is bounded and is continuous in all but the first coordinate, so that we may apply the first part of the lemma to conclude $\mathbb{E}[f(\phi(A_{n},\boldsymbol{B}_{n}))]\rightarrow\mathbb{E}[f(\phi(A,\boldsymbol{B}))]$. ∎ We will sometimes use the following alternative form of recursion (5) defining the lower bound in the high-dimensional regression model. ###### Lemma 4. Consider a family, indexed by $x\in\mathbb{R}$, of bounded probability densities $p(\cdot|x,u)$ with respect to some base measure $\mu_{Y}$. Then for $\tilde{\tau}>0$ and $\sigma\geq 0$ we have that $\frac{1}{\tilde{\tau}^{2}}\mathbb{E}[\mathbb{E}[G_{1}|Y,G_{0},U]^{2}]=\mathbb{E}_{G_{0},Y}\left[\left(\frac{\mathrm{d}\phantom{b}}{\mathrm{d}x}\log\mathbb{E}_{G_{1}}p(Y|x+\sigma G_{0}+\tilde{\tau}G_{1},U)\Big{|}_{x=0}\right)^{2}\right]\,,$ where $G_{0},G_{1}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1)$ and $Y|G_{0},G_{1},U$ had density $p(\cdot|\sigma G_{0}+\tilde{\tau}G_{1},U)$ with respect to $\mu_{Y}$. In particular, the derivatives exist. (In this case, we may equivalently generate $Y=h(\sigma G_{0}+\tilde{\tau}G_{1},\boldsymbol{W})$ for $(\boldsymbol{W},U)\sim\mu_{\boldsymbol{W},U}$). The preceding lemma applies, in particular, for $p$ as in R4. It then provides an alternative form of the second equation in recursion (5). ###### Lemma 4. We have $\mathbb{E}_{G_{1}}p(Y|x+\sigma G_{0}+\tilde{\tau}G_{1},U)=\int p(Y|\sigma G_{0}+s,U)\frac{1}{\sqrt{2\pi}\tilde{\tau}}e^{-\frac{1}{2\tilde{\tau}^{2}}(s-x)^{2}}{\mathrm{d}}g\,,$ so that $\displaystyle\frac{\mathrm{d}\phantom{b}}{\mathrm{d}x}\mathbb{E}_{G_{1}}p(Y|x+\sigma G_{0}+\tilde{\tau}G_{1},U)$ $\displaystyle=\frac{1}{\tilde{\tau}^{2}}\int p(Y|\sigma G_{0}+s,U)\frac{(s-x)}{\sqrt{2\pi}\tilde{\tau}}e^{-\frac{1}{2\tilde{\tau}^{2}}(s-x)^{2}}{\mathrm{d}}g\,,$ where the boundedness of of $p$ allows us to exchange integration and differentition. Thus, $\frac{\mathrm{d}\phantom{b}}{\mathrm{d}x}\log\mathbb{E}_{G_{1}}p(Y|x+\sigma G_{0}+\tilde{\tau}G_{1},U)=\frac{1}{\tilde{\tau}}\mathbb{E}[G_{1}|Y,G_{0},U]\,.$ The result follows. ∎ Finally, we collect some results on the Bayes risk with respect to quadratically-bounded losses $\ell:\mathbb{R}^{k}\times\mathbb{R}^{k}\rightarrow\mathbb{R}_{\geq 0}$. Recall that quadratically-bounded means that $\ell$ is pseudo-Lipschitz of order 2 and also satisfies $|\ell(\boldsymbol{\vartheta},\boldsymbol{d})-\ell(\boldsymbol{\vartheta}^{\prime},\boldsymbol{d})|\leq C\left(1+\sqrt{\ell(\boldsymbol{\vartheta},\boldsymbol{d})}+\sqrt{\ell(\boldsymbol{\vartheta}^{\prime},\boldsymbol{d})})\right)\|\boldsymbol{\vartheta}-\boldsymbol{\vartheta}^{\prime}\|\,.$ (23) We consider a setting $(\boldsymbol{\Theta},\boldsymbol{V})\sim\mu_{\boldsymbol{\Theta},\boldsymbol{V}}\in\mathscrsfs{P}_{2}(\mathbb{R}^{k}\times\mathbb{R}^{k})$, $\boldsymbol{Z}\sim{\mathsf{N}}(0,\boldsymbol{I}_{k})$ independent and $\tau,K,M\geq 0$. Define $\boldsymbol{\Theta}^{(K)}$ by $\Theta^{(K)}_{i}=\Theta_{i}\mathbf{1}\\{|\Theta_{i}|\leq K\\}$. Denote by $\mu_{\boldsymbol{\Theta}^{(K)},\boldsymbol{V}}$ the joint distribution of $\boldsymbol{\Theta}^{(K)}$ and $\boldsymbol{V}$, and by $\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}:\mathbb{R}^{k}\times\mathcal{B}\rightarrow[0,1]$ a regular conditional probability distribution for $\boldsymbol{\Theta}^{(K)}$ conditioned on $\boldsymbol{V}$. Define the posterior Bayes risk $R(\boldsymbol{y},\tau,\boldsymbol{v},K,M):=\inf_{\|\boldsymbol{d}\|_{\infty}\leq M}\int\frac{1}{Z}\ell(\boldsymbol{\vartheta},\boldsymbol{d})e^{-\frac{1}{2\tau^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})\,,$ (24) where $Z=\int e^{-\frac{1}{2\tau^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})$ is a normalization constant. It depends on $\boldsymbol{y},\tau,\boldsymbol{v},K$. When required for clarity, we write $Z(\boldsymbol{y},\tau,\boldsymbol{v},K)$. ###### Lemma 5. The following properties hold for the Bayes risk with respect to pseudo- Lipschitz losses of order 2 satsifying (23). 1. (a) For any $\tau,K,M$, with $K,M$ possibly equal to infinity, the Bayes risk is equal to the expected posterior Bayes risk. That is, $\inf_{\hat{\boldsymbol{\theta}}(\cdot)}\mathbb{E}[\ell(\boldsymbol{\Theta}^{(K)},\hat{\boldsymbol{\theta}}(\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{z},\boldsymbol{V})]=\mathbb{E}[R(\boldsymbol{Y}^{(K)},\tau,\boldsymbol{V},K,M)]\,,$ (25) where $\boldsymbol{Y}^{(K)}=\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z}$ with $\boldsymbol{Z}\sim{\mathsf{N}}(\boldsymbol{0},\boldsymbol{I}_{k})$ independent of $\boldsymbol{\Theta}^{(K)}$ and the infimum is taken over all measurable functions $(\mathbb{R}^{k})^{2}\rightarrow[-M,M]^{k}$. Moreover, $\displaystyle\mathbb{E}[R(\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z},\tau,\boldsymbol{V},K,\infty)]$ $\displaystyle=\lim_{M\rightarrow\infty}\mathbb{E}[R(\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z},\tau,\boldsymbol{V},K,M)]\,.$ (26) 2. (b) For a fixed $K<\infty$, the posterior Bayes risk is bounded: $R(\boldsymbol{y},\tau,\boldsymbol{v},K,M)\leq\bar{R}(K)$ for some function $\bar{R}$ which does not depend on $\boldsymbol{y},\tau,\boldsymbol{v},M$. Further, for $K<\infty$ the function $(\boldsymbol{y},\tau)\mapsto R(\boldsymbol{y},\tau,\boldsymbol{v},K,M)$ is continuous on $\mathbb{R}^{k}\times\mathbb{R}_{>0}$. 3. (c) The Bayes risk is jointly continuous in truncation level $K$ and noise variance $\tau$. This is true also at $K=\infty$: $\displaystyle\mathbb{E}[R(\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z},\tau,\boldsymbol{V},K,\infty)]$ $\displaystyle=\lim_{\begin{subarray}{c}K\rightarrow\infty\\\ \tau^{\prime}\rightarrow\tau\end{subarray}}\mathbb{E}[R(\boldsymbol{\Theta}^{(K)}+\tau^{\prime}\boldsymbol{Z},\tau^{\prime},\boldsymbol{V},K,\infty)]\,,$ (27) where the limit holds for any way of taking $K,\tau^{\prime}$ to their limits (ie., sequentially or simultaneously). ###### Proof of Lemma 5(a). For any measurable $\hat{\boldsymbol{\theta}}:\mathbb{R}^{k}\times\mathbb{R}^{k}\rightarrow[-M,M]^{k}$, $\displaystyle\mathbb{E}[\ell(\boldsymbol{\Theta}^{(K)},\hat{\boldsymbol{\theta}}(\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z},\boldsymbol{V}))]$ $\displaystyle=\mathbb{E}[\mathbb{E}[\ell(\boldsymbol{\Theta}^{(K)},\hat{\boldsymbol{\theta}}(\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z},\boldsymbol{V}))|\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z},\boldsymbol{V}]]$ $\displaystyle\geq\mathbb{E}[R(\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z},\tau,\boldsymbol{V},K,M)]\,.$ (28) For $M<\infty$, equality obstains. Indeed, we may define $\hat{\boldsymbol{\theta}}^{(M)}(\boldsymbol{y},\boldsymbol{v};\tau)=\arg\min_{\|\boldsymbol{d}\|_{\infty}\leq M}\int\frac{1}{Z}\ell(\boldsymbol{\vartheta},\boldsymbol{d})e^{-\frac{1}{2\tau^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})\,,$ (29) because the integral is continuous in $\boldsymbol{d}$ by dominated convergence. Then $\mathbb{E}[\ell({\boldsymbol{\Theta}^{(K)}},\hat{\boldsymbol{\theta}}^{(M)}(\boldsymbol{Y},\boldsymbol{V};\tau))]=\mathbb{E}[R(\boldsymbol{Y},\tau,\boldsymbol{V},K,M)]$ when $\boldsymbol{Y}=\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z}$. Observe $R(\boldsymbol{y},\tau,\boldsymbol{v},K,M)\downarrow R(\boldsymbol{y},\tau,\boldsymbol{v},K,\infty)$ as $M\rightarrow\infty$ with the other arguments fixed. Thus, $\mathbb{E}[R(\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z},\tau,\boldsymbol{V},K,M)]\downarrow\mathbb{E}[R(\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z},\tau,\boldsymbol{V},K,\infty)]\,$ in this limit. Because $\mathbb{E}[R(\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z},\tau,\boldsymbol{V},K,\infty)]$ is a lower bound on the Bayes risk at $M=\infty$ by (28) and we may achieve risk arbitrarily close to this lower bound by taking $M\rightarrow\infty$ in (29), we conclude (25) at $M=\infty$ as well. ∎ ###### Proof of Lemma 5(b). The quantity $R(\boldsymbol{y},\tau,\boldsymbol{v},K,M)$ is non-negative. Define $\bar{R}(K)=\max_{\|\boldsymbol{\vartheta}\|_{\infty}\leq K}\ell(\boldsymbol{\vartheta},\boldsymbol{0})$. Observe that $R(\boldsymbol{y},\tau,\boldsymbol{v},K,M)\leq\bar{R}(K)$ for all $\boldsymbol{y},\tau,\boldsymbol{v},K,M$. Let $p^{*}(\boldsymbol{\theta}|\boldsymbol{y},\tau,\boldsymbol{v},K)=\frac{1}{Z}e^{-\frac{1}{2\tau^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}$. For any fixed $\boldsymbol{d}$, we have $\displaystyle\left\|\nabla_{\boldsymbol{y}}\int\ell(\boldsymbol{\vartheta},\boldsymbol{d})p^{*}(\boldsymbol{\vartheta}|\boldsymbol{y},\tau,\boldsymbol{v},K)\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},\mathrm{d}\boldsymbol{\vartheta})\right\|$ $\displaystyle\leq\int\ell(\boldsymbol{\vartheta},\boldsymbol{d})p^{*}(\boldsymbol{\vartheta}|\boldsymbol{y},\tau,v)\left\|\nabla_{\boldsymbol{y}}\log p^{*}(\boldsymbol{\vartheta}|\boldsymbol{y},\tau,\boldsymbol{v})\right\|\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},\mathrm{d}\boldsymbol{\vartheta})$ $\displaystyle\leq\frac{2K\sqrt{k}}{\tau^{2}}\int\ell(\boldsymbol{\vartheta},\boldsymbol{d})p^{*}(\boldsymbol{\vartheta}|\boldsymbol{y},\tau,\boldsymbol{v})\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},\mathrm{d}\boldsymbol{\vartheta})\,,$ where we have used that $\|\nabla_{\boldsymbol{y}}\log p^{*}(\boldsymbol{\vartheta}|\boldsymbol{y},\tau,\boldsymbol{v})\|=\frac{1}{\tau^{2}}(\boldsymbol{\vartheta}-\mathbb{E}_{\boldsymbol{\Theta}^{(K)}}[\boldsymbol{\Theta}^{(K)}])\leq 2K\sqrt{k}/\tau^{2}$, and the expectation is taken with respect to $\boldsymbol{\Theta}^{(K)}$ having density $p^{*}(\boldsymbol{\vartheta}|\boldsymbol{y},\tau,\boldsymbol{v})$ with respect to $\mu_{\boldsymbol{\Theta}^{(K)}|V}(\boldsymbol{v},\cdot)$. Thus, for fixed $\tau,\boldsymbol{d},\boldsymbol{v}$ satisfying $\int\ell(\boldsymbol{\vartheta},\boldsymbol{d})p^{*}(\boldsymbol{\vartheta}|\boldsymbol{y},\tau,\boldsymbol{v})\mu_{\Theta|V}(\boldsymbol{v},\mathrm{d}\boldsymbol{\vartheta})\leq\bar{R}$, the function $\boldsymbol{y}\mapsto\int\ell(\boldsymbol{\vartheta},\boldsymbol{d})p^{*}(\boldsymbol{\vartheta}|\boldsymbol{y},\tau,\boldsymbol{v})\mu_{\Theta|V}(\boldsymbol{v},\mathrm{d}\boldsymbol{\vartheta})$ is $2K\sqrt{k}\bar{R}/\tau^{2}$-Lipschitz. Because the infimum defining $R$ can be taken over such $\boldsymbol{d}$ and infima retain a uniform Lipschitz property, $R(\boldsymbol{y},\tau,\boldsymbol{v},K,M)$ is $2K\sqrt{k}\bar{R}/\tau^{2}$-Lipschitz in $\boldsymbol{y}$ for fixed $\tau,\boldsymbol{v},K,M$. By a similar argument, we can establish that $R(\boldsymbol{y},\tau,\boldsymbol{v},K,M)$ is $2(K^{2}k+2\|\boldsymbol{y}\|K\sqrt{k})/\bar{\tau}^{3}$-Lipschitz in $\tau$ on the set $\tau>\bar{\tau}$ for any fixed $\bar{\tau}>0$ and any fixed $\boldsymbol{y},\boldsymbol{v},K,M$. We conclude $(\boldsymbol{y},\tau)\mapsto R(\boldsymbol{y},\tau,\boldsymbol{v},K,M)$ is continuous on $\mathbb{R}^{k}\times\mathbb{R}_{>0}$. Lemma 5(b) has been shown. ∎ ###### Proof of Lemma 5(c). Finally, we prove (27). For any $K>0$, we may write555Precisely, for any regular conditional probability distribution $\mu_{\boldsymbol{\Theta}|\boldsymbol{V}}$ for $\boldsymbol{\Theta}$ given $\boldsymbol{V}$, this formula gives a valid version of a regular conditional probability distribution for $\boldsymbol{\Theta}^{(K)}$ given $\boldsymbol{V}$. We assume we use this version throughout our proof. $\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},\cdot)=\mu_{\boldsymbol{\Theta}|\boldsymbol{V}}(\boldsymbol{v},\cdot)|_{[-K,K]^{k}}+\mu_{\boldsymbol{\Theta}|\boldsymbol{V}}(\boldsymbol{v},([-K,K]^{k})^{c})\delta_{\boldsymbol{0}}(\cdot)\,.$ (30) Choose $\bar{K},\epsilon^{\prime}>0$ such that $|\tau^{\prime}-\tau|<\epsilon^{\prime}$ implies $\int_{[-\bar{K},\bar{K}]^{k}}\frac{1}{Z(\boldsymbol{y},\tau^{\prime},\boldsymbol{v},\infty)}e^{-\frac{1}{2{\tau^{\prime}}^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})\geq\frac{1}{2}\int\frac{1}{Z(\boldsymbol{y},\tau,\boldsymbol{v},\infty)}e^{-\frac{1}{2\tau^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})\,.$ Fix $\epsilon>0$ and $K^{\prime}>K>0$ with $K^{\prime}$ possibly equal to infinity. By (24), we may choose $\boldsymbol{d}^{*}$ such that $\int\frac{1}{Z(\boldsymbol{y},\tau,\boldsymbol{v},K)}\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})e^{-\frac{1}{2\tau^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})\leq(1+\epsilon)R(\boldsymbol{y},\tau,\boldsymbol{v},K,\infty)\,.$ (31) By the definition of $\bar{K}$, there exists $\boldsymbol{\vartheta}^{*}\in[-\bar{K},\bar{K}]^{k}$ such that $\ell(\boldsymbol{\vartheta}^{*},\boldsymbol{d}^{*})\leq 2(1+\epsilon)R(\boldsymbol{y},\tau,\boldsymbol{v},K,\infty)\,.$ By (23), we conclude that $\displaystyle\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})$ $\displaystyle\leq C\left(1+\sqrt{2(1+\epsilon)R(\boldsymbol{y},\tau,\boldsymbol{v},K,\infty)}+\sqrt{\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})}\right)\|\boldsymbol{\vartheta}-\boldsymbol{\vartheta}^{*}\|\,,$ whence $\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})\leq\left(1+\sqrt{2(1+\epsilon)R(\boldsymbol{y},\tau,\boldsymbol{v},K,\infty)}+3C\|\boldsymbol{\vartheta}-\boldsymbol{\vartheta}^{*}\|\right)^{2}\,.$ (32) Then $\displaystyle\left|\int\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})e^{-\frac{1}{2{\tau^{\prime}}^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K^{\prime})}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})-\int\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})e^{-\frac{1}{2\tau^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})\right|$ $\displaystyle\qquad\leq\left|\int\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})e^{-\frac{1}{2{\tau^{\prime}}^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K^{\prime})}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})-\int\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})e^{-\frac{1}{2{\tau^{\prime}}^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})\right|$ $\displaystyle\qquad\qquad+\left|\int\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})e^{-\frac{1}{2{\tau^{\prime}}^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})-\int\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})e^{-\frac{1}{2{\tau}^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})\right|$ $\displaystyle\qquad\leq\int_{([-K,K]^{k})^{c}}\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})e^{-\frac{1}{2{\tau^{\prime}}^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})+\ell(\boldsymbol{0},\boldsymbol{d}^{*})e^{-\frac{1}{2{\tau^{\prime}}^{2}}\|\boldsymbol{y}\|^{2}}\mu_{\boldsymbol{\Theta}|\boldsymbol{V}}(\boldsymbol{v},([-K,K]^{k})^{c})$ $\displaystyle\qquad\qquad+\left|\int\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})e^{-\frac{1}{2{\tau^{\prime}}^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})-\int\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})e^{-\frac{1}{2{\tau}^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})\right|$ $\displaystyle\qquad\leq\xi(K,\tau^{\prime})(1+R(\boldsymbol{y},\tau,\boldsymbol{v},K,\infty))\,,$ for some $\xi(K,\tau^{\prime})\rightarrow 0$ as $K\rightarrow\infty$, $\tau^{\prime}\rightarrow\tau$ because the conditional measure $\mu_{\boldsymbol{\Theta}|\boldsymbol{V}}(\boldsymbol{v},\cdot)$ has finite second moment and $\ell$ is bounded by (32). Then, by (31), $\displaystyle Z(\boldsymbol{y},\tau^{\prime},\boldsymbol{v},K^{\prime})R(\boldsymbol{y},\tau^{\prime},\boldsymbol{v},K^{\prime},\infty)$ $\displaystyle\leq\int\ell(\boldsymbol{\vartheta},\boldsymbol{d}^{*})e^{-\frac{1}{2\tau^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K^{\prime})}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})$ $\displaystyle\leq(1+\epsilon)Z(\boldsymbol{y},\tau,\boldsymbol{v},K)R(\boldsymbol{y},\tau,\boldsymbol{v},K,\infty)+\xi(K,\tau^{\prime})(1+R(\boldsymbol{y},\tau,\boldsymbol{v},K,\infty))\,.$ By dominated convergence, we have that $Z(\boldsymbol{y},\tau^{\prime},\boldsymbol{v},K^{\prime})\rightarrow Z(\boldsymbol{y},\tau,\boldsymbol{v},\infty)$ as $\tau^{\prime}\rightarrow\tau,K^{\prime}\rightarrow\infty$. Also, $\bar{R}(K)=\max_{\|\boldsymbol{\vartheta}\|_{\infty}\leq K}\ell(\boldsymbol{\vartheta},\boldsymbol{0})$ cannot diverge at finite $K$. Thus, applying the previous display with $K,\epsilon$ fixed allows us to conclude that $R(\boldsymbol{y},\tau,\boldsymbol{v},K^{\prime},\infty)$ is uniformly bounded over $K^{\prime}>K$ and $\tau^{\prime}$ in a neighborhood of $\tau$. Then, taking $K^{\prime}=\infty$ and $K\rightarrow\infty$, $\tau^{\prime}\rightarrow\tau$ followed by $\epsilon\rightarrow 0$ allows us to conclude that $\displaystyle\lim_{\begin{subarray}{c}K\rightarrow\infty\\\ \tau^{\prime}\rightarrow\tau\end{subarray}}R(\boldsymbol{y},\tau^{\prime},\boldsymbol{v},K,\infty)=R(\boldsymbol{y},\tau,\boldsymbol{v},\infty,\infty)\,.$ (33) for every fixed $\boldsymbol{y},\boldsymbol{v}$. Moreover, $\displaystyle R(\boldsymbol{y},\tau,\boldsymbol{v},K,M)$ $\displaystyle=\inf_{\|\boldsymbol{d}\|_{\infty}\leq M}\int\frac{1}{Z}\ell(\boldsymbol{\vartheta},\boldsymbol{d})e^{-\frac{1}{2\tau^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})$ $\displaystyle\leq\int\frac{1}{Z}\ell(\boldsymbol{\vartheta},\boldsymbol{0})e^{-\frac{1}{2\tau^{2}}\|\boldsymbol{y}-\boldsymbol{\vartheta}\|^{2}}\mu_{\boldsymbol{\Theta}^{(K)}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})$ $\displaystyle\leq\int\frac{1}{Z}C(1+\|\boldsymbol{\Theta}^{(K)}\|^{2})e^{-\frac{1}{\tau^{2}}(\boldsymbol{y}-\boldsymbol{\vartheta})^{2}}\mu_{{\boldsymbol{\Theta}^{(K)}}|\boldsymbol{V}}(\boldsymbol{v},{\mathrm{d}}\boldsymbol{\vartheta})$ $\displaystyle=C(1+\mathbb{E}[\|{\boldsymbol{\Theta}^{(K)}}\|^{2}|{\boldsymbol{\Theta}^{(K)}}+\tau\boldsymbol{G}=\boldsymbol{y},\boldsymbol{V}=\boldsymbol{v}])\,.$ Thus, $R(\boldsymbol{\Theta}^{(K)}+\tau\boldsymbol{Z},\tau,\boldsymbol{V},K,M)$ is uniformly integrable as we vary $\tau,K,M$. Because the total variation distance between $(\boldsymbol{\Theta}^{(K)}+\tau^{\prime}\boldsymbol{Z},\boldsymbol{V})$ and $(\boldsymbol{\Theta}+\tau\boldsymbol{Z},\boldsymbol{V}))$ goes to 0 as $K\rightarrow\infty$ and $\tau^{\prime}\rightarrow\tau$, for any discrete sequence $(K,\tau^{\prime})\rightarrow(\infty,\tau)$, there exists a probability space containing variables $\tilde{\boldsymbol{Y}}^{(K,\tau^{\prime})},\tilde{\boldsymbol{V}},\tilde{\boldsymbol{Y}}$ such that $(\tilde{\boldsymbol{Y}}^{(K,\tau^{\prime})},\tilde{\boldsymbol{V}})=(\tilde{\boldsymbol{Y}},\tilde{\boldsymbol{V}})$ eventually. Thus, Eq. (33) and uniform integrability imply (27). ∎ ## Appendix B Proof for reduction from GFOMs to AMP (Lemma 1) In this section, we prove Lemma 1. ### B.1 A general change of variables For any GFOM (1), there is a collection of GFOMs to which it is, up to a change of variabes, equivalent. In this section, we specify these GFOMs and the corresponding changes of variables. The change of variables is determined by a collection of $r\times r$ matrices $(\boldsymbol{\xi}_{t,s})_{t\geq 1,1\leq s\leq t}$, $(\boldsymbol{\zeta}_{t,s})_{t\geq 1,0\leq s<t}$. We will often omit subscripts outside of the parentheses. Define recursively the functions $(f_{t})_{t\geq 0}$, $(\phi_{t})_{t\geq 1}$ $\displaystyle f_{t}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{t};y,\boldsymbol{u})$ $\displaystyle=F_{t}^{(1)}(\phi_{1}(\boldsymbol{b}^{1};y,\boldsymbol{u}),\ldots,\phi_{t}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{t};y,\boldsymbol{u});y,\boldsymbol{u})$ (34a) $\displaystyle\phi_{t}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{t};y,\boldsymbol{u})$ $\displaystyle=\boldsymbol{b}^{t}+\sum_{s=0}^{t-1}f_{s}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{s};y,\boldsymbol{u})\boldsymbol{\zeta}_{t,s}^{\mathsf{T}}$ $\displaystyle\qquad\qquad+G_{t}^{(2)}(\phi_{1}(\boldsymbol{b}^{1};y,\boldsymbol{u}),\ldots,\phi_{t-1}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{t-1};y,\boldsymbol{u});y,\boldsymbol{u}),$ initialized by $f_{0}(y,\boldsymbol{u})=F_{0}^{(1)}(y,\boldsymbol{u})$ (here $\boldsymbol{b}^{s},\boldsymbol{u}\in\mathbb{R}^{r}$), and define recursively the functions $(g_{t})_{t\geq 1}$, $(\varphi_{t})_{t\geq 1}$ $\displaystyle\varphi_{t+1}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{t+1};\boldsymbol{v})$ $\displaystyle=\boldsymbol{a}^{t+1}+\sum_{s=1}^{t}g_{s}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{t+1};\boldsymbol{v})\boldsymbol{\xi}_{t,s}^{\mathsf{T}}$ (34b) $\displaystyle\qquad\qquad+F_{t}^{(2)}(\phi_{1}(\boldsymbol{a}^{1};\boldsymbol{v}),\ldots,\phi_{t}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{t};\boldsymbol{v});\boldsymbol{v}),$ $\displaystyle\qquad g_{t}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{t};\boldsymbol{v})$ $\displaystyle=G_{t}^{(1)}(\varphi_{1}(\boldsymbol{a}^{1};\boldsymbol{v}),\ldots,\varphi_{t}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{t};\boldsymbol{v});\boldsymbol{v}),$ initialized by $\varphi_{1}(\boldsymbol{a}^{1};\boldsymbol{v})=a^{1}+F_{0}^{(2)}(\boldsymbol{v})$ (here $\boldsymbol{a}^{s},\boldsymbol{v}\in\mathbb{R}^{r}$). Algebraic manipulation verifies that the iteration $\displaystyle\boldsymbol{a}^{t+1}$ $\displaystyle=\boldsymbol{X}^{\mathsf{T}}f_{t}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{t};y,\boldsymbol{u})-\sum_{s=1}^{t}g_{s}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{s};\boldsymbol{v})\boldsymbol{\xi}_{t,s}^{\mathsf{T}},$ (35) $\displaystyle\boldsymbol{b}^{t}$ $\displaystyle=\boldsymbol{X}g_{t}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{t};\boldsymbol{v})-\sum_{s=0}^{t-1}f_{s}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{s};y,\boldsymbol{u})\boldsymbol{\zeta}_{t,s}^{\mathsf{T}}$ initialized by $\boldsymbol{a}^{1}=\boldsymbol{X}^{\mathsf{T}}f_{0}(y,\boldsymbol{u})$ generates sequences $(\boldsymbol{a}^{t})_{t\geq 1}$, $(\boldsymbol{b}^{t})_{t\geq 1}$ which satisfy $\displaystyle\boldsymbol{v}^{t}=\varphi_{t}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{t};\boldsymbol{v}),\quad t\geq 1,$ $\displaystyle\boldsymbol{u}^{t}=\phi_{t}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{t};y,\boldsymbol{u}),\quad t\geq 1.$ Thus, $(\boldsymbol{\xi}_{t,s})$, $(\boldsymbol{\zeta}_{t,s})$ index a collection of GFOMs which, up to a change of variables, are equivalent. ### B.2 Approximate message passing and state evolution We call the iteration (35) an approximate message passing algorithm if the matrices $(\boldsymbol{\xi}_{t,s}),(\boldsymbol{\zeta}_{t,s})$ satisfy a certain model-specific recursion involving the functions $f_{t},g_{t}$. The state evolution characterization of the iterates (see Eq. (17)) holds whenever the matrices $\boldsymbol{\xi}_{t,s}$, $\boldsymbol{\zeta}_{t,s}$ satisfy this recursion. In this section, we specify this recursion and the parameters $(\boldsymbol{\alpha}_{s}),(\boldsymbol{T}_{s,s^{\prime}})$ in both the high- dimensional regression and low-rank matrix estimation models. #### B.2.1 High-dimensional regression AMP In the high-dimensional regression model, $r=1$ and $\xi_{t,s}$, $\zeta_{t,s}$, $\alpha_{t}$, and $T_{s,s^{\prime}}$ will be scalars (hence, written with non-bold font). The recursion defining $\xi_{t,s}$, $\zeta_{t,s}$ also defines $(\alpha_{t})$, $T_{s,s^{\prime}}$ as well as a collection of scalars $(\Sigma_{s,t})_{s,t\geq 0}$ which did not appear in the statement of Lemma 1. The recursion, whose lines are implemented in the order in which they appear, is $\begin{split}\xi_{t,s}&=\mathbb{E}[\partial_{B^{s}}f_{t}(B^{1},\ldots,B^{t};h(B^{0},W),U)],\quad 1\leq s\leq t,\\\ \alpha_{t+1}&=\mathbb{E}[\partial_{B^{0}}f_{t}(B^{1},\ldots,B^{t};h(B^{0},W),U)],\\\ T_{s+1,t+1}&=\mathbb{E}[f_{s}(B^{1},\ldots,B^{s};h(B^{0},W),U)f_{t}(B^{1},\ldots,B^{t};h(B^{0},W),U)],\quad 0\leq s\leq t,\\\ \zeta_{t,s}&=\frac{1}{\delta}\mathbb{E}[\partial_{Z^{s+1}}g_{t}(\alpha_{1}\Theta+Z^{1},\ldots,\alpha_{t}\Theta+Z^{t};V)],\quad 0\leq s\leq t-1,\\\ \Sigma_{0,t}&=\frac{1}{\delta}\mathbb{E}[\Theta g_{t}(\alpha_{1}\Theta+Z^{1},\ldots,\alpha_{t}\Theta+Z^{t};V)],\\\ \Sigma_{s,t}&=\frac{1}{\delta}\mathbb{E}[g_{s}(\alpha_{1}\Theta+Z^{1},\ldots,\alpha_{t}\Theta+Z^{s};V)g_{t}(\alpha_{1}\Theta+Z^{1},\ldots,\alpha_{t}\Theta+Z^{t};V)],\quad 1\leq s\leq t,\end{split}$ (36) where $\Theta\sim\mu_{\Theta}$, $U\sim\mu_{U}$, $V\sim\mu_{V}$, $W\sim\mu_{W}$, $(B^{0},\ldots,B^{t})\sim\mathsf{N}(\boldsymbol{0},\boldsymbol{\Sigma}_{[0:t]})$, $(Z^{1},\ldots,Z^{t})\sim\mathsf{N}(\boldsymbol{0},\boldsymbol{T}_{[1:t]})$, all independent. We initialize just before the second line with $\Sigma_{0,0}=\mathbb{E}[\Theta^{2}]$. Eq. (17) for $(\alpha_{s}),(T_{s,s^{\prime}})$ defined in this way is a special case of Proposition 5 of [JM13], as we now explain. We fix iteration $t$ design an algorithm that agrees, after a change of variables, with iteration (16) up to iteration $t$ and to which we can apply the results of [JM13]. Because we take $n,p\rightarrow\infty$ before $t\rightarrow\infty$, this establishes the result. We view the first $t$ iterations of (16) as acting on matrices $\tilde{\boldsymbol{a}}^{s}\in\mathbb{R}^{p\times(t+1)}$ and $\tilde{\boldsymbol{b}}^{s}\in\mathbb{R}^{n\times(t+1)}$ as follows. Define $\tilde{\boldsymbol{a}}^{s}$ to be the matrix whose first column is $\boldsymbol{\theta}$ and whose $i^{\text{th}}$ column is $\boldsymbol{a}^{i-1}$ for $2\leq i\leq s+1$ and is $\boldsymbol{0}$ for $i>s+1$; define $\tilde{\boldsymbol{b}}^{s}$ to be the matrix whose first column is $\boldsymbol{X}\boldsymbol{\theta}$ and whose $i^{\text{th}}$ column is $\boldsymbol{b}^{i=1}$ for $2\leq i\leq s+1$ and is $\boldsymbol{0}$ for $i>s+1$. The following change of variables transforms (16) into equations (28) and (29) of Proposition 5 in [JM13]. Our notation is on the right and is separated from the notation of [JM13] by the symbol “$\leftarrow$”. $\displaystyle\tilde{A}\leftarrow X,$ $\displaystyle u^{s}(i)\leftarrow\begin{cases}\boldsymbol{X}\boldsymbol{\theta}&i=1,\\\ \boldsymbol{b}^{i-1}&2\leq i\leq s+1,\\\ \boldsymbol{0}&\text{otherwise},\end{cases}\qquad\text{and}\qquad v^{s}(i)\leftarrow\begin{cases}\boldsymbol{a}^{i-1}-\alpha_{i-1}\boldsymbol{\theta}&2\leq i\leq s+1,\\\ \boldsymbol{0}&\text{otherwise},\end{cases}$ $\displaystyle y(i)\leftarrow\begin{cases}\boldsymbol{v}&i=1,\\\ \boldsymbol{\theta}&i=2,\\\ \boldsymbol{0}&\text{otherwise},\end{cases}\qquad\text{and}\qquad w(i)\leftarrow\begin{cases}\boldsymbol{u}&i=1,\\\ \boldsymbol{w}&i=2,\\\ \boldsymbol{0}&\text{otherwise},\end{cases}$ $\displaystyle\widehat{e}(v,y;s)(i)\leftarrow\begin{cases}y(2)&i=1,\\\ g_{i-1}(v(2)+\alpha_{1}y(2),\ldots,v(i+1)+\alpha_{i}y(2);y(1))&2\leq i\leq s+1,\\\ \boldsymbol{0}&\text{otherwise},\end{cases}$ $\displaystyle\widehat{h}(u,w;s)(i)\leftarrow\begin{cases}f_{i-1}(u(2),\ldots,u(i+1);h(u(1),w(2)),w(1)),&1\leq i\leq s+1,\\\ \boldsymbol{0}&\text{otherwise},\end{cases}$ where the “$(i)$” notation indexes columns of a matrix. The Onsager correction coefficients $(\xi_{t,s})$ and $(\zeta_{t,s})$ correspond, after a change of variables, to entries in the matrices $\mathsf{D}_{s}$ and $\mathsf{B}_{s}$ in [JM13]. $\displaystyle(\mathsf{D}_{s})_{i,j}=\mathbb{E}[\partial_{u(j)}\widehat{h}(U,W;s)]\leftarrow\begin{cases}\mathbb{E}[\partial_{B^{j-1}}f_{i-1}(B^{1},\ldots,B^{i};h(B^{0},W),U)]&1\leq j-1\leq i\leq s+1,\\\ 0&\text{otherwise},\end{cases},$ $\displaystyle(\mathsf{B}_{s})_{i,j}=\frac{1}{\delta}\mathbb{E}[\partial_{v(j)}\widehat{e}(V,Y;i)]\leftarrow\begin{cases}0&i=1\text{ or }j=1,\\\ \frac{1}{\delta}\mathbb{E}[\partial_{Z^{j-1}}g_{i}(\alpha_{1}\Theta+Z^{1},\ldots,\alpha_{i}\Theta+Z^{i};V)]&2\leq j\leq i+1\leq s+2,\\\ 0&\text{otherwise}.\end{cases}$ The Onsager coefficients and state evolution coefficients are arrived at through the change of variables: $\displaystyle(\mathsf{B}_{s})_{s+1,s^{\prime}+2}\leftarrow\zeta_{s,s^{\prime}},\;\;\;\;(\mathsf{D}_{s})_{s+1,s^{\prime}+1}\leftarrow\xi_{s,s^{\prime}}\;\;\;\;(\mathsf{D}_{s})_{s+1,1}\leftarrow\alpha_{s}.$ We remark that in [JM13] the quantities $(\mathsf{B}_{s})_{s+1,s^{\prime}+2}$, $(\mathsf{D}_{s})_{s+1,s^{\prime}+1}$, and $(\mathsf{D}_{s})_{s+1,1}$ are empirical averages. Because they concentration well on their population averages, we may replace them with their population averages, as we do here, without affecting the validity of state evolution. This observation is common in the AMP literature: see, for example, the relationship between Theorem 1 and Corollary 2 of [BMN19]. The state evolution matrices now correspond to $\displaystyle\mathbb{E}[V^{s+1}(s+1)V^{s+1}(s^{\prime}+1)]$ $\displaystyle=\mathbb{E}[\widehat{h}(U,W;s)(s)\widehat{h}(U,W;s)(s^{\prime})]$ $\displaystyle\leftarrow\mathbb{E}[f_{s-1}(B^{1},\ldots,B^{s-1};h(B^{0},W),U)f_{s^{\prime}-1}(B^{1},\ldots,B^{s^{\prime}-1};h(B^{0},W),U)]$ $\displaystyle=T_{s,s^{\prime}},$ $\displaystyle\mathbb{E}[U^{s+1}(s+1)U^{s+1}(s^{\prime}+1)]$ $\displaystyle=\frac{1}{\delta}\mathbb{E}[\widehat{e}(V,Y;s+1)(s+1)\widehat{e}(V,Y;s+1)(s^{\prime}+1)]$ $\displaystyle\leftarrow\frac{1}{\delta}\mathbb{E}[g_{s}(\alpha_{1}\Theta+Z^{1},\ldots,\alpha_{s}\Theta+Z^{s};V)g_{s^{\prime}}(\alpha_{1}\Theta+Z^{1},\ldots,\alpha_{s^{\prime}}\Theta+Z^{s^{\prime}};V)]$ $\displaystyle=\Sigma_{s,s^{\prime}}.$ From these changes of variables, Eq. (17) holds in the high-dimensional regression model from Theorem 1 and Proposition 5 of [JM13]. #### B.2.2 Low-rank matrix estimation AMP In the low-ank matrix estimation model, the recrusion defining $(\boldsymbol{x}_{t,x})$, $(\boldsymbol{\zeta}_{t,s})$ also defines $(\boldsymbol{\alpha}_{t})$, $(\boldsymbol{T}_{s,t})_{s,t\geq 1}$ as well as collections of $r\times r$ matrices $(\boldsymbol{\gamma}_{t})_{t\geq 1}$, $(\boldsymbol{\Sigma}_{s,t})_{s,t\geq 0}$ which did not appear in Lemma 1. The recursion, whose lines are implemented in the order in which they appear, is $\begin{split}\boldsymbol{\xi}_{t,s}&=\mathbb{E}[\nabla_{\tilde{\boldsymbol{Z}}^{s}}f_{t}(\boldsymbol{\gamma}_{1}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{1},\ldots,\boldsymbol{\gamma}_{t}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{t};0,\boldsymbol{U})],\quad 1\leq s\leq t,\\\ \boldsymbol{\alpha}_{t+1}&=\mathbb{E}[f_{t}(\boldsymbol{\gamma}_{1}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{1},\ldots,\boldsymbol{\gamma}_{t}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{t};0,\boldsymbol{U})\boldsymbol{\Lambda}^{\mathsf{T}}],\\\ \boldsymbol{T}_{s+1,t+1}&=\mathbb{E}[f_{s}(\boldsymbol{\gamma}_{1}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{1},\ldots,\boldsymbol{\gamma}_{t}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{s};0,\boldsymbol{U})f_{t}(\boldsymbol{\gamma}_{1}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{1},\ldots,\boldsymbol{\gamma}_{t}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{t};0,\boldsymbol{U})^{\mathsf{T}}],\;s\leq t,\\\ \boldsymbol{\zeta}_{t,s}&=\frac{1}{\delta}\mathbb{E}[\nabla_{\boldsymbol{Z}^{s+1}}g_{t}(\boldsymbol{\alpha}_{1}\boldsymbol{\Theta}+Z^{1},\ldots,\boldsymbol{\alpha}_{t}\boldsymbol{\Theta}+\boldsymbol{Z}^{t};\boldsymbol{V})],\quad 0\leq s\leq t-1,\\\ \boldsymbol{\gamma}_{t}&=\frac{1}{\delta}\mathbb{E}[g_{t}(\boldsymbol{\alpha}_{1}\boldsymbol{\Theta}+\boldsymbol{Z}^{1},\ldots,\boldsymbol{\alpha}_{t}\boldsymbol{\Theta}+\boldsymbol{Z}^{t};\boldsymbol{V})\boldsymbol{\Theta}^{\mathsf{T}}],\\\ \boldsymbol{\Sigma}_{s,t}&=\frac{1}{\delta}\mathbb{E}[g_{s}(\boldsymbol{\alpha}_{1}\boldsymbol{\Theta}+\boldsymbol{Z}^{1},\ldots,\boldsymbol{\alpha}_{t}\boldsymbol{\Theta}+\boldsymbol{Z}^{s};\boldsymbol{V})g_{t}(\boldsymbol{\alpha}_{1}\boldsymbol{\Theta}+\boldsymbol{Z}^{1},\ldots,\boldsymbol{\alpha}_{t}\boldsymbol{\Theta}+\boldsymbol{Z}^{t};\boldsymbol{V})^{\mathsf{T}}],\quad 1\leq s\leq t,\end{split}$ (37) where $\boldsymbol{\Lambda}\sim\mu_{\boldsymbol{\Lambda}}$ $\boldsymbol{U}\sim\mu_{\boldsymbol{U}}$, $\boldsymbol{\Theta}\sim\mu_{\boldsymbol{\Theta}}$, $\boldsymbol{V}\sim\mu_{\boldsymbol{V}}$, $(\tilde{\boldsymbol{Z}}^{1},\ldots,\tilde{\boldsymbol{Z}}^{t})\sim{\mathsf{N}}(\boldsymbol{0},\boldsymbol{\Sigma}_{[1:t]})$, and $(\boldsymbol{Z}^{1},\ldots,\boldsymbol{Z}^{t})\sim{\mathsf{N}}(\boldsymbol{0},\boldsymbol{T}_{[1:t]})$, all independent. Here $\nabla$ denotes the Jacobian with respect to subscripted (vectorial) argument, which exists almost everywhere because the functions involved are Lipschitz and the random variables have density with respect to Lebesgue measure [EG15, pg. 81]. As with $\boldsymbol{T}_{[1:t]}$, we define $\boldsymbol{\Sigma}_{[1:t]}$ to be the $rt\times rt$ block matrix with block $(s,t)$ given by $\boldsymbol{\Sigma}_{s,t}$. We initialize at the second line with $\boldsymbol{\alpha}_{1}=\mathbb{E}[f_{0}(0,\boldsymbol{U})\boldsymbol{\Lambda}^{\mathsf{T}}]$. In addition to (17), we have $\frac{1}{n}\sum_{i=1}^{n}\psi(\boldsymbol{b}^{1}_{i},\ldots,\boldsymbol{b}^{t}_{i},\boldsymbol{u}_{i},\boldsymbol{\lambda}_{i})\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\mathbb{E}[\psi(\boldsymbol{\gamma}_{1}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{1},\ldots,\boldsymbol{\gamma}_{t}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{t},\boldsymbol{U},\boldsymbol{\Lambda})],$ where we remind the reader that $\psi:\mathbb{R}^{r(t+2)}\rightarrow\mathbb{R}$ is any pseudo-Lipschitz function of order 2. We now show Eq. (17) for $(\alpha_{s}),(T_{s,s^{\prime}})$ defined in this way. We consider the $r=1$ case, as $r>1$ is similar by requires more notational overhead. Because $\boldsymbol{X}=\frac{1}{n}\boldsymbol{\lambda}\boldsymbol{\theta}^{\mathsf{T}}+\boldsymbol{Z}$, we have $\displaystyle\boldsymbol{a}^{t+1}-\frac{1}{n}\langle\boldsymbol{\lambda},f_{t}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{t},0,\boldsymbol{u})\rangle\boldsymbol{\theta}$ $\displaystyle=\boldsymbol{Z}^{\mathsf{T}}f_{t}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{t},0,\boldsymbol{u})-\sum\limits_{s=1}^{t}\xi_{t,s}g_{s}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{s},\boldsymbol{v}),$ $\displaystyle\boldsymbol{b}^{t}-\frac{1}{n}\langle\boldsymbol{\theta},g_{t}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{t},\boldsymbol{v})\rangle\boldsymbol{\lambda}$ $\displaystyle=\boldsymbol{Z}g_{t}(\boldsymbol{a}^{1},\ldots,\boldsymbol{a}^{t},\boldsymbol{v})-\sum\limits_{s=0}^{t-1}\zeta_{t,s}f_{s}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{s},\boldsymbol{y},\boldsymbol{u}).$ We introduce a change of variables: $\displaystyle\hat{f}_{t}(d^{1},\ldots,d^{t},u,\lambda)\overset{\Delta}{=}f_{t}(d^{1}+\gamma_{1}\lambda,\ldots,d^{t}+\gamma_{t}\lambda,0,u),$ $\displaystyle\boldsymbol{d}^{t}=\boldsymbol{b}^{t}-\gamma_{t}\boldsymbol{\lambda}\in\mathbb{R}^{n},$ $\displaystyle\hat{g}_{t}(c^{1},\ldots,c^{t},v,\theta)\overset{\Delta}{=}g_{t}(c^{1}+\alpha_{1}\theta,\ldots,c^{t}+\alpha_{t}\theta,v),$ $\displaystyle\boldsymbol{c}^{t}=\boldsymbol{a}^{t}-\alpha_{t}\boldsymbol{\theta}\in\mathbb{R}^{p}.$ Because $f_{t}$, $g_{t}$ are Lipschitz continuous, so too are $\hat{f}_{t}$, $\hat{g}_{t}$. We have $\displaystyle\boldsymbol{a}^{t+1}-\frac{1}{n}\langle\boldsymbol{\lambda},\hat{f}_{t}(\boldsymbol{d}^{1},\ldots,\boldsymbol{d}^{t},\boldsymbol{u},\boldsymbol{\lambda})\rangle\boldsymbol{\theta}$ $\displaystyle=\boldsymbol{Z}^{\mathsf{T}}\hat{f}_{t}(\boldsymbol{d}^{1},\ldots,\boldsymbol{d}^{t},\boldsymbol{u},\boldsymbol{\lambda})-\sum\limits_{s=1}^{t}\xi_{t,s}\hat{g}_{s}(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{s},\boldsymbol{v},\boldsymbol{\theta}),$ $\displaystyle\boldsymbol{b}^{t}-\frac{1}{n}\langle\boldsymbol{\theta},\hat{g}_{t}(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{t},\boldsymbol{v},\boldsymbol{\theta})\rangle\boldsymbol{\lambda}$ $\displaystyle=\boldsymbol{Z}\hat{g}_{t}(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{t},\boldsymbol{v},\boldsymbol{\theta})-\sum\limits_{s=0}^{t-1}\zeta_{t,s}\hat{f}_{s}(\boldsymbol{b}^{1},\ldots,\boldsymbol{b}^{s},\boldsymbol{u},\boldsymbol{\lambda}).$ Define $\displaystyle\hat{\boldsymbol{c}}^{t+1}$ $\displaystyle=\boldsymbol{Z}^{\mathsf{T}}\hat{f}_{t}(\hat{\boldsymbol{d}}^{1},\ldots,\hat{\boldsymbol{d}}^{t},\boldsymbol{u},\boldsymbol{\lambda})-\sum\limits_{s=1}^{t}\xi_{t,s}\hat{g}_{s}(\hat{\boldsymbol{c}}^{1},\ldots,\hat{\boldsymbol{c}}^{t},\boldsymbol{v},\boldsymbol{\theta}),$ $\displaystyle\hat{\boldsymbol{d}}^{t}$ $\displaystyle=\boldsymbol{Z}\hat{g}_{t}(\hat{\boldsymbol{c}}^{1},\ldots,\hat{\boldsymbol{c}}^{t},\boldsymbol{v},\boldsymbol{\theta})-\sum\limits_{s=0}^{t-1}\zeta_{t,s}\hat{f}_{s}(\hat{\boldsymbol{d}}^{1},\ldots,\hat{\boldsymbol{d}}^{t},\boldsymbol{u},\boldsymbol{\lambda}).$ We can analzye this iteration via the same techniques we used to analyze AMP in the high-dimensional regression model in the previous section [JM13]. In particular, for any pseudo-Lipschitz function $\psi:\mathbb{R}^{t+2}\rightarrow\mathbb{R}$ of order 2, we have $\displaystyle\frac{1}{p}\sum\limits_{j=1}^{p}\psi(\hat{c}_{j}^{1},\ldots,\hat{c}_{j}^{t},v_{j},\theta_{j})$ $\displaystyle\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\mathbb{E}[\psi(Z^{1},\ldots,Z^{t},V,\Theta)],$ (38) $\displaystyle\frac{1}{n}\sum\limits_{i=1}^{n}\psi(\hat{d}^{1}_{i},\ldots,\hat{d}^{t}_{i},u_{i},\lambda_{i})$ $\displaystyle\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\mathbb{E}[\psi(\tilde{Z}^{1},\ldots,\tilde{Z}^{t},U,\Lambda)].$ Now, tøestablish (17), it suffices to show $\frac{1}{n}\|\hat{\boldsymbol{c}}^{t}-\boldsymbol{c}^{t}\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0,\qquad\frac{1}{n}\|\hat{\boldsymbol{d}}^{t}-\boldsymbol{d}^{t}\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0.$ (39) We proceed by induction. By the weak law of large numbers, we have that $\frac{1}{n}\langle\boldsymbol{\lambda},\hat{f}_{0}(\boldsymbol{\lambda},\boldsymbol{u})\rangle=\frac{1}{n}\langle\boldsymbol{\lambda},f_{0}(0,\boldsymbol{u})\rangle\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\alpha_{1}$. Therefore, $\boldsymbol{c}^{1}=\boldsymbol{Z}^{\mathsf{T}}\hat{f}_{0}(\boldsymbol{\lambda},\boldsymbol{u})+o_{p}(1)\boldsymbol{\theta}=\hat{\boldsymbol{c}}^{1}+o_{p}(1)\boldsymbol{\theta}$. Since $\frac{1}{p}\|\boldsymbol{\theta}\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\mathbb{E}[\Theta^{2}]$, we have that $\frac{1}{n}\|\boldsymbol{c}^{1}-\hat{\boldsymbol{c}}^{1}\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0$. Because $\hat{g}_{1}$ is Lipschitz and $\frac{1}{p}\|\boldsymbol{\theta}\|^{2}=O_{p}(1)$, we have $|\frac{1}{n}\langle\boldsymbol{\theta},\hat{g}_{1}(\boldsymbol{c}^{1},\boldsymbol{\theta},\boldsymbol{v})\rangle-\frac{1}{n}\langle\boldsymbol{\theta},\hat{g}_{1}(\hat{\boldsymbol{c}}^{1},\boldsymbol{\theta},\boldsymbol{v})\rangle|\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0$. By (38), we have that $\frac{1}{n}\langle\boldsymbol{\theta},\hat{g}_{1}(\hat{\boldsymbol{c}}^{1},\boldsymbol{\theta},\boldsymbol{v})\rangle\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\gamma_{1}$. We have $\frac{1}{n}\|\hat{g}_{1}(\boldsymbol{c}^{1},\boldsymbol{v},\boldsymbol{\theta})-\hat{g}_{1}(\hat{\boldsymbol{c}}^{1},\boldsymbol{v},\boldsymbol{\theta})\|_{2}^{2}\leq\frac{1}{n}L^{2}\|\boldsymbol{c}^{1}-\hat{\boldsymbol{c}}^{1}\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0,$ wehre $L$ is a Lipschitz constant for $\hat{g}_{1}$. By [BY08], the maximal singular value of $\boldsymbol{Z}^{T}\boldsymbol{Z}$ is $O_{p}(1)$. Therefore, $\frac{1}{n}\|\boldsymbol{Z}\hat{g}_{1}(\boldsymbol{c}^{1},\boldsymbol{v},\boldsymbol{\theta})-\boldsymbol{Z}\hat{g}_{1}(\hat{\boldsymbol{c}}^{1},\boldsymbol{v},\boldsymbol{\theta})\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0$. As a result, and using that $\frac{1}{n}\|\boldsymbol{\lambda}\|_{2}^{2}$ converges almost surely to a constant, $\frac{1}{n}\|\hat{\boldsymbol{d}}^{1}-\boldsymbol{d}^{1}\|_{2}^{2}=\frac{1}{n}\|\boldsymbol{Z}\hat{g}_{t}(\hat{\boldsymbol{c}}^{1},\boldsymbol{v},\boldsymbol{\theta})-\boldsymbol{Z}\hat{g}_{t}(\boldsymbol{c}^{1},\boldsymbol{v},\boldsymbol{\theta})+(\frac{1}{n}\langle\boldsymbol{\theta},\hat{g}_{1}(\boldsymbol{c}^{1},\boldsymbol{\theta},\boldsymbol{v})\rangle-\gamma_{1})\boldsymbol{\lambda}\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0.$ Now assume that (39) holds for $1,2,\ldots,t$. For the $(t+1)$-th iteration, we have $|\frac{1}{n}\langle\boldsymbol{\lambda},\hat{f}_{t}(\boldsymbol{d}^{1},\ldots,\boldsymbol{d}^{t},\boldsymbol{u},\boldsymbol{\lambda})\rangle-\frac{1}{n}\langle\boldsymbol{\lambda},\hat{f}_{t}(\hat{\boldsymbol{d}}^{1},\ldots,\hat{\boldsymbol{d}}^{t},\boldsymbol{u},\boldsymbol{\lambda})\rangle|\leq\frac{L}{n}\|\boldsymbol{\lambda}\|_{2}\sum\limits_{s=1}^{t}\|\boldsymbol{d}^{s}-\hat{\boldsymbol{d}}^{s}\|_{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0.$ where $L$ is a Lipschitz constant for $\hat{f}$. By (38), we have $\frac{1}{n}\langle\boldsymbol{\lambda},\hat{f}_{t}(\hat{\boldsymbol{d}}^{1},\ldots,\hat{\boldsymbol{d}}^{t},\boldsymbol{\lambda},\boldsymbol{u})\rangle\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\alpha_{t+1}$. As a result, we have $\frac{1}{n}\langle\boldsymbol{\lambda},\hat{f}_{t}(\boldsymbol{d}^{1},\ldots,\boldsymbol{d}^{t},\boldsymbol{u},\boldsymbol{\lambda})\rangle\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\alpha_{t+1}$. Furthermore, for any $1\leq s\leq t$, we have $\displaystyle\frac{1}{n}\|\hat{f}_{s}(\boldsymbol{d}^{1},\ldots,\boldsymbol{d}^{s},\boldsymbol{u},\boldsymbol{\lambda})-\hat{f}_{s}(\hat{\boldsymbol{d}}^{1},\ldots,\hat{\boldsymbol{d}}^{s},\boldsymbol{u},\boldsymbol{\lambda})\|_{2}^{2}$ $\displaystyle\leq\frac{\hat{L}_{t}^{2}}{n}\sum\limits_{i=1}^{s}\|\boldsymbol{d}^{i}-\hat{\boldsymbol{d}}^{i}\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0,$ $\displaystyle\frac{1}{n}\|\hat{g}_{s}(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{s},\boldsymbol{v},\boldsymbol{\theta})-\hat{g}_{s}(\hat{\boldsymbol{c}}^{1},\ldots,\hat{\boldsymbol{c}}^{s},\boldsymbol{v},\boldsymbol{\theta})\|_{2}^{2}$ $\displaystyle\leq\frac{\hat{L}_{t}^{2}}{n}\sum\limits_{i=1}^{s}\|\boldsymbol{c}^{i}-\hat{\boldsymbol{c}}^{i}\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0.$ Again using that the maximal singular value of $\boldsymbol{Z}^{\mathsf{T}}\boldsymbol{Z}$ is $O_{p}(1)$, we have $\frac{1}{n}\|\boldsymbol{Z}^{\mathsf{T}}\hat{f}_{t}(\hat{\boldsymbol{d}}^{1},\ldots,\hat{\boldsymbol{d}}^{t},\boldsymbol{u},\boldsymbol{\lambda})-\boldsymbol{Z}^{\mathsf{T}}\hat{f}_{t}(\boldsymbol{d}^{1},\ldots,\boldsymbol{d}^{t},\boldsymbol{u},\boldsymbol{\lambda})\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0\,.$ As a result, we have $\displaystyle\frac{1}{n}\|\hat{\boldsymbol{c}}^{t+1}-\boldsymbol{c}^{t+1}\|_{2}^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{n}\|(\frac{1}{n}\langle\boldsymbol{\lambda},\hat{f}_{t}(\boldsymbol{d}^{1},\ldots,\boldsymbol{d}^{t},\boldsymbol{u},\boldsymbol{\lambda})-\alpha_{t+1})\boldsymbol{\theta}+\boldsymbol{Z}^{\mathsf{T}}(\hat{f}_{t}(\hat{\boldsymbol{d}}^{1},\ldots,\hat{\boldsymbol{d}}^{t},\boldsymbol{u},\boldsymbol{\lambda})-\hat{f}_{t}(\boldsymbol{d}^{1},\ldots,\boldsymbol{d}^{t},\boldsymbol{u},\boldsymbol{\lambda}))-$ $\displaystyle\sum\limits_{s=1}^{t}\xi_{t,s}(\hat{g}_{s}(\hat{\boldsymbol{c}}^{1},\ldots,\hat{\boldsymbol{c}}^{s},\boldsymbol{v},\boldsymbol{\theta})-\hat{g}_{s}(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{s},\boldsymbol{\theta},\boldsymbol{v}))\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0.$ Similarly, we have $|\frac{1}{n}\langle\boldsymbol{\theta},\hat{g}_{t+1}(\hat{\boldsymbol{c}}^{1},\ldots,\hat{\boldsymbol{c}}^{t+1},\boldsymbol{v},\boldsymbol{\theta})\rangle-\frac{1}{n}\langle\boldsymbol{\theta},\hat{g}_{t+1}(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{t+1},\boldsymbol{v},\boldsymbol{\theta})\rangle|\leq\frac{L}{n}\|\boldsymbol{\theta}\|_{2}\sum\limits_{s=1}^{t+1}\|\hat{\boldsymbol{c}}^{t+1}-\boldsymbol{c}^{t+1}\|_{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0,$ where $L$ is a Lipschitz constant for $\hat{g}_{t+1}$. By (38), we have that $\frac{1}{n}\langle\boldsymbol{\theta},\hat{g}_{t+1}(\hat{\boldsymbol{c}}^{1},\ldots,\hat{\boldsymbol{c}}^{t+1},\boldsymbol{v},\boldsymbol{\theta})\rangle\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\gamma_{t+1}$. As a result, we have that $\frac{1}{n}\langle\boldsymbol{\theta},\hat{g}_{t+1}(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{t+1},\boldsymbol{v},\boldsymbol{\theta})\rangle\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\gamma_{t+1}$. Furthermore, for any $1\leq s\leq t$, we have $\frac{1}{n}\|\hat{f}_{s}(\boldsymbol{d}^{1},\ldots,\boldsymbol{d}^{s},\boldsymbol{u},\boldsymbol{\lambda})-\hat{f}_{s}(\hat{\boldsymbol{d}}^{1},\ldots,\hat{\boldsymbol{d}}^{s},\boldsymbol{u},\boldsymbol{\lambda})\|_{2}^{2}\leq\frac{L^{2}}{n}\sum\limits_{i=1}^{s}\|\boldsymbol{d}^{i}-\hat{\boldsymbol{d}}^{i}\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0.$ Also, for any $1\leq s\leq t+1$, we have $\frac{1}{n}\|\hat{g}_{s}(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{s},\boldsymbol{v},\boldsymbol{\theta})-\hat{g}_{s}(\hat{\boldsymbol{c}}^{1},\ldots,\hat{\boldsymbol{c}}^{s},\boldsymbol{v},\boldsymbol{\theta})\|_{2}^{2}\leq\frac{L^{2}}{n}\sum\limits_{i=1}^{s}\|\boldsymbol{c}^{i}-\hat{\boldsymbol{c}}^{i}\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0.$ Then $\frac{1}{n}\|\boldsymbol{Z}\hat{g}_{t+1}(\hat{\boldsymbol{c}}^{1},\ldots,\hat{\boldsymbol{c}}^{t+1},\boldsymbol{v},\boldsymbol{\theta})-\boldsymbol{Z}\hat{g}_{t+1}(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{t+1},\boldsymbol{v},\boldsymbol{\theta})\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0$. As a result, we have $\displaystyle\frac{1}{n}\|\hat{\boldsymbol{d}}^{t+1}-\boldsymbol{d}^{t+1}\|_{2}^{2}$ $\displaystyle\qquad=\frac{1}{n}\|(\frac{1}{n}\langle\boldsymbol{\theta},\hat{g}_{t+1}(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{t+1},\boldsymbol{v},\boldsymbol{\theta})\rangle-\gamma_{t+1})\boldsymbol{\lambda}$ $\displaystyle\qquad+\boldsymbol{Z}(\hat{g}_{t+1}(\hat{\boldsymbol{c}}^{1},\ldots,\hat{\boldsymbol{c}}^{t+1},\boldsymbol{v},\boldsymbol{\theta})-\hat{g}_{t+1}(\boldsymbol{c}^{1},\ldots,\boldsymbol{c}^{t+1},\boldsymbol{v},\boldsymbol{\theta}))$ $\displaystyle\qquad-\sum\limits_{s=0}^{t}\zeta_{t,s}(\hat{f}_{s}(\hat{\boldsymbol{d}}^{1},\ldots,\hat{\boldsymbol{d}}^{s},\boldsymbol{u},\boldsymbol{\lambda})-\hat{f}_{s}(\boldsymbol{d}^{1},\ldots,\boldsymbol{d}^{s},\boldsymbol{u},\boldsymbol{\lambda}))\|_{2}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0.$ Thus, we have proved (39). Therefore, for all pseudo-Lipschitz function $\psi$ of order 2, we have that there exists a numerical constant $C$ such that $\displaystyle\left|\frac{1}{p}\sum\limits_{j=1}^{p}\psi(c_{j}^{1}+\alpha_{1}\theta_{j},\ldots,c_{j}^{t}+\alpha_{t}\theta_{j},v_{j},\theta_{j})-\frac{1}{p}\sum\limits_{j=1}^{p}\psi(\hat{c}_{j}^{1}+\alpha_{1}\theta_{j},\ldots,\hat{c}_{j}^{t}+\alpha_{t}\theta_{j},v_{j},\theta_{j})\right|$ $\displaystyle\qquad\qquad\leq L_{\psi}(1+\sum\limits_{s=1}^{t}\|\boldsymbol{a}^{s}\|_{2}+\|\boldsymbol{\theta}\|_{2}+\|\boldsymbol{v}\|_{2})\sum\limits_{s=1}^{t}\|\hat{\boldsymbol{c}}^{s}-\boldsymbol{c}^{s}\|_{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0.$ By (38), $\dfrac{1}{p}\sum\limits_{j=1}^{p}\psi(\hat{c}_{j}^{1}+\alpha_{1}\theta_{j},\ldots,\hat{c}_{j}^{t}+\alpha_{t}\theta_{j},v_{j},\theta_{j})\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\mathbb{E}[\psi(\boldsymbol{\alpha}_{1}\boldsymbol{\Theta}+\boldsymbol{Z}^{1},\ldots,\boldsymbol{\alpha}_{t}\boldsymbol{\Theta}+\boldsymbol{Z}^{t},\boldsymbol{V},\boldsymbol{\Theta})].$ Therefore, $\frac{1}{p}\sum\limits_{j=1}^{p}\psi(a_{j}^{1},\ldots,a_{j}^{t},v_{j},\theta_{j})\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\mathbb{E}[\psi(\boldsymbol{\alpha}_{1}\boldsymbol{\Theta}+\boldsymbol{Z}^{1},\ldots,\boldsymbol{\alpha}_{t}\boldsymbol{\Theta}+\boldsymbol{Z}^{t},\boldsymbol{V},\boldsymbol{\Theta})]$. Similarly, we can show that $\frac{1}{n}\sum_{i=1}^{n}\psi(\boldsymbol{b}^{1}_{i},\ldots,\boldsymbol{b}^{t}_{i},\boldsymbol{u}_{i},\boldsymbol{\lambda}_{i})\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\mathbb{E}[\psi(\boldsymbol{\gamma}_{1}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{1},\ldots,\boldsymbol{\gamma}_{t}\boldsymbol{\Lambda}+\tilde{\boldsymbol{Z}}^{t},\boldsymbol{U},\boldsymbol{\Lambda})]$. Thus we have finished the proof. ### B.3 The AMP change of variables To prove Lemma 1, all that remains is to show that for any GFOM (1), at least one of the change-of-variables in Eqs. (34) generates an iteration (35) which is an AMP iteration. That is, in addition to satisfying Eq. (34), the matrices $(\boldsymbol{\xi}_{t,s})$, $(\boldsymbol{\zeta}_{t,s})$ and functions $(f_{t})$, $(g_{t})$ satisfy Eqs. (36) and (37) in the high-dimensional regression and low-rank matrix estimation models respectively. To construct such a choice of scalars, we may define $(\boldsymbol{\xi}_{t,s}),(\boldsymbol{\zeta}_{t,s}),(f_{t}),(g_{t})$ in a single recursion by interlacing definition (34) with either (36) or (37). Specifically, in the high-dimensional regression model, we place (34a) before the first line of (36) and (34b) before the fourth line of (36). In the combined recursion, all quantities are defined in terms of previously defined quantities, yielding choices for $(\boldsymbol{\xi}_{t,s}),(\boldsymbol{\zeta}_{t,s}),(f_{t}),(g_{t})$ which simultaneously satisfy (34) and (36). Thus, in the high-dimensional regression model every GFOM is equivalent, up to a change of variables, to a certain AMP algorithm. The construction in the low-rank matrix estimation model is analogous: we place (34a) before the first line of (37) and (34b) before the fourth line of (37). The proof of Lemma 1 is complete. ## Appendix C Proof of state evolution for message passing (Lemma 2) In this section, we prove Lemma 2. We restrict ourselves to the case $r=1$ and $k=1$ (with $k$ the dimensionality of $\boldsymbol{W}$) because the proof for $r>1$ or $k>1$ is completely analogous but would complicate notation. Let $\mathcal{T}_{v\rightarrow f}=(\mathcal{V}_{v\rightarrow f},\mathcal{F}_{v\rightarrow f},\mathcal{E}_{v\rightarrow f})$ be the tree consisting of edges and nodes in $\mathcal{T}$ which are separated from $f$ by $v$. By convention, $\mathcal{T}_{v\rightarrow f}$ will also contain the node $v$. In particular, $f\not\in\mathcal{F}_{v\rightarrow f}$ and $(f,v)\not\in\mathcal{E}_{v\rightarrow f}$, but $v\in\mathcal{V}_{v\rightarrow f}$, and $f^{\prime}\in\mathcal{F}_{v\rightarrow f}$ and $(v,f^{\prime})\in\mathcal{E}_{v\rightarrow f}$ for $f^{\prime}\in\partial v\setminus f$. We define $\mathcal{T}_{f\rightarrow v},\mathcal{V}_{f\rightarrow v},\mathcal{F}_{f\rightarrow v},\mathcal{E}_{f\rightarrow v}$ similarly. With some abuse of notation, we will sometimes use $\mathcal{T}_{f\rightarrow v},\mathcal{V}_{f\rightarrow v},\mathcal{F}_{f\rightarrow v},\mathcal{E}_{f\rightarrow v}$ to denote either the collection of observations corresponding to nodes and edges in these sets or the $\sigma$-algebra generated by these obervations. No confusion should result. Which random variables we consider to be “observed” will vary with the model, and will be explicitly described in each part of the proof to avoid potential ambiguity. ### C.1 Gaussian message passing We first introduce a message passing algorithm whose behavior is particularly easy to analyze. We call this message passing algorithm a _Gaussian message passing_ algorithm. We will see that in both the high-dimensional regression and low-rank matrix estimation models, the message passing algorithm (19) approximates a certain Gaussian message passing algorithm. Gaussian message passing algorithms operate on a computation tree with associated random variables $\\{(\theta_{v},v_{v})\\}_{v\in\mathcal{V}}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\Theta,V}$, $\\{(w_{f},u_{f})\\}_{f\in\mathcal{F}}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{W,U}$, and $\\{z_{fv}\\}_{(f,v)\in\mathcal{E}}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1/n)$, all independent, where $\mu_{\Theta,V},\mu_{W,U}\in\mathscrsfs{P}_{4}(\mathbb{R}^{2})$.666We believe that only $\mu_{\Theta,V},\mu_{W,U}\in\mathscrsfs{P}_{2}(\mathbb{R}^{2})$ is needed, but the analysis under this weaker assumption would be substantially more complicated, and the weaker assumptions are not necessary for our purposes. Gaussian message passing algorithms access all these random variables, so that all are considered to be “observed.” Thus, for example, $\mathcal{V}_{f\rightarrow v}$ contains $\theta_{v^{\prime}},v_{v^{\prime}}$ for all nodes $v^{\prime}$ separated from $f$ by $v$ (including, by convention, $v$). Gaussian message passing algorithms are defined by sequences of Lipschitz functions $(\tilde{f}_{t}:\mathbb{R}^{t+3}\rightarrow\mathbb{R})_{t\geq 0}$, $(\tilde{g}_{t}:\mathbb{R}^{t+2}\rightarrow\mathbb{R})_{t\geq 0}$. We initialize the indexing differently than with Gaussian message passing algorithms than with the message passing algorithms in Section 5 in anticipation of notational simplifications that will occur later. For every pair of neighboring nodes $v,f$, we generate sequences of messages $(\tilde{a}_{v\rightarrow f}^{t})_{t\geq 1}$, $(\tilde{q}_{v\rightarrow f}^{t})_{t\geq 0}$, $(\tilde{b}_{f\rightarrow v}^{t})_{t\geq 0}$, $(\tilde{r}_{f\rightarrow v}^{t})_{t\geq 0}$ according to the iteration $\displaystyle\tilde{a}_{v\rightarrow f}^{t+1}=\sum_{f^{\prime}\in\partial v\setminus f}z_{f^{\prime}v}\tilde{r}_{f^{\prime}\rightarrow v}^{t},\qquad\tilde{r}_{f\rightarrow v}^{t}=\tilde{f}_{t}(\tilde{b}_{f\rightarrow v}^{0},\ldots,\tilde{b}_{f\rightarrow v}^{t};w_{f},u_{f}),$ (40a) $\displaystyle\tilde{b}_{f\rightarrow v}^{t}=\sum_{v^{\prime}\in\partial f\setminus v}z_{fv^{\prime}}\tilde{q}_{v^{\prime}\rightarrow f}^{t},\qquad\tilde{q}_{v\rightarrow f}^{t}=\tilde{g}_{t}(\tilde{a}_{v\rightarrow f}^{1},\ldots,\tilde{a}_{v\rightarrow f}^{t};\theta_{v},v_{v}),$ (40b) with initialization $\tilde{q}_{v\rightarrow f}^{0}=g_{0}(\theta_{v},v_{v})$. For $t\geq 0$, define the node beliefs $\tilde{a}_{v}^{t+1}=\sum_{f\in\partial v}z_{fv}\tilde{r}_{f\rightarrow v}^{t},\qquad\tilde{b}_{f}^{t}=\sum_{v\in\partial f}z_{fv}\tilde{q}_{v\rightarrow f}^{t}.$ (41) To compactify notation, denote $\tilde{\boldsymbol{a}}_{v}^{t}=(\tilde{a}_{v}^{1},\ldots,\tilde{a}_{v}^{t})^{\mathsf{T}}$, and likewise for $\tilde{\boldsymbol{a}}_{v\rightarrow f}^{t}$, $\tilde{\boldsymbol{q}}_{v\rightarrow f}^{t}$, $\tilde{\boldsymbol{b}}_{f}^{t}$, $\tilde{\boldsymbol{b}}_{f\rightarrow v}^{t},\tilde{\boldsymbol{r}}_{f\rightarrow v}^{t}$ (where the first two of these are $t$-dimensional, and the last three are $(t+1)$-dimensional). We will often write $\tilde{f}_{t}(\tilde{\boldsymbol{b}}_{f\rightarrow v}^{t};w_{f},u_{f})$ in place of $\tilde{f}_{t}(\tilde{b}_{f\rightarrow v}^{0},\ldots,b_{f\rightarrow v}^{t};w_{f},u_{f})$, and similarly for $\tilde{g}_{t}$. The reader should not confuse the bold font here with that in Section 5, in which, for example, $\boldsymbol{a}_{v\rightarrow f}^{t}$ denotes the vectorial message at time $t$ rather than the collection of scalar messages prior to and including time $t$. Gaussian message passing obeys a Gaussian state evolution, defined by covariance matrices $\Sigma_{s,s^{\prime}}=\mathbb{E}[\tilde{g}_{s}(\tilde{\boldsymbol{A}}^{s};\Theta,V)\tilde{g}_{s^{\prime}}(\tilde{\boldsymbol{A}}^{s^{\prime}};\Theta,V)],\;\;\;T_{s+1,s^{\prime}+1}=\mathbb{E}[\tilde{f}_{s}(\tilde{\boldsymbol{B}}^{s};W,U)\tilde{f}_{s^{\prime}}(\tilde{\boldsymbol{B}}^{s^{\prime}};W,U)],$ (42) where $s,s^{\prime}\geq 0$, $\tilde{\boldsymbol{A}}^{s}\sim{\mathsf{N}}(\boldsymbol{0}_{s},\boldsymbol{T}_{[1{:}s]})$, $\tilde{\boldsymbol{B}}^{s}\sim{\mathsf{N}}(\boldsymbol{0}_{s+1},\boldsymbol{\Sigma}_{[0{:}s]})$, and $(\Theta,V)\sim\mu_{\Theta,V}$, $(W,U)\sim\mu_{W,U}$ independent of $\tilde{\boldsymbol{A}}^{s},\tilde{\boldsymbol{B}}^{s}$. The iteration is initialized by $\Sigma_{0,0}=\mathbb{E}[\tilde{g}_{0}(\Theta,V)^{2}]$. ###### Lemma 1. If we choose a variable node $v$ and factor node $f$ independently of the randomness in our model, then for fixed $t$ and for $n,p\rightarrow\infty$, $n/p\rightarrow\delta$ we have $\displaystyle(\tilde{\boldsymbol{a}}_{v}^{t},\theta_{v},v_{v})\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}\mathsf{N}(\boldsymbol{0}_{t},\boldsymbol{T}_{[1{:}t]})\otimes\mu_{\Theta,V}\;\;\text{and}\;\;(\tilde{\boldsymbol{a}}_{v\rightarrow f}^{t},\theta_{v},v_{v})\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}\mathsf{N}(\boldsymbol{0}_{t},\boldsymbol{T}_{[1{:}t]})\otimes\mu_{\Theta,V},$ (43a) $\displaystyle(\tilde{\boldsymbol{b}}_{f}^{t},w_{f},u_{f})\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}\mathsf{N}(\boldsymbol{0}_{t+1},\boldsymbol{\Sigma}_{[0{:}t]})\otimes\mu_{W,U}\;\;\text{and}\;\;(\tilde{\boldsymbol{b}}_{f\rightarrow v}^{t},w_{f},u_{f})\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}\mathsf{N}(\boldsymbol{0}_{t+1},\boldsymbol{\Sigma}_{[0{:}t]})\otimes\mu_{W,U}.$ (43b) Further, all the random variables in the preceding displays have bounded fourth moments and $\mathbb{E}[\|\tilde{\boldsymbol{a}}_{v}^{t}-\tilde{\boldsymbol{a}}_{v\rightarrow f}^{t}\|^{2}]\rightarrow 0$ and $\mathbb{E}[\|\tilde{\boldsymbol{b}}_{f}^{t}-\tilde{\boldsymbol{b}}_{f\rightarrow v}^{t}\|^{2}]\rightarrow 0$. The analysis of message passing on the tree is facilitated by the many independence relationships between messages, which follow from the following lemma. ###### Lemma 2. For all $(f,v)\in\mathcal{E}$ and all $t$, the messages $\tilde{r}_{f\rightarrow v}^{t},\tilde{b}_{f\rightarrow v}^{t}$ are $\mathcal{T}_{f\rightarrow v}$-measurable, and the messages $\tilde{q}_{v\rightarrow f}^{t},\tilde{a}_{a\rightarrow f}^{t}$ is $\mathcal{T}_{v\rightarrow f}$-measurable. ###### Lemma 2. The proof is by induction. The base case is that $\tilde{q}_{v\rightarrow f}^{0}=g_{0}(\theta_{v},v_{v})$ is $\mathcal{T}_{v\rightarrow f}$-measurable. Then, if $\tilde{q}_{v\rightarrow f}^{s}$ are $\mathcal{T}_{v\rightarrow f}$-measurable and $\tilde{b}_{f\rightarrow v}^{s}$ are $\mathcal{T}_{f\rightarrow v}$-measurable for $0\leq s\leq t$ and all $(f,v)\in\mathcal{E}$, then $\tilde{b}_{f\rightarrow v}^{t},\tilde{r}_{f\rightarrow v}^{t}$ are $\mathcal{T}_{f\rightarrow v}$-measurable by (40). Similarly, if $\tilde{r}_{f\rightarrow v}^{s}$ are $\mathcal{T}_{f\rightarrow v}$-measurable and $\tilde{a}_{v\rightarrow f}^{s}$ are $\mathcal{T}_{v\rightarrow r}$-measurable for $0\leq s\leq t$ and all $(f,v)\in\mathcal{E}$, then $\tilde{a}_{f\rightarrow v}^{t+1},\tilde{r}_{f\rightarrow v}^{t+1}$ are $\mathcal{T}_{v\rightarrow f}$-measurable by (40). The induction is complete. ∎ We now prove Lemma 1. ###### Lemma 1. The proof is by induction. Base case: $(\theta_{v},v_{f})\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}\mu_{\Theta,V}$. This is the exact distribution in finite samples by assumption. Inductive step 1: Eq. (43a) at $t$, bounded fourth moments of $\tilde{\boldsymbol{a}}_{v}^{t},\tilde{\boldsymbol{a}}_{v\rightarrow f}^{t}$, and $\mathbb{E}[\|\tilde{\boldsymbol{a}}_{v}^{t}-\tilde{\boldsymbol{a}}_{v\rightarrow f}^{t}\|^{2}]\rightarrow 0$ imply Eq. (43b) at $t$, bounded fourth moments of $\tilde{\boldsymbol{b}}_{f}^{t},\tilde{\boldsymbol{b}}_{f\rightarrow v}^{t}$, and $\mathbb{E}[\|\tilde{\boldsymbol{b}}_{f}^{t}-\tilde{\boldsymbol{b}}_{f\rightarrow v}^{t}\|^{2}]\rightarrow 0$. The $\sigma$-algebras $(\mathcal{T}_{v\rightarrow f})_{v\in\partial f}$ are independent of $(z_{fv})_{v\in\partial f}$, which are mutually independent of each other. Thus, by (41), conditional on $\sigma((\mathcal{T}_{v\rightarrow f})_{v\in\partial f})$ the beliefs $\tilde{\boldsymbol{b}}_{f}^{t}$ are jointly normal with covariance $\widehat{\boldsymbol{\Sigma}}_{[0{:}t]}:=\frac{1}{n}\sum_{v\in\partial f}\tilde{\boldsymbol{q}}_{v\rightarrow f}^{t}(\tilde{\boldsymbol{q}}_{v\rightarrow f}^{t})^{\mathsf{T}}$. That is, $\tilde{\boldsymbol{b}}_{f}^{t}\bigm{|}\sigma((\mathcal{T}_{v\rightarrow f})_{v\in\partial f})\sim\mathsf{N}(\boldsymbol{0}_{t+1},\widehat{\boldsymbol{\Sigma}}_{[0{:}t]}).$ Because $(\tilde{\boldsymbol{a}}_{v\rightarrow f}^{t},\theta_{v},v_{v})\mapsto\tilde{g}_{s}(\tilde{\boldsymbol{a}}_{v\rightarrow f}^{s};\theta_{v},v_{v})\tilde{g}_{s^{\prime}}(\tilde{\boldsymbol{a}}_{v\rightarrow f}^{s^{\prime}};\theta_{v},v_{v})$ is uniformly pseudo-Lipschitz of order 2 by Lemma 1, we have $\mathbb{E}[\widehat{\Sigma}_{s,s^{\prime}}]=\mathbb{E}[\tilde{q}_{v\rightarrow f}^{s}\tilde{q}_{v\rightarrow f}^{s^{\prime}}]=\mathbb{E}[\tilde{g}_{s}(\tilde{\boldsymbol{a}}_{v\rightarrow f}^{s};\theta_{v},v_{v})\tilde{g}_{s^{\prime}}(\tilde{\boldsymbol{a}}_{v\rightarrow f}^{s^{\prime}};\theta_{v},v_{v})]\rightarrow\Sigma_{s,s^{\prime}}$ by the inductive hypothesis, Lemma 2, and (42). The terms in the sum defining $\widehat{\boldsymbol{\Sigma}}_{[0{:}t]}$ are mutually independent by Lemma 2 and have bounded second moments by the inductive hypothesis and the Lipschitz continuity of the functions $(\tilde{g}_{s})_{0\leq s\leq t}$. By the weak law of large numbers, $\widehat{\boldsymbol{\Sigma}}_{[0{:}t]}\stackrel{{\scriptstyle L_{1}}}{{\rightarrow}}\boldsymbol{\Sigma}_{[0{:}t]}$, whence by Slutsky’s theorem, $\tilde{\boldsymbol{b}}_{f}^{t}\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}{\mathsf{N}}(\boldsymbol{0}_{t+1},\boldsymbol{\Sigma}_{[0{:}t]})$. Further, $\mathbb{E}[\tilde{\boldsymbol{b}}_{f}^{t}(\tilde{\boldsymbol{b}}_{f}^{t})^{\mathsf{T}}]=\mathbb{E}[\widehat{\boldsymbol{\Sigma}}_{[0{:}t]}]\rightarrow\boldsymbol{\Sigma}_{[0{:}t]}$. Convergence in distribution and in second moment implies convergence in the Wasserstein space of order 2 [Vil10, Theorem 6.9], so $\tilde{\boldsymbol{b}}_{f}^{t}\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}{\mathsf{N}}(\boldsymbol{0}_{t+1},\boldsymbol{\Sigma}_{[0{:}t]})$. To bound the fourth moments of $\tilde{b}_{f}^{t}$, we compute $\mathbb{E}[(\tilde{b}_{f}^{t})^{4}]=\mathbb{E}[\widehat{\Sigma}_{t,t}^{2}]=\frac{1}{n^{2}}\sum_{v\in\partial f}\mathbb{E}[(\tilde{q}_{v\rightarrow f}^{t})^{4}]+\frac{1}{n^{2}}\sum_{v\neq v^{\prime}\in\partial f}\mathbb{E}[(\tilde{q}_{v\rightarrow f}^{t})^{2}]\mathbb{E}[(\tilde{q}_{v^{\prime}\rightarrow f}^{t})^{2}]\rightarrow\Sigma_{t,t},$ where the first term goes to 0 because the fourth moments of $\tilde{q}_{v\rightarrow f}^{t}$ are bounded by the inductive hypothesis and Lipschitz continuity of $\tilde{g}_{t}$, and the second term goes to $\mathbb{E}[(\tilde{q}_{v\rightarrow f}^{t})^{2}]$ by the same argument in the preceding paragraph. The boundedness of the fourth moments of $\tilde{b}_{f}^{s}$ holds similarly (and, anyway, will have been established earlier in the induction). Finally, observe $\tilde{b}_{f}^{t}-\tilde{b}_{f\rightarrow v}^{t}=z_{fv}\tilde{q}_{v\rightarrow f}^{t}$ and $\mathbb{E}[(z_{fv}\tilde{q}_{v\rightarrow f}^{t})^{2}]=\mathbb{E}[\tilde{q}_{v\rightarrow f}^{t})^{2}]/n\rightarrow 0$, where $\mathbb{E}[\tilde{q}_{v\rightarrow f}^{t})^{2}]$ is bounded by the inductive hypothesis and Lipschitz continuity of $\tilde{g}_{t}$. The convergence $\mathbb{E}[(\tilde{b}_{f}^{t}-\tilde{b}_{f\rightarrow v}^{s})^{2}]\rightarrow 0$ for $s<t$ holds similarly (and, anyway, will have been established earlier in the induction). The Wasserstein convergence of $(\tilde{\boldsymbol{b}}_{v\rightarrow f}^{t},\theta_{v},v_{v})$ now follows. The bounded fourth moments of $\tilde{\boldsymbol{b}}_{v\rightarrow f}^{t}$ hold similarly. Inductive step 2: Eq. (43) at $t$, bounded fourth moments of $\tilde{\boldsymbol{b}}_{f}^{t},\tilde{\boldsymbol{b}}_{f\rightarrow v}^{t}$, and $\mathbb{E}[\|\tilde{\boldsymbol{b}}_{f}^{t}-\tilde{\boldsymbol{b}}_{f\rightarrow v}^{t}\|^{2}]\rightarrow 0$ imply Eq. (43) at $t+1$, bounded fourth moments of $\tilde{\boldsymbol{a}}_{v}^{t},\tilde{\boldsymbol{a}}_{v\rightarrow f}^{t+1}$, and $\mathbb{E}[\|\tilde{\boldsymbol{a}}_{v}^{t+1}-\tilde{\boldsymbol{a}}_{v\rightarrow f}^{t+1}\|^{2}]\rightarrow 0$. This follows by exactly the same argument as in inductive step 1. The induction is complete, and Lemma 1 follows. ∎ ### C.2 Message passing in the high-dimensional regression model We prove Lemma 2 for the high-dimensional regression model by showing that the iteration (19) is well approximated by a Gaussian message passing algorithm after a change of variables. The functions $\tilde{f}_{t},\tilde{g}_{t}$ in the Gaussian message passing algorithm are defined in terms of the functions $f_{t},g_{t}$ of the original message passing algorithm (19) and the function $h$ used to define the high-dimensional regression model. $\displaystyle\tilde{f}_{t}(\tilde{b}^{0},\cdots,\tilde{b}^{t},w,u):=f_{t}(\tilde{b}^{1},\cdots,\tilde{b}^{t};h(\tilde{b}^{0},w),u),\;\;t\geq 0,$ $\displaystyle\tilde{g}_{0}(\theta,v)=\theta,\;\;\tilde{g}_{t}(\tilde{a}^{1},\cdots,\tilde{a}^{t};\theta,v):=g_{t}(\alpha_{1}\theta+\tilde{a}^{1},\cdots,\alpha_{1}\theta+\tilde{a}^{t};v),\;\;t\geq 1.$ Define $(\tilde{a}_{v\rightarrow f}^{t})_{t\geq 1}$, $(\tilde{a}_{v}^{t})_{t\geq 1}$, $(\tilde{q}_{v\rightarrow f}^{t})_{t\geq 0}$, $(\tilde{b}_{f\rightarrow v}^{t})_{t\geq 0}$, $(\tilde{b}_{f}^{t})_{t\geq 0}$, $(\tilde{r}_{f\rightarrow v}^{t})_{t\geq 0}$ via the Gaussian message passing algorithm (40) with initial data $\theta_{v},v_{v},w_{f},u_{f}$ and with $z_{fv}=x_{fv}$. Because $f_{t}$, $g_{t}$, and $h$ are Lipschitz, so too are $\tilde{f}_{t}$ and $\tilde{g}_{t}$. Under the function definitions $\tilde{f}_{t},\tilde{g}_{t}$ given above, the definitions of $\Sigma_{s,s}$ and $T_{s,s^{\prime}}$ in (42) and (36) are equivalent. Thus, Lemma 1 holds for the iterates of this Gaussian message passing algorithm with the $\boldsymbol{T}_{[1{:}t]}$, $\boldsymbol{\Sigma}_{[0{:}t]}$ defined by (36). We claim that for fixed $s\geq 1$, as $n\rightarrow\infty$ we have $\mathbb{E}[(\alpha_{s}\theta_{v}+\tilde{a}_{v\rightarrow f}^{s}-a_{v\rightarrow f}^{s})^{2}]\rightarrow 0\;\;\text{and}\;\;\mathbb{E}[(\tilde{b}_{f\rightarrow v}^{s}-b_{f\rightarrow v}^{s})^{2}]\rightarrow 0,$ (44a) and $\mathbb{E}[(a_{v\rightarrow f}^{s})^{4}]\;\;\text{and}\;\;\mathbb{E}[(b_{f\rightarrow v}^{s})^{4}]\;\;\mbox{are uniformly bounded with respect to $n$},$ (44b) where $(\alpha_{s})$ are defined by (36). These are the same coefficients appearing in the AMP state evolution (Lemma 1), as claimed. We show (44) by induction. There is no base case because the inductive steps work for $t=0$ as written. Inductive step 1: If (44) holds for $1\leq s\leq t$, then (44a) holds for $s=t+1$. We expand $\displaystyle\alpha_{t+1}\theta_{v}+\tilde{a}_{v\rightarrow f}^{t+1}-a_{v\rightarrow f}^{t+1}$ $\displaystyle=\alpha_{t+1}\theta_{v}+\sum_{f^{\prime}\in\partial v\setminus f}z_{f^{\prime}v}(\tilde{f}_{t}(\tilde{\boldsymbol{b}}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})-f_{t}(\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};y_{f^{\prime}},u_{f^{\prime}}))$ $\displaystyle=\alpha_{t+1}\theta_{v}+\sum_{f^{\prime}\in\partial v\setminus f}z_{f^{\prime}v}(\tilde{f}_{t}(\tilde{\boldsymbol{b}}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})-\tilde{f}_{t}(\tilde{b}_{f^{\prime}\rightarrow v}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}}))$ $\displaystyle\quad\quad+\sum_{f^{\prime}\in\partial v\setminus f}z_{f^{\prime}v}(\tilde{f}_{t}(\tilde{b}_{f^{\prime}\rightarrow v}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})-\tilde{f}_{t}(\tilde{b}_{f^{\prime}}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}}))$ $\displaystyle=:\alpha_{t+1}\theta_{v}+\mathsf{I}+\mathsf{II}.$ (Note that $\tilde{\boldsymbol{b}}_{f^{\prime}\rightarrow v}^{t}$ is $(t+1)$-dimensional and $\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t}$ is $t$-dimensional). First we analyze $\mathsf{I}$. We have $\displaystyle|\tilde{f}_{t}(\tilde{\boldsymbol{b}}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})-\tilde{f}_{t}(\tilde{b}_{f^{\prime}\rightarrow v}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})|\leq L\sum_{s=1}^{t}|\tilde{b}_{f^{\prime}\rightarrow v}^{s}-b_{f^{\prime}\rightarrow v}^{s}|,$ where $L$ is a Lipschitz constant of $\tilde{f}_{t}$. The terms in the sum defining $\mathsf{I}$ are mutually independent, and $\tilde{b}_{f^{\prime}\rightarrow v}^{s},b_{f^{\prime}\rightarrow v}^{s}$ are independent of $z_{f^{\prime}v}$. Thus, $\displaystyle\mathbb{E}[\mathsf{I}^{2}]$ $\displaystyle=\frac{n-1}{n}\mathbb{E}[(\tilde{f}_{t}(\tilde{\boldsymbol{b}}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})-\tilde{f}_{t}(\tilde{b}_{f^{\prime}\rightarrow v}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}}))^{2}]$ $\displaystyle\leq\frac{L^{2}(n-1)t}{n}\sum_{s=1}^{t}\mathbb{E}[(\tilde{b}_{f^{\prime}\rightarrow v}^{s}-b_{f^{\prime}\rightarrow v}^{s})^{2}]\rightarrow 0,$ by the inductive hypothesis. Next we analyze $\mathsf{II}$. Note that all arguments to the functions in the sum defining $\mathsf{II}$ are independent of $z_{f^{\prime}v}$ and $\theta_{v}$ except for $\tilde{b}_{f^{\prime}}^{0}=z_{f^{\prime}v}\theta_{v}+\sum_{v^{\prime}\in\partial f^{\prime}\setminus v}z_{f^{\prime}v^{\prime}}\theta_{v^{\prime}}$. Because $\tilde{f}_{t}$ is Lipschitz, we may apply Stein’s lemma (ie., Gaussian integration by parts) [Ste81] to get $\displaystyle\mathbb{E}[\alpha_{t+1}\theta_{v}+\mathsf{II}\bigm{|}\theta_{v},\sigma((\mathcal{T}_{v^{\prime\prime}\rightarrow f^{\prime}})_{v^{\prime\prime}\in\partial f^{\prime}\backslash v})]$ $\displaystyle=\alpha_{t+1}\theta_{v}+(n-1)\mathbb{E}\big{[}z_{f^{\prime}v}(\tilde{f}_{t}(\tilde{b}_{f^{\prime}\rightarrow v}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})-\tilde{f}_{t}(\tilde{b}_{f^{\prime}}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}}))\bigm{|}\theta_{v}\big{]}$ $\displaystyle=\theta_{v}\left(\alpha_{t+1}-\frac{n-1}{n}\mathbb{E}[\partial_{\tilde{b}^{0}}\tilde{f}_{t}(\tilde{b}_{f^{\prime}}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})\bigm{|}\theta_{v}]\right),$ where $\partial_{\tilde{b}^{0}}\tilde{f}_{t}$ is the weak-derivative of $\tilde{f}_{t}$ with respect to its first argument, which is defined almost everywhere with respect to Lebesgue measure because $\tilde{f}_{t}$ is Lipschitz [EG15, pg. 81]. We claim the right-hand side of the preceding display converges in $L_{2}$ to 0, as we now show. The random variable $\mathbb{E}[\partial_{\tilde{b}^{0}}\tilde{f}_{t}(\tilde{b}_{f^{\prime}}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})|\theta_{v},(\mathcal{T}_{v^{\prime\prime}\rightarrow f^{\prime}})_{v^{\prime\prime}\in\partial f^{\prime}\backslash v}]$ is almost- surely bounded because $\tilde{f}_{t}$ is Lipschitz. It converges in probability to $\alpha_{t+1}$. The random vector $(\tilde{b}_{f^{\prime}}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t})$ has a Gaussian distribution conditional on $\sigma((\mathcal{T}_{v^{\prime\prime}\rightarrow f^{\prime}})_{v^{\prime\prime}\in\partial f^{\prime}\backslash v})$ and $\theta_{v}$; in particular, $\displaystyle(\tilde{b}^{0}_{f^{\prime}\rightarrow v}+z_{f^{\prime}v}\theta_{v},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t})|\theta_{v},\sigma((\mathcal{T}_{v^{\prime\prime}\rightarrow f^{\prime}})_{v^{\prime\prime}\in\partial f^{\prime}\backslash v})\stackrel{{\scriptstyle\mathrm{d}}}{{=}}{\mathsf{N}}(\mathbf{0},\widehat{\boldsymbol{\Sigma}}),$ where we define $\widehat{\boldsymbol{\Sigma}}\in\mathbb{R}^{(t+1)\times(t+1)}$ by $\displaystyle\widehat{\Sigma}_{0,0}=\dfrac{1}{n}\sum\limits_{v^{\prime}\in\partial f^{\prime}}\theta_{v^{\prime}}^{2}\;\;\text{and}\;\;\widehat{\Sigma}_{s,s^{\prime}}=\dfrac{1}{n}\sum\limits_{v^{\prime}\in\partial f^{\prime}\setminus v}q_{v^{\prime}\rightarrow f^{\prime}}^{s}q_{v^{\prime}\rightarrow f^{\prime}}^{s^{\prime}}\;\;\text{for}\;s\geq 1\mbox{ or }s^{\prime}\geq 1,$ where for the purposes of the preceding display we set $q_{v^{\prime}\rightarrow f^{\prime}}^{0}=\theta_{v^{\prime}}$. By the Lipschitz continuity of the functions $(g_{s})$, Lemmas 1 and 2, and the inductive hypothesis, we have $\mathbb{E}[\widehat{\boldsymbol{\Sigma}}]\rightarrow\boldsymbol{\Sigma}_{[0{:}t]}$. The terms in the sums in the previous display have bounded second moments by the inductive hypthesis (44b) and the Lipschitz continuity of the functions $(g_{s})$. By the weak law of large numbers, we conclude $\widehat{\boldsymbol{\Sigma}}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\boldsymbol{\Sigma}_{[0:t+1]}$. Observe that $\mathbb{E}[\partial_{\tilde{b}^{0}}\tilde{f}_{t}(\tilde{b}_{f^{\prime}}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})|\theta_{v},(\mathcal{T}_{v^{\prime\prime}\rightarrow f^{\prime}})_{v^{\prime\prime}\in\partial f^{\prime}\backslash v}]=\mathbb{E}[\partial_{\tilde{b}^{0}}\tilde{f}_{t}(\widehat{\boldsymbol{\Sigma}}^{1/2}\boldsymbol{Z};W,U)]$, where on the right-hand side the expectation is with respect to $(W,U)\sim\mu_{W,U}$ and $\boldsymbol{Z}\sim{\mathsf{N}}(\boldsymbol{0}_{t+1},\boldsymbol{I}_{t+1})$ independent. Because $\partial_{\tilde{b}^{0}}\tilde{f}_{t}$ is almost surely bounded, by the dominated convergence theorem, the right-hand side is continuous in $\widehat{\boldsymbol{\Sigma}}$. By the continuous mapping theorem and (36), we conclude $\mathbb{E}[\partial_{\tilde{b}^{0}}\tilde{f}_{t}(\tilde{b}_{f^{\prime}}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})|\theta_{v},(\mathcal{T}_{v^{\prime\prime}\rightarrow f^{\prime}})_{v^{\prime\prime}\in\partial f^{\prime}\backslash v}]\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\alpha_{t+1}$. Then, by dominated convergence, $\mathbb{E}[\alpha_{t+1}\theta_{v}+\mathsf{II}\bigm{|}\theta_{v}]\stackrel{{\scriptstyle L_{2}}}{{\rightarrow}}0$. Moreover, because the terms in the sum defining $\mathsf{II}$ are mutually independent given $\theta_{v}$ $\displaystyle\operatorname{Var}(\alpha_{t+1}\theta_{v}+\mathsf{II}\mid\theta_{v})$ $\displaystyle\leq(n-1)\mathbb{E}\left[z_{f^{\prime}v}^{2}(\tilde{f}_{t}(\tilde{b}_{f^{\prime}\rightarrow v}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}})-\tilde{f}_{t}(\tilde{b}_{f^{\prime}}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};w_{f^{\prime}},u_{f^{\prime}}))^{2}\mid\theta_{v}\right]$ $\displaystyle\leq L^{2}(n-1)\mathbb{E}[z_{f^{\prime}v}^{4}\theta_{v}^{2}\mid\theta_{v}]\leq 3\theta_{v}^{2}/n,$ where $L$ is the Lipschitz constant of $\tilde{f}_{t}$. We conclude that $\mathbb{E}[\operatorname{Var}(\alpha_{t+1}\theta_{v}+\mathsf{II}\mid\theta_{v})]\rightarrow 0$. Combined with $\mathbb{E}[\alpha_{t+1}\theta_{v}+\mathsf{II}\bigm{|}\theta_{v}]\stackrel{{\scriptstyle L_{2}}}{{\rightarrow}}0$, we get $\operatorname{Var}(\alpha_{t+1}\theta_{v}+\mathsf{II})=\operatorname{Var}(\mathbb{E}[\alpha_{t+1}\theta_{v}+\mathsf{II}|\theta_{v}])+\mathbb{E}[\operatorname{Var}(\alpha_{t+1}\theta_{v}+\mathsf{II}|\theta_{v})]\rightarrow 0$, so that $\alpha_{t+1}\theta_{v}+\mathsf{II}\stackrel{{\scriptstyle L_{2}}}{{\rightarrow}}0$. Combining $\mathsf{I}\stackrel{{\scriptstyle L_{2}}}{{\rightarrow}}0$ and $\alpha_{t+1}\theta_{v}+\mathsf{II}\stackrel{{\scriptstyle L_{2}}}{{\rightarrow}}0$ gives $\mathbb{E}[(\alpha_{t+1}\theta_{v}+\tilde{a}_{v\rightarrow f}^{t+1}-a_{v\rightarrow f}^{t+1})^{2}]\rightarrow 0$, as desired. We now expand $\tilde{b}_{f\rightarrow v}^{t+1}-b_{f\rightarrow v}^{t+1}=\sum_{v^{\prime}\in\partial f\setminus v}z_{fv^{\prime}}(g_{t}(\boldsymbol{\alpha}_{t+1}\theta_{v^{\prime}}+\tilde{\boldsymbol{a}}_{v^{\prime}\rightarrow f}^{t+1};v_{v^{\prime}})-g_{t}(\boldsymbol{a}_{v^{\prime}\rightarrow f}^{t+1};v_{v^{\prime}})).$ The terms in this sum are mutually independent, and $\tilde{\boldsymbol{a}}_{v^{\prime}\rightarrow f}^{t+1},\boldsymbol{a}_{v^{\prime}\rightarrow f}^{t+1},\theta_{v^{\prime}}$ are independent of $z_{f^{\prime}v}$. Thus, $\displaystyle\mathbb{E}[(\tilde{b}_{f\rightarrow v}^{t+1}-b_{f\rightarrow v}^{t+1})^{2}]$ $\displaystyle=\frac{p-1}{n}\mathbb{E}[(g_{t}(\boldsymbol{\alpha}_{t+1}\theta_{v^{\prime}}+\tilde{\boldsymbol{a}}_{v^{\prime}\rightarrow f}^{t+1};v_{v^{\prime}})-g_{t}(\boldsymbol{a}_{v^{\prime}\rightarrow f}^{t+1};v_{v^{\prime}}))^{2}]$ $\displaystyle\leq\frac{L^{2}(p-1)(t+1)}{n}\sum_{s=1}^{t+1}\mathbb{E}[(\alpha_{s}\theta_{v^{\prime}}+\tilde{a}_{v^{\prime}\rightarrow f}^{s}-a_{v^{\prime}\rightarrow f}^{s})^{2}]\rightarrow 0.$ This completes the proof of (44a) at $s=t+1$. Inductive step 2: If (44) holds for $1\leq s\leq t$, then (44b) holds for $s=t+1$. By Lipschitz continuity, $\displaystyle\left|a_{v\rightarrow f}^{t+1}-\sum_{f^{\prime}\in\partial v\setminus f}z_{f^{\prime}v}\tilde{f}_{t}(\tilde{b}_{f^{\prime}\rightarrow v}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t},u_{f^{\prime}},w_{f^{\prime}})\right|\leq L|\theta_{v}|\sum_{f^{\prime}\in\partial v\setminus f}|z_{f^{\prime}v}|,$ where $L$ is a Lipschitz constant for $\tilde{f}_{t}$. The right-hand side has bounded fourth moment, so we must only show that the sum in the previous display has bounded fourth moment. The quantity $\tilde{f}_{t}(\tilde{b}_{f^{\prime}\rightarrow v}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t},u_{f^{\prime}},w_{f^{\prime}})$ has bounded fourth moment by the inductive hypothesis and Lipschitz continuity of $\tilde{f}_{t}$. Because $z_{f^{\prime}v}$ is independent of the argument to $\tilde{f}_{t}$ and has fourth moment $3/n^{2}$, the product $z_{f^{\prime}v}\tilde{f}_{t}(\tilde{b}_{f^{\prime}\rightarrow v}^{0},\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t},u_{f^{\prime}},w_{f^{\prime}})$ has mean 0 and fourth moment $O(1/n^{2})$. Because these products are mean zero and independent across $f^{\prime}$, their sum has bounded fourth moment. We conclude $a_{v\rightarrow f}^{t+1}$ has bounded fourth moment as well. Recall $b_{f\rightarrow v}^{t+1}=\sum_{v^{\prime}\in\partial f\setminus v}z_{fv^{\prime}}g_{t}(\boldsymbol{a}_{v^{\prime}\rightarrow f}^{t+1};v_{v}^{\prime})$. The terms in the sum are independent, and $z_{fv^{\prime}}$ is independent of $\boldsymbol{a}_{v^{\prime}\rightarrow f}^{t+1};v_{v}^{\prime}$. Using the Lipschitz continuity of $g_{t}$ and the inductive hypothesis, we conclude $b_{f\rightarrow v}^{t+1}$ has bounded fourth moment by the same argument as in the preceding paragraph. We conclude (44b) at $s=t+1$. The induction is complete, and (44a) holds for all $s\geq 1$. Lemma 2 follows by combining Lemma 1 and Eq. (44a). ### C.3 Message passing in the low-rank matrix estimation model Like in the preceding section, we prove Lemma 2 for the low-rank matrix estimation model by showing that the iteration (19) is well approximated by a Gaussian message passing algorithm after a change of variables. The functions in the Gaussian message passing algorithm are defined in terms of the functions $f_{t},g_{t}$ of the original message passing algorithm (19). $\displaystyle\tilde{f}_{t}(\tilde{b}^{0},\cdots,\tilde{b}^{t},w,u):=f_{t}(\tilde{b}^{1}+\gamma_{1}w,\cdots,\tilde{b}^{t}+\gamma_{t}w;0,u),$ $\displaystyle\tilde{g}_{t}(\tilde{a}^{1},\cdots,\tilde{a}^{t};\theta,v):=g_{t}(\tilde{a}^{1}+\alpha_{1}\theta,\cdots,\tilde{a}^{t}+\alpha_{t}\theta;v).$ Note that here $\tilde{f}_{t}$ does not depend on $\tilde{b}^{0}$ is never used, and we may define $\tilde{g}_{0}$ arbitrarily without affecting later iterates.777The iterate $\tilde{b}^{0}$ only played a role in approximating the high-dimensional regression message passing algorithm by a Gaussian message passing algorithm. Define $(\tilde{a}_{v\rightarrow f}^{t})_{t\geq 1}$, $(\tilde{a}_{v}^{t})_{t\geq 1}$, $(\tilde{q}_{v\rightarrow f}^{t})_{t\geq 0}$, $(\tilde{b}_{f\rightarrow v}^{t})_{t\geq 0}$, $(\tilde{b}_{f}^{t})_{t\geq 0}$, $(\tilde{r}_{f\rightarrow v}^{t})_{t\geq 0}$ via the Gaussian message passing algorithm (40) with initial data $\theta_{v},v_{v},u_{f},z_{fv}$ and $w_{f}=\lambda_{f}$. Because $f_{t}$, $g_{t}$, and $h$ are Lipschitz, so too are $\tilde{f}_{t}$ and $\tilde{g}_{t}$. Under the function definitions $\tilde{f}_{t},\tilde{g}_{t}$ given above and the change of variables $w_{f}=\lambda_{f}$, the definitions of $\Sigma_{s,s}$ and $T_{s,s^{\prime}}$ in (42) and (37) are equivalent. Thus, Lemma 1 holds for the iterates of this Gaussian message passing algorithm with the $\boldsymbol{T}_{[1{:}t]}$, $\boldsymbol{\Sigma}_{[0{:}t]}$ defined by (37). We claim that for fixed $s\geq 1$, as $n\rightarrow\infty$ we have $\mathbb{E}[(\alpha_{s}\theta_{v}+\tilde{a}_{v\rightarrow f}^{s}-a_{v\rightarrow f}^{s})^{2}]\rightarrow 0\;\;\text{and}\;\;\mathbb{E}[(\gamma_{s}\lambda_{f}+\tilde{b}_{f\rightarrow v}^{s}-b_{f\rightarrow v}^{s})^{2}]\rightarrow 0,$ (45a) and $\mathbb{E}[\theta_{v}^{2}(a_{v\rightarrow f}^{s})^{2}]\;\;\text{and}\;\;\mathbb{E}[\lambda_{f}^{2}(b_{f\rightarrow v}^{s})^{2}]\;\;\text{are bounded for fixed $s$.}$ (45b) We show this by induction. There is no base case because the inductive step works for $t=0$ as written. Inductive step: If (45) holds for $1\leq s\leq t$, then (45) holds for $s=t+1$. We expand $\displaystyle\alpha_{t+1}\theta_{v}+\tilde{a}_{v\rightarrow f}^{t+1}-a_{v\rightarrow f}^{t+1}$ $\displaystyle=\alpha_{t+1}\theta_{v}+\sum_{f^{\prime}\in\partial v\setminus f}z_{f^{\prime}v}(f_{t}(\tilde{\boldsymbol{b}}_{f^{\prime}\rightarrow v}^{t}+\boldsymbol{\gamma}_{t}\lambda_{f^{\prime}};0,u_{f^{\prime}})-f_{t}(\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};0,u_{f^{\prime}}))$ $\displaystyle\quad\quad-\frac{1}{n}\theta_{v}\sum_{f^{\prime}\in\partial v\setminus f}\lambda_{f^{\prime}}f_{t}(\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};0,u_{f^{\prime}})$ $\displaystyle=:\alpha_{t+1}\theta_{v}+\mathsf{I}+\mathsf{II},$ where $\tilde{\boldsymbol{b}}_{f^{\prime}\rightarrow v}^{t}=(\tilde{b}_{f^{\prime}\rightarrow v}^{1},\ldots,\tilde{b}_{f^{\prime}\rightarrow v}^{t})$ and $\boldsymbol{\gamma}_{t}=(\gamma_{1},\ldots,\gamma_{t})$ (note that $\tilde{b}_{f^{\prime}\rightarrow v}^{0}$ is excluded, which differs from the notation used in the proof of Lemma 2). First we analyze $\mathsf{I}$. The terms in the sum defining $\mathsf{I}$ are mutually independent, and $\tilde{b}_{f^{\prime}\rightarrow v}^{s}$, $b_{f^{\prime}\rightarrow v}^{s}$, $\lambda_{f^{\prime}}$, $u_{f^{\prime}}$ are independent of $z_{f^{\prime}v}$. Thus, $\displaystyle\mathbb{E}[\mathsf{I}^{2}]$ $\displaystyle=\frac{n-1}{n}\mathbb{E}[(f_{t}(\tilde{\boldsymbol{b}}_{f^{\prime}\rightarrow v}^{t}+\boldsymbol{\gamma}_{t}\lambda_{f^{\prime}};0,u_{f^{\prime}})-f_{t}(\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};0,u_{f^{\prime}})^{2}]$ $\displaystyle\leq\frac{L^{2}(n-1)t}{n}\sum_{s=1}^{t}\mathbb{E}[(\tilde{b}_{f^{\prime}\rightarrow v}^{s}+\gamma_{s}\lambda_{f^{\prime}}-b_{f^{\prime}\rightarrow v}^{s})^{2}]\rightarrow 0,$ by the inductive hypothesis, where $L$ is a Lipschitz constant of $f_{t}$. Moreover, because $\theta_{v}$ is independent of $\mathsf{I}$ and has bounded fourth moment, $\mathbb{E}[\theta_{v}^{2}\mathsf{I}^{2}]\rightarrow 0$ as well. Next we analyze $\mathsf{II}$. By the inductive hypothesis and Lemma 1, $(\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t},\lambda_{f^{\prime}},u_{f^{\prime}})\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}(\boldsymbol{\gamma}_{t}\Lambda+\tilde{B}^{t},\Lambda,U),$ where $(\Lambda,U)\sim\mu_{\Lambda,U}$ and $\tilde{B}^{t}\sim{\mathsf{N}}(\boldsymbol{0}_{t},\boldsymbol{\Sigma}_{[1{:}t]})$ independent. Because $(\boldsymbol{b}^{t},\lambda,u)\mapsto\lambda f_{t}(\boldsymbol{b}^{t};0,u)$ is uniformly pseudo-Lipschitz of order 2 by Lemma 1, we have $\mathbb{E}[\lambda_{f^{\prime}}f_{t}(\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};0,u_{f^{\prime}})]\rightarrow\alpha_{t+1}$ by Lemma 2 and the state evolution recursion (37). Moreover, because $f_{t}$ is Lipschitz, for some constant $C$ $\displaystyle\mathbb{E}[\lambda_{f^{\prime}}^{2}f_{t}(\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};0,u_{f^{\prime}})^{2}]$ $\displaystyle\leq C\mathbb{E}\left[\lambda_{f^{\prime}}^{2}\left(1+\sum_{s=1}^{t}(b_{f^{\prime}\rightarrow v}^{s})^{2}+u_{f^{\prime}}^{2}\right)\right]$ $\displaystyle=C\left(\mathbb{E}[\lambda_{f^{\prime}}^{2}]+\sum_{s=1}^{t}\mathbb{E}[\lambda_{f^{\prime}}^{2}(b_{f^{\prime}\rightarrow v}^{s})^{2}]+\mathbb{E}[\lambda_{f^{\prime}}^{2}u_{f^{\prime}}^{2}]\right),$ which bounded by the inductive hypothesis and the fourth moment assumption on $\mu_{\Lambda,U}$. Because the terms in the sum defining $\mathsf{II}$ are mutually independent, by the weak law of large numbers the preceding observations imply $\frac{1}{n}\sum_{f^{\prime}\in\partial v\setminus f}\lambda_{f^{\prime}}f_{t}(\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};0,u_{f^{\prime}})\stackrel{{\scriptstyle L_{2}}}{{\rightarrow}}\alpha_{t+1}.$ Because $\theta_{v}$ is independent of this sum and has bounded second moment, we conclude that $\alpha_{t+1}\theta_{v}+\mathsf{II}=\theta_{v}\left(\alpha_{t+1}-\frac{1}{n}\sum_{f^{\prime}\in\partial v\setminus f}\lambda_{f^{\prime}}f_{t}(\boldsymbol{b}_{f^{\prime}\rightarrow v}^{t};0,u_{f^{\prime}})\right)\stackrel{{\scriptstyle L_{2}}}{{\rightarrow}}0.$ Moreover, because $\theta_{v}$ is independent of the term in parentheses and has bounded fourth moment, $\mathbb{E}[\theta_{v}^{2}(\alpha_{t+1}\theta_{v}+\mathsf{II})^{2}]\rightarrow 0$. Combining the preceding results, we have that $\mathbb{E}[(\alpha_{t+1}\theta_{v}+\tilde{a}_{v\rightarrow f}^{t+1}-a_{v\rightarrow f}^{t+1})^{2}]\rightarrow 0$ and $\mathbb{E}[\theta_{v}^{2}(\alpha_{t+1}\theta_{v}+\tilde{a}_{v\rightarrow f}^{t+1}-a_{v\rightarrow f}^{t+1})^{2}]$ is bounded. Because $\theta_{v}$ is independent of $\tilde{a}_{v\rightarrow f}^{t+1}$, the term $\mathbb{E}[\theta_{v}^{2}(\tilde{a}_{v\rightarrow f}^{t+1})^{2}]$ is bounded, so also $\mathbb{E}[\theta_{v}^{2}(a_{v\rightarrow f}^{t+1})^{2}]$ is bounded, as desired. The argument establishing that $\mathbb{E}[(\gamma_{t+1}\lambda_{f}+\tilde{b}_{f\rightarrow v}^{t+1}-b_{f\rightarrow v}^{t+1})^{2}]\rightarrow 0$ and that $\mathbb{E}[\lambda_{f}^{2}(b_{f\rightarrow v}^{t+1})^{2}]$ is bounded is equivalent. The induction is complete, and (45) holds for all $s$. Lemma 2 follows by combining Lemma 1 and Eq. (45). ## Appendix D Proof of information-theoretic lower bounds on the computation tree (Lemma 3) In this section, we prove Lemma 3 in both the high-dimensional regression and low-rank matrix estimation models. We restrict ourselves to the case $r=1$ and $k=1$ (with $k$ the dimensionality of $\boldsymbol{W}$) because the proof for $r>1$ or $k>1$ is completely analogous but would complicate notation. For any pair of nodes $u,u^{\prime}$ in the tree $\mathcal{T}$, let $d(u,u^{\prime})$ denote the length (number of edges) of the shortest path between nodes $u$ and $u^{\prime}$ in the tree. Let $\mathcal{T}_{u,k}=(\mathcal{V}_{u,k},\mathcal{F}_{u,k},\mathcal{E}_{u,k})$ be the radius-$k$ neighborhood of node $u$; that is, $\displaystyle\mathcal{V}_{u,k}=\\{v\in\mathcal{V}\mid d(u,v)\leq k\\},$ $\displaystyle\mathcal{F}_{u,k}=\\{f\in\mathcal{F}\mid d(u,f)\leq k\\},$ $\displaystyle\mathcal{E}_{u,k}=\\{(f,v)\in\mathcal{E}\mid\max\\{d(u,f),d(u,v)\\}\leq k\\}.$ With some abuse of notation, we will often use $\mathcal{T}_{u,k},\mathcal{V}_{u,k},\mathcal{F}_{u,k},\mathcal{E}_{u,k}$ to denote either the collection of observations corresponding to nodes and edges in these sets or the $\sigma$-algebra generated by these obervations. No confusion should result. Note, our convention is that when used to denote a $\sigma$-algebra or collection of random variables, only observed random variables are in include. Thus, in the high-dimensional regression model, $\mathcal{T}_{u,k}$ is the $\sigma$-algebra generated by the local observations $x_{fv}$, $y_{f}$, $v_{v}$, and $u_{f}$; in the low-rank matrix estimation, it is the $\sigma$-algebra genreated by the local observations $x_{fv}$, $v_{v}$, and $u_{f}$. We also denote by $\mathcal{T}_{v\rightarrow f}^{t,k}$ the collection of observations associated to edges or nodes of $\mathcal{T}$ which are separated from $f$ by $v$ by at least $k$ intervening edges and at most $t$ intervening edges. For example, $\mathcal{T}_{v\rightarrow f}^{1,1}$ contains only $(y_{f^{\prime}})_{f^{\prime}\in\partial v\setminus f}$, and $\mathcal{T}_{v\rightarrow f}^{2,1}$ contains additional the observations $v_{v^{\prime}}$ and $x_{f^{\prime}v^{\prime}}$ for $v^{\prime}\in\partial f^{\prime}\setminus v$ for some some $f^{\prime}\in\partial v\setminus f$. The collections (or $\sigma$-algebras) $\mathcal{V}_{v\rightarrow f}^{t,k}$, $\mathcal{F}_{v\rightarrow f}^{t,k}$, $\mathcal{E}_{v\rightarrow f}^{t,k}$ are defined similarly, as are the versions of these where the roles of $v$ and $f$ are reversed. ### D.1 Information-theoretic lower bound in the high-dimensional regression model In this section, we prove Lemma 3 in the high-dimensional regression model. Note that conditions on the conditional density in assumption R4 are equivalent positivity, boundedness, and the existence finite, non-negative constants $q_{k}^{\prime}$ such that $\frac{|\partial_{x}^{k}p(y|x)|}{p(y|x)}\leq q_{k}^{\prime}$ for $1\leq k\leq 5$. We will often use this form of the assumption without further comment. This implies that for any random variable $A$ $\frac{|\partial_{x}^{k}\mathbb{E}[p(y|x+A)]|}{\mathbb{E}[p(y|x+A)]}\leq\int\frac{|\partial_{x}^{k}p(y|x+a)|}{p(y|x+a)}\frac{p(y|x+a)}{\mathbb{E}[p(y|x+A)]}\mu_{A}({\mathrm{d}}a)\leq q_{k}^{\prime},$ (46) because $p(y|x+a)/\mathbb{E}[p(y|x+A)]$ is a probability density with respect to $\mu_{A}$, the distribution of $A$. Denote the regular conditional probability of $\Theta$ conditional on $V$ for the measure $\mu_{\Theta,V}$ by $\mu_{\Theta|V}:\mathbb{R}\times\mathcal{B}\rightarrow[0,1]$, where $\mathcal{B}$ denotes the Borel $\sigma$-algebra on $\mathbb{R}$. The posterior of $\theta_{v}$ given $\mathcal{T}_{v,2t}$ has density with respect to $\mu_{\Theta|V}(v_{v},\cdot)$ given by $p_{v}(\vartheta|\mathcal{T}_{v,2t})\propto\int\prod_{f\in\mathcal{F}_{v,2t}}p(y_{f}\mid\sum_{v^{\prime}\in\partial f}\vartheta_{v^{\prime}}X_{v^{\prime}f},u_{f})\prod_{v^{\prime}\in\mathcal{V}_{v,2t}\setminus v}\mu_{\Theta|V}(v_{v^{\prime}},\mathrm{d}\vartheta_{v^{\prime}}).$ Asymptotically, the posterior density with respect to $\mu_{\Theta|V}(v_{v},\cdot)$ behaves like that produced by a Gaussian observation of $\theta_{v}$ with variance $\tau_{t}^{2}$, where $\tau_{t}$ is defined by (5). ###### Lemma 1. In the high-dimensional regression model, there exist $\mathcal{T}_{v,2t}$-measurable random variables $\tau_{v,t},\chi_{v,t}$ such that $p_{v}(\vartheta|\mathcal{T}_{v,2t})\propto\exp\left(-\frac{1}{2\tau_{v,t}^{2}}(\chi_{v,t}-\vartheta)^{2}+o_{p}(1)\right),$ where $o_{p}(1)$ has no $\vartheta$ dependence. Moreover, $(\chi_{v,t},\tau_{v,t},\theta_{v},v_{v})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Theta+\tau_{t}G,\tau_{t},\Theta,V)$ where $(\Theta,V)\sim\mu_{\Theta,V}$, $G\sim{\mathsf{N}}(0,1)$ independent of $\Theta,V$, and $\tau_{t}$ is given by (5). ###### Lemma 1. We compute the posterior density $p_{v}(\vartheta|\mathcal{T}_{v,2t})$ via an iteration called belief propagation. For each edge $(v,f)\in\mathcal{E}$, belief propagation generates a pair of sequences of real-valued functions $(m_{v\rightarrow f}^{t}(\vartheta))_{t\geq 0},(m_{f\rightarrow v}^{t}(\vartheta))_{t\geq 0}$. The iteration is $\displaystyle m_{v\rightarrow f}^{0}(\vartheta)=1,$ $\displaystyle m_{f\rightarrow v}^{s}(\vartheta)\propto\int p(y_{f}|X_{fv}\vartheta+\sum_{v^{\prime}\in\partial f\setminus v}X_{fv^{\prime}}\vartheta_{v^{\prime}},u_{f})\prod_{v^{\prime}\in\partial f\setminus v}m_{v^{\prime}\rightarrow f}^{s}(\vartheta_{v^{\prime}})\prod_{v^{\prime}\in\partial f\setminus v}\mu_{\Theta|V}(v_{v^{\prime}},\mathrm{d}\vartheta_{v^{\prime}}),$ $\displaystyle m_{v\rightarrow f}^{s+1}(\vartheta)\propto\prod_{f^{\prime}\in\partial v\setminus f}m_{f^{\prime}\rightarrow v}^{s}(\vartheta),$ with normalization $\int m_{f\rightarrow v}^{t}(\vartheta)\mu_{\Theta|V}(v_{v},\mathrm{d}\vartheta)=\int m_{v\rightarrow f}^{t}(\vartheta)\mu_{\Theta|V}(v_{v},\mathrm{d}\vartheta)=1$. For any variable node $v$, $p_{v}(\vartheta|\mathcal{T}_{v,2t})\propto\prod_{f\in\partial v}m_{f\rightarrow v}^{t-1}(\vartheta).$ (47) This equation is exact. We define several quantities related to the belief propagation iteration. $\displaystyle\mu_{v\rightarrow f}^{s}$ $\displaystyle=\int\vartheta m_{v\rightarrow f}^{s}(\vartheta)\mu_{\Theta|V}(v_{v},\mathrm{d}\vartheta),$ $\displaystyle(\tilde{\tau}_{v\rightarrow f}^{s})^{2}$ $\displaystyle=\int\vartheta^{2}m_{v\rightarrow f}^{s}(\vartheta)\mu_{\Theta|V}(v_{v},\mathrm{d}\vartheta)-(\mu_{v\rightarrow f}^{s})^{2},$ $\displaystyle\mu_{f\rightarrow v}^{s}$ $\displaystyle=\sum_{v^{\prime}\in\partial f\setminus v}x_{fv^{\prime}}\mu_{v^{\prime}\rightarrow f}^{s},$ $\displaystyle(\tilde{\tau}_{f\rightarrow v}^{s})^{2}$ $\displaystyle=\sum_{v^{\prime}\in\partial f\setminus v}x_{fv^{\prime}}^{2}(\tilde{\tau}_{v^{\prime}\rightarrow f}^{s})^{2},$ $\displaystyle a_{f\rightarrow v}^{s}$ $\displaystyle=\frac{1}{x_{fv}}\frac{\mathrm{d}\phantom{b}}{\mathrm{d}\vartheta}\log m_{f\rightarrow v}^{s}(\vartheta)\Big{|}_{\vartheta=0},$ $\displaystyle b_{f\rightarrow v}^{s}$ $\displaystyle=-\frac{1}{x_{fv}^{2}}\frac{\mathrm{d}^{2}\phantom{b}}{\mathrm{d}\vartheta^{2}}\log m_{f\rightarrow v}^{s}(\vartheta)\Big{|}_{\vartheta=0},$ $\displaystyle a_{v\rightarrow f}^{s}$ $\displaystyle=\frac{\mathrm{d}\phantom{b}}{\mathrm{d}\vartheta}\log m_{v\rightarrow f}^{s}(\vartheta)\Big{|}_{\vartheta=0},$ $\displaystyle b_{v\rightarrow f}^{s}$ $\displaystyle=-\frac{\mathrm{d}^{2}\phantom{b}}{\mathrm{d}\vartheta^{2}}\log m_{v\rightarrow f}^{s}(\vartheta)\Big{|}_{\vartheta=0},$ $\displaystyle\chi_{v\rightarrow f}^{s}$ $\displaystyle=a_{v\rightarrow f}^{s}/b_{v\rightarrow f}^{s},$ $\displaystyle(\tau_{v\rightarrow f}^{s})^{2}$ $\displaystyle=1/b_{v\rightarrow f}^{s}.$ Lemma 1 follows from the following asymptotic characterization of the quantities in the preceding display in the limit $n,p\rightarrow\infty$, $n/p\rightarrow\delta$: $\begin{gathered}\qquad\mathbb{E}[(\mu_{v\rightarrow f}^{s})^{2}]\rightarrow\delta\sigma_{s}^{2},\qquad\mathbb{E}[(\tilde{\tau}_{v\rightarrow f}^{s})^{2}]\rightarrow\delta\tilde{\tau}_{s}^{2},\\\ (\mu_{f\rightarrow v}^{s},u_{f})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}{\mathsf{N}}(0,\sigma_{s}^{2})\otimes\mu_{U},\qquad(\tilde{\tau}_{f\rightarrow v}^{s})^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\tilde{\tau}_{s}^{2},\\\ (\theta_{v},v_{v},a_{v\rightarrow f}^{s}/b_{v\rightarrow f}^{s},b_{v\rightarrow f}^{s})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Theta,V,\Theta+\tau_{s}G,1/\tau_{s}^{2}),\end{gathered}$ (48) where in the last line $\Theta\sim\mu_{\Theta}$, $G\sim{\mathsf{N}}(0,1)$ independent, and $\sigma_{s}^{2},\tau_{s}^{2}$ are defined in (5). By symmetry, the distribution of these quantities does not depend upon $v$ or $f$, so that the limits holds for all $v,f$ once we establish them for any $v,f$. We establish the limits inductively in $s$. Base case: $\mathbb{E}[(\mu_{v\rightarrow f}^{0})^{2}]\rightarrow\delta\sigma_{0}^{2}$ and $\mathbb{E}[(\tilde{\tau}_{v\rightarrow f}^{0})^{2}]\rightarrow\delta\tilde{\tau}_{0}^{2}$. Observe that $\mu_{v\rightarrow f}^{s}=\int\vartheta\mu_{\Theta|V}(v_{v},\mathrm{d}\vartheta)=\mathbb{E}_{\Theta,V}[\Theta|V=v_{v}]$. Because $v_{v}\sim\mu_{V}$, we have $\mathbb{E}[(\mu_{v\rightarrow f}^{1})^{2}]=\mathbb{E}_{\Theta,V}[\mathbb{E}_{\Theta,V}[\Theta|V]^{2}]=\mathbb{E}[\Theta^{2}]-\textsf{mmse}_{\Theta,V}(\infty)=\delta\sigma_{1}^{2}$. Similarly, $(\tilde{\tau}_{v\rightarrow f}^{1})^{2}=\operatorname{Var}_{\Theta,V}(\Theta|V=v_{v})$, so that $\mathbb{E}[(\tilde{\tau}_{v\rightarrow f}^{1})^{2}]=\textsf{mmse}_{\Theta,V}(\infty)=\delta\tilde{\tau}_{0}^{2}$. Inductive step 1: If $\mathbb{E}[(\mu_{v\rightarrow f}^{s})^{2}]\rightarrow\delta\sigma_{s}^{2}$, then $(\mu_{f\rightarrow v}^{s},u_{f})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}{\mathsf{N}}(0,\sigma_{s}^{2})\otimes\mu_{U}$. The quantity $\mu_{v^{\prime}\rightarrow f}^{s}$ is $\mathcal{T}_{v^{\prime}\rightarrow f}^{2s,0}$-measurable, whence it is independent of $x_{fv^{\prime}}$ and $u_{f}$. Moreover, $(\mu_{v^{\prime}\rightarrow f},x_{fv})$ are independent as we vary $v^{\prime}\in\partial f\setminus v$. Thus, $\mu_{f\rightarrow v}^{s}|\mathcal{T}_{f\rightarrow v}^{2s+1,1}\sim{\mathsf{N}}(0,\frac{1}{n}\sum_{v^{\prime}\in\partial f\setminus v}(\mu_{v^{\prime}\rightarrow f}^{s})^{2})$. Note that $\mathbb{E}[\frac{1}{n}\sum_{v^{\prime}\in\partial f\setminus v}(\mu_{v^{\prime}\rightarrow f}^{s})^{2}]=(p-1)\mathbb{E}[(\mu_{v\rightarrow f}^{s})^{2}]/n\rightarrow\sigma_{s}^{2}$ by the inductive hypothesis. Moreover, $\mu_{v\rightarrow f}^{s}$ has bounded fourth moments because it is bounded by $M$. By the weak law of large numbers, $\frac{1}{n}\sum_{v^{\prime}\in\partial f\setminus v}(\mu_{v^{\prime}\rightarrow f}^{s})^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\sigma_{s}^{2}$. We conclude by Slutsky’s theorem and independence that $(\mu_{f\rightarrow v}^{s},u_{f})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}{\mathsf{N}}(0,\sigma_{s}^{2})\otimes\mu_{U}$. Inductive step 2: If $\mathbb{E}[(\tilde{\tau}_{v\rightarrow f}^{s})^{2}]\rightarrow\delta\tilde{\tau}_{s}^{2}$, then $(\tilde{\tau}_{f\rightarrow v}^{s})^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\tilde{\tau}_{s}^{2}$. The quantity $\tilde{\tau}_{v^{\prime}\rightarrow f}^{s}$ is $\mathcal{T}_{v^{\prime}\rightarrow f}^{2s,0}$-measurable, whence it is independent of $x_{fv^{\prime}}$. Therefore, $\mathbb{E}[\sum_{v^{\prime}\in\partial f\setminus v}x_{fv^{\prime}}^{2}(\tilde{\tau}_{v^{\prime}\rightarrow f}^{s})^{2}]=(p-1)\mathbb{E}[(\tilde{\tau}_{v\rightarrow f}^{s})^{2}]/n\rightarrow\tilde{\tau}_{s}^{2}.$ Moreover, $(\tilde{\tau}_{v^{\prime}\rightarrow f},x_{fv})$ are mutually independent as we vary $v^{\prime}\in\partial f\setminus v$, and because $\tilde{\tau}_{v\rightarrow f}^{s}$ is bounded by $M$, the terms $nx_{fv^{\prime}}^{2}(\sigma_{v^{\prime}\rightarrow f}^{s})^{2}$ have bounded fourth moments. By the weak law of large numbers, $(\tilde{\tau}_{f\rightarrow v}^{s})^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\tilde{\tau}_{s}^{2}$. Inductive step 3: If $(\mu_{f\rightarrow v}^{s},u_{f},\tilde{\tau}_{f\rightarrow v}^{s})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}{\mathsf{N}}(0,\sigma_{s}^{2})\otimes\mu_{U}\otimes\delta_{\tilde{\tau}_{s}}$, then $(\theta_{v},v_{v},a_{v\rightarrow f}^{s+1}/b_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Theta,V,\Theta+\tau_{s+1}G,1/\tau_{s+1}^{2})$ where $G\sim{\mathsf{N}}(0,1)$ independent of $(\Theta,V)\sim\mu_{\Theta,V}$. For all $(f,v)\in\mathcal{E}$ and $s\geq 1$, define $p_{f\rightarrow v}^{s}(y;x)=\int p(y|x+\sum_{v^{\prime}\in\partial f\setminus v}x_{fv^{\prime}}\vartheta_{v^{\prime}},u_{f})\prod_{v^{\prime}\in\partial f\setminus v}m_{v^{\prime}\rightarrow f}^{s}(\vartheta_{v^{\prime}})\prod_{v^{\prime}\in\partial f\setminus v}\mu_{\Theta|V}(v_{v^{\prime}},\mathrm{d}\vartheta_{v^{\prime}}).$ More compactly, we may write $p_{f\rightarrow v}^{s}(y;x,u_{f})=\mathbb{E}_{\\{\Theta_{v^{\prime}}\\}}[p(y|x+\sum_{v^{\prime}\in\partial f\setminus v}x_{fv^{\prime}}\Theta_{v^{\prime}},u_{f})]$, where it is understood that the expectation is taken over $\Theta_{v^{\prime}}$ independent with densities $m_{v^{\prime}\rightarrow f}^{s}$ with respect to $\mu_{\Theta|V}(v_{v^{\prime}},\cdot)$. Note that for all $x$, we have $\int p_{f\rightarrow v}^{s}(y;x){\mathrm{d}}y=1$ everywhere. That is, $p_{f\rightarrow v}^{s}(\cdot;x)$ is a probability density with respect to Lebesgue measure. We will denote by $\dot{p}_{f\rightarrow v}^{s}(y;x)=\frac{\mathrm{d}\phantom{b}}{\mathrm{d}\xi}p_{f\rightarrow v}^{s}(y;x)\big{|}_{\xi=x}$, and likewise for higher derivatives. These derivatives exist and may be taken under the integral by R4. Define $a_{f\rightarrow v}^{s}(y)=\frac{\mathrm{d}\phantom{b}}{\mathrm{d}x}\log p_{f\rightarrow v}^{s}(y;x)\Big{|}_{x=0}\qquad\text{ and }\qquad b_{f\rightarrow v}^{s}(y)=-\frac{\mathrm{d}^{2}\phantom{b}}{\mathrm{d}x^{2}}\log p_{f\rightarrow v}^{s}(y;x)\Big{|}_{x=0}.$ For fixed $y$, the quantity $a_{f^{\prime}\rightarrow v}^{s}(y)$ is independent of $x_{f^{\prime}v}$, and $(a_{f^{\prime}\rightarrow v}^{s}(y),x_{f^{\prime}v})$ are mutually independent for $f^{\prime}\in\partial v\setminus f$. Observe that $\displaystyle a_{f\rightarrow v}^{s}=a_{f\rightarrow v}^{s}(y_{f})\qquad\text{and}\qquad a_{v\rightarrow f}^{s+1}=\sum_{f^{\prime}\in\partial v\setminus f}x_{f^{\prime}v}a_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}}),$ $\displaystyle b_{f\rightarrow v}^{s}=b_{f\rightarrow v}^{s}(y_{f})\qquad\text{and}\qquad b_{v\rightarrow f}^{s+1}=\sum_{f^{\prime}\in\partial v\setminus f}x_{f^{\prime}v}^{2}b_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}}).$ We will study the distributions of $a_{f\rightarrow v}^{s},a_{v\rightarrow f}^{s+1},b_{f\rightarrow v}^{s}$, and $b_{v\rightarrow f}^{s+1}$ under several measures, which we now introduce. Define $P_{v,\vartheta}$ to be the distribution of the regression model with $\theta_{v}$ forced to be $\theta$ and $v_{v}$ forced to be 0. That is, under $P_{v,\theta}$, we have $(\theta_{v^{\prime}},v_{v^{\prime}})\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\Theta,V}$ for $v^{\prime}\neq v$, $v_{v}=0$ and $\theta_{v}=\theta$, the features are distributed independently $x_{fv^{\prime}}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1/n)$ for all $f,v^{\prime}$, and the observations $y_{f}$ are drawn independently from $p(\cdot|\sum_{v^{\prime}\in\partial f}x_{fv^{\prime}}\theta_{v^{\prime}})$ for all $f$. We will consider the distribution of $a_{f\rightarrow v}^{s},a_{v\rightarrow f}^{s+1},b_{f\rightarrow v}^{s}$, and $b_{v\rightarrow f}^{s+1}$ under $P_{v,\theta}$ for $\theta\in[-M,M]$. We require the following lemmas, whose proofs are deferred to Section D.1.1. ###### Lemma 2. Under $P_{v,\theta}$ for any $\theta\in[-M,M]$, we have for all fixed $y$ that $\displaystyle p_{f\rightarrow v}^{s}(y;0)-\mathbb{E}_{G_{1}}[p(y|\mu_{f\rightarrow v}^{s}+\tilde{\tau}_{f\rightarrow v}^{s}G_{1},u_{f})]=o_{p}(1),$ $\displaystyle\dot{p}_{f\rightarrow v}^{s}(y;0)-\mathbb{E}_{G_{1}}[\dot{p}(y|\mu_{f\rightarrow v}^{s}+\tilde{\tau}_{f\rightarrow v}^{s}G_{1},u_{f})]=o_{p}(1),$ $\displaystyle\ddot{p}_{f\rightarrow v}^{s}(y;0)-\mathbb{E}_{G_{1}}[\ddot{p}(y|\mu_{f\rightarrow v}^{s}+\tilde{\tau}_{f\rightarrow v}^{s}G_{1},u_{f})]=o_{p}(1),$ where the expectation is over $G_{1}\sim{\mathsf{N}}(0,1)$. Further, for any $u$, the functions $(\mu,\tilde{\tau})\mapsto\mathbb{E}_{G_{1}}[p(y|\mu+\tilde{\tau}G_{1},u)]$, $(\mu,\tilde{\tau})\mapsto\mathbb{E}_{G_{1}}[\dot{p}(y|\mu+\tilde{\tau}G_{1},u)]$, and $(\mu,\tilde{\tau})\mapsto\mathbb{E}_{G_{1}}[\ddot{p}(y|\mu+\tilde{\tau}G_{1},u)]$ are continuous. ###### Lemma 3. Under $P_{v,\theta}$ for any $\theta\in[-M,M]$, we have for any fixed $s$ $\log\frac{m_{v\rightarrow f}^{s+1}(\vartheta)}{m_{v\rightarrow f}^{s+1}(0)}=\vartheta a_{v\rightarrow f}^{s+1}-\frac{1}{2}\vartheta^{2}b_{v\rightarrow f}^{s+1}+O_{p}(n^{-1/2}),$ where $O_{p}(n^{-1/2})$ has no $\vartheta$ dependence, and the statement holds for $\vartheta\in[-M,M]$. First we study the distribution of $a_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1}$ under $P_{v,0}$. Because $\mu_{f^{\prime}\rightarrow v}^{s},\tilde{\tau}_{f^{\prime}\rightarrow v}^{s}$ is independent of $\theta_{v},v_{v}$ for all $f^{\prime}\in\partial v$, its distribution is the same under $P_{v,\theta}$ for all $\theta\in[-M,M]$ and is equal to its distribution under the original model. Thus, the inductive hypothesis implies $(\mu_{f\rightarrow v}^{s},\tilde{\tau}_{f\rightarrow v}^{s})\xrightarrow[P_{v,0}]{\mathrm{d}}{\mathsf{N}}(0,\sigma_{s}^{2})\times\delta_{\tilde{\tau}_{s}}$. By Lemma 2, the inductive hypothesis, and Lemma 3, we have for fixed $y$ $\begin{pmatrix}\mathbb{E}_{G_{1}}[p(y|\mu_{f\rightarrow v}^{s}+\tilde{\tau}_{f\rightarrow v}^{s}G_{1},u_{f})]\\\ \mathbb{E}_{G_{1}}[\dot{p}(y|\mu_{f\rightarrow v}^{s}+\tilde{\tau}_{f\rightarrow v}^{s}G_{1},u_{f})]\\\ \mathbb{E}_{G_{1}}[\ddot{p}(y|\mu_{f\rightarrow v}^{s}+\tilde{\tau}_{f\rightarrow v}^{s}G_{1},u_{f})]\end{pmatrix}\xrightarrow[P_{v,0}]{\mathrm{d}}\begin{pmatrix}\mathbb{E}_{G_{1}}[p(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U]\\\ \mathbb{E}_{G_{1}}[\dot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]\\\ \mathbb{E}_{G_{1}}[\ddot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]\end{pmatrix},$ where and $G_{0},G_{1}\sim{\mathsf{N}}(0,1)$ and $U\sim\mu_{U}$ independent. Applying Lemma 2 and Slutsky’s Theorem, we have that $\begin{pmatrix}p_{f\rightarrow v}^{s}(y;0)\\\ \dot{p}_{f\rightarrow v}^{s}(y;0)\\\ \ddot{p}_{f\rightarrow v}^{s}(y;0)\end{pmatrix}\xrightarrow[P_{v,0}]{\mathrm{d}}\begin{pmatrix}\mathbb{E}_{G_{1}}[p(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]\\\ \mathbb{E}_{G_{1}}[\dot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]\\\ \mathbb{E}_{G_{1}}[\ddot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]\end{pmatrix}.$ By the Continuous Mapping Theorem, $\displaystyle p_{f\rightarrow v}^{s}(y;0)\xrightarrow[P_{v,0}]{\mathrm{d}}\mathbb{E}_{G_{1}}[p(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)],$ $\displaystyle a_{f\rightarrow v}^{s}(y)\xrightarrow[P_{v,0}]{\mathrm{d}}\frac{\mathrm{d}\phantom{b}}{\mathrm{d}x}\log\mathbb{E}_{G_{1}}[\dot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]\Big{|}_{x=0},$ $\displaystyle b_{f\rightarrow v}^{s}(y)\xrightarrow[P_{v,0}]{\mathrm{d}}-\frac{\mathrm{d}^{2}\phantom{b}}{\mathrm{d}x^{2}}\log\mathbb{E}_{G_{1}}[\dot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]\Big{|}_{x=0}.$ Because the quantity $p(y|x)$ is bounded (assumption R4) and the quantities $a_{f\rightarrow v}^{s}(y),b_{f\rightarrow v}^{s}(y)$ are bounded by (46), we have $\displaystyle\mathbb{E}_{P_{v,0}}[p_{f\rightarrow v}^{s}(y|0)]\rightarrow\mathbb{E}_{G_{0},G_{1},U}[p(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)],$ $\displaystyle\mathbb{E}_{P_{v,0}}[a_{f\rightarrow v}^{s}(y)^{2}]\rightarrow\mathbb{E}_{G_{0},U}\left[\left(\frac{\mathrm{d}\phantom{b}}{\mathrm{d}x}\log\mathbb{E}_{G_{1}}[\dot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]\Big{|}_{x=0}\right)^{2}\right],$ $\displaystyle\mathbb{E}_{P_{v,0}}[b_{f\rightarrow v}^{s}]\rightarrow-\mathbb{E}_{G_{0},U}\left[\frac{\mathrm{d}^{2}\phantom{b}}{\mathrm{d}x^{2}}\log\mathbb{E}_{G_{1}}[\dot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]\Big{|}_{x=0}\right].$ Under $P_{v,0}$, we have for all $f^{\prime}\in\partial v$ that the random variable $y_{f^{\prime}}$ is independent of $x_{f^{\prime}v}$. Thus, conditional on $\mathcal{T}_{v\rightarrow f}^{2s+2,1}$, the random variable $\sum_{f^{\prime}\in\partial v\setminus f}x_{f^{\prime}v}a_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}})$ is normally distributed. Specifically, $\sum_{f^{\prime}\in\partial v\setminus f}x_{f^{\prime}v}a_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}})\bigm{|}\mathcal{T}_{v\rightarrow f}^{2s+2,1}\underset{P_{v,0}}{\sim}{\mathsf{N}}\left(0,\frac{1}{n}\sum_{f^{\prime}\in\partial v\setminus f}(a_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}}))^{2}\right).$ Because $(a_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}}))^{2}$ is bounded by (46), if we show $\mathbb{E}_{P_{v,0}}[(a_{f\rightarrow v}^{s}(y_{f}))^{2}]\rightarrow 1/\tau_{s+1}^{2}$, then the weak law of large numbers and Slutsky’s theorem will imply that $a_{v\rightarrow f}^{s+1}=\sum_{f^{\prime}\in\partial v\setminus f}x_{f^{\prime}v}a_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}})\xrightarrow[P_{v,0}]{\mathrm{d}}{\mathsf{N}}\left(0,1/\tau_{s+1}^{2}\right).$ (49) We compute $\displaystyle\mathbb{E}_{P_{v,0}}[(a_{f\rightarrow v}^{s}(y_{f}))^{2}]$ $\displaystyle=\mathbb{E}_{P_{v,0}}[\mathbb{E}_{P_{v,0}}[(a_{f\rightarrow v}^{s}(y_{f}))^{2}|\sigma(\mathcal{T}_{f\rightarrow v}^{2s+1,1},(x_{fv^{\prime}})_{v^{\prime}\in\partial f\setminus v}),u_{f}]]$ $\displaystyle=\mathbb{E}_{P_{v,0}}\left[\int a_{f\rightarrow v}^{s}(y)^{2}p_{f\rightarrow v}^{s}(y;0){\mathrm{d}}y\right]$ $\displaystyle=\int\mathbb{E}_{P_{v,0}}\left[a_{f\rightarrow v}^{s}(y)^{2}p_{f\rightarrow v}^{s}(y;0)\right]{\mathrm{d}}y.$ where the second equation holds because under $P_{v,0}$ we have $y_{f}\mid\sigma(\mathcal{T}_{f\rightarrow v}^{2s+1,1},(x_{fv^{\prime}})_{v^{\prime}\in\partial f\setminus v},u_{f})$ has density $p_{f\rightarrow v}^{s}(\cdot;0)$ with respect to Lebesgue measure, and the last equation follows by Fubini’s theorem (using the non-negativity of the integrand). Because $a_{f\rightarrow v}^{s}(y)^{2}\leq(q_{1}^{\prime})^{2}$ and $\mathbb{E}_{P_{v,0}}[p_{f\rightarrow v}^{s}(y;0)]$ are probability densities which converge pointwise to $\mathbb{E}_{G_{1}}[p(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1})]$, we conclude that $\displaystyle\mathbb{E}_{P_{v,0}}[(a_{f\rightarrow v}^{s}(y_{f}))^{2}]$ $\displaystyle\rightarrow\int\mathbb{E}_{G_{0},U}\left[\frac{\mathbb{E}_{G_{1}}[\dot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]^{2}}{\mathbb{E}_{G_{1}}[p(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]}\right]{\mathrm{d}}y$ $\displaystyle=\mathbb{E}_{G_{0},U}\left[\int\frac{\mathbb{E}_{G_{1}}[\dot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]^{2}}{\mathbb{E}_{G_{1}}[p(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]}{\mathrm{d}}y\right]=\frac{1}{\tau_{s+1}^{2}},$ where we have used the alternative characterization of the recursion (5) from Lemma 4. We conclude (49). Now we compute the asymptotic behavior of $b_{v\rightarrow f}^{s+1}$ under $P_{v,0}$. Under $P_{v,0}$, $x_{f^{\prime}v}$ is independent of $y_{f^{\prime}}$, and $(x_{f^{\prime}v},b_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}}))$ are mutually independent for $f^{\prime}\in\partial v\setminus f$. Thus, $\mathbb{E}_{P_{v,0}}[x_{f^{\prime}v}^{2}b_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}})]=\mathbb{E}_{P_{v,0}}[b_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}})]/n$. Because $b_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}})$ is bounded by (46), if we can show that $\mathbb{E}_{P_{v,0}}[b_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}})]\rightarrow 1/\tau_{s+1}^{2}$, then $b_{v\rightarrow f}^{s+1}\xrightarrow[P_{v,0}]{\mathrm{p}}1/\tau_{s+1}^{2}$ will follow by the weak law of large numbers. We compute $\displaystyle\mathbb{E}_{P_{v,0}}[b_{f\rightarrow v}^{s}(y_{f})]$ $\displaystyle=\mathbb{E}_{P_{v,0}}[\mathbb{E}_{P_{v,0}}[b_{f\rightarrow v}^{s}(y_{f})|\sigma(\mathcal{T}_{f\rightarrow v}^{2s+1,1},(x_{fv^{\prime}})_{v^{\prime}\in\partial f\setminus v},u_{f})]]$ $\displaystyle=\mathbb{E}_{P_{v,0}}\left[\int b_{f\rightarrow v}^{s}(y)p_{f\rightarrow v}^{s}(y;0){\mathrm{d}}y\right]$ $\displaystyle=\int\mathbb{E}_{P_{v,0}}\left[b_{f\rightarrow v}^{s}(y)p_{f\rightarrow v}^{s}(y;0)\right]{\mathrm{d}}y$ where the last equation follows by Fubini’s theorem (using that the integrand is bounded by the integrable function $q_{2}\mathbb{E}_{P_{v,0}}[p_{f\rightarrow v}^{s}(y;0)]$). The integrands converge point-wise, so that $\displaystyle\mathbb{E}_{P_{v,0}}$ $\displaystyle[b_{f\rightarrow v}^{s}(y_{f})]$ $\displaystyle\rightarrow\mathbb{E}_{G_{0},U}\left[\int\frac{\mathbb{E}_{G_{1}}[\dot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]^{2}}{\mathbb{E}_{G_{1}}[p(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]}{\mathrm{d}}y\right]-\int\mathbb{E}_{G_{0},G_{1},U}[\ddot{p}(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]{\mathrm{d}}y$ $\displaystyle=\frac{1}{\tau_{s+1}^{2}},$ where we have concluded that the second integral is zero because $x\mapsto\mathbb{E}_{G_{0},G_{1},U}[p(y|\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},U)]$ parameterizes a statistical model whose scores up to order 3 are bounded by (46). Thus, we conclude that $b_{v\rightarrow f}^{s+1}\xrightarrow[P_{v,0}]{\mathrm{p}}1/\tau_{s+1}^{2}$. Now we compute the asymptotic distribution of $(a_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})$ under $P_{v,\theta}$ for any $\theta\in[-M,M]$. The log-likelihood ratio between $P_{v,\theta}$ and $P_{v,0}$ is $\displaystyle\sum_{f^{\prime}\in\partial v}\log\frac{p_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}}|x_{f^{\prime}v}\theta)}{p_{f^{\prime}\rightarrow v}^{s}(y_{f^{\prime}}|0)}$ $\displaystyle=\log\frac{m_{v\rightarrow f}^{s+1}(\theta)}{m_{v\rightarrow f}^{s+1}(0)}+\log\frac{p_{f\rightarrow v}^{s}(y_{f}|x_{fv}\theta)}{p_{f\rightarrow v}^{s}(y_{f}|0)}$ $\displaystyle=\theta a_{v\rightarrow f}^{s+1}-\frac{1}{2}\theta^{2}b_{v\rightarrow f}^{s+1}+O_{p}(n^{-1/2}),$ where we have used Lemma 3 and that $\left|\log\frac{p_{f\rightarrow v}^{s}(y_{f}|x_{fv}\theta)}{p_{f\rightarrow v}^{s}(y_{f}|0)}\right|\leq Mq_{1}|x_{fv}|=O_{p}(n^{-1/2})$. Thus, $\left(a_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1},\log\frac{P_{v,\theta}}{P_{v,0}}\right)\xrightarrow[P_{v,0}]{\mathrm{p}}\left(Z,\frac{1}{\tau_{s+1}^{2}},\theta Z-\frac{1}{2}\frac{\theta^{2}}{\tau_{s+1}^{2}}\right),$ where $Z\sim{\mathsf{N}}(0,1/\tau_{s+1}^{2})$. By Le Cam’s third lemma [Vaa98, Example 6.7], we have $(a_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})\xrightarrow[P_{v,\theta}]{\mathrm{d}}\left(Z^{\prime},\frac{1}{\tau_{s+1}^{2}}\right).$ where $Z^{\prime}\sim{\mathsf{N}}(\theta/\tau_{s+1}^{2},1/\tau_{s+1}^{2})$. By the Continuous Mapping Theorem [Vaa98, Theorem 2.3], we conclude $(a_{v\rightarrow f}^{s+1}/b_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})\xrightarrow[P_{v,\theta}]{\mathrm{d}}{\mathsf{N}}(\theta,\tau_{s+1}^{2})\otimes\delta_{1/\tau_{s+1}^{2}}$. Denote by $P^{*}$ the distribution of the the original model. Consider a continuous bounded function $f:(\theta,\nu,\chi,b)\mapsto\mathbb{R}$, and define $\hat{f}_{n}(\theta,\nu)=\mathbb{E}_{P_{v,\theta}}[f(\theta,\nu,a_{v\rightarrow f}^{s+1}/b_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})]$. Under $P^{*}$, the random variables $a_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1}$ are functions are $\theta_{v}$ and random vectors $\boldsymbol{D}:=\mathcal{T}_{v,2t}\setminus\\{\theta_{v},v_{v}\\}$, which is independent of $\theta_{v},v_{v}$. In particular, we may write $\mathbb{E}_{P^{*}}[f(\theta_{v},v_{v},a_{v\rightarrow f}^{s+1}/b_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})]=\mathbb{E}_{P^{*}}[f(\theta_{v},v_{v},\chi(\theta_{v},\boldsymbol{D}),B(\theta_{v},\boldsymbol{D}))],$ for some measurable functions $\chi,B$. We see that $\mathbb{E}_{P^{*}}[f(\theta_{v},v_{v},a_{v\rightarrow f}^{s+1}/b_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})\mid\theta_{v},v_{v}]=\hat{f}_{n}(\theta_{v},v_{v})$ where $\hat{f}_{n}(\theta,\nu)=\mathbb{E}_{\boldsymbol{D}}[f(\theta,\nu,\chi(\theta,\boldsymbol{D}),B(\theta,\boldsymbol{D}))],$ with $\boldsymbol{D}$ distributed as it is under $P^{*}$ (see e.g., [Dur10, Example 5.1.5]). Because $\boldsymbol{D}$ has the same distribution on $P^{*}$ as under $P_{v,\theta}$, we see that in fact $\hat{f}_{n}(\theta,\nu)=\mathbb{E}_{P_{v,\theta}}[f(\theta,\nu,a_{v\rightarrow f}^{s+1}/b_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})]$. Because $(a_{v\rightarrow f}^{s+1}/b_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})\xrightarrow[P_{v,\theta}]{\mathrm{d}}{\mathsf{N}}(\theta,\tau_{s+1}^{2})\otimes\delta_{1/\tau_{s+1}^{2}}$, we conclude that $\hat{f}_{n}(\theta,\nu)\rightarrow\mathbb{E}_{G}[f(\theta,\nu,\theta+\tau_{s+1}G,\tau_{s+1}^{-2})]$ for all $\theta,\nu$. By bounded convergence and the tower property, $\mathbb{E}_{\Theta,V}[\hat{f}_{n}(\Theta,V)]\rightarrow\mathbb{E}_{\Theta,V,G}[f(\theta,\nu,\theta+\tau_{s+1}G,\tau_{s+1}^{-2})]$ where $(\Theta,V)\sim\mu_{\Theta,V}$ independent of $G\sim{\mathsf{N}}(0,1)$. Also by the tower property, we have $\mathbb{E}_{\Theta,V}[\hat{f}_{n}(\Theta,V)]=\mathbb{E}_{P^{*}}[f(\theta_{v},v_{v},\chi(\theta_{v},\boldsymbol{D}),B(\theta_{v},\boldsymbol{D}))]=\mathbb{E}_{P^{*}}[f(\theta_{v},v_{v},a_{v\rightarrow f}^{s+1}/b_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})].$ We conclude $\mathbb{E}_{P^{*}}[f(\theta_{v},v_{v},a_{v\rightarrow f}^{s+1}/b_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})]\rightarrow\mathbb{E}_{\Theta,V,G}[f(\Theta,V,\Theta+\tau_{s+1}G,\tau_{s+1}^{-2})].$ Thus, we conclude that $(\theta_{v},v_{v},a_{v\rightarrow f}^{s+1}/b_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})\xrightarrow[P^{*}]{\mathrm{d}}(\Theta,V,\Theta+\tau_{s+1}G,1/\tau_{s+1}^{2})$, as desired. Inductive step 4: If $(\theta_{v},v_{v},a_{v\rightarrow f}^{s+1}/b_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Theta,V,\Theta+\tau_{s+1}G,1/\tau_{s+1}^{2})$ where $G\sim{\mathsf{N}}(0,1)$ independent of $(\Theta,V)\sim\mu_{\Theta,V}$, then $\mathbb{E}[(\mu_{v\rightarrow f}^{s})^{2}]\rightarrow\delta\sigma_{s}^{2}$ and $\mathbb{E}[(\tilde{\tau}_{v\rightarrow f}^{s})^{2}]\rightarrow\textsf{mmse}_{\Theta,V}(\tau_{s}^{2})$. Define $\epsilon_{v\rightarrow f}^{s}=\sup_{\vartheta\in[-M,M]}\left|\log\frac{m_{v\rightarrow f}^{s}(\vartheta)}{m_{v\rightarrow f}^{s}(0)}-\left(\vartheta a_{v\rightarrow f}^{s}-\frac{1}{2}\vartheta^{2}b_{v\rightarrow f}^{s}\right)\right|,$ where because all the terms are continuous in $\vartheta$, the random variable $\epsilon_{v\rightarrow f}^{s}$ is measurable and finite. We have that $\mu_{v\rightarrow f}^{s}\geq\frac{\int\vartheta\exp(\vartheta a_{v\rightarrow f}^{s}-\vartheta^{2}b_{v\rightarrow f}^{s}/2-\epsilon_{v\rightarrow f}^{s})\mu_{\Theta}(v_{v},\mathrm{d}\vartheta)}{\int\exp(\vartheta a_{v\rightarrow f}^{s}-\vartheta^{2}b_{v\rightarrow f}^{s}/2+\epsilon_{v\rightarrow f}^{s})\mu_{\Theta}(v_{v},\mathrm{d}\vartheta)}\geq e^{-2\epsilon_{v\rightarrow f}^{s}}\eta_{\Theta,V}(a_{v\rightarrow f}^{s}/b_{v\rightarrow f}^{s},v_{v};1/b_{v\rightarrow f}^{s})$ where $\eta_{\Theta,V}(y,v;\tau^{2})=\mathbb{E}_{\Theta,V,G}[\Theta|\Theta+\tau G=y;V=v]$ where $(\Theta,V)\sim\mu_{\Theta,V}$, $G\sim{\mathsf{N}}(0,1)$ independent. Likewise, $\mu_{v\rightarrow f}^{s}\leq e^{2\epsilon_{v\rightarrow f}^{s}}\eta_{\Theta,V}(a_{v\rightarrow f}^{s}/b_{v\rightarrow f}^{s},v_{v};1/b_{v\rightarrow f}^{s}).$ Because $\eta_{\Theta,V}$ takes values in the bounded interval $[-M,M]$ and $\epsilon_{v\rightarrow f}=o_{p}(1)$ by Lemma 3, we conclude that $\mu_{v\rightarrow f}^{s}=\eta_{\Theta,V}(a_{v\rightarrow f}^{s}/b_{v\rightarrow f}^{s},v_{v};1/b_{v\rightarrow f}^{s})+o_{p}(1).$ For a fixed $v_{v}$, the Bayes estimator $\eta_{\Theta,V}$ is continuous in the observation and the noise variance on $\mathbb{R}\times\mathbb{R}_{>0}$.888This commonly known fact holds, for example, by [LR05, Theorem 2.7.1] because the posterior mean can be viewed as the mean in an exponential family paramterized by the observation and noise variance. Thus, by the inductive hypothesis and the fact that $v_{v}\sim\mu_{V}$ for all $n$, we have $\mathbb{E}[\eta_{\Theta,V}(a_{v\rightarrow f}^{s}/b_{v\rightarrow f}^{s},v_{v};1/b_{v\rightarrow f}^{s})^{2}]=\mathbb{E}[\eta_{\Theta,V}(a_{v\rightarrow f}^{s}/b_{v\rightarrow f}^{s},v_{v};1/b_{v\rightarrow f}^{s})^{2}\vee M^{2}]\rightarrow\mathbb{E}_{\Theta,V,G}[\eta_{\Theta,V}(\Theta+\tau_{s}G,V;\tau_{s}^{2})]=\mathbb{E}[\Theta^{2}]-\textsf{mmse}_{\Theta,V}(\tau_{s}^{2})=\delta\sigma_{s}^{2}$ by Lemma 3. By the previous display and the boundedness of $\mu_{v\rightarrow f}^{s}$ and $\eta_{\Theta,V}$, we conclude $\mathbb{E}[(\mu_{v\rightarrow f}^{s})^{2}]\rightarrow\delta\sigma_{s}^{2}$, as desired. Similarly, we may derive that $\displaystyle e^{-2\epsilon_{v\rightarrow f}^{s}}s_{\Theta,V}^{2}(a_{v\rightarrow f}^{s}/b_{v\rightarrow f}^{s},v_{v};1/b_{v\rightarrow f}^{s})$ $\displaystyle\leq\int\vartheta^{2}m_{v\rightarrow f}^{s}(\vartheta)\mu_{\Theta}(\mathrm{d}\vartheta)$ $\displaystyle\leq e^{2\epsilon_{v\rightarrow f}^{s}}s_{\Theta,V}^{2}(a_{v\rightarrow f}^{s}/b_{v\rightarrow f}^{s},v_{v};1/b_{v\rightarrow f}^{s}),$ where $s_{\Theta,V}^{2}(y,v;\tau^{2})=\mathbb{E}_{\Theta,V,G}[\Theta^{2}|\Theta+\tau G=y,V=v]$ where $(\Theta,V)\sim\mu_{\Theta,V}$, $G\sim{\mathsf{N}}(0,1)$ independent. For fixed $v_{v}$, the the posterior second moment is continuous in the observation and the noise variance. Further, it is bounded by $M^{2}$. Thus, by exactly the same argument as in the previous paragraph, we have that $\mathbb{E}[(\tilde{\tau}_{v\rightarrow f}^{s})^{2}]\rightarrow\mathbb{E}_{\Theta,V,G}[s_{\Theta,V}^{2}(\Theta+\tau_{s}G,V;\tau_{s}^{2})-\eta_{\Theta,V}(\Theta+\tau_{s}G,V;\tau_{s}^{2})^{2}]=\textsf{mmse}_{\Theta,V}(\tau_{s}^{2})$, as desired. The inductive argument is complete, and (48) is established. To complete the proof of Lemma 1, first observe by (47) that we may express $\log p_{v}(\vartheta|\mathcal{T}_{v,2t})$ as, up to a constant, $\log\frac{m_{v\rightarrow f}^{t}(\vartheta)}{m_{v\rightarrow f}^{t}(0)}+\log\frac{m_{f\rightarrow v}^{t-1}(\vartheta)}{m_{f\rightarrow v}^{t-1}(0)}$. Note that $\left|\log\frac{m_{f\rightarrow v}^{t-1}(\vartheta_{v})}{m_{f\rightarrow v}^{t-1}(0)}\right|\leq M|x_{fv}|\sup_{x\in\mathbb{R}}\left|\frac{\dot{p}_{f\rightarrow v}^{t-1}(y_{f};x)}{p_{f\rightarrow v}^{t-1}(y_{f};x)}\right|\leq Mq_{1}|x_{fv}|=o_{p}(1).$ By Lemma 3, we have that, up to a constant, $\log\frac{m_{v\rightarrow f}^{t}(\vartheta)}{m_{v\rightarrow f}^{t}(0)}=-\frac{1}{2}b_{v\rightarrow f}^{s}\left(a_{v\rightarrow f}^{t}/b_{v\rightarrow f}^{t}-\vartheta\right)^{2}+o_{p}(1)$. The lemma follows from (48). ∎ We complete the proof of Lemma 3 for the high-dimensional regression model. Consider any estimator $\hat{\theta}:\mathcal{T}_{v,2t}\mapsto[-M,M]$ on the computation tree. We compute $\displaystyle\mathbb{E}[\ell(\theta_{v},\hat{\theta}(\mathcal{T}_{v,2t}))]=\mathbb{E}[\mathbb{E}[\ell(\theta_{v},\hat{\theta}(\mathcal{T}_{v,2t}))|\mathcal{T}_{v,2t}]]$ $\displaystyle\qquad=\mathbb{E}\left[\int\ell(\vartheta,\hat{\theta}(\mathcal{T}_{v,2t}))\frac{1}{Z(\mathcal{T}_{v,2t})}\exp\left(-\frac{1}{2\tau_{v,t}^{2}}(\chi_{v,t}-\vartheta)^{2}+o_{p}(1)\right)\mu_{\Theta|V}(v_{v},\mathrm{d}\vartheta)\right]$ $\displaystyle\qquad\geq\mathbb{E}\left[\exp(-2\epsilon_{v})\int\ell(\vartheta,\hat{\theta}(\mathcal{T}_{v,2t}))\frac{1}{Z(\chi_{v,t},\tau_{v,t},v_{v})}\exp\left(-\frac{1}{2\tau_{v,t}^{2}}(\chi_{v,t}-\vartheta)^{2}\right)\mu_{\Theta|V}(v_{v},\mathrm{d}\vartheta)\right]$ $\displaystyle\qquad\geq\mathbb{E}\left[\exp(-2\epsilon_{v})R(\chi_{v,2t},\tau_{v,2t},v_{v})\right],$ where $Z(\mathcal{T}_{v,2t})=\int\exp\left(-\frac{1}{2\tau_{v,t}^{2}}(\chi_{v,t}-\vartheta)^{2}+o_{p}(1)\right)\mu_{\Theta|V}(v_{v},\mathrm{d}\vartheta)$, $R(\chi,\tau,v):=\inf_{d\in\mathbb{R}}\int\frac{1}{Z}\ell(\vartheta,d)e^{-\frac{1}{2\tau^{2}}(\chi-\vartheta)^{2}}\mu_{\Theta|V}(v,{\mathrm{d}}\vartheta)\,,$ and $\epsilon_{v}=\sup_{\vartheta\in[-M,M]}\left|\log\frac{p(\vartheta|\mathcal{T}_{v,2t})}{p(0|\mathcal{T}_{v,2t})}+\vartheta\chi_{v,t}/\tau_{v,t}^{2}-\vartheta^{2}/(2\tau_{v,t}^{2})\right|.$ Because $\Theta$ is bounded support, by Lemma 5(b), $R(\chi,\tau,v)$ is continuous in $(\chi,\tau)$ on $\mathbb{R}\times\mathbb{R}_{>0}$. By Lemma 1, $\epsilon_{v}=o_{p}(1)$. The quantity on the right-hand side does not depend on $\hat{\theta}$, so provides a uniform lower bound over the performance of any estimator. Because $(v_{v},\chi_{v,2t},\tau_{v,2t},\epsilon_{v})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(V,\Theta+\tau_{t}G,\tau_{t},0)$, $v_{v}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}V$ for all $n$, and $\tau_{t}>0$, we have $\mathbb{E}\left[\exp(-2\epsilon_{v})R(\chi_{v,2t},\tau_{v,2t},v_{v})\right]\rightarrow\mathbb{E}[R(\Theta+\tau_{t}G,\tau_{t},V)]=\inf_{\hat{\theta}(\cdot)}\mathbb{E}[\ell(\Theta,\hat{\theta}(\Theta+\tau_{t}G,V))]$, where the convergence holds by Lemma 3 and the equality holds by Lemma 5(a). Thus, $\liminf_{n\rightarrow\infty}\inf_{\hat{\theta}(\cdot)}\mathbb{E}[\ell(\theta_{v},\hat{\theta}(\mathcal{T}_{v,2t}))]\geq\inf_{\hat{\theta}(\cdot)}\mathbb{E}[\ell(\Theta,\hat{\theta}(\Theta+\tau_{t}G))].$ The proof of Lemma 3 in the high-dimensional regression model is complete. #### D.1.1 Technical tools ###### Lemma 2. By Lindeberg’s principle (see, e.g., [Cha06]) and using that $\mu_{\Theta}$ is supported on $[-M,M]$, we have $\displaystyle|p_{f\rightarrow v}^{s}(y;0)-\mathbb{E}_{G_{1}}[p(y|\mu_{f\rightarrow v}^{s}+\tilde{\tau}_{f\rightarrow v}^{s}G_{1},u_{f})]|\leq\frac{M^{3}\sup_{x\in\mathbb{R}}|\partial_{x}^{3}p(y|x,u_{f})|}{3}\sum_{v^{\prime}\in\partial f\setminus v}|x_{fv^{\prime}}|^{3},$ $\displaystyle|\dot{p}_{f\rightarrow v}^{s}(y;0)-\mathbb{E}_{G_{1}}[\dot{p}(y|\mu_{f\rightarrow v}^{s}+\tilde{\tau}_{f\rightarrow v}^{s}G_{1},u_{f})]|\leq\frac{M^{3}\sup_{x\in\mathbb{R}}|\partial_{x}^{4}p(y|x,u_{f})|}{3}\sum_{v^{\prime}\in\partial f\setminus v}|x_{fv^{\prime}}|^{3},$ $\displaystyle|\ddot{p}_{f\rightarrow v}^{s}(y;0)-\mathbb{E}_{G_{1}}[\ddot{p}(y|\mu_{f\rightarrow v}^{s}+\tilde{\tau}_{f\rightarrow v}^{s}G_{1},u_{f})]|\leq\frac{M^{3}\sup_{x\in\mathbb{R}}|\partial_{x}^{5}p(y|x,u_{f})|}{3}\sum_{v^{\prime}\in\partial f\setminus v}|x_{fv^{\prime}}|^{3}.$ Using the $\sup_{x\in\mathbb{R}}|\partial_{x}^{k}p(y|x,u)|\leq q_{k}^{\prime}\sup_{x\in\mathbb{R}}|p(y|x,u)|<\infty$ for $k=3,4,5$ by R4, we have that for fixed $y$ the expectations on the right-hand side go to 0 as $n\rightarrow\infty$, whence the required expessions are $o_{p}(1)$. Further, $|\mathbb{E}_{G_{1}}[p(y|\mu+\tilde{\tau}G_{1},u)]-\mathbb{E}_{G_{1}}[p(y|\mu^{\prime}+\tilde{\tau}^{\prime}G_{1},u)]|\leq(|\mu-\mu^{\prime}|+|\tilde{\tau}-\tilde{\tau}^{\prime}|\sqrt{2/\pi})\sup_{x\in\mathbb{R}}|\dot{p}(y|x,u)|$, whence $\mathbb{E}_{G_{1}}[p(y|\mu+\tilde{\tau}G_{1},u)]$ is continuous in $(\mu,\tilde{\tau})$ by R4. The remaining continuity results follow similarly. ∎ ###### Lemma 3. Fix any $\vartheta\in[-M,M]$. By Taylor’s theorem, there exist $\vartheta_{i}\in[-M,M]$ (in fact, between $0$ and $\vartheta$) such that $\displaystyle\log$ $\displaystyle\frac{m_{v\rightarrow f}^{s+1}(\vartheta)}{m_{v\rightarrow f}^{s+1}(0)}=\sum_{f^{\prime}\in\partial v\setminus f}\log\frac{m_{f^{\prime}\rightarrow v}^{s}(\vartheta)}{m_{f^{\prime}\rightarrow v}^{s}(0)}$ $\displaystyle=\vartheta a_{v\rightarrow f}^{s+1}-\frac{1}{2}\vartheta^{2}b_{v\rightarrow f}^{s+1}+\frac{1}{6}\vartheta^{3}\sum_{f^{\prime}\in\partial v\setminus f}\left(\frac{\mathrm{d}^{3}}{\mathrm{d}\vartheta^{3}}\log\mathbb{E}_{\hat{G}_{f^{\prime}}}[p(y_{f^{\prime}}|x_{f^{\prime}v}\vartheta+\hat{G}_{f^{\prime}},u_{f^{\prime}})]\bigg{|}_{\vartheta=\vartheta_{i}}\right).$ where it is understood that expectation is taken with respect to $\hat{G}_{f^{\prime}}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\sum_{v^{\prime}\in\partial f^{\prime}\setminus v}x_{f^{\prime}v^{\prime}}\Theta_{v^{\prime}\rightarrow f^{\prime}}$ where $x_{f^{\prime}v^{\prime}}$ is considered fixed and $\Theta_{v^{\prime}\rightarrow f^{\prime}}$ are drawn independently with densities $m_{v^{\prime}\rightarrow f^{\prime}}^{s}$ with respect to $\mu_{\Theta|V}(v_{v^{\prime}},\cdot)$. We bound the sum using assumption R4: $\displaystyle\left|\sum_{f^{\prime}\in\partial v\setminus f}\left(\frac{\mathrm{d}^{3}}{\mathrm{d}\vartheta^{3}}\log\mathbb{E}_{\hat{G}_{f^{\prime}}}[p(y_{f}|x_{fv}\vartheta+\hat{G}_{f^{\prime}},u_{f^{\prime}})]\bigg{|}_{\vartheta=\vartheta_{i}}\right)\right|$ $\displaystyle\leq q_{3}\sum_{f^{\prime}\in\partial v\setminus f}|x_{f^{\prime}v}|^{3}=O_{p}(n^{-1/2}).$ The proof is complete. ∎ ### D.2 Information-theoretic lower bound in the low-rank matrix estimation model In this section, we prove Lemma 3 in the low-rank matrix estimation model. Recall that conditions on the conditional density in assumption R4 are equivalent positivity, boundedness, and the existence finite, non-negative constants $q_{k}^{\prime}$ such that $\frac{|\partial_{x}^{k}p(y|x)|}{p(y|x)}\leq q_{k}^{\prime}$ for $1\leq k\leq 5$. In particular, we have (46) for any random variable $A$. Denote the regular conditional probability of $\Theta$ conditional on $V$ for the measure $\mu_{\Theta,V}$ by $\mu_{\Theta|V}:\mathbb{R}\times\mathcal{B}\rightarrow[0,1]$, where $\mathcal{B}$ denotes the Borel $\sigma$-algebra on $\mathbb{R}$, similarly for $\mu_{\Lambda|U}$. The posterior density of $\theta_{v}$ given $\mathcal{T}_{v,2t-1}$ has density respect to $\mu_{\Theta|V}(v_{v},\cdot)$ given by $p_{v}(\vartheta_{v}|\mathcal{T}_{v,2t-1})\propto\int\prod\exp\left(-\frac{n}{2}(x_{f^{\prime}v^{\prime}}-\frac{1}{n}\ell_{f^{\prime}}\vartheta_{v^{\prime}})^{2}\right)\prod\mu_{\Lambda}(u_{f},\mathrm{d}\ell_{f})\prod\mu_{\Theta}(v_{v^{\prime}},\mathrm{d}\vartheta_{v^{\prime}}),$ where the produces are over $(f^{\prime},v^{\prime})\in\mathcal{E}_{v,2t-1}$, $f\in\mathcal{F}_{v,2t-1}$, and $v^{\prime}\in\mathcal{V}_{v,2t-1}$, respectively. Asymptotically, the posterior behaves like that produced by a Gaussian observation of $\theta_{v}$ with variance $\tau_{t}^{2}$. ###### Lemma 4. In the low-rank matrix estimation model, there exist $\mathcal{T}_{v,2t-1}$-measurable random variables $q_{v,t},\chi_{v,t}$ such that for fixed $t\geq 1$ $p_{v}(\vartheta|\mathcal{T}_{v,2t-1})\propto\exp\left(-\frac{1}{2}(\chi_{v,t}-q_{v,t}^{1/2}\vartheta)^{2}+o_{p}(1)\right),$ where $o_{p}(1)$ has no $\vartheta$ dependence. Moreover, $(\theta_{v},v_{v},\chi_{v,t},q_{v,t})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Theta,V,q_{t}^{1/2}\Theta+G,q_{t})$ where $(\Theta,V)\sim\mu_{\Theta,V},G\sim{\mathsf{N}}(0,1)$ independent of $\Theta,V$, and $q_{t}$ is given by (7). ###### Lemma 4. As in the proof of Lemma 1, we compute the posterior density $p_{v}(\vartheta|\mathcal{T}_{v,2t-1})$ via belief propogation. The belief propagation iteration is $\displaystyle m_{f\rightarrow v}^{0}(\ell)=1,$ $\displaystyle m_{v\rightarrow f}^{s+1}(\vartheta)\propto\int\prod_{f^{\prime}\in\partial v\setminus f}\left(\exp\left(-\frac{n}{2}(x_{f^{\prime}v}-\frac{1}{n}\ell_{f^{\prime}}\vartheta)^{2}\right)m_{f^{\prime}\rightarrow v}^{s}(\ell_{f^{\prime}})\mu_{\Lambda|U}(u_{f^{\prime}},\mathrm{d}\ell_{f^{\prime}})\right),$ $\displaystyle m_{f\rightarrow v}^{s}(\ell)\propto\int\prod_{v^{\prime}\in\partial f\setminus v}\left(\exp\left(-\frac{n}{2}(x_{fv^{\prime}}-\frac{1}{n}\ell\vartheta_{v^{\prime}})^{2}\right)m_{v^{\prime}\rightarrow f}^{s}(\vartheta_{v^{\prime}})\mu_{\Theta|V}(v_{v^{\prime}},\mathrm{d}\vartheta_{v^{\prime}})\right),$ with normalization $\int m_{f\rightarrow v}^{s}(\ell)\mu_{\Lambda|U}(u_{f},\mathrm{d}\ell)=\int m_{v\rightarrow f}^{s}(\vartheta)\mu_{\Theta|V}(v_{v},\mathrm{d}\vartheta)=1$. For $t\geq 1$ $\displaystyle p_{v}(\vartheta|\mathcal{T}_{v,2t-1})\propto\int\prod_{f\in\partial v}\left(\exp\left(-\frac{n}{2}(x_{fv}-\frac{1}{n}\ell_{f}\vartheta)^{2}\right)m_{f\rightarrow v}^{t-1}(\ell_{f})\mu_{\Lambda|U}(u_{f},\mathrm{d}\ell_{f})\right),$ This equation is exact. We define several quantities related to the belief propagation iteration. $\displaystyle\mu_{f\rightarrow v}^{s}$ $\displaystyle=\int\ell m_{f\rightarrow v}^{s}(\ell)\mu_{\Lambda|U}(u_{f},\mathrm{d}\ell),$ $\displaystyle s_{f\rightarrow v}^{s}$ $\displaystyle=\int\ell^{2}m_{f\rightarrow v}^{s}(\ell)\mu_{\Lambda|U}(u_{f},\mathrm{d}\ell),$ $\displaystyle\alpha_{v\rightarrow f}^{s+1}$ $\displaystyle=\frac{1}{n}\sum_{f^{\prime}\in\partial v\setminus f}\mu_{f^{\prime}\rightarrow v}^{s}\lambda_{f^{\prime}},$ $\displaystyle(\tau_{v\rightarrow f}^{s+1})^{2}$ $\displaystyle=\frac{1}{n}\sum_{f^{\prime}\in\partial v\setminus f}(\mu_{f^{\prime}\rightarrow v}^{s})^{2},$ $\displaystyle a_{v\rightarrow f}^{s}$ $\displaystyle=\frac{\mathrm{d}\phantom{b}}{\mathrm{d}\vartheta}\log m_{v\rightarrow f}^{s}(\vartheta)\Big{|}_{\vartheta=0},$ $\displaystyle b_{v\rightarrow f}^{s}$ $\displaystyle=-\frac{\mathrm{d}^{2}\phantom{b}}{\mathrm{d}\vartheta^{2}}\log m_{v\rightarrow f}^{s}(\vartheta)\Big{|}_{\vartheta=0},$ $\displaystyle\mu_{v\rightarrow f}^{s}$ $\displaystyle=\int\vartheta m_{v\rightarrow f}^{s}(\vartheta)\mu_{\Theta|V}(v_{v},\mathrm{d}\vartheta),$ $\displaystyle s_{v\rightarrow f}^{s}$ $\displaystyle=\int\vartheta^{2}m_{v\rightarrow f}^{s}(\vartheta)\mu_{\Theta|V}(v_{v},\mathrm{d}\vartheta),$ $\displaystyle\alpha_{f\rightarrow v}^{s}$ $\displaystyle=\frac{1}{n}\sum_{v^{\prime}\in\partial f\setminus v}\mu_{v^{\prime}\rightarrow f}^{s}\theta_{v^{\prime}},$ $\displaystyle(\hat{\tau}_{f\rightarrow v}^{s})^{2}$ $\displaystyle=\frac{1}{n}\sum_{v^{\prime}\in\partial f\setminus v}(\mu_{v^{\prime}\rightarrow f}^{s})^{2},$ $\displaystyle a_{f\rightarrow v}^{s}$ $\displaystyle=\frac{\mathrm{d}\phantom{b}}{\mathrm{d}\ell}\log m_{f\rightarrow v}^{s}(\ell)\Big{|}_{\ell=0},$ $\displaystyle b_{f\rightarrow v}^{s}$ $\displaystyle=-\frac{\mathrm{d}^{2}\phantom{b}}{\mathrm{d}\ell^{2}}\log m_{f\rightarrow v}^{s}(\ell)\Big{|}_{\ell=0},$ Lemma 4 follows from the following asymptotic characterization of the quantities in the preceding display in the limit $n,p\rightarrow\infty$, $n/p\rightarrow\delta$: $\begin{gathered}\mathbb{E}[\mu_{f\rightarrow v}^{s}\lambda_{f}]\rightarrow q_{s+1},\qquad\mathbb{E}[(\mu_{f\rightarrow v}^{s})^{2}]\rightarrow q_{s+1},\\\ \alpha_{v\rightarrow f}^{s+1}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1},\qquad(\tau_{v\rightarrow f}^{s+1})\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1},\\\ (\theta_{v},v_{v},a_{v\rightarrow f}^{s},b_{v\rightarrow f}^{s})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Theta,V,q_{s}\Theta+q_{s}^{1/2}G_{2},q_{s}),\\\ \mathbb{E}[\mu_{v\rightarrow f}^{s}\theta_{v}]\rightarrow\delta\hat{q}_{s},\qquad\mathbb{E}[(\mu_{v\rightarrow f}^{s})^{2}]\rightarrow\delta\hat{q}_{s},\\\ \alpha_{f\rightarrow v}^{s}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\hat{q}_{s},\qquad(\hat{\tau}_{f\rightarrow v}^{s+1})^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\hat{q}_{s},\\\ (\lambda_{f},u_{f},a_{f\rightarrow v}^{s},b_{f\rightarrow v}^{s})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Lambda,U,\hat{q}_{s}\Lambda+\hat{q}_{s}^{1/2}G,\hat{q}_{s}).\end{gathered}$ (50) As in the proof of Lemma 1, the distribution of these quantities does not depend upon $v$ or $f$, so that the limits hold for all $v,f$ once we establish them for any $v,f$. We establish the limits inductively in $s$. Base case: $\mathbb{E}[\mu_{f\rightarrow v}^{0}\lambda_{f}]\rightarrow q_{1}$ and $\mathbb{E}[(\mu_{f\rightarrow v}^{0})^{2}]\rightarrow q_{1}$. Note $\mu_{f\rightarrow v}^{0}=\mathbb{E}[\lambda_{f}|u_{f}]$. Thus $\mathbb{E}[\mu_{f\rightarrow v}^{0}\lambda_{f}]=\mathbb{E}[\mathbb{E}[\lambda_{f}|u_{f}]^{2}]=V_{\Lambda,U}(0)=q_{1}$ exactly in finite samples, so also asymptotically. The expectation $\mathbb{E}[(\mu_{f\rightarrow v}^{0})^{2}]$ has the same value. Inductive step 1: If $\mathbb{E}[\mu_{f\rightarrow v}^{s}\lambda_{f}]\rightarrow q_{s+1}$ and $\mathbb{E}[(\mu_{f\rightarrow v}^{s})^{2}]\rightarrow q_{s+1}$, then $\alpha_{v\rightarrow f}^{s+1}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1}$ and $(\tau_{v\rightarrow f}^{s+1})^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1}$. By the inductive hypothesis, $\mathbb{E}[\alpha_{v\rightarrow f}^{s+1}]=(n-1)\mathbb{E}[\mu_{f\rightarrow v}^{s}\lambda_{f}]/n\rightarrow q_{s+1}$ and $\mathbb{E}[(\tau_{v\rightarrow f}^{s+1})^{2}]=(n-1)\mathbb{E}[(\mu_{f\rightarrow v}^{s})^{2}]/n\rightarrow q_{s+1}$. Moreover, $\mu_{f^{\prime}\rightarrow v}^{s}\lambda_{f^{\prime}}$ are mutually independent as we vary $f^{\prime}\in\partial v\setminus f$, and likewise for $\mu_{f^{\prime}\rightarrow v}^{s}$. We have $\mathbb{E}[(\mu_{f^{\prime}\rightarrow v}^{s}\lambda_{f^{\prime}})^{2}]\leq M^{4}$ and $\mathbb{E}[(\mu_{f^{\prime}\rightarrow v}^{s})^{4}]\leq M^{4}$ because the integrands are bounded by $M^{4}$. By the weak law of large numbers, $\alpha_{v\rightarrow f}^{s+1}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1}$ and $(\tau_{v\rightarrow f}^{s+1})^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1}$. Inductive step 2: If $\alpha_{v\rightarrow f}^{s+1}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1}$ and $(\tau_{v\rightarrow f}^{s+1})^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1}$, then $(\theta_{v},v_{v},a_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Theta,V,q_{s+1}\Theta+q_{s+1}^{1/2}G,q_{s+1})$. We may express $\log m_{v\rightarrow f}^{s+1}(\vartheta)=\mathsf{const}+\sum_{f^{\prime}\in\partial v\setminus f}\log\mathbb{E}_{\Lambda_{f^{\prime}}}\left[\exp\left(-\frac{1}{2n}\Lambda_{f^{\prime}}^{2}\vartheta^{2}+x_{f^{\prime}v}\Lambda_{f^{\prime}}\vartheta\right)\right],$ where $\Lambda_{f^{\prime}}$ has density $m_{f^{\prime}\rightarrow v}^{s}$ with respect to $\mu_{\Lambda|U}(u_{f^{\prime}},\cdot)$. We compute $\displaystyle\frac{\mathrm{d}\phantom{b}}{\mathrm{d}\vartheta}\mathbb{E}_{\Lambda_{f^{\prime}}}\left[\exp\left(-\frac{1}{2n}\Lambda_{f^{\prime}}^{2}\vartheta^{2}+x_{f^{\prime}v}\Lambda_{f^{\prime}}\vartheta\right)\right]\Big{|}_{\vartheta=0}$ $\displaystyle=\mathbb{E}_{\Lambda_{f^{\prime}}}\left[x_{f^{\prime}v}\Lambda_{f^{\prime}}\right]=x_{f^{\prime}v}\mu_{f^{\prime}\rightarrow v}^{s},$ $\displaystyle\frac{\mathrm{d}^{2}\phantom{b}}{\mathrm{d}\vartheta^{2}}\mathbb{E}_{\Lambda_{f^{\prime}}}\left[\exp\left(-\frac{1}{2n}\Lambda_{f^{\prime}}^{2}\vartheta^{2}+x_{f^{\prime}v}\Lambda_{f^{\prime}}\vartheta\right)\right]\Big{|}_{\vartheta=0}$ $\displaystyle=\mathbb{E}_{\Lambda_{f^{\prime}}}\left[x_{f^{\prime}v}^{2}\Lambda_{f^{\prime}}^{2}-\frac{1}{n}\Lambda_{f^{\prime}}^{2}\right]=\left(x_{f^{\prime}v}^{2}-\frac{1}{n}\right)s_{f^{\prime}\rightarrow v}^{s}.$ Then $\displaystyle a_{v\rightarrow f}^{s+1}=\sum_{f^{\prime}\in\partial v\setminus f}x_{f^{\prime}v}\mu_{f^{\prime}\rightarrow v}^{s}\;\;\text{and}\;\;b_{v\rightarrow f}^{s+1}=\sum_{f^{\prime}\in\partial v\setminus f^{\prime}}\left(x_{f^{\prime}v}^{2}(\mu_{f^{\prime}\rightarrow v}^{s})^{2}-\left(x_{f^{\prime}v}^{2}-\frac{1}{n}\right)s_{f^{\prime}\rightarrow v}^{s}\right).$ We compute $\displaystyle a_{v\rightarrow f}^{s+1}=\left(\frac{1}{n}\sum_{f^{\prime}\in\partial v\setminus f}\mu_{f^{\prime}\rightarrow v}^{s}\lambda_{f^{\prime}}\right)\theta_{v}+\sum_{f^{\prime}\in\partial v\setminus f}z_{f^{\prime}v}\mu_{f^{\prime}\rightarrow v}^{s}.$ Because $(z_{f^{\prime}v})_{f^{\prime}\in\partial v\setminus f}$ are independent of $\mu_{f^{\prime}\rightarrow v}^{2}$ and are mutually independent from each other, conditional on $\mathcal{T}_{v\rightarrow f}^{1}$ the quantity $\sum_{f^{\prime}\in\partial v\setminus f}z_{f^{\prime}v}\mu_{f^{\prime}\rightarrow v}^{s}$ is distributed ${\mathsf{N}}(0,(\tau_{v\rightarrow f}^{s+1})^{2})$. By the inductive hypothesis, $(\tau_{f\rightarrow v}^{s+1})^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1}$, so that $\sum_{f^{\prime}\in\partial v\setminus f}z_{f^{\prime}v}\mu_{f^{\prime}\rightarrow v}^{s}\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}{\mathsf{N}}(0,q_{s+1})$. Further, $z_{f^{\prime}v}$ and $\mu_{f^{\prime}\rightarrow v}^{s}$ are independent of $\theta_{v}$, and by the inductive hypothesis, the coefficient of $\theta_{v}$ converges in probability to $q_{s+1}$. By the Continuous Mapping Theorem [Vaa98, Theorem 2.3], we conclude that $(\theta_{v},v_{v},a_{v\rightarrow f}^{s+1})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Theta,V,q_{s+1}\Theta+q_{s+1}^{1/2}G)$ where $G\sim{\mathsf{N}}(0,1)$ independent of $\Theta$, as desired. Now we show that $b_{v\rightarrow f}^{s+1}\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}q_{s+1}$. We expand $b_{v\rightarrow f}^{s+1}=A-B$ where $A=\sum_{f^{\prime}\in\partial v\setminus f}x_{f^{\prime}v}^{2}(\mu_{f^{\prime}\rightarrow v}^{s})^{2}$ and $B=\sum_{f^{\prime}\in\partial v\setminus f}(x_{f^{\prime}v}^{2}-1/n)s_{f^{\prime}\rightarrow v}^{s}$. We have $A=\frac{1}{n^{2}}\sum_{v^{\prime}\in\partial f\setminus v}\lambda_{f^{\prime}}^{2}\theta_{v}^{2}(\mu_{f^{\prime}\rightarrow v}^{s})^{2}+\frac{2}{n}\sum_{f^{\prime}\in\partial v\setminus f}\lambda_{f^{\prime}}\theta_{v}z_{f^{\prime}v}(\mu_{f^{\prime}\rightarrow v}^{s})^{2}+\sum_{v^{\prime}\in\partial f\setminus v}z_{f^{\prime}v}^{2}(\mu_{f^{\prime}\rightarrow v}^{s})^{2}.$ Observe $\mathbb{E}[\lambda_{f^{\prime}}^{2}\theta_{v}^{2}(\mu_{f^{\prime}\rightarrow v}^{s})^{2}]\leq M^{6}$, so that the expectation of the first term is bounded by $M^{6}(p-1)/n^{2}\rightarrow 0$. Thus, the first term converges to 0 in probability. Because $z_{f^{\prime}v}$ is independent of $\mu_{f^{\prime}\rightarrow v}^{s}$, $\mathbb{E}[|\lambda_{f^{\prime}}\theta_{v}z_{f^{\prime}v}(\mu_{f^{\prime}\rightarrow v}^{s})^{2}|]\leq M^{4}\sqrt{2/(\pi n)}$, so that the absolute value of the expectation of the second term is bounded by $2M^{4}\sqrt{2/(\pi n)}\rightarrow 0$. Thus, the second term converges to 0 in probability. Because $\mu_{f^{\prime}\rightarrow v}^{s}$ is independent of $z_{f^{\prime}v}$, the expectation of the last term is $(n-1)\mathbb{E}[(\mu_{f^{\prime}\rightarrow v})^{2}]/n\rightarrow q_{s+1}$ (we have used here the assumption of inductive step 1). The terms $(z_{f^{\prime}v}^{2}(\mu_{f^{\prime}\rightarrow v}^{s})^{2})_{f^{\prime}\in\partial v\setminus f}$ are mutually independent and $\mathbb{E}[z_{f^{\prime}v}^{4}(\mu_{f^{\prime}\rightarrow v}^{s})^{4}]\leq 3M^{4}/n^{2}$, so that by the weak law of large numbers we have that the last term converges to $q_{s+1}$ in probability. Thus, $A\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1}$. We have $B=\frac{1}{n^{2}}\sum_{v^{\prime}\in\partial f\setminus v}\lambda_{f}^{2}\theta_{v^{\prime}}^{2}s_{v^{\prime}\rightarrow f}^{s}+\frac{2}{n}\sum_{v^{\prime}\in\partial f\setminus v}\lambda_{f}\theta_{v^{\prime}}s_{v^{\prime}\rightarrow f}^{s}+\sum_{v^{\prime}\in\partial f\setminus v}(z_{f^{\prime}v}^{2}-1/n)s_{v^{\prime}\rightarrow f}^{s}.$ As in the analysis of the first two terms of $A$, we may use that $s_{v^{\prime}\rightarrow f}^{s}\leq M^{2}$ to argue that the first two terms of $B$ converge to 0 in probability. Further, because $z_{f^{\prime}v}$ is independent of $s_{v^{\prime}\rightarrow f}^{s}$, the expectation of the last term is 0. Further, $\mathbb{E}[(z_{f^{\prime}v}^{2}-1/n)^{2}(s_{v^{\prime}\rightarrow f}^{s})^{2}]\leq 2\mathbb{E}[(z_{f^{\prime}v}^{4}+1/n^{2})]\mathbb{E}[(s_{v^{\prime}\rightarrow f}^{s})^{2}]\leq 8M^{4}/n^{2}$, so that by the weak law of large numbers, the final term converges to 0 in probability. Thus, $B\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}0$. Because, as we have shown, $A\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1}$, we conclude $b_{v\rightarrow f}^{s+1}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}q_{s+1}$. Combining with $(\theta_{v},v_{v},a_{v\rightarrow f}^{s+1})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Theta,V,q_{s+1}\Theta+q_{s+1}^{1/2}G)$ and applying the Continuous Mapping Theorem [Vaa98, Theorem 2.3], we have $(\theta_{v},a_{v\rightarrow f}^{s+1},b_{v\rightarrow f}^{s+1})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Theta,q_{s+1}\Theta+q_{s+1}^{1/2}G,q_{s+1})$. Inductive step 3: If $(\theta_{v},v_{v},a_{v\rightarrow f}^{s},b_{v\rightarrow f}^{s})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Theta,V,q_{s}\Theta+q_{s}^{1/2}G_{1},q_{s})$, then $\mathbb{E}[\mu_{v\rightarrow f}^{s}\theta_{v}]\rightarrow\delta\hat{q}_{s}$ and $\mathbb{E}[(\mu_{v\rightarrow f}^{s})^{2}]\rightarrow\delta\hat{q}_{s}$. We will require the following lemma, whose proof is deferred to section D.2.1. ###### Lemma 5. For any fixed $s$, we have $\vartheta,\ell\in[-M,M]$ $\displaystyle\log\frac{m_{v\rightarrow f}^{s}(\vartheta)}{m_{v\rightarrow f}^{s}(0)}=\vartheta a_{v\rightarrow f}^{s}-\frac{1}{2}\vartheta^{2}b_{v\rightarrow f}^{s}+O_{p}(n^{-1/2}),$ $\displaystyle\log\frac{m_{f\rightarrow v}^{s}(\ell)}{m_{f\rightarrow v}^{s}(0)}=\ell a_{f\rightarrow v}^{s}-\frac{1}{2}\ell^{2}b_{f\rightarrow v}^{s}+O_{p}(n^{-1/2}),$ where $O_{p}(n^{-1/2})$ has no $\vartheta$ (or $\ell$) dependence. Define $\epsilon_{f\rightarrow v}^{s}=\sup_{\vartheta\in[-M,M]}\left|\log\frac{m_{v\rightarrow f}^{s}(\vartheta)}{m_{v\rightarrow f}^{s}(0)}-\left(\vartheta a_{v\rightarrow f}^{s}-\frac{1}{2}\vartheta^{2}b_{v\rightarrow f}^{s}\right)\right|.$ By Lemma 5, we have $\epsilon_{v\rightarrow f}^{s}=o_{p}(1)$. Moreover, using the same argument as in inductive step 4 of the proof of Theorem 1, we have that $\displaystyle e^{-2\epsilon_{v\rightarrow f}^{s}}\eta_{\Theta,V}(a_{v\rightarrow f}^{s}(b_{v\rightarrow f}^{s})^{-1/2}$ $\displaystyle,v_{v};b_{v\rightarrow f}^{s})\leq\mu_{v\rightarrow f}^{s}$ $\displaystyle\leq e^{2\epsilon_{v\rightarrow f}^{s}}\eta_{\Theta,V}(a_{v\rightarrow f}^{s}(b_{v\rightarrow f}^{s})^{1/2},v_{v};b_{v\rightarrow f}^{s}),$ where $\eta_{\Theta,V}(y,v;q)=\mathbb{E}_{\Theta,V,G}[\Theta|q^{1/2}\Theta+\tau G=y;V=v]$. Because $\eta_{\Theta,V}$ takes values in the bounded interval $[-M,M]$ and $\epsilon_{v\rightarrow f}^{s}=o_{p}(1)$ by Lemma 5, we conclude that $\mu_{v\rightarrow f}^{s}=\eta_{\Theta,V}(a_{v\rightarrow f}^{s}/b_{v\rightarrow f}^{s},v_{v};b_{v\rightarrow f}^{s})+o_{p}(1).$ For a fixed $v_{v}$, the Bayes estimator in the observation and coefficient $q$. Thus, by the inductive hypothesis and the fact that $v_{v}\sim\mu_{V}$ for all $n$, we have that $\mathbb{E}[\Theta\eta_{\Theta,V}(a_{v\rightarrow f}^{s}(b_{v\rightarrow f}^{s})^{1/2},v_{v};b_{v\rightarrow f}^{s})]$ has limit $\mathbb{E}[\Theta\eta_{\Theta,V}(q_{s}^{1/2}\Theta+G,V;q_{s})]=\delta\hat{q}_{s}$ and $\mathbb{E}[\eta_{\Theta,V}(q_{s}^{1/2}\Theta+G,V;q_{s})^{2}]$ has limit $\mathbb{E}_{\Theta,V,G}[\eta_{\Theta,V}(q_{s}^{1/2}\Theta+G,V;q_{s})^{2}]=\delta\hat{q}_{s}$. Because $|\theta_{v}|,|\mu_{v\rightarrow f}^{s}|,|\eta_{\Theta,V}(a_{v\rightarrow f}^{s}/b_{v\rightarrow f}^{s},v_{v};b_{v\rightarrow f}^{s})|\leq M$, by bounded convergence, we conclude $\mathbb{E}[\mu_{v\rightarrow f}^{s}\theta_{v}]\rightarrow\delta\hat{q}_{s}$ and $\mathbb{E}[(\mu_{f\rightarrow v}^{s})^{2}]\rightarrow\delta\hat{q}_{s}$. The remaining inductive steps are completely analagous to those already shown. We list them here for completeness. Inductive step 4: If $\mathbb{E}[\mu_{v\rightarrow f}^{s}\theta_{v}]\rightarrow\delta\hat{q}_{s}$ and $\mathbb{E}[(\mu_{v\rightarrow f}^{s})^{2}]\rightarrow\delta\hat{q}_{s}$, then $\alpha_{f\rightarrow v}^{s}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\hat{q}_{s}$ and $(\hat{\tau}_{f\rightarrow v}^{s+1})^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\hat{q}_{s}$. Inductive step 5: If $\alpha_{f\rightarrow v}^{s}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\hat{q}_{s}$ and $(\hat{\tau}_{f\rightarrow v}^{s})^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\hat{q}_{s}$, then $(\lambda_{f},u_{f},a_{f\rightarrow v}^{s},b_{f\rightarrow v}^{s})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Lambda,U,\hat{q}_{s}\Lambda+\hat{q}_{s}^{1/2}G,\hat{q}_{s})$. Inductive step 6: If $(\lambda_{f},u_{f},a_{f\rightarrow v}^{s},b_{f\rightarrow v}^{s})\stackrel{{\scriptstyle\mathrm{d}}}{{\rightarrow}}(\Lambda,U,\hat{q}_{s}\Lambda+\hat{q}_{s}^{1/2}G,\hat{q}_{s})$, then $\mathbb{E}[\mu_{f\rightarrow v}^{s}\lambda_{f}]\rightarrow q_{s+1}$ and $\mathbb{E}[(\mu_{f\rightarrow v}^{s})^{2}]\rightarrow q_{s+1}$. The induction is complete, and we conclude (50). To complete the proof of Lemma 4, first observe that we may express $\log\frac{p_{v}(\vartheta|\mathcal{T}_{v,2t-1})}{p_{v}(0|\mathcal{T}_{v,2t-1})}$ as $\log\frac{m_{v\rightarrow f}^{t}(\vartheta)}{m_{v\rightarrow f}^{t}(\vartheta)}+\log\mathbb{E}_{\Lambda_{f}}[\exp(\vartheta x_{fv}\Lambda_{f}-\vartheta^{2}\Lambda_{f}^{2}/(2n))]$. Note that $\left|\log\mathbb{E}_{\Lambda_{f}}[\exp(\vartheta x_{fv}\Lambda_{f}-\vartheta^{2}\Lambda_{f}^{2}/(2n))]\right|\leq M^{2}|x_{fv}|+M^{4}/2n=o_{p}(1).$ By Lemma 5, we have that, up to a constant, $\log\frac{m_{v\rightarrow f}^{t}(\vartheta)}{m_{v\rightarrow f}^{t}(\vartheta)}=-\frac{1}{2}((a_{v\rightarrow f}^{t}(b_{v\rightarrow f}^{t})^{-1/2}-b_{v\rightarrow f}^{t})^{1/2}\vartheta)^{2}+o_{p}(1)$. The lemma follows from (50) and Slutsky’s theorem. ∎ Lemma 3 in the low-rank matrix estimation model follows from Lemma 4 by exactly the same argument that derived Lemma 3 in the high-dimensional regression model from Lemma 1. #### D.2.1 Technical tools ###### Lemma 5. Fix any $\vartheta\in[-M,M]$. By Taylor’s theorem, there exist $\vartheta_{f^{\prime}}\in[-M,M]$ (in fact, between $0$ and $\vartheta$) such that $\displaystyle\log\frac{m_{v\rightarrow f}^{s}(\vartheta)}{m_{v\rightarrow f}^{s}(0)}$ $\displaystyle=\sum_{f^{\prime}\in\partial v\setminus f}\log\frac{\mathbb{E}_{\Lambda_{f^{\prime}}}[\exp(-n(x_{f^{\prime}v}-\Lambda_{f^{\prime}}\vartheta/n)^{2}/2)]}{\mathbb{E}_{\Lambda_{f^{\prime}}}[\exp(-nx_{f^{\prime}v}^{2}/2)]}$ $\displaystyle=\vartheta a_{v\rightarrow f}^{s+1}-\frac{1}{2}\vartheta^{2}b_{v\rightarrow f}^{s+1}+\frac{1}{6}\vartheta^{3}\sum_{f^{\prime}\in\partial v\setminus f}\frac{\mathrm{d}^{3}}{\mathrm{d}\vartheta^{3}}\log\mathbb{E}_{\Lambda_{f^{\prime}}}[\exp(-n(x_{f^{\prime}v}-\Lambda_{f^{\prime}}\vartheta/n)^{2}/2)]\Big{|}_{\vartheta=\vartheta_{f^{\prime}}},$ where it is understood that $\Lambda_{f^{\prime}}\sim\mu_{\Lambda|U}(u_{f^{\prime}},\cdot)$. Denote $\psi(\vartheta,\ell,x)=-n(x_{f^{\prime}v}-\ell\vartheta/n)^{2}/2$. By the same argument that allowed us to derive (46) from R4 in the proof of Lemma 3(a), we conclude $\displaystyle\frac{\mathrm{d}^{3}}{\mathrm{d}\vartheta^{3}}\log\mathbb{E}_{\Lambda}[\exp(\psi(\vartheta,\Lambda,x))]\Big{|}_{\vartheta=\vartheta_{f^{\prime}}}$ $\displaystyle\qquad\qquad\leq C\sup_{\ell,\vartheta\in[-M,M]}\max\\{|\partial_{\vartheta}\psi(\vartheta,\ell,x)|^{3},|\partial_{\vartheta}\psi(\vartheta,\ell,x)\partial_{\vartheta}^{2}\psi(\vartheta,\ell,x)|,|\partial_{\vartheta}^{3}\psi(\vartheta,\ell,x)|\\}$ $\displaystyle\qquad\qquad\leq C\max\left\\{M^{3}|M^{2}/n+x_{f^{\prime}v}|^{3},(M^{2}/n)M|M^{2}/n+x_{f^{\prime}v}|,0\right\\},$ where $C$ is a universal constant. The expectaton of the right-hand side is $O(n^{-3/2})$, whence we get $\frac{1}{6}\vartheta^{3}\sum_{f^{\prime}\in\partial v\setminus f}\frac{\mathrm{d}^{3}}{\mathrm{d}\vartheta^{3}}\log\mathbb{E}_{\Lambda_{f^{\prime}}}[\exp(-n(x_{f^{\prime}v}-\Lambda_{f^{\prime}}\vartheta/n)^{2}/2)]\Big{|}_{\vartheta=\vartheta_{f^{\prime}}}=O_{p}(n^{-1/2}),$ where because $\vartheta\in[-M,M]$, we may take $O_{p}(n^{-1/2})$ to have no $\vartheta$-dependence. The expansion of $\log\frac{m_{f\rightarrow v}^{s}(\ell)}{m_{f\rightarrow v}^{s}(0)}$ is proved similarly. ∎ ## Appendix E Weakening the assumptions Section 5 and the preceding appendices establish under the assumptions A1, A2 and either R3, R4 or M2 all claims in Theorems 1 and 2 except that the lower bound may be achieved. In this section we show that if these claims hold under assumptions A1, A2, R3, R4, then they also hold under assumptions A1, A2, R1, R2 in the high-dimensional regression model; and similarly for the low-rank matrix estimation model. In the next section we prove we can achieve the lower bounds under the weaker assumptions A1, A2 and either R1, R2 or M1. ### E.1 From strong to weak assumptions in the high-dimensional regression model To prove the reduction from the stronger assumptions in the high-dimensional regression model, we need the following lemma, whose proof is given at the end of this section. ###### Lemma 1. Consider on a single probability space random variables $A,B,(B_{n})_{n\geq 1}$, and $Z\sim{\mathsf{N}}(0,1)$ independent of the $A$’s and $B$’s, all with finite second moment. Assume $\mathbb{E}[(B-B_{n})^{2}]\rightarrow 0$. Let $Y=B+\tau Z$ and $Y_{n}=B_{n}+\tau Z$ for $\tau>0$. Then $\mathbb{E}[\mathbb{E}[A|Y_{n}]^{2}]\rightarrow\mathbb{E}[\mathbb{E}[A|Y]^{2}]\,.$ We now establish the reduction. Consider $\mu_{W,U}$, $\mu_{\Theta,V}$, and $h$ satisfying R1 and R2. For any $\epsilon>0$, we construct $\mu_{\tilde{\boldsymbol{W}},\tilde{U}}$, $\mu_{\tilde{\Theta},\tilde{V}}$, and $\tilde{h}$ satisfying R3 and R4 for $k=3$ as well as data $\boldsymbol{X}\in\mathbb{R}^{n\times p}$, $\boldsymbol{\theta},\tilde{\boldsymbol{\theta}},\boldsymbol{v},\tilde{\boldsymbol{v}}\in\mathbb{R}^{p}$, and $\boldsymbol{y},\tilde{\boldsymbol{y}},\boldsymbol{w},\boldsymbol{u},\tilde{\boldsymbol{u}}\in\mathbb{R}^{n}$ and $\tilde{\boldsymbol{w}}\in\mathbb{R}^{n\times 3}$ such that the following all hold. 1. 1. $(\boldsymbol{X},\boldsymbol{\theta},\boldsymbol{v},\boldsymbol{u},\boldsymbol{w},\boldsymbol{y})$ and $(\boldsymbol{X},\tilde{\boldsymbol{\theta}},\tilde{\boldsymbol{v}},\tilde{\boldsymbol{u}},\tilde{\boldsymbol{w}},\tilde{\boldsymbol{y}})$ are generated according to their respective regression models: namely, $(\theta_{j},v_{j})\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\Theta,V}$ and $(w_{i},u_{i})\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{W,U}$ independent; $(\tilde{\theta}_{j},\tilde{v}_{j})\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\tilde{\Theta},\tilde{V}}$ and $(\tilde{\boldsymbol{w}}_{i},\tilde{\boldsymbol{u}}_{i})\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\tilde{\boldsymbol{W}},\tilde{U}}$ independent; $x_{ij}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1/n)$ independent of everything else; and $\boldsymbol{y}=h(\boldsymbol{X}\boldsymbol{\theta},\boldsymbol{w})$ and $\tilde{\boldsymbol{y}}=\tilde{h}(\boldsymbol{X}\tilde{\boldsymbol{\theta}},\tilde{\boldsymbol{v}})$. Here $\tilde{\boldsymbol{w}}_{i}^{\mathsf{T}}$ is the $i^{\text{th}}$ row of $\tilde{\boldsymbol{w}}$. We emphasize that the data from the two models are not independent. 2. 2. We have $\displaystyle\mathbb{P}\left(\frac{1}{n}\|\boldsymbol{y}-\tilde{\boldsymbol{y}}\|^{2}>\epsilon\right)\rightarrow 0,\;\mathbb{P}\left(\frac{1}{p}\|\boldsymbol{v}-\tilde{\boldsymbol{v}}\|^{2}>\epsilon\right)\rightarrow 0,\;\mathbb{P}\left(\frac{1}{n}\|\boldsymbol{u}-\tilde{\boldsymbol{u}}\|^{2}>\epsilon\right)\rightarrow 0\,.$ (51) Note that because in any GFOM the functions $F_{t}^{(1)},F_{t}^{(2)},G_{t}^{(1)},G_{t}^{(2)},G_{*}$ are Lipschitz and $\|\boldsymbol{X}\|_{\mathsf{op}}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}C_{\delta}<\infty$ as $n,p\rightarrow\infty,n/p\rightarrow 0$ [Ver12, Theorem 5.31], the previous display and the iteration (1) imply $\mathbb{P}\left(\frac{1}{p}\|\hat{\boldsymbol{\theta}}^{t}-\tilde{\hat{\boldsymbol{\theta}}}^{t}\|^{2}>c(\epsilon,t)\right)\rightarrow 0\,,$ (52) for some $c(\epsilon,t)<\infty$ which goes to 0 as $\epsilon\rightarrow 0$ for fixed $t$. 3. 3. We have $\displaystyle|\textsf{mmse}_{\Theta,V}(\tau_{s}^{2})-\textsf{mmse}_{\tilde{\Theta},\tilde{V}}(\tau_{s})^{2}|<\epsilon,$ (53) $\displaystyle\left|\mathbb{E}\left[\mathbb{E}[G_{1}|h(G,W)+\epsilon^{1/2}Z,G_{0}]^{2}\right]-\mathbb{E}\left[\mathbb{E}[G_{1}|\tilde{h}(G,\tilde{\boldsymbol{W}}),G_{0}]^{2}\right]\right|<\tilde{\tau}_{s}^{2}\epsilon\,,$ (54) for all $s\leq t$ where $G_{0},G_{1},Z\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1)$, $W\sim\mu_{W}$, and $\tilde{\boldsymbol{W}}\sim\mu_{\tilde{\boldsymbol{W}}}$ independent, and $G=\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1}$. We now describe the construction described and prove it has the desired properties. Let $\mu_{A}$ be a smoothed Laplace distribution with mean zero and variance 1; namely, $\mu_{A}$ has a $C_{\infty}$ positive density $p_{A}(\cdot)$ with respect to Lebesgue measure which satisfies $\partial_{a}\log p_{A}(a)=c\cdot\mathsf{sgn}(a)$ when $|x|>1$ for some positive constant $c$. This implies that $|\partial_{a}^{k}\log p_{A}(a)|\leq q_{k}$ for all $k$ and some constants $q_{k}$, and that $\mu_{A}$ has moments of all orders. First we construct $\tilde{h}$ and $\tilde{\boldsymbol{W}}$. For a $\xi>0$ to be chosen, let $\hat{h}$ be a Lipschitz function such that $\mathbb{E}[(\hat{h}(G,W)-h(G,W))^{2}]<\xi$ for $(G,W)$ as above, which is permitted by assumption R2. Let $L>0$ be a Lipschitz constant for $\hat{h}$. Choose $M>0$ such that $\mathbb{E}[W^{2}\mathbf{1}\\{|W|>M\\}]<\xi/L^{2}$. Define $\bar{W}=W\mathbf{1}\\{|W|\leq M\\}$. Note that $\mathbb{E}[(h(G,W)-\hat{h}(G+\xi^{1/2}A,\bar{W}))^{2}]\leq 2\mathbb{E}[(h(G,W)-\hat{h}(G,W))^{2}]+2\mathbb{E}[(\hat{h}(G,W)-\hat{h}(G+\xi^{1/2}A,\bar{W}))^{2}]<4\xi$. By Lemma 1, we may pick $0<\xi<\min\\{\epsilon/4,\epsilon/L^{2}\\}$ sufficiently small that $\left|\mathbb{E}\left[\mathbb{E}[G_{1}|h(G,W)+\epsilon^{1/2}Z,G_{0}]^{2}\right]-\mathbb{E}\left[\mathbb{E}[G_{1}|\hat{h}(G+\xi^{1/2}A,\bar{W})+\epsilon^{1/2}Z,G_{0}]^{2}\right]\right|<\tilde{\tau}_{s}^{2}\epsilon\,.$ In fact, because $t$ is finite, we may choose $\xi>0$ small enough that this holds for all $s\leq t$. Define $\tilde{\boldsymbol{W}}=(\bar{W},A,Z)$ and $\tilde{h}(x,\tilde{\boldsymbol{w}})=\hat{h}(x+\xi^{1/2}a,\bar{w})+\epsilon^{1/2}z$ where $\tilde{\boldsymbol{w}}=(\bar{w},a,z)$. Then $\tilde{h}$ is Lipschitz, Eq. (54) holds for all $s\leq t$, and $\mathbb{E}[(h(G,W)-\tilde{h}(G,\tilde{\boldsymbol{W}}))^{2}]<\epsilon$ (the last because $\xi<\epsilon/4$). Now choose $K>0$ large enough that $\displaystyle\mathbb{E}[\Theta^{2}\mathbf{1}\\{|\Theta|>K\\}]<\delta\epsilon/L^{2},\;\;\mathbb{E}[U^{2}\mathbf{1}\\{|U|>K\\}]<\epsilon/2,\;\;\mathbb{E}[V^{2}\mathbf{1}\\{|V|>K\\}]<\epsilon/2\,.$ (55) Define $\tilde{\Theta}=\bar{\Theta}=\Theta\mathbf{1}\\{|\Theta|\leq K\\}$, $\tilde{V}=\bar{V}=V\mathbf{1}\\{|V|\leq K\\}$, $\tilde{U}=\bar{U}=U\mathbf{1}\\{|U|\leq K\\}$, and let $\mu_{\tilde{\Theta},\tilde{V}},\mu_{\tilde{\boldsymbol{W}},\tilde{U}}$ be the corresponding distributions; namely, $\mu_{\tilde{\Theta},\tilde{V}}$ is the distribution of $(\Theta\mathbf{1}\\{|\Theta|\leq K\\},V\mathbf{1}\\{|V|\leq K\\})$ when $(\Theta,V)\sim\mu_{\Theta,V}$, and $\mu_{\tilde{\boldsymbol{W}},\tilde{U}}$ is the distribution of $(W\mathbf{1}\\{|W|\leq M\\},A,Z)$ when $(W,U)\sim\mu_{W,U}$ and $(A,Z)\sim\mu_{A}\otimes{\mathsf{N}}(0,1)$ independent. Because the Bayes risk converges as $K\rightarrow\infty$ to the Bayes risk with respect to the untruncated prior, we may choose $K$ large enough that also (53) holds for these truncated distributions. The distributions $\mu_{\tilde{\Theta},\tilde{V}},\mu_{\tilde{\boldsymbol{W}},\tilde{U}}$ satisfy assumption R3. We now show that $\tilde{h}$ and $\tilde{\boldsymbol{W}}$ constructed in this way satisfy assumption R4. The function $\tilde{h}$ is Lipschitz because $\hat{h}$ is Lipschitz. The random variable $\tilde{Y}:=\hat{h}(x+\xi^{1/2}A,\bar{W})+\epsilon^{1/2}Z$ has density with respect to Lebesgue measure given by $p(y|x)=\int\int p_{\xi^{1/2}A}\left(s-x\right)p_{{\mathsf{N}}(0,\epsilon)}(y-\hat{h}(s,\bar{w}))\mu_{\bar{W}}({\mathrm{d}}\bar{w}){\mathrm{d}}s,$ where $p_{{\mathsf{N}}(0,\epsilon)}$ is the density of ${\mathsf{N}}(0,\epsilon)$ and $p_{\xi^{1/2}A}\left(s-x\right)$ the density of $\xi^{1/2}A$ with respect to Lebesgue measure. We have $p(y|x)\leq\sup_{y}p_{{\mathsf{N}}(0,\epsilon)}(y)=1/\sqrt{2\pi\epsilon}$, so is bounded, as desired. Moreover $\left|\frac{\int\int\partial_{x}p_{\xi^{1/2}A}\left(s-x\right)p_{{\mathsf{N}}(0,\epsilon)}(y-\hat{h}(s,\bar{w}))\mu_{\bar{W}}({\mathrm{d}}\bar{w}){\mathrm{d}}s}{p(y|x)}\right|\leq\sup_{s}\left|\frac{\dot{p}_{\xi^{1/2}A}(s)}{p_{\xi^{1/2}A}(s)}\right|.$ Because $A$ has a smoothed Laplace distribution, the right-hand side is finite. Thus, by bounded convergence, we may exchange differentiation and integration and the preceding display is equal to $\partial_{x}\log p(y|x)$. We conclude that $|\partial_{x}\log p(y|x)|$ is bounded. The boundededness of all higher derivatives holds similarly. Thus, R4 holds. We now generate the appropriate joint distribution over $(\boldsymbol{X},\boldsymbol{\theta},\boldsymbol{v},\boldsymbol{u},\boldsymbol{w},\boldsymbol{y})$ and $(\boldsymbol{X},\tilde{\boldsymbol{\theta}},\tilde{\boldsymbol{v}},\tilde{\boldsymbol{u}},\tilde{\boldsymbol{w}},\tilde{\boldsymbol{y}})$. First, generate $(\boldsymbol{X},\boldsymbol{\theta},\boldsymbol{v},\boldsymbol{u},\boldsymbol{w},\boldsymbol{y})$ from original the high-dimensional regression model. Then generate $\boldsymbol{a},\boldsymbol{z}$ independent and with entries $a_{i}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{A}$ and $z_{i}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1)$. Define $\tilde{\boldsymbol{\theta}},\tilde{\boldsymbol{v}},\tilde{\boldsymbol{u}}$ by truncating $\boldsymbol{\theta},\boldsymbol{v},\boldsymbol{u}$ at threshold $K$; define $\tilde{\boldsymbol{w}}$ by truncating $\boldsymbol{w}$ at threshold $M$ to form $\bar{\boldsymbol{w}}$ and concatenating to it the vectors $\boldsymbol{a},\boldsymbol{z}$ to form a matrix in $\mathbb{R}^{n\times 3}$; and define $\tilde{\boldsymbol{y}}=\tilde{h}(\boldsymbol{X}\tilde{\boldsymbol{\theta}},\tilde{\boldsymbol{w}})$. All that remains is to show (51) holds for the model generated in this way. The bounds on $\|\boldsymbol{v}-\tilde{\boldsymbol{v}}\|^{2}$ and $\|\boldsymbol{u}-\tilde{\boldsymbol{u}}\|^{2}$ hold by the weak law of large numbers and (55). To control $\|\boldsymbol{y}-\tilde{\boldsymbol{y}}\|$, we bound $\displaystyle\|\boldsymbol{y}-\tilde{\boldsymbol{y}}\|=\|h(\boldsymbol{X}\boldsymbol{\theta},\boldsymbol{w})-\tilde{h}(\boldsymbol{X}\tilde{\boldsymbol{\theta}},\tilde{\boldsymbol{w}})\|$ $\displaystyle\qquad\leq\|h(\boldsymbol{X}\boldsymbol{\theta},\boldsymbol{w})-\hat{h}(\boldsymbol{X}\boldsymbol{\theta},\boldsymbol{w})\|+\|\hat{h}(\boldsymbol{X}\boldsymbol{\theta},\boldsymbol{w})-\hat{h}(\boldsymbol{X}\tilde{\boldsymbol{\theta}},\boldsymbol{w})\|+\|\hat{h}(\boldsymbol{X}\tilde{\boldsymbol{\theta}},\boldsymbol{w})-\tilde{h}(\boldsymbol{X}\tilde{\boldsymbol{\theta}},\tilde{\boldsymbol{w}})$ $\displaystyle\qquad\leq\|h(\boldsymbol{X}\boldsymbol{\theta},\boldsymbol{w})-\hat{h}(\boldsymbol{X}\boldsymbol{\theta},\boldsymbol{w})\|+L\|\boldsymbol{X}(\boldsymbol{\theta}-\tilde{\boldsymbol{\theta}})\|+L\xi^{1/2}\|\boldsymbol{a}\|+L\|\boldsymbol{w}-\bar{\boldsymbol{w}}\|+\epsilon^{1/2}\|\boldsymbol{z}\|\,.$ Because $|h(x,w)|\leq C(1+|x|+|w|)$ by R2 and $\hat{h}$ is Lipschitz, there exist $C>0$ such that $|h(x,w)-\hat{h}(x,w)|\leq C(1+|x|+|w|)$. Then, $\mathbb{E}[(h(\tau Z,w)-\hat{h}(\tau Z,w))^{2}]=\int(h(x,w)-\hat{h}(x,w))^{2}\frac{1}{\sqrt{2\pi}\tau}e^{-\frac{1}{2\tau^{2}}x^{2}}{\mathrm{d}}x<C(1+\tau^{2}+w^{2})$ and is continuous in $\tau^{2}$ for $\tau>0$ by dominated convergence convergence, and is uniformly continuous for $\tau$ bounded away from 0 and infinity and $w_{i}$ restricted to a compact set. Because $\boldsymbol{x}_{i}^{\mathsf{T}}\boldsymbol{\theta}|\boldsymbol{\theta}\sim{\mathsf{N}}(0,\|\boldsymbol{\theta}\|^{2}/n)$ and $\|\boldsymbol{\theta}\|^{2}/n\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\tau_{\Theta}^{2}/\delta$, we have that $\mathbb{E}[(h(\boldsymbol{x}_{i}^{\mathsf{T}}\boldsymbol{\theta},w_{i})-\hat{h}(\boldsymbol{x}_{i}^{\mathsf{T}}\boldsymbol{\theta},w_{i}))^{2}|\boldsymbol{\theta},w_{i}]=\mathbb{E}[(h(\tau_{\Theta}\boldsymbol{x}_{i}^{\mathsf{T}}\boldsymbol{\theta}/\|\boldsymbol{\theta}\|,w_{i})-\hat{h}(\tau_{\Theta}\boldsymbol{x}_{i}^{\mathsf{T}}\boldsymbol{\theta}/\|\boldsymbol{\theta}\|,w_{i}))^{2}|\boldsymbol{\theta},w_{i}]+o_{p}(1)\,.$ The right-hand side is a constant equal to $\mathbb{E}[(h(G,W)-\hat{h}(G,W))^{2}]$ and the left-hand side is uniformly integrable. Thus, $\limsup_{n\rightarrow\infty}\mathbb{E}[(h(\boldsymbol{x}_{i}^{\mathsf{T}}\boldsymbol{\theta},w_{i})-\hat{h}(\boldsymbol{x}_{i}^{\mathsf{T}}\boldsymbol{\theta},w_{i}))^{2}]\leq\mathbb{E}[(h(G,w_{i})-\hat{h}(G,w_{i}))^{2}]<\xi\,.$ Markov’s inequality proves the the first convergence in (51) because $\xi<\epsilon$. Further, by the weak law of large numbers $\frac{L^{2}}{n}\|\boldsymbol{X}(\boldsymbol{\theta}-\tilde{\boldsymbol{\theta}})\|^{2}\leq\frac{L^{2}\|\boldsymbol{X}\|_{\mathsf{op}}^{2}}{n}\|\boldsymbol{\theta}-\tilde{\boldsymbol{\theta}}\|^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}L^{2}C_{\delta}\delta^{-1}\mathbb{E}[\Theta^{2}\mathbf{1}\\{|\Theta|>M\\}]<C_{\delta}\epsilon\,,$ where $C_{\delta}$ is the constant satisfying $\|\boldsymbol{X}\|_{\mathsf{op}}^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}C_{\delta}$ [Ver12, Theorem 5.31]. Similarly, by the weak law of large numbers $\frac{L^{2}\xi}{n}\|\boldsymbol{a}\|^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}L^{2}\xi<\epsilon,\;\;\frac{L^{2}}{n}\|\boldsymbol{w}-\bar{\boldsymbol{w}}\|^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}L^{2}\mathbb{E}[W^{2}\mathbf{1}\\{|W|>M\\}]<\xi<\epsilon,\;\;\frac{\epsilon}{n}\|\boldsymbol{z}\|^{2}\stackrel{{\scriptstyle\mathrm{p}}}{{\rightarrow}}\epsilon\,.$ We conclude that $\mathbb{P}\left(\frac{1}{n}\|\boldsymbol{y}-\tilde{\boldsymbol{y}}\|^{2}>5(C_{\delta}+4)\epsilon\right)\rightarrow 0.$ Becuse $\epsilon$ was arbitrary, we can in fact achieve (51) by considering a smaller $\epsilon$ (without affecting the validity of (53)). This completes the construction. To summarize, we have two models: the first satisfying R1 and R2, and the second satisfying R3 and R4. With the construction now complete, we explain why it establishes the reduction. Let $\tau_{s}^{(\epsilon)},\tilde{\tau}_{s}^{(\epsilon)}$ be the state evolution parameters generated by (5) with $\mu_{\tilde{\boldsymbol{W}},\tilde{U}}$, $\mu_{\tilde{\Theta},\tilde{V}}$, and $\tilde{h}$ in place of $\mu_{W,U},\mu_{\Theta,V}$, and $h$. First, we claim that Eqs. (53) and (54) imply, by induction, that as $\epsilon\rightarrow 0$, we have $\tau_{t}^{(\epsilon)}\rightarrow\tau_{t}.$ Indeed, to show this, we must only establish that $\mathbb{E}\left[\mathbb{E}[G_{1}|h(G,W)+\epsilon^{1/2}Z,G_{0}]^{2}\right]$ converges to $\mathbb{E}[\mathbb{E}[G_{1}|h(G,W),G_{0}]^{2}]$ as $\epsilon\rightarrow 0$. Without loss of generality, we may assume that on the same probability space there exists a Brownian motion $(B_{\epsilon})_{\epsilon>0}$ independent of everything else. We see that $\mathbb{E}[G_{1}|h(G,W)+\epsilon^{1/2}Z,G_{0}]^{2}]\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\mathbb{E}[G_{1}|h(G,W)+B_{\epsilon},G_{0}]=\mathbb{E}[G_{1}|(h(G,W)+B_{s})_{s\geq\epsilon},G_{0}]$. By Lévy’s upward theorem [Dur10, Theorem 5.5.7], we have that $\mathbb{E}[G_{1}|(h(G,W)+B_{s})_{s\geq\epsilon},G_{0}]$ converges to $\mathbb{E}[G_{1}|(h(G,W)+B_{s})_{s\geq 0},G_{0}]=\mathbb{E}[G_{1}|h(G,W),G_{0}]$ almost surely. By uniform integrability, we conclude that $\mathbb{E}[\mathbb{E}[G_{1}|(h(G,W)+B_{s})_{s\geq\epsilon},G_{0}]^{2}]\rightarrow\mathbb{E}[\mathbb{E}[G_{1}|h(G,W),G_{0}]^{2}]$, as claimed. Thus, we conclude the previous display. We now show that as $\epsilon\rightarrow 0$, we have $\inf_{\hat{\theta}(\cdot)}\mathbb{E}[\ell(\tilde{\Theta},\hat{\theta}(\tilde{\Theta}+\tau_{t}^{(\epsilon)}G,V))]\rightarrow\inf_{\hat{\theta}(\cdot)}\mathbb{E}[\ell(\Theta,\hat{\theta}(\Theta+\tau_{t}G,V))]\,.$ Because the truncation level $K$ can be taken to $\infty$ as $\epsilon\rightarrow 0$, this holds by combining Lemma 5(a) and (c), and specifically, Eqs. (25) and (27). Because the lower bound of Theorem 1 holds under assumptions R3 and R4, which are satisfied by $\mu_{\tilde{\boldsymbol{W}},\tilde{U}}$, $\mu_{\tilde{\Theta},\tilde{V}}$, and $\tilde{h}$, we conclude that $\lim_{n\rightarrow\infty}\frac{1}{p}\sum_{j=1}^{p}\ell(\theta_{j},\hat{\theta}_{j}^{t})\geq\inf_{\hat{\theta}(\cdot)}\mathbb{E}[\ell(\tilde{\Theta},\hat{\theta}(\tilde{\Theta}+\tau_{t}^{(\epsilon)}G,V))].$ Taking $\epsilon\rightarrow 0$ and applying (52), we conclude that (6) holds for $\hat{\boldsymbol{\theta}}^{t}$, as desired. The reduction in the high-dimensional regression model is complete. ###### Lemma 1. It is enough to prove the result for $\tau=1$. Note $\mathbb{E}[A|Y=y]=\frac{\int ae^{-(y-b)^{2}}\mu({\mathrm{d}}a,{\mathrm{d}}b)}{\int e^{-(y-b)^{2}}\mu({\mathrm{d}}a,{\mathrm{d}}b)},\qquad\mathbb{E}[A|Y_{n}=y]=\frac{\int ae^{-(y-b)^{2}}\mu_{n}({\mathrm{d}}a,{\mathrm{d}}b)}{\int e^{-(y-b)^{2}}\mu_{n}({\mathrm{d}}a,{\mathrm{d}}b)}.$ Because $\mu_{n}\stackrel{{\scriptstyle\mathrm{W}}}{{\rightarrow}}\mu$, we have $\frac{\int ae^{-(y-b)^{2}}\mu_{n}({\mathrm{d}}a,{\mathrm{d}}b)}{\int e^{-(y-b)^{2}}\mu_{n}({\mathrm{d}}a,{\mathrm{d}}b)}\rightarrow\frac{\int ae^{-(y-b)^{2}}\mu({\mathrm{d}}a,{\mathrm{d}}b)}{\int e^{-(y-b)^{2}}\mu({\mathrm{d}}a,{\mathrm{d}}b)},$ for all $y$, and moreover, this convergence is uniform on compact sets. Moreover, one can check that the stated functions are Lipschitz (with uniform Lipschitz constant) in $y$ on compact sets. This implies that $\mathbb{E}[A|Y_{n}]\rightarrow\mathbb{E}[A|Y]$ almost surely. Because the $\mathbb{E}[A|Y_{n}]^{2}$ are uniformly integrable, the lemma follows. ∎ ### E.2 From strong to weak assumptions in the low-rank matrix estimation model Consider $\mu_{\boldsymbol{\Lambda},\boldsymbol{U}},\mu_{\boldsymbol{\Theta},\boldsymbol{V}}$ satisfying M1. Fix $M>0$. For $(\boldsymbol{\Lambda},\boldsymbol{U})\sim\mu_{\boldsymbol{\Lambda},\boldsymbol{U}}$, define $\tilde{\Lambda}$ by setting $\tilde{\Lambda}_{i}=\Lambda_{i}\mathbf{1}\\{|\Lambda_{i}|\leq M\\}$ for $1\leq i\leq k$. Define $\tilde{\boldsymbol{U}}$ similarly, and let $\mu_{\tilde{\boldsymbol{\Lambda}},\tilde{\boldsymbol{U}}}$ be the distribution of $(\tilde{\boldsymbol{\Lambda}},\tilde{\boldsymbol{U}})$ so constructed. Define $\mu_{\tilde{\boldsymbol{\Theta}},\tilde{\boldsymbol{V}}}$ similarly. Consider $\\{(\boldsymbol{\lambda}_{i},\boldsymbol{u}_{i})\\}_{i\leq n}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\boldsymbol{\Lambda},\boldsymbol{U}}$ and $\\{(\boldsymbol{\theta}_{j},\boldsymbol{v}_{j})\\}_{j\leq p}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}\mu_{\boldsymbol{\Theta},\boldsymbol{V}}$ and $\boldsymbol{Z}\in\mathbb{R}^{n\times p}$ independent with $z_{ij}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1/n)$. Constructe $\tilde{\boldsymbol{\lambda}}_{i},\tilde{\boldsymbol{u}}_{i},\tilde{\boldsymbol{\theta}}_{j},\tilde{\boldsymbol{v}}_{j}$ by truncated each coordinate at level $M$ as above. Define $\boldsymbol{X},\tilde{\boldsymbol{X}}\in\mathbb{R}^{n\times p}$ by $x_{ij}=\frac{1}{n}\boldsymbol{\lambda}_{i}^{\mathsf{T}}\boldsymbol{\theta}_{j}+z_{ij}$ and $\tilde{z}_{ij}=\frac{1}{n}\tilde{\boldsymbol{\lambda}}_{i}^{\mathsf{T}}\tilde{\boldsymbol{\theta}}+z_{ij}$. As in the previous section, we have for any $\epsilon>0$ that $\mathbb{P}(\|\boldsymbol{X}-\tilde{\boldsymbol{X}}\|_{\mathsf{op}}>\epsilon)\rightarrow 0,\;\;\mathbb{P}\left(\frac{1}{p}\|\boldsymbol{v}-\tilde{\boldsymbol{v}}\|^{2}>\epsilon\right)\rightarrow 0,\;\;\mathbb{P}\left(\frac{1}{p}\|\boldsymbol{u}-\tilde{\boldsymbol{u}}\|^{2}>\epsilon\right)\rightarrow 0.$ As in the previous section, this implies that the iterates of the GFOMs before and after the truncation become arbitrarily close with high probability at a fixed iterate $t$ as we take $M\rightarrow\infty$. Further, as $M\rightarrow\infty$ we have $\boldsymbol{V}_{\tilde{\boldsymbol{\Theta}},\tilde{\boldsymbol{V}}}(\boldsymbol{Q})\rightarrow\boldsymbol{V}_{\boldsymbol{\Theta},\boldsymbol{V}}(\boldsymbol{Q})$ for all $\boldsymbol{Q}$, and likewise for $\tilde{\boldsymbol{\Lambda}},\tilde{\boldsymbol{U}}$. Further, $\boldsymbol{V}_{\tilde{\boldsymbol{\Theta}},\tilde{\boldsymbol{V}}}(\boldsymbol{Q})$ is jointly continuous in $\boldsymbol{Q}$ and $M$ (where $M$ is implicit in the truncation used to generate $\tilde{\boldsymbol{\Theta}},\tilde{\boldsymbol{V}}$). Thus, as we take $M\rightarrow\infty$, the state evolution (7) after the truncation converges to the state evolution with no truncation. The reduction now occurs exactly as in the previous section. ## Appendix F Achieving the bound All that remains to prove Theorems 1 and 2 under assumptions A1, A2 and either R1, R2 or M1, respectively, is to show that the lower bounds in Eqs. (6) and (8) can be achieved. In both cases, we can achieve the bound up to tolerance $\epsilon$ using a certain AMP algorithm. ### F.1 Achieving the bound in the high-dimensional regression model We first derive certain monotonicity properies of the parameters $\tau_{s},\sigma_{s},\tilde{\tau}_{s}$ defined in the state evolution recursion (5). As we saw in Appendix D.1 and in particular, in Lemma 1, the posterior of $\theta_{v}$ on the computation tree given observations in the local neighborhood $T_{v,2s}$ behaves like that from an observation under Gaussian noise with variance $\tau_{s}^{2}$. This is made precise in Lemma 1. Moreover, we saw in the same section that a consequence of Lemma 1 is that the asymptotic limiting Bayes risk with respect to loss $\ell$ for estimation $\theta_{v}$ given observations in $\mathcal{T}_{v,2s}$ is given by the corresponding risk for estimating $\Theta$ given $\Theta+\tau_{s}G$, $V$ with $(\Theta,V)\sim\mu_{\Theta,V}$ and $G\sim{\mathsf{N}}(0,1)$ independent. In particular, this applies to the minimum mean square error. On the computation tree, minimum mean square error can only decrease as $s$ grows because as $s$ grows we receive strictly more information. If $\mathbb{E}[\operatorname{Var}(\Theta|V)]>0$, then $\textsf{mmse}_{\Theta,V}(\tau^{2})$ is strictly increasing in $\tau$, so that we conclude that $\tau_{s}$ is non-increasing in $s$. Thus, by (5), we have also $\tilde{\tau}_{s}$ is non-increasing in $s$ and $\sigma_{s}$ is non- decreasing in $s$. In the complementary case that $\mathbb{E}[\operatorname{Var}(\Theta|V)]=0$, we compute $\sigma_{s}^{2}=\tau_{\Theta}^{2}/\delta$ and $\tilde{\tau}_{s}^{2}=0$ for all $s\geq 0$, and $\tau_{s}^{2}=0$ for all $s\geq 1$. Thus, the same monotoncity results hold in this case. These monotonicity results will imply the needed structural properties of the state evolution matrices $(T_{s,s^{\prime}}),(\Sigma_{s,s^{\prime}})$ used below. For all $s\leq t$, define $\alpha_{s}=\frac{1}{\tilde{\tau}_{s}}\mathbb{E}[\mathbb{E}[G_{1}|Y,G_{0},U]^{2}],\;\;T_{s,t}=\mathbb{E}[\mathbb{E}[G_{1}|Y,G_{0},U]^{2}],\;\;\Sigma_{s,t}=\sigma_{t}^{2},$ where $Y=h(\sigma_{s}G_{0}+\tilde{\tau}_{s}G_{1},W)$ and $G_{0},G_{1}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1)$ and $W\sim\mu_{W}$ independent. By the monotoncity properties stated, $(T_{s,t}),(\Sigma_{s,t})$ define positive definite arrays. Define $\displaystyle f_{t}(b^{t};y,u)=\mathbb{E}[B^{0}-B^{t}|h(B^{0},W)=y,\,B^{t}=b^{t},\,U=u]/\tilde{\tau}_{t},$ $\displaystyle g_{t}(a^{t};v)=\mathbb{E}[\Theta|V=v,\,\alpha_{t}\Theta+Z^{t}=a^{t}],$ where $(\Theta,V)\sim\mu_{\Theta,V})$, $(W,U)\sim\mu_{W,U}$, $(B^{0},\ldots,B^{t})\sim{\mathsf{N}}(\boldsymbol{0},\boldsymbol{\Sigma}_{[0{:}t]})$, $(Z^{1},\ldots,Z^{t})\sim{\mathsf{N}}(\boldsymbol{0},\boldsymbol{T}_{[1{:}t]})$, all independent. With these definitions, $(B^{t},B^{0}-B^{t})\stackrel{{\scriptstyle\mathrm{d}}}{{=}}(\sigma_{t}G_{0},\tilde{\tau}_{t}G_{1})$ where $G_{0},G_{1}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}{\mathsf{N}}(0,1)$. In particular, $(B^{t})$ form a backwards Gaussian random walk. We thus compute $\displaystyle\mathbb{E}[(B^{0}-B^{t})f_{t}(B^{t};h(B^{0},W),U)]/\tilde{\tau}_{t}^{2}=\mathbb{E}[(\mathbb{E}[B^{0}-B^{t}|Y,B^{t},U]/\tilde{\tau}_{t})^{2}]/\tilde{\tau}_{t}=\alpha_{t},$ $\displaystyle\mathbb{E}[f_{s}(B^{s};h(B^{0},W),U)f_{t}(B^{t};h(B^{0},W),U)]$ $\displaystyle\qquad\qquad=\mathbb{E}[\mathbb{E}[B^{0}-B^{s}|Y,B^{s},U]\mathbb{E}[B^{0}-B^{t}|Y,B^{t},U]]/\tilde{\tau}_{t}^{2}$ $\displaystyle\qquad\qquad=\mathbb{E}[(B^{0}-B^{t})^{2}|Y,B^{t},U]/\tilde{\tau}_{t}^{2}=T_{s,t},$ $\displaystyle\frac{1}{\delta}\mathbb{E}[\Theta g_{t}(\alpha_{t}\Theta+Z^{t};V)]=\frac{1}{\delta}\mathbb{E}[\mathbb{E}[\Theta|\Theta+Z^{t}/\alpha_{t},V]^{2}]=\sigma_{t}^{2},$ $\displaystyle\frac{1}{\delta}\mathbb{E}[g_{s}(\alpha_{s}\Theta+Z^{s};V)g_{t}(\alpha_{t}\Theta+Z^{t};V)]=\frac{1}{\delta}\mathbb{E}[\mathbb{E}[\Theta|\Theta+Z^{t}/\alpha_{t},V]^{2}].$ If $f_{t},g_{t}$ are Lipschitz, then, because $h$ is also Lipschitz, Stein’s lemma [Ste81] implies that the first line is equivalent to $\mathbb{E}[\partial_{B^{0}}f_{t}(B^{t};h(B^{0},W),U)]=\alpha_{t}$. (Here, we have used that $B^{0}-B^{t}$ is independent of $B^{t}$). Thus, $(\alpha_{s}),(T_{s,t}),(\Sigma_{s,t})$ are exactly the state evolution parameters determined by (36), and Lemma 1 implies that AMP with these $(f_{s}),(g_{s})$ achieves the lower bound. If the $f_{t},g_{t}$ are not Lipschitz, we proceed as follows. Fix $\epsilon>0$. First, pick Lipschitz $\hat{f}_{0}$ such that $\mathbb{E}[(\hat{f}_{0}(B^{0},W)-f_{0}(B^{0},W))^{2}]<\epsilon$, which is possibly because Lipschitz functions are dense in $L_{2}$. Define $\hat{\alpha}_{0}$ and $\hat{T}_{1,1}$ via (36) with $\hat{f}_{0}$ in place of $f_{0}$. Note that $\lim_{\epsilon\rightarrow 0}\hat{\alpha}_{0}=\alpha_{0}$ and $\lim_{\epsilon\rightarrow 0}\hat{T}_{1,1}=T_{1,1}$. Next, pick Lipschitz $\hat{g}_{0}$ such that $\mathbb{E}[(\hat{g}_{0}(\hat{\alpha}_{0}\Theta+\hat{T}_{1,1}^{1/2}G;V)-\mathbb{E}[\Theta|\hat{\alpha}_{0}+\Theta+\hat{T}_{1,1}^{1/2}G;V)])^{2}]<\epsilon$, which is again possibly because Lipschitz functions are dense in $L_{2}$. Define $\hat{\Sigma}_{0,1}=\frac{1}{\delta}\mathbb{E}[\Theta\hat{g}_{t}(\hat{\alpha}\Theta+\hat{T}_{1,1}^{1/2}G;V)]$ and $\hat{\Sigma}_{1,1}=\frac{1}{\delta}\mathbb{E}[\hat{g}_{t}(\hat{\alpha}\Theta+\hat{T}_{1,1}^{1/2}G;V)^{2}]$. Because as $\alpha\rightarrow\alpha_{0}$ and $\tau\rightarrow T_{0,0}^{1/2}$, we have $\mathbb{E}[\Theta|\alpha\Theta+\tau G;V)]\stackrel{{\scriptstyle L_{2}}}{{\rightarrow}}\mathbb{E}[\Theta|\alpha_{0}\Theta+T_{0,0}^{1/2}G;V)]$, we conclude that as $\epsilon\rightarrow 0$ that $\hat{\Sigma}_{0,1}\rightarrow\Sigma_{1,1}$ and $\hat{\Sigma}_{1,1}\rightarrow\Sigma_{1,1}$. Continuing in this way, we are able to by taking $\epsilon$ sufficiently small construct Lipschitz functions $(\hat{f}_{t}),(\hat{g}_{t})$ which track the state evolutoin of the previous paragraph arbitrarily closely up to a fixed time $t^{*}$. Thus, we may come arbitrarily close to achieving the lower bound of Theorem 1. ### F.2 Achieving the bound in the low-rank matrix estimation model Let $\boldsymbol{\gamma}_{t}=\hat{\boldsymbol{Q}}_{t}$ for $t\geq 0$ and $\boldsymbol{\alpha}_{t}=\boldsymbol{Q}_{t}$, $\boldsymbol{\Sigma}_{t,t}=\hat{\boldsymbol{Q}}_{t}$, $\boldsymbol{T}_{t,t}=\boldsymbol{Q}_{t}$ for $t\geq 1$. Define $\displaystyle f_{t}(\boldsymbol{b}^{t};\boldsymbol{u})=\mathbb{E}[\boldsymbol{\Lambda}|\boldsymbol{\gamma}_{t}\boldsymbol{\Lambda}+\boldsymbol{\Sigma}_{t,t}^{1/2}\boldsymbol{G}=\boldsymbol{b}^{t};\boldsymbol{U}],$ $\displaystyle g_{t}(\boldsymbol{a}^{t};\boldsymbol{v})=\mathbb{E}[\boldsymbol{\Theta}|\boldsymbol{\alpha}_{t}\boldsymbol{\Theta}+\boldsymbol{T}_{t,t}^{1/2}\boldsymbol{G}=\boldsymbol{a}^{t};\boldsymbol{V}].$ We check that the parameters so defined satisfy the AMP state evolution (37). Note that by (7), $\displaystyle\boldsymbol{T}_{t+1,t+1}$ $\displaystyle=\boldsymbol{Q}_{t+1}=\mathbb{E}[\mathbb{E}[\boldsymbol{\Lambda}|\hat{\boldsymbol{Q}}_{t}^{1/2}\boldsymbol{\Lambda}+\boldsymbol{G};\boldsymbol{U}]\mathbb{E}[\boldsymbol{\Lambda}|\hat{\boldsymbol{Q}}_{t}^{1/2}\boldsymbol{\Lambda}+\boldsymbol{G};\boldsymbol{U}]^{\mathsf{T}}]$ $\displaystyle=\mathbb{E}[\mathbb{E}[\boldsymbol{\Lambda}|\hat{\boldsymbol{Q}}_{t}\boldsymbol{\Lambda}+\hat{\boldsymbol{Q}}_{t}^{1/2}\boldsymbol{G};\boldsymbol{U}]\mathbb{E}[\boldsymbol{\Lambda}|\hat{\boldsymbol{Q}}_{t}\boldsymbol{\Lambda}+\hat{\boldsymbol{Q}}_{t}^{1/2}\boldsymbol{G};\boldsymbol{U}]^{\mathsf{T}}]$ $\displaystyle=\mathbb{E}[\mathbb{E}[\boldsymbol{\Lambda}|\boldsymbol{\gamma}_{t}\boldsymbol{\Lambda}+\boldsymbol{\Sigma}_{t,t}^{1/2}\boldsymbol{G};\boldsymbol{U}]\mathbb{E}[\boldsymbol{\Lambda}|\boldsymbol{\gamma}_{t}\boldsymbol{\Lambda}+\boldsymbol{\Sigma}_{t,t}^{1/2}\boldsymbol{G};\boldsymbol{U}]^{\mathsf{T}}],$ $\displaystyle\boldsymbol{\alpha}_{t+1}$ $\displaystyle=\mathbb{E}[\mathbb{E}[\boldsymbol{\Lambda}|\hat{\boldsymbol{Q}}_{t}^{1/2}\boldsymbol{\Lambda}+\boldsymbol{G};\boldsymbol{U}]\mathbb{E}[\boldsymbol{\Lambda}|\hat{\boldsymbol{Q}}_{t}^{1/2}\boldsymbol{\Lambda}+\boldsymbol{G};\boldsymbol{U}]^{\mathsf{T}}]$ $\displaystyle=\mathbb{E}[\mathbb{E}[\boldsymbol{\Lambda}|\boldsymbol{\gamma}_{t}\boldsymbol{\Lambda}+\boldsymbol{\Sigma}_{t,t}^{1/2}\boldsymbol{G};\boldsymbol{U}]\boldsymbol{\Lambda}^{\mathsf{T}}]$ where $(\boldsymbol{\Theta},\boldsymbol{V})\sim\mu_{\boldsymbol{\Theta},\boldsymbol{V}}$ and $(\boldsymbol{\Lambda},\boldsymbol{U})\sim\mu_{\boldsymbol{\Lambda},\boldsymbol{U}}$. The state evolution equations (7) for $\boldsymbol{\Sigma}_{t,t}$ and $\boldsymbol{\gamma}_{t}$ hold similarly. If $f_{t},g_{t}$ so defined are Lipschitz, then $(\alpha_{s}),(\boldsymbol{T}_{s,t}),(\boldsymbol{\Sigma}_{s,t})$ are exactly the state evolution parameters determined by (36), and Lemma 1 implies that AMP with these $(f_{s}),(g_{s})$ achieves the lower bound. If the $f_{t},g_{t}$ so defined are not Lipschitz, then the same strategy used in the previous section allows us to achieve the lower bound within tolerance $\epsilon>0$. ## Appendix G Proofs for sparse phase retrieval and sparse PCA ### G.1 Proof of Lemma 1 Note that $\|\overline{\boldsymbol{\theta}}_{0}\|_{2}$ is tightly concentrated around $\mu^{2}\varepsilon$. As a consequence, we can replace the side information $\overline{\boldsymbol{v}}$ by $\boldsymbol{v}=\sqrt{\tilde{\alpha}}\boldsymbol{\theta}_{0}+\boldsymbol{g}$. We apply Theorem 2 with $r=1$, and loss $\ell_{\lambda}(\theta,\hat{\theta})=(\hat{\theta}-\theta_{0}/\lambda)^{2}$, where $\lambda\in\mathbb{R}_{\geq 0}$ will be adjusted below. Setting $\boldsymbol{Q}_{t}=q_{t}$, $\hat{\boldsymbol{Q}}_{t}=\hat{q}_{t}$, we obtain the iteration $\displaystyle q_{t+1}=\frac{\hat{q}_{t}}{1+\hat{q}_{t}}\,,\;\;\;\;\hat{q}_{t}=\frac{1}{\delta}\mathbb{E}\big{\\{}\mathbb{E}[\sqrt{\delta}\Theta_{0}|(\delta q_{t})^{1/2}\Theta_{0}+G;V]^{2}\big{\\}}\ ,$ (56) where $\Theta_{0}\sim\mu_{\theta}$,and $V=\sqrt{\delta\tilde{\alpha}}+G^{\prime}$, $G^{\prime}\sim{\mathsf{N}}(0,1)$. Notice that the additional factors $\sqrt{\delta}$ are due to the different normalization of the vector $\boldsymbol{\theta}_{0}$ with respect to the statement in Theorem 2. Also note that the second moment of the conditional expectation bove is equal to $\mathbb{E}\big{\\{}\mathbb{E}[\sqrt{\delta}\Theta_{0}|(\delta(q_{t}+\tilde{\alpha}))^{1/2}\Theta_{0}+G]^{2}\big{\\}}$ and a simple calculation yields $\displaystyle\hat{q}_{t+1}=V_{\pm}(q_{t}+\tilde{\alpha})\,,\;\;\;\;q_{t}=\frac{\hat{q}_{t}}{1+\hat{q}_{t}}\,,$ (57) which is equivalent to Eqs. (12), (13). Let $Y=\sqrt{\delta(q_{t}+\tilde{\alpha})}\Theta_{0}+G$, $G\sim{\mathsf{N}}(0,1)$. Theorem 2 then yields $\displaystyle\frac{1}{p}\|\hat{\boldsymbol{\theta}}^{t}-\boldsymbol{\theta}_{0}/\lambda\|_{2}^{2}$ $\displaystyle\geq\inf_{\hat{\theta}(\,\cdot\,)}\mathbb{E}\big{\\{}\big{(}\hat{\theta}(Y)-\Theta_{0}/\lambda\big{)}^{2}\big{\\}}+o_{p}(1)$ (58) $\displaystyle=\frac{1}{\lambda^{2}}\mathbb{E}\big{\\{}\big{(}\mathbb{E}(\Theta_{0}|Y)-\Theta_{0}\big{)}^{2}\big{\\}}+o_{p}(1)\,.$ (59) In order to prove the upper bound (14), it is sufficient to consider $\|\hat{\boldsymbol{\theta}}^{t}\|^{2}_{2}\leq p$. Then, for any $\lambda\geq 0$, $\displaystyle\frac{1}{p}\langle\hat{\boldsymbol{\theta}}^{t},\boldsymbol{\theta}_{0}\rangle$ $\displaystyle\leq\frac{1}{p}\langle\hat{\boldsymbol{\theta}}^{t},\boldsymbol{\theta}_{0}\rangle-\frac{\lambda}{2p}(\|\hat{\boldsymbol{\theta}}^{t}\|^{2}_{2}-p)$ (60) $\displaystyle=\frac{\lambda}{2}+\frac{1}{2\lambda p}\|\boldsymbol{\theta}_{0}\|_{2}^{2}-\frac{\lambda}{2p}\|\hat{\boldsymbol{\theta}}^{t}-\boldsymbol{\theta}_{0}/\lambda\|_{2}^{2}$ (61) $\displaystyle\leq\frac{\lambda}{2}+\frac{1}{2\lambda}\mathbb{E}\\{\Theta_{0}^{2}\\}-\frac{1}{2\lambda}\mathbb{E}\big{\\{}\big{(}\mathbb{E}(\Theta_{0}|Y)-\Theta_{0}\big{)}^{2}\big{\\}}+o(1)$ (62) $\displaystyle\leq\frac{\lambda}{2}+\frac{1}{2\lambda}V_{\pm}(q_{t}+\tilde{\alpha})+o(1)\,.$ (63) The claim follows by choosing $\lambda=V_{\pm}(q_{t}+\tilde{\alpha})^{1/2}$, and noting that $\|\boldsymbol{\theta}_{0}\|^{2}_{2}/p\to\mu^{2}\varepsilon$, almost surely. ### G.2 Proof of Corollary 2 Choose $\mu=R/\sqrt{\varepsilon}$, and let $\mu^{\prime}<\mu$, $\varepsilon^{\prime}<\varepsilon$, $R^{\prime}=\mu^{\prime}\sqrt{\varepsilon^{\prime}}$. Draw the coordinates of $\boldsymbol{\theta}_{0}=\overline{\boldsymbol{\theta}}_{0}\sqrt{p}$ according to the three points distribution with parameters $\mu^{\prime},\varepsilon^{\prime}$. Then, with probability one, we have $\overline{\boldsymbol{\theta}}_{0}\in\mathscrsfs{T}(\varepsilon,R)$ for all $n$ large enough. Applying Lemma 1, we get $\displaystyle\lim_{n\to\infty}\inf_{\overline{\boldsymbol{\theta}}_{0}\in\mathscrsfs{T}(\varepsilon,R)}\mathbb{E}\left\\{\frac{\langle\overline{\boldsymbol{\theta}}_{0},\hat{\boldsymbol{\theta}}^{t}\rangle}{\|\overline{\boldsymbol{\theta}}_{0}\|_{2}\|\hat{\boldsymbol{\theta}}^{t}\|_{2}}\right\\}\leq\sqrt{\frac{V_{\pm}(q^{\prime}_{t}+\tilde{\alpha}^{\prime})}{(\mu^{\prime})^{2}\varepsilon^{\prime}}}\,,$ (64) ahere we used dominated convergence to pass from the limit in probability to limit in expectation, and $q^{\prime}_{t},\tilde{\alpha}^{\prime}$ are computed with parameters $\mu^{\prime}$, $\varepsilon^{\prime}$. By letting $\varepsilon^{\prime}\to\varepsilon$, $\mu^{\prime}\to\mu$, and since $\tilde{\alpha}^{\prime},q^{\prime}_{t}$ are continuous in these parameters by an induction argument, Eq. (64) also holds with $\mu^{\prime}$, $\varepsilon^{\prime}$, $q^{\prime}_{t}$ replaced by $\mu$, $\varepsilon$, $q_{t}$: $\displaystyle\lim_{n\to\infty}\inf_{\overline{\boldsymbol{\theta}}_{0}\in\mathscrsfs{T}(\varepsilon,R)}\mathbb{E}\left\\{\frac{\langle\overline{\boldsymbol{\theta}}_{0},\hat{\boldsymbol{\theta}}^{t}\rangle}{\|\overline{\boldsymbol{\theta}}_{0}\|_{2}\|\hat{\boldsymbol{\theta}}^{t}\|_{2}}\right\\}\leq\sqrt{\frac{V_{\pm}(q_{t}+\tilde{\alpha})}{\mu^{2}\varepsilon}}\,,$ (65) Claims $(a)$ and $(b)$ follow by upper bounding the right-hand side of the last equation. First notice that $V_{\pm}(q)=\mu^{4}\varepsilon^{2}\delta\,q+O(q^{2})$ and hence Eqs. (12), (13) imply that, for any $\eta>0$ there exists $q_{*}>0$ such that, if $q_{t}+\tilde{\alpha}\leq q_{*}$, then $\displaystyle q_{t+1}\leq(\mu^{4}\varepsilon^{2}\delta+\eta)(q_{t}+\tilde{\alpha})\,.$ (66) If $\mu^{4}\varepsilon^{2}\delta<1$, choosing $\eta=(1-\mu^{4}\varepsilon^{2}\delta)/2$, this inequality implies $q_{t}\leq 2\tilde{\alpha}/(1-\mu^{4}\varepsilon^{2}\delta)$, which proves claim $(a)$. For the second claim, we use the bounds $e^{-\delta q\mu^{2}/2}\cosh(\mu\sqrt{\delta q}G)\geq 0$ and $x/(1+x)\leq x$ in Eq. (13) to get $q_{t}\leq\overline{q}_{t}$ for all $t$, where $\overline{q}_{0}=0$ and $\displaystyle\overline{q}_{t+1}$ $\displaystyle=F_{0}(\overline{q}_{t}+\tilde{\alpha})\,,\;\;\;\;\;\;\;F_{0}(q):=\frac{\mu^{2}\varepsilon^{2}}{1-\varepsilon}\sinh(\mu^{2}\delta q)\,.$ (67) Further Eq. (65) implies $\displaystyle\lim_{n\to\infty}\inf_{\overline{\boldsymbol{\theta}}_{0}\in\mathscrsfs{T}(\varepsilon,R)}\mathbb{E}\left\\{\frac{\langle\overline{\boldsymbol{\theta}}_{0},\hat{\boldsymbol{\theta}}^{t}\rangle}{\|\overline{\boldsymbol{\theta}}_{0}\|_{2}\|\hat{\boldsymbol{\theta}}^{t}\|_{2}}\right\\}\leq\sqrt{\frac{\overline{q}_{t+1}}{\mu^{2}\varepsilon}}\,.$ (68) Define $x_{t}:=\mu^{2}\delta\overline{q}_{t}$, $a:=\mu^{4}\varepsilon^{2}\delta/(1-\varepsilon)$, $b:=\mu^{2}\delta\tilde{\alpha}=(\delta/\varepsilon)(\alpha/(1-\alpha))$. Then $x_{t}$ obeys the recursion $\displaystyle x_{t+1}=a\sinh(x_{t}+b)\,.$ (69) Since $a=R^{4}\delta/(1-\varepsilon)$, we know that $a<1/4$. Using the fact that $\sinh(u)\leq 2u$ for $u\leq 1$, this implies $x_{t}\leq b$ for all $t$ provided $b<1/2$. Subsitiuting this bound in Eq. (68), we obtain the desired claim. ### G.3 Proof of Corollary 1 Consider first the case of a random vector $\boldsymbol{\theta}_{0}$ with i.i.d. entries $\theta_{0,i}\sim\mu_{\theta}$. Define, for $\Theta_{0}\sim\mu_{\theta}$, $\displaystyle F_{\varepsilon}(q)$ $\displaystyle:=\mathbb{E}\big{\\{}\mathbb{E}[\Theta_{0}|\sqrt{q}\Theta_{0}+G]^{2}\big{\\}}$ (70) $\displaystyle=e^{-q\mu^{2}}\mu^{2}\varepsilon^{2}\mathbb{E}\left\\{\frac{\sinh(\mu\sqrt{q}G)^{2}}{1-\varepsilon+\varepsilon e^{-q\mu^{2}/2}\cosh(\mu\sqrt{q}G)}\right\\}\,.$ (71) Setting $q_{t}=\tau_{t}^{-2}$, $\hat{q}_{t}=\sigma_{t}^{2}$, and $\tilde{\alpha}=\alpha/(1-\alpha)$, and referring to Lemma 4, the state evolution recursion (5) takes the form $\displaystyle\hat{q}_{t}$ $\displaystyle=F_{\varepsilon}(q_{t}+\tilde{\alpha})\,,\;\;\;\;q_{t+1}=\delta\,H(\hat{q}_{t})\,,$ (72) $\displaystyle H(q)$ $\displaystyle:=\mathbb{E}_{G_{0},Y}\left[\left(\frac{\mathbb{E}_{G_{1}}\partial_{x}p(Y|\sqrt{q}\,G_{0}+\sqrt{1-q}G_{1})}{\mathbb{E}_{G_{1}}p(Y|\sqrt{q}\,G_{0}+\sqrt{1-q}G_{1}}\right)^{2}\right]\,.$ (73) Notice the change in factors $\delta$ with respect to Eq. (5), which is due to the different normalization of the design matrix. By the same argument used in the proof of Lemma 1, Theorem 1 implies that, for any GFOM, with output $\hat{\boldsymbol{\theta}}_{t}$, we have $\displaystyle\lim_{n,p\to\infty}\mathbb{E}\frac{\langle\overline{\boldsymbol{\theta}}_{0},\hat{\boldsymbol{\theta}}^{t}\rangle}{\|\overline{\boldsymbol{\theta}}_{0}\|_{2}\|\hat{\boldsymbol{\theta}}^{t}\|_{2}}\leq\sqrt{\hat{q}_{t}}\,.$ (74) We next compute the first order Taylor-expansion of the iteration (72), and obtain $F_{\varepsilon}(q)=q+O(q^{2})$, $H(q)=q/\delta_{\mbox{\tiny{sp}}}+O(q^{2})$ (the first order Taylor expanson of $H(q)$ was already computed in [MM19]). As a consequence, for any $\eta>0$, there exists $\alpha_{0}$ such that, if $\tilde{\alpha}<\alpha_{0}$, $q_{t}<\alpha_{0}$, then $\displaystyle q_{t+1}\leq(\frac{\delta}{\delta_{\mbox{\tiny{sp}}}}+\eta)(q_{t}+\tilde{\alpha})\,.$ The claim follows by taking $\eta=\eta(\delta):=(\delta_{\mbox{\tiny{sp}}}-\delta)/(2\delta_{\mbox{\tiny{sp}}})$, whence $q_{t}\leq\tilde{\alpha}/\eta(\delta)$ for all $t$, provided $\tilde{\alpha}<\alpha_{*}:=\alpha_{0}\eta(\delta)$. The deterministic argument follows in the same way as Corollary 2.

2024-09-04T02:54:55.653372

2002.12907

arxiv-papers

# Two types of alternating spin-$\frac{1}{2}$ chains and their field-induced transitions in $\varepsilon$-LiVOPO4 Prashanta K. Mukharjee School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram-695551, India K. M. Ranjith M. Baenitz Max Planck Institute for Chemical Physics of Solids, N$\ddot{o}$thnitzer Str. 40, 01187 Dresden, Germany Y. Skourski Dresden High Magnetic Field Laboratory (HLD-EMFL), Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany A. A. Tsirlin<EMAIL_ADDRESS>Experimental Physics VI, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany R. Nath <EMAIL_ADDRESS>School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram-695551, India ###### Abstract Thermodynamic properties, 31P nuclear magnetic resonance (NMR) measurements, and density-functional band-structure calculations for $\varepsilon$-LiVOPO4 are reported. This quantum magnet features a singlet ground state and comprises two types of alternating spin-$\frac{1}{2}$ chains that manifest themselves by the double maxima in the susceptibility and magnetic specific heat, and by the two-step magnetization process with an intermediate $\frac{1}{2}$-plateau. From thermodynamic data and band-structure calculations, we estimate the leading couplings of $J_{1}\simeq 20$ K and $J_{2}\simeq 60$ K and the alternation ratios of $\alpha_{1}=J_{1}^{\prime}/J_{1}\simeq 0.6$ and $\alpha_{2}=J_{2}^{\prime}/J_{2}\simeq 0.3$ within the two chains, respectively. The zero-field spin gap $\Delta_{0}/k_{\rm B}\simeq 7.3$ K probed by thermodynamic and NMR measurements is caused by the $J_{1}$-$J_{1}^{\prime}$ spin chains and can be closed in the applied field of $\mu_{0}H_{\rm c1}\simeq 5.6$ T, giving rise to a field-induced long-range order. The NMR data reveal predominant three-dimensional spin-spin correlations at low temperatures. Field-induced magnetic ordering transition observed above $H_{c1}$ is attributed to the Bose-Einstein condensation of triplons in the sublattice formed by the $J_{1}$-$J_{1}^{\prime}$ chains with weaker exchange couplings. ## I Introduction Field-induced quantum phase transitions in magnets set a link between fermionic spin systems and lattice boson gas Matsubara and Matsuda (1956); Giamarchi _et al._ (2008); Zapf _et al._ (2014). In this context, spin-dimer compounds possessing a gap in the excitation spectrum are extensively studied Giamarchi _et al._ (2008); Zapf _et al._ (2014). Their triplet excitations (triplons) are equivalent to lattice bosons and can be tuned by applying magnetic field. Gap closing in the applied field will usually lead to magnetic ordering that can be understood as Bose-Einstein condensation (BEC) of triplons Mukhopadhyay _et al._ (2012); Sebastian _et al._ (2006); Giamarchi _et al._ (2008); Nikuni _et al._ (2000). If interactions between the dimers are one-dimensional (1D) in nature, one further expects a non-Fermi-liquid- type Tomonaga-Luttinger Liquid (TLL) state realized at intermediate temperatures before the BEC state is reached Klanjšek _et al._ (2008); Matsushita _et al._ (2017); Willenberg _et al._ (2015); Thielemann _et al._ (2009). The ground-state spin configuration in the applied field may also depend on the delicate balance between the kinetic energy and repulsive interaction of the triplons or $S_{\rm z}=+1$ bosons Rice (2002). The dominance of repulsive interaction would lead to the formation of superlattices, which result in magnetization plateaus Narumi _et al._ (1998); Shiramura _et al._ (1998). This has been experimentally verified in the celebrated Shastry-Sutherland compound SrCu2(BO3)2 Kodama _et al._ (2002); Kageyama _et al._ (1999). On the other hand, when the kinetic energy dominates over repulsive interactions, the triplons become delocalized, and the ground state is a superposition of singlet-triplet states, which can be approximated as a BEC of triplons. The phenomenon of BEC has been studied in detail for spin-dimer compounds TlCuCl3Rüegg _et al._ (2003), BaCuSi2O6Jaime _et al._ (2004), (Ba,Sr)3Cr2O8 Aczel _et al._ (2009a, b), etc. The transition from TLL as a 1D quantum critical state to the three-dimensional (3D) BEC state has often been observed in quasi-1D spin systems e.g. spin-$1/2$ ladders (C7H10N)2CuBr4 Jeong _et al._ (2013, 2017); Möller _et al._ (2017) and (C5H12N)2CuBr4 Thielemann _et al._ (2009); Rüegg _et al._ (2008); Klanjšek _et al._ (2008) and spin-$1/2$ alternating spin chains Cu(NO3)${}_{2}\cdot$2.5D2O and F5PNN Willenberg _et al._ (2015); Matsushita _et al._ (2017). Thus, quasi-1D spin gap materials provide ample opportunities to tune the spin gap and study the field-induced quantum phase transitions. The V4+-based compounds offer an excellent playground to study gapped quantum magnets and related phenomena. Several of these compounds were already reported in the past in the context of spin gap physics Ueda (1998); Yamauchi _et al._ (1999); Johnston _et al._ (1987); Ghoshray _et al._ (2005). Recently, we studied magnetic properties of the $A$VO$X$O4 series ($A$ = Na, Ag; $X$ = P, As), where all compounds were found to host alternating spin-$\frac{1}{2}$ chains not matching the structural chains of the VO6 octahedra Mukharjee _et al._ (2019); Arjun _et al._ (2019); Ahmed _et al._ (2017); Tsirlin _et al._ (2011). In these systems, long-range superexchange couplings via two oxygen atoms play a central role. They are highly tunable and allow the variation of the zero-field spin gap $\Delta_{\rm 0}/k_{\rm B}$ from 21 K in NaVOAsO4 to $\sim 2$ K in NaVOPO4. External magnetic field closes the gap and leads to a field-induced magnetic transition, which is explained in terms of the triplon BEC Mukharjee _et al._ (2019); Weickert _et al._ (2019). Herein, we report ground-state properties of the chemically similar, but structurally different LiVOPO4. Unlike the $A$VO$X$O4 compounds, which are all monoclinic ($P2_{1}/c$), LiVOPO4 crystallizes in several polymorphs with different symmetries and atomic arrangements Harrison and Manthiram (2013); Hidalgo _et al._ (2019). We shall focus on the triclinic $\varepsilon$-LiVOPO4 ($P\bar{1}$) that can be seen as a distorted version of monoclinic $A$VO$X$O4 111Triclinic crystal structure is derived from $\varepsilon$-VOPO4. Alternatively, triclinic LiVOPO4 is sometimes referred to as the $\alpha$-phase, because it was the earliest discovered LiVOPO4 polymorph. and has been actively studied in the context of battery research Yang _et al._ (2008); Quackenbush _et al._ (2015); Lin _et al._ (2016); Shi _et al._ (2018); Chung _et al._ (2019), but not as a quantum magnet. In its crystal structure, each of the Li, V, and P reside at two nonequivalent sites, whereas the O atoms have ten nonequivalent sites. The magnetic V4+ ions form chains of the VO6 octahedra with the alternation of V1 and V2, leading to two alternating V–V distances of 3.599 and 3.629 Å along these structural chains (Fig. 1). Assuming that strongest magnetic interactions run along the structural chains, one expects the magnetic behavior of alternating spin-$\frac{1}{2}$ chains that was indeed proposed by Onoda and Ikeda Onoda and Ikeda (2013) who reported, among others Yang _et al._ (2008); Chung _et al._ (2019), magnetic susceptibility of $\varepsilon$-LiVOPO4. On the other hand, our recent results for the monoclinic $A$VO$X$O4 compounds suggest that spin chains may not coincide with the structural chains, because leading interactions occur through the double bridges of the $X$O4 tetrahedra. In this case, $\varepsilon$-LiVOPO4 should feature two types of alternating spin-$\frac{1}{2}$ chains, one formed by V(1) and the other one formed by V(2), each with different interactions and different spin gaps (Fig. 1). Below, we report experimental fingerprints of these two nonequivalent spin chains and thus directly confirm the proposed microscopic scenario. Moreover, we detect a field-induced phase transition, which is reminiscent of triplon BEC. Table 1: Atomic distances and bond angles along the superexchange paths involving P(1) and P(2) in the two alternating spin chains of $\varepsilon$-LiVOPO4 at $300$ K to highlight the coupling of P atoms with V atoms. Site | Bond Length (Å) | Angle (deg.) ---|---|--- P(1) | | V(1)-O(1) = 1.96 --- O(1)-P(1) = 1.53 P(1)-O(2) = 1.54 O(2)-V(1) = 1.97 V(2)-O(7) = 1.98 O(7)-P(1) = 1.53 P(1)-O(8) = 1.53 O(8)-V(2) = 2.01 | $\angle$V(1)-O(1)-O(2) = 152.16 --- $\angle$V(1)-O(2)-O(1) = 109.37 $\angle$V(2)-O(8)-O(7) = 140.38 $\angle$V(2)-O(7)-O(8) = 124.64 Average value $\simeq$ 131.63 P(2) | | V(1)-O(3) = 1.96 --- O(3)-P(2) = 1.53 P(2)-O(4) = 1.55 O(4)-V(1) = 2.02 V(2)-O(9) = 1.98 O(9)-P(2) = 1.53 P(2)-O(10) = 1.52 O(10)-V(2) = 1.94 | $\angle$V(1)-O(4)-O(3) = 118.59 --- $\angle$V(1)-O(3)-O(4) = 149.06 $\angle$V(2)-O(9)-O(10) = 134.87 $\angle$V(2)-O(10)-O(9) = 132.66 Average value $\simeq$ 133.79 Figure 1: (a) Crystal structure of $\varepsilon$-LiVOPO4 projected onto the $ac$-plane. The deep blue and light blue solid circles represent the V(1) and V(2) sites, respectively. Chain 1 with the couplings $J_{1}$ and $J_{1}^{\prime}$ is formed by V(1) and chain 2 is formed by V(2) atoms with the couplings $J_{2}$ and $J_{2}^{\prime}$ via the extended V-O$\ldots$O-V paths. These chains are nearly orthogonal to each other, whereas the structural chains are parallel comprising both V1 and V2 atoms at the same time. These chains run perpendicular to the $ac$-plane. (b) A segment of the chain 1 formed by the V(1)O6 octahedra with the intrachain couplings $J_{1}$ and $J_{1}^{\prime}$. The distances $d_{1}$ and $d_{1}^{\prime}$ are the lateral displacements of the VO6 octahedra in chain 1. (c) A section of the structural chain with the V(1)-O-V(2) paths along the $b$-axis. (d) An empirical/qualitative sketch of the spin model with all possible exchange interactions involving chain 1 ($J_{1}$, $J_{1}^{\prime}$) and chain 2 ($J_{2},J_{2}^{\prime}$). ## II Methods Polycrystalline sample of $\varepsilon$-LiVOPO4 was prepared by the conventional solid-state reaction method from stoichiometric mixtures of LiPO3 and VO2 (Aldrich, 99.995%). LiPO3 was obtained by heating LiH2PO4.H2O (Aldrich, 99.995%) for 4 h at 400 ∘C in air. The reactants were ground thoroughly, pelletized, and fired at 740 ∘C for two days in flowing argon atmosphere with two intermediate grindings. Phase purity of the sample was confirmed by powder x-ray diffraction (XRD) recorded at room temperature using a PANalytical powder diffractometer (CuKα radiation, $\lambda_{\rm avg}\simeq 1.5418$ Å). Rietveld refinement of the acquired data was performed using FULLPROF software package Carvajal (1993) taking the initial cell parameters from Ref. [A. Mba _et al._ , 2012]. The low-temperature XRD data down to $15$ K were recorded using a low-temperature attachment (Oxford Phenix) to the x-ray diffractometer. Magnetization ($M$) was measured as a function of temperature (2 K $\leq T\leq$ 380 K) using the vibrating sample magnetometer (VSM) attachment to the Physical Property Measurement System [PPMS, Quantum Design]. A 3He attachment to the SQUID [MPMS-7T, Quantum Design] magnetometer was used for magnetization measurements in the low-temperature range ($0.5$ K $\leq T\leq$ $2$ K). Specific heat ($C_{\rm p}$) as a function of temperature was measured down to $0.35$ K using the thermal relaxation technique in PPMS under magnetic fields up to $14$ T. For $T\leq 2$ K, measurements were performed using an additional 3He attachment to PPMS. High-field magnetization was measured in pulsed magnetic field up to $60$ T at the Dresden High Magnetic Field Laboratory Tsirlin _et al._ (2009); Skourski _et al._ (2011). The NMR experiments on the 31P nucleus (nuclear spin $I=1/2$ and gyromagnetic ratio $\gamma/2\pi=17.235$ MHz/T) were carried out using pulsed NMR technique in the temperature range 1.6 K $\leq T\leq$ 230 K. The 31P NMR spectra as a function of temperature were obtained either by sweeping the field at a fixed frequency or by taking the Fourier transform (FT) of the echo signal, keeping the magnetic field fixed. The NMR shift $K(T)$ = [$H_{\rm ref}-H(T)$]/$H(T)$ was determined by measuring the resonant field $H(T)$ of the sample with respect to the standard H3PO4 sample (resonance frequency $H_{\rm ref}$). The 31P nuclear spin-lattice relaxation rate ($1/T_{1}$) was measured using the inversion recovery technique at different temperatures. Details of the positions of both P sites with respect to the magnetic V4+ centers are given in Table 1. Density-functional (DFT) band-structure calculations were performed in the FPLO code Koepernik and Eschrig (1999) using experimental crystal structure from Ref. [Lavrov _et al._ , 1982] and the Perdew-Burke-Ernzerhof (PBE) flavor of the exchange-correlation potential Perdew _et al._ (1996). Exchange couplings were obtained within superexchange theory or by a mapping procedure Tsirlin (2014) using total energies of collinear spin configurations calculated within DFT+$U$, where the on-site Coulomb repulsion $U_{d}=5$ eV and Hund’s coupling $J_{d}=1$ eV Nath _et al._ (2008a); Tsirlin _et al._ (2008) were used to account for strong correlations in the V $3d$ shell. All calculations are performed for the $4\times 4\times 4$ $k$-mesh. To resolve all pertinent exchange couplings, the $(\mathbf{a}_{t}+\mathbf{c}_{t})\times\mathbf{b}_{t}\times(\mathbf{a}_{t}-\mathbf{c}_{t})$, $\mathbf{a}_{t}\times 2\mathbf{b}_{t}\times\mathbf{c}_{t}$, and $(\mathbf{a}_{t}+\mathbf{b}_{t})\times 2\mathbf{b}_{t}\times\mathbf{c}_{t}$ supercells were used, where $\mathbf{a}_{t}$, $\mathbf{b}_{t}$, and $\mathbf{c}_{t}$ are lattice vectors of the triclinic unit cell given in Ref. [Lavrov _et al._ , 1982]. ## III Results ### III.1 X-ray Diffraction Figure 2: X-ray powder diffraction patterns of $\varepsilon$-LiVOPO4 at two different temperatures ($T=300$ K and $15$ K). The solid lines denote the Rietveld refinement fit of the data. The Bragg-peak positions are indicated by green vertical bars and bottom blue line indicates the difference between the experimental and calculated intensities. Figure 3: Lattice parameters (a) $a$, (b) $b$, (c) $c$, (d) $\alpha$, (e) $\beta$, and (f) $\gamma$ vs temperature. The unit cell volume ($V_{\rm cell}$) is plotted as a function of temperature in right $y$-axis of (c) and the solid line represents the fit using Eq. (1). In order to confirm the phase purity and to study the temperature variation of the crystal structure, the powder XRD patterns are analyzed for $15$ K $\leq$ $T$ $\leq$ $300$ K. The XRD patterns at two end temperatures, $T=300$ K and 15 K, along with the refinement are shown in Fig. 2. At room temperature, all the peaks could be indexed based on the triclinic (space group: $P\bar{1}$) structure, implying phase purity of the sample. The refined lattice parameters [$a=6.729(1)$ Å, $b=7.194(1)$ Å, $c=7.920(2)$ Å, $\alpha=89.82(2)^{\circ}$, $\beta=91.2288(2)^{\circ}$, and $\gamma=116.8799(2)^{\circ}$] at room temperature are in good agreement with the previous report A. Mba _et al._ (2012). Identical XRD patterns with no extra peaks are observed in the whole measured temperature range, which excludes any structural phase transition or lattice deformation in $\varepsilon$-LiVOPO4, unlike other spin gap compounds NaTiSi2O6 Isobe _et al._ (2002); Popović _et al._ (2004), CuGeO3 Hirota _et al._ (1994), NaV2O5Fujii _et al._ (1997), and K-TCNQ Lépine _et al._ (1978). The variation of both lattice parameters and unit cell volume ($V_{\rm cell}$) as a function of temperature are shown in Fig. 3. The cell parameters $b$ and $c$ decrease systematically, while the rest of the parameters ($a$, $\alpha$, $\beta$, and $\gamma$) rise with decreasing temperature. However, the overall unit cell volume shrinks upon cooling. The temperature variation of $V_{\rm cell}$ was fitted by the equation Kittel (1986); Bag _et al._ (2018) $V_{\rm cell}(T)=\gamma U(T)/K_{0}+V_{0},$ (1) where $V_{0}$ is the unit cell volume at $T$ = $0$ K, $K_{0}$ is the bulk modulus, and $\gamma$ is the Gr$\ddot{\rm u}$neisen parameter. The internal energy $U(T)$ can be expressed in terms of the Debye approximation as $U(T)=9pk_{\rm B}T\left(\frac{T}{\theta_{\rm D}}\right)^{3}\int_{0}^{\theta_{\rm D}/T}\dfrac{x^{3}}{e^{x}-1}dx.$ (2) In the above, $p$ stands for the total number of atoms in the unit cell and $k_{\rm B}$ is the Boltzmann constant. The best fit of the data down to 15 K [see Fig. 3(c)] was obtained with the Debye temperature $\theta_{D}\simeq 530$ K, $\frac{\gamma}{K_{0}}\simeq 5.78\times 10^{-5}$ Pa-1, and $V_{0}\simeq 340.5$ Å3. ### III.2 Magnetization Figure 4: Upper panel: $\chi(T)$ measured in $\mu_{0}H=0.5$ T. The two shoulders of the broad maximum are indicated by the vertical arrows. Inset: low-$T$ $\chi(T)$ measured in different applied fields. Lower panel: $1/\chi$ vs $T$ and the solid line is the CW fit using Eq. (3). The temperature-dependent bulk magnetic susceptibility $\chi~{}(\equiv M/H$) of $\varepsilon$-LiVOPO4 measured in an applied field of $\mu_{0}H=0.5$ T is shown in the upper panel of Fig. 4. It is consistent with earlier reports Chung _et al._ (2019); Yang _et al._ (2008); Onoda and Ikeda (2013) and exhibits a very broad maximum with two shoulders at around $T_{\chi}^{\rm max1}\simeq 11$ K and $T_{\chi}^{\rm max2}\simeq 27$ K. Such a broad maximum should not be mistaken with a magnetic transition Chung _et al._ (2019) and mimics the short-range ordering of a low-dimensional quantum magnet. However, already the fact that two shoulders are observed in the susceptibility indicates the presence of two nonequivalent spin chains in $\varepsilon$-LiVOPO4. Since peak position is related to the intrachain exchange coupling Johnston _et al._ (2000), we can estimate relative strengths of the interactions in the two chains, $\bar{J}_{2}/\bar{J}_{1}\simeq 2.45$, where $\bar{J}_{1}=(J_{1}+J_{1}^{\prime})/2$ and $\bar{J}_{2}=(J_{2}+J_{2}^{\prime})/2$ are average couplings in the chain 1 and chain 2, respectively. The susceptibility alone does not give information on which of the chains features stronger couplings, but our DFT calculations (Sec. III.5) suggest that stronger interactions occur within chain 2. Below $T_{\chi}^{\rm max1}$, $\chi$ decreases rapidly suggesting the opening of a spin gap. However, below $2$ K, a large upturn is observed, which can be attributed to the effect of extrinsic paramagnetic contributions. As shown in the inset of the upper panel of Fig. 4, with the application of magnetic field this low-temperature Curie tail gets suppressed. Moreover, our powder XRD suggests high purity of the sample. Therefore, this low-temperature upturn in $\chi(T)$ may be due to the uncorrelated V4+ free spins or chain-end effects that largely affect $\chi(T)$ at low temperatures Eggert and Affleck (1995). To extract the magnetic parameters, we first fitted the $1/\chi(T)$ data (see the lower panel of Fig. 4) above 150 K by the modified Curie-Weiss (CW) law, $\chi(T)=\chi_{0}+\frac{C}{T-\theta_{\rm CW}},$ (3) where $\chi_{\rm 0}$ represents the temperature-independent susceptibility, which includes the Van Vleck paramagnetic and core diamagnetic contributions, $C$ is the Curie constant, and $\theta_{\rm CW}$ is the CW temperature. The resulting fitting parameters are $\chi_{0}\simeq 7.76\times 10^{-5}$ cm3/mol-V4+, $C\simeq 0.383$ cm3K/mol-V4+, and $\theta_{\rm CW}\simeq-13.4$ K. Negative value of $\theta_{\rm CW}$ indicates that the dominant interactions between the V4+ spins are antiferromagnetic (AFM) in nature. The effective moment was calculated by using the experimental value of $C$ in the relation $\mu_{\rm eff}=\sqrt{3k_{\rm B}C/N_{\rm A}}$, where $N_{\rm A}$ is the Avogadro’s number. The calculated value of $\mu_{\rm eff}\simeq 1.72\mu_{\rm B}$/V4+ is very close to the theoretical spin-only value [$\mu_{\rm eff}=g\sqrt{S(S+1)}\simeq 1.73\mu_{\rm B}$] for $S=1/2$, assuming the Land$\acute{e}$ $g$ factor $g=2$. The experimental value of $\mu_{\rm eff}$ corresponds to a slightly reduced $g$ value of $g\simeq 1.98$ which is identical to the case of other V4+-based compounds Mukharjee _et al._ (2019); Tsirlin _et al._ (2011); Yogi _et al._ (2015). The core diamagnetic susceptibility ($\chi_{\rm core}$) of $\varepsilon$-LiVOPO4 was estimated to be $-6.8\times 10^{-5}$ cm3/mol by adding the $\chi_{\rm core}$ of the individual ions Li1+, V4+, P5+, and O2- Selwood (2013). The Van Vleck paramagnetic susceptibility ($\chi_{\rm VV}$) of $\sim 14.6\times 10^{-5}$ cm3/mol is obtained by subtracting $\chi_{\rm core}$ from $\chi_{\rm 0}$. Figure 5: Magnetization vs field measured at $T=1.5$ K using pulsed magnetic field. Inset: $dM$/$dH$ vs $H$ to highlight the critical fields $H_{\rm c1}$, $H_{\rm c2}$, and $H_{\rm c3}$. Two alternating spin chains with different exchange parameters should also manifest themselves in the magnetization process. Indeed, experimental magnetization curve measured at the base temperature of $T=1.5$ K (Fig. 5) shows a kink due to the gap closing at $\mu_{0}H_{\rm c1}\simeq 5.6$ T that corresponds to the spin gap of $\Delta_{\rm 0}^{\rm M}/k_{\rm B}=g\mu_{0}\mu_{\rm B}H_{\rm c1}/k_{\rm B}\simeq 7.3$ K. At $\mu_{0}H_{\rm c2}\simeq 25$ T, half of the spins are saturated, with the $\frac{1}{2}$-plateau observed up to $\mu_{0}H_{\rm c3}\simeq 35$ T. At even higher fields, the remaining spins are gradually polarized with nearly 80 % of the saturation magnetization reached at 58 T, the highest field of our experiment. The two-step increase of the magnetization with the spin gap and intermediate $\frac{1}{2}$-plateau is common for dimers built by two spin-1 ions Samulon _et al._ (2009). However, $\varepsilon$-LiVOPO4 contains spins-$\frac{1}{2}$, so this behavior should have a different origin. We infer that at $\mu_{0}H_{\rm c1}$ the spin gap corresponding to chain 1 is closed. This chain is fully polarized at $\mu_{0}H_{\rm c2}$, whereas at $\mu_{0}H_{\rm c3}$ the gap corresponding to chain 2 is closed, and the magnetization increases further. Such a scenario is confirmed by our quantitative analysis in Sec. III.5. ### III.3 Specific heat Figure 6: Upper panel: specific heat $C_{\rm p}$ vs $T$ of $\varepsilon$-LiVOPO4 in zero applied field. The dashed line stands for the phonon contribution to the specific heat ($C_{\rm ph}$) using Debye fit [Eq. (4)]. The solid line is the magnetic contribution to the specific heat ($C_{\rm mag}$). Inset: $C_{\rm mag}$ vs $T$ in the low-temperature region. The solid line is an exponential fit as described in the text. Lower panel: $C_{\rm mag}/T$ and $S_{\rm mag}$ vs $T$ in the left and right $y$-axes, respectively. Inset: $C_{\rm p}/T$ vs $T$ in different magnetic fields in the low-temperature regime. The specific-heat ($C_{\rm p}$) data measured under zero field are shown in the upper panel of Fig. 6. No sharp anomaly/peak was noticed down to $T=0.35$ K, thus ruling out the possibility of any magnetic or structural transition. A broad hump associated with low-dimensional short-range order is observed around $T_{C}^{\rm max}$ $\simeq 6$ K which moves weakly toward low temperatures with increasing magnetic field (see the inset of the lower panel of Fig. 6), reflecting the closing of the spin gap. The gap was estimated by fitting the data below 4 K with the activated behavior, $C_{\rm mag}\propto\exp(-\Delta_{0}^{\rm C}/k_{\rm B}T)$. The fit, as illustrated in the inset of the upper panel of Fig. 6, returns the zero-field spin gap of $\Delta_{0}^{\rm C}/k_{\rm B}\simeq 7.3$ K that matches nicely the value obtained from the high-field magnetization data. Typically, in magnetic insulators, the high-temperature part of $C_{\rm p}$ is dominated by the phonon contribution, whereas the magnetic contribution becomes prominent at low temperatures. To estimate the phonon contribution ($C_{\rm ph}$), the experimental data at high temperatures (60 K $\leq T\leq$ 110 K) were fitted by a linear combination of three Debye functions Nath _et al._ (2008b) $C_{\rm ph}(T)=9R\displaystyle\sum\limits_{\rm n=1}^{3}c_{\rm n}\left(\frac{T}{\theta_{\rm Dn}}\right)^{3}\int_{0}^{\frac{\theta_{\rm Dn}}{T}}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx.$ (4) In the above, $R$ is the universal gas constant, the coefficients $c_{\rm n}$ stand for the groups of different atoms present in the crystal, and $\theta_{\rm Dn}$ are the corresponding Debye temperatures. The $C_{\rm ph}$ was extrapolated down to low temperatures and subtracted from the total specific heat to obtain the magnetic contribution to the specific heat ($C_{\rm mag}$). The obtained $C_{\rm mag}(T)$ is presented in the upper panel of Fig. 6. The accuracy of the above fitting procedure was further verified by calculating the magnetic entropy $S_{\rm mag}$ obtained by integrating $C_{\rm mag}/T$ (see the lower panel of Fig. 6). The value of $S_{\rm mag}$ is calculated to be $\sim 5.9$ J/mol K at $T\simeq 100$ K, which is close to $S_{\rm mag}=R\ln 2=5.76$ J/mol K expected for spin $\frac{1}{2}$. As shown in the upper panel of Fig. 6, $C_{\rm mag}$ develops two broad maxima at $T_{C}^{\rm max1}\simeq 7$ K and $T_{C}^{\rm max2}\simeq 27$ K, similar to the two shoulders in the $\chi(T)$ data. The clear separation of these maxima indicates the different interaction strength in chains 1 and 2. Figure 7: $C_{\rm p}/T$ vs $T$ measured in different applied magnetic fields up to 14 T. With the spin gap closed around $\mu_{0}H_{\rm c1}\simeq 5.6$ T, the system may enter a long-range order (LRO) state. This state is indeed observed in the specific-heat data measured above $\mu_{0}H_{\rm c1}$. As shown in Fig. 7, no anomaly in $C_{\rm p}$ is found down to $T=0.35$ K and up to $\mu_{0}H=5$ T. However, for $\mu_{0}H>7$ T a clear anomaly appears indicating the onset of a field-induced magnetic LRO ($T_{\rm N}$). This peak shifts toward higher temperature with increasing magnetic field. ### III.4 31P NMR As mentioned previously, $\chi(T)$ does not show an exponential decrease at low temperatures anticipated for a gapped spin system, and may be influenced by extrinsic contributions. To access the intrinsic susceptibility of $\varepsilon$-LiVOPO4, we performed NMR measurements on the 31P nuclei. #### III.4.1 NMR Shift Figure 8: Typical 31P field-sweep NMR spectra of $\varepsilon$-LiVOPO4 measured at 24.96 MHz at different temperatures. The dashed lines track the shifting of NMR line positions. The vertical line corresponds to the 31P reference line ($H_{\rm ref}$) measured for H3PO4. The solid line is the fit of the spectra at $T=50$ K using a double Gaussian function. Figure 8 presents the field-sweep 31P NMR spectra measured over a wide temperature range. At high temperatures, a narrow and symmetric spectral line typical for a $I=1/2$ nucleus is observed. As the temperature is lowered, the line shape becomes asymmetric, followed by a complete splitting of the two spectral lines below about 120 K. This suggests the existence of two nonequivalent P sites, P(1) and P(2), with a different crystallographic environment, which is consistent with the structural data (Table 1). Both lines shift with temperature and merge below about 4 K. The absence of drastic line broadening and/or line splitting down to 1.6 K, except for the one caused by the two nonequivalent P sites, rules out the occurrence of any structural and magnetic transitions. The line with a lower intensity shifts more strongly than the one with the higher intensity. The former can be assigned to P(2) and the latter to P(1), because the P(2)O4 tetrahedra mediate stronger exchange interactions, $J_{1}$ in chain 1 and $J_{2}$ in chain 2, whereas the P(1)O4 tetrahedra mediate weaker interactions $J_{1}^{\prime}$ and $J_{2}^{\prime}$, respectively (see Sec. III.5). At $T=1.6$ K, the position of the peak is very close to the zero shift value suggesting that the ground state of $\varepsilon$-LiVOPO4 is nonmagnetic. The temperature-dependent NMR shift $K(T)$ for both 31P sites was extracted by fitting each spectrum to a sum of two Gaussian functions. The results are shown in the upper panel of Fig. 9. One advantage of the NMR shift over the bulk $\chi(T)$ is that the Curie-Weiss term due to foreign phases and/or defects does not appear. The randomly distributed defects/impurities only broaden the NMR line but do not contribute to the NMR shift Walstedt and Walker (1974). Therefore, $K(T)$ is more favorable than bulk $\chi(T)$ data for a reliable determination of magnetic parameters. The $K(T)$’s corresponding to the P(1) and P(2) sites pass through a very broad maximum, similar to the $\chi(T)$ data. The overall temperature dependence is similar for P(1) and P(2), but the absolute values differ due to the different hyperfine couplings. The presence of several distinct P sites in the structure is reminiscent of the ambient-pressure polymorph of (VO)2P2O7 with its two non-equivalent alternating spin-$\frac{1}{2}$ chains. In that case, different phosphorous sites probe the behavior of different spin chains in the structure Yamauchi _et al._ (1999); Kikuchi _et al._ (1999). In contrast, each of the P sites in $\varepsilon$-LiVOPO4 is coupled to both spin chains, so it is not possible to probe $K(T)$ separately. Two distinct shoulders are observed in $K(T)$ at $T_{K}^{\rm max1}\simeq 10$ K and $T_{K}^{\rm max2}\simeq 26$ K and closely resemble bulk magnetic susceptibility (Fig. 4). At low temperatures, both shifts decrease rapidly toward zero, suggesting the opening of a spin gap between the singlet ($S=0$) ground state and triplet ($S=1$) excited states. As NMR shift is insensitive to the impurities and defects, one can use it to accurately measure the intrinsic spin susceptibility. In the upper panel of Fig. 9, we show that for both P sites, $K(T)$ decreases toward zero, which is in contrast to the upturn observed in the low-temperature $\chi(T)$ data. This confirms the extrinsic nature of the low-temperature upturn observed in $\chi(T)$. In powder samples, the defects often break spin chains, with the unpaired spins at the ends of finite chains giving rise to the staggered magnetization, which also appears as the low- temperature Curie tail in $\chi(T)$. Figure 9: Upper panel: temperature-dependent 31P NMR shift $K(T)$ measured at 24.96 MHz as a function of temperature for both P(1) and P(2) sites. Lower panel: $K$ vs $\chi$ (measured at 1.5 T) with temperature as an implicit parameter. The solid lines are the linear fits. The direct relation between $K(T)$ and spin susceptibility $\chi_{\rm spin}$ can be written as $K(T)=K_{0}+\frac{A_{\rm hf}}{N_{A}\mu_{\rm B}}\chi_{\rm spin},$ (5) where $K_{0}$ is the temperature-independent NMR shift and $A_{\rm hf}$ is the total hyperfine coupling constant between the 31P nuclei and V4+ spins. The $A_{\rm hf}$ consists of the contributions due to transferred hyperfine coupling and nuclear dipolar coupling constants. Since both of the aforementioned couplings are temperature-independent, $K(T)$ is a direct measure of $\chi_{\rm spin}$. Using Eq. (5), $A_{\rm hf}$ can be calculated from the slope of the linear $K$ vs $\chi$ plot with temperature as an implicit parameter. The lower panel of Fig. 9 presents the $K$ vs $\chi$ plots for both P sites showing linear behavior at high temperatures. From the linear fit, the hyperfine coupling constants $A_{\rm hf}^{\rm P(1)}\simeq 3290$ Oe/$\mu_{\rm B}$ and $A_{\rm hf}^{\rm P(2)}\simeq 7068$ Oe/$\mu_{\rm B}$ are estimated for the P(1) and P(2) sites, respectively. Thus, the P(2) site is coupled with the V4+ ions twice stronger than the P(1) site. These values are comparable to the $A_{\rm hf}$ values reported for other spin chains with similar interaction geometry Mukharjee _et al._ (2019); Nath _et al._ (2008c, 2005). Figure 10: Upper panel: $K$ vs $T$ for both P(1) and P(2) sites. The solid lines are the fits below 5 K by $K=K_{0}+mT^{(d/2-1)}e^{-\Delta/k_{\rm B}T}$, fixing $\Delta/k_{\rm B}\simeq 5.5$ K. Lower panel: $(K-K_{0})e^{5.5/T}$ vs $T$ is shown in the low-temperature regime. The solid lines are the fits with $d\simeq 2.83$ and 2.73 for the P(1) and P(2) sites, respectively. It is also possible to estimate the value of spin gap and the effective lattice dimensionality ($d$) by analyzing the low-temperature $K(T)$ data. Typically, the non-negligible interchain couplings are inevitably present in real materials and significantly influence the $K(T)$ data at low temperatures. For a $d$-dimensional system, the susceptibility at $k_{\rm B}T\ll\Delta$ can be approximated as Taniguchi _et al._ (1995) $\chi_{d}\propto T^{(d/2)-1}\times e^{-\Delta/k_{\rm B}T}.$ (6) Our NMR experiments were carried out in the magnetic field of $\mu_{0}H=1.4$ T. Assuming a linear variation of $\Delta/k_{\rm B}$ with $H$, the zero-field spin gap $\Delta_{\rm 0}/k_{\rm B}\simeq 7.3$ K determined from the specific heat and magnetization is expected to be reduced to $\Delta_{\rm 1.4~{}T}/k_{\rm B}\simeq 5.5$ K at $\mu_{0}H=1.4$ T. In the upper panel of Fig. 10, the $K(T)$ data below 5 K are fitted by $K=K_{0}+mT^{(d/2-1)}e^{-\Delta/k_{\rm B}T}$, fixing $\Delta/k_{\rm B}\simeq 5.5$ K where $m$ is a proportionality constant. In order to highlight the low- temperature linear regime and the fit, in the lower panel of Fig. 10, we have plotted $(K-K_{0})e^{5.5/T}$ vs $T$ in the log-log plot. The fit in this region returns $K_{0}\simeq 0.04694$%, $m\simeq 0.4654$ %/K1/2, and $d\simeq 2.83$ for the P(1) site and $K_{0}\simeq 0.0462$%, $m\simeq 1.1663$ %/K1/2, and $d\simeq 2.73$ for the P(2) site. The average value of $d\simeq 2.78$, which is very close to 3, suggests the dominant role of 3D spin-spin correlations at low temperatures. Indeed, we find rather strong interchain couplings from DFT (Sec. III.5). #### III.4.2 Spin-lattice relaxation rate $1/T_{1}$ Figure 11: Upper panel: recovery of the longitudinal magnetization as a function of waiting time $t$ at three different temperatures for the P(1) site. Solid lines are the fits using Eq. (7). Lower panel: temperature variation of $1/T_{1}$ measured in different magnetic fields. For $\mu_{0}H=1.4$ T, measurements are done on both P(1) and P(2) sites while for other fields, only P(1) site is probed. The solid line represents the fit using $1/T_{1}\propto T^{\alpha}$ to the 10 T data at low temperatures with $\alpha\simeq-0.6$. The spin-lattice relaxation rate $1/T_{1}$ measures the dynamic susceptibility, which provides direct access to the low-energy spin excitations or spin-spin correlation function Moriya (1956). $1/T_{1}$ was measured at the central peak position of the spectra at each temperature using an inversion pulse sequence down to $T=1.6$ K. Since 31P has the nuclear spin $I=1/2$, the value of $T_{1}$ at each temperature was estimated by fitting the recovery curve of the longitudinal magnetization to a single exponential function $\frac{1}{2}\left[\frac{M(0)-M(t)}{M(0)}\right]=Ae^{-(t/T_{1})}.$ (7) Here, $M(t)$ is the nuclear magnetization at a time $t$ after the inversion pulse and $M(0)$ is the equilibrium magnetization. The upper panel of Fig. 11 shows the recovery curves at three different temperatures probed for the P(1) site at $\mu_{0}H=1.4$ T. The recovery curves show linearity over one and half decades when the $y$ axis is plotted in log scale. $1/T_{1}$ was estimated by fitting Eq. (7) in this linear regime. $1/T_{1}$ estimated from the above fit is shown in the lower panel of Fig. 11. Our measurements are done at different field values ranging from 1.4 T to 10 T, above and below $H_{\rm c1}$. For $\mu_{0}H=1.4$ T, the measurements are done at both P(1) and P(2) sites and over the whole temperature range, while for other fields only the P(1) site is probed and the measurements are restricted to low temperatures ($T<30$ K). Since there is a large difference in the magnitude of $A_{\rm hf}$ for the P(1) and P(2) sites, they experience different local magnetic fields induced by the V4+ spins. Therefore, it is expected that the resulting temperature-dependent $1/T_{1}$ will have different values accordingly. Indeed, for $\mu_{0}H=1.4$ T, $1/T_{1}$ of the P(2) site has larger magnitude than that of the P(1) site, as $A_{\rm hf}$ of P(2) is larger than that of P(1). For both the P-sites, $1/T_{1}$ follows a temperature-independent behavior due to the random fluctuation of the paramagnetic moments at high temperatures Moriya (1956). At lower temperatures, $1/T_{1}$ starts to decrease and below about 10 K it drops rapidly towards zero. The 1.6 K value is almost two orders of magnitude lower than the room-temperature one, indicating the opening of a spin gap. In higher fields, the low-temperature values of $1/T_{1}$ increase and show an upward curvature. The spin gap should be closed at $H_{\rm c1}$ and thereafter an AFM LRO sets in. Therefore, we measured $1/T_{1}$ at different fields above $H_{\rm c1}$. The increase in the low-temperature values of $1/T_{1}$ confirms the closing of the spin gap and the growth of 3D AFM correlations due to the field-induced LRO Klanjšek _et al._ (2008). Since our measurements are limited down to 1.6 K, we are unable to detect the field-induced LRO from the $1/T_{1}$ data. Nevertheless, the systematic increase of $1/T_{1}$ with field implies that $T_{\rm N}$ shifts toward higher temperatures with increasing $H$, in good agreement with our $C_{\rm p}(T)$ data, where field-induced LRO is detected above $H_{\rm c1}$. The data sets for the P(1) site at various field values exhibit a fanlike pattern in the low-temperature regime. Similar behavior has been reported for spin-$1/2$ ladder compound (C5H12N)2CuBr4 (BPCB) and spin-1 chain compound NiCl2-4SC(NH2)2 (DTN) Mukhopadhyay _et al._ (2012). It is expected that $1/T_{1}(T)$ data for different fields, around the quantum critical point (QCP) (i.e., around $H_{\rm c1}$) should follow a power-law behavior, $1/T_{1}\propto T^{\alpha}$. The exponent $\alpha$ should vary across $H_{\rm c1}$, resulting in a fanlike pattern of the $1/T_{1}$ data. For instance, in the gapless TLL region ($H>H_{\rm c1}$), the power law has the form $1/T_{1}\propto T^{1/(2K)-1}$, with $K$ being the TLL exponent Klanjšek _et al._ (2008); Giamarchi and Tsvelik (1999). The value of $K$ increases gradually as one approaches QCP from the TLL regime and reaches the maximum value 1 that corresponds to $\alpha[=1/(2K)-1]=-0.5$. In order to test this formalism, we fitted the $1/T_{1}$ data below 5 K measured at 10 T by the power law. As shown in the lower panel of Fig. 11, our fit yields $\alpha\simeq-0.6$, which is very close to the predicted value ($\alpha=-0.5$) in the TLL regime. This indicates the pronounced 1D character of the compound. Figure 12: $1/T_{1}e^{5.5/T}$ vs $T$ in the log-log plot. The solid lines are the fits, as described in the text with $d\simeq 2.74$ and 3.2, for the P(1) and P(2) sites, respectively. On the other hand, from analyzing the $K(T)$ data in magnetic fields below $H_{c1}$ we already inferred that 3D spin-spin correlations caused by the interchain coupling may be relevant. According to Mukhopadhyay et al. Mukhopadhyay _et al._ (2012), in the gapped region ($H\leq H_{c1}$) with 3D correlations $1/T_{1}$ follows an activated behavior of the form, $1/T_{1}\propto T^{\alpha_{\rm 0}}\exp\left[\frac{g\mu_{B}(H-H_{\rm c1})}{k_{\rm B}T}\right].$ (8) The exponent $\alpha_{0}$ in the above equation depends on the effective dimension of the magnon dispersion relation as set by the thermal fluctuations $k_{\rm B}T$. With increasing temperature, $\alpha_{0}$ slowly decreases from 2 for $k_{\rm B}T<J_{\rm 3D}$ (3D regime) to 0 for $J_{3D}<k_{\rm B}T<J_{1D}$ (1D regime). In order to detect the effective dimensionality of the spin-spin correlations, Eq. (8) was further simplified as $1/T_{1}\propto T^{d-1}e^{-\Delta_{\rm 1.4\,T}/k_{\rm B}T}$ by putting $\alpha_{\rm 0}=d-1$ and $\Delta_{\rm 1.4~{}T}/k_{\rm B}=g\mu_{\rm B}(H_{\rm c1}-H)/k_{\rm B}$ and fitted to the low temperature $1/T_{1}$ data. The fit for $T\leq 5$ K, taking $\Delta_{\rm 1.4\,T}/k_{\rm B}\simeq 5.5$ K results in $d\simeq 2.74$ and 3.2 for the P(1) and P(2) sites, respectively. The average value of $d\simeq 2.97$ is consistent with the value obtained from the $K(T)$ analysis. This further confirms the importance of interchain couplings at low temperatures, where activated behavior is observed. Figure 12 presents the $1/T_{1}e^{5.5/T}$ vs $T$ plot for the data measured at $\mu_{0}H=1.4$ T along with the fit using Eq. (8). Both the $x$ and $y$ axes are shown in log scale to highlight the power-law prefactor to the activated behavior, at low temperatures. ### III.5 Microscopic magnetic model Similar to Refs. Tsirlin _et al._ (2011); Arjun _et al._ (2019); Mukharjee _et al._ (2019), we use two complementary computational methods to derive exchange couplings in $\varepsilon$-LiVOPO4. For a single magnetic orbital of V4+, superexchange theory yields antiferromagnetic exchange couplings $J_{i}^{\rm AFM}=4t_{i}^{2}/U_{\rm eff}$, where $t_{i}$ are V–V hoppings extracted from the uncorrelated (PBE) band structure, and $U_{\rm eff}$ is an effective Coulomb repulsion in the V $3d$ bands. On the other hand, exchange couplings $J_{i}$ can be obtained from DFT+$U$ by a mapping procedure, where both ferromagnetic and antiferromagnetic contributions are taken into account. Table 2: Exchange couplings in $\varepsilon$-LiVOPO4. The $t_{i}$ values are the V–V hoppings extracted from the uncorrelated band structure, and show relative strengths of the AFM contributions to the exchange couplings $J_{i}^{\rm AFM}\sim t_{i}^{2}$. The $J_{i}$ are exchange interactions obtained by the mapping procedure within DFT+$U$. | $d_{\rm V-V}$ (Å) | | $t_{i}$ (meV) | $J_{i}$ (K) ---|---|---|---|--- $J_{1}$ | 5.250 | V1–V1 | $-72$ | 33 $J_{1}^{\prime}$ | 5.101 | V1–V1 | $-55$ | 23 $J_{2}$ | 5.275 | V2–V2 | $-117$ | 63 $J_{2}^{\prime}$ | 5.303 | V2–V2 | $-78$ | 22 $J_{c1}$ | 3.599 | V1–V2 | 0 | $-15$ $J_{c2}$ | 3.629 | V1–V2 | 0 | $-15$ $J_{a1}$ | 6.018 | V1–V2 | $-21$ | 12 $J_{a2}$ | 6.070 | V1–V2 | $-32$ | 7 In Table 2, we list the $t_{i}$ values for the uncorrelated band structure and the exchange couplings $J_{i}$ obtained from DFT+$U$. The two methods are in excellent agreement and consistently show the stronger couplings within chain 2. Moreover, we find that within each spin chain the stronger couplings involve the P(2) bridges and the weaker couplings involve the P(1) bridges. On the structural level, this difference should be traced back to the lateral displacements $d_{i}$ of the VO6 octahedra within the spin chain [Fig. 1(b)], where smaller displacement leads to a stronger coupling Mukharjee _et al._ (2019); Roca _et al._ (1998). Indeed, we find $d_{1}=0.71$ Å for $J_{1}$ vs $d_{1}^{\prime}=0.97$ Å for $J_{1}^{\prime}$ and $d_{2}=0.08$ Å for $J_{2}$ vs $d_{2}^{\prime}=0.23$ Å for $J_{2}^{\prime}$. The smaller lateral displacements $d_{2}$ and $d_{2}^{\prime}$ could also explain the overall stronger couplings in chain 2, although in this case other geometrical parameters Roca _et al._ (1998) are relevant as well, because $J_{1}$ is about as strong as $J_{2}^{\prime}$, despite the fact that $d_{1}>d_{2}^{\prime}$. Regarding the interchain couplings, the microscopic scenario is very similar to that of monoclinic $A$VO$X$O4 with $A$ = Ag, Na and $X$ = P, As Tsirlin _et al._ (2011); Arjun _et al._ (2019); Mukharjee _et al._ (2019). Shorter V–O–V bridges render $J_{c1}$ and $J_{c2}$ ferromagnetic, whereas the long- range couplings $J_{a1}$ and $J_{a2}$ are weakly antiferromagnetic. These ferromagnetic and antiferromagnetic interactions compete and make the spin lattice of $\varepsilon$-LiVOPO4 frustrated. Table 3: Exchange couplings (in K) extracted from the $\chi(T)$ and $M(H)$ data using the fits shown in Fig. 13. The susceptibility fit using Eq. (9) returns $\chi_{0}=2.7\times 10^{-5}$ cm3/mol, $C_{\rm imp}=0.013$ cm3K/mol (3.5% of paramagnetic impurities), $\theta_{\rm imp}=0.9$ K, and $g=2.03$. This $g$ value is slightly higher than 1.98 obtained from the Curie-Weiss fit, probably because the interchain couplings were not taken into account. For magnetization data, $g=1.98$ has been used as a fixed parameter. | $J_{1}$ | $J_{1}^{\prime}$ | $J_{2}$ | $J_{2}^{\prime}$ ---|---|---|---|--- $\chi(T)$ | 20 | 12 | 70 | 20 $M(H)$ | 19 | 12 | 63 | 22 Our DFT results suggest that chain 1 shows only a moderate degree of alternation ($\alpha_{1}=J_{1}^{\prime}/J_{1}\simeq 0.6$) that, together with the lower energy scale of the couplings, leads to a relatively small spin gap closed at $H_{\rm c1}$. In contrast, the alternation ratio of $\alpha_{2}=J_{2}^{\prime}/J_{2}\simeq 0.3$ renders chain 2 strongly dimerized with a larger spin gap that is closed at the much higher field $H_{\rm c3}$. The model of two alternating spin-$\frac{1}{2}$ chains was further used to calculate temperature-dependent magnetic susceptibility and field-dependent magnetization of $\varepsilon$-LiVOPO4 (Fig. 13). For the susceptibility, we used the analytical expression from Ref. Johnston _et al._ (2000) augmented by additional terms that account for the temperature-independent ($\chi_{0}$) and Curie-type impurity contributions, $\chi=\chi_{0}+\frac{C_{\rm imp}}{T+\theta_{\rm imp}}+\chi_{\rm ch1}+\chi_{\rm ch2},$ (9) where $\chi_{\rm ch1}$ and $\chi_{\rm ch2}$ are susceptibilities of two nonequivalent alternating spin-$\frac{1}{2}$ chains with the interaction parameters $J_{1}$, $J_{1}^{\prime}$ and $J_{2}$, $J_{2}^{\prime}$, respectively (same $g$ factor is used for both chains). Magnetization curve of $\varepsilon$-LiVOPO4 was modeled by the sum of simulated magnetization curves for the V(1) and V(2) sublattices obtained by the method described in Ref. Tsirlin _et al._ (2011). Figure 13: Temperature-dependent susceptibility (top) and field-dependent magnetization (bottom) of $\varepsilon$-LiVOPO4 fitted using the model of two non-equivalent alternating spin-$\frac{1}{2}$ chains, as explained in the text. See Table 3 for the fitting parameters. The fitting results listed in Table 3 show good agreement between the fits to the susceptibility and magnetization. They are also consistent with the exchange parameters calculated by DFT (Table 2). We chose not to include interchain couplings into the susceptibility fit, which became ambiguous when more than four exchange parameters were involved. The effect of the interchain couplings can be seen from the fact that for an isolated chain 1 one finds, using $J_{1}$ and $J_{1}^{\prime}$ from the susceptibility fit, the zero-field spin gap of 11.3 K, which is somewhat larger than 7.3 K obtained experimentally 222Similar to Ref. Tsirlin _et al._ (2011), magnetization curve was simulated by including a weak interchain coupling of $J_{\perp}/J_{1}=-0.05$ in order to reproduce the experimental data around $H_{\rm c1}$. This interchain coupling is much lower than estimated by DFT (Table 2), possibly because $J_{\perp}$ reflects a cumulative effect of the frustrated couplings $J_{a1}$, $J_{a2}$ and $J_{c1}$, $J_{c2}$. . ## IV Discussion and summary $\varepsilon$-LiVOPO4 features two types of alternating spin-$\frac{1}{2}$ chains that manifest themselves in the double maxima of the susceptibility and magnetic specific heat and in the two-step magnetization process. This unusual microscopic scenario is reminiscent of the ambient-pressure polymorph of (VO)2P2O7 Johnston _et al._ (2001), where two spin gaps corresponding to two types of spin chains were directly observed by NMR Yamauchi _et al._ (1999) and inelastic neutron scattering Garrett _et al._ (1997). On the other hand, large size of these gaps (35 K and 68 K, respectively) and high critical fields associated with them preclude experimental access to field-induced transitions, where triplon excitations of the spin chains consecutively condense leading to a long-range magnetic order. Figure 14: $H-T$ phase diagram of $\varepsilon$-LiVOPO4 obtained using the data points from $C_{\rm p}(T)$ measurements. The solid line corresponds to Eq. (10). Inset: $T_{\rm N}$ vs ($H-H_{\rm c1})^{1/1.5}$ to highlight the low- temperature linear regime and the agreement of the simulated curve with the experimental data points. With its lower critical fields, $\varepsilon$-LiVOPO4 offers a much better platform for studying these transitions experimentally. Indeed, we observed field-induced magnetic order already above $\mu_{0}H_{\rm c1}\simeq 5.6$ T. Transition temperature systematically increases with field and tracks the $H$-$T$ phase boundary shown in Fig. 14. The field-induced transition in gapped quantum magnets is often understood as triplon BEC. In this case, the phase boundary close to $H_{\rm c1}$ should follow the universal power law Zapf _et al._ (2014); Giamarchi and Tsvelik (1999); Nohadani _et al._ (2004), $T_{\rm N}\propto(H-H_{\rm c1})^{\frac{1}{\phi}},$ (10) where $\phi=d/2$ is the critical exponent reflecting the universality class of the quantum phase transition at $H_{\rm c1}$, and $d$ is the lattice dimensionality. In the absence of the low-temperature data immediately above $H_{\rm c1}$, we simulate the anticipated critical behavior by choosing $d=3$, $\phi=\frac{3}{2}$, and $\mu_{0}H_{\rm c1}\simeq 5.6$ T in Eq. (10). The resulting curves match our data up to 1.2 K (Fig. 14), although this agreement may be partly accidental, because the temperature of 1.2 K is rather high compared to the temperature scale of the BEC transition given by $T_{N}^{\max}$ at the tip of the BEC dome. Since 14 T is roughly the midpoint between $H_{\rm c1}$ and $H_{\rm c2}$, we expect $T_{N}^{\max}\simeq 1.7$ K, and 1.2 K is more than half of this temperature. Further measurements of the phase boundary around $H_{\rm c1}$ and below 0.7 K would be clearly interesting to confirm the tentative BEC critical behavior in $\varepsilon$-LiVOPO4. Impurity effects witnessed by the low-temperature upturn in the magnetic susceptibility (Fig. 4) may also be important. The fate of the ordered phase in fields above 14 T is of significant interest too. The two-step increase in the magnetization suggests that the transition at $H_{\rm c1}$ corresponds to chain 1 and will lead to a BEC dome between $H_{\rm c1}$ and $H_{\rm c2}$, while chain 2 remains in the singlet state up to $H_{\rm c3}$, where another BEC dome should appear. Phenomenologically, this behavior would be similar to the two-dome $H$-$T$ phase diagrams of spin-1 dimer magnets Samulon _et al._ (2009), although in $\varepsilon$-LiVOPO4 with local spin $\frac{1}{2}$ it should have a very different microscopic origin. It does in fact arise from the coexistence of the nonequivalent magnetic sublattices. We also point out that the $\frac{1}{2}$-magnetization plateau in this compound is not caused by Wigner crystallization and is thus dissimilar to the magnetization plateaus in SrCu2(BO${}_{3})_{2}$. Because $H_{\rm c3}$ lies above $H_{\rm c2}$, magnetic orders related to chain 1 and chain 2 should be mostly decoupled. This is different from other systems with multiple spin gaps, where intertwined BEC transitions lead to an unusual critical behavior, as in the dimer magnet BaCuSi2O6 Mazurenko _et al._ (2014); Allenspach _et al._ (2020). In summary, we have shown that $\varepsilon$-LiVOPO4 is a gapped quantum magnet that features singlet ground state in zero field. With two non- equivalent alternating spin-$\frac{1}{2}$ chains, it shows double maxima in the susceptibility and magnetic specific heat and a two-step increase in the magnetization. Chain 1 features weaker couplings and a weaker alternation ($J_{1}\simeq 20$ K, $\alpha_{1}\simeq 0.6$), whereas chain 2 reveals stronger couplings and lies closer to the dimer limit ($J_{2}\simeq 60$ K, $\alpha_{2}\simeq 0.3$). The zero-field spin gap of $\Delta_{0}/k_{B}\simeq 7.3$ K is closed at $\mu_{0}H_{\rm c1}\simeq 5.6$ T. The magnetization increases up to $\mu_{0}H_{\rm c2}\simeq 25$ T, flattens out within the $\frac{1}{2}$ plateau, and increases again above $\mu_{0}H_{\rm c3}\simeq 35$ T. The gap closing above $H_{\rm c1}$ leads to a field-induced LRO that can be understood as Bose-Einstein condensation of triplons. ###### Acknowledgements. 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2024-09-04T02:54:55.679893

2003.00008

arxiv-papers

spacing=nonfrench # Reduction theory for connections over the formal punctured disc Andres Fernandez Herrero ###### Abstract We give a purely algebraic treatment of reduction theory for connections over the formal punctured disc. Our proofs apply to arbitrary connected linear algebraic groups over an algebraically closed field of characteristic $0$. We also state and prove some new quantitative results. ###### Contents 1. 1 Introduction 2. 2 Some notation and definitions 1. 2.1 Preliminaries on formal connections 2. 2.2 Adjoint orbits in semisimple Lie algebras 3. 3 Regular connections 1. 3.1 Regular connections for semisimple groups 2. 3.2 Regular connections for tori and reductive groups 3. 3.3 Connections for unipotent groups 4. 3.4 Regular connections for solvable groups 5. 3.5 Regular connections for arbitrary linear algebraic groups 6. 3.6 Descent for gauge equivalence classes 4. 4 Irregular connections for $\mathbf{G}$ reductive 1. 4.1 Connections in canonical form 2. 4.2 The case when $A_{r}$ is not nilpotent 3. 4.3 The case when $A_{r}$ is nilpotent 4. 4.4 Algorithm for reductive groups and some quantitative results 5. 5 Irregular connections for arbitrary linear algebraic groups 1. 5.1 Irregular connections for solvable groups 2. 5.2 Irregular connections for arbitrary linear algebraic groups 3. 5.3 Galois cohomology for irregular connections ## 1 Introduction Let $\mathbf{k}$ be an algebraically closed field of characteristic $0$. Fix a connected linear algebraic group $\mathbf{G}$ over $\mathbf{k}$. Let $D^{*}\vcentcolon=\text{Spec}\,\mathbf{k}(({t}))$ denote the formal punctured disc over $\mathbf{k}$. In this paper we give an algebraic classification of formal $\mathbf{G}$-connections over $D^{*}$ up to gauge equivalence. The regular singular case is more explicit and is presented first (see Subsection 3.6). The more general irregular case is Proposition 5.9. These constitute the two main results of the paper. In order to get our parametrizations, we first prove that every connection can be put into canonical form [BV83] after passing to a ramified cover (Theorems 3.32 and 5.3). Next, we describe a set of representatives for canonical forms for which we can develop a clean description of Galois cohomology cocycles (see the set of conditions in Theorem 5.3). We then proceed to describe the Galois cocycles in Subsections 3.6 and 5.3. As a consequence of our arguments we obtain some new quantitative results in Subsection 4.4. Our approach to the existence of canonical forms is based on the work of Babbitt and Varadarajan [BV83]. Some of the crucial parts in their argument are analytic in nature, so they only apply when the ground field is $\mathbb{C}$. We sidestep those parts to provide a completely algebraic proof. In addition, we simplify the global structure of their inductive arguments. Our treatment of uniqueness of canonical forms is substantially different from the one in [BV83]. We choose a different set of representatives for canonical classes in order to set up our Galois cohomology argument (see the list of properties in Theorem 5.3). This allows us to avoid the use of the complex exponential map. In our setup the proof of uniqueness and the identification of the gauge transformation centralizer become elementary power series arguments. We develop separate treatments of reduction theory depending on whether $\mathbf{G}$ is reductive, unipotent or solvable. This allows us to give sharper separate statements, including some new determinacy results in the unipotent and solvable cases (see Propositions 3.25, 3.29 and 5.2). There is some related work by Schnürer on the subject. [Sch07] gives a purely algebraic proof of the reduction of formal connections when the group $\mathbf{G}$ is reductive and the connection has a regular singularity. In contrast, our arguments apply more generally to arbitrary linear algebraic groups and irregular singular connections. Let us briefly describe the body of the paper. Section 2 sets up the notation and some of the facts that are used throughout. In Section 3 we develop reduction theory for regular singular connections. Section 3 culminates with the first main result of this paper: an explicit parametrization of regular singular connections over the formal punctured disc. This is achieved in Subsection 3.6. This parametrization is rather concrete in the case of classical groups, see Example 3.42. Section 4 treats reduction theory of irregular connections for reductive groups. Section 4 also includes some of the quantitative results mentioned in the abstract. In Subsection 4.4 we give an explicit description of the reduction algorithm for reductive groups. An analysis of this algorithm yields determinacy results for both the irregular and the regular part of the canonical form in the reductive case (Proposition 4.21). We also prove a new uniform bound on the ramification needed to put connections into canonical form (see Propositions 4.19 and 4.23). Section 5 develops reduction theory for irregular connections in the case of an arbitrary linear algebraic group. Subsection 5.3 gives a parametrization of irregular connections over $D^{*}$ up to gauge equivalence; this is the second main result of this paper. Precisely, this parametrization consists of two pieces of data. The first is a connection $B$ in canonical form satisfying all five conditions listed in Theorem 5.3. $B$ determines the connection up to ramified gauge equivalence (i.e. when we allow passing to ramified covers of the formal disc). The second piece of data is a $B$-twisted $\mu_{b}$-cocycle, which is an element in the centralizer of the residue of $B$ satisfying certain additional condition. See Definition 5.7 and the paragraph following it for an explanation. Section 5 ends with Proposition 5.10, where we use this parametrization to give another proof of a result found in [CK17]. ## 2 Some notation and definitions ### 2.1 Preliminaries on formal connections We will always work over a fixed algebraically closed field $\mathbf{k}$ of characteristic $0$. An undecorated product of $\mathbf{k}$-schemes (e.g. $X\times S$) should always be interpreted as a fiber product over $\mathbf{k}$. $\mathbf{G}$ will be a connected linear algebraic group over $\mathbf{k}$ and $\mathfrak{g}=\text{Lie}(\mathbf{G})$ will be the corresponding Lie algebra. We let $\mathcal{O}=\mathbf{k}[[{t}]]$ denote the ring of formal power series over $\mathbf{k}$ and $F=\mathbf{k}(({t}))$ denote the corresponding field of Laurent series. $\mathcal{O}$ is a discrete valuation ring with maximal ideal $t\mathcal{O}$. Recall that the module of Kähler differentials $\Omega^{1}_{\mathcal{O}/\mathbf{k}}$ classifies $\mathbf{k}$-derivations from $\mathcal{O}$. It is spanned as an $\mathcal{O}$-module by formal elements $df$ for every $f\in\mathcal{O}$, subject to the relations $d(fg)=fdg+gdf$. We will work with the module of continuous Kähler differentials $\hat{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}$ , which is defined to be the completion $\hat{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}\vcentcolon=\underset{n}{\varprojlim}\,\,\,{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}\,/\,\,t^{n}\,{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}$ This is a free $\mathcal{O}$-module of rank $1$. The natural completion map $\widehat{(-)}:\Omega_{\mathcal{O}/\mathbf{k}}^{1}\rightarrow\hat{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}$ can be thought of as the projection onto the quotient obtained by adding the extra relations coming from allowing termwise differentiation of power series. ###### Remark 2.1. The module of ordinary Kähler differentials $\Omega^{1}_{\mathcal{O}/\mathbf{k}}$ is not finitely generated as an $\mathcal{O}$-module. We don’t want to work with $\Omega^{1}_{\mathcal{O}/\mathbf{k}}$, because the relations above do not include classical intuitive identities like $d(e^{t})=e^{t}dt$. That is the reason why we use continuous Kähler differentials instead. For any positive natural number $b$, let $F_{b}\vcentcolon=\mathbf{k}(({t^{\frac{1}{b}}}))$. This is a finite Galois extension of $F$ with Galois group canonically isomorphic to $\mathbf{\mu}_{b}$, the group of $b$-roots of unity in $\mathbf{k}$. Under this isomorphism, we have that $\gamma\in\mu_{b}$ acts by $\gamma\cdot t^{\frac{1}{b}}=\gamma^{-1}t^{\frac{1}{b}}$. Notice that the choice of a primitive root of unity yields an identification $\mu_{b}\cong\mathbb{Z}/b\,\mathbb{Z}$, since we are working in characteristic $0$. A well known theorem of Puiseux states that the algebraic closure of $F$ is $\overline{F}=\bigcup_{b\geq 1}F_{b}$. In this paper we will work with a (right) $\mathbf{G}$-torsor $P$ over the formal punctured disc $D^{*}\vcentcolon=\text{Spec}\,F$. We know that $P$ can be trivialized, meaning that $P\cong\text{Spec}\,F\times\mathbf{G}$ as right $\mathbf{G}$-torsors. This follows from theorems of Tsen and Springer, see [Ser02] page 80 - 3.3(b) and page 132 - 2.3(c). A formal connection $A$ on $P$ is a function from the set of trivializations of $P$ into $\mathfrak{g}\,\otimes_{\mathbf{k}}\,\hat{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}\left[\frac{1}{t}\right]$ that satisfies a certain transformation law. In order to describe the transformation law we need some notation. Let $T_{\mathbf{G}}$ be the tangent sheaf of $\mathbf{G}$. There is a natural trivialization $T_{\mathbf{G}}\cong\mathfrak{g}\,\otimes_{\mathbf{k}}\mathcal{O}_{\mathbf{G}}$ given by left translation. Therefore, we get an isomorphism $\mathfrak{g}\,\otimes_{\mathbf{k}}\,\Omega^{1}_{\mathbf{G}/\mathbf{k}}\cong\mathfrak{g}\,\otimes_{\mathbf{k}}\,\text{Hom}_{\mathcal{O}_{\mathbf{G}}}(T_{\mathbf{G}},\mathcal{O}_{\mathbf{G}})\cong\text{Hom}_{\mathbf{k}}(\mathfrak{g},\mathfrak{g})\otimes_{\mathbf{k}}\,\mathcal{O}_{\mathbf{G}}$. The invariant $\mathfrak{g}$-valued 1-form on $\mathbf{G}$ that corresponds to $\text{id}_{\mathfrak{g}}\otimes 1$ under this isomorphism is called the Maurer-Cartan form. We will denote it by $\omega\in\mathfrak{g}\,\otimes_{\mathbf{k}}\Omega^{1}_{\mathbf{G}/\mathbf{k}}$. Suppose that we are given an element $g\in\mathbf{G}(F)$. We can think of it as a map $g:\text{Spec}\,F\longrightarrow\mathbf{G}$. We can use $g$ to pull back the Maurer-Cartan form to $\text{Spec}\,F$ in order to obtain $g^{*}\omega\in\mathfrak{g}\,\otimes_{\mathbf{k}}\,\Omega_{F/\mathbf{k}}^{1}=\mathfrak{g}\,\otimes_{\mathbf{k}}\,\Omega_{\mathcal{O}/\mathbf{k}}^{1}\left[\frac{1}{t}\right]$. By applying the completion map $\widehat{(-)}:\Omega_{\mathcal{O}/\mathbf{k}}^{1}\rightarrow\hat{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}$, we get an element $\widehat{{g}^{*}\omega}\in\mathfrak{g}\,\otimes_{\mathbf{k}}\,\hat{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}\left[\frac{1}{t}\right]$. Now we can define the gauge action of $\mathbf{G}\left(F\right)$ on $\mathfrak{g}\,\otimes_{\mathbf{k}}\,\hat{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}\left[\frac{1}{t}\right]$. For any $g\in\mathbf{G}\left(F\right)$ and $B\in\mathfrak{g}\,\otimes_{\mathbf{k}}\,\hat{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}\left[\frac{1}{t}\right]$, we set $g\cdot B\vcentcolon=\text{Ad}(g)B+\widehat{{g}^{*}\omega}$ . ###### Definition 2.2. By a formal connection $A$ for $P$ we mean a function $A\;:\;\left\\{\text{trivializations}\;\;P\xrightarrow{\sim}\text{Spec}\,F\times\mathbf{G}\right\\}\;\;\longrightarrow\;\;\mathfrak{g}\,\otimes_{\mathbf{k}}\,\hat{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}\left[\frac{1}{t}\right]$ satisfying the following transformation law. Let $\phi_{1},\,\phi_{2}\,:\,P\xrightarrow{\sim}\text{Spec}\,F\times\mathbf{G}$ be two trivializations of $P$. We know that $\phi_{2}\circ\phi_{1}^{-1}$ is given by left multiplication by a unique element $g\in\mathbf{G}\left(F\right)$. We then require $A(\phi_{2})=g\cdot A(\phi_{1})$. ###### Remark 2.3. Th reader might have encountered a different definition of formal connection. Using the action of $\mathbf{G}$ on $\mathfrak{g}\otimes_{\mathbf{k}}\Omega^{1}_{\mathcal{O}/\mathbf{k}}\left[\frac{1}{t}\right]$ one can define a formal version of the Atiyah sequence [Ati57]. Splittings of such sequence will correspond to formal connections as we have defined them. Such a connection $A$ is completely determined by its value at any given trivialization. We will often assume that we have chosen a fixed trivialization of $P$. Hence we can think of $P$ as the trivial bundle, and think of $A$ as the element of $\mathfrak{g}\,\otimes_{\mathbf{k}}\,\hat{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}\left[\frac{1}{t}\right]$ given by the image of this trivialization. Note that we have implicitly fixed a choice of uniformizer $t$ for $\mathcal{O}$. This yields an isomorphism $\hat{\Omega}_{\mathcal{O}/\mathbf{k}}^{1}=\mathcal{O}\,dt\cong\mathcal{O}$. We will often think of connections as elements of $\mathfrak{g}_{F}\vcentcolon=\mathfrak{g}\otimes_{\mathbf{k}}F$ obtained under the induced isomorphism $\Omega_{\mathcal{O}/\mathbf{k}}^{1}\left[\frac{1}{t}\right]=F\,dt\cong F$. All of the discussion above also applies over any finite field extension $F_{b}$ of $F$. The choice of a uniformizer $u\vcentcolon=t^{\frac{1}{b}}$ for $F_{b}$ yields an isomorphism from $F$ onto $F_{b}$ sending $t$ to $u$. This allows us to“lift” $\mathbf{G}$-bundles and trivializations from $\text{Spec}\,F_{b}$ to $\text{Spec}\,F$ by transport of structure. We can therefore lift connections from $F_{b}$ to $F$. There are some subtleties for the lift of connections when we think of them as elements of $\mathfrak{g}_{F}$. We generally take derivatives with respect to $t$, and not $u=t^{\frac{1}{b}}$. That is, we fix the trancendental element $t=u^{b}$ of $F_{b}$ in order to get the isomorphism $\hat{\Omega}_{\mathcal{O}_{b}/\mathbf{k}}^{1}\left[\frac{1}{u}\right]=\left(\mathcal{O}_{b}\,dt\right)\left[\frac{1}{u}\right]\cong\mathcal{O}_{b}\left[\frac{1}{u}\right]=F_{b}$. Under this identification, the lift of a $\mathbf{G}$-connection is not the obvious one given by replacing $u$ by $t$. Instead, the lift of a connection $A=\sum_{j=r}^{\infty}A_{j}\,t^{\frac{j}{b}}\in\mathfrak{g}_{F_{b}}$ is given by $\tilde{A}\vcentcolon=bt^{b-1}\sum_{j=r}^{\infty}A_{j}\,t^{j}$. This is called the $b$-lift of the connection. Let $\mathbf{T}\subset\mathbf{G}$ be a maximal torus in $\mathbf{G}$. We will denote by $X_{*}(\mathbf{T})$ (resp. $X^{*}(\mathbf{T})$) the cocharacter (resp. character) lattice of $\mathbf{T}$. We will write $\langle-,-\rangle:X_{*}(\mathbf{T})\otimes X^{*}(\mathbf{T})\longrightarrow\mathbb{Z}$ for the canonical pairing. There is a natural inclusion $X_{*}(\mathbf{T})\subset\text{Lie}(\mathbf{T})$ given by taking differentials at the identity. We will freely use this identification without further notice. Note that a cocharacter $\lambda:\mathbb{G}_{m}\longrightarrow\mathbf{T}\subset\mathbf{G}$ yields a point $\lambda\in\mathbf{G}(\mathbf{k}[t,t^{-1}])$. We denote by $t^{\lambda}$ the element of $\mathbf{G}(F)$ obtained via the natural inclusion $\mathbf{k}[t,t^{-1}]\hookrightarrow F$. We will make use of the algebraic exponential map, as in [DG80] pg. 315. For $X\in t\mathfrak{gl}_{n}(\mathcal{O})$ we have an exponential $\text{exp}(X)\in\mathbf{GL_{n}}(\mathcal{O})$ defined by $\text{exp}(X)\vcentcolon=\sum_{i=0}^{\infty}\frac{1}{i!}X^{i}$. By choosing a closed embedding $\mathbf{G}\hookrightarrow\mathbf{GL_{n}}$ we can similarly define an exponential map $\text{exp}\vcentcolon t\mathfrak{g}(\mathcal{O})\longrightarrow\mathbf{G}(\mathcal{O})$. It can be checked that this does not depend on the choice of embedding. We will only use one property of this map: for any $X\in\mathfrak{g}$, the image of $\text{exp}(t^{n}\,X)$ when we reduce modulo $t^{n+1}$ is given by $1+t^{n}X\in\mathbf{G}\left(\mathcal{O}/t^{n+1}\mathcal{O}\right)$. ### 2.2 Adjoint orbits in semisimple Lie algebras Here we include some facts about semisimple algebraic groups and their Lie algebras. Most of these results are standard and can be found in the book [CM93]. For the rest of this section we will assume that $\mathbf{G}$ is connected semisimple. Recall that an element of a semisimple Lie algebra is called semisimple (resp. nilpotent) if the the image under the adjoint representation is semisimple (resp. nilpotent) as a linear transformation of $\mathfrak{g}$. It turns out that we can check these conditions on any faithful representation. This fact follows from the following theorem. ###### Theorem 2.4 (Additive Jordan Decomposition). Let $\mathfrak{g}$ semisimple. For any $A\in\mathfrak{g}$ there exist unique a $A_{s}$ semisimple and $A_{n}$ nilpotent such that 1. (i) $A=A_{s}+A_{n}$ 2. (ii) $[A_{s},A_{n}]=0$ ###### Remark 2.5. For a reductive Lie algebra, all elements of the center are considered semisimple. For the Lie algebra of an arbitrary linear algebraic group, we will usually fix a Levi subgroup $\mathbf{L}$ and speak of semisimple elements inside $\text{Lie}(\mathbf{L})$. Recall that $\mathfrak{sl}_{2}=\\{X\in\mathfrak{gl}_{2}\mid\text{tr}(X)=0\\}$. The Lie bracket is given by the matrix commutator. Define $H=\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}$, $X=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix}$ and $Y=\begin{bmatrix}0&0\\\ 1&0\end{bmatrix}$. Then we have $\mathfrak{sl}_{2}=\mathbf{k}H\oplus\mathbf{k}X\oplus\mathbf{k}Y$ as a $\mathbf{k}$-vector space. ###### Definition 2.6. A $\mathfrak{sl}_{2}$-triple in $\mathfrak{g}$ is a nonzero Lie algebra map $\phi:\mathfrak{sl}_{2}\longrightarrow\mathfrak{g}$. We will often abuse notation and denote the images of $H,X,Y$ with the same letters. ###### Theorem 2.7 (Jacobson-Morozov). Let $\mathbf{G}$ be a connected semisimple algebraic group with Lie algebra $\mathfrak{g}$. Let $U\in\mathfrak{g}$ be a nilpotent element. Then there exists a homomorphism $\Phi\vcentcolon SL_{2}\longrightarrow G$ such that the $\mathfrak{sl}_{2}$-triple corresponding to the differential $d\Phi:\mathfrak{sl}_{2}\longrightarrow\mathfrak{g}$ satisfies $d\Phi(Y)=U$. Moreover such a homomorphism is uniquely determined up to conjugation by an element of the centralizer $Z_{\mathbf{k}}(U)(\mathbf{k})$. If $Y\neq 0$ is a nilpotent element in $\mathfrak{g}$, we will denote by $(H,X,Y)$ the $\mathfrak{sl}_{2}$-triple granted by Jacobson-Morozov. For any element $X\in\mathfrak{g}$, we will write $\mathfrak{g}_{X}$ for the centralizer of $X$ in $\mathfrak{g}$. Let $G=\mathbf{G}(\mathbf{k})$ denote the $\mathbf{k}$-rational points of $\mathbf{G}$. Recall that for any $Y\in\mathfrak{g}$, the orbit under the adjoint action $\mathbf{G}\cdot Y$ can be equipped with the structure of a smooth locally closed subvariety of $\mathfrak{g}$. We will often harmlessly identify it with its closed points $G\cdot Y$. The following proposition is going to be the essential technical tool for the induction argument in the reductive case. The proof can be found in [BV83] pages 17-18. ###### Proposition 2.8. Let $Y\neq 0$ be nilpotent in $\mathfrak{g}$. Let $(H,X,Y)$ be the corresponding $\mathfrak{sl}_{2}$-triple. Then the affine space $Y+\mathfrak{g}_{X}$ meets the orbit $G\cdot Y$ exactly at $Y$. For any other nilpotent $U\in Y+\mathfrak{g}_{X}$ with $U\neq Y$, we have $\text{dim}(G\cdot U)>\text{dim}(G\cdot Y)$. ###### Example 2.9. If $Y$ is regular nilpotent, then it is the unique nilpotent element in $Y+\mathfrak{g}_{X}$. Fix a maximal torus $\mathbf{T}\subset\mathbf{G}$. Let $\Phi$ be the set of roots of $\mathbf{G}$ with respect to $\mathbf{T}$. The coweight lattice $Q_{\mathbf{G}}$ of $\mathbf{G}$ with respect to $\mathbf{T}$ is defined to be $Q_{\mathbf{G}}\vcentcolon=\text{Hom}(\mathbb{Z}\Phi,\,\mathbb{Z})$. Since $\mathbf{G}$ is semisimple, the cocharacter lattice $X_{*}(\mathbf{T})$ has finite index in the coweight lattice $Q_{\mathbf{G}}$. ###### Definition 2.10. The index $I(\mathbf{G})$ is defined to be the exponent of the finite group $Q_{\mathbf{G}}/\,X_{*}(\mathbf{T})$. Let $\Phi^{\vee}$ be the set of coroots of $\mathbf{G}$ with respect to $\mathbf{T}$. We have the following chain of inclusions $\mathbb{Z}\Phi^{\vee}\,\subset\,X_{*}(\mathbf{T})\subset\,Q_{\mathbf{G}}$ ###### Definition 2.11. $J(\mathbf{G})$ is the exponent of the finite group $Q_{\mathbf{G}}/\,\mathbb{Z}\Phi^{\vee}$. ###### Remark 2.12. Since all maximal tori in $\mathbf{G}$ are conjugate, both $I(\mathbf{G})$ and $J(\mathbf{G})$ do not depend on the choice of $\mathbf{T}$. Let us fix a Borel subgroup $\mathbf{B}\subset\mathbf{G}$ containing $\mathbf{T}$. This amounts a choice of positive roots $\Phi^{+}$. We let $\Delta$ be the corresponding subset of simple roots. ###### Definition 2.13. Let $\alpha=\sum_{\beta\in\Delta}m_{\beta}\beta$ be a positive root. The height of $\alpha$ is defined to be $\text{hgt}\,(\alpha)\vcentcolon=\sum_{\beta\in\Delta}\;m_{\beta}$. The height of the Lie algebra $\mathfrak{g}$ is $\text{hgt}\,(\mathfrak{g})\vcentcolon=\text{sup}_{\alpha\in\Phi^{+}}\;\text{hgt}\,(\alpha)$. To conclude this section, we define a function that measures the “size” of the semisimple element $H$ in the Jacobson-Morozov triple corresponding to a nilpotent $Y\in\mathfrak{g}$. We can always arrange $H\in X_{*}(\mathbf{T})$. We will implicitly assume this from now on. ###### Definition 2.14. Let $Y\in\mathfrak{g}$ be a nilpotent element. Let $H$ be the corresponding semisimple element in the Jacobson-Morozov triple of $Y$. Then, we define $\Lambda(Y)\vcentcolon=\text{sup}_{\alpha\in\Phi}\;\left(\frac{1}{2}\alpha(H)+1\right)$. This function $\Lambda$ is constant on nilpotent orbits. ###### Example 2.15. Suppose that $Y$ is regular nilpotent. We can choose $H$ so that $\alpha(H)=2$ for every $\alpha\in\Delta$ (see [CM93] Chapter 3). Therefore, $\Lambda(Y)=\text{hgt}\,(\mathfrak{g})+1$ in this case. It turns out that this is the biggest possible value for $\Lambda$. In other words $\Lambda(Y)\leq\text{hgt}\,(\mathfrak{g})+1$ for any nilpotent $Y\in\mathfrak{g}$. ## 3 Regular connections Fix a connected linear algebraic group $\mathbf{G}$ over $\mathbf{k}$. What we call regular connections are also known as connections with at worst regular singularities. ###### Definition 3.1. A connection $A=\sum_{j=r}^{\infty}A_{j}\,t^{j}\in\mathfrak{g}_{F}$ is said to be of the first kind if if it has at worst a simple pole (i.e. $r\geq-1$). A connection $A$ is called regular if there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot A$ is of the first kind. In the analytic context, regular connections are classified by topological data. Indeed, such connections are determined by their monodromy representation. Our goal in this section is to classify formal regular connections over an arbitrary ground field. This will be achieved in Subsection 3.6. It should be noted that the development of the regular case is a necessary preliminary step in our treatment of the general irregular singular case in Sections 4 and 5. As mentioned in the introduction, the regular singular case for $\mathbf{k}=\mathbb{C}$ is treated in [BV83] using transcendental methods. The case of the group $\text{GL}_{n}$ was known well before. See Deligne’s book [Del70][II §1] for a discussion of regular singular connections for $\text{GL}_{n}$ before the paper [BV83]. It should be noted that Levelt [Lev75] gave a proof of the existence of canonical forms for $\text{GL}_{n}$ that applies to any algebraically closed field $\mathbf{k}$. ### 3.1 Regular connections for semisimple groups We will start with the semisimple case. A regular connection is said to be in canonical form if it can be written as $t^{-1}C$ for some $C\in\mathfrak{g}$. In order to prove Theorem 3.6 we can assume that $A$ is of first kind, because of the definition of regular connection. We first need the following definition and lemma from [BV83, 8.5], which actually work for arbitrary $\mathbf{G}$. We include a detailed proof of the lemma in order to keep the exposition self-contained. ###### Definition 3.2. Let $\mathbf{G}$ be a connected linear algebraic group. Let $A=\sum_{j=-1}^{\infty}A_{j}\,t^{j}$ be a connection of the first kind in $\mathfrak{g}_{F}$. The endomorphism $\text{ad}\,(A_{-1})\,\in\text{GL}_{n}(\mathfrak{g})$ yields a decomposition of $\mathfrak{g}$ into generalized eigenspaces $\mathfrak{g}=\bigoplus_{\lambda}\mathfrak{g}_{\lambda}$. We say that $A$ is aligned if $A_{j}\in\mathfrak{g}_{j+1}$ for all $j$. ###### Lemma 3.3. Let $\mathbf{G}$ be a connected linear algebraic group and $A=\sum_{j=-1}^{\infty}A_{j}\,t^{j}$ a formal connection of the first kind in $\mathfrak{g}_{F}$. Then there exist $x\in\mathbf{G}(\mathcal{O})$ such that $x\cdot A$ is aligned. ###### Proof. We will inductively build a sequence $(B_{j})_{j=1}^{\infty}$ of elements of $\mathfrak{g}$ such that the change of trivialization by $x\vcentcolon=\lim_{n\rightarrow\infty}\prod_{j=0}^{n-1}\text{exp}(t^{n-j}\,B_{n-j})$ puts $A$ in aligned form. Let $k\in\mathbb{N}$. Suppose that we have chosen $B_{j}\in\mathfrak{g}$ for all $j\leq k$ such that the connection $A^{(k)}=\sum_{l=-1}^{\infty}A^{(k)}_{l}\,t^{l}$ defined by $A^{(k)}\vcentcolon=\prod_{j=0}^{k-1}\text{exp}(t^{k-j}\,B_{k-j})\cdot A$ satisfies $A_{l}^{(k)}\in\mathfrak{g}_{l+1}$ for all $l<k$. Notice that the base case $k=0$ is trivial and that we have $A^{(k)}_{-1}=A_{-1}$. Let’s try to determine $B_{k+1}$. Recall that $\text{exp}(t^{k+1}\,B_{k+1})\equiv 1+t^{k+1}\,B_{k+1}\;(\text{mod}\;t^{k+2})$. By an elementary matrix computation (choose an embedding of $\mathbf{G}\hookrightarrow\mathbf{\text{GL}_{n}}$), we can see that $\text{exp}(t^{k+1}B_{k+1})\cdot A^{(k)}\equiv\sum_{l=-1}^{k-1}A^{(k)}_{l}\,t^{l}+[A^{(k)}_{k}-(ad(A_{-1})-(k+1))B_{k+1}]\,t^{k}\;\;(\text{mod}\;t^{k+1})$ Decompose $\mathfrak{g}$ into generalized $ad(A_{-1})$ eigenspaces $\mathfrak{g}=\bigoplus_{\lambda}\mathfrak{g}_{\lambda}$. By definition the operator $ad(A_{-1})-(k+1)$ restricts to an automorphism of $\mathfrak{g}_{\lambda}$ for all $\lambda\neq k+1$. In particular, we can choose $B_{k+1}\in\mathfrak{g}$ such that $A^{(k)}_{k}-(ad(A_{-1})-(k+1))B_{k+1}$ is in $\mathfrak{g}_{k+1}$. This concludes the induction step. It follows by construction that the gauge transformation by $x\vcentcolon=\lim_{n\rightarrow\infty}\prod_{j=0}^{n-1}\text{exp}(t^{n-j}\,B_{n-j})$ puts $A$ in aligned form. ∎ ###### Remark 3.4. Any aligned connection is actually in $\mathfrak{g}\otimes\mathbf{k}[t,t^{-1}]$. The coefficient with largest exponent is $\left(x\cdot A\right)_{j}t^{j}$, where $j+1$ is the biggest integer eigenvalue of $ad\left(\,(A_{-1})_{s}\,\right)$. We denote this number by $k(A_{-1})\vcentcolon=j+1$ for further reference. In order to determine the resulting aligned connection, we only need to multiply by $k(A_{-1})$-many exponentials in the proof above. Therefore the aligned form only depends on $A_{j}$ for $-1\leq j\leq k(A_{-1})$. Note that $k(A_{-1})$ can drastically change if we multiply $A$ by a scalar in $\mathbf{k}$. This reflects the fact that gauge transformations are not $\mathbf{k}$-linear. ###### Example 3.5. Suppose that $\text{ad}\left(\,\left(A_{-1}\right)_{s}\,\right)$ does not have any integer eigenvalues. Then the aligned connection will be in canonical form. ###### Theorem 3.6. Let $\mathbf{G}$ be a connected semisimple algebraic group. Let $A\in\mathfrak{g}_{F}$ be a regular connection. Then, there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot A=t^{-1}C$ for some $C\in\mathfrak{g}$. ###### Proof. This is a special case of [Sch07][Thm. 4.2]. We include the proof with some modifications that will suit our needs in subsequent subsections. By Lemma 3.3, we can assume that $A$ is an aligned connection in $\mathfrak{g}_{F}$. Let $(A_{-1})_{s}$ be the semisimple part of $A_{-1}$. Choose a maximal torus $\mathbf{T}$ of $\mathbf{G}$ such that the corresponding Cartan subalgebra $\text{Lie}(\mathbf{T})$ contains $(A_{-1})_{s}$. Fix a choice of positive roots $\Phi^{+}$ of $\mathbf{G}$ relative to $\mathbf{T}$. Let $\Delta$ be the subset of simple roots. Choose a basis for $\mathbf{k}$ as a vector space over $\mathbb{Q}$. Suppose that $1$ is one of the basis elements. Let $\pi\vcentcolon\mathbf{k}\longrightarrow\mathbb{Q}$ be the corresponding projection. We can define $\tau$ in $\text{Lie}(\mathbf{T})$ given by $\tau(\alpha)=\pi(\alpha((A_{-1})_{s}))$ for all $\alpha\in\Delta$. There exists $b\in\mathbb{N}$ such that $b\tau$ is in the cocharacter lattice of $\mathbf{T}$. We let $\mu\vcentcolon=b\tau$ be the corresponding cocharacter. Recall from the preliminaries that we have a $b$-lift $\tilde{A}=\sum_{j=-1}^{\infty}bA_{j}\,t^{bj+b-1}$. We can assume that we are working with $\tilde{A}$ by passing to the $b$-ramified cover. We claim that $t^{-\mu}\cdot\tilde{A}$ is in canonical form. In order to show this, it is convenient to use the $ad$ representation and view everything as matrices in $\text{End}(\mathfrak{g})$. The root decomposition $\mathfrak{g}=\bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha}$ gives us the spectral decomposition of $({A}_{-1})_{s}$. We can view $\text{Ad}(t^{-\mu})$ as a matrix in $\text{GL}(\mathfrak{g}_{F})$. $\text{Ad}(t^{-\mu})$ acts as the scalar $t^{-\langle\mu,\beta\rangle}$ on the root space $\mathfrak{g}_{\beta}$. By assumption $A$ is aligned. This means that $A_{j}$ is in a sum of root spaces $\mathfrak{g}_{\beta}$ where $(A_{-1})_{j+1}$ has eigenvalue $j+1$. These are the root spaces $\mathfrak{g}_{\beta}$ where $\beta((A_{-1})_{s})=j+1$. By the construction of $\mu$, we know that $\langle\mu,\beta\rangle=b\beta((A_{-1})_{s})$ whenever $\beta((A_{-1})_{s})$ is an integer. Therefore, $\text{Ad}(t^{-\mu})\,A_{j}=t^{-bj-b}A_{j}$. We conclude that $t^{-\mu}\cdot\tilde{A}=t^{-\mu}\cdot\left(\sum_{j=-1}^{\infty}bA_{j}\,t^{bj+b-1}\right)=\left(\sum_{j=-1}^{\infty}bA_{j}\right)\,t^{-1}\,+\,\frac{d}{dt}\,\left(t^{-\mu}\right)\,t^{\mu}$ For the last term $\frac{d}{dt}\,\left(t^{-\mu}\right)\,t^{\mu}$ we are performing the calculation in $\text{End}(\mathfrak{g}_{F})$. A matrix computation yields $\frac{d}{dt}\,\left(t^{-\mu}\right)\,t^{\mu}=-\mu\,t^{-1}$. The theorem follows. ∎ ###### Remark 3.7. Babbitt and Varadarajan prove the theorem over the ground field $\mathbf{k}=\mathbb{C}$ using the analytic theory of regular singular connections (see section 8 of [BV83]). In the analytic setting, we need to pass to a ramified cover only when the monodromy class of the connection is not in the image of the exponential map. For example all conjugacy classes in $\mathbf{GL}_{n}$ are exponential, so we don’t need to pass to a ramified cover to reduce regular $\mathbf{GL_{n}}$-connections. This latter fact can also be proven algebraically using the center of $\mathbf{GL}_{n}$. See the argument in pages 19-22 of [BV83]. ###### Remark 3.8. We only need to fix a rational basis of $\text{span}_{\mathbb{Q}}\\{\alpha((A_{-1})_{s})\,\vcentcolon\,\alpha\in\Delta\\}$ in the proof above. So the argument is constructive. We can be a bit more careful in the proof of Theorem 3.6. This way we can get a uniform bound for the ramification needed. We record this as a small lemma. ###### Lemma 3.9. We can always choose $b\leq\text{hgt}(\mathfrak{g})\cdot I(\mathbf{G})$ in the proof of Theorem 3.6. ###### Proof. Set $\tau(\alpha)$ to be the best approximation of $\pi(\alpha((A_{-1})_{s}))$ in $\frac{1}{\text{hgt}(\mathfrak{g})}\mathbb{Z}$. By the definition of $\text{hgt}(\mathfrak{g})$, it follows that $\tau(\beta)=\beta((A_{-1})_{s})$ whenever $\beta((A_{-1})_{s})$ is an integer. So the proof of Theorem 3.6 still goes through with this choice of $\tau$. By construction we have $\text{hgt}(\mathfrak{g})\tau\in Q_{\mathbf{G}}$. Then the definition of $I(\mathbf{G})$ implies that $\text{hgt}(\mathfrak{g})I(\mathbf{G})\tau\in X_{*}(\mathbf{T})$. Hence we can choose $b=\text{hgt}(\mathfrak{g})I(\mathbf{G})$. ∎ Choose a maximal torus $\mathbf{T}\subset\mathbf{G}$. Let $W$ be the Weyl group of $\mathbf{G}$ with respect $\mathbf{T}$. Fix a projection $\pi:\mathbf{k}\longrightarrow\mathbb{Q}$ as in the proof above. We can extend this projection to a natural map $\pi:\text{Lie}(\mathbf{T})\cong X_{*}(\mathbf{T})\otimes\mathbf{k}\longrightarrow X_{*}(\mathbf{T})\otimes\mathbb{Q}$. We will once and for all fix a fundamental domain $\mathfrak{D}$ for the action of $W$ on the set $\Xi\vcentcolon=\left\\{C\in\text{Lie}(\mathbf{T})\,\mid\,\pi(C)=0\right\\}$. Notice that $\Xi$ is a set of representatives for the quotient $\text{Lie}(\mathbf{T})/\,X_{*}(\mathbf{G})\otimes\mathbb{Q}$. It turns out that we can always choose $x$ in Theorem 3.6 so that the semisimple part $C_{s}$ is in $\mathfrak{D}$. ###### Corollary 3.10. Let $\mathbf{G}$ be connected semisimple with a choice of maximal torus $\mathbf{T}\subset\mathbf{G}$. Let $A\in\mathfrak{g}_{F}$ be a regular connection. Then, there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot A=t^{-1}C$ for some $C\in\mathfrak{g}$ satisfying $C_{s}\in\mathfrak{D}$. ###### Proof. By Theorem 3.6, we can assume that $A=t^{-1}\,C$ for some $C\in\mathfrak{g}$. Since $\mathbf{k}$ is algebraically closed, we can conjugate the semisimple element $C_{s}$ to the torus $\mathbf{T}$. By applying the gauge transformation $t^{-\pi(C_{s})}$, we can assume that $\pi(C_{s})=0$. Finally, we can conjugate by an element of $W$ to obtain $C_{s}\in\mathfrak{D}$. ∎ The following proposition will be crucial in establishing uniqueness of canonical reductions in general. ###### Proposition 3.11. Let $\mathbf{G}$ be connected semisimple with a choice of maximal torus $\mathbf{T}\subset\mathbf{G}$. Let $C,D\in\mathfrak{g}$ with $C_{s},D_{s}\in\mathfrak{D}$. Suppose that there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot\left(t^{-1}\,C\right)=t^{-1}\,D$. Then we have $C_{s}=D_{s}$. Moreover $x$ is a $\mathbf{k}$-point in the centralizer $Z_{\mathbf{G}}(C_{s})(\mathbf{k})$. ###### Proof. By lifting everything to a ramified cover, we can assume for simplicity that $x\in\mathbf{G}(F)$. Choose a faithful representation $\mathbf{G}\hookrightarrow\mathbf{\text{GL}_{n}}$. We can view $x\in\mathbf{\text{GL}_{n}}(F)$ and $C,D\in\mathfrak{gl}_{n}$. Let’s consider the linear transformation $U$ in $\text{End}(\mathfrak{gl}_{n})$ given by $U\,v=Dv\,-\,vC$ for all $v\in\mathfrak{gl}_{n}$. Notice that we can write $U=U_{s}+U_{n}$, where $\displaystyle U_{s}\,v\vcentcolon=D_{s}v-vC_{s}$ $\displaystyle U_{n}\,v\vcentcolon=D_{n}v-vC_{n}$ We know that $C_{s}$ and $D_{s}$ can be simultaneously diagonalized. Therefore $U_{s}$ is semisimple. The eigenvalues of $U_{s}$ are differences of eigenvalues of $C_{s}$ and $D_{s}$. Since $\pi(C_{s})=\pi(D_{s})=0$, we conclude that $0$ is the only possible rational eigenvalue of $U_{s}$. By definition, we have that $U_{n}$ is nilpotent and $[U_{s},U_{n}]=0$. We conclude that $U=U_{s}+U_{n}$ is the additive Jordan decomposition of $U$. In particular the set of eigenvalues of $U$ is the same as the set of eigenvalues of $U_{s}$. Therefore, $0$ is the only possible rational eigenvalue of $U$. The condition $x\cdot\left(t^{-1}\,C\right)=t^{-1}\,D$ can be expressed as $\frac{d}{dt}x=t^{-1}U\,x$. Here we are viewing $x$ as an invertible matrix in $\mathfrak{gl}_{n}(F)$. Set $x=\sum_{j=r}^{\infty}x_{j}\,t^{j}$. Then this condition reads $\sum_{j=r}^{\infty}jx_{j}\,t^{j-1}=\sum_{j=r}^{\infty}U\,x_{j}\,t^{j-1}$ Hence we have $jx_{j}=U\,x_{j}$ for all $j$. Since $0$ is the only possible rational eigenvalue of $U$, we conclude that $x_{j}=0$ for all $j\neq 0$. Therefore, $x=x_{0}\in\mathbf{G}(\mathbf{k})$. Hence the relation $x\cdot\left(t^{-1}\,C\right)=t^{-1}\,D$ implies that $\text{Ad}(x)\,C=D$. By uniqueness of Jordan decomposition for $\mathbf{\text{GL}_{n}}$, this means that $\text{Ad}(x)\,C_{s}=D_{s}$. It is well-known that $\text{Lie}(\mathbf{T})/W$ parametrizes semisimple conjugacy classes in $\mathfrak{g}$ (see [CM93] Chapter 2). In particular, $\mathfrak{D}$ is a set of representatives of conjugacy classes of semisimple elements that map to $0$ under $\pi$. We conclude that we must have $C_{s}=D_{s}$. Then $\text{Ad}(x)\,C_{s}=D_{s}$ implies that $x\in Z_{\mathbf{G}}(C_{s})(\mathbf{k})$. ∎ ### 3.2 Regular connections for tori and reductive groups ###### Proposition 3.12. Let $\mathbf{G}$ be a torus and $A=\sum_{j=-1}^{\infty}A_{j}\,t^{j}$ a formal connection of the first kind. Then there exists $x\in\mathbf{G}(\mathcal{O})$ such that $x\cdot A=t^{-1}A_{-1}$. Moreover, there is a unique such $x$ with $x\equiv 1\,\left(mod\;t\right)$. ###### Proof. Since $\mathbf{k}$ is algebraically closed, $\mathbf{G}$ is split. Therefore the theorem follows from the special case $\mathbf{G}=\mathbb{G}_{m}$. We are reduced to an elementary computation. Let $v=\sum_{j=0}^{\infty}A_{j}t^{j}\;\in\,\mathcal{O}$. It suffices to find $u=\sum_{j=0}^{\infty}B_{j}t^{j}\in\mathcal{O}^{\times}$ with $\frac{d}{dt}(u)=-vu$. By expanding we see that we want $(j+1)B_{j+1}=-\sum_{l=0}^{j}A_{l}B_{j-l}$ for all $j\geq 0$. This is a recurrence we can solve, because we are in characteristic $0$. We can set the initial condition $B_{0}=1$ and then the rest of the coefficients are uniquely determined. ∎ ###### Example 3.13. For $\mathbf{G}=\mathbb{G}_{m}$ we can phrase this result concretely in terms of differential equations. In this case we have an equation $\frac{d}{dt}x=Ax$, where $A\in\mathbf{k}(({t}))$ is a Laurent series with at worst a simple pole. The statement says that we can do a multiplicative change of variables $y=Bx$ for some power series $B\in\mathcal{O}^{\times}$ such that the equation becomes $\frac{d}{dt}y=\frac{a}{t}y$ for some scalar $a\in\mathbf{k}$. Let’s state a uniqueness result for canonical forms of regular formal connections for tori. ###### Proposition 3.14. Let $\mathbf{G}$ be a torus, and let $C_{1},C_{2}\in\mathfrak{g}$. Suppose that there exists $x\in\mathbf{G}(F)$ with $x\cdot\left(t^{-1}\,C_{1}\right)=t^{-1}\,C_{2}$. Then, we have $x=g\,t^{\mu}$ for some cocharacter $\mu\in X_{*}(\mathbf{G})$ and some $g\in\mathbf{G}(\mathbf{k})$. Moreover $C_{1}=C_{2}-\mu$. ###### Proof. We will do the computation for $\mathbf{G}=\mathbb{G}_{m}$. The general case follows from the same argument. Write $x=k\,t^{r}\,y$, where $k\in\mathbf{k}^{\times}$ and $y=1+\sum_{j=1}^{\infty}a_{j}\,t^{j}$. Then, $x\cdot\left(t^{-1}\,C_{1}\right)\;=\;t^{-1}\;C_{1}+rt^{-1}+dy\,y^{-1}\;=\;t^{-1}\,C_{2}$ Notice that $dy\,y^{-1}$ is in $\mathbf{k}[[{t}]]$. By looking at the nonnegative coefficients in the equation above, we conclude that $dy=0$. Therefore we have $y=1$. Hence $x=k\,t^{r}$, and the result follows. ∎ We can patch together some of the previous of results to get canonical forms for regular connections when the group is reductive. The following corollary is [Sch07][Thm. 4.2]. ###### Corollary 3.15. Let $\mathbf{G}$ be reductive and $A\in\mathfrak{g}_{F}$ a regular formal connection. Then there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot A=t^{-1}C$ for some $C\in\mathfrak{g}$. ###### Proof. For completness we explain how to deduce the corollary from previous propositions. We can assume that $A$ is of the first kind. Let $\mathbf{Z}^{0}_{G}$ be the neutral component of the center of $\mathbf{G}$. Set $\mathfrak{z}\vcentcolon=\text{Lie}(\mathbf{Z}^{0}_{\mathbf{G}})$. Let $\mathbf{G}_{\text{der}}$ the derived subgroup of $\mathbf{G}$. $\mathbf{G}_{der}$ is semisimple with Lie algebra $\mathfrak{g}_{\text{der}}\vcentcolon=[\mathfrak{g},\mathfrak{g}]$. We have $\mathfrak{g}=\mathfrak{g}_{\text{der}}\oplus\mathfrak{z}$. Decompose $A=A_{\mathfrak{g}_{\text{der}}}+A_{\mathfrak{z}}$. By the semisimple case there exists $x\in\mathbf{G}_{\text{der}}(\overline{F})$ such that $x\cdot A_{\mathfrak{g}_{\text{der}}}$ is in canonical form. Now $x\cdot A=x\cdot A_{\mathfrak{g}_{\text{der}}}+A_{\mathfrak{z}}$. Use the result for tori to put $A_{\mathfrak{z}}$ in canonical form and conclude. ∎ ###### Remark 3.16. By Remark 3.4, we only need to know $k\left(\,(A_{\mathfrak{g}_{\text{der}}})_{-1}\,\right)$-many coefficients of a connection of the first kind in order to determine its canonical form. The bound for the ramification needed in this case is reduced to the bound for the semisimple group $\mathbf{G}_{\text{der}}$ as explained in Lemma 3.9. Notice that the setup before Corollary 3.10 applies to the reductive case. We formulate the analogous statement for convenience. ###### Corollary 3.17. Let $\mathbf{G}$ connected reductive with maximal torus $\mathbf{T}\subset\mathbf{G}$. 1. (i) Let $A\in\mathfrak{g}_{F}$ be a regular connection. Then there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot A=t^{-1}C$ for some $C\in\mathfrak{g}$ satisfying $C_{s}\in\mathfrak{D}$. 2. (ii) Assume that $C,D\in\mathfrak{g}$ satisfy $C_{s},D_{s}\in\mathfrak{D}$. Suppose that there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot\left(t^{-1}\,C\right)=t^{-1}\,D$. Then, we have $C_{s}=D_{s}$. Moreover $x$ is in the centralizer $Z_{\mathbf{G}}(C_{s})(\mathbf{k})$. ###### Proof. Part (i) follows by combining Proposition 3.12 for tori and Corollary 3.10 for semisimple groups. Part (ii) follows from the same argument as in Proposition 3.11. ∎ This corollary allows us to give a concrete parametrization of regular $\mathbf{G}(\overline{F})$-gauge equivalence classes of formal connections. Let $A\in\mathfrak{g}_{F}$ be a regular formal connection. Suppose that $B=t^{-1}C$ is a connection in canonical form that is $\mathbf{G}(\overline{F})$-gauge equivalent to $A$. Assume that $C_{s}\in\mathfrak{D}$. By Corollary 3.17, $C_{s}$ does not depend on the choice of canonical form $B$. Let $W$ denote the Weyl group of $\mathbf{G}$ with respect to $\mathbf{T}$. Recall that $\mathfrak{D}$ is a set of representatives for $\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)/\,W$. In particular we get a well defined element in $\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)/\,W$ corresponding to $C_{s}$. ###### Definition 3.18. Let $A\in\mathfrak{g}_{F}$ be a regular formal connection as above. We define the semisimple $\overline{F}$-monodromy of $A$ to be the element $m^{s}_{A,\,\overline{F}}\in\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)/\,W$ corresponding to $C_{s}$ as described above. Let $m\in\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)/\,W$. We define $Z_{\mathbf{G}}(m)$ to be the centralizer in $\mathbf{G}$ of the unique representative of $m$ in $\mathfrak{D}$. It turns out that $Z_{\mathbf{G}}(m)$ is a Levi subgroup of a parabolic in $\mathbf{G}$. It is well-known that the Lie algebra centralizer of a semisimple element $\text{Lie}(Z_{\mathbf{G}}(m))=\mathfrak{g}_{m}$ is the Levi component of a parabolic subalgebra of $\mathfrak{g}$. For connectedness, we can pass to an isogenous cover $p:\tilde{\mathbf{G}}\longrightarrow\mathbf{G}$ with simply connected derived subgroup. Notice that $p(Z_{\tilde{\mathbf{G}}}(m))=Z_{\mathbf{G}}(m)$. So it suffices to prove connectedness of $Z_{\tilde{\mathbf{G}}}(m)$, which follows from [Hum95] pg. 33. Note that the isomorphism class of $Z_{\mathbf{G}}(m)$ does not depend on the choice of projection $\pi:\mathbf{k}\longrightarrow\mathbb{Q}$ and fundamental domain $\mathfrak{D}$. In fact, $Z_{\mathbf{G}}(m)\cong Z_{\mathbf{G}}(C)$ for any representative $C$ such that $\text{ad}(C)$ has no rational eigenvalues. Corollary 3.17 implies that the nilpotent part of a canonical form $B$ that is $\mathbf{G}(\overline{F})$-gauge equivalent to $A$ is uniquely determined up to $Z_{\mathbf{G}}(m^{s}_{A,\,\overline{F}})$-conjugacy. We record this as a corollary. ###### Corollary 3.19. Fix $m\in\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)/\,W$. Let $\mathcal{N}_{Z_{\mathbf{G}}(m)}$ denote the nilpotent cone in the Lie algebra of $Z_{\mathbf{G}}(m)$. There is a natural correspondence $\left\\{\text{regular}\;A\in\mathfrak{g}_{\overline{F}}\;\text{with}\;m^{s}_{A,\,\overline{F}}=m\right\\}/\,\mathbf{G}(\overline{F})\;\;\;\longleftrightarrow{\;\;\;\;}\mathcal{N}_{Z_{\mathbf{G}}(m)}/\,Z_{\mathbf{G}}(m)$ Let $S\subset\Delta$ be a subset of simple roots. Each $\alpha\in S$ induces a linear functional $X_{*}(\mathbf{T})\otimes\mathbf{k}\longrightarrow\mathbf{k}$. Let $H_{\alpha}$ be the hyperplane of $X_{*}(\mathbf{T})\otimes\mathbf{k}$ where this functional vanishes. We denote by $\overline{H}_{\alpha}$ the image of $H_{\alpha}$ in $\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)/\,W$. Define $\overline{H}_{S}\vcentcolon=\bigcap_{\alpha\in S}\overline{H}_{\alpha}$. We say that $m\in\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)/\,W$ is of type $S$ if we have $m\in\overline{H}_{S}$ and $m\notin\overline{H}_{V}$ for all $S\subsetneq V\subset\Delta$. Let $Q_{S}$ be the set of all $m$ of type $S$. For any $m\in Q_{S}$, the centralizer $Z_{\mathbf{G}}(m)$ is conjugate to the standard Levi $\mathbf{M}_{S}$ associated to the subset of simple roots $S\subset\Delta$. We get the following rewording of the corollary above. ###### Corollary 3.20. There is a natural correspondence $\left\\{\text{regular formal connections}\right\\}/\,\mathbf{G}(\overline{F})\;\;\;\longleftrightarrow{\;\;\;\;}\bigsqcup_{S\subset\Delta}Q_{S}\times\mathcal{N}_{\mathbf{M}_{S}}/\,\mathbf{M}_{S}$ This gives us a procedure to describe all regular $\mathbf{G}(\overline{F})$-gauge equivalence classes of formal connections. For each $S\subset\Delta$, the group $\mathbf{M}_{S}$ is connected and reductive. The set of nilpotent orbits $\mathcal{N}_{\mathbf{M}_{S}}/\,\mathbf{M}_{S}$ is a finite set which admits many well studied parametrizations. For example, nilpotent orbits for classical groups can be classified by partition diagrams as in [CM93] Chapter 5. This yields an explicit canonical block decompositions for $\mathbf{G}(\overline{F})$-gauge equivalence classes of regular connections for classical groups. ### 3.3 Connections for unipotent groups All connections in a unipotent group are regular. They admit canonical forms. ###### Proposition 3.21. Let $\mathbf{G}$ be connected unipotent and let $A\in\mathfrak{g}_{F}$ be a connection. Then, there exists $x\in\mathbf{G}(F)$ such that $x\cdot A=t^{-1}C$ for some $C\in\mathfrak{g}$. ###### Proof. We proceed by induction on $\text{dim}(\mathbf{G})$. Suppose that $\text{dim}(\mathbf{G})=1$. Since $\text{char}(\mathbf{k})=0$, we know that $\mathbf{G}\cong\mathbb{G}_{a}$. In this case the theorem follows from an elementary computation. See Example 3.23 below. Now assume $\text{dim}\,(\mathbf{G})\geq 2$. Since $\mathbf{k}$ is of characteristic 0, $\mathbf{G}$ is split unipotent. In particular, the center $\mathbf{Z}_{\mathbf{G}}$ contains a closed subgroup $\mathbf{H}$ isomorphic to $\mathbb{G}_{a}$. Let $\text{Lie}(\mathbf{H})=\mathfrak{h}$. By induction there exists $\overline{x}\in\mathbf{G}/\mathbf{H}(F)$ such that $\overline{x}\cdot\overline{A}\in\mathfrak{g}/\mathfrak{h}$ is in canonical form. We can lift $\overline{x}$ to an element $x\in\mathbf{G}(F)$ because $H^{1}\left(F,\,\mathbf{H}(\overline{F})\right)=H^{1}\left(F,\,\overline{F}\right)=0$ (the additive version of Hilbert’s Theorem 90). By construction, we have $x\cdot A=t^{-1}C+B$ for some $C\in\mathfrak{g}$ and $B\in\mathfrak{h}_{F}$. Now we can use the base case for $\mathbf{H}\cong\mathbb{G}_{a}$ to put $B$ into regular canonical form. ∎ ###### Remark 3.22. Here we didn’t need to pass to a ramified cover in order to find a good trivialization. ###### Example 3.23. In the case of $\mathbf{G}=\mathbb{G}_{a}$, we can phrase this result concretely in terms of differential equations. We use the embedding $\mathbb{G}_{a}\hookrightarrow\text{GL}_{2}$, so that we can interpret the connection as system of differential equations $\displaystyle\frac{d}{dt}x_{1}$ $\displaystyle=Ax_{2}$ $\displaystyle\frac{d}{dt}x_{2}$ $\displaystyle=0$ Set $x_{2}=c$, where $c\in\mathbf{k}$ is a constant. We are left with the nonhomogeneous equation $\frac{d}{dt}x_{1}=cA$ for some Laurent series $cA$. The statement of the proposition reduces to the obvious fact that we can find a formal antiderivative for any Laurent series with residue $0$ (i.e. $A_{-1}=0$). We now prove uniqueness of the canonical form up to conjugacy. ###### Proposition 3.24. Let $\mathbf{U}$ be a unipotent group. Let $C_{1},C_{2}$ be two elements of the Lie algebra $\mathfrak{u}\vcentcolon=\text{Lie}(\mathbf{U})$. Suppose that there exists $x\in\mathbf{U}(F)$ such that $x\cdot\left(t^{-1}C_{1}\right)=t^{-1}C_{2}$. Then, we have $x\in\mathbf{U}(\mathbf{k})$. ###### Proof. We will argue by induction on the dimension of $\mathbf{U}$. If $\text{dim}(\mathbf{U})=1$, then $\mathbf{U}\cong\mathbb{G}_{a}$. We can write $x=\sum_{j=m}^{\infty}a_{j}\,t^{j}$ for some $a_{j}\in\mathbf{k}$. The hypothesis then becomes $x\cdot\left(t^{-1}C_{1}\right)=t^{-1}C_{1}+dx=t^{-1}C_{2}$ This means that $dx=\sum_{j=m}^{\infty}ja_{j}\,t^{j-1}=t^{-1}\left(C_{2}-C_{1}\right)$. In particular we must have $ja_{j}=0$ for all $j\neq 0$. Hence $x=a_{0}\in\mathbf{k}$. Suppose that $\mathbf{U}$ is an arbitrary unipotent group. Assume that the result holds for all unipotent groups of smaller dimension. Let $\mathbf{H}$ be a subgroup of the center $Z_{\mathbf{U}}$ of $\mathbf{U}$ such that $\mathbf{H}\cong\mathbb{G}_{a}$ (this is possible because $\text{char}(\mathbf{k})=0$) . Let $\overline{x}\in\mathbf{U}/\mathbf{H}\,(F)$ be the image of $x$ in the quotient. By the induction hypothesis, the proposition holds for $\mathbf{U}/\,\mathbf{H}$. Hence we have that $\overline{x}\in\mathbf{U}/\mathbf{H}\,(\mathbf{k})$. We can lift $\overline{x}$ to an element $v\in\mathbf{U}(\mathbf{k})$, since $\mathbf{k}$ is algebraically closed. We can therefore write $x=vu$, with $u\in\mathbf{H}(F)$. Our assumption thus becomes $x\cdot\left(t^{-1}C_{1}\right)\;=\;t^{-1}\text{Ad}(v)\,\text{Ad}(u)\,C_{1}+\text{Ad}(v)du\;=\;t^{-1}C_{2}$ Since $u\in\mathbf{H}(F)\subset Z_{\mathbf{U}}(F)$, we have $\text{Ad}(u)C_{1}=C_{1}$. After rearranging we get $du\;=\;t^{-1}\left(\text{Ad}(v^{-1})\,C_{2}-C_{1}\right)$ The computation for $\mathbb{G}_{a}$ above implies that $u\in\mathbf{H}(\mathbf{k})$. Therefore $x\in\mathbf{U}(\mathbf{k})$. ∎ We end this section with a determinacy result for canonical forms in the unipotent case. Recall that the nilpotency class of a unipotent group is the length of the upper central series. For example, a commutative unipotent group has nilpotency class $0$. ###### Proposition 3.25. Let $\mathbf{U}$ be a unipotent group of nilpotency class $n$. Let $A=\sum_{j=m}^{\infty}A_{j}\,t^{j}\in\mathfrak{u}_{F}$ be a connection with $A_{m}\neq 0$. 1. (i) If $m>-1$, then there exists $x\in\mathbf{U}(\mathcal{O})$ such that $x\equiv 1\;\left(mod\;t^{m+1}\right)$ and $x\cdot A=0$. 2. (ii) If $m\leq-1$, then the gauge equivalence class of $A$ is determined by $A_{j}$ for $m\leq j<n(|m|-1)$. More precisely, suppose that $B$ is another connection with $B\equiv A\;\left(mod\;t^{k}\right)$ for some $k\geq n(|m|-1)$. Then there exists $x\in\mathbf{U}(\mathcal{O})$ with $x\equiv 1\left(mod\;t^{k-n|m|+n+1}\right)$ such that $x\cdot A=B$. ###### Proof. We will induct on the nilpotency class $n$. The base case $n=0$ means that $\mathbf{U}\cong\mathbb{G}_{a}^{l}$ for some $l$. Here we can make use of the explicit computation we have done for $\mathbb{G}_{a}$ a few times already. Define $u_{A}\vcentcolon=-\sum_{j=0}^{\infty}\frac{1}{j+1}A_{j+1}\,t^{j+1}$. We have $u_{A}\cdot A=A+du_{A}=\sum_{j=m}^{-1}A_{j}\,t^{j}$ Now both (i) and (ii) are clear by taking $x=-u_{B}+u_{A}$ (we use $B=0$ for part (i)). For the induction step, let $\mathbf{U}$ be an arbitrary unipotent group of nilpotency class $n$. We will think of $\mathbf{U}$ as embedded in the group of upper triangular matrices of $\text{GL}_{p}$ for some $p$. By definition, the quotient $\mathbf{U}/\,Z_{\mathbf{U}}$ of $\mathbf{U}$ by its center $Z_{\mathbf{U}}$ has nilpotency class $n-1$. It follows from the matrix description that we can choose a section $s$ over $\mathbf{k}$ for the $Z_{\mathbf{U}}$-torsor $\mathbf{U}\longrightarrow\mathbf{U}/\,Z_{\mathbf{U}}$ such that $s(1)=1$ (this is a section as $\mathbf{k}$-schemes, it is not a homomorphism). Let’s address part (i). Let $\overline{A}$ be the image of $A$ in the quotient $\text{Lie}(\mathbf{U}/\,Z_{\mathbf{U}})_{F}$. By the induction hypothesis, there exists $\overline{x}\in\mathbf{U}/\,Z_{\mathbf{U}}(\mathcal{O})$ such that $\overline{x}\equiv 1\;\left(mod\;t^{m+1}\right)$ and $\overline{x}\cdot\overline{A}=0$. Therefore, we have $s(\overline{x})\cdot A\in\text{Lie}(Z_{\mathbf{U}})_{F}$. Notice that we have $s(\overline{x})\equiv s(\overline{x})^{-1}\equiv 1\;\left(mod\;t^{m+1}\right)$. It follows that $s(\overline{x})\cdot A\;=\;s(\overline{x})\,A\,s(\overline{x})^{-1}\;+\;d\,s(\overline{x})\;s(\overline{x})^{-1}\;\equiv 0\;\left(mod\;t^{m}\right)$ Now we can conclude by using the base case for $Z_{\mathbf{U}}$. For part (ii), let $\overline{A}$ and $\overline{B}$ denote the images of $A$ and $B$ in the quotient. By the induction hypothesis, there exists $\overline{x}\in\mathbf{U}/\,Z_{\mathbf{U}}(\mathcal{O})$ with $\overline{x}\equiv 1\;\left(mod\;t^{k-(n-1)|m|+n}\right)$ such that $\overline{x}\cdot\overline{A}=\overline{B}$. We can now write $s(\overline{x})\cdot A=ds\left(\overline{x}\cdot\overline{A}\right)+C$ and $B=ds\left(\overline{x}\cdot\overline{A}\right)+D$ for some $C,D\in\text{Lie}(Z_{\mathbf{U}})_{F}$. Notice that $s(\overline{x})\equiv s(\overline{x})^{-1}\equiv 1\;\left(mod\;t^{k-(n-1)|m|+n}\right)$. Therefore, $s(\overline{x})\cdot A\;=\;s(\overline{x})\,A\,s(\overline{x})^{-1}\;+\;d\,s(\overline{x})\;s(\overline{x})^{-1}\;\equiv\;A\;\left(mod\;t^{k-n|m|+n}\right)$ Since $A\equiv B\;\left(mod\;t^{k-n|m|+n}\right)$, it follows that $C\equiv D\;\left(mod\;t^{k-n|m|+n}\right)$. Now by the base case we can find $y\in Z_{\mathbf{U}}(\mathcal{O})$ with $y\equiv 1\;\left(mod\;t^{k-n|m|+n+1}\right)$ such that $y\cdot C=D$. We conclude that $y\,s(\overline{x})\cdot A=B$, because $y$ is in the center. We clearly have $y\,s(\overline{x})\equiv 1\;\left(mod\;t^{k-n|m|+n+1}\right)$, as desired. ∎ ### 3.4 Regular connections for solvable groups We fix a projection $\pi:\mathbf{k}\longrightarrow\mathbb{Q}$ as in the proof of Proposition 3.6. For $\mathbf{T}$ a torus, we extend this projection to a map $\pi:\text{Lie}(\mathbf{\mathbf{T}})\cong X_{*}(\mathbf{T})\otimes\mathbf{k}\longrightarrow X_{*}(\mathbf{T})\otimes\mathbb{Q}$. ###### Proposition 3.26. Let $\mathbf{G}$ be of the form $\mathbf{T}\ltimes\mathbf{U}$, where $\mathbf{T}$ is a torus and $\mathbf{U}$ is unipotent. Let $A=A_{\mathbf{T}}+A_{\mathbf{U}}$ be a formal connection with $A_{\mathbf{T}}\in\text{Lie}(\mathbf{T})_{F}$ a connection of the first kind and $A_{\mathbf{U}}\in\text{Lie}(\mathbf{U})_{F}$. Let $b$ be a positive integer such that $b\,\pi\left(\,(A_{\mathbf{T}})_{-1}\,\right)\in X_{*}(\mathbf{T})$. Then there exists $x\in\mathbf{G}(F_{b})$ such that $x\cdot A=t^{-1}\,C_{\mathbf{T}}+t^{-1}\,C_{\mathbf{U}}$ for some $C_{\mathbf{T}}\in\text{Lie}(\mathbf{T})$ and $C_{\mathbf{U}}\in\text{Lie}(\mathbf{U})$. Moreover, we can arrange that $\pi(C_{\mathbf{T}})=0$ and $[C_{\mathbf{T}},C_{\mathbf{U}}]=0$. ###### Proof. By the proof of Proposition 3.12, we can find $g\in\mathbf{T}(F)$ with $g\cdot A_{\mathbf{T}}=t^{-1}\,(A_{\mathbf{T}})_{-1}$. Set $\mu\vcentcolon=b\,\pi\left(\,(A_{\mathbf{T}})_{-1}\,\right)\in X_{*}(\mathbf{T})$. Then we have $(t^{\frac{1}{b}\mu}\,g)\cdot A_{\mathbf{T}}=t^{-1}\,C_{\mathbf{T}}$ for some $C_{\mathbf{T}}\in\text{Lie}(\mathbf{T})$ with $\pi(C_{\mathbf{T}})=0$. We can replace $A$ with $B\vcentcolon=(t^{\frac{1}{b}\,\mu}\,g)\cdot A$. We know that $B=t^{-1}\,C_{\mathbf{T}}+B_{\mathbf{U}}$ for some $B_{\mathbf{U}}\in\text{Lie}(\mathbf{U})_{F_{b}}$. By lifting to the $b$-ramified cover, we can assume that $B_{\mathbf{U}}\in\text{Lie}(\mathbf{U})_{F}$. We claim that we can find $u\in\mathbf{U}(F)$ such that $u\cdot B=t^{-1}\,C_{\mathbf{T}}+t^{-1}\,C_{\mathbf{U}}$ with $C_{\mathbf{U}}\in\text{Lie}(\mathbf{U})$ and $[C_{\mathbf{T}},C_{\mathbf{U}}]=0$. We will show this by induction on the dimension of $\mathbf{U}$. The base case is $\mathbf{U}=\mathbb{G}_{a}$. Then, $\mathbf{T}$ acts on $\mathbf{U}$ by a character $\chi:\mathbf{T}\longrightarrow\mathbb{G}_{m}$. Write $B_{\mathbf{U}}=\sum_{j=r}^{\infty}\left(B_{\mathbf{U}}\right)_{j}\,t^{j}$. For any $u=\sum_{j=r}^{\infty}u_{j}\,t^{j}\in\mathbf{U}(F)$, we have $u\cdot B=t^{-1}\,C_{\mathbf{T}}\,+\,B_{\mathbf{U}}\,-\,\sum_{j=r}^{\infty}\left(d\chi(C_{\mathbf{T}})-j\right)u_{j}\,t^{j-1}$ Since $\pi(C_{\mathbf{T}})=0$, we have $\pi\left(d\chi(C_{\mathbf{T}})\right)=0$. There are two options: 1. (1) $d\chi(C_{\mathbf{T}})\notin\mathbb{Q}$. Then, setting $u_{j}=\frac{1}{d\chi(C_{\mathbf{T}})-j}\left(B_{\mathbf{U}}\right)_{j-1}$ we get $u\cdot B=t^{-1}\,C_{\mathbf{T}}$. 2. (2) $d\chi(C_{\mathbf{T}})=0$. We can set $u_{j}=\frac{1}{d\chi(C_{\mathbf{T}})\,-\,j}\left(B_{\mathbf{U}}\right)_{j-1}$ for $j\neq 0$ and $u_{0}=0$. Then $u\cdot B=t^{-1}\,C_{\mathbf{T}}+t^{-1}\,\left(B_{\mathbf{U}}\right)_{-1}$. Notice that we have $[C_{\mathbf{T}},\,\left(B_{\mathbf{U}}\right)_{-1}]=d\chi(C_{\mathbf{T}})\left(B_{\mathbf{U}}\right)_{-1}=0$. This concludes the proof of the base case. Let’s proceed with the induction step. We can decompose the action of the split torus $\mathbf{T}$ on the vector space $Z_{\mathbf{U}}$ into one- dimensional spaces. Let $\mathbf{H}\cong\mathbb{G}_{a}\leq Z_{\mathbf{U}}$ be one of these eigenspaces. The eigenspace decomposition yields a natural $\mathbf{T}$-equivariant section of the quotient map $Z_{\mathbf{U}}\longrightarrow Z_{\mathbf{U}}/\,\mathbf{H}$. We claim that we can extend this to a $\mathbf{T}$-equivariant section $s$ of the morphism of schemes $\mathbf{U}\longrightarrow\mathbf{U}/\,\mathbf{H}$. In order to see this claim, we can use induction on the nilpotency class to reduce to the case when $\mathbf{U}$ has nilpotency class $1$. Notice that we can find a section which is not necessarily $\mathbf{T}$-equivariant, since everything is isomorphic to an affine space. Then we can use the argument in Lemma 9.4 of [BS68] to obtain a $\mathbf{T}$-equivariant section. We can arrange so that $s$ preserves the identities by substracting the image $s(1_{\mathbf{U}/\,\mathbf{H}})$. Let us denote by $ds$ the induced map of tangent spaces at the identity. Let $\overline{B}$ be the image of $B$ in the quotient $\text{Lie}(\mathbf{T}\ltimes\mathbf{U}/\,\mathbf{H})_{F}$. By the induction hypothesis, we can find $\overline{u}\in\mathbf{U}/\,\mathbf{H}(F)$ such that $\overline{u}\cdot\overline{B}=t^{-1}\,C_{\mathbf{T}}+t^{-1}\,\overline{D}$ for some $\overline{D}\in\text{Lie}\left(\mathbf{U}/\,\mathbf{H}\right)$ with $[C_{\mathbf{T}},\overline{D}]=0$. We can then write $s(\overline{u})\cdot B=t^{-1}\,C_{\mathbf{T}}+t^{-1}\,ds(\overline{D})+B_{\mathbf{H}}$ for some $B_{\mathbf{H}}\in\text{Lie}(\mathbf{H})_{F}$. Since $s$ is $\mathbf{T}$-equivariant, we have $[ds(\overline{D}),\,C_{\mathbf{T}}]=0$. We can now use the base case for $\mathbf{H}$ in order to conclude. ∎ ###### Remark 3.27. We can decompose the Lie algebra $\mathfrak{u}\vcentcolon=\text{Lie}(\mathbf{U})$ into weight spaces $\mathfrak{u}=\bigoplus_{i}\mathfrak{u}_{\chi_{i}}$. Here each $\chi_{i}$ is a character of $\mathbf{T}$. Fix a basis $\\{\alpha_{j}\\}$ for the character lattice $X^{*}(\mathbf{T})$. For each $i$ we can write $\chi_{i}=\sum_{j}m^{i}_{j}\alpha_{j}$ for some integers $m^{i}_{j}$. Define $\text{hgt}(\chi_{i})=\sum_{j}|m^{i}_{j}|$. Set $b=\underset{1\leq i\leq l}{\text{max}}\,\text{hgt}(\chi_{i})$. If we don’t require $\pi(C_{\mathbf{T}})$ and $[C_{\mathbf{T}},C_{\mathbf{U}}]=0$ in Proposition 3.26, then it suffices to pass to a $b$-ramified cover. So there is a uniform upper bound on the ramification needed to put any regular $\mathbf{G}$-connection into canonical form. It only depends on the solvable group $\mathbf{G}$. Let us prove a uniqueness result for regular canonical forms in the solvable case. ###### Proposition 3.28. Let $\mathbf{G}$ be of the form $\mathbf{T}\ltimes\mathbf{U}$ as above. Let $C=t^{-1}\,C_{\mathbf{T}}+t^{-1}\,C_{\mathbf{U}}$ and $D=t^{-1}\,D_{\mathbf{T}}+t^{-1}\,D_{\mathbf{U}}$ be two regular canonical connections with $C_{\mathbf{T}},D_{\mathbf{T}}\in\text{Lie}(\mathbf{T})$ and $C_{\mathbf{U}},D_{\mathbf{U}}\in\text{Lie}(\mathbf{U})$. Suppose that $\pi(C_{\mathbf{T}})=\pi(D_{\mathbf{T}})=0$ and $[C_{\mathbf{T}},C_{\mathbf{U}}]=[D_{\mathbf{T}},D_{\mathbf{U}}]=0$. If there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot C=D$, then in fact $C_{\mathbf{T}}=D_{\mathbf{T}}$. Moreover, $x$ is in the centralizer $Z_{\mathbf{G}}(C_{\mathbf{T}})(\mathbf{k})$ of $C_{\mathbf{T}}$. ###### Proof. By lifting to a ramified cover, we can assume that $x\in\mathbf{G}(F)$. Write $x=x_{\mathbf{U}}\,x_{\mathbf{T}}$ with $x_{\mathbf{U}}\in\mathbf{U}(F)$ and $x_{\mathbf{T}}\in\mathbf{T}(F)$. By the computation in Proposition 3.14 applied to $\mathbf{T}$, we get that $x_{\mathbf{T}}\in\mathbf{T}(\mathbf{k})$ and $C_{\mathbf{T}}=D_{\mathbf{T}}$. The same proof of Proposition 3.11 implies that $x\in\mathbf{G}(\mathbf{k})$ and $\text{Ad}(x)C_{\mathbf{T}}=D_{\mathbf{T}}$. Since $C_{\mathbf{T}}=D_{\mathbf{T}}$, this means that $x\in Z_{\mathbf{G}}(C_{\mathbf{T}})(\mathbf{k})$. ∎ We conclude this section with a determinacy result for regular connections in the case of solvable groups. But first we need to setup some notation. Let $\mathbf{G}=\mathbf{T}\ltimes\mathbf{U}$ solvable. We have an action of the split torus $\mathbf{T}$ on the Lie algebra $\mathfrak{u}\vcentcolon=\text{Lie}(\mathbf{U})$ via the adjoint representation. We can decompose this representation into weight spaces $\mathfrak{u}=\bigoplus_{i=1}^{l}\mathfrak{u}_{\chi_{i}}$ for some finite set $\\{\chi_{1},\chi_{2},...,\chi_{l}\\}$ of charaters $\chi_{i}:\mathbf{T}\longrightarrow\mathbb{G}_{m}$. Suppose that we have a formal connection $A=A^{\mathbf{T}}+A^{\mathbf{U}}$, with $A^{\mathbf{T}}\in\text{Lie}(\mathbf{T})_{F}$ a connection of the first kind and $A^{\mathbf{U}}\in\text{Lie}(\mathbf{U})_{F}$. We can write $A^{\mathbf{T}}=t^{-1}A^{\mathbf{T}}_{-1}\,+\,\sum_{j=p}^{\infty}A^{\mathbf{T}}_{j}\,t^{j}$ for some $p\geq 0$ and $A^{\mathbf{U}}=\sum_{j=m}^{\infty}A^{\mathbf{U}}_{j}\,t^{j}$ for some $m\in\mathbb{Z}$. Let $b$ be a positive integer such that $\mu\vcentcolon=b\,\pi\left(\,A^{\mathbf{T}}_{-1}\,\right)$ is in $X_{*}(\mathbf{T})$. Define $L$ to be $L\vcentcolon=\text{max}\left(\left\\{\frac{1}{b}\langle\mu,\,\chi_{i}\rangle\right\\}_{i=1}^{l}\cup\\{0\\}\right)$ ###### Proposition 3.29. Keep the same notation as in the paraggraph above. Assume that $\mathbf{U}$ has nilpotency class $n$. 1. (i) Suppose that $m>L-1$. Then there exists $x\in\mathbf{G}(\mathcal{O})$ with $x\cdot A=t^{-1}A^{\mathbf{T}}_{-1}$. More precisely, there exists $x_{\mathbf{T}}\in\mathbf{T}(\mathcal{O})$ with $x_{\mathbf{T}}\equiv 1_{\mathbf{T}}\,\left(mod\;t^{p+1}\right)$ and $x_{\mathbf{U}}\in\mathbf{U}(\mathcal{O})$ with $x_{\mathbf{U}}\equiv 1_{\mathbf{U}}\,\left(mod\;t^{m+1}\right)$ such that $(x_{\mathbf{U}}x_{\mathbf{T}})\cdot A=t^{-1}A^{\mathbf{T}}_{-1}$. 2. (ii) Suppose that $m\leq L-1$. The $\mathbf{G}(F)$-gauge equivalence class of $A$ is determined by the coefficients $A_{j}^{\mathbf{T}}$ for $-1\leq j<(n+1)(|m|-1)+L$ and $A_{j}^{\mathbf{U}}$ for $m\leq j<n(|m|-1)+L$. More precisely, suppose that there is another connection $B$ and positive integer $k\geq n(|m|-1)+L$ with $B^{\mathbf{T}}\equiv A^{\mathbf{T}}\,\left(mod\;t^{k+|m|}\right)$ and $B^{\mathbf{U}}\equiv A^{\mathbf{U}}\,\left(mod\;t^{k}\right)$. Then, there exists $x\in\mathbf{G}(\mathcal{O})$ with $x\equiv 1\,\left(mod\;t^{k-n|m|+n+1}\right)$ such that $x\cdot A=B$. ###### Proof. 1. (i) By assumption $A^{\mathbf{T}}\equiv t^{-1}A^{\mathbf{T}}_{-1}\,\left(mod\;t^{p}\right)$. The proof of Proposition 3.12 shows that there exists $x_{\mathbf{T}}\in\mathbf{T}(\mathcal{O})$ with $x_{\mathbf{T}}\equiv 1\,\left(mod\;t^{k+1}\right)$ such that $x_{\mathbf{T}}\cdot A^{\mathbf{T}}=t^{-1}A^{\mathbf{T}}_{-1}$. Set $C\vcentcolon=x_{\mathbf{T}}\cdot A$. We can write $C=t^{-1}A_{-1}^{\mathbf{T}}+\text{Ad}(x_{\mathbf{T}})A^{\mathbf{U}}$. In order to ease notation, set $C^{\mathbf{U}}\vcentcolon=\text{Ad}(x_{\mathbf{T}})A^{\mathbf{U}}$. Since $A^{\mathbf{U}}\equiv 0\,\left(mod\;t^{m}\right)$, we have $C^{\mathbf{U}}\equiv 0\,\left(mod\;t^{m}\right)$. We claim that there exists $x\in\mathbf{U}(\mathcal{O})$ with $x\equiv 1_{\mathbf{U}}\,\left(mod\;t^{m+1}\right)$ such that $x\cdot C=t^{-1}A^{\mathbf{T}}_{-1}$. This claim finishes the proof of part (i). In order to prove the claim, we will induct on the nilpotency class of $\mathbf{U}$. The base case $n=0$ means that $\mathbf{U}\cong\mathbb{G}_{a}^{d}$ for some $d$. We can decompose into eigenvalues and look at each coordinate separately in order to reduce to the case $d=1$. Then there is a single weight space $\mathfrak{u}_{\chi_{i}}$. This case amounts to solving a recurrence as in the base case for the proof of Proposition 3.26. We want to find $x=\sum_{j=0}^{\infty}t^{j}u_{j}$ satisfying $C^{\mathbf{U}}_{j-1}=\left(d\chi_{i}(A_{-1}^{\mathbf{T}})-j\right)u_{j}$ If $j\leq m$ then $C^{\mathbf{U}}_{j-1}=0$ by assumption. So we can set $u_{j}=0$. If $j\geq m+1$, then we have $\pi\left(d\chi_{i}(A^{\mathbf{T}}_{-1})\right)-j=\frac{1}{b}\langle\mu,\chi_{i}\rangle-j\leq L-m-1$ By assumption $L-m-1<0$, so we must have $d\chi_{i}(A^{\mathbf{T}}_{-1})-j\neq 0$. Hence we can set $u_{j}=\frac{1}{d\chi_{i}(A^{\mathbf{T}}_{-1})\,-\,j}\,C^{\mathbf{U}}_{j-1}$. The base case follows. For the induction step, let $Z_{\mathbf{U}}$ denote the center of $\mathbf{U}$. Let $s$ be a $\mathbf{T}$-equivariant section of the quotient $\mathbf{U}\longrightarrow\mathbf{U}/\,Z_{\mathbf{U}}$, as in the proof of Proposition 3.26. Let $\overline{C}$ be the image of $C$ in the quotient $\text{Lie}(\mathbf{T}\ltimes\mathbf{U}/\,Z_{\mathbf{U}})_{F}$. By the induction hypothesis, there exists $\overline{x}\in\mathbf{U}/\,Z_{\mathbf{U}}(\mathcal{O})$ such that $\overline{x}\equiv 1\;\left(mod\;t^{m+1}\right)$ and $\overline{x}\cdot\overline{C}=t^{-1}A^{\mathbf{T}}_{-1}$. We must then have $s(\overline{x})\cdot C=t^{-1}A^{\mathbf{T}}_{-1}+D_{Z_{\mathbf{U}}}$ for some $D_{Z_{\mathbf{U}}}\in\text{Lie}(Z_{\mathbf{U}})_{F}$. By definition $s(\overline{x})\cdot C\;=\;t^{-1}\text{Ad}(s(\overline{x}))A^{\mathbf{T}}_{-1}\,+\,\text{Ad}(s(\overline{x}))C^{\mathbf{U}}\,+\,ds(\overline{x})s(\overline{x})^{-1}$ We know that $s(\overline{x})\equiv s(\overline{x})^{-1}\equiv 1\;\left(mod\;t^{m+1}\right)$. Also by assumption $C^{\mathbf{U}}\in\mathfrak{u}_{\mathcal{O}}$. It follows that $s(\overline{x})\cdot C\;\equiv\;t^{-1}A^{\mathbf{T}}_{-1}\,+\,C^{\mathbf{U}}\;\equiv\;t^{-1}C_{\mathbf{T}}\;\left(mod\;t^{m}\right)$ Therefore $D_{Z_{\mathbf{U}}}\equiv 0\;\left(mod\;t^{m}\right)$. Now we can conclude by using the base case for $Z_{\mathbf{U}}$. 2. (ii) The hypothesis implies that $B^{\mathbf{T}}_{-1}=A^{\mathbf{T}}_{-1}$. The proof of Proposition 3.12 shows that there exist $x_{\mathbf{T}}\in\mathbf{T}(\mathcal{O})$ with $x_{\mathbf{T}}\equiv 1\,\left(mod\;t^{k+|m|}\right)$ such that $x_{\mathbf{T}}\cdot A^{\mathbf{T}}=B^{\mathbf{T}}$. Set $C\vcentcolon=x_{\mathbf{T}}\cdot A$. We have $C=B^{\mathbf{T}}+\text{Ad}(x_{\mathbf{T}})A^{\mathbf{U}}$. Define $C^{\mathbf{U}}\vcentcolon=\text{Ad}(x_{\mathbf{T}})A^{\mathbf{U}}$. We know that $C^{\mathbf{U}}\equiv A^{\mathbf{U}}\,\left(mod\;t^{k}\right)$, because $x_{\mathbf{T}}\equiv 1\,\left(mod\;t^{k+|m|}\right)$ and $A^{\mathbf{U}}\in t^{m}\mathfrak{u}_{\mathcal{O}}$. Therefore $C^{\mathbf{U}}\equiv B^{\mathbf{U}}\,\left(mod\;t^{k}\right)$ by assumption. We claim that there exists $u\in\mathbf{U}(\mathcal{O})$ with $u\equiv 1\;\left(mod\;t^{k-n|m|+n+1}\right)$ such that $u\cdot C=B$. This claim concludes the proof of part (ii). In order to prove the claim, we will induct on the nilpotency class of $\mathbf{U}$. The base case $n=0$ follows from an argument similar to the one for part (i), we omit the details. For the induction step, let $Z_{\mathbf{U}}$ and $s$ be as in part (i). Let $\overline{C}$ and $\overline{B}$ denote the images of $C$ and $B$ in the quotient $\text{Lie}(\mathbf{T}\ltimes\mathbf{U}/\,Z_{\mathbf{U}})_{F}$. By the induction hypothesis, there exists $\overline{x}\in\mathbf{U}/\,Z_{\mathbf{U}}(\mathcal{O})$ with $\overline{x}\equiv 1\;\left(mod\;t^{k-(n-1)|m|+n}\right)$ such that $\overline{x}\cdot\overline{C}=\overline{B}$. We can now write $s(\overline{x})\cdot C=ds\left(\overline{B}\right)+E_{Z_{\mathbf{U}}}$ and $B=ds\left(\overline{B}\right)+K_{Z_{\mathbf{U}}}$ for some $E_{Z_{\mathbf{U}}},F_{Z_{\mathbf{U}}}\in\text{Lie}(Z_{\mathbf{U}})_{F_{b}}$. By definition $s(\overline{x})\cdot C\;=\;t^{-1}\text{Ad}(s(\overline{x}))B^{\mathbf{T}}\,+\,\text{Ad}(s(\overline{x}))C^{\mathbf{U}}\,+\,ds(\overline{x})s(\overline{x})^{-1}$ We know that $s(\overline{x})\equiv s(\overline{x})^{-1}\equiv 1\;\left(mod\;t^{k-(n-1)|m|+n}\right)$. Since $|m|\geq 1$, we conclude that $ds\left(\overline{B}\right)+E_{Z_{\mathbf{U}}}\;=\;s(\overline{x})\cdot C\;\equiv\;B^{\mathbf{T}}\,+\,C^{\mathbf{U}}\;=\;C\;\left(mod\;t^{k-n|m|+n}\right)$ Since $k\geq k-n|m|+n$, we have $C\equiv B\;\left(mod\;t^{k-n|m|+n}\right)$. It follows that $E_{Z_{\mathbf{U}}}\equiv K_{Z_{\mathbf{U}}}\;\left(mod\;t^{k-n|m|+n}\right)$. Now by the base case we can find $y\in Z_{\mathbf{U}}(\mathcal{O})$ with $y\equiv 1\;\left(mod\;t^{k-n|m|+n+1}\right)$ such that $(y\,s(\overline{x}))\cdot C=B$. We have $y\,s(\overline{x})\equiv 1\;\left(mod\;t^{k-n|m|+n+1}\right)$, as desired. ∎ ###### Remark 3.30. Suppose that $\langle\mu,\chi_{i}\rangle>0$ for all $i$. It follows from the proof above that we can relax further the conditions on the coefficients of $A^{\mathbf{U}}$. Similarly, we can obtain sharper conditions for the coefficients of $A^{\mathbf{T}}$ in the case $0\leq m\leq L-1$. We leave the details of these refinements to the interested reader. ###### Remark 3.31. If $L=0$, then the statement simplifies and we recover conditions similar to the unipotent case (Proposition 3.25). ### 3.5 Regular connections for arbitrary linear algebraic groups ###### Theorem 3.32. Let $\mathbf{G}$ be a connected linear algebraic group. Let $A\in\mathfrak{g}_{F}$ be a regular connection. Fix a Levi subgroup $\mathbf{L}$ and maximal torus $\mathbf{T}\subset\mathbf{L}$. Then there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot A=t^{-1}C$ for some $C\in\mathfrak{g}$. Moreover, such $x$ can be chosen so that the semisimple part $C_{s}$ of the Levi component satisfies $C_{s}\in\mathfrak{D}$ and $[C_{s},C]=0$. ###### Proof. Assume that $A$ is of the first kind. Let $\mathbf{U}\subset\mathbf{G}$ be the unipotent radical of $\mathbf{G}$ with Lie algebra $\mathfrak{u}$. Let $\mathfrak{l}$ be the Lie algebra of $\mathbf{L}$. We know that $\mathbf{G}=\mathbf{L}\ltimes\mathbf{U}$, and so $\mathfrak{g}=\mathfrak{l}\oplus\mathfrak{u}$. Decompose $A=A_{\mathfrak{l}}+A_{\mathfrak{u}}$. By the reductive group case, there exists $x\in\mathbf{L}(\overline{F})$ such that $x\cdot A_{\mathfrak{l}}=t^{-1}C$ for some $C\in\mathfrak{l}$ satisfying $C_{s}\in\mathfrak{D}$. Let $C_{n}\in\mathfrak{l}$ denote the nilpotent part of $C$. Let $\mathbf{E}$ be the neutral component of the centralizer $Z_{\mathbf{T}}(C_{n})$ of $C_{n}$ in $\mathbf{T}$. Note that $\mathbf{E}$ is a subtorus of $\mathbf{T}$ and $C_{s}\in\text{Lie}(\mathbf{E})$. Since $\text{char}(\mathbf{k})=0$, there is a unique connected one-dimensional unipotent subgroup $\mathbf{N}$ of $\mathbf{L}$ with $C_{n}\in\text{Lie}(\mathbf{N})$. We have that $x\cdot A$ is a formal connection for the solvable group $(\mathbf{E}\times\mathbf{N})\ltimes\mathbf{U}$. Now the result follows from the solvable case (Proposition 3.26). ∎ ###### Remark 3.33. In the beginning of the proof above, let $X$ denote the semisimple part of $(A_{\mathfrak{l}})_{-1}$. After conjugating by an element of $\mathbf{L}(\mathbf{k})$, we can suppose that $X\in\text{Lie}(\mathbf{T})$. Let $b$ be a positive integer such that $\mu\vcentcolon=b\,\pi(X)$ is in $X_{*}(\mathbf{T})$. Then, we can take $x\in G(F_{b})$ in the proof above. In order to see this we can first apply $t^{-\frac{1}{b}\,\mu}$. So we can assume that $\pi(X)=0$. By the proofs of Theorem 3.6 and Proposition 3.26, it follows that we don’t need any further ramification to put $A$ into canonical form. ###### Proposition 3.34. Let $\mathbf{G}$ be a connected linear algebraic group. Fix a Levi subgroup $\mathbf{L}$ and maximal torus $\mathbf{T}\subset\mathbf{L}$. Let $C,D\in\mathfrak{g}$. Write $C_{s},D_{s}$ for the semisimple parts of the Levi components $C_{\mathfrak{l}},D_{\mathfrak{l}}$. Assume that $C_{s},D_{s}\in\mathfrak{D}$ and $[C_{s},C]=[D_{s},D]=0$. Suppose that there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot\left(t^{-1}\,C\right)=t^{-1}\,D$. Then, we have $C_{s}=D_{s}$. Moreover $x$ is in the centralizer $Z_{\mathbf{G}}(C_{s})(\mathbf{k})$. ###### Proof. Write $x=x_{\mathbf{U}}\,x_{\mathbf{L}}$ with $x_{\mathbf{U}}\in\mathbf{U}(\overline{F})$ and $x_{\mathbf{L}}\in\mathbf{L}(\overline{F})$. By Corollary 3.17 applied to $\mathbf{L}$, we get that $x_{\mathbf{L}}\in Z_{\mathbf{L}}(C_{s})(\mathbf{k})$ and $C_{s}=D_{s}$. The same proof as in Proposition 3.11 shows that $x\in\mathbf{G}(\mathbf{k})$ and $\text{Ad}(x)C_{s}=D_{s}$. Since $C_{s}=D_{s}$, we conclude that $x\in Z_{\mathbf{G}}(C_{s})(\mathbf{k})$. ∎ Let $A\in\mathfrak{g}_{F}$ be a regular formal connection. Proposition 3.34 implies that we can define the semisimple $\overline{F}$-monodromy $m^{s}_{A}\in\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)/\,W$ just as we did in Definition 3.18. The same reasoning as in the reductive case yields the following. ###### Corollary 3.35. Fix $m\in\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)/\,W$. Let $\mathcal{N}_{Z_{\mathbf{G}}(m)}$ denote the nilpotent cone in the Lie algebra of $Z_{\mathbf{G}}(m)$. There is a natural correspondence $\left\\{\text{regular}\;A\in\mathfrak{g}_{\overline{F}}\;\text{with}\;m^{s}_{A}=m\right\\}/\,\mathbf{G}(\overline{F})\;\;\;\longleftrightarrow{\;\;\;\;}\mathcal{N}_{Z_{\mathbf{G}}(m)}/\,Z_{\mathbf{G}}(m)$ Since $\mathbf{T}$ is a split torus, we have a weight decomposition $\mathfrak{g}=\bigoplus_{\chi\in V}\mathfrak{g}_{\chi}$ of $\mathfrak{g}$ under the adjoint action of $\mathbf{T}$. Here $V$ is a set of characters of $\mathbf{T}$. Let $W$ be the Weyl group of $\mathbf{L}$ with respect to $\mathbf{T}$. There is a natural action of $W$ on $V$. For any subset $S$ of the quotient $V/\,W$, we can define the set $Q_{S}$ of elements in $\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)/\,W$ of type $S$, just as we did for reductive groups. Let $Z_{\mathbf{G}}(S)$ be the centralizer of any element in $Q_{S}$. The same reasoning as in the reductive case yields the following concrete parametrization. ###### Corollary 3.36. There is a natural correspondence $\left\\{\text{regular formal connections}\right\\}/\,\mathbf{G}(\overline{F})\;\;\;\longleftrightarrow{\;\;\;\;}\bigsqcup_{S\subset V/\,W}Q_{S}\times\mathcal{N}_{Z_{\mathbf{G}}(S)}/\,Z_{\mathbf{G}}(S)$ ### 3.6 Descent for gauge equivalence classes All of the theorems above give us a description of connections up to trivializations over $\text{Spec}\,\overline{F}$. We would like to get a classification over $\text{Spec}\,F$. This amounts to a problem in Galois cohomology. We have an action of $\mathbf{G}(\overline{F})$ on $\mathfrak{g}_{\overline{F}}$ that is compatible with the action of the absolute Galois group $\text{Gal}(F)$. Choose a regular connection in canonical form $B=t^{-1}C$ with $C_{s}\in\mathfrak{D}$ and $[C_{s},C]=0$. It is a direct consequence of Proposition 3.34 that the centralizer of $B$ in $\mathbf{G}(\overline{F})$ is $Z_{G}(C)\vcentcolon=Z_{\mathbf{G}}(C)(\mathbf{k})$. Therefore, we get an exact sequence of sheaves of sets over the etale site of $\text{Spec}\,F$ $1\longrightarrow Z_{G}(C)\longrightarrow\mathbf{G}\longrightarrow\mathbf{G}\cdot B\longrightarrow 1$ Here $Z_{G}(C)$ is the constant sheaf associated to the group of $\mathbf{k}$-points of the centralizer $Z_{\mathbf{G}}(C)$ of $C$. This yields a long exact sequence of pointed sets: $1\longrightarrow Z_{G}(C)\longrightarrow\mathbf{G}(F)\longrightarrow\mathbf{G}\cdot B(F)\longrightarrow H^{1}_{\text{Gal}(F)}(Z_{G}(C))\longrightarrow H^{1}_{\text{Gal}(F)}(\mathbf{G})$ The theorems of Tsen and Springer mentioned in the preliminaries imply that the right-most Galois cohomology group vanishes. This means that the set of connections over $\text{Spec}\,F$ that admit a trivialization over $\text{Spec}\,\overline{F}$ with canonical form $t^{-1}C$ is in bijection with $H^{1}_{\text{Gal}(F)}(Z_{G}(C))$. Since the action of $\text{Gal}(F)$ on $Z_{G}(C)$ is trivial, $H^{1}_{\text{Gal}(F)}(Z_{G}(C))$ is in (noncanonical) bijection with the set conjugacy classes of elements of finite order in $Z_{G}(C)$. Such bijection comes from the choice of a topological generator of $\text{Gal}(F)\cong\hat{\mathbb{Z}}$. Such a generator corresponds to the compatible choice of a generator $\omega_{b}$ of $\mu_{b}$ for all positive integers $b$. Here is a summary of the classification we have obtained. ###### Proposition 3.37 (Regular Connections over $D^{*}$). Let $B=t^{-1}\,C$ be a regular canonical connection with $C_{s}\in\mathfrak{D}$ and $[C_{s},C]=0$. The set of $\mathbf{G}$-connections over $\text{Spec}(F)$ that become gauge equivalent to $B$ over $\text{Spec}(\overline{F})$ is in (noncanonical) bijection with the set of conjugacy classes of elements of finite order in $Z_{\mathbf{G}}(C)(\mathbf{k})$ as described below. The correspondence goes as follows. Let $x\in Z_{\mathbf{G}}(C)(\mathbf{k})$ of order $b$. By the vanishing of $H^{1}_{\text{Gal}(F)}(\mathbf{G})$, we can find an element $y\in\mathbf{G}(F_{b})$ such that $\omega_{b}\cdot y=y\,x$. The connection associated to $x$ will be $A=y\cdot\left(t^{-1}\,C\right)\;\in\mathfrak{g}_{F}$. Conversely, suppose that $A=y\cdot B$ is a connection in $\mathfrak{g}_{F}$ for some $y\in\mathbf{G}(F_{b})$. We set $x\vcentcolon=y^{-1}\,\left(\omega_{b}\cdot y\right)$. Using the descriptions of regular $\mathbf{G}(\overline{F})$\- gauge equivalence classes we have given previously, we can parametrize regular formal connections. Let $(m,u)$ be a pair with $m\in\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)$ and $u$ a nilpotent element in $\mathcal{N}_{Z_{\mathbf{G}}(m)}$. A cohomology cocycle as described above is given by an element $t$ of finite order in the centralizer $Z_{\mathbf{G}}(m,u)$ of $u$ in $Z_{\mathbf{G}}(m)$. Since $Z_{\mathbf{G}}(m)$ is connected, we can conjugate by an element of $Z_{\mathbf{G}}(m)$ in order to assume that the semisimple element $t$ lies in $\mathbf{T}\subset Z_{\mathbf{G}}(m)$. It follows that the set of regular formal $\mathbf{G}$ connections over $D^{*}$ is in natural correspondence with equivalence classes of triples $(m,x,u)$, where 1. (i) $m\in\left(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q}\right)$. 2. (ii) $x$ is an element of finite order in $\mathbf{T}(\mathbf{k})$. 3. (iii) $u\in\mathcal{N}_{Z_{\mathbf{G}}(m)}$ with $\text{Ad}(t)(u)=u$. Two such triples are considered equivalent if they can be conjugated by an element of $\mathbf{G}(\mathbf{k})$. Recall that there is a canonical isomorphism $\mathbf{T}(\mathbf{k})\cong X_{*}(\mathbf{T})\otimes\mathbf{k}^{\times}$. Under this identification, the set $\mathbf{T}(\mathbf{k})^{tor}$ of elements of finite order in $\mathbf{T}(\mathbf{k})$ correspond to $X_{*}(\mathbf{T})\otimes\mu_{\infty}$. The compatible choice of primitive roots of unity $\omega_{b}$ yields an isomorphism $\mu_{\infty}\cong\mathbb{Q}/\,\mathbb{Z}$. Hence we get an identification $\mathbf{T}(\mathbf{k})^{tor}\cong X_{*}(\mathbf{T})\otimes\,\mathbb{Q}/\,\mathbb{Z}$. This means that the set of pairs $(m,x)\in(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q})\times\mathbf{T}(\mathbf{k})^{tor}$ is in natural bijection with $X_{*}(\mathbf{T})\otimes\,\mathbf{k}/\,\mathbb{Z}$. For an element $v\in X_{*}(\mathbf{T})\otimes\,\mathbf{k}/\,\mathbb{Z}$, we will let $Z_{\mathbf{G}}(v)$ denote the centralizer $Z_{\mathbf{G}}(m,x)$ of the corresponding pair $(m,x)\in(X_{*}(\mathbf{T})\otimes\mathbf{k}/\,\mathbb{Q})\times\mathbf{T}(\mathbf{k})^{tor}$. Conjugate elements of $X_{*}(\mathbf{T})\otimes\,\mathbf{k}/\,\mathbb{Z}$ yield isomorphic centralizers, so it makes sense to define $Z_{\mathbf{G}}(v)$ for $v\in(X_{*}(\mathbf{T})\otimes\,\mathbf{k}/\,\mathbb{Z})/\,W$. We end up the following parametrization of regular formal connections. ###### Corollary 3.38. There is a natural bijection between regular formal connections over $D^{*}$ and pairs $(v,O)$, where 1. (i) $v\in(X_{*}(\mathbf{T})\otimes\,\mathbf{k}/\,\mathbb{Z})/\,W$. 2. (ii) $O$ is a nilpotent orbit in $\mathcal{N}_{Z_{\mathbf{G}}(v)}/\,Z_{\mathbf{G}}(v)$. ###### Definition 3.39. Let $A$ be a regular formal connection over $D^{*}$. We will denote by $(m_{A}^{s},m_{A}^{n})$ the corresponding pair granted by Corollary 3.38. $m_{A}^{s}$ (resp. $m_{A}^{n}$) is called the semisimple (resp. unipotent) monodromy of $A$. ###### Example 3.40. Suppose that $\mathbf{k}=\mathbb{C}$. The set of pairs $(m_{A}^{s},m_{A}^{n})$ as above is in correspondence with the set conjugacy classes in $\mathbf{G}(\mathbb{C})$. For a representative $(C,U)\in\text{Lie}(\mathbf{T})\times\mathcal{N}_{\mathbf{G}}$ of the pair $(m_{A}^{s},m_{A}^{n})$, the corresponding element of $\mathbf{G}(\mathbb{C})$ is given by $\text{exp}(2\pi iC+U)$. This just the monodromy class of the regular formal connection. We can use the theorem in [Hum95] pg. 26 to give a description of $Z_{\mathbf{G}}(v)$. We can decompose $\mathfrak{g}=\text{Lie}(\mathbf{T})\oplus\bigoplus_{i=1}^{l}\mathfrak{u}_{\chi_{i}}$, where each $\mathfrak{u}_{i}$ is a one dimensional eigenspace of $\mathbf{T}$ consisting of nilpotent elements (we allow repetition in the $\chi_{i}$). Suppose that $\mathbf{T}$ acts on $\mathfrak{u}_{i}$ through the character $\chi_{i}:\mathbf{T}\longrightarrow\mathbb{G}_{m}$. Since we are working in characteristic $0$, each $\mathfrak{u}_{i}$ is the Lie algebra of a unique unipotent subgroup $\mathbf{U}_{i}$ isomorphic to $\mathbb{G}_{a}$. Let $v\in(X_{*}(\mathbf{T})\otimes\,\mathbf{k}/\,\mathbb{Z})$. For each character $\chi\in X^{*}(\mathbf{T})$ it makes sense to ask whether $\langle v,\chi\rangle\in\mathbb{Z}$, even though $v$ is only defined up to an element of $X_{*}(\mathbf{T})$. The connected component of $Z_{\mathbf{G}}(v)$ is generated by $\mathbf{T}$ and those unipotent weight spaces $\mathbf{U}_{i}$ such that $\langle v,\chi_{i}\rangle\in\mathbb{Z}$. The full group $Z_{\mathbf{G}}(v)$ is generated by its neutral component and the reflections $w_{\alpha}$ in the Weyl group $W$ of $\mathbf{L}$ corresponding to roots $\alpha\in\Phi$ such that $\langle v,\,\alpha\rangle\in\mathbb{Z}$. In order to classify formal regular connections, it is convenient to group them depending on the type of their semisimple monodromy. For each $v\in(X_{*}(\mathbf{T})\otimes\,\mathbf{k}/\,\mathbb{Z})/\,W$, we can define the type of $v$ to be a subset $S\subset V/\,W$ just as we did when working over $\overline{F}$. For any $S\subset V/\,W$, let us denote by $P_{S}\subset(X_{*}(\mathbf{T})\otimes\,\mathbf{k}/\,\mathbb{Z})/\,W$ the set of all element of type $S$. By the description given in the last paragraph, it follows that the isomorphism class of $Z_{\mathbf{G}}(v)$ is the same for all $v\in P_{S}$. We will denote this group by $Z_{\mathbf{G}}(S)_{F}$. We can now rewrite the last corollary. ###### Corollary 3.41. There is a natural correspondence $\left\\{\text{regular formal connections over $D^{*}$}\right\\}\;\;\;\longleftrightarrow{\;\;\;\;}\bigsqcup_{S\subset V/\,W}P_{S}\times\mathcal{N}_{Z_{\mathbf{G}}(S)_{F}}/\,Z_{\mathbf{G}}(S)_{F}$ This yields procedure to describe all regular $\mathbf{G}(F)$-gauge equivalence classes of formal connections. The different types $S$ can be established by declaring a subsets of characters that evaluate to integer eigenvalues. For each $S$, one needs to parametrize the nilpotent orbits in the Lie algebra of $\mathbf{M}_{S}$. This works especially well when the group is reductive. ###### Example 3.42. Suppose that $\mathbf{G}$ is reductive. Then each $\mathbf{M}_{S}$ is connected reductive. If the group $\mathbf{G}$ is classical, we can parametrize the finite set of nilpotent orbits $\mathcal{N}_{\mathbf{M}_{S}}/\,\mathbf{M}_{S}$ using partition diagrams as in [CM93] Chapter 5. This yields explicit canonical block decompositions of regular connections for classical groups. ## 4 Irregular connections for $\mathbf{G}$ reductive ### 4.1 Connections in canonical form Let $\mathbf{G}$ be connected reductive over $\mathbf{k}$. ###### Definition 4.1 (Canonical form). A connection $B\in\mathfrak{g}_{\overline{F}}$ is said to be in canonical form if we have $B=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}+t^{-1}\,C$, where 1. (1) $r_{j}\in\mathbb{Q}$ for all $j$, and they satisfy satisfy $r_{1}<r_{2}<..r_{l}<-1$. 2. (2) $D_{j}\neq 0$ is a semisimple element in $\mathfrak{g}$ for all $j$. 3. (3) $D_{1},D_{2},...,C$ are pairwise commuting (the brackets vanish). The $r_{j}$ above are called the levels of the canonical form. The smallest of them $r_{1}$ is called the principal level. The initial sum $\sum_{j=1}^{l}D_{j}\,t^{r_{j}}$ is called the irregular part of the connection; we denote it by $B_{\text{irr}}$. ###### Remark 4.2. The irregular part could be $0$ if the summation is empty. We then recover the notion of canonical form for a regular connection. We prove the existence of canonical form for reductive groups first. ###### Theorem 4.3 (Reduction Theory for Reductive Groups). Let $\mathbf{G}$ be connected reductive and $A\in\mathfrak{g}_{\overline{F}}$. Then there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot A=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}+t^{-1}\,C$ is in canonical form. The argument proceeds by induction on $\text{dim}\,\mathbf{G}$. The base case (when $\mathbf{G}=\mathbb{G}_{m}$) follows from the computation done in the proof of Proposition 3.12. We state this result for future reference. ###### Proposition 4.4. Let $\mathbf{G}$ be a tori. 1. (a) Let $A=\sum_{j=r}^{\infty}A_{j}\,t^{j}$ a formal connection in $\mathfrak{g}_{F}$. Then there exists $x\in\mathbf{G}(\mathcal{O})$ such that $x\cdot A=\sum_{j=r}^{-1}A_{j}\,t^{j}$. Moreover there is a unique such $x$ with $x\equiv 1\,\left(mod\;t\right)$. 2. (b) Let $B=\sum_{j=r}^{-1}B_{j}\,t^{j}$ and $C=\sum_{j=r}^{-1}C_{j}\,t^{j}$ be two connections in canonical form. Suppose that there exists $x\in\mathbf{G}(F)$ such that $x\cdot C=B$. Then $x=g\,t^{\mu}$ for some cocharacter $\mu\in X_{*}(\mathbf{G})$ and some $g\in\mathbf{G}(\mathbf{k})$. In this case, we have $B_{j}=C_{j}$ for all $r\leq j<-1$ and $C_{-1}=B_{-1}-\mu$. ###### Proof. This is the same computation as in Propositions 3.12 and 3.14. We omit the details. ∎ ###### Remark 4.5. In particular we see that two canonical connections $B=\sum_{j=r}^{-1}B_{j}\,t^{j}$ and $C=\sum_{j=r}^{-1}C_{j}\,t^{j}$ for a torus are gauge equivalent over $F$ if and only if $B_{j}=C_{j}$ for all $r\leq j<-1$ and $C_{-1}-B_{-1}\in X_{*}(\mathbf{G})$. By lifting, we conclude that they are equivalent over $\overline{F}$ if and only if $B_{j}=C_{j}$ for all $r\leq j<-1$ and $C_{-1}-B_{-1}\in X_{*}(\mathbf{G})\otimes\mathbb{Q}$. Let’s get started with the argument for Theorem 4.3. By the structure theory of reductive groups, we know that $\mathbf{G}$ admits an isogeny from the product of its maximal central torus and its derived subgroup $\mathbf{G}_{der}$. By Proposition 4.4, we can deal with the central part. We may therefore assume that $\mathbf{G}$ is semisimple. By lifting to a ramified cover, we can assume $A=\sum_{j=r}^{\infty}A_{j}\,t^{j}\in\mathfrak{g}_{F}$ with $A_{r}\neq 0$. If $r\geq-1$ we can use the theory for regular connections developed in Section 2. So we can assume $r<-1$. There are two substantially different possibilities: $A_{r}$ could be nilpotent or not. The case when $A_{r}$ is not nilpotent turns out to be the easiest; we do it first. ### 4.2 The case when $A_{r}$ is not nilpotent We need the following lemma (cf. [BV83, 9.3]). ###### Lemma 4.6. Let $A=\sum_{j=r}A_{j}\,t^{j}$ a connection in $\mathfrak{g}_{F}$ with $r<-1$. Let $V=(A_{r})_{s}$ be the semisimple part of $A_{r}$. Then, there exist $x\in\mathbf{G}(F)$ such that $x\cdot A$ is in $\mathfrak{g}_{V}(F)$. ###### Proof. This is very similar to Lemma 3.3. We will inductively build a sequence $(B_{j})_{j=1}^{\infty}$ of elements of $\mathfrak{g}$ such that the gauge transformation $x\vcentcolon=\lim_{n\rightarrow\infty}\prod_{j=0}^{n-1}\text{exp}(t^{n-j}\,B_{n-j})$ satisfies the conclusion of the lemma. Suppose that we have chosen $B_{j}$ for $j\leq k$ such that the connection $A^{(k)}=\sum_{l=r}^{\infty}A^{(k)}_{l}\,t^{l}$ defined by $A^{(k)}\vcentcolon=\prod_{j=0}^{k-1}\text{exp}(t^{n-j}\,B_{k-n})\cdot A$ satisfies $A_{l}^{(k)}\in\mathfrak{g}_{V}$ for all $l\leq k+r$. Notice that the base case $k=0$ is trivial and $A^{(k)}_{r}=A_{r}$. Let’s try to determine $B_{k+1}$. Recall that $\text{exp}(t^{k+1}\,B_{k+1})\equiv 1+t^{k+1}\,B_{k+1}\;(\text{mod}\;t^{k+2})$. By an elementary matrix computation (choose an embedding of $\mathbf{G}\hookrightarrow\mathbf{\text{GL}_{n}}$), one can see that $\text{exp}(t^{k+1}B_{k+1})\cdot A^{(k)}\equiv\sum_{l=r}^{k+r}A^{(k)}_{l}\,t^{l}+[A^{(k)}_{k+1+r}-ad(A_{r})B_{k+1}]\,t^{k+1+r}\;\;(\text{mod}\;t^{k+2+r})$ Let $\mathfrak{g}=\bigoplus_{\lambda}\mathfrak{g}_{\lambda}$ be the spectral decomposition of $ad(X)=(ad(A_{r}))_{s}$. By definition the operator $ad(A_{-r})$ restricts to an automorphism of $\mathfrak{g}_{\lambda}$ for all $\lambda\neq 0$. In particular, we can choose $B_{k+1}\in\mathfrak{g}$ such that $A^{(k)}_{k+1-r}-ad(A_{-r})B_{k+1}$ is in $\mathfrak{g}_{0}=\mathfrak{g}_{V}$. This concludes the construction of the sequence $(B_{j})_{j=1}^{\infty}$. It follows from the construction that $x\vcentcolon=\lim_{n\rightarrow\infty}\prod_{j=0}^{n-1}\text{exp}(t^{n-j}\,B_{n-j})$ satisfies $x\cdot A\;\in\,\mathfrak{g}_{V}(F)$. ∎ Let us continue with the proof of Theorem 4.3. Suppose that $A_{r}$ is not nilpotent. Then the semisimple part $X=(A_{r})_{s}$ is not $0$. Since we are assuming that $\mathbf{G}$ is semisimple, the connected reductive centralizer $\mathbf{Z}_{G}(X)$ is a proper subgroup of $\mathbf{G}$. By Lemma 4.6 we can assume that $A\in\mathfrak{g}_{V}(F)$. We win by induction, because $\text{dim}\,\mathbf{Z}_{G}(V)\,<\,\text{dim}\,\mathbf{G}$. This settles the case when $A_{r}$ is not nilpotent in the proof of Theorem 4.3. Recall that the principal level of a canonical connection is defined to be the order $r_{1}$ (see the paragraph after Definition 4.1). We can define the principal level of a connection $A$ to be the principal level of any canonical connection equivalent to $A$. This is well defined by Lemma 5.4 in the next section. The inductive argument given above implies the following interesting fact. ###### Proposition 4.7. Suppose that $A=\sum_{j=r}^{\infty}A_{j}t^{j}$ with $r<-1$ and $A_{r}$ not nilpotent. Then $r$ is the principal level of A. ###### Proof. We induct on the dimension of the group. The base case $\mathbb{G}_{m}$ is clear by direct computation. Notice that in the proof above we have that $A_{r}$ is still not nilpotent in the smaller group $\mathbf{Z}_{G}(V)$, since its semisimple part $V$ is not $0$. We can then conclude by induction. ∎ ###### Remark 4.8. As we will soon see, this is not necessarily the case when $A_{r}$ is nilpotent. In the nilpotent case the principal level can be larger than $r$. ### 4.3 The case when $A_{r}$ is nilpotent In this section we conclude the proof of Theorem 4.3 by dealing with the case when $A_{r}$ is nilpotent. We closely follow an argument sketched in [BV83], which we include here for the sake of completeness. For this section $\mathbf{G}$ will be semisimple, as we may assume for our proof of Theorem 4.3. Let’s set up the notation we will use. Suppose that we have $A=\sum_{j=r}^{\infty}A_{j}\,t^{j}$ with $A_{r}$ nilpotent and $r<-1$. Let $(H,X,\,Y=A_{r})$ be a Jacobson-Morozov $\mathfrak{sl}_{2}$-triple coming from an algebraic homomorphism $\Phi:\text{SL}_{2}\longrightarrow\mathbf{G}$. For an integer $n$, we will denote by $t^{nH}$ the element $t^{\mu}$, where $\mu$ is the natural composition $\mathbb{G}_{m}\xrightarrow{[n]}\mathbb{G}_{m}\hookrightarrow\text{SL}_{2}\xrightarrow{\Phi}\mathbf{G}$. ###### Lemma 4.9. With notation as above, there exists $x\in\mathbf{G}(F)$ such that $x\cdot A=\sum_{j=r}^{\infty}B_{j}\,t^{j}$ satisfies 1. (1) $B_{r}=A_{r}$. 2. (2) $B_{j}\in\mathfrak{g}_{X}$ for all $j>r$. ###### Proof. This is a carbon copy of the proof of Lemma 4.6. The only difference is that in the last paragraph we have to use the fact that the range of $ad(A_{r})$ is complementary to $\mathfrak{g}_{X}$. This follows from the theory of representations of $\mathfrak{sl}_{2}$. We omit the details. ∎ By Lemma 4.9, we can assume that $A_{j}\in\mathfrak{g}_{X}$ for $j>r$. For the purpose of having an algorithm that works in finitely many steps, we won’t actually use the full force of the lemma. Instead, we will use a weaker hypothesis as an input for the next proposition. Let $\Lambda\vcentcolon=\Lambda\left(A_{r}\right)$ be as in Definition 2.14. We will henceforth suppose that $A_{r+m}\in\mathfrak{g}_{X}$ for $1\leq m<\Lambda(|r|-1)$. Let $(Z_{l})_{l=1}^{q}$ be a basis of eigenvectors of $ad(H)$ acting on $\mathfrak{g}_{X}$. This means that $\mathfrak{g}_{X}=\bigoplus_{l=1}^{q}\mathbf{k}\,Z_{l}$ and that there exist $\lambda_{l}$ such that $[H,Z_{l}]=\lambda_{l}\,Z_{l}$. It turns out that the $\lambda_{l}$s are nonnegative integers, by the theory of representations of $\mathfrak{sl}_{2}$. By the assumption on $A$, we can write $A_{r+m}=\sum_{l=1}^{q}a_{r+m,\,l}\,Z_{l}$ for all $1\leq m\leq\Lambda(|r|-1)$ and some constants $a_{r+m,\,l}\in\mathbf{k}$. ###### Definition 4.10. In the situation above, define $\delta=\delta(A)$ to be given by: $\delta=\text{inf}\;\left\\{\frac{m}{\frac{1}{2}\lambda_{l}+1}\;\vcentcolon\;\;1\leq m<\Lambda(|r|-1),\;\;1\leq l\leq q,\;\;a_{r+m,\,l}\neq 0\right\\}$ We set $\delta=\infty$ if $A_{r+m}=0$ for all $1\leq m<\Lambda$. We also define the set $P\vcentcolon=\left\\{(m,l)\;\vcentcolon\;\;1\leq m<\Lambda(|r|-1),\;\;1\leq l\leq q,\;\;a_{r+m,\,l}\neq 0,\;\;\frac{m}{\frac{1}{2}\lambda_{l}+1}=\delta\right\\}$ In plain words, $P$ is the set of pairs $(m,l)$ of indices in the definition of $\delta$ where the infimum is actually achieved. ###### Remark 4.11. By the definition of $\Lambda(A_{r})$, it follows that the denominators appearing in the set defining $\delta$ are always less than $\Lambda$. This implies that there exists a positive integer $b\leq 2\Lambda-1$ such that $b\delta\in\mathbb{Z}$. This fact will be used later to determine a bound for the ramification needed to put $A$ in canonical form. The following proposition is one of the main steps in the argument in [BV83]. We are going to get a step closer to canonical form by applying a transformation of the type $t^{nH}$. These elements are called shearing transformations. The statement and proof in the case of $\mathbf{GL_{n}}$ can be found in [BV83] pages 33-34. We have decided to include a detailed treatment of the general case for the convenience of the reader. ###### Proposition 4.12 (Main Proposition for the Induction Step). Let the notation/set-up be as discussed above. 1. (C1) Suppose $|r|-1\leq\delta\leq\infty$. Let $\tilde{A}$ be the $2$-lift of $A$. Then $B\vcentcolon=t^{(r+1)H}\cdot\tilde{A}$ is of the first kind, and $B_{-1}$ only depends on $A_{r+m}$ for $0\leq m\leq\Lambda(|r|-1)$. 2. (C2) Suppose $0<\delta<|r|-1$. We know that $b\delta\in\mathbb{Z}$ for some $b\in\mathbb{N}$. Let $\tilde{A}$ be the $2b$-lift of $A$. We have that $B\vcentcolon=t^{-b\delta H}\cdot\tilde{A}$ has order $r^{\prime}\vcentcolon=2br+2b\delta+2b-1<-1$. Moreover, $B_{r^{\prime}}=2bA_{r}+2b\sum_{(m,l)\in P}a_{r+m,\,l}\,Z_{l}\;\neq\;2bA_{r}$ In particular we have that $B_{r^{\prime}}$ is determined by $A_{r+m}$ for $0\leq m<\Lambda(|r|-1)$. If $B_{r^{\prime}}$ is nilpotent, then $\text{dim}\,(G\cdot B_{r^{\prime}})\,>\,\text{dim}\,(G\cdot A_{r})$. ###### Proof. The computation is similar to the one we did in the proof of Theorem 3.6. Recall from the discussion in that proof that for all $W\in\mathfrak{g}_{\beta}$ we have $\text{Ad}(t^{nH})\,W=t^{n\beta(H)}W\;\;\;(*)$ 1. (C1) By using the definitions and expanding $\displaystyle t^{(r+1)H}\cdot\tilde{A}$ $\displaystyle=\;\;2\sum_{m=0}^{\Lambda(|r|-1)-1}\text{Ad}(t^{(r+1)H})A_{r+m}\,t^{2(r+m)+1}$ $\displaystyle\quad\;+\,\text{Ad}(t^{(r+1)H})A_{r+\Lambda(|r|-1)}\,t^{2(r+\Lambda(|r|-1))+1}$ $\displaystyle\quad+\,2\sum_{m=\Lambda(|r|-1)+1}^{\infty}\text{Ad}(t^{(r+1)H})A_{r+m}\,t^{2(r+m)+1}$ $\displaystyle\quad\;+\,\frac{d}{dt}\,(t^{(r+1)H})\,t^{-(r+1)H}$ The fourth summand is just $(r+1)Ht^{-1}$, which is of the first kind. We can see that the third summand is actually in $\mathfrak{g}(\mathcal{O})$ by using $(*)$ and the fact that $(r+1)\beta(H)\geq(2\Lambda-2)(r+1)$ for all roots $\beta$. The same reasoning implies that the second summand is of the first kind. For the first summand, we can write $A_{r+m}=\sum_{l=1}^{q}a_{r+m,\,l}\,Z_{l}$. We can expand and use $(*)$ plus the definition of $\lambda_{l}$. We get that the first summand is: $2\sum_{m=0}^{\Lambda(|r|-1)-1}\sum_{l=1}^{q}a_{r+m,\,l}\,Z_{l}\,t^{2(r+m)+1+(r+1)\lambda_{l}}$ This expression is also of the first kind. This can be shown by doing some algebra with the exponents of $t$, keeping in mind the definition of $\delta$ and the fact that $\delta\geq|r|-1$. The remark about $B_{-1}$ follows plainly from the argument, because the third summand did not contribute to $B_{-1}$. 2. (C2) This is very similar to the first case. We expand: $\displaystyle t^{-b\delta H}\cdot\tilde{A}$ $\displaystyle=\;\;2b\sum_{m=0}^{\Lambda(|r|-1)-1}\text{Ad}(t^{-b\delta H})A_{r+m}\,t^{2b(r+m)+2b-1}$ $\displaystyle\quad+\,2b\sum_{m=\Lambda(|r|-1)}^{\infty}\text{Ad}(t^{-b\delta H})A_{r+m}\,t^{2b(r+m)+2b-1}$ $\displaystyle\quad\;+\,\frac{d}{dt}\,(t^{(-b\delta)H})\,t^{(b\delta)H}$ The third summand is $-b\delta Ht^{-1}$, which is of the first kind. We can therefore ignore the third summand. The bound $-b\delta\beta(H)\geq-2b\delta(\Lambda-1)$ and equation $(*)$ show that the order of the second summand is at least $r^{\prime}+1=2br+2b\delta+2b$. The computation is almost the same as for the third summand in Case 1 above. For the first summand, we can again use $A_{r+m}=\sum_{l=1}^{q}a_{r+m,\,l}\,Z_{l}$ and expand using $(*)$ to get: $2b\sum_{m=0}^{\Lambda(|r|-1)-1}\sum_{l=1}^{q}a_{r+m,\,l}\,Z_{l}\,t^{2b(r+m)+2b-1-b\delta\lambda_{l}}$ We are reduced to check that the exponent of $t$ in the sum above has minimal value $r^{\prime}=2br+2b\delta+2b-1$ exactly for the pairs $(m,l)$ in $P$. This is an exercise in elementary algebra. The claim about $B_{r^{\prime}}$ follows from the argument, because the second summand does not contribute to $B_{r^{\prime}}$. The claim about the increase of the dimension of nipotent orbits is a direct consequence of Proposition 2.8. ∎ ###### Remark 4.13. The claim about the dimension of the orbit in $C2$ is essential. This guarantees that the process of applying shearing transformations eventually stops. Hence we are provided with a terminating algorithm. See the proof of Theorem 4.3 given below for details. ###### Example 4.14. Let’s see how this works in the case of $\text{SL}_{2}$. Up to inner automorphism, we can assume that $A_{r}=Y=\begin{bmatrix}0&0\\\ 1&0\end{bmatrix}$. Then $X=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix}$ and $H=\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}$. In this case $\Lambda=2$, and $\mathfrak{g}_{X}=\mathbf{k}X$. So there is a single eigenvalue $\lambda=2$. Our assumption just says that $A$ is of the form $A=Y\,t^{r}+\sum_{m=1}^{2|r|-3}a_{r+m}X\,t^{r+m}+\text{higher order terms}$ We have that $\delta$ is $\frac{n}{2}$, where $n$ is the smallest index such that $a_{r+n}\neq 0$. The set $P$ only contains this index $n$. So in fact $A$ can be written in the form $A=Y\,t^{r}+\sum_{m=n}^{2|r|-3}a_{r+m}X\,t^{r+m}+\text{higher order terms}$ There are two cases. 1. (C1) The first case is $n\geq 2(|r|-1)$. This just means that all $a_{i}$ above are $0$. Then we can use the change of trivialization $t^{\frac{r+1}{2}H}=\begin{bmatrix}t^{\frac{r+1}{2}}&0\\\ 0&t^{-\frac{r+1}{2}}\end{bmatrix}$ to transform $A$ into a connection of the first kind. 2. (C2) The second case is when $n<2(|r|-1)$. Here at least some of the $a_{i}$ are not $0$. We can apply the transformation $t^{-\frac{n}{4}H}=\begin{bmatrix}t^{-\frac{n}{4}}&0\\\ 0&t^{\frac{n}{4}}\end{bmatrix}$. The resulting connection will have order $r+\frac{n}{2}$. The principal term will be $B_{r+\frac{n}{2}}=Y+a_{r+n}X$, which is semisimple. Hence we can use Lemma 4.6 to reduce to the group $\mathbb{G}_{m}$. We can then apply Proposition 4.4 to find the canonical form. ###### Proof of Theorem 4.3. By Lemma 4.9, we can put ourselves in the situation of Proposition 4.12 above. We have three possibilities: 1. (i) If $|r|-1\leq\delta\leq\infty$, then we can use Proposition 4.12 Case 1. We are done by the theory of regular connections we have already developed. 2. (ii) If $0<\delta<|r|-1$, we can use 4.12 Case 2. Suppose that $B_{r^{\prime}}$ is not nilpotent. Then we are in the case worked out in Subsection 4.2. 3. (iii) Suppose that $0<\delta<|r|-1$ and $B_{r^{\prime}}$ is nilpotent with $\text{dim}\,(G\cdot B_{r^{\prime}})\,>\,\text{dim}\,(G\cdot A_{r})$. We can apply Proposition 4.12 Case 2 again with $B$ instead of $A$. We can keep iterating this procedure until we are in one of the first two possibilities above. Notice that this process cannot go on indefinitely, because the dimensions of nilpotent orbits in $\mathbf{G}$ are bounded. ∎ ###### Remark 4.15. The dimension of adjoint nilpotent orbits in $\mathbf{G}$ is always even [CM93]. Therefore we need to apply at most $\left\lfloor{\frac{1}{2}\text{dim}(\mathbf{G})}\right\rfloor$ shearing transformations as in Proposition 4.12 Case 2 before we land in one of the first two possibilities. ### 4.4 Algorithm for reductive groups and some quantitative results Let’s give a detailed description of the reduction algorithm that we obtain from the proof of Theorem 4.3. ###### Algorithm 4.16 (Algorithm for reduction of a formal connection for a reductive group). There is a set of six possible operations that we will use as steps in our algorithm. 1. (i) Apply Lemma 4.6. 2. (ii) Apply Lemma 4.9. 3. (iii) Apply Proposition 4.12 Case 1. 4. (iv) Apply Proposition 4.12 Case 2. 5. (v) Find the canonical form of a connection in a torus. 6. (vi) Find the canonical form for a connection of the first kind in a semisimple group (as in Theorem 3.6 of Section 3). The algorithm proceeds as follows. The input is a given reductive group $\mathbf{G}$ and a formal connection $A=\sum_{j=r}^{\infty}A_{j}\,t^{j}$. First, we know that $\mathbf{G}$ is isogenous to the product $Z^{0}\left(\mathbf{G}\right)\times\mathbf{G}_{\text{der}}$ of its maximal central torus and its derived subgroup. Apply operation (v) to the central part of the connection $A_{Z^{0}(\mathbf{G})}$. We can record the (canonical form) output of (v) and ignore it from now on, since it is not going to be altered by the subsequent steps in the algorithm. Replace $\mathbf{G}$ by $\mathbf{G}_{\text{der}}$ and $A$ by $A_{\text{der}}$. We have two cases. 1. (1) If $A_{der}$ is of the first kind, apply step (vi) to reduce this connection to canonical form. Add any “central” parts we might have split off earlier in the algorithm and output the result. End of the algorithm. 2. (2) If $A_{der}$ is not of the first kind, check whether $A_{r}$ is nilpotent or not. There are now two ways to proceed: 1. (2-1) If $A_{r}$ is not nilpotent, use operation (i). Replace $\mathbf{G}$ by $Z_{\mathbf{G}}\left(\,\left(A_{r}\right)_{s}\,\right)$ and replace $A$ by the output of operation (i). Return to the beginning of the algorithm with this new input. 2. (2-2) If $A_{r}$ is nilpotent, compute $\Lambda\left(A_{r}\right)$. Apply operation $(ii)$ and replace $A$ with the output. Now compute $\delta$. 1. (2-2a) If $|r|-1\leq\delta$, apply operation (iii) and replace $A$ with the output. This is a connection of the first kind. Return to the beginning of the algorithm. 2. (2-2b) If $\delta<|r|-1$, apply operation (iv). Go to the beginning of the algorithm. ###### Remark 4.17. In the algorithm above the order of the pole of $A$ only ever gets smaller (taking into account $b$-lifting whenever passing to a ramified cover). This shows that the principal level of $A$ determines the mildest pole in the $\mathbf{G}(\overline{F})$-gauge equivalence class of $A$. A careful analysis of Algorithm 4.16 yields a bound for the ramification needed to put a given connection into canonical form. We can get a uniform bound that only depends on the group, and not on the connection. Before giving the proof of such bound, we need a lemma. Note that in each step of Algorithm 4.16 we are ultimately working with a semisimple subgroup of $\mathbf{G}$. These subgroups are of the form $\mathbf{H}_{der}$, where $\mathbf{H}$ is the centralizer of a finite set $\\{D_{1},D_{2},...,D_{l}\\}$ of pairwise commuting semisimple elements in $\mathfrak{g}$. We will first try to understand $J(\mathbf{H}_{der})$ (see Definition 2.11). ###### Lemma 4.18. Let $\mathbf{G}$ be a connected semisimple group. Let $R$ be the rank of $\mathbf{G}$. Suppose that $\\{D_{1},D_{2},...,D_{l}\\}$ is a finite set of pairwise commuting semisimple elements in $\mathfrak{g}$. Set $\mathbf{H}=Z_{\mathbf{G}}\left(\\{D_{1},D_{2},...,D_{l}\\}\right)$, the centralizer of all $D_{i}$. Let $\mathbf{H}_{der}$ be the derived subgroup of $\mathbf{H}$. Then we have $J(\mathbf{H}_{der})\leq\text{hgt}(\mathfrak{g})^{2R-2}\cdot J(\mathbf{G})$. ###### Proof. The lemma is clearly true if $\mathbf{H}=\mathbf{G}$. We can therefore assume that $\mathbf{H}\neq\mathbf{G}$. Let $\mathbf{T}$ be a maximal torus of $\mathbf{G}$ such that $\text{Lie}(\mathbf{T})$ contains the set $\\{D_{i}\\}$ of pairwise commuting semisimple elements. Note that $\mathbf{T}\subset\mathbf{H}$ is a maximal torus. Let $\Phi$ (resp. $\Sigma$) be the set of roots of $\mathbf{G}$ (resp. $\mathbf{H}$) with respect to $\mathbf{T}$. We will denote by $\Sigma^{\vee}$ and $\Phi^{\vee}$ the corresponding sets of coroots. It follows by definition that $\Sigma\subset\Phi$ and $\Sigma^{\vee}\subset\Phi^{\vee}$. Write $Q_{\mathbf{H}_{der}}$ for the coweight lattice of $\mathbf{H}_{der}$. Let $\lambda\in Q_{\mathbf{H}_{der}}$. We want to show that there exists $b\leq\text{hgt}(\mathfrak{g})^{2R-1}\cdot J(\mathbf{G}_{\text{der}})$ such that $b\lambda\in\mathbb{Z}\Sigma^{\vee}$. Fix a choice of positive roots $\Phi^{+}$ in $\Phi$. Let $\Delta_{\Phi}$ be the corresponding set of simple roots. By definition $|\Delta_{\Phi}|=R$. Notice that this induces a set of positive roots $\Sigma^{+}\vcentcolon=\Phi^{+}\cap\Sigma$. Let $\Delta_{\Sigma}$ be the corresponding set of simple roots in $\Sigma$. Set $c\vcentcolon=|\Delta_{\Sigma}|$. We know that $c\leq R-1$ because $\mathbf{H}\neq\mathbf{G}$. Consider the short exact sequence $0\longrightarrow\mathbb{Z}\Sigma\xrightarrow{\;\;M\;\;}\mathbb{Z}\Phi\longrightarrow\mathbb{Z}\Phi/\,\mathbb{Z}\Sigma\longrightarrow 0$ The theory of Smith normal form implies that $\mathbb{Z}\Phi/\,\mathbb{Z}\Sigma\cong E\oplus\mathbb{Z}^{R-c}$, where $E$ is a finite group. The exponent of $E$ is given by the biggest elementary divisor $d$ of the inclusion $M$ of free $\mathbb{Z}$-modules. Applying the functor $\text{Hom}(-,\mathbb{Z})$ to the short exact sequence yields an exact sequence $0\longrightarrow\mathbb{Z}^{R-c}\longrightarrow Q_{\mathbf{G}}\longrightarrow Q_{\mathbf{H}_{der}}\longrightarrow E\longrightarrow 0$ Hence we have that $d\lambda$ can be extended to an element of $Q_{\mathbf{G}}$. By the definition of $J(\mathbf{G})$, it follows that $d\,J(\mathbf{G})\,\lambda$ extends to an element of $\mathbb{Z}\Phi^{\vee}$. Let $\varphi:\mathbb{Z}\Phi^{\vee}\longrightarrow Q_{\mathbf{H}_{der}}$ be the composition $\varphi:\mathbb{Z}\Phi^{\vee}\,\hookrightarrow\,Q_{\mathbf{G}}\,\longrightarrow Q_{\mathbf{H}_{der}}$ Set $L\vcentcolon=\text{Im}\,\varphi$ and $K\vcentcolon=\text{Ker}\,\varphi$. The discussion above implies that the exponent of the finite group $Q_{\mathbf{H}_{der}}/\,L$ is bounded by $d\,J(\mathbf{G})$. By definition we have a short exact sequence $0\,\longrightarrow\,K\,\longrightarrow\,\mathbb{Z}\Phi^{\vee}\,\longrightarrow\,L\,\longrightarrow\,0$ Since $L$ is a torsion-free $\mathbb{Z}$-module, the above exact sequence splits. Fix a splitting $\mathbb{Z}\Phi^{\vee}\cong K\oplus L$. Let’s look now at the inclusion of lattices $\mathbb{Z}\Sigma^{\vee}\subset\mathbb{Z}\Phi^{\vee}$. The composition $\mathbb{Z}\Sigma^{\vee}\,\hookrightarrow\,\mathbb{Z}\Phi^{\vee}\,\xlongequal{\;}\,K\oplus L\,\xrightarrow{\;\;pr_{2}\;\;}\,L\,\xhookrightarrow{\;\;}\,Q_{\mathbf{H}_{der}}$ is the natural inclusion $\mathbb{Z}\Sigma^{\vee}\hookrightarrow Q_{\mathbf{H}_{der}}$. Hence the morphism $\psi$ given by the composition $\psi\,:\;\;\mathbb{Z}\Sigma^{\vee}\,\hookrightarrow\,\mathbb{Z}\Phi^{\vee}\,\xlongequal{\;}\,K\oplus L\,\xrightarrow{\;\;pr_{2}\;\;}\,L$ is injective. So we have an inclusion $\psi:\mathbb{Z}\Sigma^{\vee}\hookrightarrow L$. Let $e$ denote the exponent of the finite group $L/\,\mathbb{Z}\Sigma^{\vee}$. By definition $e$ is the biggest elementary divisor of the inclusion $\psi:\mathbb{Z}\Sigma^{\vee}\hookrightarrow L$. Notice that this is also the biggest elementary divisor of the natural inclusion $\mathbb{Z}\Sigma^{\vee}\subset\mathbb{Z}\Phi^{\vee}\xlongequal{}K\oplus L$. The discussion up to now implies that $J(\mathbf{H}_{der})\leq e\,d\,J(\mathbf{G})$. We are left to compute the elementary divisors $d$ and $e$ of the inclusions of the root and coroot lattices. We first claim that $d\leq\text{hgt}(\mathfrak{g})^{R-1}$. In order to prove the claim, we will use $\Delta_{\Sigma}$ and $\Delta_{\Phi}$ as bases for the root lattices. For each $\alpha\in\Delta_{\Sigma}$, we can write $\alpha=\sum_{\beta\in\Delta_{\Phi}}m_{\beta}^{\alpha}\,\beta$ for some nonnegative integers $m_{\beta}^{\alpha}$. Set $M\vcentcolon=(m_{\beta}^{\alpha})_{\beta\in\Delta_{\Phi},\,\alpha\in\Delta_{\Sigma}}$. This is a $R\times c$ matrix representing the inclusion $\mathbb{Z}\Sigma\hookrightarrow\mathbb{Z}\Phi$. By the theory of Smith normal form, $d$ divides all $c\times c$-minors of $M$. Since all $m_{\beta}^{\alpha}$ are nonnegative, such $c\times c$-minor is bounded by $\prod_{\alpha\in\Delta_{\Sigma}}\left(\sum_{\beta\in\Delta_{\Phi}}m_{\beta}^{\alpha}\right)\,\leq\,\text{hgt}(\mathfrak{g})^{c}\,\leq\,\text{hgt}(\mathfrak{g})^{R-1}$ The claim $d\leq\text{hgt}(\mathfrak{g})^{R-1}$ follows. We can apply the same argument to the inclusion $\mathbb{Z}\Sigma^{\vee}\subset\mathbb{Z}\Phi^{\vee}$. The maximal height in the dual root system $\Phi^{\vee}$ is also $\text{hgt}(\mathfrak{g})$. Therefore the same proof yields a bound $e\leq\text{hght}(\mathfrak{g})^{R-1}$. This implies $J(\mathbf{H}_{der})\,\leq\,e\,d\,J(\mathbf{G})\,\leq\,\text{hgt}(\mathfrak{g})^{2R-2}\cdot J(\mathbf{G})$ ∎ ###### Proposition 4.19. Let $\mathbf{G}$ be connected reductive. Let $A\in\mathfrak{g}_{F}$ be a connection. Then there exist $x\in\mathbf{G}(F_{b})$ for some positive integer $b$ such that $x\cdot A$ is in canonical form. If we set $R\vcentcolon=\text{rank}(\mathbf{G}_{der})$, then $b$ can be chosen so that $b\;\leq\;2\,\text{hgt}(\mathfrak{g})^{2R-1}\cdot J(\mathbf{G}_{\text{der}})\;\cdot\prod_{j=0}^{\left\lfloor{\frac{\text{dim}(\mathbf{G}_{\text{der}})}{3}}\right\rfloor}\left(4\,\text{hgt}(\mathfrak{g})+2\right)^{\left\lfloor{}\frac{1}{2}\left(\text{dim}(\mathbf{G}_{\text{der}})-3j\right)\right\rfloor}$ ###### Proof. We have to keep track of how much ramification is needed to perform each of the steps in Algorithm 4.16. Recall our six operations: 1. (i) Apply Lemma 4.6. No ramification is needed for this operation, as is apparent from the proof of the lemma. 2. (ii) Apply Lemma 4.9. No ramification is needed for this operation. This also follows directly from the proof of the lemma. 3. (iii) Apply Proposition 4.12 Case 1. We need to pass to a $2$-cover. 4. (iv) Apply Proposition 4.12 Case 2. We need to pass to a $2b$-cover, where $b$ is such that $b\delta\in\mathbb{Z}$. By Remark 4.11, we know that we can choose $b\leq 2\Lambda-1\leq 2\text{hgt}(\mathfrak{g})+1$. 5. (v) Find the canonical form of a connection in a torus. No ramification is needed to perform this operation, by the proof of Proposition 3.12. 6. (vi) Find the canonical form for a connection of the first kind (as in Theorem 3.6). By Lemma 3.9, we can perform this operation after passing to a $b$-cover with $b\leq\text{hgt}(\mathfrak{g})\cdot I(\mathbf{G})$. We know that operations (iii) and (vi) will be used only once, at the end of the algorithm. This gives us a factor of $2\,\text{hgt}(\mathfrak{g})\cdot I(\mathbf{H}_{\text{der}})$, where $\mathbf{H}$ is the centralizer $Z_{\mathbf{G}_{der}}\left(\\{D_{1},D_{2},...,D_{l}\\}\right)$ of a finite set of pariwise commuting semisimple elements $D_{i}$ in $\mathfrak{g}_{der}$. Since $I(\mathbf{H}_{der})\leq J(\mathbf{H}_{der})$, this is bounded by $2\,\text{hgt}(\mathfrak{g})\cdot J(\mathbf{H}_{\text{der}})$. By Lemma 4.18 we known that $J(\mathbf{H}_{\text{der}})\leq\text{hgt}(\mathfrak{g})^{2R-2}\cdot J(\mathbf{G}_{\text{der}})$. This yields the first factor in the bound above. We are now left to count the amount of times that we need to apply operation (iv) in our algorithm. Each time we apply it we need to pass to a cover of ramification at most $4\,\text{hgt}(\mathfrak{g})+2$. By the remark after the proof of Theorem 4.3, we need to apply operation (iv) at most $\left\lfloor{\frac{1}{2}\text{dim}(\mathbf{G}_{\text{der}})}\right\rfloor$ times before we are in the case when $A_{r}$ is not nilpotent. We therefore pick up a ramification of at most $\left(4\,\text{hgt}(\mathfrak{g})+2\right)^{\left\lfloor{}\frac{1}{2}\text{dim}(\mathbf{G}_{\text{der}})\right\rfloor}$. After that we change our group. We can apply operation (i) and split off the central part in order to pass to a proper semisimple subgroup $\mathbf{H}_{der}\vcentcolon=\left(Z_{\mathbf{G}}\left(\,(A_{r})_{s}\,\right)\right)_{\text{der}}$. Notice that $\text{dim}(\mathbf{H}_{der})\leq\text{dim}(\mathbf{G}_{\text{der}})-3$, because we are removing at least two root spaces (positive and negative pair) and the nontrivial central torus of the centralizer $\mathbf{H}$. Now we start all over again. We know that we need to apply operation (iv) at most $\left\lfloor{\frac{1}{2}\left(\text{dim}(\mathbf{G}_{\text{der}})-3\right)}\right\rfloor$-many times until $A_{r}$ is not nilpotent. So we pick up a ramification of at most $\left(4\,\text{hgt}(\mathfrak{g})+2\right)^{\left\lfloor{\frac{1}{2}\left(\text{dim}(\mathbf{G}_{\text{der}})-3\right)}\right\rfloor}$. Iterating this procedure, we get the product appearing in the bound above. ∎ ###### Remark 4.20. In terms of dimension, the right hand side is $J(\mathbf{G}_{\text{der}})\cdot e^{O\left(\text{dim}(\mathbf{G}_{\text{der}})^{2}\,\text{log}\,\text{dim}(\mathbf{G}_{\text{der}})\right)}$. We proceed to establish a quantitative refinement of Theorem 4.3. It essentially follows from keeping track of some indices in the operations for the algorithm above. It makes sense once we know that the irregular part of a canonical form is unique up to conjugacy, as stated in Theorem 5.5 below. The following result over $\mathbb{C}$ can already be found in the work of Babbitt and Varadarajan [BV83, 9.7]. ###### Proposition 4.21 (Determinacy for the Irregular part of the Canonical Form). Let $\mathbf{G}$ be connected reductive. Let $A=\sum_{j=r}^{\infty}A_{j}t^{j}$ be a connection in $\mathfrak{g}_{F}$. The irregular part of the canonical form of $A$ depends only on $A_{r+m}$ for $0\leq m<\left(\text{hgt}(\mathfrak{g})+1\right)(|r|-1)$. ###### Proof. It suffices to check the steps in Algorithm 4.16. In some steps we replace the group $\mathbf{G}$ by a proper subgroup (either a centralizer or the derived subgroup). This can only decrease the quantity $\left(\text{hgt}(\mathfrak{g})+1\right)(|r|-1)$, so we can safely ignore these changes of groups. We are left to study the effect of operations (i)-(vi) in Algorithm 4.16. The last operation (vi) has no effect on the irregular part of the connection, so there is nothing to do here. Step (v) takes a connection $A=\sum^{\infty}_{j=r}A_{j}\,t^{j}$ and outputs its truncation $A=\sum^{-1}_{j=r}A_{j}\,t^{j}$ (see the proof of Proposition 3.12). The output is therefore determined by the coefficients given in the statement of the proposition. Step (iii) outputs a connection with no irregular part. Notice that in Proposition 4.12 we can determine whether we are in Case 1 (i.e. when we have to perform Step (iii)) based on the value of $\delta$. This depends only on the $A_{r+m}$ for $0\leq m<\Lambda\left(A_{r}\right)(|r|-1)$. Since $\Lambda\left(A_{r}\right)\leq\text{hgt}(\mathfrak{g})+1$ by Example 2.15, this case can be determined by the coefficients provided. For the remaining operations (i), (ii) and (iv), we start with a given connection $A$ with lowest coefficient $A_{r}$ and output an irregular connection $B$ with lowest coefficient $B_{r^{\prime}}$. The proposition will follow if we can prove that for each of these operations the coefficients $B_{r^{\prime}+m}$ for $0\leq m<\left(\text{hght}+1\right)\left(|r^{\prime}|-1\right)$ are completely determined by the coefficients $A_{r+m}$ for $0\leq m<\left(\text{hgt}(\mathfrak{g})+1\right)(|r|-1)$. Operations (i) and (ii) are very similar. In this case we have $r=r^{\prime}$. Let $m$ be an integer. From the proofs of Lemma 4.6 and Lemma 4.9, it follows that $B_{m}$ is determined by $A_{j}$ for $j\leq m$. So we are done with these operations. We are left with operation (iv). Recall from the proof of Proposition 4.12 Case 2 that we have $\displaystyle B=t^{-b\delta H}\cdot\tilde{A}$ $\displaystyle=\;\;2b\sum_{m=0}^{\infty}\text{Ad}(t^{-b\delta H})A_{r+m}\,t^{2b(r+m)+2b-1}\;-\;b\delta Ht^{-1}$ The term $b\delta Ht^{-1}$ is determined by the knowledge of $\delta$ and $H=\left(A_{r})\right)_{s}$. For the infinite sum, we can write the root decompositions $A_{r+m}=\sum_{\beta\in\Phi}A^{\beta}_{r+m}$ and use that $\text{Ad}(t^{-b\delta H})\,A^{\beta}_{r+m}=t^{-b\delta\beta(H)}\,A^{\beta}_{r+m}$ in order to get $\displaystyle\text{Ad}(t^{-b\delta H})A^{\beta}_{r+m}\,t^{2b(r+m)+2b-1}\;=A^{\beta}_{r+m}\,t^{-b\delta\beta(H)+2b(r+m)+2b-1}=A^{\beta}_{r+m}\,t^{r^{\prime}+2bm-b\delta\beta(H)}$ By Example 2.15 we know $\beta\left(H\right)\leq 2\,\text{hgt}(\mathfrak{g})$. Suppose that a positive integer $m$ satisfies $2bm-b\delta\beta(H)<\left(\text{hgt}(\mathfrak{g})+1\right)(|r^{\prime}|-1)$. Some algebraic manipulations show that $m<\left(\text{hght}+1\right)\left(|r|-1\right)$. So indeed the coefficients $B_{r^{\prime}+m}$ for $0\leq m<\left(\text{hght}+1\right)\left(|r^{\prime}|-1\right)$ are completely determined by the coefficients $A_{r+m}$ for $0\leq m<\left(\text{hgt}(\mathfrak{g})+1\right)(|r|-1)$. ∎ ###### Remark 4.22. We can think of Proposition 4.21 as a continuity statement. It says that a small perturbation of the original connection will not alter the irregular part of its canonical form. This is analogous to the finite determinacy theorem for analytic singularities, as in [dJP00] Theorem 9.4 in page 313. One can obtain a similar continuity statement for the residue of the canonical connection (i.e. the coefficient of $t^{-1}$ in the canonical form). However the explicit bound for the number of terms needed is complicated and not very illuminating. We refrain from including a formula for the bound. ###### Proposition 4.23. Let $\mathbf{G}$ be connected reductive and let $A\in\mathfrak{g}_{F}$ be a connection. There exist a positive integer $n$ such that all connections $C\in\mathfrak{g}_{F}$ satisfying $C\equiv A\;(mod\;t^{n})$ are $\mathbf{G}(\overline{F})$-gauge equivalent to $A$. ###### Proof. In Algorithm 4.16 we apply operations (iii) and (vi) exactly once at the very end. Suppose that we are given the coefficients $A_{r+m}$ for $0\leq m\leq\left(\text{hgt}(\mathfrak{g})+1\right)(|r|-1)$ in a given connection $A$. Let $D$ be the output of applying one of the operations (i), (ii), (iv) or (v) to $A$. The proof of Proposition 4.21 implies that we can determine the corresponding coefficients $D_{r^{\prime}+m}$ for $0\leq m\leq\left(\text{hgt}(\mathfrak{g})+1\right)(|r^{\prime}|-1)$. We can iterate this reasoning. Suppose that $D=\sum_{j=r^{\prime}}^{\infty}D_{j}\,t^{j}$ is the ouput of the algorithm before applying the last two steps (operations (iii) and (vi)). Then we see that the coefficients $0\leq m<\left(\text{hgt}(\mathfrak{g})+1\right)(|r^{\prime}|-1)$ are completely determined by $A_{r+m}$ for $0\leq m\leq\left(\text{hgt}(\mathfrak{g})+1\right)(|r|-1)$, where $A$ is the original connection we start with. The number of steps needed in the algorithm is also completely determined. By the statement of Proposition 4.12 Case 1, we will be able to determine the residue (i.e. the coefficient of $t^{-1}$) when we apply operation (iii) to $D$. The output of operation (iii) will be a connection of the first kind $B=\sum_{j=-1}^{\infty}B_{j}\,t^{j}$, and we can compute $k\left(B_{-1}\right)$ (see the remark after Lemma 3.3). Now we need to determine the result of applying operation (vi) to $B$ as above. By Remark 3.4, this will be determined by $B_{j}$ for $-1\leq j\leq k(B_{-1})$. We can then work backwards using an argument similar to the proof of Proposition 4.21 to find a number $n$ big enough so that the coefficients $B_{j}$ for $-1\leq j\leq k(B_{-1})$ are determined by $A_{j}$ for $r\leq j<n$. ∎ ## 5 Irregular connections for arbitrary linear algebraic groups We proceed as in the regular case. Just as before, we start with solvable groups. ### 5.1 Irregular connections for solvable groups We will again make use of the map $\pi:\text{Lie}(\mathbf{\mathbf{T}})\cong X_{*}(\mathbf{T})\otimes\mathbf{k}\longrightarrow X_{*}(\mathbf{T})\otimes\mathbb{Q}$ as in Proposition 3.26. ###### Proposition 5.1. Let $\mathbf{G}$ be of the form $\mathbf{T}\ltimes\mathbf{U}$, where $\mathbf{T}$ is a torus and $\mathbf{U}$ is unipotent. Let $A\in\mathfrak{g}_{F}$ be a formal connection. Write $A=A_{\mathbf{T}}+A_{\mathbf{U}}$ for some $A_{\mathbf{T}}\in\text{Lie}(\mathbf{T})_{F}$ and $A_{\mathbf{U}}\in\text{Lie}(\mathbf{U})_{F}$. Let $b$ be a positive integer such that $b\,\pi\left(\,(A_{\mathbf{T}})_{-1}\,\right)\in X_{*}(\mathbf{T})$. Then there exists $x\in\mathbf{G}(F_{b})$ such that $x\cdot A=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}\,+\,t^{-1}\,C$ with: 1. (1) $r_{j}\in\mathbb{\mathbb{Z}}_{<-1}$ for all $j$. 2. (2) $D_{j}\in\text{Lie}(\mathbf{T})$ for all $j$. 3. (3) $[D_{j},\,C]=0$ for all $j$. 4. (4) $\pi(C_{s})=0$. ###### Proof. The global structure of the proof is very similar to the argument in Proposition 3.26. Write $A_{\mathbf{T}}=\sum_{j=-q}^{\infty}t^{j}\,D_{j}$. By Proposition 4.4 (a), we can find $g\in\mathbf{T}(F)$ with $g\cdot A_{\mathbf{T}}=\sum_{j=-q}^{-1}t^{j}\,D_{j}$. Set $\mu\vcentcolon=b\,\pi\left(D_{-1}\right)\in X_{*}(\mathbf{T})$. Then we have $(t^{\frac{1}{b}\mu}\,g)\cdot A_{\mathbf{T}}=\sum_{j=-q}^{-2}D_{j}\,t^{j}\,+\,t^{-1}\,C_{\mathbf{T}}$ for some $C_{\mathbf{T}}\in\text{Lie}(\mathbf{T})$ with $\pi(C_{\mathbf{T}})=0$. Replace $A$ with $B\vcentcolon=(t^{\frac{1}{b}\,\mu}\,g)\cdot A$. We know that $B=\sum_{j=-q}^{-2}D_{j}\,t^{j}\,+\,t^{-1}\,C_{\mathbf{T}}+B_{\mathbf{U}}$ for some $B_{\mathbf{U}}\in\text{Lie}(\mathbf{U})_{F_{b}}$. By lifting to the $b$-ramified cover, we can assume that $B_{\mathbf{U}}\in\text{Lie}(\mathbf{U})_{F}$. We claim that we can find $u\in\mathbf{U}(F)$ such that $u\cdot B=\sum_{j=-q}^{-2}D_{j}\,t^{j}\,+\,t^{-1}\,C_{\mathbf{T}}+t^{-1}\,C_{\mathbf{U}}$ with $C_{\mathbf{U}}\in\text{Lie}(\mathbf{U})$ and $[C_{\mathbf{T}},C_{\mathbf{U}}]=[D_{j},C_{\mathbf{U}}]=0$ for all $j$. We will show this by induction on the dimension of $\mathbf{U}$. The base case is $\mathbf{U}=\mathbb{G}_{a}$. Then, $\mathbf{T}$ acts on $\mathbf{U}$ by a character $\chi:\mathbf{T}\longrightarrow\mathbb{G}_{m}$. For $u=\sum_{j=r}^{\infty}u_{j}\,t^{j}\in\mathbf{U}(F)$, we have $u\cdot B=t^{-1}\,C_{\mathbf{T}}\,+\,B_{\mathbf{U}}\,-\,\sum_{j=r-q}^{\infty}\left[\left(d\chi(C_{\mathbf{T}})-j\right)u_{j}+\sum_{i=2}^{q}d\chi(D_{i})u_{j+i-1}\right]\,t^{j-1}$ We have two cases 1. (1) Suppose that $d\chi(D_{j})\neq 0$ for some $j$. Then, we can solve the recurrence $\left(d\chi(C_{\mathbf{T}})-j\right)u_{j}+\sum_{i=2}^{q}d\chi(D_{i})u_{j+i}=B_{j-1}$ with initial values $u_{j}=0$ for $j\ll 0$. This yields an element $u\in\mathbf{U}(F)$ with $u\cdot B=\sum_{j=-q}^{-2}D_{j}\,t^{j}\,+\,t^{-1}\,C_{\mathbf{T}}$. 2. (2) Suppose that $d\chi(D_{j})=0$ for all $j$. The argument for the base case in Proposition 3.26 shows that there is an element $u\in\mathbf{U}(F)$ such that $u\cdot B=\sum_{j=-q}^{-2}D_{j}\,t^{j}\,+\,t^{-1}\,C_{\mathbf{T}}\,+\,t^{-1}\,C_{\mathbf{U}}$ for some $C_{\mathbf{U}}\in\text{Lie}(\mathbf{U})$ satisfying $[C_{\mathbf{T}},C_{\mathbf{U}}]=0$. Notice that we have $[D_{j},\,C_{\mathbf{U}}]=d\chi(D_{j})\,C_{\mathbf{U}}=0$ by assumption. So we are done in this case. This concludes the proof of the base case. Let’s proceed with the induction step. We can decompose the action of the split torus $\mathbf{T}$ on the vector space $Z_{\mathbf{U}}$ into one- dimensional spaces. Let $\mathbf{H}\cong\mathbb{G}_{a}\leq Z_{\mathbf{U}}$ be one of these eigenspaces. Let $s$ be a $\mathbf{T}$-equivariant section of the morphism of schemes $\mathbf{U}\longrightarrow\mathbf{U}/\,\mathbf{H}$ as in the proof of Proposition 3.26. Let $\overline{B}$ be the image of $B$ in the quotient $\text{Lie}(\mathbf{U}/\,\mathbf{H})_{F}$. By the induction hypothesis, we can find $\overline{u}\in\mathbf{U}/\,\mathbf{H}(F)$ such that $\overline{u}\cdot\overline{B}=\sum_{j=-q}^{-2}D_{j}\,t^{j}\,+\,t^{-1}\,C_{\mathbf{T}}+t^{-1}\,\overline{E}$ for some $\overline{E}\in\text{Lie}\left(\mathbf{U}/\,\mathbf{H}\right)$ with $[D_{j},\overline{E}]=[C_{\mathbf{T}},\overline{E}]=0$. We can then write $s(\overline{u})\cdot B=\sum_{j=-q}^{-2}D_{j}\,t^{j}\,+\,t^{-1}\,C_{\mathbf{T}}+t^{-1}\,ds(\overline{E})+B_{\mathbf{H}}$ for some $B_{\mathbf{H}}\in\text{Lie}(\mathbf{H})_{F}$. Since $s$ is $\mathbf{T}$-equivariant, we have $[ds(\overline{E}),D_{j}]=[ds(\overline{E}),\,C_{\mathbf{T}}]=0$. We can use the base case for $\mathbf{H}$ in order to conclude. ∎ We end with a generalization of Proposition 3.29. We will use the same notation as in the regular case. Let $A=A^{\mathbf{T}}+A^{\mathbf{U}}$ be a formal connection with $A^{\mathbf{T}}\in\text{Lie}(\mathbf{T})_{F}$ and $A^{\mathbf{U}}\in\text{Lie}(\mathbf{U})_{F}$. Write $A^{\mathbf{T}}=\sum_{j=-q}^{-1}A^{\mathbf{T}}_{j}\,t^{j}\,+\,\sum_{j=p}^{\infty}A^{\mathbf{T}}_{j}\,t^{j}$ for some $q,p\geq 0$. Also, write $A^{\mathbf{U}}=\sum_{j=m}^{\infty}A^{\mathbf{U}}_{j}\,t^{j}$. ###### Proposition 5.2. Keep the same notation as above. Assume that $\mathbf{U}$ has nilpotency class $n$. 1. (i) Suppose that $m>L-1$. Then there exists $x\in\mathbf{G}(\mathcal{O})$ such that $x\cdot A=\sum_{-q}^{-1}A^{\mathbf{T}}_{j}\,t^{j}$. More precisely, there exist $x_{\mathbf{T}}\in\mathbf{T}(\mathcal{O})$ with $x_{\mathbf{T}}\equiv 1_{\mathbf{T}}\,\left(mod\;t^{p+1}\right)$ and $x_{\mathbf{U}}\in\mathbf{U}(\mathcal{O})$ with $x_{\mathbf{U}}\equiv 1_{\mathbf{U}}\,\left(mod\;t^{m+1}\right)$ such that $(x_{\mathbf{U}}x_{\mathbf{T}})\cdot A=\sum_{-q}^{-1}A^{\mathbf{T}}_{j}\,t^{j}$. 2. (ii) Suppose that $m\leq L-1$. Then the $\mathbf{G}(F)$-gauge equivalence class of $A$ is determined by the coefficients $A^{\mathbf{T}}_{j}$ for $-q\leq j<(n+1)(|m|-1)+L$ and $A^{\mathbf{U}}_{j}$ for $-q\leq j<n(|m|-1)+L$. More precisely, suppose that there is another connection $B$ and an integer $k\geq n(|m|-1)+L$ satisfying $A^{\mathbf{T}}\equiv B^{\mathbf{T}}\,\left(mod\;t^{k+|m|-1}\right)$ and $A^{\mathbf{U}}\equiv B^{\mathbf{U}}\,\left(mod\;t^{k}\right)$. Then, there exists $x\in\mathbf{G}(\mathcal{O})$ with $x\equiv 1\,\left(mod\;t^{k-n|m|+n+1}\right)$ such that $x\cdot A=B$. ###### Proof. The proof is similar in spirit to the argument for Proposition 3.29, but it involves an extra subtle twist to deal with the negative powers. 1. (i) Just as in Proposition 3.29, we can find $x_{\mathbf{T}}\in\mathbf{T}(\mathcal{O})$ with $x_{\mathbf{T}}\equiv 1_{\mathbf{T}}\,\left(mod\;t^{p+1}\right)$ such that $C\vcentcolon=x_{\mathbf{T}}\cdot A=\sum_{-q}^{-1}A^{\mathbf{T}}_{j}\,t^{j}\,+\,C^{\mathbf{U}}$ for some $C^{\mathbf{U}}\in\mathfrak{u}_{\mathcal{O}}$. Moreover we have $C^{\mathbf{U}}\equiv 0\,\left(mod\;t^{m}\right)$. We claim that there exists $u\in\mathbf{U}(\mathcal{O})$ with $u\equiv 1_{\mathbf{U}}\,\left(mod\;t^{m+1}\right)$ such that $u\cdot C=\sum_{-q}^{-1}A^{\mathbf{T}}_{j}\,t^{j}$. This claim finishes the proof of part (i). In order to prove the claim, we will actually show something stronger. Let us fix some notation. By [BS68] Corollary 9.12, there is a $\mathbf{T}$-equivariant map of $\mathbf{k}$-schemes $\psi_{\mathbf{U}}:\mathbf{U}\longrightarrow\mathfrak{u}$. We can define this map so that the following diagram commutes $\textstyle{\mathbf{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{\mathbf{U}}}$$\textstyle{\mathbf{U}/\,Z_{\mathbf{U}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{\mathbf{U}/\,Z_{\mathbf{U}}}}$$\textstyle{\mathfrak{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{u}/\,\mathfrak{z}}$ Here $Z_{\mathbf{U}}$ is the center of $\mathbf{U}$ and $\mathfrak{z}=\text{Lie}(Z_{\mathbf{U}})$. Notice that $Z_{\mathbf{U}}$ is just a direct sum of copies of $\mathbb{G}_{a}$. The corresponding map $\psi_{Z_{\mathbf{U}}}$ can be taken to be the usual identification of a vector space with its tangent space at the identity. By iterating, we can arrange so that we get a corresponding compatibility at each step of the upper central series of $\mathbf{U}$. Recall that we have a weight decomposition $\mathfrak{u}=\bigoplus_{i=1}^{l}\mathfrak{u}_{\chi_{i}}$. Via the isomorphism $\psi_{\mathbf{U}}$, we can get a decomposition $\mathbf{U}=\prod_{\chi_{i}}\mathbf{U}_{\chi_{i}}$ as a product of schemes. For $u\in\mathbf{U}(\mathbf{k})$, we will denote by $u_{\chi_{i}}$ the corresponding component in $\mathbf{U}_{\chi_{i}}$. For each $i$, define $a_{i}$ to be the biggest positive integer $j$ such that $d\chi_{i}\left(A^{\mathbf{T}}_{-j}\right)\neq 0$. If $d\chi_{i}\left(A^{\mathbf{T}}_{-j}\right)=0$ for all $j>0$, we set $a_{i}=1$. Then, we claim that we can find $u\in\mathbf{U}(\mathcal{O})$ with $u_{\chi_{i}}\equiv 1_{\mathbf{U}}\,\left(mod\;t^{m+a_{i}}\right)$ such that $u\cdot C=\sum_{-q}^{-1}A^{\mathbf{T}}_{j}\,t^{j}$. We will prove this stronger claim by induction on the nilpotency class of $\mathbf{U}$. For the base case $n=0$, we have $\mathbf{U}\cong\mathbb{G}_{a}^{d}$ for some $d$. By decomposing into one-dimensional $\mathbf{T}$-modules and looking at each coordinate, we can reduce to the case $d=1$. So we have a single weight space $\mathfrak{u}_{\chi_{i}}$. This case amounts to solving a recurrence as in the computation for the base case in Proposition 5.1. We want to find $u=\sum_{j=0}^{\infty}u_{j}\,t^{j}$ satisfying $\left(d\chi_{i}(A^{\mathbf{T}}_{-1})-j\right)u_{j}+\sum_{k=2}^{q}d\chi_{i}(A^{\mathbf{T}}_{-k})u_{j+k}=C^{\mathbf{U}}_{j-1}$ By the definition of $a_{i}$, this is the same as $\left(d\chi_{i}(A^{\mathbf{T}}_{-1})-j\right)u_{j}+\sum_{k=2}^{a_{i}}d\chi_{i}(A^{\mathbf{T}}_{-k})u_{j+k}=C^{\mathbf{U}}_{j-1}$ There are two different cases. 1. (1) If $a_{i}=1$, then the recurrence reduces to $\left(d\chi_{i}(A^{\mathbf{T}}_{-1})-j\right)u_{j}=C^{\mathbf{U}}_{j-1}$ The claim follows by the argument for the base case in Proposition 3.29. 2. (2) Suppose that $a_{i}\neq 1$. We know that $d\chi_{i}(A^{\mathbf{T}}_{-a_{i}})\neq 0$. We can solve the recurrence by rewriting $d\chi_{i}(A^{\mathbf{T}}_{-a_{i}})\,u_{j+a_{i}}=C^{\mathbf{U}}_{j-1}-\left(d\chi_{i}(A^{\mathbf{T}}_{-1})-j\right)u_{j}-\sum_{k=2}^{a_{i}-1}d\chi_{i}(A^{\mathbf{T}}_{-k})u_{j+k}$ Since $C^{\mathbf{U}}_{j}=0$ for all $j\leq m-1$, we can set $u_{j}=0$ for all $j\leq m+a_{i}$. Then we can solve for the rest of the $u_{j}$ using the recursion formula above. Let’s proceed with the induction step. Notice that $\mathfrak{z}$ is a direct sum of some one-dimensional $\mathbf{T}$-submodules of $\mathfrak{u}$. We can get an identification of $\mathfrak{u}/\,\mathfrak{z}$ with the direct sum of some choice of remaining one-dimensional $\mathbf{T}$-submodules. This way we get a $\mathbf{T}$-equivariant inclusion $\mathfrak{u}/\,\mathfrak{z}\hookrightarrow\mathfrak{u}$. We can get a $\mathbf{T}$-equivariant section $s:\mathbf{U}/\,Z_{\mathbf{U}}\longrightarrow\mathbf{U}$ defined by the composition $s:\mathbf{U}/\,Z_{\mathbf{U}}\,\xrightarrow{\;\psi_{\mathbf{U}/\,Z_{\mathbf{U}}}\;}\,\mathfrak{u}/\,\mathfrak{z}\,\xhookrightarrow{\;\;\;\;}\,\mathfrak{u}\,\xrightarrow{\;\psi_{\mathbf{U}}^{-1}\;}\,\mathbf{U}$ Let $\overline{C}$ be the image of $C$ in the quotient $\text{Lie}(\mathbf{T}\ltimes\mathbf{U}/\,Z_{\mathbf{U}})_{F_{b}}$. By the induction hypothesis, there exists $\overline{x}\in\mathbf{U}/\,Z_{\mathbf{U}}(\mathcal{O})$ such that $\overline{x}_{\chi_{i}}\equiv 1\;\left(mod\;t^{m+a_{i}}\right)$ and $\overline{x}\cdot\overline{C}=\sum_{-q}^{-1}A^{\mathbf{T}}_{j}\,t^{j}$. By the $\mathbf{T}$-equivariance of $s$, we must then have $s(\overline{x})\cdot C=\sum_{-q}^{-1}A^{\mathbf{T}}_{j}\,t^{j}\,+\,D_{Z_{\mathbf{U}}}$ for some $D_{Z_{\mathbf{U}}}\in\text{Lie}(Z_{\mathbf{U}})_{F}$. By definition $s(\overline{x})\cdot C\;=\;\sum_{-q}^{-1}t^{j}\text{Ad}(s(\overline{x}))A^{\mathbf{T}}_{j}\,+\,\text{Ad}(s(\overline{x}))C^{\mathbf{U}}\,+\,ds(\overline{x})s(\overline{x})^{-1}$ Since $s(\overline{x})\equiv s(\overline{x})^{-1}\equiv 1\;\left(mod\;t^{m+1}\right)$, it follows that $ds(\overline{x})s(\overline{x})^{-1}\equiv 0\,\left(mod\;t^{m+1}\right)$. Also $\text{Ad}(s(\overline{x}))C^{\mathbf{U}}\equiv C^{\mathbf{U}}\,\left(mod\;t^{m+1}\right)$, because by assumption $C_{\mathbf{U}}\in\mathfrak{u}_{\mathcal{O}}$. We are left to study $\text{Ad}(s(\overline{x}))A^{\mathbf{T}}_{j}$. Consider the map of $\mathbf{k}$-schemes $\varphi_{j}:\mathbf{U}\longrightarrow\mathfrak{u}$ given by $\varphi_{j}(u)\vcentcolon=\text{Ad}(u)A^{\mathbf{T}}_{j}-A^{\mathbf{T}}_{j}$. By construction $\varphi_{j}$ is $\mathbf{T}$-equivariant. This means that it must respect the decomposition into weight spaces. In other words, the $\chi_{i}$-coordinate of $\varphi_{j}(u)$ is given by $\varphi_{j}(u_{\chi_{i}})$. In particular, this means that $\text{Ad}(s(\overline{x}))A^{\mathbf{T}}_{j}=A^{\mathbf{T}}_{j}\,+\,\sum_{i=1}^{l}\left(\,\text{Ad}(s(\overline{x})_{\chi_{i}})A^{\mathbf{T}}_{j}-A^{\mathbf{T}}_{j}\,\right)$ We have that $\text{Ad}(s(\overline{x})_{\chi_{i}})A^{\mathbf{T}}_{j}=A^{\mathbf{T}}_{j}$ whenever $d\chi_{i}\left(A^{\mathbf{T}}_{j}\right)=0$. By definition this happens whenever $-j>a_{i}$. So we get $\text{Ad}(s(\overline{x}))A^{\mathbf{T}}_{j}=A^{\mathbf{T}}_{j}\,+\,\sum_{-j\leq a_{i}}\left(\,\text{Ad}(s(\overline{x})_{\chi_{i}})A^{\mathbf{T}}_{j}-A^{\mathbf{T}}_{j}\,\right)$ Suppose that $-j\leq a_{i}$. By assumption $s(\overline{x})_{\chi_{i}}\equiv 1\;\left(mod\;t^{m+a_{i}}\right)$, so in particular $s(\overline{x})_{\chi_{i}}\equiv 1\;\left(mod\;t^{m-j}\right)$. Hence we have $\text{Ad}(s(\overline{x})_{\chi_{i}})A^{\mathbf{T}}_{j}\,\equiv\,A^{\mathbf{T}}_{j}\,\left(mod\;t^{m-j}\right)$. The sum above becomes $\text{Ad}(s(\overline{x}))A^{\mathbf{T}}_{j}\;\equiv\;A^{\mathbf{T}}_{j}\;\left(mod\;t^{m-j}\right)$ Hence $t^{j}\,\text{Ad}(s(\overline{x})A^{\mathbf{T}}_{j}\,\equiv\,t^{j}\,A^{\mathbf{T}}_{j}\;\left(mod\;t^{m}\right)$. We can put together all of the discussion above to conclude that $s(\overline{x})\cdot C\;\equiv\;\sum_{-q}^{-1}A^{\mathbf{T}}_{j}\,t^{j}\,+C^{\mathbf{U}}\;=\;C\;\left(mod\;t^{m}\right)$ Therefore $D_{Z_{\mathbf{U}}}\equiv 0\;\left(mod\;t^{m}\right)$. Now we can conclude by using the base case for $Z_{\mathbf{U}}$. 2. (ii) The hypothesis implies that we have equality of singular parts $\sum_{j=-q}^{-1}B^{\mathbf{T}}_{j}\,t^{j}=\sum_{j=-q}^{-1}A^{\mathbf{T}}_{j}\,t^{j}$. The proof of Proposition 3.12 shows that there exist $x_{\mathbf{T}}\in\mathbf{T}(\mathcal{O})$ with $x_{\mathbf{T}}\equiv 1_{\mathbf{T}}\,\left(mod\;t^{p+1}\right)$ such that $x_{\mathbf{T}}\cdot A^{\mathbf{T}}=B^{\mathbf{T}}$. Set $C\vcentcolon=x_{\mathbf{T}}\cdot A$. We have $C=B^{\mathbf{T}}\,+\,\text{Ad}(x_{\mathbf{T}})A^{\mathbf{U}}$. Define $C^{\mathbf{U}}\vcentcolon=\text{Ad}(x_{\mathbf{T}})A^{\mathbf{U}}$. We know that $C^{\mathbf{U}}\equiv A^{\mathbf{U}}\,\left(mod\;t^{k}\right)$, because $x_{\mathbf{T}}\equiv 1\,\left(mod\;t^{k+|m|}\right)$ and $A^{\mathbf{U}}\in t^{m}\mathfrak{u}_{\mathcal{O}}$. Therefore $C^{\mathbf{U}}\equiv B^{\mathbf{U}}\ \left(mod\;t^{k}\right)$ by assumption. Let $s$, $\mathbf{U}_{\chi_{i}}$ and $a_{i}$ be defined as in part (i). We claim that there exists $u\in\mathbf{U}(\mathcal{O})$ with $u_{\chi_{i}}\equiv 1\;\left(mod\;t^{k-n|m|+n+a_{i}}\right)$ such that $u\cdot C=B$. This implies that $u\equiv 1\,\left(mod\;t^{k-n|m|+n+1}\right)$, so this claim concludes the proof of part (ii). In order to prove the claim, we will induct on the nilpotency class of $\mathbf{U}$. The base case $n=0$ follows again from the explicit computation done in Proposition 5.1, we omit the details. Let’s proceed with the induction step. Let $\overline{C}$ and $\overline{B}$ denote the images of $C$ and $B$ in the quotient $\text{Lie}(\mathbf{T}\ltimes\mathbf{U}/\,Z_{\mathbf{U}})_{F}$. By the induction hypothesis, there exists $\overline{x}\in\mathbf{U}/\,Z_{\mathbf{U}}(\mathcal{O})$ with $\overline{x}_{\chi_{i}}\equiv 1\;\left(mod\;t^{k-(n-1)|m|+n-1+a_{i}}\right)$ such that $\overline{x}\cdot\overline{C}=\overline{B}$. We can now write $s(\overline{x})\cdot C=ds\left(\overline{B}\right)+E_{Z_{\mathbf{U}}}$ and $B=ds\left(\overline{B}\right)+K_{Z_{\mathbf{U}}}$ for some $E_{Z_{\mathbf{U}}},F_{Z_{\mathbf{U}}}\in\text{Lie}(Z_{\mathbf{U}})_{F}$. By definition $s(\overline{x})\cdot C\;=\;\sum_{j=-q}^{\infty}t^{j}\text{Ad}(s(\overline{x}))B^{\mathbf{T}}_{j}\,+\,\text{Ad}(s(\overline{x}))C^{\mathbf{U}}\,+\,ds(\overline{x})s(\overline{x})^{-1}$ Since $s(\overline{x})\equiv 1\;\left(mod\;t^{k-(n-1)|m|+n}\right)$, it follows that $t^{j}\,\text{Ad}(s(\overline{x}))B^{\mathbf{T}}_{j}\,\equiv\,t^{j}\,B^{\mathbf{T}}_{j}\,\left(mod\;t^{k-(n-1)|m|+n}\right)$ for all $j\geq 0$. The same reasoning as in part (i) shows that $t^{j}\,\text{Ad}(s(\overline{x}))B^{\mathbf{T}}_{j}\,\equiv\,t^{j}\,B^{\mathbf{T}}_{j}\,\left(mod\;t^{k-(n-1)|m|+n-1}\right)$ for all $j<0$. Also we know that $\text{Ad}(s(\overline{x})C^{\mathbf{U}}\,\equiv\,C^{\mathbf{U}}\,\left(mod\;t^{k-n|m|+n}\right)$, because $s(\overline{x})\equiv 1\;\left(mod\;t^{k-(n-1)|m|+n}\right)$ and $C_{\mathbf{U}}\in t^{m}\mathfrak{u}_{\mathcal{O}}$. We conclude that $ds\left(\overline{B}\right)+E_{Z_{\mathbf{U}}}\;=\;s(\overline{x})\cdot C\;\equiv\;B^{\mathbf{T}}\,+\,C^{\mathbf{U}}\;=\;C\;\left(mod\;t^{k-n|m|+n}\right)$ Since $k\geq k-n|m|$, we have $C\equiv B\;\left(mod\;t^{k-n|m|}\right)$. It follows that $E_{Z_{\mathbf{U}}}\equiv K_{Z_{\mathbf{U}}}\;\left(mod\;t^{k-n|m|+n}\right)$. Now by the base case we can find $y\in Z_{\mathbf{U}}(\mathcal{O})$ with $y_{\chi_{i}}\equiv 1\;\left(mod\;t^{k-n|m|+n+a_{i}}\right)$ such that $(y\,s(\overline{x}))\cdot C=B$. By the definition of $\mathbf{U}_{\chi_{i}}$ and its compatibility with the center, we can see that $(y\,s(\overline{x}))_{\chi_{i}}\equiv 1\;\left(mod\;t^{k-n|m|+n+a_{i}}\right)$. The claim follows. ∎ ### 5.2 Irregular connections for arbitrary linear algebraic groups ###### Theorem 5.3. Let $\mathbf{G}$ be a connected linear algebraic group. Fix a Levi subgroup $\mathbf{L}$ and a maximal torus $\mathbf{T}\subset\mathbf{L}$. Let $A\in\mathfrak{g}_{\overline{F}}$ be a formal connection. Then there exists $x\in\mathbf{G}(\overline{F})$ such that $x\cdot A=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}\,+\,t^{-1}\,C$ with 1. (1) $r_{j}\in\mathbb{Q}_{<-1}$ for all $j$. 2. (2) $D_{j}\in\text{Lie}(\mathbf{T})$ for all $j$. 3. (3) $[D_{j},\,C]=0$ for all $j$. 4. (4) $C_{s}\in\mathfrak{D}$. 5. (5) $[C_{s},C]=0$. ###### Proof. The same steps as in the proof of Theorem 3.32 reduce the result to the solvable case (Proposition 5.1). ∎ A connection of the form $B=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}\,+\,t^{-1}\,C$ satisfying conditions (1)-(3) above is said to be in canonical form. Let’s formulate some uniqueness results for such irregular canonical forms. Before doing this, we need a lemma. ###### Lemma 5.4. Let $B=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}+t^{-1}\,C$ and $B^{\prime}=\sum_{j=1}^{s}D^{\prime}_{j}\,t^{r^{\prime}_{j}}+t^{-1}\,C^{\prime}$ be two connections in canonical form. Suppose that $x\in\mathbf{G}(\overline{F})$ satisfies $x\cdot B=B^{\prime}$. Then all the following statements are true 1. (1) $l=s$ and $r_{j}=r^{\prime}_{j}$. 2. (2) $\text{Ad}(x)D_{j}=D^{\prime}_{j}$ for all $j$. 3. (3) $x\cdot\,(t^{-1}\,C)=t^{-1}\,C^{\prime}$. ###### Proof. If we know both $(1)$ and $(2)$, then part $(3)$ follows. So we will focus on the first couple of statements. By lifting everything to a ramified cover, we can assume that $x\in\mathbf{G}(F)$. Choose a faithful representation $\mathbf{G}\hookrightarrow\mathbf{\text{GL}_{n}}$. We can view $x\in\mathbf{\text{GL}_{n}}(F)$ and $B,B^{\prime}\in\mathfrak{gl}_{n}(\overline{F})$. To simplify notation, let us add some trivial $D_{j}$ and $D^{\prime}_{j}$ so that we have the same indexes and exponents for both $B_{\text{irr}}$ and $B^{\prime}_{\text{irr}}$. We therefore write $B=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}+t^{-1}\,C$ and $B^{\prime}=\sum_{j=1}^{l}D^{\prime}_{j}\,t^{r_{j}}+t^{-1}\,C^{\prime}$. Now $D_{j}$ and $D^{\prime}_{j}$ are (possibly $0$) semisimple elements in $\mathfrak{g}$. We claim that $\text{Ad}(x)D_{j}=D^{\prime}_{j}$ for all $j$. This claim would imply that none of the new $D_{j}$ and $D^{\prime}_{j}$ are $0$. This would mean that we didn’t actually add any extra terms. So both (1) and (2) would follow. We are left to show the claim. Let us consider the linear transformation $W$ in $\text{End}(\mathfrak{gl}_{n})\,(\overline{F})$ given by $W\,v\vcentcolon=B^{\prime}v\,-\,vB$ for all $v\in\mathfrak{gl}_{n}$. We can write $W=\sum_{j=1}^{l}W_{j}\,t^{r_{j}}\,+\,t^{-1}\,U$, where $\displaystyle W_{j}\in\text{End}(\mathfrak{gl}_{n})\;\text{is given by}\;W_{j}\,v\vcentcolon=D^{\prime}_{j}v-vD_{j}$ $\displaystyle U\in\text{End}(\mathfrak{gl}_{n})\;\text{is given by}\;U\,v\vcentcolon=C^{\prime}v-vC$ Each $W_{j}$ is semisimple by definition. Also we have that the $W_{j}$s and $U$ pairwise commute. Therefore there is a simultaneous spectral decomposition $\mathfrak{gl}_{n}=\bigoplus_{\vec{\lambda}}(\mathfrak{gl}_{n})_{\vec{\lambda}}$ for the $W_{j}$s, where $\vec{\lambda}=(\lambda_{j})_{j=1}^{l}$ ranges over a set of $l$-tuples of eigenvalues of the $W_{j}$s. Note that $W$ preserves this spectral decomposition, because $U$ commutes with all $W_{j}$s. The condition $x\cdot B=B^{\prime}$ can be expressed as $\frac{d}{dt}x=W\,x$. Here we are viewing $x$ as an invertible matrix in $\mathfrak{gl}_{n}(F)$. We can restrict to the $\vec{\lambda}$-eigenspace and use the decomposition for $W$ in order to see that the component $x_{\vec{\lambda}}\in(\mathfrak{gl}_{n})_{\vec{\lambda}}$ of $x$ satisfies $\frac{d}{dt}x_{\vec{\lambda}}=\sum_{j=1}^{l}\lambda_{j}\,t^{r_{j}}\,x_{\vec{\lambda}}\,+\,t^{-1}\,U\,x_{\vec{\lambda}}$ Recall that $r_{j}<-1$ for all $j$. By comparing the smallest exponent of $t$ in both sides, we conclude that $x_{\vec{\lambda}}=0$ unless $\vec{\lambda}=\vec{0}$. Hence $x\in(\mathfrak{gl}_{n})_{\vec{0}}\,(F)$. This means that $\text{Ad}(x)D_{j}=D^{\prime}_{j}$ for all $j$. ∎ As a consequence, we get the following uniqueness result for all irregular canonical forms that satisfy (1)-(5) as in Therorem 5.3. ###### Theorem 5.5. Let $\mathbf{G}$ be a connected linear algebraic group. Fix a Levi subgroup $\mathbf{L}$ and a maximal torus $\mathbf{T}\subset\mathbf{L}$. Let $A=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}\,+\,t^{-1}\,C$ and $B=\sum_{j=1}^{l}D^{\prime}_{j}\,t^{r^{\prime}_{j}}\,+\,t^{-1}\,C^{\prime}$ be two connections in canonical form. Suppose that $C_{s},\,C^{\prime}_{s}\in\mathfrak{D}$ and $[C_{s},C]=[C^{\prime}_{s},C^{\prime}]=0$. If there exists $x\in\mathbf{G}(\overline{F})$ with $x\cdot A=B$, then we have 1. (1) $C_{s}=C^{\prime}_{s}$. 2. (2) $x\in Z_{\mathbf{G}}(C_{s})(\mathbf{k})$. 3. (3) $\text{Ad}(x)\,D_{j}=D^{\prime}_{j}$ for all $j$. ###### Proof. This follows from Lemma 5.4 combined with Proposition 3.34. ∎ We conclude this section with a determinacy result for arbitrary linear algebraic groups. ###### Proposition 5.6. Let $\mathbf{G}$ be connected linear algebraic group. Let $A\in\mathfrak{g}_{F}$ be a connection. There exist a positive integer $n$ such that all connections $C\in\mathfrak{g}_{F}$ satisfying $C\equiv A\;(mod\;t^{n})$ are $\mathbf{G}(\overline{F})$-gauge equivalent to $A$. ###### Proof. This follows from the corresponding determinacy results for reductive groups (Proposition 4.23) and solvable groups (Proposition 5.2) via a reduction as in the proof of Theorem 5.3. ∎ ### 5.3 Galois cohomology for irregular connections For this section $\mathbf{G}$ will be a connected linear algebraic group. We will fix a choice of Levi subgroup $\mathbf{L}\subset\mathbf{G}$ and maximal torus $\mathbf{T}\subset\mathbf{L}$. Let $B=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}+t^{-1}C$ be a connection in canonical form with $C_{s}\in\mathfrak{D}$ and $[C_{s},C]=0$, as in the statement of Theorem 5.5. If $B_{\text{irr}}\neq 0$, then we don’t necessarily have $B\in\mathfrak{g}_{F}$. Suppose that $B$ is in $\mathfrak{g}_{F_{b}}$, with $b$ a given positive integer. Then we have a Galois action of $\mathbf{\mu}_{b}\cong\text{Gal}(F_{b}/\,F)$ on $B$ by the formula $\gamma\cdot B=\sum_{j=1}^{l}\gamma^{-br_{j}}\,D_{j}\,t^{r_{j}}\,+\,t^{-1}\,C$. Because this action is not necessarily trivial, we have to consider twisted cocyles in order to classify connections over $\text{Spec}\,F$ with canonical form $B$. ###### Definition 5.7. Let $b$ be a natural number. Let $B=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}+t^{-1}C\;\in\mathfrak{g}_{F_{b}}$ be a connection in canonical form. A $B$-twisted $\mu_{b}$-cocycle is a map $\phi\vcentcolon\mathbf{\mu}_{b}\longrightarrow Z_{G}(C)$ satisfying 1. (i) $\text{Ad}(\phi_{\gamma})B=\gamma\cdot B$ for all $\gamma\in\mathbf{\mu}_{b}$. 2. (ii) $\phi_{\gamma\gamma^{\prime}}=\phi_{\gamma}\phi_{\gamma^{\prime}}$ for all $\gamma,\gamma^{\prime}\in\mathbf{\mu}_{b}$. Fix a compatible choice of generators $\omega_{b}$ of $\mu_{b}$ for all $b$ positive, just as we did in the regular case. Note that a $B$-twisted $\mu_{b}$ cocycle $\phi$ is completely determined by $\phi_{\omega_{b}}\in Z_{G}(C)$. This is a an element of finite order dividing $b$, and it satisfies $\text{Ad}(\phi_{\omega_{b}})B=\omega_{b}\cdot B$. Conversely, for any element $\phi_{\omega_{b}}\in Z_{G}(C)$ satisfying $\text{Ad}(\phi_{\omega_{b}})B=\omega_{b}\cdot B$ we can get a corresponding $B$-twisted cocycle. Notice that the centralizer $Z_{G}(\\{D_{1},...,D_{l},C\\})$ acts on the set of $B$-twisted $\mu_{b}$-cocycles by conjugation. By the same type of general Galois cohomology argument as in the regular case, we get the following couple of propositions. The proofs are omitted. ###### Proposition 5.8 (Criterion for Descent to $D^{*}$). Let $b$ be a natural number. Let $B=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}+t^{-1}C\;\in\mathfrak{g}_{F_{b}}$ be a connection in canonical form with $C_{s}\in\mathfrak{D}$ and $[C_{s},C]=0$. Then $B$ is equivalent to a connection in $\mathfrak{g}_{F}$ via an element of $\mathbf{G}(F_{b})$ if and only there exists a $B$-twisted $\mu_{b}$-cocycle. ###### Proposition 5.9 (Classification of Connections over $D^{*}$). Let $B=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}+t^{-1}C\;\in\mathfrak{g}_{F_{b}}$ be a connection in canonical form with $C_{s}\in\mathfrak{D}$ and $[C_{s},C]=0$. Suppose that $B$ satisfies the equivalent statements in Proposition 5.8 above for some $b$. Then the set of equivalence classes of $\mathbf{G}$-connections over $D^{*}$ that become gauge equivalent over $\text{Spec}\,F_{b}$ are in bijection with the set of $B$-twisted $\mu_{b}$-cocycles up to $Z_{G}(\\{D_{1},\,...,D_{l},\,C\\})$-conjugacy. The correspondence in Proposition 5.9 can be described as follows. Let $\phi_{\omega_{b}}\in Z_{\mathbf{G}}(C)(\mathbf{k})$ be such that $\text{Ad}(\phi_{\omega_{b}})B=\omega_{b}\cdot B$. By the vanishing of $H^{1}_{\text{Gal}(F)}(\mathbf{G})$, we can find an element $y\in\mathbf{G}(F_{b})$ such that $\omega_{b}\cdot y=y\,\phi_{\omega_{b}}$. Then the connection associated to $\phi_{\omega_{b}}$ will be $A=y\cdot B\;\in\mathfrak{g}_{F}$. Conversely, suppose that $A=y\cdot B$ is a connection in $\mathfrak{g}_{F}$ for some $y\in\mathbf{G}(F_{b})$. We set $\phi_{\omega_{b}}\vcentcolon=y^{-1}\,\left(\omega_{b}\cdot y\right)$. As a consequence of this Galois cohomology classification, we can put a bound on the denominators of the levels $r_{j}$ of a canonical form for a connection in $\mathfrak{g}_{F}$. Let $W$ denote the Weyl group of $\mathbf{L}$ with respect to $\mathbf{T}$. A Coxeter element of $W$ is an element of largest length in $W$. All Coxeter elements are conjugate to each other. The Coxeter number $h_{\mathbf{L}}$ of $\mathbf{L}$ is the order of a Coxeter element in $W$. ###### Proposition 5.10. Let $A\in\mathfrak{g}_{F}$ be a formal connection. Let $B=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}+t^{-1}C$ be a connection in canonical form with $C_{s}\in\mathcal{D}$ and $[C_{s},C]=0$. Suppose that B is $\mathbf{G}(\overline{F})$-gauge equivalent to $A$. Let $b$ be the smallest positive integer such that $B\in\mathfrak{g}_{F_{b}}$. Then 1. (1) $b$ divides a fundamental degree of $\text{Lie}(\textbf{L})$. In particular $b\leq h_{\mathbf{L}}$. 2. (2) If $b=h_{\mathbf{L}}$, then $C_{s}\in\text{Lie}(Z_{\mathbf{L}}^{0})$. ###### Proof. Recall the notation $B_{irr}=\sum_{j=1}^{l}D_{j}\,t^{r_{j}}$. We have $\mathbf{G}=\mathbf{L}\ltimes\mathbf{U}$, where $\mathbf{U}$ is the unipotent radical. Write $\mathfrak{l}\vcentcolon=\text{Lie}(\mathbf{L})$ and $\mathfrak{u}\vcentcolon=\mathbf{U}$. We can decompose $A=A_{\mathfrak{l}}+A_{\mathfrak{u}}$. It follows from the proof of Theorem 5.3 that $B_{irr}$ is given by the irregular part of the canonical form of $A_{\mathfrak{l}}$. Therefore, we can assume without loss of generality that $\mathbf{G}=\mathbf{L}$. By assumption, we have $B=\mathbf{G}(F_{d})\cdot A$ for some $d$ dividing $b$. By Proposition 5.8, we know that there exists a $B$-twisted $\mu_{d}$-cocycle $\phi$. This means in particular that $\text{Ad}(\phi_{\omega_{d}})(B_{irr}+t^{-1}C_{s})=\omega_{d}\cdot B_{irr}+t^{-1}C_{s}$. We can consider $B_{irr}+t^{-1}C_{s}$ as an element of $\text{Lie}(\mathbf{T}_{\overline{F}})$. Also $\phi_{\omega_{d}}$ can be viewed as an element of $\mathbf{G}(\overline{F})$. This means that $B_{irr}+t^{-1}$ and $\omega_{d}\cdot B_{irr}+t^{-1}$ are $\mathbf{G}(\overline{F})$-conjugate elements of $\text{Lie}(\mathbf{T}_{\overline{F}})$. By [CM93] Chapter 2, there is an element $w\in W$ such that $w\cdot(B_{irr}t^{-1}C_{s})=\omega_{d}\cdot B_{irr}+t^{-1}C_{s}$. By definition, $b$ is the least positive integer such that $(\omega_{d})^{b}\cdot B_{irr}=B_{irr}$. We conclude that some of the eigenvalues of $w$ are primitive $b$ roots of unity. It follows that $b$ divides a fundamental degree of $\mathfrak{l}$ by [Spr74] Theorem 3.4. If $b=h_{\mathbf{L}}$, then $w$ must be a Coxeter element by [Kan01] Theorem 32.2-C. Since $w\cdot C_{s}=C_{s}$, we must have $C_{s}\in\text{Lie}(Z_{\mathbf{L}}^{0})$ by the lemma in page 76 of [Hum90]. ∎ ###### Remark 5.11. This does not yield a bound on the ramification needed to put $A$ into canonical form. For example, if $A$ is regular then $B\in\mathfrak{g}_{F}$. But we have seen that it is sometimes necessary to pass to a ramified cover in order to put a a regular connection into canonical form. We remark that part (i) of Proposition 5.10 was proven in [CK17] via the existence of oper structures for any connection [FZ10]. Here we note that there is a direct argument using some facts about Coxeter groups. ### Acknowledgements This paper grew out of a suggestion from Nicolas Templier to write a modern exposition to [BV83]. I am happy to thank him for his very valuable input on the redaction of the manuscript. ## References * [Ati57] M. F. Atiyah. Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc., 85:181–207, 1957. * [BS68] A. Borel and T. A. Springer. Rationality properties of linear algebraic groups. II. Tohoku Math. J. (2), 20:443–497, 1968. * [BV83] Donald G. Babbitt and V. S. Varadarajan. Formal reduction theory of meromorphic differential equations: a group theoretic view. Pacific J. Math., 109(1):1–80, 1983. * [CK17] Tsao-Hsien Chen and Masoud Kamgarpour. Preservation of depth in the local geometric Langlands correspondence. Trans. Amer. Math. Soc., 369(2):1345–1364, 2017. * [CM93] David H. Collingwood and William M. McGovern. Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993. * [Del70] Pierre Deligne. Équations différentielles à points singuliers réguliers. Lecture Notes in Mathematics, Vol. 163. Springer-Verlag, Berlin-New York, 1970. * [DG80] Michel Demazure and Peter Gabriel. Introduction to algebraic geometry and algebraic groups, volume 39 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from the French by J. Bell. * [dJP00] Theo de Jong and Gerhard Pfister. Local analytic geometry. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 2000. Basic theory and applications. * [FZ10] Edward Frenkel and Xinwen Zhu. Any flat bundle on a punctured disc has an oper structure. Math. Res. Lett., 17(1):27–37, 2010. * [Hum90] James E. Humphreys. Reflection groups and Coxeter groups, volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1990. * [Hum95] James E. Humphreys. Conjugacy classes in semisimple algebraic groups, volume 43 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1995. * [Kan01] Richard Kane. Reflection groups and invariant theory, volume 5 of CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer-Verlag, New York, 2001. * [Lev75] A. H. M. Levelt. Jordan decomposition for a class of singular differential operators. Ark. Mat., 13:1–27, 1975. * [Sch07] Olaf M. Schnürer. Regular connections on principal fiber bundles over the infinitesimal punctured disc. J. Lie Theory, 17(2):427–448, 2007. * [Ser02] Jean-Pierre Serre. Galois cohomology. Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, 2002. Translated from the French by Patrick Ion and revised by the author. * [Spr74] T. A. Springer. Regular elements of finite reflection groups. Invent. Math., 25:159–198, 1974.

2024-09-04T02:54:55.704509

2003.00045

arxiv-papers

Library Adoption Dynamics in Software Teams Pamela Bilo Thomas University of Notre Dame Rachel Krohn University of Notre Dame Tim Weninger University of Notre Dame When a group of people strives to understand new information, struggle ensues as various ideas compete for attention. Steep learning curves are surmounted as teams learn together. To understand how these team dynamics play out in software development, we explore Git logs, which provide a complete change history of software repositories. In these repositories, we observe code additions, which represent successfully implemented ideas, and code deletions, which represent ideas that have failed or been superseded. By examining the patterns between these commit types, we can begin to understand how teams adopt new information. We specifically study what happens after a software library is adopted by a project, , when a library is used for the first time in the project. We find that a variety of factors, including team size, library popularity, and prevalence on Stack Overflow are associated with how quickly teams learn and successfully adopt new software libraries. § INTRODUCTION The process of learning new information and adopting new technology can be challenging. When new information is acquired, a learning curve often exists until an individual or group becomes proficient in the new technology. These challenges are present throughout society but are especially prevalent in computing and software development, where existing technologies are ever-changing and new innovations are constantly being developed. Further complicating these issues, most major software development occurs in teams where multiple parties collaborate on a project together, and where teammates may or may not know each other. This can lead to teammates having communication issues, which could result in teams which are inefficient and uncooperative. Online collaboration systems like GitHub have provided a powerful setting in which to study this process, because through analyzing the history of commits, whether removed, added, or otherwise, we can reconstruct a story of what happened between the teammates to eventually create the finalized code. By retelling this story, we can further investigate the struggles that occurred as the teammates tried to learn new information together, and see how long it takes for a team to become totally proficient at working together using the same technology. The present work introduces a new approach to study this problem. By investigating how software developers adopt and use software libraries in this specific context, we may better understand how humans learn new technical information and incorporate concepts previously unknown to the user or group. The findings from this study can be generalized to understand how humans work, learn, and fight together, since GitHub provides a rich dataset which approximates the collaborative process. Previous work by Kula et al found that programmers adopt new libraries only when they can be assured of the library's quality and functional correctness [1]. But what happens after the adoption event? When are other group member receptive to new libraries; and when do they resist? What tools can help a team find and learn how to use new libraries? How long does it take a group of people to be successful at learning new information together? And finally, what does it even mean for that group to be successful? To answer these and other questions, we explore the circumstances surrounding a library adoption, including the number of commits, the size of commits (measured by lines of code), and other related information, including availability of online resources, such as Stack Overflow. We present an in-depth look at commit addition and deletions, and analyze these questions from a variety of different angles. Additionally, we ask if there exists any competition among team members over the inclusion of a library. When teammates have disagreements about what should be included in a GitHub project, there will ultimately be a winner. Uncovering which user eventually wins these code fights is an interesting research question. We explore competition by examining code fights where two users revert each other’s code over the course of several commits. Like edit-wars on Wikipedia [2], by looking at the users and libraries that participate in code fights we can learn a great deal about the adoption and diffusion of information. Although the present work focuses specifically on code contributed to public Python projects hosted on GitHub, we hope that other domains can use the methodology of the present work in other explorations of information adoption generally. In addition to competition, we also aim to find the characteristics of fast adoption. It is preferable to have a short adoption period - in that way, teams can become productive more quickly, since a longer adoption time results in periods of lost production. By finding what combinations of repositories and libraries result in quick adoption times, we can begin to understand how team members work together to learn new information. Therefore, we hope to identify the qualities that create a good team, and what the characteristics of a bad team are. We hope that the findings from our research can be applied to help improve team dynamics and create groups that can work well together. To summarize, this work aims to answer the following research questions: What are the events that happen when a team adopts a library for the first time? Learning new information together can be challenging. As a team strives to use a new library together efficiently and correctly, it is one of the aims of this work to uncover the events that happen as the group learns, and what causes groups to work together well (or poorly!). In the data, we hope to find when library adoptions are successful - and what leads team members to abandon library adoptions as they decide to use other libraries instead. Are commits containing new libraries more likely to have deletions than other types of commits? One of our research questions is to understand how teams learn to use these new libraries. We expect to find that as teams struggle to use new libraries, commits that contain new libraries will contain many deletions as users attempt to understand how the library works. Eventually, as team members become proficient in using a library, and the library becomes established in a project, we expect to see that there will be less deletions in the project repository code relative to when the library was first adopted. This ratio of positive to negative commits will help us define what it means for a library to be adopted. Do the answers to these questions vary by library type, team size, or the amount of information available on Stack Overflow? Not all libraries and teams are alike, and we expect to find that as the types of the teams and the libraries change, so will the adoption time. Therefore, we will analyze library adoptions along the axes of team size and library type, and discover the relationship between adoption speeds and resources available on places like StackOverflow. We attempt to show how the speed of library adoptions changes with the addition of easy to use online resources, which we use as a proxy to represent library popularity. We hypothesize that if there exist few resources to learn how to use a library, the time to adoption for that library will be longer than for those libraries that are more well documented. Do team members fight over the adoption and usage of a new library? We cannot assume that all teams work together well. Fights between team members might result as teammates have differing opinions about what libraries to use or how to implement new code. Therefore, we analyze code fights as part of this work. We wish to uncover what happens in teams when teammates remove code, or remove entire libraries. We examine which libraries are most often fought over, and which team members end up winning code fights. We hypothesize that more experienced team members will win these code fights. To answer these questions we cloned 259,613 repositories from Github. This data provided us entire commit histories with userids, timestamps and code-diffs across all of the projects. We also downloaded, indexed, and cross-referenced library usage within Stack Overflow. Library adoption in software repositories reveals a natural experiment from which causal relationships between inferred. By following a careful extraction and experimental methodology, the following pages presents answers to these questions and an exploration of this rich resource. An example of a Git history log where lines which begin with a `+' represent additions, and lines which begin with a `-' represent deletions. § DATA AND METHODOLOGY Characteristics of GitHub. Since GitHub's rise to popularity and the growth of the open source software movement, developers have become increasingly comfortable contributing to public repositories [3]. Riding this wave of easily accessible data, researchers have done considerable investigation into GitHub, but many of these studies are limited to a few dozen project repositories in small-scale experiments, thereby limiting their potential generalizability [4]. Instead, our goal is to gather as much data as possible, which, as we shall see, presents different challenges. We summarize our findings from this large-scale collection of data in the subsequent sections of this paper. As future work, further, more specific analysis can be done on smaller datasets, but for the purposes of our research, we wish to treat this work as a big data problem, and try to ingest as much data as possible to answer these questions. Project Repositories on GitHub The confluence of GitHub's online social system and Git's complete history of pull and commit behavior provides an unprecedented source of information on adoption behavior [5, 6]. We focus on commits that contain imported libraries and their functions. The example code snippet in Fig. <ref> shows how a single commit can add and/or remove one or more lines of code. In this example, the 5 additions and 2 deletions result in a net commit size of +3 lines, including the adoption of . We consider usages of libraries $\ell$ (, , , ) that are imported into Python code through an statement. An important assumption that we make is that submodule statements represent the parent library. Therefore, submodules like are considered equivalent to in the present work, because the function is included as a part of the parent library, and we are interested in functions which refer to the adopted library. We also consider direct function calls on the imported library. It is important to carefully consider the scoping of library aliases (, , , in Fig. <ref>) so that we can mine these library functions accurately. Thankfully, the library aliases are included in the import statement, so we can easily find these aliases in the code. In the above example, the functions from the library ( and ) are referenced in two of the added lines. Indirect library usage is exemplified by the function of the object and by the function of the object in Fig. <ref>. These objects were created from some ancestor call to the library (, ) in some other location, but do not directly reference the library themselves. We do not consider indirect library usage in the present work, and only focus on direct library calls. Data Collection First, we issued a query to the GitHub search API for projects written primarily in Python. GitHub returned repository IDs of the 1,000 most popular Python projects on the site. We then found all GitHub users who made at least one commit to a repository in this set and retrieved all their Python projects. We did this breadth-first style crawling two additional times, culminating in 259,923 projects with 89,311 contributing GitHub users. Of these, we were able to clone 259,613 Python repositories to disk; the remainder were made private or deleted between the time we crawled their project URL and the time that we performed the clone operation. Each cloned repository includes all files, branches, and versions, along with a complete edit history. These repositories constitute about 13% of all Python repositories on GitHub as of September 2018. The full dataset of cloned projects occupies about 8 TB of disk space. For analysis, we parsed the commits to only contain imported functions and libraries, which drastically reduced the size of the dataset. Because we sampled 13% of the total available public Python projects available on GitHub, it is important to be wary of sampling biases. Our initial GitHub query returned the most popular projects, so our dataset may over-represent highly active repositories compared to the population. It is not our intention to faithfully survey GitHub use nor to represent all Python projects; instead, our goal is to understand how programmers adopt and use new software libraries. Our findings can be applied to projects of all sizes, and small projects are well-represented in our data. However, it is important to remember that private software repositories are not included in our dataset, so we can only investigate team interactions in a public environment. Additionally, we downloaded all question posts from Stack Overflow from its inception until September 2018. Appropriate tagging tends to increase viewership of questions [7], so we filtered out any posts that were not tagged as Python posts, then extracted all libraries from any code block using the same pattern matching technique as used in the library extraction from Python projects. Only top-level libraries were included. Because Stack Overflow is a free-form text entry platform, this pattern matching procedure may occasionally count extraneous words and misspellings as libraries. Additionally, it is possible that some libraries might not be returned by our query because the library name might have been misspelled, or the question did not include Python code containing a library import. Recreating the Project Workflow Each project repository contains files, commits, and users. Each commit is marked with a timestamp $t$, a message, one (or many) parent-commits (in the event of a merge operation), and a specifying lines that were added, edited, or deleted within each file. An important complication that arises in Git commit histories is that stashes, reverts, branches, and other Git commands can result in a non-monotonic chronology of edits. Because of this, we should not compare commits across different branches to each other until they are merged. Reversions introduce an additional complication. For example, if a user (1) commits code on Monday, (2) commits some new code on Wednesday, and then (3) on Friday reverts the project's code to Monday's commit, then the chronology of the repository flows non-monotonically from Monday to Wednesday to Monday again despite the reversion being made on Friday. Fortunately, each commit keeps a pointer to its parent(s), so we can create the actual lineage of the commits by following the graph of each commit's parent rather than blindly following the commit timestamps. Because the order of actions is more important than exact times, we enforce a monotonic chronology according to the commit graph. Future work can attempt to explore how this problem can be approached by using timestamps to analyze how the project changes over time. (a) Number of commits per project follows a shifted power law distribution, , most projects have few commits, and few projects have many commits. (b) Number of libraries used per project also follows a shifted power law distribution, , most projects adopt few libraries, and few libraries are adopted by many repositories. (c) When libraries are adopted into repositories, it tends to occur early in the repository history. Text Mining for Libraries After downloading the data, we combed through each Git commit log to find which libraries had been imported by using Regex pattern matching to identify import statements, such as `from $\ell$ import $f$" and `import $\ell$ as $f$". Once we identified which libraries were used in a Git project, we searched the log to find the lines which referenced the functions contained in a library import. To do this, we used pattern matching to search for the libraries $\ell$ and functions $f$, along with indicators such as and , to indicate that the library or function was being used. We gathered the author name, and then stored all this information, including library used, author name, and commit type, in a database for further analysis. This parsing of the data allowed us to perform a relatively quick analysis on a smaller dataset than the entire commit log. § LIBRARY ADOPTION Formally, for a project $p$, a library $\ell$, and a time $t$, we define a library adoption to be an event ($p$, $\ell$, $t$) representing the first time $t$ that $\ell$ is found in $p$. Some project repositories are simple, containing only a single commit, while others are extremely complex with multiple branches, hundreds of users, and thousands of commits. In 2014, Kalliamvakou et al found that the median number of commits per project on a small sample of GitHub projects was 6, and 90 percent of projects had less than 50 commits [5]. Our dataset shows a slightly larger distribution of commits, though as mentioned earlier, it is possible that our GitHub data is over-represented by active projects. The distribution of commit-activity per project, illustrated in Fig. <ref>, resembles a shifted power law distribution. Because of this dynamic, 50% of projects were found to have 10 or fewer commits (, median of 10) and 90% of projects have 100 or fewer commits. The distribution of the number of libraries adopted per project, illustrated in Fig. <ref>, also resembles a shifted power law distribution, albeit with a larger offset than the commit-activity distribution of Fig. <ref>. However, the number of adoptions is less evenly distributed: 54% of projects adopted 10 or fewer distinct libraries and 98% of projects adopted 100 or fewer libraries. Across all commits of all projects, we find that library adoptions occur more frequently within the first few commits. Figure <ref> shows that a project's first commit adopts 6.4 libraries on average (with a median of 2 not illustrated). A project's second through fifth commits adopt 3.3, 1.1, 0.8, and 0.65 libraries on average (with median values of 0 not illustrated). In general, the average number of adoptions per commit appears to follow a Zipfian decay, and commits tend to occur early in a project repository history. § ACTIVITY CHANGE AFTER ADOPTION A simple (albeit poor) indicator of productivity in software projects is the number of lines of code (LOC) that are added and/or removed in a commit. While it can be argued that not every line added to a repository is important, productive, or useful, and that inefficient code tends to have more lines, we go with this indicator because it is the simplest to understand, and provides a good summary statistic. Further analysis can be done on measuring the effectiveness of lines of code added to a repository. Within a single project the addition of code lines typically indicates positive activity like the addition of new functionality or features. Conversely, the removal of code typically indicates some negative activity like reversions or mistakes that are removed by the user. Oftentimes software bugs or other types of mistakes occur requiring edits that both add and remove code. Our data will contain the number of lines that are added or deleted in each commit, along with the libraries involved. We begin our analysis of library adoptions by parsing each code commit over each project repository. For each project starting from the first commit, we retrieve all imported libraries, their commit number, and the scope of any aliases. Users reference libraries and use them in several different ways. We define two classes of library use for the purposes of this study: (1) direct use, and (2) indirect use. Direct library use, as the name implies, references the library name or alias directly. From the example in Fig. <ref> above, the function call directly references the alias of the library; this is an example of direct use. Indirect library use references library calls made through a variable or other intermediary. Again from the example in Fig. <ref> above, the function call is an indirect use of the library because it references the library's functions through the variable. Taking an accurate accounting of indirect library use, especially in a language as fluid and dynamic as Python, would require an interpretation or partial-compile of each of the 23 million commits in our dataset. Therefore, in the present work, we limit our analysis to the import and direct use of libraries. [legend columns=-1, legend style= column sep=1ex, legend entries=Additions, Deletions, Net Use] green, mark=* red, mark=* black, mark=* Library additions, deletions and net-usage (in LOC) after the adoption event. Statistics surrounding newly adopted library $\ell$ Avg LOC that reference $\ell$ 31.34 Median LOC that reference $\ell$ 4 Avg insert LOC after 1st adoption to ref $\ell$ 2.09 Avg deleted LOC after 1st adoption to ref $\ell$ 1.62 Table <ref> shows there is a wide gap between the average LOC and median lines of code that represent a library $\ell$, indicating skew caused by large commits. This matches with our analysis which showed earlier than many of the statistics surrounding commits follow a power law distribution. Additionally, average LOC drops quickly after the first commit. Fig. <ref> shows the average direct use of an adopted library in lines added and deleted (in green and red respectively) as well as the net change, , insertions minus deletions (in black). After the initial commit, we find that most of the following commits have only a small positive net productivity. We also find that the volume of activity of lines of code referencing $\ell$ in Fig. <ref> tends towards zero rather quickly after the adoption. This indicates that, on average, the activity surrounding the adoption of a library is brief and oftentimes contradicted quickly. Recall from Fig. <ref> that most repositories only adopt a few libraries, with more than half adopting 10 or fewer libraries. Therefore, we can safely deduce that in most repositories, when adoptions occur, they occur early in a repository's history. Stack Overflow The popularity of software-centric question-and-answer Web sites has spread rapidly. Stack Overflow was created in 2008 [8], which is the same year that GitHub was launched [9]. Because these resources have grown in popularity together, we expect that they have influenced each other in some way. Much of the previous work in this area has focused on understanding user behaviors across Stack Overflow and GitHub [10, 11], modelling user interest [12], or mining for code that users have imported from Stack Overflow posts into GitHub projects [13]. In other cases, researchers aim to leverage posts and their answers in order to build tools that aid software development [14, 15]. Further research needs to be cone to understand data flows from Stack Overflow to GitHub, and vice-versa. Number of users of $\ell$ in GitHub dataset as a function of the number of Stack Overflow posts about $\ell$, showing that each library type has a statistically significant positive correlation. Here we see how the library adoption has become faster as the number of posts that appear on Stack Overflow have increased. Here we see how the library adoption has become faster as the number of posts that appear on Stack Overflow have increased. Growth of library usage after adoption grouped by Stack Overflow usage. Q1, Median, and Q3 growth in lines of code referencing $\ell$ are represented from bottom to top respectively. We plot the number of users of $\ell$ by the mean average number of Stack Overflow posts (across all adoption times) in Fig. <ref> that existed when $\ell$ was referenced. This illustration also groups libraries that are (1) included in PyPi, the default library repository used by the pip installer, (2) part of Python's standard suite of libraries, , os, json, time, and (3) all other libraries. We observe a strong positive correlation between the number of library users and the number of Stack Overflow mentions for standard libraries ($R^2$=0.625, $p<$0.001) and PyPi libraries ($R^2$=0.410, $p<$0.001). There is a small positive correlation between usage of unknown libraries and Stack Overflow posts ($R^2$=0.08, $p<$0.001), most likely due to individuals naming libraries like words and phrases that also happen to appear on Stack Overflow, or perhaps users sharing GitHub repositories that have not made it to PyPi. Exemplar libraries are called out in Fig. <ref>. For example, the standard libraries , , and indicate some of the most widely used libraries on GitHub. The , , and libraries represent three libraries found on PyPi. The library is used to create source code documentation files; it is relatively popular on GitHub but has only a few dozen posts on Stack Overflow. This seems to indicate that users have few questions about this library relative to its use (perhaps the library that produces source code documentation is well-documented!). Conversely, the library, which is used to crawl the Web, and the library, which is used for data analysis, have many questions, potentially indicating that the library is complicated to use. Despite being rather popular on Stack Overflow and GitHub, the “library” is not found in the standard Python libraries nor PyPi. This is a bit of a misnomer because the use of a “contrib” folder/module is a standard way to encourage open source developers to contribute to various projects. As a result, the contrib module is indicated as a common library simply because of its use as a naming convention in many distinct projects, so we see that it is mentioned quite frequently on Stack Overflow as a “local” library, but the functions defined in local “contrib” libraries across Github would yield very different functions since each user is writing their own “contrib” library for different purposes. Our next task is to understand what differences, if any, exist in the net productivity of these various libraries. To help understand the dynamics of library adoption, we calculate the median percentage growth (in LOC) of an adopted library, , the change in the number of added lines containing $\ell$ in a commit minus the number of deleted lines containing $\ell$ in a commit for the first 100 commits after the adoption. This provides us with a simple way to compare growth across teams which are different sizes. Formally, we compute the growth of a library $\ell$ within a project as follows. If $x=0$, then let $y_{x} = 1$; otherwise $$y_{x} = y_{x-1} \left(\frac{\sum\limits_{i=0}^{x-1}{(n_i)} + n_x}{\sum\limits_{i=0}^{x-1}{(n_i)}}\right),$$ where $n_i$ is the number of changed lines of code (net additions minus deletions) in commit $i$ that contain $\ell$. From this equation $y_x$ contains the percentage change at commit $x$ relative to the adoption event ($x=0$). Consider as an example the following series of commits $n$ = [+2, +1, +4, -1]. Here, the adoption commit ($x=0$) introduces two lines of code that reference $\ell$. We set $y_{0} = 1$. The next commit contains a net change of +1, rendering $y_1 = 1((2 + 1)/2) = 1.5$. In other words, in the second commit the number of lines of code referencing $\ell$ within this example project grew such that it is now 150% of its original size. The next commit contains a net change of +3, rendering $y_2 = 1.5((3 + 4)/3) = 3.5$ indicating that after the third commit the use of $\ell$ within this example project grew to be 350% of its size over the initial commit, , from 2 lines to 7. The final commit contains a net change of -1, rendering $y_3 = 3.5((7 - 1 )/7) = 3$. This normalization of median growth rate helps us compare larger teams to smaller ones. We plot the median growth (in LOC referencing $\ell$) as a function of the number of commits after the adoption in Fig. <ref>. Note that the adoption commit is not shown; instead, each commit either net-adds or net-subtracts from the initial adoption. Columns represent four groups of Stack Overflow mention sizes: no mention, between 1 and 100, 100 and 1000, and greater than 1000 from left to right respectively. Within each plot, solid lines represent the median, and the dashed lines on top and bottom represent the 3rd and 1st quartiles respectively. We observe that the use of an adopted library has a complicated relationship with its popularity on Stack Overflow. The primary distinction is in the growth rates for libraries with more than 1000 Stack Overflow posts. 100 commits after the adoption, the adoption of a highly mentioned library on Stack Overflow will have approximately 350% growth (on average) compared to only 250% growth for less mentioned libraries. We can assume that having over 1000 Stack Overflow posts means that the library is highly successful, and highly popular. Over time, it might be easier for users to find new resources about the library online - and add more functionality as the library becomes fully integrated into the project. When a library does not appear on Stack Overflow, the growth rate is similar to libraries that have over 1000 posts. Libraries that do not appear at all in Stack Overflow mostly consist of libraries that were written by developers who are also the authors committing the library to the repository. This may explain why growth is large in unknown libraries – the adopters know how to use the library because they wrote it. We could also propose that programmers who are using libraries for which there are no online resources available might be more experienced than those that are using more popular libraries, so their growth rate is faster. Project Team Size Next, we investigate differences in library adoptions as a function of team size. Because library adoptions occur from the perspective of a project, studying how various team sizes adopt libraries is important as we attempt to understand how teams form and work together. Researchers have studied GitHub previously for its team formation attributes. Git and GitHub directly store how team members collaborate and the types of activities that they perform [16]. For example, researchers have found that diversity and team makeup have a significant impact on the productivity of GitHub teams [17], and larger teams tend to process more pull requests on average [18]. Like the commit and adoption distributions illustrated in Figs. <ref> and <ref>, the team size distribution follows a power law. Median percentage change in lines of code referencing $\ell$ after adoption. We calculate a project's team size by counting the number of distinct committers. We observe in Fig. <ref> that the distribution of team sizes has a power law-like heavy tail wherein 59% of projects have only a single committer; 24% and 7% of projects have two and three distinct committers respectively. Projects with small teams therefore dominate the GitHub community. For the 59% of projects with a single committer, we do not have to even consider the team dynamics when a library adoption occurs, because only one individual is adopting a new library for use in the project. Like in the Stack Overflow analysis, we calculate the median growth over the first 100 commits after a library adoption for various team sizes, which we can see in Figure <ref>. We see that smaller teams add more lines of code after the first adoption event than larger teams. A possible explanation for the slower growth of library usage in larger teams is because of perspective differences between two or more committers to a project. Users might feel more comfortable making more commits or experimenting with newly adopted libraries in smaller teams, or if they are working alone, because there are fewer team members to consult with before a commit is made - and also a greater need to ensure that all team members understand the purpose of the commit. Also, more communication might be necessary between teammates before large commits are made, which would appear to cause slower growth rates for bigger teams. It is possible that in larger teams, the first adoption event is more substantial - and then grow more slowly afterwards. § CODE FIGHTS Fights between committers to a project occur whenever there is a disagreement about how others should structure code, how they should implement features, or any other decision impacting code production. We use this analysis of code fight to understand who wins these arguments by tracking who eventually commits code which eventually stays in the project - or is ultimately removed. Researchers have long analyzed the diffusion and change of information and conventions including work on the adoption of word use in offline textbooks [19], on Twitter [20], and other domains. The experience of the individuals in the group also plays a key role in what ideas are adopted offline [21] and in online social systems [22]. For example, Sumi et al describe edit wars on Wikipedia where two or more Wikipedia editors constantly edit each other's changes to some article [23]. Investigators have found fights in collaborative science writing where researchers often use and adapt various  macros and vocabulary. Specifically, based on files obtained from ArXiV, Rotabi et al showed that user experience is a large factor in determining who will win a fight. Less experienced researchers tended to win invisible fights, , fights over -macros that did not have a high visibility. More experienced researchers, , advisors and senior PIs, tended to win highly visible fights such as fights over the conventions used in the title of the paper [24]. In the software development paradigm, norms tend to develop in software teams, to which developers eventually learn and conform [25]. In the context of library adoptions in collaborative projects, we informally define a code fight as a series of commits that include back-and-forth additions and deletions of the same code containing a newly adopted $\ell$. For clarity, in the current work we restrict a fight to occur between two committers $u$ and $v$, but we encourage follow-up research that lifts this restriction in future analysis. In this context, a fight occurs when user $v$ removes all (or almost all) of the code that user $u$ committed that references $\ell$. Occasionally, the adopting user $u$ will recommit the original code, which $v$ may then revert. A user may add or remove code over a series of contiguous commits, rather than in a single large commit. Therefore, rather than thinking of fights as one commit after another, we model a fight in rounds. A round is a series of commits by one user that is uninterrupted by the other user. For example, if $v$ deletes 5 lines of code in commit 2 and then another 6 lines of code in commit 3, then we represent these two commits as a single round with -11 lines deleted. We formally define a fight as follows. Let $n^{(r)}$ represent the net change in lines of code referencing $\ell$ in round $r$; $r=0$ indicates the round of the adoption event. Also let $n^{\le r}$ be the sum of all lines of code referencing $\ell$ up to and including $r$, , the running total. A code fight occurs if there exists any $r$ such that $(1-\epsilon)n^{\le (r-1)} \le n^{\le r}$, where we set $\epsilon$ to represent the percent reduction that must occur, with $\epsilon \in \{.10, .20, .30, .40, .50\}$. Once a fight starts it will continue until there are no more rounds regardless of the size of the change in each round, , further rounds within the same fight do not have to necessarily add or remove $1-\epsilon$ LOC. The probability of a fight, for various sizes of $\epsilon$ and by project team size, is illustrated in Fig. <ref>. We observe that fights are relatively rare, occurring between 1 and 3 times for every 100,000 commits on average. We also observe that the choice for $\epsilon$ has a limited effect on the probability of a fight. The probability of a fight increases with team size, but with diminishing returns that resemble a Zipfian Distribution. In other words, because there are more interactions in a larger team, it is more likely that a fight will occur. Next, we analyze what happens during a two-person fight. Technically speaking, the first round of a fight is the adoption event and the second round of the fight is the removal of at least 100(1 - $\epsilon$) percent lines of the adopter's code. After this point, the two fighters (the adopter and the deleter) may continue with more rounds of back and forth commits. Despite the dropoff in number of fights, the adopter tends to fight back with more lines of code. In Fig. <ref> we observe that odd-numbered rounds, corresponding to the adopter, have more net LOC referencing $\ell$ per round, than the deleter's round that comes afterwards. Also, we see that the larger the original deletion of the code was, the less likely the adoptor is to fight back with lines of code. We can see that in the majority of fights, the fighters, who are the people who originally delete code from a repository, win the fight as the last committer. Non-adopters, who are parties who were not involved as either the deleter of code or the adopter, win fights as someone who enters a fight later slightly more often than the adopter themselves. We define a fight's winner as the user who was the last commiter referencing $\ell$. In some cases, the adopter may acquiesce to the deleter and allow the code to remain deleted. In other cases, the deleter may allow the adopter's reassertion of the library's addition to stand. By our definition of rounds, it is clear that the deleter wins approximately 90% of the fights because the adopter only fights back 10% of the time. This shows that it is relatively rare for users to counter and re-add code. Perhaps in some cases, the “fight” was not contentious and the result of a mutual agreement to remove a library. Further research is needed to find out how many of these fights are a result of team friction. What role does experience play in winning a fight? To answer this question, we must first ask how to best define experience. Two options are to count 1) the number of commits of a user (in any project) or 2) the time elapsed since the users first commit (in any project). Although these two options obtain similar results, the current work maintains the standard set by prior studies [24] and therefore defines experience as the time since the user's first commit (in any project). We observe in Fig. <ref>, which plots only results from $\epsilon=0.1$, that the more experienced committer wins the fights between 70% and 80% of the time. Results from alternative $\epsilon$ values were nearly identical to $\epsilon = 0.1$ and are omitted for clarity and brevity. The experience difference groupings were selected so that each contained a similar number of fights. Interestingly, the more experienced users have about a 75% win probability even when the experience differences are less than a week or even a day (not illustrated). This suggests that even slightly more familiarity with the project (perhaps an indication of project leadership) results in the more experienced user winning the fight. It appears that it is more common for new people working on a team to be subservient to the person who has been working in the codebase the most. Probability of a fight as a function of team size for various $\epsilon$. An increase in team size is directly correlated with the probability of a fight (Spearman $R^2=0.14$, p$\le0.01$ if $\epsilon$ = 0.1). When a two-person fight occurs, the adopter $u$ (indicated by odd-numbered $x$) tends to commit more code than $v$ on average The more experienced user is more likely to win a code fight regardless of the experience gap. Finally, we ask: which libraries are the most fought over? To answer this question, we counted the occurrences of $\ell$ within each two-person fight and normalized by the number of times $\ell$ was adopted. Here we set $\epsilon=0.1$, but results for other $\epsilon$ were similar. Table <ref> shows the top eight libraries involved in the most fights. Library Prob in Fight pdb 12.4 pprint 11.3 telnetlib 10.5 syslog 10.3 distutils 9.1 glob 9.0 poplib 8.4 imp 7.8 Libraries most likely to be involved in a fight. We observe that common debugging libraries , , and comprise three of the top four most common causes of fights. In these instances, we assume that adopters might be implementing or testing some functionality, they use the debugging libraries, and commit changes with the debugging code still implemented. This code then is then removed by another team member thereby instigating a fight. It is not surprising to see the library counted among the top fight starters. This particular library is used to generate source code distributions , code releases, but it is strongly encouraged that users use the library instead. So most cases importing is likely an error, and the usage of that library was deleted by others who are updating their code base, which appears in our analysis as a code “fight.” However, we can suppose that for some teams, the discussion of using in favor of is indeed a fight, especially if a team member is reluctant to switch libraries. Fight Threshold % Median Net Commit Size 0.5 24.58 0.4 28.49 0.3 27.22 0.2 24.09 0.1 18.12 Various values for median net commit lines before a fight occurs, in which the next commit results in net lines of code that are below the set fight threshold. § DISCUSSION Let's return to our original questions and summarize our results based on our findings. What does it look like when a team adopts a library for the first time? In Fig. <ref> we observe that library adoptions tend to happen early in a project's history. Hence, the probability of a new library being adopted later is lower. We can expect that it is difficult to adopt a new library once a project has matured. Perhaps this is because new libraries may introduce instability into a repository, or because the primary innovation within a project occurs early on in its lifespan. Further research is needed to understand this more clearly. While Fig. <ref> shows us that these adoptions happen early on, it would also be interesting to approach this problem from the perspective of percent of project repository history. We see that the commit distribution follows a power law distribution, and many projects have few commits. Therefore, while we measure the commits in a sequential order in Fig. <ref>, further research should be done to see when library adoptions occur as a measure of project completeness. Early in a project history, do we see that it takes more time for a library to be adopted? Or do we see that a team takes longer to adopt a new project after they have been working using established methods? Further research needs to be done to answer these questions. Despite these questions, we can see that library adoptions tend to be events that happen in the first few commits - though further research can help us understand when they occur in relation to the length of a project. That would give us another dimension to analyze this problem. Are commits containing new libraries more likely to have deletions than other types of commits? Once an adoption has occurred, we track how long it takes for library usage to become stable, or adopted, within the project by examining how many additions and deletions occur in the commits after a library is first used. In Fig. <ref>, we observe that activity involving a newly adopted library is relatively high after a commit occurs. Over time, the number of lines of code referencing the adopted library stabilizes, which indicates that the library has been fully adopted and incorporated into the project. We can safely conclude that users tend to write most lines of code that involve a newly adopted library relatively soon (within 10-15 commits) after library adoption. However, in many instances we also find that some libraries never stabilize and end up being deleted. While we currently measure when adoptions occur, it would also be interesting to see when libraries are completely deleted from project repositories. When would those deletions occur? Would we find that libraries are more likely to be deleted early on in a project repository history, or later? Would these deletions follow a similar distribution over time as library adoptions? To answer these questions, we need more research from the perspective of library deletions. As with Research Question 1, while we are currently asking this question from the perspective of commit number, we can also use the percentage of project completeness to understand when these deletions are occurring over the lifespan of a project. An analysis such as this would not skew the averages towards smaller projects since the project timeline would be measured from the perspective of percentage of a project completed, instead of absolute commit number. Additionally, we could ask further questions about the coding history of users that are on the project. To add to this work, we could track the libraries used by the individuals who are contributing to these projects. Other work as shown that user activity rates are higher soon after an adoption event [26]. Tracking user history across GitHub repositories could give us more information about what occurs in an adoption event, and help us learn more about how individuals learn and retain new information. Do the answers to these questions vary by library type, team size, or the amount of information available on Stack Overflow? When team sizes are larger, the lines of library code do not grow as quickly relative to the first adoption commit as they do when team sizes are smaller. This may be because larger team projects require more communication and planning and are therefore less agile than small teams or individual projects. As mentioned previously, this is one of the drawbacks of using lines of code as a measurement tool, since it could be argued that the code committed by larger teams is more valuable than the lines of code committed by smaller teams. Additionally, this work only looked at public software repositories. Perhaps an analysis of private software repositories would lead to different conclusions about how various team sizes work together - because the interaction between teammates who know each other personally may be different than those who do not, or may be operating in a public environment. Further in-depth research is necessary here. While we defined `large' team sizes as those with 10+ members, it would be interesting to see how teams whose members number in the hundreds or thousands would differ. We would expect these teams to have different adoption behavior. Also, a team size and lifespan analysis would provide unique insights into how long large team repositories survive versus smaller teams. What is the distribution of smaller team projects' lifespans compared to larger teams? In addition to team size, we showed that the number of times a library appears in Stack Overflow is highly correlated with the number of adoptions in our data set. Further questions about StackOverflow still need to be answered. In particular, more questions need to be answered about which way StackOverflow and GitHub grow - does StackOverflow influence GitHub, does GitHub influence StackOverflow, or is there information flowing in both directions? How do individuals find new libraries on StackOverflow? What is the distribution of attention given to different StackOverflow questions? Each of these questions requires further analysis as we attempt to understand how groups utilize outside resources to learn about new information. What does it look like when team members fight over new library usage? When working on a team, there is bound to be conflict. Different team members have various opinions about which library is best to use in a repository. The probability of these fights occurring increases with team size, as shown in Fig. <ref>. The winner of these fights tends to be more experienced as shown in Fig. <ref>. While we have done an analysis of team size, length of experience, and libraries used in fights, there are still questions to be answered regarding fights. Further research directions could include analyzing when fights occur in a project history. It would be interesting to see if fights are something that occur early or late in a project repository history. Additionally, this research was focused solely on fights between two individuals. Further analysis needs to be done on larger team sizes. Would an analysis of large team fights yield groups of people who are fighting for control on a project? Would larger teams result in more or less fights? It makes intuitive sense that more experienced teammates would win GitHub fights, since they have more seniority in a project. However, it would also be interesting to investigate the projects in which the individual with less project experience wins the fight. What characteristics do these new, inexperienced team members have that causes them to win code fights? Is there another interesting quality that results in them having more influence over their teammates? More research is needed to answer these questions. Additionally, in future work we could to uncover who is bringing new libraries into a project. We have seen that more experienced programmers tend to win these code fights. However, we could learn more information about how teammates work together by uncovering which team members are the first to bring in a new library. Would the teammates who have worked longer on the project be the ones to try new approaches with other libraries? Or would new teammates be the ones to suggest new ideas? Additionally, is there a divide between who is using new libraries and the popularity of that library on StackOverflow? We could hypothesize that common libraries, such as , would be used for the first time equally by both experienced and inexperienced programmers, but a more specialized library like would only be used by veteran team members. Since we have suggested libraries such as have steep learning curves due to the number of posts on StackOverflow about that particular library, we could also track how team members fight over how to use complicated libraries, and if those libraries have a longer time to adoption than others. There are some important caveats to the findings presented in the present work. We were only able to crawl 13% of all public Python GitHub projects; even if we could obtain all Python projects, these public projects only represent a subset of all Git projects. Therefore, we must temper conclusions to represent a case study of this domain and we caution the reader against drawing broad conclusions about user behavior outside of Python projects from public GitHub repositories. We can hypothesize that private software repositories are written by teams that have higher interpersonal relationships, and library adoption will appear to be different in these groups. Therefore, our conclusions cannot be generalized to all GitHub projects, though they provide a good overview of how libraries are adopted in public GitHub repositories. In particular, fights might look different in private repositories because we can assume that there would be some sort of relationship between the committers in those repositories, which might effect who wins those code fights. Additionally, we might find that people who post only in private repositories might be very different from those who have public GitHub accounts. Understanding the differences between these types of users would yield more interesting takeaways about how individuals work together in teams, as we analyze the difference between people who are more guarded of their work to those who are more open. From our work, we found some interesting takeaways. We see that the number of commits and adoptions per project, along with team size, follow a power law distribution. We can conclude that power law distributions are common as we understand social behavior - there are many people who contribute little, and only a few that contribute in large amounts. We found positive correlations between the number of times libraries appear in StackOverflow and GitHub. We discovered that popular libraries on StackOverflow have faster rates of adoption for projects in Git. Additionally, smaller teams are more agile and can grow more quickly than larger teams, when productivity is measured as a function of median percentage growth. We also find that code fights are rare, but when they occur, they tend to be won by more experienced coders, and involve libraries which are used for debugging purposes. We can therefore conclude that the availability and proliferation of online resources has helped improve productivity for programmer, and that team characteristics, including size and seniority, have interesting implications for team dynamics. We can only expect resources such as StackOverflow and GitHub to continue growing as they attract new users. However, just analyzing commit histories of varied team sizes do not tell the whole story. While we see in Figure <ref> that larger teams do not grow as quickly, further research needs to be done to conclude if these larger teams are actually more efficient. These smaller teams may be `moving fast and breaking things,' while larger teams could be more cautious in their execution due to the fact that larger teams by definition have more interpersonal relationships which need to be managed. It could be that larger team sizes have a higher percentage of their code that makes its way into the production version, while the multiple, minor commits that smaller teams make might be junk code that winds up being deleted. Therefore, this research needs to be continued to investigate this problem of team productivity from many different angles. This begs the quesiton of what `productivity' means, which could fuel many different research questions. While this work has focused on the library-centric approach to understanding adoptions, more work needs to be done to understand how individuals work to adopt information. Epidemiological models have been used to understand how information spreads across a social network [27], with a SIR model (suspectible, infected, recovered) being used to mimic one's potential to become `infected' (or viewing a post) and `infecting' it (by sharing it). In these models, individuals have varying levels of susceptibility. Further research could apply these models to GitHub users. Do we see that there are some individuals which have high susceptibility rates, which could enable them to adopt to new libraries more quickly? Also are there some users who are more `infecting' than others, and when they are present on a project, their introduced libraries are adopted more quickly than those introduced by individuals who are less `good' at spreading `infections.' This could be an interesting research topic that could attempt to apply epidemiological models to GitHub projects, where a library could be considered a virus. This type of approach could also help us create a model where we find highly influential GitHub users that are highly successful in implementing new libraries in varied projects. We could also view new libraries as a `virus' that spread throughout GitHub - and track the users that are instrumental in helping them spread. From this work, we can conclude that the proliferation of online coding resources such as GitHub and StackOverflow have been a positive development for programmers who wish to learn how to use new libraries to accomplish their coding tasks. Observing the growth of individuals who use StackOverflow and GitHub over time shows that the platforms have had tremendous growth over time for those hoping to learn to program. We expect continued high growth rates for both programs. Future Work We uncovered some interesting findings, but ultimately end up with more questions than conclusive answers. We encourage the community to explore specific questions raised by our results using the methodology developed in the present work. Specifically, we encourage further probing into how Stack Overflow contributes to the growth of library adoption and popularity. We have uncovered patterns that exist when team members fight over library adoption, and we look forward to further research which investigates code fights at an even greater depth. We present several facets of analysis, but due to the complexity of varying size of teams and GitHub repositories, there exists virtually limitless possibilities in exploring the data. Since the commit and adoption distribution follow a power law, it might be interesting to investigate what occurs only in projects where these values are very large. This paper attempted to account for this variety of team sizes, but more focused work on investigating either small or large team sizes would yield very interesting results. Additionally, tracking the growth of team sizes could yield some very interesting research. While this work ignores when commits occurred as a matter of date and time, gathering time stamps of commits might also yield interesting research. There are several questions that could be answered by analyzing time stamps, such as which time of year projects are more productive (students trying to finish semester projects? Teams being more productive at the end of the month?). Additionally, this work does not attempt to track users across projects. Further research could be done to discover if a person's prior programming experience results in lower time to adoption, or if sufficiently complex libraries remain hard even for experienced programmers. Some of these questions have been answered by Krohn et. al [26]. 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2024-09-04T02:54:55.722760

2003.00063

arxiv-papers

# Bio-Inspired Modality Fusion for Active Speaker Detection Gustavo Assunção1,2,†, Nuno Gonçalves1,2, and Paulo Menezes1,2 This work was supported by OE - national funds of FCT/MCTES (PIDDAC) under project UID/EEA/00048/2019.1Institute of Systems and Robotics (ISR), University of Coimbra, R. Silvio Lima, 3030-194 Coimbra, Portugal2Department of Electrical and Computer Engineering, University of Coimbra, Portugal†Corresponding author <EMAIL_ADDRESS> ###### Abstract Human beings have developed fantastic abilities to integrate information from various sensory sources exploring their inherent complementarity. Perceptual capabilities are therefore heightened enabling, for instance, the well-known ”cocktail party” and McGurk effects, i.e. speech disambiguation from a panoply of sound signals. This fusion ability is also key in refining the perception of sound source location, as in distinguishing whose voice is being heard in a group conversation. Furthermore, Neuroscience has successfully identified the superior colliculus region in the brain as the one responsible for this modality fusion, with a handful of biological models having been proposed to approach its underlying neurophysiological process. Deriving inspiration from one of these models, this paper presents a methodology for effectively fusing correlated auditory and visual information for active speaker detection. Such an ability can have a wide range of applications, from teleconferencing systems to social robotics. The detection approach initially routes auditory and visual information through two specialized neural network structures. The resulting embeddings are fused via a novel layer based on the superior colliculus, whose topological structure emulates spatial neuron cross-mapping of unimodal perceptual fields. The validation process employed two publicly available datasets, with achieved results confirming and greatly surpassing initial expectations. ## I INTRODUCTION The ability to discern between activity and passivity in humans may well benefit social robots and autonomous cognition systems by enabling a more accurate and natural response in interactive scenarios. Accordingly, machines capable of this task would be able to measure some user’s engagement level and infer valuable conclusions. Moreover, successfully solving this task would expedite a research shift towards other problems whose solutions depend on the output of a human activity-passivity recognizer. Recent work on data mining and collection [1] and speaker diarization [2], which heavily rely on the use of active speaker detection are exemplary of the topic’s importance. Active speaker detection (ASD), or the ability to recognize who in a group is speaking at any given time, is a subset of human activity-passivity recognition. This subset focuses on mapping utterances that make up a speech signal, such as in a conversation, to either a closed or open set of intervening speakers. In turn, techniques from the subset may take advantage of audio-only, visual-only or audiovisual information depending on their nature. Though studies do exist taking advantage of other types of perceptual input, such as [3], these are far more uncommon and somewhat less successful. In the past, extensive research effort has been put towards the audio-only section of ASD as a side tool for other problems [4], [5]. Techniques generally relying on sequential speaker clustering [6] or speech source localization [7], [8] were often enough to achieve real-time diarization, some occasionally aided by supervised learning. Unfortunately these were constrained by predetermined assumptions – mic location, static speakers, controlled environment. Furthermore, audio-only approaches remain unable to match segmented and clustered speech with visual speakers without additional data or restrictions, limiting their use on audiovisual data. Hence, visual approaches to the topic become useful, though not without their own drawbacks. Unreservedly interpreting lip movement [9], or other facial feature variations [10], [11] as vocal activity without accounting for similar motions stemming from normal human interaction – emotional expression, coughing, chewing – will of course negatively impact the detection process. An exemplary output of an active speaker detection system is shown in Fig. 1, for easier understanding. Figure 1: Exemplary output of an multi-speaker ASD system where the intervening speaker(s) is(are) denoted in green and speaker passiveness is encircled in red. In terms of multi-modal approaches, by means of complementing a visual approach with its audio counterpart or vice-versa, performances are generally higher thanks to increased robustness. This improvement is obtained by exploiting correlations between the two types of data (e.g. [12], [13]) which may decrease ambiguous situations prone to cause model confusion – speaker overlap, unrelated movement. Self-evidently, techniques requiring both a visual and audio input are unable to deal with situations where either of the modalities is missing. These situations however are scarce, an do not correspond to the immensely vast amount of audiovisual data available nowadays. Thus multi-modality remains as the preferred manner of addressing active speaker detection [14], [15], [16]. In spite of the recent technological advancements which have allowed deep learning research to flourish, such improvements would likely not have been possible without key biological contributions to the area. The success of remarkable architectures such as convolutional neural networks, whose earliest iterations [17] strived to emulate neuron behavior in the visual cortex of primates [18], would likely have been halted without this biological inspiration. However, Engineering maintains a collaborative yet stubbornly segregated relationship with Biology by disregarding several other promising models which the latter has to offer. In this study our contribution is two-fold as we propose a new multi-modal approach to visual ASD as well as a novel bio-inspired modality fusion layer which strives to emulate the superior colliculus structure of the brain. The approach is designed to work in real-time and makes no prior assumptions regarding noise, number of participants or spatial configuration. This is because it was developed also with the intent of being used in parallel projects, where it is required to recognize speakers and the moments they intervene in. The newly designed modality fusion layer performs successful integration of multi-source uni-sensory information through feedback stimulation of spatially proximal neural regions. The obtained results were highly positive, having exceeded expectations in terms of current state-of- the-art standards. The presented paper is structured as follows. Initially, examination and insight into previous research is shown in Section II, inducting the reader into the topic and contextualizing our approach. The proposed technique is described in Section III, followed by an overview of the experiments carried out in Section IV and the obtained results. These are then examined and critically discussed in Section V. Finally, a conclusion is provided in Section VI. ## II RELATED WORK The overview presented is branched into two sub-sections. To begin with, recent research on the topic is summarized in order to give further insight into active speaker detection (ASD), and contextualize our own approach to the matter. Following that, we analyze a number of the few available datasets and their suitability as an integrating part of ASD techniques. ### II-A ASD Recent Research Given the broad array of early audio-centered approaches, and their lingering success with nonvisual data, current efforts have turned more towards vision- oriented issues. This is not to say audio has stopped being a key aspect of current approaches to ASD. In fact, multi-modal state-of-the-art techniques have used machine learning to realize audiovisual correlations in data. Such techniques are successful, for instance, as a means to recognize speech related facial motion while also discerning it from other unrelated movements. In [16], [19] Stefanov et al. used the outputs of a face detector, along with audio-derived voice activity detection (VAD) labels, to train a CNN task specific feature extractor together with a Perceptron classifier. For transfer learning comparison, known CNN architectures were used as pre-trained feature extractors whose outputs were employed in training temporal (LSTM net) and nontemporal (Perceptron) classifiers. In a related work [20], Ren et al. tackled clutter and unrelated motion fallible situations by extending conventional LSTM models to share learned weights across the audio and visual modalities. Thus learning audiovisual correlations to improve model robustness. With a real implementation in a NAO robot, Cech et al. [21] used time difference of arrival (TDOA) over a set of microphone pairs to simultaneously evince potential sound sources in 2D space. A facial detector then estimated mouth locations in 3D space, to be statistically mapped to the former 2D sound sources and TDOAs. In a similar fashion, Gebru et al. [22] mapped located sound sources to the 2D image plane as well as estimated potential speaker lip positions. However, this cross-modal weighting approach employed a GMM variation to assign weights to audio observations based on their spatial proximity to visual observations, and vice-versa. Audiovisual clustering then achieved active speaker detection. To give another example, Hoover et al. [23] attempted correlating the degree of occurrence of individual faces in a video with single speaker clusters of speech segments, obtained using vectors of locally aggregated descriptors (VLADs) [24]. In addition to the usual reliance on high-end equipment or ideal settings – consistent number of speakers, frontal facing and nonoverlapping speech, mic/camera location – which is common to clustering and other methods, further issues also occur with current state-of-the-art techniques. Particularly, unidirectional dependency between modalities is recurrent among machine learning approaches. The lack of carefully labeled big data on both the audio and visual modalities for ASD evidently forces resourcing to alternatives which transfer labels generated from one modality to its sibling (typically from audio to video). Such processes unavoidably lead to propagation of error, originating from the single modality prediction techniques, which could be avoided were large ASD datasets existent and applicable to robust machine learning architectures. ### II-B Datasets In order to validate audiovisual ASD techniques (and train when required), some data encompassing all considered modalities must be available. Furthermore, such data is required to include some form of labeling so as to enable its application in the aforementioned techniques. Unfortunately, the multi-modal nature and challenges inherent to audiovisual ASD constitute a major setback in terms of acquiring labeled data. As such, small and constrained datasets are frequently built by research teams just to evaluate their own approaches, which of course restricts their comparability and reduces the credibility of their success. Yet very recently a few datasets have been created and made available, which address this current necessity in ASD research. In [2] the AVSpeech dataset was presented, an automatically collected large- scale corpus made up of several lecture recordings available on YouTube. These recordings, amounting to around 4700 hours of audiovisual data, always show a single clearly visible face to which the ongoing speech corresponds. This dataset could be labeled for active speaker detection using some voice activity detection (VAD) or speech-silence discerning technique, and then applied for training and testing of ASD approaches. However, this may lead to model confusion later on in multi-speaker scenarios and as such AVSpeech is perhaps best left for ASD testing. A potentially better example of a suitable corpus is given in [15], where the Columbia dataset was first introduced. Here, the recording of a panel discussion between 7 speakers was overlapped with upper body bounding boxes for all participants, each labeled with speak/no-speak markers. By always including more than one speaker in each frame, this dataset has the benefit over AVSpeech to enable dealing with multi-speaker situations in inference models. As for its downside, the dataset is of reduced size and its data is considerably homogeneous. This evidently means it may be harder to extrapolate relevant features only and generalize a model trained on this data, which constrains the dataset’s usage in machine learning approaches. The largest and most heterogeneous active speaker detection dataset available at the moment is presumably AVA-ActiveSpeaker [25], developed by Google for the 4th ActivityNet challenge at CVPR 2019. This hand labeled set of segments obtained from 160 YouTube videos amounts to around 38.5 hours of audiovisual data, where each of the covered 3.65 million frames is detailed with bounding box locations for all detected speaker faces as well as with three types of labels for each present bounding box – speaking and audible, speaking but not audible, not speaking. Such a corpus should therefore be perfect for learning ASD as it covers all previously mentioned requirements for a suitable audiovisual dataset and presents no major drawbacks. To support this, the team behind AVA-ActiveSpeaker used it to train and test both static and recurrent prediction models, applying modality fusion and consistently achieving great performances. Unfortunately however, this dataset encompasses an extremely high amount of low quality videos in terms of image resolution, audio intelligibility and noise. The creators of the dataset even went so far as to include several voice-over videos and dubbed versions of movies not in their original language. This naturally means a copious amount audio clips are out of sync with their supposedly respective facial image sequences, besides also the lengths of these clips frequently not pairing with those of their sequences – the same phrase usually varies in duration for different languages. Taking this into account each of the 160 videos was manually inspected for quality assessment, as though they may be appropriate for a challenge in the same conditions, it would be senseless to use such great amounts of noisy incorrect data to train a model and expect it to perform successfully with correct data and in the real world. A total of only 20 videos, whose list is available on our github repository111https://github.com/gustavomiguelsa/SCF, were deliberated to meet today’s audiovisual capture standards. The repository also contains an exemplary demonstration video of the technique described in this paper, for easier understanding. ## III METHODOLOGY This section describes the path taken to develop the proposed technique aimed at performing active speaker detection, including detailed descriptions of each individual component of the overall system. The developed method is capable of dealing with either audio data, visual data or audiovisual data thanks to its split structure. To put simply, each modality is dealt with independently through separate preprocessing and the use of dedicated architectures for feature extraction. In order to achieve this, audiovisual raw data is divided into audio segments for which narrowband spectrograms are generated and propagated through the VGGVox model [1], and corresponding sequences of facial images, whose features and corresponding temporal correlations are obtained through the employment of an extended ResNet50 architecture [26] with incorporated recurrence henceforth designated as RN-LSTM. The convolutional stage of the former has been pretrained with the VGGFace2 dataset [27] and is available in [28]. Detail is further provided for each model component in the following sections. ### III-A Audio Clips and VGGVox In [1], the VGGVox model was first introduced as a modified VGG-M CNN architecture [29] aimed at speaker recognition through examination of spectral representations of audio clips. Given its extensive training with over 100,000 utterances by 7000+ distinct speakers, the observed excellent performance has been recognized as a state-of-the-art standard in recent research. Thus, the model is ideal for extrapolating speaker specific cues and discern multiple distinct speakers in an active speaker detection scenario. In terms of actual audio clips to be analyzed, they should be required to undergo minimal preprocessing only while still being able to progress through a machine learning model. In the past, many techniques have been proposed to obtain audio representations which achieve this goal such as the known Mel- frequency cepstral coefficients [30]. Despite their success, such techniques are noted to introduce increased overhead and require a manual tuning which somewhat invalidates the posterior model’s positive performance. Therefore for each raw audio clip in our approach, a sliding Hamming window of width 25ms was applied with a step of 10ms in order to generate a respective narrowband spectrogram, as well as means and variances being normalized at each frequency bin of the spectrum. These operations were the single ones performed on the audio data, following VGGVox’s preprocessing stage. ### III-B Image Sequences and RN-LSTM The ResNet-50 residual network architecture is one based on the deep residual learning process proposed in the work [26], which may shortcut the connections between the inputs and outputs of a network’s intermediate layers – creating identity mappings. This procedure is performed in order to attempt approximation of residual functions in lieu of fitting a more complex mapping, thanks to the ease of learning advantage caused by occasional skipping of layers considered less relevant. Through skipping, increasingly deeper models may be created without the inherent accuracy degradation previously observed in [31] and still benefiting from improved generalization. In addition, the identity mappings also imply a loss similar to that of their shallow non- residual equivalent models. For these reasons, residual networks should be capable of achieving the level of generalization necessary for extracting activity/inactivity cues from face images. Whilst residual networks may be suitable for the task, these were not originally trained for facial analysis. Furthermore, ASD approaches must unavoidably deal with human faces in real world interactions which are obviously unconstrained by pose or expression, besides potentially encompassing several and extremely diversified physical attributes. As a means of dealing with these face issues and other common image problems (e.g. varying light/exposure) while still benefiting from the advantages of residual architectures, the ResNet-50 model trained with VGGFace2 made available by [27] was chosen for our approach. Considering how little to no conclusions pertaining to speech activity may be drawn from solo face images, analyzing the sequential patterns between them becomes a requirement for success. Undoubtedly, recurrent neural networks and more specifically LSTMs are highly suitable for approaching that requirement given their demonstrated past success when dealing with the temporal correlation of facial embeddings [32], [33]. Thus a wide double-layered LSTM was appended to the trained ResNet-50 architecture creating the mentioned RN- LSTM. This new recurrent section was trained freely by freezing the weights of its preceding ResNet-50 layers. ### III-C Bio-Inspired Embedding Fusion Integration of multi-modal information is a considerably complex task characterized by deeply convoluted data correlations. In the animal kingdom, such a task is unconsciously performed on certain brain regions, allowing for beings and equally humans to derive multi-sensory conclusions. Inspired by the work of Ursino et al. in [34] and their model of the brain’s superior colliculus (SC) for multi-sensory integration, we present a novel bio-inspired neural network layer for audio and visual embedding fusion. This section describes its structure and behavior. The proposed SC fusion layer, henceforth abbreviated to SCF, is composed by two upstream uni-modal neural areas corresponding to the audio and visual modalities, plus the additional downstream multi-modal area in charge of audiovisual integration. Each upstream area comprises $N\times M$ basic neuron units interlinked within that same area, but not between upstream areas, and topologically organized to communicate inputs to corresponding downstream neurons whose receptive fields (RF) – kernels – are in the same spatial positions. The downstream area in turn, also made up of $N\times M$ neurons, feedbacks to each upstream area in the same neuron spatial fashion. This serves to emulate how proximal biological neurons respond to stimuli in proximal position of space, and allows for visual area neurons to indirectly excite audio area neurons or vice-versa through the multi-modal area. Lateral synapses between intra-area neurons are modelled after a mexican hat (Ricker) wavelet, so they may be both excitatory (central region) and inhibitory (surrounding region). The overall described structure is depicted in Fig. 2. Figure 2: Overall SCF layer structure, where the synapses of the central darker neuron (representing neuron $ij$ in a $N\times M$ neural area) are modelled after a Mexican Hat wavelet. Each neuron is laterally linked to its area $s$ neighbors by excitatory $ex$ and inhibitory $in$ synapses (red arrows). The described spatially conditional connection between uni-modal neurons (visual/auditory) and their multi-modal counterparts is shown on the right. Considering a neural area $s\in S=\\{a,v,m\\}$, pertaining to the auditory, visual and multi-modal factors. Asserting $s$ to be composed of $N\times M$ neurons as previously described, each neuron $ij$ or $hk$ ($i,h=1,\ldots,N;j,k=1,\ldots,M$) receives a composite input $u^{s}_{ij}[n]$ as in (1). $u^{s}_{ij}[n]=\left\\{\begin{array}[]{ll}r^{s}_{ij}[n]+l^{s}_{ij}[n]+f^{s}_{ij}[n]\hskip 28.45274pts=\\{a,v\\}\vspace{0.5cm}\\\ k^{a}\cdot z^{a}_{ij}[n]+k^{v}\cdot z^{v}_{ij}[n]+l^{s}_{ij}[n]\hskip 8.5359pts=m\\\ \end{array}\right.$ (1) Where $r^{s}_{ij}[n]$ denotes the input at neuron $ij$ triggered by an external stimulus, $l^{s}_{ij}[n]$ represents the input caused by lateral synapses between intra-area neurons, $f^{s}_{ij}[n]$ is the feedback input from the multi-modal SC neurons up to the uni-modal neurons, and $z^{s}_{ij}[n]$ describes an arbitrary neuron’s activity whose continuous differential model is discretized in our implementation as the signal in (2) for any $s\in S$. $\begin{array}[]{ll}\tau_{s}z^{s}_{ij}[n+1]=(\tau_{s}-1)z^{s}_{ij}[n]+\phi(p^{s}(u^{s}_{ij}[n]-\vartheta^{s}))\end{array}$ (2) Figure 3: Structure overview of the proposed model. From a video, the image sequence of a face is extracted as well as the corresponding audio. A narrowband spectrogram is generated from the audio and progressed through VGGVox. The image sequence is progressed through the RN-LSTM without any preprocessing. The obtained audio and visual embeddings are then classified either separately or merged by concatenation or the SCF layer. For such a response $\tau_{s}$ refers to the speed with which a neuron responds to some stimulus, and $\phi$ is the sigmoidal activation of some input $u^{s}_{ij}[n]$ normalized by $\vartheta^{s}$ and $p^{s}$ – respectively the input value at the central point of neuron activity and its slope. As a consequence, the input to a multi-modal neuron can be seen as the weighted sum of the input activities of each of its spatial uni-sensory counterparts ($k^{a}$ and $k^{v}$ being inter-area synaptic connection intensity) plus the area’s own lateral synapses. The laterally synaptic input received by some neuron $ij$ of an area $s$ is defined simply as the weighted sum of the neighboring neurons activities, following (3). Here $L^{s}_{ij,hk}$ represents the strength of the synaptic connection from neuron $hk$ to neuron $ij$, which follows the mexican hat wavelet in (4) as previously explained. $\begin{array}[]{ll}l^{s}_{ij}[n]=\sum_{h,k}L^{s}_{ij,hk}\cdot z^{s}_{hk}[n]&s\in S\end{array}$ (3) $\begin{array}[]{ll}L^{s}_{ij,hk}=L_{ex}^{s}\cdot e^{-\frac{[d_{x}^{2}+d_{y}^{2}]}{2\cdot(\sigma_{ex}^{s})^{2}}}-L_{in}^{s}\cdot e^{-\frac{[d_{x}^{2}+d_{y}^{2}]}{2\cdot(\sigma_{in}^{s})^{2}}}&s\in S\end{array}$ (4) Where the $L_{ex},\sigma_{ex}$ and $L_{in},\sigma_{in}$ parameters define the strength and spatial disposition of excitatory and inhibitory lateral synapses, respectively. As for $d_{x}$ and $d_{y}$ these refer to the inter- area distances between vertically and horizontally intervening neurons, being calculated circularly to prevent border issues. In terms of the external stimulus input at each neuron $r^{s}_{ij}[n]$, this was obtained through the inner product of the stimulus $I_{s}$ with the neuron’s own receptive field $R_{ij}^{s}$, made trainable and adaptive at each epoch, as in (5). This was also the case for the feedback input $f^{s}_{ij}[n]$, being calculated as the inner product of a multi-modal neuron’s activity $z^{m}_{ij}[n]$ and $F_{s}$, the strength of the feedback synaptic connection, as in (6). $\begin{array}[]{ll}r^{s}_{ij}[n]=R_{ij}^{s}\cdot I_{s}&s=\\{a,v\\}\end{array}$ (5) $\begin{array}[]{ll}f^{s}_{ij}[n]=F_{s}\cdot z^{m}_{ij}[n]&s=\\{a,v\\}\end{array}$ (6) The motivation behind the proposed layer structure comes from the need to effectively fuse single modality embeddings. The ideal result is an enhanced multi-modal embedding of the most useful audio-video correlations and features. This is achieved thanks to the SC emulation area which upon receiving a multi-modal stimulus reinforces each of its uni-sensory fractions upstream to spatially correlated neuron regions. This concept has greater reach than the one demonstrated on this paper, as there is absolutely no restrictions in terms of number of modalities for fusion – a posture analysis neural area could be added by extending (1), behaving similarly to the auditory and visual areas. Moreover, prior sub-fusion areas could also be integrated in the model to combine more closely correlated information – the vision neural area could be divided in two in the case of using a stereo rather than mono visual capture technique. Finally, the structure’s described non-trainable parameters were made fine tunable to application specific goals but we kept their default values in our implementation, being supported by the literature pointed in [34]. Evidently the embeddings used for the concatenation approach as well as the input auditory + visual stimulus for the novel SCF layer method were obtained using the previously described VGGVox and RN-LSTM networks. A comprehensive diagram of the overall structure is shown in Fig. 3. ## IV EXPERIMENTS The experimental portion of our work consisted of two distinct parts: (1) data preparation, given how the condition of the available data was not suitable for proper immediate testing; (2) modality experimentation, where each modality was tested first individually and then fused together. ### IV-A Data preparation The evaluation of the proposed ASD system required a considerably large amount of data, preferably with a high degree of heterogeneity and with respect to natural conversations. Yet as previously noted, no single dataset encompasses all these characteristics and so the testing phase examined data from two sources: the Columbia dataset [15]; and the 20 videos considered acceptable from the AVA-ActiveSpeaker dataset [25]. All of the latter’s sequences were already under the 10 second mark and so were left untouched, as opposed to the former’s which were originally only 136 and ranged from less than 1 second to several minutes in terms of duration. Consequently and considering the standard 30 fps frame rate of the original video, all instances above the 30 frame mark were segmented into 30 frame long sequences. This was performed in order to obtain a much larger number of instances and take better advantage of all the available data for training and testing. Sequences from both datasets below the 30 frame mark were discarded as it was considered samples under 1 second in length would likely suffer from the same problems as static images with respect to active speaker detection. Given how ultimately the obtained sequences for training and testing were of varying lengths, masking was employed in order to enable all available sequences to be accepted by the RN- LSTM. Considering this, training was independent of frame window size, making it arbitrary (within 1 and 10 seconds) for model applications. As for the AVA- ActiveSpeaker sequences pertaining to the third label ’SPEAKING_NOT_AUDIBLE’, these were considered for the visual only portion of the experiments and disregarded for the rest. The overall characteristics of the evaluated data are specified in Table I. TABLE I: Characteristics of the prepared data Dataset | Sequences | Frames | | Unique --- Facial Trackings 222This factor is provided in place of number of speakers given the fact that AVA-ActiveSpeaker does not inform on speaker amount. However, the same speaker has been manually verified to not exist in two different videos. | Image --- Conditions Columbia | 4914 | 30 | 34 (6 speakers) | Identical AVA-ActiveSpeaker | | 5510 (3 labels) --- 5470 (2 labels) [30,300] | 4321 (20 videos) | Varied ### IV-B Modality Experimentation Before attempting classification of the fused multi-modal embeddings, an initial testing of each individual modality was performed to assess the extrapolation quality and obtain a term of comparison for later on. Accordingly, multilayer perceptrons (MLP) were attached to the end of the feature extraction layers of VGGVox and the developed RN-LSTM, for classification. Additionally, level of model generalization was also assessed by comparing dependence and independence scenarios in terms of speaker for the Columbia dataset and in terms of video for AVA-ActiveSpeaker – a solution to the unknown speaker amount issue22footnotemark: 2. Essentially, for dependent scenario experiments (closed set) the test data includes instances from speakers who are also represented in the training data, albeit by completely distinct instances. On the contrary, for independent scenario experiments (open set) it is guaranteed that the speakers from the test data are completely different and never before seen by the model during training. These two types of testing allow for a better evaluation of the model’s generalization ability, by examining how much the model’s performance decreases from dependency to independency. The obtained results are presented in Tables II and III, respectively. TABLE II: Audio/video results using the Columbia dataset Mean Accuracy (Standard Deviation) | Speaker Dependent | Speaker Independent ---|---|--- | $VGGVox+MLP_{2}$ --- (Audio Only) 47.37 (0.52) [%] | 17.34 (9.14) [%] | $RN$-$LSTM+MLP_{1}$ --- (Video Only) 97.68 (0.39) [%] | 64.89 (14.44) [%] TABLE III: Audio/video results using the AVA-ActiveSpeaker videos Mean Accuracy (Standard Deviation) | Video Dependent | Video Independent ---|---|--- | $VGGVox+MLP_{2}$ --- (Audio Only) 87.73 (0.87) [%] | 84.96 (1.78) [%] | $RN$-$LSTM+MLP_{1}$ --- (Video Only) 78.68 (1.79) [%] | 72.48 (2.17) [%] The novel SCF layer, as well as the entirety of the implemented experiments, were realized using the Keras framework [35]. In order to assess its superiority against state-of-the-art standards and suitability to the task at hand, we compared its performance to a naive baseline. This was composed of an audiovisual concatenated embedding array fed directly to a multi-layer perceptron, as seen in Fig. 3. The results of this experiment and those obtained with the SCF layer alternative are shown in Tables IV and V for the Columbia and AVA datasets, respectively. TABLE IV: Audiovisual results using the Columbia dataset Mean Accuracy (Standard Deviation) | Speaker Dependent | Speaker Independent ---|---|--- | $Concat+MLP_{3}$ --- (Baseline) 97.60 (0.49) [%] | 41.78 (15.46) [%] | $SCF+MLP_{3}$ --- (New Approach) 98.53 (0.21) [%] | 58.50 (7.71) [%] TABLE V: Audiovisual results using the AVA-ActiveSpeaker videos Mean Accuracy (Standard Deviation) | Video Dependent | Video Independent ---|---|--- | $Concat+MLP_{3}$ --- (Baseline) 89.10 (0.41) [%] | 87.52 (2.49) [%] | $SCF+MLP_{3}$ --- (New Approach) 91.68 (0.51) [%] | 88.46 (2.06) [%] The SCF layer’s neural areas were kept at a $17\times 17$ dimension, for the sake of computational simplicity. Each of the presented values in all tables was obtained using 5-fold cross validation for increased robustness. Furthermore, the adaptive moment estimation (Adam) optimizer [36] was employed during the training phase of the models for weight adjustment. Batch normalization [37] was also integrated in the training, in addition to minor dropout [38]. The number of epochs was maintained constant across transversal testing, having each ideal value been obtained through early stopping mechanism for the three experimental phases (video-only, audio-only and audiovisual). ## V DISCUSSION As a first observation, it can be noted how the audio-only approach to active speaker detection (Tables II, III) performed surprisingly well for the AVA- ActiveSpeaker videos but was rather unsuccessful for the Columbia dataset. This is attributed to the mentioned data homogeneity of the latter, considering it encompasses a single speaker panel video. The fact that there is constant speech in the video (one speaker picks up after the other) prevents the model from learning even the most basic factor of discerning silence from speech. Hence results matching random decision for speaker dependency but failing with the independent case. Still regarding the Columbia dataset’s homogeneity, it proves to be advantageous for the dependent section of the video-only testing phase, but again not so much for the independent one. Thus the model naturally performs better with previously seen subjects but fails to extrapolate suitable ASD features by training with such a reduced number of speakers. Not unlike the audio testing however, the AVA videos led to a considerably high performance using images only, proving data heterogeneity and speaker abundance to be ultimately better for training as expected. Audiovisual results greatly surpassed expectations, with the baseline concatenation approach already demonstrating state-of-the-art performance for either dataset considered. Increase in terms of standard deviation from the dependent to the independent testing was observed as expected, considering how the model suffers a natural decrease of confidence in its predictions when analyzing previously unseen speakers. Relatively poor results are of course still obtained for the independent part of the Columbia dataset testing due to the constant speech issue, which as expected severely hinders the model’s performance. Nonetheless, results are remarkably positive for either concatenation or SCF layer approaches with respect to speaker dependent testing. Moreover, the SCF layer still showed its superiority by majorly improving over concatenation and even surpassing a random choice performance despite having to deal with crippling audio data. As for the AVA video testing, both the concatenation and SCF results beat those presented in the dataset paper [25]333Even though testing was carried out with only 20 of the original videos, the performance gain was still exemplary in terms of result excellence.. Even more so, the SCF layer successfully improved the already excellent baseline performance. This serves to validate its bio-inspired concept – mutual neuron excitation of spatially proximal multi-modal neural regions for integration of uni-sensory information – here emulated for modality fusion. Undoubtedly even greater performance gains could be achieved if all the layer’s hyperparameters were finely tuned to our application, rather than using default values. Nonetheless, results were highly successful and warrant further research in this and other areas encompassing several types of modalities. ## VI CONCLUSIONS This work proposed a novel modality fusion layer (SCF), bio-inspired by the brain’s superior colliculus region and incorporated into a new method for active speaker detection (ASD) in real-world scenarios. This proposed method deals with audio and video information jointly and makes no assumptions regarding input data. The audiovisual technique was extensively tested against data from two tried and tested ASD databases, specifically the Columbia [15] and the AVA-ActiveSpeaker [25] datasets. The obtained results were compared to those of audio-only and video-only approaches, as well as those of a naive concatenation multi-modal baseline. In any case the SCF technique always demonstrated state-of-the-art performance, surpassing its baseline and uni- modal counterparts. It was shown how mutual neuron excitation of a spatially proximal multi-modal region by uni-modal neurons can be a remarkably successful modality fusion technique. This observed SCF layer’s success in terms of uni-sensory data integration, even despite doing so with default settings, evidently warrants a greater exploration of its capabilities. As such, in the future we intend to further examine the layer’s behavior in terms of not only audiovisual fusion but also combination of data of other natures (e.g. tactile). In addition, the development of a real-time ASD demo based on the system presented in this work is underway. ## References * [1] A. Nagrani, J. S. Chung, and A. Zisserman. Voxceleb: a large-scale speaker identification dataset. InProc. Inter-speech, 2018. * [2] A. Ephrat, I. Mosseri, O. Lang, T. Dekel, K. Wilson, A. Hassidim, W. T. Freeman, and M. Rubinstein. Looking to listen at the cocktail party: A speaker-independent audio-visual model for speech separation. ACT Trans. 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2024-09-04T02:54:55.735176

2003.00083

arxiv-papers

Nonparametric Estimation in the Dynamic Bradley-Terry Model Heejong Bong* Wanshan Li* Shamindra Shrotriya* Alessandro Rinaldo Department of Statistics and Data Science Carnegie Mellon University ###### Abstract We propose a time-varying generalization of the Bradley-Terry model that allows for nonparametric modeling of dynamic global rankings of distinct teams. We develop a novel estimator that relies on kernel smoothing to pre- process the pairwise comparisons over time and is applicable in sparse settings where the Bradley-Terry may not be fit. We obtain necessary and sufficient conditions for the existence and uniqueness of our estimator. We also derive time-varying oracle bounds for both the estimation error and the excess risk in the model-agnostic setting where the Bradley-Terry model is not necessarily the true data generating process. We thoroughly test the practical effectiveness of our model using both simulated and real world data and suggest an efficient data-driven approach for bandwidth tuning. ## 1 Introduction and Prior Work ### 1.1 Pairwise Comparison Data and the Bradley-Terry Model Pairwise comparison data are very common in daily life, especially in cases where the goal is to rank several objects. Rather than directly ranking all objects simultaneously, it is usually much easier and more efficient to first obtain results of pairwise comparisons and then use them to derive a global ranking across all individuals in a principled manner. Since the required global rankings are not directly observable, developing a statistical framework for the estimating rankings is a challenging unsupervised learning problem. One such statistical model for deriving global rankings using pairwise comparisons was presented in the classic paper (Bradley and Terry,, 1952), and thereafter commonly referred to as the Bradley-Terry model in the literature. A similar model was also studied by Zermelo (Zermelo,, 1929). The Bradley-Terry model is one of the most popular models to analyze paired comparison data due to its interpretability and computational efficiency in parameter estimation. The Bradley-Terry model along with its variants has been studied and applied in various ranking applications across many domains. This includes the ranking of sports teams (Masarotto and Varin,, 2012; Cattelan et al.,, 2013; Fahrmeir and Tutz,, 1994), scientific journals (Stigler,, 1994; Varin et al.,, 2016), and the quality of several brands (Agresti,, 2013; Radlinski and Joachims,, 2007), to name a few. In order to introduce the Bradley-Terry model, suppose that we have $N$ distinct teams, each with a positive score $s_{i}$, $i\in[N]:=\\{1,\ldots,N\\}$, quantifying its propensity to be picked or win over other items. The model postulates that the comparisons between different pairs are independent and the results of the comparisons between a given pair, say team $i$ and team $j$, are independent and identically distributed Bernoulli random variables, with winning probability defined as $\mathbb{P}\\!\left(i\ \text{ beats }j\right)=\frac{s_{i}}{s_{i}+s_{j}},\>\forall\;i,j\in[N]$ (1) A common way to parametrize the model is to set, for each $i$, $s_{i}=\exp(\beta_{i})$, where $(\beta_{1},\ldots,\beta_{N})$ are real parameters such that $\sum_{i\in[N]}\beta_{i}=0$ (this latter condition is to ensure identifiability). In this case, equation (1) is usually expressed as ${\sf{logit}}(\mathbb{P}\\!\left(i\ \text{ beats }j\right))=\beta_{i}-\beta_{j}$, where, for $x\in(0,1)$, ${\sf{logit}}(x):=\log\frac{x}{1-x}$. ### 1.2 The time-varying (dynamic) Bradley-Terry Model In many applications it is very common to observe paired comparison data spanning over multiple (discrete) time periods. A natural question of interest is then to understand how the global rankings vary over time i.e. dynamically. For example, in sports analytics the performance of teams often changes across match rounds and thus explicitly incorporating the time-varying dependence into the model is crucial. In particular the paper (Fahrmeir and Tutz,, 1994) considers a state-space generalization of the Bradley-Terry model to analyze sports tournaments data. In a similar manner, bayesian frameworks for the dynamic Bradley-Terry model are studied further in (Glickman,, 1993; Glickman and Stern,, 1998; Lopez et al.,, 2018). Such dynamic ranking analysis is becoming increasingly important because of the rapid growth of openly available time-dependent paired comparison data. Our main focus in this paper is to tackle the problem of generalizing the Bradley-Terry model to the time-varying setting with statistical guarantees. Our approach estimates the changes in the Bradley-Terry model parameters over time nonparametrically. This enables the derivation of time-varying global dynamic rankings with minimal parametric assumptions. Unlike previous time- varying Bradley-Terry estimation approaches, we seek to establish guarantees in the model-agnostic setting where the Bradley-Terry model is not the true data generating process. This is in contrast to more assumption-heavy parametric frequentist dynamic Bradley-Terry models including (Cattelan et al.,, 2013). Our method is also computationally efficient compared to some state-space methods including but not limited to (Fahrmeir and Tutz,, 1994; Glickman and Stern,, 1998; Maystre et al.,, 2019). ## 2 Time-varying Bradley-Terry Model ### 2.1 Model Setup In our time-varying generalization of the original Bradley-Terry model we assume that $N$ distinct teams play against each other at possibly different times, over a given time interval which, without loss of generality, is taken to be $[0,1]$. The result of a game between team $i$ and team $j$ at time $t$ is determined by the timestamped winning probability $p_{ij}(t):=\mathbb{P}\\!\left(i\text{ defeats }j\text{ at time }t\right)$ which we assume arises from a distinct Bradley-Terry model, one for each time point. In detail, for each $i$, $j$, and $t$ ${\sf{logit}}(p_{ij}(t))=\beta_{i}(t)-\beta_{j}(t),\>\forall\;i,j\in[N],t\in[0,1]$ (2) where $\bm{\beta}(t)=\text{vec}(\beta_{1}(t),\beta_{2}(t),\dots,\beta_{N}(t))\in\mathbb{R}^{N}$ is an unknown vector such that $\sum_{i}\beta_{i}(t)=0$. We observe the outcomes of $M$ timestamped pairwise matches among the $N$ teams $\\{(i_{m},j_{m},t_{m}):m\in[M]\\}$. Here $(i_{m},j_{m},t_{m})$ indicates that team $i_{m}$ and team $j_{m}$ played at time $t_{m}$, where $t_{1}\leq t_{2}\leq\ldots\leq t_{M}$. The result of the $m$-th match can be expressed in a $N\times N$ data matrix $X^{(m)}$ as follows: $X_{ij}^{(m)}=\begin{cases}\begin{split}&\mathbf{1}(i\text{ defeats }j\text{ at }t_{m})\\\ &\sim\text{Bernoulli}(p_{ij}(t_{m}))\end{split}&\text{for }i=i_{m}\text{ and }j=j_{m}\\\ 1-X_{ji}^{(m)}&\text{for }i=j_{m}\text{ and }j=i_{m}\\\ 0&\text{elsewhere}\end{cases}$ (3) Our goal is to estimate the underlying parameters $\bm{\beta}(t)$ where $t\in[0,1]$, and then derive the corresponding global ranking of teams. In order to make the estimation problem tractable we assume that the parameters $\bm{\beta}(t)$ vary smoothly as a function of time $t\in[0,1]$. It is worth noting that the naive strategy of estimating the model parameters separately at each observed discrete time point on the original data will suffer from two major drawbacks: (i) it will in general not guarantee smoothly varying estimates and, perhaps more importantly, (ii) computing the maximum likelihood estimator (MLE) of the parameters in the static Bradley-Terry model may be infeasible due to sparsity in the data (e.g., at each time point we may observe only one match), as we discuss below in 4. To overcome these issues we propose a nonparametric methodology which involves kernel-smoothing the observed data over time. ### 2.2 Estimation Our approach in estimating time-varying global rankings is described in the following three step procedure: 1. 1. Data pre-processing: Kernel smooth the pairwise comparison data across all time periods. This is used to obtain the smoothed pairwise data at each time $t$: $\tilde{X}(t)=\sum_{m=1}^{M}W_{h}(t_{m},t)X^{(m)},$ (4) where $W_{h}$ is an appropriate kernel function with bandwidth $h$, which controls the extent of data smoothing. The higher the value of $h$ is, the smoother $\tilde{X}_{ij}(t)$ is over time. 2. 2. Model fitting: Fit the regular Bradley-Terry model on the smoothed data $\tilde{X}_{ij}(t)$. The model estimates the performance of each team at time $t$ using the estimated score vector $\widehat{\bm{\beta}}(t)={\arg\min}_{\bm{\beta}:\sum_{i}\beta_{i}(t)=0}\widehat{\mathcal{R}}(\bm{\beta};t)$ (5) minimizing the negative log-likelihood risk $\widehat{\mathcal{R}}(\bm{\beta};t)=\sum_{i,j:i\neq j}\frac{\tilde{X}_{ij}(t)\log(1+\exp(\beta_{j}(t)-\beta_{i}(t)))}{\sum_{i,j:i\neq j}\tilde{X}_{ij}(t)}$ (6) 3. 3. Derive global rankings: Rank each team at time $t$ by its score from $\hat{\bm{\beta}}(t)$. We observe that if $t=t_{1}=t_{2}=\dots=t_{M}$ then step 1 reduces to the original (static) Bradley-Terry model. In this case, $\begin{split}\tilde{X}_{ij}(t)&=W_{h}(t,t)\sum_{m=1}^{M}\mathbf{1}(i_{m}=i,j_{m}=j)\\\ &=W_{h}(t,t)X_{ij}\end{split}$ (7) where $X_{ij}=\\#\\{i\text{ defeated }j\\}$. Thus, fitting the model on $\tilde{X}(t)$ in step 2 is equivalent to the original method on data $X$. In this sense our proposed ranking approach represents a time-varying generalization of the original Bradley-Terry model. This data pre-processing is a critical step in our method and is similar to the approach adopted in (Zhou et al.,, 2010) where it was used in the context of estimating smoothed time varying undirected graphs. This approach has two main advantages. First, applying kernel smoothing on the input pairwise comparison data enables borrowing of information across timepoints. In sparse settings, this reduces the data requirement at each time point to meet necessary and sufficient conditions required for the Bradley-Terry model to have a unique solution as detailed in Section 4. Second, kernel smoothing is computationally efficient in a single dimension and is readily available in open source scientific libraries. ## 3 Our Contributions Our main contributions in this paper are summarized as follows: 1. 1. We obtain necessary and sufficient conditions for the existence and uniqueness for our time-varying estimator $\widehat{\bm{\beta}}(t)$ for each $t\in[0,1]$ simulataneously. See Section 4. 2. 2. We extend the results of Simons and Yao, (1999) and obtain statistical guarantees for our proposed method in the form of convergence results of the estimated model parameters uniformly over all times. We express such guarantees in the form of oracle inequalities in the model-agnostic setting where the Bradley-Terry model is not necessarily the true data generating process. See Section 5. 3. 3. We apply our estimator with an data-driven tuned (by LOOCV) hyperparameter successfully to simulations and to real life applications including a comparison to 5 seasons of NFL ELO ratings. See Section 6 and Section 7. ## 4 Existence and uniqueness of solution The existence and uniqueness of solutions for model (5) is not guaranteed in general. This is an innate property of the original Bradley-Terry model (Bradley and Terry,, 1952). As pointed out in (Ford,, 1957) existence of the MLE for the Bradley-Terry model parameters demands a sufficient amount of pairwise comparison data so that there is enough information of relative performance between any pair of two teams for parameter estimation purposes. For example, if there is a team which has never been compared to the others, there is no information which the model can exploit to assign a score for the team. As such its derived rank could be arbitrary. In addition if there are several teams which have never outperformed the others then the Bradley-Terry model would assign negative infinity for the performance of these teams. It would not be possible to compare amongst them for global ranking purposes. In all such cases, the model parameters are not estimable. Ford, (1957) derived the necessary and sufficient condition for the existence and uniqueness of the MLE in the original Bradley-Terry model. Below we show how this condition can also be adapted to guarantee the existence and uniqueness of the solution in our time-varying Bradley-Terry model. The condition can be stated for each time $t$ in terms of the corresponding kernel-smoothed data $\tilde{X}(t)$ as follows: ###### Condition 4.1. In every possible partition of the teams into two nonempty subsets, some team $i$ in the first set and some team $j$ in the second set satisfy $\tilde{X}_{ij}(t)>0$. Or equivalently, for each ordered pair $(i,j)$, there exists a sequence of indices $i_{0}=i,i_{1},\dots,i_{n}=j$ such that $\tilde{X}_{i_{k-1}i_{k}}(t)>0$ for $k=1,\dots,n$. ###### Remark 1. If we regard $[|\tilde{X}(t)|_{ij}]$ as the adjacency matrix of a (weighted) directed graph, then 4.1 is equivalent to the strong connectivity of the graph. Under condition 4.1 we obtain the following existence and uniqueness theorem on the solution set of the time-varying Bradley-Terry model. ###### Theorem 4.1. If the smoothed data $\tilde{X}(t)$ satisfies Condition 4.1, then the solution of (5) uniquely exists at time $t$. Hence, in the proposed time-varying Bradley-Terry model we do not require the strong conditions of (Ford,, 1957) to be met at each time point, but simply require the aggregated conditions in Theorem 4.1 to hold. This is a significant weakening of the data requirement. For example, even if one team did not play any game in a match round – a situation that would preclude the MLE in the standard Bradley-Terry model – it is still possible to assign a rank to this team in their missing round, as long a game is recorded in another round (with at least one win and one loss). In this sense, the kernel- smoothing of the data in our time-varying Bradley-Terry model reduces the required sample complexity for a unique solution. ## 5 Statistical Properties of the Time-varying Bradley-Terry Model ### 5.1 Preliminaries Existing results (Simons and Yao,, 1999; Negahban et al.,, 2017) demonstrate the consistency of the estimated static Bradley-Terry scores provided that the data were generated from the Bradley-Terry model. However this assumption may be too restrictive for data generation processes in real world applications. In the rest of this section, we will consider model-agnostic time-varying settings where the Bradley-Terry model is not necessarily the true pairwise data generating model. In order to investigate the statistical properties of the proposed estimator, we impose the following relatively mild assumptions. ###### Assumption 5.1. Each pair of teams $(i,j)$ play $T^{(i,j)}$ times at time points $\\{t_{k}^{(i,j)},k=1,2,\dots,T^{(i,j)}\\}$, where each $T^{(i,j)}>0$ satisfy the following conditions, for fixed constants $T>0$ and $0<D_{m}\leq 1\leq D_{M}$: 1. 1. $T^{(i,j)}>T$ ; 2. 2. for every interval $(a,b)\subset[0,1]$, $\begin{split}\lfloor D_{m}(b-a)T^{(i,j)}\rfloor\leq&\lvert\\{k:t_{k}^{(i,j)}\in(a,b)\\}\rvert\\\ \leq&\lceil D_{M}(b-a)T^{(i,j)}\rceil.\end{split}$ (8) We remark that the second condition further implies that $\begin{split}&t_{1}^{(i,j)}\leq\frac{1}{D_{m}T^{(i,j)}},\enspace t_{T^{(i,j)}}^{(i,j)}\geq 1-\frac{1}{D_{m}T^{(i,j)}},\\\ &\frac{1}{D_{M}T^{(i,j)}}\leq t_{k+1}^{(i,j)}-t_{k}^{(i,j)}\leq\frac{1}{D_{m}T^{(i,j)}}\end{split}$ (9) for $k=1,2,\dots,T^{(i,j)}-1$. 5.1 allows for different team pairs to play against each other a different number of times and for the game times to be spaced irregularly, though in a controlled manner. To enable statistical analyses on time-varying quantities, we further require that the winning probabilities satisfy a minimal degree of smoothness and that their rate of decay is controlled. ###### Assumption 5.2. For any $i,j$, the function $t\in[0,1]\mapsto p_{ij}(t)$ is Lipschitz with universal constant $L_{p}$ and uniformly bounded below $p_{\min}>0$ which is dependent to $N$ and $T$. The quantity $p_{\min}$ need not be bounded away from $0$ as function of $T$ and $N$. However, in order to guarantee estimability of the model parameters in time-varying settings, we will need to control the rate at which it is allowed to vanish. See Theorem 5.1 below. Finally, we assume that the kernel used to smooth over time satisfy the following regularity conditions, which are quite standard in the nonparametric literature. ###### Assumption 5.3. The kernel function $W:(-\infty,\infty)\rightarrow(0,\infty)$ is a symmetric function such that $\begin{split}\int_{-\infty}^{\infty}W(x)dx=1,&\enspace\int_{-\infty}^{\infty}|x|W(x)dx<\infty\\\ \mathcal{V}(W)<\infty,&\enspace\mathcal{V}(|\cdot|W)<\infty\\\ \end{split}$ (10) where $\mathcal{V}(f(x))$ is the total variation of a function $f$. For each $s,t\in[0,1]$ we further write $W_{h}(s,t)=\frac{1}{h}W\left(\frac{s-t}{h}\right).$ (11) It is easy to see that these conditions imply $\|W\|_{\infty}=\sup_{x}W(x)<\infty.$ (12) Thus, without loss of generality, we assume that $\|W\|_{\infty}\leq 1$; the general case can be handled by modifying the constants accordingly. The use of kernels satisfying the above assumptions is standard in nonparametric problems involving Hölder continuous functions of order $1$, such as the winning probabilities function of Assumption 5.2. ### 5.2 Existence and uniqueness of solutions Simons and Yao, (1999) showed that the necessary and sufficient condition for the existence and uniqueness of the MLE in the original Bradley-Terry model is satisfied asymptotically almost surely under minimal assumptions. Below, we show that this type of result can be extended to our more general time-varying settings. ###### Theorem 5.1. $\begin{split}&\mathbb{P}(\text{Condition~{}\ref{cond:nec_suff_bt_1} is satisfied at every }t)\\\ &\geq 1-4N\exp\left(-\frac{NT}{2}p_{\text{min}}\right).\end{split}$ (13) ###### Remark 2. As we remarked above, $p_{\min}$ needs not be bounded away from zero, but is allowed to vanish slowly enough in relation to $N$ and $T$ so that condition (5.1) is fulfilled as long as $\frac{1}{p_{\text{min}}}=o\left(\frac{NT}{2\log N}\right)$. ### 5.3 Oracle Properties In our general agnostic time-varying setting the Bradley-Terry model is not assumed to be the true data generating process. It follows that, for each $t$, there is no true parameter to which to compare the estimator defined in (5). Instead, we may compare it to the projection parameter $\bm{\beta}^{*}(t)\in\mathbb{R}^{N}$, which is the best approximation to the winning probabilities at time $t$ using the dynamic Bradley Terry model; see (2). In detail, the oracle parameter is defined as $\bm{\beta}^{*}(t)={\arg\min}_{\bm{\beta}:\sum_{i}\beta_{i}(t)=0}\mathcal{R}(\bm{\beta};t)$ (14) where $\mathcal{R}(\bm{\beta};t)=\frac{1}{\binom{N}{2}}\sum_{i,j:i\neq j}p_{ij}(t)\log(1+\exp(\beta_{j}(t)-\beta_{i}(t)))$ (15) We note that, when the winning probabilities obey a Bradley-Terry model, the projection parameter corresponds to the true model parameters. Next, for each fixed time $t\in(0,1)$ and $h>0$, we introduce two quantities, namely $M(t)$ and $\delta_{h}(t)$, that can be thought of as conditions numbers of sort, affecting directly both the estimation and prediction accuracy of the proposed estimator. In detail, we set $M(t)=\max_{i,j:i\neq j}\exp(\beta_{i}^{*}(t)-\beta_{j}^{*}(t))$ and $\delta_{h}(t)=\max_{i}\sum_{j:j\neq i}\left|\frac{\tilde{T}_{ij}(t)}{\tilde{T}_{i}(t)}-\frac{1}{N-1}\right|.$ where $\tilde{T}_{ij}(t)=\tilde{X}_{ij}(t)+\tilde{X}_{ji}(t)$ and $\tilde{T}_{i}(t)=\sum_{j:j\neq i}\tilde{T}_{ij}(t)$. The ratio $M(t)$ quantifies the maximal discrepancy in winning scores among all possible pairs at time $t$, and, as shown in Simons and Yao, (1999), determines the consistency rate of the MLE in the traditional Bradely-Terry model (see also the comments following 3 below). The quantity $\delta_{h}(t)$ is instead a measure of regularity in how evenly the teams play against each other. In particular $\delta_{h}(t)=0$, for all $t$ and $h$ when there is a constant number of matches among each pair of teams, for each time. Since we allow for the possibility of different number of matches between teams and across time, it becomes necessary to quantity such degree of design irregularity. In order to verify the quality of the proposed estimator $\widehat{\bm{\beta}}(t)$, we will consider high-probability oracle bound on estimation error $\lVert\widehat{\bm{\beta}}(t)-\bm{\beta}^{*}(t)\rVert_{\infty}$. In the following theorems, we present both a point-wise and a uniform in $t$ version of this bound in the asymptotic regime of $T,N\rightarrow\infty$ and under only minimal assumptions on the ground-truth winning probabilities $p_{ij}(t)$’s. ###### Theorem 5.2. Let $\gamma=\gamma(T,N,p_{\min})$ be the probability that 4.1 fails and suppose that the kernel bandwidth is chosen as $\displaystyle h=\max\left\\{\frac{1}{T^{1+\eta}},\left(\frac{36(1-p_{\min})\log N}{C_{s}^{2}D_{m}(N-1)T}\right)^{\frac{1}{3}}\right\\},$ for any $\eta>0$ and some universal constant $C_{s}$ depending only to $D_{m}$, $D_{M}$, and $W$. Then, for each fixed time $t\in(0,1)$ and sufficiently large $N$ and $T$, $\begin{split}&\|\hat{\bm{\beta}}(t)-\bm{\beta}^{*}(t)\|_{\infty}\leq 48M(t)\left(\begin{split}&\delta_{h}(t)+C_{s}h\end{split}\right)\end{split}$ (16) with probability at least $1-\frac{2}{N}-\gamma$ as long as the right hand side is smaller than $\frac{1}{3}$. Next, we strengthen our previous result, which is valid for each fixed time $t$, to a uniform guarantee over the entire time course. ###### Theorem 5.3. Let $\gamma=\gamma(T,N,p_{\min})$ be the probability that 4.1 fails and suppose that the kernel bandwidth is chosen as $\displaystyle h=\max\left\\{\frac{1}{T^{1+\eta}},\left(\frac{36(1-p_{\min})\log(NT^{3+3\eta})}{C_{s}^{2}D_{m}(N-1)T}\right)^{\frac{1}{3}}\right\\}$ and that $W$ is $L_{W}$-Lipschitz. Then, for sufficiently large $N$ and $T$, $\begin{split}&\sup_{t\in[0,1]}\|\hat{\bm{\beta}}(t)-\bm{\beta}(t)^{*}\|_{\infty}\leq 48\sup_{t\in[0,1]}M(t)\left(\begin{split}&\delta_{h}(t)+C_{s}h\end{split}\right)\end{split}$ (17) with probability at least $1-\frac{2h^{3}}{N}-\gamma$ as long as the right hand side is smaller than $\frac{1}{3}$. ###### Remark 3. The rate of point-wise convergence for the estimation error implied by the previous result is $\begin{split}&\begin{split}&M(t)\left(\delta_{h}(t)+\max\left\\{\frac{1}{T^{1+\eta}},\left(\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{\log N}{NT}\,}$}\lower 0.4pt\hbox{\vrule height=12.30554pt,depth=-9.84448pt}}}{{\hbox{$\textstyle\sqrt{\frac{\log N}{NT}\,}$}\lower 0.4pt\hbox{\vrule height=8.61386pt,depth=-6.89113pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{\log N}{NT}\,}$}\lower 0.4pt\hbox{\vrule height=6.15276pt,depth=-4.92223pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{\log N}{NT}\,}$}\lower 0.4pt\hbox{\vrule height=6.15276pt,depth=-4.92223pt}}}\right)^{\frac{2}{3}}\right\\}\right)\end{split},\end{split}$ (18) while the rate for uniform convergence is $\begin{split}&\begin{split}&M(t)\left(\delta_{h}(t)+\max\left\\{\frac{1}{T^{\frac{1}{2}+\eta}},\left(\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{\log(NT^{1+\eta})}{NT}\,}$}\lower 0.4pt\hbox{\vrule height=14.55441pt,depth=-11.64359pt}}}{{\hbox{$\textstyle\sqrt{\frac{\log(NT^{1+\eta})}{NT}\,}$}\lower 0.4pt\hbox{\vrule height=10.21387pt,depth=-8.17113pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{\log(NT^{1+\eta})}{NT}\,}$}\lower 0.4pt\hbox{\vrule height=7.66386pt,depth=-6.13112pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{\log(NT^{1+\eta})}{NT}\,}$}\lower 0.4pt\hbox{\vrule height=7.66386pt,depth=-6.13112pt}}}\right)^{\frac{2}{3}}\right\\}\right)\end{split}.\end{split}$ (19) Importantly, as we can see in the previous results, the proposed time varying estimator $\hat{\bm{\beta}}(t)$ is consistent only provided that the design regularity parameter $\delta_{h}(t)$ goes to zero. Of course, if all teams play each other a constant number of times, then $\delta_{h}(t)=0$ automatically. In general, however, the impact of the design on the estimation accuracy needs to be assessed on a case-by-case basis. The rate (18) should be compared with the convergence rate to the true parameters under the static Bradley Terry model, which (Simons and Yao,, 1999) show to be $O_{p}(\max_{i,j:i\neq j}\exp(\beta_{i}^{*}-\beta_{j}^{*})\mathchoice{{\hbox{$\displaystyle\sqrt{\log N/NT\,}$}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{\log N/NT\,}$}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{\log N/NT\,}$}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\log N/NT\,}$}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}})$. Thus, not surprisingly, in the more challenging dynamic settings with smoothly varying winning probabilities the estimation accuracy decreases. The exponent of $\frac{2}{3}$ in the rate (18) matches the familiar rate for estimating Hölder continuous function of order $1$. From (18) and (19) we observe that the desired oracle property on estimated parameters requires rate constraints on $M(t)$. These constraints appear to be strong assumptions without a direct connection to $p_{ij}(t)$’s in our model- agnostic setting. Instead, we circumvent this issue by introducing a more interpretable condition number $K$ (or $p_{\min}$), dependent on $N,T$, given by $\displaystyle K=$ $\displaystyle\exp\left(\frac{1}{p_{\min}}\right)$ and proving that for each fixed time $t\in(0,1)$ our desired oracle property follows from a bound on $M(t)$. ###### Theorem 5.4. Under the conditions in Theorem 5.2 $\begin{split}&\|\hat{\bm{\beta}}(t)-\bm{\beta}^{*}(t)\|_{\infty}\leq 72K\left(\begin{split}&\delta_{h}(t)+C_{s}h\end{split}\right)\end{split}$ (20) and under the conditions Theorem 5.3 $\begin{split}&\sup_{t\in[0,1]}\|\hat{\bm{\beta}}(t)-\bm{\beta}(t)^{*}\|_{\infty}\leq 72K\sup_{t\in[0,1]}\left(\begin{split}&\delta_{h}(t)+C_{s}h\end{split}\right)\end{split}$ (21) with probability at least $1-\frac{2}{N}-\gamma$. ###### Remark 4. We note that assuming $p_{\text{min}}$ to be bounded away from $0$ ensures $\sup_{t\in[0,1]}\|\hat{\bm{\beta}}(t)-\bm{\beta}^{*}(t)\|_{\infty}\to 0$, with high probability. This assumption means that no team is uniformly dominated by or dominates others (since this implies $1-p_{\text{min}}(t)$ is bounded away from 1). This is a reasonable assumption in real-world data such as sports match histories where teams are screened to be competitive with each other. Therefore it is reasonable to only consider matches between teams that do not have vastly different skills, which is reflected in winning probabilities that are bounded away from $\\{0,1\\}$. In summary, our proposed method achieves high-probability oracle bounds on the estimation error in our general model-agnostic time-varying setting. We provide the proofs for the stated theorems in Appendix Section 9.1. ## 6 Experiments We compare our method with some other methods on synthetic data***Code available at https://github.com/shamindras/bttv-aistats2020. We consider both cases where the pairwise comparison data are generated from the Bradley-Terry model and from a different, nonparametric model. ### 6.1 Bradley-Terry Model as the True Model First we conduct simulation experiments in which the Bradley-Terry model is the true model. Given the number of teams $N$ and the number of time points $M$, the synthetic data generation process is as follows: 1. 1. For $i\in[N]$, simulate $\bm{\beta}_{i}\in\mathbb{R}^{M}$ as described below; 2. 2. For $1\leq i<j\leq N$ and $t\in[M]$, set $n_{ij}{(t)}$ and simulate $X{(t)}$ by $x_{ij}{(t)}\sim\text{Binom}\Big{(}n_{ij}{(t)},{1}/\\{(1+\exp[\beta_{j}{(t)}-\beta_{i}{(t)}]\\}\Big{)}$ and $x_{ji}{(t)}=n_{ij}{(t)}-x_{ij}{(t)}$. For each $i\in[N]$, we generate $\bm{\beta}_{i}\in\mathbb{R}^{M}$ from a Gaussian process ${\rm GP}({\mu}_{i}(t),\sigma_{i}(t,s))$ as follows: 1. 1. Set the values of the mean process ${\mu}_{i}(t)$ for all $t\in[M]$ and get mean vector $\bm{\mu}_{i}=(\mu_{i}(1),\ldots,\mu_{i}(M))$; 2. 2. Set the values of the variance process $\sigma_{i}(t,s)$ at $(s,t)\in[M]^{2}$, to derive $\Sigma_{i}\in\mathbb{R}^{M\times M}$; 3. 3. Generate a sample $\bm{\beta}_{i}$ from ${\rm Normal}(\bm{\mu}_{i},\Sigma_{i})$. In our experiment, we generate the parameter $\bm{\beta}$ via a Gaussian process for $N=50$, $M=50$ and $n_{ij}(t)=1$ for all $t$. See appendix for full details. We compare the true $\bm{\beta}$, the win rate, and $\hat{\bm{\beta}}$ by different methods in Fig. 1. Figure 1: Comparison of $\bm{\beta}$ and different estimators. First row: true $\bm{\beta}$ (left), $\hat{\bm{\beta}}$ with our dynamic BT (right); second row: win rate (left), original BT (right). [HTML]FFFFC7 Estimator | Rank Diff | LOO Prob | LOO nll ---|---|---|--- [HTML]DAE8FCWin Rate | 3.75 | 0.44 | - [HTML]DAE8FCOriginal BT | 3.75 | 0.37 | 0.56 [HTML]DAE8FCDynamic BT | 2.29 | 0.37 | 0.55 Table 1: Comparison of Different estimators with results based on 20 repeats. Rank Diff. is the average absolute difference between the estimated and true rankings of teams. LOO Prob means average leave-one-out prediction error of win/loss. LOO nll means average leave-one-out negative log-likelihood. We use LOOCV to select the kernel parameter $h$. The CV curve can be found in Section 6 of Appendix. In this relatively sparse simulated data we see that our dynamic Bradley-Terry estimator $\hat{\bm{\beta}}$ recovers the comprehensive global ranking of each team in line with the original Bradley- Terry model. We also observe that due to the kernel-smoothing that our estimator has relatively more stable paths over time. Table 1 compares the three estimators across key metrics. The results are averaged over 20 repeats. As expected, in this sparse data setting, our dynamic Bradley-Terry method performs better than the original Bradley-Terry model. ### 6.2 Model-Agnostic Setting In our second experiment we adopt a model-agnostic setting where we assume that Bradley-Terry model is not necessarily the true data generating model, as described in Section 5.1. With the same notation, for $1\leq i<j\leq N$ and $t\in[M]$ we first simulate $p_{ij}(t)$, and then set $n_{ij}{(t)}$ and simulate $X{(t)}$ by $x_{ij}{(t)}\sim\text{Binom}\Big{(}n_{ij}{(t)},p_{ij}(t)\Big{)}$ and $x_{ji}{(t)}=n_{ij}{(t)}-x_{ij}{(t)}$. To generate a smoothly changing $p_{ij}(t)$, we again use Gaussian process. Specifically, first we generate $p_{ij}(t)$ for $1\leq i<j\leq N$ and $t\in[M]$ from a Gaussian process. Then we scale those $p_{ij}(t)$’s uniformly to make the values fall within a range $[p_{l},p_{u}]$. In our experiment we set $M=50$, $N=50$, $[p_{l},p_{u}]=[0.05,0.95]$ and $n_{ij}(t)=1$ for all $t$. The projection parameter $\bm{\beta}^{*}$, the win rate, and $\widehat{\bm{\beta}}$ by different methods are compared in Fig. 2. Again the kernel parameter $h$ in our model is selected by LOOCV. Figure 2: Comparison of $\bm{\beta}^{*}$ and different estimators when the underlying model is not the Bradley-Terry model. First row: projection $\bm{\beta}^{*}$ (left), our dynamic BT (right); second row: win rate (left), original BT (right). [HTML]FFFFC7 Estimator | Rank Diff | LOO Prob | LOO nll ---|---|---|--- [HTML]DAE8FCWin Rate | 10.68 | 0.49 | - [HTML]DAE8FCOriginal BT | 10.70 | 0.49 | 0.71 [HTML]DAE8FCDynamic BT | 5.48 | 0.49 | 0.68 Table 2: Comparison of different estimators when the underlying model is not the Bradley-Terry model. By comparing curves in Fig. 2, we note that our estimator $\hat{\bm{\beta}}$ recovers the global rankings better than the win rate and the original Bradley-Terry model, and produces relatively more stable paths over time. The same conclusion is confirmed by Table 2, which compares the three estimators in some metrics with 20 repetitions. ###### Remark 5. In this sparse data setting where $n_{ij}(t)$ is fairly small, the original Bradley-Terry model performs worse than our model for two reasons: 1. 4.1 can fail to hold at some time points, whence the MLE does not exist; 2. even when the MLE exists, it can fluctuate significantly over time because of the relatively small sample size at each time point. As we show in Section 9.3.4 in the appendix, when $M=50$, $N=50$ and $n_{ij}(t)=1$, the MLE exists with fairly high frequency. Still, our model performs much better than the original Bradley-Terry model. ###### Remark 6. Since our method is aimed at obtaining accurate estimates of smoothly changing beta/rankings with strong prediction power, it may fail to capture some changes in rankings, especially when these changes are relatively small (as in the present case). However the winrate and original Bradley-Terry methods perform much worse, as they appear to miss some true ranking changes while returning many false change points. Additional details about experiments, including running time efficiency in simulated settings, can be found in the Appendix Section 9.3. rank | 2011 | 2012 | 2013 | 2014 | 2015 ---|---|---|---|---|--- ELO | BT | ELO | BT | ELO | BT | ELO | BT | ELO | BT 1 | blue!25GB | blue!25GB | yellow!25NE | yellow!25HOU | blue!25SEA | blue!25SEA | yellow!25SEA | yellow!25DEN | yellow!25SEA | yellow!25CAR 2 | yellow!25NE | yellow!25SF | yellow!25DEN | yellow!25ATL | yellow!25SF | yellow!25DEN | yellow!25NE | yellow!25ARI | yellow!25CAR | yellow!25DEN 3 | blue!25NO | blue!25NO | yellow!25GB | yellow!25SF | yellow!25NE | yellow!25 NO | yellow!25DEN | yellow!25NE | yellow!25ARI | yellow!25NE 4 | yellow!25PIT | yellow!25NE | yellow!25SF | CHI | yellow!25DEN | yellow!25KC | yellow!25GB | yellow!25SEA | yellow!25KC | yellow!25CIN 5 | yellow!25BAL | yellow!25DET | yellow!25ATL | yellow!25GB | yellow!25CAR | yellow!25SF | blue!25DAL | blue!25DAL | yellow!25DEN | yellow!25ARI 6 | yellow!25SF | yellow!25BAL | yellow!25SEA | yellow!25NE | yellow!25CIN | yellow!25NE | PIT | yellow!25GB | yellow!25NE | yellow!25GB 7 | yellow!25ATL | yellow!25PIT | NYG | yellow!25DEN | yellow!25NO | yellow!25IND | BAL | PHI | yellow!25PIT | yellow!25MIN 8 | PHI | yellow!25HOU | CIN | yellow!25SEA | yellow!25ARI | yellow!25CAR | IND | SD | yellow!25CIN | yellow!25KC 9 | yellow!25SD | CHI | blue!25BAL | blue!25BAL | yellow!25IND | yellow!25ARI | yellow!25ARI | DET | yellow!25GB | yellow!25PIT 10 | yellow!25HOU | yellow!25ATL | yellow!25HOU | IND | yellow!25SD | yellow!25CIN | CIN | KC | yellow!25MIN | yellow!25SEA Av. Diff. | 4.2 | 5.0 | 3.5 | 4.3 | 3.4 Table 3: BT within season vs. ELO NFL top 10 rankings. Blue: perfect match, yellow: top 10 match. Our dynamic BT model is fitted on 16 rounds of each season, and the ranking of a season is based on the ranking at the last round. ## 7 Application - NFL Data In order to test our model in practical settings we consider ranking National Football League (NFL) teams over multiple seasons. Specifically we source 5 seasons of openly available NFL data from 2011-2015 (inclusive) using the nflWAR package (Yurko et al.,, 2018). Each NFL season is comprised of $N=32$ teams playing $M=16$ games each over the season. This means that at each point in time $t$ the pairwise comparison matrix based on scores across all 32 teams is sparsely populated with only 16 entries. We fit our time-varying Bradley Terry estimator over all 16 rounds in the season using a standard Gaussian Kernel and tune $h$ using the LOOCV approach described in section 9.2. In order to gauge whether the rankings produced by our model are reasonable we compare our season-ending rankings (fit over all games played in that season) with the relevant openly available NFL ELO ratings (Paine,, 2015). The top 10 season-ending rankings from each method across NFL seasons 2011-2015 are summarized in Table 3. Based on Table 3 we observe that we roughly match between 6 to 10 of the top 10 ELO teams consistently over all 5 seasons. There is often misalignment with specific ranking values across both ranking methods. We note that the unlike our method, the NFL ELO rankings use pairwise match data and also additional features including an adjustment for margin of victory. This demonstrates an advantage of our model in only requiring the minimal time-varying pairwise match data and smoothness assumptions to deliver comparable results to this more feature rich ELO ranking method. Furthermore, since our model aggregates data across time it can provide a reasonable minimalist ranking benchmark in modern sparse time-varying data settings with limited “expert knowledge” e.g. e-sports. ## 8 Conclusion We propose a time-varying generalization of the Bradley-Terry model that captures temporal dependencies in a nonparametric fashion. This enables the modeling of dynamic global rankings of distinct teams in settings in which the parameteres of the ordinary Bradley Terry model would not be estimable. From a theoretical perspective we adapt the results of (Ford,, 1957) to obtain the necessary and sufficient condition for the existence and uniqueness of our Bradley-Terry estimator in the time-varying setting. We extend the previous analysis of (Simons and Yao,, 1999) to derive oracle inequalities on for our proposed method for both the estimation error and the excess risk under smoothness conditions on the winning probabilities. The resulting rates of consistency are of nonparametric type. From an implementation perspective we provide a general strategy for tuning the kernel bandwidth hyperparameter using an efficient data-driven approach specific to our unsupervised time-varying setting. Finally, we test the practical effectiveness of our estimator under both simulated and real world settings. In the latter case we separately rank 5 consecutive seasons of open National Football League (NFL) team data (Yurko et al.,, 2018) from 2011-2015. Our NFL ranking results compare favourably to the well-accepted NFL ELO model rankings (Paine,, 2015). We thus view our nonparametric time-varying Bradley- Terry estimator as a useful benchmarking tool for other feature-rich time- varying ranking models since it simply relies on the minimalist time-varying score information for modeling. ## References * Agresti, (2013) Agresti, A. (2013). Categorical data analysis. Wiley Series in Probability and Statistics. 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Learn., 80(2-3):295–319. ## 9 Appendices ### 9.1 Proofs of Theorems #### 9.1.1 Proof of Theorem 4.1 ##### Uniqueness of the solution The elements of the Hessian for $\hat{\mathcal{R}}(\bm{\beta};t)$ in (6) are: $\begin{split}H(\hat{\mathcal{R}})_{ii}=&\sum_{j:j\neq i}(\tilde{X}_{ij}(t)+\tilde{X}_{ji}(t))\frac{\exp{\beta_{i}}\exp{\beta_{j}}}{(\exp{\beta_{i}}+\exp{\beta_{j}})^{2}}\\\ H(\hat{\mathcal{R}})_{ij}=&-(\tilde{X}_{ij}(t)+\tilde{X}_{ji}(t))\frac{\exp{\beta_{i}}\exp{\beta_{j}}}{(\exp{\beta_{i}}+\exp{\beta_{j}})^{2}}\end{split}$ (22) Note that the Hessian has positive diagonal elements, non-positive off- diagonal elements, and zero column sums. With Condition 4.1, this implies that the Hessian can be regarded as a graph Laplacian for a connected graph. Following the classical proof of the property of graph Laplacian (von Luxburg,, 2007), $v^{T}H(\hat{\mathcal{R}})v=\sum_{i<j}\frac{|\tilde{X}_{ij}(t)+\tilde{X}_{ji}(t)|}{2}(v_{i}-v_{j})^{2}\geq 0$ (23) Then, Condition 4.1 guarantees that “=” is achieved if and only if $v=c\mathbf{1}$. This proves the uniqueness up to constant shifts. ##### Existence of solution Plugging in $\bm{\beta}=\mathbf{0}$, we get an upperbound for the minimum loss function $\hat{\mathcal{R}}^{\star}(t):=\hat{\mathcal{R}}(\hat{\bm{\beta}};t)$: $\hat{\mathcal{R}}^{\star}(t)\leq\log 2$ (24) As $\hat{\mathcal{R}}(\bm{\beta};t)$ is continuous with respect to $\bm{\beta}$, it suffices to show that the level set of $\hat{\mathcal{R}}(\cdot;t)$ at $\log 2$ within $\\{\bm{\beta}:\sum_{i=1}^{N}\beta_{i}=0\\}$ is bounded so that it is compact. Suppose that $\bm{\beta}$ is in the intersection between the levelset and $\\{\bm{\beta}:\sum_{i=1}^{N}\beta_{i}=0\\}$. Since each summand of $\hat{\mathcal{R}}(\bm{\beta})$ in (6) is non-negative, i.e., $\frac{\tilde{X}_{ij}(t)}{\sum_{i^{\prime},j^{\prime}:i^{\prime}\neq j^{\prime}}\tilde{X}_{i^{\prime}j^{\prime}}(t)}\log(1+\exp(\beta_{j}-\beta_{i}))\geq 0$ (25) if $i$ and $j$ satisfies $\tilde{X}_{ij}(t)>0$ then the corresponding summand should be smaller than $\log 2$ so that: $\begin{split}\beta_{j}-\beta_{i}\leq&\log(1+\exp(\beta_{j}-\beta_{i}))\\\ \leq&\log 2\frac{\sum_{i^{\prime},j^{\prime}:i^{\prime}\neq j^{\prime}}\tilde{X}_{i^{\prime}j^{\prime}}(t)}{\tilde{X}_{ij}(t)}\end{split}$ (26) By Condition 4.1, for any distinct $i$ and $j$, there exists an index sequence $(i=i_{0},i_{1},\dots,i_{n}=j$ such that $X_{i_{k-1}i_{k}}>0$ for $k=1,2,\dots,n$. Hence, $\begin{split}\beta_{j}-\beta_{i}\leq&\log 2\sum_{k=1}^{n}\frac{\sum_{i^{\prime},j^{\prime}:i^{\prime}\neq j^{\prime}}\tilde{X}_{i^{\prime}j^{\prime}}(t)}{\tilde{X}_{i_{k-1}i_{k}}(t)}\\\ \leq&\log 2\sum_{i^{\prime},j^{\prime}:i^{\prime}\neq j^{\prime}}\tilde{X}_{i^{\prime}j^{\prime}}(t)\sum_{i^{\prime},j^{\prime}:i^{\prime}\neq j^{\prime}}\frac{1}{\tilde{X}_{i^{\prime}j^{\prime}}(t)}=:B\end{split}$ (27) where $B\in(0,\infty)$. In sum, $\|\bm{\beta}\|_{\infty}\leq\max_{i,j:i\neq j}|\beta_{i}-\beta_{j}|\leq B$ (28) and this proves the existence part of the theorem. #### 9.1.2 Proof of Theorem 5.1 The proof of this theorem is based on the proof of Lemma 1 in Simons and Yao, (1999). Since the kernel function $W$ in Assumption 5.3 has support $(-\infty,\infty)$, $\tilde{X}_{ij}(t)>0$ if and only if team $i$ defeated team $j$ at least once any time. In other words, if Condition 4.1 holds for at least one time point, then so it does for every time point. Here, we prove that the probability of Condition 4.1 to hold at at least one time point converge to $1$ as $N,T\rightarrow\infty$. Given $p_{\text{min}}$ instead of $\max_{i,j:i\neq j}\exp(\beta^{*}_{i}-\beta^{*}_{j})$, the probability of the event $\mathcal{S}$ that no team in a subset $S$ loses against a team not of $S$ is no larger than $(1-p_{\text{min}})^{|S|(N-|S|-1)T}$ (29) Hence, we bound the probability that data does not meet Condition 4.1 by a union bound $\begin{split}&\mathbb{P}\\!\left(\text{Condition~{}\ref{cond:nec_suff_bt_1} fails}\right)\leq\sum_{S\subset[N]:S\neq\emptyset}\mathbb{P}\\!\left(\mathcal{S}\right)\\\ &\leq\sum_{n=1}^{N-1}\binom{N}{n}(1-p_{\text{min}})^{n(N-n-1)T}\\\ &\leq 2\sum_{n=1}^{\lceil N/2\rceil}\binom{N}{n}(1-p_{\text{min}})^{n(N-n-1)T}\\\ &\leq 2\sum_{n=1}^{\lceil N/2\rceil}\binom{N}{n}(1-p_{\text{min}})^{nNT/2}\\\ &\leq 2\left[(1+(1-p_{\text{min}})^{NT/2})^{N}-1\right]\\\ &\leq 2\left[(1+e^{-\frac{NTp_{\text{min}}}{2}})^{N}-1\right]\\\ &\leq 4Ne^{-\frac{NTp_{\text{min}}}{2}}\end{split}$ (30) as long as $e^{-\frac{NTp_{\text{min}}}{2}}\leq\frac{\log 2}{N}$. We note that $(1+\frac{\log 2}{N})^{N}\leq e^{\log 2}=2\leq 2\log 2+1=2N\frac{\log 2}{N}+1$. Hence, $\begin{split}\mathbb{P}\\!\left(\text{Condition~{}\ref{cond:nec_suff_bt_1} fails}\right)\leq 4Ne^{-\frac{NTp_{\text{min}}}{2}}\end{split}$ (31) as long as $Ne^{-\frac{NTp_{\text{min}}}{2}}\leq\log 2$. Since $Ne^{-\frac{NTp_{\text{min}}}{2}}\geq\log 2$ implies $4Ne^{-\frac{NTp_{\text{min}}}{2}}$ to be larger than $1$, the probability bound holds for any $N$, $T$, and $p_{\text{min}}$. #### 9.1.3 Proof of Theorem 5.2 For readability, in our notation we will omit the dependence on the time point $t$ in the expressions for $\hat{\bm{\beta}}(t)$ and $\bm{\beta}^{*}(t)$, unless required for clarity. In our proofs we rely on the results and arguments of Simons and Yao, (1999) to demonstrate consistency for the maximum likelihood estimator in the static Bradley-Terry model with an increasing number of parameters. In that setting, the authors parametrize the winning probabilities as $p_{i,j}=\frac{u^{*}_{i}}{u^{*}_{i}+u^{*}_{j}}$, where $u^{*}_{i}\equiv\exp(\beta^{*}_{i})$, with $\bm{\beta}^{*}\in\mathbb{R}^{N}$ such that $\beta^{*}_{1}=0$. Then, setting $\Delta u_{i}=\frac{\hat{u}_{i}-u^{*}_{i}}{u^{*}_{i}}$, where $\hat{u}_{i}$ is the MLE of $u^{*}_{i}$ (with $\hat{u}_{1}=0$ by convention), it follows from the proof of Lemma 3 of Simons and Yao, (1999) that $\begin{split}&\max_{i}\frac{\lvert\Delta u_{i}\rvert}{\lvert\Delta u_{i}\rvert+1}\\\ &\leq\frac{8}{N-1}\max_{i,j}\frac{u_{i}^{*}}{u_{j}^{*}}\max_{i}\sum_{j:j\neq i}\left\\{\frac{\hat{u}_{i}}{\hat{u}_{i}+\hat{u}_{j}}-\frac{u^{*}_{i}}{u^{*}_{i}+u^{*}_{j}}\right\\}\\\ \end{split}$ (32) where $u^{*}_{i}=\exp(\beta_{i}^{*})$. Next, the authors derived a high probability upper bound on $\begin{split}&\max_{i}\sum_{j:j\neq i}\left\\{\frac{\hat{u}_{i}}{\hat{u}_{i}+\hat{u}_{j}}-\frac{u^{*}_{i}}{u^{*}_{i}+u^{*}_{j}}\right\\}\end{split}$ (33) using the facts that $\sum_{j:j\neq i}p_{ij}=\sum_{j:j\neq i}\frac{u^{*}_{i}}{u^{*}_{i}+u^{*}_{j}}$ (34) and $\begin{split}\sum_{j:j\neq i}\frac{X_{ij}}{T}=\sum_{j:j\neq i}\frac{\hat{u}_{i}}{\hat{u}_{i}+\hat{u}_{j}},\end{split}$ (35) where $X_{ij}$ is the number of matches in which $i$ defeated $j$. The second identity comes from the first order optimality condition of $\hat{\bm{\beta}}$. In our time-varying setting, however, the subgradient optimality of $\hat{\bm{\beta}}(t)$ for $\hat{\mathcal{R}}(\bm{\beta};t)$ only imply that, for each $j$, $\sum_{j:j\neq i}\tilde{X}_{ij}(t)=\sum_{j:j\neq i}\tilde{T}_{ij}(t)\frac{e^{\hat{\beta}_{i}}}{e^{\hat{\beta}_{j}}+e^{\hat{\beta}_{i}}}.$ (36) Thus, Eq. 35 does not hold in the dynamic setting, due to different $\tilde{X}_{ij}(t)+\tilde{X}_{ji}(t)$ across all $j\neq i$. Instead, we have that $\begin{split}&\frac{1}{N-1}\left(\sum_{j:j\neq i}\frac{\tilde{X}_{ij}(t)}{\tilde{T}_{ij}(t)}-\sum_{j:j\neq i}\frac{e^{\hat{\beta}_{i}}}{e^{\hat{\beta}_{j}}+e^{\hat{\beta}_{i}}}\right)\\\ &=\left(\begin{split}&\sum_{j:j\neq i}\left(\frac{1}{N-1}-\frac{\tilde{T}_{ij}(t)}{\tilde{T}_{i}(t)}\right)\frac{\tilde{X}_{ij}(t)}{\tilde{T}_{ij}(t)}\\\ &+\sum_{j:j\neq i}\left(\frac{\tilde{T}_{ij}(t)}{\tilde{T}_{i}(t)}-\frac{1}{N-1}\right)\frac{e^{\hat{\beta}_{i}}}{e^{\hat{\beta}_{j}}+e^{\hat{\beta}_{i}}}\end{split}\right)\end{split}$ (37) Since $\frac{\tilde{X}_{ij}(t)}{\tilde{T}_{ij}(t)},\frac{e^{\hat{\beta}_{i}}}{e^{\hat{\beta}_{j}}+e^{\hat{\beta}_{i}}}<1$, $\begin{split}&\left|\begin{split}&\frac{1}{N-1}\left(\sum_{j:j\neq i}\frac{\tilde{X}_{ij}(t)}{\tilde{T}_{ij}(t)}-\sum_{j:j\neq i}\frac{e^{\hat{\beta}_{i}}}{e^{\hat{\beta}_{j}}+e^{\hat{\beta}_{i}}}\right)\end{split}\right|\\\ &\leq 2\delta_{h}(t)\end{split}$ (38) and $\begin{split}&\frac{1}{N-1}\sum_{j:j\neq i}\left\\{\frac{e^{\hat{\beta}_{i}}}{e^{\hat{\beta}_{i}}+e^{\hat{\beta}_{j}}}-\frac{e^{\beta^{*}_{i}}}{e^{\beta^{*}_{i}}+e^{\beta^{*}_{j}}}\right\\}\\\ &\leq 2\delta_{h}(t)+\frac{1}{N-1}\sum_{j:j\neq i}\left\\{\frac{\tilde{X}_{ij}(t)}{\tilde{T}_{ij}(t)}-p_{ij}(t)\right\\}.\end{split}$ (39) To make the bias-variance trade-off due to kernel smoothing more explicit, we decompose the term $\begin{split}\sum_{j:j\neq i}\left\\{\frac{\tilde{X}_{ij}(t)}{\tilde{T}_{ij}(t)}-p_{ij}(t)\right\\}\end{split}$ (40) as $\begin{split}&\sum_{j:j\neq i}\left(\begin{split}&\frac{\sum_{k}W_{h}(t_{k},t)(\mathbf{1}_{ij}(t_{k})-p_{ij}(t_{k}))}{\sum_{k}W_{h}(t_{k},t)}\\\ \end{split}\right)\\\ &+\sum_{j:j\neq i}\left(\frac{\sum_{k}W_{h}(t_{k},t)p_{ij}(t_{k})}{\sum_{k}W_{h}(t_{k},t)}-p_{ij}(t)\right)\\\ &=:\Delta^{(var)}_{i}+\Delta^{(bias)}_{i}\end{split}$ (41) where, for brevity, $t_{k}$ and $\mathbf{1}_{ij}(t_{k})$ here stand for $t_{k}^{(i,j)}$ and $\mathbf{1}(i\text{ defeats }j\text{ at }t_{k})$, respectively. For the first term, we have that $\begin{split}&\mathbb{P}\\!\left(\left|\Delta^{(var)}_{i}\right|\geq\epsilon\right)\\\ &=\mathbb{P}\\!\left(\left|\sum_{j:j\neq i}\frac{\sum_{k}W_{h}(t_{k},t)(\mathbf{1}_{ij}(t_{k})-p_{ij}(t_{k}))}{\sum_{k}W_{h}(t_{k},t)}\right|\geq\epsilon\right)\\\ &=\mathbb{P}\\!\left(\begin{split}&\left|\sum_{j,k}hW_{h}(t_{k},t)\frac{s_{\min}}{s_{j}}(\mathbf{1}_{ij}(t_{k})-p_{ij}(t_{k}))\right|\\\ &\geq\epsilon\cdot h\cdot s_{\min}\end{split}\right)\\\ \end{split}$ (42) where $s_{j}=\sum_{k}W_{h}(t_{k},t)$ and $s_{\min}=\min_{j:j\neq i}~{}s_{j}$. Next, $hW_{h}(t_{k},t)\frac{s_{\min}}{s_{j}}=W\left(\frac{t_{k}-t}{h}\right)\frac{s_{\min}}{s_{j}}\leq 1$ and hence that multiplicative Chernoff bound (see, e.g. Raghavan,, 1988) yields that $\begin{split}&\mathbb{P}\\!\left(\left|\Delta^{(var)}_{i}\right|\geq\epsilon\right)\\\ &\leq 2\exp\left(-\frac{(\epsilon\cdot h\cdot s_{\min})^{2}}{3\sum_{j,k}hW_{h}(t_{k},t)\frac{s_{\min}}{s_{j}}p_{ij}(t_{k})}\right)\\\ &\leq 2\exp\left(-\frac{\epsilon^{2}hD_{m}T}{18(N-1)(1-p_{\text{min}})}\right)\end{split}$ (43) for each $i$ as long as $\begin{split}\frac{\epsilon}{\sum_{j,k}\frac{W_{h}(t_{k},t)}{\sum_{k^{\prime}}W_{h}(t_{k^{\prime}},t)}p_{ij}(t_{k})}\leq 1.\end{split}$ (44) This condition holds for $\epsilon\leq p_{\min}$. We note that we have also used the bounds $\frac{1}{6}D_{m}T\leq\sum_{k}W_{h}(t_{k},t)\leq D_{M}T$ (45) for any $i,j$ and sufficiently small $h$, which were shown in Section 9.1.4. Then using the union bound, $\begin{split}&\mathbb{P}\\!\left(\max_{i}\left|\Delta^{(var)}_{i}\right|\geq\epsilon\right)\\\ &\leq 2N\exp\left(-\frac{\epsilon^{2}hD_{m}T}{18(1-p_{\text{min}})(N-1)}\right)\end{split}$ (46) Hence, plugging in $\epsilon=\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=14.075pt,depth=-11.26006pt}}}{{\hbox{$\textstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=9.8611pt,depth=-7.88892pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=7.25238pt,depth=-5.80194pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=7.25238pt,depth=-5.80194pt}}}$, we get that, with probability at least $1-\frac{2}{N}$, $\begin{split}\max_{i}|\Delta^{(var)}_{i}|\leq&\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=14.075pt,depth=-11.26006pt}}}{{\hbox{$\textstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=9.8611pt,depth=-7.88892pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=7.25238pt,depth=-5.80194pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=7.25238pt,depth=-5.80194pt}}}\end{split}$ (47) To handle the deterministic bias terms $\Delta^{(bias)}_{i}$, we rely on the following bound, whose proof is given below in Section 9.1.4. ###### Lemma 9.1. Suppose that 1. 1. $t_{1},t_{2},\dots,t_{T}$ satisfies Eq. 8 and 2. 2. $\frac{1}{T}=o(h)$ as $T\rightarrow\infty$. Then, for a $L_{f}$-Lipschitz function $f:[0,1]\rightarrow\mathbb{R}$, $\begin{split}&\sup_{t\in[0,1]}\left|\sum_{k=1}^{T}\frac{W_{h}(t_{k},t)}{\sum_{k^{\prime}}W_{h}(t_{k^{\prime}},t)}f(t_{k})-f(t)\right|\leq C_{s}h\end{split}$ (48) with a universal constant $C_{s}$ depending only on $D_{m},D_{M},W$ and $L_{f}$. Accordingly, $\begin{split}&\max_{i}|\Delta^{(bias)}_{i}|\\\ &\leq\max_{i}\sum_{j:j\neq i}\left\\{\frac{\sum_{k}W_{h}(t_{k},t)p_{ij}(t_{k})}{\sum_{k}W_{h}(t_{k},t)}-p_{ij}(t)\right\\}\\\ &\leq C_{s}(N-1)h\end{split}$ (49) for some constant $C_{s}$ depending only on $D_{m},D_{M},W$ and $L_{p}$. Thus, combining all the pieces, $\begin{split}&\max_{i}\frac{|e^{\hat{\beta}_{i}-\beta^{*}_{i}}-1|}{|e^{\hat{\beta}_{i}-\beta^{*}_{i}}-1|+1}\\\ &\leq 8M(t)\left(2\delta_{h}(t)+\max_{i}\frac{|\Delta_{i}^{(var)}(t)|+|\Delta_{i}^{(bias)}(t)|}{N-1}\right)\\\ &\leq 8M(t)\left(2\delta_{h}(t)+\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{36(1-p_{\min})\log N}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=15.0pt,depth=-12.00005pt}}}{{\hbox{$\textstyle\sqrt{\frac{36(1-p_{\min})\log N}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=10.5pt,depth=-8.40004pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{36(1-p_{\min})\log N}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=7.58572pt,depth=-6.0686pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{36(1-p_{\min})\log N}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=7.58572pt,depth=-6.0686pt}}}+C_{s}h\right)\end{split}$ (50) with probability at least $1-\frac{2}{N}$ as long as $\epsilon\leq p_{\min}$. Plugging in $h=\max\left\\{\left(\frac{1}{T}\right)^{1+\eta},\left(\frac{36(1-p_{\min})\log N}{C_{s}^{2}D_{m}(N-1)T}\right)^{\frac{1}{3}}\right\\}$ leads to the bound $\begin{split}&\max_{i}\frac{|e^{\hat{\beta}_{i}-\beta^{*}_{i}}-1|}{|e^{\hat{\beta}_{i}-\beta^{*}_{i}}-1|+1}\\\ &\leq 8M(t)\left(\begin{split}&2\delta_{h}(t)+\left(\frac{36C_{s}(1-p_{\min})\log N}{D_{m}(N-1)T}\right)^{\frac{1}{3}}\\\ &+C_{s}h\end{split}\right)\\\ &\leq 16M(t)\left(\begin{split}&\delta_{h}(t)+C_{s}h\end{split}\right)\end{split}$ (51) with probability at least $1-\frac{2}{N}$ when $\epsilon\leq p_{\min}$. We note that, given our choice for $h$, $\epsilon=\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=14.075pt,depth=-11.26006pt}}}{{\hbox{$\textstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=9.8611pt,depth=-7.88892pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=7.25238pt,depth=-5.80194pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log N}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=7.25238pt,depth=-5.80194pt}}}\leq C_{s}h$. Hence, for a sufficiently small $h$, if the right hand side is smaller than, say, $\frac{1}{3}$ then $\begin{split}&\|\hat{\bm{\beta}}-\bm{\beta}^{*}\|_{\infty}\leq 3\max_{i}\frac{|e^{\hat{\beta}_{i}-\beta^{*}_{i}}-1|}{|e^{\hat{\beta}_{i}-\beta^{*}_{i}}-1|+1}\\\ &\leq 48M(t)\left(\begin{split}&\delta_{h}(t)+C_{s}h\end{split}\right)\end{split}$ (52) with probability at least $1-\frac{2}{N}$ since $\frac{|e^{x}-1|}{|e^{x}-1|+1}\geq\frac{|x|}{3}$ for $|x|\leq 1$. #### 9.1.4 Proof of Lemma 9.1 Since $f$ is $L_{f}$-Lipschitz, $\begin{split}&\left|\sum_{k=1}^{T}\frac{W_{h}(t_{k},t)}{\sum_{k^{\prime}}W_{h}(t_{k^{\prime}},t)}f(t_{k})-f(t)\right|\\\ &\leq\sum_{k=1}^{T}\frac{W_{h}(t_{k},t)}{\sum_{k^{\prime}}W_{h}(t_{k^{\prime}},t)}|f(t_{k})-f(t)|\\\ &\leq L_{f}\sum_{k=1}^{T}\frac{W_{h}(t_{k},t)}{\sum_{k^{\prime}}W_{h}(t_{k^{\prime}},t)}|t_{k}-t|.\end{split}$ (53) Let $I_{1}=\left[0,\frac{t_{1}+t_{2}}{2}\right],I_{2}=\left[\frac{t_{1}+t_{2}}{2},\frac{t_{2}+t_{3}}{2}\right],\dots,I_{T}=\left[\frac{t_{T-1}+t_{T}}{2},1\right]$ and $l_{k}$ be the length of $I_{k}$. We note that $\frac{1}{D_{M}T}\leq l_{k}\leq\frac{2}{D_{m}T}$ by Eq. 9. Then, $\begin{split}&\int_{0}^{1}|x-t|W_{h}(x,t)dx=\sum_{k}\int_{I_{k}}|x-t|W_{h}(x,t)\\\ &=\left(\begin{split}&\sum_{k}l_{k}|t_{k}-t|W_{h}(t_{k},t)\\\ &+\sum_{k}\int_{I_{k}}\left(\begin{split}&\left|\frac{x-t}{h}\right|W\left(\frac{x-t}{h}\right)\\\ &-\left|\frac{t_{k}-t}{h}\right|W\left(\frac{t_{k}-t}{h}\right)\end{split}\right)dx\\\ \end{split}\right).\end{split}$ (54) Since $|\cdot|W$ has a finite total variation, $\begin{split}&\sum_{k}\int_{I_{k}}\left|\begin{split}&\left|\frac{x-t}{h}\right|W\left(\frac{x-t}{h}\right)\\\ &-\left|\frac{t_{k}-t}{h}\right|W\left(\frac{t_{k}-t}{h}\right)\end{split}\right|dx\\\ &\leq\sum_{k}\frac{1}{D_{m}T}\sup_{x,y\in I_{k}}\left|\begin{split}&\left|\frac{x-t}{h}\right|W\left(\frac{x-t}{h}\right)\\\ &-\left|\frac{y-t}{h}\right|W\left(\frac{t_{k}-t}{h}\right)\end{split}\right|\\\ &\leq\frac{\mathcal{V}(|\cdot|W)}{D_{m}T}.\end{split}$ (55) Hence, $\begin{split}&\int_{0}^{1}|x-t|W_{h}(x,t)dx\\\ &\geq\left(\begin{split}&\frac{1}{D_{M}T}\sum_{k}|t_{k}-t|W_{h}(t_{k},t)\\\ &-\frac{\mathcal{V}(|\cdot|W)}{D_{m}T}\end{split}\right).\end{split}$ (56) As a result, $\begin{split}&\sum_{k}|t_{k}-t|W_{h}(t_{k},t)\\\ &\leq D_{M}Th\int_{-\infty}^{\infty}|x|W(x)dx+\frac{D_{M}\mathcal{V}(|\cdot|W)}{D_{m}}.\end{split}$ (57) On the other hand, with a similar argument, $\begin{split}&\int_{0}^{1}W_{h}(x,t)dx=\sum_{k}\int_{I_{k}}W_{h}(x,t)\\\ &=\left(\begin{split}&\sum_{k}l_{k}W_{h}(t_{k},t)\\\ &+\sum_{k}\int_{I_{k}}\left(\frac{1}{h}W\left(\frac{x-t}{h}\right)-\frac{1}{h}W\left(\frac{t_{k}-t}{h}\right)\right)dx\\\ \end{split}\right)\\\ &\leq\frac{2}{D_{m}T}\sum_{k}W_{h}(t_{k},t)+\frac{\mathcal{V}(W)}{D_{m}Th},\end{split}$ (58) implying that $\begin{split}\sum_{k}W_{h}(t_{k},t)&\geq\frac{D_{m}T}{2}\int_{-t/h}^{(1-t)/h}W(x)dx-\frac{D_{M}\mathcal{V}(W)}{2D_{m}h}.\\\ \end{split}$ (59) As long as $h\rightarrow 0$ and $\frac{1}{T}=o(h)$, $\inf_{t\in[0,1]}\int_{-t/h}^{(1-t)/h}W(x)dx$ (60) is bounded below from $0$ (in particular, we consider a small enough $h$ so that it is bounded below by, say, $\frac{1}{3}$), and the term $\frac{D_{M}\mathcal{V}(|\cdot|W)}{D_{m}}$ and $\frac{D_{M}\mathcal{V}(W(x)}{2D_{m}h}$ in Eqs. 57 and 59 become asymptotically negligible. As a result, $\begin{split}\sum_{k=1}^{T}\frac{W_{h}(t_{k},t)}{\sum_{k^{\prime}}W_{h}(t_{k^{\prime}},t)}|t_{k}-t|\leq C^{\prime}h\end{split}$ (61) where $C^{\prime}$ is a universal constant depending only on $D_{m}$, $D_{M}$, and $W$, and furthermore $\begin{split}\left|\sum_{k=1}^{T}\frac{W_{h}(t_{k},t)}{\sum_{k^{\prime}}W_{h}(t_{k^{\prime}},t)}f(t_{k})-f(t)\right|\leq C_{s}h\end{split}$ (62) for a univeral constant $C_{s}$ depending only on $D_{m}$, $D_{M}$, $W$, and $L_{f}$. #### 9.1.5 Proof of Theorem 5.3 In Section 9.1.3, we showed that $\begin{split}&\max_{i}\frac{|e^{\hat{\beta}_{i}(t)-\beta^{*}_{i}(t)}-1|}{|e^{\hat{\beta}_{i}(t)-\beta^{*}_{i}(t)}-1|+1}\\\ &\leq 8M(t)\left(2\delta_{h}(t)+\max_{i}\frac{|\Delta_{i}^{(var)}(t)|+|\Delta_{i}^{(bias)}(t)|}{N-1}\right)\\\ \end{split}$ (63) Since the bound for $\Delta^{(bias)}_{i}(t)$ depends only on $D_{m}$, $D_{M}$, $W$, and $L_{f}$, it is sufficient to find a bound for $\sup_{t\in[0,1]}\max_{i}|\Delta^{(var)}_{i}(t)|.$ (64) We use a covering approach. For $L\in\mathbb{N}$, let $\overline{t}_{l}=\frac{2l-1}{2L}$ for $l=1,2,\dots,L$. Then for any $t\in[0,1]$ there exists $l^{*}$ such that $|t-\overline{t}_{l^{*}}|\leq\frac{1}{2L}$ and $\begin{split}&\max_{i}\left\\{\Delta^{(var)}_{i}(t)\right\\}\\\ &\leq\max_{i}\left\\{\begin{split}&\sum_{j:j\neq i}\frac{\sum_{k}W_{h}(t_{k},t)\mathbf{1}(i\text{ defeats }j\text{ at }t_{k})}{\sum_{k}W_{h}(t_{k},t)}\\\ &-\sum_{j:j\neq i}\frac{\sum_{k}W_{h}(t_{k},\overline{t}_{l^{*}(t)})\mathbf{1}(i\text{ defeats }j\text{ at }t_{k})}{\sum_{k}W_{h}(t_{k},\overline{t}_{l^{*}(t)})}\end{split}\right\\}\\\ &+\max_{i}\Delta^{(var)}_{i}(\overline{t}_{l^{*}})\end{split}$ (65) where $t_{k}$ here stands $t_{k}^{(i,j)}$ for brevity. In order to bound the second term in the curly brackets, we bound each of its summands as follows: $\begin{split}&\left|\frac{W_{h}(t_{k},t)}{\sum_{k}W_{h}(t_{k},t)}-\frac{W_{h}(t_{k},\overline{t}_{l^{*}(t)})}{\sum_{k}W_{h}(t_{k},\overline{t}_{l^{*}(t)})}\right|\\\ &\leq\left|\frac{W_{t}S_{\overline{t}}-W_{\overline{t}}S_{t}}{S_{t}S_{\overline{t}}}\right|\\\ &\leq\frac{|W_{t}-W_{\overline{t}}|}{S_{\overline{t}}}+\frac{|S_{\overline{t}}-S_{t}|W_{\overline{t}}}{S_{t}S_{\overline{t}}}\\\ \end{split}$ (66) where we denote $W_{t}=W_{h}(t_{k},t)$, $W_{\overline{t}}=W_{h}(t_{k},\overline{t}_{l^{*}(t)})$, $S_{t}=\sum_{k}W_{h}(t_{k},t)$, and $S_{\overline{t}}=\sum_{k}W_{h}(t_{k},\overline{t}_{l^{*}(t)})$ for brevity. We have seen in Section 9.1.4 that, for any a sufficiently small $h$, $S_{t},~{}S_{\overline{t}}\geq\frac{D_{m}T}{6}.$ (67) Thus, $\begin{split}&\frac{|W_{t}-W_{\overline{t}}|}{S_{\overline{t}}}+\frac{|S_{\overline{t}}-S_{t}|W_{\overline{t}}}{S_{t}S_{\overline{t}}}\\\ &\leq\frac{6}{D_{m}T}\frac{L_{W}}{h}|t-\overline{t}_{l^{*}(t)}|+\left(\frac{6}{D_{m}T}\right)^{2}T\frac{L_{W}}{h}|t-\overline{t}_{l^{*}(t)}|\\\ &\leq\frac{36}{D_{m}^{2}Lh^{2}T}\end{split}$ (68) as $D_{m}\leq 1$ and $W$ is $L_{W}$-Lipschitz by assumption. Hence, $\begin{split}&\max_{i}\left\\{\begin{split}&\sum_{j:j\neq i}\frac{\sum_{k}W_{h}(t_{k},t)\mathbf{1}(i\text{ defeats }j\text{ at }t_{k})}{\sum_{k}W_{h}(t_{k},t)}\\\ &-\sum_{j:j\neq i}\frac{\sum_{k}W_{h}(t_{k},\overline{t}_{l^{*}(t)})\mathbf{1}(i\text{ defeats }j\text{ at }t_{k})}{\sum_{k}W_{h}(t_{k},\overline{t}_{l^{*}(t)})}\end{split}\right\\}\\\ &\leq\frac{36(N-1)}{D_{m}^{2}Lh^{2}}\end{split}$ (69) On the other hand, $\begin{split}&\max_{i}\Delta^{(var)}_{i}(\overline{t}_{l^{*}})\leq\max_{l,i}\Delta^{(var)}_{i}(\overline{t}_{l})\\\ &=\max_{l,i}\left\\{\sum_{j:j\neq i}\left(\begin{split}&\frac{\sum_{k}W_{h}(t_{k},t)(\mathbf{1}_{ij}(t_{k})-p_{ij}(t_{k}))}{\sum_{k}W_{h}(t_{k},t)}\\\ \end{split}\right)\right\\}\end{split}$ (70) where, again,$\mathbf{1}_{ij}(t_{k})$ here stands $\mathbf{1}(i\text{ defeats }j\text{ at }t_{k})$ for simplicity. Using Eq. 42 and a union bound, we get that $\begin{split}&\mathbb{P}\\!\left(\max_{l,i}\Delta^{(var)}_{i}(\overline{t}_{l})\geq\epsilon\right)\\\ &\leq 2NL\exp\left(-\frac{\epsilon^{2}hD_{m}T}{18(1-p_{\text{min}})(N-1)}\right),\end{split}$ (71) for $\epsilon\leq p_{\min}$. Next we plug in $\epsilon=\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{36(1-p_{\min})(N-1)\log(NL)}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=14.075pt,depth=-11.26006pt}}}{{\hbox{$\textstyle\sqrt{\frac{36(1-p_{\min})(N-1)\log(NL)}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=9.8611pt,depth=-7.88892pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{36(1-p_{\min})(N-1)\log(NL)}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=7.25238pt,depth=-5.80194pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{36(1-p_{\min})(N-1)\log(NL)}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=7.25238pt,depth=-5.80194pt}}}$ to obtain the bounds $\begin{split}&\left|\max_{l,i}\Delta^{(var)}_{i}(\overline{t}_{l^{*}})\right|\leq\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log(NL)}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=14.075pt,depth=-11.26006pt}}}{{\hbox{$\textstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log(NL)}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=9.8611pt,depth=-7.88892pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log(NL)}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=7.25238pt,depth=-5.80194pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{36(1-p_{\text{min}})(N-1)\log(NL)}{hD_{m}T}\,}$}\lower 0.4pt\hbox{\vrule height=7.25238pt,depth=-5.80194pt}}}\end{split}$ (72) and, in turn, $\begin{split}&\sup_{t,i}\frac{|e^{\hat{\beta}_{i}(t)-\beta^{*}_{i}(t)}-1|}{|e^{\hat{\beta}_{i}(t)-\beta^{*}_{i}(t)}-1|+1}\\\ &\leq\sup_{t}8M(t)\left(\begin{split}&2\delta_{h}(t)+\frac{36}{D_{m}^{2}Lh^{2}}\\\ &+\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{36(1-p_{\min})\log(NL)}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=15.0pt,depth=-12.00005pt}}}{{\hbox{$\textstyle\sqrt{\frac{36(1-p_{\min})\log(NL)}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=10.5pt,depth=-8.40004pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{36(1-p_{\min})\log(NL)}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=7.58572pt,depth=-6.0686pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{36(1-p_{\min})\log(NL)}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=7.58572pt,depth=-6.0686pt}}}+C_{s}h\end{split}\right)\end{split}$ (73) with probability at least $1-\frac{2}{NL}$ and as long as $\epsilon\leq p_{\min}$. Plugging in $h=\max\left\\{\left(\frac{1}{T}\right)^{1+\eta},\left(\frac{36(1-p_{\min})\log(NT^{3+3\eta})}{C_{s}^{2}D_{m}(N-1)T}\right)^{\frac{1}{3}}\right\\}$ and $L=\lceil h^{-3}\rceil$, we conclude that $\begin{split}&\sup_{t,i}\frac{|e^{\hat{\beta}_{i}(t)-\beta^{*}_{i}(t)}-1|}{|e^{\hat{\beta}_{i}(t)-\beta^{*}_{i}(t)}-1|+1}\\\ &\leq\sup_{t}8M(t)\left(\begin{split}&2\delta_{h}(t)+\frac{36}{D_{m}^{2}Lh^{2}}+C_{s}h\\\ &+\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{36(1-p_{\min})\log(NL)}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=15.0pt,depth=-12.00005pt}}}{{\hbox{$\textstyle\sqrt{\frac{36(1-p_{\min})\log(NL)}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=10.5pt,depth=-8.40004pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{36(1-p_{\min})\log(NL)}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=7.58572pt,depth=-6.0686pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{36(1-p_{\min})\log(NL)}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=7.58572pt,depth=-6.0686pt}}}\end{split}\right)\\\ &\leq\sup_{t}8M(t)\left(\begin{split}&2\delta_{h}(t)+\left(C_{s}+\frac{72}{D_{m}^{2}}\right)h\\\ &+\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{36(1-p_{\min})\log(Nh^{-3})}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=16.24889pt,depth=-12.99916pt}}}{{\hbox{$\textstyle\sqrt{\frac{36(1-p_{\min})\log(Nh^{-3})}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=11.39998pt,depth=-9.12003pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{36(1-p_{\min})\log(Nh^{-3})}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=8.59682pt,depth=-6.87749pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{36(1-p_{\min})\log(Nh^{-3})}{hD_{m}(N-1)T}\,}$}\lower 0.4pt\hbox{\vrule height=8.59682pt,depth=-6.87749pt}}}\end{split}\right)\\\ &\leq\sup_{t}16M(t)\left(\begin{split}&\delta_{h}(t)+\left(C_{s}+\frac{36}{D_{m}^{2}}\right)h\\\ \end{split}\right)\\\ \end{split}$ (74) with probability at least $1-\frac{2h^{3}}{N}$ when $\epsilon\leq p_{\min}$. Since $\epsilon\leq\mathchoice{{\hbox{$\displaystyle\sqrt{3(1+\eta)\,}$}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{3(1+\eta)\,}$}\lower 0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{3(1+\eta)\,}$}\lower 0.4pt\hbox{\vrule height=5.25pt,depth=-4.20003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{3(1+\eta)\,}$}\lower 0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}}C_{s}h$ given the choice of $h$, this bound holds for all sufficiently small $h$. #### 9.1.6 Proof of Theorem 5.4 For convenience, we omit the time index $t$ for $\hat{\bm{\beta}}(t)$, $\bm{\beta}^{*}(t)$, and $p_{\text{min}}(t)$, unless it is required for clarification. We seek to replace $M(t)$ in Eq. 16 by a term of $p_{\text{min}}$. This requires $\exp(\beta^{*}_{i}-\beta^{*}_{j})$ to be bounded above by a function of $p_{\text{min}}$. The following lemma provides the desired bound. The proof is in Section 9.1.7. ###### Lemma 9.2. $\max_{i,j:i\neq j}|\beta_{i}^{*}-\beta_{j}^{*}|-\frac{1}{p_{\text{min}}}$ (75) is upper-bounded by a universal constant, and hence $\max_{i,j:i\neq j}\exp(|\beta_{i}^{*}-\beta_{j}^{*}|)\leq C_{p}\exp\left(\frac{1}{p_{\text{min}}}\right)$ (76) for some universal constant $1<C_{p}<1.5$. Plugging in the new bound on $\exp(\beta^{*}_{i}-\beta^{*}_{j})$, we get $\begin{split}&\|\hat{\bm{\beta}}(t)-\bm{\beta}^{*}(t)\|_{\infty}\leq 72K\left(\begin{split}&\delta_{h}(t)+C_{s}h\end{split}\right)\end{split}$ (77) instead of $48M(t)\left(\delta_{h}(t)+C_{s}h\right)$ in Eq. 16 This result easily extends to the uniform case Eq. 21. #### 9.1.7 Proof of Lemma 9.2 Let $d_{0}$ be the difference in scores which implies a bias of probability $\frac{p_{\text{min}}}{2}$: $\frac{1}{1+\exp(d_{0})}=\frac{p_{\text{min}}}{2}$ (78) Suppose that $i_{\text{max}}={\arg\max}_{i}\beta_{i}^{*}\text{ and }i_{\text{min}}={\arg\min}_{i}\beta_{i}^{*}$ (79) and that $\beta^{*}_{\text{max}}=\max_{i}\beta_{i}^{*}\text{ and }\beta^{*}_{\text{min}}=\min_{i}\beta_{i}^{*}$ (80) Then, the maximal difference between $\bm{\beta}_{i}^{*}$’s $d_{\text{max}}$ is $d_{\text{max}}=\max_{i,j:i\neq j}\beta_{i}^{*}-\beta_{j}^{*}=\beta^{*}_{\text{max}}-\beta^{*}_{\text{min}}$ (81) Let $I_{1}=\\{i:\beta_{i}<\beta_{\text{min}}+d_{0}\\}$. Plugged in $i=i_{\text{min}}$, Eq. 34 implies $\begin{split}&(N-1)p_{\text{min}}\leq\sum_{j:j\neq i_{\text{min}}}p_{i_{\text{min}}j}(t)\\\ &=\sum_{j:j\neq i_{\text{min}}}\frac{1}{1+\exp(\beta^{*}_{\text{j}}-\beta^{*}_{\text{min}})}\\\ &\leq\frac{|I_{1}|-1}{2}+(N-1)\frac{p_{\text{min}}}{2}\end{split}$ (82) Hence, $|I_{1}|\geq(N-1)p_{\text{min}}+1$. Now, let $I_{2}=\\{i:\beta_{i}<\beta_{\text{min}}+2d_{0}\\}$. Summing Eq. 34 plugged in $i\in I$, we get $\begin{split}&(N-|I_{1}|)|I_{1}|p_{\text{min}}\leq\sum_{j\in I_{1}^{C}}\sum_{i\in I_{1}}p_{ij}(t)\\\ &=\sum_{j\in I_{1}^{C}}\sum_{i\in I_{1}}\frac{1}{1+\exp(\beta^{*}_{j}-\beta^{*}_{i})}\\\ &\leq\frac{(|I_{2}|-|I_{1}|)|I_{1}|}{2}+(N-|I_{1}|)|I_{1}|\frac{p_{\text{min}}}{2}\end{split}$ (83) and hence $\begin{split}|I_{2}|\geq&|I_{1}|+(N-|I_{1}|)p_{\text{min}}\\\ \geq&Np_{\text{min}}+|I_{1}|(1-p_{\text{min}})\\\ \geq&(N-1)(1-(1-p_{\text{min}})^{2})+1\\\ \end{split}$ (84) Similarly for $I_{k}=\\{i:\beta_{i}<\beta_{\text{min}}+kd_{0}\\}$ and $J_{k}=\\{j:\beta_{j}>\beta_{\text{max}}-kd_{0}\\}$, $\begin{split}&|I_{k}|\geq(N-1)(1-(1-p_{\text{min}})^{k})+1,\\\ &|J_{k}|\geq(N-1)(1-(1-p_{\text{min}})^{k})+1.\end{split}$ (85) Now, without loss of generality we assume that $d_{\text{max}}>2kd_{0}$. Then, by the optimality of $\bm{\beta}^{*}$ for $\mathcal{R}(\bm{\beta})$, $\begin{split}\log 2=&\mathcal{R}(\mathbf{0})\geq\mathcal{R}(\beta^{*})\\\ =&\frac{1}{\binom{N}{2}}\sum_{i,j:i\neq j}p_{ij}(t)\log(1+\exp(\beta_{j}^{*}-\beta_{i}^{*}))\\\ \geq&\frac{1}{\binom{N}{2}}\sum_{i\in I_{k}}\sum_{j\in J_{k}}p_{\text{min}}\log(1+\exp(d_{\text{max}}-2kd_{0}))\\\ \geq&2p_{\text{min}}(1-(1-p_{\text{min}})^{k})^{2}(d_{\text{max}}-2kd_{0}).\end{split}$ (86) Thus, $d_{\text{max}}\leq\frac{\log 2}{2p_{\text{min}}(1-(1-p_{\text{min}})^{k})^{2}}+2kd_{0}$ for any $k$. Plugging in $k=\lceil\log(\frac{1}{p_{\text{min}}})\rceil$, we get that $\begin{split}d_{\text{max}}\leq&\frac{\log 2}{2(1-1/e)^{2}p_{\text{min}}}\\\ &+2\log\left(\frac{2}{p_{\text{min}}}-1\right)\left(\log\left(\frac{1}{p_{\text{min}}}\right)+1\right)\\\ \leq&\frac{1}{p_{\text{min}}}+C,\end{split}$ (87) for some universal constant $C$ since the derivative of $2\log\left(2x-1\right)\left(\log x+1\right)$ is positive and converges to $0$ as $x\rightarrow\infty$. Then $2\log\left(2x-1\right)\left(\log x+1\right)$ has a upper-bounding tangent line with slope $1-\frac{\log 2}{2(1-1/e)^{2}}$, and $C$ is its y-intercept. This also yields that $\max_{i,j:i\neq j}\exp(\beta^{*}_{i}-\beta^{*}_{j})\leq C_{p}\exp\left(\frac{1}{p_{\text{min}}}\right),$ (88) for some universal constant $C_{p}$. In particular, $1<C_{p}<1.5$. ### 9.2 Tuning kernel bandwidth in practical settings As noted in Section 2.2, the bandwidth $h\in\mathbb{R}_{>0}$ of the kernel function serves as an effective global smoothing parameter between subsequent time periods and allows to borrow information across contiguous time points. Increasing $h$, all else held constant, leads to parameter estimates (and hence the derived global rankings) becoming “smoothed” together across time. Naturally the question remains on how to tune $h$ in practical applications in a principled data-driven manner. This is a fundamentally challenging question not just in our problem but, more generally, in nonparametric regression. Here we present a way to tuning $h$ with some degree of objectivity based on leave- one-out cross-validation (LOOCV). In general settings where we have independent and identically distributed (i.i.d.) samples, LOOCV assesses the performance of a predictive model on a single held-out i.i.d. sample. In our case, each pairwise comparison can be considered an i.i.d. sample if we take the compared teams and the time point on which they are compared as covariates. Remember that $(i_{m},j_{m},t_{m})$ denotes $m$-th pairwise comparison where team $i_{m}$ won against team $j_{m}$ at time point $t_{m}$ for $m=1,\dots,M$. Then, for a given smoothing penalty parameter $h$, LOOCV is adapted to our estimation approach as follows: 1. 1. For $m=1,\dots,M$, given $h>0$: 1. (a) fit our model with kernel bandwidth $h$ on the dataset with the $m$-th comparison held-out; 2. (b) calculate the negative log-likelihood (nll) of the previous solution to $(i_{m},j_{m},t_{m})$. 2. 2. Take the average of the negative log-likelihoods to obtain $\text{nll}_{h}$ as a loss in the predictive performance of time-varying Bradley-Terry estimator for given $h$ on our dataset. 3. 3. Choose the bandwith $h^{*}$ with the smallest $\text{nll}_{h}$ value. We apply this data-driven methodology to the experiments and real-life application in Section 6 and Section 7. ### 9.3 Details of Experiments Here we explain some details of the setting of the numerical experiments in Section 6. #### 9.3.1 Bradley-Terry Model as the True Model We set the number of teams $N=50$ and the number of time points $M=50$. We set $n_{ij}(t)=1$ for all $i,j\in[N]$ and $t\in[M]$. For the Gaussian process to generate a path for $\bm{\beta}^{*}_{i}$ at $t=1,\ldots,M$, we use the same covariance matrix $\Sigma_{i}=\Sigma\in\mathbb{R}^{M\times M}$ for all $i\in[N]$, and $\Sigma$ is set to be a Teoplitz matrix defined by $\Sigma_{ij}=1-M^{-\alpha}|i-j|^{r},$ and in our experiment we set $(\alpha,r)=(1,1)$. The mean vector is set to be a constant over time, i.e., $\mu_{i}(t)=u_{i}$ for $t=1,\ldots,M$, and $u_{1},\ldots,u_{N}$ are $i.i.d.$ generated from uniform distribution on $[0,1]$. Figure 3: LOOCV curve of our Dynamic Bradley-Terry model fitted with Gaussian kernel. $y$-axis: averaged negative log-likelihood. The optimal $h^{*}$ is 0.03. Fig. 3 shows the curve of LOOCV of our dynamic Bradley-Terry model fitted with a Gaussian kernel in one repetition of our experiment. The curve is for the setting here and for the agnostic model setting the CV curve has similar shape. The curve shows a typical shape of CV curve for tuning parameter. The kernel bandwidth, $h$, with smallest $\mathrm{nll}_{h}$ is $h^{*}=0.03$. The LOOCV procedure is described in Section 9.2. #### 9.3.2 Agnostic Model Setting Again we set the number of teams $N=50$, the number of time points $M=50$, and $n_{ij}(t)=1$ for all $i,j\in[N]$ and $t\in[M]$. The covariance matrix is also the same as section 9.3.1. The only difference lies in the mean vector. Now the mean vector is still constant over time, or $\mu_{i}(t)=u_{i}$ for $t=1,\ldots,M$, but $u_{i}$’s are generated in a following group-wise way: 1. 1. Set the number of groups $G$ and the index set of each group $I_{1},\ldots,I_{G}$ so that $\sum_{i}|I_{i}|=N$. Set the base support to be $[0,b]$ and the group gap to be $a$. 2. 2. For each $i\in\\{1,\ldots,G\\}$, generate $u_{j}$ from ${\rm Uniform}(a(i-1),a(i-1)+b)$ for all $j\in I_{i}$. In our experiment we set $G=5$ with each group containing two randomly picked indices, $b=0.5$ and $a=1.5$. Such group-wise generation intends to ensure that different teams have distinguishable perofrmance in pairwise comparison so that the ranking is more reasonable. #### 9.3.3 Running Time Fig. 4 compares the time it takes to fit our model and the original Bradley- Terry model under 3 different settings, where $N$ is the number of teams and $M$ is the number of time points: * • Fix $N$, vary $M$. * • Fix $M$, vary $N$. * • Vary $N$ and $M$ together while keeping $N=M$. Figure 4: Comparison of running time of original Bradley Terry model (oBT) and our Dynamic Bradley Terry model (DBT). The values are averaged over 20 repetitions. For our dynamic Bradley-Terry model, the running time here is measured for fitting the model with a given kernel parameter $h$, hence it contains the time cost of kernel smoothing step and the optimization step. In real applications, if one wants to select the best $h$ from a range of values with cross-validation, then the total computation time would be approximately the running time here multiplied by the number of cross-validations. The results in Fig. 4 shows that with all advantages our model can bring with, it does not cost much more in terms of computation time. Furthermore, when the number of time points $M$ is large while $N$ is relatively small, our model can cost even less time than the original Bradley-Terry model. If one wants to do LOOCV to select $h$ when $N$ and $M$ are huge, then it could take a long time to finish the whole procedure. However, in this case we observed in some extended experiments that with a pre-determined $h$ in a reasonable range, our model can give fairly good estimate close to the one given by the best $h^{*}$ selected by LOOCV. The supporting files of our experiments can be found in our GitHub repository†††Code available at https://github.com/shamindras/bttv-aistats2020. #### 9.3.4 MLE of the Bradley-Terry Model Table 4 shows the frequency with which 4.1 holds at a single time point for the original pairwise comparison data for different $M$ and $N$, where $n_{ij}(t)=1$ for all $i,j,t$. To be clear, here we just regard the matrix $\tilde{X}(t)$ in 4.1 as the original data rather than the smoothed data, as it originally was in Ford, (1957). Given $\\{X(t),t\in[M]\\}$, the frequency here refers to $\\#\\{t:\text{The condition holds for }X(t)\\}/M$. The data are generated as described in Section 6.1, and the frequency is averaged over 50 repetitions. When $N=M=10$ and $n_{ij}(t)=4$ for all $i,j,t$, the frequency arises to 0.988, illustrating how $n_{ij}(t)$ controls the sparsity of the game matrix and consequently whether 4.1 holds or not. $\mathbf{(N,M)}$ | (5,5) | (10,10) | (20,10) | (30,10) | (40,10) | (50,10) ---|---|---|---|---|---|--- Freq. | 0.248 | 0.622 | 0.902 | 0.950 | 0.984 | 0.984 Table 4: Frequency that 4.1 holds at a single time point for the original pairwise comparison data. $n_{ij}(t)=1$. As a comparison, under the same setting, for the kernel-smoothed pairwise comparison data, 4.1 always holds in the experiment. This fact demonstrates the advantage of using kernel-smooth, and partly explains why in our experiments where the data is sparse our model performs the best. The frequencies in Table 4 seem high for $N>20$, but from a global perspective, the induced frequency that 4.1 holds for all $M$ time points could be much lower. Table 5 shows such frequency in some settings. Again the values are averaged over 50 repetitions. Remember that in these settings the condition always holds for kernel-smoothed data. $\mathbf{(N,M)}$ | (10,10) | (20,10) | (30,10) | (40,10) | (50,10) | (60,10) ---|---|---|---|---|---|--- Freq. | 0.02 | 0.44 | 0.62 | 0.86 | 0.84 | 0.88 Table 5: Frequency that 4.1 holds for all $M$ time points for the original pairwise comparison data. $n_{ij}(t)=1$. To make it clearer how $n_{ij}(t)$ affects the global connectivity, we make Table 6. In the table we fix $(N,M)=(10,10)$. $n_{ij}(t)$ | 1 | 2 | 4 | 6 | 8 | 10 ---|---|---|---|---|---|--- Freq. | 0.02 | 0.48 | 0.92 | 0.94 | 0.96 | 1.0 Table 6: Frequency that 4.1 holds for all $M$ time points for the original pairwise comparison data. $(N,M)=(10,10)$. Figure 5: Divergence of max$|\hat{\beta}|$ when MLE does not exist at some time points for the original Bradley-Terry model. By direct inspection of the likelihood of the original Bradley-Terry model, it can be seen that, when the MLE does not exist, the norm of $\hat{\beta}$ will go to infinity if one uses gradient descent without any regularization. Fig. 5 shows an example where $N=M=10$ and $n_{ij}(t)=1$ for all $i,j,t$.

2024-09-04T02:54:55.749879

2003.00084

arxiv-papers

Convexity in Multivalued Harmonic Functions IMMANUEL BEN PORAT We investigate variants of a Three Circles type Theorem in the context of $\mathcal{Q}-$valued functions. We prove some convexity inequalities related to the $L^{2}$ growth function in the $\mathcal{Q}-$valued settings. Optimality of these inequalities and comparsion to the case of real valued harmonic functions is also discussed. § INTRODUCTION §.§ Background The study of multivalued harmonic functions was originated in the pioneering work of Almgren [5] on Plateau's problem, which asks for a surface of minimal area among all surfaces stretched accross a given closed contour. Almgren's theory was further extended and simplified in [1]. The profound geometric applications of Almgren's theory to minimal surfaces are not addressed here. Instead, we shall connect the theory of $\mathcal{Q}-$valued functions to a classical results from complex analysis, which has some modern reflections. Let us begin by providing some background to the material that motivated this work. Let $0\neq u$ be harmonic on the unit ball $B_{1}(0)$. Then one can associate to $u$ real valued functions $H_{u},D_{u},\overline{H}_{u},I_{u}:\mathrm{(0,1)\rightarrow\mathbb{R}}$ by letting \[ H_{u}(r)=\underset{\partial B_{r}(0)}{\int}u^{2}(x)d\sigma,D_{u}(r)=\underset{B_{r}(0)}{\int}|\nabla u|^{2}dx,\overline{H}_{u}(r)=\frac{1}{|\partial B_{r}(0)|}H_{u}(r),I_{u}(r)=\frac{rD_{u}(r)}{H_{\mathit{u}}(r)} \] $\overline{H}_{u}(r)$ is called the $L^{2}-$growth function of $u$ and $I_{u}(r)$ is called the frequency function of $u$. The motivation behind the definition of the function $\overline{H}_{u}$ comes from a classsical result in complex analysis known as the Three Circles Given a holomorphic function $f$ on the unit ball $B_{1}$, let $M(r)=\underset{B_{r}(0)}{\max}|f|$. The Three Circles Theorem, proved by Hadamard, states that the function $\widetilde{M}:(-\infty,0)\rightarrow\mathbb{R}$ defined by $\widetilde{M}(t)=M(e^{t})$ is $\log$ convex, that is $\log\widetilde{M}(t)$ is convex. It is therefore natural to seek for a Three Circles type Theorem for real harmonic functions $u:B_{1}(0)\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$. It was first observed by Agmon [6] that such a theorem holds if the function $M$ is replaced by an appropriate $L^{2}$-version on the sphere. Namely, Agmon proves that the function $t\mapsto\overline{H}_{u}(e^{t})$ is $\log$ convex. In 2015, Lippner and Mangoubi observed the following stronger result: . Let $u:B_{1}(0)\rightarrow\mathbb{R}$ be harmonic. Then $\overline{H}_{u}$ is absolutely monotonic, that is $\overline{H}_{u}^{(k)}\geq0$ for all $k\in\mathbb{N}$. In particular, $H_{u}^{(k)}\geq0$ for all Since Theorem <ref> in [3] is carried out in a discrete setting, for the sake of completeness a proof of the continuous version (as stated above) is presented in the appendix. The second statement in Theorem <ref> is an immediate consequence of the first one. It is an exercise to verify that absolute monotinicity of $\overline{H}$ implies $\log$ convexity of $t\mapsto\overline{H}(e^{t})$ (see [7], II, Problem 123). Roughly speaking, we are interested in the question whether a Lippner-Mangoubi type theorem can be obtained in the more general setting of multivalued harmonic functions. Let us emphasize this could have fascinating applications in the regularity theory of these objects, since absolutely monotonic functions are real analytic (due to a celebrated theorem of Bernstein. See [2]). In some sense, the nonlinear nature of the problem is the main obstacle in obtaining elliptic regularity type results for multivalued harmonic functions. We hope that approaching the problem via Bernstein's theorem may be useful in overcoming some of the difficulties that are created by the lack of linearity. §.§ Main Results Given some $P\in\mathbb{R}^{n}$ we denote by $[[P]]$ the Dirac mass in $P\in\mathbb{R}^{n}$ and define \[ \mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n}):=\{\stackrel[i=1]{\mathfrak{\mathcal{Q}}}{\sum}[[P_{i}]]|P_{i}\in\mathbb{R}^{n},1\le i\le\mathcal{Q}\} \] The set $\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$ is endowed with a metric $\mathcal{G}$, not specified for the moment, such that the space $(\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}),\mathcal{G})$ is a complete metric space. We then consider functions $f:\Omega\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$, where $\Omega$ is some domain in $\mathbb{R}^{m}$. We call such functions $\mathcal{Q}$-valued functions. One key fact is the existence of a notion of a harmonic $\mathcal{Q}$-valued function. We adapt the terminology of [1] and call such functions Dir-minimizing. As their name suggests, Dir-minimizing functions are defined as functions minimizing a certain class of integrals, by analogy with the classical Dirichlet principle. For each $f:B_{1}(0)\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n})$ Dir-minimizing we associate a real valued function $\overline{H}_{f}:(0,1)\rightarrow\mathbb{R}$ by letting: $\overline{H}_{f}(r)=\frac{1}{|\partial B_{r}(0)|}\underset{\partial B_{r}(0)}{\int}|f|{}^{2}d\sigma$. The function $\overline{H}_{f}$ is a generalization of the function introduced in the beginning. Our first aim would be to generlize Agmon's Theorem to the multivalued case. We will prove Let $f:B_{1}(0)\subset\mathbb{R^{\mathit{m}}}\rightarrow\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$ be Dir minimizing such that $H(r)>0$. Define $a:(-\infty,0)\rightarrow\mathbb{R}$ by $a(t)=\log\overline{H}(e^{t})$. Then $a'(t)\geq0$ for all $t\in(-\infty,0)$ for a.e. Furthermore, $a$ is convex. Since (ii) holds merely up to a null set, the convexity of $a$ does not follow directly. This requires an additional consideration. The following theorem, is the main result of this work Let $f:B_{1}(0)\subset\mathbb{R}^{2}\rightarrow\mathcal{A_{Q}}(\mathbb{R}^{n})$ be a Dir minimizing function. Suppose $f|_{\partial B_{r}}\in W^{1,2}(\partial B_{r}(0),\mathcal{A_{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$ for a.e. $0<r<1$. For each $N>0$ define $\overline{h}_{N,f}:(0,1)\rightarrow\mathbb{R}$ by $\overline{h}_{N,f}(r)=\overline{H}_{f}(r^{N}).$ $\overline{h}'_{\frac{\mathcal{Q}}{2},f}(r)\geq0$ for all $r\in(0,1)$ for a.e. $r\in(0,1)$ Furthermore, $\overline{h}_{\frac{\mathcal{Q}}{2}}$is convex. The proof of Theorem <ref> will be followed by a higher dimensional version, that is, when the domain is the $m-$dimensional unit baḷl for arbitrary $m>2$. In the higher dimensional version, the constant $\frac{\mathcal{Q}}{2}$ will be replaced by some constant depending on $m$ which does not have a simple closed formula. It should be remarked that unlike in the scenario of Theorem <ref>, the fact that $\overline{H}_{f}$ (and hence $a$ and $\overline{h}_{N,f}$ ) is a.e. twice differentiable (and moreover $C^{1}$) is nontrivial. A naive version of Theorem <ref> for $\mathcal{Q}$-valued functions is not valid, as wittnesed by the example $f(z)=\underset{w^{3}=z}{\sum}[[w]]$, for which the associated $\overline{H}$ function has a negative second derivative for all $0<r<1$. In addition, the following proposition demonstrates that we do not have an obvious third derivative version of Theorem <ref>: Define $f:B_{1}(0)\subset\mathbb{R}^{2}\mathbb{\rightarrow\mathcal{A}_{\mathrm{2}}\mathrm{(\mathbb{R^{\mathrm{2}}\mathrm{)}}}}$ by $f(z)=\underset{w^{2}=2z-1}{\sum}[[w]]$. Then $f$ is Dir minimizing,$f|_{\partial B_{r}}\in W^{1,2}(\partial B_{r},\mathcal{A_{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$ for all $r\neq\frac{1}{2}$ and $\overline{h}'''_{1,f}(r)=\mathcal{\mathit{\overline{H}_{f}'''}}(r)<0$ for all $\frac{1}{2}<r<1$. Both Proposition <ref> and the abovementioned example will be proved and explained in section <ref>. In Proposition <ref>, our main conribution is performing the formal computation which shows that $\overline{H}_{f}'''<0$ for all $\frac{1}{2}<r<1$ and showing that the boundary condition $f|_{\text{\ensuremath{\partial}}B_{r}}\text{\ensuremath{\in}}W^{1,2}(\text{\ensuremath{\partial}}B_{r},\mathcal{A_{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$ is indeed satisfied for all $r\neq\frac{1}{2}$. Proving that $f$ is Dir minimizing is a difficult task, and relies on some rather heavy machinery from geometric measure theory. It should be emphasized that the domain of $f$ in both counterexamples is the planar unit disk. Thus, we did not rule out the possibillity that in higher dimensions the $L^{2}$- growth function of $f$ is more well behaved. Organization of the paper. In Section <ref>, we fix some notation and briefly review the frequency function and its relatives. A detailed exposition may be found in [1]. Section <ref> is devoted mainly to the proof of Proposition <ref>, Theorem <ref> and other related convexity inequalities. In Section <ref> we preform the calculations required to establish the counterexample given in Proposition <ref>. In the same context we will prove that the boundary condition $f|_{\text{\ensuremath{\partial}}B_{r}}\text{\ensuremath{\in}}W^{1,2}(\text{\ensuremath{\partial}}B_{r}(0),\mathcal{A_{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$ in Theorem <ref> is in fact verified for a certain class of Dir-minimizing functions on the unit disk. § ACKNOWLEDGEMENT This work is part of the author's MA thesis. I wish to express my most deep gratitude towards my supervisor Dan Mangoubi for suggesting and guiding me in this problem, for many fruitful and stimulating disscusions and for carefully reading previous versions of this manuscript and providing insightful comments. In particular, for explaining the proof of Theorem <ref>. I would also like to thank Camillo De Lellis for his interest in this work. In particular, for suggesting the counterexample that appears in Proposition <ref> and for explaining the proof of Theorem <ref>. § PRELIMINARIES §.§ Notations $d\sigma=$The surface measure $\Delta u=$The Laplacian of $u$ $\cdot=$The standard scalar product on $\mathbb{R}^{m}$ $\nu=$The unit normal to the sphere $C_{m}=$Surface area of the $m-$dimensional unit sphere $B_{R}(x)=$The ball of radius $R$ centered at $x$ We assume that the reader is familiar with the basic theory of $\mathcal{Q}$-valued functions. Following [1], we recall some basic notions and terminology. Given some $P\in\mathbb{R}^{n}$ we denote by $[[P]]$ the Dirac mass in $P\in\mathbb{R}^{n}$ and define $\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n}):=\{\stackrel[i=1]{\mathcal{Q}}{\sum}[[P_{i}]]|P_{i}\in\mathbb{R}^{n},1\le i\le\mathcal{Q}\}$. We endow $\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$ with a metric $\mathcal{G}$, defined as follows: For each $T_{1},T_{2}\in\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$ with $T_{1}=\stackrel[\mathit{i=\mathrm{1}}]{\mathcal{Q}}{\sum}[[P_{i}]],T_{2}=\stackrel[\mathit{i=\mathrm{1}}]{\mathcal{Q}}{\sum}[[S_{i}]]$ we define $\mathcal{G\mathrm{(\mathit{T_{\mathrm{1}},T_{\mathrm{2}}})=\underset{\sigma\in\mathscr{P_{\mathcal{Q}}}}{min}\sqrt{\stackrel[\mathit{i=\mathrm{1}}]{\mathcal{Q}}{\sum}|\mathit{P_{i}-S_{\sigma\mathrm{(}i\mathrm{)}}}|^{2}}}}$, where $\mathscr{P_{\mathcal{Q}}}$ is the permutation group on $\{1,...,\mathcal{Q}\}$. The space $(\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}),\mathcal{G})$ is a complete metric space. A $\mathcal{Q}$- valued function is a function $f:\Omega\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$. Of course, this formalism was designed to capture the notion of a function attaining multiple values at each point. A regularity theory can be developed for $\mathcal{Q}$-valued functions. In particular, the notion of a Sobolev space $W^{1,p}(\Omega,\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$ and the notion of an approximate differential denoted by $Df$. Suppose $\Omega\subset\mathbb{R}^{m}$ is a bounded domain with smooth boundary. By analogy with the Dirichlet principle we say that a function is Dir-minimizing if $f\in W^{1,2}(\Omega,\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$ \[ \underset{\Omega}{\int}|Df|^{2}\leq\underset{\Omega}{\int}|Dg|^{2} \] for all $g\in W^{1,2}(\Omega,\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$ whose trace on $\partial\Omega$ agrees with that of $f$. We shall allways assume $m\geq2$. 3.2. Frequency Function. We recall the frequency function and its relatives in the context of $\mathcal{Q}-$valued functions. We have the following Holder regularity type theorem for Dir-minimizing functions: There are constants $\alpha=\alpha(m,\mathcal{Q})\in(0,1)$ and $C=C(m,n,\mathcal{Q\mathrm{,}\delta\mathit{\mathrm{)}}}$ with the following property. If $f:B_{1}(0)\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}\mathrm{(\mathbb{R}^{\mathit{n}})}$ is Dir-minimizing then: $\underset{x\neq y\in\overline{B_{\delta}(0)}}{\sup}\frac{\mathcal{G\mathrm{(\mathit{f\mathrm{(\mathit{x\mathrm{),\mathit{f\mathrm{(\mathit{y}))}}}}}}}}}{|x-y|^{\alpha}}$$\leq CDir(f)^{\frac{1}{2}}$ for all $0<\delta<1$. § PROOF OF MAIN RESULTS §.§ Variants of the Three Circles Theorem In this section we give a proof of Proposition <ref> and Theorem <ref>. The following identities will play a crucial role in the study of convexity of the frequency function and its relatives Let $f:B_{R}(x)\subset\mathbb{R^{\mathit{m}}\rightarrow\mathcal{A_{Q}\mathrm{(\mathbb{R}^{\mathit{n}})}}}$ be Dir-minimizing. Then for a.e. $0<r<R$ we have: (i) $(m-2)$$\underset{B_{r}(x)}{\int}|Df|^{2}dx=r\underset{\partial B_{r}(x)}{\int}|Df|^{2}d\sigma-2r\underset{\partial B_{r}(x)}{\int}\stackrel[i=1]{\mathcal{Q}}{\sum}|\partial_{\nu}f_{i}|^{2}d\sigma$ (ii) $\underset{B_{r}(x)}{\int}|Df|^{2}dx=\underset{\partial B_{r}(x)}{\int}\stackrel[i=1]{\mathcal{Q}}{\sum}(\partial_{\nu}f_{i})\cdot f_{i}d\sigma$ Our starting point is the following theorem be Dir minimizing. Then: (a) $H\in C^{1}(0,1)$ and the following identity holds for all $r\in(0,1):$ \begin{equation} \end{equation} (b) If $H(r)>0$ then $I(r)$ is absolutely continuous and nondecreasing. In particular, $I'(r)\geq0$ for a.e. $r$. Statement (b) is known as Almgren's monotonicity formula. Since $D$ is absolutely continuous, it is a.e. differentiable. Therefore, in view of equation <ref>, we see that $H'$ is a.e. differentiable. Otherwise put, the second derivative of $H$ exists a.e. The regularity properties of $H$ clearly apply for $\overline{H}$ as well. With the aid of Almgren's monotonicity formula we are able to extend Agmon's convexity result ([6]) in the context of $\mathcal{Q}-$valued functions. This method of proof differs from Agmon's original approach for real valued harmonic functions, which involves ODEs on Banach Proof of Proposition 1.2. In light of the above discussion it is clear that $a$ is $C^{1}(-\infty,0)$ and that $a''$ exists almost everywhere. For (i), note that \[ \] \[ \frac{1}{r^{m-1}}(\frac{m-1}{r}H(r)+2D(r))-(m-1)\frac{1}{r^{m}}H(r)=\frac{2D(r)}{r^{m-1}}\geq0 \] Where the second equality is due to equation <ref>. So \begin{equation} \overline{H}'(r)=\frac{2D(r)}{C_{m}r^{m-1}}\geq0\label{eq:7} \end{equation} Therefore, for all $t\in(-\infty,0)$ \begin{equation} \end{equation} For (ii), start by noting that $a(t)=\log(\overline{H}(e^{t}))=\log(H(e^{t}))-\log(C_{m}e^{t(m-1)})=\log(H(e^{t}))-\log(C_{m})-(m-1)t$. Therefore, to prove $a''(t)\geq0$ it suffices to prove $(\log(H(e^{t})))''\geq0$. By Theorem <ref>, we have that $I'(r)\geq0$ for a.e. $0<r<1.$ To spare some space, all equalities and inequalities from now on should be interpreted up to a null set. By virtue of equation \[ 0\leq I'(r)=(\frac{rD(r)}{H(r)})'=(\frac{D(r)}{r^{m-2}\overline{H}(r)})'=\frac{1}{2}(\frac{r(H'(r)-(\frac{m-1}{r})H(r))}{H(r)})'=\frac{1}{2}(\frac{rH'(r)-(m-1)H(r)}{H(r)})'= \] \[ \frac{1}{2}(\frac{rH'(r)}{H(r)})'=\frac{1}{2}(\frac{(H'(r)+rH''(r))H(r)-r(H'(r))^{2}}{H^{2}(r)}) \] Thus, we get \begin{equation} \end{equation} On the other hand by a straightforward calculation: \begin{equation} \end{equation} Combining inequality <ref> with equation <ref> we arrive at $e^{-t}(\log(H(e^{t}))''\geq0$ which is the same as $(\log(H(e^{t}))''\geq0$. We are left to explain why $a$ is convex. It is classical that a continuously differentiable function is convex iff its derivative is nondecreasing, and so our task reduces to showing that $a'$ is nondecreasing. By equations <ref> and <ref> we get $a'(t)=\frac{\overline{H}'(e^{t})e^{t}}{\overline{H}(e^{t})}=\frac{2D(e^{t})}{C_{m}e^{t(m-2)}\overline{H}(e^{t})}$. Since $D(e^{t})$ is a composition of an absolutely continuous function with a nondecreasing smooth function, it is absolutely continuous. In addition, $\frac{1}{C_{m}e^{t(m-2)}\overline{H}(e^{t})}$ is differentiable. So $a'(t)$ is absolutely continuous function on any closed subinterval of $(-\infty,0)$, as a product of such function. Therefore, the fundamental theorem of calculus is applicable: if $t_{1},t_{2}\in(-\infty,0),t_{1}<t_{2}$ then $a'(t_{2})-a'(t_{1})=\stackrel[t_{1}]{t_{2}}{\int}a''(t)dt\geq0$. We draw the reader's attention to a somewhat delicate point, which will be also relevant in what will come next . The implication “nonnegative derivative a.e.$\Rightarrow$nondecreasing” is not true in general. In Proposition <ref> we employed the fact that the first derivative of $a$ is absolutely continuous in order to deduce that it is convex. The absolute continuity of the derivatives is a consequence of equation <ref>. Hence, we implicitly relied here on the Dir-minimization property. In the case of 1 valued harmonic functions, this technicality is not created because all functions involved are smooth. To the best of our knowledge, improved regularity for the frequency function and its relatives in the multivalued settings is still an open problem. The inequality “$\overline{H}''(r)\geq0$ a.e.” is not true in general, as witnessed by the counterexample in section <ref>. Nevertheless, we are still able to obtain a convexity result by reducing the power of the normalization of $H$. More precisely we observe the weaker Let $f:B_{1}(0)\subset\mathbb{R^{\mathit{m}}}\rightarrow\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n})$ be Dir minimizing. Then (i) $(r\overline{H}(r))'\geq0$ for all $r\in(0,1)$ (ii) $(r\overline{H}(r))''\geq0$ for a.e. $r\in(0,1)$. Furthermore $r\mapsto r\overline{H}(r)$ is Proof. As for (i) we can in fact derive a stronger result, namely $\overline{H}'(r)\geq0$ for all $r\in(0,1)$. We compute: \begin{equation} \overline{H}'(r)=\frac{1}{C_{m}}[-(m-1)r^{-m}H(r)+r^{-(m-1)}H'(r)]=\frac{r^{-(m-1)}}{C_{m}}[H'(r)-\frac{m-1}{r}H(r)]=\frac{2D(r)r^{-(m-1)}}{C_{m}}\geq0\label{eq:3.4} \end{equation} where the last equality is by equation <ref>. For (ii), start by noting that equation <ref> implies the following equality for a.e. $r$ \begin{equation} \overline{H}''(r)=\frac{2}{C_{m}r^{m-1}}[D'(r)-\frac{(m-1)D(r)}{r}]\label{eq:3.4'} \end{equation} Gathering our calculations one readily checks that $(r\overline{H}(r))''\geq0\iff rD'(r)-(m-3)D(r)\geq0$. \[ rD'(r)-(m-3)D(r)\geq rD'(r)-(m-2)D(r)= \] \[ r\underset{\partial B_{r}(0)}{\int}|Df|^{2}-(m-2)\underset{B_{r}(0)}{\int}|Df|^{2}=2r\underset{\partial B_{r}(0)}{\int}\stackrel[i=1]{\mathcal{Q}}{\sum}|\partial_{\nu}f_{i}|^{2}\geq0 \] Where the last equality is thanks to (i), <ref>. The convexity of $r\mapsto r\overline{H}(r)$ follows by a similar argument to the one demonstrated in Proposition <ref>. It is clear that Proposition <ref> implies in particular that the same conclusion holds true for $H$ (and this can also be derived directly from iterating equation <ref>). We present now a proof of Theorem <ref>, including a higher dimensional analog. The main ingredients of the proof are the variational formulas provided by Proposition <ref> and the following estimates Let $f:B_{1}(0)\subset\mathbb{R^{\mathit{m}}}\rightarrow\mathcal{A}_{\mathcal{Q}}\mathrm{(\mathbb{R}^{\mathit{n}})}$ be Dir-minimizing and suppose that $g_{r}:=f|_{\partial B_{r}(0)}\in W^{1,2}(\partial B_{r}(0),\mathcal{A}_{\mathcal{Q}}\mathrm{(\mathbb{R}^{\mathit{n}}))}$ for a.e $0<r<1$. Then for a.e $r$: (i) If $m=2$, $\mathrm{Dir}(f,B_{r}(0))\leq\mathcal{Q}r\mathrm{Dir}(g_{r},\partial B_{r}(0))$ (ii) If $m>2$, $\mathrm{Dir}(f,B_{r}(0))\leq c(m)r\mathrm{Dir}(g_{r},\partial B_{r}(0))$, where $c(m)<\frac{1}{m-2}$. Before going into the proof of Theorem <ref>, let us heuristically explain why it is reasonable the expect the validity of such a result through the following example. We recall that a function $f:B_{1}(0)\rightarrow\mathcal{A}_{\mathcal{Q}}\mathrm{(\mathbb{R}^{\mathit{n}})}$ is $\alpha-$homogeneous ($\alpha>0)$ if $\forall y\in B_{1},y\neq0:f(y)=|y|^{\alpha}f(\frac{y}{|y|})$. Denote by $\eta:\mathcal{A}_{\mathcal{Q}}\mathrm{(\mathbb{R}^{\mathit{n}})}\rightarrow\mathbb{R}^{n}$ the center of mass map defined by $\eta(\stackrel[i=1]{\mathcal{Q}}{\sum}[[P_{i}]])=\frac{\stackrel[i=1]{\mathcal{Q}}{\sum}P_{i}}{\mathcal{Q}}$. We can derive an explicit formula for the $L^{2}-$growth function of a continuous $\alpha-$homogeneous map: \[ H(r)=\underset{\partial B_{r}}{\int}|f|^{2}d\sigma=r^{m-1}\underset{\partial B_{1}}{\int}|f|^{2}(ry)d\sigma= \] \[ r^{m-1}\underset{\partial B_{1}}{\int}\stackrel[i=1]{\mathcal{Q}}{\sum}|f_{i}|^{2}(ry)d\sigma=r^{2\alpha+m-1}\underset{\partial B_{1}}{\int}\stackrel[i=1]{\mathcal{Q}}{\sum}|f_{i}|^{2}(y)d\sigma=\kappa r^{2\alpha+m-1} \] For some constant $\kappa\geq0$. Hence $\overline{H}$ takes the form $\overline{H}(r)=\kappa r^{2\alpha}$. Assume now that $m=2$, is a Dir-minimizing, nontrivial, $\alpha-$homogeneous map with $\eta\circ f=0$ (the simplest example of such a map the 2-valued function $f:B_{1}(0)\rightarrow\mathcal{A}_{2}(\mathbb{R}^{2})$ defined by $f(z)=\underset{w^{2}=z}{\sum}[[w]]$. See Theorem <ref>). It is proved in [1], Proposition 8.2 that in this case necessarily $\alpha=\frac{p}{q}\in\mathbb{Q}$ for some $q\leq\mathcal{Q}$. So $\overline{H}(r)=\kappa r^{\frac{2p}{q}}$, which implies $\overline{H}(r^{\frac{q}{2}})=\kappa r^{p}$, and the latter function is obviously absolutely monotonic. We are thus lead to spectulate that more generally, composing $\overline{H}$ with some suitable $\frac{1}{2}[\mathcal{Q}]$-power produces a function which is more well behaved. Theorem <ref> partially confirms this speculation. Proof of Theorem <ref>. That $\overline{h}'_{\frac{\mathcal{Q}}{2}}(r)\geq0$ follows immediately from equation <ref>. Henceforth all equalities and inequlities should be interpeted up to a null set. A direct calculation Writing $\xi=r^{N}$, we see that $\overline{h}_{N}''(\xi)\geq0\Leftrightarrow(\frac{N-1}{N})\overline{H}'(\xi)+\overline{H}''(\xi)\xi\geq0$. Taking $N=\frac{\mathcal{Q}}{2}$ we get \[ \overline{h}_{\frac{\mathcal{Q}}{2}}''(\xi)\geq0\Leftrightarrow(\mathcal{Q}-2)\overline{H}'(\xi)+\mathcal{Q}\overline{H}''(\xi)\xi\geq0 \] Owing to equation <ref> we can express $\overline{H}',\overline{H}''$ \[ \overline{H}'(\xi)=\frac{2}{C\xi}\underset{B_{\xi}(0)}{\int}|Df|^{2} \] \[ \overline{H}''(\xi)=\frac{2}{C\xi}[\underset{\partial B_{\xi}(0)}{\int}|Df|^{2}-\frac{1}{\xi}\underset{B_{\xi}(0)}{\int}|Df|^{2}] \] We now have \begin{equation} C\xi[(\mathcal{Q}-2)\overline{H}'(\xi)+\mathcal{Q}\overline{H}''(\xi)\xi]=2(\mathcal{Q}-2)\underset{B_{\xi}(0)}{\int}|Df|^{2}+2\mathcal{Q}\xi[\underset{\partial B_{\xi}(0)}{\int}|Df|^{2}-\frac{1}{\xi}\underset{B_{\xi}(0)}{\int}|Df|^{2}]=2\mathcal{Q}\xi\underset{\partial B_{\xi}(0)}{\int}|Df|^{2}-4\underset{B_{\xi}(0)}{\int}|Df|^{2}\label{eq:3.7} \end{equation} Proposition <ref>, (i) combined with Proposition <ref> yield the following estimate: \[ \underset{B_{\xi}(0)}{\int}|Df|^{2}dx\leq\xi\mathcal{Q}\mathrm{Dir}(g_{\xi},\partial B_{\xi}(0)) \] \[ 2\underset{B_{\xi}(0)}{\int}|Df|^{2}\leq2\xi\mathcal{Q}\mathrm{Dir}(g_{\xi},\partial B_{\xi}(0))=2\xi\mathcal{Q}[\underset{\partial B_{\xi}(0)}{\int}|Df|^{2}-\stackrel[i=1]{\mathcal{Q}}{\sum}|\partial_{\nu}f_{i}|^{2}d\sigma]=\mathcal{Q}\xi\underset{\partial B_{\xi}(0)}{\int}|Df|^{2} \] Where the last equality is due to <ref>, (i). Combining the last inequality with equation <ref> gives $(\mathcal{Q}-2)\overline{H}'(\xi)+\mathcal{Q}\overline{H}''(\xi)\xi\geq0$, as wanted. Finally, note that $\overline{h}'_{\frac{\mathcal{Q}}{2}}(r)=\frac{2D(r^{\frac{\mathcal{Q}}{2}})r^{-\frac{\mathcal{Q}}{2}(m-1)}}{C_{m}}r^{\frac{\mathcal{Q}}{2}-1}=\frac{2D(r^{\frac{\mathcal{Q}}{2}})r^{\frac{\mathcal{Q}}{2}(2-m)-1}}{C_{m}}$. Since $D(r^{\frac{\mathcal{Q}}{2}})$ is a composition of absolutely continuous nondecreasing functions, it is absolutely continuous. In view of the previous equation we see that $\overline{h}'_{\frac{\mathcal{Q}}{2}}$ is absolutely continuous. Thus, proceeding as in Proposition <ref>, it follows that $\overline{h}{}_{\frac{\mathcal{Q}}{2}}$ is convex. We finish this section by stating an higher dimensional analog of Let $m>2$ and $f:B_{1}(0)\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{\mathit{n}})$ be a Dir minimizing function. Suppose $f|_{\partial B_{r}}\in W^{1,2}(\partial B_{r},\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$ for a.e. $0<r<1$. Let $c(m)=\frac{1}{m-2}-\epsilon_{m}$ , $0<\epsilon_{m}<\frac{1}{m-2}$ be the constant obtained via proposition <ref>. Let $\alpha_{m}=\frac{1-\epsilon_{m}(m-2)}{\epsilon_{m}(m-2)^{2}}$. for all $r\in(0,1)$ for a.e. $r\in(0,1)$ The proof is identical to that of Theorem <ref>, using estiamte (ii) in Proposition <ref> instead of (i). We can now conclude that nontrivial Dir minimizing, $\alpha-$homogeneous function have exponents $\alpha$ far away from $0$. There are constants $\beta_{m}>0$ with the following property. Let $f:B_{1}(0)\subset\mathbb{R}^{m}\rightarrow\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{\mathit{n}})$ be a nontrivial Dir minimizing, $\alpha-$homogeneous function. Suppose $f|_{\partial B_{r}}\in W^{1,2}(\partial B_{r},\mathcal{A}_{\mathcal{Q}}(\text{\ensuremath{\mathbb{R}}}^{n}))$ for a.e. $0<r<1$. Then $\alpha\geq\beta_{m}$. Put $\beta_{2}=\frac{1}{\mathcal{Q}}$ and $\beta_{m}=\frac{1}{\alpha_{m}},m>2$. According to the computation performed in Example <ref>, $\overline{H}(r)=\kappa r^{2\alpha}$ for some $\kappa>0$. According to Theorem <ref> and Theorem <ref> we see that $0\leq\kappa\frac{\alpha}{\beta_{m}}(\frac{\alpha}{\beta_{m}}-1)r^{\frac{\alpha}{\beta_{m}}-2}$ for a.e. $r$, which in particular gives $\alpha\geq\beta_{m}$. Note that in the specific case that $m=2$ and $\eta\circ f=0$, Corollary <ref> recovers a weaker form of Proposition 8.2, [1] which was already mentioned in Example <ref>. § COUNTEREXAMPLES The following theorem allows us to produce nontrivial examples of Dir-minimizing functions: $a\neq0,b\in\mathbb{R}$. Define $u:B_{1}(0)\subset\mathbb{R}^{2}\mathbb{\rightarrow}\mathcal{A_{\mathcal{Q}}}(\mathbb{R}^{2})$ by $u(z)=\underset{w^{\mathcal{Q}}=az+b}{\sum}[[w]]$. Then $u$ is Define $f:B_{1}(0)\subset\mathbb{R}^{2}\mathbb{\rightarrow\mathcal{A}_{\mathrm{3}}\mathrm{(\mathbb{R^{\mathrm{2}}\mathrm{)}}}}$ by $f(z)=\underset{w^{3}=z}{\sum}[[w]]$. That $f$ is Dir minimizing follows from Theorem <ref>. We compute: \[ \underset{B_{\rho}(0)}{\int}|f|^{2}dx=\stackrel[0]{2\pi}{\int}\stackrel[0]{\rho}{\int}3r^{\frac{5}{3}}drd\theta=3\stackrel[0]{2\pi}{\int}\stackrel[0]{\rho}{\int}r^{\frac{5}{3}}drd\theta=\frac{9\pi\rho^{\frac{8}{3}}}{4}=C\rho^{\frac{8}{3}} \] where $C>0$. Therefore, $H(\rho)=C\rho^{\frac{5}{3}}$ for some constant $C>0$, and so $\overline{H}(\rho)=\frac{C\rho^{\frac{5}{3}}}{2\pi\rho}=C\rho^{\frac{2}{3}}$, hence $\overline{H}''(\rho)<0$. Our next aim is to show that the boundary regularity condition appearing in Proposition <ref> is indeed verified for a certain Dir-minimizing functions on the planar unit disk. This will be the content of Lemma <ref>, which is of interest by its own right. Any $z\in\mathbb{C}-\{0\}$ admits a representation of the form $z=Re^{i\omega}$ for some $R>0,\omega\in[0,2\pi)$. We shall use the convention $\sqrt{z}=\sqrt{R}e^{\frac{i\omega}{2}}$. Fix $r\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$ and define $h:(0,2\pi)\rightarrow\mathbb{C}$ by $h(\theta)=\sqrt{2re^{i\theta}-1}$. Then $h\in W^{1,2}((0,2\pi),\mathbb{C})$. Proof. $\square$ We recall that $f\in W^{1,2}(\Omega,\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n}))$ if there are $\{\varphi_{j}\}_{j=1}^{m}\subset L^{2}(\Omega,\mathbb{R}_{\geq0})$ such that 1. $x\mapsto\mathcal{G}(f(x),T)\in W^{1,2}(\Omega)$ for all $T\in\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n})$ and 2. $|\partial_{j}\mathcal{G}(f(x),T)|\leq\varphi_{j}$ a.e. for all $T\in\mathcal{A}_{\mathcal{Q}}(\mathbb{R}^{n})$ and $1\leq j\leq m$. Fix $r\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$ and define $f:(0,2\pi)\rightarrow\mathcal{A}_{2}(\mathbb{R}^{2})$ by $f(\theta)=\underset{z^{2}=2re^{i\theta}-1}{\sum}[[z]]$. Then $f\in W^{1,2}((0,2\pi),\mathcal{A}_{2}(\mathbb{R}^{2}))$. Proof. Let $T=\stackrel[i=1]{2}{\sum}[[T_{i}]]\in\mathcal{A}_{2}(\mathbb{R}^{2})$. Set $h(\theta)=\sqrt{2re^{i\theta}-1}$ and for $P=(P_{1},P_{2})\in\mathbb{R}^{2}\times\mathbb{R}^{2}$ denote by $d_{P}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ the function $d_{P}(x)=\sqrt{|x-P_{1}|^{2}+|x-P_{2}|^{2}}$. It is not difficult to see that for any fixed $P$, $d_{P}$ is Lipschitz. Put $\alpha_{T}(\theta)=(d_{(-T_{1},T_{2})}\circ h)(\theta),\beta_{T}(\theta)=(d_{(T_{1},-T_{2})}\circ h)(\theta)$. Note that \begin{equation} \mathcal{G}(f(\theta),T)=\frac{\alpha_{T}(\theta)+\beta_{T}(\theta)-|\alpha_{T}(\theta)-\beta_{T}(\theta)|}{2}\label{eq:} \end{equation} By Lemma <ref>, $\alpha_{T},\beta_{T}$ are a composition of a Lipschitz function on a $W^{1,2}(0,2\pi)$ function. Therefore $\alpha_{T},\beta_{T}$ are also $W^{1,2}(0,2\pi)$. In view of Equation <ref> it is now apparent that $\theta\mapsto\mathcal{G}(f(\theta),T)\in W^{1,2}(0,2\pi)$. In addition, there is at most one $\theta_{0}\in(0,2\pi)$ for which $\alpha_{T}(\theta_{0})=0$. An elementary calculation shows that the following estimate is obeyed for $\theta\in(0,2\pi)-\{\theta_{0}\}$: and $\partial_{\theta}h_{i}\in L^{2}(0,2\pi)$, $i=1,2$ by Lemma The same is true for $\beta_{T}$. Combining these estimates with equation <ref> we see that \[ |\partial_{\theta}\mathcal{G}(f(\theta),T)|\leq8(|(\partial_{\theta}h_{1})(\theta)|+|(\partial_{\theta}h_{2})(\theta)|)\in L^{2}(0,2\pi) \] for all but finitely many $\theta\in(0,2\pi)$. So strictly by definition, $f\in W^{1,2}((0,2\pi),\mathcal{A}_{2}(\mathbb{R}^{2}))$. $\square$ Proof of Proposition <ref>. That $f$ is Dir minimizing follows from Theorem <ref>. Furthermore, by Lemma <ref> $f|_{\partial B_{r}}\in W^{1,2}(\partial B_{r},\mathcal{A}_{\mathrm{2}}\mathrm{(\mathbb{R^{\mathrm{2}}\mathrm{)}}})$ for all $r\neq\frac{1}{2}$. We compute: \[ \underset{B_{\rho}(0)}{\int}|f|^{2}dx=2\stackrel[0]{\rho}{\int}\stackrel[0]{2\pi}{\int}r\sqrt{(2r\cos\theta-1)^{2}+4r^{2}\sin^{2}(\theta)}d\theta dr \] \[ \] which implies \[ \overline{H}(\rho)=\frac{1}{\pi}\stackrel[0]{2\pi}{\int}\sqrt{(2\rho\cos\theta-1)^{2}+4\rho^{2}\sin^{2}(\theta)}d\theta=\frac{1}{\pi}\stackrel[0]{2\pi}{\int}\sqrt{1+4\rho^{2}-4\rho\cos\theta}d\theta\coloneqq\frac{1}{\pi}A(\rho) \] We compute $A'''(\rho)$ for all $\frac{1}{2}<\rho<1$: \[ \] \[ \] \[ \stackrel[0]{2\pi}{\int}\frac{4(1+4\rho^{2}-4\rho\cos\theta)-(4\rho-2\cos\theta)^{2}}{(1+4\rho^{2}-4\rho\cos\theta)^{\frac{3}{2}}}d\theta=\stackrel[0]{2\pi}{\int}\frac{4\sin^{2}\theta}{(1+4\rho^{2}-4\rho\cos\theta)^{\frac{3}{2}}}d\theta \] \begin{equation} \end{equation} \[ \] We note that as long as $\frac{1}{2}<\rho<1$, the RHS of equation <ref> is strictly negative, hence the claim. $\square$ § APPENDIX This section was explained to me by Dan Mangoubi. We present here a proof of Theorem <ref>. The converse statement of Theorem <ref> is clearly not true. As an example we can take $m=1$ and $u(x)=x^{2}$. Clearly $u$ is not harmonic. However, which is obviously absolutely monotonic. Let $u$ be harmonic on $B_{1}(0)$. Let $\phi:[0,1]\rightarrow\mathbb{R}$ be smooth and nondecreasing. Define $d(r)=\underset{B_{1}(0)}{\int}u^{2}(rx)\phi'(||x||\text{\texttwosuperior})dx$. Then $d$ is absolutely monotonic. We denote by $u_{i}$ the derivative of $u$ with respect to the $i-$th variable. Define $\psi(\xi)=\phi(\frac{||\xi||^{2}}{r^{2}})$. \[ d'(r)=\underset{\text{\ensuremath{B_{1}(0)}}}{{\normalcolor \int}}[2\stackrel[\text{\ensuremath{i}=1}]{m}{\sum}u(rx)u_{i}(rx)x_{i}]\phi'(||x||\text{\texttwosuperior})dx=\underset{B_{r}(0)}{\int}[2\stackrel[\text{\ensuremath{i}=1}]{m}{\sum}u(\xi)u_{i}(\xi)\frac{\xi_{i}}{r^{m+1}}]\phi'(\frac{||\xi||^{2}}{r^{2}})d\xi= \] \[ =\frac{1}{r^{m-1}}\underset{B_{r}(0)}{\int}[2\stackrel[\text{\ensuremath{i}=1}]{m}{\sum}u(\xi)u_{i}(\xi)\frac{\xi_{i}}{r^{2}}]\phi'(\frac{||\xi||^{2}}{r^{2}})d\xi=\frac{1}{2r^{m-1}}\underset{B_{r}(0)}{\int}\nabla u^{2}\cdot\nabla\psi d\xi\overset{\ast}{=} \] \[ \frac{1}{2r^{m-1}}[\underset{\partial B_{r}(0)}{\int}\psi\nabla_{\nu}u^{2}d\sigma-\underset{B_{r}(0)}{\int}\psi\Delta u^{2}d\xi]=\frac{1}{2r^{m-1}}[\underset{\partial B_{r}(0)}{\int}\phi(1)\nabla_{\nu}u^{2}d\sigma-\underset{B_{r}(0)}{\int}\psi\Delta u^{2}d\xi]\overset{\ast\ast}{=} \] \[ =\frac{1}{2r^{m-1}}[\underset{B_{r}(0)}{\int}\phi(1)\text{\textgreek{D}}u^{2}d\xi-\underset{B_{r}(0)}{\int}\psi\Delta u^{2}d\xi]=\frac{1}{2r^{m-1}}\underset{B_{r}(0)}{\int}(\phi(1)-\psi)\Delta u^{2}d\xi \] The equality $\ast$ is by Green's identity and the equality $\ast\ast$ is by the divergence theorem. Taking advantage of the identity $\Delta u^{2}=|\nabla u|^{2}$ for $u$ harmonic we obtain \[ d'(r)=\frac{1}{2r^{m-1}}\underset{B_{r}(0)}{\int}(\phi(1)-\psi)|\nabla u|^{2}d\xi=\frac{1}{2}r\underset{B_{1}(0)}{\int}(\phi(1)-\phi(||x||^{2}))|\nabla u|^{2}(rx)dx \] Let $\Phi$ be some anti derivative of $\phi$ and define $\varphi:[0,1]\rightarrow$ by $\varphi(\rho)=\phi(1)\rho-\Phi(\rho)$. Evidently $\varphi$ is nondecreasing and according to the computation we preformed \[ \] Since each $u_{i}$ is harmonic we can iterate the the same argument in order to obtain $d^{(k)}(r)\ge0$ for all $k$. As a corollary we obtain a proof of theorem <ref>: Let $u$ be harmonic on $B_{1}(0)$. Then $\overline{H}$ is absolutely monotonic. \[ \overline{H}(r)=\frac{1}{C_{m}r^{m-1}}\underset{\partial B_{r}(0)}{\int}u^{2}(x)d\sigma=\frac{1}{C_{m}r^{m-1}}\frac{d}{dr}[\underset{B_{r}(0)}{\int}u^{2}(x)dx]=\frac{1}{C_{m}r^{m-1}}\frac{d}{dr}[\underset{B_{r}(0)}{\int}u^{2}(x)dx] \] \[ \] \[ \frac{1}{C_{m}}[m\underset{B_{1}(0)}{\int}u^{2}(rx)dx+r\frac{d}{dr}[\underset{B_{1}(0)}{\int}u^{2}(rx)dx]] \] Taking $\phi(t)=t$ in Proposition <ref>, we see that last expression is a sum of absolutely monotonic functions, and hence absolutely monotonic. $\square$ [1] Camillo De Lellis and Emanuele Nunzio Spadaro. Q-valued functions and approximation of minimal currents. PhD Thesis. 2009. [2] Sergei Bernstein. Sur la definition et les proprietes des fonctions analytiques d’unevariable reelle. Math. Ann.75(1914), no. 4, 449–468 (French). [3] Gabor Lippner and Dan Mangoubi. Harmonic functions on the lattice: Absolute monotonicity and propagation of smallness. Duke Math. J. 164, no. 13. 2015. [4] Camillo De Lellis. Private [5] Frederick J. Almgren. Almgren's big regularity paper.World Scientific. 2000. [6] S. Agmon. Unicit´e et convexit´e dans les probl`emes diff´erentiels. S´eminaire de Math´ematiques Sup´erieures, No. 13 (´Et´e, 1965). Les Presses de l’Universit´e de Montr´eal, Montreal, Que. 1966. [7] G. Po´lya and G. Szego. Problems and theorems in analysis. Vol. I: Series, integral calculus, theory of functions, Springer-Verlag, New York, 1972. Translated from the German by D. Aeppli; Die Grundlehren der mathematischen Wissenschaften, Band 193. Immanuel Ben Porat Einstein Institute of Mathematics, The Hebrew University of Jerusalem Edmond J. Safra Campus; Givat Ram, Jerusalem, Israel; 9190401

2024-09-04T02:54:55.763060

2003.00094

arxiv-papers

Improved Algorithm for Min-Cuts in Distributed Networks COMPUTER SCIENCE AND ENGINEERING # Improved Algorithm for Min-Cuts in Distributed Networks Mohit Daga (OCTOBER 2017) ###### Abstract KEYWORDS: Connectivity, Reliability, Edge Min-Cuts In this thesis, we present fast deterministic algorithm to find small cuts in distributed networks. Finding small min-cuts for a network is essential for ensuring the quality of service and reliability. Throughout this thesis, we use the CONGEST model which is a typical message passing model used to design and analyze algorithms in distributed networks. We survey various algorithmic techniques in the CONGEST model and give an overview of the recent results to find cuts. We also describe elegant graph theoretic ideas like cut spaces and cycle spaces that provide useful intuition upon which our work is built. Our contribution is a novel fast algorithm to find small cuts. Our algorithm relies on a new characterization of trees and cuts introduced in this thesis. Our algorithm is built upon several new algorithmic ideas that, when coupled with our characterization of trees and cuts, help us to find the required min- cuts. Our novel techniques include a _tree restricted semigroup function (TRSF)_ , a novel _sketching_ technique, and a _layered algorithm_. TRSF is defined with respect to a spanning tree and is based on a commutative semigroup. This simple yet powerful technique helps us to deterministically find min-cuts of size one (bridges) and min-cuts of size two optimally. Our sketching technique samples a small but relevant vertex set which is enough to find small min-cuts in certain cases. Our layered algorithm finds min-cuts in smaller sub-graphs _pivoted_ by nodes at different levels in a spanning tree and uses them to make the decision about the min-cuts in the complete graph. This is interesting because it enables us to show that even for a global property like finding min-cuts, local information can be exploited in a coordinated manner. This is to certify that the thesis titled Improved Algorithm for Min-Cuts in Distributed Networks, submitted by Mohit Daga, to the Indian Institute of Technology, Madras, for the award of the degree of Master of Science, is a bona fide record of the research work done by him under my supervision. The contents of this thesis, in full or in parts, have not been submitted to any other Institute or University for the award of any degree or diploma. Dr. John Augustine Research Guide Associate Professor Dept. of CSE IIT-Madras, 600 036 Place: Chennai Date: ###### Acknowledgements. First and foremost, I would like to extend my sincere gratitude to my thesis supervisor Dr. John Augustine. I am thankful to John for his constant encouragement, guidance and optimism. John actively helped me in improving my technical writing and presentation with lot of patience. Our countless discussions about research (we literally burnt the midnight oil) enabled the timely completion of this thesis. His opinion and vision about a career in research and academia are well placed. I also closely collaborated with Dr. Danupon Nanongkai, who introduced me to the problem covered in this thesis. I thank him for inviting me to KTH in April’17. Danupon also reviewed a draft of this work and gave me valuable inputs. During the initial days at IIT-Madras, I interacted with Dr. Rajseakar Manokaran, in his capacity as my faculty adviser. He inspired me to pursue research in Theoretical Computer Science. I thank Dr. N.S. Narayanaswamy, chairperson of my General Test Committee (GTC) for being critical of my seminar presentations and helping me to grow as a researcher. I also thank Dr. Krishna Jagannathan and Dr. Shweta Agrawal for accepting to be part of my GTC. I feel proud to be part of IIT Madras and CSE Department. I thank the administration here for establishing tranquility and serenity of campus after the devastating 2016 cyclone. I thank the support staff of my department. Special thanks to Mr. Mani from the CSE office for scheduling the required meetings and helping me to understand the procedures and requirements with patience. During my stay at IIT Madras, I was part of two groups: ACT Lab and TCS Lab. I thank my lab mates for maintaining good work culture. My various discussions on a wide range of topics with my lab mates were enjoyable and enriching. This experience will be helpful when I move to Stockholm for my Ph.D. studies. Last but not the least, I thank my mother Beena, my father Manohar and my sisters: Kaushal and Sheetal. This thesis is dedicated to them. ###### Contents 1. NOTATION 2. 1 Introduction 1. 1.1 Notations and Conventions 2. 1.2 Distributed Model 3. 1.3 Classification of network problems based on locality 4. 1.4 Overview of past results 5. 1.5 Organization of this thesis 3. 2 Preliminaries and Background 1. 2.1 Simple algorithms in Distributed Networks 1. 2.1.1 Tree Constructions and Notations 2. 2.1.2 Convergecast and Broadcast 2. 2.2 Cuts Space and Cycle Space 3. 2.3 Min-Cuts in Distributed Networks 1. 2.3.1 Using Greedy Tree Packing 2. 2.3.2 Using Random Circulations 4. 3 Technical Overview 1. 3.1 Characterization of Trees and Cuts 2. 3.2 Our Contribution 5. 4 Tree Restricted Semigroup Function 1. 4.1 Min-Cuts of size 1 2. 4.2 Min-Cuts of size 2 6. 5 Min-Cuts of size three 1. 5.1 Graph Sketching 1. 5.1.1 Definition of Sketch 2. 5.1.2 Algorithm to Compute Sketch 3. 5.1.3 Application of graph sketch 2. 5.2 Layered Algorithm 1. 5.2.1 Definition of Layered Algorithm 2. 5.2.2 Application of layered algorithm 3. 5.3 Summary 7. 6 Future Work ###### List of Tables 1. 5.1 Overview of the case-structure of min-cut of size $3$ ###### List of Figures 1. 3.1 Roadmap for finding small min-cuts 2. 5.1 Different cases of min-cuts of size three 3. (a) CASE-2 (Either of $\alpha$ or $\beta$ is 1 and the other is $0$) 4. (b) CASE-3 (Either of $\alpha$ or $\beta$ is 1 and the other is $0$) 5. (c) CASE-4 (Both $\alpha$ and $\beta$ are non-zero) 6. (d) CASE-5 (At least two of $\alpha$, $\beta$ and $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})$ are non-zero) 7. (e) CASE-6 (at least two of $\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow})$, $\gamma({v_{3}}^{\downarrow},{v_{2}}^{\downarrow})$, $\gamma({v_{1}}^{\downarrow},{v_{3}}^{\downarrow})$ are non-zero) 8. (f) CASE-7 (Both $\alpha$ and $\beta$ are non-zero) 9. 5.2 Motivation for Sketching Technique in CASE-3 10. 5.3 Motivation for Sketching Technique in CASE-7 11. 5.4 Branching Number 12. 5.5 Different types of sub-cases which occur when there exists a min-cut of size $3$ as in CASE-3 13. 5.6 Two different non-isomorphic sub-cases of a min-cut as given by CASE-6 14. (a) One of $\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow}),\gamma({v_{1}}^{\downarrow},{v_{3}}^{\downarrow}),\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})$ is zero. WLOG let $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})=0$ 15. (b) All three of $\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow}),\gamma({v_{1}}^{\downarrow},{v_{3}}^{\downarrow}),\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})$ are non-zero 16. 5.7 3-Sketch of different cases as viewed from node $v$ 17. 5.8 Reduced sketch ${\mathcal{S}}^{2}(v_{2},v_{3})$ when for some $v_{1},v_{2},v_{3}\in V\setminus r$ as in CASE-7 18. 5.9 Min-cut of size $3$ as given by CASE-5 IITM | Indian Institute of Technology, Madras ---|--- TRSF | Tree Restricted Semigroup function ## NOTATION ${v}^{\downarrow{T}}$ | set of descendants of vertex $v$ (including itself) in some tree $T$ ---|--- $\mathcal{A}_{T}\left(v\right)$ | set of ancestors of vertex $v$ (including itself) in some tree $T$ $\pi_{{T}}\left(v\right)$ | parent of vertex $v$ in some tree $T$ $\operatorname{children}_{T}(v)$ | children of node $v$ in some tree $T$ $\delta(A)$ | cut set induced by vertex set $A$ $\eta_{T}(v)$ | $|\delta({v}^{\downarrow{T}})|$ $\gamma(A,B)$ | $|\delta(A)\cap\delta(B)|$, where $A,B$ are vertex set $\ell_{T}\left(v\right)$ | level of node $v$ in a rooted tree $T$, root at level $0$ $\alpha_{T}(x,l)$ | ancestor of vertex $x$ at level $l$ $R_{T}(v)$ | canonical tree of a node $v$ with respect to the spanning $T$ $\mathcal{S}_{T}^{k}(v)$ | $k$-Sketch of node $v$ w.r.t a spanning tree $T$ $\mathcal{\xi}_{{T}}\left(v\right)$ | branching number of node $v$ w.r.t some tree $T$ ${\mathcal{S}}_{T}^{k}(v,a)$ | reduced $k$-Sketch of a node $v$ and some $a\in{v}^{\downarrow{T}}\setminus v$ w.r.t a spanning tree $T$ $\rho_{T}(v)$ | a path from the root to node $v$ in the tree $T$ $\operatorname{LCA}_{T}(v_{1},v_{2})$ | Lowest Common Ancestor of two node $v_{1},v_{2}$ in some tree T $\overline{\mathcal{N}}_{T}({A})$ | $\left\\{y\mid x\in A,y\notin A,(x,y)\in E,(x,y)\ \text{is a non-tree edge in tree }T\right\\}$ $\mathcal{P}_{T}(A)$ | $\left\\{\rho_{T}(a)\mid a\in A\right\\}$ ## Chapter 1 Introduction Networks of massive sizes are all around us. From large computer networks which are the backbone of the Internet to sensor networks of any manufacturing unit, the ubiquity of these networks is unquestionable. Each node in these kinds of networks is an independent compute node or abstracted as one. Further, nodes in these networks only have knowledge of their immediate neighbors, but not the whole topology. There are many applications which require us to solve problems related to such networks. To solve such problems, a trivial idea is as follows: Each node sends its adjacency list to one particular node through the underlying communication network; that node then solves the required problem using the corresponding sequential algorithm. Such an idea may not be cost effective due to the constraints of the capacity of links and large sizes of these networks. The question then is: Does there exist a faster cost-effective algorithm through which nodes in a network can coordinate among each other to find the required network property? Such a question is part of a more generalized spectrum of _Distributed Algorithms_. In this thesis, we give fast distributed algorithm for finding small min-cuts. An edge cut is the set of edges whose removal disconnects the network and the one with a minimum number of edges is called as a _min-cut_. Finding small min-cuts has applications in ensuring reliability and quality of service. This enables the network administrator to know about the critical edges in the network. Our approach to finding the min-cut is based on our novel distributed algorithmic techniques and backed by new characterizations of cuts and trees. Our algorithmic techniques are as follows: * • _Tree Restricted Semigroup Function_ wherein a semigroup function is computed for each node in the network based on a rooted spanning tree * • _Sketching Technique_ : which allows us to collect small but relevant information of the network in a fast and efficient manner * • _Layered Algorithm_ which finds min-cuts in smaller sub-graphs and strategically uses the information to make the decision about min-cuts in the complete graph. ### 1.1 Notations and Conventions We consider an undirected, unweighted and simple graph $G=(V,E)$ where $V$ is the vertex set and $E$ is the edge set. We use $n$ and $m$ to denote the cardinalities of $V$ and $E$ respectively. Edges are the underlying physical links and vertices are the compute nodes in the network. We will use the term vertex and node interchangeably. Similarly, the terms edges and links will be used interchangeably. A cut $C=(A,V\setminus A)$, where $\emptyset\subsetneq A\subsetneq V$ is a partition of the vertex set into two parts. The cut-set is the edges crossing these two parts. A min-cut is a cut-set with a minimum number of edges. Let $d_{G}(u,v)$ be the distance between any two nodes $u,v$ in the network which is defined by the number of edges in the shortest path between them. We will consider connected networks, so $d_{G}(u,v)$ is guaranteed to be well-defined and finite. The diameter of the graph $G$ is defined as $\max\limits_{u,v\in V}d_{G}(u,v)$. We will use $D$ to denote the diameter of the graph and $n$ for the size of the vertex set and $m$ for the size of edge set. In a typical communication network, the diameter $D$ is much smaller than the number of vertices $n$. This is because the networks are often designed in such a way that there is an assurance that a message sent from any node to any other node should not be required to pass through too many links. ### 1.2 Distributed Model Different types of communication networks have different parameters. This is primarily based on how communication takes place between any two nodes in a distributed system. For the algorithm designers, these details are hidden and abstracted by appropriate modeling of the real world scenario. An established book in this domain titled ‘Distributed Computing: A Locality-Sensitive Approach’ [23] remarks: “ _The number of different models considered in the literature is only slightly lower than the number of papers published in this field_ ”. But certainly, over the years, the research community working around this area have narrowed down to a handful of models. Out of them, the CONGEST model has been quite prominent and captures several key aspects of typical communication networks. In the CONGEST model, as in most other distributed networks, each vertex acts as an independent compute node. Two nodes may communicate to each other in synchronous rounds only if they have a physical link between them. The message size is restricted to $O(\log n)$ bits across each link in any given round. We will focus our attention on communication which is more expensive than computation. In the CONGEST model, the complexity of an algorithm is typically measured in terms of the number of rounds required for an algorithm to terminate. Parallels of this could be drawn to the time complexity of sequential algorithms. Much of the focus has been on the study of the distributed round complexity of algorithms but recently there has been some renewed interest to study the message complexity (the sum total of messages transmitted across all edges) as well [22, 4]. Apart from these some recent research is also being directed towards the study of asymptotic memory required at each node during the execution of the algorithm [5]. In this thesis, we will limit our focus on the analysis of round complexity. ### 1.3 Classification of network problems based on locality Any problem pertaining to a communication network may depend either on local information or global information. Local information of a node comprises knowledge of immediate neighbors or nodes which are just a constant number of hops away. Global information includes knowledge of nodes across the network. One of the prominent examples of a problem which just requires local information is of finding a _maximal independent set_ (MIS). An _independent set_ is a set of vertices such that no two vertices in the set have an edge between them. An MIS is an independent set such that no vertex can be added into it without violating independence. It has been shown by [18] that finding MIS requires $O(\log n)$ rounds. On the other side, to gather global information at all nodes, it takes at least $D$ rounds. This is because the farthest node could be as much as $D$ distance away. Thus the complexity of algorithms which require global information is $\Omega(D)$. Simple problems like finding the count of the number of nodes in the network, constructing a BFS tree, leader election etc are problems which require global information and takes $O(D)$ rounds. Apart from this, there are some problems which require global information and have been proved to take at least $\Omega(D+\sqrt{n})$ rounds. A prominent example is that of finding a Minimum Spanning Tree (MST). It has been shown that to even find an approximate MST we require $\Omega(D+\sqrt{n})$ rounds [3]. One of the key idea used in most of the algorithms which take $O(D+\sqrt{n})$ is a result from [17], which gives a distributed algorithm that can partition nodes of any spanning tree $T$ into $O(\sqrt{n})$ subtrees such that each subtree has $O(\sqrt{n})$ diameter. Basically, this approach can be considered as a classical divide and conquer paradigm of distributed algorithms. To solve any problem using this idea, nodes are divided into $O(\sqrt{n})$ clusters and then the problem is solved in each of the clusters and subsequently, a mechanism is given to combine the results. ### 1.4 Overview of past results In this section, we will give an overview of the current progress in finding a min-cut. A more detailed technical review of the relevant historical results will be given in next chapter. In the centralized setting, the problem of finding min-cut has seen a lot of advances [11, 13, 12, 15, 20, 19, 6, 27, 14]. Recently, in the centralized setting even a linear time algorithm have been proposed for finding min-cut by [16] and further improved by [9]. A few attempts at finding min-cuts in the distributed setting (in the CONGEST model in particular) have been made in past. These include work by [1, 29, 30, 24, 25, 21, 7]. A decade-old result by [25], currently stands out as the most efficient way to find min-cut of small sizes in particular of size one. They gave an $O(D)$ round algorithm using random circulation technique for finding min-cuts of size $1$. Their result of finding a min-cut of size one has also been extended to give a Las Vegas type randomized algorithms to find min-cut of size 2 in $O(D)$ rounds. But a deterministic analogue of the same has so far been elusive. More recently [21] have given a $O((\sqrt{n}\log^{*}n+D)k^{4}\log^{2}n)$111$\log^{*}n$ (usually read ”log star”), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to $1$ rounds algorithm to find min-cut of size $k$. There have also been results to find an approximate min-cut. [8] gave a distributed algorithm that, for any weighted graph and any $\epsilon\in(0,1)$, with high probability finds a cut of size at most $O(\epsilon^{-1}k)$ in $O(D)+\tilde{O}(\sqrt{n})$ 222$\widetilde{O}$ hides logarithmic factors rounds, where $k$ is the size of the minimum cut. [21] improves this aproximation factor and give a $(1+\epsilon)$ approximation algorithm in $O((\sqrt{n}\log^{*}n+D)\epsilon^{-5}\log^{3}n)$ time. Also, [7] have given a min-cut algorithm for planar graphs which finds a $(1-\epsilon)$ approximate min-cut in $\widetilde{O}(D)$ rounds. While a linear time algorithm for finding min-cut exists in the centralized setting, in distributed setting the best known exact algorithm still takes $O((\sqrt{n}\log^{*}n+D)k^{4}\log^{2}n)$ rounds to deterministically find a min-cut of size $k>1$. Moreover, a $\widetilde{O}(D)$ round algorithm exists to find a min-cut in planar graphs. The following question arises: Is $\Omega(\sqrt{n}+D)$ a lower bound for finding min-cuts or can we save the large factor of $\sqrt{n}$ as known for planar graphs for at least the small min-cuts. We answer this question through our thesis. We present a new algorithm to deterministically find all the min-cuts of size one, two, and three. For min-cut of size either one or two our algorithm takes $O(D)$ rounds. For min-cuts of size three our algorithm takes $O(D^{2})$ rounds. ### 1.5 Organization of this thesis In Chapter 2, we will survey simple distributed algorithms for the CONGEST model and also review graph properties like the cuts spaces and cycle spaces. We also give a brief overview of previously know techniques to find min-cut in distributed setting, in particular the techniques based on greedy tree packing and random circulations. In Chapter 3, we will present our characterization of trees and cuts. Here we will also give a road-map of the novel distributed techniques introduced in this thesis. In Chapter 4 and Chapter 5, we will present distributed algorithms which will ensure that whenever there exists a min-cut of the kind the required quantities by our charecterization will be communicated to at least one node in network. In Chapter 4, we introduce _Tree Restricted Semigroup Function_ and show that this is enough for finding min-cuts of size $1$ and $2$ optimally. Further, in Chapter 5, we will introduce two new techniques: _Sketching_ and _Layered Algorithm_ to find min-cuts of size $3$. ## Chapter 2 Preliminaries and Background In this chapter, we will review fundamental network algorithms in the distributed setting. Among the distributed algorithms, we will discuss the construction of various types of spanning trees including Breadth First Search (BFS) Tree, Depth First Search (DFS) tree and Minimum Spanning Tree (MST). Further, we will discuss some simple tree based algorithmic paradigms generally used in distributed networks. We also review important and relevant concepts like cut spaces and cycle spaces and show that these are vector spaces and orthogonal to each other. Finally, we will survey two important recent techniques used to find min-cuts in distributed networks. ### 2.1 Simple algorithms in Distributed Networks As mentioned in the earlier chapter, we use the CONGEST model of distributed networks which considers point-to-point, synchronous communications between any two nodes in a network. In any given synchronous round, communication is only allowed between nodes with a physical link and is restricted to $O(\log n)$ bits. The network will be modeled as a simple undirected graph $G=(V,E)$. As stated earlier, we will use $n$ to denote the number of nodes, $m$ to denote the number of edges and $D$ as the diameter of the network. Each node in this network has a unique $id$ which is of size $O(\log n)$ bits. Whenever we say node $x$ in the network, we mean a node with $x$ as its $id$. Various algorithms in distributed setting start with constructing a tree. A tree gives structure to the problem by defining a relationship between nodes as decedents or as ancestors or neither of them. Moreover, it also serves as a medium for information flow between nodes in a structured way using the tree edges. In this section, we state results about the construction of various types of spanning trees and review simple distributed algorithms on trees. #### 2.1.1 Tree Constructions and Notations Let ${T}$ be any spanning tree rooted at node $r_{T}$ (or just $r$ when clear from the context). Let $\ell_{T}\left(x\right)$ be the level of node $x$ in $T$ such that the root $r$ is at level $0$ and for all other nodes $x$, $\ell_{T}\left(x\right)$ is just the distance from root following the tree edges. For $x\neq r$, let $\pi_{{T}}\left(x\right)$ denote the parent of the vertex $x$ in $T$. For every node $v$, let $\operatorname{children}_{T}(v)$ be the set of children of $v$ in the tree $T$. Further for any node $v$, we will use $\mathcal{A}_{T}\left(v\right)$ to denote the set of all ancestors of node $v$ in tree $T$ including $v$ and ${v}^{\downarrow{T}}$ as the set of vertices which are descendants of $v$ in the tree $T$ including $v$. We will briefly review construction of different types of trees. For details the reader is referred to [23]. ##### BFS Tree. Our approach to find min-cuts uses an arbitrary BFS tree. Although, our technique is invariant of the type of the spanning tree chosen but a BFS tree is ideal because it can be constructed in just $O(D)$ rounds. The following lemma states the guarantees regarding construction of a BFS trees ###### Lemma 2.1. There exists a distributed algorithm for construction of a BFS tree which requires at most $O(D)$ rounds. A brief description of an algorithm to construct BFS tree as follows: an arbitrary node $r$ initiates the construction of BFS tree, it assigns itself a level $\ell\left(r\right)=0$. Then the node $r$ advertises to all its adjacent neighbors about its level, who then joins the BFS tree at level $1$ as children of $r$. Subsequently, each node at level $1$ also advertises its level to all its neighbors, but only those nodes join the BFS tree as children of nodes at level $1$ who are not part of the BFS tree already. Here, ties are broken arbitrarily. This process goes on until all the nodes are part of the BFS tree. ##### DFS Tree. Unlike BFS tree, the depth of a DFS tree is oblivious of the diameter. In fact, two of the early approaches to find cut edges in distributed networks used a DFS tree [10, 1]. The following lemma states the guarantees regarding construction of a DFS trees. ###### Lemma 2.2. There exists a distributed algorithm for construction of a DFS tree which requires at most $O(n)$ rounds. ##### MST Tree. The following lemma is known regarding the complexity to construct an MST tree. In fact, it is also known that we cannot do better in terms of round complexity for construction of the MST tree. [3]. ###### Lemma 2.3. For a weighted network, there exists a distributed algorithm for construction of MST tree which requires $O(\sqrt{n}+D)$ rounds. A recent technique to find min-cuts given by [21] assigns weights to each edge in an unweighted graph and compute MST tree, further these weights are updated and a new MST tree is constructed. This goes on for $\text{polylog }n$ iterations. We review this technique in Section 2.3. #### 2.1.2 Convergecast and Broadcast We will now give brief details about simple distributed algorithms. In particular, we will describe _Broadcasts_ and _Convergecasts_. These are mechanisms to move data along the edges of any fixed tree. In this thesis, we will often use these techniques to compute properties of the given network which will, in turn, help us to find min-cuts. ##### Convergecasts In distributed network algorithms, there are applications where it is required to collect information upwards on some tree, in particular, this is a flow of information from nodes to their ancestors in the tree. This type of a technique is called as _Convergecast_. In a BFS tree $T$, any node has at most $D$ ancestors, but for any node $v$ the number of nodes in descendants set ${v}^{\downarrow{T}}$ is not bounded by $D$. Thus during convergecast, all nodes in the set ${v}^{\downarrow{T}}$ might want to send information upwards to one of the ancestor node $v$. This may cause a lot of contentions and is a costly endeavor. In this thesis, whenever we use a covergecast technique, we couple it with either an _aggregation based strategy_ or a _contention resolution mechanism_. In an aggregation based strategy, during covergecast, the flow of information from nodes at a deeper level is synchronized and aggregated together to be sent upwards. In Chapter 4, we introduce _tree restricted semigroup function_ which is such an aggregation based technique. Moreover, our algorithm to compute the specially designed _Sketch_ also uses this kind of convergecasting. Further, in a contention resolution mechanism, only certain information are allowed to move upwards based on some specific criteria, thus limiting contention. In Chapter 5, we use such contention resolution mechanism to convergecast details found by our _Layered Algorithm_. ##### Broadcasts In this technique, the flow of information takes place in the downward direction along the tree edges. When such an information is aimed to be communicated to all the nodes in the decedent set then it is called _broadcast_. In the general broadcast algorithm, for some spanning tree $T$, the root is required to send a message of size $O(\log n)$ bits to all the nodes. We state the following lemma from [23] whose proof uses a simple flooding based algorithm. ###### Lemma 2.4 (Lemma 3.2.1 from [23]). For any spanning tree $T$, let the root have a message of size $O(\log n)$ bits, then it can be communicated to all the nodes in $O(Depth(T))$ rounds. ###### Proof. Here we assume that there exists a spanning tree $T$ and each node is aware of the spanning tree edges incident to it. Now, all we need is to flood the message to the entire tree. In the first round, the root sends this message to all its children, who are at level $1$. Further, in the second round, nodes at level $1$ sends the message to all their children who are at level $2$ and so on. Thus, in all, we will require $O(Depth(T))$ rounds. ∎ At various stages of the algorithms presented in this thesis, we require two variants of the general broadcast techniques. They are described below. We also give the respective algorithm for them in Lemma 2.5. 1. _Broadcast Type-1_ Given a spanning tree $T$, all nodes $v\in V$ have a message of size $O(\log n)$ bits which is required to be communicated to all the nodes in the vertex set ${v}^{\downarrow{T}}$. 2. _Broadcast Type-2_ Given a spanning tree $T$, all nodes $v\in V$ have $Depth(T)$ messages of size $O(\log n)$ bits which is required to be communicated to all the nodes in the vertex set ${v}^{\downarrow{T}}$. ###### Lemma 2.5. There exist distributed algorithms which require $O(Depth(T))$ and $O(Depth(T)^{2})$ rounds respectively for Broadcast Type-1 and Broadcast Type-2. ###### Proof. First, let us consider _Broadcast Type-1_. Here we will use Lemma 2.4 coupled with _pipelining_. Each node $v$ initiates a broadcast of its $O(\log n)$ bits sized message to the subtree rooted at node $v$. using the algorithm from Lemma 2.4 in the first round. If $v$ is not the root of the spanning tree, then in the first round it will have received the message broadcasted by its parent and it is imperative to node $v$ to send this message to all its children. Note that this does not hinder with the broadcast initiated by node $v$ to the subtree rooted at it. In all the subsequent rounds a train of messages is _pipelined_ one after another through the tree edges, enabling broadcasts initiated by nodes to run without any hindrance or congestion. Thus it only takes $O(Depth(T))$ rounds for this type of broadcast. For _Broadcast Type-2_ , we just need to run $Depth(T)$ iterations of _Broadcast Type-1_ taking $O(Depth(T)^{2})$ rounds. ∎ ### 2.2 Cuts Space and Cycle Space In this section, we will discuss about the _cut space_ and _cycle spaces_. We will first define _cut spaces_ , _cycle spaces_ and prove that, they are vector spaces. Further, we will show that they are orthogonal. All these facts are well known and have been part of standard graph theory books, for example, see [2]. As defined earlier, $E$ and $V$ are edge set and vertex set respectively. We will work with the field $\mathbb{Z}_{2}$ and the space $\mathbb{Z}_{2}^{|E|}$. Let $S\subseteq E$, then its characteristics vector is defined as follows: $\chi_{e}^{S}=1$ (at coordinate $e$) for $e\in S$ and $\chi_{f}^{S}=0$ for $f\notin E$. Here we will use the operator $\oplus$ (symmetric difference or XOR) for the space $\mathbb{Z}_{2}^{|E|}$. Let $R,S\subseteq E$. Note that $R,S$ are edge sets whereas $\chi^{R}$ and $\chi^{S}$ are binary vectors in $\mathbb{Z}_{2}^{|E|}$. When we use $R\oplus S$, we will mean symmetric difference of the two sets and when we use $\chi^{R}\oplus\chi^{S}$, we will mean XOR operation between the two vectors. It is easy to see that $\chi^{R}\oplus\chi^{S}=\chi^{R\oplus S}$ Let $\phi\subseteq E$. If the subgraph $(V,\phi)$ has all the vertex with even degrees then $\phi$ is a _binary circulation_. The set of all binary circulations of a graph is called the _cycle space_ of the graph. Similarly, let $A\subseteq V$. Define an induced cut $\delta(A)$ as the set of edges with exactly one endpoint in $A$ or essentially the cut set $(A,V\setminus A)$. The set of all the induced cuts is called the _cut space_ of the graph. ###### Theorem 2.6. The cut space and cycle space are vector spaces of $\mathbb{Z}_{2}^{|E|}$ ###### Proof. To show that both the structures are vector spaces it is sufficient to show that there exists a $0$ for the vector space, an addition operator and inverse of each element. We will show that $\emptyset_{E}$ is the required $0$ of the vector space and $\oplus$ is the operator. For cut space take a set $A=\emptyset$. Then $\delta(A)=\emptyset_{E}$ that is $\chi^{\delta(A)}=\vec{0}_{2}$. Similarly for cycle space if take $\phi=\emptyset\subset E$ then all the vertex in the graph $(V,\phi)$ have degree as $0$ which is even. Let $A,B\subseteq V$. Then $\delta(A)$ and $\delta(B)$ are induced cut defined for A,B. Let $A\oplus B$ be the symmetric difference of the set $A$ and $B$. As per the definition $\delta(A)$ is the set of edges which have exactly one end in $A$ and similarly for $\delta(B)$ which are the edges with one end in $B$. Now $A\oplus B$ is a symmetric difference between the set $A,B$ and $\delta(A\oplus B)$ is the set of edges with one end in $A\oplus B$. To see that $\delta(A)\oplus\delta(B)$ is equal to $\delta(A\oplus B)$ just take two cases where $A\cap B=\emptyset$ and $A\cap B\not=\emptyset$. Now let us turn our attention to the cycle space. Let $R,S\subseteq E$. And let $G_{1}=(V,R)$ and $G_{2}=(V,S)$ be two sub-graphs such that all vertices are of even degrees in them; that is both $R,S$ are binary circulations. Now consider the subgraph $G_{1,2}=(V,R\oplus S)$ and a vertex $v\in V$. The degree of vertex $v$ in $G_{1,2}$ is $\deg_{R}(v)+\deg_{S}(v)-2\deg_{R\cap S}(v)$ which is even. Hence $R\oplus S$ is also a binary circulation. ∎ The following corollary will be useful for proving our characterization of cuts and trees given in Chapter 3. ###### Corollary 2.7. Let $A_{1},A_{2},\ldots,A_{j}$ be set of vertices such that $\forall i\ A_{i}\subset V$ then $\delta(A_{1}\oplus A_{2}\oplus\ldots\oplus A_{j})=\delta(A_{1})\oplus\delta(A_{2})\oplus\ldots\oplus\delta(A_{j})$ ###### Theorem 2.8. The cut space and cycle space are orthogonal. ###### Proof. Let $A\subset V$ and $\phi$ be a binary circulation. Let $\chi^{\phi}$ and $\chi^{\delta(A)}$ be the vectors corresponding to the $\phi$ and induced cut $\delta(A)$. We have to show that $\chi^{\delta(A)}.\chi^{\phi}=0\left(\mod 2\right)$ which is equivalent to showing that $|\chi^{\phi}\cap\chi^{\delta(A)}|$ is even. Now $\sum_{a\in A}\deg_{\phi}(a)=\sum_{a\in A}|\phi\cap\delta(a)|$. Observe that $\sum_{a}\deg_{\phi}(a)$ is even because $\phi$ is a binary circulation. Further $\sum_{a\in A}|\phi\cap\delta(a)|$ counts the edges with both ends in $A$ twice and every edge in $|\phi\cap\delta(A)|$ once and does not count any other edge. Hence $|\phi\cap\delta(A)|$ is even. ∎ ### 2.3 Min-Cuts in Distributed Networks In this section we will review some past results for finding min-cuts, in particular, the ideas employed in two recent works [26, 21]. While [21] relied on greedy tree packing introduced by [28]; [26] gave a novel technique of random circulations. #### 2.3.1 Using Greedy Tree Packing First, we will define the concept of greedy tree packing. A _tree packing_ $\mathbb{T}$ is a multiset of spanning trees. Let the load of any edge $e$ w.r.t a tree packing $\mathbb{T}$, denoted by $\mathcal{L}^{\mathbb{T}}(e)$, be defined as the number of trees in $\mathbb{T}$ containing $e$. A tree packing $\mathbb{T}=\left\\{T_{1},\ldots,T_{j}\right\\}$ is greedy if each $T_{i}$ is a minimum spanning tree (MST) with respect to the loads induced by $\left\\{T_{1},\ldots,T_{i-1}\right\\}$. [28] has given the following results related to tree packing: ###### Theorem 2.9 ([28]). Let $G=(V,E)$ be an unweighted graph. Let $m$ be the number of edges in $G$ and $k$ be the size of min-cut. Then a greedy tree-packing with $\omega(k^{7}\log^{3}m)$ contains a tree crossing some min-cut exactly once. Based on the above theorem, [21] construct a greedy tree packing of $\omega(k^{7}\log^{3}m)$ trees. There is a well-known algorithm to find MST in $O(D+\sqrt{n})$ rounds. Further, in each tree, the authors find the size of the smallest edge cut which shares only one edge with the tree. They give an algorithm to do the same in $O(D+\sqrt{n})$ rounds for each tree. Note that they cannot find all the min-cuts because of the limitations of Theorem 2.9 which only guarantees that there will exists some (but not all) min-cut which will share exactly one edge with at least one tree in tree packing $\mathbb{T}$. #### 2.3.2 Using Random Circulations The random circulation technique has its foundation based on a well-known fact that cut spaces and cycle spaces are orthogonal to each other [2]. Based on this technique [25] gave the following result ###### Theorem 2.10 ([25]). There is a Las Vegas distributed algorithm to compute all the min-cuts of size 1 and 2 (cut edges and cut pairs) in $O(D)$ time. The above result cannot be made deterministic for min-cuts of size 2. Moreover extending the random circulation technique for min-cuts of size $3$ or more do not seem plausible due to its fundamental limitations. But as shown in [24], 1-cuts can be found deterministically in $O(D)$ rounds. ## Chapter 3 Technical Overview We give deterministic algorithm for finding min-cuts of size one, two, and three. We find the min-cut of size one and two in $O(D)$ time and of size three in $O(D^{2})$ time. We recreate the optimal result for min-cut of size one. For min-cuts of size two and three our results resolve the open problem from [26]. We give a new characterization involving trees and min-cuts. We also introduce a new algorithmic technique named as the Tree restricted Semigroup function (TRSF) which is defined with respect to a tree $T$. The TRSF is a natural approach to find min-cuts of size one. We also show a non- trivial application of TRSF in finding min-cuts of size two, which is quite intriguing. For finding, if there exists a min-cut of size $3$ we introduce a new _sketching technique_ where each node finds a view of the graph with respect to a tree by getting rid of edges which may not be a part of a min- cut. The sketching idea coupled with our characterization results helps us to find the required min-cut. ### 3.1 Characterization of Trees and Cuts In this subsection, we establish some fundamental relationships between trees and cuts which will help us find the required min-cuts. When a min-cut shares $k$ edges with a spanning tree then we say that it $k$-respects the tree. Each min-cut will at least share one edge with a spanning tree otherwise the said min-cut will not be a cut because there will exist a spanning tree which connects all the nodes. We give the following simple observation in this regard. ###### Observation 3.1. For any induced cut of size k and a spanning tree $T$, there exists some $j\in[1,\min(k,n)]$ such that induced cut $j-$ respects the tree $T$. The above simple observation lets us break the problem of finding small min- cuts into different cases. A min-cut of size $1$ will always share one edge with any spanning tree. A min-cut of size $2$ shares either $1$ or $2$ edges with any spanning tree. Similarly a min-cut of size $3$ either shares $1,2$ or $3$ edges with any spanning tree. We now make a simple observation along with a quick proof sketch ###### Observation 3.2. When an induced cut of size $k$ 1-respects a spanning tree $T$, then there exists at least one node $v$ such that $|\delta({v}^{\downarrow{T}})|=k$. ###### Proof Sketch. Consider the cut edge $(u,v)$ that is in $T$ with $u$ being closer to the root $r_{T}$. Since the tree only 1-respects the cut, ${v}^{\downarrow{T}}$ remains on one side of the cut, while $r_{T}$ is on the other. Moreover, $V\setminus{v}^{\downarrow{T}}$ fully remains on the side with $r_{T}$; otherwise, $T$ will not be limited to 1-respecting the cut. Thus, ${v}^{\downarrow{T}}$ will serve our required purpose. ∎ We will now give a distributed algorithm such that each node $v$ finds $|\delta({v}^{\downarrow{T}})|$ in $O(D)$ rounds. For the remaining cases, we will give lemmas to characterize the induced cuts and tree. These characterizations follow from Corollary 2.7. We begin with the characterization of an induced cut of size 2 when it 2-respects a tree $T$. ###### Lemma 3.3 (2-cut, 2-respects a spanning tree $T$). Let $T$ be any spanning tree. When $u\neq v$ and $u,v\in V\setminus r$. Then the following two statements are equivalent. 1. $P_{1}:$ $\left\\{(\pi_{{T}}\left(u\right),u),(\pi_{{T}}\left(v\right),v)\right\\}$ is a cut set induced by ${{u}^{\downarrow{T}}}\oplus{{v}^{\downarrow{T}}}$. 2. $P_{2}:$ $|\delta({u}^{\downarrow{T}})|=|\delta({v}^{\downarrow{T}})|=|\delta({u}^{\downarrow{T}})\cap\delta\left({v}^{\downarrow{T}}\right)|+1$. To prove the above lemma the following simple observation will be helpful. ###### Observation 3.4. For any $u\neq v$ and $u,v\in V\setminus r$, and a spanning tree $T$ the edge $(\pi_{{T}}\left(v\right),v)\in\delta({{v}^{\downarrow{T}}})$ but $(\pi_{{T}}\left(v\right),v)\notin\delta({{u}^{\downarrow{T}}})$; also $(\pi_{{T}}\left(u\right),u)\in\delta({{u}^{\downarrow{T}}})$ but $(\pi_{{T}}\left(u\right),u)\notin\delta({{v}^{\downarrow{T}}})$. ###### Proof. Consider the given spanning tree $T$. Here the edge $(\pi_{{T}}\left(u\right),u)$ has one end in the vertex set ${u}^{\downarrow{T}}$ and the other end in the vertex set $V\setminus{u}^{\downarrow{T}}$. Thus the edge $(u,\pi_{{T}}\left(u\right))$ is part of the cut set $({u}^{\downarrow{T}},V\setminus{u}^{\downarrow{T}})=\delta({u}^{\downarrow{T}})$. Now, consider any other node $v$. Either $v\in{u}^{\downarrow{T}}$ or it does not. Let’s take the first case when $v\in{u}^{\downarrow{T}}$. Since $v\neq u$, thus both the end points of the edge $(\pi_{{T}}\left(u\right),u)$ are outside the set ${v}^{\downarrow{T}}$ hence $(\pi_{{T}}\left(u\right),u)\notin\delta({v}^{\downarrow{T}})$. When $v\notin{u}^{\downarrow{T}}$, then also similar argument holds. If $v\in\mathcal{A}_{T}\left(u\right)$, then both the endpoints are in the set ${v}^{\downarrow{T}}$, otherwise both of them are outside the set ${v}^{\downarrow{T}}$. In either of them the edge $(\pi_{{T}}\left(u\right),u)\notin\delta({v}^{\downarrow{T}})$. ∎ ###### Proof of Lemma 3.3. Consider the forward direction, $\left\\{(\pi_{{T}}\left(u\right),u),(\pi_{{T}}\left(v\right),v)\right\\}$ is a cut set induced by ${{u}^{\downarrow{T}}}\oplus{{v}^{\downarrow{T}}}$. Therefore $\delta({{v}^{\downarrow{T}}}\oplus{{u}^{\downarrow{T}}})=\left\\{(\pi_{{T}}\left(u\right),u),(\pi_{{T}}\left(v\right),v)\right\\}$. Further from Corollary 2.7, we have $\delta({{v}^{\downarrow{T}}}\oplus{{u}^{\downarrow{T}}})=\delta({{u}^{\downarrow{T}}})\oplus\delta({{v}^{\downarrow{T}}})$. Therefore $\left\\{(\pi_{{T}}\left(u\right),u),(\pi_{{T}}\left(v\right),v)\right\\}=\delta({{u}^{\downarrow{T}}})\oplus\delta({{v}^{\downarrow{T}}})$ which along with Observation 3.4 implies that $\delta({{u}^{\downarrow{T}}})\setminus(\pi_{{T}}\left(u\right),u)=\delta({{v}^{\downarrow{T}}})\setminus(\pi_{{T}}\left(v\right),v)$ thus $|\delta({{v}^{\downarrow{T}}})|-1=|\delta({{v}^{\downarrow{T}}})\cap\delta({{u}^{\downarrow{T}}})|$ and $|\delta({{u}^{\downarrow{T}}})|-1=|\delta({{v}^{\downarrow{T}}})\cap\delta({{u}^{\downarrow{T}}})|$. For the other direction, given that $|\delta({{u}^{\downarrow{T}}})\cap\delta({{v}^{\downarrow{T}}})|+1=|\delta({{v}^{\downarrow{T}}})|=|\delta({{u}^{\downarrow{T}}})|$, which along with Observation 3.4 implies that $\delta({{u}^{\downarrow{T}}})\setminus(\pi_{{T}}\left(u\right),u)=\delta({{v}^{\downarrow{T}}})\setminus(\pi_{{T}}\left(v\right),v)$. Hence $\delta({{u}^{\downarrow{T}}})\oplus\delta({{v}^{\downarrow{T}}})=\left\\{(\pi_{{T}}\left(u\right),u),(\pi_{{T}}\left(v\right),v)\right\\}$. Using the fact that $\delta({{v}^{\downarrow{T}}})\oplus\delta({{u}^{\downarrow{T}}})=\delta({{v}^{\downarrow{T}}}\oplus{{u}^{\downarrow{T}}})$, the edge set $\left\\{(\pi_{{T}}\left(u\right),u),(\pi_{{T}}\left(v\right),v)\right\\}$ is a cut induced by ${{v}^{\downarrow{T}}}\oplus{{u}^{\downarrow{T}}}$. ∎ When an induced cut of size $2$, 2-respects a spanning tree there could be two sub-cases. Consider nodes $u,v$ as in the above lemma. Here either ${u}^{\downarrow{T}}\cap{v}^{\downarrow{T}}=\emptyset$ or ${u}^{\downarrow{T}}\subseteq{v}^{\downarrow{T}}$ (which is same as ${v}^{\downarrow{T}}\subseteq{u}^{\downarrow{T}}$). The above lemma unifies these two cases. In the next chapter, we will show that how one of the nodes in the network finds all the three quantities as required by the above lemma when there exists a min-cut of this kind. Now we will see the similar characterizations for an induced cut of size 3 when it 2 respects the cut and 3-respects the cut. ###### Lemma 3.5 (3-cut, 2-respects a spanning tree $T$). Let $T$ be any tree. Let $v_{1},v_{2}\in V\setminus r$ be two different nodes and $e$ be a non-tree edge, then the following two are equivalent. 1. $P_{1}:$ $\left\\{(\pi_{{T}}\left(v_{1}\right),v_{1}),(\pi_{{T}}\left(v_{2}\right),v_{2}),e\right\\}$ is a cut set induced by the vertex set ${v_{1}}^{\downarrow{T}}\oplus{v_{2}}^{\downarrow{T}}$. 2. $P_{2}:$ $|\delta({v_{1}}^{\downarrow{T}})|-2=|\delta({v_{2}}^{\downarrow{T}})|-1=\gamma({{v_{1}}^{\downarrow{T}}},{{v_{2}}^{\downarrow{T}}})$ or $\delta({v_{1}}^{\downarrow{T}})-1=|\delta({v_{2}}^{\downarrow{T}})|-2=\gamma({{v_{1}}^{\downarrow{T}}},{{v_{2}}^{\downarrow{T}}})$. ###### Proof. The proof is similar to Lemma 3.3. Consider the forward direction, $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),e\right\\}$ is a cut set induced by ${{v_{1}}^{\downarrow}}\oplus{{v_{2}}^{\downarrow}}$. Therefore $\delta({{v_{2}}^{\downarrow}}\oplus{{v_{1}}^{\downarrow}})=\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),e\right\\}$. Further from Corollary 2.7, we have $\delta({{v_{1}}^{\downarrow}}\oplus{{v_{2}}^{\downarrow}})=\delta({{v_{1}}^{\downarrow}})\oplus\delta({{v_{2}}^{\downarrow}})$. Therefore $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),e\right\\}=\delta({{v_{1}}^{\downarrow}})\oplus\delta({{v_{2}}^{\downarrow}})$. Since $e\in\delta({v_{1}}^{\downarrow})\oplus\delta({v_{2}}^{\downarrow})$. Thus $e\in\delta({v_{1}}^{\downarrow})$ or $e\in\delta({v_{2}}^{\downarrow})$ but not in both due to the symmetric difference operator. Without loss in generality, let $e\in\delta({v_{1}}^{\downarrow})$. Now Observation 3.4 implies that $\delta({{v_{1}}^{\downarrow}})\setminus\left\\{(\pi\left(v_{1}\right),v_{1}),e\right\\}=\delta({{v_{2}}^{\downarrow}})\setminus(\pi\left(v_{2}\right),v_{2})$ thus $|\delta({{v_{1}}^{\downarrow}})|-2=|\delta({{v_{2}}^{\downarrow}})\cap\delta({{v_{2}}^{\downarrow}})|$ and $|\delta({{v_{2}}^{\downarrow}})|-1=|\delta({{v_{1}}^{\downarrow}})\cap\delta({{v_{2}}^{\downarrow}})|$. Similarly when $e\in\delta({v_{2}}^{\downarrow})$ then $|\delta({{v_{1}}^{\downarrow}})|-1=|\delta({{v_{2}}^{\downarrow}})\cap\delta({{v_{2}}^{\downarrow}})|$ and $|\delta({{v_{2}}^{\downarrow}})|-2=|\delta({{v_{1}}^{\downarrow}})\cap\delta({{v_{2}}^{\downarrow}})|$ For the other direction, without loss in generality, let us choose one of the two statements. In particular, let $|\delta({{v_{1}}^{\downarrow}})\cap\delta({{v_{2}}^{\downarrow}})|=|\delta({{v_{1}}^{\downarrow}})|-1=|\delta({{v_{2}}^{\downarrow}})|-2$, which along with Observation 3.4 implies that $|\delta({{v_{1}}^{\downarrow}})\setminus(\pi\left(v_{1}\right),v_{1})|=|\delta({{v_{2}}^{\downarrow}})\setminus(\pi\left(v_{2}\right),v_{2})|-1$. Thus there exists exactly one edge $e$ which is not in $\delta({v_{1}}^{\downarrow})$ but in $\delta({v_{2}}^{\downarrow})$ Hence $\delta({{v_{1}}^{\downarrow}})\oplus\delta({{v_{2}}^{\downarrow}})=\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),e\right\\}$. Using the fact that $\delta({{v_{2}}^{\downarrow}})\oplus\delta({{v_{1}}^{\downarrow}})=\delta({{v_{2}}^{\downarrow}}\oplus{{v_{1}}^{\downarrow}})$; the edge set $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),e\right\\}$ is a cut induced by ${{v_{1}}^{\downarrow}}\oplus{{v_{2}}^{\downarrow}}$. ∎ ###### Lemma 3.6 (3-cut, 3-respects a spanning tree $T$). Let $T$ be any tree. Let $v_{1},v_{2},v_{3}\in V\setminus r$, further each of $v_{1},v_{2}$ and $v_{3}$ are different then the following two statements are equivalent: 1. $P_{1}:$ $\left\\{(v_{1},\pi_{{T}}\left(v_{1}\right)),(v_{2},\pi_{{T}}\left(v_{2}\right)),(v_{3},\pi_{{T}}\left(v_{3}\right))\right\\}$ is a cut-set induced by ${v_{1}}^{\downarrow{T}}\oplus{v_{2}}^{\downarrow{T}}\oplus{v_{3}}^{\downarrow{T}}$. 2. $P_{2}:$ $|\delta({v_{i}}^{\downarrow{T}})|-1=\sum\limits_{j\in\left\\{1,2,3\right\\}\setminus\left\\{i\right\\}}\gamma({v_{i}}^{\downarrow{T}},{v_{j}}^{\downarrow{T}}),\ \forall i\in\left\\{1,2,3\right\\}$. ###### Proof. Consider the forward direction $\delta({v_{1}}^{\downarrow}\ \oplus\ {v_{2}}^{\downarrow}\ \oplus\ {v_{3}}^{\downarrow})=\\{(v_{1},\pi\left(v_{1}\right)),(v_{2},\pi\left(v_{2}\right)),$ $(v_{3},\pi\left(v_{3}\right))\\}$. Also, by Corollary 2.7, we know that $\delta({v_{1}}^{\downarrow})\oplus\delta({v_{2}}^{\downarrow})\oplus\delta({v_{3}}^{\downarrow})=\delta({v_{1}}^{\downarrow}\oplus{v_{2}}^{\downarrow}\oplus{v_{3}}^{\downarrow})$. Notice that there are 3 equations in $P_{2}$ when $i\in[1,3]$. We will show for $i=1$ and the argument for the rest will follow similarly. For all edge $e\neq(v_{1},\pi\left(v_{1}\right))$ and $e\in\delta({v_{1}}^{\downarrow})$ then $e$ must be present exactly once in either of $\delta({v_{2}}^{\downarrow})$ or $\delta({v_{3}}^{\downarrow})$. Because if it is present in none of them then among the three sets $\delta({v_{1}}^{\downarrow}),\delta({v_{2}}^{\downarrow})$ and $\delta({v_{3}}^{\downarrow})$, $e$ is exactly in one that is $\delta({v_{1}}^{\downarrow})$. Thus $e\in\delta({v_{1}}^{\downarrow}\oplus{v_{2}}^{\downarrow}\oplus{v_{3}}^{\downarrow})$ which is not true. Also if it is present in both $\delta({v_{2}}^{\downarrow})$ and $\delta({v_{3}}^{\downarrow})$ then similarly $e\in\delta({v_{1}}^{\downarrow}\oplus{v_{2}}^{\downarrow}\oplus{v_{3}}^{\downarrow})$, which again is not true. Thus apart from the edge $(\pi\left(v_{1}\right),v_{1})$ whenever there exists an $e\in\delta({v_{1}}^{\downarrow})$ then it is also present in exactly one of $\delta({v_{2}}^{\downarrow})$ or $\delta({v_{3}}^{\downarrow})$. Thus $|\delta({v_{1}}^{\downarrow})|-1=|\delta({v_{1}}^{\downarrow})\cap\delta({v_{2}}^{\downarrow})|+|\delta({v_{1}}^{\downarrow})\cap\delta({v_{3}}^{\downarrow})|$. For the backward direction, we will again focus our attention to three different edge sets $\delta({v_{1}}^{\downarrow}),\delta({v_{2}}^{\downarrow})$ and $\delta({v_{3}}^{\downarrow})$. We know that the edge $(\pi\left(v_{1}\right),v_{1})\in\delta({v_{1}}^{\downarrow})$ but not in either of $\delta({v_{2}}^{\downarrow})$ and $\delta({v_{3}}^{\downarrow})$. 3 Similarly for $(\pi\left(v_{2}\right),v_{2})$ and $(\pi\left(v_{3}\right),v_{3})$. Lets the edge set $C=\delta({v_{1}}^{\downarrow})\oplus\delta({v_{2}}^{\downarrow})\oplus\delta({v_{3}}^{\downarrow})$. Thus $C$ is sure to contain the edges $(v_{1},\pi\left(v_{1}\right)),(v_{2},\pi\left(v_{2}\right))$ and $(v_{3},\pi\left(v_{3}\right))$ because of the symmetric difference operator between the three sets. Now based on the condition in $P_{2}$ we will show that there is no other edge $e$ in $C$. Imagine there is only one edge $e\neq(\pi\left(v_{1}\right),v_{1})$ and $e\in\delta({v_{1}}^{\downarrow})$ and $e\in C$. Thus $e$ could either be in both $\delta({v_{2}}^{\downarrow})$ and $\delta({v_{3}}^{\downarrow})$ or none of them. Imagine $e$ is in both $\delta({v_{2}}^{\downarrow})$ and $\delta({v_{3}}^{\downarrow})$, then $|\delta({v_{1}}^{\downarrow})|-2=|\delta({v_{1}}^{\downarrow})\cap\delta({v_{2}}^{\downarrow})|+|\delta({v_{1}}^{\downarrow})\cap\delta({v_{3}}^{\downarrow})|$ which contradicts $P_{2}$. Imagine $e$ is in none of $\delta({v_{2}}^{\downarrow})$ and $\delta({v_{3}}^{\downarrow})$, then $|\delta({v_{1}}^{\downarrow})|=|\delta({v_{1}}^{\downarrow})\cap\delta({v_{2}}^{\downarrow})|+|\delta({v_{1}}^{\downarrow})\cap\delta({v_{3}}^{\downarrow})|$ which again contradicts $P_{2}$. ∎ Lemma 3.5 and 3.6 characterize an induced cut of size 3 when it shares 2 edges and 3 edges with a spanning tree respectively. Similar to a min-cut of size 2, here again we have few different cases. These lemmas unify all the different cases. We will discuss these different cases in Chapter 5. We will work with an arbitrary BFS tree $\mathcal{T}$ of the given network. Whenever a quantity is computed with respect to this BFS tree $\mathcal{T}$, we will skip the redundant $\mathcal{T}$ from the subscript or superscript, for example $\pi\left(v\right)$ instead of $\pi_{{\mathcal{T}}}\left(v\right)$ and ${v}^{\downarrow}$ instead of ${v}^{\downarrow{\mathcal{T}}}$. At the beginning each node $v\neq r$ knows $\ell\left(v\right)$, $\pi\left(v\right)$ and the ancestor set $\mathcal{A}\left(v\right)$. The BFS tree and these associated quantities can be computed in the beginning in $O(D)$ rounds. For any node $v$ which is not the root of the BFS tree, $\ell\left(v\right)$ and $\pi\left(v\right)$ are known to $v$ at the time of construction of the BFS tree as shown in the proof of the Lemma 2.1. Further the ancestor set $\mathcal{A}\left(v\right)$ can also be known to node $v$ in $O(D)$ rounds using _Broadcast Type - 1_. And can be done as follows: each node $a$ tells all the nodes in the set ${a}^{\downarrow}$ about it being one the ancestor. This information is just of size $O(\log n)$ bits. ### 3.2 Our Contribution In this thesis, we present deterministic algorithm to find min-cuts of size 1,2 and 3. Figure 3.1: Roadmap for finding small min-cuts For min-cuts of size $1$ and $2$, our algorithm takes $O(D)$ rounds. For min- cuts of size $3$, our algorithm takes $O(D^{2})$ rounds. We use our characterization given in Lemma 3.3, Lemma 3.5 and Lemma 3.6. Further, in subsequent chapters, we will give communication strategies which will ensure that at least one node in the network knows about the required information as per these lemmas whenever a min-cut of the kind exists. The idea is basically to break the problem into smaller parts. Recall from Observation 3.1 that a min-cut of size 1 shares an edge with any spanning tree, a min-cut of size 2 may share 1 or 2 edges and finally a min-cut of size $3$ may share 1,2 or 3 edges with the tree. In each of these cases, we our communication strategies ensure that at least one of the node will have the required information as per the aforementioned characterization lemmas. We give a road-map of this division in Figure 3.1. Our results of finding min-cuts are split into two chapters. In Chapter 4, we present a new technique called the _tree restricted semigroup function_. We will show that this is enough to optimally find the min-cuts of size $1$ and $2$. In Chapter 5, we will give algorithm to find min-cut of size $3$. We introduce two techniques: _sketching_ and _layered algorithm_ to do the same. ## Chapter 4 Tree Restricted Semigroup Function In this chapter, we will define tree restricted semigroup function and demonstrate its utility in finding the induced-cuts of size 1 and 2. A tree restricted semigroup function is defined with respect to a tree $T$ and is based on a commutative semigroup. Recall that a commutative semigroup is a set $\mathcal{X}$ together with a commutative and associative binary operation $\boxplus$ which is a function $\boxplus:\mathcal{X}\times\mathcal{X}\rightarrow\mathcal{X}$. Tree restricted semigroup function is inspired by semigroup function defined in [23, Def. 3.4.3]. Let $\mathcal{X}$ be a commutative semigroup with operator $\boxplus$. Let $T$ be any spanning tree. We will formally define $f:V\rightarrow\mathcal{X}$ to be a tree restricted semigroup function with respect to the tree $T$ in Definition 4.1. For any node $v$, $f(v)$ will only depend on the vertex set ${v}^{\downarrow{T}}$. Let $a\in V$ be any node, for each $v\in\mathcal{A}_{T}\left(a\right)$, node $a$ computes the value $X_{a}^{v}$ through a pre-processing step which is the contribution by node $a$ for calculating $f(v)$. ###### Definition 4.1 (Tree Restricted Semigroup function). Let $\mathcal{X}$ be a commutative semigroup with the operator $\boxplus$. Then, the function $f:V\rightarrow\mathcal{X}$ is a tree restricted semigroup function if $f(v)=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits\limits_{a\in{v}^{\downarrow{T}}}X_{a}^{v}$ Further for any $a\in V$ and $v\in\mathcal{A}_{T}\left(a\right)$, define $X_{{a}^{\downarrow{T}}}^{v}=\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits\limits_{a^{\prime}\in{a}^{\downarrow{T}}}X_{a^{\prime}}^{v}$. Note that $X_{{a}^{\downarrow{T}}}^{v}\in\mathcal{X}$. We will say that $X_{{a}^{\downarrow{T}}}^{v}$ is the contribution of nodes in the set ${a}^{\downarrow{T}}$ for computing $f(v)$. ###### Observation 4.2. Let $a$ be an internal node in the tree $T$ and $v\in\mathcal{A}_{T}\left(a\right)$ then $X_{{a}^{\downarrow{T}}}^{v}=X_{a}^{v}\boxplus\left(\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits\limits_{c\in\operatorname{children}_{T}(a)}X_{{c}^{\downarrow{T}}}^{v}\right)$ ###### Observation 4.3. For any internal node $v$ $f(v)=X_{v}^{v}\boxplus\left(\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits\limits_{c\in\operatorname{children}_{T}(v)}X_{c^{\downarrow}}^{v}\right)$ 1 2 Pre-Processing __ 3 For any node $a,\forall v\in\mathcal{A}_{T}\left(a\right)$, node $a$ knows $X_{a}^{v}$ through a pre-processing 4 1 Phase _1 : Aggregation phase run on all node $a$, aggregates $X_{{a}^{\downarrow{T}}}^{v}\ \forall v\in\mathcal{A}\left(a\right)$_ 2 3 for _rounds $t=1$ to $Depth(T)-\ell_{T}\left(a\right)$_ wait 4 $l\leftarrow 0$ 5 for _rounds $t=Depth(T)-\ell_{T}\left(a\right)+1$ to $h\left(T\right)$_ do 6 $v\leftarrow$ ancestor of node $a$ at level $l$ 7 if _$a$ is leaf node_ then $X_{a^{\downarrow T}}^{v}\leftarrow X_{a}^{v}$ 8 else 9 for _$c\in\operatorname{children}_{T}(a)$_ parallely collect $\langle l,X_{c^{\downarrow T}}^{v}\rangle$ 10 $X_{a^{\downarrow T}}^{v}\leftarrow X_{a}^{v}\boxplus\left(\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits\limits_{c\in\operatorname{children}_{T}(a)}X_{c^{\downarrow T}}^{v}\right)$ 11 12 send to the parent $\langle l,X_{a^{\downarrow T}}^{v}\rangle$ 13 $l\leftarrow l+1$ 14 15 1 Phase _2: Computation Phase (run on all node $v\in V$), finds f(v)_ 2 Available Info: __Each node $v$ knows $X_{c^{\downarrow T}}^{v}$ for all $c\in\operatorname{children}(v)$ 3 if _$v$ is a leaf node_ then $f(v)\leftarrow X_{v}^{v}$ 4 else 5 $f(v)\leftarrow X_{v}^{v}\boxplus\left(\mathop{\vphantom{\bigoplus}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxplus$}}}}}\displaylimits\limits_{c\in\operatorname{children}_{T}(v)}X_{c^{\downarrow T}}^{v}\right)$ 6 Algorithm 1 Tree Restricted Semigroup function ###### Definition 4.4. A tree restricted semigroup function $f(\cdot)$ is efficiently computable if for all $v$, $f(v)$ can be computed in $O(Depth(T))$ time in the CONGEST model. In Lemma 4.5, we will give sufficient conditions for a tree restricted semigroup function to be efficiently computable. ###### Lemma 4.5. For all $a\in V$ and $v\in\mathcal{A}_{T}\left(a\right)$ if $X_{a}^{v},X_{{a}^{\downarrow{T}}}^{v},f(v)$ are all of size $O(\log n)$ bits and if $X_{a}^{v}$ can be computed in $O(Depth(T))$ time by node $a$ then the tree restricted semigroup function $f(\cdot)$ is efficiently computable. ###### Proof. For any node $v$, tree restricted function $f(v)$ depends on the value $X_{a}^{v}$ for all $a\in{v}^{\downarrow{T}}$, thus each such node $a$ _convergecasts_ (see Sec. 2.1.2) the required information up the tree which is supported by aggregation of the values. We will give an algorithmic proof for this lemma. The algorithm to compute a efficiently computable semigroup function $f(\cdot)$ is given in Algorithm 1. The aggregation phase of the algorithm given in Phase 1 runs for at most $Depth(T)$ time and facilitates a coordinated aggregation of the required values and convergecasts them in a synchronized fashion. Each node $a$ in Phase 2, sends $\ell_{T}\left(a\right)$ messages of size $O(\log n)$ to its parent, each message include $X_{a^{\downarrow T}}^{v}$ where $v\in\mathcal{A}_{T}\left(a\right)$; which as defined earlier is the contribution of nodes in $a^{\downarrow T}$ to $f(v)$. This message passing takes $O(1)$ time since $X_{a^{\downarrow T}}^{v}\in\mathcal{X}$ is of size $O(\log n)$ bits. For brevity, we assume this takes exactly $1$ round, this enables us to talk about each round more appropriately as follows: Any node $a$ at level $\ell_{T}\left(a\right)$ waits for round $t=1$ to $Depth(T)-\ell_{T}\left(a\right)$. For any $l\in[0,\ell_{T}\left(a\right)-1]$, in round $t=Depth(T)-\ell_{T}\left(a\right)+l+1$ node $a$ sends to its parent $\langle l,X_{a^{\downarrow T}}^{v}\rangle$ where $v$ is the ancestor of $a$ at level $l$. When node $a$ is an internal node then as per Observation 4.2, $X_{{a}^{\downarrow{T}}}^{v}$ depends on $X_{a}^{v}$ which can be pre- calculated in $O(Depth(T))$ rounds. Also, $X_{a^{\downarrow T}}^{v}$ depends on $X_{c^{\downarrow T}}^{v}$ for all $c\in\operatorname{children}_{T}(a)$ which are at level $\ell_{T}\left(a\right)+1$ and have send to $a$ (which is their parent) the message $\langle l,X^{v}_{c^{\downarrow T}}\rangle$ in the $(Depth(T)-\ell_{T}\left(a\right)+l)^{\text{th}}$ round. For a leaf node $a$, $X_{a^{\downarrow T}}^{v}=X_{a}^{v}$ which again is covered in pre-processing step. In Phase 2, node $v$ computes tree restricted semigroup function $f(v)$. As per Observation 4.3 of the algorithm each internal node $v$ requires $X_{{c}^{\downarrow{T}}}^{v}\ \forall c\in\operatorname{children}_{T}(v)$ and $X_{v}^{v}$. $X_{v}^{v}$ is computed in the pre-processing step. And $X_{{c}^{\downarrow{T}}}^{v}$ is received by $v$ in the aggregation phase. When node $v$ is a leaf node, $f(v)$ depends only on $X_{v}^{v}$. ∎ For any node $v$ and a tree $T$, let us define $\eta_{T}(v)=|\delta({v}^{\downarrow{T}})|$. In Subsection 4.1, we will show that $\eta_{T}(\cdot)$ is an efficiently computable tree restricted semigroup function. This will be enough to find if there exists a min-cut of size $1$ or a min-cut of size $2,3$ which 1-respects our fixed spanning tree. Further in Subsection 4.2, we will find induced cuts of size 2 and define a tree restricted semigroup function $\zeta_{T}(\cdot)$ which will enable us to find induced-cuts of size 2. As mentioned earlier, we will work with a fixed BFS tree $\mathcal{T}$. We will skip the redundant $\mathcal{T}$ in the notations. For example $\eta(\cdot)$ instead of $\eta_{T}(\cdot)$ and $\zeta(\cdot)$ instead of $\zeta_{T}(\cdot)$. ### 4.1 Min-Cuts of size 1 In this subsection we will prove that the function $\eta(\cdot)$ is an efficiently computable tree restricted semigroup function. Here we will work with our fixed BFS tree $\mathcal{T}$. Recall that the function $\eta:V\rightarrow[m]$ where $\eta(v)=|\delta(v^{\downarrow})|$ and $\delta(v^{\downarrow})$ is the cut induced by the vertex set $v^{\downarrow}$. For any node $a\in v^{\downarrow}$, we will use $H_{a}^{v}\triangleq|\delta(a)\cap\delta({v}^{\downarrow})|$. This is equal to the number of vertices adjacent to $a$ which are not in vertex set $v^{\downarrow}$. The commutative semigroup associated here is the set of all positive integers $\mathbb{Z}^{+}$ with ‘addition’operator. Thus for a node $a\in\mathcal{A}\left(v\right)$, $H_{{a}^{\downarrow}}^{v}=\sum_{a^{\prime}\in{a}^{\downarrow}}H_{a^{\prime}}^{v}$. We give the pre-processing steps in Algorithm 2 which calculates $H_{a}^{v}$ for all $v\in\mathcal{A}\left(a\right)$ 1 for _$b$ adjacent to $a$_ parallely send the ancestor set $\mathcal{A}\left(a\right)$ to $b$ 2 for _$b$ adjacent to $a$_ parallely receive the ancestor set $\mathcal{A}\left(b\right)$ 3 for _$v\in\mathcal{A}\left(a\right)$_ do 4 $H_{a}^{v}\leftarrow|\left\\{b\mid(a,b)\in E,\ v\notin\mathcal{A}\left(b\right)\right\\}|$ /* node $a$ can execute the above step locally because it knows $\mathcal{A}\left(b\right)$, $\forall\ (a,b)\in E$ */ 5 Algorithm 2 Pre-Processing for computing the function $\eta(\cdot)$ (run at all node $a$ for finding $H^{v}_{a}\ \forall\ v\in\mathcal{A}\left(a\right)$) ###### Observation 4.6. Pre processing as given in Algorithm 2 takes $O(D)$ time. ###### Proof. Any node $a$ has at most $D$ ancestors in $\mathcal{T}$. So for every node $b$ adjacent to $a$, it takes $O(D)$ time to communicate the set $\mathcal{A}\left(a\right)$ to it. Similarly, it takes $O(D)$ time to receive the set $\mathcal{A}\left(b\right)$ from node $b$. Now $H_{a}^{v}$ for any $v\in\mathcal{A}\left(a\right)$ will just be an internal computation at node $a$. ∎ ###### Lemma 4.7. $\eta(\cdot)$ is an efficiently computable tree restricted semigroup function. ###### Proof. Let $v\in V$, $\eta(v)$ is the number of edges going out of the vertex set ${v}^{\downarrow}$. That is $\forall a\in{v}^{\downarrow}$, $\eta(v)$ is the sum of the number of vertices incident to $a$ which are not in the set ${v}^{\downarrow}$. Thus $\eta(v)=\sum_{a\in{v}^{\downarrow}}H_{a}^{v}$. By Observation 4.6, for all $v\in\mathcal{A}\left(a\right)$, $H_{a}^{v}$ can be computed in $O(D)$ time. Also for any $v$, $\eta(v)$ could be as big as the number of edges $m$. Therefore, for any $a\in V$ and $v\in\mathcal{A}\left(a\right)$ we have, $0\leq H_{a}^{v}\leq H_{{a}^{\downarrow}}^{v}\leq\eta(v)\leq m$. Thus $H_{a}^{v}$,$H_{{a}^{\downarrow}}^{v}$ and $f(v)$ can be represented in $O(\log n)$ bits. Hence by Lemma 4.5, $\eta(\cdot)$ is an efficiently computable tree restricted semigroup function. ∎ To compute $\eta(\cdot)$, we use Algorithm 1 given in Lemma 4.5. Further during the computation of $\eta(\cdot)$, each node $a$ computes $H_{{a}^{\downarrow}}^{v}$ for all $v\in\mathcal{A}\left(a\right)$ which is the aggregated value of all the decendents of node $a$. We summarize these results in the following Lemmas. ###### Lemma 4.8. The function $\eta(\cdot)$ can be computed in $O(D)$ time for every node $v\in V$. ###### Proof. In Lemma 4.7, we show that $\eta(\cdot)$ is an efficiently computable tree restricted semigroup function defined with respect to the BFS tree $\mathcal{T}$. Also, $Depth(\mathcal{T})=O(D)$. Thus by Definition 4.1, $\eta(\cdot)$ can be computed in $O(D)$ ∎ ###### Lemma 4.9. For each node $a$ and $v\in\mathcal{A}\left(a\right)$, node a knows $H_{{a}^{\downarrow}}^{v}=|\delta\left({a}^{\downarrow}\right)\cap\delta\left({v}^{\downarrow}\right)|$ in $O(D)$ rounds. ###### Proof. During the computation of the tree restricted semigroup function $\eta(\cdot)$, each node $a$ for every $v\in\mathcal{A}\left(a\right)$, computes the aggregated value $H_{{a}^{\downarrow}}^{v}=\sum_{a^{\prime}\in{a}^{\downarrow}}H_{a}^{v}$ which is the contribution of nodes in ${a}^{\downarrow}$ towards the computation of $\eta(v)$. Thus the lemma follows. ∎ ###### Theorem 4.10. Min-cut of size one (bridge edges) can be found in $O(D)$ time. ###### Proof. For any node $v\neq r$, the edge $(\pi\left(v\right),v)\in\delta(v^{\downarrow})$. Thus when $\eta(v)=1$, then $(\pi\left(v\right),v)$ is the only edge in $\delta(v^{\downarrow})$ and is a cut edge. ∎ ###### Lemma 4.11. If $\eta(v)=k$ then $\delta(v^{\downarrow})$ is a cut-set of size $k$. Having found $\eta(\cdot)$ for all the nodes we downcast them. That is each node $v$, downcasts its $\eta(v)$ value to the vertex set ${v}^{\downarrow}$. This kind of broadcast is similar to _Broadcast Type -1_ defined in Chapter 2. Thus by Lemma 2.4, this can be done in $O(D)$ time. That is all nodes $a\in{v}^{\downarrow}$ will have the value of $\eta(v)$ in $O(D)$ time. This will be useful for the algorithm to find induced cuts of size 2 given in next subsection. We quantify the same in the following lemma. ###### Lemma 4.12. For any node $v$, the nodes in the vertex set ${v}^{\downarrow}$ know $\eta(v)$ in $O(D)$ rounds. ### 4.2 Min-Cuts of size 2 In this subsection, we will give an algorithm to find min-cuts of size 2. The theme here will be the use of tree restricted semigroup function. We will define a new tree restricted semigroup function $\zeta(\cdot)$ which will be based on a specially designed semigroup. Before we get into details of $\zeta(\cdot)$, we will give details about cuts of size $2$. Let $A\subset V$ and $|\delta(A)|=2$. The question here is to find $\delta(A)$. Here, $\delta(A)$ could share either one edge with the tree $T$, or it could share $2$ edges with the tree. When $\delta(A)$ shares one edge with the tree then it can be found in $O(D)$ rounds as described by Lemma 4.11. We make the following observation for the case when $\delta(A)$, 2-respects the tree. In this case, $\delta(A)=\left\\{(\pi\left(a\right),a),(\pi\left(w\right),w)\right\\}$ for some $a,w\in V\setminus r$ and $a\neq w$. ###### Observation 4.13. Let $A\subset V$ and $|\delta(A)|=2$. Also, let $\delta(A)$ 2-respect the tree $\mathcal{T}$ then $\delta(A)=\delta({a}^{\downarrow})\oplus\delta({w}^{\downarrow})$ for some $a\neq w$ and $a,w\in V\setminus r$. Further either ${{a}^{\downarrow}}\subset{{w}^{\downarrow}}$ (nested) or ${{a}^{\downarrow}}\cap{{w}^{\downarrow}}=\emptyset$ (mutually disjoint). ###### Proof. Let $\delta(A)=\left\\{(\pi\left(a\right),a),(\pi\left(w\right),w)\right\\}$ such that $a\neq w$ and $a,w\in V\setminus r$. WLOG let $\ell\left(a\right)\geq\ell\left(w\right)$. Because of the tree structure, either ${a}^{\downarrow}\subset{w}^{\downarrow}$ or ${a}^{\downarrow}\cap{w}^{\downarrow}=\emptyset$. When ${a}^{\downarrow}\subset{w}^{\downarrow}$ then the induced cut $\delta(A)$ is the cut set $({w}^{\downarrow}\setminus{a}^{\downarrow},V\setminus({w}^{\downarrow}\setminus{a}^{\downarrow}))=\delta({w}^{\downarrow}\setminus{a}^{\downarrow})$. Since ${a}^{\downarrow}\subset{w}^{\downarrow}$ thus ${w}^{\downarrow}\setminus{a}^{\downarrow}={w}^{\downarrow}\oplus{a}^{\downarrow}$. Hence $\delta({w}^{\downarrow}\setminus{a}^{\downarrow})=\delta({w}^{\downarrow}\oplus{a}^{\downarrow})$ and finally by Corollary 2.7, we have $\delta({a}^{\downarrow}\oplus{w}^{\downarrow})=\delta({a}^{\downarrow})\oplus\delta({w}^{\downarrow})$. Similarly, when ${a}^{\downarrow}\cap{w}^{\downarrow}=\emptyset$ then the induced cut $\delta(A)$ is the cut set $({a}^{\downarrow}\cup{w}^{\downarrow},V\setminus{a}^{\downarrow}\cup{w}^{\downarrow})=\delta({a}^{\downarrow}\cup{w}^{\downarrow})=\delta({a}^{\downarrow}\oplus{w}^{\downarrow})=\delta({a}^{\downarrow})\oplus\delta({w}^{\downarrow})$. ∎ The above observation states that we have two different cases when an induced- cut of size $2$ shares both the edges with the tree. For any $A,B\subseteq V$, let $\gamma(A,B)=|\delta(A)\cap\delta(B)|$. In this subsection, we will prove the following lemma. This lemma will be enough to prove that the min-cuts of size $2$ can be deterministically found in $O(D)$ rounds. Moreover, it will also help us find min-cuts of size 3 as given in Chapter 5. ###### Lemma 4.14. Let $a,w$ be two nodes. WLOG let $\ell\left(a\right)\geq\ell\left(w\right)$. If $\left\\{(\pi\left(a\right),a),(\pi\left(w\right),w)\right\\}$ is a cut set induced by ${a}^{\downarrow}\oplus{w}^{\downarrow}$ and if $\gamma({a}^{\downarrow},{w}^{\downarrow})>0$ then such an induced cut can be found in $O(D)$ rounds by node $a$. Using the above lemma we now prove the following theorem. ###### Theorem 4.15. Min-cuts of size $2$ can be found in $O(D)$ rounds. ###### Proof. When there is a min-cut of size 2, it either 1-respects the tree or 2-respects the tree. When it 1-respects the tree, by Lemma 4.11, we know that it can be found in $O(D)$ rounds because this only requires computation of $\eta(\cdot)$. When a min-cut of size 2, 2-respects the tree then by Lemma 4.13 we know that the cut is of the form $\delta({w}^{\downarrow})\oplus\delta({a}^{\downarrow})$ for some nodes $a$ and $w$. Moreover $\gamma({a}^{\downarrow},{w}^{\downarrow})>0$ since there is not cut of size $1$. From Lemma 4.14 we know that this can be found in $O(D)$ rounds. ∎ We will now prove Lemma 4.14. When the induced cut of size 2, 2-respects the tree, we know by Observation 4.13 that there could be two different cases. For the easy case when ${{a}^{\downarrow}}\subset{{w}^{\downarrow}}$, we know from Lemma 4.9 that $H_{{a}^{\downarrow}}^{w}=|\delta({a}^{\downarrow})\cap\delta({w}^{\downarrow})|=\gamma({{a}^{\downarrow}},{{w}^{\downarrow}})$ is known by node $a$ in $O(D)$ rounds. Also from Lemma 4.12, $\eta(w)$ is known by all the vertices $a\in{{w}^{\downarrow}}$. Hence, if $\left\\{(\pi\left(w\right),w),(\pi\left(a\right),a)\right\\}$ is an induced cut and $a\in{w}^{\downarrow}$ and $a\neq w$, then as per Lemma 3.3 node $a$ has the required information to find the induced cut. The following lemma summarizes this. ###### Lemma 4.16. Let $a,w$ be two vertices such that ${a}^{\downarrow}\subset{w}^{\downarrow}$. Let $\delta({a}^{\downarrow}\oplus{w}^{\downarrow})=\left\\{(\pi\left(a\right),a),(\pi\left(w\right),w)\right\\}$ be an induced cut, then node $a$ can find such a cut in $O(D)$ rounds. The other case is when ${a}^{\downarrow}$ and ${w}^{\downarrow}$ are disjoint sets. This is the non-trivial part for finding an induced cut of size 2. Recall from Lemma 3.3, that to make a decision about an induced cut of the form $\left\\{(\pi\left(a\right),a),(\pi\left(w\right),w)\right\\}$ we require one of the node in the network to know $\gamma({a}^{\downarrow},{w}^{\downarrow}),\eta(a)$ and $\eta(w)$. The idea here is for node $a$ (when $\ell\left(a\right)\geq\ell\left(w\right)$) to find $\eta(w)$ and $\gamma({a}^{\downarrow},{w}^{\downarrow})$ which is quite a challenge because there does not exists a straightforward way through broadcast or convergecast. Moreover, there may not even exist an edge between the vertices $a$ and $w$. To deal with this, we introduce a new tree restricted semigroup function $\zeta(\cdot)$ The tree restricted semigroup function $\zeta(\cdot)$ is based on a specially defined semigroup $\mathcal{Z}$. Before we give further details and intuition about the function $\zeta(\cdot)$, let us define the semigroup $\mathcal{Z}$. There are two special elements $\mathbb{e}$ and $\nparallel$ in the semigroup $\mathcal{Z}$. Apart from these special elements all other $Z\in\mathcal{Z}$ are a four tuple. Let the four-tuple be given as $Z=\langle Z[1],Z[2],Z[3],Z[4]\rangle$, here $Z[1],Z[2]\in V$ and $Z[3],Z[4]\in\mathbb{Z}^{+}$. The operator associated with the semigroup $\mathcal{Z}$ is $\odot$. Special elements $\mathbb{e}$ and $\nparallel$ are the identity and the absorbing element with respect to the operator $\odot$ of the semigroup $\mathcal{Z}$. These special elements are defined in such a way that for any $Z\in\mathcal{Z}$, $Z\odot\mathbb{e}=Z=\mathbb{e}\odot Z$ and $Z\ \odot\nparallel\ =\ \nparallel\ =\ \nparallel\odot Z$. (Here the symbols $\mathbb{0},\mathbb{1}$ are not chosen as identity and absorbing elements, because $\mathbb{e}$ corresponds to zero edges and $\nparallel$ is considered as a zero of the semigroup which we will see in Property 4.18). In Algorithm 3, we define the operator $\odot$. // for $i\in\left\\{1,2,3,4\right\\}$ let $Z_{1}[i]$ be the $i^{\text{th}}$ element in 4-tuple of $Z_{1}$. Similarly for $Z_{2}$ 1 if _one of $Z_{1},Z_{2}$ is $\nparallel$_ then return $\nparallel$ 2 else if _$Z_{1}=\mathbb{e}$_ then return $Z_{2}$ 3 else if _$Z_{2}=\mathbb{e}$_ then return $Z_{1}$ 4 else if _$Z_{1}[1:3]=Z_{2}[1:3]$_ then return $\langle Z_{1}[1],Z_{1}[2],Z_{1}[3],Z_{1}[4]+Z_{2}[4]\rangle$ 5 else return $\nparallel$ Algorithm 3 $Z_{1}\odot Z_{2}$ (Both $Z_{1},Z_{2}\in\mathcal{Z}$) ###### Observation 4.17. $\odot\ (\texttt{zeta-operator})$ is commutative and associative. ###### Proof. Commutativity of $\odot$ is straightforward and is implied by the commutativity of addition over $\mathbb{Z}^{+}$. For associativity, let’s imagine there exist $Z_{1},Z_{2},Z_{3}\in\mathcal{Z}$. Further let’s imagine that $Z_{1},Z_{2},Z_{3}\notin\left\\{\nparallel,\mathbb{e}\right\\}$. Now when the first two elements of each of the four tuples $Z_{1},Z_{2},Z_{3}$ is same, then associativity is trivial and implied by the associativity of the addition operator over $\mathcal{Z}^{+}$. When the first two elements are not equal in $Z_{1},Z_{2},Z_{3}$ then $Z_{1}\odot(Z_{2}\odot Z_{3})=(Z_{1}\odot Z_{2})\odot Z_{3}=\ \nparallel$. Similarly, If one of $Z_{1},Z_{2},Z_{3}$ is $\nparallel$, then also $Z_{1}\odot(Z_{2}\odot Z_{3})=(Z_{1}\odot Z_{2})\odot Z_{3}=\ \nparallel$, hence associativity is implied. And lastly when one of $Z_{1},Z_{2},Z_{3}$ is $\mathbb{e}$, then $Z_{1}\odot(Z_{2}\odot Z_{3})$ becomes an operation between just two elements and associativity is implied directly. ∎ For any node $v$, define the edge set $E\left\\{\zeta(v)\right\\}\triangleq\delta({v}^{\downarrow})\setminus(\pi(v),v)$. The value of $\zeta(v)$ depends on the edge set $E\left\\{\zeta(v)\right\\}$. For any node $a$ and $v\in\mathcal{A}\left(a\right)$, let $E\left\\{Z_{a}^{v}\right\\}\triangleq E\left\\{\zeta(v)\right\\}\cap\delta(a)$. We define $Z_{a}^{v}$ as per the Property 4.18. ###### Property 4.18. For any node $a$ and $v\in\mathcal{A}\left(a\right)$, $Z_{a}^{v}$ takes one of the following three values:- 1. i. $\mathbb{e}$ when $E\left\\{Z_{a}^{v}\right\\}=\emptyset$ 2. ii. ${\langle w,\pi\left(w\right),\eta(w),\gamma(a,{{w}^{\downarrow}})\rangle}$ when there exists a node $w$ at level $\ell\left(v\right)$ such that all the edges in $E\left\\{Z_{a}^{v}\right\\}$ have one endpoint in ${{w}^{\downarrow}}$ 3. iii. $\nparallel$ otherwise Similar to the previous subsection the semigroup function $\zeta(\cdot)$ is defined as $\zeta(v)\triangleq\bigodot_{a\in{v}^{\downarrow}}Z_{a}^{v}$. We will say that $Z_{{{a}^{\downarrow}}}^{v}$ is the contribution of nodes in vertex set ${a}^{\downarrow}$ to compute $\zeta(v)$. We will now prove that for all node $a\in V$ and $v\in\mathcal{A}\left(a\right)$, $Z_{a}^{v}$ can be computed in $O(D)$ time in Lemma 4.19, then using Definition 4.1 and Lemma 4.5, we will prove that $\zeta(\cdot)$ is an efficiently computable tree restricted semigroup function. For any $l<\ell\left(a\right)$, let $\alpha(a,l)$ be the ancestor of node $a$ at level $l$. For notational convenience let $\alpha(a,\ell\left(a\right))=a$. ###### Lemma 4.19. For all node $a\in V$ and $v\in\mathcal{A}\left(a\right)$, $Z_{a}^{v}$ can be computed in $O(D)$ time as defined in Property 4.18. ###### Proof. Here we will give an algorithmic proof, We describe the steps in Algorithm 4. Further we will prove that this algorithm correctly finds $Z_{a}^{v}$ and takes $O(D)$ time. 1 $\mathcal{N}(a)\leftarrow\left\\{b\mid(a,b)\in E,(a,b)\ \text{is a non-tree edge}\right\\}$ 2 for _all $b\in\mathcal{N}(a)$_ parallely send tuples $\langle\ell\left(u\right),\eta(u),u\rangle$ for all $u\in\mathcal{A}\left(a\right)$ to $b$ 3 for _all $b\in\mathcal{N}(a)$_ parallely receive tuples $\langle\ell\left(u\right),\eta(u),u\rangle$ for all $u\in\mathcal{A}\left(b\right)$ 4 $l_{\min}\leftarrow\min(\left\\{\ell\left(b\right)\ |\ b\in\mathcal{N}(a)\right\\}\cup\ell(a))$ 5 if _$l_{\min}\neq\ell(a)$_ then 6 for _$v\in\left\\{\alpha(a,l)\mid l\in(l_{\min},\ell(a)]\right\\}$_ do $Z_{a}^{v}=\ \nparallel$ 7 8for _$l=l_{\min}$ to $1$_ do 9 $v\leftarrow\alpha(a,l)$ // $A^{l}$ is set of ancestors at level $l$ of nodes in $\mathcal{N}(a)$ except $v$ 10 $A^{l}\leftarrow\left\\{\alpha(b,l)\mid b\in\mathcal{N}(a)\right\\}\setminus v$ 11 if _$A^{l}=\emptyset$_ then $Z^{v}_{a}\leftarrow\mathbb{e}$ 12 else if _$|A^{l}|=1$_ then 13 $w\leftarrow$ element in singleton $A^{l}$ 14 $\gamma({{w}^{\downarrow}},a)\leftarrow|\left\\{b\mid b\in\mathcal{N}(a),\alpha(b,l)=w\right\\}|$ 15 $Z^{v}_{a}\leftarrow\langle w,\pi\left(w\right),\eta(w),\gamma({{w}^{\downarrow}},a)\rangle$ 16 else $Z^{v}_{a}\leftarrow\nparallel$ 17 Algorithm 4 Pre-Processing step for $\zeta$ (run at all node $a$ for finding $Z_{a}^{v}$ for all $v\in\mathcal{A}\left(a\right)$) As per Property 4.18, $Z_{a}^{v}$ just depend on the nodes adjacent to the node $a$. The required information is received and sent from the adjacent nodes in Algorithm 4 at line 4 and 4. Each of these takes only $O(D)$ time because at max $O(D)$ messages of $O(\log n)$ bits are communicated. $\mathcal{N}(a)$ is the non-tree neighbors of node $a$ found in line 4. In Algorithm 4, decision in regard to $Z_{a}^{v}$ is taken based on $A^{\ell(v)}$ which is the set of ancestors at level $\ell\left(v\right)$ of nodes in $\mathcal{N}(a)$ except the node $v$. When $A^{\ell(v)}=\emptyset$, then in line 4, $Z_{a}^{v}$ is set to $\mathbb{e}$. For bullet (i) of Property 4.18, we need to prove that $E\left\\{Z_{a}^{v}\right\\}=\emptyset\implies A^{\ell(v)}=\emptyset$. Here $E\left\\{Z_{a}^{v}\right\\}=\delta({v}^{\downarrow})\setminus(\pi\left(v\right),v)\cup\delta(a)$. When $a\neq v$, then $E\left\\{Z_{a}^{v}\right\\}$ is the set of all the edges which are incident on $a$ and goes out of the vertex set ${v}^{\downarrow}$. Now when $E\left\\{Z_{a}^{v}\right\\}=\emptyset$, then all edges incident on $a$ have the other endpoint in the vertex set ${v}^{\downarrow}$. Thus $A^{\ell(v)}=\emptyset$. When $a=v$, here $E\left\\{Z_{v}^{v}\right\\}$ is the set of all edges other then $(\pi\left(v\right),v)$ which are incident on $v$ and goes out of the vertex set ${v}^{\downarrow}$. Note that $(\pi\left(v\right),v)$ is a tree-edge thus $\pi\left(v\right)\notin\mathcal{N}(a)$. Thus $E\left\\{Z_{v}^{v}\right\\}=\emptyset\implies A^{\ell(v)}=\emptyset$. Now for correctness, if $E\left\\{Z_{a}^{v}\right\\}=\emptyset$ and $v\neq a$ then no edge incident on node $a$ goes out of ${v}^{\downarrow}$, hence the set $A^{\ell(v)}=\emptyset$. When $v=a$, we know that $\mathcal{N}(v)$ (in line 4) contains only non-tree neighbors, thus $\pi\left(v\right)\notin\mathcal{N}(v)$. We know that edge $(\pi\left(v\right),v)\notin E\left\\{Z_{v}^{v}\right\\}$. Thus here when $E\left\\{Z_{v}^{v}\right\\}=\emptyset$ then no edge other than $(\pi\left(v\right),v)$ incident on $v$ goes out of ${v}^{\downarrow}$, thus $A^{\ell(v)}=\emptyset$. Hence in both the cases $Z_{a}^{v}$ is correctly set to $\mathbb{e}$. (in line 4) For the other two bullet’s in Property 4.18, we employ the same idea. If $A^{\ell\left(v\right)}=\left\\{w\right\\}$ (for some node $w$) then it implies that all the neighbours adjacent to node $a$ have a common ancestor $w$ other than $v$ and thus $Z_{a}^{v}$ captures the required information about node $w$. And when $|A^{\ell\left(v\right)}|>1$ it simply means there are more than one node at level $\ell\left(v\right)$ as given in bullet $(iii)$ of Property 4.18. ∎ Similar to previous section, $Z_{{a}^{\downarrow}}^{{v}}\triangleq\bigodot_{a^{\prime}\in{a}^{\downarrow}}Z_{a^{\prime}}^{v}$. Since for any $a^{\prime}\in{a}^{\downarrow}$, $Z_{{{a^{\prime}}}}^{v}$ depends on $E\left\\{\zeta(v)\right\\}\cap\delta({a^{\prime}})$ thus $Z_{{{a}^{\downarrow}}}^{v}$ depends on $E\left\\{Z_{{{a}^{\downarrow}}}^{v}\right\\}=\bigcup_{a^{\prime}\in{a}^{\downarrow}}E\left\\{Z_{{{a^{\prime}}}}^{v}\right\\}=E\left\\{\zeta(v)\right\\}\cap\delta({a}^{\downarrow})$. The following lemma about $Z_{{a}^{\downarrow}}^{v}$ is now implicit and immediately follows from Property 4.18. ###### Lemma 4.20. For any node $a$ and $v\in\mathcal{A}\left(a\right)$, $Z_{{a}^{\downarrow}}^{v}$ depends on the edge set $E\left\\{Z_{{a}^{\downarrow}}^{v}\right\\}=\bigcup\limits_{a^{\prime}\in{a}^{\downarrow}}E\left\\{Z_{a^{\prime}}^{v}\right\\}$ and takes one of the following values 1. a) $\mathbb{e}$ when $E\left\\{Z_{{a}^{\downarrow}}^{v}\right\\}=\emptyset$ 2. b) ${\langle w,\pi\left(w\right),\eta(w),\gamma({a}^{\downarrow},{{w}^{\downarrow}})\rangle}$ when there exists a node $w$ at level $\ell\left(v\right)$ such that all the edges in $E\left\\{Z_{{a}^{\downarrow}}^{v}\right\\}$ have one endpoint in ${{w}^{\downarrow}}$ 3. c) $\nparallel$ otherwise Basically, $Z_{{a}^{\downarrow}}^{v}$ captures if there exists some node $w$ at level $\ell\left(v\right)$ such that all the edges which are in $\delta({a}^{\downarrow})$ and go out of the vertex set ${v}^{\downarrow}$, have the other end point in the vertex set ${w}^{\downarrow}$. ###### Lemma 4.21. $\zeta(\cdot)$ is an efficiently computable semigroup function. ###### Proof. In Lemma 4.19, it was shown that there exists a $O(D)$ time algorithm to compute $Z_{a}^{v}$ for any node $a$ and $v\in\mathcal{A}\left(a\right)$. Further each such $Z_{a}^{v}$ is either a special symbol among $\mathbb{e},\nparallel$ or a four tuple. It is easy to see that this four tuple is of $O(\log n)$ bits because the first two elements in the four tuple are node ids and the last two are integers which cannot exceed the number of edges and we require $O(\log n)$ bits to represent them. Now invoking Lemma 4.5, we know that $\zeta(\cdot)$ is an efficiently computable tree restricted semigroup function. ∎ ###### Lemma 4.22. For all nodes $a$ and $v\in\mathcal{A}\left(a\right)$, $Z_{{a}^{\downarrow}}^{v}$ can be computed in $O(D)$ time. ###### Proof. Since $\zeta(\cdot)$ is an efficiently computable semigroup function and can be computed in $O(D)$ rounds. Also, during the computation of $\zeta(\cdot)$, for all node $a$ and $v\in\mathcal{A}\left(a\right)$, $Z_{{a}^{\downarrow}}^{v}$ is also computed. ∎ ###### Lemma 4.23. Let $a,w$ be two vertices such that ${w}^{\downarrow}\cap{a}^{\downarrow}=\emptyset$. WLOG, let $\ell\left(a\right)\geq\ell\left(w\right)$. Let $\left\\{(\pi\left(a\right),a),(\pi\left(w\right),w)\right\\}$ be an induced cut such that $\gamma({w}^{\downarrow},{a}^{\downarrow})>0$, then node $a$ can find it in $O(D)$ rounds. ###### Proof. Here we have to prove that node $a$ will have access to $\eta(w)$ and $\gamma({w}^{\downarrow},{a}^{\downarrow})$ if such a min cut occurs. Then confirming the min-cut is easy by Lemma 3.3. Let $v$ be the ancestor of node $a$ at level $\ell\left(w\right)$. By Observation 4.20, we know that $Z_{{a}^{\downarrow}}^{v}=\langle w,\pi\left(w\right),\eta(w),\gamma({w}^{\downarrow},{a}^{\downarrow})\rangle$ in this case. Thus, the required information will be available at node $a$. ∎ ###### Proof of Lemma 4.14. When the induced cut 2-respects the tree that is it is a symmetric difference of two tree cuts $\delta({{a}^{\downarrow}})$ and $\delta({{w}^{\downarrow}})$ for some $a,w\in V\setminus r$. As per Observation 4.13 here two cases could occur. The nested case when ${{a}^{\downarrow}}\subset{{w}^{\downarrow}}$ and the mutually disjoint case when ${{a}^{\downarrow}}\cap{{w}^{\downarrow}}$, results regarding them are given in Lemma 4.16 and Lemma 4.23. In line 5 and 5 of Algorithm 5, we give details about the actual search. 1 Available Info: __Each node $a$, $\forall v\in\mathcal{A}\left(a\right)$ knows $\eta(v),H_{{a}^{\downarrow}}^{v}$ and $Z_{{a}^{\downarrow}}^{v}$ (Lemma 4.12,4.9,4.22) 1 for _$l=1$ to $\ell(a)$_ do $v\leftarrow$ $\alpha(a,l)$ // ancestor of node $a$ at level $l$ 2 if _$Z_{{{a}^{\downarrow}}}^{v}\notin\left\\{\mathbb{e},\nparallel\right\\}$_ then 3 Let $Z_{{{a}^{\downarrow}}}^{v}=\langle w,\pi\left(w\right),\eta(w),\gamma({{w}^{\downarrow}},{{a}^{\downarrow}})\rangle$ 4 if _$\eta(a)-\gamma({{w}^{\downarrow}},{{a}^{\downarrow}})=1$ & $\eta(w)-\gamma({{w}^{\downarrow}},{{a}^{\downarrow}})=1$_ then 5 $\left\\{(\pi\left(w\right),w),(\pi\left(a\right),a)\right\\}$ is a 2-cut 6 7 if _$v\neq a$ & $\eta(v)-H_{{{a}^{\downarrow}}}^{v}=1$ & $\eta(a)-H_{{{a}^{\downarrow}}}^{v}=1$_ then 8 $\left\\{(\pi\left(v\right),v),(\pi\left(a\right),a)\right\\}$ is a 2-cut 9 Algorithm 5 Algorithm to find 2-cut for node $a$ ∎ In this chapter, we formally defined tree restricted semigroup function. Further, we introduced two different types of tree restricted semigroup function $\eta(\cdot)$ and $\zeta(\cdot)$. We showed that these are enough to find min-cuts of size 1 and 2. In the next chapter, we will give algorithms to find min-cuts of size $3$. ## Chapter 5 Min-Cuts of size three In this chapter, we will give an algorithm to find a min-cut of size three. The idea here is to use Lemma 3.5 and Lemma 3.6 given in Chapter 3 which characterizes the min-cut of size $3$. Having laid down these characterization lemmas, the critical aspect which remains to find the min-cut of size 3 (if it exists) is to communicate the required quantities by the characterizing lemmas to at least one node in the network whenever a min-cut of the kind occurs. For this chapter, we will assume that a min-cut of size $1,2$ do not occur. Recall that we have fixed a BFS tree $\mathcal{T}$ in the beginning. If there exists a min-cut in the network, there could be 7 different cases. These cases are different to each other based on the relation of the min-cut to the fixed BFS tree $\mathcal{T}$. We enumerate these cases in Lemma 5.1. Further, give an algorithmic outline about how these cases can be found in Table 5.1. ###### Lemma 5.1. If there exists a min-cut of size 3 then the following cases may arise for some $v_{1},v_{2},v_{3}\in V\setminus r$ and $e,f$ as non-tree edge. 1. CASE-1 $\left\\{(\pi\left(v_{1}\right),v_{1}),e,f\right\\}$ is a min-cut 2. CASE-2 $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),e\right\\}$ is a min-cut such that ${v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ 3. CASE-3 $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),e\right\\}$ is a min-cut such that ${v_{2}}^{\downarrow}\cap{v_{1}}^{\downarrow}=\emptyset$ 4. CASE-4 $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut and ${v_{3}}^{\downarrow}\subset{v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ 5. CASE-5 $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut ${v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{3}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{2}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$ 6. CASE-6 $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut and ${v_{3}}^{\downarrow},{v_{2}}^{\downarrow}$ and ${v_{1}}^{\downarrow}$ are pairwise mutually disjoint 7. CASE-7 $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut and ${v_{3}}^{\downarrow}\subset{v_{2}}^{\downarrow}$, ${v_{1}}^{\downarrow}\cap{v_{2}}^{\downarrow}=\emptyset$ and ${v_{1}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$ ###### Proof. From Observation 3.1, we know that that if there exists a min-cut of size $3$ then it shares either $1,2$ or $3$ edges with the tree $\mathcal{T}$. When a min-cut of size $3$ shares one edge with the tree then CASE-1 applies. Here two edges are non-tree edges. When a min-cut of size $3$ shares 2-edges with the tree $\mathcal{T}$ then there exists two tree edges. For some node $v_{1},v_{2}\in V\setminus r$ let these edges be $(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2})$. WLOG let $\ell\left(v_{1}\right)\leq\ell\left(v_{2}\right)$. Similar to Observation 4.13, there could be two cases here either $v_{2}\subset v_{1}$ or $v_{1}\cap v_{2}=\emptyset$. Both of which are described in CASE-2 and CASE-3 respectively. The non-trivial part here is when the min-cut of size 3 shares all 3 edges with the tree. Here we will have 4 different cases. Let these cut edges be $(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2})$ and $(\pi\left(v_{3}\right),v_{3})$. Also in the beginning let $\ell\left(v_{1}\right)<\ell\left(v_{2}\right)<\ell\left(v_{3}\right)$. We start with the case when ${v_{3}}^{\downarrow}\subset{v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ which is described in CASE-4. Now when ${v_{3}}^{\downarrow}\not\subset{v_{2}}^{\downarrow}$ then since $v_{2},v_{3}$ are different thus we have ${v_{2}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$ which is CASE-5. Notice that in both CASE-4 and CASE-5 we have $({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow})\subset{v_{1}}^{\downarrow}$. Now we move to the different scenario where $({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow})\not\subset{v_{2}}^{\downarrow}$. Here also we may have two cases when ${v_{2}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$ then we get CASE-6 and when ${v_{2}}^{\downarrow}\subset{v_{3}}^{\downarrow}$ then we get CASE-7. Note that for CASE-6 and CASE-7 we may not require $\ell\left(v_{1}\right)\leq\ell\left(v_{2}\right)$ and $\ell\left(v_{1}\right)\leq\ell\left(v_{3}\right)$ ∎ | Characterization | Case | Technique ---|---|---|--- 1-respects $\mathcal{T}$ | - | CASE-1 | check if for some $v$, $\eta(v)=3$ 2-respects $\mathcal{T}$ | Lemma 3.5 | CASE-2 | _Broadcast Type - 1_ Section 2.1.2 CASE-3 | 2-Sketch Section 5.1 3-respects $\mathcal{T}$ | Lemma 3.6 | CASE-4 | _Broadcast Type - 2_ Section 2.1.2 CASE-5 | Layered Algorithm Section 5.2 CASE-6 | 3-Sketch Section 5.1 CASE-7 | Reduced 2-Sketch Section 5.1 Table 5.1: Overview of the case-structure of min-cut of size $3$. A min-cut of size $3$ if exists may share $1,2$ or $3$ edges with the fixed BFS tree $\mathcal{T}$. The different cases as mentioned in Lemma 5.1 are pictorially shown in Figure 5.1. (a) CASE-2 (Either of $\alpha$ or $\beta$ is 1 and the other is $0$) (b) CASE-3 (Either of $\alpha$ or $\beta$ is 1 and the other is $0$) (c) CASE-4 (Both $\alpha$ and $\beta$ are non-zero) (d) CASE-5 (At least two of $\alpha$, $\beta$ and $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})$ are non-zero) (e) CASE-6 (at least two of $\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow})$, $\gamma({v_{3}}^{\downarrow},{v_{2}}^{\downarrow})$, $\gamma({v_{1}}^{\downarrow},{v_{3}}^{\downarrow})$ are non-zero) (f) CASE-7 (Both $\alpha$ and $\beta$ are non-zero) Figure 5.1: Different cases of an min cut of size three. Each figure in the above examples is a snippet of the tree and shows a different case of min-cut of size 3. Red edges are cut edges. Each shaded region in the above figures correspond to a vertex set. Sets with same color in the shaded region correspond to one side of cut. Thick edges with arrow ends may correspond to zero or more edges between two vertex set. The label on these edges represent the actual number of edges. Among these cases, CASE-1 is simple. Here for some node $v_{1}$ the induced cut is $(V\setminus{v_{1}}^{\downarrow},{v_{1}}^{\downarrow})$. Here we just need to find the size of tree cut $\eta(v_{1})$ and if there exists such a min cut of size $3$, then $\eta(v_{1})=3$. This takes $O(D)$ time as shown in Lemma 4.11. Among the other cases only CASE-2 and CASE-4 have a simple broadcast based algorithm which is enough to let at least one of the node know the required quantities as per Lemma 3.5 and Lemma 3.6. We give the details in the following lemmas. ###### Lemma 5.2. When there exists a min-cut of size 3 as given in CASE-2 (for some nodes $v_{1},v_{2}\in V\setminus r$ and a non-tree edge $e$ $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),e\right\\}$ is a min-cut such that ${v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$), then the cut edges can be found in $O(D)$ time. ###### Proof. From Chapter 4, we know that each node $v$ knows $\eta(v)=|\delta({v}^{\downarrow})|$. Also for each $u\in\mathcal{A}\left(v\right)$, $v$ knows $\eta(u)$ and $H^{u}_{{v}^{\downarrow}}$ by Lemma 4.12 and 4.9. Thus if there exists an induced cut as in $\textbf{CASE}-2$ then we just need to make a simple evaluation based on Lemma 3.5. We give the details of the evaluation in Algorithm 6. 1 Available Info: __Each node $x$, $\forall v\in\mathcal{A}\left(x\right)$ knows $\eta(v),H_{{x}^{\downarrow}}^{v}$ (Lemma 4.12 ,4.9) 1 for _$v\in\mathcal{A}\left(x\right)\setminus\left\\{x,r\right\\}$_ do 2 if _$\eta(v)-1=H_{{x}^{\downarrow}}^{v}=\eta(x)-2$ OR $\eta(v)-2=H_{{x}^{\downarrow}}^{v}=\eta(x)-1$_ then 3 $\delta({v}^{\downarrow}\oplus{x}^{\downarrow})$ is an induced cut of size 3 4 5 Algorithm 6 Algorithm to find an induced cut of size $3$ as given in CASE-2 (for some nodes $v_{1},v_{2}\in V\setminus r$ and a non-tree edge $e$ $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),e\right\\}$ is a min-cut such that ${v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$) run on all node $x\in V\setminus r$ ∎ ###### Lemma 5.3. When there exists a min-cut of size 3 as given in CASE-4 (For some $v_{1},v_{2},v_{3}\in V\setminus r$, $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut and ${v_{3}}^{\downarrow}\subset{v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$) then the cut edges can be found in $O(D^{2})$ time. ###### Proof. By Lemma 4.9, each node $a$ knows $H_{{a}^{\downarrow}}^{v}=\gamma({a}^{\downarrow},{v}^{\downarrow})$ for all $v\in\mathcal{A}\left(a\right)$. Further, for any node $a\in V\setminus r$ we want to make sure that for any two nodes $v,u\in\mathcal{A}\left(v\right)\setminus\left\\{a,r\right\\}$, node $a$ knows $H_{{u}^{\downarrow}}^{v}$ if $\ell\left(u\right)>\ell\left(v\right)$. For this, each node $x$ at level $i$ has $i-1$ such quantities to broadcast to its descendants in ${x}^{\downarrow}$. This is similar to _Broadcast Type-2_ and takes $O(D^{2})$ rounds. After this step every node $x$ knows $H_{{y}^{\downarrow}}^{z}$ for all $y,z\in\mathcal{A}\left(x\right)\setminus\left\\{r,x\right\\}$ and $\ell\left(y\right)>\ell\left(z\right)$. Now at each node $x$, to determine if $\left\\{(\pi\left(x\right),x),(\pi\left(y\right),y),(\pi\left(z\right),z)\right\\}$ is a min-cut, node $x$, checks if $\eta(x)-1=H_{{x}^{\downarrow}}^{y}+H_{{x}^{\downarrow}}^{z}$ and $\eta(y)-1=H_{{y}^{\downarrow}}^{z}+H_{{x}^{\downarrow}}^{y}$ and $\eta(z)-1=H_{{x}^{\downarrow}}^{z}+H_{{y}^{\downarrow}}^{z}$ ∎ For other cases the problem boils down to efficiently computing the required quantities as per Lemma 3.5 and 3.6 and communicating them to at least one node in the network, then this node can make the required decision about the min-cut of size 3. Unfortunately, simple broadcast and convergecast techniques do not seem plausible for the cases which are left. This is because of the arbitrary placements of the nodes in the tree. In the remaining part of this chapter, we introduce two new techniques which will take care of this. In Section 5.1, we give Sketching Technique for CASE-3,CASE-6 and CASE-7. Further, in Section 5.2, we give Layered Algorithm which is enough to find the min-cut as given by CASE-5 if it exists. ### 5.1 Graph Sketching In this section, we will introduce our graph sketching technique. Recall from Lemma 3.5 and Lemma 3.6 that to make a decision about min-cut of size $3$, a node requires certain information about other nodes. The whole graph has as many as $n$ nodes. It will be cost ineffective for every node to know the details about each of $n$ nodes. We introduce sketching technique, which reduces the number of nodes, any particular node has to scan, in order to find the required quantities to make a decision about the min-cut. A sketch is defined for all the nodes in the network. If there exists a min- cut, at least some node $v$, can make the decision regarding it using its sketch or appropriate sketch of other nodes communicated to it. In this section, first we will give the motivation behind the use of sketch, then in Subsection 5.1.1, we will define sketch and introduce related notations. Here we will prove that the size of the sketch is not large. Later in Subsection 5.1.2, we will give the algorithm to compute the sketch and in Subsection 5.1.3, we will showcase the application of graph sketch in finding a min-cut as given by CASE-3, CASE-6 and CASE-7. Before going to the formal notation of the sketch definition, we will describe algorithmic idea for these cases. First, we begin with CASE-3. Imagine that $\left\\{(\pi\left(u\right),u),(\pi\left(v\right),v),e\right\\}$ is a min-cut of size $3$, for some nodes $v,u\in V\setminus r$ such that ${v}^{\downarrow}\cap{u}^{\downarrow}=\emptyset$ and a non-tree edge $e$ as given by CASE-3 and shown in Fig. 5.2 (A). Now imagine node $u$ has to make a decision of this min-cut, then as per Lemma 3.5, it will require information about $\eta(v),\gamma({u}^{\downarrow},{v}^{\downarrow})$. But upfront, node $u$ has no idea that it is part of such a min-cut and there exists some other node $v$, because it only has local information. Moreover, there are, as many as $n$ nodes, in the whole network which is lot of information for node $u$ to look at and is cost inefficient. Our sketching technique brings down the size of the set of nodes which any node $u$ has to scan to make a decision about a min-cut as give by CASE-3. Let $\overline{\mathcal{N}}(A)\triangleq\left\\{y\mid x\in A,y\notin A,(x,y)\in E\ \text{is a non-tree edge}\right\\}$. We make two simple observations: 1. i) node $u$ needs to search only the paths from the root $r$ to all nodes $y\in\overline{\mathcal{N}}({u}^{\downarrow})$ shown in Fig. 5.2 (B). 2. ii) such paths can be significantly trimmed: for instance as shown in Fig. 5.2 (A), $\left\\{(\pi\left(u\right),u),(v,v_{1}),e\right\\}$ cannot be a min-cut because removing these edges does not partition the vertex set into two. Figure 5.2: Demonstration of Sketching Technique for CASE-3. Each part in the above figure is a snippet of the tree. Edges with dashed stroke style are non- tree edges. Red edges are cut edges. Shaded region with vertices denote vertex sets. Based on the above two simple observation, we can see that node $u$ can limit its scan of finding some node $x$ and subsequently $\eta(x),\gamma({x}^{\downarrow},{u}^{\downarrow})$ to the bold path shown in Fig. 5.2 (C). Our sketch exactly computes this. We will give details about it in the later part of this section. The idea for CASE-6 is similar to the one demonstrated here. To find a min-cut as given by CASE-7, we will use a different idea called _reduced sketch_. Recall that, a min-cut as given by CASE-7 is as follows: for $v_{1},v_{2},v_{3}\in V\setminus r$, $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut such that ${v_{3}}^{\downarrow}\subset{v_{2}}^{\downarrow}$, ${v_{1}}^{\downarrow}\cap{v_{2}}^{\downarrow}=\emptyset$ such that ${v_{1}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$. Here we use the characterization given in Lemma 3.6 which requires that at least one node knows 6 quantities $\eta(v_{1}),\eta(v_{2}),\eta(v_{3}),\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow}),\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow}),\gamma({v_{1}}^{\downarrow},{v_{3}}^{\downarrow})$. In this case, we use a modified sketch. For any node $v$, our algorithm ensures that each node $c\in{v}^{\downarrow}$ have information about strategically truncated and trimmed paths from root $r$ to all the vertices in the set $\overline{\mathcal{N}}({v}^{\downarrow}\setminus{{c}^{\downarrow}})$. The same is illustrated in Fig. 5.3. The pictorial representation of this case shown in Fig. 5.3 (A). Our algorithm makes sure that node $v_{3}$ (see that $v_{3}\in v_{2}^{\downarrow}$) has information about the nodes in the bold path (specially truncated and trimmed paths from root to nodes in $\overline{\mathcal{N}}(v_{2}^{\downarrow}\setminus v_{3}^{\downarrow})$) shown in Fig. 5.3 (C). The intermediate step is shown in Fig. 5.3 (B). Also, coupling it with Lemma 4.9 node $v_{3}$ knows $H_{{v_{3}}^{\downarrow}}^{v_{2}}=\gamma({v_{3}}^{\downarrow},{v_{2}}^{\downarrow})$. Figure 5.3: Motivation for Sketching Technique in CASE-7. Edges with dashed stroke style are non-tree edges. Shaded region with vertices denote vertex sets. In the next subsection, we will give a formal definition of _sketch_ and _reduced sketch_. We will give the definition for a general spanning tree $T$. The sketch is defined for a parameter $k$ which governs the number of branches which can be included in the sketch. Further, we will give distributed algorithms to compute sketch and reduced sketch. #### 5.1.1 Definition of Sketch For any node $x$, let $\rho_{T}(x)$ represent the unique path from root $r$ to the node $x$ in tree $T$. Further for any vertex set $A\subseteq V$, let $\mathcal{P}_{T}(A)\triangleq\left\\{\rho_{T}(x)\mid x\in A\right\\}$. Basically, $\mathcal{P}_{T}(A)$ is a set of paths. We say that a tree path $\rho_{T}(x)$ is parallel to a tree edge $e=(a,b)$, if ${x}^{\downarrow{T}}\cap{a}^{\downarrow{T}}=\emptyset$ and ${x}^{\downarrow{T}}\cap{b}^{\downarrow{T}}=\emptyset$. Also, for any vertex set $A\subset V$ and a tree $T$ recall that $\overline{\mathcal{N}}_{T}(A)=\left\\{y\mid x\in A,(x,y)\in E,(x,y)\text{ is a non-tree edge}\right\\}$. Now, we define canonical tree which is the first structure towards defining sketch. The sketch which we will define is nothing but a truncation of this canonical tree. For any node $v$, the canonical tree is the graph-union of paths from the root $r$ to non-tree neighbors of node $v$. This notation for a canonical tree is also overloaded for a vertex set as well and formally defined below. ###### Definition 5.4 (Canonical Tree). Canonical tree of a node $v$ is a subtree of some spanning tree $T$, denoted by $R_{T}(v)$ and formed by union (graph-union operation) of tree paths in $\mathcal{P}_{T}\left(\left\\{v\cup\overline{\mathcal{N}}_{T}\left(\left\\{v\right\\}\right)\right\\}\right)$. Canonical tree of a vertex set ${v}^{\downarrow{T}}$ is denoted by $R_{T}({v}^{\downarrow{T}})$ and formed by union of the paths in $\mathcal{P}_{T}\left(\left\\{v\cup\overline{\mathcal{N}}_{T}({v}^{\downarrow{T}})\right\\}\right)$. Further, we also define a reduced canonical tree. We use the same notation since the idea is same. ###### Definition 5.5. Let $v$ be an internal (non-leaf and non-root) node of a tree $T$. Let $c\in{v}^{\downarrow{T}}$, we define the reduced canonical tree denoted by $R_{T}({v}^{\downarrow{T}}\setminus{c}^{\downarrow{T}})$ and formed by union of the paths in $\mathcal{P}_{T}\left(\left\\{v\cup\overline{\mathcal{N}}_{T}({v}^{\downarrow{T}}\setminus{c}^{\downarrow{T}})\right\\}\right)$ We define sketch as the truncation of the canonical tree and give an algorithm that can compute it in $O(D^{2})$ rounds. A canonical tree could be of very large size. Thus its truncation is required. To characterize this truncation we will use branching number as defined in Definition 5.6. Let $\operatorname{firstBranchNode}(T^{\prime})$ of any rooted tree $T^{\prime}$ be the branch node (a node which has at least two children in a tree $T^{\prime}$) closest to root. If the tree $T^{\prime}$ has no branch node then $\operatorname{firstBranchNode}(T^{\prime})$ is the root itself. ###### Definition 5.6 (Branching Number). For any tree $T^{\prime}$, the branching number of a node $b$ in tree $T^{\prime}$ is denoted by $\mathcal{\xi}_{{T^{\prime}}}\left(b\right)$. It is defined as $\mathcal{\xi}_{{T^{\prime}}}\left(b\right)\triangleq\begin{cases}1&\ell_{T^{\prime}}\left(b\right)\leq\ell_{T^{\prime}}\left(x\right)\ \&\ x\neq root(T^{\prime})\\\ 2&{b}={x}=root(T^{\prime})\\\ deg_{{}_{T^{\prime}}}(\pi_{{T^{\prime}}}\left(b\right))+\mathcal{\xi}_{{T^{\prime}}}\left(\pi_{{T^{\prime}}}\left(b\right)\right)-2&\ell_{T^{\prime}}\left(b\right)>\ell_{T^{\prime}}\left(x\right)\\\ \end{cases}$ where $x=\operatorname{firstBranchNode}(T^{\prime})$. The aforementioned definition is illustrated through examples in Figure 5.4. Basically, for any given tree $T^{\prime}$, branching number of any node in the tree is a function of number of splits in paths from root to that node. Figure 5.4: Illustration of branching number on two separate trees $T_{1}$ and $T_{2}$ We will now make a simple observation about branching number and give a characterizing lemma regarding the size of the canonical tree. ###### Observation 5.7. Let $v$ be a node and $c\in\operatorname{children}_{T}(v)$. Let $b\in R_{T}({c}^{\downarrow{T}})$, then $\xi_{R_{T}({c}^{\downarrow{T}})}(b)\leq\xi_{R_{T}({v}^{\downarrow{T}})}(b)$. ###### Proof. The canonical tree $R_{T}({c}^{\downarrow{T}})$ of any node $c\in\operatorname{children}_{T}(v)$ is the subtree of the canonical tree $R_{T}({v}^{\downarrow{T}})$. Hence the observation. ∎ ###### Lemma 5.8. For any tree $T^{\prime}$, the number of nodes in the tree that has branching number less than $k$ is $O(2^{k}Depth(T^{\prime}))$. ###### Proof. In the worst case $T^{\prime}$ may be a binary tree. Then each branching node in the tree will have a degree $3$ and on the path beyond that branching node will have branching number one more than the parent (by the definition of branching node). On every branching node there are two paths which split in the tree $T^{\prime}$. Thus we may have as many as $O(2^{k})$ different branching paths. Each path may be $O(Depth(T^{\prime}))$ long thus we have $O(2^{k}Depth(T^{\prime}))$ such nodes. ∎ We now define graph-sketch of a node which is defined based on the canonical tree and comes with a parameter $k$ on which the truncation is based. We define the truncation of a tree as below. ###### Definition 5.9. For some tree $T^{\prime}$ and a number $k$, $\operatorname{trunc}(T^{\prime},k)$ is an sub-tree of $T^{\prime}$ induced by vertices with branching number less than or equal to $k$. Our graph sketch will be called as $k$-Sketch because it comes with a parameter $k$ on which the truncation is based. For every node in the $k$-sketch, meta information is also added. We define the $k$-Sketch of a node as below. ###### Definition 5.10 ($k$-Sketch). For any node $v$ and a spanning tree $T$, let $R^{\prime}=\operatorname{trunc}(R_{T}({v}^{\downarrow{T}}),k)$. The $k$-Sketch of a node $v$ w.r.t. the spanning tree $T$ is denoted as $\mathcal{S}_{T}^{k}(v)$ and it is defined as $\mathcal{S}_{T}^{k}(v)\triangleq\left\\{R^{\prime}\cup\left\\{\langle u:\eta_{T}(u),\pi_{{T}}\left(u\right),\gamma\left({v}^{\downarrow{T}},{u}^{\downarrow{T}}\right)\rangle\ \forall u\in R^{\prime}\right\\}\right\\}$ Basically $k$-Sketch is a truncated canonical tree packaged along with the meta information $\langle u:\eta_{T}(u),\pi_{{T}}\left(u\right),\gamma\left({v}^{\downarrow{T}},{u}^{\downarrow{T}}\right)\rangle$ for each node $u$ in the truncated tree. We will give an algorithm to compute $k$-Sketch for every node $v\in V$ in the next sub-section and further showcase the application of $k$-Sketch to find a min-cut if it exists as given by CASE-3 and CASE-6. Similar to $k$-Sketch of a node $v$ we define the reduced $k$-Sketch which is based on the reduced canonical tree. This will be used to find a min-cut if it exists as given by CASE-7. ###### Definition 5.11 (Reduced $k$-Sketch). Let v be an internal node of a spanning tree $T$. Let $c\in{v}^{\downarrow{T}}$ and let $R^{\prime}=\operatorname{trunc}(R_{T}({v}^{\downarrow{T}}\setminus{c}^{\downarrow{T}}),k)$. The reduced $k$-Sketch of a node $v$ and $c$ w.r.t. the spanning tree $T$ is denoted as ${\mathcal{S}}_{T}^{k}(v,c)$ and it is defined as ${\mathcal{S}}_{T}^{k}(v,c)\triangleq\left\\{R^{\prime}\cup\left\\{\langle u:\eta_{T}(u),\pi_{{T}}\left(u\right),\gamma\left({v}^{\downarrow{T}}\setminus{c}^{\downarrow{T}},{u}^{\downarrow{T}}\right)\rangle\ \forall u\in R^{\prime}\right\\}\right\\}$ We know give the following Lemma about the size of the $k$-Sketch. ###### Lemma 5.12. For any spanning tree $T$, the k-Sketch of a node $v$, $\mathcal{S}_{T}^{k}(v)$ w.r.t. $T$ is of size $O(2^{k}Depth(T)\log n)$ bits. ###### Proof. For any arbitrary node $u\in\mathcal{S}_{T}^{k}(v)$, the sketch contains a three tuple $\langle\eta_{T}(u),\pi_{{T}}\left(u\right),\gamma\left({v}^{\downarrow{T}},{u}^{\downarrow{T}}\right)\rangle$. Here $\eta_{T}(u),\gamma\left({v}^{\downarrow{T}},{u}^{\downarrow{T}}\right)\leq|E|=O(n^{2})$ and can be represented in $O(\log n)$ bits. Thus the three tuple is of $O(\log n)$ bits. Now by Lemma 5.8 it is clear that $\mathcal{S}_{T}^{k}(v)$ is of size $O(2^{k}Depth(T)\log n)$ bits. ∎ ###### Corollary 5.13. For any spanning tree $T$ and an internal node $v$ and some $c\in{v}^{\downarrow{T}}$ the reduced $k$-Sketch ${\mathcal{S}}_{T}^{k}(v,c)$ w.r.t. $T$ is of size $O(2^{k}Depth(T)\log n)$ bits. The $k$-Sketch of a node will be used to find a min-cut as given by CASE-3 and CASE-6. Whereas the reduced $k$-Sketch will be used to find a min-cut as given by CASE-7. In the subsequent section, we will give algorithms to compute $k$-Sketch and the reduced $k$-Sketch. We will work with the fixed BFS tree $\mathcal{T}$ and for simplicity in the notations the $\mathcal{T}$ will be skipped from subscript or superscript. #### 5.1.2 Algorithm to Compute Sketch In this subsection, we will give distributed algorithms to compute $k$-Sketch and the reduced $k$-Sketch. We will prove that our algorithm takes $O(D^{2})$ rounds. The idea to compute sketch is as follows: Each node computes its own $k$-Sketch (which is of size $O(D)\log n$ bits) and communicates the same to the parent. The parent node after receiving the sketch from all the children computes its own sketch and communicates the same further up. This process continues and at the end each node has its $k$-Sketch. Here we will use Observation 5.7, to argue that the sketch received from children is enough for a node to compute its sketch. ###### Lemma 5.14. For all $v\in V$, $\mathcal{S}^{k}(v)$ can be computed in $O(D^{2})$ rounds. ###### Proof. We describe a detailed algorithm to compute this in Algorithm 7. In Line 7, Algorithm 7 calculates $\mathcal{P}\left(\left\\{v\cup\overline{\mathcal{N}}(v)\right\\}\right)$ which as per the definitions is the tree paths of the non-tree neighbors of $v$ including $\rho(v)$. 1 _Algorithm to be run on each node $a$_ Output: k-Sketch $\mathcal{S}^{k}(a)$ 2 _Past Knowledge_ 3 Each node $u\in V$ knows $\ell\left(u\right)$ and $\eta(u)$ from previous section 4 For each $u\in\mathcal{A}\left(a\right)$, $a$ knows $H_{{a}}^{u}$ and $\eta(u)$ 5 $\overline{\mathcal{N}}(a)\leftarrow\left\\{b\mid(a,b)\in E,(a,b)\ \text{is a non-tree edge}\right\\}$ 6 for _all $b\in\overline{\mathcal{N}}(a)$_ parallely do 7 send tuples $\langle\ell\left(u\right),\eta(u),u\rangle$ for all $u\in\mathcal{A}\left(a\right)$ to $b$ $\mathcal{P}\leftarrow\left\\{\rho(a)\right\\}$ // $\rho(a)$ can be computed easily because $a$ has all nodes in $\mathcal{A}\left(a\right)$ 8 for _all $b\in\overline{\mathcal{N}}(a)$_ parallely do 9 for _all $u\in\mathcal{A}\left(b\right)$_ do receive tuples $\langle\ell\left(u\right),\eta(u),u\rangle$ 10 Construct the path $\rho(b)$ using the information 11 $\mathcal{P}\leftarrow\mathcal{P}\cup\rho(b)$ 12 13 perform graph union of all the paths in $\mathcal{P}$ and form canonical tree $R(a)$ 14 for _all nodes $u\in R(a)$_ do 15 if _$u\in\mathcal{A}\left(a\right)$_ then $\gamma(a,{u}^{\downarrow})=H_{a}^{u}$ 16 else $\gamma(a,{u}^{\downarrow})=|\left\\{b\mid b\in\overline{\mathcal{N}}(a),u\in\mathcal{A}\left(b\right)\right\\}|$ 17 include the tuple $\langle u:\eta(u),\pi\left(u\right),\gamma(a,{u}^{\downarrow})\rangle$ for the node $u\in R(a)$ 18 if _$a$ is an internal node_ then wait until $\mathcal{S}^{k}(c)$ is received for all $c\in\operatorname{children}(a)$ 19 $S\leftarrow R(a)\cup_{c\in\operatorname{children}(a)}\mathcal{S}^{k}(c)$ 20 perform graph union of all trees in $S$ to form a tree $T$ 21 for _node $u\in T$_ do 22 compute branching number $\mathcal{\xi}_{{T}}\left(u\right)$ as per the definition 23 if _$\mathcal{\xi}_{{T}}\left(u\right) >k$ and $u\in{v}^{\downarrow}$_ then 24 remove node $u$ from $T$ 25 26 for _node $u\in T$_ do 27 compute $\gamma({a}^{\downarrow},{u}^{\downarrow})$ by adding appropriately from all $T^{\prime}\in S$ 28 29 Construct $\mathcal{S}^{k}(a)$ using $T$ by including $\langle u:\eta(u),\pi\left(u\right),\gamma({a}^{\downarrow},{u}^{\downarrow})\rangle$ for all node $u\in T$ 30 Send $\mathcal{S}^{k}(a)$ to $\pi\left(a\right)$ Algorithm 7 Distributed Algorithm for Computing $k$-Sketch This can be computed easily because each non-tree neighbor sends all its ancestors and their associated meta information. Having calculated $\mathcal{P}\left(\left\\{v\cup\overline{\mathcal{N}}(v)\right\\}\right)$, then it is easy to compute the sketch tree $R(v)$ by Definition 5.4. Now if a node is an internal node we again need to perform a graph union of other sketches received from children. This is also trivial and it is guaranteed that we will not lose any node here because of Observation 5.7. Further branching number is computed for all the nodes and those nodes which do not satisfy the condition of branching number are removed to form the sketch. Also among the three tuples of meta information two remain fixed from the sketch of children and $\gamma({u}^{\downarrow},{v}^{\downarrow})$ for a node $u\in\mathcal{S}^{k}(v)$ can be computed by appropriate addition. ##### Time Requirements In Algorithm 7, communication between any two nodes occur only in line 7, 7 and 7. In line 7, only $O(D)$ rounds are taken because a node has at most $O(D)$ ancestors in the BFS tree $\mathcal{T}$. Similarly, line 7 also takes $O(D)$ rounds because each of the neighbors also has $O(D)$ ancestors and the transfer of the three tuples from each of the neighbors happen in parallel. Further in line 7, a node $a$ waits to recieve all the sketch from its children. From Lemma 5.12, we know that the size of the sketch is $O(D\log n)$ bits when the tree in action is a BFS tree. Now a node at level $l$ will wait for all the sketch from its children which are at level $l+1$ and they inturn depend on all the children which are at level $l+2$ and so on. Thus line 7 takes atmost $O(D^{2})$ rounds. ∎ In Lemma 5.14, we gave an algorithm for computing the $k$-Sketch. We will now move toward an algorithm to find a reduced sketch. There are two steps towards these details of which are given in Observation 5.15 and Lemma 5.16. ###### Observation 5.15. For a constant $k$. For any internal node $a$, and $c^{\prime}\in\operatorname{children}(a)$, ${\mathcal{S}}^{k}(a,c^{\prime})$ can be computed at node $a$ in $O(D^{2})$ rounds. ###### Proof. Basically, the idea here is not to include the sketch received from child $c$, while computing the sketch of node $a$ and the resultant becomes the reduced sketch ${\mathcal{S}}^{k}(v,c)$. To do this, we just need to change Line 7 in Algorithm 7 to $\mathcal{S}\leftarrow R(a)\cup_{c\in\operatorname{children}(a)\setminus\left\\{c^{\prime}\right\\}}\mathcal{S}^{k}(c)$ for each $c^{\prime}\in\operatorname{children}(a)$ and this enables the computation ${\mathcal{S}}^{k}(a,c^{\prime})$ at node $a$. ∎ ###### Lemma 5.16. For a fixed $k$, there exists a $O(D^{2})$ round algorithm such that all nodes $x$ can compute ${\mathcal{S}}^{k}(v,c)$ for all $v\in\mathcal{A}\left(x\right)$. ###### Proof. For any internal node $v\in V$, we ensure that for all $c\in\operatorname{children}(v)$ the reduced k-Sketch ${\mathcal{S}}^{k}(v,c)$ is downcasted to all the nodes in ${c}^{\downarrow}$. This takes $O(D^{2})$ rounds. After this step every node $x$ at level $\ell\left(x\right)$ has the sketch ${\mathcal{S}}^{k}(a,b)$ for any node $a,b$ at level $i$ and $i+1$ for all $i\in[1,l-1]$ and $a,b\in\mathcal{A}\left(a\right)$. Now based on this we will show that node $x$ can compute ${\mathcal{S}}^{k}(v,x)$ for all $v\in\mathcal{A}\left(x\right)$ as per Algorithm 8. Each node $x\in V\setminus r$ has to perform some local computation based on the various reduced sketch received earlier. For $1\leq i\leq\ell\left(x\right)-1$ let $\mathcal{S}_{i}={\mathcal{S}}^{k}(a,b)$ when $a=\alpha(x,i)$ and $b=\alpha(x,i+1)$ (recall that $\alpha(x,l)$ is the ancestor of node $x$ at level $l$). For some node $v=\alpha(x,l)$ which is the ancestor of node $x$ at some level $l<\ell\left(x\right)$; to compute ${\mathcal{S}}^{k}(v,x)$ node $x$ uses $\mathcal{S}=\cup_{l\leq j\leq\ell\left(x\right)-1}\mathcal{S}_{j}$. Here $\mathcal{S}$ is basically a set of Sketches. Note that any two sketch in the set $\mathcal{S}=\cup_{l\leq j\leq\ell\left(x\right)-1}\mathcal{S}_{j}$ are overlapping. That is they have information about disjoint vertex sets. The rest of the steps are exactly same as given in Algorithm 7 and are described in detail in Algorithm 8 Input: ${\mathcal{S}}^{k}(a,b)$ for any node $a,b$ at level $i$ and $i+1$ such that $1\leq i\leq\ell\left(x\right)-1$ Output: ${\mathcal{S}}^{k}(v,x)$ for all $v\in\mathcal{A}\left(x\right)$ 1 for _$1\leq i\leq\ell\left(x\right)-1$_ do $\mathcal{S}_{i}\leftarrow{\mathcal{S}}^{k}(a,b)$ when $a=\alpha(x,i)$ and $b=\alpha(x,i+1)$ // recall that $\alpha(x,l)$ is the ancestor of node $x$ at level $l$ 2 for _$1\leq i\leq\ell\left(x\right)-1$_ do 3 $v=\alpha(x,i)$ 4 $S\leftarrow\cup_{i\leq j\leq\ell\left(x\right)-1}\mathcal{S}_{j}$ 5 perform graph union of all trees in $\mathcal{S}$ to form a tree $T$ 6 compute branching number $\mathcal{\xi}_{{T}}\left(u\right)\ \forall u\in T$ as per the definition 7 for _node $u\in T$_ do 8 if _$\mathcal{\xi}_{{T}}\left(u\right) >k$ and $u\in{v}^{\downarrow}$_ then 9 remove $u$ from $T$ 10 11 for _node $u\in T$_ do 12 compute $\gamma({v}^{\downarrow}\setminus{x}^{\downarrow},{u}^{\downarrow})$ by adding appropriately from all $T^{\prime}\in S$ 13 14 15 Construct ${\mathcal{S}}^{k}(v,x)$ using $T$ by including $\langle u:\eta(u),\pi\left(u\right),\gamma({v}^{\downarrow}\setminus{x}^{\downarrow},{u}^{\downarrow})\rangle$ for all node $u\in T$ 16 Algorithm 8 To be run on all node $x\in V\setminus\left\\{r\right\\}$ ∎ #### 5.1.3 Application of graph sketch Now we will describe the application of graph sketch to find a min-cut of size $3$ if it exists as given by CASE-3, CASE-6 and CASE-7. We prove the same in Lemma 5.17. Here, CASE-3 is a direct application of $3$-Sketch; in CASE-6 we will be required to move $3$-Sketch in a strategic way and in CASE-7 reduced $2$-Sketch will be used. We give further details of each of the cases in Lemma 5.17, 5.18 and 5.20. For any two nodes $v_{1},v_{2}$, let $\operatorname{LCA}_{T}(v_{1},v_{2})$ be the lowest common ancestor of node $v_{1}$ and $v_{2}$ in tree $T$. ###### Lemma 5.17. For some $v_{1},v_{2}\in V\setminus r$ and a non-tree edge $e$, if $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),e\right\\}$ is a min-cut as given in CASE-3, then node $v_{1}$ can make a decision about the said min-cut using $\mathcal{S}^{3}(v_{1})$. ###### Proof. As per Lemma 3.5, we know that node $v_{1}$ can decide for such a min-cut if it knows three quantities: $\eta(v_{1}),\eta(v_{2})$ and $\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow})$. Also, for any node $u,v$, we know that if there is a node $u\in\mathcal{S}^{3}(v)$ then the sketch also contains $\eta(u),\gamma({u}^{\downarrow},{v}^{\downarrow})$. And every node $v$ in the network knows $\eta(v)$ from previous section. Thus here to prove this lemma we have to prove that node $v_{2}\in\mathcal{S}^{3}(v_{1})$. Then node $v_{1}$ can enumerate through all the nodes in $\mathcal{S}^{3}(v_{1})$ which are not in $\mathcal{A}\left(v\right)$ and apply the condition of Lemma 3.5 to test if there exists such a min-cut. Figure 5.5: Illustrating different types of sub-cases which might occur when there exists a min-cut of size $3$ as in CASE-3. The figures presented here are tree snippets. In each of the sub-cases we demonstrate the $3$-Sketch as computed by the node $v_{1}$. Solid black line denote the tree paths in Sketch of $v_{1}$. Dashed line represent other paths in tree $\mathcal{T}$. Edges in red are cut edges. Here all the edges that go out of the vertex set ${v_{1}}^{\downarrow}$ have their other end points in the vertex set ${v_{2}}^{\downarrow}$ barring one non-tree edge $e$ in (B) and (C). The important fact here is that in all the different cases $v_{2}\in\mathcal{S}^{3}(v_{1})$ . We will now show that if such a min-cut exists then node $v_{2}\in\mathcal{S}^{3}(v_{1})$. As illustrated in Fig. 5.5 for all the different sub-cases node $v_{2}\in\mathcal{S}^{3}(v_{1})$. In (A) when the other non-tree edge $e$ has one end-point in ${v_{2}}^{\downarrow}$ then the branching number of $v_{2}$ in the sketch-tree $R({v_{1}}^{\downarrow})$ is $\mathcal{\xi}_{{R({v_{1}}^{\downarrow})}}\left(v_{2}\right)=2$ thus $v_{2}\in\mathcal{S}^{3}(v_{1})$. Both (B) and (C) are similar in terms of the fact that the non-tree cut edge $e$ has the other endpoint in ${v_{1}}^{\downarrow}$ but differ in terms of the branching node. Nevertheless here also $\mathcal{\xi}_{{R({v_{1}}^{\downarrow})}}\left(v_{2}\right)=3$ thus $v_{2}\in\mathcal{S}^{3}(v_{1})$ ∎ ###### Lemma 5.18. For some $v_{1},v_{2},v_{3}\in V\setminus r$, let ${v_{1}}^{\downarrow},{v_{2}}^{\downarrow},{v_{3}}^{\downarrow}$ be pair wise mutually disjoint. If there exists a min cut as in CASE-6 such that $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut, then it can be found in $O(D^{2})$ time. ###### Proof. For such a min-cut to exists then at least two of $\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow}),\gamma({v_{1}}^{\downarrow},{v_{3}}^{\downarrow}),\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})$ need to be non-zero otherwise the vertex sets ${v_{1}}^{\downarrow}\cup{v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow}$ may not form a connected component. WLOG here we may have two non-isomorphic cases as illustrated in Fig. 5.6. (a) One of $\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow}),\gamma({v_{1}}^{\downarrow},{v_{3}}^{\downarrow}),\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})$ is zero. WLOG let $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})=0$ (b) All three of $\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow}),\gamma({v_{1}}^{\downarrow},{v_{3}}^{\downarrow}),\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})$ are non-zero Figure 5.6: Two different non-isomorphic sub-cases of a min-cut as in CASE-6. Here a circle correspond to a vertex set. And an double arrow line between them indicate that there are some edges which have an end point in each of the vertex set. Also, as per Lemma 3.6 we need $\eta(v_{1}),\eta(v_{2}),\eta(v_{3}),\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow}),\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})$ and $\gamma({v_{1}}^{\downarrow},{v_{3}}^{\downarrow})$ at at least one node to decide for a min-cut as given by CASE-6. We will first work with the case in Fig. 5.6(a). Here, $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})=0$. In this sub-case node $v_{1}$ can make the decision based on $\mathcal{S}^{3}(v_{1})$. We will prove that if such a min-cut exists then $v_{2},v_{3}\in\mathcal{S}^{3}(v_{1})$. We demonstrate the different cases in Figure 5.7. The three different cases are in terms of the intersection of the paths $\rho(v_{1}),\rho(v_{2}),\rho(v_{3})$. In all the sub-cases demonstrated in the Fig. 5.7 we can see that the branching number $\mathcal{\xi}\left(v_{1}\right){v_{2}},\mathcal{\xi}\left(v_{1}\right){v_{3}}\leq 3$. Thus $v_{2},v_{3}\in\mathcal{S}^{3}(v_{1})$. Here, $v_{1}$ just need to pick pairs of nodes $a,b\in\mathcal{S}^{3}(v_{1})$ such that ${a}^{\downarrow},{b}^{\downarrow},{v_{1}}^{\downarrow}$ are pair-wise mutually disjoint and compare $\eta(a),\eta(b),\eta(v_{1}),\gamma({a}^{\downarrow},{v_{1}}^{\downarrow})$ and $\gamma({b}^{\downarrow},{v_{1}}^{\downarrow})$ as per Lemma 3.6. Figure 5.7: Illustrating different types of sub-cases. We demonstrate the $3$-Sketch as per the node $v_{1}$. Solid black line denote the tree paths in Sketch of $v_{1}$. Dashed line represent other paths in tree $\mathcal{T}$. Edges in red are cut edges. The different sub-cases occur based on the intersection of three paths $\rho(v_{1}),\rho(v_{2}),\rho(v_{3})$. In sub-case (A), $\operatorname{LCA}(v_{1},v_{3})=\operatorname{LCA}(v_{2},v_{3})\neq\operatorname{LCA}(v_{1},v_{2})$. In sub-case (B), $\operatorname{LCA}(v_{1},v_{2})=\operatorname{LCA}(v_{1},v_{3})\neq\operatorname{LCA}(v_{2},v_{3})$. In sub-case (C), $\operatorname{LCA}(v_{1},v_{2})=\operatorname{LCA}(v_{1},v_{3})=\operatorname{LCA}(v_{2},v_{3})$. Now, we will take the case as mentioned in Fig. 5.6(b). By the same argument as above we can say that $v_{1},v_{2}\in\mathcal{S}^{3}(v_{3})$ and $v_{1},v_{3}\in\mathcal{S}^{3}(v_{2})$. In this case $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})\neq 0$ and $\mathcal{S}^{3}(v_{1})$ does not have $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})$ which is required to make a decision about the said min-cut as per Lemma 3.6. Thus node $v_{1}$ cannot make the required decision based on only $\mathcal{S}^{3}(v_{1})$. Amidst, the above mentioned challenge, there is an easy way around. Each node $v\in V$ downcasts $\mathcal{S}^{3}(v)$ (its 3-Sketch) to all nodes in ${v}^{\downarrow}$. Since 3-Sketch is of size $O(D\log n)$ thus this takes $O(D^{2})$ rounds. After this step every node $u$ has $\mathcal{S}^{3}(v)\ \forall v\in\mathcal{A}\left(u\right)$. Now each node $u$ sends $\mathcal{S}^{3}(a)\ \forall a\in\mathcal{A}\left(u\right)$ to its non-tree neighbors. Since there are as many as $D$ such sketches thus this takes $O(D^{2})$ time. Also, $\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow})>0$ thus we know that there is at least one node node $v_{1}^{\prime}\in{v_{1}}^{\downarrow}$ and $v_{2}^{\prime}\in{v_{2}}^{\downarrow}$ such that $(v_{1}^{\prime},v_{2}^{\prime})$ is a non-tree edge. After the above mentioned steps node $v_{1}^{\prime}$ as well as node $v_{2}^{\prime}$ will have both $\mathcal{S}^{3}(v_{1})$ and $\mathcal{S}^{3}(v_{2})$. Now, both ${v_{1}^{\prime}}$ and ${v_{2}^{\prime}}$ can make the decision about the said min-cut. We will discuss steps which may be undertaken at node $v_{1}^{\prime}$ for the sane, for each $v\in\mathcal{A}\left(v_{1}^{\prime}\right)$, $v_{1}^{\prime}$ locally iterates through all the parallel paths of the $\mathcal{S}^{3}(v)$ and picks all possible $x,y$ such that ${x}^{\downarrow},{y}^{\downarrow},{v}^{\downarrow}$ are mutually disjoint. Notice that through $\mathcal{S}^{3}(v)$, $v_{1}^{\prime}$ has $\gamma({x}^{\downarrow},{v}^{\downarrow}),\gamma({y}^{\downarrow},{v}^{\downarrow}),\eta(x),\eta(y)$. Also it has $\eta(v)$ from previous calculations. Now from the sketches received through its non-tree neighbors $v_{1}^{\prime}$ looks for $\gamma({x}^{\downarrow},{y}^{\downarrow})$. If $\gamma({x}^{\downarrow},{v}^{\downarrow}),\gamma({y}^{\downarrow},{v}^{\downarrow}),\eta(x),\eta(y)$ and $\gamma({x}^{\downarrow},{y}^{\downarrow})$ then satisfies Lemma 3.6 then it can make the decision about the required min-cut. ∎ Next, we move to CASE-7. Here we will use reduced sketch as given in Definition 5.11. First we the use the reduced $2$-sketch. ###### Observation 5.19. For some $v_{1},v_{2},v_{3}\in V\setminus r$, let ${v_{3}}^{\downarrow}\subset{v_{2}}^{\downarrow}$, ${v_{1}}^{\downarrow}\cap{v_{2}}^{\downarrow}=\emptyset$ and ${v_{1}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$. If there exists a min cut as in CASE-7 such that $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut then $v_{1}\in{\mathcal{S}}^{2}(v_{2},v_{3})$ ###### Proof. If such a min-cut exists then all the edges going out of the vertex set ${v_{2}}^{\downarrow}\setminus{v_{3}}^{\downarrow}$ apart from $(\pi\left(v_{2}\right),v_{2})$ and $((\pi\left(v_{3}\right),v_{3}))$ goes to the vertex set ${v_{1}}^{\downarrow}$. Thus the reduced canonical tree $R({v_{2}}^{\downarrow}\setminus{v_{3}}^{\downarrow})$ contains node $v_{1}$. We showcase this scenario in Fig. 5.20. Also the branching number of $v_{1}$ will be $\mathcal{\xi}_{{R({v_{2}}^{\downarrow}\setminus{v_{3}}^{\downarrow})}}\left(v_{1}\right)=2$ based on the definition. Thus $v_{1}\in{\mathcal{S}}^{2}(v_{2},v_{3})$ Figure 5.8: Reduced sketch ${\mathcal{S}}^{2}(v_{2},v_{3})$ when for some $v_{1},v_{2},v_{3}\in V\setminus r$, let ${v_{3}}^{\downarrow}\subset{v_{2}}^{\downarrow}$, ${v_{1}}^{\downarrow}\cap{v_{2}}^{\downarrow}=\emptyset$ and ${v_{1}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$ and $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut ∎ ###### Lemma 5.20. For some $v_{1},v_{2},v_{3}\in V\setminus r$, let ${v_{3}}^{\downarrow}\subset{v_{2}}^{\downarrow}$, ${v_{1}}^{\downarrow}\cap{v_{2}}^{\downarrow}=\emptyset$ and ${v_{1}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$. If there exists a min cut as in CASE-7 such that $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut then it can be found in $O(D^{2})$ time. ###### Proof. Here we run Algorithm 9 on each node. 1 _Past Knowledge_ 2 ${\mathcal{S}}^{2}(v,x)$ for all $v\in\mathcal{A}\left(x\right)$ using Lemma 5.16 3 For each $u\in\mathcal{A}\left(x\right)$, $x$ knows $H_{{x}^{\downarrow}}^{u}$ and $\eta(u)$ 1 for _$v\in\mathcal{A}\left(x\right)\setminus x$_ do 2 for _$u\in{\mathcal{S}}^{2}(v,x)\ \ &\ u\notin\mathcal{A}\left(x\right)$_ do 3 if _$\eta(v)-1=H_{{x}^{\downarrow}}^{v}+\gamma({u}^{\downarrow},{v}^{\downarrow}\setminus{x}^{\downarrow})\ \ &$ $\eta(u)-1=\gamma({u}^{\downarrow},{v}^{\downarrow}\setminus{x}^{\downarrow})\ \&$ $\eta(x)-1=H_{{x}^{\downarrow}}^{v}$_ then 4 $\left\\{(\pi\left(x\right),x),(\pi\left(v\right),v),(\pi\left(u\right),u)\right\\}$ is a min-cut 5 6 Algorithm 9 Algorithm to find an induced cut fo size $3$ as given in CASE-7 to be run on all node $x\in V\setminus r$ By Lemma 3.6 the reported min-cut is correct. Further when Algorithm 9 is run on node $v_{3}$ it will be able to make decision about the required min-cut because by Observation 5.19 if there exists such a min-cut then $v_{1}\in{\mathcal{S}}^{2}(v_{2},v_{3})$. Also, this just requires $O(D^{2})$ time because computing reduced sketch ${\mathcal{S}}^{2}(v,x)$ at any node $x$ for all $v\in\mathcal{A}\left(x\right)$ just takes $O(D^{2})$ rounds as per Lemma 5.16. ∎ ### 5.2 Layered Algorithm In the last section, we gave a technique to find a min-cut of size 3 as in CASE-3, CASE-6 and CASE-7 using a special graph-sketch. In this section, we will give an algorithm to find the min-cut as given by CASE-5. A $k$-Sketch cannot be used to find a min-cut as in CASE-5 because here it is challenging for one of the nodes to know the $6$ quantities as required by Lemma 3.6 using a $k$-sketch. To resolve this challenge, we give layered algorithm where we solve for min-cut iteratively many times. Recall a min-cut as given by CASE-5 is as follows: for some node $v_{1},v_{2},v_{3}\in V\setminus r$, $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut such that ${v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{3}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{2}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$. For the introduction of layered algorithm, let us assume that such a min-cut exists. Further let $v_{1},v_{2}$ and $v_{3}$ be these specific nodes. In Fig. 5.9, we show a pictorial representation of such a min-cut. Figure 5.9: Min-cut of size $3$ as given by CASE-5 Similar to the previous section, we will use the characterization Lemma 3.6 which requires six quantities to make a decision about the min-cut. These are $\eta(v_{1}),\eta(v_{2}),\eta(v_{3}),\gamma({v_{1}}^{\downarrow},{v_{2}}^{\downarrow}),\gamma({v_{3}}^{\downarrow},{v_{2}}^{\downarrow})$ and $\gamma({v_{1}}^{\downarrow},{v_{3}}^{\downarrow})$. In this subsection, we will show that node $x=\operatorname{LCA}(v_{2},v_{3})$ can find all these six quantities. The challenge here is the fact that some of these quantities have to come from node $v_{2}$ and/or $v_{3}$. Since node $x$ is higher up in the tree $\mathcal{T}$ than $v_{2},v_{3}$ thus the information from node $v_{2},v_{3}$ when it travels up the tree may face contention from other nodes in ${x}^{\downarrow}$. From Recall (from chapter 2) that convergecast is such a technique to send data to nodes which are higher up in the tree from nodes at a lower level. For convergecast to be efficient here we need to couple it with a contention resolution mechanism. The idea is as follows: * • We calculate min-cuts of size one and two in sub-graphs _pivoted_ at nodes at different level in the tree $\mathcal{T}$ (details are deferred to Section 5.2.1) * • After the above step, each node computes its _one-cut detail_ and _two-cut detail_ (definitions are deferred to Section 5.2.2) * • Based on the above computation, if a min-cut exists as given by $\textbf{CASE}-5$, then node $v_{2},v_{3}$ can designate themselves as special nodes using Lemma 5.24. Thus, the information from node $v_{2}$ and node $v_{3}$ could reach $LCA(v_{2},v_{3})$ efficiently. #### 5.2.1 Definition of Layered Algorithm For any node $v\in V\setminus r$, let ${E}_{{v}^{\downarrow}}$ be the set of edges whose both endpoints are in the vertex set $v^{\downarrow}$. Let subgraph pivoted at any vertex $v\neq r$ be $G_{v}\triangleq(v^{\downarrow}\cup\pi\left(v\right),{E}_{{v}^{\downarrow}}\cup(\pi\left(v\right),v))$. Note that this definition is different from the subgraph rooted at $v$ because here we add the extra edge $(\pi\left(v\right),v)$ and vertex $\pi(v)$. _Layered Algorithm_ is a special type of algorithm where we will solve for induced-cut of size one and two in a layered fashion. As usual, we will work with the fixed BFS tree $\mathcal{T}$. Here we will solve for induced-cut of size $1$ and $2$ repeatedly for $Depth(\mathcal{T})=D$ times. In the first iteration, our algorithm solves for these induced cuts in the whole graph $G$. Then it solves for the min-cuts of size $1$ and $2$ in all the sub-graph pivoted at nodes at level $1$; that is in all sub-graphs in the set $\left\\{G_{v}\mid\ell\left(v\right)=1\right\\}$, subsequently in the sub- graphs for all nodes pivoted at level $2$ and so on until level $D$. This is the _Layered Algorithm for min-cut_. We will showcase the utility of this algorithm later. In Observation 5.21, we will argue that the layered algorithm for min-cut takes $O(D^{2})$ rounds. ###### Observation 5.21. The layered algorithm for min-cut runs in $O(D^{2})$ rounds. ###### Proof. From Theorem 4.10 and Lemma 4.14, we know that cuts of size 1,2 can be found in $O(D)$ rounds. For all graphs $\left\\{G_{v}\mid\ell\left(v\right)=i\right\\}$ pivoted at any level $1\leq i\leq D$ the algorithm to compute min-cut of size $1,2$ can be run independently because none of the sub-graphs in the set $\left\\{G_{v}\mid\ell\left(v\right)=i\right\\}$ share an edge. Thus it takes $O(D)$ round to run the min-cut algorithm in all such graphs. Now to run for all graphs pivoted at nodes at all levels it takes $O(D^{2})$ rounds. ∎ We will now define the notation for the sequence of subgraphs. This is a short hand notation to say that a property is satisfied for a set of pivoted sub- graphs. Let $r\rightarrow w_{1}\rightarrow w_{2}\rightarrow\cdots\rightarrow w_{t}$ be a path in the BFS tree. Then we know that $G_{w_{1}},G_{w_{2}},\ldots,G_{w_{t}}$ are subgraphs of $G$ as defined earlier. Let $i,j\in[1,t]$ and $i\leq j$, then let $G_{\left({w_{i}},{w_{j}}\right)}$ be a set of subgraphs such that $G_{\left({w_{i}},{w_{j}}\right)}\triangleq\left\\{G_{w_{h}}\mid i\leq h\leq j\right\\}$. We say a property is true for $G_{\left({w_{i}},{w_{j}}\right)}$ if it is true for all the subgraphs in the set. ###### Observation 5.22. For some nodes $x$ and $v$, let $x\in{v}^{\downarrow}$. If $(\pi\left(x\right),x)$ is a min-cut for the sub-graph $G_{v}$ then $(\pi\left(x\right),x)$ is a min-cut for all sub-graphs in $G_{\left({x},{v}\right)}$. Similarly, let $a,b$ be two nodes. Let node $z$ be the LCA of node $a,b$ in $tree$ and $v$ be some node such that ${v}\in\mathcal{A}\left(z\right)$. If $\left\\{(\pi\left(a\right),a),(\pi\left(b\right),b)\right\\}$ is a min cut for the sub-graph $G_{v}$ then it a min-cut for all the sub-graphs in $G_{\left({z},{v}\right)}$. #### 5.2.2 Application of layered algorithm We will now showcase the application of the above-mentioned layered algorithm in finding a min-cut as given by CASE-5. First, we will give a simple process in Algorithm 10 to be run on individual nodes after the run of the _Layered Algorithm for min-cut_. For each node $a$, this collects two quantities 1-cut detail $\mathcal{D}^{1}(a)$ and 2-cut detail $\mathcal{D}^{2}(a)$ which relevant information. After this information is collected by each node $a$ it will be sent upwards via a convergecast technique described later. Further, we will characterize CASE-5 in Lemma 5.24. Recall that in CASE-5, for some node $v_{1},v_{2},v_{3}\in V\setminus r$, $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a min-cut such that ${v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{3}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{2}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$. This will help us prove that one of the information from $\left\\{\mathcal{D}^{2}\left(v_{2}\right),\mathcal{D}^{2}\left(v_{3}\right),\left\\{\mathcal{D}^{1}\left(v_{3}\right),\mathcal{D}^{1}\left(v_{2}\right)\right\\}\right\\}$ will be received by $\operatorname{LCA}(v_{2},v_{3})$ after the convergecast algorithm. And this will be enough to make a decision about the min-cut based on the characterization given in Lemma 3.6. 1 _computing 1-cut detail_ 2 $v\leftarrow$ closest node to $r$ such that $(\pi\left(a\right),a)$ is a min-cut in $G_{v}$ 3 $\mathcal{D}^{1}(a)=\langle node:a,parent:\pi\left(a\right),eta:\eta(a),pivotnode:v,numedges:H_{{a}^{\downarrow}}^{v},pivotnodelevel:\ell\left(v\right)\rangle$ 1 _computing 2-cut detail_ 2 $vSet\leftarrow\\{u\mid\exists\text{ node }x\ s.t.\ {x}^{\downarrow}\cap{a}^{\downarrow}=\emptyset;\ \gamma({x}^{\downarrow},{a}^{\downarrow})>0;\ \ell\left(a\right)\geq\ell\left(x\right);\left\\{(\pi\left(a\right),a),(\pi\left(x\right),x)\right\\}$ $\text{ is a induced cut in }G_{u}\\}$ 3 if _ $vSet=\emptyset$ _ then $\mathcal{D}^{2}(a)=\phi$ 4 else 5 $v\leftarrow\underset{u\in vSet}{argmin}\ \ell\left(u\right)$ 6 7 $bSet\leftarrow\\{x\mid{x}^{\downarrow}\cap{a}^{\downarrow}=\emptyset;\ \gamma({x}^{\downarrow},{a}^{\downarrow})>0;\ \ell\left(a\right)\geq\ell\left(x\right);\left\\{(\pi\left(a\right),a),(\pi\left(x\right),x)\right\\}$ $\text{ is a induced cut in }G_{v}\\}$ 8 $b\leftarrow\underset{x\in bSet}{argmin}\ \ell\left(x\right)$ 9 $z\leftarrow\operatorname{LCA}\left(a,b\right)$ 10 $\mathcal{D}^{2}(a)\triangleq\langle node1:a,parent1:\pi\left(a\right),eta1:\eta(a),node1PivotOutEdges:H^{v}_{{a}^{\downarrow}},node2:b,parent2:\pi\left(b\right),eta2:\eta(b),node2PivotOutEdges:H^{v}_{{b}^{\downarrow}},betweenedges:\gamma({a}^{\downarrow},{b}^{\downarrow}),lca:z,lcalevel:\ell\left(z\right),pivotnode:v,pivotenodelevel:\ell\left(v\right)\rangle$ 11 Algorithm 10 1-cut details and 2-cut details (to be run after layered algorithm for min-cut on all nodes $a$) ###### Observation 5.23. For each node $x$, $\mathcal{D}^{1}(x)$ (1-cut detail) and $\mathcal{D}^{2}(x)$ (2-cut detail) can be computed in $O(D^{2})$ time as given in Algorithm 10 ###### Proof. In Observation 5.21, we saw that the layered algorithm for min-cut runs in $O(D^{2})$ time. We claim that after the layered algorithm for min-cut is executed we have all the information to compute both 1-cut detail and 2-cut detail. For 1-cut detail of a node $a$ we require $\eta(a)$ which node $a$ knows from the first section. Further, we require node $v$ closest to the root such that the edge $(\pi\left(a\right),a)$ persists as a min-cut in the sub- graph $G_{v}$. This node can be found after layered algorithm for min-cut is run. Also $H_{{a}^{\downarrow}}^{v}$ is available with node $a$ from Section 4.1. Similarly for 2-cut detail. Here the layered algorithm uses Lemma 4.23, which takes $O(D)$ rounds. At the end of the layered algorithm for the min-cut every node $a$ knows all the required induced-cut of size $2$ that it participates in subgraph $G_{v}$ for all $v\in\mathcal{A}\left(a\right)$. Based on that, it can calculate 2-cut detail. ∎ Now, we will give characterization of $\textbf{CASE}-5$. This characterization will help us to desginate node $v_{2},v_{3}$ as special nodes in ${x}^{\downarrow}$. ###### Lemma 5.24. Let the graph $G$ be 3-connected (there are no min-cuts of size $1$ and $2$ in $G$). Let $v_{1},v_{2},v_{3}\in V\setminus\left\\{r\right\\}$, ${v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{3}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{2}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$, let $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ be a min-cut (as in CASE-5). Then the following statements are true 1. i) Let $x\in{v_{1}}^{\downarrow}\setminus\left({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow}\right)$ and $x$ is not on either of the path $\rho(v_{2}),\rho(v_{3})$ then the edge $(\pi\left(x\right),x)$ is not a min-cut in $G_{v_{1}}$ 2. ii) when $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})=0$, $(\pi\left(v_{2}\right),v_{2})$ and $(\pi\left(v_{3}\right),v_{3})$ are 1-cuts for sub-graph in $G_{\left({v_{2}},{v_{1}}\right)}$ and $G_{\left({v_{3}},{v_{1}}\right)}$ respectively 3. iii) Let $x\in{v_{1}}^{\downarrow}\setminus\left({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow}\right)$ be a node and let $x$ not be on either of the two paths $\rho(v_{2}),\rho(v_{3})$ and also let $\ell\left(x\right)\in\left\\{\ell\left(v_{2}\right),\ell\left(v_{3}\right)\right\\}$. When $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})>0$, then there does not exists a node $y$ such that ${x}^{\downarrow}\cap{y}^{\downarrow}=\emptyset$, $\gamma({x}^{\downarrow},{y}^{\downarrow})>0$ and the edge set $\left\\{(\pi\left(x\right),x),(\pi\left(y\right),y)\right\\}$ is a induced cut of size $2$ in $G_{v_{1}}$ 4. iv) Let node $z$ be the LCA of $v_{2},v_{3}$. When $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})>0$ then $\left\\{(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a 2-cut for sub-graph sequence $G_{\left({z},{v_{1}}\right)}$ 5. v) When $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})>0$ then there does not exists a node $x$ on the path $\rho(v_{2})$ such that $\ell\left(x\right)<\ell\left(v_{2}\right)$, ${x}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$ and $\left\\{(\pi\left(x\right),x),(\pi\left(v_{3}\right),v_{3})\right\\}$ is an induced cut of $G_{v_{1}}$ ###### Observation 5.25. If there is an induced cut as given in Lemma 5.24, then no node in the set ${v_{1}}^{\downarrow}\setminus\left({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow}\right)$ is connected to any node in $V\setminus{v_{1}}^{\downarrow}$. Also apart from the edges $(\pi\left(v_{2}\right),v_{2})$ and $(\pi\left(v_{3}\right),v_{3})$ node in $\left({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow}\right)$ do not have any edge with nodes in ${v_{1}}^{\downarrow}\setminus\left({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow}\right)$. ###### Proof. If such was the case then $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ would not have been a min-cut. We will prove the above claims using this simple observation. ∎ ###### Proof of Lemma 5.24. 1. i) As per given condition $v_{2}\notin{x}^{\downarrow}$ and $v_{3}\notin{x}^{\downarrow}$. By Observation 5.25, the only nodes in ${v_{1}}^{\downarrow}$ which have an adjacent edge to the nodes in $V\setminus{v_{1}}^{\downarrow}$ are in the vertex sets ${v_{2}}^{\downarrow}$ or ${v_{3}}^{\downarrow}$. Now since $(\pi\left(x\right),x)$ is a min-cut in $G_{v_{1}}$ thus no node in ${x}^{\downarrow}$ has an adjacent edge with a node in $V\setminus{v_{1}}^{\downarrow}$ . Thus if $(\pi\left(x\right),x)$ is a min-cut for the sub-graph $G_{v_{1}}$ then it is also a min-cut edge in the whole graph $G$ which is a contradiction because the graph is 3-connected. 2. ii) Since $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})=0$ thus there are no edges between vertices in the sets ${v_{2}}^{\downarrow}$ and ${v_{3}}^{\downarrow}$. Also there are no edges between vertices in the vertex set ${v_{1}}^{\downarrow}\setminus{v_{2}}^{\downarrow}$ and ${v_{2}}^{\downarrow}$. Thus $(\pi\left(v_{2}\right),v_{2})$ is a cut-edge in subgraph $G_{v_{1}}$ and in all the subgraphs in $G_{\left({v_{2}},{v_{1}}\right)}$ by Observation 5.22. Similarly for the edge $(\pi\left(v_{3}\right),v_{3})$. 3. iii) Lets consider the simple case when $y$ is in either ${v_{2}}^{\downarrow}$ or ${v_{3}}^{\downarrow}$. Here $\gamma\left({x}^{\downarrow},{y}^{\downarrow}\right)=0$ by Observation 5.25 The other case when $(\pi\left(y\right),y)$ is on one of the paths out of $\rho(v_{2})$ or $\rho(v_{3})$. WLOG let $(\pi\left(y\right),y)$ be on the path $\rho(v_{2})$. Thus ${v_{2}}^{\downarrow}\subseteq{y}^{\downarrow}$. Also given that $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})>0$ thus $\left\\{(\pi\left(x\right),x),(\pi\left(y\right),y)\right\\}$ cannot be a cut induced by $\delta({x}^{\downarrow}\oplus{y}^{\downarrow})$ in $G_{v_{1}}$ since ${v_{2}}^{\downarrow}\subseteq{y}^{\downarrow}$ and ${v_{3}}^{\downarrow}\subseteq{v_{1}}^{\downarrow}\setminus({x}^{\downarrow}\cup{y}^{\downarrow})$ thus the vertex set ${x}^{\downarrow}\cup{y}^{\downarrow}$ and ${v_{1}}^{\downarrow}\setminus({x}^{\downarrow}\cup{y}^{\downarrow})$ have at least one edge between them Now let $y$ be not on either of the path $\rho(v_{2})$ or $\rho(v_{3})$. Also $y$ is not in either of the vertex set ${v_{2}}^{\downarrow}$ and ${v_{3}}^{\downarrow}$. Now, as per given condition, $(\pi\left(x\right),x)$ is not on either of the path $\rho(v_{2})$ or $\rho(v_{3})$. Since both the edges are not on the paths $\rho(v_{2})$ and $\rho(v_{3})$ thus neither of $v_{2}$ or $v_{3}$ are in the vertex set ${x}^{\downarrow}$ or ${y}^{\downarrow}$. If here $\left\\{(\pi\left(x\right),x),(\pi\left(y\right),y)\right\\}$ is a min-cut then there are edges between vertices of the two sets ${x}^{\downarrow}$ and ${y}^{\downarrow}$ and they do not have any other edge to the vertices in ${v_{1}}^{\downarrow}\setminus({x}^{\downarrow}\cup{y}^{\downarrow})$. Also here by Observation 5.25 nodes in ${x}^{\downarrow}\cup{y}^{\downarrow}$ do not have any edge in $V\setminus{v_{1}}^{\downarrow}$. Thus by the above two statements nodes in ${x}^{\downarrow}\cup{y}^{\downarrow}$ do not have any adjacent edges to nodes in $V\setminus({x}^{\downarrow}\cup{y}^{\downarrow})$ hence, the edge set $\left\\{(\pi\left(x\right),x),(\pi\left(y\right),y)\right\\}$ persists as a min-cut in the whole graph which is a contradiction because the graph is 3-connected. 4. iv) Since $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})>0$ thus there are some edges between vertices of the sets ${v_{2}}^{\downarrow}$ and ${v_{3}}^{\downarrow}$. Also by Observation 5.25 there are no edges between vertices in the vertex set ${v_{1}}^{\downarrow}\setminus({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow})$ and ${v_{2}}^{\downarrow}$. Similarly there are no edges between vertices in the vertex set ${v_{1}}^{\downarrow}\setminus({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow})$ and ${v_{3}}^{\downarrow}$. Here $z$ is the LCA of $v_{2},v_{3}$ and there are edges which go between ${v_{2}}^{\downarrow}$ and ${v_{3}}^{\downarrow}$. Also, by Observation 5.25 no edges other than $\left\\{(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ connect nodes in ${z}^{\downarrow}/({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow})$ to nodes in ${v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow}$. Thus $\left\\{(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ is a 2-cut in $G_{z}$. Similar argument can be given for all sub-graphs in the graph sequence $G_{\left({z},{v_{1}}\right)}$ 5. v) As per the given condition $x\in{v_{1}}^{\downarrow}\setminus({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow})$ and $v_{2}\in{x}^{\downarrow}$. We will focus on the vertex set ${x}^{\downarrow}\setminus{v_{2}}^{\downarrow}$ which is not empty because $\ell\left(x\right)<\ell\left(v_{2}\right)$. If $\left\\{(\pi\left(x\right),x),(\pi\left(v_{3}\right),v_{3})\right\\}$ is an induced cut of $G_{v_{1}}$ then vertices in the set ${x}^{\downarrow}\cup{v_{3}}^{\downarrow}$ do not have an adjacent edge to vertices in ${v_{1}}^{\downarrow}\setminus({x}^{\downarrow}\cup{v_{3}}^{\downarrow})$. Thus vertices in ${x}^{\downarrow}\setminus{v_{2}}^{\downarrow}$ does not have an adjacent edge to vertices in ${v_{1}}^{\downarrow}\setminus({x}^{\downarrow}\cup{v_{3}}^{\downarrow})$. Further ${x}^{\downarrow}\setminus{v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}\setminus({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow})$ thus by Observation 5.25 vertices in ${x}^{\downarrow}\setminus{v_{2}}^{\downarrow}$ do not have an adjacent edge to vertices in $V\setminus{v_{1}}^{\downarrow}$. Hence $\delta({x}^{\downarrow}\setminus{v_{2}}^{\downarrow})=\left\\{(\pi\left(x\right),{x}),(\pi\left(v_{2}\right),{v_{2}})\right\\}$ is an induced cut in the whole graph which is a contradiction. ∎ Now we will give a variant of a convergecast algorithm for communicating 1-cut details and 2-cut detail of a node to the ancestors in Algorithm 11. Similarly for 2-cut detail. Note that for any node $v$ it will not receive all $\mathcal{D}^{1}(x)$ and $\mathcal{D}^{2}(x)$ for all $x\in{v}^{\downarrow}$ but the ones which are relevant among others. We will characterize this in Observation 5.26. 1 _convergecasting 1-cut detail_ 2 if _node $a$ is a leaf node_ then 3 send $<\mathcal{D}^{1}(a),\ell\left(a\right)>$ to $\pi\left(a\right)$ 4 else if _node $a$ is a internal node_ then 5 for _round $t=0$_ do send $<\mathcal{D}^{1}(a),\ell\left(a\right)>$ to $\pi\left(a\right)$ 6 for _each subsequet round $t=1$ to $t=D-\ell\left(a\right)$_ do 7 collect message $\operatorname{msg}_{c}$ from all node $c\in\operatorname{children}(a)$ 8 if _$\operatorname{msg}_{c}=\phi\ \forall c\in\operatorname{children}(a)$_ then send $<\phi,\ell\left(a\right)+t>$ to $\pi\left(a\right)$ 9 else 10 $\mathcal{D}\leftarrow$ among $\operatorname{msg}_{c}\forall c\in\operatorname{children}(a)$ choose the 1-cut detail from which 1-cut persists for the sub-graph closest to the root that is the lowest $6^{th}$ element (last) of the 1-cut detail tuple 11 send $<\mathcal{D},\ell\left(a\right)+t>$ to $\pi\left(a\right)$ 12 13 14 1 2 _convergecasting 2-cut detail_ 3 if _node $a$ is a leaf node_ then 4 send $<\mathcal{D}^{2}(a),\ell\left(a\right)>$ to $\pi\left(a\right)$ 5 else if _node $a$ is a internal node_ then 6 for _round $t=0$_ do send $<\mathcal{D}^{2}(a),\ell\left(a\right)>$ to $\pi\left(a\right)$ 7 for _each subsequet round $t=1$ to $t=D-\ell\left(a\right)$_ do 8 collect message $\operatorname{msg}_{c}$ from all node $c\in\operatorname{children}(a)$ 9 if _$\operatorname{msg}_{c}=\phi\ \forall c\in\operatorname{children}(a)$_ then send $<\phi,\ell\left(a\right)+t>$ to $\pi\left(a\right)$ 10 else 11 $\mathcal{D}\leftarrow$ among $\operatorname{msg}_{c}\forall c\in\operatorname{children}(a)$ choose the 2-cut detail for which 2-cut persists for the sub-graph closest to the root that is the lowest $11^{th}$ element (last) of the 2-cut detail tuple 12 send $<\mathcal{D},\ell\left(a\right)+t>$ to $\pi\left(a\right)$ 13 14 15 Algorithm 11 Converge-cast algorithm to be run at every node $a$ ###### Observation 5.26. For some nodes $v_{1},v_{2},v_{3}\in V\setminus\left\\{r\right\\}$, let ${v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{3}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{2}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$. Let $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ be a min-cut (as in CASE-5). If $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})=0$ then at the end of the Algorithm 11, $\operatorname{LCA}(v_{2},v_{3})$ has both ${\mathcal{D}^{1}(v_{2})},\mathcal{D}^{1}(v_{3})$. And when $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})>0$ then it has either of $\mathcal{D}^{2}(v_{2})$, $\left\\{\mathcal{D}^{2}(v_{3})\right\\}$. Also Algorithm 11 takes $O(D)$ rounds. ###### Proof. Let $z$ be the LCA of node $v_{2},v_{3}$. Lets first consider $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})=0$. Here based on Lemma 5.24 (i and ii) we can say that there does not exists a node $x\in{v_{1}}^{\downarrow}\setminus\left({v_{2}}^{\downarrow}\cup{v_{3}}^{\downarrow}\right)$ such that $(\pi\left(x\right),x)$ persists as a min-cut in $G_{v_{1}}$. Thus at line 11 of Algorithm 11, $\mathcal{D}^{1}(v_{2})$ will not have a contention at any node $u\in\mathcal{A}\left(v_{2}\right)$ such that $\ell\left(u\right)>\ell\left(z\right)$. Similarly for $v_{3}$. The first contention happen at node $z$ (LCA of $v_{2}$ and $v_{3}$). Let $z$ be the LCA of node $v_{2},v_{3}$. Here $z$ will have both $\mathcal{D}^{1}(v_{2})$ and $\mathcal{D}^{1}(v_{3})$ received from two different children $c_{1},c_{2}\in\operatorname{children}(z)$. Now when $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})>0$. Here by Lemma 5.24 (pt. iv) we know that $\left\\{(\pi\left(v_{2}\right),{v_{2}}),(\pi\left(v_{3}\right),{v_{3}})\right\\}$ is an induced 2-cut. WLOG Lets assume $\ell\left(v_{2}\right)\leq\ell\left(v_{3}\right)$. Thus using Algorithm 7 node $v_{2}$ can make a decision about the said induced cut of size 2. Also $\mathcal{D}^{2}(v_{2})$ (2-cut detail of node $v_{2}$) contains information about the induced cut $\left\\{(\pi\left(v_{2}\right),{v_{2}}),(\pi\left(v_{3}\right),{v_{3}})\right\\}$ because there does not exists any other node $x$ as per Lemma 5.24 (pt. v) such that $\ell\left(x\right)<\ell\left(v_{3}\right)$ and $\left\\{(\pi\left(x\right),{x}),(\pi\left(v_{2}\right),{v_{2}})\right\\}$ is an induced cut of size 2 in $G_{v_{1}}$. Further using the Lemma 5.24 (pt. ii) there will not be any contention for $\mathcal{D}^{2}(v_{2})$ to reach to $\operatorname{LCA}(v_{2},v_{3})$. Also both the converge-casting modules in Algorithm 11 run for $O(D)$ time because both 1-cut detail and 2-cut detail are of the size $O(\log n)$ bits and for each node the modules run for atmost $O(D)$ rounds. ∎ ###### Lemma 5.27. For some nodes $v_{1},v_{2},v_{3}\in V\setminus\left\\{r\right\\}$, let ${v_{2}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{3}}^{\downarrow}\subset{v_{1}}^{\downarrow}$ and ${v_{2}}^{\downarrow}\cap{v_{3}}^{\downarrow}=\emptyset$ . Let $\left\\{(\pi\left(v_{1}\right),v_{1}),(\pi\left(v_{2}\right),v_{2}),(\pi\left(v_{3}\right),v_{3})\right\\}$ be a min-cut (as in CASE-5). Then we can find this min-cut in $O(D^{2})$ rounds. ###### Proof. Here we run Algorithm 12 on each node $x$. This algorithm is run after the convergecast of 1-cut details and 2-cut details. This algorithm has two parts: finding min-cut from 1-cut detail and from 2-cut detail as given in Algorithm 11. We will argue that all the min-cuts of the form as given by CASE-5 are correctly found by this algorithm. And also prove that this process takes $O(D^{2})$ rounds. We claim that when Algorithm 12 is run on LCA of $v_{2},v_{3}$ then it can find the required min-cut. Firstly, let see the case where $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})=0$ here as per Observation 5.26 the LCA of $v_{2},v_{3}$ has both $\mathcal{D}^{1}(v_{2})$ and $\mathcal{D}^{1}(v_{3})$. Let $v$ be the pivotnode from $\mathcal{D}^{1}(v_{2})$ and $u$ be the pivotnode from $\mathcal{D}^{1}(v_{3})$. The information regarding $H_{{v_{2}}^{\downarrow}}^{v}$ and $H_{{v_{3}}^{\downarrow}}^{u}$ is available from the respective one cut detail. WLOG let $\ell\left(v\right)\geq\ell\left(u\right)$. Thus here $H_{{v_{3}}^{\downarrow}}^{u}=H_{{v_{3}}^{\downarrow}}^{v}$ because the number of edges going out of the vertex set ${v_{3}}^{\downarrow}$ which also goes out of the vertex ${u}^{\downarrow}$ is same as the number of edges going out of the vertex set ${v_{3}}^{\downarrow}$ which also goes out of the vertex ${v}^{\downarrow}$. Now if the condition in Line 12 is satisfied then the min- cut is correctly found as per Lemma 3.6. Further, moving to case when $\gamma({v_{2}}^{\downarrow},{v_{3}}^{\downarrow})>0$. Here as per Observation 5.26 one of $\mathcal{D}^{2}(v_{2})$ or $\mathcal{D}^{2}(v_{3})$ is present at node $v_{3}$. And when condition at Line 12 then the min-cut is correctly found as per Lemma 3.6. 1 _from 1-cut detail_ 2 $nodePair\leftarrow\left\\{(a,b)\mid\mathcal{D}^{1}(a),\mathcal{D}^{1}(b)\text{ where received from two nodes }c_{1},c_{2}\in\operatorname{children}(v)\right\\}$ 3 for _each $(a,b)\in nodePair$_ do 4 $v\leftarrow pivotnode$ from $\mathcal{D}^{1}(a),u\leftarrow pivotnode$ from $\mathcal{D}^{1}(b)$ 5 $z\leftarrow\underset{z^{\prime}\in\left\\{v,u\right\\}}{argmax\ }\ell\left(z^{\prime}\right)$ 6 if _$\eta(z)-1=H_{{a}^{\downarrow}}^{z}+H_{{b}^{\downarrow}}^{z}$ & $\eta(a)-1=H_{{a}^{\downarrow}}^{z}$ & $\eta(b)-1=H_{{b}^{\downarrow}}^{z}$_ then 7 $\left\\{(\pi\left(z\right),{z}),(\pi\left(a\right),{a}),(\pi\left(b\right),{b})\right\\}$ is a min-cut 8 9 1 2 _from 2-cut detail_ 3 for _all node $a$ such that $\mathcal{D}^{2}(a)$ was received_ do 4 $a,b$ be $node1$ and $node2$ in $\mathcal{D}^{2}(a)$; $v$ be the pivotNode from $\mathcal{D}^{2}(a)$ 5 if _$\eta(v)-1=H_{{a}^{\downarrow}}^{v}+H_{{b}^{\downarrow}}^{v}$ & $\eta(a)-1=H_{{a}^{\downarrow}}^{v}+\gamma({a}^{\downarrow},{b}^{\downarrow})$& $\eta(b)-1=H_{{b}^{\downarrow}}^{v}+\gamma({a}^{\downarrow},{b}^{\downarrow})$_ then 6 $\left\\{(\pi\left(v\right),{v}),(\pi\left(a\right),{a}),(\pi\left(b\right),{b})\right\\}$ is a min-cut 7 8 Algorithm 12 Algorithm to find min-cut as given by CASE-5 (to be run at each internal node $x$) Also, this whole process just takes $O(D^{2})$ rounds because the layered algorithm for min-cut takes $O(D^{2})$ rounds and the covergecast algorithm given in Algorithm 11 takes just $O(D)$ rounds. ∎ ### 5.3 Summary In this chapter, we gave noval deterministic algorithm to find min-cuts of size $3$. The backbone of our algorithm is the characterization presented in Lemma 3.5 and Lemma 3.6. In this chapter, we showed that whenever there exists a min-cut of size $3$ at least one node in the whole network will receive the required quantities as per the characterization in $O(D^{2})$ rounds and thus the min-cut will be found. The communication strategies introduced in this chapter are of two flavours: _Sketching_ technique where we collected small but relevant information and _Layered Algorithm_ which is a convergecast technique based on a contention resolution mechanism. ## Chapter 6 Future Work We have discussed techniques for finding min-cut. Before our result, a decade old work existed to find small min-cuts by [25]. They gave an algebraic technique of random circulation for finding min-cut of size $1$ and $2$. On the contrary, we give deterministic and purely combinatorial techniques for finding min-cuts of size $1,2$ and $3$. 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An optimal distributed edge-biconnectivity algorithm. Accepted as a Poster at 25th ACM symposium on Principles of distributed computing. PODC.. URL http://arxiv.org/abs/cs/0602013. * Pritchard [2008] Pritchard, D., Fast distributed computation of cuts via random circulations. In International Colloquium on Automata, Languages, and Programming, ICALP. 2008. * Pritchard and Thurimella [2011] Pritchard, D. and R. Thurimella (2011). Fast computation of small cuts via cycle space sampling. ACM Transactions on Algorithms (TALG), 7(4), 46. * Stoer and Wagner [1997] Stoer, M. and F. Wagner (1997). A simple min-cut algorithm. J. ACM, 44(4), 585–591. ISSN 0004-5411. URL http://doi.acm.org/10.1145/263867.263872. * Thorup [2001] Thorup, M., Fully-dynamic min-cut. In Proceedings of the thirty-third annual ACM symposium on Theory of computing. ACM, 2001. * Thurimella [1995] Thurimella, R., Sub-linear distributed algorithms for sparse certificates and biconnected components. In Proceedings of the fourteenth annual ACM symposium on Principles of distributed computing, PODC. 1995. * Tsin [2006] Tsin, Y. H. (2006). An efficient distributed algorithm for 3-edge-connectivity. International Journal of Foundations of Computer Science, 17(03), 677–701. ## CURRICULUM VITAE 1. | NAME | : | Mohit Daga ---|---|---|--- 2. | DATE OF BIRTH | : | July 5, 1993 3. | EDUCATION QUALIFICATIONS | | 1. (a) Bachelor of Technology Institute | : | The LNM Institute of Information Technology, Jaipur ---|---|--- Specialization | : | Communication and Computer Engineering 2. (b) M.S. By Research Institute | : | Indian Institute of Technology - Madras ---|---|--- Registration Date | : | January 6, 2016 ## GENERAL TEST COMMITTEE CHAIRPERSON | : | Prof. N.S. Narayanaswamy ---|---|--- | | Professor | | Department of Computer Science and Engineering GUIDE | : | Dr. John Augustine | | Associate Professor | | Department of Computer Science and Engineering MEMBERS | : | Dr. Shweta Agrawal | | Assistant Professor | | Department of Computer Science and Engineering | | Dr. Krishna Jagannathan | | Assistant Professor | | Department of Electrical Engineering

2024-09-04T02:54:55.786571

2003.00102

arxiv-papers

# Loxodromic elements in big mapping class groups via the Hooper-Thurston- Veech construction Israel Morales and Ferrán Valdez Ferrán Valdez Centro de Ciencias Matemáticas, UNAM, Campus Morelia, C.P. 58190, Morelia, Michoacán, México<EMAIL_ADDRESS>Israel Morales Centro de Ciencias Matemáticas, UNAM, Campus Morelia, C.P. 58190, Morelia, Michoacán, México<EMAIL_ADDRESS> ###### Abstract. Let $S$ be an infinite-type surface and $p\in S$. We show that the Thurston- Veech construction for pseudo-Anosov elements, adapted for infinite-type surfaces, produces infinitely many loxodromic elements for the action of $\mathrm{Mod}(S;p)$ on the loop graph $L(S;p)$ that do not leave any finite- type subsurface $S^{\prime}\subset S$ invariant. Moreover, in the language of [BaWa18B], Thurston-Veech’s construction produces loxodromic elements of any weight. As a consequence of Bavard and Walker’s work, any subgroup of $\mathrm{Mod}(S;p)$ containing two "Thurston-Veech loxodromics" of different weight has an infinite-dimensional space of non-trivial quasimorphisms. ## 1\. Introduction Let $S$ be an orientable infinite-type surface, $p\in S$ a marked point111Through this text we think of $p$ as either a marked point or a marked puncture. and $\mathrm{Mod}(S;p)$ the quotient of ${\rm Homeo}^{+}(S;p)$ by isotopies which fix $p$ for all times. This group is related to the (big) mapping class group222$\mathrm{Mod}(S)$ denotes ${\rm Homeo}^{+}(S;p)$ modulo isotopy. of $S$ via Birman’s exact sequence: $1\longrightarrow\pi_{1}(S,p)\overset{Push}{\longrightarrow}\mathrm{Mod}(S;p)\overset{Forget}{\longrightarrow}\mathrm{Mod}(S)\longrightarrow 1.$ The group $\mathrm{Mod}(S;p)$ acts by isometries on the (Gromov-hyperbolic, infinite-diameter) loop graph $L(S;p)$, see [BaWa18A] and [BaWa18B]. Up to date the only known examples of loxodromic elements for this action are: 1. (1) The loxodromic element $h\in S=\mathbb{S}^{2}\setminus\\{p=\infty\cup\rm Cantor\\}$ defined by J. Bavard in [Bavard16]. In this case $\mathrm{Mod}(S;p)=\mathrm{Mod}(S)$, $h$ is not in the pure mapping class group $\mathrm{PMod}(S)$, does not preserve any finite type subsurface and, in the language of [BaWa18B], has weight 1. 2. (2) Elements defined by pseudo-Anosov homeomorphisms supported in a finite-type subsurface $S^{\prime}\subset S$ containing $p$. All these live in $\mathrm{PMod}(S,p)$. Moreover, for any given $m\in\mathbb{N}$, $S^{\prime}$ can be chosen so that the loxodromic in question has weight $m$. In [BaWa18B], the authors remark that _it would be interesting to construct examples of loxodromic elements of weight greater than 1 which do not preserve any finite type subsurface (up to isotopy)_. The purpose of this article is to show that such examples can be obtained by adapting the Thurston-Veech construction for pseudo-Anosov elements (see [Thurston88], [Veech89] or [FarbMArgalit12]) to the context of infinite-type surfaces. This adaptation is an extension of Thurston and Veech’s ideas built upon previous work by Hooper [Hooper15], hence we call it the Hooper-Thurston- Veech construction. Roughly speaking, we show that if one takes as input an appropiate pair of multicurves $\alpha$, $\beta$ whose union fills $S$, then the subgroup of $\mathrm{Mod}(S,p)$ generated by the (right) multitwists $T_{\alpha}$ and $T_{\beta}$ contains infinitely many loxodromic elements. More precisely, our main result is: ###### Theorem 1.1. Let $S$ be an orientable infinite-type surface, $p\in S$ a marked point and $m\in\mathbb{N}$. Let $\alpha=\\{\alpha_{i}\\}_{i\in I}$ and $\beta=\\{\beta_{j}\\}_{j\in J}$ be multicurves in minimal position whose union fills $S$ and such that: 1. (1) the configuration graph333The vertices of this graph are $I\cup J$. There is an edge between $i\in I$ and $j\in J$ for every point of intersection between $\alpha_{i}$ and $\beta_{j}$. $\mathcal{G}(\alpha\cup\beta)$ is of finite valence444Valence here means the supremum of $degree(v)$ where $v$ is a vertex of the graph in question., 2. (2) for some fixed $N\in\mathbb{N}$, every connected component $D$ of $S\setminus\alpha\cup\beta$ is a polygon or a once-punctured polygon555The boundary of any connected component of $S\setminus\alpha\cup\beta$ is formed by arcs contained in the curves forming $\alpha\cup\beta$ and hence we can think of them as (topological) polygons. with at most $N$ sides and 3. (3) the connected component of $S\setminus\alpha\cup\beta$ containing $p$ is a $2m$-polygon. If $T_{\alpha}$, $T_{\beta}\in\mathrm{Mod}(S;p)$ are the (right) multitwists w.r.t. $\alpha$ and $\beta$ respectively then any $f\in\mathrm{Mod}(S;p)$ in the positive semigroup generated by $T_{\alpha}$ and $T_{\beta}^{-1}$ given by a word on which both generators appear is a loxodromic element of weight $m$ for the action of $\mathrm{Mod}(S;p)$ on the loop graph $L(S;p)$. ###### Remark 1.2. During the 2019 AIM-Workshop on Surfaces of Infinite Type we learned that Abbott, Miller and Patel have also a construction of loxodromic mapping classes whose support is an infinite type surface [AbMiPa]. Their examples are obtained via a composition of handle shifts and live in the complement of the closure of the subgroup of $\mathrm{Mod}(S,p)$ defined by homeomorphisms with compact support. In contrast, loxodromic elements given by Theorem 1.1 are limits of compactly supported mapping classes. As we explain in Section 2.3, the weight of a loxodromic element $f\in\mathrm{Mod}(S;p)$ is defined by Bavard and Walker using a precise description of the (Gromov) boundary $\partial L(S;p)$ of the loop graph. For finite-type surfaces, if $f$ is a pseudo-Anosov having a singularity at $p$, this number corresponds to the number of separatrices based at $p$ of an $f$-invariant transverse measured foliation. This quantity is relevant because, as shown in [BaWa18B] and using the language of Bestvina and Fujiwara [BestvinaFujiwara02], loxodromic elements with different weights are independent and anti-aligned. This has several consequences, for example the work of Bavard-Walker [BaWa18A] gives for free the following: ###### Corollary 1.3. Let $f,g$ be two loxodromic elements in $\mathrm{Mod}(S;p)$ as in Theorem 1.1 and suppose that their weights are $m\neq m^{\prime}$. Then any subgroup of $\mathrm{Mod}(S;p)$ containing them has an infinite-dimensional space of non- trivial quasimorphisms. Applications for random-walks on the loop graph with respect to probability distributions supported on countable subgroups of $\mathrm{Mod}(S;p)$ can be easily deduced from recent work of Maher-Tiozzo [MaherTiozzo18]. On the other hand, recent work by Rasmussen [Rass19] implies that the mapping classes given by Theorem 1.1 are not WWPD in the language of Bestvina- Bromberg-Fujiwara [BeBroFu15]. _About the proof of Theorem 1.1_. As in the case of Thurston’s work, our proof relies on the existence of a flat surface structure $M$ on $S$, having a conical singularity at $p$, for which the Dehn-twists $T_{\alpha}$ and $T_{\beta}$ are affine automorphisms. In the case of finite-type surfaces, this structure is unique (up to scaling) and its existence is guaranteed by the Perron-Frobenius theorem. For infinite-type surfaces the presence of such a flat surface structure is guaranteed once one can find a positive eigenfunction of the adjacency operator on the (infinite bipartite) configuration graph $\mathcal{G}(\alpha\cup\beta)$. Luckly, the spectral theory of infinite graphs in this context secures the existence of uncountably many flat surface structures (which are not related by scaling) on which the Dehn-twists $T_{\alpha}$ and $T_{\beta}$ are affine automorphisms. The main difficulty we encounter is that the description of the Gromov boundary $\partial L(S;p)$ needed to certify that $f$ is a loxodromic element depends on a hyperbolic structure on $S$ which, a priori, is not quasi-isometric to any of the aforementioned flat surface structures. To overcome this difficulty we propose arguments which are mostly of topological nature. We strongly believe that Theorem 1.1 does not describe all possible loxodromics living in the group generated by $T_{\alpha}$ and $T_{\beta}$. ###### Question 1.4. Let $\alpha$ and $\beta$ be as in Theorem 1.1. Is every element $f$ in the group generated by $T_{\alpha}$ and $T_{\beta}$ given by a word on which both generators appear loxodromic? In particular, is $T_{\alpha}T_{\beta}$ loxodromic? We spend a considerable part of this text in the proof of the next result, which guarantees that Theorem 1.1 is not vacuous. ###### Theorem 1.5. Let $S$ be an infinite-type surface, $p\in S$ a marked point and $m\in\mathbb{N}$. Then there exist two multicurves $\alpha$ and $\beta$ whose union fills $S$ and such that: 1. (1) the configuration graph $\mathcal{G}(\alpha\cup\beta)$ is of finite valence, 2. (2) every connected component $D_{i}$ of $S\setminus\alpha\cup\beta$ which does not contain the pont $p$ is a polygon or a once-punctured polygon with $n_{i}$ sides, where $n_{i}\leq N:=\max\\{8,m\\}$, and 3. (3) $p$ is contained in a connected component of $D_{j}$ of $S\setminus\alpha\cup\beta$ whose boundary $\partial D_{j}$ is a $2m$-polygon. A crucial part on the proof of this result is to find, for _any_ infinite-type surface $S$, a simple way to portrait $S$. We call this a topological normal form. Once this is achieved, we give a recipe to construct the curves $\alpha$ and $\beta$ explicitly. On the other hand, we find phenomena proper to big mapping class groups: ###### Corollary 1.6. Let $S$ be the Loch-Ness monster666$S$ has infinite genus and only one end. and consider the action of $\mathrm{Mod}(S;p)$ on the loop graph. Then there exist a sequence of loxodromic elements $(f_{n})$ in $\mathrm{Mod}(S;p)$ which converge in the compact-open topology to a non-trivial elliptic element. ###### Theorem 1.7. There exits a family of translation surface structures $\\{M_{\lambda}\\}_{\lambda\in[2,+\infty]}$ on a Loch Ness monster $S$ and $f\in\mathrm{Mod}(S)$ such that: * • For every $k\in\mathbb{N}$, $f^{k}$ does not fix any isotopy class of essential simple closed curve in $S$. * • If $\tau_{\lambda}:\mathrm{Aff}(M_{\lambda})\hookrightarrow\mathrm{Mod}(S)$ is the natural map sending an affine homeomorphism to is mapping class, then $D\tau_{\lambda}^{-1}(f)\in\mathrm{PSL}(2,\mathbb{R})$ is parabolic if $\lambda=2$ and hyperbolic for every $\lambda>2$. Recall that for finite-type surfaces $S$ a class $f\in\mathrm{Mod}(S)$ such that for every $k\in\mathbb{N}$, $f^{k}$ does not fix any isotopy class of essential simple closed curve in $S$ is necessarily pseudo-Anosov. In particular, the derivative of any affine representative $\phi\in f$ is hyperbolic. We want to stress that for many infinite-type surfaces $\mathrm{Mod}(S)$ does not admit an action on a metric space with unbounded orbits. For a more detailed discussion on this fact and the large scale geometry of big mapping class groups we refer the reader to [DurhamFanoniVlamis18], [MahnRafi20] and references therein. _Outline_. Section 2 is devoted to preliminaries about the loop graph, its boundary and infinite-type flat surfaces. In Section 3 we present the Hooper- Thurston-Veech construction777More precisely, the construction here presented is a particular case of a more general construction built upon work of P. Hooper by the second author and V. Delecroix. The first version of this more general construction had mistakes that were pointed out by the first author.. In Section 3 we also proof Theorem 1.7. Finally, Section 4 is devoted to the proof of Theorems 1.1, 1.5 and Corollary 1.6 (in this order). _Acknowledgements_. We are greatful to Vincent Delecroix, Alexander Rasmussen and Patrick Hooper for providing crucial remarks that lead to the proof of Theorem 1.1. We also want to thank Vincent Delecroix for letting us include a particular case of the Hooper-Thurston-Veech’s construction and Figures 2, 3 and 4, which are work in collaboration with the second author (see [DHV19]). We are greatful to Juliette Bavard and Alden Walker for taking the time to explain the details of their work, and to Chris Leininger for answering all our questions. We want to thank the following institutions and grants for their hospitality and financial support: Max-Planck Institut für Mathematik, American Institut of Mathematics, PAPIIT IN102018, UNAM, PASDA de la DGAPA, CONACYT and Fondo CONACYT-Ciencia Básica’s project 283960. ## 2\. Preliminaries ### 2.1. Infinite-type surfaces Any orientable (topological) surface $S$ with empty boundary is determiner up to homeomorphism by its genus (possibly infinite) and a pair of nested, totally disconnected, separable, metrizable topological spaces $\mathrm{Ends}_{\infty}(S)\subset\mathrm{Ends}(S)$ called the space of ends accumulated by genus and the space of ends of $S$, respectively. Any such pair of nested topological spaces occurs as the space of ends of some orientable surface, see [Richards63]. On the other hand, $S\cup\mathrm{Ends}(S)$ can be endowed with a natural topology which makes the correspoding space compact. This space is called the Freudenthal end-point compactification of $S$, see [Raymond60]. A surface $S$ is of finite (topological) type if its fundamental group is finitely generated. In any other case we say that $S$ is an _infinite-type_ surface. All surfaces considered henceforth are orientable. ### 2.2. Multicurves Let $S$ be an infinite-type surface. A collection of essential curves $l$ in $S$ is locally finite if for every $x\in S$ there exist a neighborhood $U$ of $x$ which intersects a finitely many elements of $l$. As surfaces are second- countable topological spaces, any locally finite collection of essential curves is countable. A _multicurve_ in $S$ is a locally finite, pairwise disjoint, and pairwise non-isotopic collection of essential curves in $S$. Let $\alpha$ be a multicurve in $S$. We say that $\alpha$ _bounds a subsurface_ $\Sigma$ of $S$, if the elements of $\alpha$ are exactly all the boundary curves of the closure of $\Sigma$ in $S$. Also, we say that $\Sigma$ is _induced_ by $\alpha$ if there exists a subset $\alpha^{\prime}\subset\alpha$ that bounds $\Sigma$ and there are no elements of $\alpha\smallsetminus\alpha^{\prime}$ in its interior. A multicurve $\alpha$ in $S$ is of _finite type_ if every connected component of $S\smallsetminus\alpha$ is a finite-type surface. Finite multicurves (that is, with formed by a finite number of curves) are not necessarily of finite type. On the other hand, there are infinite multicurves which are not of finite type, _e.g._ the blue ("vertical") curves in the right-hand side of Figure 1. Let $\alpha$ and $\beta$ be two multicurves in $S$. We say that $\alpha$ and $\beta$ are in _minimal position_ if for every $\alpha_{i}\in\alpha$ and $\beta_{j}\in\beta$, $|\alpha_{i}\cap\beta_{j}|$ realizes the minimal number of (geometric) intersection points between a representative in the free isotopy class of $\alpha_{i}$ and a representative in the homotopy class of $\beta_{j}$. For every pair of multicurves one can find representatives in their isotopy classes which are in minimal position. Let $\alpha$ and $\beta$ be two multicurves in $S$ in minimal position. We say that $\alpha$ and $\beta$ _fill_ $S$ if every connected component of $S\smallsetminus\left(\alpha\cup\beta\right)$ is either a disk or a once- punctured disk. ###### Remark 2.1. Let $\alpha$ and $\beta$ be multicurves. Then: 1. (1) If $\alpha$ and $\beta$ are of finite type and fill $S$, then every complementary connected component of $\alpha\cup\beta$ in $S$ is a polygon with finitely many sides. The converse is not true, see the left-hand side of Figure 1. 2. (2) There are pair of multicurves $\alpha$ and $\beta$ so that $S\smallsetminus\left(\alpha\cup\beta\right)$ has a connected component that is a polygon with infinitely many sides, see the right-hand side of Figure 1. Figure 1. ### 2.3. The loop graph and its boundary In [BaWa18A] and [BaWa18B], Bavard and Walker introduced the loop graph $L(S;p)$ and prove that it is hyperbolic graph on which $\mathrm{Mod}(S,p)$ acts by isometries. Moreover, they made a precise description of the Gromov boundary of $L(S;p)$ in terms of (hyperbolic) geodesics on the Poincaré disk. We recall the general aspects of their work in what follows. The exposition follows largely [BaWa18A], [BaWa18B] and [Rass19]. _The loop graph_. Let $S$ be an infinite type surface and $p\in S$. In what follows we think of $p$ as a marked puncture in $S$. The isotopy class of a topological embedding of $\gamma:(0,1)\hookrightarrow S$ is said to be a _loop_ if it can be continuously extended to the end-point Freudenthal compactification $S\cup\mathrm{Ends}(S)$ of $S$ with $\gamma(0)=\gamma(1)=p$. On the other hand, if the continuous extension of $\gamma$ satifies that $\gamma(0)=p$ and $\gamma(1)\in\mathrm{Ends}(P)\setminus\\{p\\}$ we call it a short ray. The loop graph has as vertex set isotopy classes (relative to $p$) of loops and adjacency is defined by disjointness (modulo isotopy) and it is hyperbolic w.r.t to the combinatorial distance, see [BaWa18B]. In order to describe the Gromov boundary of $L(S;p)$ we need to introduce the completed ray graph. _Long rays and the completed ray graph_. From now on we fix a hyperbolic metric $\mu$ on $S$ of the first kind888That is, the Fuchsian group appearing in the uniformization $\mathbb{D}\to S$ has as limit set the whole boundary of the Poincaré disk. for which the marked point $p$ is a cusp. Every short ray or loop has a unique geodesic representative in its isotopy class. We denote by $\pi:\hat{S}\to S$ the infinite cyclic cover of $S$ defined by the (conjugacy class of) cyclic subgroup of $\pi_{1}(S,\cdot)$ generated by a simple loop around the cusp $p$ and call it _the conical cover of $S$_. The surface $\hat{S}$ is conformally equivalent to a punctured disc and its unique cusp $\hat{p}$ projects to $p$. We denote by $\partial\hat{S}$ the Gromov boundary $\partial\hat{S}$ from which $\hat{p}$ has been removed. This cover is usefull because for every geodesic representative of a short ray or loop in $S$ there is a unique lift to $\hat{S}$ which is a geodesic with one end in $\hat{p}$ and the other in $\partial\hat{S}$. A long ray on S is a _simple_ bi-infinite geodesic of the form $\pi(\hat{\delta})$, where $\hat{\delta}\subset\hat{S}$ is a geodesic from $\hat{p}$ to $\hat{S}$, which is not a short ray or a loop. By definition, each long ray limits to p at one end and does not limit to any point of $\mathrm{Ends}(S)$ on the other end. The vertices of the completed ray graph $\mathcal{R}(S;p)$ are isotopy classes of loops and short rays, and long rays. Two vertices are adjacent if their geodesic representatives in $(S,\mu)$ are disjoint. As before, we consider the combinatorial metric on $\mathcal{R}(S;p)$ defined by declaring that each edge has length 1. ###### Theorem 2.2. [BaWa18B] The completed ray graph $\mathcal{R}(S;p)$ is disconnected. There exist a component containg all loops and short rays, which is of infinite diameter and quasi-isometric to the loop graph. All other connected components are (possibly infinite) cliques and each of them is formed exclusively by long rays. The component of $\mathcal{R}(S;p)$ containg all loops and short rays is called the _main component_ of the completed ray graph. Long rays not contained in the main component are called _high-filling_ rays and they each one of they form cliques formed exclusively of high-filling rays. _The Gromov boundary of the loop graph_. Let us denote by $\mathcal{H}(S,p)$ the set of all high-filling rays in $\mathcal{R}(S;p)$. Bavard and Walker endow $\mathcal{H}(S,p)$ with a topology. This topology is based on the notion of two rays $k$-beggining like each other, see Section 4.1 and Definition 5.2.4 in [BaWa18B]. On the other hand, they define a $\mathrm{Mod}(S;p)$-action on $\mathcal{H}(S;p)$ by homeomorphisms. We sketch this action in what follows. First they show that endpoints of lifts of loops and short rays to the conical cover $\hat{S}$ are dense in $\partial\hat{S}$. Using this, and the fact that mapping classes in $\mathrm{Mod}(S;p)$ permute loops and short rays, they prove that any $\phi\in\mathrm{Mod}(S;p)$ lifts to a homeomorphism of $\hat{S}$ which admits a unique continuous extension to a homeomorphism of $\partial\hat{S}$. Finally, they show that this extension preserves the subset of $\partial\hat{S}$ formed by endpoints of high-filling rays, hence inducing the aforementioned action by homeomorphisms. ###### Theorem 2.3. Let $\mathcal{E}(S;p)=\mathcal{H}(S;p)/\sim$, where $\sim$ identifies all high-filling rays in the same clique, and endow this set with the quotient topology. Then there exists a $\mathrm{Mod}(S;p)$-equivariant homeomorphism $F:\mathcal{E}(S;p)\to\partial L(S;p)$, where $\partial L(S;p)$ is the Gromov boundary of the loop graph. In consequence any loxodromic element $\phi\in\mathrm{Mod}(S;p)$ fixes two cliques of high-filling rays $\mathcal{C}^{-}(\phi)$ and $\mathcal{C}^{+}(\phi)$. ###### Theorem 2.4 ([BaWa18B], Theorem 7.1.1). The cliques $\mathcal{C}^{-}(\phi)$ and $\mathcal{C}^{+}(\phi)$ are finite and of the same cardinality. This allows to define the _weight_ of a loxodromic element $\phi$ as the cardinality of either $\mathcal{C}^{-}(\phi)$ or $\mathcal{C}^{+}(\phi)$. As said in the introduction, the importance of the weight of a loxodromic elements is given by the following fact: ###### Lemma 2.5 ([BaWa18B], Lemma 9.2.7). Let $g,h\in\mathrm{Mod}(S;p)$ be two loxodromic elements with different weights. Then in the language of Bestvina-Fujiwara [BestvinaFujiwara02], $g$ and $h$ are independent and anti-aligned. ### 2.4. Flat surfaces In this section we recall only basic concepts about flat surfaces needed for the rest of the paper. It is important to remark that most of the flat surfaces considered in this text are of infinite type. For a detailed discussion on infinite-type flat surfaces we refer the reader to [DHV19]. We use $x,y$ for the standard coordinates in $\mathbb{R}^{2}$, $z=x+iy$ the corresponding number in $\mathbb{C}$ and $(r,\theta)$ for polar coordinates $x=r\cos(\theta)$ and $y=r\sin(\theta)$ (or $z=r\exp(i\theta)$). The Euclidean metric $dx^{2}+dy^{2}$ can also be written as $(dr)^{2}+(rd\theta)^{2}$. Let $S$ be an orientable surface and $g$ is a metric defined on the complement of a discrete set $\Sigma\subset S$. A point $p\in S$ is called a _conical singularity of angle $\pi n$_ for some $n\in\mathbb{N}$ if there exists an open neighbourhood $U$ of $p$ such that $(U,g)$ is isometric to $(V,g_{n})$, where $V\subset\mathbb{C}^{*}$ is a (punctured) neighborhood of the origin and $g_{n}=(dr)^{2}+(nrd\theta)^{2}$. If $n=2$ we call $p$ a _regular point_. In general, regular points are not considered as singularities, though as we see in the proof of Theorem 1.1 sometimes it is convenient to think of them as marked points. A _flat surface structure_ on a topological surface $S$ is the maximal atlas $\mathcal{T}=\\{\phi_{i}:U_{i}\to\mathbb{C}\\}$ where $(U_{i})_{i\in\mathbb{N}}$ forms an open covering of $S$, each $\phi_{i}$ is a homeomorphism from $U_{i}$ to $\phi(U_{i})$ and for each $i,j$ the transition map $\phi_{j}\circ\phi_{i}^{-1}:\phi_{i}(U_{i}\cap U_{j})\to\phi_{j}(U_{i}\cap U_{j})$ is a translation in $\mathbb{C}$ or a map of the form999These kind of maps are also called _half-translations_ and for this reason flat surfaces containing half-translation are also known as half-translation surfaces. $z\to-z+\lambda$ for some $\lambda\in\mathbb{C}$. ###### Definition 2.6. A _flat surface_ is a pair $M=(S,\mathcal{T})$ made of a connected topological surface $S$ and a flat surface structure $\mathcal{T}$ on $S\setminus\Sigma$, where: 1. (1) $\Sigma$ is a discrete subset of $S$ and 2. (2) every $z\in\Sigma$ is a conical singularity. If the transition functions of $\mathcal{T}$ are all translations we call the pair $(S,\mathcal{T})$ a translation surface. ###### Remark 2.7. In the preceding definition $S$ can be of infinite topological type and $\Sigma$ can be infinite. All points in $M\setminus\Sigma$ are regular. Every flat surface carries a flat metric given by pulling back the Euclidean metric in $\mathbb{C}$. We denote by $\widehat{M}$ the corresponding metric completion and ${\rm Sing}(M)\subset\widehat{M}$ the set of non-regular points, which can be thought as singularities of the flat metric. We stress that the structure of $M$ near a non-regular point is not well understood in full generality, see [BowmanValdez13] and [Randecker18]. Every flat surface $M$ which is not a translation surface has a (ramified) double covering $\pi:\widetilde{M}\to M$ such that $\widetilde{M}$ is a translation surface whose atlas is obtained by pulling back via $\pi$ the flat surface structure of $M$. This is called the orientation double covering. If $z_{0}\in M$ is a conical singularity of angle $n\pi$ then, if $n$ is even, $\pi^{-1}(z_{0})$ is formed by two conical singularities of total angle $n\pi$; whereas if $n$ is odd, $\pi^{-1}(z_{0})$ is a conical singularity of total angle $2n\pi$. Hence the branching points of the orientation double covering are the conical singularities in $M$ of angle $n\pi$, with $n$ odd. This will be used in the proof of Theorem 1.1. On the other hand, flat surfaces can be defined using the language of complex geometry in terms of Riemann surfaces and quadratic differentials or by glueing (possibly infinite) families of polygons along their edges. A detailed discussion on these other definitions can be found in the first Chapter of [DHV19]. Affine maps. A map $f\in Homeo(M)$ with $f(\Sigma)\subset\Sigma$ is called an _affine automorphisms_ if the restriction $f:M\setminus\Sigma\to M\setminus\Sigma$ to flat charts is an $\mathbb{R}$-affine map. We denote by $\mathrm{Aff}(M)$ the group of affine homeomorphisms of $M$ and by $\mathrm{Aff}^{+}(M)$ the subgroup of $\mathrm{Aff}(M)$ made of orientation preserving affine automorphisms (_i.e_ their linear part has positive determinant). Remark that the derivative $Df$ of an element $f\in\mathrm{Aff}(M)$ is an element of ${\mathrm{GL}}_{2}(\mathbb{R})/\pm Id$. Translation flows. For each direction $\theta\in\mathbb{R}/2\pi\mathbb{Z}$ we have a well-defined translation flow $F_{\mathbb{C},\theta}^{t}:\mathbb{C}\to\mathbb{C}$ given by $F_{\mathbb{C},\theta}^{t}(z)=z+te^{i\theta}$. This flow defines a constant vector field $X_{\mathbb{C},\theta}(z):=\frac{\partial F_{\mathbb{C},\theta}^{t}}{\partial t}|_{t=0}(z)$. Now let $M$ be a translation surface and $X_{M,\theta}$ the vector field on $\widehat{M}\setminus{\rm Sing}(M)$ obtained by pulling back $X_{\mathbb{C},\theta}$ using the charts of the structure. For every $z\in M\setminus\mathrm{Sing}(M)$ let us denote by $\gamma_{z}:I\to M$, where $I\subset\mathbb{R}$ is an interval containing zero, the maximal integral curve of $X_{M,\theta}$ with initial condition $\gamma_{z}(0)=z$. We define $F_{M,\theta}^{t}(z):=\gamma_{z}(t)$ and call it the _translation flow_ on $M$ in direction $\theta$. Let us remark that formally speaking $F^{t}_{M,\theta}$ is not a flow because the trajectory of the curve $\gamma_{z}(t)$ may reach a point of $\mathrm{Sing}(M)$ in finite time. A trajectory of the translation flow whose maximal domain of definition is a bounded interval (on both sides) is called a _saddle connection_. When there is no need to distinguish translation flows in different translation surfaces we abbreviate $F_{M,\theta}^{t}$ and $X_{M,\theta}$ by $F_{\theta}^{t}$ and $X_{\theta}$ respectively. If $M$ is a flat surface but not a translation surface the pull back of the vector field $e^{i\theta}$ to $M$ does not define a global vector field on $M$ but it does define a _direction field_. In both cases integral curves defines a foliation $\mathcal{F}_{\theta}$ on $M\setminus\mathrm{Sing}(M)$. ###### Definition 2.8 (Cylinders and strips). A _horizontal cylinder_ $C_{c,I}$ is a translation surface of the form $([0,c]\times I)/\sim$, where $I\subset\mathbb{R}$ is open (but not necessarily bounded), connected, and where $(0,s)$ is identified with $(c,s)$ for all $s\in I$. The numbers $c$ and $h=|I|$ are called the _circumference_ and _height_ of the cylinder respectively. The _modulus_ of $C_{c,I}$ is the number $\frac{h}{c}$. A _horizontal strip_ $C_{\infty,I}$ is a translation surface of the form $\mathbb{R}\times I$, where $I$ is a bounded open interval. Analogously, the height of the horizontal strip is $h=|I|$. An open subset of a translation surface $M$ is called a _cylinder_ (respectively a _strip_) in direction $\theta$ (or parallel to $\theta$) if it is isomorphic, as a translation surface, to $e^{-i\theta}C_{c,I}$ (respect. to $e^{-i\theta}C_{\infty,I}$). One can think of strips as cylinders of infinite circumference and finite height. For flat surfaces which are not translation surfaces the definition of cylinder still makes sense, though its direction is well defined only up to change of sign. ###### Definition 2.9. Let $M$ be a flat surface and $\theta\in\mathbb{R}/2\pi\mathbb{Z}$ a fixed direction. A collection of maximal cylinders $\\{C_{i}\\}_{i\in I}$ parallel to $\theta$ such that $\cup_{i\in I}C_{i}$ is dense in $M$ is called a _cylinder decomposition_ in direction $\theta$. ## 3\. The Hooper-Thurston-Veech construction The main result of this section is a generalization of the Thurston-Veech construction for infinite-type surfaces. The key ingredient for this generalization is the following: ###### Lemma 3.1. Let $M$ be a flat surface for which there is a cylinder decomposition in the horizontal direction. Suppose that every maximal cylinder in this decomposition has modulus equal to $\frac{1}{\lambda}$ for some $\lambda>0$. Then there exists a unique affine automorphism $\phi_{h}$ which fixes the boundaries of the cylinders and whose derivative (in $\mathrm{PSL}(2,\mathbb{R})$) is given by the matrix $\left(\begin{smallmatrix}1&\lambda\\\ 0&1\end{smallmatrix}\right)$. Moreover, the automorphism $\phi_{h}$ acts as a Dehn twist along the core curve of each cylinder. In general, if $M$ is a flat surface having a cylinder decomposition in direction $\theta\in\mathbb{R}/2\pi\mathbb{Z}$ for which every cylinder has modulus equal to $\frac{1}{\lambda}$ for some $\lambda\in\mathbb{R}^{*}$, one can apply to $M$ the rotation $R_{\theta}\in\mathrm{SO}(2,\mathbb{R})$ that takes $\theta$ to the horizontal direction and apply the preceding lemma. For example, if $\theta=\frac{\pi}{2}$, then there exists a unique affine automorphism $\varphi_{v}$ which fixes the boundaries of the vertical cylinders, acting on each cylinder as a Dehn twists and with derivative (in $\mathrm{PSL}(2,\mathbb{R})$) is given by the matrix $\left(\begin{smallmatrix}1&0\\\ -\lambda&1\end{smallmatrix}\right)$. In particular, if $M$ is a flat surface having cylinder decompositions in the horizontal and vertical directions such that each cylinder involved have modulus $\frac{1}{\lambda}$, then $\mathrm{Aff}(M)$ has two affine multitwists $\phi_{h}$ and $\phi_{v}$ with $D\phi_{h}=\left(\begin{smallmatrix}1&\lambda\\\ 0&1\end{smallmatrix}\right)$ and $D\phi_{v}=\left(\begin{smallmatrix}1&0\\\ -\lambda&1\end{smallmatrix}\right)$ in $\mathrm{PSL}(2,\mathbb{R})$. Let us recall now the Thurston-Veech construction (see [Farb-Margalit], Theorem 14.1): ###### Theorem 3.2 (Thurston-Veech construction). Let $\alpha=\\{\alpha_{i}\\}_{i=1}^{n}$ and $\beta=\\{\beta_{j}\\}_{j=1}^{m}$ be two multicurves filling a finite type surface $S$. Then there exists $\lambda=\lambda(a,b)\in\mathbb{R}^{*}$ and a representation $\rho:\langle T_{\alpha},T_{\beta}\rangle\to\mathrm{PSL}(2,\mathbb{R})$ given by: $T_{\alpha}\to\begin{pmatrix}1&\lambda\\\ 0&1\end{pmatrix},\hskip 14.22636ptT_{\beta}\to=\begin{pmatrix}1&0\\\ -\lambda&1\end{pmatrix}.$ Moreover, an element $f\in\langle T_{\alpha},T_{\beta}\rangle$ is periodic, reducible or pseudo-Anosov according to whether $\rho(f)$ is elliptic, parabolic or hyperbolic. The proof of Theorem 3.2 uses Lemma 3.1. More precisely, one needs to find a flat surface structure on $S$ which admits horizontal and vertical cylinder decompositions $\\{H_{i}\\}_{i=1}^{n}$ and $\\{V_{j}\\}_{j=1}^{m}$ such that each cylinders has modulus equal to $\frac{1}{\lambda}$ and for which $\alpha_{i}$ and $\beta_{j}$ are the core curves of $H_{i}$ and $V_{j}$ for each $i=1,\ldots,n$ and $j=1,\ldots,m$, respectively. By the Perron-Frobenius theorem, such a flat structure always exists and is unique up to scaling. ###### Definition 3.3. Let $\alpha=\cup_{i\in I}\alpha_{i}$ and $\beta=\cup_{j\in J}\beta_{j}$ be two multicurves in a topological surface $S$ (in minimal position, not necessarily filling). The _configuration graph_ of the pair $(\alpha,\beta)$ is the bipartite graph $\mathcal{G}(\alpha\cup\beta)$ whose vertex set is $I\sqcup J$ and where there is an edge between two vertices $i\in I$ and $j\in J$ for every intersection point between $\alpha_{i}$ and $\beta_{j}$. Cylinder decompositions, bipartite graphs and harmonic functions. Let $M$ be a flat surface having horizontal and vertical cylinder decompositions $\mathcal{H}=\\{H_{i}\\}_{i\in I}$ and $\mathcal{V}=\\{V_{j}\\}_{j\in J}$ such that each cylinder has modulus $\frac{1}{\lambda}$ for some $\lambda>0$. For every $i\in I$ let $\alpha_{i}$ be the core curve of $H_{i}$ and for every $j\in J$ let $\beta_{j}$ be the core curve of $V_{j}$. Then $\alpha=\\{\alpha_{i}\\}_{i\in I}$ and $\beta=\\{\beta_{j}\\}_{j\in J}$ are multicurves whose union fills $M$. Let $\textbf{h}:I\cup J\to\mathbb{R}_{>0}$ be the function which to an index associates the height of the corresponding cylinder. Then $A\textbf{h}=\lambda\textbf{h}$ where $A$ is the adjacency operator of the graph $\mathcal{G}(\alpha\cup\beta)$, that is: (1) $(A\textbf{h})(v):=\sum_{w\sim v}\textbf{h}(w)$ where the sum above is taken over over edges $\\{v,w\\}$ having $v$ as one endpoint, that is, the summand $\textbf{h}(w)$ appears as many times as there are edges between the vertices $v$ and $w$. ###### Definition 3.4. Let $\mathcal{G}=(V,E)$ be a graph with vertices of finite degree and $A:V^{\mathbb{R}}\to\mathbb{R}$ as in (1). A function $\textbf{h}:V\to\mathbb{R}$ that satisfies $A\textbf{h}=\lambda\textbf{h}$ is called a _$\lambda$ -harmonic function_ of $\mathcal{G}$. In summary: the existence of a horizontal and a vertical cylinder decomposition where each cylinders has modulus $\frac{1}{\lambda}$ implies the existence of a _positive_ $\lambda$-harmonic function of configuration graph of the multicurves given by the core curves of the cylinders in the decomposition. The idea to generalize Thurston-Veech’s construction for infinite-type surfaces is to reverse this process: given a pair of multicurves $\alpha$ and $\beta$ whose union fills $S$, every positive $\lambda$-harmonic function of $\mathcal{G}(\alpha\cup\beta)$ can be used to construct construct horizontal and vertical cylinder decompositions of $S$ where all cylinders have the same modulus. ###### Theorem 3.5 (Hooper-Thurson-Veech construction). Let $S$ be an infinite-type surface. Suppose that there exist two multicurves $\alpha=\\{\alpha_{i}\\}_{i\in I}$ and $\beta=\\{\beta_{j}\\}_{j\in J}$ filling $S$ such that: 1. (1) there is an uniform upper bound on the degree of the vertices of the configuration graph $\mathcal{G}(\alpha\cup\beta)$ and 2. (2) every component of the complement of $S\setminus\alpha\cup\beta$ is a polygon with a finite number of sides101010That is, each component of $S\setminus\alpha\cup\beta$ is a disc whose boundary consists of finitely man subarcs of curves in $\alpha\cup\beta$.. Then there exists $\lambda_{0}\geq 2$ such that for every $\lambda\geq\lambda_{0}$ there exists a positive $\lambda$-harmonic function h on $\mathcal{G}(\alpha\cup\beta)$ which defines a flat surface structure $M=M(\alpha,\beta,\textbf{h})$ on $S$ admiting horizontal and vertical cylinder decompositions $\mathcal{H}=\\{H_{i}\\}_{i\in I}$ and $\mathcal{V}=\\{V_{j}\\}_{j\in J}$ where each cylinder has modulus $\frac{1}{\lambda}$. Moreover, for every $i\in I$ and $j\in J$ the core curves of $H_{i}$ and $V_{j}$ are $\alpha_{i}$ and $\beta_{j}$, respectively. In particular, we have (right) multitwists $T_{\alpha}$ and $T_{\beta}$ in $\mathrm{Aff}(M)$ which fix the boundary of each cylinder in $\mathcal{H}$ and $\mathcal{V}$, respectively. For each $\lambda\geq\lambda_{0}$, the derivatives of these multitwists define a representation $\rho:\langle T_{\alpha},T_{\beta}\rangle\to\mathrm{PSL}(2,\mathbb{R})$ given by: $T_{\alpha}\to\begin{pmatrix}1&\lambda\\\ 0&1\end{pmatrix},\hskip 14.22636ptT_{\beta}\to\begin{pmatrix}1&0\\\ -\lambda&1\end{pmatrix}.$ ###### Remark 3.6. Theorem 3.5 is a particular case of a more general version of Hooper-Thurston- Veech’s construction due to V. Delecroix and the second author whose final form was achieved after discussions with the first author, see [DHV19]. We do not need this more general version for the proof of our main results. On the other hand, the second assumption on the multicurves in Theorem 3.5 makes the proof simpler that in the general case and for this reason we decided to sketch it. Many of the key ideas in the proof of the result above and its general version appear already in the work of P. Hooper [Hooper15]. The main difference is that P. Hooper starts with a bipartite infinite graph with uniformly bounded valence and then, using a positive harmonic function $h$ on that graph, constructs a translation surface. We take as input an infinite- type topological surface $S$ and a pair of filling multicurves to construct a flat surface structure on $S$, which is a not _a priori_ a translation surface structure. Proof of Theorem 3.5. The union $\alpha\cup\beta$ of the multicurves $\alpha$ and $\beta$ defines a graph embedded in $S$: the vertices are points in $\bigcup_{(i,j)\in I\times J}\alpha_{i}\cap\beta_{j}$ and edges are the segments forming the connected components of $\alpha\cup\beta\setminus\bigcup_{(i,j)\in I\times J}\alpha_{i}\cap\beta_{j}$. Abusing notation we write $\alpha\cup\beta$ to refer to this graph. It is important not to confuse the (geometric) graph $\alpha\cup\beta$ with the (abstract) configuration graph $\mathcal{G}(\alpha\cup\beta)$. To define the flat structure $M$ on $S$ we consider a dual graph $(\alpha\cup\beta)^{*}$ defined as follows. If $S$ had no punctures then $(\alpha\cup\beta)^{*}$ is just the dual graph of $\alpha\cup\beta$. If $S$ has punctures111111We think of punctures also as isolated ends or points at infinity. we make the following convention to define the vertices of $(\alpha\cup\beta)^{*}$: for every connected component $D$ of $S\setminus\alpha\cup\beta$ homeomorphic to a disc choose a unique point $v_{D}$ inside the connected component. If $D$ is a punctured disc, then choose $v_{D}$ to be the puncture. Figure 2. The graph $(\alpha\cup\beta)^{*}$. The points $v_{D}$ chosen above are the vertices of $(\alpha\cup\beta)^{*}$. Vertices in this graph are joined by an edge in $S$ if the closures of the corresponding connected components of $S\setminus\alpha\cup\beta$ share an edge of $\alpha\cup\beta$. Edges are chosen to be pairwise disjoint. Remark that $(\alpha\cup\beta)^{*}$ might have loops. See Figure 2. Given that $\alpha\cup\beta$ fills, every connected component $S\setminus(\alpha\cup\beta)^{*}$ is a topological quadrilateral which contains a unique vertex of $\alpha\cup\beta$. Hence there is a well defined bijection between edges in the abstract graph $\mathcal{G}(\alpha\cup\beta)$ and the set of these quadrilaterals. This way, for every edge $e\in E(\mathcal{G}(\alpha\cup\beta))$ we denote by $R_{e}$ the closure in $S$ of the corresponding topological quadrilateral with the convention to add to $R_{e}$ vertices $v_{D}$ corresponding to punctures in $S$. Note that there are only two sides of $R_{e}$ intersecting the multicurve $\alpha$, which henceforth are called _vertical sides_. The other two sides are in consequence called _horizontal_ , see Figure 3. We now build a flat surface structure on $S$ by declaring the topological quadrilaterals $R_{e}$ of the dual graph $(\alpha\cup\beta)^{*}$ to be Euclidean rectangles. Given that there is a uniform upper bound on the degree of the vertices of the configuration graph $\mathcal{G}(\alpha\cup\beta)$ there exists121212For a more detailed discussion on $\lambda$-harmonic functions we recommend Appendix C in [Hooper15] and reference therein. $\lambda_{0}\geq 2$ such that for every $\lambda\geq\lambda_{0}$ there exists a positive $\lambda$-harmonic function $\textbf{h}:\mathcal{G}(\alpha\cup\beta)\to\mathbb{R}_{>0}$. We use this function to define compatible heights of horizontal and vertical cylinders. More precisely, let us define the maps: $p_{\alpha}:E(\mathcal{G}(\alpha\cup\beta))\to V(\mathcal{G}(\alpha\cup\beta))\qquad\text{and}\qquad p_{\beta}:E(\mathcal{G}(\alpha\cup\beta))\to V(\mathcal{G}(\alpha\cup\beta))$ which to an edge $e$ of the configuration graph $\mathcal{G}(\alpha\cup\beta)$ associate its endpoints $p_{\alpha}(e)$ in $I$ and $p_{\beta}(e)$ in $J$. The desired flat structure is defined by declaring131313For a formal description on how to identify $R_{e}$ with $[0,\textbf{h}\circ p_{\beta}(e)]\times[0,\textbf{h}\circ p_{\alpha}(e)]$ we refer the reader to [DHV19]. $R_{e}$ to be the rectangle $[0,\textbf{h}\circ p_{\beta}(e)]\times[0,\textbf{h}\circ p_{\alpha}(e)]$, see Figure 3. Figure 3. Transforming topological rectangles into Euclidean rectangles. We denote the resulting flat surface $M(\alpha,\beta,\textbf{h})$. Remark that by contruction a vertex $v_{D}$ of the dual graph $(\alpha\cup\beta)^{*}$ of valence $k$ defines a conical singularity of angle $\frac{\pi k}{2}$ in the metric completion of $M(\alpha,\beta,\textbf{h})$. Given that $k$ is always an even number we have that $M(\alpha,\beta,\textbf{h})$ is a half-translation surface (_i.e._ given by a quadratic differential) when $k=2(2n-1)$ for some $n\in\mathbb{Z}_{\geq 1}$. Now, for every $i\in I$, the curve $\alpha_{i}$ is the core curve of the horizontal cylinder $H_{i}:=\cup_{e\in p_{\alpha}^{-1}(i)}R_{e}$. Because h is $\lambda$-harmonic we have $\sum_{e\in p_{\alpha}^{-1}(i)}\textbf{h}(p_{\beta}(e))=\sum_{j\sim i}\textbf{h}(j)=\lambda\textbf{h}(i).$ This equations say that the circumference $\sum_{e\in p_{\alpha}^{-1}(i)}\textbf{h}(p_{\beta}(e))$ of $H_{i}$ is $\lambda$ times its height $\textbf{h}(i)$, hence the modulus of $H_{i}$ is equal to $\frac{1}{\lambda}$. The same computation with $\beta_{j}$ shows that the vertical cylinders $V_{j}:=\cup_{e\in p_{\beta}^{-1}(j)}R_{e}$ have core curve $\beta_{j}$ and modulus $\frac{1}{\lambda}$.∎ ###### Remark 3.7. As said before, Hooper-Thurston-Veech construction can be applied to more general pairs of multicurves $\alpha,\beta$. Consider for example the case in the Loch Ness monster depicted in Figure 4: here the graph $\mathcal{G}(\alpha\cup\beta)$ has finite valence but there exist four connected componets $\\{C_{i}\\}_{i=1}^{4}$ of $S\setminus\alpha\cup\beta$ which are infinite polygons, that is, whose boundary is formed by infinitely many segments belonging to curves in $\alpha$ and in $\beta$. In this situation the convention is to consider vertices in the dual graph $(\alpha\cup\beta)^{*}$ of infinite degree as points at infinity (that is, not in $S$). With this convention the Hooper-Thuston-Veech construction produces a translation surface structure on $S$, because each $\partial C_{i}$ is connected. In Figure 4 we illustrate the case of a $2$-harmonic function; the resulting flat surface is a translation surface known as the infinite staircase. (a) Two oriented multicurves $\alpha$ (in blue) and $\beta$ (in red) in the Loch Ness monster for which the Hooper-Thurston-Veech construction produces the infinite staircase. (b) The graph $\mathcal{G}(\alpha\cup\beta)$ Figure 4. The infinite staircase as a Hooper-Thurson-Veech surface. _Proof of Theorem 1.7_. Let $\lambda\geq 2$ and consider the subgroup of $\mathrm{SL}(2,\mathbb{R})$: (2) $G_{\lambda}:=\langle\left(\begin{smallmatrix}1&\lambda\\\ 0&1\end{smallmatrix}\right),\left(\begin{smallmatrix}1&0\\\ -\lambda&1\end{smallmatrix}\right)\rangle$ This group is free and its elements are matrices of the form $\left(\begin{smallmatrix}1+k_{11}\lambda^{2}&k_{12}\lambda\\\ k_{21}\lambda&1+k_{22}\lambda^{2}\end{smallmatrix}\right)$, $k_{ij}\in\mathbb{Z}$, such that the determinant is 1 and $|\frac{1+k_{11}\lambda^{2}}{k_{12}\lambda}|$ does not belong to the interval $(t^{-1},t)$, where $t=\frac{1}{2}(\lambda+\sqrt{\lambda^{2}-4})$, see [Brenner55]. On the other hand, since $\\{-\frac{\lambda}{2}<\Im(z)\leq\frac{\lambda}{2}\\}\cap\\{|z+\frac{1}{2\lambda}|>\frac{1}{2\lambda}\\}\cap\\{|z-\frac{1}{2\lambda}|\geq\frac{1}{2\lambda}\\}\subset\mathbb{H}^{2}$ is a fundamental domain in the hyperbolic plane for $G_{\lambda}$, this group has no elliptic elements. Morevoer, if $\lambda>2$ there are only two conjugacy classes of parabolics (correspoding to the generators of $G_{\lambda}$) and if $\lambda=2$ then $\left(\begin{smallmatrix}1&\lambda\\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0\\\ -\lambda&1\end{smallmatrix}\right)=\left(\begin{smallmatrix}1-\lambda^{2}&\lambda\\\ -\lambda&1\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1&-\lambda\\\ 0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0\\\ \lambda&1\end{smallmatrix}\right)=\left(\begin{smallmatrix}1-\lambda^{2}&-\lambda\\\ \lambda&1\end{smallmatrix}\right)$ determine, together with the generators of $G_{\lambda}$, the only 4 conjugacy classes of parabolics in $G_{\lambda}$. Remark that $\left(\begin{smallmatrix}1-\lambda^{2}&\lambda\\\ -\lambda&1\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1-\lambda^{2}&-\lambda\\\ \lambda&1\end{smallmatrix}\right)$ are hyperbolic if $\lambda>2$. If $\alpha$ and $\beta$ are the multicurves depicted in Figure 4 (A) on the Loch Ness monster then $\mathcal{G}(\alpha\cup\beta)$ is the infinite bipartite graph on Figure 4 (B). Let us index the vertices of this graph by the integers as in the Figure so that $\textbf{h}_{2}(n)=1$, for all $\hskip 2.84526ptn\in\mathbb{Z}$, is a positive $2$-harmonic function on $\mathcal{G}(\alpha\cup\beta)$. If $\lambda>2$ and $r_{+}=\frac{\lambda+\sqrt{\lambda^{2}-4}}{2}$ the positive function $\textbf{h}_{\lambda}(n)=r_{+}^{n},\hskip 2.84526ptn\in\mathbb{Z}$ is $\lambda$-harmonic on $\mathcal{G}(\alpha\cup\beta)$. The desired family of translation surfaces $\\{M_{\lambda}\\}_{\lambda\in[2,+\infty)}$, is obtained by applying Hooper-Thuston-Veech’s construction to the multicurves $\alpha$, $\beta$ and the family of positive $\lambda$-harmonic functions $\\{\textbf{h}_{\lambda}\\}_{\lambda\in[2,+\infty)}$. The desired class $f\in\mathrm{Mod}(S)$ is given by the product of (right) multitwist $T_{\alpha}T_{\beta}$. No positive power of $f$ fixes an isotopy class of simple closed curve in $S$ because on one hand if $\lambda=2$ the translation flow on eigendirection of the parabolic matrix $D\tau_{2}^{-1}(f)$ decomposes $M_{2}$ in two strips and, on the other if $\lambda>2$, then $D\tau_{\lambda}^{-1}(f)$ is hyperbolic. ∎ ###### Remark 3.8. There are only 3 infinite graphs which admit $2$-harmonic functions, these are depicted in Figure 5 together with their correspoding positive $2$-harmonic functions (which are unique up to rescaling). None of them comes from a pair of multicurves satifying (2) in Theorem 3.5. Figure 5. Graphs with $2$-harmonic functions. Renormalizable directions. The main results on Hooper’s work [Hooper15] deal with the dynamical properties of the translation flow in _renormalizable directions_. ###### Definition 3.9. Consider the action of $G_{\lambda}$ as defined in (2) by homographies on the real projective line $\mathbb{RP}^{1}$. We say that a direction $\theta\in\mathbb{R}/2\pi\mathbb{Z}$ is $\lambda$-renormalizable if its projectivization lies in the limit set of $G_{\lambda}$ and is not an eigendirection of any matrix conjugated in $G_{\lambda}$ to a matrix of the form: $\begin{pmatrix}1&\lambda\\\ 0&1\end{pmatrix},\hskip 5.69054pt\begin{pmatrix}1&0\\\ \lambda&1\end{pmatrix}\hskip 5.69054ptor\hskip 5.69054pt\begin{pmatrix}1&0\\\ -\lambda&1\end{pmatrix}\cdot\begin{pmatrix}1&\lambda\\\ 0&1\end{pmatrix}$ We use two of Hooper’s results in the proof of Theorem 3.5. Recall that in Hooper’s work one takes as input an infinite bipartite graph and a positive $\lambda$-harmonic function on this graph to produce a translation surface. ###### Theorem 3.10 (Theorem 6.2, [Hooper15]). Let $M$ be a translation surface obtained from an infinite bipartite graph as in [_Ibid_.] using a positive $\lambda$-harmonic function and let $\theta$ be a $\lambda$-renormalizable direction. Then the translation flow $F_{\theta}^{t}$ on $M$ does not have saddle connections. ###### Theorem 3.11 (Theorem 6.4, [Hooper15]). Let $M$ be a translation surface obtained from an infinite bipartite graph as in [_Ibid_.] using a positive $\lambda$-harmonic function and let $\theta$ be a $\lambda$-renormalizable direction. Then the translation flow $F_{\theta}^{t}$ is conservative, that is, given $A\subset M$ of positive measure and any $T>0$, for Lebesgue almost every $x\in M$ there is a $t>T$ such that $F_{\theta}^{t}(x)\in A$ ## 4\. Proof of results ### 4.1. Proof of Theorem 1.1 The proof is divided in two parts. In the first part we use the Hooper- Thurston-Veech construction (see Section 3) to find two transverse measured $f$-invariant foliations $\mathcal{F}^{u}$ and $\mathcal{F}^{s}$ on $S$ for which $p$ is a singular point and for which each foliation has $m$ separatrices based at $p$. We prove that each separatrix based at $p$ is dense in $S$. Then, we consider a hyperbolic metric on $S$ of the first kind (allowing us to talk about the completed ray graph $R(S;p)$). We stretch each separatrix of $\mathcal{F}^{u}$ and $\mathcal{F}^{s}$ based at $p$ to a geodesic with respect to this metric. This defines two sets $\Gamma^{+}$ and $\Gamma^{-}$ of geodesics, each having cardinality $m$. In the second part of the proof, we show that $\Gamma^{+}$ and $\Gamma^{-}$ are the only cliques of high-filling rays fixed by $f$ in the Gromov boundary of the loop graph. _Flat structures_. We use the Hooper-Thurston-Veech construction (Section 3) for this part of the proof. Let $\alpha$ and $\beta$ be two multicurves satisfying the hypothesis of Theorem 1.1. Fix $\mathbf{h}:\mathcal{G}(\alpha\cup\beta)\to\mathbb{R}_{>0}$ a positive $\lambda$-harmonic function on the configuration graph $\mathcal{G}(\alpha\cup\beta)$. Let $M=M(\alpha,\beta,\mathbf{h})$ be the flat structure on $S$ given by the Hooper-Thurston-Veech construction and $\rho:\langle T_{\alpha},T_{\beta}\rangle\to\mathrm{PSL}(2,\mathbb{R})$ the corresponding presentation. Here, we have chosen $p$ as one of the vertices of the dual graph $(\alpha\cup\beta)^{*}$ (see the proof of Theorem 3.5) and therefore it makes sense to consider the classes that the affine multitwists $T_{\alpha}$, $T_{\beta}$ define in $\mathrm{Mod}(S;p)$. We abuse notation and denote also by $T_{\alpha}$, $T_{\beta}$ these classes. The eigenspaces of the hyperbolic matrix $\rho(f)$ define two transverse ($f$-invariant) measured foliations $(\mathcal{F}^{u},\mu_{u})$ and $(\mathcal{F}^{s},\mu_{s})$ (for unstable and stable, respectively) on $M$ . Moreover, we have that $f\cdot(\mathcal{F}^{u},\mu_{u})=(\mathcal{F}^{u},\eta\mu_{u})$ and $f\cdot(\mathcal{F}^{s},\mu_{s})=(\mathcal{F}^{s},\eta^{-1}\mu_{u})$, where $\eta>1$ is (up to sign) an eingenvalue of $\rho(f)$. For simplicity we abbreviate the notation for these foliations by $\mathcal{F}^{u}$ and $\mathcal{F}^{s}$. _The set $\mathfrak{V}$_. Recall that $M=M(\alpha,\beta,\textbf{h})$ is constructed by glueing a family of rectangles $\\{R_{e}\\}_{e\in E}$, where $E=E(\mathcal{G}(\alpha\cup\beta))$ is the set of edges of the configuration graph $\mathcal{G}(\alpha\cup\beta)$, along their edges using translations and half-translations. By the way the Hooper-Thurston-Veech construction is carried out, sometimes the corners of these rectangles are not part of the surface $S$: this is the case when there are connected components of $S\setminus\alpha\cup\beta$ which are punctured discs. However, every corner of a rectangle $\\{R_{e}\\}_{e\in E}$ belongs to the metric completion $\widehat{M}$ of $M$ (w.r.t. the natural flat metric). We define $\mathfrak{V}\subset\widehat{M}$ to be the set of points that are corners of rectangles in $\\{R_{e}\\}_{e\in E}$ (after glueings). Remark that since all connected components of $S\setminus\alpha\cup\beta$ are (topological) polygons with an uniformly bounded number of sides, points in $\mathfrak{V}$ are regular points or conical singularities of $\widehat{M}$ whose total angle is uniformly bounded. Moreover the set $\mathrm{Fix}(f)$ of fixed points of the continuous extension of $f$ to $\widehat{M}$ contains $\mathfrak{V}$. Indeed, if $\mathcal{H}=\\{H_{i}\\}$ and $\mathcal{V}=\\{V_{j}\\}$ denote the horizontal and vertical (maximal) cylinder decompositions of $M$, then $\mathfrak{V}=\cup_{i\neq j}(\partial H_{i}\cap\partial V_{j})$, where the boundary of each cylinder is taken in the metric completion $\widehat{M}$. The claim follows from the fact that for every $i\in I$ and $j\in J$, $T_{\alpha}$ fixes $\partial H_{i}$ and $T_{\beta}$ fixes $\partial V_{j}$. For each $q\in\mathfrak{V}$ we denote by $\mathrm{Sep}_{q}(*)$ the set of leaves of $*\in\\{\mathcal{F}^{u},\mathcal{F}^{s}\\}$ based at $q$. We call such a leaf a _separatrix based at $q$_. Remark that if the total angle of the flat structure $M$ around $q$ is $k\pi$ then $|\mathrm{Sep}_{q}(\mathcal{F}^{u})|=|\mathrm{Sep}_{q}(\mathcal{F}^{s})|=k$. The following fact is essential for the second part of the proof. ###### Proposition 4.1. Let $q\in\mathfrak{V}$. Then any separatrix in $\mathrm{Sep}_{q}(\mathcal{F}^{u})\cup\mathrm{Sep}_{q}(\mathcal{F}^{s})$ is dense in $M$. ###### Proof. We consider first the case when $M$ is a translation surface. At the end of the proof we deal with the case when $M$ is a half-translation surface. We show that any separatrix in $\mathrm{Sep}_{q}(\mathcal{F}^{u})$ is dense. The arguments for separatrices in $\mathrm{Sep}_{q}(\mathcal{F}^{s})$ are analogous. _Claim_ : $\cup_{q\in\mathfrak{V}}\mathrm{Sep}_{q}(\mathcal{F}^{u})$, the union of all separatrices of $\mathcal{F}^{u}$, is dense in $M$. To prove this claim we strongly use the work of Hooper [Hooper15]141414Hooper only deals with the case when $M$ is a translation surface. This is why when $M$ is a half-translation surface we consider its orientation double cover.. In particular, we use the fact that leaves in $\mathcal{F}^{u}$ are parallel to a renormalizable direction, see Definition 3.9. We proceed by contradiction by assuming that the complement of the closure of $\cup_{q\in V}\mathrm{Sep}_{q}(\mathcal{F}^{u})$ in $\widehat{M}$ is non-empty. Let $U$ be a connected component of this complement. Then $U$ is $\mathcal{F}^{u}$-invariant. If $U$ contains a closed leaf of $\mathcal{F}^{u}$ then it has to be a cylinder, but this cannot happen because there are no saddle connections parallel to renormalizable directions, see Theorem 6.2 in [Hooper15]. Then $U$ contains a transversal to $\mathcal{F}^{u}$ to which leaves never return. In other words, $U$ contains an infinite strip, _i.e._ a set which (up to rotation) is isometric to $(a,b)\times\mathbb{R}$ for some $a<b$. This is impossible since the translation flow on $M$ in a renormalizable direction is conservative151515The translation flow $F_{\theta}^{t}$ is called conservative if given $A\subset M$ of positive measure and any $T>0$, for Lebesgue almost every $x\in M$ there is a $t>T$ such that $F_{\theta}^{t}(x)\in A$., see Theorem 6.4 in [Hooper15]. The claim follows. We strongly recommend that the reader uses Figure 6 as a guide for the next paragraphs. Figure 6. Henceforth if $\gamma$ is a separatrix of $\mathcal{F}^{u}$, we denote by $\gamma(t)$, $t>0$ the parametrization for which $\lim_{t\to 0}\gamma(t)\in\mathfrak{V}$ and such that $|\gamma^{\prime}|=1$ (w.r.t. to the flat metric on $M$). For each horizontal cylinder $H_{k}$ in $M$ and $\xi\in\mathfrak{V}\cap\partial H_{k}$ we denote by $\gamma^{u}_{\xi,H_{k}}\subset M$ (respectively $\gamma^{s}_{\xi,H_{k}}$) the unique separatrix of $\mathcal{F}^{u}$ (respect. of $\mathcal{F}^{s}$) based at $\xi$ within $H_{k}$, that is, for which $\gamma^{u}_{\xi,H_{k}}(t)\in H_{k}$ for all $t$ in a small neighbourhood of $0$. For a vertical cylinder $V_{l}$, $\gamma^{u}_{\xi,V_{l}}$ and $\gamma^{s}_{\xi,V_{l}}$ are defined in a similar way. Let $\mathfrak{V}^{b}(H_{k})$ and $\mathfrak{V}^{t}(H_{k})$ denote the points in $\mathfrak{V}\cap H_{k}$ in the bottom and in the top161616We pull back the standard orientation of the Euclidean plane to $M$ to make sense of the east-west and bottom-top sides of a cylinder. connected component of $\partial H_{k}$ respectively; and for any vertical cylinder $V_{l}$ let $\mathfrak{V}^{e}(V_{l})$ and $\mathfrak{V}^{w}(V_{l})$ denote the points in $\mathfrak{V}\cap V_{l}$ in the east and west connected component of $\partial V_{l}$ respectively. Without loss of generality we suppose that $q\in V^{b}(H_{k})\cap V^{e}(V_{l})$. We denote by $\omega(\gamma_{q,H_{k}}^{u})$ the $\omega$-limit set of $\gamma_{q,H_{k}}^{u}$. _Claim_ : the union of all separatrices of $\mathcal{F}^{u}$ based at points in $\partial H_{k}\cup\partial V_{l}$ within $H_{k}$ and $V_{l}$ respectively (3) $\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial H_{k}}\gamma^{u}_{\xi,H_{k}}\right)\cup\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial V_{l}}\gamma^{u}_{\xi,V_{l}}\right)$ is contained in $\omega(\gamma_{q,H_{k}}^{u})$. _Proof of claim_. Remark that since $H_{k}$ is tiled by rectangles corresponding to points of intersection of the core curve $\alpha_{k}$ with curves in $\beta$, $|\mathfrak{V}^{b}(H_{k})|=|\mathfrak{V}^{t}(H_{k})|$ and for each $\xi\in\mathfrak{V}^{b}(H_{k})$ there is exactly one point in $\mathfrak{V}^{t}(H_{k})$ just above. Hence, using the east-west orientation of $H_{k}$, we can order the elements of $\mathfrak{V}^{b}(H_{k})\cup\mathfrak{V}^{t}(H_{k})$ cyclically: we write $\mathfrak{V}^{b}(H_{k})=\\{q_{j}^{b}\\}_{j\in\mathbb{Z}/N\mathbb{Z}}$, $\mathfrak{V}^{t}(H_{k})=\\{q_{j}^{t}\\}_{j\in\mathbb{Z}/N\mathbb{Z}}$ for some $N\geq 1$. The sets $\mathfrak{V}^{e}(V_{l})=\\{q_{j}^{e}\\}_{j\in\mathbb{Z}/M\mathbb{Z}}$, $\mathfrak{V}^{w}(V_{l})=\\{q_{j}^{w}\\}_{j\in\mathbb{Z}/M\mathbb{Z}}$, for some $M\geq 1$, are defined in a similar way. We suppose that the labeling is such that above $q_{j}^{b}$ lies $q_{j}^{t}$ for all $j\in\mathbb{Z}/N\mathbb{Z}$, and that $q=q_{0}^{b}=q_{0}^{e}$. Recall that $DT_{\alpha}=\left(\begin{smallmatrix}1&\lambda\\\ 0&1\end{smallmatrix}\right)$, $DT_{\beta}^{-1}=\left(\begin{smallmatrix}1&0\\\ \lambda&1\end{smallmatrix}\right)$, hence $Df=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)$, with $a,b,c,d\in\mathbb{R}_{>0}$. In particular $Df$ sends the positive quadrant $\mathbb{R}_{x\geq 0,y\geq 0}$ into itself. If we suppose, without loss of generality, that the unstable eingenspace of $Df$ (without its zero) lies in the interior of $\mathbb{R}_{x\geq 0,y\geq 0}\cup\mathbb{R}_{x\leq 0,y\leq 0}$, then the stable eigenspace of $Df$ (without its zero) has to lie in the interior of $\mathbb{R}_{x\geq 0,y\leq 0}\cup\mathbb{R}_{x\leq 0,y\geq 0}$. Hence, for every $j\in\mathbb{Z}/N\mathbb{Z}$ we have that $\gamma^{u}_{q_{j}^{b},H_{k}}$ intersects $\gamma^{s}_{\xi,H_{k}}$, for every $\xi\in\\{q_{j}^{t},q_{j+1}^{b}\\}$ and $\gamma^{s}_{q_{j+1}^{t},V_{l^{\prime}}}$, where $V_{l^{\prime}}$ is the vertical cylinder intersecting $H_{k}$ and having $\\{q_{j}^{b},q_{j}^{t},q_{j+1}^{b},q_{j+1}^{t}\\}$ in its boundary171717Remark that in $M$ these points need not to be all different from each other. For example $q_{j}^{b}=q_{j}^{t}$ and $q_{j+1}^{b}=q_{j+1}^{t}$ if the core curve of $V_{l^{\prime}}$ only intersects the core curve of $H_{k}$. In any case the claims remain valid.. From Figure 6 we can see that some of these points of intersection are actually in $H_{k}\cup V_{l^{\prime}}$. By applying repeatedly $f$ to all these points of intersection of separatrices we obtain that $\xi\in\omega(\gamma_{q_{j}^{b}}^{u},H_{k})$ for every $j\in\mathbb{Z}/N\mathbb{Z}$ and $\xi\in\\{q_{j+1}^{b},q_{j}^{t},q_{j+1}^{t}\\}$. This implies that $\gamma^{u}_{\xi,H_{k}}\subset\omega(\gamma_{q_{j}^{b},H_{k}}^{u})$ for every $j\in\mathbb{Z}/N\mathbb{Z}$ and $\xi\in\\{q_{j+1}^{b},q_{j}^{t},q_{j+1}^{t}\\}$. In particular, we get that $\omega(\gamma_{q=q_{0}^{b},H_{k}}^{u})$ contains $\gamma^{u}_{\xi,H_{k}}$ for every $\xi\in\\{q_{1}^{b},q_{1}^{t},q_{0}^{t}\\}$. As a consequence, we have that $\omega(\gamma_{q,H_{k}}^{u})$ contains181818Here we are using the following general principle: if $\gamma_{1},\gamma_{2}$ are trayectories of a vector field on a translation surface and $\gamma_{1}$ is contained in $\omega(\gamma_{2})$, then $\omega(\gamma_{1})\subset\omega(\gamma_{2})$. $\omega(\gamma^{u}_{q_{1}^{b},H_{k}})$, which in turn contains $\\{\gamma^{u}_{q_{2}^{b},H_{k}},\gamma^{u}_{q_{1}^{t},H_{k}},\gamma^{u}_{q_{2}^{t},H_{k}}\\}$. Proceeding inductively we get that $\omega(\gamma_{q,H_{k}}^{u})$ contains $\bigcup_{\xi\in\mathfrak{V}\cap\partial H_{k}}\gamma^{u}_{\xi,H_{k}}.$ The positivity of the matrix $Df$ and the fact that its unstable eigenspace lies in $\mathbb{R}_{x\geq 0,y\geq 0}\cup\mathbb{R}_{x\leq 0,y\leq 0}$ also imply that for every $j\in\mathbb{Z}/M\mathbb{Z}$ the separatrix $\gamma_{q_{j}^{e},V_{l}}^{u}$ intersects $\gamma_{\xi,V_{l}}^{s}$ for $\xi\in\\{q_{j}^{w},q_{j+1}^{e},q_{j+1}^{w}\\}$. From here on, the logic to show that $\omega(\gamma_{q,H_{k}}^{u})$ contains $\bigcup_{\xi\in\mathfrak{V}\cap\partial V_{l}}\gamma^{u}_{\xi,V_{l}}$ is the same as the one presented in the preeceding paragraph and the claim follows. The arguments in the proof of the preceding claim are local so they can be used to show that: * • For every $j\in\mathbb{Z}/N\mathbb{Z}$, the limit set $\omega(\gamma_{q_{j}^{b},H_{k}}^{u})$ contains all separatrices: $\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial H_{k}}\gamma^{u}_{\xi,H_{k}}\right)\cup\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial V_{l^{\prime}}}\gamma^{u}_{\xi,V_{l^{\prime}}}\right)$ where $V_{l^{\prime}}$ is such that $q_{j}^{b}=\partial H_{k}\cap\partial V_{l^{\prime}}$. * • For every $j\in\mathbb{Z}/M\mathbb{Z}$, the limit set $\omega(\gamma_{q_{j}^{e},H_{k}}^{u})$ contains all separatrices: $\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial V_{l}}\gamma^{u}_{\xi,V_{l}}\right)\cup\left(\bigcup_{\xi\in\mathfrak{V}\cap\partial H_{k^{\prime}}}\gamma^{u}_{\xi,H_{k^{\prime}}}\right)$ where $H_{k^{\prime}}$ is a horizontal cylinder such that $q_{j}^{e}=\partial V_{l}\cap\partial H_{k^{\prime}}$. If we now denote by $\alpha_{k}\in\alpha$ the core curve of $H_{k}$ then the preceding discussion can be summarized as follows: $\omega(\gamma_{q,H_{k}}^{u})$ contains all separatrices of $\mathcal{F}^{u}$ based at points in the boundary of cylinders (and stemming within those cylinders) whose core curves belong to the link of $\alpha_{k}$ in the configuration graph $\mathcal{G}(\alpha\cup\beta)$; moreover, if $\beta_{l}\in{\rm link}(\alpha_{k})$ then $\omega(\gamma_{q,H_{k}}^{u})$ contains all separatrices of $\mathcal{F}^{u}$ based at points in the boundary of cylinders (and stemming within those cylinders) whose core curves belong to ${\rm link}(\beta_{l})$. This way we can extend the arguments above to the whole configuration graph $\mathcal{G}(\alpha\cup\beta)$ to conclude that $\omega(\gamma_{q,H_{k}}^{u})$ contains $\cup_{q\in\mathfrak{V}}\mathrm{Sep}_{q}(\mathcal{F}^{u})$. Since the later is dense in $M$ we conclude that $\gamma_{q,H_{k}}^{u}$ is dense in $M$. We now suppose that $M$ is a half-translation surface (_i.e._ given by a quadratic differential). Let $\pi:\widetilde{M}\to M$ the orientation double cover of $M$. We claim that for every horizontal cylinder $H_{i}$ in $M$ the lift $\pi^{-1}(H_{i})$ is formed by two disjoint isometric copies $\widetilde{H}_{i_{1}}$, $\widetilde{H}_{i_{2}}$ of $H_{i}$ and these are maximal horizontal cylinders in $\widetilde{M}$. Recall that if $p\in\mathfrak{V}$ is a conic singularity of angle $n\pi$, then $\pi^{-1}(p)$ is formed by two conical singularities of angle $n\pi$ if $n$ is even, whereas if $n$ is odd $\pi^{-1}(p)$ is a conical singularity of angle $2n\pi$. Given that the multicurves $\alpha$ and $\beta$ are in minimal position, points in $\mathfrak{V}\cap\partial H_{i}$ which are conical singularities of angle $\pi$ are actually punctures of $S$. This implies that $\widetilde{H}_{i_{1}}\cup\widetilde{H}_{i_{2}}$ cannot be merged within $\widetilde{M}$ into a flat cylinder. The same conclusion holds when $\mathfrak{V}\cap\partial H_{i}$ has conical singularities of angle different from $\pi$ and the claim follows. Analogously, we have that for every vertical cylinder $V_{j}$ in $M$ the lift $\pi^{-1}(V_{j})$ is formed by two disjoint isometric copies $\widetilde{V}_{j_{1}}$, $\widetilde{V}_{j_{2}}$ of $V_{j}$ and these are maximal vertical cylinders in $\widetilde{M}$. The families $\widetilde{\mathcal{H}}=\\{\widetilde{\mathcal{H}}_{i_{1}},\widetilde{\mathcal{H}}_{i_{2}}\\}$ and $\widetilde{\mathcal{V}}=\\{\widetilde{\mathcal{V}}_{j_{1}},\widetilde{\mathcal{V}}_{j_{2}}\\}$ define horizontal and vertical (maximal) cylinder decompositions of $\widetilde{M}$ respectively. Let $\tilde{\alpha}$, $\tilde{\beta}$ denote the lifts to the orientation double cover of $\alpha$ and $\beta$ respectively. Given that the moduli of cylinders downstairs and upstairs is the same, we have a pair of affine multitwists $\widetilde{T_{\tilde{\alpha}}},\widetilde{T_{\tilde{\beta}}}\in\mathrm{Aff}(\widetilde{M})$ with $DT_{\alpha}=D\widetilde{T_{\tilde{\alpha}}}$ and $DT_{\beta}=D\widetilde{T_{\tilde{\beta}}}$ in $\mathrm{PSL}(2,\mathbb{R})$. If we rewrite the word defining $f$ replacing each appearence of $T_{\alpha}$ with $\widetilde{T_{\tilde{\alpha}}}$ and each appearence of $T_{\beta}^{-1}$ with $\widetilde{T_{\tilde{\beta}}}^{-1}$ the result is an affine multitwist $\tilde{f}$ on $\widetilde{M}$ with $D\tilde{f}=Df$ in $\mathrm{PSL}(2,\mathbb{R})$. The eigendirections of $\tilde{f}$ define a pair of transverse $\tilde{f}$-invariant measured foliations $\widetilde{\mathcal{F}}_{u}$ and $\widetilde{\mathcal{F}}_{s}$. Moreover, we have that $\widetilde{\mathcal{F}}_{u}=\pi^{-1}(\mathcal{F}_{u})$ and $\widetilde{\mathcal{F}}_{s}=\pi^{-1}(\mathcal{F}_{s})$ (_i.e._ the projection $\pi$ sends leaves to leaves). Let $\hat{\pi}:\widehat{\widetilde{M}}\to\widehat{M}$ be the continuous extension of the projection $\pi$ to the metric completions of $M$ and $\widetilde{M}$ and define $\tilde{\mathfrak{V}}:=\hat{\pi}^{-1}(\mathfrak{V})$. Remark that $\tilde{\mathfrak{V}}=\cup(\partial\widetilde{H_{i}}\cap\partial\widetilde{V_{j}})$, where the boundaries of the cylinders are taken in $\widehat{\widetilde{M}}$. As with $M$, for every $q\in\widetilde{\mathfrak{V}}$ we define $\mathrm{Sep}_{q}(*)$ as the set of leaves of $*\in\\{\widetilde{\mathcal{F}}^{u},\widetilde{\mathcal{F}}^{s}\\}$ based at $q$. In this context, the proof of Proposition 4.1 for translation surfaces then applies to $\widetilde{M}$ and we get the following: ###### Corollary 4.2. Let $q\in\widetilde{\mathfrak{V}}$. Then any separatrix in $\mathrm{Sep}_{q}(\widetilde{\mathcal{F}}^{u})\cup\mathrm{Sep}_{q}(\widetilde{\mathcal{F}}^{s})$ is dense in $\widetilde{M}$. If separatrices are dense upstairs they are dense downstairs. This ends the proof of Proposition 4.1. ∎ Let now $p\in S$ be the marked puncture and $\mathrm{Sep}_{p}(\mathcal{F}^{u})=\\{\gamma_{1},\ldots,\gamma_{m}\\}$. We denote by $S_{\mu}$ a fixed complete hyperbolic structure on $S$ ($\mu$ stands for the metric) of the first kind and define the completed ray graph $R(S;p)$ with respect to $\mu$. Remark that in $S_{\mu}$ the point $p$ becomes a cusp, _i.e._ a point at infinity. In what follows we associate to each $\gamma_{i}$ a simple geodesic in $S_{\mu}$ based at $p$. For elements in $\mathrm{Sep}_{p}(\mathcal{F}^{s})$ the arguments are analogous. The ideas we present are largely inspired by the work of P. Levitt [Levitt83]. Henceforth $\pi:\mathbb{D}\to S_{\mu}$ denotes the universal cover, $\Gamma<{\rm PSL(2,\mathbb{R})}$ the Fuchsian group for which $S_{\mu}=\mathbb{D}/\Gamma$, $\tilde{p}\in\partial\mathbb{D}$ a chosen point in lift of the cusp $p$ to $\partial\mathbb{D}$ and $\tilde{\gamma_{i}}=\tilde{\gamma_{i}}(\tilde{p})$ the unique lift of $\gamma_{i}$ to $\mathbb{D}$ based at $\tilde{p}$. _Claim:_ $\tilde{\gamma_{i}}$ converges to two distincs points in $\partial\mathbb{D}$. First remark that since $\gamma_{i}$ is not a loop, then it is sufficient to show that $\tilde{\gamma_{i}}(t)$ converges to a point when considering the parametrization that begins at $\tilde{p}$ and $t\to\infty$. Recall that in $S$, the point $p$ is in a region bounded by a $2m$-polygon whose sides belong to closed curves ${\alpha_{1},\ldots,\alpha_{m},\beta_{1},\ldots,\beta_{m}}$ in $\alpha\cup\beta$. Each of these curves is transverse to the leaves of $\mathcal{F}^{u}$ and of $\mathcal{F}^{s}$. Up to making an isotopy, we can suppose without loss of generality that the first element in ${\alpha_{1},\ldots,\alpha_{m},\beta_{1},\ldots,\beta_{m}}$ intersected by $\gamma_{i}(t)$ (for the natural parametrization used in the proof of Proposition 4.1) is $\alpha_{j}$. Given that $\alpha_{j}$ is transverse to $\mathcal{F}^{u}$ and $\gamma_{i}$ is dense in $M$ we have that $\gamma_{i}\cap\alpha_{j}$ is dense in $\alpha_{j}$. In consequence $\tilde{\gamma_{i}}$ intersects $\pi^{-1}(\alpha_{j})$ infinitely often. Remark that $\tilde{\gamma_{i}}$ intersects a connected component of $\pi^{-1}(\alpha_{j})$ at most once. Indeed, if this was not the case there would exists a disc $D$ embedded in $\mathbb{D}$ whose boundary $\partial D$ is formed by an arc in $\tilde{\gamma_{i}}$ and an arc contained in $\pi^{-1}(\alpha_{j})$ transverse to $\widetilde{\mathcal{F}^{u}}:=\pi^{-1}(\mathcal{F}^{u})$. This is impossible because all singularities of $\widetilde{\mathcal{F}^{u}}$ are saddles (in particular only a finite number of separatrices can steam from each one of them) and there is a finite number of them inside $D$. Then, all limit points in $\widetilde{\gamma_{i}}$ in $\mathbb{D}\cup\partial\mathbb{D}$ different from $\tilde{p}$ are in the intersection of an infinite family of nest domains $\mathbb{D}\cup\partial\mathbb{D}$ whose boundaries in $\mathbb{D}$ are components of $\pi^{-1}(\alpha_{j})$. Moreover, this intersection is a single point $q_{i}\in\partial\mathbb{D}$ because it has to be connected and the endpoints in $\partial\mathbb{D}$ of components in $\pi^{-1}(\alpha_{j})$ are dense in $\mathbb{D}$ since $\Gamma$ is Fuchsian of the first kind. This finishes the proof of our claim above. We define thus $\widetilde{\delta_{i}}=\widetilde{\delta_{i}}(\tilde{p})$ to be the geodesic in $\mathbb{D}$ whose endpoints are $\tilde{p}$ and $q_{i}$ as above and $\delta_{i}:=\pi(\widetilde{\delta_{i}})$. The geodesic $\delta_{i}$ is well defined: it does not depend on the lift of $\gamma_{i}$ based at $\tilde{p}$ we have chosen and if we changed $\tilde{p}$ by some $\tilde{p}^{\prime}=gp$ for some $g\in\Gamma$ then by continuity $q_{i}^{\prime}=gq_{i}$. On the other hand $\delta_{i}$ is simple: if this was not the case two components of $\pi^{-1}(\delta_{i})$ would intersect and this implies that two components of $\pi^{-1}(\gamma_{i})$ intersect, which is imposible since $\widetilde{\mathcal{F}^{u}}$ is a foliation. Remark that if $\gamma_{i}\neq\gamma_{j}$ then $\delta_{i}$ and $\delta_{j}$ are disjoint. If this was not the case then there would be a geodesic in $\pi^{-1}(\delta_{i})$ intersecting a geodesic in $\pi^{-1}(\delta_{j})$, but this would imply that a connected component of $\pi^{-1}(\gamma_{i})$ intersects a connected component of $\pi^{-1}(\gamma_{j})$, which is impossible since $\widetilde{\mathcal{F}}^{u}$ is a foliation. Hence we can associate to the set of separatrices $\\{\gamma_{1},\ldots,\gamma_{m}\\}$ a set of pairwise distinct simple geodesics $\\{\delta_{1},\ldots,\delta_{m}\\}$ based at $p$. Remark that by construction this set is $f$-invariant. In what follows we show that $\\{\delta_{1},\ldots,\delta_{m}\\}$ is a clique of high-filling rays. By applying the same arguments to the separatrices of $\mathcal{F}^{u}$ based at $p$ one obtains a different $f$-invariant clique of $m$ high-filling rays. These correspond to the only two points in the Gromov boundary of the loop graph $L(S;p)$ fixed by $f$. Let $\tilde{\delta}$ be a geodesic in $\mathbb{D}$ based at $\tilde{p}$ such that $\delta:=\pi(\tilde{\delta})$ is a simple geodesic in $S_{\mu}$ which does not belong to $\\{\delta_{1},\ldots,\delta_{m}\\}$. We denote by $q$ the endpoint of $\tilde{\delta}$ which is different from $\tilde{p}$. Since every short ray or loop has a geodesic representative, it is sufficient to show that for every $i=1,\ldots,m$ there exists a (geodesic) component of $\pi^{-1}(\delta_{i})$ which intersects $\tilde{\delta}$. We recommend to use Figure 7 as a guide for the next paragraph. Figure 7. All components of $\pi^{-1}(\mathrm{Sep}_{p}(\mathcal{F}^{u}))$ with one endpoint in $\tilde{p}$ are of the form $\\{g^{k}\tilde{\gamma}_{1},\ldots g^{k}\tilde{\gamma}_{m}\\}_{k\in\mathbb{Z}}$, with $g\in\Gamma$ parabolic fixing $\tilde{p}$. Hence, there exists a closed disc $D\subset\mathbb{D}\cup\partial\mathbb{D}$ whose boundary is formed by $\tilde{p}\cup\tilde{\gamma}_{k}\cup\tilde{\gamma}_{l}\cup A$, where $A$ is a closed arc in $\partial\mathbb{D}$ containing $q$; $\tilde{\gamma}_{k}$, $\tilde{\gamma}_{l}$ are (lifts of) separatrices and there is no element of $\pi^{-1}(\mathrm{Sep}_{p}(\mathcal{F}^{u}))$ with one endpoint in $\tilde{p}$ in the interior of $D$. Remark that the endpoints of $A$ are $q_{k}$ and $q_{l}$ (the endpoints of $\tilde{\gamma}_{k}$ and $\tilde{\gamma}_{l}$ respectively). Given that $p$ is a saddle-type singularity of the foliation $\mathcal{F}^{u}$ there exists a neighbourhood of $\tilde{p}$ in $D$ which contains a segment191919As a matter of fact this segment can be taked to live in one of the (lifts of) the curves in $\alpha\cup\beta$ forming the boundary of the disc in $S\setminus\alpha\cup\beta$ containing $p$. $\Sigma$ with one endpoint in $\tilde{\gamma}_{k}$ and the other in $\tilde{\gamma}_{l}$, and which is transverse to $\widetilde{\mathcal{F}^{u}}$ except at one point $\xi$ in its interior. Moreover, since $p$ is an isolated singularity of $\mathcal{F}^{u}$, we can suppose that the closure of the connected component of $D\setminus\Sigma$ in $\mathbb{D}$ having $\tilde{p}$ in its boundary does not contain singular points of $\widetilde{\mathcal{F}^{u}}$ different from $\tilde{p}$. The point $\xi$ divides $\Sigma$ in two connected components $\Sigma_{L}$ and $\Sigma_{R}$. On the other hand $q$ divides $A$ in two connected components $A_{L}$ and $A_{R}$. Now let $i=1,\ldots,m$ be fixed. Since $\gamma_{i}$ is dense in $M$ we have that $\pi^{-1}(\gamma_{i})$ is dense in $\mathbb{D}$ and in particular $\pi^{-1}(\gamma_{i})\cap\Sigma$ is dense in $\Sigma$. Hence we can pick a leaf $\tilde{\gamma}_{i}^{\prime}$ in $\pi^{-1}(\gamma_{i})$ passing through a point $\xi_{L}\in\Sigma_{L}$ and suppose without loss of generality that one of its endpoints $q_{i}^{\prime}$ is in $A_{L}$. Then $\tilde{\gamma}_{i}^{\prime}\cap\Sigma=\\{\xi_{L},\xi_{R}\\}$ with $\xi_{R}\in\Sigma_{R}$. Again, given that $\pi^{-1}(\gamma_{i})\cap\Sigma$ is dense in $\Sigma$, we can find a leaf $\tilde{\gamma_{i}}^{\prime\prime}\in\pi^{-1}(\gamma_{i})$ which intersects $\Sigma$ transversally at a point $\eta_{R}$ between $\xi_{R}$ and $\tilde{\gamma}_{l}$, and which has an enpoint $t_{R}$ in $A_{R}$ arbitrarly close to $q_{l}$. This is true because of the way $q_{l}$ was found: there is a family of connected components of $\pi^{-1}(\alpha)$, for some closed curve $\alpha$ in $M$ transverse to $\mathcal{F}^{u}$, bounding domains in $\mathbb{D}\cup\partial\mathbb{D}$ whose intersection is $q_{l}$. Now, by the way $\Sigma$ was chosen we have that $\tilde{\gamma}_{i}^{\prime\prime}\cap\Sigma=\\{\eta_{L},\eta_{R}\\}$ with $\eta_{L}\in\Sigma_{L}$. This implies that $\tilde{\gamma}_{i}^{\prime\prime}$ has and endpoint $t_{L}$ in $A_{L}$. Hence the geodesic $\tilde{\delta}^{\prime}$ determined by the endpoints $t_{L}$ and $t_{R}$ intersects $\tilde{\delta}$ and $\delta_{i}=\pi(\tilde{\delta}^{\prime})$ intersects $\delta=\pi(\tilde{\delta})$.∎ ###### Remark 4.3. In the proof of Theorem 1.1 we made use of the fact that every short-ray or loop has a geodesic representative, but this is not necessary. As a matter of fact the following is true: if $\tilde{\delta}$ is any curve in $\mathbb{D}$ based at $\widetilde{p}$ whose extremities define two different points in $\partial\mathbb{D}$ and such that $\pi(\tilde{\delta})=\delta$ is simple and does not belong to the set of separatrices $\\{\gamma_{1},\ldots,\gamma_{m}\\}$, then for any $j=1,\ldots,m$ the geodesic $\delta_{j}$ intersects $\delta$. On the other hand, in the proof of Theorem 1.1 the density of each separatrix of $\mathcal{F}^{u}$ or $\mathcal{F}^{s}$ on the _whole_ surface $S$ is not used. The proof remains valid if we only require separetrices of $\mathcal{F}^{u}$ and $\mathcal{F}^{s}$ to be dense on a subsurface $S^{\prime}\subset S$ of finite type with enough topology, _e.g._ such that all curves defining the polygon on which $p$ lives are essential in $S^{\prime}$. In particular we have the following: ###### Corollary 4.4. Let $S^{\prime}\subset S$ be an essential subsurface of finite topological type containing $p$ and $h\in\mathrm{Mod}(S^{\prime})$ a pseudo-Anosov element for which $p$ is a $k$-prong for some $k\in\mathbb{N}$. Let $\hat{h}$ be the extension (as the identity) of $h$ to $S$. Then $\hat{h}$ is a loxodromic element of weight $k$. Moreover, the separatrices of the invariant transverse measured foliations of $h$ based at $p$ define202020By a stretching process as described in the proof of Theorem 1.1 the cliques of high-filling rays fixed by $\hat{h}$. This result already appears in the work of Bavard and Walker [BaWa18B], see Lemma 7.2.1 and Theorem 8.3.1. ### 4.2. Proof of Theorem 1.5 #### 4.2.1. Preliminaries In this section we present, for each infinite type surface, a model that is convenient for the proof of Theorem 1.5. Normal forms for infinite-type surfaces. In what follows we detail how to construct, for any infinite type surface $S$, a graph $T(S)\subset\mathbb{H}^{3}$ having a regular neighbourhood whose boundary is homeomorphic to $S$ (our choice of ambient space obeys illustrative purposes only). There are many ways to construct such graph. The one we present is intended to make the proof of Theorem 1.5 more transparent. Let $2^{\mathbb{N}}:=\prod_{j\in\mathbb{N}}\\{0,1\\}_{j}$ be the Cantor set. In general terms, the construction is as follows. We consider a rooted binary tree $T2^{\mathbb{N}}$, a homeomorphism $f:\prod_{j\in\mathbb{N}}\\{0,1\\}\to\mathrm{Ends}(T2^{\mathbb{N}})$ from the standard binary Cantor set to the space of ends of this tree, and we choose a topological embedding $i:\mathrm{Ends}(S)\hookrightarrow\prod_{j\in\mathbb{N}}\\{0,1\\}$. We show that there exists a subtree of $T2^{\mathbb{N}}$ whose space of ends is precisely $f\circ i(\mathrm{Ends}(S))$. For our purposes it is important that this subtree is _simple_ (see Definition 4.5 below). Then, if $S$ has genus, we perform a surgery on vertices of the aforementioned subtree of of $T2^{\mathbb{N}}$ belonging to rays starting at the root and having one end on $f\circ i(\mathrm{Ends}_{\infty}(S))$. The rooted binary tree. For every $n\in\mathbb{N}$ let $2^{(n)}:=\prod_{i=1}^{n}\\{0,1\\}$ and $\pi_{i}:2^{(n)}\to\\{0,1\\}$ the projection on to the $i^{th}$ coordinate. The rooted binary tree is the graph $T2^{\mathbb{N}}$ whose vertex set $V(T2^{\mathbb{N}})$ is the union of the simbol $\mathfrak{r}$ (this will be the root of the tree) with the set $\\{D:D\in 2^{(n)}\hskip 2.84526pt\text{for some $n\in\mathbb{N}$}\\}$. The edges $E(T2^{\mathbb{N}})$ are $\\{(\mathfrak{r},0)$, $(\mathfrak{r},1)\\}$ together with: $\\{(D,D^{\prime}):D\in 2^{(n)}\hskip 2.84526ptD^{\prime}\in 2^{(n+1)}\hskip 2.84526pt\text{for some $n\in\mathbb{N}$},\hskip 2.84526pt\text{and}\hskip 2.84526pt\pi_{i}(D_{s})=\pi_{i}(D_{t})\hskip 2.84526pt\forall 1\leq i\leq n\\}$ Henceforth $T2^{\mathbb{N}}$ is endowed with the combinatorial distance. For every $\hat{x}=(x_{n})\in 2^{\mathbb{N}}$ we define $r(\hat{x})=(\mathfrak{r},a_{1},\ldots,a_{n}):=(x_{1},\ldots,x_{n},\ldots)$ to be the infinite geodesic ray in $T2^{N}$ starting from $\mathfrak{r}$ and ending in $\mathrm{Ends}(T2^{\mathbb{N}})$. Then, the map (4) $f:\prod_{i\in\mathbb{N}}\\{0,1\\}\to\mathrm{Ends}(T2^{\mathbb{N}})$ which associates to each infinite sequence $\hat{x}=(x_{n})_{n\in\mathbb{N}}$ the end $f(\hat{x})$ of $T2^{\mathbb{N}}$ defined by the infinite geodesic ray $r(\hat{x})$ is a homeomorphism. ###### Definition 4.5. Let $v$ and $v^{*}$ two different vertices in a subtree $\mathcal{T}$ of $T2^{\mathbb{N}}$. If $v$ is contained in the geodesic which connects $v^{*}$ with $\mathfrak{r}$, then we say that $v^{*}$ is a _descendant_ of $v$. A connected rooted subtree of $\mathcal{T}$ without leaves is _simple_ if all descendants of a vertex $v\neq\mathfrak{r}$ of degree two, have also degree two. ###### Lemma 4.6. Let $F\subset\mathrm{Ends}(T2^{\mathbb{N}})$ be closed. Then there exists a _simple_ subtree $T$ of $T2^{\mathbb{N}}$ rooted at $\mathfrak{r}$ such that $F$ is homeomorphic to $\mathrm{Ends}(T)$. We postpone the proof of this lemma to the end of the section. ###### Definition 4.7. Given a subset $F$ of $\mathrm{Ends}(T2^{\mathbb{N}})$ we define $T_{F}:=\bigcup_{\hat{x}\in f^{-1}(F)}r(\hat{x})\subseteq T2^{\mathbb{N}}$ and call it the _tree induced by $F$_. Surgery. Let $T$ be a subtree of $T2^{\mathbb{N}}$ rooted at $\mathfrak{r}$ and having no leaves different from this vertex, if any. Let $L$ be a subset of the vertex set of $T$. We denote by $\Gamma_{T,L}$ the graph obtained from $T$ and $L$ after performing the following operations on each vertex $v\in L$: 1. (1) If $v$ has degree 3 with adjacent descendants $v^{\prime},v^{\prime\prime}$ we delete first the edges $\\{(v,v^{\prime}),(v,v^{\prime\prime},)\\}$. Then we add to $L$ two vertices $v_{*}^{\prime},v_{*}^{\prime\prime}$ and the edges $\\{(v,v_{*}^{\prime}),(v,v_{*}^{\prime\prime}),(v_{*}^{\prime},v^{\prime}),(v_{*}^{\prime\prime},v^{\prime\prime}),(v^{\prime}_{*},v_{*}^{\prime\prime})\\}$. 2. (2) If $v$ has degree 2 and $v^{\prime}$ is its adjacent descendant, we delete first the edge $(v,v^{\prime})$. Then we add to $L$ two vertices $v_{*}^{\prime},v_{*}^{\prime\prime}$ and the edges $\\{(v,v_{*}^{\prime}),(v,v_{*}^{\prime\prime}),(v_{*}^{\prime},v^{\prime}),(v^{\prime}_{*},v_{*}^{\prime\prime})\\}$. ###### Definition 4.8. Let $S$ be a surface of infinite type of genus $g\in\mathbb{N}\cup\\{\infty\\}$ and $\mathrm{Ends}_{\infty}(S)\subset\mathrm{Ends}(S)\subset 2^{\mathbb{N}}$ its space of ends accumulated by genus and space of ends, respectively. We define the graph $T(S)$ according to the following cases. In all of them we suppose w.l.o.g that $T_{f(\mathrm{Ends}(S))}$ is simple. 1. (1) If $g=0$ let $T(S):=T_{f(\mathrm{Ends}(S))}$, 2. (2) if $g\in\mathbb{Z}_{>0}$ let $T(S):=\Gamma_{T,L}$ where $T=T_{f(\mathrm{Ends}(S))}$ and $L={a_{1},a_{2},\ldots,a_{g}}$ and $(\mathfrak{r},a_{1},\ldots,a_{g},\ldots)=r(\hat{x})$ for some $\hat{x}\in\mathrm{Ends}(S)\subset 2^{\mathbb{N}}$, and 3. (3) if $g=\infty$ let $T(S):=\Gamma_{T,L}$ where $T=T_{f(\mathrm{Ends}(S))}$ and $L$ is the set of vertices of the subtree $T_{f(\mathrm{Ends}_{\infty}(S))}\subset T_{f(\mathrm{Ends}(S))}$. By construction, there exists a geometric realization for $T(S)$ as a graph in the plane $\\{(x,0,z)\in\mathbb{H}^{3}:z>0\\}$ in 3-dimensional hyperbolic space, which we denote again by $T(S)$. Moreover there exists a closed regular neighbourhood $N(T(S))$ so that $S$ is homeomorphic to $S^{\prime}=\partial N(T(S))$, see Figure 8. Observe that $T(S)$ is a strong deformation retract of $N(T(S))$. We identify $S$ with $S^{\prime}$, and we say that $S$ is in _normal form_ , and $T(S)$ is the _underlying_ graph that induces $S$. Figure 8. Embedding of the tree $T(S)$. ###### Remark 4.9. In [BaWa18B], A. Walker and J. Bavard carry out a similar construction. For this they introduce the notion of _rooted core tree_ $T$ from which they construct a surface $\Sigma(T)$ homeomorphic to a given infinite type surface $S$, see Lemma 2.3.1 in [BaWa18B]. The graph $T_{f(\mathrm{Ends}(S))}$ (see Definition 4.8) is turned into a rooted core tree by declaring that the vertices of $T_{f(\mathrm{Ends}_{\infty}(S))}$ are the marked vertices. The main difference with the work of Walker and Bavard is that the normal form we are proposing comes from a _simple_ tree. This property is strongly used in the proof of Theorem 1.5. _Proof of Lemma 4.6_. Let $T_{F}$ be the subtree of $T2^{\mathbb{N}}$ induced by $F$. Let $V^{\prime}$ be the set of vertices of $T_{F}$ of degree 2, different from $\mathfrak{r}$, having at least one descendant of degree 3. Then $V^{\prime}=\sqcup_{i\in I}V^{\prime}_{i}$, where: 1. (1) $V_{i}^{\prime}$ is a subset of the vertices of a ray $r(\hat{x})$ for some $\hat{x}\in f^{-1}(F)$, 2. (2) for every $i\in I$, one can label $V^{\prime}_{i}=\\{a_{i,1},\ldots,a_{i,k_{i}}\\}$ so that $a_{i,l+1}$ is a descendant of $a_{i,l}$ adjacent to $a_{i,l}$. 3. (3) for every $i\in I$, the vertex $A_{i}$ of $T_{F}$ adjacent to $a_{i,1}$ other than $a_{i,2}$ is either the root $\mathfrak{r}$ or a vertex of degree 3. Similarly, the vertex $B_{i}$ of $T_{F}$ adjacent to $a_{i,k_{i}}$ other than $a_{i,k_{i}-1}$ is of degree 3. Replacing the finite simple path from $A_{i}$ to $B_{i}$ by an edge $(A_{i},B_{i})$ does not modify the space of ends of $T_{F}$. By doing this for every $i\in I$ we obtain a simple tree as desired. ∎ _Proof of Theorem 1.5_. This proof is divided in two parts. First we show the existence of a pair of multicurves of finite type whose union fills $S$ and which satisfy (1) and (2). In the second part we use these to construct the desired multicurves $\alpha$ and $\beta$. First part: Let $S$ be an infinite-type surface in its normal form, and $T(S)\subset\mathbb{H}^{3}$ the underlying graph that induces $S$. We are supposing that $T(S)$ is obtained after surgery from a simple tree as described above. The idea here is to construct two disjoint collections $A$ (blue curves) and $B$ (red curves) of pairwise disjoint curves in $S$ such that after forgetting the non-essential curves in $A\cup B$, we get the pair of multicurves $\alpha$ and $\beta$ which satisfy (1) and (2) as in Theorem 1.5. Let $T_{g}(S)$ be the full subgraph of $T(S)$ generated by all the vertices which define a triangle in $T(S)$. Observe that, since $T(S)$ is constructed by performing a surgery on a simple tree, the graph $T_{g}(S)$ is connected. Let $T_{g}^{\prime}(S)$ be the subset obtained as the union of $T_{g}(S)$ with all the edges in $T(S)$ adjacent to $T_{g}(S)$. As $T_{g}(S)$ is connected, then $T_{g}^{\prime}(S)$ is also connected. Let $\Delta$ be a triangle in $T_{g}(S)$, and $\Delta^{\prime}$ be the disjoint union of $\Delta$ with all the edges in $T_{g}^{\prime}(S)$ adjacent to $\Delta$. We notice that $\Delta^{\prime}$ is one of the following two possibilities: (1) the disjoint union of $\Delta$ with exactly three edges adjacent to it, or (2) the disjoint union of $\Delta$ with exactly two edges adjacent to it. For each case, we choose blue a red curves in $S$ as indicated in the Figure 9. Figure 9. Blue and red curves associated to the neighborhood $\Delta^{\prime}$ of a triangle $\Delta$. For each edge $e$ in $T_{g}^{\prime}(S)$ which connects two triangles in $T_{g}^{\prime}(S)$, we choose a blue curve in $S$ as indicated in Figure 10. Figure 10. Blue curve associated to an edge which connects two triangles. We consider the following cases. $\mathrm{Ends}(S)=\mathrm{Ends}_{\infty}(S)$. In this case $A$ and $B$ are the multicurves formed by the blue and red curves as chosen above respectively. $\mathrm{Ends}(S)\neq\mathrm{Ends}_{\infty}(S)$. Let $C$ be a connected component of $T(S)-T_{g}^{\prime}(S)$. Given that $T(S)$ is obtained from a simple tree, $C$ is a tree with infinitely many vertices. Let $v$ be the only vertex in $C$ which is adjacent to an edge $e(v)$ in $T_{g}^{\prime}(S)$. If $v$ has degree one in $C$, then every vertex of $C$ different from $v$ has degree two because $T_{f(\mathrm{Ends}(S))}$ is a simple subtree of $T2^{\mathbb{N}}$. In this case, we have that the subsurface $S(C)\subset S$ induced by $C$ is homeomorphic to a punctured disc. In particular, the red curve in $S$ associated to the edge $e(v)$ chosen as depicted in Figure 9 is not essential in $S$. Suppose now that $v$ has degree two in $C$. We color with blue all the edges in $C$ having vertices at combinatorial distances $k$ and $k+1$ from $v$ for every even $k\in\mathbb{Z}_{\geq 0}$. We color all other edges in $C$ in red, see the left-hand side in Figure 11. Let $e$ and $e^{\prime}$ be two edges in $C$ of the same color and suppose that they shares a vertex $v$. Suppose that all vertices of $e\cup e^{\prime}$ different from $v$ have degree three. If $e$ and $e^{\prime}$ are marked with blue color (respectively red color), we choose the red curve (respect. blue curve) in $S$ as in the right-hand side of Figure 11. Figure 11. (Left) Edges of $C$ colored with blue and red alternating in levels. (Right) The corresponding curve in $S$ for the pair of edges $e$ and $e^{\prime}$ marked with blue color. For the edge $e(v)\in T^{\prime}_{g}(S)$, we choose the blue curve in $S$ as in the left-hand side of Figure 12. Finally, for each edge $e$ of $C$, we take a curve in $S$ with the same marked color of $e$ as is showed in the right- hand side of Figure 12. Figure 12. (Left) The corresponding curve for the $e$ which connects $F$ with $T_{g}^{\prime}(S)$. (Right) Blue (red) curve associated to an edge $e$ of $F$. If $\mathrm{Ends}(S)$ has at most one isolated planar end here ends the construction of the multicurves $A$ and $B$. Now suppose that $\mathrm{Ends}(S)$ has more that one isolated planar end, that is, $S$ has at least two punctures. Let $R$ be the full subgraph of $T(S)$ generated by all the vertices of degree 3 in $T(S)$ together with the root vertex $\mathfrak{r}$, and define $T_{g}^{\prime\prime}(S)$ as the full subgraph of $T(S)$ generated by all the vertices in $T(S)$ at distance at most 1 from $R$. The graph $T_{g}^{\prime\prime}(S)$ is connected (again, because $T_{f(\mathrm{Ends}(S))}$ is simple) and contains $T_{g}^{\prime}(S)$ . It also has at least two leaves, i.e., vertices of degree one which we denote by $v_{1}$ and $v_{2}$. If $v_{1}$ and $v_{2}$ are at distance 3 in $T_{g}^{\prime\prime}(S)$ there exist a single edge $e$ in $T_{g}^{\prime\prime}(S)$ whose adjacent vertices are at distance 1 from $v_{1}$ or $v_{2}$. Let us suppose that this is not an edge of a triangle in $T(S)$. Then $e$ is contained in a connected component of $T(S)\smallsetminus T_{g}^{\prime}(S)$. In this case, if $e$ is marked with red color (blue color), we choose the blue curve (red curve) in $S$ as is shown in Figure 13. In all other cases we do nothing and this finishes the construction of the multicurves $A$ and $B$. Figure 13. The corresponding blue curve in $S$ for the red edge $e$. We define $\alpha=\\{\alpha_{i}\\}_{i\in I}$ and $\beta=\\{\beta_{j}\\}_{j\in J}$ as the set of essential curves in $A$ and $B$, respectively. By construction, $\alpha$ and $\beta$ are multicurves of finite type in minimal position and $\rm{i}(\alpha_{i},\beta_{j})\leq 2$ for every $i\in I$ and $j\in J$. Upon investigation, one can observe that each connected component of $S\setminus\alpha\cup\beta$ is a disc or a punctured disc whose boundary is formed by 2, 4, 6 or 8 segments. Second part: Let $m\in\mathbb{N}$. Take a finite multicurve $\delta$ in $S$ such that, if $Q$ is the connected component of $S\smallsetminus\delta$ which contains $p$, we have that $Q\setminus p$ is homeomorphic to $S_{0}^{m+3}$, i.e., a genus zero surfaces with $m+3$ punctures. In $Q$ we choose blue and red curves to form a chain as in Figure 14 and color them in blue and red so that no two curves of the same color intersect. We denote the blue and red curves in $Q$ by $\alpha^{\prime}$ and $\beta^{\prime}$ respectively. Remark that the connected component of $Q\smallsetminus(\alpha^{\prime}\cup\beta^{\prime})$ containing the point $p$ is a $2m-$polygon. Figure 14. The multicurves $\alpha^{\prime}$ (blue) and $\beta^{\prime}$ (red). The idea is to extend $\alpha^{\prime}$ and $\beta^{\prime}$ to multicurves $\alpha$ and $\beta$, respectively, which satisfy all the desired properties. We consider two cases for the rest of the proof. $m$ even. Without loss of generality we suppose that all punctures in $Q$ different from $p$ are encircled by elements of $\alpha^{\prime}$. Now, let $F$ be a connected component of $S\smallsetminus\alpha^{\prime}$ not containing the point $p$. Then the closure $\overline{F}$ of $F$ in $S$ is a surface with $b>0$ boundary components. Moreover, $\overline{F}\cap\beta^{\prime}$ is a finite collection of disjoint essential arcs in $\overline{F}$ with end points in $\partial\overline{F}$, and the end points of an arc in $\overline{F}\cap\beta^{\prime}$ are in a common connected component of $\partial\overline{F}$. We denote by $\theta_{F}$ the collection of arcs in $\overline{F}\cap\beta^{\prime}$, and by $\delta_{F}$ to be the set of curves in $\delta$ contained in $F$. Claim: There exists a pair of multicurves $\alpha_{F}^{\prime\prime}$ and $\beta_{F}^{\prime\prime}$ whose union fills $F$, which satisfy (1) & (2) in Theorem 1.5 and such that $\theta_{F}\cap\beta_{F}^{\prime\prime}=\emptyset$. Remark that if we define $\alpha:=\alpha^{\prime}\bigcup\left(\bigcup_{F\subset S\setminus\alpha^{\prime}}\alpha_{F}^{\prime\prime}\right)$ and $\beta:=\beta^{\prime}\bigcup\left(\bigcup_{F\subset S\setminus\alpha^{\prime}}\beta_{F}^{\prime\prime}\right)$, then $\alpha$ and $\beta$ are the desired pair of multicurves. We divided the proof of our claim in two cases: $b=1$ and $b>1$. Case $b=1$. If $F$ is a finite-type surface is it not difficult to find the multicurves $\alpha_{F}^{\prime\prime}$ and $\beta_{F}^{\prime\prime}$. If $F$ is of infinite-type let $\alpha^{\prime\prime}_{F}$ and $\beta^{\prime\prime}_{F}$ be the blue and red curves obtained from applying the first part of the proof of Theorem 1.5 to $F$. Remark that by construction all arcs in $\theta_{F}$ intersect only one curve in $\alpha_{F}^{\prime\prime}\cup\beta_{F}^{\prime\prime}$, hence, up to interchanging the colors of $\alpha^{\prime\prime}_{F}$ and $\beta_{F}^{\prime\prime}$, we can get that $\beta_{F}^{\prime\prime}\cap\theta_{F}=\emptyset$. Case $b>1$. Again, the case when $F$ is a finite-type surface is left to the reader. If $F$ is of infinite type, let $\gamma$ be the separating curve in $\overline{F}$ which bounds a subsurface $W\subset\overline{F}$ of genus 0 with one puncture, $b$ boundary components and such that $\partial W=\partial\overline{F}$ and write $\overline{F}\smallsetminus\gamma=W\sqcup F_{1}$. Let $\theta_{F_{1}}$ to be the set of arcs given by $\theta_{F}\cap\overline{F_{1}}$. Let $\eta_{1}$ and $\eta_{2}$ be two curves in $F_{1}$ (not necessary essential) such that $\gamma,\eta_{1}$ and $\eta_{2}$ bounds a pair of pants $P$ in $F_{1}$. If an element of $\theta_{F}$ intersects $\eta_{1}\cup\eta_{2}$ then we replace it with one that doesn’t and which is disjoint from all other arcs in $\theta_{F}$. Up to making these replacements, we can assume that $\theta_{F}$ does not intersects $\eta_{1}\cup\eta_{2}$, see Figure 15. Hence, $\theta_{F_{1}}\subseteq\overline{P}$. As $\overline{F_{1}}$ has one boundary component, by the case $b=1$ above, there exist a pair of multicurves $\alpha_{F_{1}}^{\prime\prime}$ and $\beta_{F_{1}}^{\prime\prime}$ which fills $F_{1}$ and such that $\theta_{F_{1}}\cap\beta_{F_{1}}^{\prime\prime}=\emptyset$. Define then $\alpha_{F}^{{}^{\prime\prime}}:=\\{\gamma\\}\cup\alpha_{F_{1}}^{{}^{\prime\prime}}$ and $\beta_{F}^{{}^{\prime\prime}}:=\beta_{F_{2}}^{{}^{\prime\prime}}$. Figure 15. Case $b>1$. m odd. Without loss of generality we suppose that $\alpha^{\prime}$ encircles all punctures in $Q$ different from $p$ except one. We add the curves $\alpha_{1}$ and $\beta_{1}$ to $\alpha^{\prime}$ and $\beta^{\prime}$ as depicted in Figure 16 respectively. Then we consider each connected component $F$ of $S\setminus\alpha^{\prime}$ and proceed as in the preceding case. ∎ Figure 16. Case $m$ odd. ###### Remark 4.10. If $\alpha$ and $\beta$ are multicurves as constructed in the proof of Theorem 1.5, then $S\setminus\alpha\cup\beta$ is a family of polygons, each of which has either 2, 4, 6 or 8 sides. Hence, if $M=M(\alpha,\beta,\textbf{h})$ is given by the Hooper-Thurston-Veech construction, then the set $\mathfrak{V}$ defined in the proof of Theorem 1.1 is formed by regular points and conic singularities of total angle $\pi$, $3\pi$ or $4\pi$. ### 4.3. Proof of Corollary 1.6 Our arguments use Figure 17. Let $\alpha$ and $\beta$ be the multicurves in blue and red illustrated in the figure; the union of these fills the surface $S$ in question (a Loch Ness monster). Let us write $\beta=\beta^{\prime}\sqcup\\{a_{i}\\}_{i\in\mathbb{N}}\sqcup\\{b_{i}\\}_{i\in\mathbb{N}}$, and for each $n\in\mathbb{N}$ let $\beta_{n}:=\beta^{\prime}\sqcup\\{a_{i}\\}_{i\geq n}\sqcup\\{b_{i}\\}_{i\geq n}$. Theorem 1.1 implies that $f_{n}:=T_{\alpha}\circ T_{\beta_{n}}^{-1}$ acts loxodromically on the loop graph $L(S;p)$ and hence it acts loxodromically on the main component of the completed ray graph $\mathcal{R}(S;p)$ (see Theorem 2.2). Remark that $f_{n}$ converges to $f=T_{\alpha}\circ T_{\beta^{\prime}}^{-1}$ in the compact-open topology. On the other hand, $f$ fixes the short rays $l$ and $l^{\prime}$ and hence it acts elliptically on both $\mathcal{R}(S;p)$ and $L(S;p)$. ∎ ###### Remark 4.11. Using techniques similar to the ones presented in the proof of Theorem 1.5 one can construct explicit sequences $(f_{n})$ as above for any infinite-type surface $S$. 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2024-09-04T02:54:55.811777

2003.00196

arxiv-papers

# First Order Motion Model for Image Animation Aliaksandr Siarohin DISI, University of Trento <EMAIL_ADDRESS>and Stéphane Lathuilière DISI, University of Trento LTCI, Télécom Paris, Institut polytechnique de Paris <EMAIL_ADDRESS>and Sergey Tulyakov Snap Inc. <EMAIL_ADDRESS>and Elisa Ricci DISI, University of Trento Fondazione Bruno Kessler <EMAIL_ADDRESS> and Nicu Sebe DISI, University of Trento Huawei Technologies Ireland <EMAIL_ADDRESS> ###### Abstract Image animation consists of generating a video sequence so that an object in a source image is animated according to the motion of a driving video. Our framework addresses this problem without using any annotation or prior information about the specific object to animate. Once trained on a set of videos depicting objects of the same category (e.g. faces, human bodies), our method can be applied to any object of this class. To achieve this, we decouple appearance and motion information using a self-supervised formulation. To support complex motions, we use a representation consisting of a set of learned keypoints along with their local affine transformations. A generator network models occlusions arising during target motions and combines the appearance extracted from the source image and the motion derived from the driving video. Our framework scores best on diverse benchmarks and on a variety of object categories. Our source code is publicly available111 https://github.com/AliaksandrSiarohin/first-order-model. ## 1 Introduction Generating videos by animating objects in still images has countless applications across areas of interest including movie production, photography, and e-commerce. More precisely, image animation refers to the task of automatically synthesizing videos by combining the appearance extracted from a source image with motion patterns derived from a driving video. For instance, a face image of a certain person can be animated following the facial expressions of another individual (see Fig. 1). In the literature, most methods tackle this problem by assuming strong priors on the object representation (e.g. 3D model) [4] and resorting to computer graphics techniques [6, 34]. These approaches can be referred to as object-specific methods, as they assume knowledge about the model of the specific object to animate. Recently, deep generative models have emerged as effective techniques for image animation and video retargeting [2, 42, 3, 43, 28, 29, 38, 41, 32, 22]. In particular, Generative Adversarial Networks (GANs) [14] and Variational Auto-Encoders (VAEs) [21] have been used to transfer facial expressions [38] or motion patterns [3] between human subjects in videos. Nevertheless, these approaches usually rely on pre-trained models in order to extract object- specific representations such as keypoint locations. Unfortunately, these pre- trained models are built using costly ground-truth data annotations [2, 28, 32] and are not available in general for an arbitrary object category. To address this issues, recently Siarohin et al. [29] introduced Monkey-Net, the first object-agnostic deep model for image animation. Monkey-Net encodes motion information via keypoints learned in a self-supervised fashion. At test time, the source image is animated according to the corresponding keypoint trajectories estimated in the driving video. The major weakness of Monkey-Net is that it poorly models object appearance transformations in the keypoint neighborhoods assuming a zeroth order model (as we show in Sec. 3.1). This leads to poor generation quality in the case of large object pose changes (see Fig. 4). To tackle this issue, we propose to use a set of self-learned keypoints together with local affine transformations to model complex motions. We therefore call our method a first-order motion model. Second, we introduce an occlusion-aware generator, which adopts an occlusion mask automatically estimated to indicate object parts that are not visible in the source image and that should be inferred from the context. This is especially needed when the driving video contains large motion patterns and occlusions are typical. Third, we extend the equivariance loss commonly used for keypoints detector training [18, 45], to improve the estimation of local affine transformations. Fourth, we experimentally show that our method significantly outperforms state-of-the-art image animation methods and can handle high-resolution datasets where other approaches generally fail. Finally, we release a new high resolution dataset, Thai-Chi-HD, which we believe could become a reference benchmark for evaluating frameworks for image animation and video generation. | | | ---|---|---|--- | | | | | | | | | ---|---|---|--- | | | | | | | | | ---|---|---|--- | | | | | | | | | ---|---|---|--- | | | | | | Figure 1: Example animations produced by our method trained on different datasets: _VoxCeleb_ [23] (top left), _Tai-Chi-HD_ (top right), _Fashion- Videos_ [42] (bottom left) and _MGif_ [29] (bottom right). We use relative motion transfer for _VoxCeleb_ and _Fashion-Videos_ and absolute transfer for _MGif_ and _Tai-Chi-HD_ see Sec. 3.4. Check our project page for more qualitative results222https://aliaksandrsiarohin.github.io/first-order-model- website/. ## 2 Related work Video Generation. Earlier works on deep video generation discussed how spatio- temporal neural networks could render video frames from noise vectors [37, 27]. More recently, several approaches tackled the problem of conditional video generation. For instance, Wang et al. [39] combine a recurrent neural network with a VAE in order to generate face videos. Considering a wider range of applications, Tulyakov et al. [35] introduced MoCoGAN, a recurrent architecture adversarially trained in order to synthesize videos from noise, categorical labels or static images. Another typical case of conditional generation is the problem of future frame prediction, in which the generated video is conditioned on the initial frame [12, 24, 31, 36, 45]. Note that in this task, realistic predictions can be obtained by simply warping the initial video frame [1, 12, 36]. Our approach is closely related to these previous works since we use a warping formulation to generate video sequences. However, in the case of image animation, the applied spatial deformations are not predicted but given by the driving video. Image Animation. Traditional approaches for image animation and video re- targeting [6, 34, 13] were designed for specific domains such as faces [46, 43], human silhouettes [8, 38, 28] or gestures [32] and required a strong prior of the animated object. For example, in face animation, method of Zollhofer et al. [46] produced realistic results at expense of relying on a 3D morphable model of the face. In many applications, however, such models are not available. Image animation can also be treated as a translation problem from one visual domain to another. For instance, Wang et al. [38] transferred human motion using the image-to-image translation framework of Isola et al. [16]. Similarly, Bansal et al. [3] extended conditional GANs by incorporating spatio-temporal cues in order to improve video translation between two given domains. Such approaches in order to animate a single person require hours of videos of that person labelled with semantic information, and therefore have to be retrained for each individual. In contrast to these works, we neither rely on labels, prior information about the animated objects, nor on specific training procedures for each object instance. Furthermore, our approach can be applied to any object within the same category (e.g., faces, human bodies, robot arms etc). Several approaches were proposed that do not require priors about the object. X2Face [41] uses a dense motion field in order to generate the output video via image warping. Similarly to us they employ a reference pose that is used to obtain a canonical representation of the object. In our formulation, we do not require an explicit reference pose, leading to significantly simpler optimization and improved image quality. Siarohin et al. [29] introduced Monkey-Net, a self-supervised framework for animating arbitrary objects by using sparse keypoint trajectories. In this work, we also employ sparse trajectories induced by self-supervised keypoints. However, we model object motion in the neighbourhood of each predicted keypoint by a local affine transformation. Additionally, we explicitly model occlusions in order to indicate to the generator network the image regions that can be generated by warping the source image and the occluded areas that need to be inpainted. ## 3 Method We are interested in animating an object depicted in a source image $\mathbf{S}$ based on the motion of a similar object in a driving video $\mathcal{D}$. Since direct supervision is not available (pairs of videos in which objects move similarly), we follow a self-supervised strategy inspired from Monkey-Net [29]. For training, we employ a large collection of video sequences containing objects of the same object category. Our model is trained to reconstruct the training videos by combining a single frame and a learned latent representation of the motion in the video. Observing frame pairs, each extracted from the same video, it learns to encode motion as a combination of motion-specific keypoint displacements and local affine transformations. At test time we apply our model to pairs composed of the source image and of each frame of the driving video and perform image animation of the source object. Figure 2: Overview of our approach. Our method assumes a source image $\mathbf{S}$ and a frame of a driving video frame $\mathbf{D}$ as inputs. The unsupervised keypoint detector extracts first order motion representation consisting of sparse keypoints and local affine transformations with respect to the reference frame $\mathbf{R}$. The dense motion network uses the motion representation to generate dense optical flow $\hat{\mathcal{T}}_{\mathbf{S}\leftarrow\mathbf{D}}$ from $\mathbf{D}$ to $\mathbf{S}$ and occlusion map $\hat{\mathcal{O}}_{\mathbf{S}\leftarrow\mathbf{D}}$. The source image and the outputs of the dense motion network are used by the generator to render the target image. An overview of our approach is presented in Fig. 2. Our framework is composed of two main modules: the motion estimation module and the image generation module. The purpose of the motion estimation module is to predict a dense motion field from a frame $\mathbf{D}\in\mathbb{R}^{3\times H\times W}$ of dimension $H\times W$ of the driving video $\mathcal{D}$ to the source frame $\mathbf{S}\in\mathbb{R}^{3\times H\times W}$. The dense motion field is later used to align the feature maps computed from $\mathbf{S}$ with the object pose in $\mathbf{D}$. The motion field is modeled by a function $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ that maps each pixel location in $\mathbf{D}$ with its corresponding location in $\mathbf{S}$. $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}$ is often referred to as backward optical flow. We employ backward optical flow, rather than forward optical flow, since back-warping can be implemented efficiently in a differentiable manner using bilinear sampling [17]. We assume there exists an abstract reference frame $\mathbf{R}$. We independently estimate two transformations: from $\mathbf{R}$ to $\mathbf{S}$ ($\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}$) and from $\mathbf{R}$ to $\mathbf{D}$ ($\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}$). Note that unlike X2Face [41] the reference frame is an abstract concept that cancels out in our derivations later. Therefore it is never explicitly computed and cannot be visualized. This choice allows us to independently process $\mathbf{D}$ and $\mathbf{S}$. This is desired since, at test time the model receives pairs of the source image and driving frames sampled from a different video, which can be very different visually. Instead of directly predicting $\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}$ and $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}$, the motion estimator module proceeds in two steps. In the first step, we approximate both transformations from sets of sparse trajectories, obtained by using keypoints learned in a self-supervised way. The locations of the keypoints in $\mathbf{D}$ and $\mathbf{S}$ are separately predicted by an encoder-decoder network. The keypoint representation acts as a bottleneck resulting in a compact motion representation. As shown by Siarohin et al. [29], such sparse motion representation is well-suited for animation as at test time, the keypoints of the source image can be moved using the keypoints trajectories in the driving video. We model motion in the neighbourhood of each keypoint using local affine transformations. Compared to using keypoint displacements only, the local affine transformations allow us to model a larger family of transformations. We use Taylor expansion to represent $\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}$ by a set of keypoint locations and affine transformations. To this end, the keypoint detector network outputs keypoint locations as well as the parameters of each affine transformation. During the second step, a dense motion network combines the local approximations to obtain the resulting dense motion field $\hat{\mathcal{T}}_{\mathbf{S}\leftarrow\mathbf{D}}$. Furthermore, in addition to the dense motion field, this network outputs an occlusion mask $\hat{\mathcal{O}}_{\mathbf{S}\leftarrow\mathbf{D}}$ that indicates which image parts of $\mathbf{D}$ can be reconstructed by warping of the source image and which parts should be inpainted, i.e.inferred from the context. Finally, the generation module renders an image of the source object moving as provided in the driving video. Here, we use a generator network $G$ that warps the source image according to $\hat{\mathcal{T}}_{\mathbf{S}\leftarrow\mathbf{D}}$ and inpaints the image parts that are occluded in the source image. In the following sections we detail each of these step and the training procedure. ### 3.1 Local Affine Transformations for Approximate Motion Description The motion estimation module estimates the backward optical flow $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}$ from a driving frame $\mathbf{D}$ to the source frame $\mathbf{S}$. As discussed above, we propose to approximate $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}$ by its first order Taylor expansion in a neighborhood of the keypoint locations. In the rest of this section, we describe the motivation behind this choice, and detail the proposed approximation of $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}$. We assume there exist an abstract reference frame $\mathbf{R}$. Therefore, estimating $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}$ consists in estimating $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}$ and $\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{D}}$. Furthermore, given a frame $\mathbf{X}$, we estimate each transformation $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}$ in the neighbourhood of the learned keypoints. Formally, given a transformation $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}$, we consider its first order Taylor expansions in $K$ keypoints $p_{1},\dots p_{K}$. Here, $p_{1},\dots p_{K}$ denote the coordinates of the keypoints in the reference frame $\mathbf{R}$. Note that for the sake of simplicity in the following the point locations in the reference pose space are all denoted by $p$ while the point locations in the $\mathbf{X}$, $\mathbf{S}$ or $\mathbf{D}$ pose spaces are denoted by $z$. We obtain: $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p)=\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p_{k})+\left(\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)(p-p_{k})+o(\|p-p_{k}\|),$ (1) In this formulation, the motion function $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}$ is represented by its values in each keypoint $p_{k}$ and its Jacobians computed in each $p_{k}$ location: $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p)\simeq\left\\{\left\\{\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p_{1}),\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{1}}\right\\},\dots\left\\{\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p_{k}),\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{K}}\right\\}\right\\}.$ (2) Furthermore, in order to estimate $\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{X}}=\mathcal{T}^{-1}_{\mathbf{X}\leftarrow\mathbf{R}}$, we assume that $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}$ is locally bijective in the neighbourhood of each keypoint. We need to estimate $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}$ near the keypoint $z_{k}$ in $\mathbf{D}$, given that $z_{k}$ is the pixel location corresponding to the keypoint location $p_{k}$ in $\mathbf{R}$. To do so, we first estimate the transformation $\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{D}}$ near the point $z_{k}$ in the driving frame $\mathbf{D}$, e.g. $p_{k}=\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{D}}(z_{k})$. Then we estimate the transformation $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}$ near $p_{k}$ in the reference $\mathbf{R}$. Finally $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}$ is obtained as follows: $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}=\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}\circ\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{D}}=\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}\circ\mathcal{T}^{-1}_{\mathbf{D}\leftarrow\mathbf{R}},$ (3) After computing again the first order Taylor expansion of Eq. (3) (see Sup. Mat.), we obtain: $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}(z)\approx\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p_{k})+J_{k}(z-\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}(p_{k}))$ (4) with: $J_{k}=\left(\frac{d}{dp}\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)\left(\frac{d}{dp}\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)^{-1}$ (5) In practice, $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p_{k})$ and $\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}(p_{k})$ in Eq. (4) are predicted by the keypoint predictor. More precisely, we employ the standard U-Net architecture that estimates $K$ heatmaps, one for each keypoint. The last layer of the decoder uses softmax activations in order to predict heatmaps that can be interpreted as keypoint detection confidence map. Each expected keypoint location is estimated using the average operation as in [29, 25]. Note if we set $J_{k}=\mathbb{1}$ ($\mathbb{1}$ is $2\times 2$ identity matrix), we get the motion model of Monkey-Net. Therefore Monkey-Net uses a zeroth-order approximation of $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}(z)-z$. For both frames $\mathbf{S}$ and $\mathbf{D}$, the keypoint predictor network also outputs four additional channels for each keypoint. From these channels, we obtain the coefficients of the matrices $\frac{d}{dp}\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p)|_{p=p_{k}}$ and $\frac{d}{dp}\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p)|_{p=p_{k}}$ in Eq. (5) by computing spatial weighted average using as weights the corresponding keypoint confidence map. Combining Local Motions. We employ a convolutional network $P$ to estimate $\mathcal{\hat{T}}_{\mathbf{S}\leftarrow\mathbf{D}}$ from the set of Taylor approximations of $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}(z)$ in the keypoints and the original source frame $\mathbf{S}$. Importantly, since $\mathcal{\hat{T}}_{\mathbf{S}\leftarrow\mathbf{D}}$ maps each pixel location in $\mathbf{D}$ with its corresponding location in $\mathbf{S}$, the local patterns in $\mathcal{\hat{T}}_{\mathbf{S}\leftarrow\mathbf{D}}$, such as edges or texture, are pixel-to-pixel aligned with $\mathbf{D}$ but not with $\mathbf{S}$. This misalignment issue makes the task harder for the network to predict $\mathcal{\hat{T}}_{\mathbf{S}\leftarrow\mathbf{D}}$ from $\mathbf{S}$. In order to provide inputs already roughly aligned with $\mathcal{\hat{T}}_{\mathbf{S}\leftarrow\mathbf{D}}$, we warp the source frame $\mathbf{S}$ according to local transformations estimated in Eq. (4). Thus, we obtain $K$ transformed images $\mathbf{S}^{1},\dots\mathbf{S}^{K}$ that are each aligned with $\mathcal{\hat{T}}_{\mathbf{S}\leftarrow\mathbf{D}}$ in the neighbourhood of a keypoint. Importantly, we also consider an additional image $\mathbf{S}^{0}=\mathbf{S}$ for the background. For each keypoint $p_{k}$ we additionally compute heatmaps $\mathbf{H}_{k}$ indicating to the dense motion network where each transformation happens. Each $\mathbf{H}_{k}(z)$ is implemented as the difference of two heatmaps centered in $\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}(p_{k})$ and $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p_{k})$: $\mathbf{H}_{k}(z)=exp\left(\frac{\left(\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}(p_{k})-z\right)^{2}}{\sigma}\right)-exp\left(\frac{\left(\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p_{k})-z\right)^{2}}{\sigma}\right).$ (6) In all our experiments, we employ $\sigma=0.01$ following Jakab et al. [18]. The heatmaps $\mathbf{H}_{k}$ and the transformed images $\mathbf{S}^{0},\dots\mathbf{S}^{K}$ are concatenated and processed by a U-Net [26]. $\mathcal{\hat{T}}_{\mathbf{S}\leftarrow\mathbf{D}}$ is estimated using a part-based model inspired by Monkey-Net [29]. We assume that an object is composed of $K$ rigid parts and that each part is moved according to Eq. (4). Therefore we estimate $K$+1 masks $\mathbf{M}_{k},k=0,\dots K$ that indicate where each local transformation holds. The final dense motion prediction $\mathcal{\hat{T}}_{\mathbf{S}\leftarrow\mathbf{D}}(z)$ is given by: $\mathcal{\hat{T}}_{\mathbf{S}\leftarrow\mathbf{D}}(z)=\mathbf{M}_{0}z+\sum_{k=1}^{K}{\mathbf{M}_{k}\left(\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p_{k})+J_{k}(z-\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}(p_{k}))\right)}$ (7) Note that, the term $\mathbf{M}_{0}z$ is considered in order to model non- moving parts such as background. ### 3.2 Occlusion-aware Image Generation As mentioned in Sec.3, the source image $\mathbf{S}$ is not pixel-to-pixel aligned with the image to be generated $\hat{\mathbf{D}}$. In order to handle this misalignment, we use a feature warping strategy similar to [30, 29, 15]. More precisely, after two down-sampling convolutional blocks, we obtain a feature map $\boldsymbol{\xi}\in\mathbb{R}^{H^{\prime}\times W^{\prime}}$ of dimension $H^{\prime}\times W^{\prime}$. We then warp $\boldsymbol{\xi}$ according to $\mathcal{\hat{T}}_{\mathbf{S}\leftarrow\mathbf{D}}$. In the presence of occlusions in $\mathbf{S}$, optical flow may not be sufficient to generate $\hat{\mathbf{D}}$. Indeed, the occluded parts in $\mathbf{S}$ cannot be recovered by image-warping and thus should be inpainted. Consequently, we introduce an occlusion map $\mathcal{\hat{O}}_{\mathbf{S}\leftarrow\mathbf{D}}\in[0,1]^{H^{\prime}\times W^{\prime}}$ to mask out the feature map regions that should be inpainted. Thus, the occlusion mask diminishes the impact of the features corresponding to the occluded parts. The transformed feature map is written as: $\boldsymbol{\xi}^{\prime}=\mathcal{\hat{O}}_{\mathbf{S}\leftarrow\mathbf{D}}\odot f_{w}(\boldsymbol{\xi},\mathcal{\hat{T}}_{\mathbf{S}\leftarrow\mathbf{D}})$ (8) where $f_{w}(\cdot,\cdot)$ denotes the back-warping operation and $\odot$ denotes the Hadamard product. We estimate the occlusion mask from our sparse keypoint representation, by adding a channel to the final layer of the dense motion network. Finally, the transformed feature map $\boldsymbol{\xi}^{\prime}$ is fed to subsequent network layers of the generation module (see _Sup. Mat._) to render the sought image. ### 3.3 Training Losses We train our system in an end-to-end fashion combining several losses. First, we use the reconstruction loss based on the perceptual loss of Johnson et al. [19] using the pre-trained VGG-19 network as our main driving loss. The loss is based on implementation of Wang et al. [38]. With the input driving frame $\mathbf{D}$ and the corresponding reconstructed frame $\mathbf{\hat{D}}$, the reconstruction loss is written as: $L_{rec}(\mathbf{\hat{D}},\mathbf{D})=\sum_{i=1}^{I}\left|N_{i}(\mathbf{\hat{D}})-N_{i}(\mathbf{D})\right|,$ (9) where $N_{i}(\cdot)$ is the $i^{th}$ channel feature extracted from a specific VGG-19 layer and $I$ is the number of feature channels in this layer. Additionally we propose to use this loss on a number of resolutions, forming a pyramid obtained by down-sampling $\mathbf{\hat{D}}$ and $\mathbf{D}$, similarly to MS-SSIM [40, 33]. The resolutions are $256\times 256$, $128\times 128$, $64\times 64$ and $32\times 32$. There are 20 loss terms in total. Imposing Equivariance Constraint. Our keypoint predictor does not require any keypoint annotations during training. This may lead to unstable performance. Equivariance constraint is one of the most important factors driving the discovery of unsupervised keypoints [18, 44]. It forces the model to predict consistent keypoints with respect to known geometric transformations. We use thin plate splines deformations as they were previously used in unsupervised keypoint detection [18, 44] and are similar to natural image deformations. Since our motion estimator does not only predict the keypoints, but also the Jacobians, we extend the well-known equivariance loss to additionally include constraints on the Jacobians. We assume that an image $\mathbf{X}$ undergoes a known spatial deformation $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}$. In this case $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}$ can be an affine transformation or a thin plane spline deformation. After this deformation we obtain a new image $\mathbf{Y}$. Now by applying our extended motion estimator to both images, we obtain a set of local approximations for $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}$ and $\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}$. The standard equivariance constraint writes as: $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}\equiv\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}\circ\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}$ (10) After computing the first order Taylor expansions of both sides, we obtain the following constraints (see derivation details in Sup. Mat.): $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p_{k})\equiv\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}\circ\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p_{k}),$ (11) $\left(\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)\equiv\left(\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}(p)\middle|_{p=\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p_{k})}\right)\left(\frac{d}{dp}\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right),$ (12) Note that the constraint Eq. (11) is strictly the same as the standard equivariance constraint for the keypoints [18, 44]. During training, we constrain every keypoint location using a simple $L_{1}$ loss between the two sides of Eq. (11). However, implementing the second constraint from Eq. (12) with $L_{1}$ would force the magnitude of the Jacobians to zero and would lead to numerical problems. To this end, we reformulate this constraint in the following way: $\mathbb{1}\equiv\left(\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)^{-1}\left(\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}(p)\middle|_{p=\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p_{k})}\right)\left(\frac{d}{dp}\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right),$ (13) where $\mathbb{1}$ is $2\times 2$ identity matrix. Then, $L_{1}$ loss is employed similarly to the keypoint location constraint. Finally, in our preliminary experiments, we observed that our model shows low sensitivity to the relative weights of the reconstruction and the two equivariance losses. Therefore, we use equal loss weights in all our experiments. ### 3.4 Testing Stage: Relative Motion Transfer At this stage our goal is to animate an object in a source frame $\mathbf{S}_{1}$ using the driving video $\mathbf{D}_{1},\dots\mathbf{D}_{T}$. Each frame $\mathbf{D}_{t}$ is independently processed to obtain $\mathbf{S}_{t}$. Rather than transferring the motion encoded in $\mathcal{T}_{\mathbf{S}_{1}\leftarrow\mathbf{D}_{t}}(p_{k})$ to $\mathbf{S}$, we transfer the relative motion between $\mathbf{D}_{1}$ and $\mathbf{D}_{t}$ to $\mathbf{S}_{1}$. In other words, we apply a transformation $\mathcal{T}_{\mathbf{D}_{t}\leftarrow\mathbf{D}_{1}}(p)$ to the neighbourhood of each keypoint $p_{k}$: $\mathcal{T}_{\mathbf{S}_{1}\leftarrow\mathbf{S}_{t}}(z)\approx\mathcal{T}_{\mathbf{S}_{1}\leftarrow\mathbf{R}}(p_{k})+J_{k}(z-\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p_{k})+\mathcal{T}_{\mathbf{D}_{1}\leftarrow\mathbf{R}}(p_{k})-\mathcal{T}_{\mathbf{D}_{t}\leftarrow\mathbf{R}}(p_{k}))$ (14) with $J_{k}=\left(\frac{d}{dp}\mathcal{T}_{\mathbf{D}_{1}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)\left(\frac{d}{dp}\mathcal{T}_{\mathbf{D}_{t}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)^{-1}$ (15) Detailed mathematical derivations are provided in Sup. Mat.. Intuitively, we transform the neighbourhood of each keypoint $p_{k}$ in $\mathbf{S}_{1}$ according to its local deformation in the driving video. Indeed, transferring relative motion over absolute coordinates allows to transfer only relevant motion patterns, while preserving global object geometry. Conversely, when transferring absolute coordinates, as in X2Face [41], the generated frame inherits the object proportions of the driving video. It’s important to note that one limitation of transferring relative motion is that we need to assume that the objects in $\mathbf{S}_{1}$ and $\mathbf{D}_{1}$ have similar poses (see [29]). Without initial rough alignment, Eq. (14) may lead to absolute keypoint locations physically impossible for the object of interest. ## 4 Experiments Datasets. We train and test our method on four different datasets containing various objects. Our model is capable of rendering videos of much higher resolution compared to [29] in all our experiments. * • The _VoxCeleb_ dataset [23] is a face dataset of 22496 videos, extracted from YouTube videos. For pre-processing, we extract an initial bounding box in the first video frame. We track this face until it is too far away from the initial position. Then, we crop the video frames using the smallest crop containing all the bounding boxes. The process is repeated until the end of the sequence. We filter out sequences that have resolution lower than $256\times 256$ and the remaining videos are resized to $256\times 256$ preserving the aspect ratio. It’s important to note that compared to X2Face [41], we obtain more natural videos where faces move freely within the bounding box. Overall, we obtain 19522 training videos and 525 test videos, with lengths varying from 64 to 1024 frames. Table 1: Quantitative ablation study for video reconstruction on _Tai-Chi-HD_. | _Tai-Chi-HD_ ---|--- | $\mathcal{L}_{1}$ | (AKD, MKR) | AED _Baseline_ | 0.073 | (8.945, 0.099) | 0.235 _Pyr._ | 0.069 | (9.407, 0.065) | 0.213 _Pyr._ +$\mathcal{O}_{\mathbf{S}\leftarrow\mathbf{D}}$ | 0.069 | (8.773, 0.050) | 0.205 _Jac. w/o Eq. ( 12)_ | 0.073 | (9.887, 0.052) | 0.220 _Full_ | 0.063 | (6.862, 0.036) | 0.179 Table 2: Paired user study: user preferences in favour of our approach. | X2Face [41] | Monkey-Net [29] ---|---|--- _Tai-Chi-HD_ | 92.0% | 80.6% _VoxCeleb_ | 95.8% | 68.4% _Nemo_ | 79.8% | 60.6% _Bair_ | 95.0% | 67.0% Input $\mathbf{D}$ | | | | | ---|---|---|---|---|--- _Baseline_ | | | | | _Pyr._ | | | | | _Pyr._ +$\mathcal{O}_{\mathbf{S}\leftarrow\mathbf{D}}$ | | | | | _Jac. w/o Eq. ( 12)_ | | | | | _Full_ | | | | | Figure 3: Qualitative ablation on _Tai-Chi-HD_. * • The UvA-_Nemo_ dataset [9] is a facial analysis dataset that consists of 1240 videos. We apply the exact same pre-processing as for _VoxCeleb_. Each video starts with a neutral expression. Similar to Wang et al. [39], we use 1116 videos for training and 124 for evaluation. * • The _BAIR_ robot pushing dataset [10] contains videos collected by a Sawyer robotic arm pushing diverse objects over a table. It consists of 42880 training and 128 test videos. Each video is 30 frame long and has a $256\times 256$ resolution. * • Following Tulyakov et al. [35], we collected 280 tai-chi videos from YouTube. We use 252 videos for training and 28 for testing. Each video is split in short clips as described in pre-processing of _VoxCeleb_ dataset. We retain only high quality videos and resized all the clips to $256\times 256$ pixels (instead of $64\times 64$ pixels in [35]). Finally, we obtain 3049 and 285 video chunks for training and testing respectively with video length varying from 128 to 1024 frames. This dataset is referred to as the _Tai-Chi-HD_ dataset. The dataset will be made publicly available. Evaluation Protocol. Evaluating the quality of image animation is not obvious, since ground truth animations are not available. We follow the evaluation protocol of Monkey-Net [29]. First, we quantitatively evaluate each method on the "proxy" task of video reconstruction. This task consists of reconstructing the input video from a representation in which appearance and motion are decoupled. In our case, we reconstruct the input video by combining the sparse motion representation in (2) of each frame and the first video frame. Second, we evaluate our model on image animation according to a user-study. In all experiments we use $K$=10 as in [29]. Other implementation details are given in _Sup. Mat._ Metrics. To evaluate video reconstruction, we adopt the metrics proposed in Monkey-Net [29]: * • $\mathcal{L}_{1}$. We report the average $\mathcal{L}_{1}$ distance between the generated and the ground-truth videos. * • _Average Keypoint Distance (AKD)_. For the _Tai-Chi-HD_ , _VoxCeleb_ and _Nemo_ datasets, we use 3rd-party pre-trained keypoint detectors in order to evaluate whether the motion of the input video is preserved. For the _VoxCeleb_ and _Nemo_ datasets we use the facial landmark detector of Bulat et al. [5]. For the _Tai-Chi-HD_ dataset, we employ the human-pose estimator of Cao et al. [7]. These keypoints are independently computed for each frame. AKD is obtained by computing the average distance between the detected keypoints of the ground truth and of the generated video. * • _Missing Keypoint Rate (MKR)_. In the case of _Tai-Chi-HD_ , the human-pose estimator returns an additional binary label for each keypoint indicating whether or not the keypoints were successfully detected. Therefore, we also report the MKR defined as the percentage of keypoints that are detected in the ground truth frame but not in the generated one. This metric assesses the appearance quality of each generated frame. * • _Average Euclidean Distance (AED)_. Considering an externally trained image representation, we report the average euclidean distance between the ground truth and generated frame representation, similarly to Esser et al. [11]. We employ the feature embedding used in Monkey-Net [29]. Ablation Study. We compare the following variants of our model. _Baseline_ : the simplest model trained without using the occlusion mask ($\mathcal{O}_{\mathbf{S}\leftarrow\mathbf{D}}$=1 in Eq. (8)), jacobians ($J_{k}=\mathbb{1}$ in Eq. (4)) and is supervised with $L_{rec}$ at the highest resolution only; _Pyr._ : the pyramid loss is added to _Baseline_ ; _Pyr._ +$\mathcal{O}_{\mathbf{S}\leftarrow\mathbf{D}}$: with respect to _Pyr._ , we replace the generator network with the occlusion-aware network; _Jac. w/o Eq. ( 12)_ our model with local affine transformations but without equivariance constraints on jacobians Eq. (12); _Full_ : the full model including local affine transformations described in Sec. 3.1. In Fig. 3, we report the qualitative ablation. First, the pyramid loss leads to better results according to all the metrics except _AKD_. Second, adding $\mathcal{O}_{\mathbf{S}\leftarrow\mathbf{D}}$ to the model consistently improves all the metrics with respect to _Pyr._. This illustrates the benefit of explicitly modeling occlusions. We found that without equivariance constraint over the jacobians, $J_{k}$ becomes unstable which leads to poor motion estimations. Finally, our _Full_ model further improves all the metrics. In particular, we note that, with respect to the _Baseline_ model, the MKR of the full model is smaller by the factor of 2.75. It shows that our rich motion representation helps generate more realistic images. These results are confirmed by our qualitative evaluation in Tab. 1 where we compare the _Baseline_ and the _Full_ models. In these experiments, each frame $\mathbf{D}$ of the input video is reconstructed from its first frame (first column) and the estimated keypoint trajectories. We note that the _Baseline_ model does not locate any keypoints in the arms area. Consequently, when the pose difference with the initial pose increases, the model cannot reconstruct the video (columns 3,4 and 5). In contrast, the _Full_ model learns to detect a keypoint on each arm, and therefore, to more accurately reconstruct the input video even in the case of complex motion. Comparison with State of the Art. We now compare our method with state of the art for the video reconstruction task as in [29]. To the best of our knowledge, X2Face [41] and Monkey-Net [29] are the only previous approaches for model-free image animation. Quantitative results are reported in Tab. 3. We observe that our approach consistently improves every single metric for each of the four different datasets. Even on the two face datasets, _VoxCeleb_ and _Nemo_ datasets, our approach clearly outperforms X2Face that was originally proposed for face generation. The better performance of our approach compared to X2Face is especially impressive X2Face exploits a larger motion embedding (128 floats) than our approach (60=K*(2+4) floats). Compared to Monkey-Net that uses a motion representation with a similar dimension (50=K*(2+3)), the advantages of our approach are clearly visible on the _Tai- Chi-HD_ dataset that contains highly non-rigid objects (i.e.human body). Table 3: Video reconstruction: comparison with the state of the art on four different datasets. | _Tai-Chi-HD_ | _VoxCeleb_ | _Nemo_ | _Bair_ ---|---|---|---|--- | $\mathcal{L}_{1}$ | (AKD, MKR) | AED | $\mathcal{L}_{1}$ | AKD | AED | $\mathcal{L}_{1}$ | AKD | AED | $\mathcal{L}_{1}$ X2Face [41] | 0.080 | (17.654, 0.109) | 0.272 | 0.078 | 7.687 | 0.405 | 0.031 | 3.539 | 0.221 | 0.065 Monkey-Net [29] | 0.077 | (10.798, 0.059) | 0.228 | 0.049 | 1.878 | 0.199 | 0.018 | 1.285 | 0.077 | 0.034 Ours | 0.063 | (6.862, 0.036) | 0.179 | 0.043 | 1.294 | 0.140 | 0.016 | 1.119 | 0.048 | 0.027 We now report a qualitative comparison for image animation. Generated sequences are reported in Fig. 4. The results are well in line with the quantitative evaluation in Tab. 3. Indeed, in both examples, X2Face and Monkey-Net are not able to correctly transfer the body notion in the driving video, instead warping the human body in the source image as a blob. Conversely, our approach is able to generate significantly better looking videos in which each body part is independently animated. This qualitative evaluation illustrates the potential of our rich motion description. We complete our evaluation with a user study. We ask users to select the most realistic image animation. Each question consists of the source image, the driving video, and the corresponding results of our method and a competitive method. We require each question to be answered by 10 AMT worker. This evaluation is repeated on 50 different input pairs. Results are reported in Tab. • ‣ 4. We observe that our method is clearly preferred over the competitor methods. Interestingly, the largest difference with the state of the art is obtained on _Tai-Chi-HD_ : the most challenging dataset in our evaluation due to its rich motions. | | | | | | ---|---|---|---|---|---|--- X2Face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | | | | | | ---|---|---|---|---|--- | | | | | | | | | | | | | | | Figure 4: Qualitative comparison with state of the art for the task of image animation on two sequences and two source images from the _Tai-Chi-HD_ dataset. ## 5 Conclusions We presented a novel approach for image animation based on keypoints and local affine transformations. Our novel mathematical formulation describes the motion field between two frames and is efficiently computed by deriving a first order Taylor expansion approximation. In this way, motion is described as a set of keypoints displacements and local affine transformations. A generator network combines the appearance of the source image and the motion representation of the driving video. In addition, we proposed to explicitly model occlusions in order to indicate to the generator network which image parts should be inpainted. We evaluated the proposed method both quantitatively and qualitatively and showed that our approach clearly outperforms state of the art on all the benchmarks. ## References * [1] Mohammad Babaeizadeh, Chelsea Finn, Dumitru Erhan, Roy H Campbell, and Sergey Levine. Stochastic variational video prediction. In ICLR, 2017. * [2] Guha Balakrishnan, Amy Zhao, Adrian V Dalca, Fredo Durand, and John Guttag. Synthesizing images of humans in unseen poses. In CVPR, 2018. * [3] Aayush Bansal, Shugao Ma, Deva Ramanan, and Yaser Sheikh. Recycle-gan: Unsupervised video retargeting. In ECCV, 2018. * [4] Volker Blanz and Thomas Vetter. A morphable model for the synthesis of 3d faces. In SIGGRAPH, 1999. * [5] Adrian Bulat and Georgios Tzimiropoulos. 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Generating videos with scene dynamics. In NIPS, 2016. * [38] Ting-Chun Wang, Ming-Yu Liu, Jun-Yan Zhu, Guilin Liu, Andrew Tao, Jan Kautz, and Bryan Catanzaro. Video-to-video synthesis. In NIPS, 2018. * [39] Wei Wang, Xavier Alameda-Pineda, Dan Xu, Pascal Fua, Elisa Ricci, and Nicu Sebe. Every smile is unique: Landmark-guided diverse smile generation. In CVPR, 2018. * [40] Zhou Wang, Eero P Simoncelli, and Alan C Bovik. Multiscale structural similarity for image quality assessment. In ACSSC, 2003. * [41] Olivia Wiles, A Sophia Koepke, and Andrew Zisserman. X2face: A network for controlling face generation using images, audio, and pose codes. In ECCV, 2018. * [42] Polina Zablotskaia, Aliaksandr Siarohin, Bo Zhao, and Leonid Sigal. Dwnet: Dense warp-based network for pose-guided human video generation. In BMVC, 2019. * [43] Egor Zakharov, Aliaksandra Shysheya, Egor Burkov, and Victor Lempitsky. Few-shot adversarial learning of realistic neural talking head models. In ICCV, 2019. * [44] Yuting Zhang, Yijie Guo, Yixin Jin, Yijun Luo, Zhiyuan He, and Honglak Lee. Unsupervised discovery of object landmarks as structural representations. In CVPR, 2018. * [45] Long Zhao, Xi Peng, Yu Tian, Mubbasir Kapadia, and Dimitris Metaxas. Learning to forecast and refine residual motion for image-to-video generation. In ECCV, 2018. * [46] Michael Zollhöfer, Justus Thies, Pablo Garrido, Derek Bradley, Thabo Beeler, Patrick Pérez, Marc Stamminger, Matthias Nießner, and Christian Theobalt. State of the art on monocular 3d face reconstruction, tracking, and applications. In Computer Graphics Forum, 2018. ## A Detailed Derivations ### A.1 Approximating Motion with Local Affine Transformations Here, we detail the derivation leading to the approximation of $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}$ near the keypoint $z_{k}$ in Eq. (4). Using first order Taylor expansion we can obtain: $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}(z)=\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}(z_{k})+\left(\frac{d}{dz}\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}(z)\middle|_{z=z_{k}}\right)(z-z_{k})+o(\|z-z_{k}\|)$ (16) $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}$ can be written as the composition of two transformations: $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}=\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}\circ\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{D}}$ (17) In order to compute the zeroth order term, we estimate the transformation $\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{D}}$ near the point $z_{k}$ in the driving frame $\mathbf{D}$, e.g $p_{k}=\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{D}}(z_{k})$. Then we can estimate the transformation $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}$ near $p_{k}$ in the reference $\mathbf{R}$. Since $p_{k}=\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{D}}(z_{k})$ and $\mathcal{T}^{-1}_{\mathbf{R}\leftarrow\mathbf{D}}=\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}$, we can write $z_{k}=\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}(p_{k})$. Consequently, we obtain: $\displaystyle\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}(z_{k})$ $\displaystyle=\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}\circ\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{D}}(z_{k})$ $\displaystyle=\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}\circ\mathcal{T}^{-1}_{\mathbf{D}\leftarrow\mathbf{R}}(z_{k})$ $\displaystyle=\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}\circ\mathcal{T}^{-1}_{\mathbf{D}\leftarrow\mathbf{R}}\circ\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}(p_{k})$ $\displaystyle=\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p_{k}).$ (18) Concerning the first order term, we apply the function composition rule in Eq. (17) and obtain: $\left(\frac{d}{dz}\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}(z)\middle|_{z=z_{k}}\right)=\left(\frac{d}{dp}\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p)\middle|_{p=\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{D}}(z_{k})}\right)\left(\frac{d}{dz}\mathcal{T}^{-1}_{\mathbf{D}\leftarrow\mathbf{R}}(z)\middle|_{z=z_{k}}\right)$ (19) Since the matrix inverse of the Jacobian is equal to the Jacobian of the inverse function, and since $p_{k}=\mathcal{T}_{\mathbf{R}\leftarrow\mathbf{D}}(z_{k})$, Eq. (19) can be rewritten: $\left(\frac{d}{dz}\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}(z)\middle|_{z=z_{k}}\right)=\left(\frac{d}{dp}\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)\left(\frac{d}{dp}\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)^{-1}$ (20) After injecting Eqs. (A.1) and (20) into (16), we finally obtain: $\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{D}}(z)\approx\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p_{k})+\left(\frac{d}{dp}\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)\left(\frac{d}{dp}\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)^{-1}(z-\mathcal{T}_{\mathbf{D}\leftarrow\mathbf{R}}(p_{k}))$ (21) ### A.2 Equivariance Loss At training time, we use equivariance constraints that enforces: $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}\equiv\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}\circ\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}$ (22) After applying first order Taylor expansion on the left-hand side, we obtain: $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p)=\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p_{k})+\left(\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)(p-p_{k})+o(\|p-p_{k}\|).$ (23) After applying first order Taylor expansion on the right-hand side in Eq. (22), we obtain: $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}\circ\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p)=\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}\circ\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p_{k})+\left(\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}\circ\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}\middle|_{p=p_{k}}\right)(p-p_{k})+o(\|p-p_{k}\|),$ (24) We can further simplify this expression using derivative of function composition: $\left(\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}\circ\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}\middle|_{p=p_{k}}\right)=\left(\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}(p)\middle|_{p=\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p_{k})}\right)\left(\frac{d}{dp}\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right).$ (25) Eq. (22) holds only when every coefficient in Taylor expansion of the right and left sides are equal. Thus, it leads us to the following constaints: $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p_{k})\equiv\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}\circ\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p_{k}),$ (26) and $\left(\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)\equiv\left(\frac{d}{dp}\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}(p)\middle|_{p=\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p_{k})}\right)\left(\frac{d}{dp}\mathcal{T}_{\mathbf{Y}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right).$ (27) ### A.3 Transferring Relative Motion In order to transfer only relative motion patterns, we propose to estimate $\mathcal{T}_{\mathbf{S}_{t}\leftarrow\mathbf{R}}(p)$ near the keypoint $p_{k}$ by shifting the motion in the driving video to the location of keypoint $p_{k}$ in the source. To this aim, we introduce $\mathcal{V}_{\mathbf{S}_{1}\leftarrow\mathbf{D_{1}}}(p_{k})=\mathcal{T}_{\mathbf{S}_{1}\leftarrow\mathbf{R}}(p_{k})-\mathcal{T}_{\mathbf{D}_{1}\leftarrow\mathbf{R}}(p_{k})\in\mathbb{R}^{2}$ that is the 2D vector from the landmark position $p_{k}$ in $\mathbf{D}_{1}$ to its position in $\mathbf{S}_{1}$. We proceed as follows. First, we shift point coordinates according to $-\mathcal{V}_{\mathbf{S}_{1}\leftarrow\mathbf{D_{1}}}(p_{k})$ in order to obtain coordinates in $\mathbf{D_{1}}$. Second, we apply the transformation $\mathcal{T}_{\mathbf{D}_{t}\leftarrow\mathbf{D}_{1}}$. Finally, we translate the points back in the original coordinate space using $\mathcal{V}_{\mathbf{S}_{1}\leftarrow\mathbf{D_{1}}}(p_{k})$. Formally, it can be written: $\mathcal{T}_{\mathbf{S}_{t}\leftarrow\mathbf{R}}(p)=\mathcal{T}_{\mathbf{D}_{t}\leftarrow\mathbf{D}_{1}}\big{(}\mathcal{T}_{\mathbf{S}_{1}\leftarrow\mathbf{R}}(p)-\mathcal{V}_{\mathbf{S}_{1}\leftarrow\mathbf{D}_{1}}(p_{k})\big{)}+\mathcal{V}_{\mathbf{S}_{1}\leftarrow\mathbf{D}_{1}}(p_{k})$ Now, we can compute the value and Jacobian in the $p_{k}$: $\mathcal{T}_{\mathbf{S}_{t}\leftarrow\mathbf{R}}(p_{k})=\mathcal{T}_{\mathbf{D}_{t}\leftarrow\mathbf{D}_{1}}\circ\mathcal{T}_{\mathbf{D}_{1}\leftarrow\mathbf{R}}(p_{k})-\mathcal{T}_{\mathbf{D}_{1}\leftarrow\mathbf{R}}(p_{k})+\mathcal{T}_{\mathbf{S}_{1}\leftarrow\mathbf{R}}(p_{k})$ and: $\left(\frac{d}{dp}\mathcal{T}_{\mathbf{S}_{t}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)=\left(\frac{d}{dp}\mathcal{T}_{\mathbf{D}_{t}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)\left(\frac{d}{dp}\mathcal{T}_{\mathbf{D}_{1}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)^{-1}\left(\frac{d}{dp}\mathcal{T}_{\mathbf{S}_{1}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right).$ Now using Eq. (21) and treating $\mathbf{S}_{1}$ as source and $\mathbf{S}_{t}$ as driving frame, we obtain: $\mathcal{T}_{\mathbf{S}_{1}\leftarrow\mathbf{S}_{t}}(z)\approx\mathcal{T}_{\mathbf{S}_{1}\leftarrow\mathbf{R}}(p_{k})+J_{k}(z-\mathcal{T}_{\mathbf{S}\leftarrow\mathbf{R}}(p_{k})+\mathcal{T}_{\mathbf{D}_{1}\leftarrow\mathbf{R}}(p_{k})-\mathcal{T}_{\mathbf{D}_{t}\leftarrow\mathbf{R}}(p_{k}))$ (28) with $J_{k}=\left(\frac{d}{dp}\mathcal{T}_{\mathbf{D}_{1}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)\left(\frac{d}{dp}\mathcal{T}_{\mathbf{D}_{t}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)^{-1}.$ (29) Note that, here, $\left(\frac{d}{dp}\mathcal{T}_{\mathbf{S}_{1}\leftarrow\mathbf{R}}(p)\middle|_{p=p_{k}}\right)$ canceled out. ## B Implementation details ### B.1 Architecture details In order to reduce memory and computational requirements of our model, the keypoint detector and dense motion predictor both work on resolution of $64\times 64$ (instead of $256\times 256$). For the two networks of the motion module, we employ an architecture based on U-Net [26] with five $conv_{3\times 3}$ \- $bn$ \- $relu$ \- $avg-pool_{2\times 2}$ blocks in the encoders and five $upsample_{2\times 2}$ \- $conv_{3\times 3}$ \- $bn$ \- $relu$ blocks in the decoders. In the generator network, we use the Johnson architecture [19] with two down-sampling blocks, six residual-blocks and two up-sampling blocks. We train our network using Adam [20] optimizer with learning rate $2e-4$ and batch size 20. We employ learning decay by dropping the learning rate at $\frac{T}{2}$ and $\frac{3T}{4}$ iterations, where T is total number of iteration. We chose $T\approx 100k$ for _Tai-Chi-HD_ and _VoxCeleb_ , and $T\approx 40k$ for _Nemo_ and _Bair_. The model converges in approximately 2 days using 2 TitanX gpus for _Tai-Chi-HD_ and _VoxCeleb_. ### B.2 Equivariance loss implementation As explained above our equivariance losses force the keypoint detector to be equivariant to some transformations $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}$. In our experiments $\mathcal{T}_{\mathbf{X}\leftarrow\mathbf{Y}}$ is implemented using randomly sampled thin plate splines. We sample spline parameters from normal distributions with zero mean and variance equal to 0.005 for deformation component and 0.05 for the affine component. For deformation component we use uniform $5\times 5$ grid. ## C Additional experiments ### C.1 Image Animation In this section, we report additional qualitative results. We compare our approach with X2face [41] and Monkey-Net [29]. In Fig. 5, we show three animation examples from the _VoxCeleb_ dataset. First, X2face is not capable of generating realistic video sequences as we can see, for instance in the last frame of the last sequence. Then, Monkey-Net generates realistic frames but fails to generate specific facial expressions as in the third frame of the first sequence or in transferring the eye movements as in the last two frames of the second sequence. In Fig. 6, we show three animation examples from the _Nemo_ dataset. First, we observe that this dataset is simpler than _VoxCeleb_ since the persons are facing a uniformly black background. With this simpler dataset, X2Face generates realistic videos. However, it is not capable of inpainting image parts that are not visible in the source image. For instance, X2Face does not generate the teeth. Our approach also perform better than Monkey-Net as we can see by comparing the generate teeth in the first sequence or the closed eyes in the fourth frames of the second and third sequences. In Fig. 6, we report additional examples for the _Tai-Chi-HD_ dataset. These examples are well in line with what is reported in the main paper. Both X2Face and Monkey-Net completely fail to generate realistic videos. The source images are warped without respecting human body structure. Conversely, our approach is able to deform the person in foreground without affecting the background. Even though we can see few minor artifacts, our model is able to move each body part independently following the body motion in the driving video. Finally, in Fig. 8 we show three image animation examples on the _Bair_ dataset. Again, we see that X2Face is not able to transfer motion since it constantly returns frames almost identical with the source images. Compared to Monkey-Net, our approach performs slightly better since it preserves better the robot arm as we can see in the second frame of the first sequence or in the fourth frame of the last sequence. ### C.2 Keypoint detection We now illustrate the keypoints that are learned by our self-supervised approach in Fig. 9. On the _Tai-Chi-HD_ dataset, the keypoints are semantically consistent since each of them corresponds to a body part: light green for the right foot, and blue and red for the face for instance. Note that, a light green keypoint is constantly located in the bottom left corner in order to model background or camera motion. On _VoxCeleb_ , we observe that, overall, the obtained keypoints are semantically consistent except for the yellow and green keypoints. For instance, the red and purple keypoints constantly correspond to the nose and the chin respectively. We observe a similar consistency for the _Nemo_ dataset. For the Bair dataset, we note that two keypoints (dark blue and light green) correspond to the robotic arm. ### C.3 Visualizing occlusion masks In Fig. 10, we visualize the predicted occlusion masks $\hat{\mathcal{O}}_{\mathbf{S}\leftarrow\mathbf{D}}$ on the _Tai-Chi-HD_ , _VoxCeleb_ and _Nemo_ datasets. In the first sequence, when the person in the driving video is moving backward (second to fourth frames), the occlusion mask becomes black (corresponding to 0) in the background regions that are occluded in the source frame. It indicates that these parts cannot be generated by warping the source image features and must be inpainted. A similar observation can be made on the example sequence of _VoxCeleb_. Indeed, we see that when the face is rotating, the mask has low values (dark grey) in the neck region and in the right face side (in the left-hand side of the image) that are not visible in the source Frame. Then, since the driving video example from _Nemo_ contains only little motion, the predicted mask is almost completely white. Overall, these three examples show that the occlusion masks truly indicate occluded regions even if no specific training loss is employed in order to lead to this behaviour. Finally, the predicted occlusion masks are more difficult to interpret in the case of the _Bair_ dataset. Indeed, the robotic arm is masked out in every frame whereas we could expect that the model generates it by warping. A possible explanation is that, since in this particular dataset, the moving object is always the same, the network can generate without warping the source image. We observe also that masks have low values for the regions corresponding to the arm shadow. It is explained by the fact that shadows cannot be obtained by image warping and that they need to be added by the generator. | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | Figure 5: Qualitative comparison with state of the art for the task of image animation on different sequences from the _VoxCeleb_ dataset. | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | Figure 6: Qualitative comparison with state of the art for the task of image animation on different sequences from the _Nemo_ dataset. | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | Figure 7: Qualitative comparison with state of the art for the task of image animation on different sequences from the _Tai-Chi-HD_ dataset. | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | | | | | | | ---|---|---|---|---|---|--- X2face [41] | | | | | | Monkey-Net [29] | | | | | | Ours | | | | | | Figure 8: Qualitative comparison with state of the art for the task of image animation on different sequences from the _Bair_ dataset. | | | | | | ---|---|---|---|---|---|--- _Tai-Chi-HD_ | | | | | | | | | | | | | | | | | | _VoxCeleb_ | | | | | | | | | | | | | | | | | | _Nemo_ | | | | | | | | | | | | | | | | | | _Bair_ | | | | | | | | | | | | Figure 9: Keypoint visualization for the four datasets. | | | | | | ---|---|---|---|---|---|--- | | | | | | Occlusion | | | | | | Output | | | | | | | | | | | | ---|---|---|---|---|---|--- Occlusion | | | | | | Output | | | | | | | | | | | | ---|---|---|---|---|---|--- Occlusion | | | | | | Output | | | | | | | | | | | | ---|---|---|---|---|---|--- Occlusion | | | | | | Output | | | | | | Figure 10: Visualization of occlusion masks and images obtained after deformation on _Tai-Chi-HD_ , _VoxCeleb_ , _Nemo_ and _Bair_ datasets.

2024-09-04T02:54:55.836853

2003.00244

arxiv-papers

Braiding quantum gates from partition algebras ††⋆ On leave of absence from the Institute of Physics at the University of São Paulo, São Paulo, Brazil. aCenter for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon, South Korea bCenter for Theoretical Physics of the Universe, Institute for Basic Science, Daejeon, South Korea cDipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, via Campi 213/A, 41125 Modena, Italy & INFN Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy pramod23phys, fusugino<EMAIL_ADDRESS> ###### Contents 1. 1 Introduction 2. 2 Set partitions and partition algebras 1. 2.1 Representations 3. 3 Equivalence classes of $R$-matrices and SLOCC classes 4. 4 Generalized $R$-matrices 1. 4.1 2-qubits 2. 4.2 3-qubits 3. 4.3 4-qubits 4. 4.4 4-qubits from 2-qubits via Temperley-Lieb generators 5. 4.5 Algorithm for multi-qubit generalized $R$-matrices 5. 5 Comparison with known generalized $R$-matrices 6. 6 Outlook ## 1 Introduction The fragile nature of quantum entanglement is a central issue in quantum computing, which can in principle be alleviated by the use of topology. Drawing inspiration from the Aravind hypothesis [1, 2, 3], it has been proposed that braiding operators – operators that obey braiding relations and create knots from unlinked strands – could be thought of as quantum entanglers, i.e. gates that create entanglement from product states [4, 5, 6, 7, 8]. These initial studies about the relation between entangling gates and knots were then pushed forward in [9, 10, 11, 12, 13], paving the way to the proposal of topological quantum circuits with gates given by braiding operators [14, 15]. It is expected that a physical realization of these braiding operators/entangling gates could be obtained using anyons. One way to get braiding operators is by solving the parameter-independent Yang-Baxter equation (YBE), which we briefly review.111For details about the case in which the YBE depends on a so-called spectral parameter, see e.g. [16] and references therein. The YBE is an operator equation for an invertible matrix, $R:V\otimes V\rightarrow V\otimes V$, given by $\left(R\otimes I\right)\left(I\otimes R\right)\left(R\otimes I\right)=\left(I\otimes R\right)\left(R\otimes I\right)\left(I\otimes R\right),$ (1.1) where $V$ is a $d$-dimensional complex vector space and $I$ is the identity operator on $V$. We use the terms Yang-Baxter operator and R-matrix interchangeably for the operator $R$. Solutions to (1.1) for some cases are presented in [17, 18, 19]. The $R$-matrix can be seen as a generalization of the permutation operator that swaps two vector spaces. This point of view is useful if one notices that the $R$-matrices can be used to construct representations of the Artin braid group $B_{n}$ on $n$-strands, with generators $\sigma_{i}$ satisfying $\displaystyle\sigma_{i}\sigma_{i+1}\sigma_{i}$ $\displaystyle=$ $\displaystyle\sigma_{i+1}\sigma_{i}\sigma_{i+1},$ (1.2) $\displaystyle\sigma_{i}\sigma_{j}$ $\displaystyle=$ $\displaystyle\sigma_{j}\sigma_{i},~{}~{}|i-j|>1,$ (1.3) for $i=1,\ldots,n-1$. The first relation above is called the braid relation, whereas the second relation is the far-commutativity condition. Representations for $\sigma_{i}$ can be constructed out of the $R$-matrices that solve (1.1) as follows $\rho(\sigma_{i})=I^{\otimes i-1}\otimes R_{i,i+1}\otimes I^{\otimes n-i-1}.$ (1.4) Notice that this representation satisfies far-commutativity trivially. This implies that every $R$-matrix that solves (1.1) can be used to construct a representation of the braid group, denoted a braiding operator. The distinction between $R$-matrices and braiding operators become essential when introducing a natural generalization of the YBE [10, 11] which involves two extra parameters, $m$ and $l$. The linear invertible operator $R:V^{\otimes m}\rightarrow V^{\otimes m}$ now acts on $m$ copies of the $d$-dimensional vector space $V$ with $l$ identity operators, and obeys $\left(R\otimes I^{\otimes l}\right)\left(I^{\otimes l}\otimes R\right)\left(R\otimes I^{\otimes l}\right)=\left(I^{\otimes l}\otimes R\right)\left(R\otimes I^{\otimes l}\right)\left(I^{\otimes l}\otimes R\right),$ (1.5) prompting the notation $(d,m,l)$-gYBE, as used in [21]. We dub this generalized $R$-operator as either the generalized Yang-Baxter operator or the generalized $R$-matrix. This generalization is important for quantum information processes that involve more than two qubits. Unlike the $R$-matrix that solves the YBE in (1.1), not all generalized $R$-matrices that solve the $(d,m,l)$-gYBE in (1.5) provide a representation of the braid group, as they do not always satisfy the far-commutativity condition in (1.3). However, for the cases when $2l\geq m$ (assuming $l<m$) far-commutativity is trivially satisfied, just as in the case of the Yang- Baxter operators. This is seen through the representations of the braid group given in terms of the generalized $R$-matrices by $\rho(\sigma_{i})=\left(I^{\otimes l}\right)^{\otimes i-1}\otimes R_{i,\cdots,i+m-1}\otimes\left(I^{\otimes l}\right)^{\otimes n-i-m+1}.$ (1.6) We will then be interested in finding the generalized Yang-Baxter operators that satisfies the $(d,m,l)$-gYBE when $2l\geq m$, thus automatically leading to representations of the braid group.222A different approach to representations to the braid group is discussed in [22]. The $(d,m,l)$-gYBE in (1.5) involves $d^{2m+2l}$ cubic polynomial relations for $d^{2m}$ unknowns (the entries of the generalized $R$-matrix) and is in general hard to solve. In this work we use algebraic methods to solve for the $R$-matrices and generalized $R$-matrices using partition algebras [23, 24, 25, 26, 27, 28]. We obtain families of both unitary and non-unitary operators, with the former finding use as quantum gates in quantum computing and the latter being useful to investigate topological aspects of the gYBE that include the study of knot invariants. We focus on the quantum entangling aspects of these operators. The paper is structured as follows. Set partitions and partition algebras are reviewed in Sec. 2, along with representations (the Qubit and the Temperley- Lieb representations) of their modified versions. In Sec. 3 we recall the equivalence classes of the Yang-Baxter operators and discuss how they relate to the notion of Stochastic Local Operations and Classical Communication (SLOCC) classes of entangled quantum states in quantum information theory. Our main results are in Sec. 4, where we obtain and discuss in detail $R$-matrices for the 2-, 3-, and 4-qubit cases. We also study the SLOCC classes of entangled states generated by these matrices. The structure of these generalized $R$-matrices allows to find an algorithm to systematically generate solutions of the $(d,m,l)$-gYBE. There are three kinds of generalized $R$-matrices known in the 3-qubit case [10, 21, 12]. In Sec. 5 we show that the 3-qubit solutions obtained in this paper are inequivalent to the known solutions. We conclude with some open questions and an outlook in Sec. 6. ## 2 Set partitions and partition algebras We review the notion of set partitions and partition algebras following [29]. We present just the bare minimum needed in this work, pointing the reader to that reference for more details. The elements of set partition, denoted $A_{k}$, are the partitions of two copies of a set: $\\{1,2,\cdots,k\\}$ and $\\{1^{\prime},2^{\prime},\cdots,k^{\prime}\\}$. As an example consider the following diagram showing the partition of a set with $k=7$, this represents the set partition $\left\\{\left\\{1,3,5,4^{\prime},5^{\prime}\right\\},\left\\{2,3^{\prime}\right\\},\left\\{4,6,7,6^{\prime}\right\\},\left\\{1^{\prime},2^{\prime}\right\\},\left\\{7^{\prime}\right\\}\right\\}.$ Figure 1: A diagram representing $\left\\{\left\\{1,3,5,4^{\prime},5^{\prime}\right\\},\left\\{2,3^{\prime}\right\\},\left\\{4,6,7,6^{\prime}\right\\},\left\\{1^{\prime},2^{\prime}\right\\},\left\\{7^{\prime}\right\\}\right\\}\in A_{7}$. In the diagram, vertices $i$ and $j$ are connected by a path if $i$ and $j$ belong to the same block of the partition. Note that the diagram for a given element of $A_{k}$ is not unique. For example the diagram in Fig. 2 also represents the same element represented in the diagram Fig. 1. Figure 2: Another diagram representing the same element of $A_{7}$ shown in Fig. 1. To compose two diagrams, $d_{1}\circ d_{2}$, place $d_{1}$ above $d_{2}$ and then trace the lines to obtain the new partition. An example of such a composition is given for the case of $k=6$ in Fig. 3. Figure 3: Composition of elements of $A_{6}$. The elements of $A_{k}$ are generated by successive compositions of $\displaystyle p_{i},\;\;\;\;\qquad\textrm{for}~{}i\in\\{1,\cdots,k\\},$ (2.1) $\displaystyle p_{i+\frac{1}{2}},\qquad\textrm{for}~{}i\in\\{1,\cdots,k-1\\},$ (2.2) $\displaystyle s_{i},\;\;\;\;\qquad\textrm{for}~{}i\in\\{1,\cdots,k-1\\},$ (2.3) whose action can be represented diagrammatically, see Fig. 4. Figure 4: Generators of $A_{k}$. An example of composition of these generators is shown in Fig. 5. Figure 5: Example of composition in $A_{k}$. The nodes $1,\cdots i-1,i+3,\cdots,k$ on which the generators act trivially are suppressed. Using these diagrams one can easily verify that the generators satisfy the following relations $p_{i}^{2}=p_{i},\qquad p^{2}_{i+\frac{1}{2}}=p_{i+\frac{1}{2}},$ (2.4) $p_{i}p_{i\pm\frac{1}{2}}p_{i}=p_{i},\qquad p_{i\pm\frac{1}{2}}p_{i}p_{i\pm\frac{1}{2}}=p_{i\pm\frac{1}{2}},$ (2.5) $p_{i}p_{j}=p_{j}p_{i},~{}\textrm{for}~{}|i-j|>\frac{1}{2},$ (2.6) $s_{i}^{2}=1,\qquad s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1},\qquad s_{i}s_{j}=s_{j}s_{i},~{}\textrm{for}~{}|i-j|>1.$ (2.7) Here and below, we simply write $d_{1}\circ d_{2}$ as $d_{1}d_{2}$ for notational simplicity. Note that $p_{i}$ and $p_{i+\frac{1}{2}}$ generate planar diagrams. Non- planarity is introduced by the permutation group generators $s_{i}$. The mixed relations are $s_{i}p_{i}p_{i+1}=p_{i}p_{i+1}s_{i}=p_{i}p_{i+1},\quad s_{i}p_{i}s_{i}=p_{i+1},\quad s_{i}p_{i+j}=p_{i+j}s_{i},~{}\textrm{for}~{}j\neq 0,1,$ (2.8) and $s_{i}p_{i+\frac{1}{2}}=p_{i+\frac{1}{2}}s_{i}=p_{i+\frac{1}{2}},\quad s_{i}s_{i+1}p_{i+\frac{1}{2}}s_{i+1}s_{i}=p_{i+\frac{3}{2}},\quad s_{i}p_{i+j}=p_{i+j}s_{i},~{}\textrm{for}~{}j\neq-\frac{1}{2},\frac{3}{2}.$ (2.9) To emphasize that $s_{i}$ swaps the elements on the vector spaces at sites $i$ and $i+1$, one could also write it as $s_{i,i+1}$, but we will stick to the notation $s_{i}$ to avoid cluttering. The second relations in (2.8)-(2.9) can be understood as the fundamental property of the permutation operator: $\displaystyle s_{i}p_{i}s_{i}=p_{i+1},\qquad s_{i+1}p_{i+\frac{1}{2}}s_{i+1}=s_{i}p_{i+\frac{3}{2}}s_{i},$ (2.10) The index swapping is obvious in the first relation and it becomes obvious also in the second one, when one notices that $p_{i+\frac{1}{2}}$ has non- trivial support on sites $i$ and $i+1$ whereas $p_{i+\frac{3}{2}}$ has non- trivial support on sites $i+1$ and $i+2$. To make this more transparent, one can change notation by identifying $p_{i+\frac{1}{2}}$ with $p_{i,i+1}$ and $p_{i+\frac{3}{2}}$ with $p_{i+1,i+2}$, prompting the definition $s_{i+1}p_{i,i+1}s_{i+1}=s_{i}p_{i+1,i+2}s_{i}\equiv p_{i,i+2},$ (2.11) whose diagrammatic representation is shown in Fig. 6 and can be worked out using the composition laws in Fig. 5. Figure 6: The element $p_{i,i+2}$. The figure suggests one can generalize the $p_{i+\frac{1}{2}}\equiv p_{i,i+1}$ operators to the cases with support on two arbitrary sites, $i$ and $i+j$, as $p_{i,i+j}=s_{i+j-1}s_{i+j-2}\cdots s_{i+1}p_{i,i+1}s_{i+1}\cdots s_{i+j-2}s_{i+j-1},$ (2.12) satisfying the relations $\displaystyle p_{i,i+j}^{2}$ $\displaystyle=$ $\displaystyle p_{i,i+j},$ (2.13) $\displaystyle p_{i,i+j_{1}}p_{i,i+j_{2}}$ $\displaystyle=$ $\displaystyle p_{i,i+j_{1}}p_{i+j_{1},i+j_{2}},~{}~{}j_{1}<j_{2},$ (2.14) $\displaystyle p_{i+l,i+j}p_{i,i+j}$ $\displaystyle=$ $\displaystyle p_{i,i+j}p_{i+l,i+j}=p_{i,i+l}p_{i+l,i+j},~{}~{}l<j,$ (2.15) which can be verified diagrammatically. Henceforth, we shall use $p_{i,i+1}$ instead of $p_{i+\frac{1}{2}}$. Linear combinations of elements of $A_{k}$ with coefficients being complex numbers form the partition algebra $\mathbb{C}A_{k}(1)$. ### 2.1 Representations We use a slightly modified form of the relations in (2.4) where we either scale one or both of the relations by a factor, $d$. We denote them asymmetric or symmetric scaling respectively. To this end we employ hermitian representations for the generators, which come in two kinds: the generators of the planar diagrams can be rescaled either asymmetrically (Qubit representation) or symmetrically (Temperley-Lieb representation). Strictly speaking, these representations do not give the representations of the relations (2.4)-(2.9), but the representations of their deformed versions. ##### Qubit representation In this representation one of the relations satisfied by $p_{i,i+1}$ is modified to $p_{i,i+1}^{2}=d~{}p_{i,i+1},$ (2.16) with the other relations in (2.4), (2.5) and (2.6) unchanged. Here $d$ is the dimension of the local Hilbert space on the site $i$ acted upon by the generators of $A_{k}$. The relations in the non-planar part of $A_{k}$ involving $p_{i,i+1}$ and $s_{i}$, see (2.9), and the relations in (2.14) and (2.15) are unchanged. The relation in (2.13) is modified to $p_{i,i+j}^{2}=d~{}p_{i,i+j},$ (2.17) corresponding to the scaling of $p_{i,i+1}$. In this paper we deal with qubits and hence the case of $d=2$. The qudit realizations can be similarly obtained through an appropriate generalization. Qubit representations are given by $p_{i}=\frac{1+Z_{i}}{2},\qquad p_{i,j}=1+X_{i}X_{j},\qquad s_{i}=\frac{1+X_{i}X_{i+1}+Y_{i}Y_{i+1}+Z_{i}Z_{i+1}}{2},$ (2.18) where 1 is the identity operator acting on the relevant Hilbert space and $X_{i},Y_{i},Z_{i}$ are the usual Pauli matrices acting on the qubit space on site $i$, $X=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right),\quad Y=\left(\begin{array}[]{cc}0&-\mathrm{i}\\\ \mathrm{i}&0\end{array}\right),\quad Z=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),$ (2.19) written in the basis $\\{{\left|{0}\right>}$, ${\left|{1}\right>}\\}$ where $Z$ is diagonal. Another representation which is unitarily equivalent to the above is given by $p_{i}=\frac{1+X_{i}}{2},\qquad p_{i,j}=1+Z_{i}Z_{j},\qquad s_{i}=\frac{1+X_{i}X_{i+1}+Y_{i}Y_{i+1}+Z_{i}Z_{i+1}}{2}.$ (2.20) The qubit representation gives the representation of the relations (2.4)-(2.9) with the normalization of $p_{i,i+1}\equiv p_{i+\frac{1}{2}}$ changed to (2.16) with $d=2$. ##### Temperley-Lieb representation Now both generators of the planar diagrams are rescaled by the same factor, $p_{i,i+1}^{2}=(Q+Q^{-1})p_{i,i+1},\qquad p_{i}^{2}=(Q+Q^{-1})p_{i},\qquad Q\in\mathbb{R}-\\{0\\}$ (2.21) with the rest of the relations of the planar part of the partition algebra unchanged. The planar part of $A_{k}$ can be realized using the generators of the Temperley-Lieb algebra with doubled dimensions, $e_{1},\cdots,e_{2k-1}\in{\rm TL}_{2k}$, $p_{i}=e_{2i-1},\qquad p_{i,i+1}=e_{2i},$ (2.22) which satisfy the relations in (2.5)-(2.6), see [29]. Notice the doubling of the number of sites in this realization, as shown in Fig. 7. Figure 7: Temperley-Lieb representation of $p_{i}$ and $p_{i+1}$. The white dots, obtained by doubling the original sites for $A_{k}$ (the black dots), are sites on which the Temperley-Lieb generators ($e_{2i-1},e_{2i+1}$) act. In this representation the introduction of non-planarity through the permutation generators $s_{i}$ affects some of the mixed relations (2.8) and (2.9). Let us consider the case that $s_{i}$ is realized as an appropriate permutation operator given by $s_{i}=s_{2i-1,2i+1}~{}s_{2i,2i+2},$ (2.23) or by the unitarily equivalent $s_{i}=s_{2i-1,2i+2}~{}s_{2i,2i+1},$ (2.24) with $s_{i,j}$ being the operator that swaps the indices $i$ and $j$. This realization lives on the doubled lattice as shown in Fig. 8. Figure 8: Temperley-Lieb representation of $s_{i}$ in (2.23). Using this diagram the partition algebra in (2.8) can be easily verified, whereas (2.9) does not hold. Thus the Temperley-Lieb representation gives the representation of the relations (2.21) and (2.5)-(2.8). The doubling of the sites in this representation implies that one shall obtain $R$-matrices and generalized $R$-matrices on twice the number of sites, i.e. if one obtains the generalized $R$-matrix that solves the $(d,m,l)$-gYBE, then this representation will yield another solution that solves the $(d,2m,2l)$-gYBE. In section 4, generalized $R$-matrices are constructed as linear combinations of the above representations of deformed set partitions that are analogous to elements of the partition algebras. ## 3 Equivalence classes of $R$-matrices and SLOCC classes In quantum information theory the idea of SLOCC was introduced to classify entangled states. It states that two quantum states are equivalent when there exists an Invertible Local Operator (ILO) that maps one state into the other: ${\left|{\psi_{1}}\right>}=\left(A_{1}\otimes\cdots\otimes A_{n}\right){\left|{\psi_{2}}\right>},$ (3.1) where the states ${\left|{\psi_{1}}\right>}$ and ${\left|{\psi_{2}}\right>}$ live in the Hilbert space $\otimes_{i=1}^{n}~{}\mathcal{H}_{i}$ [31]. $A_{i}$ is an ILO acting only at the site $i$. This equivalence relation appeals to the intuition of entangled states, as one expects local operations to not disturb the non-local entanglement property of the state. One can also define an equivalence class of the $R$-matrices satisfying the parameter-independent YBE. To identify this class, one observes that if $R$ is a solution to the $(d,m,l)$-gYBE, then so are $\alpha R$ (with $\alpha$ a constant), $R^{-1}$, and $\left(A_{1}\otimes\cdots\otimes A_{m}\right)R\left(A_{1}^{-1}\otimes\cdots\otimes A^{-1}_{m}\right)$, where $A_{1},\cdots,A_{m}$ is an ILO that also appears in the definition of the SLOCC classes of quantum states. We can now prove the following theorem: ##### Theorem Two entangling $R$-matrices $R_{1}$ and $R_{2}$, which are equivalent under ILO, produce entangled states of the same SLOCC class. ##### Proof $R_{1}$ produces an entangled state ${\left|{E}\right>}$ acting on the product state ${\left|{P}\right>}$, $R_{1}~{}{\left|{P}\right>}={\left|{E}\right>}.$ (3.2) By assumption, one can express $R_{1}=AR_{2}A^{-1}$, where $A$ is an ILO, so that $AR_{2}A^{-1}~{}{\left|{P}\right>}={\left|{E}\right>}.$ (3.3) This means $R_{2}A^{-1}~{}{\left|{P}\right>}=A^{-1}~{}{\left|{E}\right>}$, by definition, both $A^{-1}~{}{\left|{P}\right>}$ and $A^{-1}~{}{\left|{E}\right>}$ are in the same SLOCC classes as ${\left|{P}\right>}$ and ${\left|{E}\right>}$, respectively, hence proving the assertion. $\blacksquare$ This theorem naturally implies that if two $R$-matrices produce states of two different SLOCC classes, then they cannot be related by an ILO. However, the converse of the theorem is not always true: if two entangled states ${\left|{E_{1}}\right>}$ and ${\left|{E_{2}}\right>}$ belonging to the same SLOCC class are generated by two entangling $R$-matrices $R_{1}$ and $R_{2}$, respectively, then they need not be related by an ILO. One has in fact $\displaystyle R_{1}~{}{\left|{P_{1}}\right>}={\left|{E_{1}}\right>},\qquad R_{2}~{}{\left|{P_{2}}\right>}={\left|{E_{2}}\right>}.$ (3.4) As ${\left|{E_{2}}\right>}=A{\left|{E_{1}}\right>}$ and ${\left|{P_{2}}\right>}=L{\left|{P_{1}}\right>}$, where $A$ and $L$ are two ILOs, one obtains $A^{-1}R_{2}L~{}{\left|{P_{1}}\right>}={\left|{E_{1}}\right>}.$ Note that the ILOs $A$ and $L$ need not be the same. For unitary $R$-matrices, this relation holds on all the product states that span the Hilbert space, so that one can identify $R_{1}=A^{-1}R_{2}L.$ We shall use the definitions and this result to determine the classes of $R$-matrices. ##### 2-qubit SLOCC classes There are two SLOCC classes in the 2-qubit case, the Bell state class and the product state class. ##### 3-qubit SLOCC classes There are six SLOCC classes in the 3-qubit case [31]. Two tri-partite entangled classes, GHZ state class and W state class, also denoted as $ABC$ to symbolize the three parties of the state. Three partially entangled state classes, $A-BC,AC-B,AB-C$ and finally the product state class, $A-B-C$. ##### 4-qubit SLOCC classes In the 4-qubit case, it was discussed in [31] that there are infinitely many SLOCC classes. Later, it was shown in [32] that there are nine families in the sense of nine different ways of entangling 4-qubits. On the other hand, it was reported in [33, 34] that the number of the families is eight instead of nine for genuinely entangled states.333The classification in [32] contains no genuinely entangled state with canonical state ${\left|{0000}\right>}+{\left|{0111}\right>}$. Due to this difference, [33, 34] does not contradict [32]. Furthermore, it was discovered in [35] that the nine families in [32] further split into 49 SLOCC entanglement classes by looking at SLOCC invariant quantities. ## 4 Generalized $R$-matrices The generators of the permutation group $s_{i}$ solve the $(d,2,1)$-gYBE. In fact, the transposition operators $s_{i,i+l}$ solve the $(d,m,l)$-gYBE, assuming $l\leq m$, with non-trivial support on the sites $i$ and $i+l$. The ansatze with non-trivial support on all the $m$ sites used in this paper are modifications of these transposition operators, with generators given by the planar part of the partition algebra. In the language of quantum gates, these ansatze are generalized SWAP gates. In the following we discuss the 2-qubit and 3-qubit cases in detail before writing down the answers for the 4-qubit case and outlining an algorithm for an arbitrary multi-qubit generalized $R$-matrix. ### 4.1 2-qubits On two sites $i$ and $i+1$, there are various choices of the generators $p_{i}$, $p_{i+1}$, $p_{i,i+1}$ and $s_{i}$ to construct the $R$-matrices. We consider the different possibilities separately. ##### Using $s_{i}$, $p_{i}$ and $p_{i+1}$ Consider the following ansatz for the Yang-Baxter operator with support on sites $i$ and $i+1$, $R_{i}=s_{i}\left(1+\alpha~{}p_{i}+\beta~{}p_{i+1}+\gamma~{}p_{i}p_{i+1}\right),$ (4.1) with constants $\alpha,\beta,\gamma\in\mathbb{C}$. This operator satisfies the $(d,2,1)$-gYBE for all $\alpha,\beta,\gamma$, as seen by evaluating the two sides of the YBE $\displaystyle R_{i}R_{i+1}R_{i}$ $\displaystyle=$ $\displaystyle\left(1+\alpha~{}p_{i+1}+\beta~{}p_{i}+\gamma~{}p_{i}p_{i+1}\right)\left(1+\alpha~{}p_{i+2}+\beta~{}p_{i}+\gamma~{}p_{i}p_{i+2}\right)$ (4.3) $\displaystyle\times\left(1+\alpha~{}p_{i+2}+\beta~{}p_{i+1}+\gamma~{}p_{i+1}p_{i+2}\right)\left(s_{i}s_{i+1}s_{i}\right),$ where repeated use of the permutation operator given in (2.10) has been applied. In a similar manner one can compute the right hand side $R_{i+1}R_{i}R_{i+1}$, which turns out to be equal to (4.3). Using $p_{i}^{2}=p_{i}$, $p_{i}p_{i+1}=p_{i+1}p_{i}$ and $s_{i}^{2}=1$, we can show that the inverse is given by $R_{i}^{-1}=\left(1-\frac{\alpha}{1+\alpha}~{}p_{i}-\frac{\beta}{1+\beta}~{}p_{i+1}+\frac{\alpha\beta(2+\alpha+\beta+\gamma)-\gamma}{(1+\alpha)(1+\beta)(1+\alpha+\beta+\gamma)}~{}p_{i}p_{i+1}\right)s_{i}.$ (4.4) This expression is needed to check for which values of the parameters the $R$-matrix is unitary. It is also easy to check that the operators in (4.1) satisfy far-commutativity for braid operators, $\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}\;(|i-j|>1)$, by noting that $R_{i+j}=s_{i+j}\left(1+\alpha~{}p_{i+j}+\beta~{}p_{i+j+1}+\gamma~{}p_{i+j}p_{i+j+1}\right)$ (4.5) has trivial common support with the operator in (4.1) for all $j>1$. In general, these solutions are non-unitary and generate the infinite- dimensional braid group, i.e. the image of braid group representations built using these $R$-matrices is infinite. This is seen by computing the powers of $R_{i}$, $R_{i}^{n}=s_{i}^{n}\left(1+\alpha_{n}~{}p_{i}+\beta_{n}~{}p_{i+1}+\gamma_{n}~{}p_{i}p_{i+1}\right),$ (4.6) where the parameters are defined recursively as $\displaystyle\alpha_{n}$ $\displaystyle=$ $\displaystyle\alpha_{1}+\beta_{n-1}+\alpha_{1}\beta_{n-1},\qquad\beta_{n}=\alpha_{n-1}+\beta_{1}+\alpha_{n-1}\beta_{1},$ $\displaystyle\gamma_{n}$ $\displaystyle=$ $\displaystyle\alpha_{1}\alpha_{n-1}+\beta_{1}\beta_{n-1}+\gamma_{1}\gamma_{n-1}+\gamma_{1}\left(1+\alpha_{n-1}+\beta_{n-1}\right)+\gamma_{n-1}\left(1+\alpha_{1}+\beta_{1}\right),$ (4.7) after identifying $\alpha_{1}$, $\beta_{1}$ and $\gamma_{1}$ with $\alpha$, $\beta$ and $\gamma$ in (4.1). By equating (4.6) and $R_{i}^{\dagger}$, the conditions $\displaystyle\alpha^{*}=-\frac{\alpha}{1+\alpha},\qquad\beta^{*}=-\frac{\beta}{1+\beta},\qquad\gamma^{*}=\frac{\alpha\beta(2+\alpha+\beta+\gamma)-\gamma}{(1+\alpha)(1+\beta)(1+\alpha+\beta+\gamma)}.$ (4.8) give a family of unitary solutions that generate the infinite-dimensional braid group just as the non-unitary case. (4.8) are explicitly solved by $\alpha=e^{\mathrm{i}\theta}-1,\qquad\beta=e^{\mathrm{i}\varphi}-1,\qquad\gamma=e^{\mathrm{i}\phi}-e^{\mathrm{i}\theta}-e^{\mathrm{i}\varphi}+1$ (4.9) with $\theta$, $\varphi$ and $\phi$ angles between 0 and $2\pi$. The recursive definitions of the parameters of $R_{i}^{n}$ (4.7) show that $R_{i}^{n}\neq 1$ for any finite $n$ when $\theta$, $\varphi$ and $\phi$ are generic. This can also be seen in the eigenvalues $\\{1,\pm e^{\frac{\mathrm{i}}{2}(\theta+\varphi)},e^{\mathrm{i}\phi}\\}$ at the unitary solutions. There are eight real unitary solutions, four of which are shown in Table 1 in the qubit representations of (2.18). (The remaining four generate the same SLOCC classes as these four.) | $(\alpha,\beta,\gamma)$ | $R_{i}$ | Eigenvalues | $n|R^{n}=1$ ---|---|---|---|--- 1. | $(0,0,0)$ | $\tiny{\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\ 0&0&0&1\end{array}\right)}$ | $(-1_{(1)},1_{(3)})$ | 2 2. | $(0,0,-2)$ | $\tiny{\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\ 0&0&0&1\end{array}\right)}$ | $(-1_{(2)},1_{(2)})$ | 2 3. | $(0,-2,0)$ | $\tiny{\left(\begin{array}[]{cccc}-1&0&0&0\\\ 0&0&-1&0\\\ 0&1&0&0\\\ 0&0&0&1\end{array}\right)}$ | $(-1,\mathrm{i},-\mathrm{i},1)$ | 4 4. | $(0,-2,2)$ | $\tiny{\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&-1&0\\\ 0&1&0&0\\\ 0&0&0&1\end{array}\right)}$ | $(\mathrm{i},-\mathrm{i},1_{(2)})$ | 4 Table 1: Unitary solutions in the 2-qubit case using operators $s_{i}$, $p_{i}$ and $p_{i+1}$. The $k$ in $a_{(k)}$ denotes the multiplicity of the eigenvalue $a$. In the last column, $n|R^{n}=1$ means the lowest positive integer $n$ satisfying $R^{n}=1$. In the qubit representation, the $(2,2,1)$-Yang-Baxter operator takes the explicit form $R_{i}=\left(\begin{array}[]{cccc}1+\alpha+\beta+\gamma&0&0&0\\\ 0&0&1+\beta&0\\\ 0&1+\alpha&0&0\\\ 0&0&0&1\end{array}\right).$ (4.10) The unitary operators can act as quantum gates, however not all may lead to a universal set of gates. According to a theorem by Brylinski [30] for a 2-qubit space, a gate helps building a universal set if and only if it is entangling. We can use this criterion to check which of the operators in Table 1 are entangling and can potentially lead to a universal set. A quantum gate is entangling if there is a vector ${\left|{v_{1}}\right>}\otimes{\left|{v_{2}}\right>}\in\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ that gets mapped to an entangled state by the quantum gate. With this definition the gates corresponding to $(0,0,-2)$ and $(0,-2,2)$ are entangling. This assertion can be checked by seeing that these gates map the most general product state in $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$, given by $a_{1}a_{2}{\left|{00}\right>}+a_{1}b_{2}{\left|{01}\right>}+b_{1}a_{2}{\left|{10}\right>}+b_{1}b_{2}{\left|{11}\right>}$, to an entangled state. For example, the operator corresponding to $(0,0,-2)$ maps this product state to $-a_{1}a_{2}{\left|{00}\right>}+a_{1}b_{2}{\left|{01}\right>}+b_{1}a_{2}{\left|{10}\right>}+b_{1}b_{2}{\left|{11}\right>}$, which is entangled. ##### Using $s_{i}$ and $p_{i,i+1}$ The operator $R_{i}=s_{i}\left(1+\alpha~{}p_{i,i+1}\right)$ (4.11) satisfies the $(d,2,1)$-gYBE for all values of $\alpha\in\mathbb{C}$, as can be checked by computing $R_{i}R_{i+1}R_{i}$ and $R_{i+1}R_{i}R_{i+1}$. The inverse, when $d=2$, is given by $R_{i}^{-1}=s_{i}\left(1-\frac{\alpha}{1+2\alpha}~{}p_{i,i+1}\right).$ (4.12) The image of the braid group representation built using the $(d,2,1)$-$R$-matrix in (4.11) is infinite, as seen through its powers: $R_{i}^{n}=s_{i}^{n}\left(1+\alpha_{n}~{}p_{i,i+1}\right),$ (4.13) with the parameters, when $d=2$, defined recursively as $\alpha_{n}=\alpha_{1}+\alpha_{n-1}+2\alpha_{1}\alpha_{n-1},$ after identifying $\alpha_{1}$ with $\alpha$ in (4.11). This is unitary for $\alpha^{*}=-\frac{\alpha}{1+2\alpha}$, i.e. $\alpha=\frac{1}{2}(e^{\mathrm{i}\theta}-1)$ for arbitrary angle $\theta$, which gives real solutions $\alpha=0,-1$. From the above recursion formula, unitary solutions with generic $\theta$ generate the infinite-dimensional braid group ($R_{i}^{n}\neq 1$ for any finite $n$). For the representation of the partition algebra generators in (2.18) one obtains $R_{i}=\left(\begin{array}[]{cccc}1+\alpha&0&0&\alpha\\\ 0&\alpha&1+\alpha&0\\\ 0&1+\alpha&\alpha&0\\\ \alpha&0&0&1+\alpha\end{array}\right).$ (4.14) The case $\alpha=0$ is just the permutation operator $s_{i}$, which is not an entangler by previous considerations. That leaves us with $\alpha=-1$, which is not an entangler either. ##### Comparison of the unitary solutions to known cases in [19, 20] There are five families of unitary 2-qubit solutions to the Yang-Baxter equation as found in [19] and analyzed in [20]. The solutions in (4.1) is mapped to one of the five families whose representative is given by $\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&\psi_{1}&0\\\ 0&\psi_{2}&0&0\\\ 0&0&0&\psi_{3}\end{array}\right)\qquad\mbox{with}\qquad|\psi_{1}|=|\psi_{2}|=|\psi_{3}|=1.$ (4.15) Actually, the solution of the form (4.10) with (4.9) becomes $\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&e^{\mathrm{i}(\varphi-\phi)}&0\\\ 0&e^{\mathrm{i}(\theta-\phi)}&0&0\\\ 0&0&0&e^{-\mathrm{i}\phi}\end{array}\right)$ after scaling with $e^{-\mathrm{i}\phi}$, which implies that the solution belongs to the family (4.15) with $\psi_{1}=e^{\mathrm{i}(\varphi-\phi)}$, $\psi_{2}=e^{\mathrm{i}(\theta-\phi)}$ and $\psi_{3}=e^{-\mathrm{i}\phi}$. In particular, the four real unitary solutions listed in Table 1 belong to the same family with $\psi_{1}=\psi_{2}=\psi_{3}=1$, $\psi_{1}=\psi_{2}=\psi_{3}=-1$ (after scaling with $-1$), $\psi_{1}=1,\psi_{2}=\psi_{3}=-1$ (after scaling with $-1$) and $\psi_{1}=-1,\psi_{2}=\psi_{3}=1$, respectively. In addition, the unitary solution in (4.11), more explicitly (4.14) with $\alpha=\frac{1}{2}(e^{\mathrm{i}\theta}-1)$, can also be mapped to (4.15) up to the overall phase factor $e^{\mathrm{i}\theta}$, where $\psi_{1}=\psi_{2}=e^{-\mathrm{i}\theta}$, $\psi_{3}=1$, and the mapping is done by the ILO $Q\otimes Q$ with $Q=\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}$. Thus, all the unitary 2-qubit solutions we obtained belong to the single family (4.15) among the five described in [19, 20]. ### 4.2 3-qubits The number of possible operators on three sites $i$, $i+1$ and $i+2$ are $p_{i}$, $p_{i+1}$, $p_{i+2}$, $p_{i,i+1}$, $p_{i+1,i+2}$ and $p_{i,i+2}$, along with the permutation generators $s_{i}$ and $s_{i+1}$. In order to obtain valid representations of the braid group we obtain solutions to the $(d,3,2)$-gYBE. ##### Using $s_{i}$, $p_{i}$, $p_{i+1}$ and $p_{i+2}$ As ansatz we propose the natural generalization of (4.1) from the 2-qubit case: $R_{i}=s_{i,i+2}\left(1+\alpha_{1}~{}p_{i}+\alpha_{2}~{}p_{i+1}+\alpha_{3}~{}p_{i+2}+\beta_{1}~{}p_{i}p_{i+1}+\beta_{2}~{}p_{i+1}p_{i+2}+\beta_{3}~{}p_{i}p_{i+2}+\gamma~{}p_{i}p_{i+1}p_{i+2}\right),$ (4.16) where $s_{i,i+2}=s_{i}s_{i+1}s_{i}$ and the parameters are complex. This operator does not satisfy the $(d,3,2)$-gYBE for all values of the parameters. One can however use the identities in (2.8) to check that $R_{i}R_{i+2}R_{i}=R_{i+2}R_{i}R_{i+2}$ when $\alpha_{2}=0,~{}~{}\beta_{2}=-\frac{\beta_{1}\left(1+\alpha_{3}\right)}{1+\alpha_{1}+\beta_{1}},~{}~{}\gamma=\frac{\beta_{1}\left(\alpha_{3}-\alpha_{1}-\beta_{1}\right)}{1+\alpha_{1}+\beta_{1}}.$ (4.17) The inverse is given by $R_{i}^{-1}=\left(1+\alpha_{1}^{\prime}~{}p_{i}+\alpha_{3}^{\prime}~{}p_{i+2}+\beta_{1}^{\prime}~{}p_{i}p_{i+1}+\beta_{2}^{\prime}~{}p_{i+1}p_{i+2}+\beta_{3}^{\prime}~{}p_{i}p_{i+2}+\gamma^{\prime}~{}p_{i}p_{i+1}p_{i+2}\right)s_{i,i+2},$ (4.18) where $\displaystyle\alpha_{1}^{\prime}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{1}}{1+\alpha_{1}},~{}~{}\alpha_{3}^{\prime}=-\frac{\alpha_{3}}{1+\alpha_{3}},$ $\displaystyle\beta_{1}^{\prime}$ $\displaystyle=$ $\displaystyle-\frac{\beta_{1}}{(1+\alpha_{1})(1+\alpha_{1}+\beta_{1})},~{}~{}\beta_{2}^{\prime}=-\frac{\beta_{2}}{(1+\alpha_{3})(1+\alpha_{3}+\beta_{2})},$ $\displaystyle\beta_{3}^{\prime}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{1}\alpha_{3}(2+\alpha_{1}+\alpha_{3})-\beta_{3}(1-\alpha_{1}\alpha_{3})}{(1+\alpha_{1})(1+\alpha_{3})(1+\alpha_{1}+\alpha_{3}+\beta_{3})},$ $\displaystyle\gamma^{\prime}$ $\displaystyle=$ $\displaystyle-\frac{\beta_{1}(\alpha_{1}-\alpha_{3}+\beta_{1})}{(1+\alpha_{1})(1+\alpha_{3})(1+\alpha_{1}+\beta_{1})}.$ (4.19) The image of the braid group representation constructed out of (4.16) with parameters satisfying (4.17) is infinite, as seen by computing the powers of the generalized $R$-matrix $R_{i}^{n}=s_{i,i+2}^{n}\left(1+\alpha_{1}^{(n)}~{}p_{i}+\alpha_{3}^{(n)}~{}p_{i+2}+\beta_{1}^{(n)}~{}p_{i}p_{i+1}+\beta_{2}^{(n)}~{}p_{i+1}p_{i+2}+\beta_{3}^{(n)}~{}p_{i}p_{i+2}+\gamma^{(n)}~{}p_{i}p_{i+1}p_{i+2}\right),$ (4.20) with the parameters defined recursively as $\displaystyle\alpha_{1}^{(n)}=\alpha_{1}^{(1)}+\alpha_{3}^{(n-1)}+\alpha_{1}^{(1)}\alpha_{3}^{(n-1)},\qquad\alpha_{3}^{(n)}=\alpha_{1}^{(n-1)}+\alpha_{3}^{(1)}+\alpha_{1}^{(n-1)}\alpha_{3}^{(1)},$ (4.21) and $\displaystyle\beta_{1}^{(n)}$ $\displaystyle=$ $\displaystyle\beta_{1}^{(1)}+\beta_{2}^{(n-1)}+\beta_{1}^{(1)}\beta_{2}^{(n-1)}+\alpha_{1}^{(1)}\beta_{2}^{(n-1)}+\alpha_{3}^{(n-1)}\beta_{1}^{(1)},$ $\displaystyle\beta_{2}^{(n)}$ $\displaystyle=$ $\displaystyle\beta_{1}^{(n-1)}+\beta_{2}^{(1)}+\beta_{1}^{(n-1)}\beta_{2}^{(1)}+\alpha_{3}^{(1)}\beta_{1}^{(n-1)}+\alpha_{1}^{(n-1)}\beta_{2}^{(1)},$ $\displaystyle\beta_{3}^{(n)}$ $\displaystyle=$ $\displaystyle\beta_{3}^{(1)}+\beta_{3}^{(n-1)}+\beta_{3}^{(1)}\beta_{3}^{(n-1)}+\beta_{3}^{(1)}\left(\alpha_{1}^{(n-1)}+\alpha_{3}^{(n-1)}\right)+\beta_{3}^{(n-1)}\left(\alpha_{1}^{(1)}+\alpha_{3}^{(1)}\right),$ $\displaystyle\gamma^{(n)}$ $\displaystyle=$ $\displaystyle\gamma^{(1)}+\gamma^{(n-1)}+\gamma^{(1)}\gamma^{(n-1)}+\gamma^{(1)}\left(\alpha_{1}^{(n-1)}+\alpha_{3}^{(n-1)}+\beta_{1}^{(n-1)}+\beta_{2}^{(n-1)}+\beta_{3}^{(n-1)}\right)$ (4.22) $\displaystyle+\gamma^{(n-1)}\left(\alpha_{1}^{(1)}+\alpha_{3}^{(1)}+\beta_{1}^{(1)}+\beta_{2}^{(1)}+\beta_{3}^{(1)}\right)$ $\displaystyle+\beta_{1}^{(n-1)}\left(\beta_{1}^{(1)}+\beta_{3}^{(1)}\right)+\beta_{2}^{(n-1)}\left(\beta_{2}^{(1)}+\beta_{3}^{(1)}\right)+\beta_{3}^{(n-1)}\left(\beta_{1}^{(1)}+\beta_{2}^{(1)}\right)$ $\displaystyle+\alpha_{1}^{(n-1)}\beta_{1}^{(1)}+\alpha_{3}^{(n-1)}\beta_{2}^{(1)}+\alpha_{1}^{(1)}\beta_{1}^{(n-1)}+\alpha_{3}^{(1)}\beta_{2}^{(n-1)},$ after identifying $\alpha_{1}^{(1)}$, $\alpha_{3}^{(1)}$, $\beta_{1}^{(1)}$, $\beta_{2}^{(1)}$, $\beta_{3}^{(1)}$ and $\gamma^{(1)}$ with $\alpha_{1}$, $\alpha_{3}$, $\beta_{1}$, $\beta_{2}$, $\beta_{3}$ and $\gamma$ in (4.16). Unitary solutions occur when $\alpha_{1}^{\prime}=\alpha_{1}^{*}$, $\alpha_{3}^{\prime}=\alpha_{3}^{*}$, $\beta_{1}^{\prime}=\beta_{1}^{*}$, $\beta_{2}^{\prime}=\beta_{2}^{*}$, $\beta_{3}^{\prime}=\beta_{3}^{*}$, and $\gamma^{\prime}=\gamma^{*}$ with $\alpha_{1}^{\prime},\cdots,\gamma^{\prime}$ given by (4.19). Their explicit form is given by $\alpha_{1}=e^{\mathrm{i}\theta_{1}}-1,\qquad\alpha_{3}=e^{\mathrm{i}\theta_{3}}-1,\qquad\beta_{1}=e^{\mathrm{i}\varphi_{1}}-e^{\mathrm{i}\theta_{1}},\qquad\beta_{3}=e^{\mathrm{i}\varphi_{3}}-e^{\mathrm{i}\theta_{1}}-e^{\mathrm{i}\theta_{3}}+1,$ (4.23) where $\theta_{1}$, $\theta_{3}$, $\varphi_{1}$ and $\varphi_{3}$ are arbitrary angles, and $\beta_{2}$ and $\gamma$ are obtained from (4.17). These operators with generic angles generate the infinite-dimensional braid group just as the non-unitary operators. This is further seen from the eigenvalues at these unitary solutions given by $\\{e^{-\mathrm{i}\varphi}_{(2)},\pm e^{-\frac{\mathrm{i}}{2}\left(\theta_{1}+\theta_{3}\right)}_{(2)},1_{(2)}\\}$, with the $n$ in $a_{(n)}$ denoting the multiplicity of the eigenvalue $a$. There are 16 real unitary points for the parameters in (4.17), of which we discuss eight. (The remaining eight fall into the same SLOCC classes as the chosen eight.) These eight unitary solutions are not equivalent to each other and generate the GHZ, $AC-B$ and product state SLOCC classes as shown in Tables 2, 3 and 4, respectively. | $(\alpha_{1},\alpha_{3},\beta_{1},\beta_{3})$ | $R$ | Eigenvalues | $n|R^{n}=1$ ---|---|---|---|--- 1. | $(-2,-2,2,2)$ | $\frac{1}{2}\tiny{\left(\begin{array}[]{cccccccc}0&-1&1&0&-1&0&0&-1\\\ -1&0&0&-1&0&-1&1&0\\\ 1&0&0&-1&0&-1&-1&0\\\ 0&-1&-1&0&1&0&0&-1\\\ -1&0&0&1&0&-1&-1&0\\\ 0&-1&-1&0&-1&0&0&1\\\ 0&1&-1&0&-1&0&0&-1\\\ -1&0&0&-1&0&1&-1&0\end{array}\right)}$ | $(-1_{(4)},1_{(4)})$ | 2 2. | $(-2,-2,2,4)$ | $\frac{1}{2}\tiny{\left(\begin{array}[]{cccccccc}1&0&1&0&0&1&0&-1\\\ 0&1&0&-1&1&0&1&0\\\ 1&0&1&0&0&-1&0&1\\\ 0&-1&0&1&1&0&1&0\\\ 0&1&0&1&1&0&-1&0\\\ 1&0&-1&0&0&1&0&1\\\ 0&1&0&1&-1&0&1&0\\\ -1&0&1&0&0&1&0&1\end{array}\right)}$ | $(-1_{(2)},1_{(6)})$ | 2 3. | $(-2,0,2,0)$ | $\frac{1}{2}\tiny{\left(\begin{array}[]{cccccccc}0&-1&0&-1&-1&0&1&0\\\ -1&0&1&0&0&-1&0&-1\\\ 0&-1&0&-1&1&0&-1&0\\\ 1&0&-1&0&0&-1&0&-1\\\ -1&0&-1&0&0&-1&0&1\\\ 0&-1&0&1&-1&0&-1&0\\\ -1&0&-1&0&0&1&0&-1\\\ 0&1&0&-1&-1&0&-1&0\end{array}\right)}$ | $(-1_{(2)},1_{(2)},\mathrm{i}_{(2)},-\mathrm{i}_{(2)})$ | 4 4. | $(-2,0,2,2)$ | $\frac{1}{2}\tiny{\left(\begin{array}[]{cccccccc}1&0&0&-1&0&1&1&0\\\ 0&1&1&0&1&0&0&-1\\\ 0&-1&1&0&1&0&0&1\\\ 1&0&0&1&0&-1&1&0\\\ 0&1&-1&0&1&0&0&1\\\ 1&0&0&1&0&1&-1&0\\\ -1&0&0&1&0&1&1&0\\\ 0&1&1&0&-1&0&0&1\end{array}\right)}$ | $(\mathrm{i}_{(2)},-\mathrm{i}_{(2)},1_{(4)})$ | 4 Table 2: 3-qubit unitary generalized $R$-matrices generating the GHZ SLOCC class. The generalized $R$-matrices in Table 2 generate the following entangled states in the GHZ SLOCC class $\displaystyle\frac{1}{2}\left[-{\left|{001}\right>}+{\left|{010}\right>}-{\left|{100}\right>}-{\left|{111}\right>}\right],\quad\frac{1}{2}\left[{\left|{000}\right>}+{\left|{010}\right>}+{\left|{101}\right>}-{\left|{111}\right>}\right],$ $\displaystyle\frac{1}{2}\left[-{\left|{001}\right>}+{\left|{011}\right>}-{\left|{100}\right>}-{\left|{110}\right>}\right],\quad\frac{1}{2}\left[{\left|{000}\right>}+{\left|{011}\right>}+{\left|{101}\right>}-{\left|{110}\right>}\right],$ (4.24) which are equivalent to the standard GHZ state, ${\left|{000}\right>}+{\left|{111}\right>}$, by the application of appropriate ILOs, such as, for example, $\left(\begin{array}[]{cc}a_{1}&b_{1}\\\ \mathrm{i}a_{1}&-\mathrm{i}b_{1}\end{array}\right)\otimes\left(\begin{array}[]{cc}a_{2}&b_{2}\\\ -\mathrm{i}a_{2}&\mathrm{i}b_{2}\end{array}\right)\otimes\left(\begin{array}[]{cc}\frac{\mathrm{i}}{4a_{1}a_{2}}&-\frac{\mathrm{i}}{4b_{1}b_{2}}\\\ -\frac{1}{4a_{1}a_{2}}&-\frac{1}{4b_{1}b_{2}}\end{array}\right).$ The generalized $R$-matrices in Table 3 generate the partially entangled states $AC-B$ given by $\displaystyle\frac{1}{2}\left[-{\left|{000}\right>}-{\left|{001}\right>}-{\left|{100}\right>}+{\left|{101}\right>}\right],\quad\frac{1}{2}\left[{\left|{000}\right>}-{\left|{001}\right>}+{\left|{100}\right>}+{\left|{101}\right>}\right],$ (4.25) respectively. The product state SLOCC class is reported instead on Table 4. | $(\alpha_{1},\alpha_{3},\beta_{1},\beta_{3})$ | $R$ | Eigenvalues | $n|R^{n}=1$ ---|---|---|---|--- 1. | $(-2,-2,0,2)$ | $\frac{1}{2}\tiny{\left(\begin{array}[]{cccccccc}-1&-1&0&0&-1&1&0&0\\\ -1&1&0&0&-1&-1&0&0\\\ 0&0&-1&-1&0&0&-1&1\\\ 0&0&-1&1&0&0&-1&-1\\\ -1&-1&0&0&1&-1&0&0\\\ 1&-1&0&0&-1&-1&0&0\\\ 0&0&-1&-1&0&0&1&-1\\\ 0&0&1&-1&0&0&-1&-1\end{array}\right)}$ | $(-1_{(4)},1_{(4)})$ | 2 2. | $(-2,0,0,2)$ | $\frac{1}{2}\tiny{\left(\begin{array}[]{cccccccc}1&1&0&0&-1&1&0&0\\\ -1&1&0&0&1&1&0&0\\\ 0&0&1&1&0&0&-1&1\\\ 0&0&-1&1&0&0&1&1\\\ 1&1&0&0&1&-1&0&0\\\ 1&-1&0&0&1&1&0&0\\\ 0&0&1&1&0&0&1&-1\\\ 0&0&1&-1&0&0&1&1\end{array}\right)}$ | $(\mathrm{i}_{(2)},-\mathrm{i}_{(2)},1_{(4)})$ | 4 Table 3: 3-qubit unitary generalized $R$-matrices generating the $AC-B$ SLOCC class. | $(\alpha_{1},\alpha_{3},\beta_{1},\beta_{3})$ | $R$ | Eigenvalues | $n|R^{n}=1$ ---|---|---|---|--- 1. | $(-2,-2,0,4)$ | $\tiny{\left(\begin{array}[]{cccccccc}0&0&0&0&0&1&0&0\\\ 0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\ 1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0\\\ 0&0&1&0&0&0&0&0\end{array}\right)}$ | $(-1_{(2)},1_{(6)})$ | 2 2. | $(-2,0,0,0)$ | $\tiny{\left(\begin{array}[]{cccccccc}0&0&0&0&-1&0&0&0\\\ -1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&-1&0\\\ 0&0&-1&0&0&0&0&0\\\ 0&0&0&0&0&-1&0&0\\\ 0&-1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&-1\\\ 0&0&0&-1&0&0&0&0\end{array}\right)}$ | $(\mathrm{i}_{(2)},-\mathrm{i}_{(2)},-1_{(2)},1_{(2)})$ | 4 Table 4: 3-qubit unitary generalized $R$-matrices generating the product state SLOCC class. ##### Using $s_{i}$, $p_{i,i+1}$ and $p_{i+1,i+2}$ The ansatz $R_{i}=s_{i,i+2}\left(1+\alpha~{}p_{i,i+1}+\beta~{}p_{i+1,i+2}+\gamma~{}p_{i,i+1}p_{i+1,i+2}+\delta~{}p_{i,i+2}\right),$ (4.26) with $s_{i,i+2}=s_{i}s_{i+1}s_{i}$, satisfies the $(d,3,2)$-gYBE when $\gamma=-\frac{\alpha+\beta}{2}$. This is seen after simplifying the expressions in the $(d,3,2)$-gYBE using the swapping property of the permutation operator in (2.10), as done before. The inverse, when $d=2$ and $\gamma=-\frac{\alpha+\beta}{2}$ is given by $R_{i}^{-1}=\left(1+\alpha^{\prime}~{}p_{i,i+1}+\beta^{\prime}~{}p_{i+1,i+2}+\gamma^{\prime}~{}p_{i,i+1}p_{i+1,i+2}+\delta^{\prime}~{}p_{i,i+2}\right)s_{i,i+2},$ (4.27) with $\displaystyle\alpha^{\prime}=-\frac{\alpha}{1+2\alpha},\quad\beta^{\prime}=-\frac{\beta}{1+2\beta},\quad\delta^{\prime}=-\frac{\delta}{1+2\delta},\quad\gamma^{\prime}=\frac{\alpha+\beta+4\alpha\beta}{2+4(\alpha+\beta+2\alpha\beta)},$ (4.28) which results in a unitary family when $\alpha^{\prime}=\alpha^{*}$, $\beta^{\prime}=\beta^{*}$, $\gamma^{\prime}=\gamma^{*}$, and $\delta^{\prime}=\delta^{*}$ that are solved by $\alpha=\frac{1}{2}(e^{\mathrm{i}\theta}-1),\qquad\beta=\frac{1}{2}(e^{\mathrm{i}\varphi}-1),\qquad\delta=\frac{1}{2}(e^{\mathrm{i}\phi}-1)$ (4.29) with $\gamma=-\frac{\alpha+\beta}{2}$ for $\theta$, $\varphi$ and $\phi$ arbitrary angles. The eigenvalues of this operator at the unitary family are given by $\\{e^{\mathrm{i}\phi}_{(4)},\pm e^{\frac{\mathrm{i}}{2}\left(\theta+\varphi\right)}_{(2)}\\}$. There are eight real unitary solutions, out of which four inequivalent unitary solutions are shown in Table 5. | $(\alpha,\beta,\delta)$ | $R_{i}$ | Eigenvalues | $n|R^{n}=1$ ---|---|---|---|--- 1. | $(-1,-1,-1)$ | $\frac{1}{2}\tiny{\left(\begin{array}[]{cccccccc}-1&0&0&0&0&0&0&0\\\ 0&0&0&0&-1&0&0&0\\\ 0&0&-1&0&0&0&0&0\\\ 0&0&0&0&0&0&-1&0\\\ 0&-1&0&0&0&0&0&0\\\ 0&0&0&0&0&-1&0&0\\\ 0&0&0&-1&0&0&0&0\\\ 0&0&0&0&0&0&0&-1\end{array}\right)}$ | $(-1_{(6)},1_{(2)})$ | 2 2. | $(-1,-1,0)$ | $\tiny{\left(\begin{array}[]{cccccccc}0&0&0&0&0&1&0&0\\\ 0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\ 1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0\\\ 0&0&1&0&0&0&0&0\end{array}\right)}$ | $(-1_{(2)},1_{(6)})$ | 2 3. | $(-1,0,-1)$ | $\frac{1}{2}\tiny{\left(\begin{array}[]{cccccccc}-1&0&0&1&0&-1&-1&0\\\ 0&-1&-1&0&-1&0&0&1\\\ 0&1&-1&0&-1&0&0&-1\\\ -1&0&0&-1&0&1&-1&0\\\ 0&-1&1&0&-1&0&0&-1\\\ -1&0&0&-1&0&-1&1&0\\\ 1&0&0&-1&0&-1&-1&0\\\ 0&-1&-1&0&1&0&0&-1\end{array}\right)}$ | $(-1_{(4)},\mathrm{i}_{(2)},-\mathrm{i}_{(2)})$ | 4 4. | $(-1,0,0)$ | $\frac{1}{2}\tiny{\left(\begin{array}[]{cccccccc}1&0&0&1&0&1&-1&0\\\ 0&1&-1&0&1&0&0&1\\\ 0&1&1&0&-1&0&0&1\\\ -1&0&0&1&0&1&1&0\\\ 0&1&1&0&1&0&0&-1\\\ 1&0&0&-1&0&1&1&0\\\ 1&0&0&1&0&-1&1&0\\\ 0&-1&1&0&1&0&0&1\end{array}\right)}$ | $(\mathrm{i}_{(2)},-\mathrm{i}_{(2)},1_{(4)})$ | 4 Table 5: Unitary solutions in the 3-qubit case for the generalized Yang–Baxter operator in (4.26) with $\gamma=-\frac{\alpha+\beta}{2}$. The image of the braid group representations constructed out of the generalized $R$-matrix in (4.26) for $d=2$ and $\gamma=-\frac{\alpha+\beta}{2}$, for both the unitary and non-unitary cases, is once again infinite as in the previous cases, as seen by computing the powers of the generalized $R$-matrix, $R_{i}^{n}=s_{i,i+2}^{n}\left(1+\alpha_{n}~{}p_{i,i+1}+\beta_{n}~{}p_{i+1,i+2}+\gamma_{n}~{}p_{i,i+1}p_{i+1,i+2}+\delta_{n}~{}p_{i,i+2}\right),$ (4.30) with the parameters defined recursively as $\displaystyle\alpha_{n}$ $\displaystyle=$ $\displaystyle\alpha_{1}+\beta_{n-1}+2\alpha_{1}\beta_{n-1},\qquad\beta_{n}=\alpha_{n-1}+\beta_{1}+2\alpha_{n-1}\beta_{1},$ (4.31) $\displaystyle\gamma_{n}$ $\displaystyle=$ $\displaystyle\gamma_{1}+\gamma_{n-1}+4\gamma_{1}\gamma_{n-1}+2\delta_{1}\gamma_{n-1}+2\gamma_{1}\delta_{n-1}+\delta_{n-1}\left(\alpha_{1}+\beta_{1}\right)+\delta_{1}\left(\alpha_{n-1}+\beta_{n-1}\right)$ (4.32) $\displaystyle+2\gamma_{n-1}\left(\alpha_{1}+\beta_{1}\right)+2\gamma_{1}\left(\alpha_{n-1}+\beta_{n-1}\right)+\alpha_{1}\alpha_{n-1}+\beta_{1}\beta_{n-1},$ $\displaystyle\delta_{n}$ $\displaystyle=$ $\displaystyle\delta_{1}+\delta_{n-1}+2\delta_{1}\delta_{n-1},$ (4.33) after identifying $\alpha_{1}$, $\beta_{1}$, $\gamma_{1}$ and $\delta_{1}$ with $\alpha$, $\beta$, $\gamma$ and $\delta$ in (4.26). The generalized $R$-matrices in the first two rows of Table 5 are not entanglers, as they generate just the product state SLOCC class. The generalized $R$-matrices of the second two rows, on the other hand, are both entanglers generating the GHZ SLOCC class. In particular the states they generate by acting on ${\left|{000}\right>}$ are $\displaystyle\frac{1}{2}\left[-{\left|{000}\right>}-{\left|{011}\right>}-{\left|{101}\right>}+{\left|{110}\right>}\right],\qquad\frac{1}{2}\left[{\left|{000}\right>}-{\left|{011}\right>}+{\left|{101}\right>}+{\left|{110}\right>}\right],$ (4.34) which are equivalent to the standard GHZ state ${\left|{000}\right>}+{\left|{111}\right>}$ by appropriate ILOs, such as, for example, $\left(\begin{array}[]{cc}a_{1}&b_{1}\\\ \mathrm{i}a_{1}&-\mathrm{i}b_{1}\end{array}\right)\otimes\left(\begin{array}[]{cc}a_{2}&b_{2}\\\ \mathrm{i}a_{2}&-\mathrm{i}b_{2}\end{array}\right)\otimes\left(\begin{array}[]{cc}-\frac{1}{4a_{1}a_{2}}&-\frac{1}{4b_{1}b_{2}}\\\ \frac{\mathrm{i}}{4a_{1}a_{2}}&-\frac{\mathrm{i}}{4b_{1}b_{2}}\end{array}\right).$ ### 4.3 4-qubits As one increases the number of qubits, the analytic computation for the generalized $R$-matrices gets more tedious. To illustrate the feasibility of the method, we write down the answers for the 4-qubit case as well. We use the operators $s_{i}$, $p_{i}$, $p_{i+1}$, $p_{i+2}$ and $p_{i+3}$ to build the generalized $R$-matrices. The generalized $R$-matrices that satisfy the $(d,4,2)$-gYBE and the $(d,4,3)$-gYBE also satisfy far-commutativity yielding braid group representations. These are the cases we focus on. #### The $(d,4,2)$-generalized $R$-matrix The generalized $R$-matrix $\displaystyle R_{i}$ $\displaystyle=$ $\displaystyle s_{i,i+2}\left(1+\alpha_{1}~{}p_{i}+\alpha_{3}~{}p_{i+2}+\beta_{1}~{}p_{i}p_{i+1}+\beta_{2}~{}p_{i}p_{i+2}\right.$ (4.35) $\displaystyle+\beta_{3}~{}p_{i}p_{i+3}+\beta_{4}~{}p_{i+1}p_{i+2}+\beta_{6}~{}p_{i+2}p_{i+3}+\gamma_{1}~{}p_{i}p_{i+1}p_{i+2}+$ $\displaystyle+\left.\gamma_{2}~{}p_{i}p_{i+1}p_{i+3}+\gamma_{3}~{}p_{i}p_{i+2}p_{i+3}+\gamma_{4}~{}p_{i+1}p_{i+2}p_{i+3}+\delta~{}p_{i}p_{i+1}p_{i+2}p_{i+3}\right),$ with $s_{i,i+2}=s_{i}s_{i+1}s_{i}$, satisfies the $(d,4,2)$-gYBE for $\displaystyle\beta_{4}$ $\displaystyle=$ $\displaystyle-\frac{\beta_{1}\left(1+\alpha_{3}\right)}{1+\alpha_{1}+\beta_{1}},~{}~{}\beta_{6}=-\frac{\beta_{3}\left(1+\alpha_{3}\right)}{1+\alpha_{1}+\beta_{3}},$ $\displaystyle\gamma_{1}$ $\displaystyle=$ $\displaystyle-\frac{\beta_{1}\left(\alpha_{1}+\beta_{1}-\alpha_{3}\right)}{1+\alpha_{1}+\beta_{1}},~{}~{}\gamma_{3}=-\frac{\beta_{3}\left(\alpha_{1}+\beta_{3}-\alpha_{3}\right)}{1+\alpha_{1}+\beta_{3}},$ $\displaystyle\gamma_{4}$ $\displaystyle=$ $\displaystyle\left(1+\alpha_{3}\right)\left[\frac{\beta_{1}}{1+\alpha_{1}+\beta_{1}}-\frac{\left(1+\alpha_{1}\right)\left(\beta_{1}+\gamma_{2}\right)}{\left(1+\alpha_{1}+\beta_{3}\right)\left(1+\alpha_{1}+\beta_{1}+\beta_{3}+\gamma_{2}\right)}\right],$ $\displaystyle\delta$ $\displaystyle=$ $\displaystyle-\gamma_{2}+\frac{\left(1+\alpha_{3}\right)\left[-\beta_{1}\beta_{3}\left(2\alpha_{1}+\beta_{1}+\beta_{3}+2\right)+\gamma_{2}\left(\left(1+\alpha_{1}\right)^{2}-\beta_{1}\beta_{3}\right)\right]}{\left(1+\alpha_{1}+\beta_{1}\right)\left(1+\alpha_{1}+\beta_{3}\right)\left(1+\alpha_{1}+\beta_{1}+\beta_{3}+\gamma_{2}\right)}.$ (4.36) The solutions become unitary when $\displaystyle\alpha_{1}^{*}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{1}}{1+\alpha_{1}},~{}~{}\alpha_{3}^{*}=-\frac{\alpha_{3}}{1+\alpha_{3}},$ $\displaystyle\beta_{1}^{*}$ $\displaystyle=$ $\displaystyle-\frac{\beta_{1}}{(1+\alpha_{1})(1+\alpha_{1}+\beta_{1})},~{}~{}\beta_{3}^{*}=-\frac{\beta_{3}}{(1+\alpha_{1})(1+\alpha_{1}+\beta_{3})},$ $\displaystyle\beta_{2}^{*}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{1}}{1+\alpha_{1}}-\frac{1}{1+\alpha_{3}}+\frac{1}{1+\alpha_{1}+\alpha_{3}+\beta_{2}},$ $\displaystyle\gamma_{2}^{*}$ $\displaystyle=$ $\displaystyle\frac{\beta_{1}}{(1+\alpha_{1})(1+\alpha_{1}+\beta_{1})}-\frac{1}{1+\alpha_{1}+\beta_{3}}+\frac{1}{1+\alpha_{1}+\beta_{1}+\beta_{3}+\gamma_{2}},$ (4.37) which is solved by $\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle e^{\mathrm{i}\theta_{1}}-1,~{}~{}\alpha_{3}=e^{\mathrm{i}\theta_{3}}-1,$ $\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle e^{\mathrm{i}\phi_{1}}-e^{\mathrm{i}\theta_{1}},~{}~{}\beta_{3}=e^{\mathrm{i}\phi_{3}}-e^{\mathrm{i}\theta_{1}},$ $\displaystyle\beta_{2}$ $\displaystyle=$ $\displaystyle e^{\mathrm{i}\phi_{2}}-e^{\mathrm{i}\theta_{1}}-e^{\mathrm{i}\theta_{3}}+1,~{}~{}\gamma_{2}=e^{\mathrm{i}\varphi_{2}}-e^{\mathrm{i}\phi_{1}}-e^{\mathrm{i}\phi_{3}}+e^{\mathrm{i}\theta_{1}},$ (4.38) for arbitrary angles $\theta_{1}$, $\theta_{3}$, $\phi_{1}$, $\phi_{3}$ and $\varphi_{2}$. The eigenvalues at these unitary solutions are given by $\\{1_{(4)},\pm e^{\frac{\mathrm{i}}{2}(\theta_{1}+\theta_{3})}_{(4)},e^{\mathrm{i}\phi_{2}}_{(4)}\\}$. There are 64 real unitary generalized $R$-matrices in this case. They encompass four sets of eigenvalues, with 16 unitary generalized $R$-matrices in each of these sets. The unitary generalized $R$-matrices are inequivalent when they belong to different eigenvalue sets, however when they belong to the same eigenvalue set they may or may not be equivalent. We write down one unitary solution from each set of eigenvalues. ##### Eigenvalues $\in\\{-1_{(8)},1_{(8)}\\}$ The solution with $\left(\alpha_{1},\alpha_{3},\beta_{1},\beta_{2},\beta_{3},\gamma_{2}\right)=\left(-2,-2,0,2,0,0\right)$ reads explicitly $\frac{1}{2}\left(\tiny{\begin{array}[]{cccccccc|cccccccc}-1&0&-1&0&0&0&0&0&-1&0&1&0&0&0&0&0\\\ 0&-1&0&-1&0&0&0&0&0&-1&0&1&0&0&0&0\\\ -1&0&1&0&0&0&0&0&-1&0&-1&0&0&0&0&0\\\ 0&-1&0&1&0&0&0&0&0&-1&0&-1&0&0&0&0\\\ 0&0&0&0&-1&0&-1&0&0&0&0&0&-1&0&1&0\\\ 0&0&0&0&0&-1&0&-1&0&0&0&0&0&-1&0&1\\\ 0&0&0&0&-1&0&1&0&0&0&0&0&-1&0&-1&0\\\ 0&0&0&0&0&-1&0&1&0&0&0&0&0&-1&0&-1\\\ \hline\cr-1&0&-1&0&0&0&0&0&1&0&-1&0&0&0&0&0\\\ 0&-1&0&-1&0&0&0&0&0&1&0&-1&0&0&0&0\\\ 1&0&-1&0&0&0&0&0&-1&0&-1&0&0&0&0&0\\\ 0&1&0&-1&0&0&0&0&0&-1&0&-1&0&0&0&0\\\ 0&0&0&0&-1&0&-1&0&0&0&0&0&1&0&-1&0\\\ 0&0&0&0&0&-1&0&-1&0&0&0&0&0&1&0&-1\\\ 0&0&0&0&1&0&-1&0&0&0&0&0&-1&0&-1&0\\\ 0&0&0&0&0&1&0&-1&0&0&0&0&0&-1&0&-1\end{array}}\right).$ (4.39) This generates $\frac{1}{2}\left[-{\left|{0000}\right>}-{\left|{0010}\right>}-{\left|{1000}\right>}+{\left|{1010}\right>}\right]$ from ${\left|{0000}\right>}$. ##### Eigenvalues $\in\\{-1_{(4)},1_{(12)}\\}$ The solution for $\left(\alpha_{1},\alpha_{3},\beta_{1},\beta_{2},\beta_{3},\gamma_{2}\right)=\left(-2,-2,0,4,0,2\right)$ reads $\frac{1}{4}\left(\tiny{\begin{array}[]{cccccccc|cccccccc}1&1&0&0&1&1&0&0&0&0&3&-1&0&0&-1&-1\\\ 1&1&0&0&1&1&0&0&0&0&-1&3&0&0&-1&-1\\\ 0&0&3&-1&0&0&-1&-1&1&1&0&0&1&1&0&0\\\ 0&0&-1&3&0&0&-1&-1&1&1&0&0&1&1&0&0\\\ 1&1&0&0&1&1&0&0&0&0&-1&-1&0&0&3&-1\\\ 1&1&0&0&1&1&0&0&0&0&-1&-1&0&0&-1&3\\\ 0&0&-1&-1&0&0&3&-1&1&1&0&0&1&1&0&0\\\ 0&0&-1&-1&0&0&-1&3&1&1&0&0&1&1&0&0\\\ \hline\cr 0&0&1&1&0&0&1&1&3&-1&0&0&-1&-1&0&0\\\ 0&0&1&1&0&0&1&1&-1&3&0&0&-1&-1&0&0\\\ 3&-1&0&0&-1&-1&0&0&0&0&1&1&0&0&1&1\\\ -1&3&0&0&-1&-1&0&0&0&0&1&1&0&0&1&1\\\ 0&0&1&1&0&0&1&1&-1&-1&0&0&3&-1&0&0\\\ 0&0&1&1&0&0&1&1&-1&-1&0&0&-1&3&0&0\\\ -1&-1&0&0&3&-1&0&0&0&0&1&1&0&0&1&1\\\ -1&-1&0&0&-1&3&0&0&0&0&1&1&0&0&1&1\end{array}}\right).$ (4.40) This generates $\frac{1}{4}\left[{\left|{0000}\right>}+{\left|{0001}\right>}+{\left|{0100}\right>}+{\left|{0101}\right>}+3{\left|{1010}\right>}-{\left|{1011}\right>}-{\left|{1110}\right>}-{\left|{1111}\right>}\right]$ from ${\left|{0000}\right>}$. ##### Eigenvalues $\in\\{-\mathrm{i}_{(4)},\mathrm{i}_{(4)},1_{(8)}\\}$ The solution with $\left(\alpha_{1},\alpha_{3},\beta_{1},\beta_{2},\beta_{3},\gamma_{2}\right)=\left(-2,0,0,2,2,0\right)$ reads $\frac{1}{2}\left(\tiny{\begin{array}[]{cccccccc|cccccccc}1&0&0&-1&0&0&0&0&0&1&1&0&0&0&0&0\\\ 0&1&-1&0&0&0&0&0&1&0&0&1&0&0&0&0\\\ 0&1&1&0&0&0&0&0&1&0&0&-1&0&0&0&0\\\ 1&0&0&1&0&0&0&0&0&1&-1&0&0&0&0&0\\\ 0&0&0&0&1&0&0&-1&0&0&0&0&0&1&1&0\\\ 0&0&0&0&0&1&-1&0&0&0&0&0&1&0&0&1\\\ 0&0&0&0&0&1&1&0&0&0&0&0&1&0&0&-1\\\ 0&0&0&0&1&0&0&1&0&0&0&0&0&1&-1&0\\\ \hline\cr 0&-1&1&0&0&0&0&0&1&0&0&1&0&0&0&0\\\ -1&0&0&1&0&0&0&0&0&1&1&0&0&0&0&0\\\ 1&0&0&1&0&0&0&0&0&-1&1&0&0&0&0&0\\\ 0&1&1&0&0&0&0&0&-1&0&0&1&0&0&0&0\\\ 0&0&0&0&0&-1&1&0&0&0&0&0&1&0&0&1\\\ 0&0&0&0&-1&0&0&1&0&0&0&0&0&1&1&0\\\ 0&0&0&0&1&0&0&1&0&0&0&0&0&-1&1&0\\\ 0&0&0&0&0&1&1&0&0&0&0&0&-1&0&0&1\end{array}}\right).$ (4.41) This generates $\frac{1}{2}\left[{\left|{0000}\right>}+{\left|{0011}\right>}-{\left|{1001}\right>}+{\left|{1010}\right>}\right]$ from ${\left|{0000}\right>}$. ##### Eigenvalues $\in\\{-\mathrm{i}_{(4)},\mathrm{i}_{(4)},-1_{(4)},1_{(4)}\\}$ The solution with $\left(\alpha_{1},\alpha_{3},\beta_{1},\beta_{2},\beta_{3},\gamma_{2}\right)=\left(-2,0,0,0,2,0\right)$ reads $\frac{1}{2}\left(\tiny{\begin{array}[]{cccccccc|cccccccc}0&0&-1&-1&0&0&0&0&-1&1&0&0&0&0&0&0\\\ 0&0&-1&-1&0&0&0&0&1&-1&0&0&0&0&0&0\\\ -1&1&0&0&0&0&0&0&0&0&-1&-1&0&0&0&0\\\ 1&-1&0&0&0&0&0&0&0&0&-1&-1&0&0&0&0\\\ 0&0&0&0&0&0&-1&-1&0&0&0&0&-1&1&0&0\\\ 0&0&0&0&0&0&-1&-1&0&0&0&0&1&-1&0&0\\\ 0&0&0&0&-1&1&0&0&0&0&0&0&0&0&-1&-1\\\ 0&0&0&0&1&-1&0&0&0&0&0&0&0&0&-1&-1\\\ \hline\cr-1&-1&0&0&0&0&0&0&0&0&-1&1&0&0&0&0\\\ -1&-1&0&0&0&0&0&0&0&0&1&-1&0&0&0&0\\\ 0&0&-1&1&0&0&0&0&-1&-1&0&0&0&0&0&0\\\ 0&0&1&-1&0&0&0&0&-1&-1&0&0&0&0&0&0\\\ 0&0&0&0&-1&-1&0&0&0&0&0&0&0&0&-1&1\\\ 0&0&0&0&-1&-1&0&0&0&0&0&0&0&0&1&-1\\\ 0&0&0&0&0&0&-1&1&0&0&0&0&-1&-1&0&0\\\ 0&0&0&0&0&0&1&-1&0&0&0&0&-1&-1&0&0\end{array}}\right).$ (4.42) This generates $\frac{1}{2}\left[-{\left|{0010}\right>}+{\left|{0011}\right>}-{\left|{1000}\right>}-{\left|{1001}\right>}\right]$ from ${\left|{0000}\right>}$. #### The $(d,4,3)$-generalized $R$-matrix The generalized $R$-matrix $\displaystyle R_{i}$ $\displaystyle=$ $\displaystyle s_{i,i+3}\left(1+\alpha_{1}~{}p_{i}+\alpha_{4}~{}p_{i+3}+\beta_{1}~{}p_{i}p_{i+1}\right.$ (4.43) $\displaystyle+\left.\beta_{2}~{}p_{i}p_{i+2}+\beta_{3}~{}p_{i}p_{i+3}+\beta_{5}~{}p_{i+1}p_{i+3}+\beta_{6}~{}p_{i+2}p_{i+3}\right.$ $\displaystyle+\left.\gamma_{1}~{}p_{i}p_{i+1}p_{i+2}+\gamma_{2}~{}p_{i}p_{i+1}p_{i+3}+\gamma_{3}~{}p_{i}p_{i+2}p_{i+3}+\gamma_{4}~{}p_{i+1}p_{i+2}p_{i+3}\right.$ $\displaystyle+\left.\delta~{}p_{i}p_{i+1}p_{i+2}p_{i+3}\right),$ with $s_{i,i+3}=s_{i+2}s_{i+1}s_{i}s_{i+1}s_{i+2}$, satisfies the $(d,4,3)$-gYBE for $\displaystyle\beta_{5}$ $\displaystyle=$ $\displaystyle-\frac{\beta_{1}\left(1+\alpha_{4}\right)}{1+\alpha_{1}+\beta_{1}},~{}~{}\beta_{6}=-\frac{\beta_{2}\left(1+\alpha_{4}\right)}{1+\alpha_{1}+\beta_{2}},$ $\displaystyle\gamma_{2}$ $\displaystyle=$ $\displaystyle-\frac{\beta_{1}\left(\alpha_{1}+\beta_{1}-\alpha_{4}\right)}{1+\alpha_{1}+\beta_{1}},~{}~{}\gamma_{3}=-\frac{\beta_{2}\left(\alpha_{1}+\beta_{2}-\alpha_{4}\right)}{1+\alpha_{1}+\beta_{2}},$ $\displaystyle\gamma_{4}$ $\displaystyle=$ $\displaystyle\left(1+\alpha_{4}\right)\left[\frac{\beta_{1}}{1+\alpha_{1}+\beta_{1}}-\frac{\left(1+\alpha_{1}\right)\left(\beta_{1}+\gamma_{1}\right)}{\left(1+\alpha_{1}+\beta_{2}\right)\left(1+\alpha_{1}+\beta_{1}+\beta_{2}+\gamma_{1}\right)}\right],$ $\displaystyle\delta$ $\displaystyle=$ $\displaystyle-\gamma_{1}+\frac{\left(1+\alpha_{4}\right)\left[-\beta_{1}\beta_{2}\left(2\alpha_{1}+\beta_{1}+\beta_{2}+2\right)+\gamma_{1}\left(\left(1+\alpha_{1}\right)^{2}-\beta_{1}\beta_{2}\right)\right]}{\left(1+\alpha_{1}+\beta_{1}\right)\left(1+\alpha_{1}+\beta_{2}\right)\left(1+\alpha_{1}+\beta_{1}+\beta_{2}+\gamma_{1}\right)}.$ (4.44) The solutions become unitary when $\displaystyle\alpha_{1}^{*}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{1}}{1+\alpha_{1}},~{}~{}\alpha_{4}^{*}=-\frac{\alpha_{4}}{1+\alpha_{4}},$ $\displaystyle\beta_{1}^{*}$ $\displaystyle=$ $\displaystyle-\frac{\beta_{1}}{(1+\alpha_{1})(1+\alpha_{1}+\beta_{1})},~{}~{}\beta_{2}^{*}=-\frac{\beta_{2}}{(1+\alpha_{1})(1+\alpha_{1}+\beta_{2})},$ $\displaystyle\beta_{3}^{*}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{1}}{1+\alpha_{1}}-\frac{1}{1+\alpha_{4}}+\frac{1}{1+\alpha_{1}+\alpha_{4}+\beta_{3}},$ $\displaystyle\gamma_{1}^{*}$ $\displaystyle=$ $\displaystyle\frac{\beta_{1}}{(1+\alpha_{1})(1+\alpha_{1}+\beta_{1})}-\frac{1}{1+\alpha_{1}+\beta_{2}}+\frac{1}{1+\alpha_{1}+\beta_{1}+\beta_{2}+\gamma_{1}},$ (4.45) which is solved by $\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle e^{\mathrm{i}\theta_{1}}-1,~{}~{}\alpha_{4}=e^{\mathrm{i}\theta_{4}}-1,$ $\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle e^{\mathrm{i}\phi_{1}}-e^{\mathrm{i}\theta_{1}},~{}~{}\beta_{2}=e^{\mathrm{i}\phi_{2}}-e^{\mathrm{i}\theta_{1}},$ $\displaystyle\beta_{3}$ $\displaystyle=$ $\displaystyle e^{\mathrm{i}\phi_{3}}-e^{\mathrm{i}\theta_{1}}-e^{\mathrm{i}\theta_{4}}+1,~{}~{}\gamma_{1}=e^{\mathrm{i}\varphi_{1}}-e^{\mathrm{i}\phi_{1}}-e^{\mathrm{i}\phi_{2}}+e^{\mathrm{i}\theta_{1}},$ (4.46) for arbitrary angles $\theta_{1}$, $\theta_{4}$, $\phi_{1}$, $\phi_{2}$ and $\varphi_{1}$. The eigenvalues at these unitary solutions are given by $\\{1_{(4)},\pm e^{\frac{\mathrm{i}}{2}(\theta_{1}+\theta_{4})}_{(4)},e^{\mathrm{i}\phi_{3}}_{(4)}\\}$. As in the $(2,4,2)$ case, there are 64 real unitary generalized $R$-matrices and we write down one unitary solution from each set of eigenvalue. When the parameters are complex we obtain a family of unitary solutions that generate the full braid group. ##### Eigenvalues $\in\\{-1_{(4)},1_{(12)}\\}$ When $\left(\alpha_{1},\alpha_{4},\beta_{1},\beta_{2},\beta_{3},\gamma_{1}\right)=\left(-2,-2,0,0,4,2\right)$, the matrix reads $\frac{1}{4}\left(\tiny{\begin{array}[]{cccccccc|cccccccc}1&0&1&0&1&0&1&0&0&3&0&-1&0&-1&0&-1\\\ 0&3&0&-1&0&-1&0&-1&1&0&1&0&1&0&1&0\\\ 1&0&1&0&1&0&1&0&0&-1&0&3&0&-1&0&-1\\\ 0&-1&0&3&0&-1&0&-1&1&0&1&0&1&0&1&0\\\ 1&0&1&0&1&0&1&0&0&-1&0&-1&0&3&0&-1\\\ 0&-1&0&-1&0&3&0&-1&1&0&1&0&1&0&1&0\\\ 1&0&1&0&1&0&1&0&0&-1&0&-1&0&-1&0&3\\\ 0&-1&0&-1&0&-1&0&3&1&0&1&0&1&0&1&0\\\ \hline\cr 0&1&0&1&0&1&0&1&3&0&-1&0&-1&0&-1&0\\\ 3&0&-1&0&-1&0&-1&0&0&1&0&1&0&1&0&1\\\ 0&1&0&1&0&1&0&1&-1&0&3&0&-1&0&-1&0\\\ -1&0&3&0&-1&0&-1&0&0&1&0&1&0&1&0&1\\\ 0&1&0&1&0&1&0&1&-1&0&-1&0&3&0&-1&0\\\ -1&0&-1&0&3&0&-1&0&0&1&0&1&0&1&0&1\\\ 0&1&0&1&0&1&0&1&-1&0&-1&0&-1&0&3&0\\\ -1&0&-1&0&-1&0&3&0&0&1&0&1&0&1&0&1\end{array}}\right).$ (4.47) This generates $\frac{1}{4}\left[{\left|{0000}\right>}+{\left|{0010}\right>}+{\left|{0100}\right>}+{\left|{0110}\right>}+3{\left|{1001}\right>}-{\left|{1011}\right>}-{\left|{1101}\right>}-{\left|{1111}\right>}\right]$ from ${\left|{0000}\right>}$. ##### Eigenvalues $\in\\{-1_{(8)},1_{(8)}\\}$ The solution for $\left(\alpha_{1},\alpha_{4},\beta_{1},\beta_{2},\beta_{3},\gamma_{1}\right)=\left(-2,-2,0,0,2,0\right)$ reads $\frac{1}{2}\left(\tiny{\begin{array}[]{cccccccc|cccccccc}-1&-1&0&0&0&0&0&0&-1&1&0&0&0&0&0&0\\\ -1&1&0&0&0&0&0&0&-1&-1&0&0&0&0&0&0\\\ 0&0&-1&-1&0&0&0&0&0&0&-1&1&0&0&0&0\\\ 0&0&-1&1&0&0&0&0&0&0&-1&-1&0&0&0&0\\\ 0&0&0&0&-1&-1&0&0&0&0&0&0&-1&1&0&0\\\ 0&0&0&0&-1&1&0&0&0&0&0&0&-1&-1&0&0\\\ 0&0&0&0&0&0&-1&-1&0&0&0&0&0&0&-1&1\\\ 0&0&0&0&0&0&-1&1&0&0&0&0&0&0&-1&-1\\\ \hline\cr-1&-1&0&0&0&0&0&0&1&-1&0&0&0&0&0&0\\\ 1&-1&0&0&0&0&0&0&-1&-1&0&0&0&0&0&0\\\ 0&0&-1&-1&0&0&0&0&0&0&1&-1&0&0&0&0\\\ 0&0&1&-1&0&0&0&0&0&0&-1&-1&0&0&0&0\\\ 0&0&0&0&-1&-1&0&0&0&0&0&0&1&-1&0&0\\\ 0&0&0&0&1&-1&0&0&0&0&0&0&-1&-1&0&0\\\ 0&0&0&0&0&0&-1&-1&0&0&0&0&0&0&1&-1\\\ 0&0&0&0&0&0&1&-1&0&0&0&0&0&0&-1&-1\end{array}}\right).$ (4.48) This generates $\frac{1}{2}\left[-{\left|{0000}\right>}-{\left|{0001}\right>}-{\left|{1000}\right>}+{\left|{1001}\right>}\right]$ from ${\left|{0000}\right>}$. ##### Eigenvalues $\in\\{-\mathrm{i}_{(4)},\mathrm{i}_{(4)},1_{(8)}\\}$ The solution for $\left(\alpha_{1},\alpha_{4},\beta_{1},\beta_{2},\beta_{3},\gamma_{1}\right)=\left(-2,0,0,0,2,2\right)$ reads $\frac{1}{4}\left(\tiny{\begin{array}[]{cccccccc|cccccccc}2&1&0&-1&0&-1&0&-1&-1&2&1&0&1&0&1&0\\\ -1&2&1&0&1&0&1&0&2&1&0&-1&0&-1&0&-1\\\ 0&-1&2&1&0&-1&0&-1&1&0&-1&2&1&0&1&0\\\ 1&0&-1&2&1&0&1&0&0&-1&2&1&0&-1&0&-1\\\ 0&-1&0&-1&2&1&0&-1&1&0&1&0&-1&2&1&0\\\ 1&0&1&0&-1&2&1&0&0&-1&0&-1&2&1&0&-1\\\ 0&-1&0&-1&0&-1&2&1&1&0&1&0&1&0&-1&2\\\ 1&0&1&0&1&0&-1&2&0&-1&0&-1&0&-1&2&1\\\ \hline\cr 1&2&-1&0&-1&0&-1&0&2&-1&0&1&0&1&0&1\\\ 2&-1&0&1&0&1&0&1&1&2&-1&0&-1&0&-1&0\\\ -1&0&1&2&-1&0&-1&0&0&1&2&-1&0&1&0&1\\\ 0&1&2&-1&0&1&0&1&-1&0&1&2&-1&0&-1&0\\\ -1&0&-1&0&1&2&-1&0&0&1&0&1&2&-1&0&1\\\ 0&1&0&1&2&-1&0&1&-1&0&-1&0&1&2&-1&0\\\ -1&0&-1&0&-1&0&1&2&0&1&0&1&0&1&2&-1\\\ 0&1&0&1&0&1&2&-1&-1&0&-1&0&-1&0&1&2\end{array}}\right).$ (4.49) This generates $\frac{1}{4}\left[2{\left|{0000}\right>}-{\left|{0001}\right>}+{\left|{0011}\right>}+{\left|{0101}\right>}+{\left|{0111}\right>}+{\left|{1000}\right>}+2{\left|{1001}\right>}-{\left|{1010}\right>}-{\left|{1100}\right>}-{\left|{1110}\right>}\right]$ from ${\left|{0000}\right>}$. ##### Eigenvalues $\in\\{-\mathrm{i}_{(4)},\mathrm{i}_{(4)},-1_{(4)},1_{(4)}\\}$ The solution for $\left(\alpha_{1},\alpha_{4},\beta_{1},\beta_{2},\beta_{3},\gamma_{1}\right)=\left(-2,0,0,2,0,0\right)$ reads $\frac{1}{2}\left(\tiny{\begin{array}[]{cccccccc|cccccccc}0&-1&0&-1&0&0&0&0&-1&0&1&0&0&0&0&0\\\ -1&0&1&0&0&0&0&0&-1&0&-1&0&0&0&0&0\\\ 0&-1&0&-1&0&0&0&0&1&0&-1&0&0&0&0&0\\\ 1&0&-1&0&0&0&0&0&0&-1&0&-1&0&0&0&0\\\ 0&0&0&0&0&-1&0&-1&0&0&0&0&-1&0&1&0\\\ 0&0&0&0&-1&0&1&0&0&0&0&0&0&-1&0&-1\\\ 0&0&0&0&0&-1&0&-1&0&0&0&0&1&0&-1&0\\\ 0&0&0&0&1&0&-1&0&0&0&0&0&0&-1&0&-1\\\ \hline\cr-1&0&-1&0&0&0&0&0&0&-1&0&1&0&0&0&0\\\ 0&-1&0&1&0&0&0&0&-1&0&-1&0&0&0&0&0\\\ -1&0&-1&0&0&0&0&0&0&1&0&-1&0&0&0&0\\\ 0&1&0&-1&0&0&0&0&-1&0&-1&0&0&0&0&0\\\ 0&0&0&0&-1&0&-1&0&0&0&0&0&0&-1&0&1\\\ 0&0&0&0&0&-1&0&1&0&0&0&0&-1&0&-1&0\\\ 0&0&0&0&-1&0&-1&0&0&0&0&0&0&1&0&-1\\\ 0&0&0&0&0&1&0&-1&0&0&0&0&-1&0&-1&0\end{array}}\right).$ (4.50) This generates $\frac{1}{2}\left[-{\left|{0001}\right>}+{\left|{0011}\right>}-{\left|{1000}\right>}-{\left|{1010}\right>}\right]$ from ${\left|{0000}\right>}$. ### 4.4 4-qubits from 2-qubits via Temperley-Lieb generators Up to this point we used the qubit representations of (2.20) in writing down the $(d,2,1)$-, $(d,3,2)$-, $(d,4,2)$\- and $(d,4,3)$-generalized Yang-Baxter operators. As noted in Sec. 2, if we instead use the Temperley-Lieb representation we would obtain the generalized Yang-Baxter operators that solve the $(d,4,2)$-, $(d,6,4)$-, $(d,8,4)$-, and $(d,8,6)$-gYBEs. In what follows we discuss the $(d,4,2)$-generalized $R$-matrices in detail. Note that far-commutativity is satisfied for each of these operators, thus leading to braid group representations. By applying the realization (2.22) and (2.23) to (4.1) for the 2-qubit case, we obtain the generalized Yang-Baxter operator for 4-qubits at the sites $2i-1,2i,2i+1,2i+2$: $R_{i}=s_{2i-1,\,2i+1}s_{2i,\,2i+2}\left(1+\alpha e_{2i-1}+\beta e_{2i+1}+\gamma e_{2i-1}e_{2i+1}\right).$ (4.51) It is clear that this solves $R_{i}R_{i+1}R_{i}=R_{i+1}R_{i}R_{i+1}$, but this should be recognized as the $(d,4,2)$\- gYBE instead of $(d,4,1)$-gYBE. Note that $R_{i}$ has a nontrivial support on the sites $2i-1$ to $2i+2$, whereas $R_{i+1}$ has that on the sites $2i+1$ to $2i+4$. A shift of the index $i$ of the $R$-matrix by one corresponds to a shift of the sites by two. The far- commutativity relation is satisfied in the sense of $R_{i}R_{j}=R_{j}R_{i}$ for $|i-j|>1$. If the Temperley-Lieb generators are expressed by hermitian matrices, we can see that the solution is unitary for $\alpha=\frac{1}{\Delta}(e^{\mathrm{i}\theta}-1),\qquad\beta=\frac{1}{\Delta}(e^{\mathrm{i}\varphi}-1),\qquad\gamma=\frac{1}{\Delta^{2}}(e^{\mathrm{i}\phi}-e^{\mathrm{i}\theta}-e^{\mathrm{i}\varphi}+1)$ (4.52) with $\Delta\equiv Q+Q^{-1}$, and $\theta$, $\varphi$ and $\phi$ angles taking any value. For real $\alpha$, $\beta$ and $\gamma$, there are 8 unitary points: $\displaystyle(\alpha,\beta,\gamma)$ $\displaystyle=$ $\displaystyle\large\\{(0,0,0),\left(0,0,-\frac{2}{\Delta^{2}}\right),\left(0,-\frac{2}{\Delta},0\right),\left(-\frac{2}{\Delta},0,0\right),$ (4.53) $\displaystyle\left(0,-\frac{2}{\Delta},\frac{2}{\Delta^{2}}\right),\left(-\frac{2}{\Delta},0,\frac{2}{\Delta^{2}}\right),\left(-\frac{2}{\Delta},-\frac{2}{\Delta},\frac{2}{\Delta^{2}}\right),\left(-\frac{2}{\Delta},-\frac{2}{\Delta},\frac{4}{\Delta^{2}}\right)\large\\}.$ As a representation of the Temperley-Lieb generators for qubits at the sites $i$ and $i+1$, we use the following matrix [36] $e_{i}=\left(Q{\left|{01}\right>}-{\left|{10}\right>}\right)\left({\left<{01}\right|}-Q^{-1}{\left<{10}\right|}\right)=\begin{pmatrix}0&0&0&0\\\ 0&Q&-1&0\\\ 0&-1&Q^{-1}&0\\\ 0&0&0&0\end{pmatrix}.$ (4.54) Then, the eigenvalues of the $R$-matrix (4.51) with (4.52) are $\\{1_{(6)},\,-1,\,e^{\mathrm{i}\phi},\,\pm e^{\frac{\mathrm{i}}{2}(\theta+\varphi)}_{(3)}\\}$, showing that the $R$-matrix generates the infinite-dimensional braid group. Note that these are independent of the parameter $Q$. In what follows, we discuss the corresponding entangled states for the non-trivial cases of (4.53). We obtain the $R$-matrices which are not equivalent with those in the $(d,4,2)$-cases in the previous subsection. ##### Case I: $\left(0,0,-\frac{2}{\Delta^{2}}\right)$ The solution satisfies $R_{i}^{2}=1$, and generates entangled states as $\displaystyle R_{i}{\left|{0101}\right>}$ $\displaystyle=$ $\displaystyle\left(1-\frac{2Q^{2}}{\Delta^{2}}\right){\left|{0101}\right>}-\frac{2}{\Delta^{2}}{\left|{1010}\right>}+\frac{2Q}{\Delta^{2}}\left({\left|{0110}\right>}+{\left|{1001}\right>}\right),$ $\displaystyle R_{i}{\left|{1010}\right>}$ $\displaystyle=$ $\displaystyle\left(1-\frac{2Q^{-2}}{\Delta^{2}}\right){\left|{1010}\right>}-\frac{2}{\Delta^{2}}{\left|{0101}\right>}+\frac{2Q^{-1}}{\Delta^{2}}\left({\left|{0110}\right>}+{\left|{1001}\right>}\right),$ $\displaystyle R_{i}{\left|{0110}\right>}$ $\displaystyle=$ $\displaystyle\frac{Q^{2}+Q^{-2}}{\Delta^{2}}{\left|{1001}\right>}-\frac{2}{\Delta^{2}}{\left|{0110}\right>}+\frac{2Q}{\Delta^{2}}{\left|{0101}\right>}+\frac{2Q^{-1}}{\Delta^{2}}{\left|{1010}\right>},$ $\displaystyle R_{i}{\left|{1001}\right>}$ $\displaystyle=$ $\displaystyle\frac{Q^{2}+Q^{-2}}{\Delta^{2}}{\left|{0110}\right>}-\frac{2}{\Delta^{2}}{\left|{1001}\right>}+\frac{2Q}{\Delta^{2}}{\left|{0101}\right>}+\frac{2Q^{-1}}{\Delta^{2}}{\left|{1010}\right>}.$ (4.55) For the other states, $R_{i}$ does not generate entanglement. We can see that each of the four states in (4.55) is SLOCC equivalent to ${\left|{0000}\right>}+{\left|{1111}\right>}+\lambda\left({\left|{0011}\right>}+{\left|{1100}\right>}\right),$ (4.56) with some coefficient $\lambda$. This falls into what is called $G_{abcd}$ in [32] with $a=1+\lambda,b=c=0,d=1-\lambda$, or into what is called the class of ${\rm span}\\{0_{k}\Psi,0_{k}\Psi\\}$ in [34]. For example, the first state in (4.55) is mapped to (4.56) by successively operating the following two ILOs: $1\otimes X\otimes 1\otimes X,\quad{\rm diag}\left(\left(1-\frac{2Q}{\Delta^{2}}\right)^{-1/4},\left(-\frac{2}{\Delta^{2}}\right)^{-1/4}\right)^{\otimes 4}.$ (4.57) The matrix $R_{i}$ has eigenvalues $1_{(9)}$ and $-1_{(7)}$, which is inequivalent to any of (4.39)-(4.42). ##### Case II: $\left(0,-\frac{2}{\Delta},0\right)$ The solution satisfies $R_{i}^{4}=1$ (but $R_{i}^{2},R_{i}^{3}\neq 1$), and generates entangled states as the form of $(\mbox{the Bell state})\otimes(\mbox{separable 2-qubit state})$. The eigenvalues of $R_{i}$ are $\\{-\mathrm{i}_{(3)},\mathrm{i}_{(3)},-1_{(4)},1_{(6)}\\}$, which is inequivalent to any of (4.39)-(4.42). ##### Case III: $\left(-\frac{2}{\Delta},0,0\right)$ This case provides essentially the same entanglement as the case II, since the two $R$-matrices are connected by swapping sites as $\left.R_{i}\right|_{{\rm case\,III}}=s_{2i-1,2i+1}s_{2i,2i+1}\left(\left.R_{i}\right|_{{\rm case\,II}}\right)s_{2i-1,2i+1}s_{2i,2i+1}.$ (4.58) ##### Case IV: $\left(0,-\frac{2}{\Delta},\frac{2}{\Delta^{2}}\right)$ The solution satisfies $R_{i}^{4}=1$ (but $R_{i}^{2},R_{i}^{3}\neq 1$), and gives entangled states as $\displaystyle R_{i}{\left|{0101}\right>}$ $\displaystyle=$ $\displaystyle\left(1-\frac{2Q}{\Delta}+\frac{2Q^{2}}{\Delta^{2}}\right){\left|{0101}\right>}+\frac{2}{\Delta^{2}}{\left|{1010}\right>}-\frac{2Q}{\Delta^{2}}{\left|{0110}\right>}+\frac{2Q^{-1}}{\Delta^{2}}{\left|{1001}\right>},$ $\displaystyle R_{i}{\left|{1010}\right>}$ $\displaystyle=$ $\displaystyle\left(1-\frac{2Q^{-1}}{\Delta}+\frac{2Q^{-2}}{\Delta^{2}}\right){\left|{1010}\right>}+\frac{2}{\Delta^{2}}{\left|{0101}\right>}+\frac{2Q}{\Delta^{2}}{\left|{0110}\right>}-\frac{2Q^{-1}}{\Delta^{2}}{\left|{1001}\right>},$ $\displaystyle R_{i}{\left|{0110}\right>}$ $\displaystyle=$ $\displaystyle\left(1-\frac{2Q^{-1}}{\Delta}+\frac{2}{\Delta^{2}}\right){\left|{1001}\right>}+\frac{2}{\Delta^{2}}{\left|{0110}\right>}+\frac{2Q^{-1}}{\Delta^{2}}\left({\left|{0101}\right>}-{\left|{1010}\right>}\right),$ $\displaystyle R_{i}{\left|{1001}\right>}$ $\displaystyle=$ $\displaystyle\left(1-\frac{2Q}{\Delta}+\frac{2}{\Delta^{2}}\right){\left|{0110}\right>}+\frac{2}{\Delta^{2}}{\left|{1001}\right>}+\frac{2Q}{\Delta^{2}}\left({\left|{1010}\right>}-{\left|{0101}\right>}\right).$ (4.59) Operating $R_{i}$ on any of the four states ${\left|{0001}\right>},{\left|{0010}\right>},{\left|{1101}\right>},{\left|{1110}\right>}$ generates entangled states like $(\mbox{the Bell state})\otimes(\mbox{separable 2-qubit state})$. For the other states, $R_{i}$ gives product states. The four states in (4.59) are SLOCC equivalent to (4.56). For example, successive operations of the three ILOs $1\otimes X\otimes 1\otimes X,\quad 1\otimes\begin{pmatrix}iQ^{-1}&\\\ &1\end{pmatrix}\otimes\begin{pmatrix}-iQ&\\\ &1\end{pmatrix}\otimes 1,\quad{\rm diag}\left(\left(1-\frac{2Q}{\Delta}+\frac{2Q^{2}}{\Delta^{2}}\right)^{-1/4},\left(\frac{2}{\Delta^{2}}\right)^{-1/4}\right)^{\otimes 4}$ (4.60) map the first state in (4.59) to (4.56). The eigenvalues of $R_{i}$ are $\\{-\mathrm{i}_{(3)},\mathrm{i}_{(3)},-1_{(3)},1_{(7)}\\}$, which is inequivalent to any of (4.39)-(4.42). ##### Case V: $\left(-\frac{2}{\Delta},0,\frac{2}{\Delta^{2}}\right)$ This case is essentially the same as the case IV, because $\left.R_{i}\right|_{{\rm case\,V}}=s_{2i-1,2i+1}s_{2i,2i+1}\left(\left.R_{i}\right|_{{\rm case\,IV}}\right)s_{2i-1,2i+1}s_{2i,2i+1}.$ (4.61) ##### Case VI: $\left(-\frac{2}{\Delta},-\frac{2}{\Delta},\frac{2}{\Delta^{2}}\right)$ The solution satisfies $R_{i}^{2}=1$, and generates entangled states as $\displaystyle R_{i}{\left|{0101}\right>}$ $\displaystyle=$ $\displaystyle\left(1-\frac{4Q}{\Delta}+\frac{2Q^{2}}{\Delta^{2}}\right){\left|{0101}\right>}+\frac{2}{\Delta^{2}}{\left|{1010}\right>}+\frac{2Q^{-1}}{\Delta^{2}}\left({\left|{0110}\right>}+{\left|{1001}\right>}\right),$ $\displaystyle R_{i}{\left|{1010}\right>}$ $\displaystyle=$ $\displaystyle\left(1-\frac{4Q^{-1}}{\Delta}+\frac{2Q^{-2}}{\Delta^{2}}\right){\left|{1010}\right>}+\frac{2}{\Delta^{2}}{\left|{0101}\right>}+\frac{2Q}{\Delta^{2}}\left({\left|{0110}\right>}+{\left|{1001}\right>}\right),$ $\displaystyle R_{i}{\left|{0110}\right>}$ $\displaystyle=$ $\displaystyle\left(-1+\frac{2}{\Delta^{2}}\right){\left|{1001}\right>}+\frac{2}{\Delta^{2}}{\left|{0110}\right>}+\frac{2Q}{\Delta^{2}}{\left|{1010}\right>}+\frac{2Q^{-1}}{\Delta^{2}}{\left|{0101}\right>},$ $\displaystyle R_{i}{\left|{1001}\right>}$ $\displaystyle=$ $\displaystyle\left(-1+\frac{2}{\Delta^{2}}\right){\left|{0110}\right>}+\frac{2}{\Delta^{2}}{\left|{1001}\right>}+\frac{2Q}{\Delta^{2}}{\left|{1010}\right>}+\frac{2Q^{-1}}{\Delta^{2}}{\left|{0101}\right>},$ (4.62) which are SLOCC equivalent to (4.56). The states ${\left|{0001}\right>}$, ${\left|{1110}\right>}$ and their permutations with respect to the sites provide the direct product of the Bell state and a separable 2-qubit state by acting with $R_{i}$. For the other states, $R_{i}$ gives product states. The $R_{i}$ has the eigenvalues $1_{(9)}$ and $-1_{(7)}$, which is inequivalent to any of (4.39)-(4.42). ##### Case VII: $\left(-\frac{2}{\Delta},-\frac{2}{\Delta},\frac{4}{\Delta^{2}}\right)$ Since the solution in this case is factorized as $R_{i}=s_{2i-1,\,2i+1}\left(1-\frac{2}{\Delta}e_{2i-1}\right)s_{2i\,2i+2}\left(1-\frac{2}{\Delta}e_{2i+1}\right),$ (4.63) it is easy to see that $R_{i}^{2}=1$, and the entangled states obtained fall into a class of the direct product of the two Bell states or the direct product of the Bell state and a separable 2-qubit state. The eigenvalues of $R_{i}$ are $1_{(10)}$ and $-1_{(6)}$, which is inequivalent to any of (4.39)-(4.42). Although various inequivalent solutions would be obtained by choosing other representations of the Temperley-Lieb algebra, we leave this issue as a future subject. ### 4.5 Algorithm for multi-qubit generalized $R$-matrices The ansatze in (4.1), (4.16), (4.35), and (4.43) act as a guide for constructing the generalized $R$-matrices in the multi-qubit case, while using the available operators: $s_{j}~{}j=i,\cdots,i+m-2$ and $p_{j}~{}j=i,\cdots,i+m-1$. Throughout we assume that $2l\geq m$ in order to ensure far-commutativity in addition to obeying the $(d,m,l)$-gYBE. The generalized $R$-matrix that satisfies the $(d,m,l)$-gYBE and is made up of products of the $p_{i}$ operators is $\displaystyle R_{i}$ $\displaystyle=$ $\displaystyle s_{i,i+l}\Big{[}1+\sum_{r=1}^{m-1}\Big{(}\sum_{\begin{subarray}{c}k_{1},\cdots,k_{r-1}=1\\\ 0<k_{1}<\cdots<k_{r-1}<l\end{subarray}}^{m-1}\alpha^{(r)}_{0,k_{1},\cdots,k_{r-1}}p_{i}\prod_{j=1}^{r-1}p_{i+k_{j}}+\sum_{\begin{subarray}{c}k_{1},\cdots,k_{r-1}=1\\\ 0<k_{1}<\cdots<k_{r-1}<l\end{subarray}}^{m-1}\alpha^{(r)}_{l,k_{1},\cdots,k_{r-1}}p_{i+l}\prod_{j=1}^{r-1}p_{i+k_{j}}$ (4.64) $\displaystyle\hskip 85.35826pt+\sum_{\begin{subarray}{c}k_{1},\cdots,k_{r-2}=1\\\ 0<k_{1}<\cdots<k_{r-2}<l\end{subarray}}^{m-1}\alpha^{(r)}_{0,k_{1},\cdots,k_{r-2},l}p_{i}p_{i+l}\prod_{j=1}^{r-2}p_{i+k_{j}}\Big{)}+\alpha^{(m)}_{0,1,\cdots,m-1}\prod_{j=0}^{m-1}p_{i+j}\Big{]},$ where $s_{i,i+l}=s_{i+l-1}\cdots s_{i+1}s_{i}s_{i+1}\cdots s_{i+l-1}$ swaps the sites $i$ and $i+l$. For $r=1$, $\prod_{j=1}^{r-1}p_{i+k_{j}}$ and $\prod_{j=1}^{r-2}p_{i+k_{j}}$ should be regarded as $1$ and $0$, respectively. The coefficients can be determined by requiring this to satisfy $R_{i}R_{i+l}R_{i}=R_{i+l}R_{i}R_{i+l}$. This computation can be done analytically, but it is tedious. At the moment we do not have a general analytic expression for the generalized $R$-matrix, but this algorithm works as studied in the cases of three and four qubits. As seen in those cases, we expect to obtain a family of both unitary and non-unitary generalized $R$-matrices. The image of the braid group representations using the non- unitary matrices and the complex unitary matrices will be infinite, whereas the image of the braid group representations of the real unitary matrices is expected to be finite. Analogous generalized $R$-matrices solving the $(d,m,l)$-gYBE can be constructed using the operators $s_{i,i+l}$ and the powers of $p_{k_{1},k_{2}}$ where at least one of $k_{1},k_{2}$ is either $i$ or $i+l$. We omit the general expression for such an operator here as it is straightforward. ## 5 Comparison with known generalized $R$-matrices The known unitary 3-qubit generalized $R$-matrices are the GHZ matrix obtained from the extraspecial 2-group generators in [10], the generalization of the Rowell solutions in [21], and the solutions obtained from ribbon fusion categories in [12, 13]. We now show that the four unitary 3-qubit solutions in Table 2 are inequivalent to all of the solutions above. ##### Non-equivalence to the GHZ matrix The GHZ matrix that solves the $(2,3,2)$-gYBE is given by $R_{\textrm{GHZ}}=\frac{1}{\sqrt{2}}~{}\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&1\\\ 0&1&0&0&0&0&1&0\\\ 0&0&1&0&&1&0&0\\\ 0&0&0&1&1&0&0&0\\\ 0&0&0&-1&1&0&0&0\\\ 0&0&-1&0&0&1&0&0\\\ 0&-1&0&0&0&0&1&0\\\ -1&0&0&0&0&0&0&1\end{array}\right)$ (5.1) and has eigenvalues $e^{\pm\mathrm{i}\frac{\pi}{4}}$, both with multiplicity 4. The eigenvalues of the complex unitary 3-qubit matrices of the form (4.16) have eigenvalues that cannot be mapped to the eigenvalues of the GHZ matrix either by an inversion or by a scalar multiplication and thus we conclude that at the unitary points these solutions are inequivalent to the GHZ matrix. At the same time the eigenvalues of the four real unitary 3-qubit matrices in Table 2 cannot be mapped to these eigenvalues by a scalar multiplication or an inversion, leading us to the conclusion that those matrices cannot be possibly equivalent to the GHZ matrix. ##### Comparison with the generalized Rowell solutions In [21] the Rowell solutions of the form $\left(\begin{array}[]{cc}X&0\\\ 0&Y\end{array}\right)=X\oplus Y$, are generalized to obtain three families of solutions. They have eigenvalues from the sets $\\{e^{-\mathrm{i}\frac{\pi}{12}},e^{-\mathrm{i}\frac{\pi}{12}},e^{\mathrm{i}\frac{7\pi}{12}},e^{\mathrm{i}\frac{7\pi}{12}}\\}$, $\\{e^{-\mathrm{i}\frac{\pi}{4}},-e^{-\mathrm{i}\frac{\pi}{4}},e^{\mathrm{i}\frac{\pi}{4}},e^{\mathrm{i}\frac{\pi}{4}}\\}$ and $\\{e^{-\mathrm{i}\frac{\pi}{4}},e^{-\mathrm{i}\frac{\pi}{4}},e^{\mathrm{i}\frac{\pi}{4}},e^{\mathrm{i}\frac{\pi}{4}}\\}$, respectively. However, these solve the $(2,3,1)$-gYBEs and hence cannot be compared to the solutions in this paper, which solve instead the $(2,3,2)$-gYBE. ##### Comparison with the Kitaev-Wang solutions The Kitaev-Wang solutions in [12] and their qubit realizations in [13] solve the $(d,3,1)$-gYBE, whereas the methods presented in this paper generate the generalized Yang-Baxter operators that solve the $(d,3,2)$-gYBE. Thus we cannot compare the two generalized $R$-matrices. ## 6 Outlook In this paper we have used the notion of partition algebras to introduce a solution-generating technique to obtain parameter-independent generalized $R$-matrices. This is quite remarkable, since solving the gYBE is a notoriously difficult task. This is especially true for the parameter- independent gYBE, for which very few solutions are known in the literature. In some recent work [16], we have focused on the parameter-dependent case, using supersymmetry algebras instead of partition algebras. In that case, the relation between $R$-matrices and braid group representations was not very clear, so that the present work should be considered as an improvement of our previous analysis. Improved as it may be, we need however to remark that the method based on partition algebras has certain limitations. The main issue is that not all SLOCC classes of entangled states seem to be obtainable from the generalized $R$-matrices via partition algebras. In particular, we do not obtain the W-state SLOCC class of a 3-qubit system in the case that $R$ is a unitary braid operator. We suspect that this absence of the W-state class is not peculiar to 3-qubit systems, and it might extend to the multi-qubit case as well. It would be interesting to check whether this is true, although it would represent a very laborious computation. Understanding such a distinction of the W states, in particular from the GHZ states, would be relevant to various applications in quantum information and quantum computing. The GHZ states become unentangled bipartite mixed states after tracing out one qubit, while the W states are still entangled. Thus, the GHZ states and their multipartite analogs are fragile against particle loss and suitable for quantum secret sharing. On the other hand, the W states and their multiqubit versions are robust against loss with possible application to quantum memories, multiparty quantum network protocols and universal quantum cloning machines. A new technique to construct solutions to generalized Yang-Baxter equation is presented via interplay with $k$-graphs in [37]. It will also interesting to do similar analysis on the entanglement generated through possible novel solutions obtained from the technique. Of course, the physical realization of braiding operators is of the utmost interest. A natural next step would then be to try to identify the anyons corresponding to these representations and study their computational power. This could possibly help identifying the unitary modular tensor categories that describe these anyons [38]. On a complementary direction, one could construct new integrable spin chains upon Baxterizing the 2-qubit $R$-matrices. In particular, using the Temperley- Lieb representations of the 2-qubit $R$-matrices one could obtain new 4-site interaction spin chains that are integrable. This is something on which we hope to report at some point in the future. ##### Note added After the conclusion of this work, we have found nonunitary braid operators corresponding to W states [39], which were left out of the present analysis. We have managed this by using partition algebras for W states in a 3-qubit space and extraspecial 2-groups for the 4-qubit space case. ### Acknowledgements We thank Z. Wang for his comments on the manuscript. 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2024-09-04T02:54:55.858575

2003.00262

arxiv-papers

# The Rate Distortion Function of Asynchronously Sampled Memoryless Cyclostationary Gaussian Processes Emeka Abakasanga, Nir Shlezinger, Ron Dabora E. Abakasanga and R. Dabora are with the department of ECE, Ben-Gurion University, Israel (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>N. Shlezinger is with the faculty of Math and CS, Weizmann Institute of Science, Israel (e-mail: [email protected]). This work was supported by the Israel Science Foundation under Grants 1685/16 and 0100101, and by the Israeli Ministry of Economy through the HERON 5G consortium. ###### Abstract Man-made communications signals are typically modelled as continuous-time (CT) wide-sense cyclostationary (WSCS) processes. As modern processing is digital, it operates on sampled versions of the CT signals. When sampling is applied to a CT WSCS process, the statistics of the resulting discrete-time (DT) process depends on the relationship between the sampling interval and the period of the statistics of the CT process: When these two parameters have a common integer factor, then the DT process is WSCS. This situation is referred to as synchronous sampling. When this is not the case, which is referred to as asynchronous sampling, the resulting DT process is wide-sense almost cyclostationary (WSACS). Such acquired CT processes are commonly encoded using a source code to facilitate storage or transmission over multi-hop networks using compress-and-forward relaying. In this work, we study the fundamental tradeoff of sources codes applied to sampled CT WSCS processes, namely, their rate-distortion function (RDF). We note that while RDF characterization for the case of synchronous sampling directly follows from classic information- theoretic tools utilizing ergodicity and the law of large numbers, when sampling is asynchronous, the resulting process is not information stable. In such cases, commonly used information-theoretic tools are inapplicable to RDF analysis, which poses a major challenge. Using the information spectrum framework, we show that the RDF for asynchronous sampling in the low distortion regime can be expressed as the limit superior of a sequence of RDFs in which each element corresponds to the RDF of a synchronously sampled WSCS process (but their limit is not guaranteed to exist). The resulting characterization allows us to introduce novel insights on the relationship between sampling synchronization and RDF. For example, we demonstrate that, differently from stationary processes, small differences in the sampling rate and the sampling time offset can notably affect the RDF of sampled CT WSCS processes. ## I Introduction Man-made signals are typically generated using a repetitive procedure, which takes place at fixed intervals. The resulting signals are thus commonly modeled as continuous-time (CT) random processes exhibiting periodic statistical properties [1, 2, 3], which are referred to as wide-sense cyclostationary (WSCS) processes. In digital communications, where the transmitted waveforms commonly obey the WSCS model [3], the received CT signal is first sampled to obtain a discrete-time (DT) received signal. In the event that the sampling interval is commensurate with the period of the statistics of the CT WSCS signal, cyclostationarity is preserved in DT [3, Sec. 3.9]. In this work, we refer to this situation as synchronous sampling. However, it is practically common to encounter scenarios in which the sampling rate at the receiver and symbol rate of the received CT WSCS process are incommensurate, which is referred to as asynchronous sampling. The resulting sampled process in such cases is a DT wide-sense almost cyclostationary (WSACS) stochastic process [3, Sec. 3.9]. This research aims at investigating lossy source coding for asynchronously sampled CT WSCS processes. In the source coding problem, every sequence of information symbols from the source is mapped into a sequence of code symbols, referred to as codewords, taken from a predefined codebook. In lossy source coding, the source sequence is recovered up to a predefined distortion constraint, within an arbitrary small tolerance of error. The figure-of-merit for lossy source coding is the rate-distortion function (RDF) which characterizes the minimum number of bits per symbol required to compress the source sequence such that it can be reconstructed at the decoder within the specified maximal distortion [4]. For an independent and identically distributed (IID) random source process, the RDF can be expressed as the minimum mutual information between the source variable and the reconstruction variable, such that for the corresponding conditional distribution of the reconstruction symbol given the source symbol, the distortion constraint is satisfied [5, Ch. 10]. The source coding problem has been further studied in multiple different scenarios, including the reconstruction of a single source at multiple destinations [6] and the reconstruction of multiple correlated stationary Gaussian sources at a single destination [7, 8, 9]. For stationary source processes, ergodicity theory and the asymptotic equipartition property (AEP) [5, Ch. 3] were applied for characterizing the RDF for different scenarios [10, Ch. 9], [4, Sec. I], [11]. However, as in a broad range of applications, including digital communication networks, most CT signals are WSCS, the sampling operation results in a DT source signal whose statistics depends on the relationship between the sampling rate and the period of the statistics of the source signal. When sampling is synchronous, the resulting DT source signal is WSCS [3, Sec. 3.9]. The RDF for lossy compression of DT WSCS Gaussian sources with memory was studied in [12]. This used the fact that any WSCS signal can be transformed into a set of stationary subprocess [2]; thereby facilitating the application of information-theoretic results obtained for multivariate stationary sources to the derivation of the RDF; Nonetheless, in many digital communications scenarios, the sampling rate and the symbol rate of the CT WSCS process are not related in any way, and are possibly incommensurate, resulting in a sampled process which is a DT WSACS stochastic process [3, Sec. 3.9]. Such situations can occur as a result of the a-priori determined values of the sampling interval and the symbol duration of the WSCS source signal, as well as due to sampling clock jitters resulting from hardware impairments. A comprehensive review of trends and applications for almost cyclostationary signals can be found in [13]. Despite their apparent frequent occurrences, the RDF for lossy compression of WSACS sources was not characterized, which is the motivation for the current research. A major challenge associated with characterizing fundamental limits for asynchronously sampled WSCS processes stems from the fact that the resulting processes are not information stable, in the sense that its conditional distribution is not ergodic [14, Page X], [15], [16]. As a result, the standard information-theoretic tools cannot be employed, making the characterization of the RDF a very challenging problem. Our recent study in [17] on channel coding reveals that for the case of additive CT WSCS Gaussian noise, capacity varies significantly with sampling rates, whether the Nyquist criterion is satisfied or not. In particular, it was observed that the capacity can change dramatically with minor variations in the sampling rate, causing it to switch from synchronous sampling to asynchronous sampling. This is in direct contrast to the results obtained for wide-sense stationary noise for which the capacity remains unchanged for any sampling rate above the Nyquist rate [18]. A natural fundamental question that arises from this result is how the RDF of a sampled Gaussian source process varies with the sampling rate. As a motivating example, one may consider compress-and-forward (CF) relaying, where the relay samples at a rate which can be incommensurate with the symbol rate of the incoming communications signal. In this work, we employ the information spectrum framework [14] in characterizing the RDF of asynchronously sampled memoryless Gaussian WSCS processes, as this framework is applicable to the information-theoretic analysis of non information-stable processes [14, Page VII]. We further note that while rate characterizations obtained using information spectrum tools and its associated quantities may be difficult to evaluate [14, Remark 1.7.3], here we obtain a numerically computable characterization of the RDF. In particular, we focus on the mean squared error (MSE) distortion measure in the low distortion regime, namely, source codes for which the average MSE of the difference between the source and the reproduction process is not larger than the minimal source variance. The results of this research lead to accurate modelling of signal compression in current and future digital communications systems. Furthermore, we utilize our characterization of the RDF RDF for a sampled CT WSCS Gaussian source with different sampling rates and sampling time offsets. We demonstrate that, differently from stationary signals, when applying a lossy source code a sampled WSCS process, the achievable rate- distortion tradeoff can be significantly affected by minor variations in the sampling time offset and the sampling rate. Our results thus allow identifying the sampling rate and sampling time offsets which minimize the RDF in systems involving asynchronously sampled WSCS processes. The rest of this work is organised as follows: Section II provides a scientific background on cyclostationary processes, and on rate-distortion analysis of DT WSCS Gaussian sources. Section III presents the problem formulation and auxiliary results, and Section IV details the main result of RDF characterization for sampled WSCS Gaussian process. Numerical examples and discussions are addressed in Section V, and Section VI concludes the paper. ## II Preliminaries and Background In the following we review the main tools and framework used in this work: In Subsection II-A, we detail the notations. In Subsection II-B we review the basics of cyclostationary processes and the statistical properties of a DT process resulting from sampling a CT WSCS process. In Subsection II-C, we recall some preliminaries in rate-distortion theory, and present the RDF for a DT WSCS Gaussian source process. This background creates a premise for the statement of the main result provided in Section IV of this paper. ### II-A Notations In this paper, random vectors are denoted by boldface uppercase letters, e.g., ${\boldsymbol{X}}$; boldface lowercase letters denote deterministic column vectors, e.g., ${\boldsymbol{x}}$. Scalar RVs and deterministic values are denoted via standard uppercase and lowercase fonts respectively, e.g., $X$ and $x$. Scalar random processes are denoted with $X(t),t\in\mathcal{R}$ for CT and with $X[n],n\in\mathcal{Z}$ for DT. Uppercase Sans-Serif fonts represent matrices, e.g., $\mathsf{A}$, and the element at the $i^{th}$ row and the $l^{th}$ column of $\mathsf{A}$ is denoted with $(\mathsf{A})_{i,l}$. We use $|\cdot|$ to denote the absolute value, $\lfloor d\rfloor,d\in\mathcal{R}$, to denote the floor function and $d^{+},d\in\mathcal{R}$, to denote the $\mathop{\max}\\{0,d\\}$. $\delta[\cdot]$ denotes the Kronecker delta function: $\delta[n]=1$ for $n=0$ and $\delta[n]=0$ otherwise, and $\mathbb{E}\\{\cdot\\}$ denotes the stochastic expectation. The sets of positive integers, integers, rational numbers, real numbers, positive numbers, and complex numbers are denoted by $\mathcal{N},\mathcal{Z}$, $\mathcal{Q}$, $\mathcal{R}$, $\mathcal{R}^{++}$, and $\mathcal{C}$, respectively. The cumulative distribution function (CDF) is denoted by $F_{X}(x)\triangleq\Pr{(X\leq x)}$ and the probability density function (PDF) of a CT random variable (RV) is denoted by $p_{X}(x)$. We represent a real Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$ by the notation $\mathcal{N}(\mu,\sigma^{2})$. All logarithms are taken to base-2, and $j=\sqrt{-1}$. Lastly, for any sequence $y[i]$, $i\in\mathcal{N}$, and positive integer $k\in\mathcal{N}$, ${{\boldsymbol{y}}}^{(k)}$ denotes the column vector $\big{(}{y}[1],\ldots,{y}[k]\big{)}^{T}$. ### II-B Wide-Sense Cyclostationary Random Processes Here, we review some preliminaries in the theory of cyclostationarity. We begin by recalling the definition of wide-sense cyclostationary processes: ###### Definition 1 (Wide-sense cyclostationary processes [2, Sec. 17.2]). A scalar stochastic process $\\{S(t)\\}_{t\in\mathcal{T}}$, where $\mathcal{T}$ is either discrete ($\mathcal{T}=\mathcal{Z}$) or continuous ($\mathcal{T}=\mathcal{R}$) is called WSCS if both its first-order and its second-order moments are periodic with respect to $t\in\mathcal{T}$ with some period $N_{p}\in\mathcal{T}$. WSCS signal are thus random processes whose first and second-order moments are periodic functions. To define WSACS signals, we first recall the definition of almost-periodic functions: ###### Definition 2 (Almost-periodic-function [19]). A function $x(t),t\in\mathcal{T}$ where $\mathcal{T}$ is either discrete ($\mathcal{T}=\mathcal{Z}$) or continuous ($\mathcal{T}=\mathcal{R}$), is called an almost-periodic function if for every $\epsilon>0$ there exists a number $l(\epsilon)>0$ with the property that any interval in $\mathcal{T}$ of length $l(\epsilon)$ contains a $\tau$, such that $|x(t+\tau)-x(t)|<\epsilon,\quad\forall t\in\mathcal{T}.$ ###### Definition 3 (Wide-sense almost-cyclostationary processes [2, Def. 17.2]). A scalar stochastic process $\left\\{S(t)\right\\}_{t\in\mathcal{T}}$ where $\mathcal{T}$ is either discrete ($\mathcal{T}=\mathcal{Z}$) or continuous ($\mathcal{T}=\mathcal{R}$), is called WSACS if its first and its second order moments are almost-periodic functions with respect to $t\in\mathcal{T}$. The DT WSCS model is commonly used in the communications literature, as it facilitates the the analysis of many problems of interest, such as fundamental rate limits analysis [20, 21, 22], channel identification [23], synchronization [24], and noise mitigation [25]. However, in many scenarios, the considered signals are WSACS rather than WSCS. To see how the WSACS model is obtained in the context of sampled signals, we briefly recall the discussion in [17] on sampled WSCS processes (please refer to [17, Sec. II.B] for more details): Consider a CT WSCS random process $S(t)$, which is sampled uniformly with a sampling interval of $T_{\rm s}$ and sampling time offset $\phi$, resulting in a DT random process $S[i]=S(i\cdot T_{\rm s}+\phi)$. It is well known that contrary to stationary processes, which have a time- invariant statistical characteristics, the values of $T_{\rm s}$ and $\phi$ have a significant effect on the statistics of sampled WSCS processes [17, Sec. II.B]. To demonstrate this point, consider a CT WSCS process with variance $\sigma_{s}^{2}(t)=\frac{1}{2}\cdot\sin\left(2\pi t/T_{\rm sym}\right)+2$ for some $T_{\rm sym}>0$. The sampled process for $\phi=0$ (no symbol time offset) and $T_{\rm s}=\frac{T_{\rm sym}}{3}$ has a variance function whose period is $N_{p}=3$: $\sigma_{s}^{2}(iT_{\rm s})=\\{2,2.433,1.567,2,2.433,1.567,\ldots\\}$, for $i=0,1,2,3,4,5,\ldots$; while the DT process obtained with the same sampling interval and the sampling time offset of $\phi=\frac{T_{\rm s}}{2\pi}$ has a periodic variance with $N_{p}=3$ with values $\sigma_{s}^{2}(iT_{\rm s}+\phi)=\\{2.155,2.335,1.510,2.155,2.335,1.510,\ldots\\}$, for $i=0,1,2,3,4,5,\ldots$, which are different from the values of the DT variance for $\phi=0$. It follows that both variances are periodic in discrete-time with the same period $N_{p}=3$, although with different values within the period, which is a result of the sampling time offset, yet, both DT processes correspond to two instances of synchronous sampling. Lastly, consider the sampled variance obtained by sampling without a time offset (i.e., $\phi=0$) at a sampling interval of $T_{\rm s}=(1+\frac{1}{2\pi})\frac{T_{\rm sym}}{3}$. For this case, $T_{\rm s}$ is not an integer divisor of $T_{\rm sym}$ or of any of its integer multiples (i.e., $\frac{T_{\rm sym}}{T_{\rm s}}=2+\frac{2\pi-2}{2\pi+1}\equiv 2+\epsilon$; where $\epsilon\not\in\mathcal{Q}$ and $\epsilon\in[0,1)$ ) resulting in the variance values $\sigma_{s}^{2}(iT_{\rm s})=\\{2,2.335,1.5027,2.405,1.896,1.75,\ldots\\}$, for $i=0,1,2,3,4,5\ldots$. For this scenario, the DT variance is not periodic but is almost-periodic, corresponding to asynchronous sampling and the resulting DT process is not WSCS but WSACS [3, Sec. 3.2]. The example above demonstrates that the statistical properties of sampled WSCS signals depend on the sampling rate and the sampling time offset, implying that the RDF of such processes should also depend on these quantities, as we demonstrate in the sequel. ### II-C The Rate-Distortion Function for DT WSCS Processes Encoder $f_{S}$Decoder $g_{S}$$\\{1,2,\ldots,2^{lR}\\}$$\\{S[i]\\}_{i=1}^{l}$$\\{\hat{S}[i]\\}_{i=1}^{l}$ Figure 1: Source coding block diagram In this subsection we review the source coding problem and the existing results on the RDF of WSCS processes. We begin by recalling the definition of a source coding scheme, see, e.g., [26, ch. 3], [5, Ch.10]: ###### Definition 4 (Source coding scheme). A source coding scheme with blocklength $l$ consists of: 1. 1. An encoder $f_{S}$ which maps a block of $l$ source samples $\\{S[i]\\}^{l}_{i=1}$ into an index from a set of $M=2^{lR}$ indexes, $f_{S}:\\{S[i]\\}_{i=1}^{l}\mapsto\\{1,2,\ldots,M\\}$. 2. 2. A decoder $g_{S}$ which maps the received index into a reconstructed sequence of length $l$, $\left\\{\hat{S}[i]\right\\}_{i=1}^{l}$, $g_{S}:\\{1,2,\ldots,M\\}\mapsto\left\\{\hat{S}[i]\right\\}_{i=1}^{l}$ The encoder-decoder pair is referred to as an $(R,l)$ source code, where $R$ is the rate of the code in bits per source sample, defined as: $R=\frac{1}{l}\log_{2}M$ (1) The RDF characterizes the minimal average number of bits per source sample, denoted $R(D)$, that can be used to encode a source process such that it can be reconstructed from its encoded representation with a recovery distortion not larger than $D>0$ [5, Sec. 10.2]. In the current work, we use the MSE distortion measure, which measures the cost of decoding a source symbol $S$ into $\hat{S}$ via $d(S,\hat{S})=\left\|S-\hat{S}\right\|^{2}$. The distortion for a sequence of source samples ${\boldsymbol{S}}^{(l)}$ decoded into a reproduction sequence $\hat{{\boldsymbol{S}}}^{(l)}$ is given by $d\left({\boldsymbol{S}}^{(l)},\hat{{\boldsymbol{S}}}^{(l)}\right)=\frac{1}{l}\sum\limits_{i=1}^{l}\left(S[i]-\hat{S}[i]\right)^{2}$ and the average distortion in decoding a random source sequence ${\boldsymbol{S}}^{(l)}$ into a random reproduction sequence $\hat{{\boldsymbol{S}}}^{(l)}$ is defined as: $\bar{d}\left({\boldsymbol{S}}^{(l)},\hat{{\boldsymbol{S}}}^{(l)}\right)\triangleq\mathds{E}\left\\{d\left({\boldsymbol{S}}^{(l)},\hat{{\boldsymbol{S}}}^{(l)}\right)\right\\}=\frac{1}{l}\sum\limits_{i=1}^{l}\mathds{E}\left\\{\left(S[i]-\hat{S}[i]\right)^{2}\right\\},$ (2) where the expectation in (2) is taken with respect to the joint probability distributions on the source $S[i]$ and its reproduction $\hat{S}[i]$. Using Def. 4 we can now formulate the achievable rate-distortion pair for a source $S[i]$, as stated in the following definition [10, Pg. 471]: ###### Definition 5 (Achievable rate-distortion pair). A rate-distortion pair $(R,D)$ is achievable for a process $\\{S[i]\\}_{i\in\mathcal{N}}$ if for any $\eta>0$ and for all sufficiently large $l$ one can construct an $\left(R_{s},l\right)$ source code such that $R_{s}\leq R+\eta.$ (3) and $\bar{d}\left({\boldsymbol{S}}^{(l)},\hat{{\boldsymbol{S}}}^{(l)}\right)\leq D+\eta.$ (4) ###### Definition 6. The rate-distortion function $R(D)$ is defined as the infimum of all achievable rates $R$ for a given maximum allowed distortion $D$. Def. 5 defines a rate-distortion pair to as that achievable using source codes with any sufficiently large blocklength. In the following lemma, which is required to characterize the RDF of DT WSCS signals, we state that it is sufficient to consider only source codes whose blocklength is an integer multiple of some fixed integer: ###### Lemma 1. Consider the process$\\{S[i]\\}_{i\in\mathcal{N}}$ with a finite and bounded variance. For a given maximum allowed distortion $D$, the optimal reproduction process $\\{\hat{S}[i]\\}_{i\in\mathcal{N}}$ is also the optimal reproduction process when restricted to using source codes whose blocklengths are integer multiples of some fixed positive integer $r$. ###### Proof. The proof of the lemma is detailed in Appendix A. ∎ This lemma facilitates switching between multivariate and scalar representations of the source and the reproduction processes. The RDF obviously depends on the distribution of the source $\\{S[i]\\}_{i\in\mathcal{N}}$. Thus, modifying the source yields a different RDF. However, when a source is scaled by some positive constant, the RDF of the scaled process with the MSE criterion can be inferred from that of the original process, as stated in the following theorem: ###### Theorem 1. Let $\\{S[i]\\}_{i\in\mathcal{N}}$ be a source process for which the rate- distortion pair $(R,D)$ is achievable under the MSE distortion. Then, for every $\alpha\in\mathcal{R}^{++}$, it holds that the rate-distortion pair $(R,\alpha^{2}\cdot D)$ is achievable for the source $\\{\alpha\cdot S[i]\\}_{i\in\mathcal{N}}$. ###### Proof. The proof to the theorem is detailed in Appendix B. ∎ Lastly, in the proof of our main result, we make use of the RDF for DT WSCS sources derived in [12, Thm. 1], repeated below for ease of reference. Prior to the statement of the theorem, we recall that for blocklenghts which are integer multiples of $N_{p}$, a WSCS process $S[i]$ with period $N_{p}>0$ can be represented as an equivalent $N_{p}$-dimensional process ${\boldsymbol{S}}^{(N_{p})}[i]$ via the decimated component decomposition (DCD) [2, Sec. 17.2]. The power spectral density (PSD) of the process ${\boldsymbol{S}}^{(N_{p})}$ is defined as [12, Sec. II]: $\bigg{(}{\boldsymbol{\rho}}_{{\boldsymbol{S}}}\left(e^{j2\pi f}\right)\bigg{)}_{u,v}=\sum\limits_{\Delta\in\mathcal{Z}}\bigg{(}\mathsf{R}_{{\boldsymbol{S}}}[\Delta]\bigg{)}_{u,v}e^{-j2\pi f\Delta}\qquad-\frac{1}{2}\leq f\leq\frac{1}{2},\quad u,v\in\\{1,2,\ldots N_{p}\\}$ (5) where $\mathsf{R}_{{\boldsymbol{S}}}[\Delta]\triangleq\mathds{E}\left\\{{\boldsymbol{S}}^{(N_{p})}[i]\cdot{\boldsymbol{S}}^{(N_{p})}[i+\Delta]\right\\}$ [2, Sec. 17.2]. We now proceed to the statement of [12, Thm. 1]: ###### Theorem 2. [12, Thm. 1] Consider a zero-mean real DT WSCS Gaussian source $S[i],i\in\mathcal{N}$ with memory, and let $N_{p}\in\mathcal{N}$ denote the period of its statistics. The RDF is expressed as: $R(D)=\frac{1}{2N_{p}}\sum\limits_{m=1}^{N_{p}}\int_{f=-0.5}^{0.5}\left(\log\left(\frac{\lambda_{m}\left(e^{j2\pi f}\right)}{\theta}\right)\right)^{+}\mathrm{d}f,$ (6a) where $\lambda_{m}\left(e^{j2\pi f}\right)$, $m=1,2,\ldots,N$ denote the eigenvalues of the PSD matrix of the process ${\boldsymbol{S}}^{(N_{p})}[i]$, which is obtained from $S[i]$ by applying $N_{p}$-dimensional DCD, and $\theta$ is selected such that $D=\frac{1}{N_{p}}\sum\limits_{m=1}^{N_{p}}\int_{f=-0.5}^{0.5}\min\left\\{\lambda_{m}\left(e^{j2\pi f}\right),\theta\right\\}\mathrm{d}f.$ (6b) We note that ${\boldsymbol{S}}^{(N_{p})}[i]$ corresponds to a vector of stationary processes whose elements are are not identically distributed; hence the variance is different for each element. Using [12, Thm. 1], we can directly obtain the RDF for the special case of a DT memoryless WSCS Gaussian process. This is stated in the following corollary: ###### Corollary 1. Let $\\{S[i]\\}_{i\in\mathcal{N}}$ be a zero-mean DT memoryless real WSCS Gaussian source with period $N_{p}\in\mathcal{N}$, and set $\sigma^{2}_{m}=\mathds{E}\\{S^{2}[m]\\}$ for $m=1,2,\ldots,N_{P}$. The RDF for compression of of $S[i]$ is stated as: $\displaystyle R(D)=\begin{cases}\frac{1}{2N_{p}}\sum\limits_{m=1}^{N_{p}}\log\left(\frac{\sigma^{2}_{m}}{D_{m}}\right)&D\leq\frac{1}{N_{p}}\sum\limits_{m=1}^{N_{p}}\sigma^{2}_{m}\\\ 0&D>\frac{1}{N_{p}}\sum\limits_{m=1}^{N_{p}}\sigma^{2}_{m},\end{cases}$ (7a) where $D_{m}\triangleq\min\left\\{\sigma^{2}_{m},\theta\right\\}$, and $\theta$ is defined such that $D=\frac{1}{N_{p}}\sum\limits_{m=1}^{N_{p}}D_{m}.$ (7b) ###### Proof. Applying Equations (6a) and (6b) to our specific case of a memoryless WSCS source, we obtain equations (7a) and (7b) as follows: First note that the corresponding DCD components for a zero-mean memoryless WSCS process are also zero-mean and memoryless; hence the PSD matrix for the multivariate process ${\boldsymbol{S}}^{(N_{p})}[i]$ is a diagonal matrix, whose eigenvalues are the constant diagonal elements such that the $m^{th}$ diagonal element is equal to the variance $\sigma_{m}^{2}$: $\lambda_{m}\left(e^{j2\pi f}\right)=\sigma_{m}^{2}$. Now, writing Eqn. (6a) for this case we obtain: $\displaystyle R(D)$ $\displaystyle=\frac{1}{2N_{p}}\sum\limits_{m=1}^{N_{p}}\int_{f=-0.5}^{0.5}\left(\log\left(\frac{\lambda_{m}\left(e^{j2\pi f}\right)}{\theta}\right)\right)^{+}\mathrm{d}f$ $\displaystyle=\frac{1}{2N_{p}}\sum\limits_{m=1}^{N_{p}}\left(\log\left(\frac{\sigma^{2}_{m}}{\theta}\right)\right)^{+}.$ (8) Since $\left(\log\left(\frac{\sigma^{2}_{m}}{\theta}\right)\right)^{+}=\max\left\\{0,\log\left(\frac{\sigma^{2}_{m}}{\theta}\right)\right\\}\equiv\log\left(\frac{\sigma^{2}_{m}}{D_{m}}\right)$ it follows that (8) coincides with (7a). Next, expressing Eqn. (6b) for the memoryless source process, we obtain: $D=\frac{1}{N_{p}}\sum\limits_{m=1}^{N_{p}}\int_{f=-0.5}^{0.5}\min\left\\{\lambda_{m}\left(e^{j2\pi f}\right),\theta\right\\}\mathrm{d}f=\frac{1}{N_{p}}\sum\limits_{m=1}^{N_{p}}\min\left\\{\sigma^{2}_{m},\theta\right\\},$ (9) proving Eqn. (7b). ∎ Now, from Lemma 1, we conclude that the RDF for compression of source sequences whose blocklength is an integer multiple of $N_{p}$ is the same as the RDF for compressing source sequences whose blocklength is arbitrary. We recall that from [5, Ch. 10.3.3] it follows that for the zero-mean memoryless Gaussian DCD vector source process ${\boldsymbol{S}}^{(N_{p})}[i]$ the optimal reproduction process which achieves the RDF is an $N_{p}\times 1$ memoryless process whose covariance matrix is diagonal with non-identically distributed elements. From [2], we can apply the inverse DCD to obtain a WSCS process. Hence, from Lemma 1 we can conclude that the optimal reproduction process for the DT WSCS Gaussian source is a DT WSCS Gaussian process. ## III Problem Formulation and Auxiliary Results Our objective is to characterize the RDF for compression of asynchronously sampled CT WSCS Gaussian sources when the sampling interval is larger than the memory of the source. In particular, we focus on the minimal rate required to achieve a high fidelity reproduction, representing the RDF curve for distortion values not larger than the variance of the source. Such characterization of the RDF for asynchronous sampling is essential for comprehending the relationship between the minimal required number of bits and the sampling rate at a given distortion. Our analysis constitutes an important step towards constructing joint source-channel coding schemes in scenarios in which the symbol rate of the transmitter is not necessarily synchronized with the sampling rate of the source to be transmitted. Such scenarios arise, for example, when recording a communications signal for storage or processing, or in compress-and-forward relaying [26, Ch. 16.7], [27] in which the relay compresses the sampled received signal, which is then forwarded to the assisted receiver. As the relay operates with its own sampling clock, which need not necessarily be synchronized with the symbol rate of the assisted transmitter, sampling at the relay may result in a DT WSACS source signal. In the following we first characterize the sampled source model in Subsection III-A. Then, as a preliminary step to our characterization the RDF for asynchronously sampled CT WSCS Gaussian processes stated in Section IV, we recall in Subsection III-B the definitions of some information-spectrum quantities used in this study. Finally, in Subsection III-C, we recall an auxiliary result relating the information spectrum quantities of a collection of sequences of RVs to the information spectrum quantities of its limit sequence of RVs. This result will be applied in the derivation of the RDF with asynchronous sampling. ### III-A Source Model Consider a real CT, zero-mean WSCS Gaussian random process $S_{c}(t)$ with period $T_{\rm ps}$. Let the variance function of $S_{c}(t)$ be defined as $\sigma^{2}_{S_{\rm c}}(t)\triangleq\mathds{E}\big{\\{}S_{\rm c}^{2}(t)\big{\\}}$, and assume it is both upper bounded and lower bounded away from zero, and that it is continuous in $t\in\mathcal{R}$. Let $\tau_{m}>0$ denote the maximal correlation length of $S_{c}(t)$, i.e., $r_{S_{c}}(t,\tau)\triangleq\mathds{E}\big{\\{}S_{c}(t)S_{c}(t-\tau)\big{\\}}=0,\forall|\tau|>\tau_{m}$. By the cyclostationarity of $S_{\rm c}(t)$, we have that $\sigma_{S_{c}}^{2}(t)=\sigma_{S_{c}}^{2}(t+T_{\rm ps}),\forall t\in\mathcal{R}$. Let $S_{c}(t)$ be sampled uniformly with the sampling interval $T_{\rm s}>0$ such that $T_{\rm ps}=(p+\epsilon)\cdot T_{\rm s}$ for $p\in\mathcal{N}$ and $\epsilon\in[0,1)$ yielding $S_{\epsilon}[i]\triangleq S_{\rm c}(i\cdot T_{\rm s})$, where $i\in\mathcal{Z}$. The variance of $S_{\epsilon}[i]$ is given by $\sigma^{2}_{S_{\epsilon}}[i]\triangleq r_{S_{\epsilon}}[i,0]=\sigma^{2}_{S_{\rm c}}\left(\frac{i\cdot T_{\rm ps}}{p+\epsilon}\right)$. In this work, as in [17], we assume that the duration of temporal correlation of the CT signal is shorter than the sampling interval $T_{\rm s}$, namely, $\tau_{m}<T_{\rm s}$. Consequently, the DT Gaussian process $S_{\epsilon}[i]$ is a memoryless zero-mean Gaussian process and its autocorrelation function is given by: $\displaystyle r_{S_{\epsilon}}[i,\Delta]$ $\displaystyle=\mathds{E}\bigg{\\{}S_{\epsilon}[i]S_{\epsilon}[i+\Delta]\bigg{\\}}$ $\displaystyle=\mathds{E}\left\\{S_{\rm c}\left(\frac{i\cdot T_{\rm ps}}{p+\epsilon}\right)\cdot S_{\rm c}\left(\frac{(i+\Delta)\cdot T_{\rm ps}}{p+\epsilon}\right)\right\\}=\sigma^{2}_{S_{\rm c}}\left(\frac{i\cdot T_{\rm ps}}{p+\epsilon}\right)\cdot\delta[\Delta]=\sigma^{2}_{S_{\epsilon}}[i]\cdot\delta[\Delta].$ (10) While we do not explicitly account for sampling time offsets in our definition of the sampled process $S_{\epsilon}[i]$, it can be incorporated by replacing $\sigma^{2}_{S_{\rm c}}(t)$ with a time-shifted version, i.e., $\sigma^{2}_{S_{\rm c}}(t-\phi)$, see also [17, Sec. II.C]. It can be noted from (III-A) that if $\epsilon$ is a rational number, i.e., $\exists u,v\in\mathcal{N}$, $u$ and $v$ are relatively prime, such that $\epsilon=\frac{u}{v}$, then $\left\\{S_{\epsilon}[i]\right\\}_{i\in\mathcal{Z}}$ is a DT memoryless WSCS process with the period $p_{u,v}=p\cdot v+u\in\mathcal{N}$ [17, Sec. II.C]. For this class of processes, the RDF can be obtained from [12, Thm. 1] as stated in Corollary 1. On the other hand, if $\epsilon$ is an irrational number, then sampling becomes asynchronous and leads to a WSACS process whose RDF has not been characterized to date. ### III-B Definitions of Relevant Information Spectrum Quantities Conventional information theoretic tools for characterizing RDFs are based on an underlying ergodicity of the source. Consequently, these techniques cannot be applied to characterize the RDF of of asynchronously sampled WSCS processes. To tackle this challenge, we use information spectrum methods. The information spectrum framework [14] can be utilized to obtain general formulas for rate limits for any arbitrary class of processes. The resulting expressions do are not restricted to specific statistical models of the considered processes, and in particular, do not require information stability or stationarity. In the following, we recall the definitions of several information-spectrum quantities used in this study, see also [14, Def. 1.3.1-1.3.2]: ###### Definition 7. The limit-inferior in probability of a sequence of real RVs $\\{Z_{k}\\}_{k\in\mathcal{N}}$ is defined as ${\rm p-}\mathop{\lim\inf}\limits_{k\rightarrow\infty}Z_{k}\triangleq\sup\left\\{\alpha\in\mathcal{R}\big{|}\mathop{\lim}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}<\alpha\right)=0\right\\}\triangleq\alpha_{0}.$ (11) Hence, $\alpha_{0}$ is the largest real number satisfying that $\forall\tilde{\alpha}<\alpha_{0}$ and $\forall\mu>0$ there exists $k_{0}(\mu,\tilde{\alpha})\in\mathcal{N}$ such that $\Pr(Z_{k}<\tilde{\alpha})<\mu$, $\forall k>k_{0}(\mu,\tilde{\alpha})$. ###### Definition 8. The limit-superior in probability of a sequence of real RVs $\\{Z_{k}\\}_{k\in\mathcal{N}}$ is defined as ${\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}Z_{k}\triangleq\inf\left\\{\beta\in\mathcal{R}\big{|}\mathop{\lim}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)=0\right\\}\triangleq\beta_{0}.$ (12) Hence, $\beta_{0}$ is the smallest real number satisfying that $\forall\tilde{\beta}>\beta_{0}$ and $\forall\mu>0$, there exists $k_{0}(\mu,\tilde{\beta})\in\mathcal{N}$, such that $\Pr(Z_{k}>\tilde{\beta})<\mu$, $\forall k>k_{0}(\mu,\tilde{\beta})$. The notion of uniform integrability of a sequence of RVs is a basic property in probability [28, Ch. 12], which is not directly related to information spectrum methods. However, since it plays an important role in the information spectrum characterization of RDFs, we include its statement in the following definition: ###### Definition 9 (Uniform Integrability [28, Def. 12.1],[14, Eqn. (5.3.2)]). The sequence of real-valued random variables $\\{Z_{k}\\}_{k=1}^{\infty}$, is said to satisfy uniform integrability if $\mathop{\lim}\limits_{u\rightarrow\infty}\mathop{\sup}\limits_{k\geq 1}\mathop{\int}\limits_{z:|z|\geq u}p_{Z_{k}}\left(z\right)|z|\mathrm{d}z=0$ (13) The aforementioned quantities allow characterizing the RDF of arbitrary sources. Consider a general source process $\\{S[i]\\}_{i=1}^{\infty}$ (stationary or non-stationary) taking values from the source alphabet $S[i]\in\mathcal{S}$ and a reproduction process $\\{\hat{S}[i]\\}_{i=1}^{\infty}$ with values from the reproduction alphabet $\hat{S}[i]\in\hat{\mathcal{S}}$. It follows from [14, Sec. 5.5] that for a distortion measure which satisfies the uniform integrability criterion, i.e., that there exists a deterministic sequence $\\{r[i]\\}_{i=1}^{\infty}$ such that the sequence of RVs $\\{d\big{(}{\boldsymbol{S}}^{(k)},{\boldsymbol{r}}^{(k)}\big{)}\\}_{k=1}^{\infty}$ satisfies Def. 9 [14, Pg. 336], then the RDF is expressed as [14, Eqn. (5.4.2)]: $R(D)=\mathop{\inf}\limits_{F_{S,\hat{S}}:\bar{d}_{S}({\boldsymbol{S}}^{(k)},\hat{{\boldsymbol{S}}}^{(k)})\leq D}\bar{I}\left({\boldsymbol{S}}^{(k)};\hat{{\boldsymbol{S}}}^{(k)}\right),$ (14) where $\bar{d}_{S}({\boldsymbol{S}}^{(k)},\hat{{\boldsymbol{S}}}^{(k)})=\mathop{\lim\sup}\limits_{k\rightarrow\infty}\frac{1}{k}\mathds{E}\left\\{d\left({\boldsymbol{S}}^{(k)},\hat{{\boldsymbol{S}}}^{(k)}\right)\right\\}$, $F_{S,\hat{S}}$ denotes the joint CDF of $\\{S[i]\\}_{i=1}^{\infty}$ and $\\{\hat{S}[i]\\}_{i=1}^{\infty}$, and $\bar{I}\left({\boldsymbol{S}}^{(k)}:\hat{{\boldsymbol{S}}}^{(k)}\right)$ represents the limit superior in probability of the mutual information rate of ${\boldsymbol{S}}^{(k)}$ and $\hat{{\boldsymbol{S}}}^{(k)}$, given by: $\bar{I}\left({\boldsymbol{S}}^{(k)};\hat{{\boldsymbol{S}}}^{(k)}\right)\triangleq{\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}\frac{1}{k}\log\frac{p_{{\boldsymbol{S}}^{(k)}|\hat{{\boldsymbol{S}}}^{(k)}}\left({\boldsymbol{S}}^{(k)}|\hat{{\boldsymbol{S}}}^{(k)}\right)}{p_{{\boldsymbol{S}}^{(k)}}\left({\boldsymbol{S}}^{(k)}\right)}$ (15) In order to use the RDF characterization in (14), the distortion measure must satisfy the uniform integrability criterion. For the considered class of sources detailed in Subsection III-A, the MSE distortion satisfies this criterion, as stated in the following lemma: ###### Lemma 2. For any real memoryless zero-mean Gaussian source $\\{S[i]\\}_{i=1}^{\infty}$ with bounded variance, i.e., $\exists\sigma_{\max}^{2}<\infty$ such that $\mathds{E}\\{S^{2}[i]\\}\leq\sigma_{\max}^{2}$ for all $i\in\mathcal{N}$, the MSE distortion satisfies the uniform integrability criterion. ###### Proof. Set the deterministic sequence $\\{{r}[i]\\}_{i=1}^{\infty}$ to be the all- zero sequence. Under this setting and the MSE distortion, it holds that $d\big{(}{\boldsymbol{S}}^{(k)},{\boldsymbol{r}}^{(k)}\big{)}=\frac{1}{k}\sum_{i=1}^{k}S^{2}[i]$. To prove the lemma, we show that the sequence of RVs $\big{\\{}d\big{(}{\boldsymbol{S}}^{(k)},{\boldsymbol{r}}^{(k)}\big{)}\big{\\}}_{k=1}^{\infty}$ has a bounded $\ell_{2}$ norm, which proves that it is uniformly integrable by [28, Cor. 12.8]. The $\ell_{2}$ norm of $d\big{(}{\boldsymbol{S}}^{(k)},{\boldsymbol{r}}^{(k)}\big{)}$ satisfies $\displaystyle\mathds{E}\big{\\{}d\big{(}{\boldsymbol{S}}^{(k)},{\boldsymbol{r}}^{(k)}\big{)}^{2}\big{\\}}$ $\displaystyle=\frac{1}{k^{2}}\mathds{E}\left\\{\sum_{i=1}^{k}S^{2}[i]\sum_{j=1}^{k}S^{2}[j]\right\\}$ $\displaystyle=\frac{1}{k^{2}}\sum_{i=1}^{k}\sum_{j=1}^{k}\mathds{E}\left\\{S^{2}[i]S^{2}[j]\right\\}\stackrel{{\scriptstyle(a)}}{{\leq}}\frac{1}{k^{2}}\sum_{i=1}^{k}\sum_{j=1}^{k}3\sigma_{\max}^{4}=3\sigma_{\max}^{4},$ (16) where $(a)$ follows since $\mathds{E}\\{S^{2}[i]S^{2}[j]\\}=\mathds{E}\\{S^{2}[i]\\}\mathds{E}\\{S^{2}[j]\\}=\sigma_{\max}^{4}$ for $i\neq j$ while $\mathds{E}\\{S^{4}[i]\\}=3\sigma_{\max}^{4}$ [29, Ch. 5.4]. Eqn. (16) proves that $d\big{(}{\boldsymbol{S}}^{(k)},{\boldsymbol{r}}^{(k)}\big{)}$ is $\ell_{2}$-bounded by $3\sigma_{\max}^{4}<\infty$ for all $k\in\mathcal{N}$, which in turn implies that the MSE distortion is uniformly integrable for the source $\\{S[i]\\}_{i=1}^{\infty}$. ∎ Since, as detailed in Subsection III-A, we focus in the following on memoryless zero-mean Gaussian sources, Lemma 2 implies that the RDF of the source can be characterized using (14). However, (14) is in general difficult to evaluate, and thus does not lead to a meaningful understanding of how the RDF of sampled WSCS sources behaves, motivating our analysis in Section IV. ### III-C Information Spectrum Limits The following theorem originally stated in [17, Thm. 1], presents a fundamental result which is directly useful for the derivation of the RDF: ###### Theorem 3. [17, Thm. 1] Let $\left\\{\tilde{Z}_{k,n}\right\\}_{n,k\in\mathcal{N}}$ be a set of sequences of real scalar RVs satisfying two assumptions: 1. AS1 For every fixed $n\in\mathcal{N}$, every convergent subsequence of $\left\\{\tilde{Z}_{k,n}\right\\}_{k\in\mathcal{N}}$ converges in distribution, as $k\rightarrow\infty$, to a finite deterministic scalar. Each subsequence may converge to a different scalar. 2. AS2 For every fixed $k\in\mathcal{N}$, the sequence $\big{\\{}\tilde{Z}_{k,n}\big{\\}}_{n\in\mathcal{N}}$ converges uniformly in distribution, as $n\rightarrow\infty$, to a scalar real-valued RV $Z_{k}$. Specifically, letting $\tilde{F}_{k,n}(\alpha)$ and $F_{k}(\alpha)$, $\alpha\in\mathcal{R}$, denote the CDFs of $\tilde{Z}_{k,n}$ and of $Z_{k}$, respectively, then by AS2 it follows that $\forall\eta>0$, there exists $n_{0}(\eta)$ such that for every $n>n_{0}(\eta)$ $\left|\tilde{F}_{k,n}(\alpha)-F_{k}(\alpha)\right|<\eta,$ for each $\alpha\in\mathcal{R}$, $k\in\mathcal{N}$. Then, for $\big{\\{}\tilde{Z}_{k,n}\big{\\}}_{n,k\in\mathcal{N}}$ it holds that $\displaystyle{\rm p-}\mathop{\lim\inf}\limits_{k\rightarrow\infty}Z_{k}$ $\displaystyle=$ $\displaystyle\mathop{\lim}\limits_{n\rightarrow\infty}\Big{(}{\rm p-}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{Z}_{k,n}\Big{)},$ (17a) $\displaystyle{\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}Z_{k}$ $\displaystyle=$ $\displaystyle\mathop{\lim}\limits_{n\rightarrow\infty}\Big{(}{\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}\tilde{Z}_{k,n}\Big{)}.$ (17b) ###### Proof. In Appendix C we explicitly prove Eqn. (17b). This complements the proof in [17, Appendix A] which explicitly considers only (17a). ∎ ## IV Rate-Distortion Characterization for Sampled CT WSCS Gaussian Sources ### IV-A Main Result Using the information spectrum based characterization of the RDF (14) combined with the characterization of the limit of a sequence of information spectrum quantities in Theorem 3, we now analyze the RDF of asynchronously sampled WSCS processes. Our analysis is based on formulating a sequence of synchronously sampled WSCS processes, whose RDF is given in Corollary 1. Then, we show that the RDF of the asynchronously sampled process can be obtained as the limit superior of the computable RDFs of the sequence of synchronously sampled processes. We begin by letting $\epsilon_{n}\triangleq\frac{\lfloor n\cdot\epsilon\rfloor}{n}$ for $n\in\mathcal{N}$ and defining a Gaussian source process $S_{n}[i]=S_{\rm c}\left(\frac{i\cdot T_{\rm ps}}{p+\epsilon_{n}}\right)$. From the discussion in Subsection III-A (see also [17, Sec. II.C]), it follows that since $\epsilon_{n}$ is rational, $S_{n}[i]$ is a WSCS process and its period is given by $p_{n}=p\cdot n+\lfloor n\cdot\epsilon\rfloor$. Accordingly, the periodic correlation function of $S_{n}[i]$ can be obtained similarly to (III-A) as: $r_{S_{n}}[i,\Delta]=\mathds{E}\bigg{\\{}S_{n}[i]S_{n}[i+\Delta]\bigg{\\}}=\sigma^{2}_{S_{\rm c}}\left(\frac{i\cdot T_{\rm ps}}{p+\epsilon_{n}}\right)\cdot\delta[\Delta].$ (18) Due to cyclostationarity of $S_{n}[i]$, we have that $r_{S_{n}}[i,\Delta]=r_{S_{n}}[i+p_{n},\Delta]$, $\forall i,\Delta\in\mathcal{Z}$, and let $\sigma^{2}_{S_{n}}[i]\triangleq r_{S_{n}}[i,0]$ denote its periodic variance. We next restate Corollary 1 in terms of $\epsilon_{n}$ as follows: ###### Proposition 1. Consider a DT, memoryless, zero-mean, WSCS Gaussian random process $S_{n}[i]$ with a variance $\sigma^{2}_{S_{n}}[i]$, obtained from $S_{\rm c}(t)$ by sampling with a sampling interval of $T_{s}(n)=\frac{T_{\rm ps}}{p+\epsilon_{n}}$. Let ${\boldsymbol{S}}^{p_{n}}[i]$ denote the memoryless stationary multivariate random process obtained by applying the DCD to $S_{n}[i]$ and let $\sigma^{2}_{S_{n}}[m]$, $m=1,2,\ldots,p_{n}$, denote the variance of the $m^{th}$ component of ${\boldsymbol{S}}^{p_{n}}[i]$. The rate- distortion function is given by: $\displaystyle R_{n}(D)=\begin{cases}\frac{1}{2p_{n}}\sum\limits_{m=1}^{p_{n}}\left(\log\left(\frac{\sigma^{2}_{S_{n}}[m]}{D_{n}[m]}\right)\right)&D\leq\frac{1}{p_{n}}\sum\limits_{m=1}^{p_{n}}\sigma^{2}_{S_{n}}[m]\\\ 0&D>\frac{1}{p_{n}}\sum\limits_{m=1}^{p_{n}}\sigma^{2}_{S_{n}}[m]\end{cases},$ (19a) where for $D\leq\frac{1}{p_{n}}\sum\limits_{m=1}^{p_{n}}\sigma^{2}_{S_{n}}[m]\;$ we let $D_{n}[m]\triangleq\min\big{\\{}\sigma^{2}_{S_{n}}[m],\theta_{n}\big{\\}}$, and $\theta_{n}$ is selected $s.t.$ $D=\frac{1}{p_{n}}\sum\limits_{m=1}^{p_{n}}D_{n}[m].$ (19b) We recall that the RDF of $S_{n}[i]$ is characterized in Corollary 1 via the RDF of the multivariate stationary process ${\boldsymbol{S}}_{n}^{(p_{n})}[i]$ obtained via a $p_{n}$-dimensional DCD applied to $S_{n}[i]$. Next, we recall that the relationship between the source process ${\boldsymbol{S}}_{n}^{(p_{n})}[i]$ and the optimal reconstruction process, denoted by $\hat{{\boldsymbol{S}}}_{n}^{(p_{n})}[i]$, is characterized in [5, Ch. 10.3.3] via a linear, multivariate, time-invariant backward channel with a $p_{n}\times 1$ additive vector noise process ${\boldsymbol{W}}_{n}^{(p_{n})}[i]$, and is given by: ${\boldsymbol{S}}_{n}^{(p_{n})}[i]=\hat{{\boldsymbol{S}}}_{n}^{(p_{n})}[i]+{\boldsymbol{W}}_{n}^{(p_{n})}[i],\quad i\in\mathcal{N}.$ (20) It also follows from [5, Sec. 10.3.3] that for the IID Gaussian multivariate process whose entries are independent and distributed via $\big{(}{\boldsymbol{S}}_{n}^{(p_{n})}[i]\big{)}_{m}\sim\mathcal{N}(0,\sigma^{2}_{S_{n}}[m])$, $m\in\\{1,2,\ldots,p_{n}\\}$, the optimal reconstruction vector process $\hat{{\boldsymbol{S}}}_{n}^{(p_{n})}[i]$ and the corresponding noise vector process ${\boldsymbol{W}}_{n}^{(p_{n})}[i]$ each follow a multivariate Gaussian distribution: $\hat{{\boldsymbol{S}}}_{n}^{(p_{n})}[i]\sim\mathcal{N}\left({\bf 0},\left[\begin{matrix}\sigma^{2}_{\hat{S}_{n}}[1]&\cdots&0\\\ \vdots&\ddots&\vdots\\\ 0&\cdots&\sigma^{2}_{\hat{S}_{n}}[p_{n}]\end{matrix}\right]\right)\quad\mathrm{and}\quad{\boldsymbol{W}}_{n}^{(p_{n})}[i]\sim\mathcal{N}\left({\bf 0},\left[\begin{matrix}D_{n}[1]&\cdots&0\\\ \vdots&\ddots&\vdots\\\ 0&\cdots&D_{n}[p_{n}]\end{matrix}\right]\right),$ where $D_{n}[m]\triangleq\min\left\\{\sigma^{2}_{S_{n}}[m],\theta_{n}\right\\}$; $\theta_{n}$ denotes the reverse waterfilling threshold defined in Prop. 1 for the index $n$, and is selected such that $D=\frac{1}{p_{n}}\sum\limits_{m=1}^{p_{n}}D_{n}[m]$. The optimal reconstruction process, $\hat{{\boldsymbol{S}}}_{n}^{(p_{n})}[i]$ and the noise process ${\boldsymbol{W}}_{n}^{(p_{n})}[i]$ are mutually independent, and for each $m\in\\{1,2,\ldots,p_{n}\\}$ it holds that $\mathds{E}\left\\{\left(S_{n}^{(p_{n})}[i]-\hat{S}_{n}^{(p_{n})}[i]\right)_{m}^{2}\right\\}=D_{n}[m]$, see [5, Ch. 10.3.2-10.3.3]. The multivariate relationship between stationary processes in (20) can be transformed into an equivalent linear relationship between cyclostationary Gaussian memoryless processes via the inverse DCD transformation [2, Sec 17.2] applied to each of the processes, resulting in: $S_{n}[i]=\hat{S}_{n}[i]+W_{n}[i]\quad i\in\mathcal{N}.$ (21) We are now ready to state our main result, which is the RDF of the asynchronously sampled DT source $S_{\epsilon}[i],\epsilon\not\in\mathcal{Q}$, in the low MSE regime, i.e., at a given distortion $D$ which is not larger than the source variance. The RDF is stated in the following theorem, which applies to both synchronous sampling as well as asynchronous sampling: ###### Theorem 4. Consider a DT source $\\{S_{\epsilon}[i]\\}_{i=1}^{\infty}$ obtained by sampling a CT WSCS source, whose period of statistics is $T_{\rm ps}$, at intervals $T_{\rm s}$. Then, for any distortion constraint $D$ such that $D<\mathop{\min}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)$ and any $\epsilon\in[0,1)$, the RDF $R_{\epsilon}(D)$ for compressing $\\{S_{\epsilon}[i]\\}_{i=1}^{\infty}$ can be obtained as the limit: $R_{\epsilon}(D)=\mathop{\lim\sup}\limits_{n\rightarrow\infty}R_{n}(D),$ (22) where $R_{n}(D)$ is defined Prop. 1. ###### Proof. The detailed proof is provided in Appendix D. Here, we give a brief outline: The derivation of the RDF with asynchronous sampling follows three steps: First, we note that sampling rate $T_{\rm s}(n)=\frac{T_{\rm ps}}{p+\epsilon_{n}}$ used to obtain the sequence of DT WSCS sources $\\{S_{n}[i]\\}_{i\in\mathcal{N},n\in\mathcal{N}}$ asymptotically approaches the sampling interval for irrational $\epsilon$ given by $T_{\rm s}=\frac{T_{\rm ps}}{p+\epsilon}$ as $n\rightarrow\infty$. We define a sequence of rational numbers $\epsilon_{n}$ $s.t.$ $\epsilon_{n}\rightarrow\epsilon$ as $n\rightarrow\infty$; Building upon this insight, we prove that the RDF with $T_{\rm s}$ can be stated as a double limit where the outer limit is with respect to the blocklength and the inner limit is with respect to $\epsilon_{n}$. Lastly, we use Theorem 3 to show that the limits can be exchanged, obtaining a limit of expressions which are computable. ∎ ###### Remark 1. Theorem 4 focuses on the low distortion regime, defined as the values of $D$ satisfying $D<\mathop{\min}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)$. This implies that $\theta_{n}$ has to be smaller than $\mathop{\min}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)$; hence, from Prop. 1 it follows that for the corresponding stationary noise vector ${\boldsymbol{W}}_{n}^{(p_{n})}[i]$ in (20), $D_{n}[m]=\min\left\\{\sigma^{2}_{S_{n}}[m],\theta_{n}\right\\}=\theta_{n}$ and $D=\frac{1}{p_{n}}\mathop{\sum}\limits_{m=1}^{p_{n}}D_{n}[m]=\theta_{n}=D_{n}[m]$. We note that since every element of the vector $\left({\boldsymbol{W}}_{n}^{(p_{n})}[i]\right)_{m}$ has the same variance $D_{n}[m]=D$ for all $n\in\mathcal{N}$ and $m=1,2,\ldots,p_{n}$ then by applying the inverse DCD to ${\boldsymbol{W}}_{n}^{(p_{n})}[i]$, the resulting scalar DT process $W_{n}[i]$ is wide sense stationary; and in fact IID with $\mathds{E}\left\\{\big{(}W_{n}[i]\big{)}^{2}\right\\}=D$. ### IV-B Discussion and Relationship with Capacity Derivation in [17] Theorem 4 provides a meaningful and computable characterization for the RDF of sampled WSCS signals. We note that the proof of the main theorem uses some of the steps used in our recent study on the capacity of memoryless channels with sampled CT WSCS Gaussian noise [17]. It should be emphasized, however, that there are several fundamental differences between the two studies, which require the introduction of new treatments and derivations original to the current work. First, it is important to note that in the study on capacity, a physical channel model exists, and therefore the conditional PDF of the output signal given the input signal can be characterized explicitly for both synchronous sampling and asynchronous sampling for every input distribution. For the current study of the RDF we note that the relationship (21), commonly referred to as the backward channel [30], [5, Ch.10.3.2], characterizes the relationship between the source process and the optimal reproduction process, and hence is valid only for synchronous sampling and the optimal reproduction process. Consequently, in the RDF analysis the limiting relationship (21) as $n\rightarrow\infty$ is not even known to exist and, in fact, we can show it exists under a rather strict condition on the distortion (namely, the condition $D<\mathop{\min}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)$ stated in Theorem 4). In particular, to prove the statement in Theorem 4, we had to show that from the backward channel (21), we can define an asymptotic relationship, as $n\rightarrow\infty$, which corresponds to the asynchronously sampled source process, denoted by $S_{\epsilon}[i]$, and relates $S_{\epsilon}[i]$ with its optimal reconstruction process $\hat{S}_{\epsilon}[i]$. This is done by showing that the PDFs for the reproduction process $\hat{S}_{n}[i]$ and noise process $W_{n}[i]$ from (21), each converge uniformly as $n\rightarrow\infty$ to a respective limiting PDF, which has to be defined as well. This enabled us to relate the RDFs for the synchronous sampling and for the asynchronous sampling cases using Theorem 3, eventually leading to (22). Accordingly, in our detailed proof of Theorem 4 given in Appendix D, Lemmas D.4 and D.6 as well as a significant part of Lemma D.2 are largely new, addressing the special aspects of the proof arising from the fundamental differences between current setup and the setup in [17], while the derivations of Lemmas D-A and D.5 follow similarly to [17, Lemma B.1] and [17, Lemma B.5], respectively, and parts of Lemma D.2 coincide with [17, Lemma B.2]. ## V Numerical Examples In this section we demonstrate the insights arising from our RDF characterization via numerical examples. Recalling that Theorem 4 states the RDF for asynchronously sampled CT WSCS Gaussian process, $R_{\epsilon}(D)$, as the limit supremum of a sequence of RDFs corresponding to DT memoryless WSCS Gaussian source processes $\left\\{R_{n}(D)\right\\}_{n\in\mathcal{N}}$, we first consider the convergence of $\\{R_{n}(D)\\}_{n\in\mathcal{N}}$ in Subsection V-A. Next, in Subsection V-B we study the variation of the RDF of the sampled CT process due to changes in the sampling rate and in the sampling time offset. Similarly to [17, Sec. IV], define a periodic continuous pulse function, denoted by $\Pi_{t_{\rm dc},t_{\rm rf}}(t)$, with equal rise/fall time $t_{\rm rf}=0.01$, duty cycle $t_{\rm dc}\in[0,0.98]$, and period of $1$, i.e., $\Pi_{t_{\rm dc},t_{\rm rf}}(t+1)=\Pi_{t_{\rm dc},t_{\rm rf}}(t)$ for all $t\in\mathcal{R}$. Specifically, for $t\in[0,1)$ the function $\Pi_{t_{\rm dc},t_{\rm rf}}(t)$ is given by $\Pi_{t_{\rm dc},t_{\rm rf}}(t)=\begin{cases}\frac{t}{t_{\rm rf}}&t\in[0,t_{\rm rf}]\\\ 1&t\in(t_{\rm rf},t_{\rm dc}+t_{\rm rf})\\\ 1-\frac{t-t_{\rm dc}-t_{\rm rf}}{t_{\rm rf}}&t\in[t_{\rm dc}+t_{\rm rf},t_{\rm dc}+2\cdot t_{\rm rf}]\\\ 0&t\in(t_{\rm dc}+2\cdot t_{\rm rf},1).\end{cases}$ (23) In the following, we model the time varying variance of the WSCS source $\sigma^{2}_{S_{\rm c}}(t)$ to be a linear periodic function of $\Pi_{t_{\rm dc},t_{\rm rf}}(t)$. To that aim, we define a time offset between the first sample and the rise start time of the periodic continuous pulse function; we denote the time offset by $\phi\in[0,1)$. This corresponds to the sampling time offset normalized to the period $T_{\rm ps}$. The variance of $S_{\rm c}(t)$ is periodic function with period $T_{\rm ps}$ which is given by $\sigma^{2}_{S_{\rm c}}(t)=0.2+4.8\cdot\Pi_{t_{\rm dc},t_{\rm rf}}\left(\frac{t}{T_{\rm ps}}-\phi\right),\qquad t\in[0,T_{\rm ps}),$ (24) with period of $T_{\rm ps}=5$ $\mu$secs. ### V-A Convergence of $R_{n}(D)$ in $n$ Figure 2: $R_{n}(D)$ versus $n$; offset $\phi=0$. Figure 3: $R_{n}(D)$ versus $n$; offset $\phi=\frac{1}{16}$. From Theorem 4 it follows that if the distortion satisfies $D<\mathop{\min}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)$, the RDF of the asynchronously sampled CT WSCS Gaussian process is given by the limit superior of the sequence $\\{R_{n}(D)\\}_{n\in\mathcal{N}}$; where $R_{n}(D)$ is obtained via Corollary 1. In this subsection, we study the sequence of RDFs $\\{R_{n}(D)\\}_{n\in\mathcal{N}}$ as $n$ increases. For this evaluation setup, we fixed the distortion constraint at $D=0.18$ and set $\epsilon=\frac{\pi}{7}$ and $p=2$. Let the variance of the CT WSCS Gaussian source process $\sigma^{2}_{S_{\rm c}}(t)$ be modelled by Eq. (24) for two sampling time offsets $\phi=\\{0,\frac{1}{16}\\}$. For each offset $\phi$, four duty cycle values were considered: $t_{\rm dc}=[20,45,75,98]\%$. For each $n$ we obtain the synchronous sampling mismatch $\epsilon_{n}\triangleq\frac{\lfloor n\cdot\epsilon\rfloor}{n}$, which approaches $\epsilon$ as $n\rightarrow\infty$, where $n\in\mathcal{N}$. Since $\epsilon_{n}$ is a rational number, corresponding to a sampling period of $T_{\rm s}(n)=\frac{T_{\rm ps}}{p+\epsilon_{n}}$, then for each $n$, the resulting DT process is WSCS with the period $p_{n}=p\cdot n+\lfloor n\cdot\epsilon\rfloor$ and its RDF follows from Corollary 1. Figures 3 and 3 depict $R_{n}(D)$ for $n\in[1,500]$ with the specified duty cycles and sampling time offsets, where in Fig. 3 there is no sampling time offset, i.e., $\phi=0$, and in Fig. 3 the sampling time offset is set to $\phi=\frac{1}{16}$. We observe that in both figures the RDF values are higher for higher $t_{\rm dc}$. This can be explained by noting that for higher $t_{\rm dc}$ values, the resulting time averaged variance of the DT source process increases, hence, a higher number of bits per sample is required to encode the source process maintaining the same distortion value. Also, in all configurations, $R_{n}(D)$ varies significantly for smaller values of $n$. Comparing Figures 3 and 3, we see that the pattern of these variations depends on the sampling time offset $\phi$. For example, when $t_{\rm dc}=45\%$ at $n\in[4,15]$, then for $\phi=0$ the RDF varies in the range $[1.032,1.143]$ bits per sample, while for $\phi=\frac{1}{16}$ the RDF varies in the range $[1.071,1.237]$ bits per sample. However, as we increase $n$ above $230$, the variations in $R_{n}(D)$ become smaller and are less dependent on the sampling time offset, and the resulting values of $R_{n}(D)$ are approximately in the same range in both Figures 3 and 3 for $n\geq 230$. This behaviour can be explained by noting that as $n$ varies, the period $p_{n}$ also varies and hence the statistics of the DT variance differs over its respective period. This consequently affects the resulting RDF (especially for small periods). As $n$ increases $\epsilon_{n}$ approaches the asynchronous sampling mismatch $\epsilon$ and the period $p_{n}$ takes a sufficiently large value such that the samples of the DT variance over the period are similarly distributed irrespective of the value of $\phi$; leading to a negligible variation in the RDF as seen in the above figures. ### V-B The Variation of the RDF with the Sampling Rate Figure 4: $R_{n}(D)$ versus $\frac{T_{\rm ps}}{T_{\rm s}}$; offset $\phi=0$. Figure 5: $R_{n}(D)$ versus $\frac{T_{\rm ps}}{T_{\rm s}}$; offset $\phi=\frac{1}{16}$. Next, we observe the dependence of the RDF for the sampled memoryless WSCS Gaussian process on the value of the sampling interval $T_{\rm s}$. For this setup, we fix the distortion constraint to $D=0.18$ and set the duty cycle in the source process (24) to $t_{\rm dc}=[45,75]\%$. Figures 5-5 demonstrate the numerically evaluated values for $R_{n}(D)$ at sampling intervals in the range $2<\frac{T_{\rm ps}}{T_{\rm s}}<4$ with the sampling time offsets $\phi=0$ and $\phi=\frac{1}{16}$, respectively. A very important insight which arises from the figures is that the sequence of RDFs $R_{n}(D)$ is not convergent; hence, for example, one cannot approach the RDF for $\frac{T_{\rm ps}}{T_{\rm s}}=2.5$ by simply taking rational values of $\frac{T_{\rm ps}}{T_{\rm s}}$ which approach $2.5$. This verifies that the RDF for asynchronous sampling cannot be obtained by straightforward application of previous results, and indeed, the entire analysis carried in the manuscript is necessary for the desired characterization. We observe in Figures 5-5 that when $\frac{T_{\rm ps}}{T_{\rm s}}$ has a fractional part with a relatively small integer denominator, the variations in the RDF are significant, and the variations depend on the sampling time offset. However, when $\frac{T_{\rm ps}}{T_{\rm s}}$ approaches an irrational number, the period of the sampled variance function becomes very long, and consequently, the RDF is approximately constant and independent of the sampling time offset. As an example, consider $\frac{T_{\rm ps}}{T_{\rm s}}=2.5$ and $t_{\rm dc}=75\%$: For sampling time offset $\phi=0$ the RDF takes a value of $1.469$ bits per sample, as shown in Figure 5 while for the offset of $\phi=\frac{1}{16}$ the RDF peaks to $1.934$ bits per sample as we see in Figure 5. On the other hand, when approaching asynchronous sampling, the RDF takes an approximately constant value $1.85$ bits per sample for all the considered values of $\frac{T_{\rm ps}}{T_{\rm s}}$ and this value is invariant to the offsets of $\phi$. This follows since when the denominator of the fractional part of $\frac{T_{\rm ps}}{T_{\rm s}}$ increases, then the DT period of the resulting sampled variance, $p_{n}$, increases and practically captures the entire set of values of the CT variance regardless of the sampling time offset. In a similar manner as with the study on capacity in [17], we conjecture that since asynchronous sampling captures the entire set of values of the CT variance, the respective RDF represents the RDF of the analog source, which does not depend on the specific sampling rate and offset. Figures 5-5 demonstrate how slight variations in the sampling rate can result in significant changes in the RDF. For instance, at $\phi=0$ we notice in Figure 5 that when the sampling rate switches from $T_{\rm s}=2.25\cdot T_{\rm ps}$ to $T_{\rm s}=2.26\cdot T_{\rm ps}$, i.e., the sampling rate switches from being synchronous to being nearly asynchronous, and the RDF changes from $1.624$ bits per channel use to $1.859$ bits per sample for $t_{\rm dc}=75\%$; also, we observe in Figure 5 for $t_{\rm dc}=45\%$, that when the sampling rate switches from $T_{\rm s}=2.5\cdot T_{\rm ps}$ to $T_{\rm s}=2.51\cdot T_{\rm ps}$, i.e., the sampling rate also switches from being synchronous to being nearly asynchronous, and the RDF changes from $1.005$ bits per source sample to $1.154$ bits per source sample. Figure 6: $R_{n}(D)$ versus $D$; offset $\phi=0$. Figure 7: $R_{n}(D)$ versus $D$; offset $\phi=\frac{1}{16}$. Lastly, Figures 7-7 numerically evaluate the RDF versus the distortion constraint $D\in[0.05,0.19]$ for the sampling time offsets of $0$ and $\frac{1}{16}$ respectively. At each $\phi$, the result is evaluated at three different values of synchronization mismatch $\epsilon$. For this setup, we fix $t_{\rm dc}=75\%$, $p=2$ and $\epsilon\in\\{0.5,\frac{5\pi}{32},0.6\\}$. The only mismatch value that refers to the asynchronous sampling case is $\epsilon=\frac{5\pi}{32}$ and its corresponding sampling interval is approximately $2.007$ $\mu$secs, which is a negligible variation from the sampling intervals corresponding to $\epsilon\in\\{0.5,0.6\\}$, which are $2.000$ $\mu$secs and $1.923$ $\mu$secs, respectively. Observing both figures, we see that the RDF may vary significantly for very slight variation in the sampling rate. For instance, as shown in Figure 7 for $\phi=0$, at $D=0.18$, a slight change in the synchronization mismatch from $\epsilon=\frac{5\pi}{32}$ (i.e., $T_{\rm s}\approx 2.007\mu$secs) to $\epsilon=0.5$ (i.e., $T_{\rm s}=2.000\mu$secs) results to approximately $20\%$ decrease in the RDF. For $\phi=\frac{1}{16}$ the same change in the sampling synchronization mismatch at $D=0.18$ results to a rise in the RDF by roughly $4\%$. These results demonstrate the unique and counter-intuitive characteristics of the RDF of sampled WSCS signals which arise from our derivation. ## VI Conclusions In this work the RDF of a sampled CT WSCS Gaussian source process was characterized for scenarios in which the resulting DT process is memoryless. This characterization shows the relationship between the sampling rate and the minimal number of bits required for compression at a given distortion. For cases in which the sampling rate is synchronized with the period of the statistics of the source process, the resulting DT process is WSCS and standard information theoretic framework can be used for deriving its RDF. For asynchronous sampling, information stability does not hold, and hence we resort to the information spectrum framework to obtain a characterization. To that aim we derived a relationship between some relevant information spectrum quantities for uniformly convergent sequences of RVs. This relationship was further applied to characterize the RDF of an asynchronously sampled CT WSCS Gaussian source process as the limit superior of a sequence of RDFs, each corresponding to the synchronous sampling of the CT WSCS Gaussian process. The results were derived in the low distortion regime, i.e., under the condition that the distortion constraint $D$ is less than the minimum variance of the source, and for sampling intervals which are larger than the correlation length of the CT process. Our numerical examples give rise to non-intuitive insights which follow from the derivations. In particular, the numerical evaluation demonstrates that the RDF for a sampled CT WSCS Gaussian source can change dramatically with minor variations in sampling rate and sampling time offset. In particular, when the sampling rate switches from being synchronous to being asynchronous and vice versa, the RDF may change considerably as the statistical model of the source switches between WSCS to WSACS. The resulting analysis enables determining the sampling system parameters in order to facilitate accurate and efficient source coding of acquired CT signals. ## Appendix A Proof of Lemma 1 ###### Proof. To prove that the minimum achievable rate at a given maximum distortion for a code with arbitrary blocklength can be achieved by considering only codes whose blocklength is an integer multiple of $r$, we apply the following approach: We first show that every rate-distortion pair achievable when restricted to using source codes whose blocklength is an integer multiple of $r$ is also achievable when using arbitrary blocklenghts; We then prove that every achievable rate-distortion pair is also achievable when restricted to using codes whose blocklength is an integer multiple of $r$. Combining these two assertions proves that the rate-distortion function of the source $\\{S[i]\\}_{i\in\mathcal{N}}$ can be obtained when restricting the blocklengths to be an integer multiple of $r$. Consequently, a reproduction signal $\\{\hat{S}[i]\\}_{i\in\mathcal{N}}$ which achieves the minimal rate for a given $D$ under the restriction to use only blocklengths which are an integer multiple of $r$ is also the reproduction signal achieving the minimal rate without this restriction, and vice versa, thus proving the lemma. To prove the first assertion, consider a rate-distortion pair $(R,D)$ which is achievable when using codes whose blocklength is an integer multiple of $r$. It thus follows directly from Def. 5 that for every $\eta>0$, $\exists b_{0}\in\mathcal{N}$ such that for all $b>b_{0}$ there exists a a source code $\left(R_{(b\cdot r)},b\cdot r\right)$ with rate $R_{(b\cdot r)}\leq R+\eta$ satisfying $\bar{d}\big{(}{\boldsymbol{S}}^{(b\cdot r)},\hat{{\boldsymbol{S}}}^{(b\cdot r)}\big{)}\leq D+\frac{\eta}{2}$. We now show that we can construct a code with an arbitrary blocklength $l=b\cdot r+j$ where $0<j<r$ (i.e., the blocklength $l$ is not an integer multiple of $r$) satisfying Def. 5 for all $j\in\\{1,\ldots,r-1\\}$ as follows: Apply the code $\left(R_{(b\cdot r)},b\cdot r\right)$ to the first $b\cdot r$ samples of $S[i]$ and then concatenate each codeword by $j$ zeros to obtain a source code having codewords of length $b\cdot r+j$. The average distortion (i.e., see (2)) of the resulting $\left(R_{(b\cdot r+j)},b\cdot r+j\right)$ code is given by: $\displaystyle\bar{d}\left({\boldsymbol{S}}^{(b\cdot r+j)},\hat{{\boldsymbol{S}}}^{(b\cdot r+j)}\right)$ $\displaystyle=\frac{1}{b\cdot r+j}\left(\sum\limits_{i=1}^{b\cdot r}\mathds{E}\left\\{\left(S[i]-\hat{S}[i]\right)^{2}\right\\}+\sum\limits_{i=b\cdot r+1}^{b\cdot r+j}\mathds{E}\left\\{\left(S[i]\right)^{2}\right\\}\right)$ $\displaystyle=\frac{1}{b\cdot r+j}\left(b\cdot r\cdot\bar{d}\left({\boldsymbol{S}}^{(b\cdot r)},\hat{{\boldsymbol{S}}}^{(b\cdot r)}\right)+\sum\limits_{i=1}^{j}\sigma_{S}^{2}[i]\right)$ $\displaystyle=\frac{b\cdot r}{b\cdot r+j}\cdot\bar{d}\left({\boldsymbol{S}}^{(b\cdot r)},\hat{{\boldsymbol{S}}}^{(b\cdot r)}\right)+\frac{1}{b\cdot r+j}\sum\limits_{i=1}^{j}\sigma_{S}^{2}[i].$ (A.1) Thus $\exists b>b_{o}$ such that $\frac{1}{b\cdot r+j}\sum\limits_{i=1}^{j}\sigma_{S}^{2}[i]<\frac{\eta}{2}$ and $\displaystyle\bar{d}\left({\boldsymbol{S}}^{(b\cdot r+j)},\hat{{\boldsymbol{S}}}^{(b\cdot r+j)}\right)$ $\displaystyle=\frac{b\cdot r}{b\cdot r+j}\cdot\bar{d}\left({\boldsymbol{S}}^{(b\cdot r)},\hat{{\boldsymbol{S}}}^{(b\cdot r)}\right)+\frac{1}{b\cdot r+j}\sum\limits_{i=1}^{j}\sigma_{S}^{2}[i]$ $\displaystyle\leq\frac{b\cdot r}{b\cdot r+j}\cdot\bar{d}\left({\boldsymbol{S}}^{(b\cdot r)},\hat{{\boldsymbol{S}}}^{(b\cdot r)}\right)+\frac{\eta}{2}$ $\displaystyle\leq\bar{d}\left({\boldsymbol{S}}^{(b\cdot r)},\hat{{\boldsymbol{S}}}^{(b\cdot r)}\right)+\frac{\eta}{2}\leq D+\eta.$ (A.2) The rate $R_{(b\cdot r+j)}$ satisfies: $R_{(b\cdot r+j)}=\frac{1}{b\cdot r+j}\cdot\log_{2}M=R_{(b\cdot r)}\cdot\frac{b\cdot r}{b\cdot r+j}\leq\left(R+\eta\right)\cdot\frac{b\cdot r}{b\cdot r+j}\leq R+\eta.$ (A.3) Consequently, any rate-distortion pair achievable with codes whose blocklength is an integer multiple of $r$ can be achieved by codes with arbitrary blocklengths. Next, we prove that any achievable rate-distortion pair $(R,D)$ can be achieved by codes whose blocklength is an integer multiple of $r$. To that aim, we fix $\eta>0$. By Def. 5, it holds that there exists a code of blocklength $l$ satisfying (3)-(4). To show that $(R,D)$ is achievable using codes whose blocklength is an integer multiple of $r$, we assume here that $l$ is not an integer multiple of $r$, hence, there exist some positive integers $b$ and $j$ such that $j<r$ and $l=b\cdot r+j$. We denote this code by $\left(R_{(b\cdot r+j)},b\cdot r+j\right)$. It follows from Def. 5 that $R_{(b\cdot r+j)}\leq R+\eta$ and $\bar{d}\left({\boldsymbol{S}}^{(b\cdot r+j)},\hat{{\boldsymbol{S}}}^{(b\cdot r+j)}\right)\leq D+\frac{\eta}{2}$. Next, we construct a code $\left(R_{(b+1)\cdot r},(b+1)\cdot r\right)$ with codewords whose length is $(b+1)\cdot r$, i.e., an integer multiple of $r$, by adding $r-j$ zeros at the end of each codeword of the code $\left(R_{(b\cdot r+j)},b\cdot r+j\right)$. The average distortion can now be computed as follows: $\displaystyle\bar{d}\left({\boldsymbol{S}}^{((b+1)\cdot r)},\hat{{\boldsymbol{S}}}^{((b+1)\cdot r)}\right)$ $\displaystyle=\frac{1}{(b+1)\cdot r}\left(\sum\limits_{i=1}^{b\cdot r+j}\mathds{E}\left\\{\left(S[i]-\hat{S}[i]\right)^{2}\right\\}+\sum\limits_{i=b\cdot r+j+1}^{(b+1)\cdot r}\mathds{E}\left\\{\left(S[i]\right)^{2}\right\\}\right)$ $\displaystyle=\frac{1}{(b+1)\cdot r}\left((b\cdot r+j)\cdot\bar{d}\left({\boldsymbol{S}}^{(b\cdot r+j)},\hat{{\boldsymbol{S}}}^{(b\cdot r+j)}\right)+\sum\limits_{i=b\cdot r+j+1}^{(b+1)\cdot r}\sigma_{S}^{2}[i]\right)$ $\displaystyle=\frac{b\cdot r+j}{(b+1)\cdot r}\cdot\bar{d}\left({\boldsymbol{S}}^{(b\cdot r+j)},\hat{{\boldsymbol{S}}}^{(b\cdot r+j)}\right)+\frac{\sum\limits_{i=b\cdot r+j+1}^{(b+1)\cdot r}\sigma_{S}^{2}[i]}{(b+1)\cdot r},$ (A.4) and again $\exists b>b_{o}$ such that $\frac{\sum\limits_{i=b\cdot r+j+1}^{(b+1)\cdot r}\sigma_{S}^{2}[i]}{(b+1)\cdot r}<\frac{\eta}{2}$, hence $\displaystyle\bar{d}\left({\boldsymbol{S}}^{((b+1)\cdot r)},\hat{{\boldsymbol{S}}}^{((b+1)\cdot r}\right)$ $\displaystyle\leq\frac{b\cdot r+j}{(b+1)\cdot r}\cdot\bar{d}\left({\boldsymbol{S}}^{(b\cdot r+j)},\hat{{\boldsymbol{S}}}^{(b\cdot r+j)}\right)+\frac{\eta}{2}$ $\displaystyle\leq\bar{d}\left({\boldsymbol{S}}^{(b\cdot r+j)},\hat{{\boldsymbol{S}}}^{(b\cdot r+j)}\right)+\frac{\eta}{2}\leq D+\eta.$ (A.5) The rate $R_{(b+1)\cdot r}$ can be expressed as follows: $R_{(b+1)\cdot r}=\frac{1}{(b+1)\cdot r}\cdot\log_{2}M=R_{(b\cdot r+j)}\cdot\frac{b\cdot r+j}{(b+1)\cdot r}\leq\left(R+\eta\right)\cdot\frac{b\cdot r+j}{(b+1)\cdot r}<R+\eta.$ (A.6) It follows that $R_{(b+1)\cdot r}\leq R+\eta$ for any arbitrary $\eta$ by selecting a sufficiently large $b$. This proves that every rate-distortion pair achievable with arbitrary blocklengths (e.g., $l=b\cdot r+j,j<r$) is also achievable when considering source codes whose blocklength is an integer multiple of $r$ (e.g., $l=b\cdot r$). This concludes the proof. ∎ ## Appendix B Proof of Theorem 1 Recall that $\alpha\in\mathcal{R}^{++}$. To prove the theorem, we fix a rate- distortion pair $(R,D)$ that is achievable for the source $\\{{S}[i]\\}_{i\in\mathcal{N}}$. By Def. 5 this implies that for all $\eta>0$ there exists $l_{0}(\eta)\in\mathcal{N}$ such that for all $l>l_{0}(\eta)$ there exists a source code $\mathcal{C}_{l}$ with rate $R_{l}\leq R+\eta$ and MSE distortion $D_{l}=\mathds{E}\big{\\{}\frac{1}{l}\big{\|}{\boldsymbol{S}}^{(l)}-\hat{{\boldsymbol{S}}}^{(l)}\big{\|}^{2}\big{\\}}\leq D+\eta$. Next, we use the code $\mathcal{C}_{l}$ to define the source code $\mathcal{C}_{l}^{(\alpha)}$, which operates in the following manner: The encoder first scales its input block by $1/\alpha$. Then, the block is encoded using the source code $\mathcal{C}_{l}$. Finally, the selected codeword is scaled by $\alpha$. Since the $\mathcal{C}_{l}^{(\alpha)}$ has the same number of codewords and the same blocklength as $\mathcal{C}_{l}$, it follows that its rate, denote $R_{l}^{(\alpha)}$, satisfied $R_{l}^{(\alpha)}=R_{l}\leq R+\eta$. Furthermore, by the construction of $\mathcal{C}_{l}^{(\alpha)}$, it holds that its reproduction vector when applied to $\alpha\cdot{\boldsymbol{S}}^{(l)}$ is equal to the output of $\mathcal{C}_{l}$ applied to ${\boldsymbol{S}}^{(l)}$ scaled by $\alpha$, i.e., $\alpha\cdot\hat{{\boldsymbol{S}}}^{(l)}$. Consequently, the MSE of $\mathcal{C}_{l}^{(\alpha)}$ when applied to the source $\\{\alpha\cdot{S}[i]\\}_{i\in\mathcal{N}}$, denoted $D_{l}^{(\alpha)}$, satisfies $D_{l}^{(\alpha)}=\mathds{E}\big{\\{}\frac{1}{l}\big{\|}\alpha\cdot{\boldsymbol{S}}^{(l)}-\alpha\cdot\hat{{\boldsymbol{S}}}^{(l)}\big{\|}^{2}\big{\\}}=\alpha^{2}D_{l}\leq\alpha^{2}D+\alpha^{2}\eta$. It thus follows that for all $\tilde{\eta}>0$ there exists $\tilde{l}_{0}(\tilde{\eta})=l_{0}\big{(}\min(\tilde{\eta},\alpha^{2}\tilde{\eta})\big{)}$ such that for all $l>\tilde{l}_{0}(\tilde{\eta})$ there exists a code $\mathcal{C}_{l}^{(\alpha)}$ with rate $R_{l}^{(\alpha)}\leq R+\tilde{\eta}$ which achieves an MSE distortion of $D_{l}^{(\alpha)}\leq\alpha^{2}\cdot D+\tilde{\eta}$ when applied to the compression of $\\{\alpha\cdot{S}[i]\\}_{i\in\mathcal{N}}$. Hence, $(R,\alpha^{2}D)$ is achievable for compression of $\\{\alpha\cdot{S}[i]\\}_{i\in\mathcal{N}}$ by Def. 5, proving the theorem. ## Appendix C Proof of Theorem 3 In this appendix, we prove (17b) by applying a similar approach as used for proving (17a) in [17, Appendix A]. We first note that Def. 8 can also be written as follows: ${\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}Z_{k}\\!\stackrel{{\scriptstyle(a)}}{{=}}\\!\inf\left\\{\beta\in\mathcal{R}\Big{|}|\mathop{\lim\sup}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)=0\right\\}\\!\stackrel{{\scriptstyle(b)}}{{=}}\\!\inf\left\\{\beta\in\mathcal{R}\Big{|}|\mathop{\lim\inf}\limits_{k\rightarrow\infty}F_{k}(\beta)=1\right\\}.$ (C.1) For the equality $(a)$, we note that the set of probabilities $\\{\Pr\left(Z_{k}>\beta\right)\\}_{k\in\mathcal{N}}$ is non-negative and bounded in $[0,1]$; hence, for any $\beta\in\mathcal{R}$ for which $\mathop{\lim\sup}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)=0$, it also holds from [31, Thm. 3.17] that the limit of any subsequence of $\left\\{\Pr\left(Z_{k}>\beta\right)\right\\}_{k\in\mathcal{N}}$ is also $0$, since non-negativity of the probability implies $\mathop{\lim\inf}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)\geq 0$. Then, combined with the relationship $\mathop{\lim\inf}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)\leq\mathop{\lim\sup}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)$, we conclude: $\displaystyle 0\leq\mathop{\lim\inf}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)\leq\mathop{\lim\sup}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)=0$ $\displaystyle\implies\mathop{\lim\inf}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)=\mathop{\lim\sup}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)\stackrel{{\scriptstyle(a)}}{{=}}\mathop{\lim}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)=0,$ where $(a)$ follows from [31, Example. 3.18(c)]. This implies $\mathop{\lim}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)$ exists and is equal to 0. In the opposite direction, if $\mathop{\lim}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)=0$ then it follows from [31, Example. 3.18(c)] that $\mathop{\lim\sup}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)=0$. Next, we note that since $F_{k}(\beta)$ is bounded in $[0,1]$ then $\mathop{\lim\inf}\limits_{k\rightarrow\infty}F_{k}(\beta)$ is finite $\forall\beta\in\mathcal{R}$, even if $\mathop{\lim}\limits_{k\rightarrow\infty}F_{k}(\beta)$ does not exist. Equality $(b)$ follows since $\mathop{\lim\sup}\limits_{k\rightarrow\infty}\Pr\left(Z_{k}>\beta\right)=\mathop{\lim\sup}\limits_{k\rightarrow\infty}\left(1-\Pr\left(Z_{k}\leq\beta\right)\right)$ which according to [32, Thm. 7.3.7] is equal to $1+\mathop{\lim\sup}\limits_{k\rightarrow\infty}\left(-\Pr\left(Z_{k}\leq\beta\right)\right)$. By [33, Ch. 1, page 29], this quantity is also equal to $1-\mathop{\lim\inf}\limits_{k\rightarrow\infty}\left(\Pr\left(Z_{k}\leq\beta\right)\right)=1-\mathop{\lim\inf}\limits_{k\rightarrow\infty}F_{k}(\beta)$. Next, we state the following lemma: ###### Lemma C.1. Given assumption AS2, for all $\beta\in\mathcal{R}$ it holds that $\mathop{\lim\inf}\limits_{k\rightarrow\infty}F_{k}(\beta)=\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta).$ (C.2) ###### Proof. To prove the lemma we first show that $\mathop{\lim\inf}\limits_{k\rightarrow\infty}F_{k}(\beta)\leq\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)$, and then we show $\mathop{\lim\inf}\limits_{k\rightarrow\infty}F_{k}(\beta)\geq\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)$. Recall that by AS2, for all $\beta\in\mathcal{R}$ and $k\in\mathcal{N}$, $\tilde{F}_{k,n}(\beta)$ converges as $n\rightarrow\infty$ to $F_{k}(\beta)$, uniformly over $k$ and $\beta$, i.e., for all $\eta>0$ there exists $n_{0}(\eta)\in\mathcal{N}$, $k_{0}\big{(}n_{0}(\eta),\eta\big{)}\in\mathcal{N}$ such that for every $n>n_{0}(\eta)$, $\beta\in\mathcal{R}$ and $k>k_{0}\big{(}n_{0}(\eta),\eta\big{)}$, it holds that $\big{|}\tilde{F}_{k,n}(\beta)-F_{k}(\beta)\big{|}<\eta$. Consequently, for every subsequence $0<k_{1}<k_{2}<\ldots$ such that $\mathop{\lim}\limits_{l\rightarrow\infty}\tilde{F}_{k_{l},n}(\beta)$ exists for any $n>n_{0}(\eta)$, it follows from [31, Thm. 7.11] 111[31, Thm. 7.11]: Suppose $f_{n}\rightarrow f$ uniformly in a set $E$ in a metric space. Let $x$ be a limit point of $E$, and suppose that $\mathop{\lim}\limits_{t\rightarrow x}f_{n}(t)=A_{n}$, $(n=1,2,3,\ldots)$. Then $A_{n}$ converges, and $\mathop{\lim}\limits_{t\rightarrow x}\mathop{\lim}\limits_{n\rightarrow\infty}f_{n}(t)=\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim}\limits_{t\rightarrow x}f_{n}(t)$ that, as the convergence over $k$ is uniform, the limits over $n$ and $l$ are interchangeable: $\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim}\limits_{l\rightarrow\infty}\tilde{F}_{k_{l},n}(\beta)=\mathop{\lim}\limits_{l\rightarrow\infty}\mathop{\lim}\limits_{n\rightarrow\infty}\tilde{F}_{k_{l},n}(\beta)=\mathop{\lim}\limits_{l\rightarrow\infty}F_{k_{l}}(\beta).$ (C.3) The existence of such a convergent subsequence is guaranteed by the Bolzano- Weierstrass theorem [31, Thm. 2.42] 222[31, Thm. 2.42]:Every bounded infinite subset of $\mathcal{R}^{k}$ has a limit point in $\mathcal{R}^{k}$ as $\tilde{F}_{k,n}(\beta)\in[0,1]$. From the properties of the limit inferior [31, Thm. 3.17] 333[31, Thm. 3.17]: Let $\\{s_{n}\\}$ be a sequence of real numbers; Let $E$ be the set of numbers $x$ (in the extended real number system) containing all limits of all subsequences of $\\{s_{n}\\}$. Then…… $\mathop{\lim\inf}\limits_{n\rightarrow\infty}s_{n}\in E$. it follows that there exists a subsequence of $\big{\\{}F_{k}(\beta)\big{\\}}_{k\in\mathcal{N}}$, denoted $\big{\\{}F_{k_{m}}(\beta)\big{\\}}_{m\in\mathcal{N}}$, such that $\mathop{\lim}\limits_{m\rightarrow\infty}F_{k_{m}}(\beta)=\mathop{\lim\inf}\limits_{k\rightarrow\infty}F_{k}(\beta)$. Consequently, $\displaystyle\mathop{\lim\inf}\limits_{k\rightarrow\infty}F_{k}(\beta)$ $\displaystyle=\mathop{\lim}\limits_{m\rightarrow\infty}F_{k_{m}}(\beta)\stackrel{{\scriptstyle(a)}}{{=}}\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim}\limits_{m\rightarrow\infty}\tilde{F}_{k_{m},n}(\beta)$ $\displaystyle\stackrel{{\scriptstyle(b)}}{{\geq}}\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta),$ (C.4) where $(a)$ follows from (C.3), and $(b)$ follows from the definition of the limit inferior [31, Def. 3.16]. Similarly, by [31, Thm. 3.17], for any $n\in\mathcal{N}$ there exists a subsequence of $\\{\tilde{F}_{k,n}(\beta)\\}_{k\in\mathcal{N}}$ which we denote by $\big{\\{}\tilde{F}_{k_{l},n}(\beta)\big{\\}}_{l\in\mathcal{N}}$ where $\\{k_{l}\\}_{l\in\mathcal{N}}$ satisfy $0<k_{1}<k_{2}<\ldots$, such that $\mathop{\lim}\limits_{l\rightarrow\infty}\tilde{F}_{k_{l},n}(\beta)=\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)$. Therefore, $\displaystyle\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)$ $\displaystyle=\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim}\limits_{l\rightarrow\infty}\tilde{F}_{k_{l},n}(\beta)$ $\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathop{\lim}\limits_{l\rightarrow\infty}F_{k_{l}}(\beta)\stackrel{{\scriptstyle(b)}}{{\geq}}\mathop{\lim\inf}\limits_{k\rightarrow\infty}F_{k}(\beta),$ (C.5) where $(a)$ follows from (C.3), and $(b)$ follows from the definition of the limit inferior [31, Def. 3.16]. Therefore, $\mathop{\lim\inf}\limits_{k\rightarrow\infty}F_{k}(\beta)\leq\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)$. Combining (C.4) and (C.5) proves (C.2) in the statement of the lemma. ∎ ###### Lemma C.2. Given assumptions AS1-AS2, the sequence of RVs $\big{\\{}\tilde{Z}_{k,n}\big{\\}}_{k,n\in\mathcal{N}}$ satisfies $\displaystyle\mathop{\lim}\limits_{n\rightarrow\infty}\left({\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}\tilde{Z}_{k,n}\right)$ $\displaystyle=\inf\left\\{\beta\in\mathcal{R}\Big{|}\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)=1\right\\}.$ (C.6) ###### Proof. Since by assumption AS1, for every $n\in\mathcal{N}$, every convergent subsequence of $\big{\\{}\tilde{Z}_{k,n}\big{\\}}_{k\in\mathcal{N}}$ converges in distribution as $k\rightarrow\infty$ to a deterministic scalar, it follows that every convergent subsequence of $\tilde{F}_{k,n}(\beta)$ converges as $k\rightarrow\infty$ to a step function, which is the CDF of the corresponding sublimit of $\tilde{Z}_{k,n}$. In particular, the limit $\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)$ is a step function representing the CDF of the deterministic scalar $\zeta_{n}$, i.e., $\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)=\begin{cases}0&\beta<\zeta_{n}\\\ 1&\beta\geq\zeta_{n}.\end{cases}$ (C.7) Since, by Lemma C.1, AS2 implies that the limit $\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)$ exists444The convergence to a discontinuous function is in the sense of [31, Ex. 7.3], then $\mathop{\lim}\limits_{n\rightarrow\infty}\zeta_{n}$ exists. Hence, we obtain that $\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)=\begin{cases}0&\beta<\mathop{\lim}\limits_{n\rightarrow\infty}\zeta_{n}\\\ 1&\beta\geq\mathop{\lim}\limits_{n\rightarrow\infty}\zeta_{n},\end{cases}$ (C.8) and from the right-hand side of (C.6) we have that $\inf\left\\{\beta\in\mathcal{R}\Big{|}\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)=1\right\\}=\mathop{\lim}\limits_{n\rightarrow\infty}\zeta_{n}.$ (C.9) Next, from (C.1) and (C.7) we note that $\displaystyle{\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}\tilde{Z}_{k,n}=\inf\left\\{\beta\in\mathcal{R}\Big{|}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)=1\right\\}=\zeta_{n}.$ Consequently, the left-hand side of (C.6) is equal to $\mathop{\lim}\limits_{n\rightarrow\infty}\zeta_{n}$. Combining with (C.9) we arrive at the equality (C.6) in the statement of the lemma. ∎ Substituting (C.2) into (C.1) results in $\displaystyle{\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}Z_{k}$ $\displaystyle=\inf\left\\{\beta\in\mathcal{R}\Big{|}\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim\inf}\limits_{k\rightarrow\infty}\tilde{F}_{k,n}(\beta)=1\right\\}\stackrel{{\scriptstyle(a)}}{{=}}\mathop{\lim}\limits_{n\rightarrow\infty}\left({\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}\tilde{Z}_{k,n}\right),$ (C.10) where $(a)$ follows from (C.6). Eq. (C.10) concludes the proof for (17b). ## Appendix D Proof of Theorem 4 In this appendix we detail the proof of Theorem 4. The outline of the proof is given as follows: * • We first show in Subsection D-A that for any $k\in\mathcal{N}$, the PDF of the random vector ${\boldsymbol{S}}_{n}^{(k)}$, representing the first $k$ samples of the CT WSCS source $S_{\rm c}(t)$ sampled at time instants $T_{\rm s}(n)=\frac{T_{\rm ps}}{p+\epsilon_{n}}$, converges in the limit as $n\rightarrow\infty$ and for any $k\in\mathcal{N}$ to the PDF of ${\boldsymbol{S}}_{\epsilon}^{(k)}$, which represents the first $k$ samples of the CT WSCS source $S_{\rm c}(t)$, sampled at time instants $T_{\rm s}=\frac{T_{\rm ps}}{p+\epsilon}$. We prove that this convergence is uniform in $k\in\mathcal{N}$ and in the realization vector ${\boldsymbol{s}}^{(k)}\in\mathcal{R}^{k}$. This is stated in Lemma D.1. * • Next, in Subsection D-B we apply Theorem 3 to relate the mutual information density rates for the random source vector ${\boldsymbol{S}}_{n}^{(k)}$ and its reproduction $\hat{{\boldsymbol{S}}}_{n}^{(k)}$ with that of the random source vector ${\boldsymbol{S}}_{\epsilon}^{(k)}$ and its reproduction $\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}$. To that aim, let the functions $F_{S_{n},\hat{S}_{n}}$ and $F_{S_{\epsilon},\hat{S}_{\epsilon}}$ denote the joint distributions of an arbitrary dimensional source and reproduction vectors corresponding to the synchronously sampled and to the asynchronously sampled source process respectively. We define the following mutual information density rates: $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}\right)\triangleq\frac{1}{k}\log\frac{p_{{\boldsymbol{S}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{S}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}\right)}{p_{{\boldsymbol{S}}_{n}^{(k)}}\left({\boldsymbol{S}}_{n}^{(k)}\right)},$ (D.1a) and $Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)\triangleq\frac{1}{k}\log\frac{p_{{\boldsymbol{S}}_{\epsilon}^{(k)}|\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left({\boldsymbol{S}}_{\epsilon}^{(k)}\big{|}\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\right)}{p_{{\boldsymbol{S}}_{\epsilon}^{(k)}}\left({\boldsymbol{S}}_{\epsilon}^{(k)}\right)},$ (D.1b) $k,n\in\mathcal{N}$. The RVs $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}\right)$ and $Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$ in (D.1) denote the mutual information density rates [14, Def. 3.2.1] between the DT source process and the corresponding reproduction process for the case of synchronous sampling and for the case of asynchronous sampling, respectively. We then show that if the pairs of source process and optimal reproduction process $\big{\\{}S_{n}[i],\hat{S}_{n}[i]\big{\\}}_{i\in\mathcal{N}}$ and $\big{\\{}S_{\epsilon}[i],\hat{S}_{\epsilon}[i]\big{\\}}_{i\in\mathcal{N}}$ satisfy that $p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)\mathop{\longrightarrow}\limits_{n\rightarrow\infty}p_{\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ uniformly with respect to $\hat{{\boldsymbol{s}}}^{(k)}\in\mathcal{R}^{k}$ and $k\in\mathcal{N}$, and that $p_{{\boldsymbol{S}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)\mathop{\longrightarrow}\limits_{n\rightarrow\infty}p_{{\boldsymbol{S}}_{\epsilon}^{(k)}|\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)$ uniformly in $\left(\big{(}\hat{{\boldsymbol{s}}}^{(k)}\big{)}^{T},\big{(}{\boldsymbol{s}}^{(k)}\big{)}^{T}\right)^{T}\in\mathcal{R}^{2k}$ and $k\in\mathcal{N}$, then $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$ uniformly in $k\in\mathcal{N}$. In addition, Lemma D.3 proves that every subsequence of $\left\\{\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}\right)\right\\}_{k\in\mathcal{N}}$ $w.r.t.$ $k$, indexed as $k_{l}$ converges in distribution, in the limit $l\rightarrow\infty$ to a deterministic scalar. * • Lastly, in Subsection D-C we combine the above results to show in Lemmas D.5 and D.6 that $R_{\epsilon}(D)\leq\mathop{\lim\sup\limits}\limits_{n\rightarrow\infty}R_{n}(D)$ and $R_{\epsilon}(D)\geq\mathop{\lim\sup}\limits_{n\rightarrow\infty}R_{n}(D)$ respectively; implying that $R_{\epsilon}(D)=\mathop{\lim\sup}\limits_{n\rightarrow\infty}R_{n}(D)$, which proves the theorem. To facilitate our proof we will need uniform convergence in $k\in\mathcal{N}$, of $p_{{\boldsymbol{S}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right)$, $p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ and $p_{{\boldsymbol{S}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)$ to $p_{{\boldsymbol{S}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right)$, $p_{\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ and $p_{{\boldsymbol{S}}_{\epsilon}^{(k)}|\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)$, respectively. To that aim, we will make the following scaling assumption w.l.o.g.: ###### Assumption D.1. The variance of the source and the allowed distortion are scaled by some factor $\alpha^{2}$ such that $\alpha^{2}\cdot\min\left\\{D,\left(\mathop{\min}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)-D\right)\right\\}>\frac{1}{2\pi}.$ (D.2) Note that this assumption has no effect on the generality of the RDF for multivariate stationary processes detailed in [5, Sec. 10.3.3], [34, Sec. IV]. Moreover, by Theorem 1, for every $\alpha>0$ it holds that when any rate $R$ achievable when compressing the original source $S_{\rm c}(t)$ with distortion not larger that $D$ is achievable when compressing the scaled source $\alpha\cdot S_{\rm c}(t)$ with distortion not larger than $\alpha^{2}\cdot D$. Note that if for the source $S_{\rm c}(t)$ the distortion satisfies $D<\mathop{\min}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)$, then for the scaled source and distortion we have $\alpha^{2}\cdot D<\mathop{\min}\limits_{0\leq t\leq T_{\rm ps}}\alpha^{2}\cdot\sigma^{2}_{S_{\rm c}}(t)$. ### D-A Convergence in Distribution of ${{\boldsymbol{S}}_{n}^{(k)}}$ to ${{\boldsymbol{S}}_{\epsilon}^{(k)}}$ Uniformly with Respect to ${k\in\mathcal{N}}$ In order to prove the uniform convergence in distribution, ${\boldsymbol{S}}_{n}^{(k)}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}{\boldsymbol{S}}_{\epsilon}^{(k)}$, uniformly with respect to $k\in\mathcal{N}$, we first prove, in Lemma D.1, that as $n\rightarrow\infty$ the sequence of PDFs of ${\boldsymbol{S}}_{n}^{(k)}$, $p_{{\boldsymbol{S}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right)$, converges to the PDF of ${\boldsymbol{S}}_{\epsilon}^{(k)}$, $p_{{\boldsymbol{S}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right)$, uniformly in ${\boldsymbol{s}}^{(k)}\in\mathcal{R}^{k}$ and in $k\in\mathcal{N}$. Next, we show in Corollary D.1 that ${\boldsymbol{S}}_{n}^{(k)}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}{\boldsymbol{S}}_{\epsilon}^{(k)}$ uniformly in $k\in\mathcal{N}$. To that aim, let us define the set $\mathcal{K}\triangleq\\{1,2,\ldots,k\\}$ and consider the $k$\- dimensional zero-mean, memoryless random vectors ${\boldsymbol{S}}_{n}^{(k)}$ and ${\boldsymbol{S}}_{\epsilon}^{(k)}$ with their respective diagonal correlation matrices expressed below: $\mathsf{R}_{n}^{(k)}\triangleq\mathds{E}\big{\\{}\big{(}{\boldsymbol{S}}_{n}^{(k)}\big{)}\big{(}{\boldsymbol{S}}_{n}^{(k)}\big{)}^{T}\big{\\}}=\textrm{diag}\big{(}\sigma^{2}_{S_{n}}[1],\ldots,\sigma^{2}_{S_{n}}[k]\big{)},$ (D.3a) $\mathsf{R}_{\epsilon}^{(k)}\triangleq\mathds{E}\big{\\{}\big{(}{\boldsymbol{S}}_{\epsilon}^{(k)}\big{)}\big{(}{\boldsymbol{S}}_{\epsilon}^{(k)}\big{)}^{T}\big{\\}}=\textrm{diag}\big{(}\sigma^{2}_{S_{\epsilon}}[1],\ldots,\sigma^{2}_{S_{\epsilon}}[k]\big{)}.$ (D.3b) Since $\epsilon_{n}\triangleq\frac{\lfloor n\cdot\epsilon\rfloor}{n}$ it holds that $\frac{n\cdot\epsilon-1}{n}\leq\epsilon_{n}\leq\frac{n\cdot\epsilon}{n}$; therefore $\mathop{\lim}\limits_{n\rightarrow\infty}\epsilon_{n}=\epsilon.$ (D.4) Now we note that since $\sigma_{S_{\rm c}}^{2}(t)$ is uniformly continuous, then by the definition of a uniformly continuous function, for each $i\in\mathcal{N}$, the limit in (D.4) implies that $\mathop{\lim}\limits_{n\rightarrow\infty}\sigma^{2}_{S_{n}}[i]\equiv\mathop{\lim}\limits_{n\rightarrow\infty}\sigma^{2}_{S_{\rm c}}\left(i\cdot\frac{T_{\rm ps}}{p+\epsilon_{n}}\right)=\sigma^{2}_{S_{\rm c}}\left(i\cdot\frac{T_{\rm ps}}{p+\epsilon}\right)\equiv\sigma^{2}_{S_{\epsilon}}[i].$ (D.5) From Assumption D.1, it follows that $\sigma^{2}_{S_{n}}[i]$ satisfies $\sigma^{2}_{S_{n}}[i]>\frac{1}{2\pi}$; Hence, we can state the following lemma: ###### Lemma D.1. The PDF of ${\boldsymbol{S}}_{n}^{(k)}$, $p_{{\boldsymbol{S}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right)$, converges as $n\rightarrow\infty$ to the PDF of ${\boldsymbol{S}}_{\epsilon}^{(k)}$, $p_{{\boldsymbol{S}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right)$, uniformly in ${\boldsymbol{s}}^{(k)}\in\mathcal{R}^{k}$ and in $k\in\mathcal{N}$: $\mathop{\lim}\limits_{n\rightarrow\infty}p_{{\boldsymbol{S}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right)=p_{{\boldsymbol{S}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right),\quad\forall{\boldsymbol{s}}^{(k)}\in\mathcal{R}^{k},\forall k\in\mathcal{N}.$ ###### Proof. The proof of the lemma directly follows from the steps in the proof of [17, Lemma B.1] 555[17, Lemma B.1] considers a CT memoryless WSCS Gaussian noise process ${{\boldsymbol{W}}}_{c}(t)$ sampled synchronously and asynchronously to yield ${{\boldsymbol{W}}}_{n}^{(k)}$ and ${{\boldsymbol{W}}}_{\epsilon}^{(k)}$ respectively, both having independent entries. This Lemma proves that, for both Gaussian vectors of independent entries, if the variance of ${{\boldsymbol{W}}}_{n}^{(k)}$ converges as $n\rightarrow\infty$ to the variance of ${{\boldsymbol{W}}}_{\epsilon}^{(k)}$, then the PDF (which in this case is a continuous mapping of the variance) of ${{\boldsymbol{W}}}_{n}^{(k)}$ also converges to the PDF of ${{\boldsymbol{W}}}_{\epsilon}^{(k)}$. Also, for simplicity, the assumption $\frac{1}{2\pi}<\sigma^{2}_{W_{\rm c}}(t)<\infty$ for all $t\in\mathcal{R}$ was used to prove uniform convergence of the PDF of ${{\boldsymbol{W}}}_{n}^{(k)}$. A similar setup is applied in this work with $\frac{1}{2\pi}<\sigma^{2}_{S_{c}}(t)<\infty$ for all $t\in\mathcal{R}$, for the memoryless CT WSCS process $S_{\rm c}(t)$; ${\boldsymbol{S}}_{n}^{(k)}$ and ${\boldsymbol{S}}_{\epsilon}^{(k)}$ also have independent entries, for the synchronously and for the asynchronously sampled case respectively. The proof for uniform convergence in $k\in\mathcal{N}$ from [17, Lemma B.1] also applies to this current work. , which was applied for a Gaussian noise process with independent entries and variance above $\frac{1}{2\pi}$. ∎ Lemma D.1 gives rise to the following corollary: ###### Corollary D.1. For any $k\in\mathcal{N}$ it holds that ${\boldsymbol{S}}_{n}^{(k)}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}{\boldsymbol{S}}_{\epsilon}^{(k)}$, and convergence is uniform over $k$. ###### Proof. The corollary holds due to [35, Thm.1] 666[35, Thm.1]: If, for a sequence $\\{p_{n}(x)\\}_{n\in\mathcal{N}}$ of densities, $\mathop{\lim}\limits_{n\rightarrow\infty}p_{n}(x)=p(x)$ for almost all $x$ in $\mathcal{R}$, then a sufficient condition that $\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\int}\limits_{\mathcal{S}}p_{n}(x)dx=\mathop{\int}\limits_{\mathcal{S}}p(x)dx$, uniformly for all Borel sets $\mathcal{S}$ in $\mathcal{R}$, is that $p(x)$ be a density. : Since $p_{{\boldsymbol{S}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right)$ converges to $p_{{\boldsymbol{S}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right)$ then ${\boldsymbol{S}}_{n}^{(k)}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}{\boldsymbol{S}}_{\epsilon}^{(k)}$. In addition, since the convergence of the PDFs is uniform in $k\in\mathcal{N}$, the convergence of the CDFs is also uniform in $k\in\mathcal{N}$. ∎ ### D-B Showing that ${\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)}$ and ${Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)}$ Satisfy the Conditions of Thm. 3 Let $F_{S_{n},\hat{S}_{n}}^{\rm opt}$ denote the joint distribution for the source process and the corresponding optimal reproduction process satisfying the distortion constraint $D$. We next prove that for $F_{S_{n},\hat{S}_{n}}^{\rm opt}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}F_{S_{\epsilon},\hat{S}_{\epsilon}}$, then $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)$ and $Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$ satisfy AS1-AS2. In particular, in Lemma D.2 we prove that $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$ uniformly in $k\in\mathcal{N}$ for the optimal zero-mean Gaussian reproduction vectors with independent entries. Lemma D.3 proves that for any fixed $n$, $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)$ converges in distribution to a deterministic scalar as $k\rightarrow\infty$. ###### Lemma D.2. Let $\\{\hat{{\boldsymbol{S}}}_{n}^{(k)}\\}_{n\in\mathcal{N}}$ and $\\{{\boldsymbol{W}}_{n}^{(k)}\\}_{n\in\mathcal{N}}$ be two sets of mutually independent sequences of $k\times 1$ zero-mean Gaussian random vectors related via the backward channel (20), each having independent entries and let PDFs $p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ and $p_{{\boldsymbol{W}}_{n}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)$, respectively, denote their PDFs. Consider two other zero-mean Gaussian random vectors $\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}$ and ${\boldsymbol{W}}_{\epsilon}^{(k)}$ each having independent entries with the PDFs $p_{\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ and $p_{{\boldsymbol{W}}_{\epsilon}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)$, respectively, such that $\mathop{\lim}\limits_{n\rightarrow\infty}p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)=p_{\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ uniformly in $\hat{{\boldsymbol{s}}}^{(k)}\in\mathcal{R}^{k}$ and uniformly with respect to $k\in\mathcal{N}$, and $\mathop{\lim}\limits_{n\rightarrow\infty}p_{{\boldsymbol{W}}_{n}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)=p_{{\boldsymbol{W}}_{\epsilon}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)$ uniformly in ${\boldsymbol{w}}^{(k)}\in\mathcal{R}^{k}$ and uniformly with respect to $k\in\mathcal{N}$. Then, the RVs $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)$ and $Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$, defined via (D.1) satisfy $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$ uniformly over $k\in\mathcal{N}$. ###### Proof. To begin the proof, for $\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)\in\mathcal{R}^{2k}$, define $f_{k,n}\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)\triangleq\frac{p_{{\boldsymbol{S}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)}{p_{{\boldsymbol{S}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right)},\qquad f_{k,\epsilon}\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)\triangleq\frac{p_{{\boldsymbol{S}}_{\epsilon}^{(k)}|\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)}{p_{{\boldsymbol{S}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\right)}.$ (D.6) Now, we recall the backward channel relationship (20): ${\boldsymbol{S}}_{n}^{(k)}=\hat{{\boldsymbol{S}}}_{n}^{(k)}+{\boldsymbol{W}}_{n}^{(k)},$ (D.7) where $\hat{{\boldsymbol{S}}}_{n}^{(k)}$ and ${\boldsymbol{W}}_{n}^{(k)}$ are mutually independent zero-mean, Gaussian random vectors with independent entries, corresponding to the optimal compression process and its respective distortion. From this relationship we obtain $\displaystyle p_{{\boldsymbol{S}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)$ $\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}+{\boldsymbol{W}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)$ $\displaystyle=p_{{\boldsymbol{W}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}-\hat{{\boldsymbol{s}}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)$ $\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}p_{{\boldsymbol{W}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}-\hat{{\boldsymbol{s}}}^{(k)}\right),$ (D.8) where $(a)$ follows since ${\boldsymbol{S}}_{n}^{(k)}=\hat{{\boldsymbol{S}}}_{n}^{(k)}+{\boldsymbol{W}}_{n}^{(k)}$, see (D.7), and $(b)$ follows since ${\boldsymbol{W}}_{n}^{(k)}$ and $\hat{{\boldsymbol{S}}}_{n}^{(k)}$ are mutually independent. The joint PDF of ${\boldsymbol{S}}_{n}^{(k)}$ and $\hat{{\boldsymbol{S}}}_{n}^{(k)}$ can be expressed via the conditional PDF as: $\displaystyle p_{{\boldsymbol{S}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)\\!=\\!p_{{\boldsymbol{S}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)\cdot p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)\\!\stackrel{{\scriptstyle(a)}}{{=}}\\!p_{{\boldsymbol{W}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\\!-\\!\hat{{\boldsymbol{s}}}^{(k)}\right)\cdot p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right),$ (D.9) where $(a)$ follows from (D.8). Since $\hat{{\boldsymbol{S}}}_{n}^{(k)}$ and ${\boldsymbol{W}}_{n}^{(k)}$ are Gaussian and mutually independent and since the product of two multivariate Gaussian PDFs is also a multivariate Gaussian PDF [36, Sec. 3], it follows from (D.9) that ${\boldsymbol{S}}_{n}^{(k)}$ and $\hat{{\boldsymbol{S}}}_{n}^{(k)}$ are jointly Gaussian. Following the mutual independence of ${\boldsymbol{W}}_{n}^{(k)}$ and $\hat{{\boldsymbol{S}}}_{n}^{(k)}$, the right hand side (RHS) of (D.9) is also equivalent to the joint PDF of $\left[\left({\boldsymbol{W}}_{n}^{(k)}\right)^{T},\left(\hat{{\boldsymbol{S}}}_{n}^{(k)}\right)^{T}\right]^{T}$ denoted by $p_{{\boldsymbol{W}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}-\hat{{\boldsymbol{s}}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)$. Now, from (D.8), the assumption $\mathop{\lim}\limits_{n\rightarrow\infty}p_{{\boldsymbol{W}}_{n}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)=p_{{\boldsymbol{W}}_{\epsilon}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)$ implies that a limit exists for the conditional PDF $p_{{\boldsymbol{S}}_{n}^{(k)}\mid\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\mid\hat{{\boldsymbol{s}}}^{(k)}\right)$, this we denote by $p_{{\boldsymbol{S}}_{\epsilon}^{(k)}\mid\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\mid\hat{{\boldsymbol{s}}}^{(k)}\right)$. Combining this with the assumption $\mathop{\lim}\limits_{n\rightarrow\infty}p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)=p_{\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$, we have that, $\displaystyle\mathop{\lim}\limits_{n\rightarrow\infty}p_{{\boldsymbol{S}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)$ $\displaystyle=\mathop{\lim}\limits_{n\rightarrow\infty}\left(p_{{\boldsymbol{S}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)\cdot p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)\right)$ $\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathop{\lim}\limits_{n\rightarrow\infty}\left(p_{{\boldsymbol{W}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}-\hat{{\boldsymbol{s}}}^{(k)}\right)\cdot p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)\right)$ $\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\mathop{\lim}\limits_{n\rightarrow\infty}\left(p_{{\boldsymbol{W}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}-\hat{{\boldsymbol{s}}}^{(k)}\right)\right)\cdot\mathop{\lim}\limits_{n\rightarrow\infty}\left(p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)\right)$ $\displaystyle=p_{{\boldsymbol{S}}_{\epsilon}^{(k)}|\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)\cdot p_{\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ $\displaystyle=p_{{\boldsymbol{S}}_{\epsilon}^{(k)},\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right),$ (D.10) where $(a)$ follows from (D.8), and $(b)$ follows since the limit for each sequence in the product exists [31, Thm. 3.3]; Convergence is uniform in $\left(\left(\hat{{\boldsymbol{s}}}^{(k)}\right)^{T},\left({\boldsymbol{s}}^{(k)}\right)^{T}\right)^{T}\in\mathcal{R}^{2k}$ and $k\in\mathcal{N}$, as each sequence converges uniformly in $k\in\mathcal{N}$ [31, Page 165] 777[31, Page 165, Ex 2]: The solution to this exercise shows that if two functions $\\{f_{n}\\}$ and $\\{g_{n}\\}$ converge uniformly on a set $E$ and both $\\{f_{n}\\}$ and $\\{g_{n}\\}$ are sequences of bounded functions then $\\{f_{n}g_{n}\\}$ converges uniformly on $E$. . Observe that the joint PDF for the zero-mean Gaussian random vectors $\left[{\boldsymbol{S}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}\right]$ is given by the general expression: $p_{{\boldsymbol{S}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}}\\!\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)\\!=\\!\Big{(}{\rm Det}\big{(}2\pi\tilde{\mathsf{C}}_{n}^{(2k)}\big{)}\Big{)}^{-\frac{1}{2}}\\!\exp\left(-\frac{1}{2}\left[\left(\hat{{\boldsymbol{s}}}^{(k)}\right)^{T}\\!\\!,\left({\boldsymbol{s}}^{(k)}\right)^{T}\right]\big{(}\tilde{\mathsf{C}}_{n}^{(2k)}\big{)}^{-1}\\!\left[\left(\hat{{\boldsymbol{s}}}^{(k)}\right)^{T}\\!\\!,\left({\boldsymbol{s}}^{(k)}\right)^{T}\right]^{T}\right),$ (D.11) where $\tilde{\mathsf{C}}_{n}^{(2k)}$ denotes the joint covariance matrix of $\left[\left(\hat{{\boldsymbol{S}}}_{n}^{(k)}\right)^{T},\left({\boldsymbol{S}}_{n}^{(k)}\right)^{T}\right]^{T}$. From (D.11) we note that $p_{{\boldsymbol{S}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)$ is a continuous mapping of $\tilde{\mathsf{C}}_{n}^{(2k)}$ with respect to the index $n$, see [17, Lemma B.1]. Hence the convergence in (D-B) of $p_{{\boldsymbol{S}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)$ as $n\rightarrow\infty$ directly implies the convergence of $\tilde{\mathsf{C}}_{n}^{(2k)}$ as $n\rightarrow\infty$ to a limit which we denote by $\tilde{\mathsf{C}}_{\epsilon}^{(2k)}$. It therefore follows that the limit function $p_{{\boldsymbol{S}}_{\epsilon}^{(k)},\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)$ corresponds to the PDF of a Gaussian vector with the covariance matrix $\tilde{\mathsf{C}}_{\epsilon}^{(2k)}$. The joint PDF for the zero-mean Gaussian random vectors $\left[{\boldsymbol{W}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}\right]$ can be obtained using their mutual independence as: $\displaystyle p_{{\boldsymbol{W}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}-\hat{{\boldsymbol{s}}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)$ $\displaystyle=\\!\Big{(}{\rm Det}\big{(}2\pi\Sigma_{n}^{(2k)}\big{)}\Big{)}^{\frac{1}{2}}\\!\exp\left(-\frac{1}{2}\left[\left({\boldsymbol{s}}^{(k)}\\!-\\!\hat{{\boldsymbol{s}}}^{(k)}\right)^{T},\left(\hat{{\boldsymbol{s}}}^{(k)}\right)^{T}\right]\big{(}\Sigma_{n}^{(2k)}\big{)}^{-\\!1}\\!\left[\left({\boldsymbol{s}}^{(k)}\\!-\\!\hat{{\boldsymbol{s}}}^{(k)}\right)^{T},\left(\hat{{\boldsymbol{s}}}^{(k)}\right)^{T}\right]^{T}\right),$ (D.12) where $\Sigma_{n}^{(2k)}$ denotes the joint covariance matrix of $\left[\left({\boldsymbol{W}}_{n}^{(k)}\right)^{T},\left(\hat{{\boldsymbol{S}}}_{n}^{(k)}\right)^{T}\right]^{T}$. Since the vectors ${\boldsymbol{W}}_{n}^{(k)}$ and $\hat{{\boldsymbol{S}}}_{n}^{(k)}$ are zero-mean, mutually independent and, by the relationship (20), each vector has independent entries, it follows that $\Sigma_{n}^{(2k)}$ is a diagonal matrix with each diagonal element taking the value of the corresponding temporal variance at the respective index $i\in\\{1,2,\ldots k\\}$. i.e., $\displaystyle\Sigma_{n}^{(2k)}$ $\displaystyle\triangleq\mathds{E}\left\\{\left(\left({\boldsymbol{W}}_{n}^{(k)}\right)^{T},\left(\hat{{\boldsymbol{S}}}_{n}^{(k)}\right)^{T}\right)^{T}\cdot\left(\left({\boldsymbol{W}}_{n}^{(k)}\right)^{T},\left(\hat{{\boldsymbol{S}}}_{n}^{(k)}\right)^{T}\right)\right\\}$ $\displaystyle=\textrm{diag}\big{(}\mathds{E}\left\\{\left(W_{n}[1]\right)^{2}\right\\},\mathds{E}\left\\{\left(W_{n}[2]\right)^{2}\right\\},\ldots,\mathds{E}\left\\{\left(W_{n}[k]\right)^{2}\right\\},\sigma^{2}_{\hat{S}_{n}}[1],\sigma^{2}_{\hat{S}_{n}}[2]\ldots,\sigma^{2}_{\hat{S}_{n}}[k]\big{)}.$ (D.13) The convergence of $p_{{\boldsymbol{W}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}-\hat{{\boldsymbol{s}}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)$, from (D-B), implies a convergence of the diagonal elements in (D-B) as $n\rightarrow\infty$. Hence $\Sigma_{n}^{(2k)}$ converges as $n\rightarrow\infty$ to a diagonal joint covariance matrix which we denote by $\Sigma_{\epsilon}^{(2k)}$. This further implies that the limiting vectors ${\boldsymbol{W}}_{\epsilon}^{(k)}$ and $\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}$ are zero-mean, mutually independent and each vector has independent entries in $i\in[1,2,\ldots,k]$. Relationship (D-B) implies that the joint limit distribution satisfies $p_{{\boldsymbol{S}}_{\epsilon}^{(k)},\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)=p_{\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)\cdot p_{{\boldsymbol{W}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}-\hat{{\boldsymbol{s}}}^{(k)}\right)$. Consequently, we can define an asymptotic backward channel that satisfies (D-B) via the expression: ${\boldsymbol{S}}_{\epsilon}^{(k)}[i]=\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}[i]+{\boldsymbol{W}}_{\epsilon}^{(k)}[i].$ (D.14) Next, by convergence of the joint PDF $p_{{\boldsymbol{W}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}-\hat{{\boldsymbol{s}}}^{(k)}\right)\cdot p_{\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ uniformly in $k\in\mathcal{N}$ and in $\left(\left({\boldsymbol{s}}^{(k)}\right)^{T},\left(\hat{{\boldsymbol{s}}}^{(k)}\right)^{T}\right)^{T}\in\mathcal{R}^{2k}$, it follows from [35, Thm.1] 888Please refer to the footnote 6 on page 6. that $\left[\big{(}\hat{{\boldsymbol{S}}}_{n}^{(k)}\big{)}^{T},\big{(}{\boldsymbol{W}}_{n}^{(k)}\big{)}^{T}\right]^{T}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}\left[\big{(}\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\big{)}^{T},\big{(}{\boldsymbol{W}}_{\epsilon}^{(k)}\big{)}^{T}\right]^{T}$ and the convergence is uniform in $k\in\mathcal{N}$ and in $\left(\left({\boldsymbol{s}}^{(k)}\right)^{T},\left(\hat{{\boldsymbol{s}}}^{(k)}\right)^{T}\right)^{T}\in\mathcal{R}^{2k}$. Then, by the continuous mapping theorem (CMT) [37, Thm. 7.7], we have $\displaystyle\left[\big{(}{\boldsymbol{S}}_{n}^{(k)}\big{)}^{T},\big{(}\hat{{\boldsymbol{S}}}_{n}^{(k)}\big{)}^{T}\right]^{T}\\!=\\!\left[\big{(}\hat{{\boldsymbol{S}}}_{n}^{(k)}\\!+\\!{\boldsymbol{W}}_{n}^{(k)}\big{)}^{T},\big{(}\hat{{\boldsymbol{S}}}_{n}^{(k)}\big{)}^{T}\right]^{T}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}\left[\big{(}\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\\!+\\!{\boldsymbol{W}}_{\epsilon}^{(k)}\big{)}^{T},\big{(}\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\big{)}^{T}\right]^{T}\\!=\\!\left[\big{(}{\boldsymbol{S}}_{\epsilon}^{(k)}\big{)}^{T},\big{(}\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\big{)}^{T}\right]^{T}.$ Now, using the extended CMT [37, Thm. 7.24] 999 [37, Thm. 7.24]: (Extended continuous mapping). Let $\mathbb{D}_{n}\subset\mathbb{D}$ and $g_{n}$ : $\mathbb{D}_{n}\mapsto\mathbb{E}$ satisfy the following: If $x_{n}\rightarrow x$ with $x_{n}\in\mathbb{D}_{n}$ for all $n\geq 1$ and $x\in\mathbb{D}_{0}$, then $g_{n}(x_{n})\rightarrow g(x)$, where $\mathbb{D}_{0}\subset\mathbb{D}$ and $g:\mathbb{D}_{0}\mapsto\mathbb{E}$. Let $X_{n}$ be maps taking values in $\mathbb{D}_{n}$, and let $X$ be Borel measurable and separable. Then (i) $X_{n}\rightsquigarrow X$ implies $g_{n}(X_{n})\rightsquigarrow g(X)$. (ii) $X_{n}\mathop{\rightarrow}\limits^{P}X$ implies $g_{n}(X_{n})\mathop{\rightarrow}\limits^{P}g(X)$. (iii) $X_{n}\mathop{\rightarrow}\limits^{as*}X$ implies $g_{n}(X_{n})\mathop{\rightarrow}\limits^{as*}g(X)$. , we will show that $f_{k,n}\big{(}{\boldsymbol{S}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}\big{)}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}f_{k,\epsilon}\big{(}{\boldsymbol{S}}_{\epsilon}^{(k)},\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\big{)}$ for each $k\in\mathcal{N}$, following the same approach of the proof for [17, Lemma B.2] 101010[17, Lemma B.2]: Consider a sequence of $k\times 1$ zero-mean Gaussian random vectors with independent entries $\\{{{\boldsymbol{X}}}_{n}^{(k)}\\}_{n\in\mathcal{N}}$ and a zero-mean Gaussian random vector with independent entries ${{\boldsymbol{X}}}^{(k)}$, such that ${{\boldsymbol{X}}}_{n}^{(k)}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}{{\boldsymbol{X}}}^{(k)}$ uniformly with respect to $k\in\mathcal{N}$. Then, the RVs $\tilde{Z}_{k,n}^{\prime}\left(F_{{{\boldsymbol{X}}}_{n}}\right)$ and $Z_{k}^{\prime}\left(F_{{{\boldsymbol{X}}}}\right)$ defined in [17, Eqn. (B.1)] satisfy $\tilde{Z}_{k,n}^{\prime}\left(F_{{{\boldsymbol{X}}}_{n}}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}Z_{k}^{\prime}\left(F_{{{\boldsymbol{X}}}}\right)$ uniformly over $k\in\mathcal{N}$. . Then, since $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)=\frac{1}{k}\log f_{k,n}\left({\boldsymbol{S}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}\right)$ and $Z_{k}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)=\frac{1}{k}\log f_{k,\epsilon}\left({\boldsymbol{S}}_{\epsilon}^{(k)},\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\right)$, we conclude that $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}Z_{k}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$, where it also follows from the proof of [17, Lemma B.2] that the convergence is uniform in $k\in\mathcal{N}$. Specifically, to prove that $f_{k,n}\left({\boldsymbol{S}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}f_{k,\epsilon}\left({\boldsymbol{S}}_{\epsilon}^{(k)},\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\right)$, we will show that the following two properties hold: 1. P1 The distribution of $\left[\left({\boldsymbol{S}}_{\epsilon}^{(k)}\right)^{T},\left(\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\right)^{T}\right]^{T}$ is separable 111111By [37, Pg. 101], an RV $X\in\mathcal{X}$ is separable if $\forall\eta>0$ there exists a compact set $\mathcal{K}(\eta)\subset\mathcal{X}$ such that $\Pr\left(X\in\mathcal{K}(\eta)\right)\geq 1-\eta$. . 2. P2 For any convergent sequence $\left(\left({\boldsymbol{s}}_{n}^{(k)}\right)^{T},\left(\hat{{\boldsymbol{s}}}_{n}^{(k)}\right)^{T}\right)^{T}\in\mathcal{R}^{2k}$ such that $\mathop{\lim}\limits_{n\rightarrow\infty}\left({\boldsymbol{{\boldsymbol{s}}}}_{n}^{(k)},\hat{{\boldsymbol{s}}}_{n}^{(k)}\right)=\left({\boldsymbol{{\boldsymbol{s}}}}_{\epsilon}^{(k)},\hat{{\boldsymbol{s}}}_{\epsilon}^{(k)}\right)$, then $\mathop{\lim}\limits_{n\rightarrow\infty}f_{k,n}\left({\boldsymbol{s}}_{n}^{(k)},\hat{{\boldsymbol{s}}}_{n}^{(k)}\right)=f_{k,\epsilon}\left({\boldsymbol{s}}_{\epsilon}^{(k)},\hat{{\boldsymbol{s}}}_{\epsilon}^{(k)}\right)$. To prove property P1, we show that ${U}^{(k)}\triangleq\left[\big{(}{\boldsymbol{S}}_{\epsilon}^{(k)}\big{)}^{T},\big{(}\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\big{)}^{T}\right]^{T}$ is 121212We point out that here, we misuse use the dimension notation as $U^{(k)}$ which denotes a $2k$ dimensional vector. Here, $k$ refers to the dimension of the compression problem and not of the vector. separable [37, Pg. 101], i.e., we show that $\forall\eta>0$, there exists $\beta>0$ such that $\Pr\left(\|U^{(k)}\|^{2}>\beta\right)<\eta$. To that aim, recall first that by Markov’s inequality [29, Pg. 114], it follows that $\Pr\left(\right\|U^{(k)}\left\|{}^{2}>\beta\right)<\frac{1}{\beta}\mathds{E}\left\\{\left\|U^{(k)}\right\|^{2}\right\\}$. For the asynchronously sampled source process, we note that $\sigma^{2}_{S_{\epsilon}}[i]\triangleq\mathds{E}\left\\{\left(S_{\epsilon}[i]\right)^{2}\right\\}\in[0,\mathop{\max}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)]$. By the independence of ${\boldsymbol{W}}_{\epsilon}^{(k)}$ and $\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}$, and by the fact that their mean is zero, we have, from (D.14) that $\mathds{E}\left\\{\left(S_{\epsilon}[i]\right)^{2}\right\\}=\mathds{E}\left\\{\left(\hat{S}_{\epsilon}[i]\right)^{2}\right\\}+\mathds{E}\left\\{\left(W_{\epsilon}[i]\right)^{2}\right\\}\leq\mathop{\max}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)$; Hence $\mathds{E}\left\\{\left(\hat{S}_{\epsilon}[i]\right)^{2}\right\\}\leq\mathop{\max}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)$, and $\mathds{E}\left\\{\left(W_{\epsilon}[i]\right)^{2}\right\\}\leq\mathop{\max}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)$. This further implies that $\mathds{E}\left\\{\left\|U^{(k)}\right\|^{2}\right\\}=\mathds{E}\left\\{\left\|\left[\big{(}{\boldsymbol{S}}_{\epsilon}^{(k)}\big{)}^{T},\big{(}\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\big{)}^{T}\right]^{T}\right\|^{2}\right\\}\leq 2\cdot k\cdot\mathop{\max}\limits_{0\leq t\leq T_{\rm ps}}\sigma^{2}_{S_{\rm c}}(t)$ ; therefore for each $\beta>\frac{1}{\eta}\mathds{E}\left\\{\left\|U^{(k)}\right\|^{2}\right\\}$ we have that $\Pr\left(\left\|U^{(k)}\right\|^{2}>\beta\right)<\eta$, and thus $U^{(k)}$ is separable. By the assumption in this lemma it follows that $\forall\eta>0$ there exists $n_{0}(\eta)>0$ such that for all $n>n_{0}(\eta)$ we have that $\forall{\boldsymbol{w}}^{(k)}\in\mathcal{R}^{k}$, $\big{|}p_{{\boldsymbol{W}}_{n}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)-p_{{\boldsymbol{W}}_{\epsilon}^{(k)}}\left(w^{(k)}\right)\big{|}<\eta$, for all sufficiently large $k\in\mathcal{N}$. Consequently, for all $\left(\left({\boldsymbol{s}}^{(k)}\right)^{T},\left(\hat{{\boldsymbol{s}}}^{(k)}\right)^{T}\right)^{T}\in\mathcal{R}^{2k}$, $n>n_{0}(\eta)$ and a sufficiently large $k\in\mathcal{N}$, it follows from (D.8) that $\displaystyle\left|p_{{\boldsymbol{S}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)-p_{{\boldsymbol{S}}_{\epsilon}^{(k)}|\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}\big{|}\hat{{\boldsymbol{s}}}^{(k)}\right)\right|$ $\displaystyle=\left|p_{{\boldsymbol{W}}_{n}^{(k)}}\left({\boldsymbol{s}}^{(k)}-\hat{{\boldsymbol{s}}}^{(k)}\right)-p_{{\boldsymbol{W}}_{\epsilon}^{(k)}}\left({\boldsymbol{s}}^{(k)}-\hat{{\boldsymbol{s}}}^{(k)}\right)\right|<\eta.$ (D.15) Following the continuity of $p_{{\boldsymbol{S}}_{n}^{(k)}|\hat{{\boldsymbol{S}}}_{n}^{(k)}}\left(s^{(k)}\big{|}\hat{s}^{(k)}\right)$ and of $p_{{\boldsymbol{S}}_{n}^{(k)}}({\boldsymbol{s}}^{(k)})$, $f_{k,n}\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)$ is also continuous [31, Thm. 4.9] 131313[31, Thm. 4.9]: Let $f$ and $g$ be complex continuous functions on a metric space $X$. Then $f+g$, $fg$ and $f/g$ are continuous on $X$. In the last case, we must assume that $g(x)\neq 0$, for all $x\in X$ ; hence, when $\mathop{\lim}\limits_{n\rightarrow\infty}\big{(}{\boldsymbol{s}}_{n}^{(k)},\hat{{\boldsymbol{s}}}_{n}^{(k)}\big{)}=\big{(}{\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\big{)}$, then $\mathop{\lim}\limits_{n\rightarrow\infty}f_{k,n}\left({\boldsymbol{s}}_{n}^{(k)},\hat{{\boldsymbol{s}}}_{n}^{(k)}\right)=f_{k,\epsilon}\left({\boldsymbol{s}}^{(k)},\hat{{\boldsymbol{s}}}^{(k)}\right)$. This satisfies condition P2 for the extended CMT; Therefore, by the extended CMT, we have that $f_{k,n}\left({\boldsymbol{S}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}f_{k,\epsilon}\left({\boldsymbol{S}}_{\epsilon}^{(k)},\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\right)$. Since the RVs $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)$ and $Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$, defined in (D.1), are also continuous mappings of $f_{k,n}\left({\boldsymbol{S}}_{n}^{(k)},\hat{{\boldsymbol{S}}}_{n}^{(k)}\right)$ and of $f_{k,\epsilon}\left({\boldsymbol{S}}_{\epsilon}^{(k)},\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\right)$, respectively, it follows from the CMT [37, Thm. 7.7] that $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$. Finally, to prove that the convergence $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$ is uniform in $k\in\mathcal{N}$, we note that as $\hat{{\boldsymbol{S}}}_{n}^{(k)}$ and $\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}$ have independent entries, and the backward channels (21) and (D.14) are memoryless. Hence, it follows from the proof of [17, Lemma B.2], that the characteristic function of the RV $k\cdot\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)$ which is denoted by $\Phi_{k\cdot\tilde{Z}_{k,n}}(\alpha)\triangleq\mathds{E}\left\\{e^{j\cdot\alpha\cdot k\cdot\tilde{Z}_{k,n}}\right\\}$ converges to the characteristic function of $k\cdot Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$, denoted by $\Phi_{k\cdot Z_{k,\epsilon}}(\alpha)$, uniformly over $k\in\mathcal{N}$. Thus, for all sufficiently small $\eta>0$, $\exists k_{0}\in\mathcal{N},n_{0}(\eta,k_{0})\in\mathcal{N}$ such that $\forall n>n_{0}(\eta,k_{0})$, and $\forall k>k_{0}$ $\big{|}\Phi_{k\cdot\tilde{Z}_{k,n}}(\alpha)-\Phi_{k\cdot Z_{k,\epsilon}}(\alpha)\big{|}<\eta,\quad\forall\alpha\cdot\in\mathcal{R}.$ (D.16) Hence, following Lévy’s convergence theorem [38, Thm. 18.1] 141414[38, Thm. 18.1]: Let $(F_{n})$ be a sequence of density functions and let $\phi_{n}$ denote the characteristic function of $F_{n}$. Suppose that $g(\theta)\coloneqq\lim\phi_{n}(\theta)$ exists for all $\theta\in\mathcal{R}$, and that $g(\cdot)$ is continuous at $0$. Then $g=\phi F$ for some distribution function $F$, and $F_{n}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}F$. we conclude that $k\cdot\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}k\cdot Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$ and that this convergence is uniform for sufficiently large $k$. Finally, since the CDFs of $k\cdot\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)$ and $k\cdot Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$ obtained at $\alpha\in\mathcal{R}$ are equivalent to the CDFs of $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)$ and $Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$ obtained at $\frac{\alpha}{k}\in\mathcal{R}$ respectively, we can conclude that $\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)$, uniformly in $k\in\mathcal{N}$. ∎ The following convergence lemma D.3 corresponds to [17, Lemma. B.3], ###### Lemma D.3. Let $n\in\mathcal{N}$ be given. Every subsequence of $\left\\{\tilde{Z}_{k,n}^{\prime}\left(F_{\hat{S}_{n},S_{n}}^{\rm opt}\right)\right\\}_{k\in\mathcal{N}}$, indexed by $k_{l}$, converges in distribution, in the limit as $l\rightarrow\infty$, to a finite deterministic scalar. ###### Proof. Recall that the RVs $\tilde{Z}_{k,n}^{\prime}\left(F_{\hat{S}_{n},S_{n}}^{\rm opt}\right)$ represent the mutual information density rate between $k$ samples of the source process $S_{n}[i]$ and the corresponding samples of its reproduction process $\hat{S}_{n}[i]$, where these processes are jointly distributed via the Gaussian distribution measure $F_{\hat{S}_{n},S_{n}}^{\rm opt}$. Further, recall that the relationship between the source signal and the reproduction process which achieves the RDF can be described via the backward channel in (21) for a Gaussian source. The channel (21) is a memoryless additive WSCS Gaussian noise channel with period $p_{n}$, thus, by [21], it can be equivalently represented as a $p_{n}\times 1$ multivariate memoryless additive stationary Gaussian noise channel, which is an information stable channel [39, Sec. 1.5]151515Information stable channels can be described as having the property that the input that maximizes mutual information and its corresponding output behave ergodically [15]. Also, the information stability was further defined in [16, Sec. IV] by applying the fact that ergodic theory is consequential to the law of large numbers. [14, Eq. (3.9.2)]: A general source $V=\left\\{V^{n}\right\\}_{n=1}^{\infty}$ is said to be information- stable if $\frac{\frac{1}{n}\log\frac{1}{P_{v^{n}}\left(V^{n}\right)}}{H_{n}\left(V^{n}\right)}\rightarrow 1$. where $H_{n}\left(V^{n}\right)=\frac{1}{n}H\left(V^{n}\right)$ and $H\left(V^{n}\right)$ stands for the entropy of $V^{n}$.. For such channels in which the source and its reproduction obey the RDF-achieving joint distribution $F_{S_{n},\hat{S}_{n}}^{\rm opt}$, the mutual information density rate converges as $k$ increases almost surely to the finite and deterministic mutual information rate [14, Thm. 5.9.1] 161616[14, Thm. 5.9.1] holds for a subadditive distortion measure [14, Eqn. (5.9.2)]; The MSE distortion measure, which was used in this research, is additive (and thus also subadditive).. Since almost sure convergence implies convergence in distribution [37, Lemma 7.21], this proves the lemma. ∎ ### D-C Showing that ${R_{\epsilon}(D)=\mathop{\lim\sup}\limits_{n\rightarrow\infty}R_{n}(D)}$ This section completes the proof to Theorem 4. We note from (14) that the RDF for the source process $S_{n}[i]$ (for fixed length coding and MSE distortion measure) is given by: $R_{n}(D)=\mathop{\inf}\limits_{F_{\hat{S}_{n},S_{n}}:\bar{d}_{S}\left(F_{\hat{S}_{n},S_{n}}\right)\leq D}\left\\{{\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}\tilde{Z}_{k,n}^{\prime}\left(F_{\hat{S}_{n},S_{n}}^{\rm opt}\right)\right\\},$ (D.17) where $\bar{d}_{S}\left(F_{\hat{S}_{n},S_{n}}\right)=\mathop{\lim\sup}\limits_{k\rightarrow\infty}\frac{1}{k}\mathds{E}\big{\\{}\big{\|}{\boldsymbol{S}}_{n}^{(k)}-\hat{{\boldsymbol{S}}}_{n}^{(k)}\big{\|}^{2}\big{\\}}$. We now state the following lemma characterizing the asymptotic statistics of the optimal reconstruction $\hat{{\boldsymbol{S}}}_{n}^{(k)}$ process and the respective noise process ${\boldsymbol{W}}_{n}^{(k)}$ used in the backward channel relationship (21): ###### Lemma D.4. Consider the RDF-achieving distribution with distortion $D$ for compression of a vector Gaussian source process ${\boldsymbol{S}}_{n}^{(k)}$ characterized by the backward channel (21). Then, there exists a subsequence in the index $n\in\mathcal{N}$ denoted $n_{1}<n_{2}<\ldots$, such that for the RDF- achieving distribution, the sequences of reproduction vectors $\\{\hat{{\boldsymbol{S}}}_{n_{l}}^{(k)}\\}_{l\in\mathcal{N}}$ and backward channel noise vectors $\\{{\boldsymbol{W}}_{n_{l}}^{(k)}\\}_{l\in\mathcal{N}}$ satisfy that $\mathop{\lim}\limits_{l\rightarrow\infty}p_{\hat{{\boldsymbol{S}}}_{n_{l}}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)=p_{\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ uniformly in ${\boldsymbol{\hat{s}}}^{(k)}\in\mathcal{R}^{k}$ and uniformly with respect to $k\in\mathcal{N}$, as well as $\mathop{\lim}\limits_{l\rightarrow\infty}p_{{\boldsymbol{W}}_{n_{l}}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)=p_{{\boldsymbol{W}}_{\epsilon}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)$ uniformly in ${\boldsymbol{w}}^{(k)}\in\mathcal{R}^{k}$ and uniformly with respect to $k\in\mathcal{N}$, where $p_{\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ and $p_{{\boldsymbol{W}}_{\epsilon}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)$ are Gaussian PDFs. ###### Proof. Recall from the analysis of the RDF for WSCS processes that for each $n\in\mathcal{N}$, the marginal distributions of the RDF-achieving reproduction process $\hat{S}_{n}[i]$ and the backward channel noise $W_{n}[i]$ is Gaussian, memoryless, zero-mean, and with variances $\sigma_{\hat{S}_{n}}^{2}[i]\triangleq\mathds{E}\left\\{\big{(}\hat{S}_{n}[i]\big{)}^{2}\right\\}$ and $\mathds{E}\left\\{\big{(}W_{n}[i]\big{)}^{2}\right\\}=\sigma_{{S}_{n}}^{2}[i]-\sigma_{\hat{S}_{n}}^{2}[i],$ (D.18) respectively. Consequently, the sequences of reproduction vectors $\\{\hat{{\boldsymbol{S}}}_{n}^{(k)}\\}_{n\in\mathcal{N}}$ and backward channel noise vectors $\\{{\boldsymbol{W}}_{n}^{(k)}\\}_{n\in\mathcal{N}}$ are zero-mean Gaussian with independent entries for each $k\in\mathcal{N}$. Since $\sigma_{{S}_{n}}^{2}[i]\leq\mathop{\max}\limits_{t\in\mathcal{R}}\sigma_{S_{c}}^{2}(t)$, then, from (D.18), it follows that $\sigma^{2}_{\hat{S}_{n}}[i]$ is also bounded in the interval $[0,\mathop{\max}\limits_{t\in\mathcal{R}}\sigma_{S_{c}}^{2}(t)]$ for all $n\in\mathcal{N}$. Therefore, by Bolzano-Weierstrass theorem [31, Thm. 2.42] 171717Every bounded infinite subset of $\mathcal{R}^{k}$ has a limit point in $\mathcal{R}^{k}$. , $\sigma^{2}_{\hat{S}_{n}}[i]$ has a convergent subsequence, and we let $n_{1}<n_{2}<\ldots$ denote the indexes of this convergent subsequence and let the limit of the subsequence be denoted by $\sigma_{\hat{S}_{\epsilon}}^{2}[i]$. From the CMT, as applied in the proof of [17, Lemma B.1], the convergence $\sigma_{\hat{S}_{n_{l}}}^{2}[i]\mathop{\longrightarrow}\limits_{l\rightarrow\infty}\sigma_{\hat{S}_{\epsilon}}^{2}[i]$ for each $i\in\mathcal{N}$ implies that the subsequence of PDFs $p_{\hat{{\boldsymbol{S}}}_{n_{l}}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ corresponding to the memoryless Gaussian random vectors $\\{\hat{{\boldsymbol{S}}}_{n_{l}}^{(k)}\\}_{l\in\mathcal{N}}$ converges as $l\rightarrow\infty$ to a Gaussian PDF which we denote by $p_{\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$, and the convergence of $p_{\hat{{\boldsymbol{S}}}_{n_{l}}^{(k)}}\left(\hat{{\boldsymbol{s}}}^{(k)}\right)$ is uniform in ${\boldsymbol{s}}^{(k)}$ for any fixed $k\in\mathcal{N}$. By Remark 1, it holds that $W_{n}[i]$ is a memoryless stationary process with variance $\mathds{E}\left\\{\left(W_{n}[i]\right)^{2}\right\\}=D$ and by Eq. (D.18), $\sigma^{2}_{\hat{S}_{n}}[i]=\sigma^{2}_{S_{n}}[i]-D$. Hence by Assumption D.1 and by the proof of [17, Lemma B.1], it follows that for a fixed $\eta>0$ and $k_{0}\in\mathcal{N}$, $\exists n_{0}(\eta,k_{0})$ such that for all $n>n_{0}(\eta,k_{0})$ and for all sufficiently large $k$, it holds that $\big{|}p_{\hat{{\boldsymbol{S}}}_{n_{l}}^{(k)}}\big{(}\hat{{\boldsymbol{s}}}^{(k)}\big{)}-p_{\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}}\big{(}\hat{{\boldsymbol{s}}}^{(k)}\big{)}\big{|}<\eta$ for every $\hat{{\boldsymbol{s}}}^{(k)}\in\mathcal{R}^{k}$. Since $n_{0}(\eta,k_{0})$ does not depend on $k$ (only on the fixed $k_{0}$), this implies that the convergence is uniform with respect to $k\in\mathcal{N}$. The fact that $W_{n}[i]$ is a zero-mean stationary Gaussian process with variance $D$ for each $n\in\mathcal{N}$, implies that the sequence of PDFs $p_{{\boldsymbol{W}}_{n}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)$ converges as $n\rightarrow\infty$ to a Gaussian PDF which we denote by $p_{{\boldsymbol{W}}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)$, hence its subsequence with indices $n_{1}<n_{2}<\ldots$ also converges to $p_{{\boldsymbol{W}}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)$. Since $D>\frac{1}{2\pi}$ by Assumption D.1 combined with the proof of [17, Lemma B.1] it follows that this convergence is uniform in ${\boldsymbol{w}}^{(k)}$ and in $k\in\mathcal{N}$ to $p_{{\boldsymbol{W}}_{\epsilon}^{(k)}}\left({\boldsymbol{w}}^{(k)}\right)$. Following the proof of Corollary D.1, it holds that the subsequences of the memoryless Gaussian random vectors $\left\\{\hat{{\boldsymbol{S}}}_{n_{l}}^{(k)}\right\\}$ and $\left\\{{\boldsymbol{W}}_{n_{l}}^{(k)}\right\\}$ converge in distribution as $l\rightarrow\infty$ to a Gaussian distribution, and the convergence is uniform in $k\in\mathcal{N}$ for any fixed $k\in\mathcal{N}$. Hence, as shown in Lemma D.2 the joint distribution $\left[\big{(}{\boldsymbol{S}}_{n_{l}}^{(k)}\big{)}^{T},\big{(}\hat{{\boldsymbol{S}}}_{n_{l}}^{(k)}\big{)}^{T}\right]^{T}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}\left[\big{(}{\boldsymbol{S}}_{\epsilon}^{(k)}\big{)}^{T},\big{(}\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)}\big{)}^{T}\right]^{T}$, and the limit distribution is jointly Gaussian. ∎ ###### Lemma D.5. The RDF of $\\{S_{\epsilon}[i]\\}$ satisfies $R_{\epsilon}(D)\leq\mathop{\lim\sup}\limits_{n\rightarrow\infty}R_{n}(D)$, and the rate $\mathop{\lim\sup}\limits_{n\rightarrow\infty}R_{n}(D)$ is achievable for the source $\\{S_{\epsilon}[i]\\}$ with distortion $D$ when the reproduction process which obeys a Gaussian distribution. ###### Proof. According to Lemma D.4, we note that the sequence of joint distributions $\\{F_{S_{n},\hat{S}_{n}}^{\rm opt}\\}_{n\in\mathcal{N}}$ has a convergent subsequence, i.e., there exists a set of indexes $n_{1}<n_{2}<\ldots$ such that the sequence of distributions with independent entries $\\{F_{S_{n_{l}},\hat{S}_{n_{l}}}^{\rm opt}\\}_{l\in\mathcal{N}}$ converges in the limit $l\rightarrow\infty$ to a joint Gaussian distribution $F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime}$ and the convergence is uniform in $k\in\mathcal{N}$. Hence, this satisfies the condition of Lemma D.2; This implies that $\tilde{Z}_{k,n_{l}}^{\prime}\left(F_{S_{n_{l}},\hat{S}_{n_{l}}}^{\rm opt}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{l\rightarrow\infty}Z_{k}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime}\right)$ uniformly in $k\in\mathcal{N}$. Also, by Lemma D.3 every subsequence of $\big{\\{}\tilde{Z}_{k,n_{l}}^{\prime}\big{(}F_{S_{n_{l}},\hat{S}_{n_{l}}}^{\rm opt}\big{)}\big{\\}}_{l\in\mathcal{N}}$ converges in distribution to a finite deterministic scalar as $k\rightarrow\infty$. Therefore, by Theorem 3 it holds that $\displaystyle\mathop{\lim}\limits_{l\rightarrow\infty}\left({\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}\tilde{Z}_{k,n_{l}}^{\prime}\left(F_{S_{n_{l}},\hat{S}_{n_{l}}}^{\rm opt}\right)\right)$ $\displaystyle={\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime}\right)$ $\displaystyle\geq\mathop{\inf}\limits_{F_{S_{\epsilon},\hat{S}_{\epsilon}}}\left\\{{\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}\right)\right\\}=R_{\epsilon}(D).$ (D.19) From (14) we have that $R_{n}(D)={\rm p-}\mathop{\lim\sup}\limits_{k\rightarrow\infty}\tilde{Z}_{k,n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)$, then from (D-C), it follows that $R_{\epsilon}(D)\leq\mathop{\lim}\limits_{l\rightarrow\infty}R_{n_{l}}(D)\stackrel{{\scriptstyle(a)}}{{\leq}}\mathop{\lim\sup}\limits_{n\rightarrow\infty}R_{n}(D),$ (D.20) where $(a)$ follows since, by [31, Def. 3.16], the limit of every subsequence is not greater than the limit superior. Noting that $F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime}$ is Gaussian by Lemma D.4 concludes the proof. ∎ ###### Lemma D.6. The RDF of $\\{S_{\epsilon}[i]\\}$ satisfies $R_{\epsilon}(D)\geq\mathop{\lim\sup}\limits_{n\rightarrow\infty}R_{n}(D)$. ###### Proof. To prove this lemma, we first show that for a joint distribution $F_{S_{\epsilon},\hat{S}_{\epsilon}}$ which achieves a rate-distortion pair $(R_{\epsilon},D)$ it holds that $R_{\epsilon}\geq\mathds{E}\\{Z_{k,\epsilon}^{\prime}(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime})\\}$: Recall that $(R_{\epsilon},D)$ is an achievable rate-distortion pair for the source $\\{S_{\epsilon}[i]\\}$, namely, there exists a sequence of codes $\\{\mathcal{C}_{l}\\}$ whose rate-distortion approach $(R_{\epsilon},D)$ when applied to $\\{S_{\epsilon}[i]\\}$, This implies that for any $\eta>0$ there exists $l_{0}(\eta)$ such that $\forall l>l_{0}(\eta)$ it holds that $\mathcal{C}_{l}$ has a code rate $R_{l}=\frac{1}{l}\log_{2}M_{l}$ satisfying $R_{l}\leq R_{\epsilon}+\eta$ by (3). Recalling Def. 4, the source code maps ${\boldsymbol{S}}_{\epsilon}^{(l)}$ into a discrete index $J_{l}\in\\{1,2,\ldots,M_{l}\\}$, which is in turn mapped into $\hat{{\boldsymbol{S}}}_{\epsilon}^{(l)}$, i.e., ${{\boldsymbol{S}}}_{\epsilon}^{(l)}\mapsto J_{l}\mapsto\hat{{\boldsymbol{S}}}_{\epsilon}^{(l)}$ form a Markov chain. Since $J_{l}$ is a discrete random variable taking values in $\\{1,2,\ldots,M_{l}\\}$, it holds that $\displaystyle\log_{2}M_{l}$ $\displaystyle\geq H(J_{l})$ $\displaystyle\stackrel{{\scriptstyle(a)}}{{\geq}}I({{\boldsymbol{S}}}_{\epsilon}^{(l)};J_{l})$ $\displaystyle\stackrel{{\scriptstyle(b)}}{{\geq}}I({{\boldsymbol{S}}}_{\epsilon}^{(l)};\hat{{\boldsymbol{S}}}_{\epsilon}^{(l)}),$ (D.21) where $(a)$ follows since $I({{\boldsymbol{S}}}_{\epsilon}^{(l)};J_{l})=H(J_{l})-H(J_{l}|{{\boldsymbol{S}}}_{\epsilon}^{(l)})$ which is not larger than $H(J_{l})$ as $J_{l}$ takes discrete values; while $(b)$ follows from the data processing inequality [5, Ch. 2.8]. Now, (D.21) implies that for each $l>l_{0}(\eta)$, the reproduction obtained using the code $\mathcal{C}_{l}$ satisfies $\frac{1}{l}I({{\boldsymbol{S}}}_{\epsilon}^{(l)};\hat{{\boldsymbol{S}}}_{\epsilon}^{(l)})\leq\frac{1}{l}\log M_{l}\leq R_{\epsilon}+\eta$. Since for every arbitrarily small $\eta\rightarrow 0$, this inequality holds for all $l>l_{0}(\eta)$, i.e., for all sufficiently large $l$, it follows that $R_{\epsilon}\geq\mathop{\lim\sup}\limits_{k\rightarrow\infty}\frac{1}{l}I({{\boldsymbol{S}}}_{\epsilon}^{(l)};\hat{{\boldsymbol{S}}}_{\epsilon}^{(l)})$. Hence, replacing the blocklength symbol from $l$ to $k$, as $\frac{1}{k}I({\boldsymbol{S}}_{\epsilon}^{(k)},\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)})=\mathds{E}\\{Z_{k,\epsilon}^{\prime}(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime})\\}$[5, Eqn. (2.3)], we conclude that $R_{\epsilon}(D)\geq\mathop{\lim\sup}\limits_{k\rightarrow\infty}\mathds{E}\\{Z_{k,\epsilon}^{\prime}(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime})\\}.$ (D.22) Next, we consider $\mathop{\lim\sup}\limits_{k\rightarrow\infty}\mathds{E}\\{Z_{k_{l},\epsilon}^{\prime}(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime})\\}$: Let $Z_{k_{l},\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime}\right)$ be a subsequence of $\mathds{E}\left\\{Z_{k,\epsilon}^{\prime}(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime})\right\\}$ with the indexes $k_{1}<k_{2}<\ldots$ such that its limit equals the limit superior. i.e., $\mathop{\lim}\limits_{l\rightarrow\infty}\mathds{E}\left\\{Z_{k_{l},\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime}\right)\right\\}=\mathop{\lim\sup}\limits_{k\rightarrow\infty}\mathds{E}\left\\{Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime}\right)\right\\}$. Since by Lemma D.2, the sequence of non-negative RVs $\left\\{\tilde{Z}_{k_{l},n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\right\\}_{n\in\mathcal{N}}$ convergences in distribution to $Z_{k_{l},\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime}\right)$ as $n\rightarrow\infty$ uniformly in $k\in\mathcal{N}$, it follows from 181818[40, Thm. 3.5] states that if $X_{n}$ are uniformly integrable and $X_{n}\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}X$ then $\mathds{E}\\{X_{n}\\}\mathop{\longrightarrow}\limits_{n\rightarrow\infty}\mathds{E}\\{X\\}$. [40, Thm. 3.5] that $\mathds{E}\left\\{Z_{k_{l},\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime}\right)\right\\}=\mathop{\lim}\limits_{n\rightarrow\infty}\mathds{E}\left\\{\tilde{Z}_{k_{l},n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\right\\}$. Also, we define a family of distributions $\mathcal{F}(D)$ such that ${\mathcal{F}(D)=\\{F_{S,\hat{S}}:\mathsf{D}\left(F_{S,\hat{S}}\right)\leq D\\}}$. Consequently, Eq. (D.22) can now be written as: $\displaystyle R_{\epsilon}(D)\geq\mathop{\lim\sup}\limits_{k\rightarrow\infty}\mathds{E}\left\\{Z_{k,\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}_{\epsilon}}^{\prime}\right)\right\\}$ $\displaystyle=\mathop{\lim}\limits_{l\rightarrow\infty}\mathop{\lim}\limits_{n\rightarrow\infty}\mathds{E}\left\\{\tilde{Z}_{k_{l},n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\right\\}$ $\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim}\limits_{l\rightarrow\infty}\mathds{E}\left\\{\tilde{Z}_{k_{l},n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\right\\}$ $\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\mathop{\lim\sup}\limits_{n\rightarrow\infty}\mathop{\lim}\limits_{l\rightarrow\infty}\mathds{E}\left\\{\tilde{Z}_{k_{l},n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\right\\}$ $\displaystyle\geq\mathop{\lim\sup}\limits_{n\rightarrow\infty}\mathop{\lim}\limits_{l\rightarrow\infty}\mathop{\inf}\limits_{F_{S,\hat{S}}\in\mathcal{F}(D)}\mathds{E}\left\\{\tilde{Z}_{k_{l},n}^{\prime}\left(F_{S,\hat{S}}\right)\right\\}$ $\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\mathop{\lim\sup}\limits_{n\rightarrow\infty}\mathop{\lim}\limits_{l\rightarrow\infty}\mathop{\inf}\limits_{F_{S,\hat{S}}\in\mathcal{F}(D)}\frac{1}{k_{l}}I\left(\hat{{\boldsymbol{S}}}_{n}^{(k_{l})};{\boldsymbol{S}}_{n}^{(k_{l})}\right),$ (D.23) where $(a)$ follows since the convergence $\tilde{Z}_{k_{l},n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\mathop{\longrightarrow}\limits^{(dist.)}_{n\rightarrow\infty}Z_{k_{l},\epsilon}^{\prime}\left(F_{S_{\epsilon},\hat{S}}^{\prime}\right)$ is uniform with respect to $k_{l}$, thus the limits are interchangeable [31, Thm. 7.11] 191919Rudin: Thm. 7.11: Suppose $f_{n}\rightarrow f$ uniformly in a set $E$ in a metric space. Let $x$ be a limit point of $E$…. $\mathop{\lim}\limits_{t\rightarrow x}\mathop{\lim}\limits_{n\rightarrow\infty}f_{n}(t)=\mathop{\lim}\limits_{n\rightarrow\infty}\mathop{\lim}\limits_{t\rightarrow x}f_{n}(t)$ ; $(b)$ follows since the limit of the subsequence $\mathds{E}\left\\{\tilde{Z}_{k_{l},n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\right\\}$ exists in the index $n$, and is therefore equivalent to the limit superior, $\mathop{\lim\sup}\limits_{n\rightarrow\infty}\mathds{E}\left\\{\tilde{Z}_{k_{l},n}^{\prime}\left(F_{S_{n},\hat{S}_{n}}^{\rm opt}\right)\right\\}$ [31, Page 57]; and $(c)$ holds since mutual information is the expected value of the mutual information density rate [5, Eqn. (2.30)]. Finally, we recall that in the proof of Lemma D.3 it was established that the backward channel for the RDF at the distortion constraint $D$, defined in (21), is information stable, hence for such backward channels, we have from [41, Thm. 1] that the minimum rate is defined as $R_{n}(D)=\mathop{\lim}\limits_{k\rightarrow\infty}\mathop{\inf}\limits_{{F_{S,\hat{S}}\in\mathcal{F}(D)}}\frac{1}{k}I\left(\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)};{\boldsymbol{S}}_{n}^{(k)}\right)$ and the limit exists; Hence, $\mathop{\lim}\limits_{k\rightarrow\infty}\mathop{\inf}\limits_{{F_{S,\hat{S}}\in\mathcal{F}(D)}}\frac{1}{k}I\left(\hat{{\boldsymbol{S}}}_{\epsilon}^{(k)};{\boldsymbol{S}}_{n}^{(k)}\right)=\mathop{\lim}\limits_{l\rightarrow\infty}\mathop{\inf}\limits_{{F_{S,\hat{S}}\in\mathcal{F}(D)}}\frac{1}{k_{l}}I\left(\hat{{\boldsymbol{S}}}^{(k_{l})};{\boldsymbol{S}}_{n}^{(k_{l})}\right)$ in the index $k$. 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2024-09-04T02:54:55.878179

2003.00274

arxiv-papers

# Causal Learning by a Robot with Semantic-Episodic Memory in an Aesop’s Fable Experiment Ajaz A. Bhat School of Psychology University of East Anglia Norwich, NR47TJ, UK <EMAIL_ADDRESS> &Vishwanathan Mohan School of Computer Science and Electronic Engineering University of Essex Wivenhoe Park Colchester, CO4 3SQ, UK <EMAIL_ADDRESS> ###### Abstract Corvids, apes, and children solve “The Crow and The Pitcher” task (from Aesop’s Fables) indicating a causal understanding of the task. By cumulatively interacting with different objects, how can cognitive agents abstract the underlying cause-effect relations to predict affordances of novel objects? We address this question by re-enacting the Aesop’s Fable task on a robot and present a) a brain-guided neural model of semantic-episodic memory; with b) four task-agnostic learning rules that compare expectations from recalled past episodes with the current scenario to progressively extract the hidden causal relations. The ensuing robot behaviours illustrate causal learning; and predictions for novel objects converge to Archimedes’ principle, independent of both the objects explored during learning and the order of their cumulative exploration. ## 1 Introduction The ability to learn causal regularities and object affordances allows organisms to exploit objects in the service of their needs and wants, an obvious adaptive advantage in the race for survival. Experiments exploring paradigms like ‘floating peanut’ task (Hanus et al., 2011), trap-tube task (Martin-Ordas et al., 2008) and more recently the Crow and the Pitcher task from Aesop’s Fable (Jelbert et al., 2014; Cheke et al., 2012) provide evidence that children, primates and corvids can reason to varying degrees about the causally task-relevant properties of objects. A pertinent question therefore is how through cumulative and explorative interactions with different objects in the world, cognitive agents learn task-relevant physical and causal relations and then exploit these flexibly in novel contexts. Accordingly, theoretical works have investigated causal learning in cognitive psychology (Gopnik et al., 2004; Griffiths & Tenenbaum, 2005), computer science (Pearl, 2009; Shimizu et al., 2006), recently in machine learning (Battaglia et al., 2016; Iten et al., 2020; Baradel et al., 2020) but less so in robotics (Xiong et al., 2016). We explore this question in the context of robotics to show how a semantic-episodic memory system along with a set of four learning rules endows a robot, iCub with the capability to cumulatively extract and infer causal relations between objects and actions in ”the Crow and the Pitcher” task. In designing this model, we connect some major trends from neuroscience on distributed hub-based semantic memory representation (Kiefer & Pulvermüller, 2012; Ralph et al., 2017), small-word properties (Sporns, 2011), episodic memory circuitry (Allen & Fortin, 2013; Lee et al., 2015) to hypothesize possible mechanisms of causal learning. ### 1.1 Causal Learning Task The task is inspired from an Aesop’s fable (see analogous empirical works (Jelbert et al., 2014; Bird & Emery, 2009)), in which a thirsty crow drops pebbles into a half-filled water pitcher, raising the water level high enough to drink. In a comparable scenario (Figure 1 A), iCub has a set of objects (on a reachable stand) and a jar of water containing a floating target (green ball) in front. With the goal to reach the green ball (which as such is unreachable), iCub explores if use of available objects can help realize its (otherwise unrealizable) goal. A priori, there is no previous experience with any of the objects. Hence, iCub knows nothing a priori concerning the causal nature of the task. Figure 1: Panel A shows the setup. Each episode involves robot either dropping a single object or making a choice between multiple objects. Objects vary in their physical properties: color, size, shape and weight. Panel B shows the block diagram of the proposed model. Panel C is a flowchart of how the learning rules are applied. ## 2 Model Description Hub-based Semantic Memory: Figure 1 B shows the proposed model. At the bottom is the sensory layer to analyze object properties: colour, shape, size and weight. Word labels are provided by the teacher either to issue goals to iCub or teach names of new objects. Output from sensory/physical layer is passed upwards to a bunch of property-specific self-organizing maps (SOMs) that encode object properties as concept features. Neural connectivity between the sensory layer and the property-specific maps is learnt using standard SOM procedure (see Bhat et al. (2016)). There onwards, perceptual analysis of an object (e.g. a large heavy blue cylinder) leads to activations in different property-specific SOM’s coding for color, shape, weight and size respectively. In this sense, layer 1 emulates the distributed property-specific organization of perceptual information in brain (Patterson et al., 2007; Martin, 2016). Activity in these maps forms the bottom up input to the layer 2 SOM i.e. the object hub. The learning rule for dual dyad (Park & Friston, 2013) connectivity matrix $W$ from SOMs to the object hub (and its counterpart $W^{\prime}$ backwards) is: if the net activity due to a neuron $i$ and a neuron $j$ winning in the maps manages to activate a neuron $k$ in the object hub, set $W_{ik}=1$ and $W_{jk}=1$. Thereby, the object hub facilitates integration and multimodal object representation, as evidenced neuro- scientifically (Ralph et al., 2017; Kiefer & Pulvermüller, 2012). This small- world network (Sporns, 2011) of maps and the object hub is complemented with a dynamics to let the neural activity in one map retro-activate other members of the network, hence allowing information to move top down, bottom up or in cross modal fashion. So, a word like “blue cylinder” activates word-map that forwards activations to the object hub, which thereafter retro-activates the shape and colour maps, as if expecting top down what a “blue cylinder” might be perceptually. In an embodied robot, _objects_ in the environment are employed via _actions_. Here, actions are represented at an abstract level (“what can be done with an object”) separated from the action planning details (“how to do”). While the former relates to the motor affordance of an object, the latter relates to motion planning details, and interested reader may refer to Bhat et al. (2017) for details related to motion planning. In layer 2, the abstract representation corresponds to the action hub and consists of single neurons coding for different action goals such as reach, grasp, etc. In this sense, action hub neurons are similar to canonical neurons in the pre-motor cortex (Rizzolatti & Craighero, 2004) that are activated at the sight of objects to which specific actions are applicable. Finally, the consequences of both perceptions and actions alter the state of the body itself. The body hub, another layer 2 neural map explicitly encodes these states of the body (like failing to reach an object etc.). Reward of an experience is either given by the user or evaluated by the robot itself though observation. In this task, reward is the volume/level of water raised by dropping an object into the jar of water. Episodic memory and its link to the Hubs: Practically, when a robot interacts with its environment, it is the ongoing sequences of actions on various objects, the ensuing consequences, internal body state, and rewards received that mainly form the content of its experiences. Thus, in our model, it is the temporal sequence of activations in the different hubs (object, action, reward, body) during an episode that make up the episodic memory content. The episodic memory is realized using a excitatory-inhibitory neural network of auto-associative memory (Mohan et al., 2014; Hopfieid, 2008). It consists of 1000 neurons, organized in a sheet like structure with 20 rows each containing 50 neurons. Every row is an event in time (indicating activation in object hub, action hub, body hub or reward) and the complete memory as an episode of experience. For example, being unable to reach a floating object in a jar of water (Body hub state), perceiving a red cylinder (Object hub state), dropping it in water (Action hub state), fetching a reward of 100 (end reward). In future, if the robot perceives the red cylinder, the object hub state serves as a partial cue to reconstruct the full experience. Importantly, in the memory network of 1000 neurons, multiple episodic memories $(\approx 230)$ can be represented and retrieved (Hopfieid, 2008). See Bhat et al. (2016) for methods to encode new experiences in episodic memory and recall past ones from partial cues. Learning Rules for Causal Abstraction: As Figure 1 C shows, these rules compare what the robot has experienced in the past against what is happening in the present situation. Let $\bigtriangleup Property$ be the difference in activity in a property-specific map when activated bottom up (through sensory layer) and when activated top down through recall of past event from episodic memory. Let $\bigtriangleup Contradiction$ be the difference between the robots anticipation of how an object might behave (expected reward due to recalled past experience) and the real observed behaviour. Then the learning rules are as follows: (E)limination rule : If ${\bigtriangleup Property}\land{\neg{\bigtriangleup}Contradiction}$, then that property is not causally dominant and hence drastically reduce the connection strength between the object hub and the associated property-specific map. (G)rowth rule : If ${\bigtriangleup Property}\land{\bigtriangleup Contradiction}$, then that property is causally dominant. Hence connectivity between the object hub and the associated property-specific is strengthened, and encode the experience in episodic memory. Contradiction in the robot’s anticipation implies that there is something new to learn. (U)ncertainty rule : If ${\neg{\bigtriangleup}Property}\land{\bigtriangleup Contradiction}$, the connection strength between the object hub and the map is marginally reduced. In this condition it is not possible to infer whether the property is causally dominant or not, unless further experience is gained. (S)tatus Quo rule : If ${\neg{\bigtriangleup}Property}\land{\neg{\bigtriangleup}Contradiction}$, nothing new to learn, so no change in connectivity. ## 3 Results Given that the robot is learning cumulatively, this section presents successive episodes of learning (a video playlist of these experiments is available _online_). All episodes share a set of computational processes, 1) bottom up experience/interaction with the world and hence activation of various maps 2) recall of past experiences from memory (if any); 3) Use of recalled past experiences to anticipate and 4) application of the learning rules. Learning that colour of objects is causally irrelevant to the Aesop’s fable task Episode 1: In the first episode, iCub is given the goal to reach the green ball (in the jar of water). The motion planning system of iCub provides the information that the goal is unreachable. A large heavy red cylinder is available and detected (see Figure 2 left, _Object 1_ ). Bottom up sensory streams activate property-specific maps related to (red) color, (cylinder) shape, $(11.5cm)$ size and $(420g)$ weight properties leading to a distributed representation of the object in the maps and object hub. The object hub activity leads to generation of a partial cue for the recall of any related past episodes. Since there is no previous experience in the episodic memory, nothing is recalled. So, there is no top down activity in the object hub nor any reward expected. With only option to explore, the robot picks and drops the object into the jar of water. The object sinks in the water displacing a volume of water of about $365cm^{3}$ enough to make the floating green sphere reachable. This experience is encoded into the episodic memory: an unreachable goal (body hub state), dropping a large heavy red cylinder (object hub activity), a volume of water displaced $365cm^{3}$ (reward) and goal realized successfully (body hub state). Note, this is a rapid one-shot event encoding into memory. Figure 2: Left panel plots growing causal knowledge as robot explores objects over successive episodes. Causal knowledge regarding a property is either Unknown (depicted by 0 in the plot); or has been learnt to be Dominant or Irrelevant (depicted by 1); or the system expects the property to be Likely Irrelevant ( a certainty value between 0 and 1). Right panel plots error in iCub’s prediction about the volume of water displaced on dropping an object against actual volume displacement calculated using Archimedes’ principle. The four curves correspond to four different orders in which the objects were presented to robot for exploration. Episode 2: iCub is presented with a blue cylinder of the same size and weight as one in episode 1. Object hub activity (with partial similarity to _Object 1_ generates a partial cue that) leads to recall of the only one past experience (i.e. episode 1: Figure 2 left, _Object 2_). So iCub anticipates that large heavy blue cylinder would displace $365cm^{3}$ of water and this turns out to be the case once the robot actually drops the object into water. Comparing the expected behaviour of object (reward hub activity in the recalled past experience) and the observed behaviour, the robot finds no contradiction. In sum, there is change in a property (colour), but it did not cause any contradiction in the expected behaviour. Elimination rule applies here implying colour is not a causally dominant property. The connectivity between the colour map and the object hub is drastically reduced, so they no longer will retro-activate each other. This episode is not encoded into episodic memory. Learning that weight is a causally dominant property Episode 3: iCub is presented with a very light cylinder $(14g)$ with other properties same as in episode 1 (see Figure 2 left, _Object 3_) Bottom-up activity recalls episode 1. A high reward of $365cm^{3}$ is anticipated. However, only a small amount $(24cm^{3})$ of water is displaced after robot’s action, leading to a contradiction between expected and observed behaviours. A comparison of the bottom up activity and the reconstructed top down activity reveals there is a difference in weight map. Growth rule is applied because a change in the weight property causes contradiction. The new experience is encoded into the episodic memory which can be recalled next time for better prediction. Furthermore, activity in shape and size maps showed no change even though there was a contradiction between the expected and the observed behaviour. Hence the Uncertainty rule applies too: as the robot still has no experience or complete knowledge of the causal relevance of object-size or shape. The robot partially believes at this point that ‘shape and size’ of the object may not be relevant in causing water rise. Accumulating over a set of such experiences (Figure 2 left) with objects of different properties, the robot grows it’s causal knowledge (certainty) of properties relevant to the task. Furthermore, in cases when the system is presented a cylinder of a weight never experienced before, all the past experiences (due to the same shape) will be recalled and a weighted averaging of the rewards expected due to these past experiences is used as an estimate of net anticipated reward (see Bhat et al. (2016) for more details). As the number of experiences with objects of different weights increases, the accuracy in the prediction of reward increases systematically. Figure 2 (right) show robot’s predictions in four different random orders of 8 objects. Results show that the causal knowledge is same at the end of explorations in all cases and error in prediction (i.e., difference between robot’s prediction and Archimedes’ principle) rapidly decreases in all cases. ## 4 Conclusion The work takes the topic of affordances from the level of object-action to the level of property-action, in line with emerging studies from neurosciences, and suggests a possible mechanism for causal learning in animals and robots. The work emphasizes that reasoning and learning always have to go hand-in-hand and grow cumulatively and continuously in lifetime of a learner, be it a natural or an artificial cognitive agent. ## References * Allen & Fortin (2013) Timothy A. Allen and Norbert J. Fortin. The evolution of episodic memory. _Proceedings of the National Academy of Sciences of the United States of America_ , 110(SUPPL2):10379–10386, 2013. ISSN 00278424. doi: 10.1073/pnas.1301199110. 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2024-09-04T02:54:55.889507

2003.00277

arxiv-papers

metadata # The imaginary part of the high-harmonic cutoff Emilio Pisanty<EMAIL_ADDRESS>ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona) Marcelo F. Ciappina ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona) Maciej Lewenstein ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona) ICREA, Passeig de Lluís Companys, 23, 08010 Barcelona, Spain (22 July 2020) ###### Abstract High-harmonic generation – the emission of high-frequency radiation by the ionization and subsequent recombination of an atomic electron driven by a strong laser field – is widely understood using a quasiclassical trajectory formalism, derived from a saddle-point approximation, where each saddle corresponds to a complex-valued trajectory whose recombination contributes to the harmonic emission. However, the classification of these saddle points into individual quantum orbits remains a high-friction part of the formalism. Here we present a scheme to classify these trajectories, based on a natural identification of the (complex) time that corresponds to the harmonic cutoff. This identification also provides a natural complex value for the cutoff energy, whose imaginary part controls the strength of quantum-path interference between the quantum orbits that meet at the cutoff. Our construction gives an efficient method to evaluate the location and brightness of the cutoff for a wide class of driver waveforms by solving a single saddle- point equation. It also allows us to explore the intricate topologies of the Riemann surfaces formed by the quantum orbits induced by nontrivial waveforms. Accepted Manuscript for J. Phys. Photonics 2, 034013 (2020), available as arXiv:2003.00277. †† © The authors, 2020. Licensed under CC BY 4.0. High-harmonic generation (HHG) is an extremely nonlinear optical process in which a strong laser field drives the emission of a train of short bursts of high-frequency radiation Krausz2009 ; Corkum2007 , which can cover hundreds of harmonic orders of the driving field, over a broad plateau that terminates at a cutoff. This emission comes from a three-step process in which the laser ionizes the target atom via tunnel ionization, and then propels the released electron back to the parent ion, where it recombines with the hole it left behind, releasing its kinetic energy as a photon Corkum1993 ; Kulander1993 . This emission can be modelled using a wide range of approaches, from classical heuristics Corkum1993 ; Kulander1993 to intensive numerical computations Scrinzi2014 , but the quantitative models that most closely follow the overall intuition are quasiclassical methods Lewenstein1994 ; Amini2019 , known as the Strong-Field Approximation (SFA), where the emission amplitude is given by a path-integral sum over discrete emission events. These are known as quantum orbits Salieres2001 ; Paulus2000 ; Kopold2002 , i.e., quasiclassical trajectories whose start and end times are complex Ivanov2014 ; Nayak2019 . The quantum-orbit formalism arises naturally under the approximation that the electron’s motion in the continuum is completely controlled by the driving laser, which allows an exact solution in terms of highly oscillatory integrals. These are then reduced, using a steepest-descent method known as the saddle-point approximation (SPA), to discrete contributions coming from the saddle points of the integrand BruijnAsymptotics ; BleisteinIntegrals ; GerlachSPAonline . These saddle points represent the quasiclassical trajectories, and they typically come in pairs – most notably the ‘short’ and ‘long’ trajectories Lewenstein1995 ; Zair2008 . Each pair of trajectories approaches each other over the harmonic plateau and then performs a Stokes transition at the harmonic cutoff Figueira2002 ; Milosevic2002 , giving way to an exponential drop where only one of the saddles is used. In practice, however, classifying the saddle points into these pairs of trajectories is one of the highest-friction points when applying this method Chipperfield2007 ; Hoffmann2011 ; Das2017 , particularly since the saddles tend to move quickly, and approach each other very closely, at the harmonic cutoff. --- (a) (b) (c) Figure 1: Solving the HHG saddle-point equations returns a discrete set of saddle points as a function of the harmonic photon energy $\Omega$, shown in (b) for the monochromatic field in (a). Our main result is the trajectory classification in (c), which consists of the harmonic-cutoff points $t_{\mathrm{hc}}$ (triangles) and the separatrices through them, which allow the cloud of saddle-point solutions in (b) to be organized into individual trajectories; the different colors correspond to different quantum orbits. We model neon in a field of wavelength $800\text{\,}\mathrm{n}\mathrm{m}$ and intensity $2\text{\times}{10}^{14}\text{\,}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$. In this work we construct a quantum-orbit classification scheme based on a natural notion of complex-valued harmonic-cutoff times $t_{\mathrm{hc}}$. These are points – given by zeros of the second derivative of the action – which sit between the two saddle points as they approach each other, and which provide an organic separation between the two. Thus, an unordered set of saddle-point solutions like the one shown in Fig. 1b can be cleanly organized programmatically into families of trajectories, as shown in Fig. 1c, in a flexible and general fashion which is robust to changes in the quantum orbits over broad families of optical fields. Once these second-order saddle points have been identified, they naturally fill in the role of the quantum orbits corresponding to the high-harmonic cutoff, and they give the photon energy at which it occurs – a quantity which can often be hard to pin down precisely – as the real part of the time derivative of the action at the harmonic-cutoff time. We benchmark this understanding of the harmonic cutoff against the standard ‘cutoff law’, which describes the scaling as $\Omega_{\mathrm{max}}\sim I_{p}+3.17U_{p}$ with the ionization potential $I_{p}$ and the ponderomotive energy $U_{p}$ of the field (previously derived both classically Corkum1993 ; Kulander1993 and via a systematic expansion in $I_{p}/U_{p}$ Lewenstein1994 ). However, our method extends trivially to drivers with higher harmonic content as well as to tailored polarizations. Moreover, the information at $t_{\mathrm{hc}}$, which can be obtained by solving a single second-order saddle point equation, is sufficient to calculate an accurate evaluation of the harmonic yield at the cutoff, as well as a good qualitative estimate (which we term the Harmonic-Cutoff Approximation) for the shape of the spectrum at the upper plateau. The efficiency of this approach makes it a good tool when optimizing both the position and the brightness of the cutoff over the high-dimensional parameter spaces available on current optical synthesizers Manzoni2015 . This understanding of the high-harmonic cutoff $\Omega_{\mathrm{hc}}$ also assigns a natural value to its imaginary part, whose direct impact is to control the closeness of the approach between the pair of saddle points at the cutoff. Since this closeness regulates, in turn, the distinguishability between the two trajectories, the imaginary part of the harmonic cutoff ultimately controls the strength of quantum-path interference (QPI) Zair2008 between the pair of orbits. In particular, the zeros of the imaginary part of the harmonic cutoff energy $\imaginary(\Omega_{\mathrm{hc}})$ pinpoint the configurations where the saddle points for the two quantum orbits have an exact coalescence at the cutoff, in which case they cannot be distinguished from each other. For tailored polarizations, as well as other polychromatic fields with one or more nontrivial shape parameters, $\imaginary(\Omega_{\mathrm{hc}})$ will generically oscillate, indicating that the quantum orbits have reconnection events – where, in essence, the evanescent post-cutoff saddle point will transfer from the ‘short’ quantum orbit to the ‘long’ one. We showcase this behaviour in bichromatic counter-rotating circularly- polarized ‘bicircular’ fields Fleischer2014 ; Kfir2015 ; Eichmann1995 ; Milosevic1996 ; Milosevic2000bicircular ; Baykusheva2016 ; Dorney2017 ; JimenezGalan2018 ; Pisanty2014 ; Pisanty2017 ; Milovsevic2019 as the relative intensity between the two components changes: the various quantum orbits then recombine with each other, making for a complicated landscape within a unified topology. This illustrates the fact that the various saddle points are simply different branches of a single, unified Riemann surface, with our harmonic- cutoff times $t_{\mathrm{hc}}$ taking on the role of the branch points of this Riemann surface. More practically, the reconnections make the quantum-orbit classification challenging, but we show that our algorithm can seamlessly handle these changes in the trajectories. In the more structural terms of catastrophe theory Poston1978 , the harmonic cutoff is a spectral caustic, in the ‘fold’ class of diffraction catastrophes Raz2012 . In this sense, the complex harmonic-cutoff energy $\Omega_{\mathrm{hc}}$ is the bifurcation set for the catastrophe – suitably generalized to the complex variables involved – and it marks the location of the caustic. The formal study of caustics has seen increasing interest within attoscience Raz2012 ; Goulielmakis2012 ; Austin2014 ; Facciala2016 ; Facciala2018 ; Hamilton2017 ; Birulia2019 ; Uzan2020 ; Kelvich2017 , and our approach provides a useful tool for exploring and describing the higher- dimensional bifurcation sets at the heart of the higher-order catastrophes that become available as our control over short pulses of light becomes more finely tuned. This work is structured as follows. In Section I we construct the harmonic- cutoff times and examine their structure, first summarizing the standard SFA in Section I.1 and constructing a model with a cubic action in Section I.2, which we explore in depth in Sections I.3 and I.4, where we construct the classification algorithm; in Section I.5 we explore the quantum-orbit Riemann surface uncovered by this perspective. In Section II we construct the Harmonic-Cutoff Approximation, by explicitly integrating the Airy integral of the cubic action of our model, and we benchmark it against the standard approximations. In Section III we examine how the complex cutoff energy $\Omega_{\mathrm{hc}}$ scales with the field parameters, and show that it agrees with the known cutoff law and that it extends it to the complex domain. Finally, in Section IV, we explore the branch-cut classification for bicircular fields, showing that our algorithm can handle the quantum-orbit reconnections, as well as how these reconnections give rise to a nontrivial topology for the quantum orbits. The functionality we describe here has been integrated into the RBSFA package for Mathematica RBSFA , and our specific implementation is available from Ref. FigureMaker, . Interactive versions of 3D Fig. 1 and 3D Fig. 2, as well as 3D-printable models of those surfaces, are available as Supplementary Material SupplementaryMaterial . ## I The complex harmonic-cutoff times ### I.1 The Strong-Field Approximation In the SFA, high-harmonic emission is calculated as the expectation value of the dipole operator, under the assumption that there is a single active electron that is either in the atomic potential’s ground state $\ket{g}$ or in a laser-driven continuum where the effect of the atomic potential is negligible. The calculation Ivanov2014 ; Nayak2019 ; Amini2019 ; Lewenstein1994 then gives the harmonic dipole in the form $\displaystyle\mathbf{D}(\Omega)$ $\displaystyle=\int_{-\infty}^{\infty}\\!\\!\\!\\!\\!\\!\textrm{d}t\int_{-\infty}^{t}\\!\\!\\!\\!\\!\\!\textrm{d}t^{\prime}\\!\int\\!\textrm{d}\mathbf{p}\,\mathbf{d}(\mathbf{p}+\mathbf{A}(t))\Upsilon(\mathbf{p}+\mathbf{A}(t^{\prime}))$ $\displaystyle\qquad\qquad\qquad\qquad\times e^{-iS_{V}\\!(\mathbf{p},t,t^{\prime})+i\Omega t}+\mathrm{c.c.},$ (1) where the three integrals over the times of ionization and recollision, $t^{\prime}$ and $t$, and the intermediate momentum, $\mathbf{p}$, form a reduced Feynman path integral summing over a restricted set of relevant orbits Salieres2001 . This Feynman sum is modulated by ionization and recollision amplitudes given by the transition dipole matrix element $\mathbf{d}(\mathbf{k})=\matrixelement{\mathbf{k}}{\hat{r}}{g}$ and the scalar function $\Upsilon(\mathbf{k})=(I_{p}+\tfrac{1}{2}\mathbf{k}^{2})\innerproduct{\mathbf{k}}{g}$, evaluated at the ionization and recollision velocities: $\mathbf{A}(t)$ is the vector potential of the laser and $\mathbf{p}$ is the canonical momentum, so $\mathbf{v}(t)=\mathbf{p}+\mathbf{A}(t)$ is the recollision velocity and $m_{e}\mathbf{v}(t)=\mathbf{v}(t)$ is the kinematic momentum. (We use atomic units with $m_{e}=\hbar=1$ throughout, and we consider neon in a linearly- polarized field $\mathbf{F}(t)=F_{0}\hat{\mathbf{e}}_{z}\cos(\omega t)$ of wavelength $800\text{\,}\mathrm{n}\mathrm{m}$ and intensity $2\text{\times}{10}^{14}\text{\,}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$ unless otherwise stated.) More importantly, the contribution from each orbit in the Feynman sum in (1) is modulated by a phase given by the total action accumulated during propagation, $\displaystyle S(\mathbf{p},t,t^{\prime})$ $\displaystyle=S_{V}(\mathbf{p},t,t^{\prime})-\Omega t$ (2a) $\displaystyle\text{for}\ S_{V}(\mathbf{p},t,t^{\prime})$ $\displaystyle=\frac{1}{2}\int_{t^{\prime}}^{t}\left(\mathbf{p}+\mathbf{A}(\tau)\right)^{2}\textrm{d}\tau+I_{p}(t-t^{\prime}),$ (2b) with $I_{p}$ the ionization potential of the atom, which is normally dominated by the Volkov component $S_{V}$. For a field with amplitude $F_{0}$ and frequency $\omega$, the action scales with the ratio $z=U_{p}/\omega$ of the field’s ponderomotive energy $U_{p}=F_{0}^{2}/4\omega^{2}$ to its frequency. This is typically large, so the exponential term $e^{-iS}$ changes much faster than the amplitudes in the prefactor, making the integral in (1) highly oscillatory. The highly oscillatory nature of this amplitude then allows us to deal with this integral using the method of steepest descents BruijnAsymptotics ; BleisteinIntegrals ; GerlachSPAonline , also known as the saddle-point approximation (SPA). In this method, we deform the integration contours in (1) into the complex plane, so that they will pass through the saddle points of the exponent (2). There the exponent is locally quadratic, so the integral can be reformulated as a gaussian and integrated exactly, under the assumption that the prefactor is slow. These points can be found through the saddle-point equations $\displaystyle 0$ $\displaystyle=\frac{\partial S}{\partial t}=\frac{1}{2}\left(\mathbf{p}+\mathbf{A}(t)\right)^{2}+I_{p}-\Omega,$ (3a) $\displaystyle 0$ $\displaystyle=\frac{\partial S}{\partial t^{\prime}}=\frac{1}{2}\left(\mathbf{p}+\mathbf{A}(t^{\prime})\right)^{2}+I_{p},$ (3b) $\displaystyle 0$ $\displaystyle=\frac{\partial S}{\partial\mathbf{p}}=\int_{t^{\prime}}^{t}\left(\mathbf{p}+\mathbf{A}(\tau)\right)\textrm{d}\tau,$ (3c) which can be interpreted on physical grounds as encoding the requirements of energy conservation at recollision and ionization ((3a) and (3b), resp.) as well as the return of the quasiclassical laser-driven trajectory $\boldsymbol{\alpha}(t)=\int_{t^{\prime}}^{t}\mathbf{v}(\tau)\textrm{d}\tau$ to the ion at the time of recollision.111We use the term ‘quasiclassical’ here to distinguish this formalism from more general semiclassical approaches, which use Feynman sums over allowed classical trajectories and add quantum corrections coming from the gaussian (or other similar) spread of the integral around them. One key feature of these conditions is that the ionization equation (3b), in particular, cannot be satisfied with real variables, forcing all of the variables involved to take complex values. Normally, the return equation (3c) is solved separately, since its linearity in $\mathbf{p}$ guarantees a unique solution, $\mathbf{p}_{s}(t,t^{\prime})=-\frac{1}{t-t^{\prime}}\int_{t^{\prime}}^{t}\mathbf{A}(\tau)\textrm{d}\tau,$ (4) for any arbitrary pair $(t,t^{\prime})$ of ionization and recollision times. Once the momentum integral has been performed in this way, the expression for the harmonic dipole takes the form $\displaystyle\mathbf{D}(\Omega)$ $\displaystyle=\int_{-\infty}^{\infty}\\!\\!\\!\\!\\!\textrm{d}t\int_{-\infty}^{t}\\!\\!\\!\\!\\!\\!\textrm{d}t^{\prime}\>\mathbf{d}(\mathbf{p}_{s}(t,t^{\prime}){+}\mathbf{A}(t))\Upsilon(\mathbf{p}_{s}(t,t^{\prime}){+}\mathbf{A}(t^{\prime}))$ $\displaystyle\qquad\times\left(\frac{2\pi}{i(t-t^{\prime})}\right)^{3/2}e^{-iS_{V}\\!(t,t^{\prime})+i\Omega t},$ (5) where the fractional power of the excursion time $\tau=t-t^{\prime}$ represents the dispersion of the released wavepacket in position space.222If the time integrals are performed numerically, this factor needs to be regularized at $\tau\to 0$, to account for a breakdown of the saddle-point approximation in that limit Pisanty2016 . If the time integrals are also evaluated via saddle-point methods, however, this is not required, as that limit is not used. We notate $S(t,t^{\prime})=S(\mathbf{p}_{s}(t,t^{\prime}),t,t^{\prime})$ where it does not lead to confusion, and we drop the added complex conjugate for simplicity. The resulting two-dimensional integral, (5), is now in its minimal form; the saddle-point equations for the amended action read $\displaystyle 0$ $\displaystyle=\frac{\partial S}{\partial t}=\frac{1}{2}\left(\mathbf{p}_{s}(t,t^{\prime})+\mathbf{A}(t)\right)^{2}+I_{p}-\Omega,$ (6a) $\displaystyle 0$ $\displaystyle=\frac{\partial S}{\partial t^{\prime}}=\frac{1}{2}\left(\mathbf{p}_{s}(t,t^{\prime})+\mathbf{A}(t^{\prime})\right)^{2}+I_{p},$ (6b) and they can only be solved numerically.333This is often done by gradient descent on the modulus of the right-hand side Ivanov2014 ; Nayak2019 , but it is also possible to use Newton’s method directly, as it readily extends to multiple complex variables Chipperfield2007 ; RBSFA . A typical set of solutions for these saddle-point equations is shown in Fig. 1b: the solutions form a discrete set of points, which shift when the harmonic frequency $\Omega$ changes. These discrete points thus trace out individual curves on the complex $t$ and $t^{\prime}$ planes, which form the individual quantum orbits. --- (a) (b) (c) Figure 2: (a,b) Detail of the first pair of quantum orbits from Fig. 1, labelled by harmonic order, $\Omega/\omega$. At the avoided crossing of the two saddles, they go through a Stokes and then an anti-Stokes transition (solid and hollow arrow, resp.), at the points where $\real(S)$ and $\imaginary(S)$ of the two are equal, resp. (c) Saddle-point approximation to the core elements of the harmonic dipole, $\tau^{-3/2}e^{-iS}$, for the first pair of quantum orbits (green and blue curves). At the anti-Stokes transition (dot-dashed line), one of the two starts growing exponentially, but it must be discarded at the Stokes transition (solid line), when the required integration-contour topology changes. This produces a discontinuous change in the total SPA dipole (dashed line), which can be fixed by using the Uniform Approximation (solid line). The oscillations over the plateau are quantum path interference between the two trajectories. Once the saddle points have been found, the SPA gives the harmonic dipole as $\displaystyle\mathbf{D}(\Omega)$ $\displaystyle=\sum_{s}\frac{2\pi}{\sqrt{-\det[S^{\prime\prime}(t_{s},t^{\prime}_{s})]}}\left(\frac{2\pi}{i(t_{s}-t^{\prime}_{s})}\right)^{3/2}$ $\displaystyle\qquad\qquad\times\mathbf{d}(\mathbf{p}_{s}{+}\mathbf{A}(t_{s}))\Upsilon(\mathbf{p}_{s}{+}\mathbf{A}(t^{\prime}_{s}))\,$ (7) $\displaystyle\qquad\qquad\times e^{-iS_{V}\\!(t_{s},t^{\prime}_{s})+i\Omega t_{s}},$ with $\det[S^{\prime\prime}]=\partial_{tt}^{2}S\,\partial_{t^{\prime}t^{\prime}}^{2}S-(\partial_{tt^{\prime}}^{2}S)^{2}$ the Hessian of the action, thus giving the harmonic yield as a coherent sum of discrete contributions coming from each of the quantum orbits. Fig. 2c shows a typical example of how the individual contributions (blue and green curves) get combined into a total harmonic yield, with quantum-path interference beatings in the plateau as the two contributions go in and out of phase Lewenstein1995 ; Zair2008 . Within the SPA expression (7), the key component is the summation: this runs over all the saddle points of the action which are compatible with a suitable steepest-descent integration contour, a property that can be nontrivial to determine. Fig. 2b shows a typical configuration, with the short- and long- trajectory saddle points approaching each other over the harmonic plateau, close to the real axis, performing an ‘avoided crossing’ at the harmonic cutoff, and then advancing into imaginary time. At this avoided crossing, the saddle-point pair experiences two important transitions, known as the Stokes and anti-Stokes lines Figueira2002 ; Milosevic2002 ; Chipperfield2007 . At the anti-Stokes transition, which is defined as the point where $\imaginary(S)$ for the two orbits is equal, the short-trajectory contribution begins to grow exponentially, and it must be discarded from the summation in order to keep reasonably physical results. This elimination is enforced by the Stokes transition, which typically happens earlier, defined as the point where $\real(S)$ for the two orbits is equal and then changes order. This is an important change, as the steepest-descent method requires integration contours to follow the contour lines of $\real(S)$: thus, the change in the ordering of $\real(S(t_{s},t_{s}^{\prime}))$ for the long- and short-trajectory saddle points means that, after the Stokes transition, the short-trajectory saddle point (in this example) can no longer form part of a suitable integration contour, and it needs to be discarded from the summation. This means, however, that the SPA harmonic yield at the cutoff is a discontinuous function of $\Omega$, coming from the discrete jump at the points where one of the trajectories is eliminated, and this discontinuity is clearly incompatible with the initial, obviously continuous, expressions for $\mathbf{D}(\Omega)$ in (1) and (5). This apparent paradox is resolved by noting that the SPA is not valid when saddles are close together, as the quadratic approximation to the exponent fails. Instead, one must use a cubic form for the exponent, which can be integrated in terms of Airy or fractional- order Bessel functions. This gives a continuous spectrum from the saddle-point pair, known as the Uniform Approximation Figueira2002 ; Milosevic2002 (UA), which is shown in Fig. 2c. From a more general viewpoint, it is important to stress that applying these considerations is only possible once the various saddle points have been classified into continuously-connected quantum orbits, as without that step it is impossible to even define the objects that will be included in the summation or discarded from it.444Similarly, keeping good track of the saddle points in continuous quantum orbits is also essential to ensure that the Hessian square root in (7), $\sqrt{-\det[S^{\prime\prime}(t_{s},t^{\prime}_{s})]}$, does not cross any branch cuts. In practice, this is a high-friction point in the calculation, requiring expensively tight grid spacings in $\Omega$ to accurately resolve the avoided crossings, or the additional design of an adaptive grid to increase the energy resolution there Chipperfield2007 ; Hoffmann2011 ; Das2017 . Moreover, this problem is compounded in more complex polychromatic fields, where the saddle-point structure changes depending on the details of the laser pulse. It is the goal of this manuscript to provide a simple and effective method to classify these quantum orbits, separating the points in Fig. 2b along the diagonal into the two curves shown. ### I.2 A model for the saddle points at the cutoff In order to build this method, we first consider a simple model for the saddle points at and near the harmonic cutoff, in order to isolate the key features of the problem. In essence, Fig. 2b shows us two saddle points approaching each other and then receding, so we consider the simplest model action with only two saddles, of the polynomial form $\frac{\textrm{d}S}{\textrm{d}t}=A(t-t_{s,1})(t-t_{s,2}).$ (8) (For simplicity, we restrict our attention for now to actions with only one variable, $t$, which corresponds to solving (6b) first for $t_{s}^{\prime}=t_{s}^{\prime}(t)$, giving $S(t)=S(t,t_{s}^{\prime}(t))$ and then examining (6a). We shall lift this restriction later.) Moreover, we use the clear symmetry evident in Fig. 2b, with both saddles (approximately) symmetrically placed about the center of the plot, and enforce the condition $t_{s,1}=-t_{s,2}$ in our model, so that its action obeys $\displaystyle\frac{\textrm{d}S}{\textrm{d}t}$ $\displaystyle=A(t-t_{s})(t+t_{s})$ $\displaystyle=A(t^{2}-t_{s}^{2}),$ (9) which can be integrated to find $S(t)=\frac{1}{3}A\,t^{3}-At_{s}^{2}\,t+C$ (10) as a cubic polynomial, where we set $C=0$ for simplicity. Turning now to the oscillatory integral for this action, we can define it in the form $\displaystyle\int e^{-iS(t)}\textrm{d}t=\int e^{-\frac{i}{3}At^{3}+iAt_{s}^{2}\,t}\,\textrm{d}t$ (11) and then compare the linear term, $e^{+iAt_{s}^{2}\,t}$, with the Fourier kernel $e^{+i\Omega t}$ from (1), which has the same structure. Thus, in order to turn (11) into a form that is clearly analogous to (1), we separate $At_{s}^{2}=\Omega-(\Omega_{c}+i\eta)$ into a variable and a constant part, and thus define $\displaystyle D(\Omega)=\int e^{-\frac{i}{3}At^{3}+i(\Omega-\Omega_{c}-i\eta)t}\,\textrm{d}t.$ (12) Here the constant part $\Omega_{\mathrm{hc}}=\Omega_{c}+i\eta$ has a nonzero imaginary part $+i\eta$, as we do not have any guarantees that $At_{s}^{2}$ is real, but its real part $\Omega_{c}$ can be set to zero if desired, since it simply acts as an offset for the variable $\Omega$. Here the functional dependence $At_{s}^{2}=\Omega-(\Omega_{c}+i\eta)=\Omega-\Omega_{\mathrm{hc}}$ (13) on $\Omega$ is an additional postulate, justified only by analogy with the Fourier kernel of the full integral which our model attempts to mimic, but, as we shall see shortly, the ‘quantum orbits’ $t_{s}(\Omega)$ that result from this identification form a good model for the avoided-crossing behaviour in Fig. 2b. In addition, if we now separate $S(t)=S_{V}(t)-\Omega t$ as we did in (2) above, the saddle-point equation (9) can now be rephrased as the requirement that $\displaystyle\frac{\textrm{d}S_{V}}{\textrm{d}t}=At^{2}+(\Omega_{c}+i\eta)=\Omega,$ (14a) with $\Omega$ running over the real axis, in direct analogy to (3a) and (6a). This is our first key insight: the curves traced out by the solutions of the saddle-point equation (9) can also be described as the contour line $\imaginary\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}\right]\mathclose{}=0$ (14b) of the derivative of the model Volkov action $S_{V}$ (and also of the full action $S$, since they differ by a real number). --- Figure 3: Contour map of $\imaginary\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}\right]\mathclose{}$ for the model action in (10), with blue (yellow) indicating negative (positive) values, and with the $\imaginary\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}\right]\mathclose{}=0$ contour highlighted in black. The gray dot-dashed lines are the contours of $\real\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}\right]\mathclose{}$ passing through the central saddle point. We set $\eta=-1$ and $A=e^{$10\text{\,}\mathrm{\SIUnitSymbolDegree}$i}$. This insight then lights the way further: our principal task is to look for objects between the two curves in Fig. 2b that will help us separate them, and the contour map of $\imaginary\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}\right]\mathclose{}$ is clearly the place to look. We show this contour map in Fig. 3, with the zero contour of (14b) highlighted as a thick black curve, clearly showing the avoided-crossing structure of Fig. 2b that we want to model. More importantly, this plot shows us our second key insight: there is indeed a nontrivial object separating the two quantum orbits, in the form of a saddle point in this contour map. This is the central object we are after: the harmonic-cutoff time $t_{\mathrm{hc}}$ for this model. Like all saddle points, this can be found as a zero of the derivative of the contour map’s original function, or in other words, via $\frac{\textrm{d}^{2}S_{V}}{\textrm{d}t^{2}}(t_{\mathrm{hc}})=0.$ (15) In the particular case of our model action (14a), for which $\frac{\textrm{d}^{2}S_{V}}{\textrm{d}t^{2}}=2A\,t$, this saddle point lies at the origin, due to the explicit choice of center of symmetry made by setting $t_{s,1}=-t_{s,2}$ above. In the general case, (15) will find the centerpoint between $\imaginary\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}\right]\mathclose{}$ contours whenever they approach each other. The appearance of a saddle point at the midpoint between contour lines in close proximity is a generic feature of contour landscapes. Similar structures have been explored previously Pisanty2016slalom ; Keil2016 , in the context of tunnel ionization. ### I.3 The landscape at the harmonic-cutoff point Having found the saddle point, we can now use it directly, since we can fully reconstruct the model action $S_{V}(t)$ using only its behaviour at the center, by taking derivatives: $\displaystyle\frac{\textrm{d}S_{V}}{\textrm{d}t}(t_{\mathrm{hc}})$ $\displaystyle=\Omega_{\mathrm{hc}}=\Omega_{c}+i\eta,$ (16a) $\displaystyle\frac{\textrm{d}^{3}S_{V}}{\textrm{d}t^{3}}(t_{\mathrm{hc}})$ $\displaystyle=2A,$ (16b) with the second derivative vanishing by construction. --- Figure 4: Contour map of $\imaginary\mathopen{}\left[\frac{\partial S_{V}}{\partial t}(t,t_{s}^{\prime}(t))\right]$ over the complex recollision- time plane for the full Volkov action from (2b). The saddle-point trajectories of Fig. 1 appear clearly as the zero contour (thick lines), i.e. as the complex times where $\frac{\partial S_{V}}{\partial t}(t,t_{s}^{\prime}(t))$ is real-valued. The harmonic-cutoff points of Fig. 1c (triangles) are, correspondingly, the saddle points of this derivative; the coloured lines show the contours at these saddle points. We begin by focusing on the first derivative, which gives the linear term in the action, whose coefficient is directly connected to the harmonic frequency via $At_{s}^{2}=\Omega-(\Omega_{c}+i\eta)$ as set above. In particular, this term directly encodes the saddle-point solutions, and we can recover these explicitly by inverting that relationship: $t_{s}=\pm\sqrt{\frac{\Omega-(\Omega_{c}+i\eta)}{A}}.$ (17) Here the square in $At_{s}^{2}$ implies that $t_{s}$ appears as an explicit square root, with the sign ambiguity giving us the two saddle-point solutions of the problem. This is also a key structural insight, which has largely remained unobserved in the literature: the various solutions of saddle-point equations are simply different branches of a unified Riemann surface, examined at the real axis of the target space. This observation is made explicit in the branch choice given by the $\pm$ sign in (17), but the same is true in the full problem: if we approach the coupled equations in (6) by solving (6b) first for $t_{s}^{\prime}=t_{s}^{\prime}(t)$, then (6a) can be expressed in the form $\frac{\partial S_{V}}{\partial t}(t,t_{s}^{\prime}(t))=\frac{1}{2}\left(\mathbf{p}_{s}(t,t_{s}^{\prime}(t))+\mathbf{A}(t)\right)^{2}+I_{p}=\Omega,$ (18) where it explicitly involves the inverse of the many-to-one, complex-valued function $\frac{\partial S_{V}}{\partial t}(t,t_{s}^{\prime}(t))$, and this is precisely the type of problem encoded by Riemann surfaces. (We explore this perspective in detail in Section I.5 below.) Moreover, as far as the full HHG problem is concerned, our insight in (14b) above that the saddle-point trajectories are simply contour lines of a derivative of the action remains unchanged, with the obvious alterations to $\imaginary\mathopen{}\left[\frac{\partial S_{V}}{\partial t}(t,t_{s}^{\prime}(t))\right]\mathclose{}=0.$ (19) We exemplify this directly in Fig. 4, showing the contour map of $\imaginary\mathopen{}\left[\frac{\partial S_{V}}{\partial t}(t,t_{s}^{\prime}(t))\right]\mathclose{}$ for the full HHG problem as in Fig. 1, and there the zero contour lines (highlighted in black) are indeed the curves followed by the typical quantum orbits shown in Fig. 1c. Similarly, the harmonic-cutoff times $t_{\mathrm{hc}}$ used there are found as the saddle points of the $\frac{\partial S_{V}}{\partial t}(t,t_{s}^{\prime}(t)){}$ landscape in Fig. 4. That said, the quantum-orbit curves of Fig. 4 are not simply unstructured curves, as they have an explicit parametrization in terms of the harmonic frequency $\Omega$, but the same is true for the contours of $\imaginary\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}\right]\mathclose{}$, which can also be seen as parametrized by the real part of the function. This brings physical content to the first part of the derivatives we found in (16), where the real part of $\frac{\textrm{d}S_{V}}{\textrm{d}t}(t_{\mathrm{hc}})$ is simply the frequency offset $\Omega$. This offset is carried from the saddle point $t_{\mathrm{hc}}$ to the quantum-orbit lines by means of the contour lines of $\real\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}\right]\mathclose{}$. These are shown dot-dashed in Fig. 4, and they clearly intersect the quantum- orbit tracks at the point where they are closest to each other, and thus at the transition itself. This is then what allows us to identify $\Omega_{c}=\real\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}(t_{\mathrm{hc}})\right]\mathclose{}$ (20a) as the frequency of the harmonic cutoff. Having made this leap, we must now confront the fact that $\frac{\textrm{d}S_{V}}{\textrm{d}t}(t_{\mathrm{hc}})$ as given in (16a) also has a nonzero imaginary part, $\eta=\imaginary\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}(t_{\mathrm{hc}})\right]\mathclose{},$ (20b) which similarly needs to be addressed. This imaginary part has already played a role, and it is explicitly shown in Fig. 3 as the height of the $\frac{\textrm{d}S_{V}}{\textrm{d}t}$ saddle with respect to the zero contour where the quantum orbits lie. As such, the imaginary part $\eta$ of the harmonic cutoff, defined by (16a), controls how closely the two quantum orbits approach each other at the cutoff, and thus how distinguishable they are. As we shall see in Section II.1 below, this distinguishability further controls the strength of quantum-path interference between them. Moreover, the sign of $\eta$ controls the direction in which the quantum orbits turn when they approach each other, and thus which of the two post- cutoff evanescent solutions, at positive and negative imaginary time, corresponds to the ‘short’ and ‘long’ trajectory to the left and right of $t_{\mathrm{hc}}$. Generally, this imaginary part of $\frac{\textrm{d}S_{V}}{\textrm{d}t}$ is fixed by the problem and it can be either positive or negative, but, as we shall see in Section IV, sign changes in $\eta$ are generic behaviour when the driving pulse shape changes depending on one or more parameters, which implies reconnections and changes in topology for the curves traced out by the quantum orbits. This further emphasizes the fact that the different saddle points are essentially one and the same object, corresponding only to different branches of one unified Riemann surface. However, this point can be observed more cleanly, by simply allowing the harmonic frequency $\Omega$ to acquire complex values, i.e., by considering the analytical continuation of $D(\Omega)$. Doing this amounts to directly affecting the value of $\eta$ in (14a), and if $\Omega$ is taken at the complex harmonic cutoff itself, $\Omega_{c}+i\eta$, then the constant term in the saddle-point equation vanishes, leaving a double zero of the form $At^{2}=0$ (21) at $t=t_{\mathrm{hc}}$. In other words, the harmonic-cutoff times $t_{\mathrm{hc}}$ are the locations where the saddle points for the two quantum orbits fully coalesce into a single point. ### I.4 The orientation of the harmonic-cutoff saddle and quantum-orbit classification Having examined the first derivative of $S_{V}$ at the harmonic-cutoff time, we now turn to the role of the third derivative, given in (16b). This term gives the coefficient of the quadratic term in $\frac{\textrm{d}S_{V}}{\textrm{d}t}$ and, as such, it controls the orientation of the saddle in $\frac{\textrm{d}S_{V}}{\textrm{d}t}$ at $t_{\mathrm{hc}}$, as shown in Fig. 3. (Indeed, we chose a nonzero phase for $A=e^{$10\text{\,}\mathrm{\SIUnitSymbolDegree}$i}$ for that configuration to emphasize that this saddle need not be neatly oriented, as observed in Fig. 4.) Retrieving this orientation is essential in order to use our harmonic- cutoff times $t_{\mathrm{hc}}$ to classify the saddle points into quantum orbits: in order to turn the clouds of points in Fig. 1b into the ordered curves of Fig. 1c, we also require the direction of the separatrix that goes through the missed approach. To obtain this direction, we look for the vector that goes from $t_{\mathrm{hc}}$ to the closest point on the $\imaginary\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}\right]\mathclose{}=0$ contour that the quantum orbits follow. As we noted above, this point occurs at $\Omega=\Omega_{c}$, which means that we simply need to solve the saddle- point equation (14a) at that frequency, i.e., $\displaystyle At_{\mathrm{sep}}^{2}+i\eta=0\ \implies\ t_{\mathrm{sep}}=\sqrt{-i\eta/A},$ (22) or, in terms of the explicit derivatives of the action and with an explicit relation to the saddle center $t_{\mathrm{hc}}$, $\displaystyle\delta t_{\mathrm{sep}}=t_{\mathrm{sep}}-t_{\mathrm{hc}}=\sqrt{-2i\frac{\imaginary\mathopen{}\left[\displaystyle\frac{\textrm{d}S_{V}}{\textrm{d}t}(t_{\mathrm{hc}})\right]\mathclose{}}{\displaystyle\frac{\textrm{d}^{3}S_{V}}{\textrm{d}t^{3}}(t_{\mathrm{hc}})}}.$ (23) We can then write the explicit condition for the separatrix by treating $\delta t_{\mathrm{sep}}$ and $t_{s}-t_{\mathrm{hc}}$ as vectors and asking for the sign their inner product. Thus, the criterion that separates the two quantum orbits on either side of a given harmonic-cutoff time $t_{\mathrm{hc}}$ is the sign comparison $\real\mathopen{}\Big{[}(t_{s}-t_{\mathrm{hc}})^{*}\,\delta t_{\mathrm{sep}}\Big{]}\mathclose{}\lessgtr 0,$ (24) with $t_{\mathrm{hc}}$ defined as in (15) and $\delta t_{\mathrm{sep}}$ defined as in (23). In a practical calculation, there will typically be multiple harmonic-cutoff times, alternating between near-threshold harmonic frequencies, where the quantum orbits first appear, and high-frequency cutoffs. To complete our classification scheme – shown as the background coloured zones in Fig. 1c, which we repeat in some additional detail in Fig. 5 – we separate the complex time plane into strips using the mid-points between the real parts of the successive harmonic-cutoff times, and then divide each strip in two using the criterion in (24). Here it is important to note that the definition in (23) contains a sign ambiguity coming from the choice of sign for the square root. In practice, the principal branch of the square root tends to work well for most cases, but the branch choice there does occasionally require dedicated attention, particularly for the first low-energy harmonic cutoff at the start of the series. The results of our classification procedure are shown in Fig. 5: the separatrices obtained from the sign-comparison criterion (24) break up the complex plane into trapezoids that contain one and only one quantum orbit each, and they can thus be used to classify the saddle-point solutions into well-defined families in a uniform and robust fashion. --- Figure 5: Results of our saddle-point classification procedure, as shown previously in Fig. 1c. We find harmonic-cutoff times $t_{\mathrm{hc}}$ (triangles) as solutions of (15), with the separatrix through each $t_{\mathrm{hc}}$ given by the sign comparison in (24) with $\delta t_{\mathrm{sep}}$ defined in (23); we apply this criterion on vertical strips obtained by taking the midpoints between the real parts of the $t_{\mathrm{hc}}$. This then defines clear regions occupied by each of the quantum orbits, which can be unambiguously labelled. --- List of tdfigures 1 Topology of the Riemann surface of the quantum orbits. This is the manifold $\mathcal{S}=\\{(t,\Omega)\in\mathbb{C}^{2}:\Omega=\frac{\textrm{d}S_{V}}{\textrm{d}t}(t)\\}$, which has complex dimension 1 and is embedded in the space $\mathbb{C}^{2}$, with real dimension 4, so we can only plot it by projecting out one component: we show its projections to $\real(\omega t)$ in (a, b) and to $\imaginary(\omega t)$ in (c-f). The projection to $\real(\omega t)$ shows the topology as a single sheet which wraps around itself, covering the $\Omega\in\mathbb{C}$ plane multiple times when the $\real(\omega t)$ coordinate is projected out, so that $\frac{\textrm{d}S_{V}}{\textrm{d}t}$ has a multi-valued inverse. To separate this multi-valued inverse into valid single-valued functions, the sheet needs to be cut into separate branches: this is the role of the separation into coloured regions from the classification scheme in Fig. 5, which carry over to the multiple coloured sections in (a). The sheets connect to each other via square-root-type branch cuts, and we show a detail of the first such connection in (b). The projection onto $\imaginary(\omega t)$ forms a self-intersecting surface which loops around itself multiple times, so we plot each of the branches separately (with half of each of its neighbours shown half-transparently) in (c-f). Interactive versions and 3D-printable models of these plots are available at imaginary- harmonic-cutoff.github.io SupplementaryMaterial . ### I.5 The quantum-orbit Riemann surface We now return to an observation we made earlier: the saddle-point equations for HHG represent the inverse of a many-to-one analytical function, $\frac{\partial S_{V}}{\partial t}$ as given in (18), evaluated on the real axis in $\Omega$, and this is precisely the definition of a Riemann surface. As such, it is pertinent to study this Riemann surface as a whole, since its topology and geometry are the key factors that govern the quantum-orbit classification. The Riemann surface here is the set $\mathcal{S}=\left\\{(t,\Omega)\in\mathbb{C}^{2}:\Omega=\frac{\textrm{d}S_{V}}{\textrm{d}t}(t)\right\\},$ (25) and it forms a manifold of complex dimension 1 (i.e., a manifold where each point has a neighbourhood homeomorphic with the complex unit disk) embedded in a space of complex dimension 2 (and therefore of real dimension 4). This space is too big to be visualized directly, so we approach it by projecting down to the real and imaginary parts of $t$, i.e., by projecting the surface to the spaces $(\real(\Omega),\imaginary(\Omega),\real(\omega t))$ and $(\real(\Omega),\imaginary(\Omega),\imaginary(\omega t))$, which we show in 3D Fig. 1. The topology is most clearly displayed in the $\real(\omega t)$ projection, shown in 3D Fig. 11: the Riemann surface consists of a series of connected sheets which connect sequentially to each other, one by one, as $\real(\omega t)$ increases. In essence, the topology and the overall geometry here are those of the Riemann surface of $\sin(z)$, with the sheets connected pairwise by square-root branch points (locally homeomorphic to the quadratic model discussed in Section I.2) as shown in detail in 3D Fig. 11. This Riemann surface carries an image of the $\omega t$ complex plane (so, in particular, the mesh on the surface represents a square grid on that plane), but it forms a multiple cover of the image plane $\Omega\in\mathbb{C}$. To get the inverse, then, the surface must be split into separate branches, and this is precisely the role of the coloured regions shown in Fig. 5 as produced by our saddle-point classification algorithm: this colouring is retained in the Riemann surface as displayed in 3D Fig. 1, and each individual region, when projected down to the $\Omega$ plane, forms a single full cover of the complex plane.555This single-cover property is approximate, as there is still a small amount of double-cover overlap in regions next to the separatrix when away from the branch points at the $t_{\mathrm{hc}}$. This can be fixed if necessary, but it is not central to our argument here. In other words, each of the coloured regions in Fig. 5 forms the image of a single-valued branch of the inverse of $\frac{\textrm{d}S_{V}}{\textrm{d}t}(t)$. The imaginary-part projection onto $\imaginary(\omega t)$ is slightly harder to represent and visualize, as the corresponding surface folds back on itself, with multiple self-intersections. However, the separation of this surface into individual branches also solves this problem: when plotting one branch at a time, as we show in 3D Fig. 11-1, the self-intersections disappear – as they must if the surface represents a single-valued function – and they are only present as intersections between different branches. Physically, the most important part of this Riemann surface is its intersection with the $\Omega\in\mathbb{R}^{+}$ positive real axis in the $\Omega$ plane, which, as discussed in Section I.3 above, forms the quantum orbits themselves. These are shown highlighted as curves in 3D Fig. 1, and their role here is most clearly apparent in 3D Fig. 11: this intersection forms the $(\real(\omega t),\real(\Omega))$ energy-time mapping for the quantum orbits, a well-known (and deeply physical) part of the theory Ivanov2014 ; Nayak2019 . The quantum-orbit Riemann surface, together with its topology, is nothing more than the analytical continuation of this standard energy-time mapping. ### I.6 Derivatives of the action in the two-variable case Before moving on, it is important to define in more detail the relationship between our simple model and the full-blown HHG integral, whose action has two variables instead of one. This is possible, as mentioned above, by using (6b) to define $t_{s}^{\prime}=t_{s}^{\prime}(t)$ and with that the single-variable $S_{V}(t)=S_{V}(t,t_{s}^{\prime}(t))$, but that only works explicitly for the values of the action, and its derivatives need to be considered more carefully. The first derivative is not affected, since $\displaystyle\frac{\textrm{d}S_{V}}{\textrm{d}t}(t)$ $\displaystyle=\frac{\textrm{d}}{\textrm{d}t}S_{V}(t,t_{s}^{\prime}(t))$ $\displaystyle=\frac{\partial S_{V}}{\partial t}(t,t_{s}^{\prime}(t))+\frac{\textrm{d}t_{s}^{\prime}}{\textrm{d}t}(t)\frac{\partial S_{V}}{\partial t^{\prime}}(t,t_{s}^{\prime}(t)),$ (26) and here the partial derivative in the second term, $\frac{\partial S_{V}}{\partial t^{\prime}}(t,t_{s}^{\prime}(t))$, vanishes by the definition of $t_{s}^{\prime}(t)$. However, if we now turn to the second derivative, the procedure no longer works, and the equivalent calculation, $\displaystyle\frac{\textrm{d}^{2}S_{V}}{\textrm{d}t^{2}}(t)$ $\displaystyle=\frac{\textrm{d}}{\textrm{d}t}\frac{\partial S_{V}}{\partial t^{\prime}}(t,t_{s}^{\prime}(t))$ $\displaystyle=\frac{\partial^{2}S_{V}}{\partial t^{2}}(t,t_{s}^{\prime}(t))+\frac{\textrm{d}t_{s}^{\prime}}{\textrm{d}t}(t)\frac{\partial^{2}S_{V}}{\partial t\,\partial t^{\prime}}(t,t_{s}^{\prime}(t)),$ (27) returns a term that includes $\frac{\textrm{d}t_{s}^{\prime}}{\textrm{d}t}(t)$. This cannot be evaluated explicitly within this system, as $t_{s}^{\prime}(t)$ itself is only defined implicitly. To resolve this, we use the implicit definition of $t_{s}^{\prime}(t)$, $\frac{\partial S_{V}}{\partial t^{\prime}}(t,t_{s}^{\prime}(t))\equiv 0,$ (28) and then differentiate with respect to $t$, to obtain $\displaystyle\frac{\partial^{2}S_{V}}{\partial t\,\partial t^{\prime}}(t,t_{s}^{\prime}(t))+\frac{\textrm{d}t_{s}^{\prime}}{\textrm{d}t}(t)\frac{\partial^{2}S_{V}}{\partial t^{\prime 2}}(t,t_{s}^{\prime}(t))\equiv 0,$ (29) which can then be solved for the implicit derivative as $\displaystyle\frac{\textrm{d}t_{s}^{\prime}}{\textrm{d}t}(t)=-\frac{\partial^{2}S_{V}}{\partial t\,\partial t^{\prime}}(t,t_{s}^{\prime}(t))\bigg{/}\frac{\partial^{2}S_{V}}{\partial t^{\prime 2}}(t,t_{s}^{\prime}(t)).$ (30) Finally, this can be substituted into (27) to give the form $\displaystyle\frac{\textrm{d}^{2}S_{V}}{\textrm{d}t^{2}}(t)$ $\displaystyle=\frac{\displaystyle\frac{\partial^{2}S_{V}}{\partial t^{2}}\frac{\partial^{2}S_{V}}{\partial t^{\prime 2}}-\left(\frac{\partial^{2}S_{V}}{\partial t\,\partial t^{\prime}}\right)^{2}}{\displaystyle\frac{\partial^{2}S_{V}}{\partial t^{\prime 2}}}.$ (31) Here we have dropped the right-hand side evaluations at $(t,t_{s}^{\prime}(t))$ for simplicity, but it is also possible to understand the right-hand side as a function of $(t,t^{\prime})$ as independent variables, whose zeros are to be found in conjunction with those of (6b), and this is an easier approach in practice. Finally, to obtain the higher-order derivatives (in particular, the third derivative, which gives the cubic coefficient $A$) we use symbolic computation RBSFA ; FigureMaker , using the substitution rule (30) at each step to retain explicit formulations. ### I.7 The threshold harmonics for linear fields For a general driving field, the harmonic cutoffs given by our definition will appear at both ends of the harmonic plateau, oscillating between low frequencies near the ionization threshold and high energies at the classical cutoff. In general, moreover, all of these harmonic cutoffs will have nonzero imaginary parts, corresponding to missed approaches at a finite distance between the quantum orbits involved. However, as can be seen in Fig. 4 and Fig. 1c, this is not the case for the linearly-polarized field we use above, since half of the harmonic-cutoff saddles $t_{\mathrm{hc}}$ lie directly on the quantum-orbit line, i.e., the quantum orbits have a full crossing there. This behaviour is generic to all linearly-polarized fields, whether monochromatic or not, and it implies that the low-energy cutoffs always lie at exactly $\Omega_{c}+i\eta=I_{p}$, and the threshold harmonics at $\Omega=I_{p}$ always involve an exact saddle coalescence. In other words, for threshold harmonics, the saddles of the action landscape are not gaussian: instead, they are exact monkey saddles, as shown in Fig. 7a below; this was noticed as early as Ref. Lewenstein1994, , but has not drawn much attention since then.666This is largely because the approximations that produced the SFA integral are known to be inaccurate for the lower plateau and threshold harmonics. To understand this behaviour, we return to the saddle-point equations (6), which for linearly-polarized fields can be rephrased in the simpler, scalar form $\displaystyle\frac{\partial S_{V}}{\partial t}=\frac{1}{2}\left(p_{s}(t,t^{\prime})+A(t)\right)^{2}+I_{p}$ $\displaystyle=\Omega,$ (32a) $\displaystyle-\frac{\partial S_{V}}{\partial t^{\prime}}=\frac{1}{2}\left(p_{s}(t,t^{\prime})+A(t^{\prime})\right)^{2}+I_{p}$ $\displaystyle=0.$ (32b) Here the first equation is the crucial one, since at $\Omega=I_{p}$ it tells us that $\frac{1}{2}v_{r}(t,t^{\prime})^{2}=\frac{1}{2}\left(p_{s}(t,t^{\prime})+A(t)\right)^{2}=0,$ (33) i.e., the recollision velocity $v_{r}(t,t^{\prime})=p_{s}(t,t^{\prime})+A(t)$ vanishes exactly, with a double zero which still vanishes after taking one derivative. This becomes particularly important when we consider the second derivative of the action, (31), whose zeros determine the harmonic-cutoff times. The partial derivatives involved in (31) can be evaluated by differentiating (32a) with respect to $t$ and $t^{\prime}$, which yields $\displaystyle\frac{\partial^{2}S_{V}}{\partial t^{2}}$ $\displaystyle=\left(p_{s}+A(t)\right)\left[\frac{\partial p_{s}}{\partial t}+\dot{A}(t)\right],$ (34a) $\displaystyle\frac{\partial S_{V}}{\partial t\,\partial t^{\prime}}$ $\displaystyle=\left(p_{s}+A(t)\right)\frac{\partial p_{s}}{\partial t^{\prime}}.$ (34b) Both of these derivatives share a common factor of $v_{r}(t,t^{\prime})=p_{s}(t,t^{\prime})+A(t)$, and this then means that the second derivative $\frac{\textrm{d}^{2}S_{V}}{\textrm{d}t^{2}}(t)$ will always vanish when evaluated at solutions of the first-order saddle-point equations (32) at $\Omega=I_{p}$, completing the proof. It is important to note that, when a saddle-point coalescence like this occurs, the classification scheme embodied by our test in (24) breaks down since, at the double zero, $\frac{\textrm{d}S_{V}}{\textrm{d}t}(t_{\mathrm{hc}})$ also vanishes, so the direction vector $\delta t_{\mathrm{sep}}$ from (23) also vanishes. In practice, this can cause numerical instability, with $\imaginary\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}(t_{\mathrm{hc}})\right]\mathclose{}$ evaluating to machine precision with a fluctuating sign, and this needs to be handled explicitly – which could be as simple as assigning an artificial nonzero $\imaginary\mathopen{}\left[\frac{\textrm{d}S_{V}}{\textrm{d}t}(t_{\mathrm{hc}})\right]\mathclose{}$ to be used there, as either direction will work well for the separatrix. (For the plots in Fig. 4 and Fig. 1c, we used a small ellipticity of $\epsilon=0.01\%$ to break the degeneracy and avoid this instability.) At an exact coalescence, it is impossible to classify the saddles in a unique way: two saddles come in and two saddles come out, but there is no unique way to match either of the outgoing saddles with either of the incoming ones. This happens for all linearly-polarized fields at the $\Omega=I_{p}$ threshold, but it can also happen at the harmonic cutoff in isolated cases when the field depends on a variable parameter, as we will exemplify in Section IV below. ## II The Harmonic-Cutoff Approximation Having examined the structure of the action at the harmonic-cutoff times $t_{\mathrm{hc}}$, in this section we will explore the behaviour of the oscillatory integral around it, which we will employ to build the Harmonic- Cutoff Approximation (HCA), an efficient method for estimating the harmonic dipole around the cutoff using only information at $t_{\mathrm{hc}}$ itself. ### II.1 Airy-function representation for the model case To do this, we return to the model integral from (12), where we set as the object of interest the integral the exponential of our model action (10), $\displaystyle D(\Omega)=\int_{C}e^{-\frac{i}{3}At^{3}+i(\Omega-\Omega_{\mathrm{hc}})t}\,\textrm{d}t,$ (35) over an integration contour $C$ which should start at the ionization time $t^{\prime}$ (or some suitably compatible valley of $\imaginary(S)$ to the left of $t_{\mathrm{hc}}$) and end at $t\to\infty$. This integral is essentially in Airy form, and it is almost structurally identical to the Airy function’s integral representation (NIST_handbook, , Eq. (9.5.E4)), $\operatorname{Ai}(z)=\frac{1}{2\pi i}\int_{\infty e^{-i\pi/3}}^{\infty e^{i\pi/3}}\exp(\tfrac{1}{3}t^{3}-zt)\textrm{d}t.$ (36) This is expected, since the harmonic cutoff is a ‘fold’ catastrophe, whose associated diffraction-catastrophe integral is the Airy function Berry1980 . To bring our representation (35) into the canonical form (36), the core transformation is to eliminate the coefficient in front of the cubic term, $-\frac{i}{3}At^{3}\longmapsto\frac{1}{3}\tilde{t}\,^{3}.$ (37) In a sense, this is relatively simple, as it boils down to a change in integration variable to $\tilde{t}=iA^{1/3}\,t,$ but there is an added complication in that the radical $A^{1/3}$ of the cubic coefficient admits three separate branches, $\tilde{t}=ie^{2\pi ik/3}A^{1/3}\,t,$ (38) for $k\in\\{0,1,2\\}$, which requires dedicated attention. The variable change itself is essentially trivial, and (35) transforms under (38) to $\displaystyle D(\Omega)$ $\displaystyle=\frac{e^{-2\pi ik/3}}{iA^{1/3}}\\!\\!\int_{\tilde{C}}\\!\exp(\frac{1}{3}\tilde{t}\,^{3}-\frac{\Omega_{\mathrm{hc}}-\Omega}{e^{2\pi ik/3}A^{1/3}}\tilde{t})\,\textrm{d}\tilde{t},$ (39a) with $\displaystyle\tilde{C}=ie^{2\pi ik/3}A^{1/3}\,C.$ (39b) This is essentially in explicit Airy form, so long as the contour is correct, and thus we can substitute in the Airy function as $\displaystyle D(\Omega)$ $\displaystyle=\frac{2\pi}{e^{2\pi ik/3}A^{1/3}}\operatorname{Ai}\mathopen{}\left(\frac{\Omega_{\mathrm{hc}}-\Omega}{e^{2\pi ik/3}A^{1/3}}\right)\mathclose{},$ (40) where $k$ needs to be chosen such that the altered contour $\tilde{C}=ie^{2\pi ik/3}A^{1/3}\,C$ is compatible with the standard contour in (36), which we depict in Fig. 6. | | ---|---|--- (a) (b) (c) Figure 6: Contour map of the exponent $\real(\tfrac{1}{3}t^{3}-zt)$ of the integral representation of the Airy function, (36), with the standard contour shown as the black arrows, for (a) coalescent saddles, (b) the oscillatory regime, and (c) the evanescent part. The details of the contour can be altered as required (such as e.g. to pass through the two saddle points in (b) along steepest-descent contours), under the constraints that the contour start at infinity at $-\tfrac{\pi}{2}<\arg(t)<-\tfrac{\pi}{6}$ and end at $\tfrac{\pi}{6}<\arg(t)<\tfrac{\pi}{2}$, i.e., that it go down the valleys that surround $e^{-i\pi/3}$ and $e^{i\pi/3}$. --- (a) (b) (c) Figure 7: Contour map of $\imaginary[S(t,t_{s}^{\prime}(t))-\Omega t]$, presented as in Fig. 4, taking a complex $\Omega$ equal to $\Omega_{c}+i\eta$ at the various harmonic-cutoff times $t_{\mathrm{hc}}$, causing the nearby saddles to fully coalesce, and marking a complete breakdown of the SPA. To apply the Harmonic-Cutoff Approximation, each monkey saddle (marked by the converging coloured contours) must be rotated by $\tilde{C}=ie^{2\pi ik/3}A^{1/3}\,C$, choosing the branch index $k$ so that the integration contour matches the canonical Airy contours of Fig. 6. The phase $\arg(A)$ is inset in (b) and (c), and it equals $-87.97\text{\,}\mathrm{\SIUnitSymbolDegree}$, $-$0.011\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and $-$0.155\text{\,}\mathrm{\SIUnitSymbolDegree}$$, resp., for the three monkey saddles shown in (a), consistently with their different orientations. In principle, a rigorous approach to the contour-choice problem requires a careful examination of the constrains on the contour of the original integral, as shown in Fig. 7, to determine the correct integration contour $C$ that is compatible with the original integration limits; this is then rotated by $iA^{1/3}$ and any additional factors of $e^{2\pi ik/3}$ necessary for the contour to be in canonical form. (Moreover, care must be taken when taking the radical $A^{1/3}$ over a parameter scan, since, as mentioned in Fig. 7, $A$ can lie very close to commonly-used choices for the branch cut of the radical.) In practice, if there are known constraints on the behaviour of the integral (in particular, exponential decay at $\Omega>\Omega_{c}$), there will typically only be one choice of $k$ compatible with the constraints, which can then be selected on physical grounds. As a general rule, exponential decay at $\Omega>\Omega_{c}$ requires $e^{2\pi ik/3}A^{1/3}$ to have a negative real part. Our result for the model integral, (40), now allows us to have a closer look at the role of the imaginary part of the complex harmonic-cutoff energy $\Omega_{\mathrm{hc}}$, which we introduced in (20b). As we mentioned then, this imaginary part controls the strength of quantum-path interference between the two quantum orbits that meet at the relevant cutoff, and we show this in Fig. 8. The HCA approximates the integral (35) as an explicit Airy function in $\Omega$ (red line), which transitions from oscillatory to exponential-decay behaviour at $\Omega_{c}=\real(\Omega_{\mathrm{hc}})$. However, in order for the interference fringes in the oscillatory regime to have a full contrast and pass through the zeros of the Airy function, the argument must be real, and this is generally impossible if $\eta=\imaginary(\Omega_{\mathrm{hc}})$ is nonzero. To increase the interference features, thus, we can artificially reduce (blue line) or eliminate (green line) this imaginary part; conversely, increasing $\eta$ (purple and magenta lines) further damps the interference, and it introduces exponential behaviour into the plateau. That said, to get full contrast in the interference it is also necessary for the denominator to be real-valued, which we show as the gray line by artificially adjusting it. The nonzero imaginary parts of the constants in the argument of the Airy function also makes this approximation distinct from previous approaches that regularize the cutoff in terms of single Airy functions Ivanov2014 ; Frolov2009 ; Frolov2010 , making it better able to capture the QPI contrast in the neighbourhood of the cutoff. On the other hand, our approximation sits one level below the Uniform Approximation Figueira2002 ; Milosevic2002 ; Wong2001 (as well as the equivalent constructions in Refs. Frolov2012, ; Sarantseva2013, ; Okajima2012, ), which expands the prefactor to subleading order and is thus able to capture variations in the QPI contrast coming from the prefactor’s effect on the saddles. That said, the HCA is, at least in principle, strictly local to the cutoff, and this makes it especially suited for efficient evaluation and clear analysis of the harmonic strength (as well as phase properties Khokhlova2016 ) at the cutoff. ### II.2 The full HHG integral We now return to the full two-dimensional integral for HHG, (5), and transform it into the model form (35) so that the HCA can be used to estimate it. For simplicity, we write the integral in explicit prefactor-action form, $\displaystyle\mathbf{D}(\Omega)$ $\displaystyle=\int_{-\infty}^{\infty}\\!\\!\\!\textrm{d}t\int_{-\infty}^{t}\\!\\!\\!\textrm{d}t^{\prime}\ \mathbf{f}(t,t^{\prime})\,e^{-iS_{V}\\!(t,t^{\prime})+i\Omega t}$ (41a) $\displaystyle\mathbf{f}(t,t^{\prime})$ $\displaystyle=\frac{(2\pi/i)^{3/2}}{(t-t^{\prime})^{3/2}}\mathbf{d}(\mathbf{p}_{s}(t,t^{\prime}){+}\mathbf{A}(t))\Upsilon(\mathbf{p}_{s}(t,t^{\prime}){+}\mathbf{A}(t^{\prime})).$ (41b) To transform this into the model form (35), we need to reduce the integral to a single dimension, and then translate the origin to the relevant harmonic- cutoff time $t_{\mathrm{hc}}$.777However, it is important to point out that the process is rather more general than this. As was pointed out in the construction of the Uniform Approximation Figueira2002 (as well as its earlier analogues in a semiclassical context Schomerus1997 ), the coordinate separation done here is in essence an application of the splitting lemma of catastrophe theory Poston1978 , which allows the ‘fold’ catastrophe encoded by the Airy function to be isolated into a single coordinate axis. As such, the simple model of Section I.2 is not a ‘toy’ model in any sense: instead, it is a universal model, which is fully capable of capturing the (local) behaviour of the integral. --- Figure 8: Control of the imaginary part of the harmonic cutoff on quantum- path interference. The red line shows the Airy-function dependence of the Harmonic-Cutoff Approximation, as in (40), for $\Omega_{\mathrm{hc}}$ and $A$ corresponding to the first-return cutoff in Fig. 1. The coloured lines have the imaginary part $\eta=\imaginary(\Omega_{\mathrm{hc}})$ artificially reduced or amplified by a variable factor $r$, which respectively amplifies or damps the QPI features; this corresponds to choosing a lower or higher $\imaginary(\frac{\partial S_{V}}{\partial t})$ contour to follow in Fig. 3. To obtain full interference, $r$ must be set to $0$, but the denominator must also be real and negative, i.e., we replace $e^{2\pi ik/3}A^{1/3}$ with $-|A|^{1/3}$. The black vertical line is at $\Omega=\Omega_{c}$. The first is achieved, as mentioned earlier, by doing a saddle-point approximation on the $t^{\prime}$ integral only, and this returns the harmonic dipole in the form $\displaystyle\mathbf{D}(\Omega)$ $\displaystyle=\sum_{s}\int_{-\infty}^{\infty}\textrm{d}t\,\sqrt{\frac{2\pi}{i\frac{\partial^{2}S_{V}}{\partial t^{\prime 2}}(t,t_{s}^{\prime}(t))}}\mathbf{f}(t,t_{s}^{\prime}(t))$ (42) $\displaystyle\qquad\qquad\qquad\qquad\times e^{-iS_{V}\\!(t,t_{s}^{\prime}(t))+i\Omega t},$ with $t^{\prime}_{s}(t)$ defined implicitly via (6b). After this, we perform a Taylor expansion of the Volkov action $S_{V}(t,t_{s}^{\prime}(t))$ at the harmonic-cutoff time $t_{\mathrm{hc}}$ up to third order, $\displaystyle\mathbf{D}(\Omega)$ $\displaystyle=\sum_{s,\mathrm{hc}}\int_{C}\\!\\!\textrm{d}t\sqrt{\frac{2\pi/i}{\frac{\partial^{2}S_{V}}{\partial t^{\prime 2}}(t,t_{s}^{\prime}(t))}}\mathbf{f}(t,t_{s}^{\prime}(t))e^{-iS_{V}\\!(t_{\mathrm{hc}},t_{s}^{\prime}(_{\mathrm{hc}}))}$ (43) $\displaystyle\qquad\times\exp(-\frac{i}{3}A(t-t_{\mathrm{hc}})^{3}-i\Omega_{\mathrm{hc}}(t-t_{\mathrm{hc}})+i\Omega t),$ where $A$ and $\Omega_{\mathrm{hc}}$ are the third and first derivatives of $S_{V}(t,t_{s}^{\prime}(t))$, as set in (16) and evaluated as constructed in Section I.6; here we have also allowed the integration contour $C$ to vary so that it can pass through $t_{\mathrm{hc}}$ as appropriate. Finally, we move the integration origin to $t_{\mathrm{hc}}$, $\displaystyle\mathbf{D}(\Omega)$ $\displaystyle=\sum_{s,\mathrm{hc}}\int_{\bar{C}}\\!\\!\textrm{d}\bar{t}\sqrt{\frac{2\pi/i}{\frac{\partial^{2}S_{V}}{\partial t^{\prime 2}}(t,t_{s}^{\prime}(t))}}\mathbf{f}(t,t_{s}^{\prime}(t))e^{-iS_{V}\\!(t_{\mathrm{hc}},t_{s}^{\prime}(_{\mathrm{hc}}))}$ (44) $\displaystyle\qquad\times\exp(-\frac{i}{3}A\bar{t}\,^{3}-i\Omega_{\mathrm{hc}}\bar{t}+i\Omega(\bar{t}+t_{\mathrm{hc}})),$ keeping the explicit $t=\bar{t}+t_{\mathrm{hc}}$ for notational simplicity. To apply the Harmonic-Cutoff Approximation, we now assume that the prefactor varies slowly enough at $t_{\mathrm{hc}}$ that it can be pulled out of the integral (i.e., that it changes slower than the decay into the valleys shown in Fig. 7),888On a more rigorous footing, this amounts to a zeroth-order Taylor expansion of the prefactor at $t_{\mathrm{hc}}$, as is done in the SPA GerlachSPAonline . Derivatives of the prefactor can be added as required, and the corresponding terms will change the Airy function to its derivatives (as can be seen by differentiating (36) with respect to $z$, which brings down a factor of $t$ into the prefactor). For the cases we plot here, these terms are about two orders of magnitude weaker than the leading-order contribution. which gives $\displaystyle\mathbf{D}(\Omega)$ $\displaystyle=\sum_{\mathrm{hc}}\sqrt{\frac{2\pi}{i\frac{\partial^{2}S_{V}}{\partial t^{\prime 2}}(t_{\mathrm{hc}}^{\phantom{a}},t^{\prime}_{\mathrm{hc}})}}\mathbf{f}(t_{\mathrm{hc}}^{\phantom{a}},t^{\prime}_{\mathrm{hc}})e^{-iS_{V}\\!(t_{\mathrm{hc}}^{\phantom{a}},t_{\mathrm{hc}}^{\prime})}$ (45) $\displaystyle\qquad\times e^{i\Omega t_{\mathrm{hc}}}\int_{\bar{C}}\exp(-\frac{i}{3}A\bar{t}\,^{3}+i(\Omega-\Omega_{\mathrm{hc}})\bar{t})\textrm{d}\bar{t}.$ The integral is now in the form of (35), with an added functional dependence on $\Omega$ coming from the term $e^{i\Omega t_{\mathrm{hc}}}$. This term results from the translation of the time origin, and it can have a sizeable effect on the harmonic yield if $\imaginary(t_{\mathrm{hc}})$ is nonzero. That said, we can now use the result (40) for the model form directly. We obtain thus our final result for the HCA, $\displaystyle\mathbf{D}(\Omega)$ $\displaystyle=\sum_{\mathrm{hc}}\sqrt{\frac{2\pi}{i\frac{\partial^{2}S_{V}}{\partial t^{\prime 2}}(t_{\mathrm{hc}}^{\phantom{a}},t^{\prime}_{\mathrm{hc}})}}\frac{2\pi}{e^{2\pi ik/3}A_{\mathrm{hc}}^{1/3}}$ (46) $\displaystyle\quad\times\mathbf{f}(t_{\mathrm{hc}}^{\phantom{a}},t^{\prime}_{\mathrm{hc}})e^{-iS_{V}\\!(t_{\mathrm{hc}}^{\phantom{a}},t_{\mathrm{hc}}^{\prime})+i\Omega t_{\mathrm{hc}}}\operatorname{Ai}\mathopen{}\left(\frac{\Omega_{\mathrm{hc}}-\Omega}{e^{2\pi ik/3}A_{\mathrm{hc}}^{1/3}}\right)\mathclose{},$ where the harmonic-cutoff time pairs $(t_{\mathrm{hc}}^{\phantom{a}},t^{\prime}_{\mathrm{hc}})$ are found by solving the simultaneous equations $\displaystyle\frac{\partial^{2}S_{V}}{\partial t^{2}}\frac{\partial^{2}S_{V}}{\partial t^{\prime 2}}-\left(\frac{\partial^{2}S_{V}}{\partial t\,\partial t^{\prime}}\right)^{2}$ $\displaystyle=0$ (47a) $\displaystyle\frac{\partial S_{V}}{\partial t^{\prime}}$ $\displaystyle=0$ (47b) and where the coefficients $\Omega_{\mathrm{hc}}$ and $A_{\mathrm{hc}}$ for each pair of harmonic-cutoff times $(t_{\mathrm{hc}}^{\phantom{a}},t^{\prime}_{\mathrm{hc}})$ are given, as in (16), by $\displaystyle\Omega_{\mathrm{hc}}=\frac{\partial S_{V}}{\partial t}(t_{\mathrm{hc}}^{\phantom{a}},t^{\prime}_{\mathrm{hc}})\ \text{and}\ A_{\mathrm{hc}}=\frac{1}{2}\frac{\textrm{d}^{3}S_{V}}{\textrm{d}t^{3}}(t_{\mathrm{hc}}^{\phantom{a}},t^{\prime}_{\mathrm{hc}}),$ (48) with the third derivative $\frac{\textrm{d}^{3}}{\textrm{d}t^{3}}$ understood in the constrained sense of Section I.6.999An added wrinkle can appear if the prefactor, and particularly the dipole moments it includes, have singularities at the solutions of (47), which is often the case in the regular SPA. If this occurs, a regularization scheme like the one described in Ref. Popruzhenko2014, will be required, but this is likely to extend relatively cleanly. Our implementation of this approximation, in the Wolfram Language, is available in the RBSFA software package RBSFA . --- (a) (b) (c) Figure 9: Approximations to (a) the pure action term $e^{-iS}$, (b) the action term with wavepacket diffusion, $\tau^{-3/2}e^{-iS}$, and (c) the full harmonic dipole, via the Saddle-Point Approximation for the long and short trajectories (green and blue lines) and their coherent combination (dashed gray lines), the Uniform Approximation (black lines), and the Harmonic-Cutoff Approximation (red lines). The HCA is quantitatively accurate at the harmonic cutoff, and reasonably qualitatively accurate in the upper plateau, but it only requires solving a single saddle-point equation, instead one per harmonic frequency over a tight grid in $\Omega$. We show the results of this approximation, compared to the SPA and the UA (as described in Refs. Figueira2002, ; Milosevic2002, ), in Fig. 9, plotted as in Fig. 2c. When the prefactor is fully ignored, as in Fig. 9a, the HCA is extremely accurate at the harmonic cutoff, and it retains a good qualitative accuracy as $\Omega$ descends from $\Omega_{c}$ and into the plateau: it shifts vertically from the UA in the lower plateau and the predicted period for the QPI beatings is too long, but the QPI contrast is mostly well reproduced. The qualitative accuracy of the HCA, particularly regarding the QPI contrast, gets degraded when the wavepacket-diffusion dilution factor of $\tau^{-3/2}$ is included, as shown in Fig. 9b. This happens because the $\tau^{-3/2}$ factor affects the long trajectories (green line) much more than it does the short ones (blue line), bringing them closer together and allowing for better interference between them; the HCA, with its information coming from a single point, is blind to this effect, which only acts as a global vertical shift on the curve. The same is true for the full harmonic dipole $\mathbf{D}(\Omega)$,101010Here we use the ground-state wavefunction and dipole transition matrix element for the $s$ state of a short-range-potential, as described in Ref. Pisanty2017, . which we plot in Fig. 9c – the HCA has good quantitative accuracy at the cutoff, but it becomes more of a qualitative estimation for the middle and lower plateau. However, this weakness of the HCA is also its biggest strength: precisely because it is able to describe the harmonic yield using only information coming from a single solution of a set of saddle-point equations (47), it can be calculated in a small fraction of the time taken to produce SPA or UA calculations of the harmonic spectrum, since those require the solution of one system of saddle-point equations – the system in (6) – for each harmonic frequency $\Omega$ of interest, and these will typically form a grid with hundreds of instances. In this sense, the HCA sits at the far end of the tradeoff spectrum between numerical accuracy and computational complexity, which is typically understood as running from full simulations of the time-dependent Schrödinger equation (TDSE) Scrinzi2014 , passing through explicit numerical integration of the SFA dipole (5), to the SPA and UA quantum-orbit approaches. Normally, the quantum- orbit methods are considered to be fast enough that no further optimization is necessary, but this is not always the case when large scans are required spanning a high-dimensional parameter space—particularly if the waveforms involved cause nontrivial behaviour in the quantum orbits—, such as when optimizing the length or strength of the harmonic plateau and cutoff Chipperfield2009 ; Chipperfield2010 ; Haessler2014 . Moreover, when the primary focus is on the harmonic cutoff, the HCA will typically be sufficiently accurate. ## III Parameter scaling and the cutoff law As we have seen, the formalism we have presented allows us to extract both a unique and natural definition for the (complex) times that correspond to the harmonic cutoff, and also a natural definition of the harmonic frequency $\Omega_{c}$ at which the cutoff occurs. This identification is a valuable connection, since the high-harmonic cutoff is often understood as a vague term, referring to a spectral region where the behaviour changes, instead of a concrete point—largely because of the difficulty in pinning down a specific frequency at which this change occurs. This is particularly important, since the scaling of the cutoff frequency with the field parameters is one of the key hallmarks of HHG, in terms of the so- called ‘cutoff law’, $\Omega_{\mathrm{max}}\approx 3.17\,U_{p}+I_{p},\phantom{\,F(I_{p}/U_{p})}$ (49) which relates the cutoff frequency $\Omega_{\mathrm{max}}$ to the target’s ionization potential $I_{p}$ and the ponderomotive energy $U_{p}=F^{2}/4\omega^{2}$ of the field. This relationship was first uncovered in numerical simulations Krause1992 and subsequently explained using the classical three-step model Corkum1993 ; Kulander1993 : the numerical factor of $3.17$ arises as the maximal kinetic energy that can be achieved at the return to the origin by an electron released at zero velocity, with the $I_{p}$ term representing quantum energy conservation at the recombination, with an essentially heuristic justification. The fully-quantum quasiclassical theory Lewenstein1994 re-derives this cutoff law, giving also a quantum correction to the $I_{p}$ term of the form $\Omega_{\mathrm{max}}\approx 3.17\,U_{p}+1.32\,I_{p}.\phantom{\,F(I_{p}/U_{p})}$ (50) To obtain this formulation, the cutoff frequency is defined as the maximum of the recollision kinetic energy, $\real(\mathbf{v}(t_{r})^{2})$, taken under the restriction that its imaginary part $\imaginary(\mathbf{v}(t_{r})^{2})$ vanish. The dynamics are then analyzed using a systematic expansion on $I_{p}/U_{p}$, using the purely-classical case at $I_{p}=0$ for the leading $3.17U_{p}$ term, after which the cutoff law can be derived in the exact form $\Omega_{\mathrm{max}}=3.17\,U_{p}+F(I_{p}/U_{p})\,I_{p},$ (51) in terms of a universal function $F(I_{p}/U_{p})$ which can be calculated numerically as $F(0)=1.32$. This prescription can be computed exactly for linearly-polarized monochromatic fields, and it can be extended to elliptical polarizations Milosevic2000 ; Flegel2004 , but it is laborious to apply for polychromatic combinations Figueira1999 (where the complex waveform makes analytical calculations challenging) and for tailored field polarizations Milosevic1996 , especially for field shapes whose recolliding orbits do not ionize at zero velocity, as required by the classical theory. In any case, this definition of the harmonic cutoff has not seen wide adoption in the literature. Instead, a wide variety of other methods have been used in its place, including relaxing the $\imaginary(\mathbf{v}(t_{r})^{2})=0$ condition to $\imaginary(t_{r})=0$ Milosevic2000bicircular , focusing only on the classical recollision velocity Chipperfield2009 ; Chipperfield2010 , examining the change in the intensity scaling of the harmonic yield at a fixed harmonic order Lewenstein1995 , the use of graphical methods based on tangents to the optical waveform Figueira1999 , and extracting the cutoff from the Bessel- function expansion of the oscillatory integral Averbukh2001 ; Avetissian2014 , as well as direct numerical methods used to extract the cutoff from the results of TDSE simulations Neufeld2019timescales . None of these definitions, however, is particularly satisfactory, and – like the original definition – none of the analytical approaches have been used very widely in the literature. --- (a) (b) (c) (d) Figure 10: Scaling of the harmonic-cutoff frequency $\Omega_{\mathrm{hc}}$ with respect to the field parameters. (a) The affine dependence of the cutoff law $\real(\Omega_{\mathrm{hc}})\approx 3.17U_{p}+I_{p}$ is reproduced well; the dots at the axis mark the position of $I_{p}$, and the mismatch to the lines comes from the quantum corrections. (b) The imaginary part $\imaginary(\Omega_{\mathrm{hc}})$ decays with $U_{p}$ as $I_{p}^{3/2}/U_{p}^{1/2}$. These scalings indicate a relationship of the form $\Omega_{\mathrm{hc}}=3.17U_{p}+F(\gamma)I_{p}$, with the real and imaginary parts of $F(\gamma)$, shown in (c) and (d), suggesting that it is a power series in $i\gamma$. As we argued earlier, the real part $\Omega_{c}$ of our complex-valued $\Omega_{\mathrm{hc}}$ forms a natural candidate as a precise definition of the harmonic-cutoff frequency $\Omega_{\mathrm{max}}$, with the added advantage that it can be trivially adapted to complicated waveforms and to tailored polarizations without significantly complicating the calculation, a useful feature as the complexity of the field shapes under consideration continues to increase Neufeld2019 ; Javsarevic2020 . To strengthen this identification, though, it is important to verify that it reproduces the cutoff law, including the quantum corrections coming from the SFA as captured by the original definition. We show this in Fig. 10a: the variation of $\real(\Omega_{\mathrm{hc}})$ with $U_{p}$ is linear with an $I_{p}$-dependent offset, which closely follows the quasiclassical quantum-corrected cutoff law (50). However, our formalism allows us to go beyond this level, since $\Omega_{\mathrm{hc}}$ also has an imaginary part, whose scaling with $U_{p}$ and $I_{p}$ is shown in Fig. 10b. As rough behaviour, $\imaginary(\Omega_{\mathrm{hc}})$ decreases with $U_{p}$ and increases with $I_{p}$, and simple testing shows that the leading asymptotic component is the behaviour $\imaginary(\Omega_{\mathrm{hc}})\sim{I_{p}^{3/2}}\big{/}{U_{p}^{1/2}}.$ (52) At first glance, this shows very different behaviour to the real part, which follows (51). However, once a factor of $I_{p}$ has been set apart, as we did for the real part, the scaling for $\imaginary(\Omega_{\mathrm{hc}})$ is simply the Keldysh parameter, $\gamma=\sqrt{I_{p}/2U_{p}}$. This is therefore indicative that the parameter in the quantum scaling law (51) should be amended from $I_{p}/U_{p}$ to its square root, so it should read $\Omega_{\mathrm{hc}}=3.17\,U_{p}+F(\gamma)\,I_{p}.$ (53) In this form, (53) essentially acts as a definition for $F(\gamma)=(\Omega_{\mathrm{hc}}-3.17U_{p})/I_{p}$, but this definition can only work if that value is independent of what combination of $I_{p}$ and $U_{p}$ gives rise to $\gamma$. This is indeed the case, as we show in Figs. 10c and 10d for the real and imaginary parts, respectively: plotting $F(\gamma)$ by changing $I_{p}$ reveals the same universal curve for a range of different values of $U_{p}$. Moreover, we can extract the low-$\gamma$ behaviour here to obtain $\displaystyle F(\gamma)$ $\displaystyle\approx 1.323+i\,0.07361\,\gamma$ (54) $\displaystyle\quad\ -0.068\,\gamma^{2}-i\,0.025\,\gamma^{3}+\cdots,$ with the first two terms shown as the gray dashed line in Figs. 10c and 10d. This understanding coincides with the existing theory (so, in particular, Fig. 10c coincides with Fig. 5 of Ref. Lewenstein1994, when plotted as a function of $\gamma^{2}$), but it also helps extend it and clarify its structure, though a formal analysis of the existence and convergence of the power series in $i\gamma$ for $F(\gamma)$, as defined here for $\Omega_{\mathrm{hc}}$, is still lacking. We summarize our parameter-scaling results, and their relationship to the existing theories, in Table 1. Theory | | Cutoff law ---|---|--- Corkum Corkum1993 | | $\Omega_{\mathrm{max}}\approx 3.17U_{p}+I_{p}$ Kulander Kulander1993 | Lewenstein Lewenstein1994 | | $\begin{aligned} \Omega_{\mathrm{max}}&=3.17U_{p}+F(I_{p}/U_{p})I_{p}\\\ F(0)&=1.32\end{aligned}$ This work | | $\begin{aligned} \Omega_{\mathrm{hc}}&=3.17U_{p}+F(\gamma)I_{p}\\\ F(\gamma)&=1.323+i\,0.07361\,\gamma\\\ &\quad\ -0.068\,\gamma^{2}-i\,0.025\,\gamma^{3}+\cdots\end{aligned}$ Table 1: Scaling of the harmonic cutoff with $I_{p}$ and $U_{p}$, for the different understandings of the cutoff. ## IV Nontrivial quantum-orbit topology for bicircular fields To conclude our (current) explorations of the role of the complex harmonic- cutoff times in HHG, we return to the classification scheme for separating saddle-point solutions into quantum orbits, which we first showcased in Fig. 1 and constructed in detail in Section I.4. One crucial aspect of this classification scheme is the orientation of the separatrix, which indicates the direction that the saddle points take in their avoided crossing at the cutoff: i.e., whether the short-trajectory saddle, coming in from the left, will go up into positive (or down into negative) imaginary time when it meets the long-trajectory saddle. Our construction shows that this direction is given, via (23), by $\displaystyle\delta t_{\mathrm{sep}}=\sqrt{-i\eta/A_{\mathrm{hc}}},$ (55) and thus that it has a sensitive dependence on the sign of $\eta=\frac{\partial S_{V}}{\partial t}(t_{\mathrm{hc}}^{\phantom{a}},t_{\mathrm{hc}}^{\prime}),$ (56) the imaginary part of the time derivative of the Volkov action at the harmonic-cutoff time. This imaginary part is essentially controlled by the internal details of the action and, as such, its sign can be either positive or negative depending on the precise particulars of the optical waveform of the driving laser. If the shape of the driving laser pulse remains essentially constant (say, when doing an intensity scan as in Fig. 10) then the sign of the imaginary part is unlikely to change (as was seen to be the case in Fig. 10b). However, if one has a more complicated driving field with a nontrivial shape parameter that affects the waveform of the pulse, and thus the details of the Volkov action, then the sign of $\eta$ will change, and the direction of the separatrix, $\delta t_{\mathrm{sep}}$, will switch to one $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ away. This will, in turn, switch the direction of the avoided crossing, and with that the identity (say, as ‘short’ or ‘long’ trajectories) of the post-cutoff branches of the orbit. This behaviour is generic and commonplace: as a simple example, it can be observed in monochromatic fields once an envelope is introduced, where changes in the carrier-envelope phase can produce this type of topological change in the quantum-orbit layout. In this section we showcase a concrete example of this behaviour, and we explore the nontrivial topologies that it induces in the set of quantum-orbit saddle points. We focus, in particular, on tailored polarization states known as ‘bicircular’ fields Fleischer2014 ; Kfir2015 ; Eichmann1995 ; Milosevic1996 ; Milosevic2000bicircular ; Baykusheva2016 ; Dorney2017 ; JimenezGalan2018 ; Pisanty2014 ; Pisanty2017 ; Milovsevic2019 , formed by the combination of two counter-rotating circularly-polarized strong laser pulses at frequencies $\omega$ and $2\omega$, $\mathbf{F}(t)=F_{1}\begin{pmatrix}\cos(\omega t)\\\ \sin(\omega t)\end{pmatrix}+F_{2}\begin{pmatrix}\phantom{-}\cos(2\omega t)\\\ -\sin(2\omega t)\end{pmatrix}.$ (57) These fields have been the focus of widespread interest in recent years because they are subject to a spin selection rule Fleischer2014 ; Pisanty2014 which ensures that the harmonics of the driver are circularly polarized Kfir2015 . For our purposes, however, we select them as an example of a reasonably well-understood waveform with a nontrivial shape parameter, namely, the relative intensity between the two pulses; this is normally kept at unity, but the effects of its variation have seen some exploration Milosevic2000bicircular ; Baykusheva2016 ; Dorney2017 ; JimenezGalan2018 . If we keep the total intensity constant, at $I_{\mathrm{tot}}=$2\text{\times}{10}^{14}\text{\,}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$$ for concreteness, then the shift in intensity is best described by the mixing angle $\theta$, which we use to define the individual field amplitudes as $\displaystyle F_{1}=F\cos(\theta),\qquad F_{2}=F\sin(\theta),$ (58) where $F=\sqrt{2}\times$0.053\text{\,}\mathrm{{a.u.}}$$ is the peak field strength. --- (a) (b) (c) Figure 11: Topological transition and reconnection in the quantum orbits of a bicircular field as the mixing angle $\theta$ changes. Here we show a detail of the quantum-orbit layout for the short and long trajectories, plotted as in Fig. 1c, with the triangles marking the harmonic-cutoff times $t_{\mathrm{hc}}$ and the coloured separatrices marking the regions occupied by the different orbits, as described in Section I.4. In (b) we show the topological transition, where $\eta=\imaginary(\tfrac{\textrm{d}S_{V}}{\textrm{d}t})$ vanishes and the separatrix becomes undefined, which we show by highlighting the $t_{\mathrm{hc}}$ with a black border and plotting both options (using an artificial $\eta=\pm 1$) as the gray cross. In this case, both choices of separatrix are acceptable for saddle classification, since the saddles fully coalesce. We show in Fig. 11 a representative topological reconnection transition, between the short and long orbits (blue and green curves, respectively) of the bicircular field. Before the transition, in Fig. 11a, the short orbit connects up to the positive-imaginary saddle, while after the transition, in Fig. 11c, it connects to negative imaginary time. Between these two there must always lie a transition, which we show in Fig. 11b: here the saddles fully coalesce, producing a monkey-saddle landscape like the ones shown in Fig. 7, with a purely-real HCA Airy function corresponding to enhanced quantum path interference between the two orbits. | --- | --- (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Figure 12: Topological transitions and reconnections in the quantum orbits of a bicircular field as a function of the mixing angle $\theta$, plotted as in Fig. 11. As $\theta$ varies over a wider range, the various pairs of neighbouring quantum orbits go through reconnection transitions with each other (with two separate transitions, in (b) and (f), for the first pair of orbits), so that all of the quantum orbits are connected into a single topology, which we show below in 3D Fig. 2. At each panel the inset at bottom right shows the shape of the electric field at the relevant mixing angle. More importantly, however, this transition separates the instances with $\eta>0$ from those with $\eta<0$, so at the transition itself $\eta$ must vanish – and, as with the exact coalescences we studied in Section I.7, no (nonzero) unique separatrix direction $\delta t_{\mathrm{sep}}$ can be found. That said, this failure is fairly benign since, at the transition, either of the separatrices obtained by the artificial choices of $\eta=\pm 1$ will work correctly. At the transition the missed approach becomes a full coalescence, and any identification of pre- and post-cutoff orbits is artificial, so both signs of $\eta$ will work equally well. In Fig. 11b, we mark the transition by showing both possible separatrices in gray, taking an arbitrary choice between the two as to which one to use for the actual classification. We also highlight the triangle at the harmonic-cutoff time $t_{\mathrm{hc}}$ with a black border for further clarity. On a more practical footing, the fact that $\eta=\imaginary(\tfrac{\textrm{d}S_{V}}{\textrm{d}t})$ vanishes at the transition is especially useful, since it can be used to look for the parameters where the transitions happen, by looking for zeros of $\eta$ as a function of $\theta$. That said, the most important aspect of these topological transitions becomes apparent when we survey the quantum-orbit landscape on a wider perspective, in terms of the multiple quantum orbits present in the dynamics, as well as for a wider interval in the mixing angle, which we show in Fig. 12. Here we see that all of the pairs of quantum orbits undergo one or more reconnection transitions with each other, so that no pair of quantum orbits, however apparently distant, can actually be separated: in the same way that the evanescent orbits of Fig. 11b have ‘confused’ identities between the short and long orbits, the short orbit ‘mingles’ at low $\Omega$ with the short-time orbit shown in cyan, at two separate transitions (shown in Fig. 12b and Fig. 12f), the long orbit connects with the second return (with the transition shown in Fig. 12d), and so on. --- List of tdfigures 2 Unified surface formed by the quantum orbits in Fig. 12, when the $\theta$ dependence is unfolded as a third dimension. (Thus, a cut through the surface at constant $\theta$ will return a panel from Fig. 12.) At the transitions, the horizontal parts of the surface (the plateau harmonics) change from going up to going down and vice versa, but from a global perspective, the saddles form a single surface with a unified topology, and the separation of this surface into individual regions corresponding to the different quantum orbits is somewhat artificial. This highlights the fact that the various quantum orbits, as inverse images under the multivalued inverse of an analytical function, are simply different branches of a single Riemann surface. An interactive version and a 3D-printable model of this plot is available at imaginary-harmonic-cutoff.github.io SupplementaryMaterial . | ---|--- | | | (a) (b) (c) (d) (e) (f) (g) (h) Figure 13: Energy-time relations (a, c, e, g) and indicative SPA harmonic spectra, including the factors from the action and the wavepacket dilution (b, d, f, h) for four of the quantum-orbit transitions from Fig. 12. For bicircular fields, attention is generally focused on the short quantum orbit (blue curve), since at equal intensities ($\theta=$45\text{\,}\mathrm{\SIUnitSymbolDegree}$$) it dominates the spectrum (as shown in (b), which closely follows Fig. 8 of Ref. Milosevic2000bicircular, ). However, this is no longer the case at mixing angles with a high intensity for the $2\omega$ field. At the long-short topological transition (e, f), where the two become indistinguishable at the cutoff, the latter is intensified by this effect, while at large mixing angles, as shown in (g, h), the contribution of the short trajectory plummets. In other words, the quantum orbits here form a single unified topology, and they should be understood as such. We show this topology in 3D Fig. 2, by unfolding the $\theta$ dependence into a third dimension for the plot: this reveals that the quantum orbits form a single surface, with discontinuous changes in colour where the (apparent) reconnection transitions force us to choose where to split this unified surface into individual components. The fundamental topological object here, as we explored in depth in Section I.5, is the Riemann surface formed by the saddle points: that is, the surface defined by the equation $\frac{\textrm{d}S_{V}}{\textrm{d}t}(t_{s})=\Omega$, encoding the multiple inverses of the action’s derivative, whose intersection with the real $\Omega$ axis gives the quantum orbits themselves. Within that perspective, the difference between the two types of connections between the quantum orbits boils down to what side of the real $\Omega$ axis the branch point (i.e., the harmonic-cutoff time $t_{\mathrm{hc}}$) falls on. For fixed field shapes, this branch-cut structure can essentially be ignored, if so desired, since we are only looking for the inverse images for $\Omega$ over the real axis, and those inverse images will normally be well separated. In 3D Fig. 2, however, as well as in any other situations where the field shape changes over a control parameter, we see how these changes to the position and shape of the Riemann surface bring these once-separate images into collision (and reconnection) with each other, and this structure can no longer be ignored. In more general terms, the reconnection phenomenon we have demonstrated in this section entails that, ultimately, attempting to attach labels to individual quantum orbits is fundamentally impossible, since they are essentially just different instances of one and the same object. Saddle-point tagging schemes of this type, in terms of multi-indices typically notated as $\alpha\beta m$, have been used widely in the literature, both for monochromatic fields Chipperfield2007 ; Milosevic2002 ; Odzak2005 ; Chipperfield2005 ; Milosevic2017chapter as well as more complex waveforms Hasovic2016 ; Milosevic2019xray ; Milovsevic2019quantum ; Milosevic2016 . The topological features we have explored here imply that these schemes should be regarded as labelling the principal branches of the quantum-orbit Riemann surface. These schemes can, in principle, be applied to changing driver waveforms with nontrivial connections between the various branches involved, but then those branch changes must be explicitly tracked – in which case, the branch-point machinery we have developed here (both for finding the branch points $t_{\mathrm{hc}}$ as well as the parameters where their branch cuts cross the real $\Omega$ axis) is essential. As a final note here, it is important to remark that, while some of the dynamics we have explored here occurs for quantum orbits with weak or negligible contributions to the spectrum, that is not always the case. Indeed, as we show in Fig. 13, the equal-intensities viewpoint that the spectrum is completely dominated by the short-trajectory contribution breaks down at large mixing angles, where the $2\omega$ contribution is significant, and where many of the topological transitions we have discussed in this section take place. ## V Discussion and conclusions As we have shown, the problem of saddle-point classification can be solved in a robust and flexible way by using the harmonic-cutoff times $t_{\mathrm{hc}}$ to locate the center of the missed approach and implement a separatrix that lies between the two quantum orbits that perform an avoided crossing at each cutoff. These harmonic-cutoff times can be easily found as the zeros of a second-order saddle-point equation. In a clear sense, the $t_{\mathrm{hc}}$’s form the centerpiece of the quasiclassical structure at the cutoff, and they are a useful and flexible tool to understand a range of questions about the quantum-orbit dynamics. The strongest implication of this is that the harmonic-cutoff times can be used to provide a natural identification for the energy of the cutoff, $\Omega_{\mathrm{hc}}=\frac{\partial S_{V}}{\partial t}(t_{\mathrm{hc}})$. This now takes on a complex value, with the imaginary part controlling the strength of quantum-path interference between the quantum orbits that meet at that cutoff, as well as the closeness and direction of the approach between them. Similarly, our approach provides an efficient method for finding the position of the cutoff as well as the harmonic yield there, which will work uniformly and efficiently for a broad range of optical waveforms for the driving laser. This extends to a full estimation of the spectrum, the Harmonic-Cutoff Approximation, which uses only quantities local to $t_{\mathrm{hc}}$, and which accurately captures the cutoff as well as the qualitative shape of the spectrum down into the middle of the plateau. On a more abstract note, our search for structures that can be used to classify the solutions of the saddle-point equations into individual quantum orbits yields a fresh perspective on the quasiclassical theory of strong-field phenomena. The quantum orbits are thus revealed as the $\imaginary(\frac{\partial S_{V}}{\partial t})=0$ contour of the time derivative of the action, with the rest of its contour map holding crucial information – most notably via its saddles, the harmonic-cutoff times $t_{\mathrm{hc}}$. Likewise, the usual saddle points are re-understood as individual branches of a larger Riemann surface, which encompasses all of the quantum orbits. The branch points that separate these branches are again the harmonic-cutoff times, which can thus be used to watch for changes in the quantum-orbit topology as the driving field’s waveform changes. The location of the cutoff at a zero of the second derivative of the action also admits a rather more pedestrian interpretation, though. If the emission is modelled using only the classical-trajectory level, then the energy of return $E_{\mathrm{return}}=\frac{\partial S}{\partial t}$ is a function of the return time, and if we want to find its extrema, then we simply need to solve the equation $\frac{\partial^{2}S}{\partial t^{2}}=0$. When the theory is upgraded to the quasiclassical formalism, that equation might seem to lose its meaning, since the action and the trajectories are complex, as would be the solutions to the extremum equation $\frac{\partial^{2}S}{\partial t^{2}}=0$. Our solution in this work is entirely in line with the spirit of the quasiclassical formalism for strong-field physics: we solve this extremum equation in the same form it takes classically, allowing for complex values wherever necessary, and then look at the underlying oscillatory integral for the correct interpretation of those complex quantities. More physically, the key structure at play is the fact that the harmonic cutoff is a caustic: the quantum-orbit analysis of the underlying matter-wave dynamics is exactly analogous to the geometric-optics analysis of wave optics in terms of rays, and in this analogy the harmonic cutoff corresponds to caustics where two families of rays interfere and then meet and fold into each other, giving way to evanescent-wave behaviour analogous to the post-cutoff decay in the harmonic spectrum. In one dimension, this caustic can be precisely localized to a point, known as the ‘fold’ (more technically, the ‘bifurcation set’), which marks the transition between the two regimes, and our harmonic-cutoff times are the precise embodiment of that understanding of the caustic. This view of the harmonic cutoff as a caustic has been used for some time Raz2012 ; Goulielmakis2012 ; Austin2014 , but recent years have seen a marked increase in interest in that perspective on the dynamics Facciala2016 ; Facciala2018 ; Hamilton2017 ; Birulia2019 ; Uzan2020 ; Kelvich2017 , as laser sources achieve better control over polychromatic combinations with high relative intensities (which is slated to continue to increase Mitra2020 ). This has opened the door to the observation of higher-order catastrophes in strong-field observables, involving higher-dimensional bifurcation sets; however, the theoretical understanding remains somewhat behind, and a quantitative understanding that reflects those structures has not yet followed. In this work we have precisely pinned down the embodiment of the bifurcation set in the quasiclassical formalism for the ‘fold’ catastrophe, and this can then be used to analyze in detail the more complicated configurations involved in newer experiments. Similarly, our work suggests that the harmonic-cutoff times we have discussed here should have analogous structures in ionization experiments (most notably high-order above-threshold ionization Figueira2019 ), where the experimentally-measurable variable, momentum, has multiple dimensions. This allows greater freedom to the theory (while at the same time substantially complicating its analysis), and this in turn allows for higher-dimensional singular matter-wave structures to be contained in the experimental results. The tools we have demonstrated here for understanding the one-dimensional caustic formed at the harmonic cutoff should then provide a useful basis for understanding those configurations. ###### Acknowledgements. We thank Przemyslaw Grzybowski for essential and generous help in understanding the structures presented here. E.P. acknowledges Cellex-ICFO-MPQ Fellowship funding. 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2024-09-04T02:54:55.909409

2003.00314

arxiv-papers

# A Complexity Chasm for Solving Sparse Polynomial Equations Over $p$-adic Fields J. Maurice Rojas<EMAIL_ADDRESS>TAMU 3368, College Station, TX 77843-3368 and Yuyu Zhu<EMAIL_ADDRESS>TAMU 3368, College Station, TX 77843-3368 ###### Abstract. We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\\!\in\\!\\{\mathbb{Q}_{2},\mathbb{Q}_{3},\mathbb{Q}_{5},\ldots\\}$, we prove that any polynomial $f\\!\in\\!\mathbb{Z}[x_{1}]$ with exactly $3$ monomial terms, degree $d$, and all coefficients having absolute value at most $H$, can be solved over $K$ in deterministic time $\log^{9}(dH)$ in the classical Turing model. (The best previous algorithms were of complexity exponential in $\log d$, even for just counting roots in $\mathbb{Q}_{p}$.) In particular, our algorithm generates approximations in $\mathbb{Q}$ with bit- length $\log^{8}(dH)$ to all the roots of $f$ in $K$, and these approximations converge quadratically under Newton iteration. On the other hand, we give a unified family of tetranomials requiring $\Omega(d\log H)$ bits to distinguish the base-$b$ expansions of their roots in $K$. Partially supported by NSF grants CCF-1900881 and CCF-1409020. Partially supported by NSF grants CCF-1900881 and CCF-1409020. ## 1\. Introduction The applications of solving systems of polynomial equations are legion: The real case permeates all of non-linear optimization as well as numerous problems in engineering. The $p$-adic case leads to many classical questions in number theory, and is close to many applications in cryptography, coding theory, and computational number theory. As such, it is important to understand the complexity of solving systems of polynomial equations over local fields. Furthermore, the complexity of solving structured systems — such as those with a fixed number of monomial terms or invariance with respect to a group action — arises naturally in many computational geometric applications and is closely related to a deeper understanding of circuit complexity (see, e.g., [16]). Clearly, if we are to fully understand the complexity of solving sparse polynomial systems, then we should at least be able to settle the univariate case, e.g., classify when it is possible to separate and approximate roots in deterministic time polynomial in the input size. Our first main result settles the univariate case, over a fundamental family of local fields admitting an analogue of Descartes’ Rule. More precisely, thanks to 17th century work of Descartes, and 20th century work of Lenstra [18] and Poonen [22], it is known that univariate polynomials with exactly $t$ monomial terms have at most $t^{O(1)}$ roots in a fixed field $K$ only when $K$ is $\mathbb{R}$ or a finite algebraic extension of $\mathbb{Q}_{p}$ for some prime $p\\!\in\\!\mathbb{N}$. For instance, $\mathbb{C}$ is ruled out because $x^{d}-1$ has just $2$ monomial terms but $d$ distinct complex roots. Also, the Laurent series fields $\mathbb{F}_{p}((\theta))$ are ruled out by an elementary calculation [22] showing that$\prod\limits_{z_{0},\ldots,z_{t-2}\in\mathbb{F}_{p}}(x_{1}-z_{0}-z_{1}\theta-\cdots- z_{t-2}\theta^{t-2})$ has exactly $t$ monomial terms as a polynomial in $\mathbb{F}_{p}[\theta][x_{1}]$, but exactly $p^{t-1}$ roots in $\mathbb{F}_{p}[\theta]$. We’ll use $|\cdot|_{p}$ for the usual absolute value on $\mathbb{C}_{p}$, normalized so that $|p|_{p}\\!=\\!\frac{1}{p}$. Recall also that for any function $f$ analytic on $K$, the corresponding Newton endomorphism is $N_{f}(z):=z-\frac{f(z)}{f^{\prime}(z)}$, and the corresponding sequence of Newton iterates of a $z_{0}\\!\in\\!K$ is the sequence $(z_{i})^{\infty}_{i=0}$ where $z_{i+1}\\!:=\\!N_{f}(z_{i})$ for all $i\\!\geq\\!0$. Finally, we call any polynomial in $\mathbb{Z}[x_{1},\ldots,x_{n}]$ having exactly $t$ terms in its monomial term expansion an $n$-variate $t$-nomial. ###### Theorem 1.1. Suppose $K\\!=\\!\mathbb{Q}_{p}$ for some fixed111We clarify the dependence of our complexity bounds on $p$ in Section 5. prime $p\\!\in\\!\mathbb{N}$. Then there is an algorithm which, for any input trinomial $f\\!\in\\!\mathbb{Z}[x_{1}]$ with degree $d$ and all coefficients of (Archimedean) absolute value at most $H$, outputs a set $\left\\{\frac{a_{1}}{b_{1}},\ldots,\frac{a_{m}}{b_{m}}\right\\}\\!\subset\\!\mathbb{Q}$ of cardinality $m\\!=\\!m(K,f)$ such that: 1\. For all $j$ we have $a_{j}\\!\neq\\!0\Longrightarrow\log|a_{j}|,\log|b_{j}|=\log^{8}(dH)$. 2\. There is a $\mu\\!=\\!\mu(d,H)\\!>\\!1$ such that for all $j$ we have that $z_{0}\\!:=\\!a_{j}/b_{j}$ implies that $f$ has a root $\zeta_{j}\in K$ such that the corresponding sequence of Newton iterates satisfies $|z_{i}-\zeta_{j}|_{p}\\!\leq\\!\mu^{-2^{i-1}}|z_{0}-\zeta_{j}|_{p}$ for all $i\\!\geq\\!1$. 3\. $m$ is exactly the number of roots of $f$ in $K$ and the cardinality of $\\{\zeta_{1},\ldots,\zeta_{m}\\}$. Moreover, the underlying algorithm (Algorithm 5.7 in Section 5 below) takes deterministic time $O\\!\left(\log^{9}(dH)\right)$ on a Turing machine. We prove Theorem 1.1 in Section 5. (An analogue of Theorem 1.1 in fact holds for $K\\!=\\!\mathbb{R}$ as well, and this will be presented in a sequel to this paper.) We will call the convergence condition on $z_{0}$ above being an approximate root (in the sense of Smale222This terminology has only been applied over $\mathbb{C}$ with $\mu\\!=\\!2$ so far [28], so we take the opportunity here to extend it to $\mathbb{Q}_{p}$ for $p$ prime.), with associated true root $\zeta_{j}$. This type of convergence provides an efficient encoding of an approximation that can be quickly tuned to any desired accuracy. ###### Remark 1.2. Defining the input size of a univariate polynomial $f(x_{1})\\!:=\\!\sum^{t}_{i=1}c_{i}x^{a_{i}}_{1}$$\in\\!\mathbb{Z}[x_{1}]$ as $\sum^{t}_{i=1}\log((|c_{i}|+2)(|a_{i}|+2))$ we see that Theorem 1.1 implies that one can solve univariate trinomial equations, over any fixed $p$-adic field, in deterministic time polynomial in the input size. $\diamond$ ###### Remark 1.3. Efficiently solving univariate $t$-nomial equations over $K$ in the sense of Theorem 1.1 is easier for $t\\!\leq\\!2$: The case $t\\!=\\!1$ is clearly trivial (with $0$ the only possible root) while the case $(K,t)\\!=\\!(\mathbb{R},2)$ is implicit in work on computer arithmetic from the 1970s (see, e.g., [7]). We review the case $(K,t)\\!=\\!(\mathbb{Q}_{p},2)$ with $p$ prime in Theorem 2.15 of Section 2 below. $\diamond$ Despite much work on factoring univariate polynomials over $\mathbb{Q}_{p}$ (see, e.g., [9, 12, 4, 5]), all known general algorithms for solving (or even just counting the solutions of) arbitrary degree $d$ polynomial equations over $\mathbb{Q}_{p}$ have complexity exponential in $\log d$. So Theorem 1.1 presents a new speed-up, and extends earlier work in [24] where it was shown that detecting roots in $\mathbb{Q}_{p}$ for univariate trinomials can be done in $\mathbf{NP}$ for fixed $p$. We’ll see in Section 3 how our speed-up depends on applying Yu’s Theorem on linear forms in $p$-adic logarithms [32]. The key new ingredient in proving Theorem 1.1 is an efficient encoding of roots in $\mathbb{Z}/\langle p^{k}\rangle$ from [17] (with an important precursor in [5]). ### 1.1. The Chasm at Four Terms via Root Separation Unfortunately, there are obstructions to attaining complexity polynomial in the input size, for solving univariate polynomial equations with sufficiently many terms. Indeed, our second main result is that the underlying root spacing changes dramatically already at $4$ terms. ###### Theorem 1.4. Consider the family of tetranomials $\displaystyle\displaystyle{f_{d,\varepsilon}(x_{1}):=x^{d}_{1}-\frac{1}{\varepsilon^{2h}}x^{2}_{1}+\frac{2}{\varepsilon^{h+1}}x_{1}-\frac{1}{\varepsilon^{2}}}$ with $h\\!\in\\!\mathbb{N}$, $h\\!\geq\\!3$, and $d\\!\in\\!\left\\{4,\ldots,\left\lfloor e^{h}\right\rfloor\right\\}$ even. Let $H\\!:=\\!\max\\{\varepsilon^{\pm 2h}\\}$. Then $f_{d,\varepsilon}$ has distinct roots $\zeta_{1},\zeta_{2}\\!\in\\!K$ with $|\log|\zeta_{1}-\zeta_{2}|_{p}|$ or $\log|\zeta_{1}-\zeta_{2}||$ of order $\Omega(d\log H)$, according as $(K,\varepsilon)\\!=\\!\left(\mathbb{R},\frac{1}{2}\right)$ or $(K,\varepsilon)\\!=\\!(\mathbb{Q}_{p},p)$. In particular, the coefficients of $f_{d,\frac{1}{2}}$ and $p^{2h}f_{d,p}$ all lie in $\mathbb{Z}$ and have bit- length $O(\log H)$. We prove Theorem 1.4 in Section 4. The special case $K\\!=\\!\mathbb{R}$ was derived earlier (in different notation) by Mignotte [20]. (See also [25].) The case $K\\!=\\!\mathbb{Q}_{p}$ with $p$ prime appears to be new, and our proof unifies the Archimedean and non-Archimedean cases via tropical geometry (Newton polygons in particular). Note that Theorem 1.4 implies that the roots in $K$ of a tetranomial can be so close that one needs $\Omega(d\log H)$ many digits to distinguish their base-$b$ expansions (for any fixed $b$) in the worst case. Mignotte used the tetranomial $f_{d,\frac{1}{2}}$ in [20] to show that an earlier root separation bound of Mahler [19], for arbitrary degree $d$ polynomials in $\mathbb{Z}[x_{1}]$, is asymptotically near-optimal. We recall the following paraphrased version: ###### Mahler’s Theorem . Suppose $f\\!\in\\!\mathbb{Z}[x_{1}]$ has degree $d$, all coefficients of (Archimedean) absolute value at most $H$, and is irreducible in $\mathbb{Z}[x_{1}]$. Let $\zeta_{1},\zeta_{2}\\!\in\\!\mathbb{C}$ be distinct roots of $f$. Then $|\log|\zeta_{1}-\zeta_{2}||\\!=\\!O(d\log(dH))$. $\blacksquare$ Our new algorithmic results are enabled by our third and final main result: Mahler’s bound is far from optimal for $t$-nomials with $t\\!\leq\\!3$. ###### Theorem 1.5. Suppose $p$ is prime and $f\\!\in\\!\mathbb{Z}[x_{1}]$ is irreducible, has exactly $3$ monomial terms, degree $d$, and all coefficients of (Archimedean) absolute value at most $H$. Let $\zeta_{1},\zeta_{2}\\!\in\\!\mathbb{C}_{p}$ be distinct roots of $f$. Then $|\log|\zeta_{1}-\zeta_{2}|_{p}|\\!=\\!O\\!\left(p\log^{4}(dH)\right)$. We prove Theorem 1.5 (which arose from the unpublished Ph.D. thesis of author Zhu) in Section 3. Theorem 1.5 is in fact a $p$-adic analogue of a separation bound of Koiran [15]. Even sharper bounds can be derived for binomials and we review these bounds in Section 2.2. ### 1.2. Arbitrary Sparse Polynomials It is worth recalling that merely deciding the existence of roots over $\mathbb{Q}_{p}$ for univariate polynomials (with an arbitrary number of monomial terms) is $\mathbf{NP}$-hard with respect to randomized ($\mathbf{ZPP}$, a.k.a. Las Vegas) reductions [24]. Put another way, this means that a polynomial-time algorithm for this problem would imply that $\mathbf{NP}\\!\subseteq\\!\mathbf{ZPP}$ — a containment doubted by complexity theorists just as much as the equality $\mathbf{P}\\!=\\!\mathbf{NP}$ [31]. Our paper is devoted to showing that a harder problem (giving a set of approximate roots over $\mathbb{Q}_{p}$ of the correct cardinality) is doable in polynomial-time for trinomials, but not for tetranomials, for fixed $p$. Based on our results, we conjecture the following: ###### Conjecture 1.6. For any fixed $p$ and $t$, there is a polynomial-time algorithm that counts the roots in $\mathbb{Q}_{p}$ of any input $t$-nomial $f\\!\in\\!\mathbb{Z}[x_{1}]$. In particular, we expect the complexity to be exponential in $t+\log p$, but polynomial in $\log(dH)$. We also note that detecting roots in $\mathbb{Q}_{p}$ for $n$-variate $(n+1)$-nomials is known to be doable in $\mathbf{NP}$ [24]. However, speeding this up to polynomial-time, even for $n\\!=\\!2$ and fixed $p$, hinges upon detecting roots in $(\mathbb{Z}/\langle p^{k}\rangle)^{2}$ for bivariate trinomials of degree $d$ in time $(k+\log d)^{O(1)}$. The latter problem remains open, but some progress has been made in author Zhu’s Ph.D. thesis. On a related note, even counting points on trinomial curves over the prime fields $\mathbb{F}_{p}$ in time $\log^{O(1)}(pd)$ remains a challenging open question. ### 1.3. Acknowledgements We thank Erich Bach and Bjorn Poonen for informative discussions on Hensel’s Lemma. Let us now review the necessary background for our proofs. ## 2\. Background ### 2.1. Newton Polygons and Newton Iteration: Archimedean and Non- Archimedean Definitive sources for $p$-adic arithmetic and analysis include [27, 26, 23]. We will only review a few facts necessary for our development. We use $\operatorname{ord}_{p}:\mathbb{C}_{p}\longrightarrow\mathbb{Q}$ for the standard $p$-adic valuation on $\mathbb{C}_{p}$, normalized so that $\operatorname{ord}_{p}p\\!=\\!1$. The most significant ($p$-adic) digit of $\sum\limits^{\infty}_{j=s}a_{j}p^{j}\\!\in\\!\mathbb{Q}_{p}$ is simply $a_{s}$, assuming $a_{j}\\!\in\\!\\{0,\ldots,p-1\\}$ for all $j$ and $a_{s}\\!\neq\\!0$, i.e., $a_{s}$ is the digit being multiplied by the power of $p$ with largest $p$-adic norm. The notion of Newton polygon goes back to 17th century work of Newton on Puiseux series solutions to polynomial equations. We will need variants of this notion over $\mathbb{C}_{p}$ and $\mathbb{C}$. (See, e.g., [30] for the $p$-adic case and [21, 1] for the complex case.): ###### Definition 2.1. Suppose $p$ is prime and $f(x)\\!:=\\!\sum_{i=1}^{t}c_{i}x^{a_{i}}_{1}\\!\in\\!\mathbb{Z}[x_{1}]$ with $c_{i}\\!\neq 0$ for all $i$ and $a_{1}\\!<\\!\cdots\\!<\\!a_{t}$. We then define $\operatorname{Newt}_{p}(f)$ (resp. $\operatorname{Newt}_{\infty}(f)$) to be the convex hull of the set of points $\\{(a_{i},\operatorname{ord}_{p}c_{i})\;|\;i\\!\in\\!\\{1,\ldots,t\\}\\}$ (resp. the convex hull of $\\{(a_{i},-\log|c_{i}|)\;|\;i\\!\in\\!\\{1,\ldots,t\\}\\}$). We call an edge $E$ of a polygon in $\mathbb{R}^{2}$ lower if and only if $E$ has an inner normal with positive last coordinate. We also define the horizontal length of a line segment $E$ connecting $(r,s)$ and $(u,v)$ to be $\lambda(E)\\!:=\\!|u-r|$. $\diamond$ ###### Theorem 2.2. Following the notation above, when $p$ is prime, the number of roots of $f$ in $\mathbb{C}_{p}$ of valuation $v$ is exactly the horizontal length of the face of $\operatorname{Newt}_{p}(f)$ with inner normal $(v,1)$. Furthermore, if $\operatorname{Newt}_{\infty}(f)$ has a lower edge $E$ with slope $v$, and no other lower edges with slope in the open interval $(v-\log 3,v+\log 3)$, then the number of roots $\zeta\\!\in\\!\mathbb{C}$ of $f$ with $\log|\zeta|\\!\in\\!(v-\log 3,v+\log 3)$ is exactly $\lambda(E)$, counting multiplicity. $\blacksquare$ The first portion of Theorem 2.2 goes back to early 20th century work of Hensel, while the second portion is an immediate consequence of [1, Thm. 1.5] (and has an important precursor in [21]). We will also use the following version of Hensel’s famous criterion for the rapid convergence of Newton’s method over $\mathbb{C}_{p}$: ###### Hensel’s Lemma . (See, e.g., [23, Thm., pg. 48, Sec. 6.4].) Suppose $p$ is prime, $f\in\mathbb{Z}[x]$, $j\\!\geq\\!1$, $\zeta\\!\in\\!\mathbb{Z}_{p}$, $\ell\\!=\\!\operatorname{ord}_{p}f^{\prime}(\zeta)\\!<\\!\infty$, and $f(\zeta)\equiv 0\mod p^{2\ell+j}$. Let $\zeta^{\prime}\\!:=\\!\zeta-\frac{f(\zeta)}{f^{\prime}(\zeta)}$. Then $f(\zeta^{\prime})\\!=\\!0$ mod $p^{2\ell+2j}$, $\operatorname{ord}_{p}f^{\prime}(\zeta^{\prime})\\!=\\!\ell$, and $\zeta\\!=\\!\zeta^{\prime}$ mod $p^{\ell+j}$. $\blacksquare$ ### 2.2. Separating Roots of Binomials When $f\\!\in\\!\mathbb{Z}[x_{1}]$ is a binomial, all of its roots in $\mathbb{C}$ are multiples of roots of unity that are evenly spaced on a circle. The same turns out to be true over $\mathbb{C}_{p}$, but the root spacing then depends more subtly on the degree and $p$. For convenience, we will sometimes write $|\cdot|_{\infty}$ instead of $|\cdot|$ for the standard norm on $\mathbb{C}$. In summary, we have the following: ###### Proposition 2.3. Suppose $f(x_{1})\\!:=\\!c_{1}+c_{2}x^{d}_{1}\\!\in\\!\mathbb{Z}[x_{1}]$, $c_{1}c_{2}\\!\neq\\!0$, and $|c_{1}|,|c_{2}|\\!\leq\\!H$. Also let $p\\!\in\\!\\{\infty,2,3,5,\ldots\\}$ and let $\overline{K}_{p}$ denote $\mathbb{C}$ or $\mathbb{C}_{p}$, according as $p\\!=\\!\infty$ or $p$ is prime. Then for any distinct roots $\zeta_{1},\zeta_{2}\\!\in\\!\overline{K}_{p}$ of $f$, we have that $|\log|\zeta_{1}-\zeta_{2}|_{p}|$ is bounded from above by $\begin{cases}\frac{3}{2}\log(3)+\log\\!\left(dH^{1/d}\right);\ \text{for }p\\!=\\!\infty\text{ and }d\\!\geq\\!3\\\ \frac{\log p}{d}\log H;\hskip 56.9055pt\text{for }p\text{ prime and }p\nmid d,\\\ \frac{\log p}{d}\log(H)+\frac{\log p}{p-1};\hskip 19.91684pt\text{for }p\text{ prime and }p|d.\ \text{$\blacksquare$}\end{cases}$ The case $p\\!=\\!\infty$ is immediate (using a very rough estimate for the distance between the vertices of a regular $d$-gon), while the case of prime $p$ follows easily from the ultrametric inequality and classical facts on the spacing of $p$-adic roots of unity (see, e.g., [23, Cor. 1, Pg. 105, Sec. 4.3 & Thm. Pg. 107, Sec. 4.4]). While we won’t need Proposition 2.3 for our upcoming algorithms, it is an important precursor to our trinomial root separation bound (Theorem 1.5), and is algorithmically relevant for more general problems like approximating roots in angular sectors in $\mathbb{C}$ or finite extensions of $\mathbb{Q}_{p}$. ### 2.3. Counting Roots of Binomials Over $\mathbb{Q}_{p}$ To efficiently solve a binomial $f$ over $\mathbb{Q}_{p}$, it helps to first find a fast way to count the roots of $f$ in $\mathbb{Q}_{p}$, and then to find a fast way to incrementally compute the base-$p$ expansions of the roots of $f$ in $\mathbb{Q}_{p}$. Counting roots over $\mathbb{Q}_{p}$ is more involved than counting roots over $\mathbb{R}$, but is still quite efficiently doable. For any ring $R$ we will let $R^{*}$ denote the multiplicatively invertible elements of $R$. ###### Lemma 2.4. Suppose $p$ is an odd prime and $f(x_{1})\\!:=\\!c_{1}+c_{2}x^{d}_{1}\\!\in\\!\mathbb{Z}[x_{1}]$ with $|c_{1}|,|c_{2}|\\!\leq\\!H$, $c_{1}c_{2}\\!\neq\\!0$, and $\ell\\!:=\\!\operatorname{ord}_{p}d$. Then the number of roots of $f$ in $\mathbb{Q}_{p}$ is either $0$ or $\gcd(d,p-1)$. In particular, $f$ has roots in $\mathbb{Q}_{p}$ if and only if both of the following conditions hold: 1\. $d|\operatorname{ord}_{p}(c_{1}/c_{2})$. 2\. $\left(-\frac{c_{1}}{c_{2}}p^{\operatorname{ord}_{p}(c_{2}/c_{1})}\right)^{p^{\ell}(p-1)/\gcd(d,p-1)}\\!=\\!1$ mod $p^{2\ell+1}$. $\blacksquare$ Lemma 2.4 is classical and follows from basic group theory and [24, Cor. 3.2]. ###### Remark 2.5. A criterion similar to Lemma 2.4 also holds for detecting roots in $\mathbb{Q}_{2}$ for binomials (see, e.g., [24, Cor. 3.2]). However, for brevity’s sake, we will not pursue the case $p\\!=\\!2$ further in the present version of this paper. $\diamond$ Since powers in finite groups are easily computable via recursive squaring (see, e.g., [3, pp. 102–103]), we easily obtain the following via Hensel’s Lemma, Proposition 2.3, Lemma 2.4, and standard complexity bounds on modular arithmetic [3, 29]: ###### Corollary 2.6. Following the notation and assumptions of Lemma 2.4, one can count exactly the number of roots of $f$ in $\mathbb{Q}_{p}$ in time $O\\!\left(\log^{3}(pdH)\right)$. Furthermore, for any such root $\zeta\\!\in\\!\mathbb{Q}_{p}$ there is an $x_{0}\\!\in\\!\mathbb{Z}\left/\left\langle p^{2\ell+1}\right\rangle\right.$ that is a root of the mod $p^{2\ell+1}$ reduction of $\frac{c_{1}}{p^{\operatorname{ord}_{p}c_{1}}}+\frac{c_{2}}{p^{\operatorname{ord}_{p}c_{2}}}x^{d}_{1}$, and with $z_{0}\\!:=\\!p^{\operatorname{ord}_{p}(c_{2}/c_{1})/d}x_{0}\\!\in\\!\mathbb{Q}$ an approximate root of $f$ with associated true root $\zeta$. In particular, the logarithmic height of $z_{0}$ is $O\\!\left(\log\left(dH^{1/d}\right)\right)$. $\blacksquare$ At this point, one may wonder how one can find a suitable $x_{0}$ as in Corollary 2.6. Naive brute-force search can take time super-linear in $p^{\ell}$, but in the next section we’ll see a combinatorial encoding enabling a much better time bound of $O\\!\left(p\log^{2}(p)\log\gcd(d,p-1)+\log^{3}(pdH)\gcd(d,p-1)\right)$. ### 2.4. Trees and Roots in $\mathbb{Z}/\langle p^{k}\rangle$ and $\mathbb{Z}_{p}$ A key tool we will need is a tree structure that will enable us to easily approximate the first few base-$p$ digits of the roots of an arbitrary univariate polynomial. ###### Definition 2.7. [17] Let $p\\!\in\\!\mathbb{N}$ be prime. For any $f\in\mathbb{Z}[x_{1}]$ let $\tilde{f}$ denote the mod $p$ reduction of $f$. For any degenerate root $\zeta_{0}$ of $\tilde{f}$, we define $\displaystyle s(f,\zeta_{0}):=\min\left\\{i+\operatorname{ord}_{p}\frac{f^{(i)}(\zeta_{0})}{i!}\right\\}.$ Fixing $k\in\mathbb{N}$, for $i\geq 1$, let us inductively define a set $T_{p,k}(f)$ of pairs $(f_{i-1,\mu},k_{i-1,\mu})$$\in\mathbb{Z}[x_{1}]\times\mathbb{N}$. We set $(f_{0,0},k_{0,0}):=(f,k)$. Then for any $i\geq 1$ with $(f_{i-1,\mu},k_{i-1,\mu})$$\in T_{p,k}(f)$, and any degenerate root $\zeta_{i-1}\in\mathbb{Z}/p\mathbb{Z}$ of $\tilde{f}_{i-1,\mu}$ with $\displaystyle s_{i-1}:=s(f_{i-1,\mu},\zeta_{i-1})\in\\{2,\cdots,k_{i-1,\mu}-1\\},$ we define $\zeta:=\mu+p^{i-1}\zeta_{i-1},k_{i,\zeta}:=k_{i-1,\mu}-s(f_{i-1,\mu},\zeta_{i-1})$, and $\displaystyle f_{i,\zeta}(x):=\left[\frac{1}{p^{s(f_{i-1,\mu},\zeta_{i-1})}}f_{i-1,\mu}(\zeta_{i-1}+px)\right]\mod p^{k_{i,\zeta}}.\text{ \ $\diamond$}$ Note that $\frac{f^{(i)}(\zeta_{0})}{i!}$ is the coefficient of $x^{i}$ in the Taylor expansion of $f(x+\zeta_{0})$ about $0$, and is thus an integer since $f\\!\in\\!\mathbb{Z}[x_{1}]$. The collection of pairs $(f_{i,\zeta},k_{i,\zeta})$ admits a tree structure that will give us a way of extending Hensel lifting to degenerate roots. ###### Definition 2.8. [17] Let us identify the elements of $T_{p,k}(f)$ with nodes of a labelled rooted directed tree $\mathcal{T}_{p,k}(f)$ defined inductively as follows333This definition differs slightly from the original in [17]: there are no edge labels in our version here since we won’t need them for root counting in $\mathbb{Z}_{p}$.: * i. We set $f_{0,0}\\!:=\\!f$, $k_{0,0}\\!:=\\!k$, and let $(f_{0,0},k_{0,0})$ be the label of the root node of $\mathcal{T}_{p,k}(f)$. * ii. The non-root nodes of $\mathcal{T}_{p,k}(f)$ are uniquely labelled by each $(f_{i,\zeta},k_{i,\zeta})$$\in T_{p,k}(f)$ with $i\\!\geq\\!1$. * iii. There is an edge from node $(f_{i^{\prime},\zeta^{\prime}},k_{i^{\prime},\zeta^{\prime}})$ to node $(f_{i,\zeta},k_{i,\zeta})$ if and only if$i^{\prime}\\!=\\!i-1$ and there is a degenerate root $\zeta_{i-1}\\!\in\\!\mathbb{Z}/(p)$ of $\tilde{f}_{i^{\prime},\zeta^{\prime}}$ with $s(f_{i^{\prime},\zeta^{\prime}},\zeta_{i-1})$$\in\\{2,\ldots,k_{i^{\prime},\zeta^{\prime}}-1\\}$ and $\zeta\\!=\\!\zeta^{\prime}+p^{i-1}\zeta_{i-1}\\!\in\\!\mathbb{Z}/(p^{i})$. In particular, the labels of the nodes lie in $\mathbb{Z}[x]\times\mathbb{N}$. $\diamond$ We call each $f_{i,\zeta}$ a nodal polynomial of $\mathcal{T}_{p,k}(f)$. It is in fact possible to easily read off the roots of $f$ in $\mathbb{Z}/\langle p^{k}\rangle$ from $\mathcal{T}_{p,k}(f)$ [17]. We will instead use $\mathcal{T}_{p,k}(f)$, with $k$ chosen via our root separation bounds, to efficiently count the roots of $f$ in $\mathbb{Z}_{p}$ (and thus in $\mathbb{Q}_{p}$ via rescaling). ###### Example 2.9. Let $f(x_{1})\\!=\\!1-x^{397}_{1}$. Then $\mathcal{T}_{17,k}(f)$, for any $k\\!\geq\\!1$, consists of a single node, labelled $(1-x^{397}_{1},k)$. Note that $1$ is the unique root of $f$ in $\mathbb{Q}_{17}$, $1$ is a non- degenerate root of $\tilde{f}$, and $1$ is the only root of $\tilde{f}$ in $\mathbb{F}_{17}$. $\diamond$ ###### Example 2.10. Let $f(x_{1})\\!=\\!1-x^{340}_{1}$. Then, when $k\\!\in\\!\\{1,2\\}$, the tree $\mathcal{T}_{17,k}(f)$ consists of a single root node, labelled $(1-x^{340}_{1},k)$. However, when $k\\!\geq\\!3$, the tree $\mathcal{T}_{17,k}(f)$ has depth $1$, and consists of the aforementioned root node and exactly $4$ child nodes. The child nodes are labelled $(f_{1,\zeta},k-2)$ where the $\tilde{f}_{1,\zeta}$ are, respectively, $3x_{1}$, $5x_{1}+7$, $12x_{1}+2$, and $14x+14$. Note that this $f$ has exactly $4$ roots in $\mathbb{Q}_{17}$: $1$, $4+2\cdot 17+\cdots$, $13+14\cdot 17+\cdots$, and $16+16\cdot 17+\cdots$. In particular, the most significant digits ($1$, $4$, $13$, and $16$) of these roots are exactly the roots in $\mathbb{F}_{17}$ of $\tilde{f}_{0,0}$ (the mod $17$ reduction of $f$); and the next digits ($0$, $2$, $14$, and $16$) of these roots in $\mathbb{Q}_{17}$ are exactly the roots in $\mathbb{F}_{17}$ of the nodal polynomials $f_{1,\zeta_{0}}$ of the $4$ child nodes. $\diamond$ It will be important to recall the following properties of nodal polynomials: ###### Lemma 2.11. [17, Lemmata 2.2 & 3.6] Following the preceding notation, suppose $i\\!\geq\\!1$, $\zeta_{i-1}\\!\in\\!\mathbb{F}_{p}$ is a degenerate root of $f_{i-1,\mu}$ with multiplicity $m$, and$\zeta\\!=\\!\mu+p^{i-1}\zeta_{i-1}$. Then $\mathcal{T}_{p,k}(f)$ has depth $\\!\leq\\!\left\lfloor(k-1)/2\right\rfloor$, $\deg\tilde{f}_{i,\zeta}\\!\leq\\!s(f_{i-1,\mu},\zeta_{i-1})$, and $s(f_{i-1,\mu},\zeta_{i-1})\\!\in\\!\\{2,\ldots,m\\}$. Also, $f_{i,\zeta}(x)\\!=\\!\frac{1}{p^{s}}f(\zeta_{0}+\zeta_{1}p+\cdots+\zeta_{i-1}p^{i-1}+p^{i}x)$ where $s\\!:=\\!\sum\limits^{i-1}_{j=0}s(f_{j,\zeta_{0}+\cdots+\zeta_{j-1}p^{j-1}},\zeta_{j})$. $\blacksquare$ Let $n_{p}(f)$ denote the number of non-degenerate roots in $\mathbb{F}_{p}$ of the mod $p$ reduction of $f$. ###### Lemma 2.12. If $f\\!\in\\!\mathbb{Z}[x_{1}]$, $D$ is the maximum of $\operatorname{ord}_{p}(\zeta_{1}-\zeta_{2})$ over all $\zeta_{1},\zeta_{2}\\!\in\\!\mathbb{Z}_{p}$ with $f(\zeta_{1})\\!=\\!f(\zeta_{2})\\!=\\!0\\!\neq\\!f^{\prime}(\zeta_{1})f^{\prime}(\zeta_{2})$, and $k\\!\geq\\!1+D$, then $f$ has exactly $\displaystyle{\sum\limits_{(f_{i,\zeta},k_{i,\zeta})\in\mathcal{T}_{p,k}(f)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!n_{p}(f_{i,\zeta})}$ non-degenerate roots in $\mathbb{Z}_{p}$. Proof: First note that $\tilde{f}_{i,\zeta}$ having $\zeta_{i}$ as a non- degenerate root in $\mathbb{F}_{p}$ implies that $\operatorname{ord}_{p}f^{\prime}_{i,\zeta}(\zeta_{i})\\!\neq\\!0$. By Lemma 2.11, $f_{i,\zeta}(x)\\!=\\!\frac{1}{p^{s}}f(\zeta_{0}+\cdots+\zeta_{i-1}p^{i-1}+p^{i}x)$. So by Hensel’s Lemma, $\zeta+\zeta_{i}p^{i}$ lifts to a unique non-degenerate root of $f$ in $\mathbb{Z}_{p}$. In particular, these lifted roots in $\mathbb{Z}_{p}$ are distinct because they differ somewhere in their $i+1$ most significant digits. In other words, we have shown that $\sum\limits_{(f_{i,\zeta},k_{i,\zeta})\in\mathcal{T}_{p,k}(f)}n_{p}(f_{i,\zeta})$ is a lower bound on the number of non-degenerate roots of $f$ in $\mathbb{Z}_{p}$. To see that we obtain all non-degenerate roots of $f$ in $\mathbb{Z}_{p}$ this way, simply note that any non-degenerate root $\rho$ of $f$ has a unique mod $p^{k}$ reduction, by our definition of $k$. So we are done. $\blacksquare$ ### 2.5. Trees and Solving Binomial Equations Digit by Digit The following proposition is elementary upon counting the powers of $p$ that divide the factors of the expression $\frac{d\cdot(d-1)\cdots(d-j+1)}{1\cdot 2\cdots j}$. ###### Proposition 2.13. If $p$ is an odd prime, $d\\!\in\\!\mathbb{N}$ with $d\\!\geq\\!2$, $\ell\\!:=\\!\operatorname{ord}_{p}d$, and $j\\!\in\\!\\{2,\ldots,d-1\\}$, then $j+\operatorname{ord}_{p}\binom{d}{j}\\!\geq\\!2+\ell$. $\blacksquare$ The following lemma is a useful property of $\mathcal{T}_{p,k}(f)$ when $f$ is a binomial. ###### Lemma 2.14. Suppose $p$ is an odd prime, $f(x_{1})\\!=\\!c_{1}+c_{2}x^{d}\\!\in\\!\mathbb{Z}[x_{1}]$ with $p\nmid c_{1}c_{2}$, and $\ell\\!:=\\!\operatorname{ord}_{p}d$. Then $k\\!\geq\\!\ell+2$ implies that $\mathcal{T}_{p,k}(f)$ has depth $0$ or $1$, according as $\ell$ is $0$ or not. Proof: Note that any root $\zeta_{0}\\!\in\\!\mathbb{F}_{p}$ of $\tilde{f}$ must be nonzero. So$\operatorname{ord}_{p}\frac{f^{(j)}(\zeta_{0})}{j!}=\operatorname{ord}_{p}\\!\left(\binom{d}{j}c_{2}\zeta_{0}^{d-j}\right)=\operatorname{ord}_{p}\binom{d}{j}$. We then obtain by Proposition 2.13 that $s(f,\zeta_{0})\\!=\\!\min\\{\operatorname{ord}_{p}f(\zeta_{0}),1+\ell\\}\\!\leq\\!k-1$, and thus $\tilde{f}_{1,\zeta_{0}}(x_{1})\\!=\\!\frac{f(\zeta_{0})}{p^{s(f,\zeta_{0})}}+\frac{c_{2}d\zeta^{d-1}_{0}}{p^{s(f,\zeta_{0})}}x_{1}$. So then, no $\tilde{f}_{1,\zeta_{0}}$ can have a degenerate root and we are done. $\blacksquare$ With our tree-based encoding of $p$-adic roots in place, we can now prove that it is easy to find approximate roots in $\mathbb{Q}_{p}$ for binomials when $p$ is fixed. ###### Theorem 2.15. Suppose $f\\!\in\\!\mathbb{Z}[x_{1}]$ is a binomial of degree $d$ with coefficients of absolute value at most $H$, $f(0)\\!\neq\\!0$, $\gamma\\!=\\!\gcd(d,p-1)$, and $\\{\zeta_{1},\ldots,\zeta_{\gamma}\\}$ is the set of roots of $f$ in $\mathbb{Q}_{p}$. Then in time $O\\!\left(p\log^{2}(p)\log\gamma+\log^{3}(pdH)\gamma\right)$, we can find, for all $j\\!\in\\!\\{1,\ldots,\gamma\\}$, a $z_{0}\\!\in\\!\mathbb{Q}$ of logarithmic height $O\\!\left(\log\left(dH^{1/d}\right)\right)$ that is an approximate root of $f$ with associated true root $\zeta_{j}$. An algorithm that proves Theorem 2.15, and more, can be outlined as follows: ###### Algorithm 2.16. (Solving Binomial Equations Over $\boldsymbol{\mathbb{Q}_{p}}$) Input. An odd prime $p$ and $c_{1},c_{2},d\\!\in\\!\mathbb{Z}\setminus\\{0\\}$ with $|c_{i}|\\!\leq\\!H$ for all $i$. Output. A true declaration that $f(x_{1})\\!:=\\!c_{1}+c_{2}x^{d}$ has no roots in $\mathbb{Q}_{p}$, or $z_{1},$$\ldots,z_{\gamma}\\!\in\\!\mathbb{Q}$ with logarithmic height $O\\!\left(\log\left(dH^{1/d}\right)\right)$ such that $\gamma\\!=\\!\gcd(d,p-1)$, $z_{j}$ is an approximate root of $f$ with associated true root $\zeta_{j}\\!\in\\!\mathbb{Q}_{p}$ for all $j$, and the $\zeta_{j}$ are pair-wise distinct. Description. 1: If $\operatorname{ord}_{p}c_{1}\\!\neq\\!\operatorname{ord}_{p}c_{2}$ mod $d$ then say ‘‘No roots in $\mathbb{Q}_{p}$!’’ and STOP. 2: Rescale roots so that we may replace $c_{i}$ by $\frac{c_{i}}{p^{\operatorname{ord}_{p}c_{i}}}$ for all $i$. 3: Invert roots if necessary so we may replace $d$ by $|d|$. 4: Let $\ell\\!:=\\!\operatorname{ord}_{p}d$. 5: If $\left(-\frac{c_{1}}{c_{2}}\right)^{p^{\ell}(p-1)/\gamma}\\!\neq\\!1$ mod $p^{2\ell+1}$ then say ‘‘No roots in $\mathbb{Q}_{p}$!’’ and STOP. 6: Let $r\\!:=\\!(d/\gamma)^{-1}$ mod $\frac{p-1}{\gamma}$, $c\\!:=\\!(-c_{1}/c_{2})^{r}$ mod $p$, and let $\tilde{h}(x_{1})\\!:=\\!x^{\gamma}_{1}-c$. 7: Find a root $x^{\prime}\\!\in\\!\\{1,\ldots,p-1\\}$ of $\tilde{h}$ via brute-force search and then let $x_{1}\\!:=\\!(x^{\prime})^{d/\gamma}$ mod $p$. 8: For all $j\\!\in\\!\\{2,\ldots,\gamma\\}$ let $x_{j}\\!:=\\!x_{j-1}g^{(p-1)/\gamma}$ mod $p$. 9: If $\ell\\!\geq\\!1$ then, for each $j\\!\in\\!\\{1,\ldots,\gamma\\}$, replace $x_{j}$ by $x_{j}-\frac{f(x_{j})/p^{\ell}}{f^{\prime}(x_{j})/p^{\ell}}\\!\in\\!\mathbb{Z}/\langle p^{2\ell+1}\rangle$. 10: Output $\left\\{x_{1}p^{\operatorname{ord}_{p}(c_{2}/c_{1})/d},\ldots,x_{\gamma}p^{\operatorname{ord}_{p}(c_{2}/c_{1})/d}\right\\}$. ###### Remark 2.17. While we have strived for simplicity in Algorithm 2.16, its complexity is obviously super-linear in $p$, vis à vis Step 7. However, since our main goal is to prove polynomial-time deterministic complexity for fixed $p$, the dependence on $p$ does not matter for our theoretical purposes. If one allows randomization then there are techniques that can considerably improve the practical complexity (see, e.g., [3, Ch. 7, Sec. 3]). $\diamond$ Proof of Theorem 2.15: It clearly suffices to prove the correctness of Algorithm 2.16, and then to establish a sufficiently good complexity bound. Correctness: Theorem 2.2 implies that Step 1 merely checks whether the valuations of the roots of $f$ in $\mathbb{C}_{p}$ in fact lie in $\mathbb{Z}$, which is necessary for $f$ to have roots in $\mathbb{Q}_{p}$. Steps 2 and 3 merely involve substitutions of the form $x_{1}\leftarrow p^{\delta}x_{1}$ and $x_{1}\leftarrow x^{-1}_{1}$, and it is elementary to check that the bit length of any resulting approximate roots in $\mathbb{Q}$ remains within $O\\!\left(\log\left(dH^{1/d}\right)\right)$. Lemma 2.4 implies that Steps 4–5 merely check that the coset of roots of $f$ in $\mathbb{C}_{p}$ intersects $\mathbb{Z}_{p}$. Step 6 is merely the application of an automorphism of $\mathbb{F}^{*}_{p}$ (that preserves the existence of roots of $\tilde{f}$ in $\mathbb{F}^{*}_{p}$) that enables us to work with a binomial of degree $\gamma$ ($<\\!d$). Steps 7–10 then clearly find the correct coset of $\mathbb{F}^{*}_{p}$ that makes $f$ vanish. In particular, since $\mathbb{F}^{*}_{p}$ is cyclic, Step 9 clearly gives the correct output if $\ell\\!=\\!0$. If $\ell\\!>\\!0$ then each nodal polynomial $f_{1,\zeta_{0}}$ is of degree $\leq\\!1$ (thanks to Lemma 2.14) and, by Definition 2.7, its unique root in $\mathbb{Z}/\langle p^{2\ell+1-s}\rangle$ determines the next $2\ell+1-s$ base-$p$ digits of a unique root of $f$ in $\mathbb{Z}_{p}$. (See also the proof of [17, Lemma 1.5].) So Steps 8–10 indeed give us suitable approximants in $\mathbb{Q}$ to all the roots in $\mathbb{Z}_{p}$ of (our renormalized) $f$ in $\mathbb{Z}_{p}$. So our algorithm is correct. Note also that the outputs, being numbers in $\mathbb{Z}/\langle p^{2\ell+1}\rangle$, rescaled by a factor of $p^{\operatorname{ord}_{p}(c_{2}/c_{1})/d}$, clearly each have bit-length $O\\!\left(\operatorname{ord}_{p}(d)\log(p)+\frac{|\log(c_{2}/c_{1})|}{d\log p}\log p\right)\\!=\\!O(\log(d)+\frac{\log H}{d})\\!=\\!O(\log(dH^{1/d}))$. $\blacksquare$ Complexity Analysis: Via Corollary 2.6, and some additional elementary bit complexity estimates for modular arithmetic, it is clear that, save for Steps 7–10, Algorithm 2.16 has complexity $O(\gamma\log^{3}(pdH))$. Evaluating a $\gamma$th power mod $p$ takes time $O(\log(\gamma)\log^{2}p)$ via recursive squaring (using grade-school multiplication). Step 7 (whose complexity dominates that of Steps 8–10), consists of no more than $p-1$ evaluations of a $\gamma$th power mod $p$. This takes time $O\\!\left(p\log^{2}(p)\log\gamma\right)$ so we are done. $\blacksquare$ ## 3\. Separating Roots of Trinomials in $\mathbb{Q}_{p}$: Proving Theorem 1.5 In the last section, we saw that we could pick a moderately-sized $\ell$ and then lift roots of a binomial in $\mathbb{Z}/\langle p^{2\ell+1}\rangle$ to roots in $\mathbb{Z}_{p}$. The same strategy will work for trinomials, but it will be much harder to prove that a moderately-sized $\ell$ exists. Toward this end, let us first recall the following version of Yu’s Theorem: ###### Theorem 3.1. [32, Cor. 1] Suppose $p$ is any prime; let $\alpha_{1},\ldots,\alpha_{n}$ be $n$ ($\geq\\!2$) nonzero rational numbers, with $\alpha_{i}=r_{i}/s_{i}$ the reduced fraction for each $i$; and $b_{1},\cdots,b_{n}$ are integers not all zero. Then $\alpha_{1}^{b_{1}}\cdots\alpha_{n}^{b_{n}}\neq 1$ implies that $\alpha_{1}^{b_{1}}\cdots\alpha_{n}^{b_{n}}-1$ has $p$-adic valuation bounded from above by $11145\left(\frac{24(n+1)^{2}}{\log p}\right)^{n+2}(p-1)\left(\prod^{n}_{i=1}\log A_{i}\right)\log(4B)\\\ \max\left\\{\log(2^{12}\cdot 3n(n+1)\log A_{n}),\frac{\log p}{n}\right\\},$ where $B\\!:=\\!\max\\{|b_{1}|,\ldots,|b_{n}|,3\\}$, and $A_{1},\ldots,A_{n}$ are real numbers such that$A_{1}\leq\cdots\leq A_{n}$ and, for each $j$, $A_{j}\geq\max\\{|r_{j}|,|s_{j}|,p\\}$. $\blacksquare$ In order to prove our separation bound for roots of trinomials, we will follow three steps for a given trinomial $f$: 1. 1. Use Yu’s Theorem on linear forms in $p$-adic logarithms to prove that for $m\in\mathbb{C}_{p}$ such that $f^{\prime}(m)=0$, $\left|f(m)\right|_{p}$ cannot be too small. 2. 2. Show that $\left|f^{\prime}(m)\right|_{p}$ cannot be too large for $\left|m\right|_{p}\leq 1$, since $f(0)\neq 0$ by construction. 3. 3. Use a $p$-adic adaption of Rolle’s Theorem to obtain our bound. To expand on this approach, first note that by a simple computation, we obtain the following useful formula: ###### Proposition 3.2. Fix a prime $p$ and consider a trinomial$f(x_{1})=c_{1}+c_{2}x^{a_{2}}_{1}+c_{3}x^{a_{3}}_{1}\\!\in\\!\mathbb{Z}_{p}[x_{1}]$ where $a_{3}\\!>\\!a_{2}\\!\geq\\!1$. If $f^{\prime}(m)=0$ at some point $m\in\mathbb{C}_{p}$, then $m^{a_{3}-a_{2}}=-\frac{a_{2}c_{2}}{a_{3}c_{3}}$. Moreover, $f(m)=c_{1}+c_{2}m^{a_{2}}\left(1-\frac{a_{2}}{a_{3}}\right)$. $\blacksquare$ We want to show that if $m$ is as in Proposition 3.2 then $\left|f(m)\right|_{p}$ cannot be too small if it is nonzero. ###### Lemma 3.3. Let $f(x_{1})=c_{1}+c_{2}x^{a_{2}}_{1}+c_{3}x^{a_{3}}_{1}\\!\in\\!\mathbb{Z}_{p}[x_{1}]$ be a square-free trinomial and $s$ the input size of $f$. If $f^{\prime}(m)=0$ at some point $m\in\mathbb{C}_{p}$ then$\left|f(m)\right|_{p}\geq\exp(-O(ph_{p}^{4}/(\log p)^{3}))$, for $h_{p}=\max\\{s,\log p\\}$. Proof: By Proposition 3.2, $\displaystyle\operatorname{ord}_{p}(f(m))$ $\displaystyle=\operatorname{ord}_{p}(c_{1}+bm^{a}_{2}(1-a_{2}/a_{3}))$ (*) $\displaystyle=\operatorname{ord}_{p}(c_{1})+\operatorname{ord}_{p}\left(\frac{-c_{2}(a_{3}-a_{2})}{c_{1}a_{3}}\left(-\frac{a_{2}c_{2}}{a_{3}c_{3}}\right)^{a_{2}/(a_{3}-a_{2})}-1\right)$ Clearly $\operatorname{ord}_{p}(c_{1})\leq s/\log p$. In order to bound the second summand of (* ‣ 3), we claim that it suffices to bound $\displaystyle M:=\operatorname{ord}_{p}\left(\left(\frac{-c_{2}(a_{3}-a_{2})}{c_{1}a_{3}}\right)^{a_{3}-a_{2}}\left(-\frac{a_{2}c_{2}}{a_{3}c_{3}}\right)^{a_{2}}-1\right).$ Let $\alpha_{1}=-c_{2}\frac{a_{3}-a_{2}}{c_{1}a_{3}}$, $\alpha_{2}=-\frac{a_{2}c_{2}}{a_{3}c_{3}}$, $b_{1}=a_{3}-a_{2}$, and $b_{2}=a_{2}$. To apply Yu’s Theorem on $M=\operatorname{ord}_{p}(\alpha_{1}^{b_{1}}\alpha_{2}^{b_{2}}-1)$, it suffices to take $n\\!=\\!2$, $A_{i}=\max\\{e^{2s},p\\}$ for $i\\!\in\\!\\{1,2\\}$, and $B=\max\\{e^{s},3\\}$. Then $\log A_{1},\log A_{2},\log B=O(h_{p})$. Thus $\displaystyle\operatorname{ord}_{p}(f(m))\leq s/\log p+M\leq s/\log p+C_{2}ph_{p}^{4}/(\log p)^{4},$ for some absolute constant $C_{2}$. So then $\left|f(m)\right|_{p}\geq p^{-\operatorname{ord}_{p}(f(m))}$$=\\!\exp(-s-C_{2}ph_{p}^{4}/(\log p)^{3})\\!\geq\\!\exp(-C_{1}ph_{p}^{4}/(\log p)^{3})$, by picking $C_{1}\geq C_{2}(1+\frac{s}{M})$. Now to prove the claim first observe that $x^{a_{3}-a_{2}}-1=\prod_{i=1}^{a_{3}-a_{2}}(x-\zeta^{j})$ for $\zeta\in\mathbb{C}_{p}$ a root of unity of order $a_{3}-a_{2}$. Taking $T=\frac{c_{2}(a_{3}-a_{2})}{c_{1}a_{3}}\left(-\frac{a_{2}c_{2}}{c_{3}a_{3}}\right)^{a_{2}/(a_{3}-a_{2})}$, we get $M=\operatorname{ord}_{p}(T^{a_{3}-a_{2}}-1)=\sum_{j=1}^{a_{3}-a_{2}}\operatorname{ord}_{p}(T-\zeta^{j})$. The the second summand of (* ‣ 3) is exactly $\operatorname{ord}_{p}(T-\zeta^{a_{3}-a_{2}})$. Suppose $\operatorname{ord}_{p}T<0$. Then for each $i\\!\in\\!\\{1,\ldots,a_{3}-a_{2}\\}$ we have $\operatorname{ord}_{p}(T-\zeta^{j})=\operatorname{ord}_{p}T<0$. This is because roots of unity always have $p$-adic valuation $0$ regardless of the choice of $p$. Then we must have $\operatorname{ord}_{p}(f(m))\leq\operatorname{ord}_{p}(a)+\operatorname{ord}_{p}(T-\zeta^{a_{3}-a_{2}})\leq s/\log p$, and the conclusion of the lemma is immediate. Now suppose $\operatorname{ord}_{p}T\geq 0$. Then for each $j$, $\operatorname{ord}_{p}(T-\zeta^{j})\geq j\operatorname{ord}_{p}(\zeta)=0$, and $M\geq\operatorname{ord}_{p}(T-\zeta^{j})$, thus proving the claim. $\blacksquare$ Thanks to the ultrametric inequality we can effectively bound $\left|f^{\prime}(x)\right|_{p}$ for $\left|x\right|_{p}\leq 1$ with $f^{\prime}(x)\neq 0$. ###### Lemma 3.4. Following the notation above, assume $f$ is a square-free trinomial. Then $\left|x\right|_{p}\leq 1\Longrightarrow\left|f^{\prime}(x)\right|_{p}\leq 1$. Proof: As $c_{2},c_{3}\\!\in\\!\mathbb{Z}_{p}$, $a_{2},a_{3}\\!\in\\!\mathbb{Z}$, $a_{3}\\!>\\!a_{2}\\!\geq\\!1$, and $\left|x\right|_{p}\leq 1$, we have $\displaystyle\operatorname{ord}_{p}(c_{2}a_{2}x^{a_{2}-1})=\operatorname{ord}_{p}(c_{2})+\operatorname{ord}_{p}(a_{2})+(a_{2}-1)\operatorname{ord}_{p}(x)\geq 0.$ Similarly, $\operatorname{ord}_{p}(c_{3}a_{3}x^{a_{3}-1})\geq 0$. Then simply observe that: $\displaystyle\left|f^{\prime}(x)\right|_{p}=\left|c_{2}a_{2}x^{a_{2}-1}+c_{3}a_{3}x^{a_{3}-1}\right|_{p}\leq\max\left\\{\left|c_{2}a_{2}m^{a_{2}-1}\right|_{p},\left|c_{3}a_{3}x^{a_{3}-1}\right|_{p}\right\\}\leq 1.\text{ $\blacksquare$}$ To conclude our approach we state the following scaled version of $p$-adic Rolle’s Theorem, which follows immediately from [23, Thm., pg. 316, Sec. 2.4]. ###### Theorem 3.5. Let $f\in\mathbb{C}_{p}[x]$ be a polynomial with two distinct roots $x_{1},x_{2}\in\mathbb{C}_{p}$, having $\left|x_{1}-x_{2}\right|_{p}=cp^{1/(p-1)}$ for some constant $c>0$. Then its derivative $f^{\prime}$ has a root $m$ within $p$-adic distance $c$ of both $x_{1}$ and $x_{2}$. $\blacksquare$ We now prove our third and final main result. Proof of Theorem 1.5: Suppose $x_{1},x_{2}$ are two distinct roots of$f(x)=a+bx^{a_{2}}+cx^{a_{3}}$. In particular, as we assume the constant term is nonzero, both the roots $x_{1},x_{2}$ are nonzero. Then we proceed by cases. Case 1: (Both roots are small: $\boldsymbol{\left|x_{1}\right|_{p},\left|x_{2}\right|_{p}\leq 1}$.) Suppose $\left|x_{1}-x_{2}\right|_{p}>p^{-2/(p-1)}$. Then $\left|x_{1}-x_{2}\right|_{p}>\exp(-2\log p/(p-1))$$>\exp(-Cph_{p}^{4}/(\log p)^{3})$ for large $C$. Now assume that $\left|x_{1}-x_{2}\right|_{p}\leq p^{-2/(p-1)}$. Then by Theorem 3.5 there exists $m\in\mathbb{C}_{p}$ such that $\left|m\right|_{p}\leq p^{-1/(p-1)}$ and $f^{\prime}(m)=0$. Moreover, $\left|x_{i}-m\right|_{p}\leq p^{1/(p-1)}\left|x_{1}-x_{2}\right|_{p}\leq p^{-1/(p-1)}$ for $i\\!\in\\!\\{1,2\\}$. Without loss of generality, assume that $i=1$. So it suffices bound $\left|m-x_{1}\right|_{p}$ from below. As we assume $f$ is square-free, $f(m)\neq 0$. Then by Lemma 3.3, $\left|f(m)\right|_{p}\geq\exp(-C_{1}ph_{p}^{4}/(\log p)^{3})$, for $h_{p}=\max\\{s,\log p\\}$ and some constant $C_{1}$. By applying Theorem 3.5 again, we see that there exists a $\zeta$ with $\left|\zeta\right|_{p}\leq 1$ and such that $\displaystyle f(m)=f(m)-f(x_{1})=f^{\prime}(\zeta)(m-x_{1}).$ As $f(m)\neq 0$, then $f^{\prime}(\zeta)\neq 0$ and $m\neq x_{1}$. Recall from Lemma 3.4 that $\left|f^{\prime}(\zeta)\right|_{p}\leq 1$, so then $\left|m-x_{1}\right|_{p}=\frac{\left|f(m)\right|_{p}}{\left|f^{\prime}(\zeta)\right|_{p}}\geq\frac{\exp(-C_{1}ph_{p}^{4}/(\log p)^{3})}{1}=\exp(-C_{1}ph_{p}^{4}/(\log p)^{3})$. This in turn implies $\left|x_{1}-x_{2}\right|_{p}\geq p^{-1/(p-1)}\left|m-x_{1}\right|_{p}\geq\exp\\!\left(-C_{1}ph_{p}^{4}/\log^{3}(p)-\frac{1}{p-1}\log p\right)\geq\exp(-C_{1}ph_{p}^{4}/\log^{3}p-1)$. By picking an appropriate $C$ depending on $C_{1}$, the conclusion follows in this case. Case 2: (Both roots are large: $\boldsymbol{\left|x_{1}\right|_{p},\left|x_{2}\right|_{p}>1}$.) In this case, consider $g(x)=x^{a_{3}}f(\frac{1}{x})=ax^{a_{3}}+bx^{a_{3}-a_{2}}+c$. Correspondingly, $\frac{1}{x_{1}},\frac{1}{x_{2}}$ are the roots of $g$ with $\left|\frac{1}{x_{1}}\right|_{p},\left|\frac{1}{x_{2}}\right|_{p}<1$. By using the same argument in Case 1, $\left|\frac{1}{x_{1}}-\frac{1}{x_{2}}\right|_{p}\geq\exp(-C^{\prime}ph_{p}^{4}/(\log p)^{3})$ for some absolute constant $C^{\prime}$. Hence $\displaystyle\left|x_{1}-x_{2}\right|_{p}=\left|x_{1}\right|_{p}\left|x_{2}\right|_{p}\left|\frac{1}{x_{1}}-\frac{1}{x_{2}}\right|_{p}\geq\exp(-Cph_{p}^{4}/(\log p)^{3})$ by picking $C=C^{\prime}$. Case 3: (Only one root has norm $\boldsymbol{>1}$.) Without loss of generality, we may assume that $|x_{1}|_{p}\\!\leq\\!1\\!<\\!|x_{2}|_{p}$. We then simply note that, as $\left|x_{1}\right|_{p}\neq\left|x_{2}\right|_{p}$, we have $\left|x_{1}-x_{2}\right|_{p}=\max\left\\{\left|x_{1}\right|_{p},\left|x_{2}\right|_{p}\right\\}\\!>\\!1$ and we are done. $\blacksquare$ ## 4\. Tetranomials: The Proof of Theorem 1.4 ### 4.1. The Case of Prime $\boldsymbol{p}$ We will see that $f(x_{1}):=f_{d,p}(x_{1})=x^{d}_{1}-\left(\frac{x_{1}}{p^{h}}-\frac{1}{p}\right)^{2}$ has two roots $x_{1},x_{2}\in\mathbb{Q}_{p}$ such that $\left|x_{1}-x_{2}\right|_{p}<p^{-\Omega(dh)}$. Let $g(x)=p^{2h}f(x+p^{h-1})=p^{2h}(x+p^{h-1})^{d}-p^{2h}\left(\frac{x+p^{h-1}}{p^{h}}-\frac{1}{p}\right)^{2}$$=p^{2h}(x+p^{h-1})^{d}-x^{2}$. Then $g$ has the same roots as $f$, save for a “small” shift by $p^{h-1}$. Rescaling, we get: $\displaystyle G(x)$ $\displaystyle:=\frac{g(p^{(h-1)d/2+h}x)}{p^{(h-1)d+2h}}$ $\displaystyle=p^{-(h-1)d-2h}\left[p^{2h}(p^{(h-1)d/2+h}x+p^{h-1})^{d}-p^{(h-1)d+2h}x^{2}\right]$ $\displaystyle=\sum_{i=0}^{d}{d\choose i}p^{(h-1)(di/2-i)+ih}x^{i}-x^{2}$ $\displaystyle=1-x^{2}\mod p^{d(h-1)/2+1},$ which is square-free for odd prime $p$. When $p\\!=\\!2$, as $h\\!>\\!2$, we have $p^{d(h-1)/2+1}\geq 8$. Then, as $G(x)=1-x^{2}=(3-x)(5-x)\mod p^{8}$, we obtain that $G$ is also square-free. Hensel’s Lemma then implies that there are roots $\zeta_{1},\zeta_{2}\in\mathbb{Z}_{p}$ of $G$ such that $\zeta_{1}\equiv 1\mod p^{n(h-1)/2+1}$ and $\zeta_{2}\equiv-1\mod p^{n(n-1)/2+1}$. Moreover, $\left|\zeta_{1}\right|_{p}=\left|\zeta_{2}\right|_{p}=1$. For each $i\\!\in\\!\\{1,2\\}$, $y_{i}=p^{(h-1)n/2+h}\zeta_{i}$ is the corresponding root of $G$, and thus of $g$. Then $x_{1}=y_{1}+p^{h-1}$ and $x_{2}=y_{2}+p^{h-1}$ are two roots of $f$ in $\mathbb{Z}_{p}$ such that $\displaystyle\left|x_{1}-x_{2}\right|_{p}$ $\displaystyle=\left|(y_{1}+p^{h-1})-(y_{2}+p^{h-1})\right|_{p}$ $\displaystyle=\left|y_{1}-y_{2}\right|_{p}\leq\max\left\\{\left|y_{1}\right|_{p},\left|y_{2}\right|_{p}\right\\}=p^{-(h-1)d/2-h}=p^{-\Omega(dh)}.\text{ $\blacksquare$}$ ### 4.2. The Case $\boldsymbol{p\\!=\\!\infty}$ Shifting by $\frac{1}{2^{h-1}}$, we get: $\displaystyle g(x)$ $\displaystyle:=f_{d,\frac{1}{2}}(x+2^{1-h})=(x+2^{1-h})^{d}-2^{2h}x^{2}$ $\displaystyle=2^{d(1-h)}+d2^{(d-1)(1-h)}x+\left({d\choose 2}2^{(d-2)(1-h)}-2^{2h}\right)x^{2}$ $\displaystyle+{d\choose 3}2^{(d-3)(1-h)}x^{3}+\cdots+x^{d}.$ When computing $\operatorname{Newt}_{\infty}(g)$, the three lowest order terms of $g$ contribute the points $\displaystyle p_{0}=(0,d(h-1)\log 2),$ $\displaystyle p_{1}=(1,(d-1)(h-1)\log 2-\log d),\text{ and }$ $\displaystyle p_{2}=\left(2,-\log\left(4^{h}-\frac{{d\choose 2}}{2^{(d-2)(h-1)}}\right)\right)$ as potential vertices of $\operatorname{Newt}_{\infty}(f)$. In particular, observe that $\frac{{d\choose 2}}{2^{(d-2)(h-1)}}\\!<\\!0.059$ for all $h\\!\geq\\!3$ and $d\\!\geq\\!4$, and thus $p_{2}$ is the only point of $\operatorname{Newt}_{\infty}(f)$ with negative $y$-coordinate. So $p_{2}$ is a vertex of $\operatorname{Newt}_{\infty}(f)$, and all edges with vertices to the right of $p_{2}$ have positive slope. Furthermore, the slopes of the line segments $\overline{p_{0}p_{1}}$ and $\overline{p_{0}p_{2}}$ are respectively $-(h-1)\log(2)-\log d$ and a number less than $-\frac{1}{2}\log(4^{h}-0.059)-\frac{1}{2}d(h-1)\log 2$. Since $2^{h-1}\\!<\\!\sqrt{4^{h}-0.059}$ and $\log d\\!<\\!\frac{1}{2}d(h-1)\log 2$ for all $d\\!\geq\\!4$ and $h\\!\geq\\!3$, we thus see that the slope of $\overline{p_{0}p_{2}}$ is more negative. So the leftmost lower edge of $\operatorname{Newt}_{\infty}(g)$ has vertices $p_{0}$ and $p_{2}$. It is easily checked that the slope of this edge is less than $-10.3$, which is in turn clearly $<\\!-2\log 3$. So by Theorem 2.2, there are two roots $z_{1},z_{2}$ of $g$ such that $\displaystyle\log|z_{i}|\leq\frac{1}{2}\left[-\log\left(2^{2h}-{d\choose 2}2^{(d-2)(1-h)}\right)-d(h-1)\log 2\right].$ These two roots thus satisfy $|z_{i}|=2^{-\Omega(dh)}$. Now, for $i\\!\in\\!\\{1,2\\}$, $\zeta_{i}=z_{i}+2^{1-h}$ yields roots of $f_{d,\frac{1}{2}}$ with $|\zeta_{1}-\zeta_{2}|=|z_{1}+2^{1-h}-(z_{2}+2^{1-h})|\leq|z_{1}|+|z_{2}|<2^{-\Omega(dh)}$. $\blacksquare$ ## 5\. Solving Trinomial Equations over $\mathbb{Q}_{p}$ Unlike the binomial case, the tree $\mathcal{T}_{p,k}(f)$ can have high depth for $f$ an arbitrary trinomial. However, its structure will nevertheless be simple: For $k$ sufficiently large, $\mathcal{T}_{p,k}(f)$ will be a subtree of the union of $L$ chains of length $\left\lfloor(k-1)/2\right\rfloor$ emanating from a single root node. The number $L$ will be the number of degenerate roots in $\mathbb{F}_{p}$ of $\tilde{f}$, and will turn out to be congruent to $0$ or $1$ mod $p-1$. The lower limit for $k$ to be “large” will be unwieldy, particulary as a function of $p$, but the lower limit will be small enough for us to obtain reasonable dependence on $d$ and $H$: We will attain complexity $p^{2+o(1)}\log^{9}(dH)$ in our algorithm. We begin with an earlier result, also derived via Yu’s Theorem: ###### Theorem 5.1. [24, Sec. 5] If $f(x_{1})\\!=\\!c_{1}+c_{2}x^{a_{2}}_{1}+c_{3}x^{a_{3}}_{1}\\!\in\\!\mathbb{Z}[x_{1}]$ is a trinomial of degree $d\\!=\\!a_{3}\\!>\\!a_{2}\\!\geq\\!1$, with coefficients of absolute value at most $H$, $p$ is any prime, $p\nmid c_{1}c_{2}$, and $p|c_{3}$, then the sum of $\operatorname{ord}_{p}f^{\prime}(\zeta)$ over all roots $\zeta\\!\in\\!\mathbb{C}_{p}$ of valuation $0$ is $O(p\log^{8}(dH))$. $\blacksquare$ We point out out that while only the case $\gcd(a_{2},a_{3})\\!=\\!1$ is derived in [24, Sec. 5], the general case follows immediately from the $\gcd(a_{2},a_{3})\\!=\\!1$ case by an application of the chain rule. It is also worth noting that while certain special cases (such as when $a_{2}$, $a_{3}-a_{2}$, and $a_{3}$ are all relatively prime to $p-1$) admit better bounds like $\log^{O(1)}(pdH)$, there are several other vexing cases where there is no obvious technique other than to apply Yu’s Theorem. ###### Lemma 5.2. Following the notation and assumptions of Theorem 5.1, suppose $D$ is the maximum of $\operatorname{ord}_{p}f^{\prime}(\zeta)$ over all roots $\zeta\\!\in\\!\mathbb{Z}_{p}$ of valuation $0$, $\tilde{f}$ has at least one degenerate root in $\mathbb{F}_{p}$, and $k\\!\geq\\!2D+1$. Then $\mathcal{T}_{p,k}(f)$ has depth $\geq\\!1$ and is a union of $L$ chains of length $\leq\\!\left\lfloor(k-1)/2\right\rfloor$, where $L$ is the number of degenerate roots of $\tilde{f}$. Moreover, the degree of any non-root nodal polynomial is at most $3$. ###### Example 5.3. Suppose $f(x_{1})=x^{10}_{1}-10x_{1}+738$ and $p\\!=\\!3$. Then $\mathcal{T}_{3,7}(f)$ is a chain with depth $2$. In particular, $f_{0,0}\\!:=\\!f$ and $\tilde{f}_{0,0}(x_{1})\\!=\\!x_{1}(x_{1}-1)^{9}$. So $1$ is a degenerate root of $\tilde{f}_{0,0}$ and $s(f_{0,0},1)\\!=\\!4$. So then $f_{1,1}(x_{1})\\!=\\!21x^{4}_{1}+13x^{3}_{1}+5x^{2}_{1}+9$ mod $27$ and $\tilde{f}_{1,1}(x_{1})\\!=x^{2}_{1}(x_{1}-1)$. We then see that $0$ is a degenerate root of $\tilde{f}_{1,1}$ and $s(f_{1,1},0)\\!=\\!2$. So $f_{2,1}(x_{1})\\!=\\!2(x_{1}-1)(x_{1}-2)$ mod $3$. There are a total of $4$ non-degenerate roots for the nodal polynomials: $1$ for $f_{0,0}$, $1$ for $f_{1,1}$, and $2$ for $f_{2,1}$. These non-degenerate roots in $\mathbb{F}_{3}$ then lift to the following roots of $f$ in $\mathbb{Z}_{3}$: $0+O(3^{1})$, $1+1\cdot 3+O(3^{2})$, $1+0\cdot 3+1\cdot 3^{2}+O(3^{3})$, and $1+0\cdot 3+2\cdot 3^{2}+O(3^{3})$. A quick calculation via Maple’s rootp command tells us that these are all the $3$-adic rational roots of $f$. As far as we are aware, this is the first example of a trinomial with $4$ roots in $\mathbb{Q}_{3}$, each with most significant digit $1$. Via a more careful look at $\mathcal{T}_{3,k}(f)$ for trinomial $f$, one can in fact prove that no trinomial in $\mathbb{Z}[x_{1}]$ has more than $4$ roots in $\mathbb{Q}_{3}$ with most significant digit $1$ (see author Zhu’s Ph.D. thesis). For $p\\!\geq\\!5$ the optimal upper bound is $3$ [2], while for $p\\!=\\!2$, the optimal upper bound is $6$ [18]. $\diamond$ ###### Remark 5.4. It is worth noting that, over a finite field, trinomials can only vanish on a small number of cosets in $\mathbb{F}_{q}$: Building on earlier results from [8, 6, 13], Kelley and Owen have proved [14, Thm. 1.2] that $c_{1}+c_{2}x^{a_{2}}_{1}+c_{3}x^{a_{3}}_{1}\\!\in\\!\mathbb{F}_{q}[x_{1}]$, with $q$ a prime power, vanishes at no more than $\left\lfloor\frac{1}{2}+\sqrt{\frac{q-1}{\delta}}\right\rfloor$ cosets of the size $\delta$ subgroup of $\mathbb{F}^{*}_{q}$, where $\delta\\!=\\!\gcd(a_{2},a_{3},q-1)$. In particular, this bound is optimal for $\mathbb{F}_{q}$ an even degree extension of a prime field. For $q$ prime, there is computational evidence (for all $q\leq\\!292,837$) that the number of such cosets might in fact just be $O(\log q)$ [10]. $\diamond$ Proof of Lemma 5.2: First note that Lemma 2.11 implies that $k\\!\geq\\!2D+1$ guarantees that $s(f_{0,0},\zeta_{0})\\!\leq\\!k-1$, so there is a child node corresponding to each degenerate root of $\tilde{f}_{0,0}$. Now suppose $\zeta_{0}$ is a nonzero degenerate root of $\tilde{f}$ of multiplicity $m\geq 2$, so $\zeta_{0}\in\\{1,\ldots,p-1\\}$. Let $r:=\min_{0\leq i\leq m}\\{\operatorname{ord}_{p}f^{(i)}(\zeta_{0})\\}$. Consider the matrix (1) $A_{m}:=\begin{bmatrix}1&1&1\\\ 0&a_{2}&a_{3}\\\ 0&a_{2}(a_{2}-1)&a_{3}(a_{3}-1)\\\ \vdots&\vdots&\vdots\\\ 0&a_{2}\cdots(a_{2}-m+2)&a_{3}\cdots(a_{3}-m+2)\end{bmatrix}\mod p.$ Then $[c_{1},c_{2}\zeta^{a_{2}}_{0},c_{3}\zeta^{a_{3}}_{0}]^{T}$ must be a right null vector of $A_{m}$ modulo $p$ and$A_{m}\begin{bmatrix}c_{1}\\\ c_{2}\zeta_{0}^{a_{2}}\\\ c_{3}\zeta_{0}^{a_{3}}\end{bmatrix}=0\mod p^{r}$. For each $i\geq 3$, $\operatorname{ord}_{p}f^{(i-1)}(\zeta_{0})\geq r$ if and only if $\operatorname{ord}_{p}(c_{3}\cdot a_{3}(a_{3}-a_{2})C_{i})\geq r$, where $C_{m}$ is such that $[0,0,a_{3}(a_{3}-a_{2})C_{m}]^{T}$ lies in the row space of $A_{m}$. (It is easily checked that $C_{3}=1$.) So $\min_{2\leq i\leq m}\operatorname{ord}_{p}(f^{(i)}(\zeta_{0}))=\operatorname{ord}_{p}f^{\prime\prime}(\zeta_{0})$, and for $i,p\geq 3$ we have $\operatorname{ord}_{p}\frac{f^{\prime\prime}(\zeta_{0})}{2}+2\leq\operatorname{ord}_{p}\frac{f^{(i)}(\zeta_{0})}{i!}+i$, with equality holding only when $p=3$. Therefore, $s_{0}:=s(f,\zeta_{0})=$ $\begin{cases}\min\\{\operatorname{ord}_{p}f^{\prime}(\zeta_{0})+1,\operatorname{ord}_{p}\frac{f^{\prime\prime}(\zeta_{0})}{2}+2,\operatorname{ord}_{p}\frac{f^{(3)}(\zeta_{0})}{3!}+3\\}&\text{if }p=3\\\ \min\\{\operatorname{ord}_{p}f(\zeta_{0}),\operatorname{ord}_{p}f^{\prime}(\zeta_{0})+1,\operatorname{ord}_{p}\frac{f^{\prime\prime}(\zeta_{0})}{2}+2\\}&\text{if }p\neq 3\end{cases}$ This says that $s_{0}$, and thus $\tilde{f}_{1,\zeta_{0}}$, is simply determined by the coefficients of the first three or four terms of the Taylor series expansion of $f(\zeta_{0}+px)$ about $0$. Moreover, $\tilde{f}_{1,\zeta_{0}}(x)$ has degree $\leq\\!2$ and at most one degenerate root $\zeta_{1}$ (of multiplicity $2$) when $p\geq 5$, or degree $\leq\\!3$ and at most one degenerate root $\zeta_{1}$ (of multiplicity $3$) when $p=3$. So by Lemma 2.11, $s_{1}:=s(f_{1,\zeta_{0}},\zeta_{1})$ is bounded from above by $2$ or $3$ respectively. So, inductively, any nodal polynomial $\tilde{f}_{i,\zeta}$ of $\mathcal{T}_{p}(f_{1,\zeta_{0}})$ has degree $\leq\\!2$ when $p\neq 3$, and $\leq\\!3$ when $p=3$. Moreover, $\tilde{f}_{1,\zeta_{0}}$ can have at most one degenerate root. So $\mathcal{T}_{p,k}(f_{1,\zeta_{0}})$ is a path and we are done. $\blacksquare$ Generalizing our automorphism trick from Algorithm 2.16 for lowering the degree of a binomial, we can efficiently do the same for trinomials if we apply a fast algorithm for the Shortest Vector Problem in a lattice (see, e.g., [11]). A very special case of this approach is the following: ###### Lemma 5.5. [6, Special Case of Lemma 1.9] Given any prime $p$, and $a_{2},a_{3}\\!\in\\!\mathbb{N}$ with $0<a_{2}<a_{3}<p-1$ and $\gamma\\!:=\\!\gcd(a_{2},a_{3},p-1)$, one can find within $\log^{1+o(1)}p$ bit operations, an integer $e$ such that for all $i\in\\{1,\ldots,t\\}$, $ea_{i}\equiv m_{i}$ mod $p-1$ and $|m_{i}|\leq\gamma\sqrt{2(p-1)}$. $\blacksquare$ It is then simple to prove that computing the nodal polynomials of $\mathcal{T}_{p,k}(f)$ can be done efficiently for trinomial $f$. ###### Lemma 5.6. Following the preceding notation, suppose $p$ is an odd prime, $\zeta_{0}$ is a nonzero degenerate root of $\tilde{f}$ mod $p$, and $\mathcal{T}_{p,k}(f_{1,\zeta_{0}})$ has depth greater than $1$. Then for each $j\geq 1$, given $f_{1,\zeta_{0}}$, we can compute any other nodal polynomial of $\mathcal{T}_{p}(f_{1,\zeta_{0}})$ in time $(\log p)^{1+o(1)}$. Proof: Suppose $\mathcal{T}_{p,k}(f_{1,\zeta_{0}})$ has depth $\delta\\!\geq\\!2$, and suppose for each integer $j\\!\in\\!\\{1,\ldots,\delta-1\\}$ that $\zeta_{j}$ is the unique degenerate root of the $(j-1)$-th nodal polynomial $\tilde{f}_{j,\sigma_{j-1}}$ of $\mathcal{T}_{p,k}(f_{1,\zeta_{0}})$, where $\sigma_{0}:=\zeta_{0}$ and $\sigma_{j}:=\sum_{1\leq i\leq j}\zeta_{i}p^{i-1}$. For each $j$ let $s_{j}:=s(f_{j,\sigma_{j-1}},\zeta_{j})$. Then $s_{j}\leq 3$ and in fact $s_{j}\\!\geq\\!2$ as otherwise, by Lemma 2.11, the node corresponding to $f_{j-1,\sigma_{j-2}}$ would have no child and then $f_{j,\sigma_{j-1}}$ wouldn’t be a nodal polynomial. Let $\ell\geq 1$, suppose $\mathcal{T}_{p,k}(f)$ has depth $>\\!\ell$, and suppose for each $j\\!\in\\!\\{0,\ldots,\ell\\}$ we have that $\zeta_{j}$ is the unique degenerate root of the $j$th nodal polynomial $\tilde{f}_{j,\zeta_{j-1}}$. Then $s(f_{j,\zeta_{j-1}},\zeta_{j})\leq 2$, and in fact $s(f_{j,\zeta_{j-1}},\zeta_{j})\\!=\\!2$ as otherwise, by the definition of the $f_{i,\zeta}$, the tree $\mathcal{T}_{p,k}(f_{1,\zeta_{0}})$ would terminate at depth $j<\delta$. So $f_{j+1,\sigma_{j}}(x)=\frac{1}{p^{s_{j}}}f_{j,\sigma_{j-1}}(\zeta_{j}+px)=\frac{1}{p^{\sum_{\i=1}^{j}s_{j}}}f_{1,\zeta_{0}}(\sigma_{j}+p^{j}x)$ and $\tilde{f}_{j+1,\sigma_{j}}(x)$ is a quadratic polynomial, and thus we can determine if a degenerate root $\zeta_{j+1}$ exists by simply computing the discriminant. Via Horner’s method and finite ring arithmetic, this takes time no worse than $(\log p)^{1+o(1)}$ (see [3]). Suppose such a $\zeta_{j+1}$ exists. Then $\displaystyle f_{j+1,\sigma_{j}}(\zeta_{j+1}+px)$ $\displaystyle=\frac{1}{p^{\sum_{\i=1}^{j}s_{j}}}f_{1,\zeta_{0}}(\sigma_{j+1}+p^{j+1}x)$ $\displaystyle=\frac{1}{p^{\sum_{\i=1}^{j+1}s_{j}}}\left[f_{1,\zeta_{0}}(\sigma_{j+1})+p^{j+1}f^{\prime}_{1,\zeta_{0}}(\sigma_{j+1})+p^{2j+2}\frac{f^{\prime\prime}_{1,\zeta_{0}}(\sigma_{j+1})}{2}x^{2}\right.$ $\displaystyle\left.+p^{3j+3}\frac{f^{(3)}_{1,\zeta_{0}}(\sigma_{j+1})}{3!}x^{3}+\text{higher powers of }p\right].$ As $\sum_{i=1}^{j+1}s_{j}\leq 3(j+1)$, and $s_{j+1}\leq 3$, to determine $\tilde{f}_{j+2,\sigma_{j+1}}$, it really suffices to compute the coefficients of the first four terms. Thus for $j>1$, the computation at the $j$-th recursion step take time no worse than $(\log p)^{1+o(1)}$. $\blacksquare$ We are now ready to outline the algorithm that proves theorem 1.1. ###### Algorithm 5.7. (Solving Trinomial Equations Over $\boldsymbol{\mathbb{Q}_{p}}$) Input. An odd prime $p$ and $c_{1},c_{2},c_{3},a_{2},a_{3}\\!\in\\!\mathbb{Z}\setminus\\{0\\}$ with $|c_{i}|\\!\leq\\!H$ for all $i$ and $1\\!\leq\\!a_{2}\\!<\\!a_{3}\\!=:\\!d$. Output. A true declaration that $f(x_{1})\\!:=\\!c_{1}+c_{2}x^{a_{2}}+c_{3}x^{a_{3}}$ has no roots in $\mathbb{Q}_{p}$, or $z_{1},\ldots,z_{m}\\!\in\\!\mathbb{Q}$ with logarithmic height $O\\!\left(p\log^{8}(dH)\right)$ such that $m$ is the number of roots of $f$ in $\mathbb{Q}_{p}$, $z_{j}$ is an approximate root of $f$ with associated true root $\zeta_{j}\\!\in\\!\mathbb{Q}_{p}$ for all $j$, and the $\zeta_{j}$ are pair-wise distinct. Description. 1: If $\operatorname{ord}_{p}c_{1}\\!\neq\\!\operatorname{ord}_{p}c_{2}$ mod $a_{2}$ and $\operatorname{ord}_{p}c_{2}\\!\neq\\!\operatorname{ord}_{p}c_{3}$ mod $a_{3}-a_{2}$ then say ‘‘No roots in $\mathbb{Q}_{p}$!’’ and STOP. 2: Rescale and invert roots if necessary, so that we may assume $p\nmid c_{1}c_{2}$ and $\operatorname{ord}_{p}c_{3}\\!\geq\\!0$. 3: Compute all the nodal polynomials for $T_{p,k}(f)$ for $k\\!=\\!2D^{\prime}+1$ where $D^{\prime}$ is at least as large as the parameter $D$ in Lemma 5.2. 4: Use Hensel Lifting to find the first $2D^{\prime}+1$ base-$p$ digits of all the non-degenerate roots of $f$ in $\mathbb{Z}_{p}$ of valuation $0$. 5: Via Algorithm 2.16 find the first $O(\log(dH))$ base-$p$ digits of all the degenerate roots of $f$. 6: If $p|c_{3}$ then rescale and invert roots to compute approximants for the remaining roots of $f$ in $\mathbb{Q}_{p}$ by computing roots of valuation $0$ for a rescaled version of $f$ with coefficients in reverse order. Proof of Theorem 1.1: We have described Algorithm 5.7 at a higher level of abstraction than Algorithm 2.16, so that we may more easily isolate the key ingredients of the proof. In particular, the height bound for our approximate roots from Assertion (1) follows directly from Lemma 5.2, which is used in Step 3. Assertion (2) follows easily from Theorem 5.1: Since Steps 3 and 4 (which use Hensel’s Lemma) will imply a decay rate of $O\\!\left(\left(\frac{1}{p}\right)^{2D+2^{i}}\right)$ for the $p$-adic distance of the $i$th Newton iterate to a true root, we merely observe that this is no worse than a decay rate of $O\\!\left(\left(\frac{1}{p^{1/(2D+1)}}\right)^{2^{i}}\right)$. So Assertion (2) holds with $\mu\\!=\\!p^{1/O(p\log^{8}(dH))}$. As for Assertion (3) on correctly counting the roots of $f$ in $\mathbb{Q}_{p}$, this follows immediately from Steps 3–5. So all that remains is to prove correctness (including elaborating Step 5) and to do a sufficiently good complexity analysis. Correctness: Thanks to Theorem 2.2, Step 1 merely guarantees that $f$ has roots of integral valuation, which is a necessary condition for their to be roots in $\mathbb{Q}_{p}$. Step 2 merely involves simple substitutions that only negligibly affect the heights of the coefficients, similar to the binomial case. Steps 3 and 4 correctly count the number of non-degenerate roots of $f$ in $\mathbb{Z}_{p}$ of valuation $0$, thanks to Lemma 2.12. Step 5 can be elaborated as follows: First note that $0$ can not be a degenerate root since $f(0)\\!\neq\\!0$. Next, rearranging the equations $f(\zeta)\\!=\\!f^{\prime}(\zeta)\\!=\\!0$, it is easy to see that $\zeta\\!\in\\!\mathbb{Q}^{*}_{p}$ is a degenerate root of $f$ if and only if $[c_{1},c_{2}\zeta^{a_{2}},c_{3}\zeta^{a_{3}}]^{T}$ is a right null-vector for the matrix $B\\!:=\\!\begin{bmatrix}1&1&1\\\ 0&a_{2}&a_{3}\end{bmatrix}$. Since the right null-space of $B$ is generated by $[a_{3}-a_{2},-a_{3},a_{2}]^{T}$, we see that such a $\zeta$ is defined by two binomial equations: $(a_{3}-a_{2})c_{2}\zeta^{a_{2}}\\!=\\!-c_{1}a_{3}$ and $-a_{3}c_{3}\zeta^{a_{3}-a_{2}}\\!=\\!c_{2}a_{2}$. Via an application of the Extended Euclidean Algorithm, we can find $R,S\\!\in\\!\mathbb{Z}$ with $Ra_{2}+S(a_{3}-a_{2})\\!=\\!\gcd(a_{2},a_{3})$ and the logarithmic heights of $R$ and $S$ of order $O(\log d)$. So by multiplying and dividing suitable powers of our binomial equations, we get that $\zeta$ must satisfy the single equation $((a_{3}-a_{2})c_{2})^{R}(-a_{3}c_{3})^{S}\zeta^{\gcd(a_{2},a_{3})}\\!=\\!(-c_{1}a_{3})^{R}(c_{2}a_{2})^{S}$. The latter equation can be solved easily, within our overall time bound, via Algorithm 2.16. Note in particular that while the coefficient heights look much larger, any root $\zeta$ ultimately satisifies the original pair of binomials, thus implying $\zeta$ must have low logarithmic height. Step 6 merely takes care of the remaining roots, at negligible affect to the coefficient heights. Note that we do need to renormalize the roots in the end, due to the various rescalings, but this adds a neglible summand of $O(\log H)$ to the logarithmic heights of the roots. So we are done. $\blacksquare$ Complexity Analysis: The only part of the algorithm going beyond basic arithmetic operations modulo $p$ or $p^{\ell}$ is Steps 3–4. In particular, Theorem 5.1 tells us that we can take $D^{\prime}\\!=\\!O\\!\left(p\log^{8}(dH)\right)$, and Lemma 5.6 tells us that the complexity of Steps 3–4 is no worse than $O\\!\left(p^{2}\log^{2}(p)\log^{9}(dH)\right)$, assuming we employ brute- force search to find the roots in $\mathbb{F}_{p}$ of the mod $p$ reductions of the nodal polynomials. So the final overall complexity of our algorithm is $O\\!\left(p^{2}\log^{2}(p)\log^{9}(dH)\right)$. $\blacksquare$ ###### Remark 5.8. We can apply degree reduction to lower the exponent of $p$ slightly in our complexity estimate. Also, if we apply randomness, we can lower the exponent a bit more. Of course, the main source of the size of our complexity bound is our use of Yu’s Theorem. With a slightly more careful application, we could possibly reduce the exponents of $9$ and $8$ to $5$. $\diamond$ ## References * [1] Martín Avendano, Roman Kogan, Mounir Nisse, and J. Maurice Rojas. Metric estimates and membership complexity for Archimedean amoebae and tropical hypersurfaces. Journal of Complexity, 46:45–65, 2018. * [2] Martín Avendaño and Teresa Krick. Sharp bounds for the number of roots of univariate fewnomials. Journal of Number Theory, 131(7):1209 – 1228, 2011. * [3] Eric Bach and Jeffrey Shallit. Algorithmic number theory, volume 1: efficient algorithms. MIT Press, Cambridge, Massachusetts, 1996. * [4] Jens-Dietrich Bauch, Enric Nart, and Hayden D. Stainsby. 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Cambridge University Press, Cambridge, 2006. An introduction to $p$-adic analysis, Reprint of the 1984 original [MR0791759]. * [27] J.-P. Serre. A course in arithmetic. Springer-Verlag, New York-Heidelberg, 1973. Translated from the French, Graduate Texts in Mathematics, No. 7. * [28] Steve Smale. Newton’s method estimates from data at one point. In The merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985), pages 185–196. Springer, New York, 1986. * [29] Joachim von zur Gathen and Jürgen Gerhard. Modern computer algebra. Cambridge University Press, Cambridge, third edition, 2013. * [30] Edwin Weiss. Algebraic number theory. International series in pure and applied mathematics. McGraw-Hill, 1963\. * [31] R. Ryan Williams. Some Estimated Likelihoods for Computational Complexity, pages 9–26. Springer International Publishing, Cham, 2019. * [32] Kunrui Yu. Linear forms in $p$-adic logarithms III. 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2003.00336

arxiv-papers

# Three-dimensional matching is NP-Hard Shrinu Kushagra111Email<EMAIL_ADDRESS>The results in this note first appeared in [Kushagra et al., 2019] where the reduction was used to prove query lower bounds for a clustering problem. ###### Abstract The standard proof of NP-Hardness of 3DM provides a power-$4$ reduction of 3SAT to 3DM. In this note, we provide a linear-time reduction. Under the exponential time hypothesis, this reduction improves the runtime lower bound from $2^{o(\sqrt[4]{m})}$ (under the standard reduction) to $2^{o(m)}$. Keywords: 3DM, 3SAT, ETH, X3C, NP-Hard ## 1 Introduction In this note, we first establish the hardness of the following decision problem. ###### Definition 1 (3DM). . Input: Sets $W,X$ and $Y$ and a set of matches $M\subseteq W\times X\times Y$ of size $m$. Output: YES if there exists $M^{\prime}\subseteq M$ such that each element of $W,X,Y$ appears exactly once in $M^{\prime}$. NO otherwise. To prove that 3DM is NP-Hard, we reduce an instance of 3SAT to the given problem. Next, we define the 3SAT decision problem. ###### Definition 2 (3-SAT). . Input: A boolean formulae $\phi$ in 3CNF form with $n$ literals and $m$ clauses. Output: YES if $\phi$ is satisfiable, NO otherwise. Given an instance of 3SAT with $n$ literals and $m$ clauses, [Garey and Johnson, 1979] construct a graph with $\Theta(nm)$ vertices and $\Theta(n^{2}m^{2})$ edges. Thus, this is a power-$4$ reduction. In this note, we use a similar but a more efficient gadget and provide a linear time reduction of the 3SAT instance to the given problem. ## 2 Hardness of 3DM ###### Theorem 3. Three-dimensional matching is an NP-Hard problem. Figure 1: Part of graph $G$ constructed for the literal $x_{1}$. The figure is an illustration for when $x_{1}$ is part of four different clauses. The triangles (or hyper-edge) $(a_{i},b_{i},c_{i})$ capture the case when $x_{1}$ is true and the other triangle $(b_{i},c_{i}^{\prime},a_{i+1})$ captures the case when $x_{1}$ is false. Assuming that a clause $C_{j}=\\{x_{1},x_{2},x_{3}\\}$, the hyper-edges containing $tf_{i},tf_{i}^{\prime}$ and $t_{1},t_{1}^{\prime}$ capture different settings. The hyper-edges containing $t_{1},t_{1}^{\prime}$ ensure that atleast one of the literals in the clause is true. The other two ensure that two variables can take either true or false values. Our reduction is described in Fig. 1. For each literal $x_{i}$, let $m_{i}$ be the number of clauses in which the the literal is present. We construct a “truth-setting” component containing $2m_{i}$ hyper-edges (or triangles). We add the following hyper-edges to $M$. $\displaystyle\\{(a_{k}[i],b_{k}[i],c_{k}[i]):1\leq k\leq m_{i}\\}$ $\displaystyle\cup\\{(a_{k+1}[i],b_{k}[i],c_{k}^{\prime}[i]):1\leq k\leq m_{i}\\}$ Note that one of $(a_{k},b_{k},c_{k})$ or $(a_{k+1},b_{k},c_{k}^{\prime})$ have to be selected in a matching $M^{\prime}$. If the former is selected, that corresponds to the variable $x_{i}$ being assigned true, the latter corresponds to false. This part is the same as the standard construction. For every clause $C_{j}=\\{x_{1},x_{2},x_{3}\\}$ we add three types of hyper- edges. The first type ensures that atleast one of the literals is true. $\\{(c_{k}[i],t_{1}[j],t_{1}^{\prime}[j]):x_{i}^{\prime}\in C_{j}\\}\cup\\{(c_{k}^{\prime}[i],t_{1}[j],t_{1}^{\prime}[j]):x_{i}\in C_{j}\\}$ The other two types of hyper-edges (conected to the $tf_{i}$’s) say that two of the literals can be either true or false. Hence, we connect them to both $c_{k}$ and $c_{k}^{\prime}$ $\displaystyle\\{(c_{k}[i],tf_{1}[j],tf_{1}^{\prime}[j]):x_{i}^{\prime}\text{ or }x_{i}\in C_{j}\\}$ $\displaystyle\cup\\{(c_{k}[i],tf_{2}[j],tf_{2}^{\prime}[j]):x_{i}\text{ or }x_{i}^{\prime}\in C_{j}\\}$ $\displaystyle\cup\\{(c_{k}^{\prime}[i],tf_{1}[j],tf_{1}^{\prime}[j]):x_{i}^{\prime}\text{ or }x_{i}\in C_{j}\\}$ $\displaystyle\cup\\{(c_{k}^{\prime}[i],tf_{2}[j],tf_{2}^{\prime}[j]):x_{i}\text{ or }x_{i}^{\prime}\in C_{j}\\}$ Note that in the construction $k$ refers to the index of the clause $C_{j}$ in the truth-setting component corresponding to the literal $x_{i}$. Using the above construction, we get that $\displaystyle W=\\{c_{k}[i],c_{k}^{\prime}[i]\\}$ $\displaystyle X=\\{a_{k}[i]\\}\cup\\{t_{1}[j],tf_{1}[j],tf_{2}[j]\\}$ $\displaystyle Y=\\{b_{k}[i]\\}\cup\\{t_{1}^{\prime}[j],tf_{1}^{\prime}[j],tf_{2}^{\prime}[j]\\}$ Hence, we see that $|W|=2\sum_{i}m_{i}=6m$. Now, $|X|=|Y|=\sum_{i}m_{i}+3m=6m$. And, we have that $|M|=2\sum_{i}m_{i}+15m=21m$. Thus, we see that this construction is linear in the number of clauses. Now, if the 3-SAT formula $\phi$ is satisfiable then there exists a matching $M^{\prime}$ for the 3DM problem. If a variable $x_{i}=T$ in the assignment then add $(c_{k}[i],a_{k}[i],b_{k}[i])$ to $M^{\prime}$ else add $(c_{k}^{\prime}[i],a_{k+1}[i],b_{k}[i])$. For every clause $C_{j}$, let $x_{i}$ (or $x_{i}^{\prime}$) be the variable which is set to true in that clause. Add $(c_{k}^{\prime}[i],t_{1}[j],t_{1}^{\prime}[j])$ (or $(c_{k}[i],t_{1}[j],t_{1}^{\prime}[j])$) to $M^{\prime}$. For the remaining two clauses, add the hyper-edges containing $tf_{1}[j]$ and $tf_{2}[j]$ depending upon their assignments. Clearly, $M^{\prime}$ is a matching. Now, the proof for the other direction is similar. If there exists a matching, then one of $(a_{k},b_{k},c_{k})$ or $(a_{k+1},b_{k},c_{k}^{\prime})$ have to be selected in a matching $M^{\prime}$. This defines a truth assignment of the variables. Now, the construction of the clause hyper-edges ensures that every clause is satisfiable. ## 3 Exponential Time Hypothesis for 3DM Before we start the discussion in the section, lets review the definition of the exponential time hypothesis. Exponential Time Hypothesis (ETH) There does not exist an algorithm which decides 3-SAT and runs in $2^{o(m)}$ time. If the exponential hypothesis is true, the standard reduction of 3-SAT to 3DM [Garey and Johnson, 1979] implies that any algorithm for 3DM runs in $2^{o(m^{1/4})}$. However, using the reduction in Section 2, we get a more tighter dependence on $m$ stated as a theorem below. ###### Theorem 4. If the exponential time hypothesis holds then there does not exist an algorithm which decides the three-dimensional matching problem (3DM) and runs in time $2^{o(m)}$. ###### Proof. For the sake on contradiction, suppose that such an algorithm $\mathcal{A}$ exists. Then, using the reduction from Section 2 and $\mathcal{A}$, we get an algorithm for 3SAT that runs in $2^{o(m)}$ time which contradicts the ET hypothesis. ∎ An immediate corollary of this result applies to another popular problem Exact Cover by 3-sets. ###### Definition 5 (X3C). . Input: $U=\\{x_{1},\ldots,x_{3q}\\}$. A collections of subsets $S=\\{S_{1},\ldots,S_{m}\\}$ such that each $S_{i}\subset U$ and contains exactly three elements. Output: YES if there exist $S^{\prime}\subseteq S$ such that each element of $U$ occurs exactly once in $S^{\prime}$, NO otherwise. ###### Corollary 6. If the exponential time hypothesis holds then there does not exist an algorithm which decides exact cover by 3-sets problem (X3C) and runs in time $2^{o(m)}$. ###### Proof. The proof follows from the trivial reduction of 3DM to X3C where $U=W\cup X\cup Y$ and $S=M$. ∎ ## References * [Garey and Johnson, 1979] Garey, M. R. and Johnson, D. S. (1979). Computers and intractability, volume 174. freeman San Francisco. * [Kushagra et al., 2019] Kushagra, S., Ben-David, S., and Ilyas, I. F. (2019). Semi-supervised clustering for de-duplication. In The 22nd International Conference on Artificial Intelligence and Statistics, AISTATS 2019, 16-18 April 2019, Naha, Okinawa, Japan, pages 1659–1667.

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2003.00341

arxiv-papers

Finite $N$ unitary matrix model Raghav G. Jha Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 🖄<EMAIL_ADDRESS> Abstract: We consider one-plaquette unitary matrix model at finite $N$ using exact expression of the partition function for both SU($N$) and U($N$) groups. ###### Contents 1. 1 Introduction 2. 2 Partition function and observables 3. 3 Results & Conclusions ## 1 Introduction The study of matrix models has been important in understanding and gaining deep insights into the structure of gauge theories. Their range of applicability is wide and some examples include – the geometry of random surfaces, two-dimensional quantum gravity, statistical mechanics of spin systems. They also have connections to supersymmetric gauge theories and non- perturbative formulations of superstring theory. A class of matrix models has also been studied in the context of exhibiting properties similar to the four- dimensional theory of strong interactions (QCD). The techniques of strong coupling expansion and lattice methods [1] along with approximate recursion relations [2] were pioneered to understand the properties of QCD. Around the same time, another important direction was pursued by ’t Hooft in finding an expansion parameter, independent of the scale to study QCD by considering 1/$N$ expansion [3]. In this setting, the Feynman diagrams drawn in the double-line notation can be organized to follow an expansion in powers of $N^{\chi=2-2g}$ and were argued to be analogous to the topological expansion in string theory. The large $N$ limit was then used to solve the two- dimensional model of QCD in the light-cone gauge which is now also known as ’t Hooft model [4]. This model has many similarities with the four-dimensional theory but is exactly solvable in the planar limit. In higher dimensions, the success has been scarce but it is at least known that the large $N$ limit does not alter a distinctive feature of QCD – ultraviolet freedom, which is obvious from the $N\to\infty$ limit of the two-loop QCD $\beta$-function which has the leading coefficient with the correct sign. Following these developments, the large $N$ limit of gauge theories have been extensively explored for a wide class of gauge theories. In this topological limit, a remarkable simplification occurs, where only the planar diagrams survive, when we keep $\lambda=g_{\text{YM}}^{2}N$ fixed and take $N\to\infty$. The interplay between the large $N$ limit of gauge theories and string theory has significantly improved our understanding of strongly coupled gauge theories and resulted in the AdS/CFT conjecture. In this work, we consider a unitary matrix model that is equivalent to two-dimensional pure QCD. The two- dimensional models such as this have rich behavior but are trivial since gauge field is non-dynamical (absence of degrees of freedom) and the expectation value of observables can be calculated exactly as done in [5, 6, 7]. In this model, the Wilson loop obeys area law for all couplings since the Coulomb potential is always linear and exhibits both ultraviolet freedom and infrared prison. The two-dimensional lattice gauge theories with Wilson action were shown to exhibit third-order phase transition at large $N$. This transition came as a surprise when it was first observed but now we have a better understanding through AdS/CFT correspondence that these phase transitions at large $N$ and finite volume occurs in several holographic models and are related to transitions between different black hole solutions in supergravity [8, 9]. The occurrence of this phase transition in the large $N$ limit signifies that non-analytic behaviour can occur even in the simplest of models which makes it clear that the strong coupling expansions cannot be usually smoothly continued to the limit of weak coupling. This two-dimensional lattice gauge theory with SU($N$) or U($N$) gauge group is often described in terms of single plaquette action over compact unitary group manifold and will hereafter be referred to as the GWW model after the authors of [6, 5]. The unitary matrix models of this kind have also been studied from the holographic point of view [10, 11]. The interest in this model stems from two main reasons, 1) This model admits an exact solution for any $N$ and coupling, 2) This model is closely related to one of the simplest strings (minimal superstring theory) with manageable non-perturbative description (Type 0 strings in one dimension) [12]. In fact, it was shown that unitary matrix models in the double scaling limit ($N\to\infty$ & $\lambda\to\lambda_{\text{critical}}$ with some well- defined relation between $N$ and $|\lambda-\lambda_{\text{critical}}|$) is described by the Painlevé II equation [13]. The GWW model and several of its modifications have been well-studied using various techniques in the $N\to\infty$ limit. The finite $N$ limit has been surprisingly less explored but recently attracted some attention [14]. Even in these explorations, the gauge group considered was U($N$) since it is a little easier to handle and in the planar limit, which is mostly of interest, there is no distinction with SU($N$) . However, at finite $N$, they have _different_ behaviour and independent studies of both $\mathbb{G}=$ SU($N$) and U($N$) gauge groups are desirable and to our knowledge no treatment of this unitary matrix model at finite $N$ for SU($N$) gauge group has been done yet. This paper aims to fill this gap in the literature. We derive an expression for the partition function considering SU($N$) gauge group and study observables in the finite $N$ limit. In case the qualitative behaviour is not severely altered by addition of matter fields i.e. presuming this phase transition is not turned into a smooth crossover, then these studies may be useful in understanding black hole solutions and stringy corrections by considering the finite $N$ limit of matrix models [15] since the transition at $\lambda_{\text{critical}}=2$ is supposed to separate the supergravity regime from the perturbative gauge theory regime. There are only a few matrix models where the finite $N$ regime is exactly solvable for any coupling and this is one such model. In this respect and various others, a finite $N$ study of these class of matrix models would be useful in probing the quantum effects of gravity using holography. We briefly outline the structure of the paper. In Section 2, we write down the exact expression of the partition function for _special unitary matrix_ model at any coupling and $N$ in terms of determinant of a Toeplitz matrix111Named after German mathematician Otto Toeplitz (1881 - 1940), each descending diagonal from left to right in this matrix is constant and matrix element $(j,k)$ depends on $j-k$ and sketch the well-known phase structure of this model at large $N$. In Section 3, we numerically calculate the corrections to the planar result for free energy in both the phases and show that it is simpler to deduce the instanton corrections in the strong coupling phase since the $1/N$ (or genus) expansion terminates at genus-zero for U($N$) unitary model. However, this is not the case for the SU($N$) model which has contributions to the free energy at small $N$ in both the phases. We also provide results for the Wilson loop in the SU($N$) matrix model using our exact expression and show that in the large $N$ limit, they converge to the known results for the U($N$) unitary model. We end with a brief mention of some open questions which can be addressed in the context of SU($N$) one- plaquette matrix model and other related unitary matrix models. ## 2 Partition function and observables The central object in the study of matrix models is the partition function which is determined by integration over some compact group manifold. However, there exists only a handful of models which can be solved exactly [16] and most of these proceeds by reduction of $N^{2}$ matrix degrees of freedom to $N$ by exploiting some symmetry in the large $N$ limit. The simplest among all such matrix models is the well-studied one-matrix model. For a nice review for unitary matrix models, we refer the reader to [17]. The lattice Wilson action in two dimensions is equivalent to one-plaquette model by considering the integration over the unitary group. This has also been studied using character expansion222It is useful to sometimes think of character expansion as Fourier expansion on a compact group manifold which is discussed in [18, 19]. It was shown [2] that in two dimensions, the expansion in terms of characters constitute the recursion relations through which one can exactly solve the model. For the large $N$ analysis, the saddle point methods [20] are often used as was done in [6]. However, the saddle point methods are not useful in extracting sub-leading orders in the 1/$N$ expansion for which the method of orthogonal polynomials is usually used as was done in [7]. The general partition function can be schematically written as: $Z=\frac{1}{\text{Vol}(\mathbb{G})}\int\prod_{\text{links}\leavevmode\nobreak\ l}\mathscr{D}U_{l}\prod_{\Box}Z_{\Box},$ (2.1) where $\Box$ denotes the plaquette on whose perimeter we take the ordered product of the link matrices $U$ of size $N\times N$. In order to compute expectation values in the two-dimensional model, one has to make a choice between one-plaquette and heat kernel actions. The partition function of the two-dimensional Yang-Mills theory based on the heat kernel action is written in terms of sum over all irreducible representations of the unitary gauge group. We will use the one-plaquette action in this work similar to [5, 6] which can be expressed as333Some authors use $g$ to denote $1/\lambda$ or $2/\lambda$. This distinction is clear from the coupling where phase transition occurs.: $S(U)=\frac{N}{\lambda}\Big{[}\mathrm{Tr}\Big{(}\prod_{\Box}U\Big{)}+\mathrm{Tr}\Big{(}\prod_{\Box}U^{\dagger}\Big{)}\Big{]},$ (2.2) where $\prod_{\Box}U$ is the product of links around a plaquette and shorthand for $U_{\mu}(\textbf{n})U_{\nu}(\mathbf{n}+\widehat{\boldsymbol{\mu}})U^{\dagger}_{\mu}(\mathbf{n}+\widehat{\boldsymbol{\mu}}+\widehat{\boldsymbol{\nu}})U^{\dagger}_{\nu}(\mathbf{n}+\widehat{\boldsymbol{\nu}})$. The convention used is such that $U_{\mu}(\textbf{n})$ denotes a link starting from site n and going in $\mu$-direction. It was found that for this model in the $N\to\infty$ limit with fixed $\lambda=g_{\text{YM}}^{2}N$, one observes a discontinuity in the third derivative of the free energy corresponding to a third-order phase transition. This means that one loses the analytic structure for even simple actions in the large $N$ limit and the continuation from the weak coupling to strong coupling is non-trivial. This transition is not like the usual phase transitions in statistical mechanics which occur in the infinite volume limit. Here, it occurs in a finite volume (single plaquette) for an infinite rank gauge group. We denote the full partition function of the model with $\mathcal{Z}$ and since in two dimensions, we can treat all the plaquettes as independent, we deal with just a single plaquette [21, 6], which we will refer to as one- plaquette partition function, $Z=\mathcal{Z}^{1/N_{s}N_{t}}$, where $N_{s}N_{t}$ are the number of nodes in the lattice and equals the number of plaquettes. It is given by: $Z=\int_{\begin{subarray}{c}{\sf U(N)}\leavevmode\nobreak\ \\\ \text{or\leavevmode\nobreak\ }{\sf SU(N)}\leavevmode\nobreak\ \end{subarray}}\mathscr{D}U\exp\Bigg{[}\frac{N}{\lambda}\mathrm{Tr}\Big{(}U^{\dagger}+U\Big{)}\Bigg{]}.$ (2.3) It is known that the partition function for the U($N$) matrix model given above can be written in terms of Toeplitz determinant444The determinant of infinite size Toeplitz matrix has a well-defined asymptotic behaviour limit due to a theorem by Szegö. given by [22, 7]: $Z(N,\lambda)=\text{Det}\Bigg{[}I_{j-k}\Big{(}\frac{2N}{\lambda}\Big{)}\Bigg{]}_{j,k=1\cdots N},$ (2.4) where $I_{\nu}(x)$ is the modified Bessel function of first kind or Bessel function of imaginary argument of order $\nu$. Note that the argument of the Bessel function is twice the prefactor of action in (2.3). The appearance of partition function in terms of determinant has deep connections to the notion of integrability and special differential equations. For instance, the determinant of Toeplitz matrices also play a role in the context of the Ising model, for a partial set of references, see [23, 24]. Instead of working with the partition function given in (2.3), one can also consider more general unitary matrix model with source arbitrary $N\times N$ matrix $A$ as was considered in [25]. The action for this model is given by: $Z=\int\mathscr{D}U\exp\mathrm{Tr}\Big{[}UA^{\dagger}+AU^{\dagger}\Big{]},$ (2.5) where $U$ is product of four ordered links. The exact form of $Z$ was derived in the large $N$ limit and it was shown that the strong and weak coupling regimes in this model are characterized by $(1/N)\mathrm{Tr}(A^{\dagger}A)^{-1/2}$. One gets back the usual GWW model by setting $A=\mathbb{I}/\lambda$. The model in (2.5) was related to a specific $N\times N$ Hermitian matrix model following some parameter tuning in [26] and an exact solution was found. In this work we will only consider (2.3) and derive the exact expression for the partition function when the integration is over SU($N$) group in (2.3). The analysis for SU($N$) is similar to the one for U($N$) except that now we have the constraint that the product of eigenvalues should satisfy $\prod_{j=1}^{N}e^{i\alpha_{j}}=1$. We start with the one-plaquette partition function: $Z=\int_{SU(N)}\mathscr{D}U\exp\Bigg{[}\frac{N}{\lambda}\Bigg{(}\mathrm{Tr}\prod_{\text{single\leavevmode\nobreak\ }\Box}U+\mathrm{Tr}\prod_{\text{single\leavevmode\nobreak\ }\Box}U^{\dagger}\Bigg{)}\Bigg{]},$ (2.6) where the measure is given by: $\mathscr{D}U=\frac{1}{N!}\prod_{j=1}^{N}\underbrace{\sum_{q}\delta\Big{(}\sum_{m=1}^{N}\alpha_{m}-2q\pi\Big{)}}_{\text{{\sf SU}($N$)\leavevmode\nobreak\ constraint}}\frac{d\alpha_{j}}{2\pi}\prod_{j<k}\Big{(}e^{i\alpha_{j}}-e^{i\alpha_{k}}\Big{)}\Big{(}e^{-i\alpha_{j}}-e^{-i\alpha_{k}}\Big{)},$ (2.7) and the constraint can be written as: $\frac{1}{2\pi}\sum_{p=-\infty}^{\infty}e^{ip\sum\alpha_{m}}\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (2.8) By using the representation of the modified Bessel function as: $I_{k-j-p}\Big{(}\frac{2N}{\lambda}\Big{)}=I_{j-k+p}\Big{(}\frac{2N}{\lambda}\Big{)}=\frac{1}{2\pi}\int_{0}^{2\pi}\leavevmode\nobreak\ e^{\frac{2N}{\lambda}\cos\alpha}e^{i(j-k+p)\alpha}d\alpha,$ (2.9) where, $e^{i\alpha}$ are the eigenvalues of $U$ and (2.9) fills the corresponding $j^{\text{th}}$ and $k^{\text{th}}$ element of the matrix $\mathcal{M}_{\alpha}=I_{j-k+\alpha}(\frac{2N}{\lambda})$ and using (2.6) through (2.9), we obtain the partition function for SU($N$) matrix model as: $\displaystyle Z(N,\lambda)=\sum_{p=-\infty}^{\infty}\text{Det}\Bigg{[}I_{j-k+p}\Big{(}\frac{2N}{\lambda}\Big{)}\Bigg{]}_{j,k=1\cdots N}.$ (2.10) In contrast to the expression for the exact partition function for U($N$) matrix model i.e. (2.4), there is an additional sum over the index $p$ from the constraint in (2.7). In this paper, we study different observables using this partition function. However, for practical purposes of computation using Mathematica, we simply replace the $\infty$ in (2.10) by a large number. We checked that $\left|p\right|\leq 15$ suffices i.e., $Z(N,\lambda)=\underbrace{\sum_{p=-15,\neq 0}^{15}\text{Det}\Bigg{[}I_{j-k+p}\Big{(}\frac{2N}{\lambda}\Big{)}\Bigg{]}}_{\text{due to {\sf SU}($N$)\leavevmode\nobreak\ }}+\text{Det}\Bigg{[}I_{j-k}\Big{(}\frac{2N}{\lambda}\Big{)}\Bigg{]}.$ (2.11) This partition function enables us to evaluate free energy which can be written as a sum over genus expansion for this model: $F(N,\lambda)=\sum_{g\in\mathbb{N}}F_{g}N^{2-2g}+\mathcal{O}(e^{-N}),$ (2.12) where $\mathbb{N}$ denotes the set of non-negative integers. The exact result for the leading coefficient, $F_{0}$, in the planar limit is given by: $F_{0}=\hskip 5.69054pt\begin{cases}\frac{-1}{\lambda^{2}}\hskip 108.12047pt\text{$\lambda\geq 2$}\\\ \frac{-2}{\lambda}-\frac{1}{2}\ln\frac{\lambda}{2}+\frac{3}{4}\hskip 48.36967pt\text{$\lambda<2$}\end{cases}.$ (2.13) Another important observable in unitary matrix models and the one we consider are the Wilson loops. The finite $N$ analysis of Wilson loops in various representations for U($N$) gauge theory was recently done in [14]. An expression for the expectation value of normalized winding Wilson loops defined as: $\mathcal{W}_{k}(N,\lambda)=\frac{1}{N}\Big{\langle}\mathrm{Tr}\Big{(}\prod_{\mathcal{C}}U\Big{)}^{k}\Big{\rangle},$ (2.14) was given in terms of $\mathrm{Tr}(\mathcal{M}_{k}/\mathcal{M}_{0})$ with $k\in\mathbb{Z}^{+}$ denoting the winding number and the expectation value is computed over a closed contour $\mathcal{C}$. A similar expression for the SU($N$) case is yet unknown. Note that like the partition function, we can write $\mathcal{W}_{k,\mathcal{C}}(N,\lambda)$ as $W_{k}(N,\lambda)^{N_{s}N_{t}}$, where the contour is of time extent $aN_{t}$ and spatial extent $aN_{s}$. The single winding ($k=1$) Wilson loop is related to the derivative of the free energy and is given by: $W_{1}(N,\lambda)=\frac{-\lambda^{2}}{2N^{2}}\frac{\partial\ln Z}{\partial\lambda}=\hskip 5.69054pt\begin{cases}\frac{1}{\lambda}\hskip 48.36967pt\text{$\lambda\geq 2$}\\\ 1-\frac{\lambda}{4}\hskip 28.45274pt\text{$\lambda\leq 2$}\end{cases}.$ (2.15) For this U($N$) matrix model as mentioned above, there is another equivalent definition of $W_{1}$ given by: $W_{1}(N,\lambda)=\mathrm{Tr}\leavevmode\nobreak\ \Big{(}\frac{\mathcal{M}_{1}}{\mathcal{M}_{0}}\Big{)},$ (2.16) where $\mathcal{M}_{\alpha}$ is defined below (2.9). We will present results for the free energy and Wilson loops in Section 3 for the SU($N$) and U($N$) models. Even though this matrix model is one of the simplest it has a wide range of interesting features. In [27], by considering the trans-series solutions of the pre-string equation to obtain instanton corrections, it was deduced that the instanton action vanishes at the critical point i.e. $\lambda=2$ and it was concluded that the GWW transition is caused by the effect of instantons555Usually, when one thinks of large $N$ limit, it seems that the instanton corrections which goes as $\exp(-A/g_{s})\sim\exp(-AN/\lambda)$ are insignificant with $A$ denoting the instanton action. In fact, a more general form is $\exp(-F(\lambda)N)$, where $F$ is some non-negative function of the coupling $\lambda$ and proportional to $A$. When $F(\lambda)=0$ corresponding to vanishing action, the exponentially suppressed instanton contribution to $1/N$ expansion becomes important. It turns out that for the GWW model this happens exactly at $\lambda_{\text{critical}}=2$ where the third-order phase transition takes place. Therefore, one also refers to these as instanton driven large $N$ phase transitions and physically relate them to condensation of instantons. We thank M. Mariño for email correspondence regarding the one- plaquette model and his book ‘Instantons and Large $N$’ for clarification regarding the instanton contributions. There exist other examples where a similar phenomenon occurs [28]. The behaviour of the corrections to the planar result due to $1/N$ and instanton have been well-studied. This model also exhibits resurgence behaviour thoroughly explored in [29]. One of the striking features of these studies, which is clearly evident in our results is that the strong coupling phase has no $1/N$ corrections [7] but only $\mathcal{O}(e^{-N})$ corrections from the contributions due to instantons. In GWW model, both gapped and ungapped phases have instanton corrections, albeit of different nature. In the ungapped phase, the eigenvalues of the holonomy matrix fill the circle while in the gapped phase they are distributed over some interval. When the eigenvalue distribution is restricted to some domain, it is called a one-cut (or single interval) distribution/solution. In the ungapped phase, the instanton contribution to the free energy can simply be evaluated by subtracting genus-zero contribution from the total free energy. As we will see, this does not hold for the SU($N$) model. The distribution of eigenvalues, which is one of the central objects in these matrix models, have been studied in both the phases for the U($N$) model and is given by: $\rho(\lambda,\theta)=\hskip 5.69054pt\begin{cases}\frac{2}{\pi\lambda}\cos\Big{(}\frac{\theta}{2}\Big{)}\sqrt{\frac{\lambda}{2}-\sin^{2}\frac{\theta}{2}}\hskip 48.36967pt\text{$\lambda<2$}\\\ \frac{1}{2\pi}\Big{(}1+\frac{2}{\lambda}\cos\theta\Big{)}\hskip 82.51299pt\text{$\lambda>2$}\end{cases}.$ (2.17) In the gapped phase, the distribution is only supported on some interval $\theta\in[-a,a]$ while it is uniform in the ungapped phase. For SU($N$) unitary matrix model, there are corrections to the distributions above which was discussed in [30] and will not be further discussed in this work. Figure 1: A telegraphic summary describing the phase diagram of the one- plaquette matrix model at large $N$ and the exact expression of the partition function for SU($N$) model as given in the main text. ## 3 Results & Conclusions In this section we present the results obtained using (2.11) and (2.4) for SU($N$) and U($N$) groups respectively. We primarily focus on the free energy for different couplings to emphasize behaviours in both phases and at the critical coupling. Our results converge to expected results in (2.13) when we take large $N$, while also probing the finite $N$ coefficients i.e. $F_{g}$ with $g\neq 0$ according to (2.12). In the weak coupling limit, $\lambda<2$, the free energy up to genus-two was calculated in [7] and given by: $\displaystyle F(\lambda,N)=F_{0}-\frac{1}{N^{2}}\Big{(}\frac{1}{12}-\ln A-\frac{1}{12}\ln N-\frac{1}{8}\ln(1-(\lambda/2))\Big{)}-\frac{1}{N^{4}}\Big{(}\frac{3\lambda^{3}}{1024}\frac{1}{(1-(\lambda/2))^{3}}+\frac{1}{240}\Big{)},$ (3.1) where, $A=2^{\frac{7}{36}}\pi^{\frac{-1}{6}}\exp\Big{[}\frac{1}{3}+\frac{2}{3}\int_{0}^{\frac{1}{2}}\ln\Gamma(1+x)dx\Big{]}=1.2824271291\cdots$ is the Glaisher-Kinkelin constant and is related to the derivative of zeta function as, $\zeta^{\prime}(-1)=\frac{1}{12}-\ln A$. A general expression for genus $g$ free energy $F_{g}(\lambda)$ is also known [7, 13, 27, 14] and can be written as: $F_{g}(\lambda)=\frac{B_{2g}}{2g(2g-2)}+\frac{1}{\Big{(}\frac{2}{\lambda}-1\Big{)}^{3g-3}}\sum_{n=0}^{g-2}C_{n}^{g}\lambda^{-n},$ (3.2) where $B_{2g}$ is the Bernoulli number666We note that if we calculate the orbifold Euler characteristic of the moduli space of Riemann surfaces of genus $g$ with $n$ marked points i.e. $\chi(\mathcal{M}_{g,n})$ using Harer-Zagier formula and set $n=0$ we also obtain $\frac{B_{2g}}{2g(2g-2)}$. A general expression for the SU($N$) case is unknown but is expected to have some similarity since the origin of these Bernoulli numbers are in the volume of the U($N$) group which is similar to that of SU($N$) . In Figure 2, we compute the free energy for $\lambda=4/3$ and show the results for SU($N$) and U($N$) models. Figure 2: The free energy (normalized by $N^{2}$) for $\lambda=4/3$ plotted against 1/$N$. The dashed green line is (3.1) from [7] and the dashed black line is the $N\to\infty$ value. The analytic expression corresponding to the dashed yellow line is yet unknown. Figure 3: The dependence of the free energy (normalized by $N^{2}$) for SU($N$) and U($N$) model at $\lambda=2$ on $N$. In the $N\to\infty$ limit, the exact value is $F_{0}=-0.25$. We then consider $\lambda_{\text{critical}}=2$ and plot the results in Figure 3. The free energy has noticeable difference for SU($N$) and U($N$) groups for $N<15$ (which is not the case for other couplings) and we have explored up to $N=100$ for this case. One plausible explanation for this behaviour is that at the critical coupling the instanton contributions are more important compared to any other $\lambda$ (for fixed $N$) and the difference between SU($N$) and U($N$) instanton sectors are therefore most significant. It might be possible to explain this by a systematic finite $N$ instanton contributions with SU($N$) group and comparing with the known results in the U($N$) models [27]. Figure 4: The dependence of the free energy (normalized by $N^{2}$) on $N$ for $\lambda=4$ (strong coupling). There are no $1/N$ corrections for the U($N$) model while for SU($N$) the genus expansion does not terminate at genus-zero. We then consider strong coupling ($\lambda>2$). This analysis is more interesting since the genus expansion terminates at genus-zero in case of U($N$) , first discussed in [7]. Our results shown in Figure 4 are in complete agreement with this and we note that there are no $1/N$ corrections in this ungapped phase when the SU($N$) constraint is not imposed. For SU($N$) model, our study of the partition function signals that there are corrections and the genus convergence is subtle (and certainly not genus-0 exact up to instanton effects) in this case and this deserves further study. Finally, in Figure 5, we study the Wilson loop by taking the numerical derivative of the free energy for a range of couplings with $N=4,10$ for both SU($N$) and U($N$) groups. Figure 5: The expectation value of Wilson loop against coupling for $N=4,10$ around the critical coupling. The dashed lines (different colours) are the planar limit result in different phases. The most distinct feature is that the planar result is approached from different sides for SU($N$) and U($N$) models (similar to free energy behaviour) and signifies that the $1/N$ corrections to planar values come with opposite sign. It will be interesting to compute the exact expression for the Wilson loop winding around $k$ times as done for the U($N$) model in [14]. A related model to the one we studied here is the ‘double trace model’ for which the action is $\mathrm{Tr}U\mathrm{Tr}{U}^{\dagger}$ and can be written in terms of the partition function of GWW model. This model is closely related to a truncated limit of $\mathcal{N}$ = 4 SYM and in the double scaling limit exhibits the Hagedorn/deconfinement phase transition. It would be interesting to understand the finite $N$ limit of this model while not restricting to U($N$) integral as done in [31]. In this paper, we have given an exact expression for the partition function of SU($N$) one-plaquette matrix model valid for all $N$ and couplings and computed exact results for free energy and Wilson loop at finite $N$ for several couplings. We concluded that the $1/N$ corrections to free energy vanish for U($N$) matrix model in the ungapped (strongly coupled) phase where the only corrections to the planar result come from the instanton correction, while for SU($N$) matrix model, the genus expansion contribution to the free energy does not terminate at genus-zero. Our results suggest that that the finite $N$ behaviour of SU($N$) is _very_ different from the U($N$) matrix model and deserves further analysis. It would be interesting to understand results for multiply winded Wilson loops and 1/$N$ expansion of the free energy for SU($N$) along the lines as done for U($N$) group. ### Acknowledgements Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade. ## References * [1] K. G. Wilson, “Confinement of Quarks,” Phys. Rev. D10 (1974) 2445–2459. * [2] A. A. Migdal, “Recursion Equations in Gauge Theories,” Sov. Phys. JETP 42 (1975) 413. * [3] G. ’t Hooft, “A Planar Diagram Theory for Strong Interactions,” Nucl. Phys. B72 (1974) 461. [,337(1973)]. * [4] G. ’t Hooft, “A Two-Dimensional Model for Mesons,” Nucl. Phys. B75 (1974) 461–470. * [5] S. R. 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Balantekin, “Character expansions, Itzykson-Zuber integrals, and the QCD partition function,” Phys. Rev. D62 (2000) 085017, arXiv:hep-th/0007161 [hep-th]. * [19] A. Bazavov, S. Catterall, R. G. Jha, and J. Unmuth-Yockey, “Tensor renormalization group study of the non-Abelian Higgs model in two dimensions,” Phys. Rev. D99 no. 11, (2019) 114507, arXiv:1901.11443 [hep-lat]. * [20] F. Green and S. Samuel, “Calculating the large- N phase transition in gauge and matrix models,” Nuclear Physics B 194 no. 1, (Jan, 1982) 107–156. * [21] J. M. Drouffe and C. Itzykson, “Lattice Gauge Fields,” Phys. Rept. 38 (1978) 133–175. * [22] I. Bars and F. Green, “Complete Integration of U ($N$) Lattice Gauge Theory in a Large $N$ Limit,” Phys. Rev. D20 (1979) 3311. * [23] P. Deift, A. Its, and I. Krasovsky, “Toeplitz matrices and toeplitz determinants under the impetus of the ising model. some history and some recent results,” arXiv:1207.4990 [math.FA]. * [24] T. T. 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Vicari, “Large N phase transition in lattice 2-d principal chiral models,” Phys. Rev. D52 (1995) 395–401, arXiv:hep-lat/9412102 [hep-lat]. * [31] H. Liu, “Fine structure of Hagedorn transitions,” arXiv:hep-th/0408001 [hep-th].

2024-09-04T02:54:55.944001

2003.00351

arxiv-papers

# Emotion Recognition System from Speech and Visual Information based on Convolutional Neural Networks Nicolae-Cătălin Ristea University Politehnica of Bucharest Bucharest, Romania <EMAIL_ADDRESS>Liviu Cristian Dutu Fotonation Romania, member of Xperi Group Bucharest, Romania <EMAIL_ADDRESS>Anamaria Radoi University Politehnica of Bucharest Bucharest, Romania <EMAIL_ADDRESS> ###### Abstract Emotion recognition has become an important field of research in the human- computer interactions domain. The latest advancements in the field show that combining visual with audio information lead to better results if compared to the case of using a single source of information separately. From a visual point of view, a human emotion can be recognized by analyzing the facial expression of the person. More precisely, the human emotion can be described through a combination of several Facial Action Units. In this paper, we propose a system that is able to recognize emotions with a high accuracy rate and in real time, based on deep Convolutional Neural Networks. In order to increase the accuracy of the recognition system, we analyze also the speech data and fuse the information coming from both sources, i.e., visual and audio. Experimental results show the effectiveness of the proposed scheme for emotion recognition and the importance of combining visual with audio data. ###### Index Terms: Emotion Recognition, Facial Action Units, Spectrogram, Convolutional Neural Network ††publicationid: pubid: 978-1-7281-0984-8/19/$31.00 ©2019 IEEE. ## I Introduction Facial attribute recognition, including facial action units and emotions, has been a topic of interest among computer vision researchers for over a decade. Being able to recognize and understand the emotion of a subject could be a key factor in a wide range of fields such as public security, healthcare (including therapeutic treatments) or entertainment. Moreover, the ability of today’s systems to recognize and express emotions would leverage the barriers in obtaining a "natural" interaction between systems and humans. The interest of researchers on this subject lead to the development of Facial Action Coding System (FACS) [1]. Following FACS encoding system, each emotion can be modelled as a finite group of Facial Action Units (FAUs). Indeed, Ekman identifies several facial attributes that allow emotion recognition from face expressions (e.g., morphology, symmetry, duration, coordination of facial muscles) [2]. The recent success of deep learning techniques in many Computer Vision-related applications influenced the emotion recognition field as well. Due to the release of labeled large datasets for emotion recognition from images, as well as the advances made in the design of convolutional neural networks, error rates have significantly dropped. An interesting approach towards emotion recognition is to use multimodal systems of recognition. It is well known that visual and audio information are very important when building emotion recognition systems because the usage of combined sound and video information leads to a better understanding of the emotion context than having access to a single source of information [3]. In this regard, several databases have been recently developed (e.g., CREMA-D [4], OMG-Emotion [5]), but still the lack of multimodal data is a major problem. Therefore, despite the success of multimodal emotion recognition systems, there are still problems regarding emotion classification in real world scenarios that benefit from the existence of both visual and audio information. In this paper, we propose a novel approach towards emotion recognition using multiple sources of information. Our approach involves both visual and speech information and it is based on convolutional neural networks. The experiments prove the effectiveness of the proposed approach and show that having both images and sounds helps to achieve high classification accuracy. In Section II, we review several of the state-of-the-art approaches in emotion recognition, whereas Section III describes several databases related to emotion recognition. The proposed approach is presented in Section IV and the corresponding results are discussed in Section V. Section VI concludes the paper. ## II Related Work ### II-A Emotion recognition from images A wide range of approaches have been developed for the recognition of emotions from still images. The recognition system proposed in [6] (called DeXpression) is based on a Convolutional Neural Network (CNN) architecture inspired by the well-known GoogleNet arhitecture [7]. The architecture contains two blocks of feature extraction composed by convolution, Max Pooling (MaxPool), Rectified Linear Units (ReLU) and concatenation layers. The end part of the network is a fully connected layer which performs the actual emotion classification. The authors use standard datasets (e.g. Extended Cohn-Kanade (CKP) and MMI Facial Expression Database) in their experimental setup and reported that the system has better performances than previous approaches. With an accuracy of 99.6% for CKP and 98.63% for MMI, the DeXpression architecture is robust, small and suitable for real-time applications when emotions are found in the pictures. However, the system fails to recognize an emotion from the first few frames, but these misclassification errors do not affect the overall accuracy of the system since the first corresponding instances can be considered as neutral. Another method, proposed by Z. Yu and C. Zhang [8], is based on learning multiple deep neural networks. The authors describe a more complex face detector composed by three state-of-the-art detectors, followed by a classification module made by combining multiple CNNs. The combining method considers minimizing both the log-likelihood loss and the hinge loss. The approach achieved state-of-the-art results on the Facial Expression Recognition (FER) Challenge 2013 dataset, whereas the classification accuracy reported on the validation and test set of Static Facial Expressions in the Wild (SFEW) dataset 2.0 was 55.96% and 61.29%, respectively. A comprehensive review of the methods related to facial expression recognition can be found in [9]. However, as the authors mention, two key issues appear when dealing with facial expression recognition system. Firstly, training deep convolutional neural networks require large volumes of annotated data. Secondly, variations such as illumination, head pose and person identity might lead to inconsistent recognition results. Bringing audio information to the recognition system may leverage several of these drawbacks. ### II-B Emotion recognition using visual and audio information Combining more sources of information leads to a higher accuracy rate if compared to the case of using a single source of information, audio or visual. EmoNets, proposed in [10], is an emotion recognition system that considers visual features extracted using Convolutional Neural Networks, whereas a deep belief network is intended for building the representation of the audio stream. The spatio-temporal relations in the video streams are tackled by using a relational autoencoder. The authors stress that combining visual and audio features leads to a classifier that achieves a high accuracy rate. The method proposed in [10] won the 2013 Emotion Recognition in the Wild Challenge (EmotiW) [11], reporting an accuracy level on the test set of 47.67% for the 2014 dataset. A different approach is presented in [12] which considers using a CNN to extract features from the speech, whilst, in order to represent the visual information, a deep residual network ResNet-50 is used. The features from the before-mentioned networks are concatenated and inserted into a two-layers Long Short-Term Memory (LSTM) module. Two continuous values are predicted at each moment, namely arousal and valence. The method outperforms other previous methods on the RECOLA database of the Audio-Visual Emotion Challenge (AVEC) 2016. Figure 1: Examples of frames from the CREMA-D dataset along with the corresponding emotion labels. ## III Database Several databases have been developed for emotion recognition purposes. However, regarding multimodal emotion recognition, the lack of data is a major problem. In this approach we focused on the CREMA-D database which is commonly used in multimodal emotion recognition [4, 13, 14]. Some examples are provided in Fig. 1. The CREMA-D database was published in 2015 and contains 7442 clips of 91 actors (48 male and 43 female) with different ethnic backgrounds, coordinated by professional theater directors. The actors were asked to convey particular emotions while producing, with different intonations, 12 particular sentences that evoke the target emotions. There were six labeled emotions (neutral, happy, anger, disgust, fear, sad) and four different emotion levels (low, medium, high, unspecified). The recordings were performed in three different modes, namely audio, visual, and audiovisual. The labels corresponding to each recording were collected using crowdsourcing. More precisely, 2443 participants were asked to label the perceived emotion and its intensity in three modes, depending on the information put at their disposal: video, audio and combined audiovisual. The human accuracy reported for each mode is presented in Table I. In this case, human training was achieved through participants’ previous experiences, whereas the results mentioned in Table I refer to the accuracy over the whole dataset. TABLE I: Human accuracy rate on CREMA-D [4] Mode | Audio | Video | Audio + Video ---|---|---|--- Accuracy | 40.9% | 58.2% | 63.6% Figure 2: Proposed scheme for emotion recognition. The first part of the network deals with visual information extraction, whilst, the second part of the network handles the sound information. The last part of the network is a stack of two fully-connected (FC) layers, representing the classifier. ## IV Proposed method ### IV-A Spectrogram The spectrogram represents a traditional approach towards the time-frequency analysis of signals with many applications in the fields of music, radar and speech processing. More precisely, the spectrogram of a signal shows the temporal evolution of its short-time spectral amplitude. From a visual representation point of view, the spectrogram is a two-dimensional plot with x axis representing time, y axis representing frequency and the magnitude of signal being encoded by the pixel value. The most common method of computing a spectrogram is with the discrete Short- Time Fourier Transform (STFT), described by the following formula: $STFT\\{x[n]\\}(m,k)=\sum_{m=-\infty}^{\infty}x[m]\cdot w[n-m]e^{-j\frac{2\pi}{N_{x}}kn}$ (1) where $N_{x}$ is the number of samples. Moreover, several others methods of computing a spectrogram have developed during time. The Wigner-Ville distribution is one of the techniques used to obtain information about a signal which varies in time [15]. Wavelet analysis, through the _continuous wavelet transform_ with different wavelet bases can also be used for this purpose. Nevertheless, in this study we restrict ourselves to the Short-Time Fourier Transform as a way to compute the spectrogram, and leave the other methods presented above for future investigations. While all these techniques extract a time-frequency representation of the signal’s energy density, we note that the resolution accuracy of all these representations is bounded by the time-frequency uncertainty principle (see [16], Chapter 3) which forbids arbitrarily accurate energy density representations in both time and frequency simultaneously. This is why we believe that in order to extract relevant features for the emotions classification task, these time-frequency representations should be followed by a learned hierarchical feature extractor, such as a convolutional neural network, as detailed below. ### IV-B Preprocessing The video sequence is divided into frames and we consider keeping a fixed number of images $N$, that are equally distanced in the video. The following step consists in detecting the face of the subjects and resizing them to have equal size. In order to prevent the overfitting effect caused by the insufficient number of training samples, data augmentation techniques are applied, e.g. rotation, random cropping, different illumination and flips. In this sense, for each video sequence in part, other 30 videos were made following the methods presented above. The raw audio signal is processed with the discrete Short-Time Fourier Transform (STFT) described above. In the majority of cases, the audio files do not have the same length. For this reason, the spectrograms do not have the same width. In order to overcome this aspect, the spectrograms are reshaped to a fix dimension (the height is the same for all signals, so a reshape was made on the $x$ axis). Two examples of spectrograms for two audio signals expressing different types of emotions are shown in Fig. 3. (a) Angry (b) Happy Figure 3: Resized spectrograms for two different types of emotions Figure 4: Loss function scores on the training set and sets. Figure 5: Accuracy scores on the training set and sets. ### IV-C Proposed Architecture In this paper, we propose a deep Convolutional Neural Network (CNN)-based architecture, composed of three parts. The first part of the network deals with the feature extraction from the image sequences, the second part with the feature extraction from the audio signals, whereas the last part performs the emotion recognition part. The entire architecture is depicted in Fig. 2. An important component of the proposed network architecture is the convolution operation, which is defined, for the audio (1-D) and visual (2-D) signals as follows: $(f*h)[i]=\sum_{k=-T}^{T}h[k]\cdot f[i-k]$ (2) $(f*h)[i,j]=\sum_{k=-T}^{T}\sum_{m=-T}^{T}h[k,m]\cdot f[i-k,j-m]$ (3) where $h[k]$ and $h[k,m]$ are the 1-D and 2-D kernels whose parameters are learned during the training phase, and $f$ is the 1-D or 2-D signal at hand. The first part of the network is represented by the lower branch of the scheme presented in Fig. 2 and it is designed to receive a sequence of images and to output abstract features which encode the information from frames. The input dimension is $B\times N\times W\times H$, where $B$ is the batch size, $N$ is the length of the video sequence, whereas $W$ is the width and $H$ is the corresponding height. The video frames are analysed one after another. The second part of the network is also a CNN-based architecture whose aim is to process the spectrograms of the audio signals and to extract meaningful features from them. Two remarks should be made at this point. Firstly, the number of convolutional layers is rather small. Secondly, the last layer contains a smaller number of neurons if compared to the first part of the network which deals with image analysis. This is argued by the fact that images contain more information than the audio signals (e.g., the position of the eyebrows, eyes, mouth or cheeks may indicate person’s emotional state). More specifically, non-verbal information is essential in discovering the emotional state of an individual. For example, happiness can be unambiguously determined from the facial expression of a person. However, vocal expression is needed to detect anger [4]. The third part of the network shown in Fig. 2 is a classifier composed of two fully-connected (FC) layers. After the inference through the before-mentioned parts of the network, the abstract features are concatenated and the resulting feature vector is introduced in the last part of the network, which provides the probability distribution of the emotion in the analyzed video. As already mentioned, the lengths of the feature vectors for frames and sound are different, in proportion of 4:1, because the importance of the visual data is greater. The output is a tensor of $L$ elements $\\{y_{x,1},y_{x,2},\ldots,y_{x,L}\\}$ that indicates the emotional state of the person (i.e., $L$ being the number of emotional states taken into consideration). In order to obtain the probability distribution for the emotional states, we consider using a SoftMax function that transforms the vector of 6 scores $\\{y_{x,1},y_{x,2},\ldots,y_{x,L}\\}$ into a normalized vector of probabilities $\\{p_{x,1},p_{x,2},\ldots,p_{x,L}\\}$, using the formula: $p_{x,c}=\frac{e^{y_{x,c}}}{\sum_{c^{\prime}=1}^{L}e^{y_{x,c^{\prime}}}}$ (4) for $c\in\\{1,...,L\\}$. The loss function considered for training is the Cross-Entropy, defined, for an observation $x$, as: $\mathcal{L}(x)=-\sum_{c=1}^{L}\delta_{x,c}\log(p_{x,c})$ (5) where $\delta_{x,c}$ is the binary indicator (0 or 1) showing if class label $c$ is correct for observation $x$ or not and $p_{x,c}$ is the predicted probability of observation $x$ pertaining to class $c$. ## V Experimental Setup and Results In order to prove the effectiveness of the proposed classification architecture that the audio data is helpful for the classification scope, we did two experiments. In the first experiment, we consider as input data just the sequence of images and we removed the part of the network which processes spectrograms. In the second experiment, we used the entire network architecture described in the previous section which receive data from two different sources, i.e., audio and visual. Several setups are considered during the implementation of the proposed neural network. As mentioned above, the spectrograms of the signals must have equal dimensions even if the analysed audio signals are different in length. We choose to resize all the spectrograms to $192\times 120$. Similarly, the images containing the detected faces are resized to images of $98\times 80$ pixels. The experiments were implemented in PyTorch on a computer with NVIDIA GTX 1080Ti graphic board and Intel i7 processor. We tried more optimizers, but the best performance regarding time of convergence and network accuracy was achieved by Adam. The learning rate was $10^{-4}$, with a weight decay of $5\cdot 10^{-5}$. At the training phase, we set the batch size to be 32 and we kept 20 frames equally distanced from every video. Therefore, the input data size is $32\times 20\times 98\times 80$ for the first part of the network and $1\times 1\times 192\times 120$ for the second part of the network. In order to learn the parameters that minimize the loss function, we have used a learning rate of $0.001$ and Adam as the method for stochastic optimization [17]. The testing method is leave-one-out. More precisely, we trained the network on all actors except one, then we tested the network on the actor that was removed from the dataset. This procedure was made for all 91 actors from database and the obtained results were averaged to obtain the final classification accuracy score. The loss function values obtained on train and test sets are showed for different epoch numbers in Fig. 4, whereas the corresponding accuracy values are given in Fig. 5. It can be easily observed that the loss function, computed for the training set, has a fast decay in the first epoch, followed by a slower decay in the next epochs, as showed in Fig. 4. We mention that the network training was stopped after four epochs in average, just before overfitting occurs. Moreover, as shown in Fig. 5, the accuracy starts to decay at the fifth epoch. Also, the validation loss was greater in some cases because the Cross-Entropy loss function is known to punish significantly the network if the classification returns unsuitable results. Because the overfitting was a constant problem in the training process, new methods of data augmentation could be added, such as near infra–red (NIR) generated data [18]. The obtained results on the test set are reported in Table II for two different modes of the recordings, namely video and audiovisual, respectively. It is worth mentioning that the accuracy is reported considering the leave- one-out strategy, whereas in Table I the human accuracy was computed over the whole dataset. TABLE II: Classification results on CREMA-D | Mean accuracy over 91 actors | Standard deviation ---|---|--- Video | 62.84% | 14.06 Audio + Video | 69.42% | 13.25 ## VI Conclusion In this article, we present a network architecture that is able to recognize emotion by combining visual with audio information. The output of the network is a distribution of probabilities for each type of emotion considered for training. The architecture is CNN-based, whilst the time series of frames are processed in a pipeline that takes into consideration multiple channels for each image. This approach introduced the data variation in time, which is a key point in emotion recognition because each emotion evolves in intensity over time. The results sustain the idea that having more information about a subject is important for determining the emotion with increased accuracy. In this sense, it can be observed that adding the audio information (i.e., which is a signature for each subject) in the network, yield an improvement of almost 6.57% in the accuracy results. Also, we can observe that having multimodal information about a subject the standard deviation of predictions is lower, which sustains the fact that the network performs better when the input is a more complex information, in our case video and audio information. Moreover, our results, in both experiments, outperform the human accuracy reported by the authors of CREMA-D dataset, in video mode with 4.64% and, in video combined with audio, with 5.82%. As future development, the emotion intensity levels reported on CREMA-D samples could be taken into consideration to further increase the accuracy of the emotion recognition system. This information may help the training process by pondering the relative importance of every sample. Other future research directions include the use different techniques for time-frequency analysis, such as the Wigner-Ville distribution or the Continuous Wavelet Transform, which could lead to improved information extraction from the audio signal. ## Acknowledgment This work has been partially supported by the Ministry of Innovation and Research, UEFISCDI, project SPIA-VA, agreement 2SOL/2017, grant PN- III-P2-2.1-SOL-2016-02-0002. ## References * [1] Y.-I. Tian, T. Kanade, and J. F. Cohn, “Recognizing action units for facial expression analysis,” _IEEE Transactions on pattern analysis and machine intelligence_ , vol. 23, no. 2, pp. 97–115, 2001. * [2] P. Ekman, “Facial expression and emotion.” _American psychologist_ , vol. 48, no. 4, p. 384, 1993. * [3] C. Marechal, D. Mikołajewski, K. Tyburek, P. Prokopowicz, L. Bougueroua, C. Ancourt, and K. Wegrzyn-Wolska, _Survey on AI-Based Multimodal Methods for Emotion Detection_. Springer International Publishing, 2019, pp. 307–324. * [4] H. Cao, D. G. Cooper, M. K. Keutmann, R. C. Gur, A. Nenkova, and R. Verma, “Crema-d: Crowd-sourced emotional multimodal actors dataset,” _IEEE transactions on affective computing_ , vol. 5, no. 4, pp. 377–390, 2014. * [5] P. Barros, N. Churamani, E. Lakomkin, H. Siqueira, A. Sutherland, and S. Wermter, “The omg-emotion behavior dataset,” in _2018 International Joint Conference on Neural Networks (IJCNN)_. IEEE, 2018, pp. 1–7. * [6] P. Burkert, F. Trier, M. Z. Afzal, A. Dengel, and M. Liwicki, “Dexpression: Deep convolutional neural network for expression recognition,” _arXiv preprint arXiv:1509.05371_ , 2015. * [7] C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich, “Going deeper with convolutions,” in _Computer Vision and Pattern Recognition (CVPR)_ , 2015. [Online]. Available: http://arxiv.org/abs/1409.4842 * [8] Z. Yu and C. Zhang, “Image based static facial expression recognition with multiple deep network learning,” in _Proceedings of the 2015 ACM on International Conference on Multimodal Interaction_. ACM, 2015, pp. 435–442. * [9] S. Li and W. Deng, “Deep facial expression recognition: A survey,” _CoRR_ , vol. abs/1804.08348, 2018. [Online]. Available: http://arxiv.org/abs/1804.08348 * [10] S. E. Kahou, X. Bouthillier, P. Lamblin, C. Gulcehre, V. Michalski, K. Konda, S. Jean, P. Froumenty, Y. Dauphin, N. Boulanger-Lewandowski _et al._ , “Emonets: Multimodal deep learning approaches for emotion recognition in video,” _Journal on Multimodal User Interfaces_ , vol. 10, no. 2, pp. 99–111, 2016. * [11] A. Dhall, R. Goecke, J. Joshi, M. Wagner, and T. Gedeon, “Emotion recognition in the wild challenge (emotiw) challenge and workshop summary,” in _Proceedings of the 15th ACM on International conference on multimodal interaction_. ACM, 2013, pp. 371–372. * [12] P. Tzirakis, G. Trigeorgis, M. A. Nicolaou, B. W. Schuller, and S. Zafeiriou, “End-to-end multimodal emotion recognition using deep neural networks,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 11, no. 8, pp. 1301–1309, Dec 2017. * [13] K. Vougioukas, S. Petridis, and M. Pantic, “Realistic speech-driven facial animation with gans,” _ArXiv_ , vol. abs/1906.06337, 2019. * [14] R. Beard, R. Das, R. W. M. Ng, P. G. K. Gopalakrishnan, L. Eerens, P. Swietojanski, and O. Miksik, “Multi-modal sequence fusion via recursive attention for emotion recognition,” in _Proceedings of the 22nd Conference on Computational Natural Language Learning_. Brussels, Belgium: Association for Computational Linguistics, Oct. 2018, pp. 251–259. [Online]. 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2024-09-04T02:54:55.966086

2003.00467

arxiv-papers

# NeuroTac: A Neuromorphic Optical Tactile Sensor applied to Texture Recognition Benjamin Ward-Cherrier, Member, IEEE, Nicholas Pestell, Student Member, IEEE, Nathan F. Lepora, Member, IEEE BWC was supported by a University of Bristol Vice-Chancellor’s fellowship, NP was supported by an EPSRC DTP studentship and NL was supported in part by a Leverhulme Trust Research Leadership Award on ’A biomimetic forebrain for robot touch’ (RL-2016-039).Authors are with the Department of Engineering Mathematics, University of Bristol and Bristol Robotics Laboratory, University of Bristol, UK. Email: {b.ward-cherrier, n.pestell<EMAIL_ADDRESS> ###### Abstract Developing artificial tactile sensing capabilities that rival human touch is a long-term goal in robotics and prosthetics. Gradually more elaborate biomimetic tactile sensors are being developed and applied to grasping and manipulation tasks to help achieve this goal. Here we present the neuroTac, a novel neuromorphic optical tactile sensor. The neuroTac combines the biomimetic hardware design from the TacTip sensor which mimicks the layered papillae structure of human glabrous skin, with an event-based camera (DAVIS240, iniVation) and algorithms which transduce contact information in the form of spike trains. The performance of the sensor is evaluated on a texture classification task, with four spike coding methods being implemented and compared: Intensive, Spatial, Temporal and Spatiotemporal. We found timing-based coding methods performed with the highest accuracy over both artificial and natural textures. The spike-based output of the neuroTac could enable the development of biomimetic tactile perception algorithms in robotics as well as non-invasive and invasive haptic feedback methods in prosthetics. ## I INTRODUCTION The long-term scientific goal of human-like artificial touch is being worked towards with the creation of gradually more elaborate biomimetic tactile sensors. The development of sensors which mimic aspects of biological touch could lead to safer robots and more intuitive and versatile prosthetic devices. An example of such sensors are neuromorphic tactile sensors, which aim to replicate the spike-based representation of information found in the nervous system. The principal objective of neuromorphic engineering is to develop technologies which can exploit efficient representations of information and operate in a rapid, power-efficient manner. Using neuromorphic technologies with event- based outputs also holds potential for integration with the human nervous system, as has been demonstrated with artificial retinas [1]. Optical tactile sensors have also recently demonstrated progress on a number of tactile tasks [2, 3], and have the advantage of capitalizing on advances in image recognition and machine learning techniques. These technologies will be combined here to develop a tactile sensor for robotic manipulation and prosthetics applications. The aims of this paper are as follows: * • Develop a novel neuromorphic optical tactile sensor to enable the development and investigation of biomimetic spike-based information processing methods. * • Validate this sensor on a texture classification task. * • Investigate 4 spike coding methods (intensive, spatial, temporal and spatiotemporal) and their effect on texture recognition performance. Figure 1: Transduction, encoding and decoding mechanisms for the neuroTac sensor. The sensor mimics biological processes by accumulating pixel events (Potentials) from an event-based camera and combining them into taxel events (Spikes). The neuroTac sensor, described here, follows the tradition of neuromorphic technologies [4] in seeking to produce and decode spike-based information (Fig. 1). The spike-based sensor output is coded using 4 different methods: intensive (overall number of spikes), spatial (number of spikes per taxel), temporal (number of spikes per time window) and spatiotemporal (Van Rossum metric [5]). Although neuromorphic devices may present certain advantages over their non-neuromorphic counterparts (speed and energy efficiency), our principal objective here is linked to biomimetism. The TacTip sensor emulates the internal structure of human skin [6], and the neuroTac adds to that biomimetic morphology by producing a neuromorphic, spike-based output. We aim to develop the neuroTac and associated spike-based information processing methods to investigate the advantages this approach might provide biological organisms. The performance of the sensor and its associated coding methods is validated on texture recognition tasks, in which artificial and natural textures are identified using a K-nearest-neighbour (KNN) algorithm. The sensor successfully performs texture recognition, with temporal methods producing the highest classification accuracy. This underlines the importance of spike timing in texture recognition tasks. ## II BACKGROUND AND RELATED WORK One of the strongest motivations for studying the human sense of touch is its important role in our interactions and in the manipulation of our environment. Artificial tactile sensing thus often mimics particular features of biological touch through biomimetic hardware or perception algorithms [7]. An emerging area of biomimetic tactile sensing aims to replicate biological spike-based signalling through event-based systems and study the encoding of tactile information [8]. This is generally referred to as neuromorphic sensing [4]. Neuromorphic sensing arose from seminal work on silicon neurons [9] and in the area of vision it has led to the successful integration of artificial retinas with the human nervous system [1]. Evidence suggests that a neuromorphic tactile sensor could similarly be used to restore a natural sense of touch to amputees [10, 11]. Large-scale event-based tactile sensors have been developed for use as robotic skins [12], with a focus on the efficient processing of large quantities of asynchronous data. Bartolozzi et. al. also developed an architecture for use with distributed off-the shelf tactile sensors, transforming raw data into event-based signals [13]. Another example of a large-scale system for the investigation of spiking outputs in neuromorphic touch is that developed by Lee et. al. [14]. These systems represent crucial technological developments in the area of event-based systems, and will be essential in the creation of rapidly reacting fully tactile robot systems. Here, our focus is not on large-scale event-based robotic skins or platforms for investigating spike-based encoding but rather on an artificial fingertip with high spatial resolution for fine in-hand manipulation tasks. Oddo et. al. developed a tactile sensor more in line with this design objective [15]. Their neuromorphic tactile fingertip comprises 16 taxels combining raw outputs from 4 microelectromechanical system (MEMS) sensors, fed to an Izhikevich model to produce spikes. This system simulates the outputs from biological SA1 afferents, and has been proven capable of accurately distinguishing natural textures [16]. A crucial difference with the sensor presented here is that the neuroTac emulates the fast-adapting responses of FA1 afferents to dynamic contact (rather than the slowly adapting SA1 afferent responses). Recently, a sensor has been developed using similar hardware, with an event-based camera capturing the deformations of a deformable membrane [17]. The system was calibrated for use as a force sensor and to measure material hardness. The neuroTac sensor design aligns more closely both with traditional tactile sensor designs comprising taxels (internal markers which transduce contact), and biological fingertip structures (with taxels functioning as artificial mechanoreceptors). We validate the neuroTac on a texture classification task, in which 3d-printed and natural textures are classified by sliding the sensor horizontally across each stimulus. Texture identification is a common task in tactile sensing due to its important implications in object recognition and manipulation. Studies on the tactile sensing of textures generally involve naturally occurring textures [18, 16, 19, 20]. Here, we initially wish to investigate the sensor’s response to highly structured textures. To achieve this, we utilize purposely designed 3d-printed stimuli with regular grids of cylindrical bumps as has been done in past studies of biological touch [21]. Following this, we evaluate the sensor’s classification performance on a set of 20 natural textures. ## III METHODS ### III-A NeuroTac design and operation Figure 2: The neuroTac sensor. The tip contains internal pins treated as mechanoreceptors, which produce pixel events at the event-based camera. These are pooled and converted to taxel events (akin to biological spikes) upstream. ##### Sensor design The NeuroTac is based on the TacTip sensor [6], a 3d-printed optical tactile sensor with a compliant dome-shaped outer membrane comprising biomimetic internal markers which emulate the internal structure of human fingertips [22]. To convert the TacTip into a neuromorphic sensor, we replace its camera module (ELP, USBFHD01M-L21) with an event-based camera (iniVation, DAVIS240) which processes only dynamic events within the image frame. The DAVIS240 is an evolution of the DVS128 [23] and comprises 240x180 pixels, which process changes in brightness through independent electronic circuits to produce events in the address-event representation (AER) format [24]. These events are combined by taxel and transmitted by the sensor, analogously to biological spike trains (see Section III-A for more details on the sensor operation). Like the TacTip, the NeuroTac is made up of 3 main hardware elements (Fig. 2): * • Tip: This is a compliant, 3d-printed modular part whose outer membrane comes into contact with the environment. Its internal surface comprises white-tipped pins which are displaced during contact. It is filled with silicone gel (Techsil, RTV27905) and covered with an acrylic lens to protect electronic components. * • LED ring: This illuminates the tip’s internal surface. * • Camera: Housed within the main part of the sensor (the 3d-printed Body), this camera is event-based (iniVation, DAVIS240) and produces AER outputs in response to movements of the sensor’s internal markers. Figure 3: Sensor operation. Pixel events produced by the camera (iniVation, DAVIS240) are initially filtered, then pooled into a single taxel event. Finally, the position of taxels is updated (see Section III-A). Figure 4: Experimental setup: The NeuroTac is attached to a 6-dof industrial robot arm (ABB, IRB120) which is used to slide the sensor horizontally across the 3d-printed textures. ##### Sensor operation As the NeuroTac’s compliant membrane deforms through contact, its 49 white- tipped internal pins deflect, and their movement triggers events in the camera (iniVation, Davis240) in the Address-Event Representation (AER) format. These events are produced by thresholding brightness changes at each photodiode in parallel [23, 24], leading to fast data transmission and high temporal precision. We designate these events ’pixel events’ as they are created at the pixel level. Data transduction by the neuroTac then occurs through the 3 following steps (illustrated in Fig. 3): * • Noise filtering: each of the sensor’s 49 pins represents a taxel, and has a receptive field assigned to it (6 px diameter). Pixel events that occur outside a taxel’s receptive field, or which do not have another pixel event occur within a given spatiotemporal window (neighbouring pixels, 5 ms) are filtered out. * • Pooling: pixel events are pooled over a short duration (20 ms) and combined into a ’taxel event’ based on the receptive field they are located within. Each taxel event is an array comprising 3 numbers: the number of pixel events it contains, their average location and their average timing. Note that each taxel event is interpreted as a spike associated with an artificial mechanoreceptor. * • Position update: Receptive fields are re-centred around each taxel by shifting towards detected pixel events, to account for each pin’s movement across the image. ### III-B Experimental setup The first experiment with artificial textures involves sliding the NeuroTac across 11 3d-printed textures. Artificial textures consist of rectangular grids of cylindrical bumps (1 mm height) with equal spacing and diameter. The texture grid size varies from 0 mm (smooth surface) to 5 mm in steps of 0.5 mm (Fig. 4). Figure 5: Natural textures used for classification. 20 textures were used, with a full list presented in Table II. Data is collected by mounting the NeuroTac on a 6-dof robotic arm (ABB, IRB120) and sliding it horizontally across the textures (Fig. 4). The robot slides the NeuroTac across each texture at a speed of 15 mm/s, over a distance of 60 mm, comprising one data sample. We collect 100 samples for each texture, to obtain a dataset of 100 (number of runs) $\times$ 11 (number of textures) $\times$ 49 (number of taxels) spike trains. In the second experiment on natural texture classification, the data collection procedure is repeated for 50 runs over 20 natural textures(see Table II). ### III-C Spike train encoding methods The multi-taxel spike trains obtained from the sensor are denoted as a matrix of spike times $t_{n}^{i}$, where $n=1,2...,N$ (N denotes the number of taxels, $N=49$ in this case) and $i=1...I_{n}$ ($I_{n}$ denotes the number of spikes for taxel $n$). A sample is denoted as the multi-taxel spike train resulting from the sensor sliding 60 mm horizontally across one texture. Following, we describe the 4 coding methods applied to transform the spike trains $t_{n}^{i}$ to the encoded representation $R$. #### III-C1 Intensive coding This encoding consists of the average spike count per taxel for a given data sample. We name it intensive, analogously to work on biological touch [25], since it can be interpreted as an overall intensity of the sample signal with an absence of any spatial or temporal resolution. The encoding produces a single average spike count per sample, which is used for texture classification. $R=\frac{1}{N}\sum_{n=1}^{N}\sum_{i=1}^{I_{n}}t_{n}^{i}$ (1) where $N$ is the total number of taxels and $I_{n}$ denotes the number of spikes for taxel n. #### III-C2 Spatial coding Here we consider the topology of the sensor, and sum the spike counts separately for each of the sensor’s 49 taxels. The resulting array of 49 spike rates is used as an encoded representation of the texture $R_{n}=\sum_{i=1}^{I_{n}}t_{n}^{i}$ (2) #### III-C3 Temporal coding Temporal coding considers a rolling window of width $\Delta t$ over each data sample, which is rolled forward over the time domain in timesteps of 1 ms. Within the window $\Delta t$, the average spikes per taxel are recorded before proceeding to the next timestep. $R_{n}(t)=\frac{1}{N}\sum_{t=t}^{t+\Delta t}t_{n}^{i}(t)$ (3) The $\Delta t$ parameter in this encoding method affects the classification accuracy, therefore we optimize it through brute-force optimization within a limited range of 1-200 ms (see Section IV-A). #### III-C4 Spatiotemporal coding Spatiotemporal coding uses the spatial and temporal features of spike trains produced by the sensor. A multi-neuron Van Rossum distance [26] is used as a metric in the texture classification, to ensure that both spatial and temporal dimensions of the data are considered. The spatiotemporal encoded representation can be considered the convolution of the multi-taxel spike train with an exponential kernel used in the Van Rossum distance calculation (see Section III-D). $R_{n}^{i}(t)=t_{n}^{i}h(t-t_{i})$ (4) where $h=\begin{cases}0,&t\leq 0\\\ \frac{1}{\tau}e^{-t/\tau},&t\geq 0\end{cases}$ (5) $\tau$ is a time constant parameter which can be optimized to create the most accurate clustering of texture representations. ### III-D Classification Texture classification is performed through a KNN algorithm (k=4), which assigns a class (texture grid size 0-5 mm in 0.5 mm steps here) to a test sample based on its 4 closest training samples. For the intensive, spatial and temporal coding methods, a standard Euclidean distance is used to calculate distances between samples. For the spatiotemporal coding, we use a multi-neuron Van Rossum distance [26] which involves the convolution of spike trains with an exponential kernel (Section III-C4) before applying a distance metric. In the original Van Rossum metric [5], which applies to single neuron spike trains, the distance metric between two spike trains is simply: $d_{2}(t_{1},t_{2};h)=\sqrt{\int dt(f_{1}-f_{2})^{2}}$ (6) where $f_{1}$ and $f_{2}$ are the convolved functions. The extension of the Van Rossum metric to multi-neuron cases, as described by Houghton and Sen [26], introduces an additional parameter $\theta$ which represents the correlation between neurons. The parameter’s effect is best described by taking its cosine, which varies from 0 to 1 with: * • $cos\theta=0$. Labelled line code: each neuron is considered independent and their distances are summed. * • $cos\theta=1$. Summed population code: spike trains are superimposed before calculating the distance metric. * • $1>cos\theta>0$: an intermediate level of inter-neuron correlation between these two extremes. The $\theta$ angle is an optimizable parameter, and thus we perform a two parameter Bayesian optimization of $\tau$ (exponential kernel time constant) and $\theta$ for spatiotemporal encoding as described in section IV-B. ## IV RESULTS ### IV-A Inspection of data - Artificial textures Figure 6: Examples of the spike trains produced by the NeuroTac when sliding across three different 3d-printed textures. From left to right, textures are 0 mm grid (smooth), 2.5 mm grid and 5 mm grid, with the corresponding spike trains, spatial and temporal distributions displayed below them. Data is gathered by sliding the sensor horizontally across the set of 11 artificial textures (see Section III-B). The output of the NeuroTac consists of 49 spike trains which represent the events being produced at each taxel. The spike trains will vary in overall intensity for different textures, as well as having distinct spatial and temporal signatures. Examples of spike trains obtained for 3 textures of grid sizes 0 mm (smooth), 2.5 mm and 5 mm are displayed here (Fig. 6, second row). It is visually noticeable that these 3 multi-neuron spike trainsdiffer, with the number of spikes produced increasing with texture coarseness. As expected, applying intensive coding to these samples gives readily distinguishable average spike counts over all taxels of $4.63$ spikes/taxel (0 mm grid), $9.84$ spikes/taxel (2.5 mm grid) and $34.73$ spikes/taxel (5 mm grid). Spatial coding (spike frequency per taxel) reveals an additional topological structure to the data (Fig. 6, third row), wherein certain taxels seem to fire at higher rates than others (taxels 10, 17 and 27 for the 2.5 mm texture for instance). This could be due to their location on the sensor being directly in the path of the texture’s raised bumps, leading to more events being produced for those taxels. We also illustrate temporal coding with a rolling window of size $\Delta t=159\,ms$ (Fig. 6, bottom row), in which the increased response with texture grid size is visibly apparent. There is also a sharp increase in activity at the beginning of each sample which corresponds to the start of the sensor’s horizontal sliding motion, and is likely linked to the coefficient of static friction between the sensor and texture. ### IV-B Texture classification - Artificial textures Figure 7: Confusion matrices for artificial texture classification with each of the 4 encoding methods: Intensive (Top Left), Spatial (Top Right), Temporal (Bottom Left) and Spatiotemporal (Bottom Right). For each encoding method (see Section III-C), the resulting data is classified with a KNN algorithm to identify the 11 3d-printed textures. Note that temporal and spatiotemporal encodings are both parameterised (see Section III-C3, III-C4), and their parameters ($\Delta t$ for temporal encoding, $cos\theta$ and $\tau$ for spatiotemporal encoding) are optimised to maximize classification accuracy. Temporal encoding contains a single parameter $\Delta t$, allowing us to perform a brute-force optimization over a range of values from 1 to 200 ms. We set an upper threshold on the $\Delta t$ parameter both to accelerate the optimization procedure and ensure the method is distinct from intensive coding (which corresponds to $\Delta t=T$, where T is the full duration of the sample). This optimization process give us a value of $\Delta t=159\,ms$. In the case of the spatiotemporal encoding, we perform a Bayesian optimization over the two parameters $cos\theta$ and $\tau$. The domain space we apply optimization over is $cos\theta=0-1$ and $\tau=10-100ms$. We place an upper limit on the $\tau$ parameter for faster convergence and to ensure spatiotemporal coding is distinct from spatial coding. The parameters converge over 1000 optimization epochs to values of $cos\theta=0.4$ and $\tau=76ms$. Coding method | Performance - Artificial textures (%) ---|--- Intensive | 78.5 $\pm$ 41.1 Spatial | 95.5 $\pm$ 20.6 Temporal | 98.1 $\pm$ 13.7 Spatiotemporal | 98.3 $\pm$ 13 TABLE I: Comparison of different encoding methods using leave-one-out cross- validation for texture classification. First, we run a simple 80/20 train-test split on the gathered dataset, and attempt to classify the textures using a KNN classifier (k=4). The results are presented in the form of confusion matrices (Fig. 7), and provide insight into each coding method’s performance. Visually, we can observe that the intensive method is the least effective, particularly for the smoother textures (bump diameter 0-2 mm). The spatial coding method seems more accurate, though there is still some slight inaccuracy in the classification of smoother textures (bump diameter 0-1.5 mm). Temporal and spatiotemporal coding appears to provide near-perfect classification accuracy, indicating that sliding the sensor over the textures produces discriminable time-dependent features. To obtain a more complete picture of the performance of each coding method, we perform a leave-one-out cross-validation over the gathered dataset. Results are presented in table I and mirror the results displayed in the train-test split confusion matrices. Intensive coding performs worst, with spatial information improving performance significantly. However both temporal and spatiotemporal coding have the highest accuracy, indicating the sensor produces features in the temporal domain which are likely most representative of the classified textures. ### IV-C Texture classification - Natural textures Here we seek to test the sensor’s performance more thoroughly on texture recognition by replacing the 3d-printed textures with natural textures, listed below in Table II. We again run a KNN classifier (k=4) over 20 runs with an 80/20 train-test split (giving 4 test runs for each texture). The results are presented in the form of confusion matrices for each coding method (Fig. 8). Figure 8: Confusion matrices for natural texture classification with each of the 4 encoding methods: Intensive (Top Left), Spatial (Top Right), Temporal (Bottom Left) and Spatiotemporal (Bottom Right). Classification accuracy appears generally strong, with Intensive coding the weakest method, and Temporal and Spatiotemporal coding performing the best, as was the case for artificial textures. This is confirmed through a leave-one- out cross-validation, with results displayed in Table III. One significant difference between the natural and artificial textures is that natural textures appear more irregular in their misclassifications. The artificial textures lie along a regularly designed gradient of coarseness, and thus classification errors occur most often between neighbouring classes. In the case of natural textures, there is no such regularity or order and classification errors appear to be spread more widely across the confusion matrices (Fig. 8). It is also interesting to note that classification errors vary between coding methods. For instance for Intensive and Spatial coding, fake fur and felt (textures 6 and 17) are confused, whereas Temporal and Spatiotemporal coding can distinguish them. Equally, liquid satin (texture 7) is misclassified as foam (texture 1) only when Spatiotemporal coding is applied (Fig. 8). The fact that these misclassifications vary between coding methods likely stems from the fact that each method produces a distinct set of features. ## V Discussion The neuroTac sensor developed here is a neuromorphic optical tactile sensor, with a data output consisting of multi-taxel spike trains. The spike trains were encoded using four different bio-inspired encoding mechanisms: Intensive, Spatial, Temporal and Spatiotemporal. We validated the sensor through a texture classification task, in which 11 3d-printed textures (grid sizes 0-5 mm in steps of 0.5 mm) and 20 natural textures were discriminated using a KNN classification algorithm. We chose texture discrimination as a validation task as it has been a key experimental procedure in psychophysical and neuroscientific studies of touch. Here we will attempt to contrast our results with existing theories in these fields, and identify limitations as well as gaps for further investigation. We found that applying spatial coding to the neuroTac data produced high classification accuracy for coarser textures (2.5-5 mm grid size), but performed less well for smoother textures (0-2.5 mm grid size). This result seems to concur with Katz’s duplex theory of human tactile perception, with spatial resolution being important for rougher textures, and vibration cues taking over as textures get smoother [27]. The stronger performance of the temporal and spatiotemporal coding methods for smooth textures could be linked to their detection of high frequency vibrational cues. Texture | Texture | Texture | Texture ---|---|---|--- number | name | number | name 1 | Foam | 11 | Flarefree net 2 | Plywood | 12 | Embroidered cotton 3 | MDF | 13 | Shirt canvas 4 | Acrylic | 14 | Fairydust organza 5 | Wool | 15 | Sequins allover 6 | Fake fur | 16 | Metallic mesh 7 | Liquid satin | 17 | Felt 8 | Shimmer organza | 18 | Needlecord 9 | Tweed | 19 | Fleece 10 | Lace | 20 | Microdot foil TABLE II: Natural textures and their corresponding number. Coding method | Performance - natural textures (%) ---|--- Intensive | 69.8 Spatial | 85.5 Temporal | 93.0 Spatiotemporal | 92.8 TABLE III: Comparison of the 4 different coding methods using leave-one-out cross-validation for texture classification of the 20 natural textures. It has also been suggested in a recent study that the specific timing of spikes carries significance for texture discrimination, as it could encode spatial frequency features of the textures being contacted [28]. The spatiotemporal coding method described here could capture these precise spike timings, with a resolution dependent on the time constant parameter $\tau$. Here $\tau$ was optimized to be 76 ms, which could indicate an approximate timescale for the frequency features of the textures considered. Human touch and proprioception provide much richer and more complex data than that explored here, and most theories suggest that human texture recognition relies on a complex coding strategy involving inputs from several mechanoreceptors (Pacinian corpuscles, Merkel cells) [28]. Further studies with the neuroTac could include methods for simulating input from these biological afferents, for instance by feeding taxel positions as an input to spiking neuron models. 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2024-09-04T02:54:55.977705

2003.00479

arxiv-papers

# The $L^{p}$-$L^{q}$ problems of Bergman-type operators Lijia Ding School of Mathematical Sciences, Peking University, Beijing, 100086, P. R. China School of Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. China<EMAIL_ADDRESS>and Kai Wang School of Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. China <EMAIL_ADDRESS> ###### Abstract. Let $\mathbb{B}^{d}$ be the unit ball on the complex space $\mathbb{C}^{d}$ with normalized Lebesgue measure $dv.$ For $\alpha\in\mathbb{R},$ denote $k_{\alpha}(z,w)=\frac{1}{(1-\langle z,w\rangle)^{\alpha}},$ the Bergman-type integral operator $K_{\alpha}$ on $L^{1}(\mathbb{B}^{d},dv)$ is defined by $K_{\alpha}f(z)=\int_{\mathbb{B}^{d}}k_{\alpha}(z,w)f(w)dv(w).$ It is an important class of operators in the holomorphic function space theory over the unit ball. We also consider the integral operator $K_{\alpha}^{+}$ on $L^{1}(\mathbb{B}^{d},dv)$ which is given by $K_{\alpha}^{+}f(z)=\int_{\mathbb{B}^{d}}|k_{\alpha}(z,w)|f(w)dv(w).$ In this paper, we completely characterize the $L^{p}$-$L^{q}$ boundedness of $K_{\alpha},K_{\alpha}^{+}$ and $L^{p}$-$L^{q}$ compactness of $K_{\alpha}.$ The results of boundedness are in fact the Hardy-Littlewood-Sobolev theorem but also prove the conjecture of [4] in the case of bounded domain $\mathbb{B}^{d}.$ Meanwhile, a trace formula and some sharp norm estimates of $K_{\alpha},K_{\alpha}^{+}$ are given. ###### Key words and phrases: Bergman projection; Embedding theorem; Compact operator; Hardy-Littlewood- Sobolev theorem; Norm estimate ###### 2010 Mathematics Subject Classification: Primary 47G10; Secondary 47A30; 47B05 The first author was partially supported by Fudan University Exchange Program (2018017). The second author was partially supported by NSFC (11722102), the Alexander von Humboldt Foundation (1151823), Shanghai Pujiang Program (16PJ1400600). ## 1\. Introduction Let $\mathbb{B}^{d}$ be the unit ball on the complex space $\mathbb{C}^{d}$ with the normalized Lebesgue measure $dv.$ For $\alpha\in\mathbb{R},$ denote $\alpha$-order Bergman-type kernel function $k_{\alpha}(z,w)$ on $\mathbb{B}^{d}\times\mathbb{B}^{d}$ by $k_{\alpha}(z,w)=\frac{1}{(1-\langle z,w\rangle)^{\alpha}}.$ Clearly the $(d+1)$-order Bergman-type kernel function $k_{d+1}(z,w)$ is the standard Bergman kernel on $\mathbb{B}^{d}.$ Denote Bergman-type integral operator $K_{\alpha}$ on $L^{1}(\mathbb{B}^{d},dv)$ by $K_{\alpha}f(z)=\int_{\mathbb{B}^{d}}k_{\alpha}(z,w)f(w)dv(w).$ Such operators $K_{\alpha}$ play an important role in complex analysis of several variables and operator theory; in particular, when $\alpha=d+1,$ $K_{d+1}$ is the standard Bergman projection over the unit ball $\mathbb{B}^{d}.$ Indeed, for any $\alpha>0,$ if restrict $K_{\alpha}$ to the holomorphic function space $H(\mathbb{B}^{d}),$ then every $K_{\alpha}$ is a spacial form of fractional radial differential operator $R^{s,t},$ which is a kind of very useful operators in the Bergman space theory on the unit ball, see Lemma 2.8; many key results on Bergman spaces can be deduced from the fractional radial differential operators, see example for [24, 25]. On the other hand, the operators $K_{\alpha}$ play a significant role in the characterization of weighted Bloch spaces and Lipschitz spaces over the unit ball $\mathbb{B}^{d},$ see [24, 25, 26]. We also consider the kernel integral operator $K_{\alpha}^{+}$ on $L^{1}(\mathbb{B}^{d},dv),$ which is given by $K_{\alpha}^{+}f(z)=\int_{\mathbb{B}^{d}}\frac{f(w)}{|1-\langle z,w\rangle|^{\alpha}}dv(w).$ The operators $K_{\alpha}^{+}$ can be regarded as Riesz potential operators over the bounded domain $\mathbb{B}^{d}.$ Comparing to the classical Riesz potential operators over real Euclidian space $\mathbb{R}^{d},$ whose basic result concerning mapping properties is the Hardy-Littlewood-Sobolev theorem, see [13, 16, 19, 21] and references therein. For convenience, we write $L^{p}(\mathbb{B}^{d},dv)$ in the simple form $L^{p}(\mathbb{B}^{d})$ or $L^{p}$ for any $1\leq p\leq\infty$ without confusion arises. In the present paper, we mainly concern the $L^{p}$-$L^{q}$ problem for $K_{\alpha}$ and $K_{\alpha}^{+},$ namely we consider the boundedness and compactness of $K_{\alpha}$ and $K_{\alpha}^{+},$ $K_{\alpha},K_{\alpha}^{+}:L^{p}\rightarrow L^{q},$ for $1\leq p,q\leq\infty.$ Indeed, the results of $L^{p}$-$L^{q}$ boundedness are the Hardy-Littlewood-Sobolev theorem with respect to $K_{\alpha}^{+}$ over the unit ball $\mathbb{B}^{d}.$ Actually, on more general bounded domain $\Omega$ with the normalized Lebesgue measure $dv$ in $\mathbb{C}^{d},$ the $L^{p}$-$L^{q}$ boundedness for Bergman- type operators and in particular $L^{p}$-$L^{p}$ boundedness for the standard Bergman projection had attracted much interest in the past decades; the target spaces are even Bloch spaces, Lipschitz spaces and Sobolev spaces [12, 18]. As we all know, it is trivial that the standard Bergman projection $P$ is bounded for any bounded domain when $p=q=2.$ However, the problem becomes very complicated for general $1\leq p,q\leq\infty.$ Nevertheless, the known results show that depends strongly on the property of the domain $\Omega.$ When $\Omega$ is a strongly pseudoconvex domain with sufficiently smooth boundary, then the standard Bergman projection $P:L^{p}(\Omega)\rightarrow L^{p}(\Omega)$ is bounded for any $1<p<\infty,$ the conclusion is also true for the Bergman-type integral operators with the order of the kernel function no more than $d+1;$ we refer the reader to [9, 15, 18] along this line. Indeed, in the case of unit ball, the boundedness of more general Bergman type integral operators were considered in [9, 23, 24, 25, 26]. However, if $\Omega$ is a bounded symmetric domain of tube type with rank $\geq 2$, the boundedness of the standard Bergman projection $P:L^{p}(\Omega)\rightarrow L^{p}(\Omega)$ is conjectured by M. Stein that $P:L^{p}(\Omega)\rightarrow L^{p}(\Omega)$ is only bounded when $p$ belongs to a finite interval around $p=2;$ we refer the reader to [1, 2] and references therein. Although the $L^{p}$-$L^{q}$ boundedness for standard Bergman projection $P$ over tube type domains with rank $\geq 2$ has been considered a long time, it is still an open problem. Now return to our unit ball setting. In [10], X. Fang and Z. Wang established a relation between the boundedness of standard Bergman projection and Berezin transform on the weighted Bergman spaces over the unit disc $\mathbb{D}=\mathbb{B}^{1}.$ The compactness of standard Bergman projection $K_{2}:L^{\infty}(\mathbb{D})\rightarrow L^{q}(\mathbb{D})$ for $1\leq q<\infty$ was observed by K. Zhu in Section 3.6 of [26]. Recently, X. Fang and G. Cheng et al [4] completely solved the $L^{p}$-$L^{q}$ boundedness problem of $K_{\alpha}$ over the unit disc $\mathbb{D};$ they also considered the $L^{p}$-$L^{q}$ boundedness of Bergman-type operator over the upper half plane $\mathbb{U}=\\{z\in\mathbb{C}:\text{Im}(z)>0\\}.$ Not long afterward, G. Cheng et al [5] solved the $L^{p}$-$L^{q}$ boundedness problem of $K_{\alpha}$ in the spacial case $\alpha=1$ over the unit ball $\mathbb{B}^{d}$ for general $d\geq 1.$ The main difficulty in the case of high dimensional ball $\mathbb{B}^{d}$ is how to determine the critical exponents $d+1$ and $d+2,$ see the following theorems. In the present paper we completely describe the $L^{p}$-$L^{q}$ boundedness of $K_{\alpha},K_{\alpha}^{+}$ but also the $L^{p}$-$L^{q}$ compactness of $K_{\alpha}$ over the unit ball $\mathbb{B}^{d}(d\geq 1).$ The results of boundedness for $K_{\alpha}$ completely prove the conjecture of [4] but also extend some classical results [7, 14, 18, 23, 25, 26] in the case unit ball, the results of boundedness for $K_{\alpha}^{+}$ are essentially the Hardy-Littlewood theorem as mentioned before; however the results of compactness are almost entirely new. Firstly, it is trivial that $K_{\alpha},K_{\alpha}^{+}:L^{p}\rightarrow L^{q}$ are compact for any $1\leq p,q\leq\infty$ when $\alpha\leq 0.$ Thus we only concern the case $\alpha>0.$ The following five theorems are our main results. Theorem 1. If $d+1<\alpha<d+2,$ then the following conditions are equivalent: 1. (1) $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded; 2. (2) $K_{\alpha}^{+}:L^{p}\rightarrow L^{q}$ is bounded; 3. (3) $K_{\alpha}:L^{p}\rightarrow L^{q}$ is compact; 4. (4) $p,q$ satisfy one of the following inequalities: 1. (a) $\frac{1}{d+2-\alpha}<p<\infty,\frac{1}{q}>\frac{1}{p}+\alpha-(d+1);$ 2. (b) $p=\infty,q<\frac{1}{\alpha-(d+1)}.$ As an consequence of Theorem 1, the following Hardy-Littlewood-Sobolev inequality (HLS) is established over the bounded domain $\mathbb{B}^{d}.$ HLS 1. For any $1<p,s<\infty,\frac{1}{s}+\frac{1}{p}+\alpha<d+2$ and $d+1<\alpha<d+2,$ then there exists a constant $C$ which depends only on $p,\alpha,d,s$ such that $\left|\int_{\mathbb{B}^{d}}\int_{\mathbb{B}^{d}}\frac{f(w)g(z)}{|1-\langle z,w\rangle|^{\alpha}}dv(w)dv(z)\right|\leq C\|f\|_{L^{p}}\|g\|_{L^{s}},$ (1.1) for all $f\in L^{p}(\mathbb{B}^{d}),g\in L^{s}(\mathbb{B}^{d}).$ Theorem 2. If $0<\alpha\leq d+1,$ then the following conditions are equivalent: 1. (1) $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded; 2. (2) $K_{\alpha}^{+}:L^{p}\rightarrow L^{q}$ is bounded; 3. (3) $p,q$ satisfy one of the following inequalities: 1. (a) $p=1,q<\frac{d+1}{\alpha};$ 2. (b) $1<p<\frac{d+1}{d+1-\alpha},\frac{1}{q}\geq\frac{1}{p}+\frac{\alpha}{d+1}-1;$ 3. (c) $p=\frac{d+1}{d+1-\alpha},q<\infty;$ 4. (d) $\frac{d+1}{d+1-\alpha}<p\leq\infty.$ In particular, $K_{\alpha},K_{\alpha}^{+}:L^{p}\rightarrow L^{p}$ are both bounded for any $1\leq p\leq\infty$ when $0<\alpha<d+1,$ which is actually a more precise conclusion than Lemma 5 of [18] in the case of unit ball. Although $K_{\alpha},K_{\alpha}^{+}:L^{1}\rightarrow L^{\frac{d+1}{\alpha}}$ are both unbounded under the condition of Theorem 2, it turns out that $K_{\alpha}$ is weak type $(1,\frac{d+1}{\alpha}),$ i.e. $K_{\alpha},K_{\alpha}^{+}:L^{1}\rightarrow L^{\frac{d+1}{\alpha},\infty}$ are both bounded over $\mathbb{B}^{d},$ see the following Corollary 4.7, which is a generalization of the result that the standard Bergman projection is weak type (1,1) over some bounded domains [7, 14]. More importantly, by Theorem 2, it implies the following the Hardy-Littlewood-Sobolev inequality over the unit ball $\mathbb{B}^{d}.$ HLS 2. For any $1<p,s<\infty,\frac{1}{s}+\frac{1}{p}+\frac{\alpha}{d+1}\leq 2$ and $\alpha\leq d+1,$ then there exists a constant $C$ that depends only on $p,\alpha,d,s$ satisfying that (1.1) holds for all $f\in L^{p}(\mathbb{B}^{d}),g\in L^{s}(\mathbb{B}^{d}).$ Comparing HLS 1 and HLS 2 to the classical Hardy-Littlewood-Sobolev inequality [13, 16, 19, 21] over $\mathbb{R}^{d},$ it is surprising that HLS 1 is a new type of Hardy-Littlewood-Sobolev inequality. Theorem 3. If $0<\alpha\leq d+1,$ then the following conditions are equivalent: 1. (1) $K_{\alpha}:L^{p}\rightarrow L^{q}$ is compact; 2. (2) $p,q$ satisfy one of the following inequalities: 1. (a) $p=1,q<\frac{d+1}{\alpha};$ 2. (b) $1<p<\frac{d+1}{d+1-\alpha},\frac{1}{q}>\frac{1}{p}+\frac{\alpha}{d+1}-1;$ 3. (c) $p=\frac{d+1}{d+1-\alpha},q<\infty;$ 4. (d) $\frac{d+1}{d+1-\alpha}<p\leq\infty.$ Theorem 4. For $\alpha\in\mathbb{R},$ then the following conditions are equivalent: 1. (1) $\alpha<d+2;$ 2. (2) there exist $1\leq p,q\leq\infty$ such that $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded; 3. (3) there exist $1\leq p,q\leq\infty$ such that $K_{\alpha}^{+}:L^{p}\rightarrow L^{q}$ is bounded; 4. (4) there exist $1\leq p,q\leq\infty$ such that $K_{\alpha}:L^{p}\rightarrow L^{q}$ is compact. Theorem 5. If $\alpha<\frac{d+2}{2},$ then the following holds. 1. (1) $K_{\alpha},K_{\alpha}^{+}:L^{2}\rightarrow L^{2}$ are Hilbert-Schmidt. 2. (2) Moreover, if $d=1$ and $0<\alpha<\frac{3}{2},$ then we have the trace formula, $Tr(K_{\alpha}^{*}K_{\alpha})=\|K_{2\alpha}^{+}\|_{L^{\infty}\rightarrow L^{1}}=\frac{1}{(\alpha-1)^{2}}\left(\frac{\Gamma(3-2\alpha)}{\Gamma^{2}(2-\alpha)}-1\right).$ where $\Gamma$ is the usual Gamma function. When $\alpha=1,$ the quantity on the right side should be interpreted as $\frac{\pi^{2}}{6}.$ The above theorems show that $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded if and only if $K_{\alpha}^{+}:L^{p}\rightarrow L^{q}$ is bounded. From Theorem 1, it is amazing to know that, when $d+1<\alpha<d+2,$ $K_{\alpha}:L^{p}\rightarrow L^{q}$ is compact if and only if $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded. However, it is very different when $0<\alpha\leq d+1$ by Theorems 2 and 3. In particular, the standard Bergman projection $K_{d+1}:L^{p}\rightarrow L^{q}$ is compact if and only if $1\leq q<p\leq\infty$ over $\mathbb{B}^{d}.$ Let us consider the above boundedness problem in the following viewpoint. Denote $G(K_{\alpha})$ by the set of $(\frac{1}{p},\frac{1}{q})\in E$ such that $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded, where $E$ is given by $E=\\{(x,y)\in\mathbb{R}^{2}:0\leq x,y\leq 1\\},$ i.e. $E$ is a unit square in the real plane $\mathbb{R}^{2}.$ Following by T. Tao [22], $G(K_{\alpha})$ is called the type diagram of the operator $K_{\alpha},$ see Figure 1. By a classical interpolation result, it implies immediately that every $G(K_{\alpha})$ is convex. The adjointness of $K_{\alpha}$ implies that $G(K_{\alpha})$ is axisymmetric on the inside of $E$. To prove the above theorems is equivalent to solve the corresponding type diagrams. The above theorems show that the type diagram $G(K_{\alpha})$ is determined by the corresponding inequalities. Conversely, the inequalities in the above theorem are determined by the type diagram $G(K_{\alpha}).$ The convexity and axisymmetry of the type diagram will make the solving process simpler. Similarly, we can define the type diagrams $G(K_{\alpha}^{+})$ for operators $K_{\alpha}^{+},$ which are also convex and is axisymmetric on the inside of $E$; see Figure 1. Note that $|K_{\alpha}(f)|\leq K_{\alpha}^{+}(|f|),$ it implies immediately that $G(K_{\alpha}^{+})\subset G(K_{\alpha}).$ Then combing with several embedding theorems of holomorphic function spaces and some estimations of Bergman kernel over the unit ball, we completely characterize $L^{p}$-$L^{q}$ boundedness and $L^{p}$-$L^{q}$ compactness of $K_{\alpha}.$ The above main theorems show in fact that $G(K_{\alpha}^{+})=G(K_{\alpha})$ for every $\alpha\in\mathbb{R}.$ After characterizing the boundedness and compactness of $K_{\alpha},$ by using of the hypergeometric function theory and the interpolation theory, we give some sharp norm estimations of $K_{\alpha},K_{\alpha}^{+}.$ It is in fact that we estimate the upper bounds of the best constant in the inequalities HLS 1 and HLS 2. The results of this paper can be generalized to cover the weighted Lesbegue integrable spaces and more general kernel operators over the unit ball. Another promising idea is the study of the boundedness of Bergman projection over the bounded symmetric domains of tube type [2] with rank $\geq 2.$ The paper is organized as follows. In Section 2, we give some basic properties of the operators $K_{\alpha}.$ In Section 3, we prove Theorem 1. The proof of Theorem 2 is given in Section 4. In Section 5, we prove Theorem 3 and Theorem 4. Finally, we give some sharp norm estimations of the operators $K_{\alpha},K_{\alpha}^{+}.$ Figure 1. Type diagrams $G(K_{\alpha}),G(K_{\alpha}^{+}).$ ## 2\. Basic properties of $K_{\alpha}$ In this section, we prove some results for latter use. We first take a rough look at the property of type diagram $G(K_{\alpha})$ of the operator $K_{\alpha}.$ We prove that every $G(K_{\alpha})$ is convex and is axisymmetric on the inside of $E$ as mentioned before. Let $l_{E}$ be the diagonal line of the square $E$ which connects points $(0,1)$ and $(1,0).$ Clearly $G(K_{\alpha})\subset E$ for any $\alpha\in\mathbb{R}.$ ###### Proposition 2.1. 1. (1) If $G(K_{\alpha})\neq\emptyset,$ then $(0,1)\in G(K_{\alpha})$; if $(1,0)\in G(K_{\alpha}),$ then $G(K_{\alpha})=E.$ 2. (2) For any $\alpha\in\mathbb{R},$ the type diagram $G(K_{\alpha})$ is convex and is axisymmetric about $l_{E}$ on the inside of $E.$ ###### Proof. (1) It comes from the following continuous embedding of $L$-integrable spaces, i.e. $L^{p}\subset L^{q}$ whenever $p\geq q.$ (2) To show that $G(K_{\alpha})$ is convex, it suffices to show that if $(\frac{1}{p_{1}},\frac{1}{q_{1}}),(\frac{1}{p_{2}},\frac{1}{q_{2}})\in G(K_{\alpha}),$ then $\theta(\frac{1}{p_{1}},\frac{1}{q_{1}})+(1-\theta)(\frac{1}{p_{2}},\frac{1}{q_{2}})\in G(K_{\alpha})$ for any $0\leq\theta\leq 1.$ Indeed, it is a direct corollary of the following Lemma 2.2, a classical complex interpolation result. Now we turn to the symmetry. By Fubini’s theorem, it implies that $K_{\alpha}$ is adjoint. Then, for $1<p,q<\infty,$ the boundedness of $K_{\alpha}:L^{p}\rightarrow L^{q}$ is equivalent to the boundedness of $K_{\alpha}:L^{q^{\prime}}\rightarrow L^{p^{\prime}},$ where $p^{\prime},q^{\prime}$ are the conjugate numbers of $p,q$, respectively. It means that $(\frac{1}{p},\frac{1}{q})\in G(K_{\alpha})$ if and only if $(\frac{1}{q^{\prime}},\frac{1}{p^{\prime}})\in G(K_{\alpha}).$ It easy to check that $(\frac{1}{p},\frac{1}{q})$ and $(\frac{1}{q^{\prime}},\frac{1}{p^{\prime}})$ are symmetric about $l_{E}$ by the conjugate relationship. ∎ ###### Lemma 2.2. [25] Suppose $1\leq p_{1},p_{2},q_{1},q_{2}\leq\infty.$ If a linear operator $T$ such that $T:L^{p_{1}}\rightarrow L^{q_{1}}$ is bounded with norm $M_{1}$ and $T:L^{p_{2}}\rightarrow L^{q_{2}}$ is bounded with norm $M_{2}.$ Then $T:L^{p}\rightarrow L^{q}$ is bounded with norm no more than $M_{1}^{\theta}M_{2}^{1-\theta},$ if there exists $\theta\in(0,1)$ such that $\frac{1}{p}=\frac{\theta}{p_{1}}+\frac{1-\theta}{p_{2}},\frac{1}{q}=\frac{\theta}{q_{1}}+\frac{1-\theta}{q_{2}}.$ ###### Remark 2.3. Proposition 2.1 shows that the type diagram $G(K_{\alpha})$ is a bounded convex set in the plane $\mathbb{R}^{2}$, so to solve $G(K_{\alpha}),$ it suffices to find out all extreme points or the boundary points of $G(K_{\alpha}).$ The symmetry of $G(K_{\alpha})$ shows that is only need to find out a half. On the other hand, Proposition 2.1 holds for more general domains and adjoint operators. ###### Corollary 2.4. 1. (1) If $G(K_{\alpha}^{+})\neq\emptyset,$ then $(0,1)\in G(K_{\alpha}^{+})$; if $(1,0)\in G(K_{\alpha}^{+}),$ then $G(K_{\alpha}^{+})=E.$ 2. (2) For any $\alpha\in\mathbb{R},$ the type diagram $G(K_{\alpha}^{+})$ is convex and is axisymmetric about $l_{E}$ on the inside of $E.$ ###### Corollary 2.5. If $\alpha\leq 0,$ then $G(K_{\alpha})=G(K_{\alpha}^{+})=E.$ Corollary 2.5 means that $K_{\alpha},K_{\alpha}^{+}:L^{p}\rightarrow L^{q}$ are bounded for any $1\leq p,q\leq\infty$ if $\alpha\leq 0.$ For any $\beta>-1,$ denote $dv_{\beta}(z)=c_{\beta}(1-|z|^{2})^{\beta}dv(z),$ where $c_{\beta}=\frac{\Gamma(d+\beta+1)}{\Gamma(d+1)\Gamma(\beta+1)}.$ For $1\leq p\leq\infty,$ let $A_{\beta}^{p}=H(\mathbb{B}^{d})\cap L^{p}(dv_{\beta})$ be the weighted Bergman space on $\mathbb{B}^{d},$ in particular, $A_{\beta}^{\infty}=H^{\infty}$ is just the bounded holomorphic function space. Recall that $K_{d+1}$ is the Bergman projection from $L^{p}$ onto $A_{0}^{p},$ a well known result is that $K_{d+1}(L^{p})=A_{0}^{p}$ for $1<p<\infty.$ Now we establish a general result for $\alpha\geq d+1.$ ###### Proposition 2.6. Suppose that $\alpha\geq d+1$ and $1<p<\infty,$ then $K_{\alpha}(L^{p})=K_{\alpha}(A_{0}^{p})=A_{p(\alpha-d-1)}^{p}.$ To prove Proposition 2.6, we need some lemmas. The following Lemma 2.7 was proved [4] in the case $d=1,$ use the same method, it can be proved in the general case, see Lemma 11 of [4] for more detail. ###### Lemma 2.7. If $\alpha>0$ and $1<p<\infty,$ then $K_{\alpha}K_{d+1}=K_{\alpha}~{}\text{on}~{}L^{p}.$ Lemma 2.7 shows that for $1<p,q<\infty$, $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded if and only if $K_{\alpha}:A_{0}^{p}\rightarrow A_{0}^{q}$ is bounded. Now we turn to the behavior of $K_{\alpha}$ on holomorphic function spaces. Recall first the definition of fractional radial differential operator $R^{s,t}$ on $H(\mathbb{B}^{d}).$ For any two real parameters $s$ and $t$ with the property that neither $d+s$ nor $d+s+t$ is a negative integer, the invertible operator $R^{s,t}$ is given by $R^{s,t}f(z)=\sum_{n=0}^{\infty}\frac{\Gamma(d+1+s)\Gamma(d+1+n+s+t)}{\Gamma(d+1+s+t)\Gamma(d+1+n+s)}f_{n}(z),$ for any $f=\sum_{n=0}^{\infty}f_{n}\in H(\mathbb{B}^{d})$ with homogeneous expansion. In fact, it can be checked by direct calculation that the invertible operator of $R^{s,t}$ is just $R^{s+t,-t}.$ Be careful of the invertible operator here merely means that is linear. ###### Lemma 2.8. For $\alpha>0$ and $1<p<\infty,$ the following holds on $A_{0}^{p},$ $K_{\alpha}=R^{0,\alpha-d-1}.$ ###### Proof. Suppose $f=\sum_{n=0}^{\infty}f_{n}\in A_{0}^{p}$ with the homogeneous expansion. By direct calculation, it implies that $K_{\alpha}f=\sum_{n=0}^{\infty}\frac{\Gamma(d+1)\Gamma(\alpha+n)}{\Gamma(\alpha)\Gamma(d+1+n)}f_{n}.$ (2.1) It leads to the desired result. ∎ Proof of Proposition 2.6. Lemma 2.7 implies that $K_{\alpha}(L^{p})=K_{\alpha}(A_{0}^{p}).$ Now we prove $K_{\alpha}(A_{0}^{p})=A_{p(\alpha-d-1)}^{p}.$ By Theorem 14 of [24], which is a characterization of Bergman space, shows that $f\in A_{0}^{p}$ if and only if $R^{0,\alpha-d-1}f\in L^{p}(dv_{p(\alpha-d-1)}),$ namely $f\in A_{0}^{p}$ if and only if $R^{0,\alpha-d-1}f\in A_{p(\alpha-d-1)}^{p}.$ Note that $K_{\alpha}=R^{0,\alpha-d-1}$ by Lemma 2.8, it follows that $f\in A_{0}^{p}$ if and only if $K_{\alpha}f\in A_{p(\alpha-d-1)}^{p}.$ It shows that $K_{\alpha}(A_{0}^{p})\subset A_{p(\alpha-d-1)}^{p}.$ To prove another direction, suppose that $g\in A_{p(\alpha-d-1)}^{p}.$ Since $K_{\alpha}=R^{0,\alpha-d-1}$ is invertible on $H(\mathbb{B}^{d}),$ i.e. there exists $f\in H(\mathbb{B}^{d})$ such that $K_{\alpha}f=R^{0,\alpha-d-1}f=g.$ From Theorem 2.19 of [25], there exists a positive constant $c$ that depends only on $\alpha,d,p$ such that $\|f\|_{L^{p}}\leq c\|g\|_{A_{p(\alpha-d-1)}^{p}}.$ It means that $f\in A_{0}^{p}.$ Thus $A_{p(\alpha-d-1)}^{p}\subset K_{\alpha}(A_{0}^{p}).$ It completes the proof. ∎ ###### Corollary 2.9. Suppose that $\alpha\geq d+1$ and $1<p<\infty,$ then for any $\gamma>-1,$ the following holds, $K_{\alpha}(L^{p}(dv_{\gamma}))=K_{\alpha}(A_{\gamma}^{p})=A_{\gamma+p(\alpha-d-1)}^{p}.$ The following Proposition 2.10 gives the image of $K_{\alpha}$ in case of $p=\infty.$ Denote $\mathcal{B}_{\beta}$ by the weighted Bloch space on $\mathbb{B}^{d},$ see definition for Section 7.1 of [25]. ###### Proposition 2.10. For $\alpha\geq d+1,$ then $K_{\alpha}(H^{\infty})\subsetneq K_{\alpha}(L^{\infty})=\mathcal{B}_{\alpha-d}.$ ###### Proof. Note that $K_{\alpha}(L^{\infty})=\mathcal{B}_{\alpha-d}$ by Theorem 7.1 of [25]. If $\alpha=d+1,$ then $K_{d+1}(H^{\infty})=H^{\infty},$ thus $K_{d+1}(H^{\infty})\subsetneq\mathcal{B}_{\alpha-d}.$ Now turn to the case $\alpha>d+1.$ Note that $K_{\alpha}(H^{\infty})\subset K_{\alpha}(A_{0}^{p})$ for any $1<p<\infty,$ then it implies by Proposition 2.6 that $K_{\alpha}(H^{\infty})\subset\bigcap_{1<p<\infty}A_{p(\alpha-d-1)}^{p}.$ (2.2) On the other hand, from Theorem 2.1 of [25], a pointwise estimates for functions in weighted Bergman spaces, we know that $A_{\gamma}^{p}\subset\mathcal{B}_{\frac{d+1+\gamma}{p}}.$ (2.3) Combing (2.2) with (2.3), it implies that $K_{\alpha}(H^{\infty})\subset\bigcap_{1<p<\infty}\mathcal{B}_{(\alpha-d)+\frac{d+1}{p}-1}.$ Together with the fact that the weighted Bloch space is strictly increased, namely $\mathcal{B}_{\beta}\subsetneq\mathcal{B}_{\beta^{\prime}}$ whenever $0<\beta<\beta^{\prime},$ it implies that $K_{\alpha}(H^{\infty})\subsetneq\mathcal{B}_{\alpha-d}.$ ∎ ###### Remark 2.11. The monotonicity of the weighted Bloch space can be obtained as follows. It is easy to see that the weighted Bloch space is increased, so it suffices to show that is strict. For any $0<\beta<\beta^{\prime},$ there exist $p>1$ and $\varepsilon>0$ such that $\beta<\beta-1+\frac{d+\varepsilon}{p}<\beta^{\prime}.$ Combing (2.3) and the following Lemma 3.2, it implies that $\mathcal{B}_{\beta}\subsetneq A_{p(\beta-1)-1+\varepsilon}^{p}\subset\mathcal{B}_{\beta^{\prime}}.$ ## 3\. Proof of Theorem 1 In this section, we prove Theorem 1. We need several embedding theorems of holomorphic function spaces on the unit ball $\mathbb{B}^{d}.$ For convenience, we state them without proof as follows. ###### Lemma 3.1. [24] Let $0<q<p<\infty$.Then $A_{\beta}^{p}\subset A_{\gamma}^{q}$ if and only if $\frac{\beta+1}{p}<\frac{\gamma+1}{q}.$ And in this case the inclusions are strict. ###### Proof. See proof of Theorem 70 of [24]. ∎ ###### Lemma 3.2. [17, 24] Suppose that $\beta>0,\gamma>-1,p\geq 1,$ then $\mathcal{B}_{\beta}\subset A_{\gamma}^{p}$ if and only if $\beta-1<\frac{1+\gamma}{p}.$ And in this case the inclusions are strict. ###### Proof. See proofs in [17] or Theorem 66 of [24]. ∎ We also needs the following lemmas. ###### Lemma 3.3. If $d+1<\alpha<d+2,$ then $K_{\alpha}:L^{\infty}\rightarrow L^{q}$ is bounded if and only if $q<\frac{1}{\alpha-(d+1)}.$ ###### Proof. We first to show that $K_{\alpha}:L^{\infty}\rightarrow L^{q}$ is bounded if $q<\frac{1}{\alpha-(d+1)}.$ Then, for $f\in L^{\infty},$ by Proposition 1.4.10 of [20] and Hölder’s inequality, it implies that $\begin{split}|K_{\alpha}f(z)|\leq\|f\|_{\infty}\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle z,w\rangle|^{\alpha}}dv(w)\leq C_{d,\alpha}\|f\|_{\infty}(1-|z|^{2})^{d+1-\alpha},|z|\rightarrow 1^{-},\end{split}$ (3.1) where $C_{d,\alpha}$ is a constant. The condition $q<\frac{1}{\alpha-(d+1)}$ means that $q((d+1)-\alpha)>-1.$ Then (3.1) implies that $K_{\alpha}f(z)\in L^{q}$ and $K_{\alpha}:L^{\infty}\rightarrow L^{q}$ is bounded. Now we turn to prove that $K_{\alpha}:L^{\infty}\rightarrow L^{q}$ is unbounded if $q\geq\frac{1}{\alpha-(d+1)}.$ By Hölder’s inequality, it is enough to prove that $K_{\alpha}:L^{\infty}\rightarrow L^{\frac{1}{\alpha-(d+1)}}$ is unbounded. It suffices to show that $K_{\alpha}(L^{\infty})\not\subset L^{\frac{1}{\alpha-(d+1)}}.$ Since $K_{\alpha}(L^{\infty})=\mathcal{B}_{\alpha-d}$, it suffices to show that $\mathcal{B}_{\alpha-d}\not\subset A_{0}^{\frac{1}{\alpha-(d+1)}}.$ Indeed, it is a fact from Lemma 3.2. ∎ ###### Corollary 3.4. If $d+1<\alpha<d+2,$ then $K_{\alpha}:L^{p}\rightarrow L^{1}$ is bounded if and only if $p>\frac{1}{(d+2)-\alpha}.$ ###### Proof. First, suppose that $p>\frac{1}{(d+2)-\alpha}.$ From Lemma 3.3 and $K_{\alpha}$ is an adjoint operator, we know that $K_{\alpha}:L^{p}\rightarrow(L^{\infty})^{*}$ is bounded if $p>\frac{1}{(d+2)-\alpha}.$ Proposition 2.6 implies that $K_{\alpha}(L^{p})=A_{p(\alpha-d-1)}^{p}.$ Since $\frac{p(\alpha-d-1)+1}{p}<(\alpha-d-1)+(d+2)-\alpha=1,$ it follows by Lemma 3.1 that $A_{p(\alpha-d-1)}^{p}\subset A_{0}^{1}.$ Thus $K_{\alpha}(L^{p})\subset L^{1}.$ Note that $L^{1}\subset(L^{\infty})^{*},$ it implies that $K_{\alpha}:L^{p}\rightarrow L^{1}$ is bounded. Conversely, suppose that $K_{\alpha}:L^{p}\rightarrow L^{1},p\neq\infty$ is bounded. Then $K_{\alpha}:L^{\infty}\rightarrow L^{p^{\prime}}$ is bounded, where $p^{\prime}=\frac{p}{p-1}.$ From Lemma 3.3, it implies that $\frac{p}{p-1}=p^{\prime}<\frac{1}{\alpha-(d+1)},$ it means that $p>\frac{1}{(d+2)-\alpha}.$ Clearly the case of $p=\infty$ is trivial by Lemma 3.3. ∎ ###### Corollary 3.5. If $d+1<\alpha<d+2,$ then 1. (1) $K_{\alpha}^{+}:L^{\infty}\rightarrow L^{q}$ is bounded if and only if $q<\frac{1}{\alpha-(d+1)};$ 2. (2) $K_{\alpha}^{+}:L^{p}\rightarrow L^{1}$ is bounded if and only if $p>\frac{1}{(d+2)-\alpha}.$ ###### Proof. (1) For $f\in L^{\infty},$ by Proposition 1.4.10 of [20] and Hölder’s inequality, it implies that $\begin{split}|K_{\alpha}^{+}f(z)|\leq\|f\|_{\infty}\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle z,w\rangle|^{\alpha}}dv(w)\leq C_{d,\alpha}\|f\|_{\infty}(1-|z|^{2})^{d+1-\alpha},|z|\rightarrow 1^{-},\end{split}$ (3.2) where $C_{d,\alpha}$ is a constant. So, if $q<\frac{1}{\alpha-(d+1)},$ i.e. $q((d+1)-\alpha)>-1,$ then (3.2) implies that $K_{\alpha}f(z)\in L^{q}$ and $K_{\alpha}^{+}:L^{\infty}\rightarrow L^{q}$ is bounded. It means that $\\{(0,\frac{1}{q}):\frac{1}{q}>\alpha-(d+1)\\}\subset G(K_{\alpha}^{+}).$ On the other hand, Lemma 3.3 implies that point $(0,\frac{1}{q})\in G(K_{\alpha})$ if and only if $\frac{1}{q}>\alpha-(d+1).$ Combing with $G(K_{\alpha}^{+})\subset G(K_{\alpha}),$ it follows that $(0,\frac{1}{q})\in G(K_{\alpha})$ if and only if $\frac{1}{q}>\alpha-(d+1).$ It leads the desired result. (2) The proof is similar to (1). ∎ ###### Lemma 3.6. Suppose that $d+1<\alpha<d+2$ and $\frac{1}{q}\leq\frac{1}{p}+\alpha-(d+1),$ then $K_{\alpha}:L^{p}\rightarrow L^{q}$ is unbounded. ###### Proof. By the continuous embedding of $L$-integrable spaces, it suffices to show that $K_{\alpha}:L^{p}\rightarrow L^{q}$ is unbounded if $d+1<\alpha<d+2,\frac{1}{q}=\frac{1}{p}+\alpha-(d+1).$ The cases of $p=\infty$ or $q=1$ had been proved in Lemma 3.3 and Corollary 3.4. For case of $1<p,q<\infty,$ it suffices to show that $K_{\alpha}(L^{p})\not\subset L^{q}.$ On the other hand, Proposition 2.6 shows that $K_{\alpha}(L^{p})=A_{p(\alpha-d-1)}^{p},$ a holomorphic function space. Thus, it suffices to show that $K_{\alpha}(L^{p})=A_{p(\alpha-d-1)}^{p}\not\subset A_{0}^{q}.$ (3.3) Since $\frac{p(\alpha-d-1)+1}{p}=\frac{1}{q},$ it follows that (3.3) holds by Lemma 3.1. It completes the proof. ∎ Proof of Theorem 1. $Step~{}1$. To prove that (1)$\Leftrightarrow$(2)$\Leftrightarrow$(4). First, we prove that (1) is equivalent to (4). As mentioned before, it is equivalent to prove that $G(K_{\alpha})$ is exactly the triangle region $D_{1}\subset E$ which determined by the equations in (4) of Theorem 1, namely $G(K_{\alpha})=D_{1}$. Lemma 3.3, Corollary 3.4 and the convexity of $G(K_{\alpha})$ imply that $D_{1}\subset G(K_{\alpha}).$ On the other hand, Lemma 3.6 and the convexity of $G(K_{\alpha})$ imply that $E-D_{1}\subset E-G(K_{\alpha}),$ it follows that $G(K_{\alpha})\subset D_{1}.$ Thus $G(K_{\alpha})=D_{1}.$ Now we turn to prove that (2) is equivalent to (4), it is equivalent to prove that $G(K_{\alpha}^{+})=D_{1}.$ Corollary 3.5 and the convexity of $G(K_{\alpha}^{+})$ implies that $D_{1}\subset G(K_{\alpha}^{+}).$ Combing the fact that $G(K_{\alpha}^{+})\subset G(K_{\alpha})=D_{1},$ then $G(K_{\alpha}^{+})=D_{1}.$ It completes the proof. $Step~{}2$. To prove that (1)$\Leftrightarrow$(3). Since compact operators must be bounded, it suffices to prove that $K_{\alpha}:L^{p}\rightarrow L^{q}~{}\text{is~{} compact},~{}\text{if }(\frac{1}{p},\frac{1}{q})\in G(K_{\alpha}).$ We first prove the following claim. $Claim:$ $K_{\alpha}:L^{\infty}\rightarrow L^{q}$ is compact if and only if $q<\frac{1}{\alpha-(d+1)}.$ If $K_{\alpha}:L^{\infty}\rightarrow L^{q}$ is compact, is immediate from Corollary 3.4 that $q<\frac{1}{\alpha-(d+1)}.$ Now we prove the reverse, that is, to prove that $K_{\alpha}:L^{\infty}\rightarrow L^{q}$ is compact if $q<\frac{1}{\alpha-(d+1)}.$ We need to show that for any bounded sequence in $L^{\infty}$, there is a subsequence such that whose image under $K_{\alpha}$ converges in $L^{q}.$ Suppose that $\\{f_{n}\\}\in L^{\infty}$ is an arbitrary bounded sequence and $K$ is an arbitrary compact subset of $\mathbb{B}^{d}.$ Moreover, we assume that $\|f_{n}\|_{\infty}\leq C$ for any $n\geq 1,$ where $C$ is a positive constant. Then we obtain $\begin{split}\sup_{z\in K}|K_{\alpha}f_{n}(z)|&\leq\|f_{n}\|_{\infty}\sup_{z\in K}\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle z,w\rangle|^{\alpha}}dv(w)\\\ &\leq\|f_{n}\|_{\infty}\sup_{z\in K}\frac{1}{(1-|z|)^{\alpha}}<\infty.\end{split}$ Combing with that the image of $K_{\alpha}$ is holomorphic, it implies that $\\{K_{\alpha}f_{n}\\}$ is a normal family. Hence $\\{f_{n}\\}$ has a subsequence $\\{f_{n_{j}}\\}$ such that $K_{\alpha}f_{n_{j}}$ converges uniformly on compact subsets of $\mathbb{B}^{d}$ to a holomorphic function $g$. By Fatou’s Lemma and boundedness of $K_{\alpha}$, it follows that $\int_{\mathbb{B}^{d}}|g|^{q}dv\leq\varliminf_{j\rightarrow\infty}\int_{\mathbb{B}^{d}}|K_{\alpha}f_{n_{j}}|^{q}dv\leq\|K_{\alpha}\|_{L^{\infty}\rightarrow L^{q}}^{q}\varliminf_{j\rightarrow\infty}\|f_{n_{j}}\|_{\infty}^{q}<\infty.$ (3.4) It means that $g\in L^{q}.$ Now we prove that there exists positive function $g_{1}\in L^{q}$ such that $|K_{\alpha}f_{n_{j}}|\leq g_{1}.$ We first observe that Proposition 1.4.10 of [20] and the condition $q<\frac{1}{\alpha-(d+1)}$ imply that $\left(\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle z,w\rangle|^{\alpha}}dv(w)\right)^{q}\in L^{1}.$ Then by easy estimation, it yields that $\begin{split}|K_{\alpha}f_{n_{j}}(z)|&\leq\|f_{n_{j}}\|_{\infty}\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle z,w\rangle|^{\alpha}}dv(w)\\\ &\leq C\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle z,w\rangle|^{\alpha}}dv(w).\\\ \end{split}$ (3.5) Thus (3.5) shows that it is enough to take $g_{1}=C\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle z,w\rangle|^{\alpha}}dv(w).$ Combing (3.4) with (3.5), it implies that $|K_{\alpha}f_{n_{j}}-g|^{q}\leq(g_{1}+|g|)^{q}\in L^{1},\forall j\geq 1.$ By dominated convergence theorem, it gives that $\lim_{j\rightarrow\infty}\|K_{\alpha}f_{n_{j}}-g\|_{q}=\lim_{j\rightarrow\infty}\left(\int_{\mathbb{B}^{d}}|K_{\alpha}f_{n_{j}}-g|^{q}dv\right)^{\frac{1}{q}}=\left(\int_{\mathbb{B}^{d}}\lim_{j\rightarrow\infty}|K_{\alpha}f_{n_{j}}-g|^{q}dv\right)^{\frac{1}{q}}=0,$ and completes the proof the claim. Combing the last claim with facts that an operator is compact if and only if its adjoint operator is still compact, thus we get that $K_{\alpha}:L^{p}\rightarrow L^{1}$ is compact if and only if $p<\frac{1}{d+2-\alpha}.$ Then by the following Lemma 3.7, an interpolation result of the compact operators, it implies that $K_{\alpha}:L^{p}\rightarrow L^{q}~{}\text{is~{} compact}~{}\text{if }(\frac{1}{p},\frac{1}{q})\in G(K_{\alpha}).$ ∎ ###### Lemma 3.7. [6, 11] Suppose that $1\leq p_{1},p_{2},q_{1},q_{2}\leq\infty$ and $q_{1}\neq\infty.$ If a linear operator $T$ such that $T:L^{p_{1}}\rightarrow L^{q_{1}}$ is bounded and $T:L^{p_{2}}\rightarrow L^{q_{2}}$ is compact, then $T:L^{p}\rightarrow L^{q}$ is compact, if there exists $\theta\in(0,1)$ such that $\frac{1}{p}=\frac{\theta}{p_{1}}+\frac{1-\theta}{p_{2}},\frac{1}{q}=\frac{\theta}{q_{1}}+\frac{1-\theta}{q_{2}}.$ ###### Remark 3.8. The compactness of $K_{\alpha}:L^{p}\rightarrow L^{q}$ for $1<p,q<\infty$ can be also proved by the Carleson type measure theory on Bergman spaces, see definition for [24, 25]. This strategy will be adopted under appropriate circumstances in Section 5. ## 4\. Proof of Theorem 2 In this section we give the proof of Theorem 2. We first establish several lemmas. Denote $k_{\alpha}(z,w)=\frac{1}{(1-\langle z,w\rangle)^{\alpha}},k_{\alpha}^{+}(z,w)=\frac{1}{|1-\langle z,w\rangle|^{\alpha}},z,w\in\mathbb{B}^{d}.$ Then $k_{\alpha},k_{\alpha}^{+}$ are integral kernel functions of integral operators $K_{\alpha},K_{\alpha}^{+}$ respectively. ###### Lemma 4.1. If $0<\alpha\leq d+1,$ then 1. (1) $K_{\alpha}:L^{1}\rightarrow L^{q}$ is bounded if and only if $q<\frac{d+1}{\alpha};$ 2. (2) $K_{\alpha}^{+}:L^{1}\rightarrow L^{q}$ is bounded if and only if $q<\frac{d+1}{\alpha}.$ ###### Proof. (1) From Proposition 5.2 of [22], we know that $\|K_{\alpha}\|_{L^{1}\rightarrow L^{q}}=\sup_{z\in\mathbb{B}^{d}}\|k_{\alpha}(z,\cdot)\|_{L^{q}}=\sup_{z\in\mathbb{B}^{d}}\left(\int\frac{dv(w)}{|1-\langle z,w\rangle|^{q\alpha}}\right)^{\frac{1}{q}}\\\ $ Then, combing with Proposition 1.4.10 of [20], we know that $\|K_{\alpha}\|_{L^{1}\rightarrow L^{q}}<\infty$ is equivalent to $q\alpha<d+1.$ It leads the desired result. (2) It is similar to (1). ∎ Dually, we have the following lemma. ###### Lemma 4.2. If $0<\alpha\leq d+1,$ then 1. (1) $K_{\alpha}:L^{p}\rightarrow L^{\infty}$ is bounded if and only if $p>\frac{d+1}{(d+1)-\alpha};$ 2. (2) $K_{\alpha}^{+}:L^{p}\rightarrow L^{\infty}$ is bounded if and only if $p>\frac{d+1}{(d+1)-\alpha}.$ ###### Proof. (1) From Proposition 5.4 of [22], we know that $\|K_{\alpha}\|_{L^{p}\rightarrow L^{\infty}}=\sup_{z\in\mathbb{B}^{d}}\|k_{\alpha}(z,\cdot)\|_{L^{p^{\prime}}}=\sup_{z\in\mathbb{B}^{d}}\left(\int\frac{dv(w)}{|1-\langle z,w\rangle|^{\frac{p\alpha}{p-1}}}\right)^{\frac{p-1}{p}}.$ (4.1) Then, combing with Proposition 1.4.10 of [20], we know that $\|K_{\alpha}\|_{L^{p}\rightarrow L^{\infty}}<\infty$ is equivalent to $\frac{p\alpha}{p-1}<d+1.$ It leads the desired result. (2) It is similar to (1). ∎ ###### Lemma 4.3. If $1<p<\frac{d+1}{d+1-\alpha},$ then $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded if and only if $\frac{1}{q}\geq\frac{1}{p}+\frac{\alpha}{d+1}-1.$ Before proving Lemma 4.3, we do some preparations. For $p\geq 1$, denote Lorentz space $L^{p,\infty}$ on $\mathbb{B}^{d}$ by $L^{p,\infty}=\\{f:\sup_{\lambda>0}\lambda d_{f}^{\frac{1}{p}}(\lambda)<\infty\\},$ where $d_{f}(\lambda)=v\\{z\in\mathbb{B}^{d}:|f(z)|>\lambda\\}.$ Note that $L^{p,\infty}\subset L^{q,\infty}$ if $p>q,$ and the inclusion is continuous. ###### Lemma 4.4. There exists a constant $C$ that only depends on $\alpha$ and $d$ such that, for every $z\in\mathbb{B}^{d},$ $\|k_{\alpha}(z,\cdot)\|_{L^{\frac{d+1}{\alpha},\infty}}=\|k_{\alpha}(\cdot,z)\|_{L^{\frac{d+1}{\alpha},\infty}}<C.$ ###### Proof. By the unitary invariance of Lebesgue measure, we only need to consider the case of $z=(|z|,0,\cdots,0).$ Note that $d_{k_{\alpha}(\cdot,z)}(\lambda)=v\\{w\in\mathbb{B}^{d}:\frac{1}{|1-\langle w,z\rangle|^{\alpha}}>\lambda\\}=v\\{w:|\frac{1}{|z|}-w_{1}|<\frac{1}{|z|}\lambda^{-\frac{1}{\alpha}}\\}$ (4.2) When $|z|<\frac{1}{2},$ then $\frac{1}{|1-\langle w,z\rangle|^{\alpha}}<2^{\alpha}.$ It follows that $d_{k_{\alpha}(\cdot,z)}(\lambda)=0,$ if $\lambda\geq 2^{\alpha}.$ Thus $\|k_{\alpha}(\cdot,z)\|_{L^{\frac{d+1}{\alpha},\infty}}\leq 2^{\alpha}.$ Now we turn to the case $\frac{1}{2}\leq|z|<1.$ The conclusion comes immediately from the following estimation, $\lambda d_{k_{\alpha}(\cdot,z)}^{\frac{\alpha}{d+1}}(\lambda)\leq\begin{cases}1,&\lambda\leq 1,\\\ (d\cdot 2^{3d-1})^{\frac{\alpha}{d+1}},&1<\lambda<\frac{1}{(1-|z|)^{\alpha}},\\\ 0,&\lambda\geq\frac{1}{(1-|z|)^{\alpha}}.\end{cases}$ (4.3) Now we prove (4.3). Let $dV(w)=(\frac{i}{2})^{d}\prod_{n=1}^{d}dw_{n}\wedge d\bar{w}_{n}.$ Then $dV=\frac{\pi^{d}}{\Gamma(d+1)}dv.$ When $\lambda\leq 1,$ then $\lambda d_{k_{\alpha}(\cdot,z)}^{\frac{\alpha}{d+1}}(\lambda)<1.$ Denote $I$ by the subset in the unit disk such that $\begin{split}I=\\{w_{1}\in\mathbb{D}:|\frac{1}{|z|}-w_{1}|<\frac{1}{|z|}\lambda^{-\frac{1}{\alpha}}\\}.\end{split}$ (4.4) When $1<\lambda<\frac{1}{(1-|z|)^{\alpha}},$ by (4.2) and Fubini’s theorem, we have that $\begin{split}d_{k_{\alpha}(\cdot,z)}(\lambda)&=v\\{w:|\frac{1}{|z|}-w_{1}|<\frac{1}{|z|}\lambda^{-\frac{1}{\alpha}}\\}\\\ &\leq\frac{\Gamma(d+1)}{\pi^{d}}(\frac{i}{2})^{d}\int_{I}dw_{1}\wedge d\bar{w}_{1}\int_{|w_{2}|^{2}+\cdots+|w_{d}|^{2}<1-|w_{1}|^{2}}\prod_{n=2}^{d}dw_{n}\wedge d\bar{w}_{n}\\\ &=d\int_{I}(1-|w_{1}|^{2})^{d-1}dv(w_{1})\\\ &<d(1-\frac{1}{|z|^{2}}+2\frac{1}{|z|^{2}}\frac{1}{\lambda^{\frac{1}{\alpha}}}-\frac{1}{|z|^{2}\lambda^{\frac{2}{\alpha}}})^{d-1}\int_{I}dv(w_{1})\\\ &<d\cdot 2^{3d-3}\frac{1}{\lambda^{\frac{d-1}{\alpha}}}\frac{4}{\lambda^{\frac{2}{\alpha}}}\\\ &=\frac{d\cdot 2^{3d-1}}{\lambda^{\frac{d+1}{\alpha}}}\\\ \end{split}$ (4.5) Then (4.5) implies that $\lambda d_{k_{\alpha}(\cdot,z)}^{\frac{\alpha}{d+1}}(\lambda)<(d\cdot 2^{3d-1})^{\frac{\alpha}{d+1}}$ if $1<\lambda<\frac{1}{(1-|z|)^{\alpha}}.$ When $\lambda\geq\frac{1}{(1-|z|)^{\alpha}},$ it is easy to see that $d_{k_{\alpha}(\cdot,z)}(\lambda)=0.$ So $\lambda d_{k_{\alpha}(\cdot,z)}^{\frac{\alpha}{d+1}}(\lambda)=0,$ if $\lambda\geq\frac{1}{(1-|z|)^{\alpha}}.$∎ ###### Corollary 4.5. There exists a constant $C$ that only depends on $\alpha$ and $d$ such that, for every $z\in\mathbb{B}^{d},$ $\|k_{\alpha}^{+}(z,\cdot)\|_{L^{\frac{d+1}{\alpha},\infty}}=\|k_{\alpha}^{+}(\cdot,z)\|_{L^{\frac{d+1}{\alpha},\infty}}<C.$ Now we modify Proposition 6.1 of [22] to suit our setting. ###### Lemma 4.6. [22] Suppose that $k:\mathbb{B}^{d}\times\mathbb{B}^{d}\rightarrow\mathbb{C}$ is measurable such that $\|k(z,\cdot)\|_{L^{r,\infty}}\leq C,z\in\mathbb{B}^{d},a.e.$ and $\|k(\cdot,w)\|_{L^{r,\infty}}\leq C,w\in\mathbb{B}^{d},a.e.$ for some $1<r<\infty$ and $C>0.$ Then the operator $T$ defined as $Tf(z)=\int_{\mathbb{B}^{d}}k(z,w)f(w)dv(w)$ is bounded from $L^{1}$ to $L^{r,\infty}.$ Moreover, if $1<p<q<\infty$ such that $\frac{1}{p}+\frac{1}{r}=\frac{1}{q}+1,$ then $T$ is bounded from $L^{p}$ to $L^{q}.$ ###### Corollary 4.7. If $0<\alpha\leq d+1,$ then $K_{\alpha},K_{\alpha}^{+}:L^{1}\rightarrow L^{\frac{d+1}{\alpha},\infty}$ are bounded. ###### Proof. When $\alpha=d+1,$ $K_{d+1}$ is the Bergman projection, then $K_{d+1}:L^{1}\rightarrow L^{1,\infty}$ is bounded by the proof of Theorem 6 of [14]. Indeed, similar to the proof of Theorem 6 of [14], by the Calderón- Zygmund decomposition, it can be proved that $K_{d+1}^{+}:L^{1}\rightarrow L^{1,\infty}$ is bounded. When $0<\alpha<d+1,$ by Lemma 4.4 and Lemma 4.6, it implies that $K_{\alpha},K_{\alpha}^{+}:L^{1}\rightarrow L^{\frac{d+1}{\alpha},\infty}$ are bounded. It completes the proof. ∎ Sufficiency part of Lemma 4.3. We need to prove that if $1<p<\frac{d+1}{d+1-\alpha},\frac{1}{q}\geq\frac{1}{p}+\frac{\alpha}{d+1}-1,$ then $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded. By the continuous embedding of $L$-integrable spaces, it suffices to show that $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded if $\frac{1}{q}=\frac{1}{p}+\frac{\alpha}{d+1}-1.$ Then Lemma 4.4 and Lemma 4.6 implies that $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded. ∎ ###### Corollary 4.8. If $1<p<\frac{d+1}{d+1-\alpha},$ then $K_{\alpha}^{+}:L^{p}\rightarrow L^{q}$ is bounded if $\frac{1}{q}\geq\frac{1}{p}+\frac{\alpha}{d+1}-1.$ For necessity part of Lemma 4.3, we need find out a function belongs to $L^{p}$ but its image under $K_{\alpha}$ is not in $L^{q}.$ So we establish an isometry from $L^{p}(\mathbb{D},dv_{d-1})$ to $L^{p}(\mathbb{B}^{d}),$ where $\mathbb{D}$ is the unit disc, i.e. $\mathbb{D}=\mathbb{B}^{1}.$ Let $I_{p}:L^{p}(\mathbb{D},dv_{d-1})\rightarrow L^{p}(\mathbb{B}^{d}),I_{p}(f)(z)=f(z_{1}).$ Denote $L_{1}^{p}(\mathbb{B}^{d})=\\{f\in L^{p}(\mathbb{B}^{d}):f(z)=f(z_{1},0\cdots,0),\forall z\in\mathbb{B}^{d}\\}.$ If $f\in L_{1}^{p}(\mathbb{B}^{d})$, we always write $f(z_{1})$ without ambiguity. Denote $A_{0,1}^{p}$ by the set of holomorphic functions in $L_{1}^{p}(\mathbb{B}^{d}).$ Then we have the following lemma. ###### Lemma 4.9. $I_{p}$ is an isometry from $A_{d-1}^{p}(\mathbb{D})$ onto $A_{0,1}^{p}(\mathbb{B}^{d}).$ ###### Proof. Suppose that $f(z_{1})\in L_{1}^{p}(\mathbb{B}^{d}),$ then $\begin{split}\|f\|_{L_{1}^{p}(\mathbb{B}^{d})}^{p}&=\int_{\mathbb{B}^{d}}|f(z_{1})|^{p}dv\\\ &=\frac{\Gamma(d+1)}{\pi^{d}}(\frac{i}{2})^{d}\int_{\mathbb{D}}|f(z_{1})|^{p}dz_{1}\wedge d\bar{z}_{1}\int_{|z_{2}|^{2}+\cdots+|z_{d}|^{2}<1-|z_{1}|^{2}}\prod_{n=2}^{d}dz_{n}\wedge d\bar{z}_{n}\\\ &=d\int_{\mathbb{D}}|f(z_{1})|^{p}(1-|z_{1}|^{2})^{d-1}dv(z_{1})\\\ &=\|f\|_{L^{p}(\mathbb{D},dv_{d-1})}^{p}.\\\ \end{split}$ (4.6) It leads to the desired result. ∎ ###### Corollary 4.10. $A_{0,1}^{p}(\mathbb{B}^{d})\simeq A_{d-1}^{p}(\mathbb{D}).$ ###### Lemma 4.11. Suppose that $t\in\mathbb{R},$ then $f(z_{1})=\sum_{n=1}^{\infty}n^{t}z_{1}^{n}\in A_{0,1}^{p}(\mathbb{B}^{d})$ if and only if $p(t+1)<d+1.$ ###### Proof. From Corollary 3.5 of [3], Corollary 4.10 and Proposition 1.4.10 of [20], it yields that $f(z_{1})=\sum_{n=1}^{\infty}n^{t}z_{1}^{n}\in A_{0,1}^{p}(\mathbb{B}^{d})\simeq A_{d-1}^{p}(\mathbb{D})$ if and only if $\sum_{n=1}^{\infty}\frac{\Gamma(n+1+t)}{\Gamma(n+1)\Gamma(t+1)}z_{1}^{n}=\frac{1}{(1-z_{1})^{t+1}}\in L_{a}^{p}(\mathbb{D},dv_{d-1})$ if and only if $p(t+1)<2+(d-1)=d+1.$ ∎ Necessity part of Lemma 4.3. We need to prove that if $1<p<\frac{d+1}{d+1-\alpha},\frac{1}{q}<\frac{1}{p}+\frac{\alpha}{d+1}-1,$ then $K_{\alpha}:L^{p}\rightarrow L^{q}$ is unbounded. Assume that $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded, it is equivalent to $K_{\alpha}:A_{0}^{p}\rightarrow A_{0}^{q}$ is bounded. Then $K_{\alpha}(A_{0}^{p})\subset A_{0}^{q}.$ Choose any $t$ such that $\frac{d+1}{q}+d-\alpha<t<\frac{d+1}{p}-1,$ (4.7) denote $f_{t}(z)=\sum_{n=1}^{\infty}n^{t}z_{1}^{n},$ then condition (4.7) and Lemma 4.11 imply that $f_{t}\in A_{0}^{p}.$ Now, assume that $K_{\alpha}f_{t}\in A_{0}^{q}.$ Then $\begin{split}K_{\alpha}f_{t}(z)&=\sum_{n=1}^{\infty}\frac{\Gamma(n+\alpha)\Gamma(d+1)}{\Gamma(n+d+1)\Gamma(\alpha)}n^{t}z_{1}^{n}\\\ &=\frac{\Gamma(d+1)}{\Gamma(\alpha)\Gamma(d+1-\alpha)}\sum_{n=1}^{\infty}\frac{\Gamma(n+\alpha)\Gamma(d+1-\alpha)}{\Gamma(n+d+1)}n^{t}z_{1}^{n}\\\ &=\frac{\Gamma(d+1)}{\Gamma(\alpha)\Gamma(d+1-\alpha)}\sum_{n=1}^{\infty}B(n+\alpha,d+1-\alpha)n^{t}z_{1}^{n}\\\ &\in A_{0,1}^{q}(\mathbb{B}^{d})\simeq A_{d-1}^{q}(\mathbb{D}),\\\ \end{split}$ (4.8) where $B(\cdot,\cdot)$ is Beta function. On the other hand, by Lemma 3.2 of [3], similar to the prove of Lemma 3.4 of [3], it can be proved that $\sum_{n=1}^{\infty}\frac{n^{\alpha-d-1}}{B(n+\alpha,d+1-\alpha)}a_{n}z^{n}\in A_{d-1}^{q}(\mathbb{D}),\forall\sum_{n=1}^{\infty}a_{n}z^{n}\in A_{d-1}^{q}(\mathbb{D}).$ (4.9) Combing (4.8) with (4.9), it implies that $\sum n^{\alpha-d-1}n^{t}z_{1}^{n}\in A_{d-1}^{q}(\mathbb{D})\simeq A_{0,1}^{q}(\mathbb{B}^{d}).$ So, by Lemma 4.11, it follows that $q(\alpha-d+t)<d+1,$ namely $t<\frac{d+1}{q}+d-\alpha,$ a contradiction to the condition $\frac{d+1}{q}+d-\alpha<t.$ It completes the proof. ∎ Proof of Theorem 2. First, we prove that (1) is equivalent to (3). Denote $D_{2}$ by the region determined by the equations in (3) of Theorem 3. It is equivalent to prove that $G(K_{\alpha})=D_{2}.$ Lemma 4.1, Lemma 4.2, Lemma 4.3 and the convexity of $G(K_{\alpha}),$ imply that $G(K_{\alpha})=D_{2}.$ Lemma 4.1, Lemma 4.2, Corollary 4.8 and the convexity of $G(K_{\alpha}^{+}),$ imply that $D_{2}\subset G(K_{\alpha}^{+}).$ From the above we know that $G(K_{\alpha}^{+})\subset G(K_{\alpha})=D_{2}.$ Then $G(K_{\alpha}^{+})=D_{2}.$ It completes the proof. ∎ ## 5\. Proofs of Theorem 3 and Theorem 4 In the previous Section 4, we have characterized completely the $L^{p}$-$L^{q}$ boundedness of $K_{\alpha},K_{\alpha}^{+}$ under the case of $0<\alpha\leq d+1.$ In the present section, we will characterize completely the $L^{p}$-$L^{q}$ compactness of $K_{\alpha}$ when $0<\alpha\leq d+1.$ It is equivalent to solve the set $F(K_{\alpha}),$ where $F(K_{\alpha})$ is defined by $F(K_{\alpha})=\\{(\frac{1}{p},\frac{1}{q})\in E:K_{\alpha}:L^{p}\rightarrow L^{q}~{}\text{is ~{}compact}\\}.$ It is easy to see that $F(K_{\alpha})$ is a subset of $G(K_{\alpha}).$ Theorem 3 in fact shows that $F(K_{\alpha})$ and $G(K_{\alpha})$ differ only by a segment on the boundary of $G(K_{\alpha}).$ Thus we always show first that $K_{\alpha}$ is compact on the other part of on the boundary of $G(K_{\alpha}).$ In the end of this section, we give the proof of Theorem 4. ###### Proposition 5.1. $K_{d+1}:L^{p}\rightarrow L^{q}$ is compact if and only if $1\leq q<p\leq\infty.$ ###### Proof. From Theorem 2, we know that $K_{d+1}:L^{p}\rightarrow L^{q}$ is bounded if and only if $q\leq p.$ Since $K_{d+1}$ is the standard Bergman projection, it is easy to see $K_{d+1}:L^{p}\rightarrow L^{p}$ is not compact for any $1<p<\infty.$ Thus it suffices to show that $K_{\alpha}:L^{p}\rightarrow L^{q}$ is compact if $q<p.$ Indeed, this can be proved by the similar method we used in the $step$ 2 of proof of Theorem 1, we omit it. ∎ Now, we recall some results on hypergeometric function theory for later use. For complex numbers $\alpha,\beta,\gamma$ and complex variable $z,$ we use the classical notation $\tensor[_{2}]{F}{{}_{1}}(\alpha,\beta;\gamma;z)$ to denote $\tensor[_{2}]{F}{{}_{1}}(\alpha,\beta;\gamma;z)=\sum_{j=0}^{\infty}\frac{(\alpha)_{j}(\beta)_{j}}{j!(\gamma)_{j}}z^{j},$ with $\gamma\neq 0,-1,-2,\ldots,$ where $(\alpha)_{j}=\Pi_{k=0}^{j-1}(\alpha+k)$ is the Pochhammer for any complex number $\alpha.$ The following lemma is in fact a restatement of Proposition 1.4.10 of [20]. ###### Lemma 5.2. [20] Suppose $\beta\in\mathbb{R}$ and $\gamma>-1,$ then $\int_{\mathbb{B}^{d}}\frac{(1-|w|^{2})^{\gamma}}{|1-\langle z,w\rangle|^{2\beta}}dv(w)=\frac{\Gamma(1+d)\Gamma(1+\gamma)}{\Gamma(1+d+\gamma)}\tensor[_{2}]{F}{{}_{1}}(\beta,\beta;1+d+\gamma;|z|^{2}).$ We also need the following lemma. ###### Lemma 5.3. [8, Chapter 2] The following three identities hold. 1. (1) $\tensor[_{2}]{F}{{}_{1}}(\alpha,\beta;\gamma;z)=(1-z)^{\gamma-\alpha-\beta}\tensor[_{2}]{F}{{}_{1}}(\gamma-\alpha,\gamma-\beta;\gamma;z);$ 2. (2) $\tensor[_{2}]{F}{{}_{1}}(\alpha,\beta;\gamma;1)=\frac{\Gamma(\gamma)\Gamma(\gamma-\alpha-\beta)}{\Gamma(\gamma-\alpha)\Gamma(\gamma-\beta)},$ if $Re(\gamma-\alpha-\beta)>0;$ 3. (3) $\frac{d}{dz}~{}\tensor[_{2}]{F}{{}_{1}}(\alpha,\beta;\gamma;z)=\frac{\alpha\beta}{\gamma}~{}\tensor[_{2}]{F}{{}_{1}}(\alpha+1,\beta+1;\gamma+1;z).$ ###### Lemma 5.4. If $0<\alpha<d+1,$ then $K_{\alpha}:L^{\infty}\rightarrow L^{q}$ is compact for any $1\leq q\leq\infty.$ ###### Proof. Since the continuous embedding of $L$-integrable spaces, it suffices to prove that $K_{\alpha}:L^{\infty}\rightarrow L^{\infty}$ is compact. We first prove that, for any $f\in L^{\infty},$ then $K_{\alpha}f\in A(\mathbb{B}^{d}),$ where $A(\mathbb{B}^{d})=H(\mathbb{B}^{d})\cap C(\overline{\mathbb{B}^{d}})$ is the ball algebra. For $f\in L^{\infty},$ it is clear that $K_{\alpha}f$ is holomorphic on the ball, i.e. $K_{\alpha}f\in H(\mathbb{B}^{d}).$ Now we prove that $K_{\alpha}f$ is also continuous on the closed ball $\overline{\mathbb{B}^{d}.}$ From Lemma 5.2 and (2) of Lemma 5.3, it implies that $K_{\alpha}f(\eta)$ exists for any $\eta\in\partial\mathbb{B}^{d}$ and $|K_{\alpha}f(\eta)|\leq\|f\|_{\infty}\frac{\Gamma(d+1)\Gamma(d+1-\alpha)}{\Gamma^{2}(d+1-\frac{\alpha}{2})}.$ We now turn to prove that $K_{\alpha}f$ is continuous on $\partial\mathbb{B}^{d}.$ It suffices to prove that, for any $\eta\in\partial\mathbb{B}^{d}$ and for any point sequence $\\{z_{n}\\}$ in $\mathbb{B}^{d}$ satisfying that $z_{n}\rightarrow\eta,$ we have $K_{\alpha}f(z_{n})\rightarrow K_{\alpha}f(\eta)$ as $n\rightarrow\infty.$ By Lemma 5.2 and (2) of Lemma 5.3 again, we have $\begin{split}|K_{\alpha}f(z)|&\leq\|f\|_{\infty}\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle z,w\rangle|^{\alpha}}dv(w)\\\ &\leq\|f\|_{\infty}\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle\eta,w\rangle|^{\alpha}}dv(w)\\\ &=\|f\|_{\infty}\frac{\Gamma(d+1)\Gamma(d+1-\alpha)}{\Gamma^{2}(d+1-\frac{\alpha}{2})},\end{split}$ (5.1) for any $z\in\mathbb{B}^{d}.$ Due to the absolute continuity of the integral, it implies that, for any $\varepsilon>0,$ there exists $0<\delta<1$, satisfying that $\int_{F}\frac{dv(w)}{|1-\langle\eta,w\rangle|^{\alpha}}\leq\frac{\varepsilon}{4},$ (5.2) whenever $v(F)<\delta.$ Denote $F_{\delta}=\\{z\in\mathbb{B}^{d}:\sqrt[d]{1-\frac{\delta}{2}}<|z|<1\\}.$ Note that $v(F_{\delta})=\frac{\delta}{2}<\delta$ and $\frac{1}{(1-\langle z_{n},w\rangle)^{\alpha}}\rightarrow\frac{1}{(1-\langle\eta,w\rangle)^{\alpha}}~{}\text{uniformly~{}on }~{}\mathbb{B}^{d}\setminus F_{\delta},\text{~{}as}~{}n\rightarrow\infty.$ Then there exists $N>0$ such that, for any $n>N,$ $\int_{\mathbb{B}^{d}\setminus F_{\delta}}\left|\frac{1}{(1-\langle z_{n},w\rangle)^{\alpha}}-\frac{1}{(1-\langle\eta,w\rangle)^{\alpha}}\right|dv(w)\leq\frac{\varepsilon}{2}.$ Combing this with (5.1), (5.2), it implies that, for any $n>N,$ $\begin{split}|K_{\alpha}f(z_{n})-K_{\alpha}f(\eta)|&\leq\|f\|_{\infty}\int_{\mathbb{B}^{d}\setminus F_{\delta}}\left|\frac{1}{(1-\langle z_{n},w\rangle)^{\alpha}}-\frac{1}{(1-\langle\eta,w\rangle)^{\alpha}}\right|dv(w)\\\ &~{}~{}~{}~{}~{}~{}~{}~{}~{}+\|f\|_{\infty}\int_{F_{\delta}}\left|\frac{1}{(1-\langle z_{n},w\rangle)^{\alpha}}-\frac{1}{(1-\langle\eta,w\rangle)^{\alpha}}\right|dv(w)\\\ &\leq\|f\|_{\infty}\int_{\mathbb{B}^{d}\setminus F_{\delta}}\left|\frac{1}{(1-\langle z_{n},w\rangle)^{\alpha}}-\frac{1}{(1-\langle\eta,w\rangle)^{\alpha}}\right|dv(w)\\\ &~{}~{}~{}~{}~{}~{}~{}~{}~{}+2\|f\|_{\infty}\int_{F_{\delta}}\frac{1}{|1-\langle\eta,w\rangle|^{\alpha}}dv(w)\\\ &\leq\|f\|_{\infty}\frac{\varepsilon}{2}+2\|f\|_{\infty}\frac{\varepsilon}{4}\\\ &=\varepsilon\|f\|_{\infty}.\\\ \end{split}$ (5.3) It completes the proof of what $K_{\alpha}f$ is continuous on the closed ball $\overline{\mathbb{B}^{d}}.$ Now we prove that, for any bounded sequence in $L^{\infty},$ there exists a subsequence satisfying that its image under $K_{\alpha}$ is convergent in $L^{\infty}.$ Suppose that $\\{f_{n}\\}$ is a bounded sequence in $L^{\infty},$ then we have $\\{K_{\alpha}f_{n}\\}$ is in $C(\overline{\mathbb{B}^{d}})$ and $\\{K_{\alpha}f_{n}\\}$ is uniformly bounded by (5.1). Now we prove that $\\{K_{\alpha}f_{n}\\}$ is also equicontinuous. From (5.3), we know that $\lim_{\mathbb{B}^{d}\ni z\rightarrow\eta}\int_{\mathbb{B}^{d}}\left|\frac{1}{(1-\langle z,w\rangle)^{\alpha}}-\frac{1}{(1-\langle\eta,w\rangle)^{\alpha}}\right|dv(w)=0,$ (5.4) for arbitrary fixed $\eta\in\partial\mathbb{B}^{d}.$ Combing (5.4) with the unitary invariance of Lebsgue measure and the symmetry of the unit ball, it implies that, for any $\epsilon>0,$ there exists $0<\delta^{\prime}<1,$ satisfying that $\int_{\mathbb{B}^{d}}\left|\frac{1}{(1-\langle z,w\rangle)^{\alpha}}-\frac{1}{(1-\langle\eta,w\rangle)^{\alpha}}\right|dv(w)\leq\frac{\epsilon}{2}$ (5.5) whenever $z\in\mathbb{B}^{d},\eta\in\partial\mathbb{B}^{d}$ and $|z-\eta|<\delta^{\prime}.$ Denote $B_{1-\frac{\delta^{\prime}}{2}}=\\{z\in\mathbb{C}^{d}:|z|\leq 1-\frac{\delta^{\prime}}{2}\\}$ and $C_{\frac{\delta^{\prime}}{2}}=\\{z\in\mathbb{C}^{d}:1-\frac{\delta^{\prime}}{2}<|z|\leq 1\\}.$ Then the closed ball $\overline{\mathbb{B}^{d}}$ has the following decomposition, $\overline{\mathbb{B}^{d}}=B_{1-\frac{\delta^{\prime}}{2}}\cup C_{\frac{\delta^{\prime}}{2}}\text{~{}and~{}}B_{1-\frac{\delta^{\prime}}{2}}\cap C_{\frac{\delta^{\prime}}{2}}=\emptyset.$ (5.6) Since the function $\frac{1}{(1-\langle z,w\rangle)^{\alpha}}$ is uniformly continuous on compact set $B_{1-\frac{\delta^{\prime}}{2}}\times\overline{\mathbb{B}^{d}},$ then there exists $0<\delta^{\prime\prime}<1$ such that $\left|\frac{1}{(1-\langle z_{1},w\rangle)^{\alpha}}-\frac{1}{(1-\langle z_{2},w\rangle)^{\alpha}}\right|\leq\epsilon,$ (5.7) whenever $(z_{1},w),(z_{2},w)\in B_{1-\frac{\delta^{\prime}}{2}}\times\overline{\mathbb{B}^{d}}$ and $|z_{1}-z_{2}|<\delta^{\prime\prime}.$ Take $\delta^{\prime\prime\prime}=\min\\{\frac{\delta^{\prime}}{2},\delta^{\prime\prime}\\}.$ Now we prove that, for any $z_{1},z_{2}\in\overline{\mathbb{B}^{d}}$ such that $|z_{1}-z_{2}|<\delta^{\prime\prime\prime},$ then we have $\int_{\mathbb{B}^{d}}\left|\frac{1}{(1-\langle z_{1},w\rangle)^{\alpha}}-\frac{1}{(1-\langle z_{2},w\rangle)^{\alpha}}\right|dv(w)\leq\epsilon.$ (5.8) In fact, there are two cases need to be considered. The first case is $z_{1}\in C_{\frac{\delta^{\prime}}{2}}$ or $z_{2}\in C_{\frac{\delta^{\prime}}{2}}.$ Without loss of generality, we can assume that $z_{1}\in C_{\frac{\delta^{\prime}}{2}},$ then there exists an $\eta\in\partial\mathbb{B}^{d}$ satisfying that $|z_{1}-\eta|<\delta^{\prime\prime\prime}\leq\frac{\delta^{\prime}}{2}.$ By triangle inequality, it implies that $|z_{2}-\eta|\leq|z_{2}-z_{1}|+|z_{1}-\eta|<\delta^{\prime}.$ Together with (5.5), it implies that $\begin{split}&\int_{\mathbb{B}^{d}}\left|\frac{1}{(1-\langle z_{1},w\rangle)^{\alpha}}-\frac{1}{(1-\langle z_{2},w\rangle)^{\alpha}}\right|dv(w)\\\ &\leq\int_{\mathbb{B}^{d}}\left|\frac{1}{(1-\langle z_{1},w\rangle)^{\alpha}}-\frac{1}{(1-\langle\eta,w\rangle)^{\alpha}}\right|dv(w)\\\ &~{}\vspace{0.2cm}+\int_{\mathbb{B}^{d}}\left|\frac{1}{(1-\langle\eta,w\rangle)^{\alpha}}-\frac{1}{(1-\langle z_{2},w\rangle)^{\alpha}}\right|dv(w)\\\ &\leq\epsilon\end{split}$ The second case is $z_{1},z_{2}\in B_{1-\frac{\delta^{\prime}}{2}}.$ By (5.7), it implies that $\begin{split}\int_{\mathbb{B}^{d}}\left|\frac{1}{(1-\langle z_{1},w\rangle)^{\alpha}}-\frac{1}{(1-\langle z_{2},w\rangle)^{\alpha}}\right|dv(w)\leq\epsilon\int_{\mathbb{B}^{d}}dv=\epsilon.\end{split}$ It proves (5.8). Combing with $|K_{\alpha}f_{n}(z_{1})-K_{\alpha}f_{n}(z_{2})|\leq\|f_{n}\|_{\infty}\int_{\mathbb{B}^{d}}\left|\frac{1}{(1-\langle z_{1},w\rangle)^{\alpha}}-\frac{1}{(1-\langle z_{2},w\rangle)^{\alpha}}\right|dv(w),$ it implies that $\\{K_{\alpha}f_{n}\\}$ is equicontinuous. Then by Arzelà- Ascoli theorem, it implies that $\\{K_{\alpha}f_{n}\\}$ has a convergency subsequence in the supremum norm. ∎ ###### Corollary 5.5. If $0<\alpha<d+1,$ then the following holds: 1. (1) $K_{\alpha}:L^{p}\rightarrow L^{1}$ is compact for any $1\leq p\leq\infty.$ 2. (2) $K_{\alpha}:L^{1}\rightarrow L^{q}$ is compact if and only if $q<\frac{d+1}{\alpha}.$ 3. (3) $K_{\alpha}:L^{p}\rightarrow L^{\infty}$ is compact if and only if $p>\frac{d+1}{d+1-\alpha}.$ ###### Proof. It comes from Lemma 3.7, Lemma 5.4 and the fact that $K_{\alpha}$ is adjoint. ∎ In the following, we deal with the case $1<p,q<\infty$ . However, we need the following result about Carleson type measures for the Bergman spaces over the unit ball. ###### Lemma 5.6. [24] Suppose $1\leq p\leq q<\infty$ and $\mu$ is a positive Borel measure on $\mathbb{B}^{d}$. Then the following conditions are equivalent: 1. (1) If $\\{f_{n}\\}$ is a bounded sequence in $A_{0}^{p}$ and $f_{n}(z)\rightarrow 0$ for every $z\in\mathbb{B}^{d}$, then $\lim_{n\rightarrow\infty}\int_{\mathbb{B}^{d}}|f_{n}|^{q}d\mu=0.$ 2. (2) For every (or some) $s>0,$ we have $\lim_{|z|\rightarrow 1^{-}}\int_{\mathbb{B}^{d}}\frac{(1-|z|^{2})^{s}}{|1-\langle z,w\rangle|^{s+\frac{q(d+1)}{p}}}d\mu(w)=0.$ The Borel measure in Lemma 5.6 is in fact the so-called vanishing Carleson measures. If denote $A^{q}(d\mu)$ by the weighted Bergman space $A^{q}(d\mu)=H(\mathbb{B}^{d})\cap L^{q}(\mathbb{B}^{d},d\mu).$ Then (1) of Lemma 5.6 guarantees ( or is equivalent to) that the embedding $Id:A^{p}_{0}\rightarrow A^{q}(d\mu)$ is compact. ###### Proposition 5.7. If $0<\alpha<d+1$ and $1<p\leq q<\infty,$ then the following conditions are equivalent. 1. (1) $K_{\alpha}:L^{p}\rightarrow L^{q}$ is compact. 2. (2) $K_{\alpha}:A_{0}^{p}\rightarrow A_{0}^{q}$ is compact. 3. (3) The embedding $Id:A_{0}^{p}\rightarrow A_{q(d+1-\alpha)}^{q}$ is compact. 4. (4) $\frac{1}{q}>\frac{1}{p}+\frac{\alpha}{d+1}-1.$ ###### Proof. We first prove that (1) is equivalent to (2). Clearly (1) implies (2). To prove the reverse, note that by $0<\alpha<d+1$ and combing this with Theorem 2 gives us that $K_{d+1}:L^{p}\rightarrow A_{0}^{p}$ is bounded. Suppose that $\\{f_{n}\\}$ is an arbitrary bounded sequence in $L^{p},$ thus we have $\\{K_{d+1}f_{n}\\}$ is a bounded sequence in $A_{0}^{p}.$ Then the compactness of operator $K_{\alpha}:A_{0}^{p}\rightarrow A_{0}^{q}$ implies that, there exists a subsequence $\\{f_{n_{j}}\\}$ such that $\\{K_{\alpha}(K_{d+1}f_{n_{j}})\\}$ is convergent in $A_{0}^{q}.$ Combing this with Lemma 2.7, yields that $\\{K_{\alpha}f_{n_{j}}\\}$ is convergent in $A_{0}^{q}.$ It proved that (2) implies (1). Now we prove that (2) is equivalent to (3). Similar to the proof of Proposition 2.6, by Theorem 14 of [24] and Theorem 2.19 of [25], it can be proved that $R^{\alpha-d-1,d+1-\alpha}:A_{0}^{q}\rightarrow A_{q(d+1-\alpha)}^{q}$ and its inverse operator are bounded. Note that $K_{\alpha}=R^{0,\alpha-d-1}$ on $A_{0}^{p}$ and $R^{\alpha-d-1,d+1-\alpha}R^{0,\alpha-d-1}f=f,\forall f\in A_{0}^{p}.$ Then we have the following decomposition for embedding $Id,$ $\begin{split}Id:A_{0}^{p}&\xrightarrow{K_{\alpha}=R^{0,\alpha-d-1}}A_{0}^{q}\xrightarrow{R^{\alpha-d-1,d+1-\alpha}}A_{q(d+1-\alpha)}^{q},\\\ &Id=R^{\alpha-d-1,d+1-\alpha}K_{\alpha}.\end{split}$ (5.9) Combing (5.9) with the fact that $R^{\alpha-d-1,d+1-\alpha}$ has bounded inverse, it implies that $K_{\alpha}:A_{0}^{p}\rightarrow A_{0}^{q}$ is compact if and only if the embedding $Id:A_{0}^{p}\rightarrow A_{q(d+1-\alpha)}^{q}$ is compact. In the next, we prove that (3) is equivalent to (4). Suppose that $\\{f_{n}\\}$ is an arbitrary bounded sequence in $A_{0}^{p},$ then by Theorem 20 of [24], the locally estimation for functions in $A_{0}^{p},$ it implies that $\\{f_{n}\\}$ is a normal family. Hence, by Fatou’s lemma, similar to the proof of Theorem 1, there exists a subsequence $\\{f_{n_{j}}\\}$ and $g\in A_{0}^{p}$ such that $f_{n_{j}}$ converges uniformly to $g$ on any compact subset of $\mathbb{B}^{d}.$ Then $\\{f_{n_{j}}-g\\}$ is in $A_{0}^{p}$ and $f_{n_{j}}-g\rightarrow 0$ pointwise as $j\rightarrow\infty.$ Together with Lemma 5.6, it yields that the embedding $Id:A_{0}^{p}\rightarrow A_{q(d+1-\alpha)}^{q}$ is compact if and only if $\lim_{|z|\rightarrow 1^{-}}\int_{\mathbb{B}^{d}}\frac{(1-|z|^{2})^{s}}{|1-\langle z,w\rangle|^{s+\frac{q(d+1)}{p}}}dv_{q(d+1-\alpha)}(w)=0,$ (5.10) for any $s>0.$ On the other hand, by Proposition 1.4.10 of [20], it implies that (5.10) is equivalent to $\frac{1}{q}>\frac{1}{p}+\frac{\alpha}{d+1}-1.$ It completes the proof. ∎ Proof of Theorem 3. When $\alpha=d+1.$ Theorem 3 degenerates into Proposition 5.1. Now we turn to the case $0<\alpha<d+1.$ We first prove that (2) implies (1). In fact, it is an immediate corollary from Lemma 3.7, Lemma 5.4 and Corollary 5.5. To see the reverse, note that by Theorem 2 and combining this with Proposition 5.7 gives that (1) is not held if (2) is not held, it implies that (1) implies (2), completing the proof. ∎ Proof of Theorem 4. By Theorem 1,2,3, it is easy to see that $(1)\Rightarrow(4)\Rightarrow(2)\Leftrightarrow(3).$ Thus we only need to show that $(2)\Rightarrow(1).$ It is equivalent to prove that $(2)$ is not held if $(1)$ is not held. It suffices to show that $K_{\alpha}:L^{\infty}\rightarrow L^{1}$ is not bounded if $\alpha\geq d+2.$ Suppose that $\alpha\geq d+2.$ In view to Theorem 7.1 of [25], it implies that $K_{\alpha}(L^{\infty})=\mathcal{B}_{\alpha-d}.$ From Lemma 3.2, we know that $\mathcal{B}_{\alpha-d}\not\subset L^{1},$ it means that $K_{\alpha}:L^{\infty}\rightarrow L^{1}$ is not bounded. It completes the proof. ∎ ## 6\. norm estimations for $K_{\alpha}.$ In the previous sections, we have completely characterized the $L^{p}$-$L^{q}$ boundedness of $K_{\alpha},K_{\alpha}^{+}$ and compactness of $K_{\alpha}.$ In the present section, we will state and prove some sharp norm estimates of $K_{\alpha},K_{\alpha}^{+},$ which gives essentially the upper bounds of the best constants in the Hardy-Littlewood-Sobolev inequalities. ###### Proposition 6.1. If $d+1<\alpha<d+2$ and $K_{\alpha}:L^{p}\rightarrow L^{q}$ is bounded, then $\|K_{\alpha}\|_{L^{p}\rightarrow L^{q}}\leq\frac{\Gamma(d+1)^{1+\frac{1}{q}-\frac{1}{p}}\Gamma(\alpha-(d+1))\Gamma(\frac{1}{q^{-1}-p^{-1}}(d+1-\alpha)+1)^{\frac{1}{q}-\frac{1}{p}}}{\Gamma(\frac{\alpha}{2})^{2}\Gamma(\frac{1}{q^{-1}-p^{-1}}(d+1-\alpha)+d+1)^{\frac{1}{q}-\frac{1}{p}}}.$ (6.1) ###### Lemma 6.2. Suppose that $d+1<\alpha<d+2$ and $(0,\frac{1}{q})\in G(K_{\alpha})=G(K_{\alpha}^{+}),$ then the following holds. 1. (1) $\|K_{\alpha}\|_{L^{\infty}\rightarrow L^{q}}\leq\|K_{\alpha}^{+}\|_{L^{\infty}\rightarrow L^{q}}=\|\int_{\mathbb{B}^{d}}k_{\alpha}^{+}(\cdot,w)dv(w)\|_{L^{q}}.$ 2. (2) In particular, when $d=1,$ $\|K_{\alpha}\|_{L^{\infty}\rightarrow L^{1}}\leq\|K_{\alpha}^{+}\|_{L^{\infty}\rightarrow L^{1}}=\frac{4}{(\alpha-2)^{2}}\left(\frac{\Gamma(3-\alpha)}{\Gamma^{2}(2-\frac{\alpha}{2})}-1\right).$ (6.2) 3. (3) For any general $(0,\frac{1}{q})\in G(K_{\alpha})=G(K_{\alpha}^{+}),$ $\|K_{\alpha}\|_{L^{\infty}\rightarrow L^{q}}\leq\|K_{\alpha}^{+}\|_{L^{\infty}\rightarrow L^{q}}\leq\frac{\Gamma(d+1)^{1+\frac{1}{q}}\Gamma(\alpha-(d+1))\Gamma(q(d+1-\alpha)+1)^{\frac{1}{q}}}{\Gamma(\frac{\alpha}{2})^{2}\Gamma(q(d+1-\alpha)+d+1)^{\frac{1}{q}}}.$ (6.3) ###### Proof. (1) Since $|K_{\alpha}(f)|\leq K_{\alpha}^{+}(|f|),$ it implies that $\|K_{\alpha}\|_{L^{\infty}\rightarrow L^{q}}\leq\|K_{\alpha}^{+}\|_{L^{\infty}\rightarrow L^{q}}$ if $K_{\alpha}$ and $K_{\alpha}^{+}$ are bounded. Note that $|K_{\alpha}^{+}f|(z)\leq\|f\|_{\infty}\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle z,w\rangle|^{\alpha}}dv(w)$ for any $f\in L^{\infty},$ it yields that $\|K_{\alpha}\|_{L^{\infty}\rightarrow L^{q}}\leq\|K_{\alpha}^{+}\|_{L^{\infty}\rightarrow L^{q}}\leq\|\int_{\mathbb{B}^{d}}k_{\alpha}^{+}(\cdot,w)dv(w)\|_{L^{q}}.$ To see the reverse, note that $\|K_{\alpha}^{+}\|_{L^{\infty}\rightarrow L^{q}}\geq\|K_{\alpha}^{+}1\|_{L^{q}}=\|\int_{\mathbb{B}^{d}}k_{\alpha}^{+}(\cdot,w)dv(w)\|_{L^{q}}.$ It leads to the desired result. (2) Now we turn to calculate the norm in the case of $d=1.$ From (2) of Lemma 5.3 and what we have proved, it follows that $\begin{split}\|K_{\alpha}\|_{L^{\infty}\rightarrow L^{1}}&\leq\|K_{\alpha}^{+}\|_{L^{\infty}\rightarrow L^{1}}\\\ &=\int_{\mathbb{B}^{d}}\tensor[_{2}]{F}{{}_{1}}(\frac{\alpha}{2},\frac{\alpha}{2};d+1;|z|^{2})dv(z)\\\ &=d\int_{0}^{1}\tensor[_{2}]{F}{{}_{1}}(\frac{\alpha}{2},\frac{\alpha}{2};d+1;r)r^{d-1}dr,\\\ \end{split}$ (6.4) in the last equality we apply the integration in polar coordinates, see Lemma 1.8 of [25], and the unitary invariance of hypergeometric function $\tensor[_{2}]{F}{{}_{1}}(\frac{\alpha}{2},\frac{\alpha}{2};d+1;|z|^{2}).$ Now we use the differential properties listed in Lemma 5.3 to calculate the integral in the case of $d=1.$ We observe (3) of Lemma 5.3, it gives that $\frac{d}{dr}\left(\tensor[_{2}]{F}{{}_{1}}(\frac{\alpha}{2}-1,\frac{\alpha}{2}-1;1;r)\right)=(\frac{\alpha}{2}-1)^{2}\tensor[_{2}]{F}{{}_{1}}(\frac{\alpha}{2},\frac{\alpha}{2};2;r).$ Then integrate the two sides of the above equality and we get $\int_{0}^{1}\tensor[_{2}]{F}{{}_{1}}(\frac{\alpha}{2},\frac{\alpha}{2};2;r)dr=\frac{4}{(\alpha-2)^{2}}\left(\tensor[_{2}]{F}{{}_{1}}(\frac{\alpha}{2}-1,\frac{\alpha}{2}-1;1;1)-1\right).$ Together with with (2) of Lemma 5.3 yields the desired result. (3) Combing (1) with Lemma 5.2 and (1),(2) of Lemma 5.3, it follows that $\begin{split}\|K_{\alpha}^{+}\|_{L^{\infty}\rightarrow L^{q}}&=\left(\int_{\mathbb{B}^{d}}\left(\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle z,w\rangle|^{\alpha}}dv(w)\right)^{q}dv(z)\right)^{\frac{1}{q}}\\\ &=\left(\int_{\mathbb{B}^{d}}\tensor[_{2}]{F}{{}_{1}}(\frac{\alpha}{2},\frac{\alpha}{2};d+1;|z|^{2})^{q}dv(z)\right)^{\frac{1}{q}}\\\ &=\left(\int_{\mathbb{B}^{d}}(1-|z|^{2})^{q(d+1-\alpha)}\tensor[_{2}]{F}{{}_{1}}(d+1-\frac{\alpha}{2},d+1-\frac{\alpha}{2};d+1;|z|^{2})^{q}dv(z)\right)^{\frac{1}{q}}\\\ &\leq\tensor[_{2}]{F}{{}_{1}}(d+1-\frac{\alpha}{2},d+1-\frac{\alpha}{2};d+1;1)\left(\int_{\mathbb{B}^{d}}(1-|z|^{2})^{q(d+1-\alpha)}dv(z)\right)^{\frac{1}{q}}\\\ &=\frac{\Gamma(d+1)^{1+\frac{1}{q}}\Gamma(\alpha-(d+1))\Gamma(q(d+1-\alpha)+1)^{\frac{1}{q}}}{\Gamma(\frac{\alpha}{2})^{2}\Gamma(q(d+1-\alpha)+d+1)^{\frac{1}{q}}}.\\\ \end{split}$ It leads to (6.3). ∎ Proof of Proposition 6.1. Suppose that $K_{\alpha}^{+}:L^{p}\rightarrow L^{q}$ is bounded, it is equivalent to $(\frac{1}{p},\frac{1}{q})\in G(K_{\alpha}^{+}).$ Then (3) of Theorem 1 guarantees $\frac{1}{q}-\frac{1}{p}>\alpha-(d+1).$ Note by (3) of Theorem 1 again yields that $(0,\frac{1}{q}-\frac{1}{p}),(1-(\frac{1}{q}-\frac{1}{p}),1)\in G(K_{\alpha}^{+})$ (6.5) and there exists $0\leq\theta\leq 1$ satisfying that $(\frac{1}{p},\frac{1}{q})=\theta\cdot(0,\frac{1}{q}-\frac{1}{p})+(1-\theta)\cdot(1-(\frac{1}{q}-\frac{1}{p}),1).$ (6.6) Combing (6.5),(6.6) with Lemma 2.2, it follows that $\|K_{\alpha}^{+}\|_{L^{p}\rightarrow L^{q}}\leq\|K_{\alpha}^{+}\|_{L^{\infty}\rightarrow L^{\frac{1}{q^{-1}-p^{-1}}}}^{\theta}\|K_{\alpha}^{+}\|_{L^{\frac{1}{1-(q^{-1}-p^{-1})}}\rightarrow L^{1}}^{1-\theta}$ (6.7) We observe that the adjoint operator of $K_{\alpha}^{+}:L^{\infty}\rightarrow L^{\frac{1}{q^{-1}-p^{-1}}}$ is exactly the operator $K_{\alpha}^{+}:L^{\frac{1}{1-(q^{-1}-p^{-1})}}\rightarrow L^{1},$ it means that $\|K_{\alpha}^{+}\|_{L^{\infty}\rightarrow L^{\frac{1}{q^{-1}-p^{-1}}}}=\|K_{\alpha}^{+}\|_{L^{\frac{1}{1-(q^{-1}-p^{-1})}}\rightarrow L^{1}}.$ Applying this to (6.7), then yields that $\|K_{\alpha}^{+}\|_{L^{p}\rightarrow L^{q}}\leq\|K_{\alpha}^{+}\|_{L^{\infty}\rightarrow L^{\frac{1}{q^{-1}-p^{-1}}}}.$ (6.8) Combing (6.8) with (6.5) and applying Lemma 6.3, it leads to the desired conclusion.∎ ###### Corollary 6.3. Suppose that $C_{1}$ is the best constant in HLS 1, then $C_{1}\leq\frac{\Gamma(d+1)^{2-\frac{1}{s}-\frac{1}{p}}\Gamma(\alpha-(d+1))\Gamma(\frac{1}{1-s^{-1}-p^{-1}}(d+1-\alpha)+1)^{1-\frac{1}{s}-\frac{1}{p}}}{\Gamma(\frac{\alpha}{2})^{2}\Gamma(\frac{1}{1-s^{-1}-p^{-1}}(d+1-\alpha)+d+1)^{1-\frac{1}{s}-\frac{1}{p}}}.$ ###### Proposition 6.4. If $0<\alpha<d+1$ and $\frac{1}{p}-(1-\frac{\alpha}{d+1})<\frac{1}{q}\leq\frac{1}{p},$ then $\|K_{\alpha}\|_{L^{p}\rightarrow L^{q}}\leq\|K_{\alpha}^{+}\|_{L^{p}\rightarrow L^{q}}\leq\left(\frac{\Gamma(d+1)\Gamma(d+1-\frac{\alpha}{1-(p^{-1}-q^{-1})})}{\Gamma^{2}(d+1-\frac{\alpha}{2(1-(p^{-1}-q^{-1}))})}\right)^{1-(\frac{1}{p}-\frac{1}{q})}.$ (6.9) In particular, when $q=\infty,$ the inequality (6.9) is an equality. ###### Proof. We first prove that (6.9) is in fact equality under the case of $q=\infty.$ From (4.1), we know that $\|K_{\alpha}\|_{L^{p}\rightarrow L^{\infty}}=\|K_{\alpha}^{+}\|_{L^{p}\rightarrow L^{\infty}}=\sup_{z\in\mathbb{B}^{d}}\left(\int\frac{dv(w)}{|1-\langle z,w\rangle|^{\frac{p\alpha}{p-1}}}\right)^{\frac{p-1}{p}}.$ (6.10) On the other hand, Lemma 5.2 and (2) of Lemma 5.3 yield $\begin{split}\int_{\mathbb{B}^{d}}\frac{dv(w)}{|1-\langle z,w\rangle|^{\frac{p\alpha}{p-1}}}&=\tensor[_{2}]{F}{{}_{1}}(\frac{p\alpha}{2(p-1)},\frac{p\alpha}{2(p-1)};d+1;|z|^{2})\\\ &\leq\tensor[_{2}]{F}{{}_{1}}(\frac{p\alpha}{2(p-1)},\frac{p\alpha}{2(p-1)};d+1;1)\\\ &=\frac{\Gamma(d+1)\Gamma(d+1-\frac{p\alpha}{p-1})}{\Gamma^{2}(d+1-\frac{p\alpha}{2(p-1)})}.\end{split}$ (6.11) Combing (6.10) and (6.11), it implies that $\|K_{\alpha}\|_{L^{p}\rightarrow L^{\infty}}=\|K_{\alpha}^{+}\|_{L^{p}\rightarrow L^{\infty}}=\left(\frac{\Gamma(d+1)\Gamma(d+1-\frac{p\alpha}{p-1})}{\Gamma^{2}(d+1-\frac{p\alpha}{2(p-1)})}\right)^{\frac{p-1}{p}}.$ (6.12) Now we turn to prove (6.9) in the general case. Note first that $|K_{\alpha}(f)|\leq K_{\alpha}^{+}(|f|),$ it implies that $\|K_{\alpha}\|_{L^{p}\rightarrow L^{q}}\leq\|K_{\alpha}^{+}\|_{L^{p}\rightarrow L^{q}}$ if $K_{\alpha}$ and $K_{\alpha}^{+}$ are bounded. Since $\frac{1}{p}-(1-\frac{\alpha}{d+1})<\frac{1}{q}\leq\frac{1}{p},$ Theorem 2 implies that $(\frac{1}{p},\frac{1}{q}),(\frac{1}{p}-\frac{1}{q},0),(1,1-(\frac{1}{p}-\frac{1}{q}))\in G(K_{\alpha}^{+})$ (6.13) and there exists $0\leq\theta\leq 1$ satisfying that $(\frac{1}{p},\frac{1}{q})=\theta\cdot(\frac{1}{p}-\frac{1}{q},0)+(1-\theta)\cdot(1,1-(\frac{1}{p}-\frac{1}{q}))$ (6.14) Combing (6.13),(6.14) with Lemma 2.2, it follows that $\|K_{\alpha}^{+}\|_{L^{p}\rightarrow L^{q}}\leq\|K_{\alpha}^{+}\|_{L^{\frac{1}{p^{-1}-q^{-1}}}\rightarrow L^{\infty}}^{\theta}\|K_{\alpha}^{+}\|_{L^{1}\rightarrow L^{\frac{1}{1-(p^{-1}-q^{-1})}}}^{1-\theta}$ (6.15) We observe that the adjoint operator of $K_{\alpha}^{+}:L^{\frac{1}{p^{-1}-q^{-1}}}\rightarrow L^{\infty}$ is exactly the operator $K_{\alpha}^{+}:L^{1}\rightarrow L^{\frac{1}{1-(p^{-1}-q^{-1})}},$ it means that $\|K_{\alpha}^{+}\|_{L^{\frac{1}{p^{-1}-q^{-1}}}\rightarrow L^{\infty}}=\|K_{\alpha}^{+}\|_{L^{1}\rightarrow L^{\frac{1}{1-(p^{-1}-q^{-1})}}}.$ (6.16) Thus by (6.15) and (6.16), it follows that $\|K_{\alpha}^{+}\|_{L^{p}\rightarrow L^{q}}\leq\|K_{\alpha}^{+}\|_{L^{\frac{1}{p^{-1}-q^{-1}}}\rightarrow L^{\infty}}.$ Together with (6.12), it completes the proof.∎ ###### Corollary 6.5. Suppose that $C_{2}$ is the best constant in HLS 2, then the following holds. 1. (1) If $\frac{1}{p}<1-\frac{1}{s},$ then $C_{2}\leq\frac{\Gamma(d+1)\Gamma(d+1-\alpha)}{\Gamma^{2}(d+1-\frac{\alpha}{2})}.$ 2. (2) If $\frac{1}{p}-(1-\frac{\alpha}{d+1})<1-\frac{1}{s}\leq\frac{1}{p},$ then $C_{2}\leq\left(\frac{\Gamma(d+1)\Gamma(d+1-\frac{\alpha}{2-p^{-1}-s^{-1}})}{\Gamma^{2}(d+1-\frac{\alpha}{2(2-p^{-1}-s^{-1})})}\right)^{2-(\frac{1}{p}-\frac{1}{s})}.$ Proof of Theorem 5. When $\alpha<\frac{d+2}{2},$ by Proposition 1.4.10 of [20], it implies that the kernel function $k_{\alpha}^{+}\in L^{2}(\mathbb{B}^{d}\times\mathbb{B}^{d},dv\times dv),$ thus $K_{\alpha},K_{\alpha}^{+}:L^{2}\rightarrow L^{2}$ are Hilbert-Schmidt. Note that $Tr(K_{\alpha}^{*}K_{\alpha})=\int_{\mathbb{B}^{d}}\int_{\mathbb{B}^{d}}\frac{1}{|1-\langle z,w\rangle|^{2\alpha}}dv(w)dv(z).$ (6.17) When $\alpha\neq 1,$ similar to (6.2), yields the trace formula. Now we deal with the spacial case $\alpha=1.$ Combing Lemma 5.2 with (6.17), it implies that $Tr(K_{1}^{*}K_{1})=\int_{0}^{1}\tensor[_{2}]{F}{{}_{1}}(1,1;2;r)dr=\sum_{j=1}^{\infty}\frac{1}{j^{2}}=\frac{\pi^{2}}{6}.$ ∎ ###### Remark 6.6. By (3) of Proposition 5.3 and inductive method, we can get explicit trace formulas for every dimension $d\geq 1.$ As a consequence of Theorem 5 we obtain the following generalized Euler-Jacobi identity. ###### Corollary 6.7. Suppose that $0<\alpha<\frac{3}{2},$ then $\sum_{j=0}^{\infty}\left(\frac{\Gamma(\alpha+j)}{\Gamma(\alpha)\Gamma(2+j)}\right)^{2}=\frac{1}{(\alpha-1)^{2}}\left(\frac{\Gamma(3-2\alpha)}{\Gamma^{2}(2-\alpha)}-1\right).$ (6.18) When $\alpha=1,$ the identity (6.18) is the well known Euler-Jacobi identity $\sum_{j=1}^{\infty}\frac{1}{j^{2}}=\frac{\pi^{2}}{6}.$ When $d=1,0<\alpha<\frac{3}{2},$ we know that $K_{\alpha}:L^{2}\rightarrow L^{2}$ is compact by Theorem 1 or Theorem 5. Thus the spectrum $\sigma(K_{\alpha})$ of the operator $K_{\alpha}$ is exactly the point spectrum. Note that every $K_{\alpha}$ is adjoint, then combing (2.1) and (6.18) with Stirling’s formula, we have the following. ###### Corollary 6.8. Suppose that $d=1$ and $0<\alpha<\frac{3}{2},$ then $K_{\alpha}:L^{2}\rightarrow L^{2}$ is compact and $\sigma(K_{\alpha})=\bigcup_{j=0}^{\infty}\\{\frac{\Gamma(\alpha+j)}{\Gamma(\alpha)\Gamma(2+j)}\\}.$ Moreover, in this case, $\|K_{\alpha}\|_{L^{2}\rightarrow L^{2}}=\max_{0\leq j\leq\infty}\frac{\Gamma(\alpha+j)}{\Gamma(\alpha)\Gamma(2+j)}.$ Acknowledgements. The first author would like to thank Professor G. Zhang for his helpful discussions and warm hospitality when the author visited Chalmers University of Technology. ## References * [1] D. Békollé, A. Bonami, _Estimates for the Bergman and Szegö projections in two symmetric domains of $\mathbb{C}^{n}$,_ Colloq. Math. 68, 81-100 (1995) * [2] A. Bonami, G. 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2024-09-04T02:54:55.992745

2003.00488

arxiv-papers

# An Algorithm for Consensus Trees Pongsaphol Pongsawakul <EMAIL_ADDRESS> ###### Abstract We consider the tree consensus problem, an important problem in bioinformatics. Given a rooted tree $t$ and another tree $T$, one would like to incorporate compatible information from $T$ to $t$. This problem is a subproblem in the tree refinement problem called the RF-Optimal Tree Refinement Problem defined by in Christensen, Molloy, Vachaspati and Warnow [WABI’19] who employ the greedy algorithm by Gawrychowski, Landau, Sung, and Weimann [ICALP’18] that runs in time $O(n^{1.5}\log n)$. We give a faster algorithm for this problem that runs in time $O(n\log n)$. Our key ingredient is a bipartition compatibility criteria based on amortized-time leaf counters. While this is an improvement, the fastest solution is an algorithm by Jansson, Shen, and Sung [JACM’16] which runs in time $O(n)$. ## 1 Introduction We consider the tree consensus problem, an important problem in bioinformatics. Given a rooted tree $t$ and another rooted tree $T$, we would like to combine “information” from $T$ into $t$. More over, we would like to only greedily take information that is currently consistent with our current $t$. (See definitions below.) This problem is a subproblem in the tree refinement problem called RF-Optimal Tree Refinement Problem defined by in Christensen, Molloy, Vachaspati and Warnow [1] who employ the greedy algorithm by Gawrychowski, Landau, Sung, and Weimann [2] that runs in time $O(n^{1.5}\log n)$. We give a faster algorithm for this problem that runs in time $O(n\log n)$. Our key ingredient is a bipartition compatibility criteria based on amortized-time leaf counters. While this is an improvement, the fastest solution is an algorithm by Jansson, Shen, and Sung [3] which runs in time $O(n)$. The algorithm by Gawrychowski et al [2] works in a more general case where the goal is the find the greedy consensus trees from $k$ trees. In this case, their algorithm runs in time $O(kn^{1.5}\log n)$, an improvement over $O(kn^{2})$ of Jansson et al [3]. For this problem, Sung [4] also present an algorithm that runs in time $O(k^{2}n)$, improving over Gawrychowski et al [2] when $k=O(\sqrt{n}\log n)$. In an earlier version, we erroneously claimed that our algorithm works for the case with many trees. We thank Pawel Gawrychowski and Oren Weiman for pointing this out. Jittat Fakcharoenphol who help advising the author on this manuscript would like to take the full responsibility for this mistake. ## 2 Definitions We start with definitions related to trees and consistency. Let $V(T),E(T)$ and $R(T)$ be vertex set, edge set and the root of tree $T$. For every vertex $u\in V(T)-\\{R(T)\\}$, let $par(u)$ be parent node of node $u$. For every vertex $u\in V(T)$, let $depth(u)$ be depth of node $u$. We can denote as $depth(u)=depth(par(u))+1$, for every vertex $u\in V(T)-\\{R(T)\\}$ and $depth(R(T))=1$. For every vertex $u\in V(T)$, let $L(u)$ be set of all leaves on subtree $u$. Let $size(u)=|L(u)|$ for each $u\in V(T)$. For each node $u\in V(T)-\\{R(T)\\}$, we call $\Lambda(u)$ be set of bipartition at edge $(u,par(u))$. For each bipartition, $\Lambda(u)$, we can represent into two clusters, $A=L(u)$ and $B=L(R(T))-L(u)$, and denoted by $A|B$. The set of bipartitions of $T$ can denoted by $C(T)=\\{\Lambda(u):u\in V(T)-\\{R(T)\\}\\}$. Let $RF(T_{a},T_{b})$ be Robinson-Foulds distance between trees $T_{a}$ and $T_{b}$. RF-distance can denoted by $RF(T_{a},T_{b})=|C(T_{a})-C(T_{b})|$. The set $S$ of bipartitions is compatible if there exists tree $T$ such that $C(T)=S$. leaves set $A$ is compatible with $t^{\prime}$ when have node $u$ that for all $v\in child(u)$ if only if $L(v)\cap S=\emptyset$ or $L(V)\subseteq S$. ## 3 The $O(n^{2})$ algorithm In this section, we describe a simpler version of the algorithm and prove its correctness and its running time (in Subsection 3.1). We improve its running time in Section 4 We assume that both $T$ and the current $t^{\prime}$ are equipped with a data structure that given an id of a bipartition $b$, find vertex $u$ in the trees such that $\Lambda(u)=b$. Since $t^{\prime}$ changes over time, we assume that our data structures can handle the tree update efficiently. We discuss this in Subsection 3.2. The main loop of the algorithm iterates over all bipartition of $T$ recursively and, if possible, add each bipartition to $t^{\prime}$. We define variables used in the main loop. Let $z$ be the node that have minimum depth in tree $t^{\prime}$ for each bipartition in $T$. The main loop is described below. Algorithm 1 Main loop 1:for each node $u\in V(T)$ do 2: ClearCounter() 3: $z\leftarrow$ UpdateCounterSubtree($u$) $\triangleright$ Update Step, referred to in Section 4 4: if IsCompatible($t,u,z$) then 5: update $t^{\prime}$ 6: end if 7:end for The main loop uses the following function. Algorithm 2 UpdateCounterSubtree($u$) 1:$z\leftarrow null$ 2:for each node $v\in L(u)$ do 3: $p\leftarrow$ UpdateCounterLeaf($v,t^{\prime}$) $\triangleright$ After this step we say that $v$ has been added. 4: if $z=null$ or $depth(p)<z$ then 5: $z\leftarrow p$ 6: end if 7:end for 8:return $z$ Our algorithm maintains variable $counter$ for each vertex in $t^{\prime}$. We also keep a list of dirty vertices so that ClearCounter can run in $O(1)$ time. Variable $counter$ is updated in function UpdateCounterLeaf. Note that $counter$ changes over time as UpdateCounterSubtree keeps adding leaves in $L(u)$. The algorithm ensures that $counter(u)$ is exactly as follows. If $u$ is a leaf vertex, we let $counter(u)=\begin{cases}1&\text{if }v\text{ has been added}\\\ 0&\text{otherwise}\end{cases}$ For other internal vertex $u$, we let $counter(u)=\sum_{v\in child(u)}\begin{cases}counter(v)&\text{if }counter(v)=|L(v)|\\\ 0&\text{otherwise}\end{cases}$ For each call this function, it can take amortized $O(1)$ time (to be proved later). The following algorithm describes function UpdateCounterLeaf. Algorithm 3 UpdateCounterLeaf($v,t^{\prime}$) 1:$counter(v)\leftarrow 1$ 2:while $counter(v)=size(v)$ do 3: $p\leftarrow par(v)$ 4: $counter(p)\leftarrow counter(p)+counter(v)$ 5: $v\leftarrow p$ 6:end while 7:return $v$ Next, we have algorithm that check that add from previous algorithm is compatible to $t^{\prime}$. We can track the node $u$ that have minimum depth and have $counter(u)>0$ in the while loop at the previous algorithm. Algorithm 4 IsCompatible($t^{\prime},u,z$) 1:if $counter(z)=|L(u)|$ then 2: return YES 3:else 4: return NO 5:end if We prove the correctness of the algorithm. We note that it considers all bipartitions. ###### Lemma 1. The main loop considers all bipartitions in $T$. ###### Proof. From definition, $\Lambda(u)$ can represent all bipartitions of $T$ and for each bipartition, we have consider all node in $\Lambda(u)$. From this, we can consider all bipartitions in $T$. ∎ For clarity, for each vertex $w\in t^{\prime}$ we denote by $L_{t^{\prime}}(w)$ its leaf set in $t^{\prime}$. Note each call to UpdateCounterLeaf increases $counter$ for each vertex by at most 1, and we call UpdateCounterLeaf exactly $|L(u)|$ times. This implies the next lemma, which can be formally proven by induction. ###### Claim 1. During the call of UpdateCounterSubtree($u$), for any vertex $v\in t^{\prime}$, $counter(v)\leq|L(u)\cap L_{t^{\prime}}(v)|$. Moreover, if $L_{t^{\prime}}(v)\subseteq L(u)$, $counter(v)=|L(u)\cap L_{t^{\prime}}(v)|=|L_{t^{\prime}}(v)|$, i.e., the counter attains its maximum. The following the key lemma. ###### Lemma 2. Let $z^{\prime}$ be the least common ancestor of leaf vertices in $L(u)$ in $t^{\prime}$. After UpdateCounterSubtree($u$) is called, leaf set $L(u)$ is compatible with the current $t^{\prime}$ if and only if $counter(z^{\prime})=|L(u)|$. ###### Proof. Note that after the call to UpdateCounterSubtree($u$), we have called UpdateCounterLeaf($v$) for every leaf $v$ in $L(u)$. From Claim 1, we only need to consider the case when $counter(z^{\prime})\leq|L(u)|$. The algorithm maintains vertex $z$, which is the vertex closest to the root that UpdateCounterLeaf has touched. We first consider the case that $z=z^{\prime}$. We show that if $counter(z^{\prime})=|L(u)|$, then for each child $w\in children(z^{\prime})$, $L_{t^{\prime}}(w)\cap L(u)=\emptyset$ or $L_{t^{\prime}}(w)\subseteq L(u)$. This implies that $L(u)$ is compatible with $t^{\prime}$. We note that $z^{\prime}$ is not a leaf. Consider each $w\in children(z^{\prime})$. We only need to consider $w$ such that $L_{t}(w)\cap L(u)\neq\emptyset$. The only way $counter(z^{\prime})=|L(u)|$ is when $w$ is “complete”, i.e., $counter(w)=|L_{t^{\prime}}(w)|$. Since $counter(w)\leq|L(u)\cap L_{t^{\prime}}(w)|$ (from Claim 1), we know that $L_{t^{\prime}}(w)\subseteq L(u)$. On the other hand, if $L(u)$ is compatible with $t^{\prime}$, we show that $counter(z^{\prime})=|L(u)|$. We prove a stronger statement: if $L(u)$ is compatible with $t^{\prime}$ for every vertex $v\in t^{\prime}$ in the subtree rooted at $z^{\prime}$, $counter(v)=|L(u)\cap L_{t^{\prime}}(v)|,$ i.e., the upper bound in Claim 1 attains its maximum. To do so, we prove inductively on the structure of $t^{\prime}$. Clearly, the claim is true when $v$ is a leaf. Consider vertex $v\neq z^{\prime}$ in the subtree of $t^{\prime}$ rooted at $z^{\prime}$. If $L_{t^{\prime}}(v)\cap L(u)=\emptyset$, $counter(v)=0$; thus the property follows. Now, consider $v$ such that $L_{t^{\prime}}(v)\cap L(u)\neq\emptyset$. Let $w$ be a child of $z^{\prime}$ such that $v$ belongs to subtree rooted at $w$. Since $L_{t^{\prime}}(v)\cap L(u)\neq\emptyset$ and $L_{t^{\prime}}(v)\subseteq L_{t^{\prime}}(w)$, we know that $L_{t^{\prime}}(w)\cap L(u)\neq\emptyset$. Since $L(u)$ is compatible with $t^{\prime}$, we have that $L_{t^{\prime}}(w)\subseteq L(u),$ implying that $L_{t^{\prime}}(v)\subseteq L(u)$; thus $counter(v)=|L_{t^{\prime}}(v)|=|L_{t^{\prime}}(v)\cap L(u)|$, from Claim 1. Finally, consider $z^{\prime}$. Note that since $z^{\prime}$ is the common ancestor of leaves in $L(u)$, $L_{t^{\prime}}(z^{\prime})\supseteq L(u)$. For each child $w$ of $z^{\prime}$, when $L_{t^{\prime}}(w)\cap L(u)\neq\emptyset$, $w$ is complete and propagate $|L_{t^{\prime}}(w)\cap L(u)|$ to $counter(z^{\prime})$. Summing all children of $z^{\prime}$, we have that $counter(z^{\prime})=|L(u)|$. This completes the proof of the lemma. ∎ ###### Lemma 3. Tree compatibility condition works From above condition, our algorithm have $add$ function $counter(u)$. for each subtree $L(u)$ have fully resolved when $counter(u)=|L(u)|$ ### 3.1 Running time analysis We first analyze the running time of the algorithm except the calls to UpdateCounterSubtree. We show that this part runs in linear time. We start with UpdateCounterLeaf. ###### Lemma 4. Function UpdateCounterLeaf($v,t^{\prime}$) runs in amortized $O(1)$ time. ###### Proof. We use the potential method. Our data structure consists of variables $counter$ for all vertices in $t^{\prime}$. Denote the data structure at time $i$ by $D_{i}$. We say that a vertex $u\in t^{\prime}$ is incomplete if $0<counter(u)<|L(u)|$. Let potential function $\Phi(D_{i})$ be the number of incomplete vertices in $t^{\prime}$ time $i$. Using the potential method, when the data structure changes from $D_{i-1}$ to $D_{i}$, the amortized cost of an operation is $\hat{c}=c+\Delta\Phi$, where $c$ is an actual cost, and $\Delta\Phi=\Phi(D_{i})-\Phi(D_{i-1})$. Let $D_{0}$ be initial data structure after ClearCounter is called; thus $\Phi(D_{0})=0$. Note that $\Phi(D_{i})\geq\Phi(D_{0})=0$ for any $i$. When invoking UpdateCounterLeaf at time $i$, let $k$ be number of times the while loop in Lines 2 - 6 is executed. Clearly, the actual cost $c$ of the operation is $k+1$. Let $\Delta\Phi=\Phi(D_{i})-\Phi(D_{i-1})$. We claim that $\Phi(D_{i})=\Phi(D_{i-1})-k+1$, i.e., the number of incomplete vertices decreases by $k-1$. Let $v^{\prime}$ be the actual leaf that UpdateCounterLeaf is called on. Note that each time the loop is executed, $counter(v)=size(v)=|L(v)|$. Except when $v=v^{\prime}$, previously at time $i-1$, we know that $0<counter(v)<|L(v)|$, because $v$ is an internal vertex with at least 2 children; hence $v$ was incomplete at time $i-1$. Since $counter(v)=|L(v)|$ at time $i$, $v$ is no longer incomplete; thus the number of incomplete vertices decreases by $k-1$ as claimed. Thus the amortized cost $\hat{c}=k+1+\Delta\Phi=k+1+(-k+1)=2=O(1)$. ∎ We now analyze the running time of UpdateCounterSubtree. ###### Lemma 5. Function UpdateCounterSubtree($u$) runs in time $O(|L(u)|)$. ###### Proof. From Lemma 4, it is clear that UpdateCounterSubtree runs in time $O(|L(u)|)=O(n)$ and it is invoked for $O(n)$ time. Therefore, the total running time of the function is $O(n^{2})$. Combining the two parts, we get that the algorithm runs in $O(n^{2})$. ∎ ### 3.2 Updating $t$ In this section, we show that when the bipartition defined by $L(u)$ is compatible with $t^{\prime}$, we can update $t^{\prime}$ to include that bipartition efficiently in time $deg(u)$. When $t^{\prime}$ is compatible with a bipartition defined by $L(u)$ for $u\in T$, to update $t^{\prime}$ we have to create a new child of $z$ that consists of only children of $z$ that corresponds to the bipartition $L(u)$. Note that these children are those “full” $counter$ that also propagate the counter to $z$. Therefore, when a child propagate a counter to any vertex, we keep a list of them. When we need to update $t^{\prime}$ at $z$, we can take every vertex in this list, and create a new child $z^{\prime}$ of $z$ with these vertices as $z^{\prime}$’s children and update their counter accordingly. This can be done in time $O(deg(u))$. ## 4 The faster algorithm: heavy child optimization In this section, we describe a simple method to speed up the algorithm from the previous section. Note that the only bottle-neck to a nearly linear time algorithm is the counting procedure. As a preprocessing, we assume that for each vertex $u\in T$, we know $|L(u)|$. This can be computed in $O(n)$ time. The improved algorithm is described below. Algorithm 5 Solve($u$) if $u$ is not leaf then let $c_{1},c_{2}$ be children of $u$ $\triangleright$ There are exactly two children, since $T$ is binary if $|L(c_{1})|>|L(c_{2})|$ then swap node $c_{1}$ and node $c_{2}$ end if Solve($c_{1}$) $\triangleright$ Solve smaller subtree ClearCounter() Solve($c_{2}$) $\triangleright$ Solve larger subtree end if $z\leftarrow$ UpdateCounterSubtree($c_{1}$) $\triangleright$ Update the counter for the smaller subtree if IsCompatible($t,u,z$) then update $t^{\prime}$ end if To see that this algorithm is the correct implementation of the Main loop, we essentially need to show that at the end of Solve($u$), variable $counter$ is exactly equal to variable $counter$ right after the “Update Step” in Line 3 in the Main loop while processing $u$, i.e., variable $counter$ is exactly equal to the case when every leaf in $L(u)$ has been added while no other leaves have been added. This can be shown by induction on the calls of Solve. We omit the proof in this version of the manuscript. We are left to analyze its running time. ###### Theorem 1. The algorithm Solve runs in $O(n\log n)$ time. ###### Proof. Note that the running time for all other operations in Solve is $O(1)$ per invocation. Since Solve is called for $O(n)$ time, the total running time of these operations is $O(n)$. Also, the running time of ClearCounter() can be amortized to the running time of UpdateCounterSubtree, where the counters are updated. Therefore, we are left to analyze the running time of UpdateCounterSubtree. Note that UpdateCounterSubtree($u$) for $u\in T$ runs in time linearly in the number of leaves, $|L(u)|$, from Lemma 5. Hence, we can charge the cost to these leaves. We analyze the running time by counting the number of times each leaf is involved in this charging scheme. Note that we only call UpdateCounterSubtree at $c_{1}$, which is the lighter subtree. Clearly, each leaf $u$ belongs to at most $O(\log n)$ light subtrees; hence, it is charged by at most $O(\log n)$ time. Summing all leaves, we have that the total running time for UpdateCounterSubtree is $O(n\log n)$. ∎ ## 5 Acknowledgements We would like to thank Pawel Gawrychowski and Oren Weiman for pointing out our erroneous claim and also give us reference to Sung’s result [4]. As mentioned earlier, Jittat Fakcharoenphol who help advising the author on this manuscript would like to take full responsibility for this mistake. We would like to thank Jittat Fakcharoenphol for suggesting this problem to work on and for his help in editing this manuscript. ## References * [1] Sarah Christensen, Erin K. Molloy, Pranjal Vachaspati, and Tandy Warnow. TRACTION: Fast Non-Parametric Improvement of Estimated Gene Trees. In Katharina T. Huber and Dan Gusfield, editors, 19th International Workshop on Algorithms in Bioinformatics (WABI 2019), volume 143 of Leibniz International Proceedings in Informatics (LIPIcs), pages 4:1–4:16, Dagstuhl, Germany, 2019. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. * [2] Pawel Gawrychowski, Gad M. Landau, Wing-Kin Sung, and Oren Weimann. A faster construction of greedy consensus trees. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 63:1–63:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. * [3] Jesper Jansson, Chuanqi Shen, and Wing-Kin Sung. Improved algorithms for constructing consensus trees. J. ACM, 63(3), June 2016. * [4] Wing-Kin Sung. Greedy consensus tree and maximum greedy consensus tree problems. In Gautam K. Das, Partha S. Mandal, Krishnendu Mukhopadhyaya, and Shin-ichi Nakano, editors, WALCOM: Algorithms and Computation, pages 305–316, Cham, 2019. Springer International Publishing.

2024-09-04T02:54:56.006207

2003.00503

arxiv-papers

JME-18-001 JME-18-001 # Pileup mitigation at CMS in 13data ###### Abstract With increasing instantaneous luminosity at the LHC come additional reconstruction challenges. At high luminosity, many collisions occur simultaneously within one proton-proton bunch crossing. The isolation of an interesting collision from the additional “pileup” collisions is needed for effective physics performance. In the CMS Collaboration, several techniques capable of mitigating the impact of these pileup collisions have been developed. Such methods include charged-hadron subtraction, pileup jet identification, isospin-based neutral particle “$\delta\beta$” correction, and, most recently, pileup per particle identification. This paper surveys the performance of these techniques for jet and missing transverse momentum reconstruction, as well as muon isolation. The analysis makes use of data corresponding to 35.9collected with the CMS experiment in 2016 at a center-of- mass energy of 13. The performance of each algorithm is discussed for up to 70 simultaneous collisions per bunch crossing. Significant improvements are found in the identification of pileup jets, the jet energy, mass, and angular resolution, missing transverse momentum resolution, and muon isolation when using pileup per particle identification. ## 0.1 Introduction At the CERN LHC, instantaneous luminosities of up to $1.5\times 10^{34}\cm^{-2}\unit{s}^{-1}$ [1] are sufficiently large for multiple proton- proton ($\Pp\Pp$) collisions to occur in the same time window in which proton bunches collide. This leads to overlapping of particle interactions in the detector. To study a specific $\Pp\Pp$ interaction, it is necessary to separate this single interaction from the overlapping ones. The additional collisions, known as pileup (PU), will result in additional particles throughout the detector that confuse the desired measurements. With PU mitigation techniques, we can minimize the impact of PU and better isolate the single collision of interest. With increasing beam intensity over the past several years, identification of interesting $\Pp\Pp$ collisions has become an ever-growing challenge at the LHC. The number of additional collisions that occur when two proton bunches collide was, on average, 23 in 2016 and subsequently increased to 32 in 2017 and 2018. At this level of collision density, the mitigation of the PU effects is necessary to enable physics analyses at the LHC. The CMS Collaboration has developed various widely used techniques for PU mitigation. One technique, charged-hadron subtraction (CHS) [2], has been the standard method to mitigate the impact of PU on the jet reconstruction for the last few years. It works by excluding charged particles associated with reconstructed vertices from PU collisions from the jet clustering procedure. In this technique, to mitigate the impact of neutral PU particles in jets, an event-by-event jet-area-based correction [3, 4, 5] is applied to the jet four- momenta. Further, a PU jet identification (PU jet ID) technique [6] is used to reject jets largely composed of particles from PU interactions. These techniques have limitations when attempting to remove PU contributions due to neutral particles. For the jet-area-based correction, the jet four- momentum correction acts on a whole jet and is therefore not capable of removing PU contributions from jet shape or jet substructure observables. To overcome this limitation, a new technique for PU mitigation, pileup per particle identification (PUPPI) [7], is introduced that operates at the particle level. The PUPPI algorithm builds on the existing CHS algorithm. In addition, it calculates a probability that each neutral particle originates from PU and scales the energy of these particles based on their probability. As a consequence, objects clustered from hadrons, such as jets, missing transverse momentum (), and lepton isolation are expected to be less susceptible to PU when PUPPI is utilized. In this paper, the performance of PU mitigation techniques, including the commissioning of PUPPI in $\Pp\Pp$ collision data, is summarized. After a short description of the CMS detector in Section 0.2 and definitions of the data set and Monte Carlo (MC) simulations used in these studies in Section 0.3, the CHS and PUPPI algorithms are described in Section 0.4. In Section 0.5.1 performance in terms of jet resolution at a high number of interactions is presented. Section 0.5.2 summarizes the impact on noise rejection of PU mitigation techniques. Section 0.5.3 presents the rejection of jets originating from PU with PU jet ID and PUPPI. Jets reconstructed with a larger cone size are often used to identify the decay of Lorentz-boosted heavy particles such as , , and Higgs bosons, and top quarks. Pileup significantly degrades the reconstruction performance, and the gain from PU mitigation techniques for such large-size jets is discussed in Section 0.6. The measurement of also benefits from PU mitigation techniques, which is discussed in Section 0.7. Mitigation of PU for muon isolation variables is presented in Section 0.8. ## 0.2 The CMS detector The central feature of the CMS apparatus is a superconducting solenoid of 6m internal diameter, providing a magnetic field of 3.8T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. The ECAL covers the pseudorapidity range $\abs{\eta}<3$, while the HCAL is extended with forward calorimeters up to $\abs{\eta}<5$. Muons are detected in gas-ionization chambers embedded in the steel flux-return yoke outside the solenoid. The silicon tracker measures charged particles within $\abs{\eta}<2.5$. It consists of 1440 silicon pixel and 15 148 silicon strip detector modules. For nonisolated particles with transverse momentum of $1<\pt<10\GeV$ and $\abs{\eta}<1.4$, the track resolutions are typically 1.5% in and 25–90 (45–150)in the transverse (longitudinal) impact parameter [8]. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [9]. The particle-flow (PF) event reconstruction [2] reconstructs and identifies each individual particle in an event, with an optimized combination of all subdetector information. In this process, the identification of the particle type (photon, electron, muon, charged or neutral hadron) plays an important role in the determination of the particle direction and energy. Photons (, coming from decays or from electron bremsstrahlung) are identified as ECAL energy clusters not linked to the extrapolation of any charged particle trajectory to the ECAL. Electrons (, coming from photon conversions in the tracker material or from hadron semileptonic decays) are identified as a primary charged-particle track and potentially many ECAL energy clusters corresponding to this track extrapolation to the ECAL and to possible bremsstrahlung photons emitted along the way through the tracker material. Muons are identified as tracks in the central tracker consistent with either tracks or several hits in the muon system, and associated with calorimeter deposits compatible with the muon hypothesis. Charged hadrons are identified as charged particle tracks neither identified as electrons, nor as muons. Finally, neutral hadrons are identified as HCAL energy clusters not linked to any charged-hadron trajectory, or as a combined ECAL and HCAL energy excess with respect to the expected charged-hadron energy deposit. The energy of photons is obtained from the ECAL measurement, corrected for zero-suppression effects. The energy of electrons is determined from a combination of the track momentum at the main interaction vertex, the corresponding ECAL cluster energy, and the energy sum of all bremsstrahlung photons attached to the track. The energy of muons is obtained from the corresponding track momentum. The energy of charged hadrons is determined from a combination of the track momentum and the corresponding ECAL and HCAL energy, corrected for zero-suppression effects and for the response function of the calorimeters to hadronic showers. Finally, the energy of neutral hadrons is obtained from the corresponding corrected ECAL and HCAL energy. The collision rate is 40MHz, and the events of interest are selected using a two-tiered trigger system [10]. The first level (L1), composed of custom hardware processors, uses information from the calorimeters and muon detectors to select events at a rate of around 100kHz within a fixed time interval of less than 4$\mu$s. The second level, known as the high-level trigger (HLT), consists of a farm of processors running a version of the full event reconstruction software optimized for fast processing, and reduces the event rate to around 1kHz before data storage. All detector subsystems have dedicated techniques to reject signals from electronic noise or from particles that do not originate from the $\Pp\Pp$ collisions in the bunch crossing of interest, such as particles arriving from $\Pp\Pp$ collisions that occur in adjacent bunch crossings before or after the bunch crossing of interest (so called out-of-time PU). While these rejection techniques are not the focus of this paper, some false signals can pass these filters and affect the PF reconstruction. Particularly relevant is residual noise from ECAL and HCAL electronics that may add to the energy of reconstructed photons, electrons, and hadrons. Algorithms for the rejection of this noise are further discussed in Section 0.5.2. ## 0.3 Data and simulated samples In this paper, data corresponding to an integrated luminosity of 35.9 [1] taken in 2016 are used. Figure 1 shows the PU conditions in the years 2016–2018. The number of $\Pp\Pp$ interactions is calculated from the instantaneous luminosity based on an estimated inelastic $\Pp\Pp$ collision cross section of 69.2mb. This number is obtained using the PU counting method described in the inelastic cross section measurements [11, 12]. In the following sections of this paper, we distinguish between two definitions: “mean number of interactions per crossing” (abbreviated “number of interactions” and denoted $\mu$) and “number of vertices” (denoted $N_{\text{vertices}}$). Vertices are reconstructed through track clustering using a deterministic annealing algorithm [8]. The number of interactions is used to estimate the amount of PU in simulation. The number of vertices can be determined in both data and simulation. Further details on the relationship between $\mu$ and $N_{\text{vertices}}$ are provided in Section 0.5.3. The studies presented in this paper focus on the PU conditions in 2016, though the trends towards higher PU scenarios with up to 70 simultaneous interactions are explored as well. The trigger paths used for the data taking are mentioned in each section. Figure 1: Distribution of the mean number of inelastic interactions per crossing (pileup) in data for $\Pp\Pp$ collisions in 2016 (dotted orange line), 2017 (dotted dashed light blue line), 2018 (dashed navy blue line), and integrated over 2016–2018 (solid grey line). A total inelastic $\Pp\Pp$ collision cross section of 69.2mbis chosen. The mean number of inelastic interactions per bunch crossing is provided in the legend for each year. Samples of simulated events are used to evaluate the performance of the PU mitigation techniques discussed in this paper. The simulation of standard model events composed uniquely of jets produced through the strong interaction, referred to as quantum chromodynamics (QCD) multijet events, is performed with v8.212 [13] in standalone mode using the Lund string fragmentation model [14, 15] for jets. For studies of lepton isolation, dedicated QCD multijet samples that are enriched in events containing electrons or muons (, from heavy-flavor meson decays) are used. The and boson production in association with jets is simulated at leading-order (LO) with the v2.2.2 [16] generator. Production of top quark-antiquark pair () events is simulated with (v2) [17, 18, 19]. Single top quark production via the $s$\- and $t$-channels, and processes are simulated at next-to-leading-order (NLO) with that is interfaced with . For Lorentz-boosted $\PW$ boson studies [20], MC simulation of high mass bulk graviton resonance [21, 22, 23] decaying to $\PW\PW$ boson pairs are generated at LO with . All parton shower simulations are performed using . For +jets production, an additional sample is generated using interfaced with v2.7.1 [24, 25] with the UE-EE-5C underlying event tune [26] to assess systematic uncertainties related to the modeling of the parton showering and hadronization. The LO and NLO NNPDF 3.0 [27] parton distribution functions (PDF) are used in all generated samples matching the QCD order of the respective process. The parameters for the underlying event are set according to the CUETP8M1 tune [28, 29], except for the $\ttbar$ sample, which uses CUETP8M2 [30]. All generated samples are passed through a detailed simulation of the CMS detector using [31]. To simulate the effect of additional $\Pp\Pp$ collisions within the same or adjacent bunch crossings, additional inelastic events are generated using with the same underlying event tune as the main interaction and superimposed on the hard-scattering events. The MC simulated events are weighted to reproduce the distribution of the number of interactions observed in data. ## 0.4 The CHS and PUPPI algorithms A detailed description of the CHS algorithm and its performance is found in Ref. [2]. In the following, we summarize the salient features and differences with respect to the PUPPI algorithm. Both algorithms use the information of vertices reconstructed from charged-particle tracks. The physics objects considered for selecting the primary $\Pp\Pp$ interaction vertex are track jets, clustered using the anti-algorithm [32, 33] with the tracks assigned to the vertex as inputs, and the associated $\vec{p}_{\mathrm{T},\text{tracks}}^{\text{miss}}$, which is the negative vector sum of those jets. The reconstructed vertex with the largest value of summed physics-object $\pt^{2}$ is selected as the primary $\Pp\Pp$ interaction vertex or “leading vertex” (LV). Other reconstructed collision vertices are referred to as PU vertices. The CHS algorithm makes use of tracking information to identify particles originating from PU after PF candidates have been reconstructed and before any jet clustering. The procedure removes charged-particle candidates that are associated with a reconstructed PU vertex. A charged particle is associated with a PU vertex if it has been used in the fit to that PU vertex [8]. Charged particles not associated with any PU vertex and all neutral particles are kept. The PUPPI [7] algorithm aims to use information related to local particle distribution, event PU properties, and tracking information to mitigate the effect of PU on observables of clustered hadrons, such as jets, , and lepton isolation. The PUPPI algorithm operates at the particle candidate level, before any clustering is performed. It calculates a weight in a range from 0 to 1 for each particle, exploiting information about the surrounding particles, where a value of 1 is assigned to particles considered to originate from the LV. These per-particle weights are used to rescale the particle four- momenta to correct for PU at particle-level, and thus reduces the contribution of PU to the observables of interest. For charged particles, the PUPPI weight is assigned based on tracking information. Charged particles used in the fit of the LV are assigned a weight of 1, while those associated with a PU vertex are assigned a weight of 0. A weight of 1 is assigned to charged particles not associated with any vertex provided the distance of closest approach to the LV along the $z$ axis ($d_{z}$) is smaller than 0.3; a weight of 0 is applied in all other scenarios. The threshold of 0.3corresponds to about 15 standard deviations of the vertex reconstruction resolution in the $z$ direction at an average PU of 10 [8], and it works as an additional filter against undesirable objects, such as accidentally reconstructed particles from detector noise. Neutral particles are assigned a weight based on a discriminating variable $\alpha$. In general, the $\alpha$ variable is used to calculate a weight, which encodes the probability that an individual particle originates from a PU collision. As discussed in Ref. [7], various definitions of $\alpha$ are possible. Within CMS, the $\alpha$ variable for a given particle $i$ is defined as $\alpha_{i}=\log\sum_{j\neq i,\,\Delta R_{ij}<R_{0}}\left(\frac{p_{\mathrm{T},\,j}}{\Delta R_{ij}}\right)^{2}\begin{cases}\text{for }\abs{\eta_{i}}<2.5,&j\text{ are charged particles from LV,}\\\ \text{for }\abs{\eta_{i}}>2.5,&j\text{ are all kinds of reconstructed particles,}\\\ \end{cases}$ (1) where $i$ refers to the particle in question, $j$ are other particles, $p_{\mathrm{T},\,j}$ is the transverse momentum of particle $j$ in , and $\Delta R_{ij}=\sqrt{\smash[b]{(\Delta\eta_{ij})^{2}+(\Delta\phi_{ij})^{2}}}$ (where $\phi$ is the azimuthal angle in radians) is the distance between the particles $i$ and $j$ in the $\eta$-$\phi$ plane. The summation runs over the particles $j$ in the cone of particle $i$ with a radius of $R_{0}=0.4$. A value of $\alpha_{i}=0$ is assigned when there are no particles in the cone. The choice of the cone radius $R_{0}$ in the range of 0.2–0.6 has a weak impact on the performance. The value of 0.4 was chosen as a compromise between the performance when used in the definition of the isolation variable (preferring larger cones) and jet performance (preferring smaller cones). In $\abs{\eta}<2.5$, where tracking information is available, only charged particles associated with the LV are included as particle $j$, whereas all particles with $\abs{\eta}>2.5$ are included. The variable $\alpha$ contrasts the collinear structure of QCD in parton showers with the soft diffuse radiation coming from PU interactions. A particle from a shower is expected to be close to other particles from the same shower, whereas PU particles can be distributed more homogeneously. The $\alpha$ variable is designed such that a particle gets a large value of $\alpha$ if it is close to either particles from the LV or, in $\abs{\eta}>2.5$, close to highly energetic particles. To translate $\alpha_{i}$ of each particle into a probability, charged particles assigned to PU vertices are used to generate the expected PU distribution in an event. From this expected distribution a median and root- mean-square (RMS) of the $\alpha$ values are computed. The $\alpha_{i}$ of each neutral particle is compared with the computed median and RMS of the $\alpha$ distribution of the charged PU particles using a signed $\chi^{2}$ approximation: $\text{signed }\chi^{2}_{i}=\frac{(\alpha_{i}-\overline{\alpha}_{\text{PU}})\abs{\alpha_{i}-\overline{\alpha}_{\text{PU}}}}{(\alpha_{\text{PU}}^{\text{RMS}})^{2}},$ (2) where $\overline{\alpha}_{\text{PU}}$ is the median value of the $\alpha_{i}$ distribution for charged PU particles in the event and $\text{RMS}_{\text{PU}}$ is the corresponding RMS. If signed $\chi^{2}_{i}$ is large, the particle most likely originates from the LV. The sign of the numerator is sensitive to the direction of the deviation of $\alpha_{i}$ from $\overline{\alpha}_{\text{PU}}$. For the detector region where $\abs{\eta}>2.5$ and tracking is not available, the values $\overline{\alpha}_{\text{PU}}$ and $\text{RMS}_{\text{PU}}$ can not be calculated directly. Therefore, $\overline{\alpha}_{\text{PU}}$ and $\text{RMS}_{\text{PU}}$ are taken from the detector region where $\abs{\eta}<2.5$ and extrapolated to the region where $\abs{\eta}>2.5$ by multiplying with transfer factors (see Tab. 0.4) derived from MC simulation. The transfer factors are necessary, since the granularity of the detector varies with $\eta$ and leads to a variation of $\alpha$ with $\eta$, particularly outside of the tracker coverage ($\abs{\eta}=2.5$) and ECAL coverage ($\abs{\eta}=3.0$). Lastly, to compute the weight of the particles, the signed $\chi^{2}_{i}$ for PU particles is assumed to be approximately distributed according to a $\chi^{2}$ distribution for $\chi^{2}_{i}>0$. The weight is given by $w_{i}=F_{\chi^{2},\,\text{NDF}=1}(\text{signed }\chi^{2}_{i})$ where $F_{\chi^{2},\,\text{NDF}=1}$ is the cumulative distribution function of the $\chi^{2}$ distribution with one degree of freedom. Particles with weights $w_{i}$ smaller than 0.01, , those with a probability greater than 99% to originate from PU are rejected; this last rejection removes remaining high-energy noise deposits. In addition, neutral particles that fulfill the following condition: $w_{i}\,p_{\mathrm{T},\,i}<(A+B\,N_{\text{vertices}})\GeV$, where $N_{\text{vertices}}$ is the number of vertices in the event, get a weight of 0. This selection reduces the residual dependence of jet energies on the number of interactions. The parameters $A$ and $B$ are tunable parameters. To perform the tuning of these parameters, jets clustered from PUPPI-weighted particles in the regions $\abs{\eta}<2.5$ and $2.5<\abs{\eta}<3.0$ are adjusted to have near-unity jet response, as a function of the number of interactions, i.e., the reconstructed jet energy matches the true jet energy regardless of the amount of PU. In the region $\abs{\eta}>3$, the parameters are chosen such that resolution is optimized. Table 0.4 summarizes the resulting parameters that have been obtained using QCD multijet simulation with an average number of interactions of 23 and a significant amount of events beyond 30 interactions reflecting the 2016 data (orange curve in Fig. 1). The parameters $A$ and $B$ are smaller in $\abs{\eta}<2.5$ (where the majority of particles are reconstructed with the tracker) than in $\abs{\eta}>2.5$ (where the measurement comes solely from the calorimeters that have a coarser granularity and thus collect more PU energy per cell). The tunable parameters of PUPPI optimized for application in 2016 data analysis. The transfer factors used to extrapolate the $\overline{\alpha}_{\text{PU}}$ and $\alpha_{\text{PU}}^{\text{RMS}}$ to $\abs{\eta}>2.5$ are denoted TF. $\abs{\eta}$ of particle $A$ [] $B$ [] TF $\overline{\alpha}_{\text{PU}}$ TF $\alpha_{\text{PU}}^{\text{RMS}}$ $[0,2.5]$ 0.2 0.015 1 1 $[2.5,3]$ 2.0 0.13 0.9 1.2 $[3,5]$ 2.0 0.13 0.75 0.95 ### 0.4.1 Data-to-simulation comparison for variables used within PUPPI Figure 2: Data-to-simulation comparison for three different variables of the PUPPI algorithm. The markers show a subset of the data taken in 2016 of the jet sample and the PU sample, while the solid lines are QCD multijet simulations or PU-only simulation. The lower panel of each plot shows the ratio of data to simulation. Only statistical uncertainties are displayed. The upper left plot shows the $\alpha$ distribution in the jet sample for charged particles associated with the LV (red triangles), charged particles associated with PU vertices (blue circles), and neutral particles (black crosses) for $\abs{\eta}<2.5$. The upper right plot shows the $\alpha$ distribution in the PU sample for charged (blue circles) and neutral (orange diamond) particles. The lower left plot shows the signed $\chi^{2}=(\alpha-\overline{\alpha}_{\text{PU}})\abs{\alpha-\overline{\alpha}_{\text{PU}}}/(\alpha_{\text{PU}}^{\text{RMS}})^{2}$ for neutral particles with $\abs{\eta}<2.5$ in the jet sample (black crosses) and in the PU sample (orange diamonds). The lower right plot shows the PUPPI weight distribution for neutral particles in the jet sample (black crosses) and the PU sample (orange diamonds). The error bars correspond to the statistical uncertainty. The behavior of the variables used in PUPPI has been studied in two complementary data samples. A subset of the data taken in 2016, corresponding to an integrated luminosity of 0.36and selected using trigger paths based on the scalar sum () of the of jets with $\pt>30\GeV$ and $\abs{\eta}<3$, requiring an offline selection of $\HT>1500\GeV$, is referred to as the jet sample. The details of jet reconstruction and performance are discussed in Section 0.5. Here, we present comparisons of data and QCD multijet simulation based on all PF candidates in the event, rather than clustered jets. As a reference, a data sample enriched in events containing mainly particles from PU collisions is compared with PU-only simulation and is referred to as the PU sample. The PU data sample is recorded with a zero-bias trigger that randomly selects a fraction of the collision events, corresponding to an integrated luminosity of 3.18. The distribution of the number of PU interactions in both subsets of data is comparable to the one in the whole data sample collected in 2016. Figure 2 shows the distribution of the three main variables used in PUPPI for data and simulation. The upper left plot presents the distribution of $\alpha$ for charged particles from the LV and the PU vertices and for neutral particles with $\abs{\eta}<2.5$ in the jet sample. The separation power of the variable $\alpha$ between particles from the LV and PU vertices for charged particles can be deduced from this figure. The majority of the charged particles from PU vertices have an $\alpha$ value below 8, whereas only a small fraction of particles have higher values. Charged particles from the LV exhibit a double-peak structure. The first peak at large $\alpha$ is characteristic of particles within jets originating from the LV. The second peak at lower $\alpha$ consists of charged particles that are isolated from other particles originating from the LV. With the exception of particles from lepton decays, which are directly addressed later, isolated particles have limited physics impact and consequently a low $\alpha$ value has a negligible impact on the algorithm performance on physics objects. The $\alpha$ distribution of neutral PU particles can be compared to charged PU particles in the PU sample shown in Figure 2 (upper right). It becomes clear that the median and RMS of the $\alpha$ distribution are similar for charged and neutral particles originating from PU. This similarity confirms one of the primary assumptions of PUPPI, namely that $\overline{\alpha}_{\mathrm{PU}}$ and $\text{RMS}_{\mathrm{PU}}$, which are computed for charged particles, can be used to compute weights for neutral particles with a discrimination power between PU and LV particles. Although the qualitative features of the $\alpha$ distribution in data are reproduced by the simulation, a disagreement between data and simulation is observed, which is most pronounced for neutral particles from PU with large values of $\alpha$. The $\chi^{2}$ distribution shown in Fig. 2 (lower left) shows two peaks for both the jet sample and the PU sample. The first peak results from particles without any neighbor and an $\alpha$ value of zero. The second peak at zero represents all PU particles. The jet sample (black curve) shows a third peak for all LV particles. Additionally, the shape of the resulting PUPPI weight distribution, shown in Fig. 2 (lower right) is well modeled by simulation for particles with high weights (, those likely originating from the LV). A considerable mismodeling is observed at low values of PUPPI weight, where low- particles from PU interactions dominate. This mismodeling does not propagate to further observables, because these particles receive small weights, and as a consequence have a negligible contribution. Although both samples have a similar distribution of number of interactions, the weight distribution of the jet sample has more events at higher values of the weight compared to the PU sample because of the selection of a high jet. ## 0.5 Jet reconstruction Jets are clustered from PF candidates using the anti-algorithm [32] with the FastJet software package [33]. Distance parameters of 0.4 and 0.8 are used for the clustering. While jets with $R=0.4$ (AK4 jets) are mainly used in CMS for reconstruction of showers from light-flavor quarks and gluons, jets with $R=0.8$ (AK8 jets) are mainly used for reconstruction of Lorentz-boosted , , and Higgs bosons, and for top quark identification, as discussed in detail in Section 0.6. Before jet clustering, CHS- or PUPPI-based PU mitigation is applied to the PF candidates. Reconstructed jets with the respective PU mitigation technique applied are referred to as CHS and PUPPI jets, respectively. Jet momentum is determined as the vectorial sum of all particle momenta in the jet, and from simulation is, on average, within 5 to 20% of the true momentum over the whole spectrum and detector acceptance. For CHS jets, an event-by- event jet-area-based correction [3, 4, 5] is applied to the jet four-momenta to remove the remaining energy due to neutral and charged particles originating from PU vertices, while no such correction is necessary for PUPPI jets. Although CHS removes charged particles associated with a PU vertex, charged particles not associated with any vertex are kept and can add charged PU energy to the jet. The remaining energy from PU particles subtracted from the jet energy is assumed proportional to the jet area and parametrized as a function of the median energy density in the event, the jet area, $\eta$, and . In addition, jet energy corrections are derived from simulation for CHS and PUPPI to bring the measured response of jets to that of generated particle- level jets on average. In situ measurements of the momentum balance in dijet, photon+jets, +jets, and multijet events are used to correct any residual differences in jet energy scale between data and simulation [5]. In the following, only jets with $\pt>15\GeV$ are used, which is the lowest jet used in physics analysis in CMS. The presentation of jet performance focuses on $\abs{\eta}<2.5$, covered by the tracking detector, ECAL, and HCAL, and the forward region, $\abs{\eta}>3$, where only the hadron forward calorimeter is present. The intermediate region, $2.5<\abs{\eta}<3.0$, which is covered by ECAL and HCAL resembles the forward region in sensitivity to PU and is not discussed in this paper. For Sec. 0.5.1 the focus is set on $\abs{\eta}<0.5$, as the region $0.5<\abs{\eta}<2.5$ provides no further information and shows a similar performance. ### 0.5.1 Jet energy and angular resolutions The performance of the jet four-momentum reconstruction is evaluated in QCD multijet simulation by comparing the kinematics of jets clustered from reconstructed PF candidates (reconstruction-level jets) to jets clustered from stable (lifetime $c\tau>1\cm$) particles excluding neutrinos before any detector simulation (particle-level jets). Particle-level jets are clustered without simulation of PU collisions whereas the reconstruction-level jets include simulation of PU collisions. Jet energy corrections are applied to the reconstruction-level jets such that the ratio of reconstruction and particle- level jet (the response) is on average 1. The jet energy resolution (JER) is defined as the spread of the response distribution, which is Gaussian to a good approximation. The resolution is defined as the $\sigma$ of a Gaussian fit to the distribution in the range $[m-2\sigma,m+2\sigma]$, where $m$ and $\sigma$ are the mean and width of the Gaussian fit, determined with an iterative procedure. The cutoff at $\pm 2\sigma$ is set so that the evaluation is not affected by outliers in the tails of the distribution. Figure 3 shows the JER as a function of jet for jets reconstructed from all of the PF candidates (PF jets), CHS jets, and PUPPI jets, simulated with on average 20–30 PU interactions. For AK4 jets, the performance of the CHS and PUPPI algorithms is similar. Jet resolution for PUPPI is slightly degraded below 30 PU, since PUPPI has been optimized for overall performance, including resolution and stability, beyond 30 PU interactions. This behavior at low PU can in principle be overcome through a special treatment in the limit of small amount of PU, where the number of particles to compute $\overline{\alpha}_{\text{PU}}$ and $\text{RMS}_{\text{PU}}$ is limited. The PF jets in the detector region of $\abs{\eta}<0.5$ exhibit a worse performance, particularly at low , since these jets are more affected by PU. In the region of $3.2<\abs{\eta}<4.7$, PF jets show the same performance as CHS jets, because no tracking is available. For AK8 jets, PUPPI provides better performance than the CHS and PF algorithms, since neutral particles from PU interactions contribute significantly to such jets. Figure 3: Jet energy resolution as a function of the particle-level jet for PF jets (orange circles), PF jets with CHS applied (red triangles), and PF jets with PUPPI applied (blue squares) in QCD multijet simulation. The number of interactions is required to be between 20 and 30. The resolution is shown for AK4 jets with $\abs{\eta}<0.5$ (upper left) and $3.2<\abs{\eta}<4.7$ (upper right), as well as for AK8 jets with $\abs{\eta}<0.5$ (lower). The error bars correspond to the statistical uncertainty in the simulation. Figure 4 demonstrates how the JER scales with the number of interactions. At more than 30 interactions, JER for AK4 jets with $\abs{\eta}<0.5$ and $\pt=30\GeV$ is better with the PUPPI than with the CHS PU mitigation. However, JER for AK4 jets with $3.2<\abs{\eta}<4.7$ and $\pt=30\GeV$ is better with the CHS than with the PUPPI PU mitigation, which is a result of the PUPPI algorithm being tuned to yield the best resolution rather than the best jet resolution in the $\abs{\eta}>3$ region. This is achieved with a low PU particle rate, rather than the best jet resolution, achieved by high LV particle efficiency. At $\pt>100\GeV$, PUPPI jets have a resolution that is slightly worse than that of CHS jets with $\abs{\eta}<0.5$, while in $3.2<\abs{\eta}<4.7$ PUPPI and CHS performances are comparable. For AK8 jets at low , PUPPI yields a better JER than CHS; this improvement is present through the high-PU scenarios, , at 50 or 60 interactions. The jet energy resolution becomes worse with PUPPI than with CHS for jets with $\pt>200\GeV$. The behavior of PUPPI at high is to a large extent limited by the quality of track-vertex association using $d_{z}$ for high-charged hadrons. The effect is not visible in CHS because the $d_{z}$ requirement for charged particles that are not associated to any vertex is not used, but instead CHS keeps all charged particles not associated with any vertex. Figure 4: Jet energy resolution as a function of the number of interactions for jets with CHS (solid red line) and with PUPPI (dashed blue line) algorithms applied in QCD multijet simulation for different jet values (different markers). The resolution is shown for AK4 jets with $\abs{\eta}<0.5$ (upper left) and $3.2<\abs{\eta}<4.7$ (upper right), as well as for AK8 jets with $\abs{\eta}<0.5$ (lower). The error bars correspond to the statistical uncertainty in the simulation. Figure 5 shows the jet $\eta$ angular resolution simulated with 20–30 interactions. The same qualitative conclusions also hold for the resolution in $\phi$, since $\phi$ and $\eta$ segmentation of the detector are similar. The resolution is evaluated as the width of a Gaussian function fit to the distribution of the $\eta$-difference between the generator- and reconstruction-level jets. The same conclusions as for JER also hold for jet angular resolution. The CHS and PUPPI algorithms perform similarly for AK4 jets with $\abs{\eta}<0.5$. However, significant improvements from PUPPI are observed for AK8 jets for $\abs{\eta}<0.5$. Angular resolution of large-size jets is particularly sensitive to PU as the clustered energy from PU particles increases with the jet size. Hence, the improvements are larger when PUPPI jets are considered. Figure 5: Jet $\eta$ resolution as a function of particle-level jet for PF jets (orange circles), PF jets with CHS applied (red triangles), and PF jets with PUPPI applied (blue squares) in QCD multijet simulation. The number of interactions is required to be between 20 and 30. The resolution is shown for AK4 jets with $\abs{\eta}<0.5$ (upper left) and $3.2<\abs{\eta}<4.7$ (upper right) as well as for AK8 jets with $\abs{\eta}<0.5$ (lower). The error bars correspond to the statistical uncertainty in the simulation. ### 0.5.2 Noise jet rejection The identification and rejection of jets originating from noise and reconstruction failures are critical to all CMS analyses where a jet or is used as part of the selection. To further reject noise after detector signal processing and jet clustering, a set of criteria on the PF candidates within a jet are applied [6]. The criteria listed in Table 0.5.2 are based on jet constituent energy fractions and multiplicities. They reject residual noise from the HCAL and ECAL, retaining 98–99% of genuine jets, , jets initiated by genuine particles rather than detector noise. Although PU mitigation algorithms are not designed to have an effect on detector noise, they could, in principle, affect the rejection capability of the noise jet ID. Figure 6 (upper left/right and lower left) shows the distribution of the charged and neutral constituent multiplicities comparing genuine jet enriched (dijet) and noise jet enriched (minimum bias) data, demonstrating the separation power. For the dijet selection, data are selected with an HLT requirement of at least one jet having a $\pt>400\GeV$, two offline reconstructed jets with greater than 60 and 30, respectively, and an opening in azimuthal angle greater than 2.7. For the minimum bias selection, jets with $\pt>30\GeV$ passing the minimum bias trigger path are used. The noise jet ID requires at least one charged constituent for jets with $\abs{\eta}<2.4$ and at least two constituents (neutral or charged) for $\abs{\eta}<2.7$. The charged constituent multiplicity is smaller for PUPPI than for CHS jets because PUPPI rejects additional charged particles by applying a $d_{z}$ requirement on tracks not associated with any vertex. The PUPPI weighted neutral constituent multiplicity, defined as the sum of PUPPI weights of all neutral particles in the jet, is also smaller than the neutral constituent multiplicity for CHS. In $3<\abs{\eta}<5$, the PUPPI neutral constituent multiplicity is significantly lower than for CHS. Thus, the ability to separate noise is reduced. With CHS, noise jets are rejected by requiring a minimum of 10 neutral particles. With PUPPI, a minimum of 3 is required for the PUPPI scaled neutral multiplicity. Figure 6 (lower right) demonstrates the PU dependence of the neutral constituent multiplicity. While for CHS, the average multiplicity changes by 30–40% going from 20–30 to 50–60 reconstructed vertices, the PUPPI scaled multiplicities do not change significantly, making noise jet rejection independent of PU. The efficiency of the jet ID criteria for genuine jets is measured in data using a tag-and-probe procedure in dijet events [6]. The background rejection is estimated using a noise-enriched minimum bias event selection. The fraction of rejected noise jets after applying jet ID criteria that yield a 99% efficiency for genuine jets is summarized in Table 0.5.2 for different regions in $\eta$. The number of noise jets reconstructed with the CHS and PUPPI algorithms is not the same, because the PUPPI reconstruction criteria reject particles that would otherwise give rise to a fraction of noise jets before jet ID criteria are applied. The absolute number of noise jets remaining after PU mitigation and jet ID together differs by less than 20% between CHS and PUPPI jets. Jet ID criteria for CHS and PUPPI jets yielding a genuine jet efficiency of 99% in different regions of $\abs{\eta}$. Region of $\abs{\eta}$ Variable Requirement (CHS) Requirement (PUPPI) $\abs{\eta}<2.4$ Charged hadron energy fraction $>$0 $>$0 Charged multiplicity $>$0 $>$0 $\abs{\eta}<2.7$ Neutral hadron energy fraction $<$0.90 $<$0.90 Neutral EM energy fraction $<$0.90 $<$0.90 Number of constituents $>$1 $>$1 $2.7<\abs{\eta}<3$ Neutral EM energy fraction $>$0.02 and $<$0.99 Number of neutral particles $>$2 Neutral hadron energy fraction $<$0.99 $\abs{\eta}>3$ Neutral EM energy fraction $<$0.90 $<$0.9 Neutral hadron energy fraction $>$0.02 $>$0.02 Number of neutral particles $>$10 $>$3 Fraction of noise jets rejected when applying jet ID criteria to PUPPI and CHS jets yielding a genuine jet efficiency of 99% in different regions of $\abs{\eta}$. Region of $\abs{\eta}$ Fraction of noise jets rejected $\abs{\eta}<$ $2.7$ 99.9% $2.7<$ $\abs{\eta}<$ $3.0$ 97.6% $3<$ $\abs{\eta}<$ $5$ 15% (PUPPI) 35% (CHS) Figure 6: The charged- and neutral-particle multiplicities for CHS and PUPPI in a dijet (genuine jets) and minimum bias (noise jets) selection in data. The multiplicities are shown for AK4 jets using CHS reconstructed real jets (red dashed), CHS reconstructed noise jets (black long dashed), PUPPI reconstructed genuine jets (blue circles), and PUPPI reconstructed noise jets (orange triangles). The upper plots show the charged (left) and neutral particle multiplicities (right) for jets with $\abs{\eta}<0.5$. The lower left plot shows the neutral particle multiplicity for jets with $3<\abs{\eta}<5$. The lower right plot shows the neutral particle multiplicity of AK4 jets with $\abs{\eta}<0.5$ in a dijet selection in data using CHS and PUPPI for 15–20 and 35–50 interactions. The error bars correspond to the statistical uncertainty. ### 0.5.3 Pileup jet rejection Particles resulting from PU collisions will introduce additional jets that do not originate from the LV. These jets are referred to as PU jets. PU jets can be classified in two categories: QCD-like PU jets, originating from PU particles from a single PU vertex, and stochastic PU jets, originating from PU particles from multiple different PU vertices. Both PU mitigation techniques, PUPPI and CHS, remove the charged tracks associated with PU vertices, reducing the of QCD-like PU jets to roughly 1/3 of their original , such that they can be largely reduced by selections on the jet . In CMS, a multivariate technique to reject the remaining PU jets (dominated by stochastic PU jets) has been developed and applied for CHS jets [6], whereas PUPPI intrinsically suppresses PU jets better by rejecting more charged and neutral particles from PU vertices before jet clustering. Both techniques suppress both QCD-like and stochastic PU jets, though the observables used for neutral particle rejection are primarily sensitive to stochastic PU jets. The performance of the PU jet rejection for both PUPPI and CHS is evaluated in +jets events in data and simulation. The jet recoiling against the boson provides a pure sample of LV jets, whereas additional jets are often from PU collisions. The +jets events are selected by requiring two oppositely charged muons with $\pt>20\GeV$ and $\abs{\eta}<2.4$ whose combined invariant mass is between 70 and 110. Jets that overlap with leptons within $\Delta R(\text{lepton},\,\text{jet})<0.4$ from the boson decay are removed from the collections of particle- and reconstruction-level jets. In simulation jets are categorized into four groups based on the separation from particle-level jets and their constituents. If a reconstruction-level jet has a particle-level jet within $\Delta R<0.4$, it is regarded as originating from the LV. Jet flavors are defined by associating generated particles to reconstructed jets. This is done by clustering a new jet with the generated and reconstructed particles together where, in this case, the four-momenta of generated particles are scaled by a very small number. Newly reconstructed jets in this way are almost identical to the original jets because the added particles, with extremely small energy, do not affect the jet reconstruction. If a jet originating from the LV contains generated quarks or gluons, it is regarded as a jet of quark or gluon origin, depending on the label of the highest particle-level particle. If a jet not originating from the LV does not contain any generated particles from the hard scattering, it is regarded as a jet originating from a PU vertex, , a PU jet. The remaining jets, which do not have nearby particle-level jets but contain particle-level particles (from LV), are labeled as unassigned. List of variables used in the PU jet ID for CHS jets. Input variable Definition $\text{LV}\sum\pt$ fraction Fraction of of charged particles associated with the LV, defined as $\sum_{i\in\text{LV}}p_{\mathrm{T},\,i}/\sum_{i}p_{\mathrm{T},\,i}$ where $i$ iterates over all charged PF particles in the jet $N_{\text{vertices}}$ Number of vertices in the event $\langle\Delta R^{2}\rangle$ Square distance from the jet axis scaled by $\pt^{2}$ average of jet constituents: $\sum_{i}\Delta R^{2}p_{\mathrm{T},\,i}^{2}/\sum_{i}p_{\mathrm{T},\,i}^{2}$ $f_{\text{ringX}}$, $X=1,2,3,\text{ and }4$ Fraction of of the constituents ($\sum p_{\mathrm{T},\,i}/\pt^{\text{jet}}$) in the region $R_{i}<\Delta R<R_{i+1}$ around the jet axis, where $R_{i}=0,0.1,0.2,$ and 0.3 for $X=1,2,3$, and 4 $\pt^{\text{lead}}/\pt^{\text{jet}}$ fraction carried by the leading PF candidate $\pt^{\text{l. ch.}}/\pt^{\text{jet}}$ fraction carried by the leading charged PF candidate $\abs{\vec{m}}$ Pull magnitude, defined as $\abs{(\sum_{i}\pt^{i}\abs{r_{i}}\vec{r}_{i})}/\pt^{\text{jet}}$ where $\vec{r_{i}}$ is the direction of the particle $i$ from the direction of the jet $N_{\text{total}}$ Number of PF candidates $N_{\text{charged}}$ Number of charged PF candidates $\sigma_{1}$ Major axis of the jet ellipsoid in the $\eta$-$\phi$ space $\sigma_{2}$ Minor axis of the jet ellipsoid in the $\eta$-$\phi$ space $\pt^{\text{D}}$ Jet fragmentation distribution, defined as $\sqrt{\sum_{i}p^{2}_{\mathrm{T},\,i}}/\sum_{i}p_{\mathrm{T},\,i}$ This identification of PU jets is based on two observations: (i) the majority of tracks associated with PU jets do not come from the LV, and (ii) PU jets contain particles originating from multiple PU collisions and therefore tend to be more broad and diffuse than jets originating from one single quark or gluon. Table 0.5.3 summarizes the input variables for a multivariate analysis. Track-based variables include the $\text{LV}\sum\pt$ fraction and $N_{\text{vertices}}$, where the $\text{LV}\sum\pt$ fraction is the summed of all charged PF candidates in the jet originating from the LV, divided by the summed of all charged candidates in the jet. The $\text{LV}\sum\pt$ fraction variable provides the strongest discrimination of any variable included in the discriminator, but is available only within the tracking volume. The inclusion of the $N_{\text{vertices}}$ variable allows the multivariate analysis to determine the optimal discriminating variables as the PU is increased. Jet shape variables included in the multivariate discriminant are as follows: $\langle\Delta R^{2}\rangle$, $f_{\text{ring0}}$, $f_{\text{ring1}}$, $f_{\text{ring2}}$, $f_{\text{ring3}}$, $\pt^{\text{lead}}/\pt^{\text{jet}}$, $\abs{\vec{m}}$, $N_{\text{total}}$, $N_{\text{charged}}$, major axis ($\sigma_{1}$), minor axis ($\sigma_{2}$), and $\pt^{\mathrm{D}}$, with their definitions given in Table 0.5.3. Pileup jets tend to have $\langle\Delta R^{2}\rangle$ of large value relative to genuine jets. For the set of $f_{\text{ringX}}$, PU jets tend to have large values for variables with large $R$, which represents the characteristic of PU jets having a large fraction of energy deposited in the outer annulus. Most of the other variables are included to distinguish quark jets from gluon jets, and thus enhance the separation from PU jets. In particular, the variable $\pt^{\mathrm{D}}$ tends to be larger for quark jets than for gluon jets, and smaller than both quark jets and gluon jets for PU jets. The $N_{\text{total}}$, $\pt^{\mathrm{D}}$ and $\sigma_{2}$ variables have previously been used for a dedicated quark- and gluon-separation technique; more details on their definition and performance are found in Ref. [6]. Figure 7 shows the distribution of the $\text{LV}\sum\pt$ fraction and the charged-particle multiplicity of jets with $30<\pt<50\GeV$ and $\abs{\eta}<1$ in data and simulation. The distributions of the variables in selected data events agree with simulation within the uncertainties, with a clear separation in the discriminating variables between LV and PU jets. Figure 7: Data-to-simulation comparison for two input variables to the PU jet ID calculation for CHS jets with $30<\pt<50\GeV$: the $\text{LV}\sum\pt$ fraction (left) and charged-particle multiplicity (right). Black markers represent the data while the colored areas are +jets simulation events. The simulation sample is split into jets originating from quarks (red), gluons (purple), PU (green), and jets that could not be assigned (gray). The distributions are normalized to unity. The shape of a sample showered with ++ is superimposed. The lower panels show the data-to-simulation ratio along with a gray band corresponding to the one-sided uncertainty, which is the difference between simulated Z+jets events showered with the PYTHIA parton shower and those showered with the HERWIG++ parton shower. Also included in the ratio panel is the PU rate uncertainty (dark gray). The set of 15 variables listed in Table 0.5.3 is used to train a boosted decision tree (BDT) algorithm, and to distinguish between jets from the LV and PU jets. For the BDT training, +jets simulation events are used. To perform the training, reconstruction-level jets that are within a distance of $\Delta R<0.4$ from any particle-level jet are regarded as jets from the LV, and the remaining jets are identified as PU jets. A jet is considered to satisfy the PU jet ID if it passes certain thresholds on the output of the BDT discriminator. This output is dependent on the $\eta$ and $\pt$ of the jet. Three working points are considered in the following resulting in different efficiencies and misidentification rates. These working points are defined by their average efficiency on quark-initiated jets. The definitions are: * • tight working point: 80% efficient for quark jets, * • medium working point: 90% efficient for quark jets, * • loose working point: 99% efficient for quark jets in $\abs{\eta}<2.5$, 95% efficient for quark jets in $\abs{\eta}>2.5$. Since 92% of the PU jets tend to occur at $\pt<50\GeV$, the contamination from PU jets with $\pt>50\GeV$ is small. Thus, the PU jet ID is designed to act only on jets with $\pt<50\GeV$. The fraction of PU jets in simulation passing this kinematic event selection is 10% for $\abs{\eta}<2.5$, 48% for $2.50<\abs{\eta}<2.75$, 59% for $2.75<\abs{\eta}<3.00$, and 65% for $3<\abs{\eta}<5$. The distribution of the output BDT discriminator in selected data events and simulation is shown in Fig. 8. Some disagreement is present between the data and simulation. This disagreement is largest for $\abs{\eta}>2.5$ and at low discrimination values, where PU jets dominate. The difference between data and simulation is roughly comparable to the total uncertainty in simulation, considering the uncertainty in the number of interactions and the difference to an alternative ++-based parton shower prediction. Figure 8: Data-to-simulation comparison of the PU jet ID boosted decision tree (BDT) output for AK4 CHS jets with $30<\pt<50\GeV$ for the detector region within the tracker volume (left) and $3<\abs{\eta}<5$ (right). Black markers represent the data while the colored areas are +jets simulation events. The simulation sample is split into jets originating from quarks (red), gluons (purple), PU (green), and jets that could not be assigned (gray). The distributions are normalized to unity. The shape of a sample showered with ++ is superimposed The lower panels show the data-to-simulation ratio along with a gray band corresponding to the one-sided uncertainty that is the difference between simulated Z+jets events showered with the PYTHIA parton shower to those showered with the HERWIG++ parton shower. Also included in the ratio panel is the PU rate uncertainty (dark gray). When studying jet performance with PU, it is clear that jet reconstruction and selection, including PU mitigation, affect the relationship between the number of reconstructed vertices and the mean number of interactions per crossing. The mean number of vertices as a function of the number of interactions can be seen in Fig. 9 (left). Without jet selection, the number of vertices is on average 30% smaller [34, 8] than the number of interactions, because the vertex reconstruction and identification efficiency is about 70% (although it is nearly 100% for hard-scattering interactions). When introducing a selection on the jet , the mean number of vertices for a given number of interactions is reduced. This effect is largest for CHS jets, where no treatment of jets composed of mostly PU particles is present. If a PU vertex is close to or overlaps with the LV, jets composed of PU particles end up in the event reconstruction and cause the observed bias. When applying a technique to reduce the number of additional jets composed of mostly PU particles (PUPPI or CHS+tight PU jet ID), the relationship shows a behavior more similar to the one without selection. The mean number of interactions as a function of the number of vertices is presented in Fig. 9 (right). This relationship depends on the assumed distribution of pileup interactions in data and is adjusted to match the 2016 data taking. The largest difference between events with and without a cut is observed for a high number of vertices, while the different PU mitigation techniques show a similar behavior. Figure 9: Left: distribution of mean number of reconstructed vertices as a function of the mean number of interactions in +jets simulation. Right: distribution of the mean number of interactions as a function of the number of vertices in +jets simulation. The black open circles show the behavior without applying any event selection, while for the other markers a selection on jets of $\pt>20\GeV$ is applied using the CHS (full red triangles), CHS+tight PU jet ID (violet open squares), and PUPPI (full blue squares) algorithms. The error bars correspond to the statistical uncertainty in the simulation. Figure 10 shows the LV jet efficiency and purity in +jets simulation as a function of the number of interactions for CHS jets, CHS jets with a PU jet ID applied, and PUPPI jets. The efficiency is defined as the fraction of particle-level jets with $\pt>30\GeV$ that match within $\Delta R<0.4$ with a reconstruction-level jet with $\pt>20\GeV$. The purity is defined as the fraction of reconstruction-level jets with $\pt>30\GeV$ that match within $\Delta R<0.4$ with a particle-level jet with $\pt>20\GeV$ from the main interaction. The cuts at reconstruction and generator level are chosen to be different to remove any significant JER effects on this measurement. For CHS jets, the efficiency is larger than 95% in entire detector region up to $\abs{\eta}<5$ regardless of the number of interactions. However, the purity drops strongly with the number of interactions down to 70 and 18% at 50 interactions for the regions of $\abs{\eta}<2.5$ and $\abs{\eta}>2.5$, respectively. The PU jet ID applied on top of CHS reduces the efficiency with respect to using only CHS, but at the same time improves the purity, especially for low-jets. In $\abs{\eta}<2.5$, the loose working point has only a slightly reduced efficiency compared to CHS alone. In $\abs{\eta}>2.5$, the efficiency drops to roughly 80% at high PU for the loose working point. In $\abs{\eta}<2.5$, the purity remains constant at around 98% over the whole range of PU scenarios. In $\abs{\eta}>2.5$, the purity is PU-dependent, but improves over CHS alone by a factor of 1.7 at high PU for the loose working point. The tight PU jet ID achieves the best purity in $\abs{\eta}>2.5$ at 40% with collisions at 50 interactions and a jet efficiency of 45%. PUPPI also reduces the efficiency with respect to CHS by removing neutral particles. At the same time, PUPPI improves the purity by removing PU jets from the event without the need of a PU jet ID. At low PU (below 10 interactions), the purity of PUPPI jets is equal to that of CHS. At high PU, the purity of PUPPI jets with respect to CHS jets is significantly higher than that of CHS jets. PUPPI has a constant efficiency above 95% in $\abs{\eta}<2.5$, and a purity compatible with the tight PU jet ID working point at high PU. In $\abs{\eta}>2.5$, above 30 interactions the efficiency of PUPPI is better than the loose PU jet ID, whereas the purity is compatible to within a few percent to the loose PU jet ID. In summary, PUPPI shows an intrinsic good balance between efficiency and purity compared to CHS, but if purity in $\abs{\eta}>2.5$ is crucial to an analysis, CHS+tight PU jet ID yields better performance. Using variables designed to distinguish quark jets from gluon jets results in a $<1\%$ difference for $20<\mathrm{PU}<30$ in efficiency for PUPPI and CHS in $\abs{\eta}<2.5$ and range up to 5% (12%) in $\abs{\eta}>3$ for PUPPI (CHS) with tight PU ID. Figure 10: The LV jet efficiency (upper) and purity (lower) in +jets simulation as a function of the number of interactions for PUPPI (blue closed squares), CHS (red closed triangles), CHS+tight PU jet ID (magenta open squares), CHS+medium PU jet ID (orange crosses), and CHS+loose PU jet ID (black triangles). Plots are shown for AK4 jets $\pt>20\GeV$, and (left) $\abs{\eta}<2.5$ and (right) $\abs{\eta}>3$. The LV jet efficiency is defined as the number of matched reconstruction-level jets with $\pt>20\GeV$ divided by the number of particle-level jets with $\pt>30\GeV$ that originate from the main interaction. For the lower plots, the purity is defined as the number of matched particle-level jets with $\pt>20\GeV$ divided by the number of reconstructed jets that have $\pt>30\GeV$. The error bars correspond to the statistical uncertainty in the simulation. To evaluate the performance of PU jet identification in data, the ratio of PU jets to genuine jets for the leading jet in the event is studied. Events are split into two categories to compare both PU and LV jets. The categorization is performed utilizing the difference between the azimuths $\phi$ of the leading jet and the boson. The PU-enriched events are required to have $\Delta\phi(\PZ\,\text{boson},\text{jet})<1.5$, while events enriched in LV jets are required to have $\Delta\phi(\PZ\,\text{boson},\text{jet})>2.5$. Figure 11 shows the rate of events in the PU-enriched region divided by the rate of events in the LV-enriched region, as a function of the number of vertices for CHS jets, CHS jets with medium PU jet ID applied, and PUPPI jets in +jets simulation and data. The rate of PU-enriched events selecting CHS jets alone exhibits a strong dependence on the number of vertices in detector regions where $\abs{\eta}<2.5$. This dependence increases from 8 to 25% when going from 5 to 40 vertices. The dependence is strongly reduced when the PU jet ID is applied or PUPPI is utilized. PUPPI shows a stable behavior across the whole range in $\abs{\eta}<2.5$ for both data and simulation. For $\abs{\eta}>2.5$, all three algorithms show a PU dependence with CHS jets having the worst performance. Furthermore, categorization with PUPPI jets has a PU-enriched rate between that of events categorized with CHS and CHS+medium PU jet ID. For reference, the rate of jets that are matched to a particle- level jet in simulation is also shown for CHS jets (simulation, CHS LV). This line shows the expected ratio of events in the two regions when only the LV jets are used for the categorization. This curve shows a slight PU dependence because of the high matching parameter of generator- with reconstruction-level jets ($\Delta R<0.4$). Scale factors for the efficiency of data and simulation for both matched jets from the LV and PU jets for various PU jet ID working points are derived using the event categories enriched in genuine jets and PU jets. Scale factors are within a few percent of unity in the detector region where $\abs{\eta}<2.5$. In $\abs{\eta}>2.5$, they are farther from unity, with differences up to 10% for jets with $2.5<\abs{\eta}<3.0$ and the tight working point applied. The scale factor for PU jets is significantly larger and leads to a visible disagreement in Fig. 11. This disagreement is found to be as large as 30% for low jets with $\abs{\eta}>2.5$. The difference in modeling when using ++ instead of for parton showering shown in the lower panel of Fig. 11 is considered as an additional uncertainty. The difference of data with respect to showered jets is contained within the total variation when considering both ++ and based parton showers. Figure 11: Rate of jets in the PU-enriched region divided by the rate of jets in the LV-enriched region as a function of the number of vertices for CHS jets (red triangles), CHS jets with medium PU jet ID applied (orange crosses) and PUPPI jets (blue squares) in +jets simulation (open markers), and data (full markers). For reference, the rate of jets that are matched to a particle-level jet in simulation is also shown for CHS jets (black solid line). The plots show the ratio for events with $\abs{\eta}<2.5$ (left) and $\abs{\eta}>2.5$ (right). The lower panels show the data-to-simulation ratio along with a gray band corresponding to the one-sided uncertainty that is the difference between simulated +jets events showered with the parton shower to those showered with the ++ parton shower. ## 0.6 , , Higgs boson, and top quark identification ### 0.6.1 Jet substructure reconstruction In various searches for new physics phenomena and measurements of standard model properties, top quarks, , , and Higgs bosons are important probes. They can be produced with a large Lorentz boost, $\gamma$, such that the direction of their decay particles becomes very collinear. The spatial separation between the decay products in the $\eta$-$\phi$ plane is approximately $\Delta R\approx 2/\gamma$. In such cases, it is difficult to reconstruct the decay products of the hadronically decaying objects of interest properly with traditional jets of size 0.4. As a result, techniques to reconstruct all decay products within one jet with a larger size of 0.8 have been widely studied and used [20, 35]. The invariant mass and substructure of the reconstructed jets are typically used to identify the different bosons and top quarks. These larger cone size jets tend to collect more PU, hence PU mitigation techniques are relevant across a larger range, extending to well beyond $\pt>100\GeV$. In addition, the jet mass and substructure variables are particularly affected by soft and wide-angle radiation. A grooming technique is applied on top of CHS and PUPPI to remove soft radiation from the jet-clustering algorithm and thereby mitigate the effects from PU, underlying event, and initial-state radiation. The main grooming algorithm used in CMS is the soft drop or modified mass drop tagger [36, 37]. It reclusters a jet with the Cambridge–Aachen algorithm [38], and splits the jet in two subjets by undoing the last step of the jet clustering. It regards the jet as the final soft drop jet if the two subjets satisfy the condition $\frac{\min(\pt^{1},\pt^{2})}{\pt^{1}+\pt^{2}}>z_{\text{cut}}\Big{(}\frac{\Delta R_{12}}{R_{0}}\Big{)}^{\beta},$ (3) where $\pt^{1}$ and $\pt^{2}$ are the transverse momenta of the two subjets, $R_{0}$ is the size parameter of the jet, $\Delta R_{12}=\sqrt{\smash[b]{(\Delta\eta_{12})^{2}+(\Delta\phi_{12})^{2}}}$ is the distance between the two subjets, and $z_{\text{cut}}$ and $\beta$ are tunable parameters of soft drop set to $z_{\text{cut}}=0.1$ and $\beta=0$ here. If the soft drop condition is not met, the declustering procedure is repeated with the subjet that has the larger of the two, and the other subjet is rejected. The soft drop jet mass is computed from the sum of the four-momenta of the constituents passing the grooming algorithm. The mass is then corrected by a factor derived in simulated boson samples to ensure a $\pt$\- and $\eta$-independent jet mass distribution [6]. Additional separation of boosted , , and Higgs bosons, and top quarks from quark and gluon jet background can be achieved with a substructure observable. A commonly used observable in CMS is $N$-subjettiness [39], defined as $\tau_{N}=\frac{1}{d_{0}}\sum_{k}p_{\mathrm{T}k}\,\min(\Delta R_{1,k},\Delta R_{2,k},\ldots,\Delta R_{N,k}),$ (4) with the normalization factor $d_{0}$: $d_{0}=\sum_{k}p_{\mathrm{T}k}\,R_{0},$ (5) where $R_{0}$ is the size parameter used in the clustering process, $p_{\mathrm{T}k}$ is the transverse momentum of the $k$-th constituent of the jet, and $\Delta R_{n,k}$ estimates the angular separation of the constituents of the jet to the closest subjet axis. We use a one-step optimization of the exclusive $\kt$ axes as a definition for the subjet axes. The ratio $\tau_{2}/\tau_{1}$, which is called $\tau_{21}$, has excellent capability in separating jets with bipolar structures, originating from boosted , , and Higgs bosons, from jets coming from quarks and gluons. The ratio $\tau_{32}=\tau_{3}/\tau_{2}$ can be used to discriminate top quark jets from , , and Higgs boson jets, or quark and gluon jets. ### 0.6.2 Identification performance and pileup The variation as a function of pileup of the median soft drop jet mass, median $\tau_{21}$, and the soft drop jet mass resolution is shown in Fig. 12 for jets from boosted bosons with $400<\pt<600\GeV$ using simulation of bulk gravitons decaying into $\PW\PW$ pairs. The soft drop jet mass resolution is defined as the spread of the ratio of reconstruction- and particle-level jet mass (the response) divided by the mean of the response. The response distribution is, to a very good approximation, Gaussian, and the resolution is determined using the same procedure as for the JER described in Section 0.5.1. The CHS jets exhibit a PU dependence for the soft drop jet mass and $\tau_{21}$ observables. The PUPPI jets, on the other hand, entirely remove the PU dependence of the soft drop jet mass and $\tau_{21}$ medians. The soft drop jet mass resolution is similar for the CHS and PUPPI algorithms, though a slightly better resolution is observed for the CHS algorithm for fewer than 20 interactions, while the PUPPI algorithm shows less dependence on PU leading to an improved resolution for more than 30 interactions, for which it has been optimized. Figure 12: Median soft drop jet mass (upper left), median $\tau_{21}$ (upper right), and soft drop jet mass resolution (lower) for AK8 jets from boosted bosons with $400<\pt<600\GeV$ for CHS (red triangles) and PUPPI (blue squares) jets in a bulk graviton decaying to signal sample, as a function of the number of vertices. The error bars correspond to the statistical uncertainty in the simulation. The performance of a typical or boson tagger with respect to the PU contamination is studied using simulation of bulk gravitons decaying into $\PW\PW$ pairs for tagging efficiency and QCD multijet production for misidentification rate. Reconstructed jets are required to have larger than 200and $\abs{\eta}<2$, and not to overlap with any well-reconstructed leptons. In addition, jets are required to have reconstructed mass compatible with the boson mass (within 65–105). Figure 13 shows the evaluated efficiency and misidentification rate of the tagger with CHS and PUPPI jets operated at two cutoff values on $\tau_{21}$ (0.6 and 0.45 for CHS jets, and 0.55 and 0.40 for PUPPI jets, which give a comparable efficiency to that for CHS jets). The tagger with PUPPI provides stable performance for both efficiency and misidentification rate, whereas the one with CHS reduces both efficiency and misidentification rate as the PU increases. This behavior of the tagger with CHS results from the linear dependence of median $\tau_{21}$ on the number of vertices for both /jets and jets (see Fig. 12). Figure 13: boson identification performance using a selection on $\tau_{21}$ for CHS (red triangles and dark red crosses) and PUPPI (blue squares and circles) AK8 jets as a function of the number of vertices for loose and tight selections, respectively. Shown on the left is the boson identification efficiency evaluated in simulation for a bulk graviton decaying to a boson pair and on the right the misidentification rate evaluated with QCD multijet simulation. The error bars correspond to the statistical uncertainty in the simulation. Figure 14: Top quark identification efficiency (left) and misidentification rate (right) as a function of the number of vertices for CHS (open symbols) and PUPPI (closed symbols) jets, using different combinations of substructure variables: soft drop mass cut between 105 and 210(blue rectangles), $\tau_{32}<0.54$ (orange circles), and both requirements together (red triangles). The error bars correspond to the statistical uncertainty in the simulation. The same stability of the PUPPI algorithm is seen in top quark jet identification, which is performed by selecting jets originating from top quarks in simulation that have a soft drop mass within 105–210and $\tau_{32}<0.54$. Figure 14 shows the tagging performance using the CHS and PUPPI algorithms with the soft drop mass and $\tau_{32}$ conditions applied separately, and with both of them together. Although the efficiency is slightly different between the application of PUPPI or CHS, the same stability is observed with respect to PU as for tagging. The performance of the boson tagger with the CHS and PUPPI algorithms is compared in data and simulation following the procedure described in Ref. [20]. The boson identification efficiency is measured in a region enriched in events, where one top quark decays to a final state with a lepton, neutrino, and a bottom quark and is used to tag the event. The other top quark is required to decay to a bottom quark and a boson that further decays to a quark-antiquark pair. The AK8 jet with the highest in the event is probed as the boson jet candidate and required to have $\pt>200\GeV$ and $\abs{\eta}<2.4$. Data collected by single-lepton triggers are compared with simulation samples of top quark pair production and backgrounds from single top, boson, and diboson production. The soft drop jet mass scale and resolution, as well as the $\tau_{21}$ selection efficiency with the CHS and PUPPI algorithms, are well modeled by the simulation. The data-to-simulation scale factors for jet mass scale, jet mass resolution, and $\tau_{21}$ selection efficiency are found in Table 0.6.2. The leading systematic effects include parton showering and variations of the fit model (treatment of nearby jets) as detailed in Ref. [20]. Data-to-simulation scale factors for the jet mass scale, jet mass resolution, and the $\tau_{21}$ selection efficiency for the CHS and PUPPI algorithms. Parameter Data/simulation CHS PUPPI Mass scale $1.007\pm 0.009\stat\pm 0.005\syst$ $0.998\pm 0.007\stat\pm 0.006\syst$ Mass resolution $1.15\pm 0.04\stat\pm 0.04\syst$ $1.08\pm 0.02\stat\pm 0.09\syst$ $\tau_{21}<0.45$ $1.00\pm 0.06\stat\pm 0.07\syst$ $\tau_{21}<0.4$ $1.01\pm 0.06\stat\pm 0.05\syst$ ## 0.7 Missing transverse momentum resolution The imbalance of momentum for all reconstructed objects in the transverse plane, called missing transverse momentum $\vec{p}_{T}^{\text{miss}}$ with magnitude , is a signature of neutrino production. It also plays an important role in searches for unknown stable neutral particles. In CMS, $\vec{p}_{T}^{\text{miss}}$ is calculated as the negative vector sum of all PF candidates (called PF $\vec{p}_{T}^{\text{miss}}$ in the following). The $\vec{p}_{T}^{\text{miss}}$ thus relies on the accurate measurement of the reconstructed physics objects, namely muons, electrons, photons, hadronically decaying taus, jets, and unclustered energy. The unclustered energy is the contribution from the PF candidates not associated with any of the previous physics objects. The CHS procedure is not suitable for $\vec{p}_{T}^{\text{miss}}$ computation since it selectively removes only particles within the tracker volume ($\abs{\eta}<2.5$). PU events that spread across the tracker volume boundary are thus partially removed leading to a degradation in the $\vec{p}_{T}^{\text{miss}}$ resolution. The $\vec{p}_{T}^{\text{miss}}$ is corrected with the difference between the vector sum of all reconstructed jets in the event calibrated to the particle level and the vector sum of all uncalibrated jet momenta (called type-1 correction), to account for the detector response of jet objects. Anomalous high-events can be due to a variety of reconstruction failures, detector malfunctions, or noncollision backgrounds. Such events are rejected by event filters that are designed to identify more than 85–90% of the spurious high- events with a mistagging rate less than 0.1% [40]. The performance of the $\vec{p}_{T}^{\text{miss}}$ reconstruction in CMS (covering $\PZ\to\Pe\Pe$, $\PZ\to\PGm\PGm$ and $\gamma$+jets data samples) is discussed in detail in Ref. [40]. The PUPPI algorithm can be used for the computation of $\vec{p}_{T}^{\text{miss}}$ by scaling the PF candidates by their PUPPI weight (PUPPI $\vec{p}_{T}^{\text{miss}}$), and then applying the type-1 correction using PUPPI jets. The PUPPI metric as defined in Eq. 1 in Section 0.4 treats charged leptons and charged hadrons in the same way, i.e., charged leptons get a weight of 0 or 1 depending on their vertex association and enter into the computation of the weight of their surrounding particles. This causes prompt leptons, e.g., leptons from the decay of the boson, to create a PU dependence by giving PU particles around the prompt lepton a higher weight. Therefore, a second PUPPI metric is defined in which charged leptons are excluded from the calculation. In this definition, it is assumed that all leptons in the event are prompt. This results in PU particles surrounding a prompt lepton having a lower weight consistent with the PU hypothesis. In the following discussion, the metric defined with the default PUPPI weight, including the leptons, is referred to as “PUPPI-with-lepton” and the metric, which excludes the leptons, as “PUPPI-no-lepton.” For the purpose of the PUPPI $\vec{p}_{T}^{\text{miss}}$ computation, PUPPI-no-lepton collection is combined with the collection of leptons given a weight of 1. In addition, a PUPPI weight of 1 is automatically assigned to photons reconstructed in the tracker region ($\abs{\eta}<2.5$) with $\pt>20\GeV$. These photons are required to pass certain identification and isolation criteria ensuring an efficiency of above 80% and a purity of above 95%. The resolution of is quantified by measuring the resolution of the hadronic recoil in boson events. The recoil is defined as the vector sum of the momenta of all the objects (with the same PU mitigation applied as for ) in the event but the boson. The transverse momenta of the recoil and of the boson are balanced against each other, such that their difference allows the determination of the momentum resolution. The momentum of the boson decaying into charged leptons can be reconstructed with high resolution such that it can serve as a reference for the measurement of the energy resolution of the hadronic recoil. The momentum of the recoil is projected to axes parallel and perpendicular to the momentum of the reconstructed boson. The resolution of the former is sensitive to the energy resolution and the latter to the PU contribution. The $\Pp\Pp$ collision data collected with a dielectron trigger are used to evaluate the performance. Events with two isolated electrons, within $\abs{\eta}<2.5$, with the leading (subleading) electron $\pt>25\,(20)\GeV$, and the invariant mass of the two electrons within a 20window centered around the boson mass are selected. The four-momentum of the boson are reconstructed from the four-momentum of the two electrons. The recoil is calculated as the vector sum of the momenta of all particles, but the two electrons. Figure 15 shows the ratio of the recoil to the boson transverse momentum ($u_{\parallel}$) as a function of the boson transverse momentum ($q_{\mathrm{T}}$) for PUPPI $\vec{p}_{T}^{\text{miss}}$ and PF $\vec{p}_{T}^{\text{miss}}$. The PUPPI tends to have a smaller response in events with low momentum recoil. This is because of the removal of PF candidates that are wrongly assigned to the PU vertex by the PUPPI algorithm. Deviations from unity indicate imperfect calibration of the hadronic energy scale. Figure 16 shows the resolution of the recoil, parallel ($\sigma_{\parallel}$), and perpendicular ($\sigma_{\perp}$) to the boson momentum, as a function of the number of vertices for PUPPI $\vec{p}_{T}^{\text{miss}}$ and PF $\vec{p}_{T}^{\text{miss}}$. The scale of the recoil is corrected as a function of the boson momentum for comparison. The PUPPI $\vec{p}_{T}^{\text{miss}}$ resolution for both components is consistently better than the PF $\vec{p}_{T}^{\text{miss}}$ resolution above a number of vertices of 10. In addition, PUPPI $\vec{p}_{T}^{\text{miss}}$ provides a more stable performance with respect to PU than $\vec{p}_{T}^{\text{miss}}$, up to at least 50 vertices. Figure 15: The hadronic recoil response ($-\langle u_{\parallel}\rangle/\langle q_{\mathrm{T}}\rangle$) of the boson computed for PUPPI and PF , as a function of $q_{\mathrm{T}}$ in $\PZ\to\Pe\Pe$ events in collision data. The lower panel shows the data-to-simulation ratio. A gray shaded band is added in the lower panel showing the systematic uncertainties resulting from jet energy scale and jet energy resolution variations, and variations in the unclustered energy added in quadrature. Figure 16: The hadronic recoil components $u_{||}$(left) and $u_{\perp}$(right) for PUPPI and PF resolution as a function of the number of vertices in $\PZ\to\Pe\Pe$ events in collision data. The lower panel shows the data-to-simulation ratio. A gray shaded band is added in the lower panel showing the systematic uncertainties resulting from jet energy scale and jet energy resolution variations, and variations in the unclustered energy added in quadrature. ## 0.8 Muon isolation Muons are reconstructed through a fit to hits in both the inner tracking system and the muon spectrometer [41, 42]. Muons must satisfy identification and reconstruction requirements on the impact parameters of the track, the number of hits reconstructed in both the silicon tracker and the muon detectors, and the uncertainty in the measurement. These quality criteria ensure a precise measurement of the four-momentum, and rejection of badly reconstructed muons. To distinguish prompt charged leptons from those originating from semileptonic decays of hadrons, the lepton isolation provides a powerful handle. Lepton isolation is defined as the sum of all surrounding particles in a cone around the lepton. In this study, PUPPI is investigated in the context of muon isolation and compared with other techniques commonly used in CMS. While not shown here, these techniques are also applicable to electron isolation. Various techniques exist to limit the impact of PU on isolation. A widely used variable within CMS is the $\delta\beta\text{-corrected}$ isolation [41]. This variable is used to estimate the contribution of neutral particles based on the nearby contributions of charged particles, defined by: $\text{$\delta\beta$-Iso}^{\mu^{i}}=\sum_{\Delta R(i,j)<0.4}^{\text{Ch- LV}}\pt^{j}+\max\left(0,\sum_{\Delta R(i,j)<0.4}^{\mathrm{Nh}}\pt^{j}+\sum_{\Delta R(i,j)<0.4}^{\mathrm{Ph}}\pt^{j}-\frac{1}{2}\sum_{\Delta R(i,j)<0.4}^{\text{Ch-PU}}\pt^{j}\right),$ (6) where each sum runs over the particles, indexed with $j$, with $\Delta R<0.4$ of the muon, $\pt^{j}$ is the transverse momentum of each surrounding particle, Ch-LV and Ch-PU are charged particles associated with the LV and PU vertices, respectively, and Nh and Ph are neutral hadrons and photons reconstructed with the PF algorithm, respectively. The subtraction by one half of the amount of Ch-PU corresponds to the subtraction of the PU contamination. It is motivated by isospin symmetry, yielding the ratio of charged to neutral pion production of two, which is responsible for the fact that jets are composed of roughly one-third neutral pions and two-thirds charged pions [5]. An alternative isolation can be constructed using PUPPI. The simplest definition of PUPPI muon isolation is: $\text{Iso}^{\mu^{i}}=\sum_{\Delta R(i,j)<0.4}\pt^{j}\omega^{j},$ (7) where $\pt^{j}$ and $\omega^{j}$ are the transverse momentum and the PUPPI weight of particle $j$, respectively. The PUPPI weight is either determined from PUPPI-with-lepton or PUPPI-no-lepton as described in Section 0.7. In addition, a combined isolation defined as the mean of the two isolation quantities is referred as “PUPPI-combined”: $\text{Iso}_{\text{combined}}=\frac{\text{Iso}_{\text{no-lepton }}+\text{Iso}_{\text{with-lepton}}}{2}.$ (8) The performance of muon isolation is tested using simulated boson (prompt muons) and QCD multijet (nonprompt muons) events with a PU distribution having a mean of 20 interactions comparable to the 2016 PU conditions. For comparison, the relative isolation algorithm, defined as the isolation divided by the muon , is used. Muons are selected if the relative isolation is below a certain threshold. The threshold value for the relative isolation (0.156 for PUPPI-combined and 0.15 for $\delta\beta\text{-corrected}$) is defined such that each isolation quantity gives an inclusive misidentification rate of 12% for the muons selected in QCD multijet simulation. The fraction of muons passing the criteria is referred to as isolation efficiency for prompt muons and as misidentification rate for nonprompt muons. The efficiency is calculated with respect to reconstructed prompt muons with $\pt>20\GeV$ and $\abs{\eta}<2.4$. As explained before, PUPPI-with-lepton has the shortcoming that PU particles around a prompt lepton get too high a weight because of the of the lepton in the $\alpha_{i}$ calculation. Therefore, the application of the weights from PUPPI-with-lepton for the muon isolation leads to a PU-dependent efficiency for prompt muons and a PU-independent misidentification rate. The misidentification rate is PU-independent, because LV particles, which drive the isolation of nonprompt leptons, get a reasonable weight. Conversely, PUPPI-no-lepton has the shortcoming that LV particles near a nonprompt lepton get a reduced weight because the of the nonprompt lepton is excluded when calculating $\alpha_{i}$ for these particles. The weight of LV particles contributing to the isolation is thus less stable against their surroundings. PU particles around leptons, however, get reasonable weights, resulting in a good estimate of the isolation for prompt leptons. Therefore, using PUPPI-no- lepton for the isolation calculation yields a stable efficiency and a less PU- resilient misidentification rate. Figure 17 shows the isolation efficiency and the misidentification rate as a function of the number of vertices. All three PUPPI isolation quantities are observed to be more stable across PU when compared with the $\delta\beta\text{-corrected}$ isolation in terms of misidentification rate. In terms of efficiency, the PUPPI-no-lepton shows a more stable behavior compared with $\delta\beta\text{-corrected}$ isolation whereas PUPPI-with- lepton shows a stronger dependence on the number of vertices. The stability of the PUPPI-combined isolation efficiency is between the two PUPPI isolation variants and similar to the $\delta\beta\text{-corrected}$ isolation. Figure 17: The identification efficiency for prompt muons in simulated +jets events (left) and the misidentification rate for nonprompt muons in QCD multijet simulated events (right) for the different definitions of the isolation: $\delta\beta\text{-corrected}$ isolation (black circles), PUPPI- with-lepton (blue triangles), PUPPI-no-lepton (red crosses), PUPPI-combined (green squares), as a function of the number of vertices. The threshold of each isolation is set to yield a 12% misidentification rate for reconstructed muons in QCD multijet simulation. The error bars correspond to the statistical uncertainty in the simulation. Figure 18 shows a receiver operating characteristic (ROC) curve, , the efficiency as a function of the misidentification rate, when using different definitions of the isolation. The combined PUPPI isolation provides the best performance over the typical analysis working points. Figure 18: The identification efficiency for prompt muons in simulated +jets events as a function of the misidentification rate for nonprompt muons in QCD multijet simulated events for the different definitions of the isolation: $\delta\beta\text{-corrected}$ isolation (black solid line), PUPPI-with-lepton (blue dashed line), PUPPI-no-lepton (red mixed dashed), PUPPI-combined (green long mixed dashed). The average number of interactions is 27. The PUPPI isolation is further investigated in collision data collected with a single-muon trigger path requiring an isolated muon with $\pt>24\GeV$. Two levels of muons are defined: loose muons are required to have $\pt>15\GeV$ and $\abs{\eta}<2.4$ with no isolation requirement and tight muons $\pt>26\GeV$ and $\abs{\eta}<2.1$ with a $\delta\beta\text{-corrected}$ isolation corresponding to an efficiency of 95% (threshold of 0.15). One tight and one loose muon, with the invariant mass of the two muons within a 10window centered around the boson mass are selected. The performance is measured using a tag-and-probe method, with the tight muon as the tag muon and the loose muon as the probe muon. The behavior of the isolation variables in data are compared with +jets simulation. Other backgrounds are neglected. Figure 19 shows the mean fractions of the contributions of charged hadrons, neutral hadrons, and photons to the relative isolation variable, as a function of the number of vertices for the two types of PUPPI isolation in data and +jets simulation. The neutral hadrons and photons make up a large contribution to the total isolation and show a clear PU dependence for the PUPPI-with- lepton isolation, whereas this is not the case for the PUPPI-no-lepton isolation. The trend in data is well described by simulation. Figure 19: Mean relative isolation for PUPPI-with-lepton (left) and PUPPI-no- lepton (right) in data compared to +jets simulation. The relative isolation is split into separate charged hadron (Ch, green squares), neutral hadron (Nh, blue circles), photon (Ph, red crosses) components, and combined (black triangles). Data and simulation are shown using full and open markers, respectively. The lower panels show the data-to-simulation ratio of each component. The error bars correspond to the statistical uncertainty. The isolation efficiency of the PUPPI-combined isolation is evaluated using the same tag-and-probe method, and is compared to the $\delta\beta\text{-corrected}$ isolation. The threshold for PUPPI-combined isolation (0.15) is chosen such that the isolation efficiencies are roughly equal for muons with $15<\pt<20\GeV$, where $\delta\beta\text{-corrected}$ isolation is applied. Figure 20: The identification efficiency for prompt muon isolation selection in $\PZ\to\PGm\PGm$ events in data compared to +jets simulation, as a function of the number of vertices for PUPPI-combined (green circles) and $\delta\beta\text{-corrected}$ isolation (black squares). Data and simulation are shown using full and open markers, respectively. The lower panel shows the data-to-simulation ratio. The error bars correspond to the statistical uncertainty. The threshold for PUPPI-combined isolation (0.15) is chosen such that the isolation efficiencies are roughly equal for muons with $15<\pt<20\GeV$, where $\delta\beta\text{-corrected}$ isolation is applied, leading to an approximately 1% higher efficiency for $\pt>15\GeV$ with variations as a function of the number of vertices. Figure 20 shows the efficiency of the chosen PUPPI and $\delta\beta\text{-corrected}$ isolation variables as a function of the number of vertices. The ratio of efficiency in data to that in simulation is 0.99. Although the PU dependence of the efficiency of the PUPPI-combined isolation is stronger than that of the $\delta\beta\text{-corrected}$ isolation, this does not mean PUPPI-combined isolation is more susceptible to PU, because the misidentification rate is stable against PU (see Fig. 21). The PUPPI-combined isolation outperforms $\delta\beta\text{-corrected}$ isolation across the PU conditions studied. The misidentification rate of the PUPPI isolation is evaluated in data by selecting $\PZ\to\PGm\PGm$ events passing a dimuon trigger path ($\pt>17$ and $8\GeV$ for the leading and subleading muons, respectively). To obtain the boson candidates, two oppositely charged muons are selected within a 10window centered around the boson mass and passing loose isolation criteria. In addition to the two muons from the boson decay, a third muon is required and labeled as the misidentified muon. This additional muon is either a third prompt muon initiated by leptonic decays of and processes or, as is usually the case, a nonprompt muon from a semileptonic hadron decay. To further reduce the prompt-muon contribution from production, the transverse mass (as defined in Ref. [40]) obtained from the muon with third-highest and needs to be less than 40. Both and production are well measured and generally well modeled. The difference in agreement between data and simulation is thus ascribed to nonprompt-lepton events. The misidentification rate shown in Fig. 21 is defined as the number of events with a third isolated muon divided by the total number of events after subtracting the background. The misidentification rate of the $\delta\beta\text{-corrected}$ isolation is $(5.4\pm 0.4)\%$ while that of PUPPI-combined isolation is $(4.2\pm 0.4)\%$. The uncertainty is statistical only. The ratio of the misidentification rate of PUPPI isolation to the $\delta\beta\text{-corrected}$ isolation is $(77\pm 4)\%$, where the correlation is included in the uncertainty computation. The performance improvements from PUPPI-combined isolation expected from simulation studies are thus confirmed by data measurements. Figure 21: The misidentification rate defined as the number of events with a third isolated muon divided by the total number of events with a third muon in $\PZ\to\PGm\PGm$ data for PUPPI-combined (blue closed circles) and $\delta\beta\text{-corrected}$ isolation (red open circles). The lower panel shows the ratio of PUPPI-combined and $\delta\beta\text{-corrected}$ isolation, taking the correlation of their uncertainties into account. The threshold for PUPPI-combined isolation (0.15) is chosen such that the isolation efficiencies are roughly equal for muons with $15<\pt<20\GeV$, where $\delta\beta\text{-corrected}$ isolation is applied. ## 0.9 Summary The impact of pileup (PU) mitigation techniques on object reconstruction performance in the CMS experiment has been presented. The main techniques under study are charged-hadron subtraction (CHS) and pileup per particle identification (PUPPI), which both exploit particle-level information. The performance of these techniques is evaluated in the context of the reconstruction of jets and missing transverse momentum (), lepton isolation, and the calculation of jet substructure observables for boosted object tagging. The CHS and PUPPI algorithms are further compared with other algorithmic approaches that act on jet, , and lepton objects. While CHS rejects charged particles associated with PU vertices, PUPPI applies a more stringent selection to charged particles and rescales the four-momentum of neutral particles according to their probability to originate from the leading vertex. Both techniques reduce the dependence on PU interactions across all objects. A stronger reduction is achieved with PUPPI, especially for events with more than 30 interactions. The PUPPI algorithm provides the best performance for jet mass and substructure observables, resolution, and rejection of misidentified muons. With respect to jet-momentum resolution and PU jet rejection, the preferred algorithm depends on the physics process under study: the PUPPI algorithm provides a better jet momentum resolution for jets with $\pt<100\GeV$, whereas CHS does so for $\pt>100\GeV$. The highest rejection rate for jets originating purely from PU is obtained when using a dedicated PU jet identification in addition to CHS. However, when a looser working point for the PU jet identification is chosen such that its efficiency for selecting jets coming from the leading vertex is similar to that of PUPPI, both provide a similar rejection power. The PU suppression techniques studied in this paper are proven to maintain reasonable object performance up to 70 interactions. Their use will be crucial for future running of the LHC, where even more challenging PU conditions up to 200 interactions per bunch crossing are expected. ###### Acknowledgements. We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMBWF and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, FAPERGS, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); SENESCYT (Ecuador); MoER, ERC IUT, PUT and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); NKFIA (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); MES (Latvia); LAS (Lithuania); MOE and UM (Malaysia); BUAP, CINVESTAV, CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MOS (Montenegro); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS, RFBR, and NRC KI (Russia); MESTD (Serbia); SEIDI, CPAN, PCTI, and FEDER (Spain); MOSTR (Sri Lanka); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR, and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU (Ukraine); STFC (United Kingdom); DOE and NSF (USA). Individuals have received support from the Marie-Curie program and the European Research Council and Horizon 2020 Grant, contract Nos. 675440, 752730, and 765710 (European Union); the Leventis Foundation; the A.P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the F.R.S.-FNRS and FWO (Belgium) under the “Excellence of Science – EOS” – be.h project n. 30820817; the Beijing Municipal Science & Technology Commission, No. Z191100007219010; the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306; the Lendület (“Momentum”) Program and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, the New National Excellence Program ÚNKP, the NKFIA research grants 123842, 123959, 124845, 124850, 125105, 128713, 128786, and 129058 (Hungary); the Council of Science and Industrial Research, India; the HOMING PLUS program of the Foundation for Polish Science, cofinanced from European Union, Regional Development Fund, the Mobility Plus program of the Ministry of Science and Higher Education, the National Science Center (Poland), contracts Harmonia 2014/14/M/ST2/00428, Opus 2014/13/B/ST2/02543, 2014/15/B/ST2/03998, and 2015/19/B/ST2/02861, Sonata-bis 2012/07/E/ST2/01406; the National Priorities Research Program by Qatar National Research Fund; the Ministry of Science and Education, grant no. 14.W03.31.0026 (Russia); the Tomsk Polytechnic University Competitiveness Enhancement Program and “Nauka” Project FSWW-2020-0008 (Russia); the Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia María de Maeztu, grant MDM-2015-0509 and the Programa Severo Ochoa del Principado de Asturias; the Thalis and Aristeia programs cofinanced by EU-ESF and the Greek NSRF; the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University and the Chulalongkorn Academic into Its 2nd Century Project Advancement Project (Thailand); the Kavli Foundation; the Nvidia Corporation; the SuperMicro Corporation; the Welch Foundation, contract C-1845; and the Weston Havens Foundation (USA). ## References * [1] CMS Collaboration, “CMS luminosity measurements for the 2016 data taking period”, CMS Physics Analysis Summary CMS-PAS-LUM-17-001, 2017. * [2] CMS Collaboration, “Particle-flow reconstruction and global event description with the CMS detector”, JINST 12 (2017) P10003, 10.1088/1748-0221/12/10/P10003, arXiv:1706.04965. * [3] M. 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Dragicevic, J. Erö, A. Escalante Del Valle, M. Flechl, R. Frühwirth1, M. Jeitler1, N. Krammer, I. Krätschmer, D. Liko, T. Madlener, I. Mikulec, N. Rad, J. Schieck1, R. Schöfbeck, M. Spanring, W. Waltenberger, C.-E. Wulz1, M. Zarucki Institute for Nuclear Problems, Minsk, Belarus V. Drugakov, V. Mossolov, J. Suarez Gonzalez Universiteit Antwerpen, Antwerpen, Belgium M.R. Darwish, E.A. De Wolf, D. Di Croce, X. Janssen, A. Lelek, M. Pieters, H. Rejeb Sfar, H. Van Haevermaet, P. Van Mechelen, S. Van Putte, N. Van Remortel Vrije Universiteit Brussel, Brussel, Belgium F. Blekman, E.S. Bols, S.S. Chhibra, J. D’Hondt, J. De Clercq, D. Lontkovskyi, S. Lowette, I. Marchesini, S. Moortgat, Q. Python, K. Skovpen, S. Tavernier, W. Van Doninck, P. Van Mulders Université Libre de Bruxelles, Bruxelles, Belgium D. Beghin, B. Bilin, B. Clerbaux, G. De Lentdecker, H. Delannoy, B. Dorney, L. Favart, A. Grebenyuk, A.K. Kalsi, L. Moureaux, A. Popov, N. Postiau, E. Starling, L. Thomas, C. Vander Velde, P. Vanlaer, D. Vannerom Ghent University, Ghent, Belgium T. Cornelis, D. Dobur, I. Khvastunov2, M. Niedziela, C. Roskas, M. Tytgat, W. Verbeke, B. Vermassen, M. Vit Université Catholique de Louvain, Louvain-la- Neuve, Belgium O. Bondu, G. Bruno, C. Caputo, P. David, C. Delaere, M. Delcourt, A. Giammanco, V. Lemaitre, J. Prisciandaro, A. Saggio, M. Vidal Marono, P. Vischia, J. Zobec Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil F.L. Alves, G.A. Alves, G. Correia Silva, C. Hensel, A. Moraes, P. Rebello Teles Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil E. Belchior Batista Das Chagas, W. Carvalho, J. Chinellato3, E. Coelho, E.M. Da Costa, G.G. Da Silveira4, D. De Jesus Damiao, C. De Oliveira Martins, S. Fonseca De Souza, L.M. Huertas Guativa, H. Malbouisson, J. Martins5, D. Matos Figueiredo, M. Medina Jaime6, M. Melo De Almeida, C. Mora Herrera, L. Mundim, H. Nogima, W.L. Prado Da Silva, L.J. Sanchez Rosas, A. Santoro, A. Sznajder, M. Thiel, E.J. Tonelli Manganote3, F. Torres Da Silva De Araujo, A. Vilela Pereira Universidade Estadual Paulista a, Universidade Federal do ABC b, São Paulo, Brazil C.A. Bernardesa, L. Calligarisa, T.R. Fernandez Perez Tomeia, E.M. Gregoresb, D.S. Lemos, P.G. Mercadanteb, S.F. Novaesa, SandraS. Padulaa Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria A. Aleksandrov, G. Antchev, R. Hadjiiska, P. Iaydjiev, M. Misheva, M. Rodozov, M. Shopova, G. Sultanov University of Sofia, Sofia, Bulgaria M. Bonchev, A. Dimitrov, T. Ivanov, L. Litov, B. Pavlov, P. Petkov, A. Petrov Beihang University, Beijing, China W. Fang7, X. Gao7, L. Yuan Department of Physics, Tsinghua University, Beijing, China M. Ahmad, Z. Hu, Y. Wang Institute of High Energy Physics, Beijing, China G.M. Chen8, H.S. Chen8, M. Chen, C.H. Jiang, D. Leggat, H. Liao, Z. Liu, A. Spiezia, J. Tao, E. Yazgan, H. Zhang, S. Zhang8, J. Zhao State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing, China A. Agapitos, Y. Ban, G. Chen, A. Levin, J. Li, L. Li, Q. Li, Y. Mao, S.J. Qian, D. Wang, Q. Wang Zhejiang University, Hangzhou, China M. Xiao Universidad de Los Andes, Bogota, Colombia C. Avila, A. Cabrera, C. Florez, C.F. González Hernández, M.A. Segura Delgado Universidad de Antioquia, Medellin, Colombia J. Mejia Guisao, J.D. Ruiz Alvarez, C.A. Salazar González, N. Vanegas Arbelaez University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Split, Croatia D. Giljanović, N. Godinovic, D. Lelas, I. Puljak, T. Sculac University of Split, Faculty of Science, Split, Croatia Z. Antunovic, M. Kovac Institute Rudjer Boskovic, Zagreb, Croatia V. Brigljevic, D. Ferencek, K. Kadija, B. Mesic, M. Roguljic, A. Starodumov9, T. Susa University of Cyprus, Nicosia, Cyprus M.W. Ather, A. Attikis, E. Erodotou, A. Ioannou, M. Kolosova, S. Konstantinou, G. Mavromanolakis, J. Mousa, C. Nicolaou, F. Ptochos, P.A. Razis, H. Rykaczewski, D. Tsiakkouri Charles University, Prague, Czech Republic M. Finger10, M. Finger Jr.10, A. Kveton, J. Tomsa Escuela Politecnica Nacional, Quito, Ecuador E. Ayala Universidad San Francisco de Quito, Quito, Ecuador E. Carrera Jarrin Academy of Scientific Research and Technology of the Arab Republic of Egypt, Egyptian Network of High Energy Physics, Cairo, Egypt Y. Assran11,12, S. Khalil13 National Institute of Chemical Physics and Biophysics, Tallinn, Estonia S. Bhowmik, A. Carvalho Antunes De Oliveira, R.K. Dewanjee, K. Ehataht, M. Kadastik, M. Raidal, C. Veelken Department of Physics, University of Helsinki, Helsinki, Finland P. Eerola, L. Forthomme, H. Kirschenmann, K. Osterberg, M. Voutilainen Helsinki Institute of Physics, Helsinki, Finland F. Garcia, J. Havukainen, J.K. Heikkilä, V. Karimäki, M.S. Kim, R. Kinnunen, T. Lampén, K. Lassila-Perini, S. Laurila, S. Lehti, T. Lindén, P. Luukka, T. Mäenpää, H. Siikonen, E. Tuominen, J. Tuominiemi Lappeenranta University of Technology, Lappeenranta, Finland T. Tuuva IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France M. Besancon, F. Couderc, M. Dejardin, D. Denegri, B. Fabbro, J.L. Faure, F. Ferri, S. Ganjour, A. Givernaud, P. Gras, G. Hamel de Monchenault, P. Jarry, C. Leloup, B. Lenzi, E. Locci, J. Malcles, J. Rander, A. Rosowsky, M.Ö. Sahin, A. Savoy-Navarro14, M. Titov, G.B. Yu Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut Polytechnique de Paris S. Ahuja, C. Amendola, F. Beaudette, P. Busson, C. Charlot, B. Diab, G. Falmagne, R. Granier de Cassagnac, I. Kucher, A. Lobanov, C. Martin Perez, M. Nguyen, C. Ochando, P. Paganini, J. Rembser, R. Salerno, J.B. Sauvan, Y. Sirois, A. Zabi, A. Zghiche Université de Strasbourg, CNRS, IPHC UMR 7178, Strasbourg, France J.-L. Agram15, J. Andrea, D. Bloch, G. Bourgatte, J.-M. Brom, E.C. Chabert, C. Collard, E. Conte15, J.-C. Fontaine15, D. Gelé, U. Goerlach, M. Jansová, A.-C. Le Bihan, N. Tonon, P. Van Hove Centre de Calcul de l’Institut National de Physique Nucleaire et de Physique des Particules, CNRS/IN2P3, Villeurbanne, France S. Gadrat Université de Lyon, Université Claude Bernard Lyon 1, CNRS-IN2P3, Institut de Physique Nucléaire de Lyon, Villeurbanne, France S. Beauceron, C. Bernet, G. Boudoul, C. Camen, A. Carle, N. Chanon, R. Chierici, D. Contardo, P. Depasse, H. El Mamouni, J. Fay, S. Gascon, M. Gouzevitch, B. Ille, Sa. Jain, F. Lagarde, I.B. Laktineh, H. Lattaud, A. Lesauvage, M. Lethuillier, L. Mirabito, S. Perries, V. Sordini, L. Torterotot, G. Touquet, M. Vander Donckt, S. Viret Georgian Technical University, Tbilisi, Georgia T. Toriashvili16 Tbilisi State University, Tbilisi, Georgia Z. Tsamalaidze10 RWTH Aachen University, I. Physikalisches Institut, Aachen, Germany C. Autermann, L. Feld, K. Klein, M. Lipinski, D. Meuser, A. Pauls, M. Preuten, M.P. Rauch, J. Schulz, M. Teroerde, B. Wittmer RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany M. Erdmann, B. Fischer, S. Ghosh, T. Hebbeker, K. Hoepfner, H. Keller, L. Mastrolorenzo, M. Merschmeyer, A. Meyer, P. Millet, G. Mocellin, S. Mondal, S. Mukherjee, D. Noll, A. Novak, T. Pook, A. Pozdnyakov, T. Quast, M. Radziej, Y. Rath, H. Reithler, J. Roemer, A. Schmidt, S.C. Schuler, A. Sharma, S. Wiedenbeck, S. Zaleski RWTH Aachen University, III. Physikalisches Institut B, Aachen, Germany G. Flügge, W. Haj Ahmad17, O. Hlushchenko, T. Kress, T. Müller, A. Nowack, C. Pistone, O. Pooth, D. Roy, H. Sert, A. Stahl18 Deutsches Elektronen- Synchrotron, Hamburg, Germany M. Aldaya Martin, P. Asmuss, I. Babounikau, H. Bakhshiansohi, K. Beernaert, O. Behnke, A. Bermúdez Martínez, D. Bertsche, A.A. Bin Anuar, K. Borras19, V. Botta, A. Campbell, A. Cardini, P. Connor, S. Consuegra Rodríguez, C. Contreras-Campana, V. Danilov, A. De Wit, M.M. Defranchis, C. Diez Pardos, D. Domínguez Damiani, G. Eckerlin, D. Eckstein, T. Eichhorn, A. Elwood, E. Eren, E. Gallo20, A. Geiser, A. Grohsjean, M. Guthoff, M. Haranko, A. Harb, A. Jafari, N.Z. Jomhari, H. Jung, A. Kasem19, M. Kasemann, H. Kaveh, J. Keaveney, C. Kleinwort, J. Knolle, D. Krücker, W. Lange, T. Lenz, J. Lidrych, K. Lipka, W. Lohmann21, R. Mankel, I.-A. Melzer-Pellmann, A.B. Meyer, M. Meyer, M. Missiroli, J. Mnich, A. Mussgiller, V. Myronenko, D. Pérez Adán, S.K. Pflitsch, D. Pitzl, A. Raspereza, A. Saibel, M. Savitskyi, V. Scheurer, P. Schütze, C. Schwanenberger, R. Shevchenko, A. Singh, H. Tholen, O. Turkot, A. Vagnerini, M. Van De Klundert, R. Walsh, Y. Wen, K. Wichmann, C. Wissing, O. Zenaiev, R. Zlebcik University of Hamburg, Hamburg, Germany R. Aggleton, S. Bein, L. Benato, A. Benecke, V. Blobel, T. Dreyer, A. Ebrahimi, F. Feindt, A. Fröhlich, C. Garbers, E. Garutti, D. Gonzalez, P. Gunnellini, J. Haller, A. Hinzmann, A. Karavdina, G. Kasieczka, R. Klanner, R. Kogler, N. Kovalchuk, S. Kurz, V. Kutzner, J. Lange, T. Lange, A. Malara, J. Multhaup, C.E.N. Niemeyer, A. Perieanu, A. Reimers, O. Rieger, C. Scharf, P. Schleper, S. Schumann, J. Schwandt, J. Sonneveld, H. Stadie, G. Steinbrück, F.M. Stober, B. Vormwald, I. Zoi Karlsruher Institut fuer Technologie, Karlsruhe, Germany M. Akbiyik, C. Barth, M. Baselga, S. Baur, T. Berger, E. Butz, R. Caspart, T. Chwalek, W. De Boer, A. Dierlamm, K. El Morabit, N. Faltermann, M. Giffels, P. Goldenzweig, A. Gottmann, M.A. Harrendorf, F. Hartmann18, U. Husemann, S. Kudella, S. Mitra, M.U. Mozer, D. Müller, Th. Müller, M. Musich, A. Nürnberg, G. Quast, K. Rabbertz, M. Schröder, I. Shvetsov, H.J. Simonis, R. Ulrich, M. Wassmer, M. Weber, C. Wöhrmann, R. Wolf, S. Wozniewski Institute of Nuclear and Particle Physics (INPP), NCSR Demokritos, Aghia Paraskevi, Greece G. Anagnostou, P. Asenov, G. Daskalakis, T. Geralis, A. Kyriakis, D. Loukas, G. Paspalaki National and Kapodistrian University of Athens, Athens, Greece M. Diamantopoulou, G. Karathanasis, P. Kontaxakis, A. Manousakis-katsikakis, A. Panagiotou, I. Papavergou, N. Saoulidou, A. Stakia, K. Theofilatos, E. Tziaferi, K. Vellidis, E. Vourliotis National Technical University of Athens, Athens, Greece G. Bakas, K. Kousouris, I. Papakrivopoulos, G. Tsipolitis University of Ioánnina, Ioánnina, Greece I. Evangelou, C. Foudas, P. Gianneios, P. Katsoulis, P. Kokkas, S. Mallios, K. Manitara, N. Manthos, I. Papadopoulos, J. Strologas, F.A. Triantis, D. Tsitsonis MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös Loránd University, Budapest, Hungary M. Bartók22, R. Chudasama, M. Csanad, P. Major, K. Mandal, A. Mehta, M.I. Nagy, G. Pasztor, O. Surányi, G.I. Veres Wigner Research Centre for Physics, Budapest, Hungary G. Bencze, C. Hajdu, D. Horvath23, F. Sikler, T.Á. Vámi, V. Veszpremi, G. Vesztergombi${}^{\textrm{\textdagger}}$ Institute of Nuclear Research ATOMKI, Debrecen, Hungary N. Beni, S. Czellar, J. Karancsi22, J. Molnar, Z. Szillasi Institute of Physics, University of Debrecen, Debrecen, Hungary P. Raics, D. Teyssier, Z.L. Trocsanyi, B. Ujvari Eszterhazy Karoly University, Karoly Robert Campus, Gyongyos, Hungary T. Csorgo, W.J. Metzger, F. Nemes, T. Novak Indian Institute of Science (IISc), Bangalore, India S. Choudhury, J.R. Komaragiri, P.C. Tiwari National Institute of Science Education and Research, HBNI, Bhubaneswar, India S. Bahinipati25, C. Kar, G. Kole, P. Mal, V.K. Muraleedharan Nair Bindhu, A. Nayak26, D.K. Sahoo25, S.K. Swain Panjab University, Chandigarh, India S. Bansal, S.B. Beri, V. Bhatnagar, S. Chauhan, N. Dhingra27, R. Gupta, A. Kaur, M. Kaur, S. Kaur, P. Kumari, M. Lohan, M. Meena, K. Sandeep, S. Sharma, J.B. Singh, A.K. Virdi, G. Walia University of Delhi, Delhi, India A. Bhardwaj, B.C. Choudhary, R.B. Garg, M. Gola, S. Keshri, Ashok Kumar, M. Naimuddin, P. Priyanka, K. Ranjan, Aashaq Shah, R. Sharma Saha Institute of Nuclear Physics, HBNI, Kolkata, India R. Bhardwaj28, M. Bharti28, R. Bhattacharya, S. Bhattacharya, U. Bhawandeep28, D. Bhowmik, S. Dutta, S. Ghosh, B. Gomber29, M. Maity30, K. Mondal, S. Nandan, A. Purohit, P.K. Rout, G. Saha, S. Sarkar, T. Sarkar30, M. Sharan, B. Singh28, S. Thakur28 Indian Institute of Technology Madras, Madras, India P.K. Behera, P. Kalbhor, A. Muhammad, P.R. Pujahari, A. Sharma, A.K. Sikdar Bhabha Atomic Research Centre, Mumbai, India D. Dutta, V. Jha, V. Kumar, D.K. Mishra, P.K. Netrakanti, L.M. Pant, P. Shukla Tata Institute of Fundamental Research-A, Mumbai, India T. Aziz, M.A. Bhat, S. Dugad, G.B. Mohanty, N. Sur, RavindraKumar Verma Tata Institute of Fundamental Research-B, Mumbai, India S. Banerjee, S. Bhattacharya, S. Chatterjee, P. Das, M. Guchait, S. Karmakar, S. Kumar, G. Majumder, K. Mazumdar, N. Sahoo, S. Sawant Indian Institute of Science Education and Research (IISER), Pune, India S. Dube, B. Kansal, A. Kapoor, K. Kothekar, S. Pandey, A. Rane, A. Rastogi, S. Sharma Institute for Research in Fundamental Sciences (IPM), Tehran, Iran S. Chenarani31, E. Eskandari Tadavani, S.M. Etesami31, M. Khakzad, M. Mohammadi Najafabadi, M. Naseri, F. Rezaei Hosseinabadi University College Dublin, Dublin, Ireland M. Felcini, M. Grunewald INFN Sezione di Bari a, Università di Bari b, Politecnico di Bari c, Bari, Italy M. Abbresciaa,b, R. Alya,b,32, C. Calabriaa,b, A. Colaleoa, D. Creanzaa,c, L. Cristellaa,b, N. De Filippisa,c, M. De Palmaa,b, A. Di Florioa,b, W. Elmetenaweea,b, L. Fiorea, A. Gelmia,b, G. Iasellia,c, M. Incea,b, S. Lezkia,b, G. Maggia,c, M. Maggia, J.A. Merlin, G. Minielloa,b, S. Mya,b, S. Nuzzoa,b, A. Pompilia,b, G. Pugliesea,c, R. Radognaa, A. Ranieria, G. Selvaggia,b, L. Silvestrisa, F.M. Simonea,b, R. Vendittia, P. Verwilligena INFN Sezione di Bologna a, Università di Bologna b, Bologna, Italy G. Abbiendia, C. Battilanaa,b, D. Bonacorsia,b, L. Borgonovia,b, S. Braibant- Giacomellia,b, R. Campaninia,b, P. Capiluppia,b, A. Castroa,b, F.R. Cavalloa, C. Cioccaa, G. Codispotia,b, M. Cuffiania,b, G.M. Dallavallea, F. Fabbria, A. Fanfania,b, E. Fontanesia,b, P. Giacomellia, C. Grandia, L. Guiduccia,b, F. Iemmia,b, S. Lo Meoa,33, S. Marcellinia, G. Masettia, F.L. Navarriaa,b, A. Perrottaa, F. Primaveraa,b, A.M. Rossia,b, T. Rovellia,b, G.P. Sirolia,b, N. Tosia INFN Sezione di Catania a, Università di Catania b, Catania, Italy S. Albergoa,b,34, S. Costaa,b, A. Di Mattiaa, R. Potenzaa,b, A. Tricomia,b,34, C. Tuvea,b INFN Sezione di Firenze a, Università di Firenze b, Firenze, Italy G. Barbaglia, A. Cassesea, R. Ceccarellia,b, V. Ciullia,b, C. Civininia, R. D’Alessandroa,b, F. Fioria,c, E. Focardia,b, G. Latinoa,b, P. Lenzia,b, M. Meschinia, S. Paolettia, G. Sguazzonia, L. Viliania INFN Laboratori Nazionali di Frascati, Frascati, Italy L. Benussi, S. Bianco, D. Piccolo INFN Sezione di Genova a, Università di Genova b, Genova, Italy M. Bozzoa,b, F. Ferroa, R. Mulargiaa,b, E. Robuttia, S. Tosia,b INFN Sezione di Milano-Bicocca a, Università di Milano-Bicocca b, Milano, Italy A. Benagliaa, A. Beschia,b, F. Brivioa,b, V. Cirioloa,b,18, M.E. Dinardoa,b, P. Dinia, S. Gennaia, A. Ghezzia,b, P. Govonia,b, L. Guzzia,b, M. Malbertia, S. Malvezzia, D. Menascea, F. Montia,b, L. Moronia, M. Paganonia,b, D. Pedrinia, S. Ragazzia,b, T. Tabarelli de Fatisa,b, D. Valsecchia,b, D. Zuoloa,b INFN Sezione di Napoli a, Università di Napoli ’Federico II’ b, Napoli, Italy, Università della Basilicata c, Potenza, Italy, Università G. Marconi d, Roma, Italy S. Buontempoa, N. Cavalloa,c, A. De Iorioa,b, A. Di Crescenzoa,b, F. Fabozzia,c, F. Fiengaa, G. Galatia, A.O.M. Iorioa,b, L. Listaa,b, S. Meolaa,d,18, P. Paoluccia,18, B. Rossia, C. Sciaccaa,b, E. Voevodinaa,b INFN Sezione di Padova a, Università di Padova b, Padova, Italy, Università di Trento c, Trento, Italy P. Azzia, N. Bacchettaa, D. Biselloa,b, A. Bolettia,b, A. Bragagnoloa,b, R. Carlina,b, P. Checchiaa, P. De Castro Manzanoa, T. Dorigoa, U. Dossellia, F. Gasparinia,b, U. Gasparinia,b, A. Gozzelinoa, S.Y. Hoha,b, P. Lujana, M. Margonia,b, A.T. Meneguzzoa,b, J. Pazzinia,b, M. Presillab, P. Ronchesea,b, R. Rossina,b, F. Simonettoa,b, A. Tikoa, M. Tosia,b, M. Zanettia,b, P. Zottoa,b, G. Zumerlea,b INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy A. Braghieria, D. Fiorinaa,b, P. Montagnaa,b, S.P. Rattia,b, V. Rea, M. Ressegottia,b, C. Riccardia,b, P. Salvinia, I. Vaia, P. Vituloa,b INFN Sezione di Perugia a, Università di Perugia b, Perugia, Italy M. Biasinia,b, G.M. Bileia, D. Ciangottinia,b, L. Fanòa,b, P. Laricciaa,b, R. Leonardia,b, E. Manonia, G. Mantovania,b, V. Mariania,b, M. Menichellia, A. Rossia,b, A. Santocchiaa,b, D. Spigaa INFN Sezione di Pisa a, Università di Pisa b, Scuola Normale Superiore di Pisa c, Pisa, Italy K. Androsova, P. Azzurria, G. Bagliesia, V. Bertacchia,c, L. Bianchinia, T. Boccalia, R. Castaldia, M.A. Cioccia,b, R. Dell’Orsoa, S. Donatoa, G. Fedia, L. Gianninia,c, A. Giassia, M.T. Grippoa, F. Ligabuea,c, E. Mancaa,c, G. Mandorlia,c, A. Messineoa,b, F. Pallaa, A. Rizzia,b, G. Rolandi35, S. Roy Chowdhury, A. Scribanoa, P. Spagnoloa, R. Tenchinia, G. Tonellia,b, N. Turinia, A. Venturia, P.G. Verdinia INFN Sezione di Roma a, Sapienza Università di Roma b, Rome, Italy F. Cavallaria, M. Cipriania,b, D. Del Rea,b, E. Di Marcoa, M. Diemoza, E. Longoa,b, P. Meridiania, G. Organtinia,b, F. Pandolfia, R. Paramattia,b, C. Quarantaa,b, S. Rahatloua,b, C. Rovellia, F. Santanastasioa,b, L. Soffia,b INFN Sezione di Torino a, Università di Torino b, Torino, Italy, Università del Piemonte Orientale c, Novara, Italy N. Amapanea,b, R. Arcidiaconoa,c, S. Argiroa,b, M. Arneodoa,c, N. Bartosika, R. Bellana,b, A. Bellora, C. Biinoa, A. Cappatia,b, N. Cartigliaa, S. Comettia, M. Costaa,b, R. Covarellia,b, N. Demariaa, B. Kiania,b, F. Legger, C. Mariottia, S. Masellia, E. Migliorea,b, V. Monacoa,b, E. Monteila,b, M. Montenoa, M.M. Obertinoa,b, G. Ortonaa,b, L. Pachera,b, N. Pastronea, M. Pelliccionia, G.L. Pinna Angionia,b, A. Romeroa,b, M. Ruspaa,c, R. Salvaticoa,b, V. Solaa, A. Solanoa,b, D. Soldia,b, A. Staianoa, D. Trocinoa,b INFN Sezione di Trieste a, Università di Trieste b, Trieste, Italy S. Belfortea, V. Candelisea,b, M. Casarsaa, F. Cossuttia, A. Da Rolda,b, G. Della Riccaa,b, F. Vazzolera,b, A. Zanettia Kyungpook National University, Daegu, Korea B. Kim, D.H. Kim, G.N. Kim, J. Lee, S.W. Lee, C.S. Moon, Y.D. Oh, S.I. Pak, S. Sekmen, D.C. Son, Y.C. Yang Chonnam National University, Institute for Universe and Elementary Particles, Kwangju, Korea H. Kim, D.H. Moon, G. Oh Hanyang University, Seoul, Korea B. Francois, T.J. Kim, J. Park Korea University, Seoul, Korea S. Cho, S. Choi, Y. Go, S. Ha, B. Hong, K. Lee, K.S. Lee, J. Lim, J. Park, S.K. Park, Y. Roh, J. Yoo Kyung Hee University, Department of Physics J. Goh Sejong University, Seoul, Korea H.S. Kim Seoul National University, Seoul, Korea J. Almond, J.H. Bhyun, J. Choi, S. Jeon, J. Kim, J.S. Kim, H. Lee, K. Lee, S. Lee, K. Nam, M. Oh, S.B. Oh, B.C. Radburn-Smith, U.K. Yang, H.D. Yoo, I. Yoon University of Seoul, Seoul, Korea D. Jeon, J.H. Kim, J.S.H. Lee, I.C. Park, I.J Watson Sungkyunkwan University, Suwon, Korea Y. Choi, C. Hwang, Y. Jeong, J. Lee, Y. Lee, I. Yu Riga Technical University, Riga, Latvia V. Veckalns36 Vilnius University, Vilnius, Lithuania V. Dudenas, A. Juodagalvis, A. Rinkevicius, G. Tamulaitis, J. Vaitkus National Centre for Particle Physics, Universiti Malaya, Kuala Lumpur, Malaysia Z.A. Ibrahim, F. Mohamad Idris37, W.A.T. Wan Abdullah, M.N. Yusli, Z. Zolkapli Universidad de Sonora (UNISON), Hermosillo, Mexico J.F. Benitez, A. Castaneda Hernandez, J.A. Murillo Quijada, L. Valencia Palomo Centro de Investigacion y de Estudios Avanzados del IPN, Mexico City, Mexico H. Castilla-Valdez, E. De La Cruz-Burelo, I. Heredia-De La Cruz38, R. Lopez- Fernandez, A. Sanchez-Hernandez Universidad Iberoamericana, Mexico City, Mexico S. Carrillo Moreno, C. Oropeza Barrera, M. Ramirez-Garcia, F. Vazquez Valencia Benemerita Universidad Autonoma de Puebla, Puebla, Mexico J. Eysermans, I. Pedraza, H.A. Salazar Ibarguen, C. Uribe Estrada Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico A. Morelos Pineda University of Montenegro, Podgorica, Montenegro J. Mijuskovic2, N. Raicevic University of Auckland, Auckland, New Zealand D. Krofcheck University of Canterbury, Christchurch, New Zealand S. Bheesette, P.H. Butler National Centre for Physics, Quaid-I-Azam University, Islamabad, Pakistan A. Ahmad, M. Ahmad, Q. Hassan, H.R. Hoorani, W.A. Khan, M.A. Shah, M. Shoaib, M. Waqas AGH University of Science and Technology Faculty of Computer Science, Electronics and Telecommunications, Krakow, Poland V. Avati, L. Grzanka, M. Malawski National Centre for Nuclear Research, Swierk, Poland H. Bialkowska, M. Bluj, B. Boimska, M. Górski, M. Kazana, M. Szleper, P. Zalewski Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland K. Bunkowski, A. Byszuk39, K. Doroba, A. Kalinowski, M. Konecki, J. Krolikowski, M. Olszewski, M. Walczak Laboratório de Instrumentação e Física Experimental de Partículas, Lisboa, Portugal M. Araujo, P. Bargassa, D. Bastos, A. Di Francesco, P. Faccioli, B. Galinhas, M. Gallinaro, J. Hollar, N. Leonardo, T. Niknejad, J. Seixas, K. Shchelina, G. Strong, O. Toldaiev, J. Varela Joint Institute for Nuclear Research, Dubna, Russia S. Afanasiev, P. Bunin, M. Gavrilenko, I. Golutvin, I. Gorbunov, A. Kamenev, V. Karjavine, A. Lanev, A. Malakhov, V. Matveev40,41, P. Moisenz, V. Palichik, V. Perelygin, M. Savina, S. Shmatov, S. Shulha, N. Skatchkov, V. Smirnov, N. Voytishin, A. Zarubin Petersburg Nuclear Physics Institute, Gatchina (St. Petersburg), Russia L. Chtchipounov, V. Golovtcov, Y. Ivanov, V. Kim42, E. Kuznetsova43, P. Levchenko, V. Murzin, V. Oreshkin, I. Smirnov, D. Sosnov, V. Sulimov, L. Uvarov, A. Vorobyev Institute for Nuclear Research, Moscow, Russia Yu. Andreev, A. Dermenev, S. Gninenko, N. Golubev, A. Karneyeu, M. Kirsanov, N. Krasnikov, A. Pashenkov, D. Tlisov, A. Toropin Institute for Theoretical and Experimental Physics named by A.I. Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia V. Epshteyn, V. Gavrilov, N. Lychkovskaya, A. Nikitenko44, V. Popov, I. Pozdnyakov, G. Safronov, A. Spiridonov, A. Stepennov, M. Toms, E. Vlasov, A. Zhokin Moscow Institute of Physics and Technology, Moscow, Russia T. Aushev National Research Nuclear University ’Moscow Engineering Physics Institute’ (MEPhI), Moscow, Russia M. Chadeeva45, P. Parygin, D. Philippov, E. Popova, V. Rusinov P.N. Lebedev Physical Institute, Moscow, Russia V. Andreev, M. Azarkin, I. Dremin, M. Kirakosyan, A. Terkulov Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia A. Belyaev, E. Boos, M. Dubinin46, L. Dudko, A. Ershov, A. Gribushin, V. Klyukhin, O. Kodolova, I. Lokhtin, S. Obraztsov, S. Petrushanko, V. Savrin, A. Snigirev Novosibirsk State University (NSU), Novosibirsk, Russia A. Barnyakov47, V. Blinov47, T. Dimova47, L. Kardapoltsev47, Y. Skovpen47 Institute for High Energy Physics of National Research Centre ‘Kurchatov Institute’, Protvino, Russia I. Azhgirey, I. Bayshev, S. Bitioukov, V. Kachanov, D. Konstantinov, P. Mandrik, V. Petrov, R. Ryutin, S. Slabospitskii, A. Sobol, S. Troshin, N. Tyurin, A. Uzunian, A. Volkov National Research Tomsk Polytechnic University, Tomsk, Russia A. Babaev, A. Iuzhakov, V. Okhotnikov Tomsk State University, Tomsk, Russia V. Borchsh, V. Ivanchenko, E. Tcherniaev University of Belgrade: Faculty of Physics and VINCA Institute of Nuclear Sciences P. Adzic48, P. Cirkovic, M. Dordevic, P. Milenovic, J. Milosevic, M. Stojanovic Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain M. Aguilar-Benitez, J. Alcaraz Maestre, A. Álvarez Fernández, I. Bachiller, M. Barrio Luna, CristinaF. Bedoya, J.A. Brochero Cifuentes, C.A. Carrillo Montoya, M. Cepeda, M. Cerrada, N. Colino, B. De La Cruz, A. Delgado Peris, J.P. Fernández Ramos, J. Flix, M.C. Fouz, O. Gonzalez Lopez, S. Goy Lopez, J.M. Hernandez, M.I. Josa, D. Moran, Á. Navarro Tobar, A. Pérez-Calero Yzquierdo, J. Puerta Pelayo, I. Redondo, L. Romero, S. Sánchez Navas, M.S. Soares, A. Triossi, C. Willmott Universidad Autónoma de Madrid, Madrid, Spain C. Albajar, J.F. de Trocóniz, R. Reyes-Almanza Universidad de Oviedo, Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias (ICTEA), Oviedo, Spain B. Alvarez Gonzalez, J. Cuevas, C. Erice, J. Fernandez Menendez, S. Folgueras, I. Gonzalez Caballero, J.R. González Fernández, E. Palencia Cortezon, V. Rodríguez Bouza, S. Sanchez Cruz Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, Santander, Spain I.J. Cabrillo, A. Calderon, B. Chazin Quero, J. Duarte Campderros, M. Fernandez, P.J. Fernández Manteca, A. García Alonso, G. Gomez, C. Martinez Rivero, P. Martinez Ruiz del Arbol, F. Matorras, J. Piedra Gomez, C. Prieels, T. Rodrigo, A. Ruiz-Jimeno, L. Russo49, L. Scodellaro, I. Vila, J.M. Vizan Garcia University of Colombo, Colombo, Sri Lanka D.U.J. Sonnadara University of Ruhuna, Department of Physics, Matara, Sri Lanka W.G.D. Dharmaratna, N. Wickramage CERN, European Organization for Nuclear Research, Geneva, Switzerland D. Abbaneo, B. Akgun, E. Auffray, G. Auzinger, J. Baechler, P. Baillon, A.H. Ball, D. Barney, J. Bendavid, M. Bianco, A. Bocci, P. Bortignon, E. Bossini, E. Brondolin, T. Camporesi, A. Caratelli, G. Cerminara, E. Chapon, G. Cucciati, D. d’Enterria, A. Dabrowski, N. Daci, V. Daponte, A. David, O. Davignon, A. De Roeck, M. Deile, M. Dobson, M. Dünser, N. Dupont, A. Elliott- Peisert, N. Emriskova, F. Fallavollita50, D. Fasanella, S. Fiorendi, G. Franzoni, J. Fulcher, W. Funk, S. Giani, D. Gigi, K. Gill, F. Glege, L. Gouskos, M. Gruchala, M. Guilbaud, D. Gulhan, J. Hegeman, C. Heidegger, Y. Iiyama, V. Innocente, T. James, P. Janot, O. Karacheban21, J. Kaspar, J. Kieseler, M. Krammer1, N. Kratochwil, C. Lange, P. Lecoq, C. Lourenço, L. Malgeri, M. Mannelli, A. Massironi, F. Meijers, S. Mersi, E. Meschi, F. Moortgat, M. Mulders, J. Ngadiuba, J. Niedziela, S. Nourbakhsh, S. Orfanelli, L. Orsini, F. Pantaleo18, L. Pape, E. Perez, M. Peruzzi, A. Petrilli, G. Petrucciani, A. Pfeiffer, M. Pierini, F.M. Pitters, D. Rabady, A. Racz, M. Rieger, M. Rovere, H. Sakulin, J. Salfeld-Nebgen, C. Schäfer, C. Schwick, M. Selvaggi, A. Sharma, P. Silva, W. Snoeys, P. Sphicas51, J. Steggemann, S. Summers, V.R. Tavolaro, D. Treille, A. Tsirou, G.P. Van Onsem, A. Vartak, M. Verzetti, W.D. Zeuner Paul Scherrer Institut, Villigen, Switzerland L. Caminada52, K. Deiters, W. Erdmann, R. Horisberger, Q. Ingram, H.C. Kaestli, D. Kotlinski, U. Langenegger, T. Rohe, S.A. Wiederkehr ETH Zurich - Institute for Particle Physics and Astrophysics (IPA), Zurich, Switzerland M. Backhaus, P. Berger, N. Chernyavskaya, G. Dissertori, M. Dittmar, M. Donegà, C. Dorfer, T.A. Gómez Espinosa, C. Grab, D. Hits, W. Lustermann, R.A. Manzoni, M.T. Meinhard, F. Micheli, P. Musella, F. Nessi-Tedaldi, F. Pauss, G. Perrin, L. Perrozzi, S. Pigazzini, M.G. Ratti, M. Reichmann, C. Reissel, T. Reitenspiess, B. Ristic, D. Ruini, D.A. Sanz Becerra, M. Schönenberger, L. Shchutska, M.L. Vesterbacka Olsson, R. Wallny, D.H. Zhu Universität Zürich, Zurich, Switzerland T.K. Aarrestad, C. Amsler53, C. Botta, D. Brzhechko, M.F. Canelli, A. De Cosa, R. Del Burgo, B. Kilminster, S. Leontsinis, V.M. Mikuni, I. Neutelings, G. Rauco, P. Robmann, K. Schweiger, C. Seitz, Y. Takahashi, S. Wertz, A. Zucchetta National Central University, Chung-Li, Taiwan T.H. Doan, C.M. Kuo, W. Lin, A. Roy, S.S. Yu National Taiwan University (NTU), Taipei, Taiwan P. Chang, Y. Chao, K.F. Chen, P.H. Chen, W.-S. Hou, Y.y. Li, R.-S. Lu, E. Paganis, A. Psallidas, A. Steen Chulalongkorn University, Faculty of Science, Department of Physics, Bangkok, Thailand B. Asavapibhop, C. Asawatangtrakuldee, N. Srimanobhas, N. Suwonjandee Çukurova University, Physics Department, Science and Art Faculty, Adana, Turkey A. Bat, F. Boran, A. Celik54, S. Damarseckin55, Z.S. Demiroglu, F. Dolek, C. Dozen56, I. Dumanoglu, G. Gokbulut, EmineGurpinar Guler57, Y. Guler, I. Hos58, C. Isik, E.E. Kangal59, O. Kara, A. Kayis Topaksu, U. Kiminsu, G. Onengut, K. Ozdemir60, S. Ozturk61, A.E. Simsek, U.G. Tok, S. Turkcapar, I.S. Zorbakir, C. Zorbilmez Middle East Technical University, Physics Department, Ankara, Turkey B. Isildak62, G. Karapinar63, M. Yalvac Bogazici University, Istanbul, Turkey I.O. Atakisi, E. Gülmez, M. Kaya64, O. Kaya65, Ö. Özçelik, S. Tekten, E.A. Yetkin66 Istanbul Technical University, Istanbul, Turkey A. Cakir, K. Cankocak67, Y. Komurcu, S. Sen68 Istanbul University, Istanbul, Turkey S. Cerci69, B. Kaynak, S. Ozkorucuklu, D. Sunar Cerci69 Institute for Scintillation Materials of National Academy of Science of Ukraine, Kharkov, Ukraine B. Grynyov National Scientific Center, Kharkov Institute of Physics and Technology, Kharkov, Ukraine L. Levchuk University of Bristol, Bristol, United Kingdom E. Bhal, S. Bologna, J.J. Brooke, D. Burns70, E. Clement, D. Cussans, H. Flacher, J. Goldstein, G.P. Heath, H.F. Heath, L. Kreczko, B. Krikler, S. Paramesvaran, B. Penning, T. Sakuma, S. Seif El Nasr-Storey, V.J. Smith, J. Taylor, A. Titterton Rutherford Appleton Laboratory, Didcot, United Kingdom K.W. Bell, A. Belyaev71, C. Brew, R.M. Brown, D.J.A. Cockerill, J.A. Coughlan, K. Harder, S. Harper, J. Linacre, K. Manolopoulos, D.M. Newbold, E. Olaiya, D. Petyt, T. Reis, T. Schuh, C.H. Shepherd-Themistocleous, A. Thea, I.R. Tomalin, T. Williams Imperial College, London, United Kingdom R. Bainbridge, P. Bloch, J. Borg, S. Breeze, O. Buchmuller, A. Bundock, GurpreetSingh CHAHAL72, D. Colling, P. Dauncey, G. Davies, M. Della Negra, R. Di Maria, P. Everaerts, G. Hall, G. Iles, M. Komm, L. Lyons, A.-M. Magnan, S. Malik, A. Martelli, V. Milosevic, A. Morton, J. Nash73, V. Palladino, M. Pesaresi, D.M. Raymond, A. Richards, A. Rose, E. Scott, C. Seez, A. Shtipliyski, M. Stoye, T. Strebler, A. Tapper, K. Uchida, T. Virdee18, N. Wardle, D. Winterbottom, A.G. Zecchinelli, S.C. Zenz Brunel University, Uxbridge, United Kingdom J.E. Cole, P.R. Hobson, A. Khan, P. Kyberd, C.K. Mackay, I.D. Reid, L. Teodorescu, S. Zahid Baylor University, Waco, USA A. Brinkerhoff, K. Call, B. Caraway, J. Dittmann, K. Hatakeyama, C. Madrid, B. McMaster, N. Pastika, C. Smith Catholic University of America, Washington, DC, USA R. Bartek, A. Dominguez, R. Uniyal, A.M. Vargas Hernandez The University of Alabama, Tuscaloosa, USA A. Buccilli, S.I. Cooper, C. Henderson, P. Rumerio, C. West Boston University, Boston, USA A. Albert, D. Arcaro, Z. Demiragli, D. Gastler, C. Richardson, J. Rohlf, D. Sperka, D. Spitzbart, I. Suarez, L. Sulak, D. Zou Brown University, Providence, USA G. Benelli, B. Burkle, X. Coubez19, D. Cutts, Y.t. Duh, M. Hadley, U. Heintz, J.M. Hogan74, K.H.M. Kwok, E. Laird, G. Landsberg, K.T. Lau, J. Lee, M. Narain, S. Sagir75, R. Syarif, E. Usai, W.Y. Wong, D. Yu, W. Zhang University of California, Davis, Davis, USA R. Band, C. Brainerd, R. Breedon, M. Calderon De La Barca Sanchez, M. Chertok, J. Conway, R. Conway, P.T. Cox, R. Erbacher, C. Flores, G. Funk, F. Jensen, W. Ko${}^{\textrm{\textdagger}}$, O. Kukral, R. Lander, M. Mulhearn, D. Pellett, J. Pilot, M. Shi, D. Taylor, K. Tos, M. Tripathi, Z. Wang, F. Zhang University of California, Los Angeles, USA M. Bachtis, C. Bravo, R. Cousins, A. Dasgupta, A. Florent, J. Hauser, M. Ignatenko, N. Mccoll, W.A. Nash, S. Regnard, D. Saltzberg, C. Schnaible, B. Stone, V. Valuev University of California, Riverside, Riverside, USA K. Burt, Y. Chen, R. Clare, J.W. Gary, S.M.A. Ghiasi Shirazi, G. Hanson, G. Karapostoli, O.R. Long, M. Olmedo Negrete, M.I. Paneva, W. Si, L. Wang, S. Wimpenny, B.R. Yates, Y. Zhang University of California, San Diego, La Jolla, USA J.G. Branson, P. Chang, S. Cittolin, S. Cooperstein, N. Deelen, M. Derdzinski, R. Gerosa, D. Gilbert, B. Hashemi, D. Klein, V. Krutelyov, J. Letts, M. Masciovecchio, S. May, S. Padhi, M. Pieri, V. Sharma, M. Tadel, F. Würthwein, A. Yagil, G. Zevi Della Porta University of California, Santa Barbara - Department of Physics, Santa Barbara, USA N. Amin, R. Bhandari, C. Campagnari, M. Citron, V. Dutta, M. Franco Sevilla, J. Incandela, B. Marsh, H. Mei, A. Ovcharova, H. Qu, J. Richman, U. Sarica, D. Stuart, S. Wang California Institute of Technology, Pasadena, USA D. Anderson, A. Bornheim, O. Cerri, I. Dutta, J.M. Lawhorn, N. Lu, J. Mao, H.B. Newman, T.Q. Nguyen, J. Pata, M. Spiropulu, J.R. Vlimant, S. Xie, Z. Zhang, R.Y. Zhu Carnegie Mellon University, Pittsburgh, USA M.B. Andrews, T. Ferguson, T. Mudholkar, M. Paulini, M. Sun, I. Vorobiev, M. Weinberg University of Colorado Boulder, Boulder, USA J.P. Cumalat, W.T. Ford, E. MacDonald, T. Mulholland, R. Patel, A. Perloff, K. Stenson, K.A. Ulmer, S.R. Wagner Cornell University, Ithaca, USA J. Alexander, Y. Cheng, J. Chu, A. Datta, A. Frankenthal, K. Mcdermott, J.R. Patterson, D. Quach, A. Ryd, S.M. Tan, Z. Tao, J. Thom, P. Wittich, M. Zientek Fermi National Accelerator Laboratory, Batavia, USA S. Abdullin, M. Albrow, M. Alyari, G. Apollinari, A. Apresyan, A. Apyan, S. Banerjee, L.A.T. Bauerdick, A. Beretvas, D. Berry, J. Berryhill, P.C. Bhat, K. Burkett, J.N. Butler, A. Canepa, G.B. Cerati, H.W.K. Cheung, F. Chlebana, M. Cremonesi, J. Duarte, V.D. Elvira, J. Freeman, Z. Gecse, E. Gottschalk, L. Gray, D. Green, S. Grünendahl, O. Gutsche, J. Hanlon, R.M. Harris, S. Hasegawa, R. Heller, J. Hirschauer, B. Jayatilaka, S. Jindariani, M. Johnson, U. Joshi, T. Klijnsma, B. Klima, M.J. Kortelainen, B. Kreis, S. Lammel, J. Lewis, D. Lincoln, R. Lipton, M. Liu, T. Liu, J. Lykken, K. Maeshima, J.M. Marraffino, D. Mason, P. McBride, P. Merkel, S. Mrenna, S. Nahn, V. O’Dell, V. Papadimitriou, K. Pedro, C. Pena, G. Rakness, F. Ravera, A. Reinsvold Hall, L. Ristori, B. Schneider, E. Sexton-Kennedy, N. Smith, A. Soha, W.J. Spalding, L. Spiegel, S. Stoynev, J. Strait, N. Strobbe, L. Taylor, S. Tkaczyk, N.V. Tran, L. Uplegger, E.W. Vaandering, C. Vernieri, R. Vidal, M. Wang, H.A. Weber University of Florida, Gainesville, USA D. Acosta, P. Avery, D. Bourilkov, L. Cadamuro, V. Cherepanov, F. Errico, R.D. Field, S.V. Gleyzer, D. Guerrero, B.M. Joshi, M. Kim, J. Konigsberg, A. Korytov, K.H. Lo, K. Matchev, N. Menendez, G. Mitselmakher, D. Rosenzweig, K. Shi, J. Wang, S. Wang, X. Zuo Florida International University, Miami, USA Y.R. Joshi Florida State University, Tallahassee, USA T. Adams, A. Askew, S. Hagopian, V. Hagopian, K.F. Johnson, R. Khurana, T. Kolberg, G. Martinez, T. Perry, H. Prosper, C. Schiber, R. Yohay, J. Zhang Florida Institute of Technology, Melbourne, USA M.M. Baarmand, M. Hohlmann, D. Noonan, M. Rahmani, M. Saunders, F. Yumiceva University of Illinois at Chicago (UIC), Chicago, USA M.R. Adams, L. Apanasevich, R.R. Betts, R. Cavanaugh, X. Chen, S. Dittmer, O. Evdokimov, C.E. Gerber, D.A. Hangal, D.J. Hofman, C. Mills, T. Roy, M.B. Tonjes, N. Varelas, J. Viinikainen, H. Wang, X. Wang, Z. Wu The University of Iowa, Iowa City, USA M. Alhusseini, B. Bilki57, K. Dilsiz76, S. Durgut, R.P. Gandrajula, M. Haytmyradov, V. Khristenko, O.K. Köseyan, J.-P. Merlo, A. Mestvirishvili77, A. Moeller, J. Nachtman, H. Ogul78, Y. Onel, F. Ozok79, A. Penzo, C. Snyder, E. Tiras, J. Wetzel Johns Hopkins University, Baltimore, USA B. Blumenfeld, A. Cocoros, N. Eminizer, A.V. Gritsan, W.T. Hung, S. Kyriacou, P. Maksimovic, J. Roskes, M. Swartz The University of Kansas, Lawrence, USA C. Baldenegro Barrera, P. Baringer, A. Bean, S. Boren, J. Bowen, A. Bylinkin, T. Isidori, S. Khalil, J. King, G. Krintiras, A. Kropivnitskaya, C. Lindsey, D. Majumder, W. Mcbrayer, N. Minafra, M. Murray, C. Rogan, C. Royon, S. Sanders, E. Schmitz, J.D. Tapia Takaki, Q. Wang, J. Williams, G. Wilson Kansas State University, Manhattan, USA S. Duric, A. Ivanov, K. Kaadze, D. Kim, Y. Maravin, D.R. Mendis, T. Mitchell, A. Modak, A. Mohammadi Lawrence Livermore National Laboratory, Livermore, USA F. Rebassoo, D. Wright University of Maryland, College Park, USA A. Baden, O. Baron, A. Belloni, S.C. Eno, Y. Feng, N.J. Hadley, S. Jabeen, G.Y. Jeng, R.G. Kellogg, A.C. Mignerey, S. Nabili, F. Ricci-Tam, M. Seidel, Y.H. Shin, A. Skuja, S.C. Tonwar, K. Wong Massachusetts Institute of Technology, Cambridge, USA D. Abercrombie, B. Allen, R. Bi, S. Brandt, W. Busza, I.A. Cali, M. D’Alfonso, G. Gomez Ceballos, M. Goncharov, P. Harris, D. Hsu, M. Hu, M. Klute, D. Kovalskyi, Y.-J. Lee, P.D. Luckey, B. Maier, A.C. Marini, C. Mcginn, C. Mironov, S. Narayanan, X. Niu, C. Paus, D. Rankin, C. Roland, G. Roland, Z. Shi, G.S.F. Stephans, K. Sumorok, K. Tatar, D. Velicanu, J. Wang, T.W. Wang, B. Wyslouch University of Minnesota, Minneapolis, USA R.M. Chatterjee, A. Evans, S. Guts${}^{\textrm{\textdagger}}$, P. Hansen, J. Hiltbrand, Sh. Jain, Y. Kubota, Z. Lesko, J. Mans, M. Revering, R. Rusack, R. Saradhy, N. Schroeder, M.A. Wadud University of Mississippi, Oxford, USA J.G. Acosta, S. Oliveros University of Nebraska-Lincoln, Lincoln, USA K. Bloom, S. Chauhan, D.R. Claes, C. Fangmeier, L. Finco, F. Golf, R. Kamalieddin, I. Kravchenko, J.E. Siado, G.R. Snow${}^{\textrm{\textdagger}}$, B. Stieger, W. Tabb State University of New York at Buffalo, Buffalo, USA G. Agarwal, C. Harrington, I. Iashvili, A. Kharchilava, C. McLean, D. Nguyen, A. Parker, J. Pekkanen, S. Rappoccio, B. Roozbahani Northeastern University, Boston, USA G. Alverson, E. Barberis, C. Freer, Y. Haddad, A. Hortiangtham, G. Madigan, B. Marzocchi, D.M. Morse, T. Orimoto, L. Skinnari, A. Tishelman-Charny, T. Wamorkar, B. Wang, A. Wisecarver, D. Wood Northwestern University, Evanston, USA S. Bhattacharya, J. Bueghly, A. Gilbert, T. Gunter, K.A. Hahn, N. Odell, M.H. Schmitt, K. Sung, M. Trovato, M. Velasco University of Notre Dame, Notre Dame, USA R. Bucci, N. Dev, R. Goldouzian, M. Hildreth, K. Hurtado Anampa, C. Jessop, D.J. Karmgard, K. Lannon, W. Li, N. Loukas, N. Marinelli, I. Mcalister, F. Meng, Y. Musienko40, R. Ruchti, P. Siddireddy, G. Smith, S. Taroni, M. Wayne, A. Wightman, M. Wolf, A. Woodard The Ohio State University, Columbus, USA J. Alimena, B. Bylsma, L.S. Durkin, B. Francis, C. Hill, W. Ji, A. Lefeld, T.Y. Ling, B.L. Winer Princeton University, Princeton, USA G. Dezoort, P. Elmer, J. Hardenbrook, N. Haubrich, S. Higginbotham, A. Kalogeropoulos, S. Kwan, D. Lange, M.T. Lucchini, J. Luo, D. Marlow, K. Mei, I. Ojalvo, J. Olsen, C. Palmer, P. Piroué, D. Stickland, C. Tully University of Puerto Rico, Mayaguez, USA S. Malik, S. Norberg Purdue University, West Lafayette, USA A. Barker, V.E. Barnes, R. Chawla, S. Das, L. Gutay, M. Jones, A.W. Jung, A. Khatiwada, B. Mahakud, D.H. Miller, G. Negro, N. Neumeister, C.C. Peng, S. Piperov, H. Qiu, J.F. Schulte, N. Trevisani, F. Wang, R. Xiao, W. Xie Purdue University Northwest, Hammond, USA T. Cheng, J. Dolen, N. Parashar Rice University, Houston, USA A. Baty, U. Behrens, S. Dildick, K.M. Ecklund, S. Freed, F.J.M. Geurts, M. Kilpatrick, Arun Kumar, W. Li, B.P. Padley, R. Redjimi, J. Roberts, J. Rorie, W. Shi, A.G. Stahl Leiton, Z. Tu, A. Zhang University of Rochester, Rochester, USA A. Bodek, P. de Barbaro, R. Demina, J.L. Dulemba, C. Fallon, T. Ferbel, M. Galanti, A. Garcia-Bellido, O. Hindrichs, A. Khukhunaishvili, E. Ranken, R. Taus Rutgers, The State University of New Jersey, Piscataway, USA B. Chiarito, J.P. Chou, A. Gandrakota, Y. Gershtein, E. Halkiadakis, A. Hart, M. Heindl, E. Hughes, S. Kaplan, I. Laflotte, A. Lath, R. Montalvo, K. Nash, M. Osherson, H. Saka, S. Salur, S. Schnetzer, S. Somalwar, R. Stone, S. Thomas University of Tennessee, Knoxville, USA H. Acharya, A.G. Delannoy, S. Spanier Texas A&M University, College Station, USA O. Bouhali80, M. Dalchenko, M. De Mattia, A. Delgado, R. Eusebi, J. Gilmore, T. Huang, T. Kamon81, H. Kim, S. Luo, S. Malhotra, D. Marley, R. Mueller, D. Overton, L. Perniè, D. Rathjens, A. Safonov Texas Tech University, Lubbock, USA N. Akchurin, J. Damgov, F. De Guio, V. Hegde, S. Kunori, K. Lamichhane, S.W. Lee, T. Mengke, S. Muthumuni, T. Peltola, S. Undleeb, I. Volobouev, Z. Wang, A. Whitbeck Vanderbilt University, Nashville, USA S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K. Padeken, F. Romeo, P. Sheldon, S. Tuo, J. Velkovska, M. Verweij University of Virginia, Charlottesville, USA M.W. Arenton, P. Barria, B. Cox, G. Cummings, J. Hakala, R. Hirosky, M. Joyce, A. Ledovskoy, C. Neu, B. Tannenwald, Y. Wang, E. Wolfe, F. Xia Wayne State University, Detroit, USA R. Harr, P.E. Karchin, N. Poudyal, J. Sturdy, P. Thapa University of Wisconsin - Madison, Madison, WI, USA T. Bose, J. Buchanan, C. Caillol, D. Carlsmith, S. Dasu, I. De Bruyn, L. Dodd, C. Galloni, H. He, M. Herndon, A. Hervé, U. Hussain, A. Lanaro, A. Loeliger, K. Long, R. Loveless, J. Madhusudanan Sreekala, A. Mallampalli, D. Pinna, T. Ruggles, A. Savin, V. Sharma, W.H. Smith, D. Teague, S. Trembath-reichert †: Deceased 1: Also at Vienna University of Technology, Vienna, Austria 2: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France 3: Also at Universidade Estadual de Campinas, Campinas, Brazil 4: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil 5: Also at UFMS, Nova Andradina, Brazil 6: Also at Universidade Federal de Pelotas, Pelotas, Brazil 7: Also at Université Libre de Bruxelles, Bruxelles, Belgium 8: Also at University of Chinese Academy of Sciences, Beijing, China 9: Also at Institute for Theoretical and Experimental Physics named by A.I. Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia 10: Also at Joint Institute for Nuclear Research, Dubna, Russia 11: Also at Suez University, Suez, Egypt 12: Now at British University in Egypt, Cairo, Egypt 13: Also at Zewail City of Science and Technology, Zewail, Egypt 14: Also at Purdue University, West Lafayette, USA 15: Also at Université de Haute Alsace, Mulhouse, France 16: Also at Tbilisi State University, Tbilisi, Georgia 17: Also at Erzincan Binali Yildirim University, Erzincan, Turkey 18: Also at CERN, European Organization for Nuclear Research, Geneva, Switzerland 19: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany 20: Also at University of Hamburg, Hamburg, Germany 21: Also at Brandenburg University of Technology, Cottbus, Germany 22: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary, Debrecen, Hungary 23: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary 24: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös Loránd University, Budapest, Hungary, Budapest, Hungary 25: Also at IIT Bhubaneswar, Bhubaneswar, India, Bhubaneswar, India 26: Also at Institute of Physics, Bhubaneswar, India 27: Also at G.H.G. Khalsa College, Punjab, India 28: Also at Shoolini University, Solan, India 29: Also at University of Hyderabad, Hyderabad, India 30: Also at University of Visva-Bharati, Santiniketan, India 31: Also at Isfahan University of Technology, Isfahan, Iran 32: Now at INFN Sezione di Bari a, Università di Bari b, Politecnico di Bari c, Bari, Italy 33: Also at Italian National Agency for New Technologies, Energy and Sustainable Economic Development, Bologna, Italy 34: Also at Centro Siciliano di Fisica Nucleare e di Struttura Della Materia, Catania, Italy 35: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy 36: Also at Riga Technical University, Riga, Latvia, Riga, Latvia 37: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia 38: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico 39: Also at Warsaw University of Technology, Institute of Electronic Systems, Warsaw, Poland 40: Also at Institute for Nuclear Research, Moscow, Russia 41: Now at National Research Nuclear University ’Moscow Engineering Physics Institute’ (MEPhI), Moscow, Russia 42: Also at St. Petersburg State Polytechnical University, St. Petersburg, Russia 43: Also at University of Florida, Gainesville, USA 44: Also at Imperial College, London, United Kingdom 45: Also at P.N. Lebedev Physical Institute, Moscow, Russia 46: Also at California Institute of Technology, Pasadena, USA 47: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia 48: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia 49: Also at Università degli Studi di Siena, Siena, Italy 50: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy, Pavia, Italy 51: Also at National and Kapodistrian University of Athens, Athens, Greece 52: Also at Universität Zürich, Zurich, Switzerland 53: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria, Vienna, Austria 54: Also at Burdur Mehmet Akif Ersoy University, BURDUR, Turkey 55: Also at Şırnak University, Sirnak, Turkey 56: Also at Department of Physics, Tsinghua University, Beijing, China, Beijing, China 57: Also at Beykent University, Istanbul, Turkey, Istanbul, Turkey 58: Also at Istanbul Aydin University, Application and Research Center for Advanced Studies (App. & Res. Cent. for Advanced Studies), Istanbul, Turkey 59: Also at Mersin University, Mersin, Turkey 60: Also at Piri Reis University, Istanbul, Turkey 61: Also at Gaziosmanpasa University, Tokat, Turkey 62: Also at Ozyegin University, Istanbul, Turkey 63: Also at Izmir Institute of Technology, Izmir, Turkey 64: Also at Marmara University, Istanbul, Turkey 65: Also at Kafkas University, Kars, Turkey 66: Also at Istanbul Bilgi University, Istanbul, Turkey 67: Also at Near East University, Research Center of Experimental Health Science, Nicosia, Turkey 68: Also at Hacettepe University, Ankara, Turkey 69: Also at Adiyaman University, Adiyaman, Turkey 70: Also at Vrije Universiteit Brussel, Brussel, Belgium 71: Also at School of Physics and Astronomy, University of Southampton, Southampton, United Kingdom 72: Also at IPPP Durham University, Durham, United Kingdom 73: Also at Monash University, Faculty of Science, Clayton, Australia 74: Also at Bethel University, St. Paul, Minneapolis, USA, St. Paul, USA 75: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey 76: Also at Bingol University, Bingol, Turkey 77: Also at Georgian Technical University, Tbilisi, Georgia 78: Also at Sinop University, Sinop, Turkey 79: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey 80: Also at Texas A&M University at Qatar, Doha, Qatar 81: Also at Kyungpook National University, Daegu, Korea, Daegu, Korea

2024-09-04T02:54:56.031280

2003.00516

arxiv-papers

# Neutrinos in curved space-time: particle mixing and flavor oscillations A. Capolupo<EMAIL_ADDRESS>Dipartimento di Fisica “E.R. Caianiello” Università di Salerno, and INFN – Gruppo Collegato di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy G. Lambiase<EMAIL_ADDRESS>Dipartimento di Fisica “E.R. Caianiello” Università di Salerno, and INFN – Gruppo Collegato di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy A. Quaranta<EMAIL_ADDRESS>Dipartimento di Fisica “E.R. Caianiello” Università di Salerno, and INFN – Gruppo Collegato di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy ###### Abstract We present a quantum field theoretical approach to the vacuum neutrino oscillations in curved space, we analyze the non–trivial interplay between quantum field mixing and field quantization in curved space and derive new oscillation formulae. We compute the formulae explicitly in the spatially flat FLRW metrics for universes dominated by a cosmological constant and by radiation. We evaluate the transition probabilities in the Schwarzschild black hole metric, and we show that the Hawking radiation affects the oscillations of neutrinos. We show that our results are consistent with those of previous analyses when the quantum mechanical limit is considered. ## I Introduction Since they were theoretically proposed by Pauli Pauli , neutrinos have proven to be among the most enigmatic particles in the universe. Until the discovery of flavor oscillations NeutrinoOscillations1 ; NeutrinoOscillations2 , whose theory was pioneered by Pontecorvo Pontecorvo ; Bilenky , neutrinos were believed to be massless. Today it is accepted that neutrinos are massive particles, and that they oscillate among three flavors $\nu_{e},\nu_{\mu},\nu_{\tau}$ corresponding to the companion charged leptons $e,\mu,\tau$. This peculiarity renders neutrinos unique among the known elementary particles and puts them beyond the scope of the standard model of particles Mohapatra . In many respects, neutrinos are forerunners of a new physics, as several issues, including the origin of their mass NeutrinoMass and their fundamental nature NeutrinoNature , are still open to the present day. On the other hand, the relevance of neutrinos in astrophyisical and cosmological contexts has grown dramatically during the last years. They figure as a valuable source of information, along with gravitational waves and electromagnetic radiation, in the ever–growing field of multi–messenger astronomy Franckowiak . The study of neutrinos of astrophysical origin can indeed provide fundamental insights on the source that produced them. In addition, neutrinos are expected to play an important role in the first phases of the universe Buchmuller ; CosmologicalNeutrinos , and the detection of the cosmic neutrino background, pursued in experiments as PTOLEMY PTOLEMY , could represent an essential test for the standard cosmological model CosmologicalNeutrinos . Mass varying neutrinos have also been proposed as a possible explanation for Dark Energy Mavans . This state of affairs requires a careful investigation of neutrino oscillations on a curved spacetime. The topic has been discussed in several works , where it was found that gravitational fields may alter both the oscillations in vacuum and in matter Grossman ; Piriz ; Cardall . Here we wish to go beyond the heuristic treatment of ref. Cardall , and present a quantum field theoretical approach, based on the field quantization in curved space-time, to evaluate the effects of gravitational fields on neutrino oscillations. We derive general oscillation formulae for flavor fields in curved space-time, which represent our main result. We discuss the particle interpretation of the fields in presence of gravity and study how the mixing changes when moving from a mass field representation to another. We demonstrate the invariance of local observables, which are represented by expectation values on flavor states of local operators constructed from the flavor fields. We show that the oscillation probabilities, on the other hand, do in general depend on the representation of the mass fields, since they are not a local observable and involve the comparison between particles in different spacetime regions. We establish the conditions which have to be satisfied in order that the resulting transition probabilities are invariant under changes of mass field representation. We also compute explicitly the oscillation formulae for two examples of spatially flat Friedmann–Lemaitre–Robertson–Walker spacetimes, corresponding to a cosmological constant–dominated and a radiation–dominated universe respectively. In these cases, exact analytical solutions to the Dirac equation are available, and the formalism here introduced can be applied directly. Moreover, we give an estimation of the oscillation formulae for neutrinos propagating from the past infinity to the future infinity in a stationary Schwarzschild spacetime. We introduce a method to extract the oscillation formulae on spacetimes with asymptotically flat regions without resorting to the exact solutions of the Dirac equation. We then employ this strategy to compute the formulae on the Schwarzschild black hole spacetime, for neutrinos propagating from the past infinity to the future infinity. We show how the Hawking radiation is naturally embedded in the resulting transition probabilities. Our results generalize those of the previous treatments Cardall , and are consistent with the latter when the suitable limits are considered. In our computation, for simplicity, we limit our analysis to the vacuum oscillations, therefore considering the sole effect of gravity. The paper is organized as follows: in section II we provide the setting for the description of the mass fields in curved space; in section III we develop field mixing and find the oscillation probabilities in curved spacetime, with a thorough analysis of their features; in section IV we apply the formalism to some spacetimes of interest, including the spatially flat FLRW metric for a radiation-dominated universe and for a cosmological constant-dominated universe, and the Schwarzschild black hole metric, where we show the impact of the Hawking effect on neutrino oscillations; finally in section V we draw our conclusions. ## II Mass neutrino fields in curved Space To evaluate the oscillation formulae for neutrinos on a curved spacetime, it is necessary to consider both the effects of curvature and mixing on the (free) mass fields. Let $M$ be a globally hyperbolic spacetime, and let $\tau\in\mathbb{R}$ label a foliation of $M$ by Cauchy surfaces. Consider the tetrad fields $e^{\mu}_{a}(x)$ satisfying $\eta^{ab}e^{\mu}_{a}(x)e^{\nu}_{b}(x)=g^{\mu\nu}(x)$. Here $\eta^{ab}\equiv diag(1,-1,-1,-1)$ is the Minkowski metric tensor, while $g^{\mu\nu}(x)$ is the contravariant metric $g^{\mu\nu}(x)g_{\nu\rho}(x)=\delta^{\mu}_{\rho}$ on $M$ in a given coordinate system. The massive neutrino fields satisfy the Dirac equations: $(i\gamma^{\mu}(x)D_{\mu}-m_{i})\psi_{i}=0$ (1) where $\gamma^{\mu}(x)=e^{\mu}_{a}(x)\gamma^{a}$, $\gamma^{a}$ being the usual flat space Dirac matrices, and $D_{\mu}=\partial_{\mu}-\frac{i}{4}\omega_{\mu}^{ab}\sigma_{ab}$. The spin connection is defined as $\omega_{\mu}^{ab}=e^{a}_{\nu}\Gamma^{\nu}_{\rho\mu}e^{\rho b}+e^{a}_{\nu}\partial_{\mu}e^{\nu b}$, whereas $\sigma_{ab}$ are the commutators of flat Dirac matrices $\sigma_{ab}=\frac{i}{2}[\gamma^{a},\gamma^{b}]$. In equation (1), the index $i=1,2,...,N$ ranges over the number of neutrino species $N$. For the sake of simplicity we focus on the case $N=2$, though the generalization to $N=3$ is straightforward. In general equation (1) cannot be solved exactly. Even if one is able to find exact solutions, these do not play the same prominent role as their flat spacetime counterpart. It is well–known, indeed, that the positive frequency solutions of equation cannot be defined univocally, and that, consequently, there is no natural (nor unique) particle interpretation for the corresponding Quantum Field Theory Birrell ; Wald . Nevertheless, the canonical quantization of the Dirac field proceeds along the same lines as in Minkowski spacetime. To perform a field expansion, one must find a set of positive $\zeta_{k,i}$ and negative $\xi_{k,i}$ frequency solutions for each of the equations (1). In general the bipartition of the solutions to eq. (1) makes sense only locally, while there is no natural global definition of positive and negative frequency modes. Anyway, one is free to choose a set of modes Note1 $\\{\zeta_{k,i},\xi_{k,i}\\}$ , deemed to be positive/negative frequency modes according to some specified observer, and expand the field with respect to them, provided that they form a complete (and orthonormal) set of solutions under the inner product $(a_{i},b_{i})=\int_{\Sigma(\tau)}\sqrt{-g}d\Sigma_{\mu}(\tau)\bar{a}_{i}\gamma^{\mu}(x)b_{i}\ $ (2) with $a_{i},b_{i}$ any solution to equation (1) with mass $m_{i}$ and $\bar{b}_{i}=b^{\dagger}_{i}\gamma^{0}(x)$. Here $d\Sigma^{\mu}(\tau)=n^{\mu}(\tau)dV_{\tau}$ denotes the volume element on the surface $\tau$ with unit timelike normal $n^{\mu}(\tau)$. This has to hold separately for each $i=1,2$. As it is easy to prove, for $a_{i},b_{i}$ solutions of the (same) Dirac equation, the inner product (2) does not depend on the hypersurface chosen for the integration. In particular, it is independent on the foliation by Cauchy hypersurfaces employed. The fields can then be expanded as $\psi_{i}(x)=\sum_{k,s}\left(\gamma_{k,s;i}\zeta_{k,s;i}(x)+\epsilon_{k,s;i}^{\dagger}\xi_{k,s;i}(x)\right)$ (3) with the operator coefficients $\gamma_{k,s,i},\epsilon_{k,s,i}$ satisfying the usual canonical anticommutation relations, $k$ momentum index and $s$ helicity index. The annihilators are also required to anticommute for $i\neq j$, and, in particular $\\{\gamma_{k,s,i},\gamma^{\dagger}_{k,s,j}\\}=\delta_{ij}$, $\\{\epsilon_{k,s,i},\epsilon^{\dagger}_{k,s,j}\\}=\delta_{ij}$. In equation (3) we prefer to keep any space-time dependence within the modes, for ease of treatment with a general metric. The expansions (3) define the mass Hilbert space $\mathcal{H}_{m}=H_{1}\otimes H_{2}$, which is constructed out of the vacuum $|0_{m}\rangle=|0_{1}\rangle\otimes|0_{2}\rangle$. Here $\ket{0_{i}}$ is defined, as usual, by $\gamma_{k,s,i}\ket{0_{i}}=0=\epsilon_{k,s,i}\ket{0_{i}}$ for each $k,s,i$. As hinted above, the field expansions (3) are somewhat arbitrary, as opposed to the flat spacetime case, where there is no ambiguity in the definition of positive and negative frequency modes. Any other basis $\\{\tilde{\zeta}_{k,s,i},\tilde{\xi}_{k,s,i}\\}$ can be used to expand the fields $\psi_{i}=\sum_{k,s}(\tilde{\gamma}_{k,s,i}\tilde{\zeta}_{k,s,i}(x)+\tilde{\epsilon}_{k,s,i}^{\dagger}\tilde{\xi}_{k,s,i}(x))$. Since both the sets $\\{\zeta_{k,s,i},\xi_{k,s,i}\\}$ and $\\{\tilde{\zeta}_{k,s,i},\tilde{\xi}_{k,s,i}\\}$ form a basis for the space of solutions of eqs. (1), one can write the modes of a set in terms of the other, for each $i$: $\displaystyle\tilde{\zeta}_{k^{\prime},s^{\prime},i}$ $\displaystyle=$ $\displaystyle\sum_{k,s}\left(\Gamma_{k^{\prime},s^{\prime};k,s;i}^{*}\zeta_{k,s,i}+\Sigma_{k^{\prime},s^{\prime};k,s;i}^{*}\xi_{k,s,i}\right)$ $\displaystyle\tilde{\xi}_{k^{\prime},s^{\prime},i}$ $\displaystyle=$ $\displaystyle\sum_{k,s}\left(\Gamma_{k^{\prime},s^{\prime};k,s;i}\xi_{k,s,i}-\Sigma_{k^{\prime},s^{\prime};k,s;i}\zeta_{k,s,i}\right)$ (4) where $\Gamma_{k^{\prime},s^{\prime};ks;i}=(\tilde{\zeta}_{k^{\prime},s^{\prime},i},\zeta_{k,s,i})=(\xi_{k,s,i},\tilde{\xi}_{k^{\prime},s^{\prime},i})$ and $\Sigma_{k^{\prime},s^{\prime};ks;i}=(\tilde{\zeta}_{k^{\prime},s^{\prime},i},\xi_{k,s,i})=-(\zeta_{k,s,i},\tilde{\xi}_{k^{\prime}s^{\prime},i})$. This is a fermionic Bogoliubov transformation, for which $\sum_{q,r}\left(\Gamma_{k,s;q,r;i}^{*}\Gamma_{k^{\prime},s^{\prime};q,r;i}+\Sigma_{k,s;q,r;i}^{*}\Sigma_{k^{\prime},s^{\prime};q,r;i}\right)=\delta_{k,k^{\prime}}\delta_{s,s^{\prime}}$ for each $i$. The corresponding relation between the two sets of annihilators is given by $\displaystyle\tilde{\gamma}_{k,s,i}$ $\displaystyle=$ $\displaystyle\sum_{k^{\prime},s^{\prime}}\left(\Gamma_{k,s;k^{\prime}s^{\prime};i}\gamma_{k^{\prime},s^{\prime},i}+\Sigma_{k,s;k^{\prime},s^{\prime};i}\epsilon^{\dagger}_{k^{\prime},s^{\prime},i}\right)$ $\displaystyle\tilde{\epsilon}_{k,s,i}$ $\displaystyle=$ $\displaystyle\sum_{k^{\prime},s^{\prime}}\left(\Gamma_{k,s;k^{\prime},s^{\prime};i}\epsilon_{k^{\prime},s^{\prime},i}-\Sigma_{k,s;k^{\prime},s^{\prime};i}\gamma^{\dagger}_{k^{\prime},s^{\prime},i}\right)\ .$ (5) It is often the case that the Bogoliubov coefficients $\Gamma_{k,s;k^{\prime},s^{\prime};i},\Sigma_{k,s;k^{\prime},s^{\prime};i}$ can be written as $\Gamma_{k,s;k^{\prime},s^{\prime};i}=\delta_{k,k^{\prime}}\delta_{s,s^{\prime}}\Gamma_{k,i}$, $\Sigma_{k,s;k^{\prime},s^{\prime};i}=\delta_{k,k^{\prime}}\delta_{s,s^{\prime}}\Sigma_{k,i}$, with $\Gamma_{k,i}$ and $\Sigma_{k,i}$ depending on $k$ alone. In this occurrence, they admit the parametrization $\Gamma_{k,i}=e^{i\eta_{k,i}}\cos(\theta_{k,i})$ ,$\Sigma_{k,i}=e^{i\phi_{k,i}}\sin(\theta_{k,i})$, with $\eta_{k,i},\phi_{k,i},\theta_{k,i}$ real functions of $k$. We remark that the Bogoliubov transformations (II) can be recast in terms of the generators $J_{i}=e^{\sum_{k,k^{\prime},s,s^{\prime}}\left[(\lambda_{k,k^{\prime},s,s^{\prime},i}^{*}\gamma_{k,s,i}^{\dagger}\epsilon_{k^{\prime}s^{\prime},i}^{\dagger}-\lambda_{k,k^{\prime},s,s^{\prime},i}\epsilon_{k,s,i}\gamma_{k^{\prime},s^{\prime},i})\right]}$, with $\lambda_{k,k^{\prime},s,s^{\prime},i}=Arctan(\frac{\Sigma_{k,s;k^{\prime},s^{\prime};i}}{\Gamma_{k,s;k^{\prime},s^{\prime};i}})$, as $\tilde{\gamma}_{k,s,i}=J_{i}^{-1}\gamma_{k,s,i}J_{i}\;,\qquad\qquad\ \ \tilde{\epsilon}_{k,s,i}=J_{i}^{-1}\epsilon_{k,s,i}J_{i}\ .$ (6) The maps $J_{i}:\tilde{\mathcal{H}}_{i}\rightarrow\mathcal{H}_{i}$ interpolate between the Fock spaces $\mathcal{H}_{i}$ built from the $\gamma_{k,s,i},\epsilon_{k,s,i}$ and the Fock space $\tilde{\mathcal{H}}_{i}$ built from the $\tilde{\gamma}_{k,s,i},\tilde{\epsilon}_{k,s,i}$. In particular, one has for the vacuum states $|\tilde{0}_{i}\rangle=J_{i}^{-1}|0_{i}\rangle$. As for the untilded representation, the mass Hilbert space in the tilded representation is the tensor product $\tilde{\mathcal{H}}_{m}=\tilde{\mathcal{H}}_{1}\otimes\tilde{\mathcal{H}}_{2}$. It is convenient to define a unique generator of Bogoliubov transformations $J:\tilde{\mathcal{H}}_{m}\longrightarrow\mathcal{H}_{m}$ on $\tilde{\mathcal{H}}_{m}$ as the tensor product $J=J_{1}\otimes J_{2}$. Then $\tilde{\gamma}_{k,s,i}=J^{-1}\gamma_{k,s,i}J\;,\qquad\qquad\ \ \tilde{\epsilon}_{k,s,i}=J^{-1}\epsilon_{k,s,i}J$ (7) for $i=1,2$. The expansions of the two fields $\psi_{1}$ and $\psi_{2}$ must be compatible with each other, i.e., each of the modes $\zeta_{k,s,2},\xi_{k,s,2}$ must be obtainable from the corresponding modes $\zeta_{k,s,1},\xi_{k,s,1}$ by the substitution $m_{1}\leftrightarrow m_{2}$, and vice–versa. In the context of mixing, this ensures that the same kind of particle, described by the same set of quantum numbers, is being mixed. This, of course, does not undermine the arbitrariness in the choice of the modes; these can be any complete set of solutions to the Dirac equation, provided that the same choice is made for the two fields. ## III Neutrino mixing and oscillation formulae in curved space-time In this Section, we show new oscillation formulae for flavor fields in curved space-time and we present general considerations on the infinitely many unitarily inequivalent representations of the canonical anticommutation relations which characterize the quantization of mixed fields and of fields in curved space. ### III.1 Oscillation Formulae As discussed above, the QFT of free Dirac fields in curved space is characterized by infinitely many unitarily inequivalent representations of the canonical anticommutation relations. The phenomenon of mixing, even in Minkowski space, suffers from an analogous ambiguity, in that the flavor and the mass representations are unitarily inequivalent Capolupo1 . The effects of such inequivalence have been analyzed in flat space time Capolupo:2006et and the possibility to reveal them in experimental setup has been recently proposed Capolupo:2019gmn . Let us start by fixing the mass field expansions (3) and describe the mixing in a given representation of the mass fields. The flavor fields are defined as $\psi_{e}=\cos(\theta)\psi_{1}+\sin(\theta)\psi_{2}$ and $\psi_{\mu}=\cos(\theta)\psi_{2}-\sin(\theta)\psi_{1}$ with $\theta$ the (2-flavor) mixing angle. Just like the Bogoliubov transformations (7), the rotation to flavor fields can be cast in terms of a generator $\mathcal{I}_{\theta}(\tau)$. This is given by $\mathcal{I}_{\theta}(\tau)=e^{\theta[(\psi_{1},\psi_{2})_{\tau}-(\psi_{2},\psi_{1})_{\tau}]}\ .$ (8) where the scalar products $(\psi_{i},\psi_{j})$ _do_ depend on the hypersurface chosen for the integration, since they are solutions to different Dirac equations. Then, by definition, the flavor fields are expressed as $\psi_{e}=\mathcal{I}_{\theta}^{-1}(\tau)\psi_{1}\mathcal{I}_{\theta}(\tau)\;,\qquad\qquad\ \ \psi_{\mu}=\mathcal{I}_{\theta}^{-1}(\tau)\psi_{2}\mathcal{I}_{\theta}(\tau).$ (9) If we let the generator (8) act on the mass annihilators, we obtain the flavor annihilators for curved space $\gamma_{k,s,e}(\tau)=\mathcal{I}_{\theta}^{-1}(\tau)\gamma_{k,s,1}\mathcal{I}_{\theta}(\tau)=\cos(\theta)\gamma_{k,s,1}+\sin(\theta)\sum_{q,r}\bigg{[}\Lambda^{*}_{q,r;k,s}(\tau)\gamma_{q,r,2}+\Xi_{q,r;k,s}(\tau)\epsilon_{q,r,2}^{\dagger}\bigg{]}\,.$ (10) And similar for $\gamma_{k,s,\mu}(\tau),\epsilon_{k,s,e}(\tau),\epsilon_{k,s,\mu}(\tau)$. The Bogoliubov coefficients are provided by the inner products of the solutions to the _curved space_ Dirac equation with mass $m_{1}$ and $m_{2}$, that is, $\Lambda_{q,r;k,s}(\tau)=(\zeta_{q,r,2},\zeta_{k,s,1})_{\tau}=(\xi_{k,s,1},\xi_{q,r,2})_{\tau}$ and $\Xi_{q,r;k,s}(\tau)=(\zeta_{k,s,1},\xi_{q,r,2})_{\tau}=-(\zeta_{q,r,2},\xi_{k,s,1})_{\tau}$. The mixing coefficients always satisfy $\sum_{q,r}\left(\Lambda_{k,s;q,r}^{*}(\tau)\Lambda_{k^{\prime},s^{\prime};q,r}(\tau)+\Xi_{k,s;q,r}^{*}(\tau)\Xi_{k^{\prime},s^{\prime};q,r}(\tau)\right)\\!=\\!\delta_{k,k^{\prime}}\delta_{s,s^{\prime}}\ .$ (11) Since the mass expansions are compatible, the mixing coefficients are often diagonal, namely of the form $\displaystyle\Lambda_{q,r;k,s}(\tau)$ $\displaystyle=$ $\displaystyle\delta_{q,k}\delta_{r,s}\Lambda_{k,s}(\tau)$ $\displaystyle\Xi_{q,r;k,s}(\tau)$ $\displaystyle=$ $\displaystyle\delta_{q,k}\delta_{r,s}\Xi_{k,s}(\tau)$ (12) with $\Lambda_{k,s}(\tau)$, $\Xi_{k,s}(\tau)$ depending on $k$ and $s$ alone Note3 . Exceptions to this arise when we consider expansions of the mass fields in terms of modes labelled by the energy. In such a case, the mixing coefficients are non–diagonal and different from zero $\Lambda_{\omega,\omega^{\prime}}\neq 0$ ,$\Xi_{\omega,\omega^{\prime}}\neq 0$, once $\omega$ is fixed, only for a specific value of $\omega^{\prime}$. $|\Lambda_{k,s}(\tau)|^{2}+|\Xi_{k,s}(\tau)|^{2}=1$ (13) for each $k,s,\tau$, and $|\Lambda_{\omega;\omega^{\prime}}(\tau)|^{2}+|\Xi_{\omega;\omega^{\prime}}(\tau)|^{2}=1$ (14) respectively. The mass and the flavor representations are unitarily inequivalent. For each $\tau$ one has a distinct flavor Fock space $\mathcal{H}_{f}(\tau)$ defined by $\gamma_{e,\mu}(\tau),\epsilon_{e,\mu}(\tau)$. The flavor vacuum $\ket{0_{f}(\tau)}=\mathcal{I}_{\theta}^{-1}(\tau)\ket{0_{m}}$ is a condensate of $\psi_{1},\psi_{2}$ particle-antiparticle pairs. In order to define the transition probabilities, we observe that the total Lagrangian is invariant under $U(1)$ gauge transformations. Therefore the total charge $Q=Q_{1}+Q_{2}=Q_{e}+Q_{\mu}$ is conserved Note4 , where $Q_{i}=\sum_{k,s}\left(\gamma_{k,s,i}^{\dagger}\gamma_{k,s,i}-\epsilon_{k,s,i}^{\dagger}\epsilon_{k,s,i}\right)$ for $i=1,2,e,\mu$. It is then meaningful to define the transition probabilities as $P^{\rho\rightarrow\sigma}_{k,s}(\tau)=\sum_{q,r}\bigg{(}\langle\nu_{\rho,k,s}(\tau_{0})|Q^{q,r}_{\sigma}(\tau)|\nu_{\rho,k,s}(\tau_{0})\rangle-\langle 0_{f}(\tau_{0})|Q^{q,r}_{\sigma}(\tau)|0_{f}(\tau_{0})\rangle\bigg{)}.$ (15) Here $\rho,\sigma=e,\mu$, the state $|\nu_{\rho,k,s}(\tau_{0})\rangle=\gamma^{\dagger}_{k,s,\rho}(\tau_{0})\ket{0_{f}(\tau_{0})}$ is the state with a single neutrino of flavor $\rho$, momentum $k$ and helicity $s$ on the reference hypersurface $\tau=\tau_{0}$ . The second term on the rhs of (15) is just the implementation of the normal ordering with respect to $\ket{0_{f}(\tau_{0})}$. By construction $P^{e\rightarrow e}_{k,s}(\tau)+P^{e\rightarrow\mu}_{k,s}(\tau)=1$ and $P^{\mu\rightarrow e}_{k,s}(\tau)+P^{\mu\rightarrow\mu}_{k,s}(\tau)=1$ for each $\tau$. A straightforward calculation yields, in the general case (accounting for both diagonal and non–diagonal mixing coefficients) the result $P^{e\rightarrow\mu}_{k,s}(\tau)=2\cos^{2}(\theta)\sin^{2}(\theta)\times\bigg{[}1-\sum_{q,r}\Re\bigg{(}\Lambda_{k,s;q,r}^{*}(\tau_{0})\Lambda_{k,s;q,r}(\tau)+\Xi_{k,s;q,r}^{*}(\tau_{0})\Xi_{k,s;q,r}(\tau)\bigg{)}\bigg{]}\ .$ (16) Equation (16) is the central result of the paper. When equations (III.1) hold, this reduces to $P^{e\rightarrow\mu}_{k,s}(\tau)=2\cos^{2}(\theta)\sin^{2}(\theta)\times\bigg{[}1-\Re\bigg{(}\Lambda_{k,s}^{*}(\tau_{0})\Lambda_{k,s}(\tau)+\Xi_{k,s}^{*}(\tau_{0})\Xi_{k,s}(\tau)\bigg{)}\bigg{]}\ .$ (17) In both cases one has $\displaystyle P^{e\rightarrow e}_{k,s}(\tau)=1-P^{e\rightarrow\mu}_{k,s}(\tau).$ (18) ### III.2 Mixing on a curved background and gravity-induced ambiguity in the particle interpretation Up to now, we have worked within a fixed, but arbitrary, representation of the mass fields. The question arises about the other possible representations, and how the mixing changes when moving from a representation to another. For the definition (15) to make sense, we must determine if and how the probabilities vary when the mass representation is changed. We take as a guideline the principle of covariance, so that the _local_ physical observables should be independent of the underlying representation. In moving from a given representation $\\{\gamma_{1},\epsilon_{1}\\},\\{\gamma_{2},\epsilon_{2}\\}$ to another $\\{\tilde{\gamma}_{1},\tilde{\epsilon}_{1}\\},\\{\tilde{\gamma}_{2},\tilde{\epsilon}_{2}\\}$, we know how to connect the mass Fock spaces, namely via the generator (7) $J^{-1}:\mathcal{H}_{m}\rightarrow\tilde{\mathcal{H}}_{m}$. For each mass representation, we can proceed as we did above and build the corresponding flavor annihilators and flavor spaces $\mathcal{H}_{f}(\tau),\tilde{\mathcal{H}}_{f}(\tau)$, together with the mixing generators $\mathcal{I}_{\theta}(\tau):\mathcal{H}_{f}(\tau)\rightarrow\mathcal{H}_{m}$, $\tilde{\mathcal{I}}_{\theta}(\tau):\tilde{\mathcal{H}}_{f}(\tau)\rightarrow\tilde{\mathcal{H}}_{m}$. It is useful, at this point, to determine the relations among the mixing coefficients $\Lambda(\tau),\Xi(\tau)$ and $\tilde{\Lambda}(\tau),\tilde{\Xi}(\tau)$ that appear in the explicit form of the two generators $\mathcal{I}_{\theta}(\tau)$ and $\tilde{\mathcal{I}}_{\theta}(\tau)$. By definition we have $\tilde{\Lambda}_{q,r;k,s}(\tau)=(\tilde{\zeta}_{q,r;2},\tilde{\zeta}_{k,s;1})_{\tau}=\sum_{q^{\prime},k^{\prime},r^{\prime},s^{\prime}}\bigg{(}\bigg{[}\Gamma_{q,r;q^{\prime},r^{\prime};2}^{*}\zeta_{q^{\prime},r^{\prime};2}+\Sigma_{q,r;q^{\prime},r^{\prime};2}^{*}\xi_{q^{\prime},r^{\prime};2}\bigg{]},\bigg{[}\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}\zeta_{k^{\prime},s^{\prime};1}+\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}\xi_{k^{\prime},s^{\prime};1}\bigg{]}\bigg{)}_{\tau}\,.$ (19) Here the first equality is just the definition of $\tilde{\Lambda}(\tau)$, the second follows from the Bogoliubov transformations (II). By using the properties of the inner product (2), in the general case (again, accounting for both the diagonal and non–diagonal mixing coefficients), we obtain $\displaystyle\tilde{\Lambda}_{q,r;k,s}(\tau)$ $\displaystyle=$ $\displaystyle\sum_{q^{\prime},k^{\prime},r^{\prime},s^{\prime}}\bigg{[}\Gamma_{q,r;q^{\prime}r^{\prime};2}\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}(\zeta_{q^{\prime},r^{\prime};2},\zeta_{k^{\prime},s^{\prime};1})_{\tau}+\ \ \Gamma_{q,r;q^{\prime},r^{\prime};2}\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}(\zeta_{q^{\prime},r^{\prime};2},\xi_{k^{\prime},s^{\prime};1})_{\tau}$ (20) $\displaystyle+$ $\displaystyle\ \ \Sigma_{q,r;q^{\prime}r^{\prime};2}\Gamma_{k,s;k^{\prime}s^{\prime};1}^{*}(\xi_{q^{\prime},r^{\prime};2},\zeta_{k^{\prime},s^{\prime};1})_{\tau}+\ \ \Sigma_{q,r;q^{\prime},r^{\prime};2}\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}(\xi_{q^{\prime},r^{\prime};2},\xi_{k^{\prime},s^{\prime};1})_{\tau}\bigg{]}\,,$ and, finally, from the definition of $\Lambda(\tau)$ and $\Xi(\tau)$, we have $\displaystyle\tilde{\Lambda}_{q,r;k,s}(\tau)=\sum_{q^{\prime},k^{\prime},r^{\prime},s^{\prime}}\bigg{[}\Gamma_{q,r,q^{\prime}r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)-\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)\right)$ $\displaystyle+\ \Sigma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime};k^{\prime}s^{\prime}}^{*}(\tau)+\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{*}(\tau)\right)\bigg{]}\ .$ (21) Similarly we have $\displaystyle\tilde{\Xi}_{q,r;k,s}(\tau)=\sum_{q^{\prime},k^{\prime};r^{\prime},s^{\prime}}\bigg{[}\Gamma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)+\Sigma_{k,s;k^{\prime},s^{\prime};1}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)\right)$ $\displaystyle-\ \Sigma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{*}(\tau)-\ \ \Sigma_{k,s;k^{\prime},s^{\prime};1}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{*}(\tau)\right)\bigg{]}\ .$ (22) When equations (III.1) hold for both the representations $\\{q,r\\},\\{q^{\prime},r^{\prime}\\}$, the equations reduce to $\displaystyle\tilde{\Lambda}_{q,r}(\tau)$ $\displaystyle=$ $\displaystyle\sum_{q^{\prime},r^{\prime}}\bigg{[}\Gamma_{q,r,q^{\prime}r^{\prime};2}\left(\Gamma_{q,r;q^{\prime},r^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime}}(\tau)-\Sigma_{q,r;q^{\prime},r^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime}}(\tau)\right)$ (23) $\displaystyle+$ $\displaystyle\ \Sigma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{q,r;q^{\prime},r^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime}}^{*}(\tau)+\Sigma_{q,r;q^{\prime},r^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime}}^{*}(\tau)\right)\bigg{]}.$ and $\displaystyle\tilde{\Xi}_{q,r}(\tau)$ $\displaystyle=$ $\displaystyle\sum_{q^{\prime},r^{\prime}}\bigg{[}\Gamma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{q,r;q^{\prime},r^{\prime};1}\Xi_{q^{\prime},r^{\prime}}(\tau)+\Sigma_{q,r;q^{\prime},r^{\prime};1}\Lambda_{q^{\prime},r^{\prime}}(\tau)\right)$ (24) $\displaystyle-$ $\displaystyle\ \Sigma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{q,r;q^{\prime},r^{\prime};1}\Lambda_{q^{\prime},r^{\prime}}^{*}(\tau)-\Sigma_{q,r;q^{\prime},r^{\prime};1}\Xi_{q^{\prime},r^{\prime}}^{*}(\tau)\right)\bigg{]}\ .$ The equations (III.2, III.2, 23, 24) provide an explicit relation between $\mathcal{I}_{\theta}(\tau)$ and $\tilde{\mathcal{I}}_{\theta}(\tau)$, and show how the mixing coefficients change, in moving from a mass representation to another, in order to ensure covariance. In particular, the tilded coefficients turn out to be a linear combination of the untilded coefficients weighted by the coefficients of the Bogoliubov transformations between the two mass representations. A slightly modified version of the eqs. (III.2) and (III.2) will be expedient in the calculation of the transition probabilities in a number of interesting cases. It remains to establish how the flavor operators $\gamma_{\rho}(\tau),\epsilon_{\rho}(\tau)$ and the flavor vacuum $|0_{f}(\tau)\rangle$ transform under a change of mass representation. We focus on the vacuum state $|0_{f}(\tau)\rangle\in\mathcal{H}_{f}(\tau)$. First we employ the generator $\mathcal{I}_{\theta}(\tau):\mathcal{H}_{f}(\tau)\rightarrow\mathcal{H}_{m}$ to get the mass vacuum $|0_{m}\rangle$. Then we apply the generator of Bogoliubov transformations (7) $J^{-1}:\mathcal{H}_{m}\rightarrow\tilde{\mathcal{H}}_{m}$ to obtain $|\tilde{0}_{m}\rangle$. Finally, the generator of mixing in the tilde representation $\tilde{\mathcal{I}}_{\theta}^{-1}(\tau):\tilde{\mathcal{H}}_{m}\rightarrow\tilde{\mathcal{H}}_{f}(\tau)$ is employed to get $|\tilde{0}_{f}(\tau)\rangle$. We conclude that the two flavor vacua are related by the transformation $|\tilde{0}_{f}(\tau)\rangle=J_{f}^{-1}(\tau)|0_{f}(\tau)\rangle\doteq\tilde{\mathcal{I}}_{\theta}^{-1}(\tau)J^{-1}\mathcal{I}_{\theta}(\tau)|0_{f}(\tau)\rangle\ $ (25) where we have defined the inverse $J_{f}^{-1}(\tau)$ for convenience. The flavor operators must then transform as $\displaystyle\gamma_{k,s,\rho}(\tau)$ $\displaystyle\rightarrow$ $\displaystyle J_{f}^{-1}(\tau)\gamma_{k,s,\rho}(\tau)J_{f}(\tau)$ $\displaystyle\epsilon_{k,s,\rho}(\tau)$ $\displaystyle\rightarrow$ $\displaystyle J_{f}^{-1}(\tau)\epsilon_{k,s,\rho}(\tau)J_{f}(\tau)$ (26) and similarly for the creation operators. Equations (25) and (III.2) ensure the invariance of local observables in the form of expectation values $\bra{\psi_{f}(\tau)}F(\psi_{e}(\tau),\psi_{\mu}(\tau))\ket{\psi_{f}(\tau)}$ with $\ket{\psi_{f}(\tau)}\in\mathcal{H}_{f}(\tau)$ and $F(\psi_{e}(\tau),\psi_{\mu}(\tau))$ any operator constructed from the fields $\psi_{e}(\tau),\psi_{\mu}(\tau)$. ### III.3 Transition probabilities and the mass representation The oscillation probabilities are not a local observable, since they involve the comparison between particles at different values of $\tau$, as it is evident from the definition (15). In general these quantities _do_ depend on the representation of the mass fields, and this is because distinct representations might assign a different meaning to the quantum numbers $k,s$. For example, one might consider two expansions of the mass fields, one in terms of plane waves, labelled by the three–momenta $\\{\boldsymbol{k}\\}$, and one in terms of localized wave packets, labelled by a suitable set of quantum numbers $\\{q\\}$. It is clear that the two expansions describe particles with different physical properties; the first describes particles with definite momentum, the second describes particles for which momentum and position are definite to some extent. Therefore the probabilities $P^{\rho\rightarrow\sigma}_{k,s}$ and $P^{\rho\rightarrow\sigma}_{q,s}$ refer to the oscillations of different particles, and have a different interpretation. It would make no sense, in such a case, to require the equivalence of the two. This, of course, would be true even in flat space. What is meaningful to require, is that the transition probabilities $P^{\rho\rightarrow\sigma}_{k,s}$ be the same for each compatible representation, i.e., for each representation that refers to the same kind of particle, and therefore agrees on the meaning of the quantum numbers $k,s$. In mathematical terms, any two such representations shall be connected by diagonal Bogoliubov transformations $\displaystyle\tilde{\zeta}_{k,s,i}$ $\displaystyle=$ $\displaystyle\Gamma_{k,s;i}^{*}\zeta_{k,s,i}+\Sigma_{k,s;i}^{*}\xi_{k,s,i}$ $\displaystyle\tilde{\xi}_{k,s,i}$ $\displaystyle=$ $\displaystyle\Gamma_{k,s;i}\xi_{k,s,i}-\Sigma_{k,s;i}\zeta_{k,s,i}\ $ (27) where it is understood that $\Gamma_{k,s;q,r;i}=\delta_{k,q}\delta_{s,r}\Gamma_{k,s;i}$ and $\Sigma_{k,s;q,r;i}=\delta_{k,q}\delta_{s,r}\Sigma_{k,s;i}$, In this case the transition probabilities $P^{\rho\rightarrow\sigma}_{k,s}$ are indeed the same, and this can be proven explicitly, by writing out equation (17) for the two representations $\displaystyle P^{e\rightarrow\mu}_{k,s}(\tau)=2\cos^{2}(\theta)\sin^{2}(\theta)\bigg{[}1-\sum_{q,r}\Re\bigg{(}\Lambda_{k,s;q,r}^{*}(\tau_{0})\Lambda_{k,s;q,r}(\tau)+$ $\displaystyle\Xi_{k,s;q,r}^{*}(\tau_{0})\Xi_{k,s;q,r}(\tau)\bigg{)}\bigg{]}$ (28) and $\displaystyle\tilde{P}^{e\rightarrow\mu}_{k,s}(\tau)=2\cos^{2}(\theta)\sin^{2}(\theta)\times\bigg{[}1-\sum_{q,r}\Re\bigg{(}\tilde{\Lambda}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Lambda}_{k,s;q,r}(\tau)+\tilde{\Xi}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Xi}_{k,s;q,r}(\tau)\bigg{)}\bigg{]}$ (29) With the aid of equations (III.2) , (III.2) and (45) we find $\displaystyle\tilde{\Lambda}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Lambda}_{k,s;q,r}(\tau)+\tilde{\Xi}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Xi}_{k,s;q,r}(\tau)=$ $\displaystyle+\left(\Lambda_{k,s;q,r}^{*}(\tau_{0})\Lambda_{k,s;q,r}(\tau)+\Xi_{k,s;q,r}^{*}(\tau_{0})\Xi_{k,s;q,r}(\tau)\right)\left[|\Gamma_{k,s,2}|^{2}|\Gamma_{q,r,1}|^{2}+|\Gamma_{k,s,2}|^{2}|\Sigma_{q,r,1}|^{2}\right]$ $\displaystyle+\left(\Lambda_{k,s;q,r}(\tau_{0})\Lambda_{k,s;q,r}^{*}(\tau)+\Xi_{k,s;q,r}(\tau_{0})\Xi_{k,s;q,r}^{*}(\tau)\right)\left[|\Sigma_{k,s,2}|^{2}|\Gamma_{q,r,1}|^{2}+|\Sigma_{k,s,2}|^{2}|\Sigma_{q,r,1}|^{2}\right]\ .$ (30) Each of the terms in the square brackets is real. Considered that $\displaystyle\Lambda_{k,s;q,r}(\tau_{0})\Lambda_{k,s;q,r}^{*}(\tau)=\left(\Lambda_{k,s;q,r}^{*}(\tau_{0})\Lambda_{k,s;q,r}(\tau)\right)^{*}\,,\qquad\Xi_{k,s;q,r}(\tau_{0})\Xi_{k,s;q,r}^{*}(\tau)=\left(\Xi_{k,s;q,r}^{*}(\tau_{0})\Xi_{k,s;q,r}(\tau)\right)^{*}\,,$ (31) and that the Bogoliubov coefficients satisfy $|\Gamma_{k,s,i}|^{2}+|\Sigma_{k,s,i}|^{2}=1$ for each $k,s,i$, we finally get $\displaystyle\Re[\tilde{\Lambda}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Lambda}_{k,s;q,r}(\tau)+\tilde{\Xi}_{k,s;q,r}^{*}(\tau_{0})\tilde{\Xi}_{k,s;q,r}(\tau)]=\Re[\Lambda_{k,s;q,r}^{*}(\tau_{0})\Lambda_{k,s;q,r}(\tau)+\Xi_{k,s;q,r}^{*}(\tau_{0})\Xi_{k,s;q,r}(\tau)]\,,$ (32) which proves the invariance of (28). In the most general case, as the quantum numbers $k,s$ and $k^{\prime},s^{\prime}$ have a different phyiscal meaning, the probabilities $P^{\rho\rightarrow\sigma}_{k,s}$ and $\tilde{P}^{\rho\rightarrow\sigma}_{k^{\prime},s^{\prime}}$ have different interpretations. Different representations of the mass fields do indeed assign a different meaning to such indices, so that the probabilities (15) have no invariant meaning. In order to make sense of the probabilities in eqs. (15) in the most general case, a representation of the mass fields must be fixed on the grounds of physical relevance. When the underlying spacetime $M$ possesses non trivial symmetries, as time translational invariance or spherical symmetry, there is no doubt that the representation should be fixed so to take them into account. In these cases ”good quantum numbers” are suggested by the symmetries themselves (for instance, the energy $\omega$ for stationary metrics, the angular momentum $l,m$ for spherically symmetric spacetimes). In any case, the probabilities in a given mass representation can always be related to the probabilites in any other mass representation with the aid of equations (23) and (24). As a final remark, we stress that the issue discussed here has nothing to do with the diffeomorphism covariance of the theory. All the probabilities (17) are (generally covariant) scalars, as it is evident from the definitions. ## IV Neutrino oscillation formulae in FLRW metrics and in presence of a Schwarzschild black hole In this section we apply the formalism developed above to some cases of interest. After an analysis of the flat space limit, we consider two cosmologically relevant FLRW metrics, corresponding to a cosmological constant–dominated and a radiation–dominated universe respectively. In these cases, exact analytical solutions to the Dirac equation are available, and it is possible to employ equation (16) directly. We then introduce a method to extract the oscillation formulae on spacetimes with asymptotically flat regions without resorting to the exact solutions of the Dirac equation. We employ this strategy to compute the formulae on the Schwarzschild black hole spacetime, for neutrinos propagating from the past infinity to the future infinity. We show how the Hawking radiation is naturally embedded in the resulting transition probabilities. ### IV.1 Flat spacetime limit As a first, trivial, application of the formulae (17), let us check the flat spacetime limit. We can see at once that the equations (17) reduce to the ordinary oscillation formulae. Indeed, in this case, we can choose the cauchy hypersurfaces to be the $t=constant$ surfaces in a given Minkowskian coordinate system, while the modes $\\{\zeta_{\boldsymbol{k},s,i}(x),\xi_{\boldsymbol{k},s,i}(x)\\}$ are just the plane wave solutions to the flat Dirac equation with definite momentum, so that $\Lambda_{q,r;k,s}(t)\rightarrow U_{q,r;k,s}(t)$ and $\Xi_{q,r;k,s}(t)\rightarrow V_{q,r;k,s}(t)$, where $U_{q,r;k,s}$ and $V_{q,r;k,s}$ are the usual mixing coefficients in flat space Capolupo1 . Assuming, without loss of generality, $k$ along the $z$ direction, the helicity indices decouple $U_{q,r;k,s}=\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\delta_{r,s}U_{\boldsymbol{k}}$, $V_{q,r;k,s}=\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\delta_{r,s}(-1)^{s}V_{\boldsymbol{k}}$. Since $U_{\boldsymbol{k}}(t)=U_{\boldsymbol{k}}(0)e^{i(\omega_{k,2}-\omega_{k,1})t}$ and $V_{\boldsymbol{k}}(t)=V_{\boldsymbol{k}}(0)e^{i(\omega_{k,2}+\omega_{k,1})t}$, we get $\displaystyle\sum_{k^{\prime},r^{\prime},q,r}\Re[U^{*}_{k,s;k^{\prime},r^{\prime}}(0)U_{k^{\prime},r^{\prime};q,r}(t)$ $\displaystyle+$ $\displaystyle V^{*}_{k,s;k^{\prime},r^{\prime}}(0)V_{k^{\prime},r^{\prime};q,r}(t)]\;=\;\Re[|U_{\boldsymbol{k}}(0)|^{2}e^{i(\omega_{k,2}-\omega_{k,1})t}+|V_{\boldsymbol{k}}(0)|^{2}e^{i(\omega_{k,2}+\omega_{k,1})t}]$ (33) $\displaystyle=$ $\displaystyle|U_{\boldsymbol{k}}(0)|^{2}\cos[(\omega_{k,2}-\omega_{k,1})t]+|V_{\boldsymbol{k}}(0)|^{2}\cos[(\omega_{k,2}+\omega_{k,1})t].$ Substituting this result in eqs. (17) yields the flat space oscillation formulae, which further reduce to the Pontecorvo oscillation formulae in the quantum mechanical limit ($V_{\boldsymbol{k}}=0$). Flat space also offers the possibility to illustrate some points discussed above in the simplest possible context. One might well expand the mass fields in terms of modes with definite energy and angular momentum $\zeta_{\omega,\kappa_{j},m_{j};i}$, $\xi_{\omega,\kappa_{j},m_{j},s;i}$ instead of considering modes with definite cartesian three–momentum $\boldsymbol{k}$. The former shall be suitable combinations of spherical spinors AngularMom . An interesting aspect, is that in such a representation, the mixing coefficients are no longer diagonal. Indeed one has $\Lambda_{\omega^{\prime}\kappa^{\prime}_{j}m^{\prime}_{j};\omega\kappa_{j}m_{j}}(t)=\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta m^{2}}}\delta_{\kappa^{\prime}_{j},\kappa_{j}}\delta_{m^{\prime}_{j},m_{j}}|U_{\omega,\omega^{\prime}}|e^{i(\omega^{\prime}-\omega)t}$ (34) with $\Delta m^{2}=m_{2}^{2}-m_{1}^{2}$ and $|U_{\omega,\omega^{\prime}}|=\sqrt{\frac{\omega^{\prime}+m_{2}}{2\omega^{\prime}}}\sqrt{\frac{\omega+m_{1}}{2\omega}}\left(1+\sqrt{\frac{(\omega^{\prime}-m_{2})(\omega- m_{1})}{(\omega+m_{1})(\omega^{\prime}+m_{2})}}\right)\ ,$ (35) and similar for $\Xi$, where the exponential is $e^{i(\omega^{\prime}+\omega)t}$ and $|U_{\omega,\omega^{\prime}}|$ is replaced by $|V_{\omega,\omega^{\prime}}|=\sqrt{\frac{\omega^{\prime}+m_{2}}{2\omega^{\prime}}}\sqrt{\frac{\omega+m_{1}}{2\omega}}\left(\sqrt{\frac{\omega^{\prime}-m_{2}}{\omega^{\prime}+m_{2}}}-\sqrt{\frac{\omega- m_{1}}{\omega+m_{1}}}\right)\ .$ (36) Here the quantum number $\kappa_{j}$ refers to a relativistic generalization of the spin–orbit operator, which enters the Dirac equation in spherical coordinates AngularMom ; Thaller , and takes into account both the orbital and spin angular momentum. The index $m_{j}$ refers to one component of the total angular momentum $\boldsymbol{J}$, and has not to be confused with the masses $m_{i}$. Without delving into the details of the calculation, the result of equation (34) can be understood as follows. The modes, apart from a normalization constant, are given by $\zeta_{\omega,\kappa_{j},m_{j};i}=e^{-i\omega t}\sqrt{\frac{\omega+m_{i}}{2\omega r^{2}}}\begin{pmatrix}P_{\kappa_{j}}(\lambda_{i}r)\Omega_{\kappa_{j},m_{j}}(\theta,\phi)\\\ \sqrt{\frac{\omega- m_{i}}{\omega+m_{i}}}P_{\kappa_{j}}(\lambda_{i}r)\Omega_{-\kappa_{j},m_{j}}(\theta,\phi)\end{pmatrix}$ (37) $\xi_{\omega,\kappa_{j},m_{j};i}=e^{i\omega t}\sqrt{\frac{\omega+m_{i}}{2\omega r^{2}}}\begin{pmatrix}-\sqrt{\frac{\omega- m_{i}}{\omega+m_{i}}}P_{\kappa_{j}}(-\lambda_{i}r)\Omega_{\kappa_{j},m_{j}}(\theta,\phi)\\\ P_{\kappa_{j}}(-\lambda_{i}r)\Omega_{-\kappa_{j},m_{j}}(\theta,\phi)\end{pmatrix}$ (38) where $\Omega_{\kappa_{j},m_{j}}(\theta,\phi)$ are spherical spinors, and $P_{\kappa_{j}}(\lambda_{i}r)$ are radial functions of the product $\lambda_{i}r$, with the radial momentum $\lambda_{i}=\sqrt{\omega^{2}-m_{i}^{2}}$. These are solutions to the radial part of the Dirac equation, which turns out to be a Riccati–Bessel equationAbramowitz . The functions $P_{\kappa_{j}}(\lambda_{i}r)$ are combinations of spherical bessel functions $j_{n}$ of the form $P_{\kappa_{j}}(\lambda_{i}r)=rj_{\kappa_{j}}(\lambda_{i}r)$. In computing the inner products, the radial integration $\int drP^{*}_{\kappa_{j};2}(\lambda_{2}r)P_{\kappa_{j};1}(\lambda_{1}r)$ will produce a factor $\delta_{\lambda_{2},\pm\lambda_{1}}$, because of the closure relation satisfied by the spherical Bessel functions. Since $\lambda_{2}=\sqrt{\omega^{{}^{\prime}2}-m_{2}^{2}}$ and $\lambda_{1}=\sqrt{\omega^{2}-m_{1}^{2}}$, this will give rise to the delta factor appearing in (34). Notice that $|U_{\omega,\omega^{\prime}}|,|V_{\omega,\omega^{\prime}}|$ are numerically the same as the usual flat space coefficients $|U_{\boldsymbol{k}}|,|V_{\boldsymbol{k}}|$, when $k^{2}=\omega^{2}-m_{1}^{2}=\omega^{{}^{\prime}2}-m_{2}^{2}$ .This shows why the mixing coefficients $\Lambda$, $\Xi$ are not generally diagonal, and the flexibility of the formalism we have employed. Indeed, the non–diagonal coefficients automatically ensure that the flavor operators $\gamma_{\omega,\rho}$, $\epsilon_{\omega,\rho}$ take into account the mass difference, involving operators with distinct energies $\omega,\omega^{\prime}$ for the fields $\psi_{1}$ and $\psi_{2}$ Note5 . In flat space, the shift between the two representations is actually of no use. However, in a non–trivial framework, the versatility of the formalism is essential, as there are instances in which the cartesian components of the momentum $k_{x},k_{y},k_{z}$ are useless, while the ”spherical” quantum numbers $\omega,l,m$ are well defined. ### IV.2 Expanding universe with exponential growth of the scale factor The simplest non–trivial application is to spatially flat Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetimes. Consider the metric $ds^{2}=dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2})$ with an exponential expansion $a(t)=e^{Ht}$, $H=constant$. This is well–suited to describe a homogeneous, spatially flat and isotropic universe dominated by a cosmological constant. The normalized solutions to the Dirac equation for this metric were derived in Barut . Assuming, without loss of generality, the momentum $k$ to be along the $z$ direction, the helicities decouple as in flat space. Choosing the Cauchy surfaces as the surfaces with $t=constant$, or equivalently with $a(t)=constant$, the mixing coefficients read $\displaystyle\Lambda_{k,s;q,r}$ $\displaystyle(t)=\delta_{s,r}\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\frac{\pi ke^{-Ht}}{2H\sqrt{\cos(\frac{i\pi m_{2}}{H})\cos(\frac{i\pi m_{1}}{H})}}\times\left[\\!J_{v_{1}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!+\\!J_{v_{1}-1}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{v_{2}-1}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\right]$ (39) $\displaystyle\Xi_{k,s;q,r}$ $\displaystyle(t)=\delta_{s,r}(-1)^{s}\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\frac{\pi ke^{-Ht}}{2H\sqrt{\cos(\frac{i\pi m_{2}}{H})\cos(\frac{i\pi m_{1}}{H})}}\left[\\!J_{v_{1}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{-v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!+\\!J_{v_{1}-1}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{1-v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\right]$ (40) where $J_{\alpha}$ denotes the $\alpha$ Bessel function and $v_{j}=\frac{1}{2}\left(1+\frac{2im_{j}}{H}\right)$ for $j=1,2$. Plugging these expressions in equations (17), we obtain $\displaystyle P^{e\rightarrow\mu}_{k,s}(t)=2\cos^{2}(\theta)\sin^{2}(\theta)\bigg{\\{}1-\frac{\pi^{2}k^{2}e^{-H(t+t_{0})}}{4H^{2}\cos(\frac{i\pi m_{2}}{H})\cos(\frac{i\pi m_{1}}{H})}$ (41) $\displaystyle\times\Re\bigg{[}\left[\\!J_{v_{1}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\\!J_{v_{2}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!+\\!J_{v_{1}-1}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\\!J_{v_{2}-1}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\right]\left[\\!J_{v_{1}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!+\\!J_{v_{1}-1}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{v_{2}-1}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\right]$ $\displaystyle+\left[\\!J_{v_{1}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\\!J_{-v_{2}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!+\\!J_{v_{1}-1}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\\!J_{1-v_{2}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht_{0}}\\!\\!\right)\\!\right]\left[\\!J_{v_{1}}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{-v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!+\\!J_{v_{1}-1}^{*}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\\!J_{1-v_{2}}\\!\\!\left(\\!\\!\frac{k}{H}e^{-Ht}\\!\\!\right)\\!\right]\bigg{]}\bigg{\\}}$ ### IV.3 Expanding universe dominated by radiation Here we consider the FLRW metric for a radiation dominated universe $a(t)=a_{0}t^{\frac{1}{2}}$. Notice that since $a(t)$ has to be adimensional, $a_{0}$ has dimension $[t]^{-\frac{1}{2}}=[m]^{\frac{1}{2}}$. As before, without loss of generality, we assume the neutrino momentum $k$ along the $z$ direction to decouple the helicities, and consider a foliation by the $t=constant$ hypersurfaces. The solutions to the Dirac equation for this metric are again found in Barut , and yield the mixing coefficients $\displaystyle\Lambda_{k,s;q,r}(t)=\delta_{s,r}\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\frac{1}{\sqrt[4]{4m_{1}m_{2}t^{2}}}e^{-\frac{\pi k^{2}(m_{1}+m_{2})}{4m_{1}m_{2}a_{0}^{2}}}\bigg{\\{}W^{*}_{\kappa_{2},\frac{1}{4}}(-2im_{2}t)W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)$ (42) $\displaystyle+\frac{4}{m_{1}m_{2}a_{0}^{2}t}\bigg{[}W^{*}_{\kappa_{2},\frac{1}{4}}(-2im_{2}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{2}a_{0}^{2}}\right)W^{*}_{\kappa_{2}-1,\frac{1}{4}}(-2im_{2}t)\bigg{]}\bigg{[}W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t)\bigg{]}\bigg{\\}}$ $\displaystyle\Xi_{k,s;q,r}(t)=\delta_{s,r}\delta^{3}(\boldsymbol{k}-\boldsymbol{q})\frac{k}{\sqrt[4]{2m_{1}(2m_{2})^{3}a_{0}^{2}t}}e^{-\frac{\pi k^{2}(m_{1}+m_{2})}{4m_{1}m_{2}a_{0}^{2}}}\bigg{\\{}W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)W_{-\kappa_{2},\frac{1}{4}}(-2im_{2}t)$ (43) $\displaystyle+\frac{1}{m_{1}m_{2}a_{0}^{2}t}\bigg{[}W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W^{*}_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t)\bigg{]}\bigg{[}W_{-\kappa_{2},\frac{1}{4}}(2im_{2}t)+\frac{2im_{2}a_{0}^{2}}{k^{2}}W_{-\kappa_{2}+1,\frac{1}{4}}(2im_{2}t)\bigg{]}\bigg{\\}}$ where $W_{\kappa,\mu}(z)$ are the Whittaker functions Abramowitz and $\kappa_{j}=\frac{1}{4}\left(1+\frac{2ik^{2}}{a_{0}^{2}m_{j}}\right)$ for $j=1,2$. Insertion in eqs. (17) gives the transition probabilities ($t_{0},t>0$) $\displaystyle P^{e\rightarrow\mu}_{k,s}(t)=2\cos^{2}(\theta)\sin^{2}(\theta)\bigg{\\{}1+\Re\bigg{[}\frac{1}{\sqrt[2]{4m_{1}m_{2}t_{0}t}}e^{-\frac{\pi k^{2}(m_{1}+m_{2})}{2m_{1}m_{2}a_{0}^{2}}}\bigg{\\{}W_{\kappa_{2},\frac{1}{4}}(-2im_{2}t_{0})W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t_{0})$ $\displaystyle+\frac{4}{m_{1}m_{2}a_{0}^{2}t_{0}}\left(W_{\kappa_{2},\frac{1}{4}}(-2im_{2}t_{0})-\frac{1}{8}\left(1+\frac{ik^{2}}{m_{2}a_{0}^{2}}\right)W_{\kappa_{2}-1,\frac{1}{4}}(-2im_{2}t_{0})\right)\left(W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t_{0})-\frac{1}{8}\left(1+\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W^{*}_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t_{0})\right)\bigg{\\}}$ $\displaystyle\times\bigg{\\{}W^{*}_{\kappa_{2},\frac{1}{4}}(-2im_{2}t)W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)$ $\displaystyle+\frac{4}{m_{1}m_{2}a_{0}^{2}t}\left(W^{*}_{\kappa_{2},\frac{1}{4}}(-2im_{2}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{2}a_{0}^{2}}\right)W^{*}_{\kappa_{2}-1,\frac{1}{4}}(-2im_{2}t)\right)\left(W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t)\right)\bigg{\\}}$ $\displaystyle+\frac{k^{2}}{\sqrt[2]{2m_{1}(2m_{2})^{3}a_{0}^{4}t_{0}t}}e^{-\frac{\pi k^{2}(m_{1}+m_{2})}{2m_{1}m_{2}a_{0}^{2}}}\bigg{\\{}W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t_{0})W^{*}_{-\kappa_{2},\frac{1}{4}}(-2im_{2}t_{0})$ $\displaystyle+\frac{1}{m_{1}m_{2}a_{0}^{2}t_{0}}\left(W_{\kappa_{1},\frac{1}{4}}(-2im_{1}t_{0})-\frac{1}{8}\left(1+\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t_{0})\right)\left(W^{*}_{-\kappa_{2},\frac{1}{4}}(2im_{2}t_{0})-\frac{2im_{2}a_{0}^{2}}{k^{2}}W^{*}_{-\kappa_{2}+1,\frac{1}{4}}(2im_{2}t_{0})\right)\bigg{\\}}$ $\displaystyle\times\bigg{\\{}W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)W_{-\kappa_{2},\frac{1}{4}}(-2im_{2}t)$ $\displaystyle+\frac{1}{m_{1}m_{2}a_{0}^{2}t}\left(W^{*}_{\kappa_{1},\frac{1}{4}}(-2im_{1}t)-\frac{1}{8}\left(1-\frac{ik^{2}}{m_{1}a_{0}^{2}}\right)W^{*}_{\kappa_{1}-1,\frac{1}{4}}(-2im_{1}t)\right)\left(W_{-\kappa_{2},\frac{1}{4}}(2im_{2}t)+\frac{2im_{2}a_{0}^{2}}{k^{2}}W_{-\kappa_{2}+1,\frac{1}{4}}(2im_{2}t)\right)\bigg{\\}}\bigg{]}\bigg{\\}}$ (44) ### IV.4 Spacetimes with asymptotically flat regions The FLRW spacetimes considered above are among the few non–trivial metrics for which the Dirac equation (1) can be solved analytically. More often one does not have an exact solution at his disposal, and the implementation of eqs. (17) is a complicated task. In many cases of interest, however, the spacetime $M$ admits asymptotically flat regions $\Omega_{A},\Omega_{B}\subset M$, usually in the far past and in the far future. $\Omega_{A}$ and $\Omega_{B}$ are separated by a region with non-trivial curvature, where the Dirac equation is usually unsolvable analytically. In $\Omega_{A}$ and $\Omega_{B}$ the solutions to the Dirac equation are the flat space modes $\\{u^{I}_{k,s,i}(x),v^{I}_{k,s,i}\\}$ with $I=A,B$, and one has a natural choice for the positive frequency modes. Because of the non-trivial curvature, the two sets of solutions $A,B$ are distinct. When limited to one of the two regions, eqs. (17) reduce to the ordinary flat space formulae. When the intermediate curvature region is involved, a direct application of eqs. (17) is prohibitive. Nevertheless, if one is able to provide the relation between the two sets $A,B$, in the form of a Bogoliubov transformation, it is possible to derive oscillation formulae for the propagation from $\Omega_{A}$ to $\Omega_{B}$. Assume that $\Omega_{A}=\bigcup_{\tau\leq\tau_{A}}\Sigma_{\tau}$ and $\Omega_{B}=\bigcup_{\tau\geq\tau_{B}}\Sigma_{\tau}$ for some values $\tau_{B}>\tau_{A}$, and consider $\tau_{0}\leq\tau_{A}$ , $\tau\geq\tau_{B}$. Suppose also that the $B$ modes are given in terms of the $A$ modes as $\displaystyle u^{B}_{k^{\prime},s^{\prime},i}$ $\displaystyle=$ $\displaystyle\sum_{k,s}\left(\Gamma_{k^{\prime},s^{\prime};k,s;i}^{*}u^{A}_{k,s,i}+\Sigma_{k^{\prime},s^{\prime};k,s;i}^{*}v^{A}_{k,s,i}\right)$ (45) $\displaystyle v^{B}_{k^{\prime}s^{\prime},i}$ $\displaystyle=$ $\displaystyle\sum_{k,s}\left(\Gamma_{k^{\prime},s^{\prime};k,s;i}v^{A}_{k,s,i}-\Sigma_{k^{\prime},s^{\prime};k,s;i}u^{A}_{k,s,i}\right)\ .$ (46) Here the Bogoliubov coefficients $\Gamma_{k^{\prime},s^{\prime};k,s;i},\Sigma_{k^{\prime},s^{\prime};k,s;i}$ are again provided by the inner products $(u^{A}_{i},u^{B}_{i}),(u^{A}_{i},v^{B}_{i})$, yet their significance is slightly different from those in (II). While equation (II) describes a general and arbitrary change of basis, the transformation of equation (45) is dictated by the circumstance of having a natural choice for the modes in $\Omega_{A}$ and $\Omega_{B}$, with a well–defined physical meaning. Indeed, the Bogoliubov coefficients take into account the effect of the curvature in the intermediate region. Then we can specialize equations (III.2), (III.2) to get $\displaystyle\Lambda^{B}_{q,r;k,s}(\tau)=\sum_{q^{\prime},k^{\prime},r^{\prime},s^{\prime}}\bigg{[}\Gamma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A}(\tau)-\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A}(\tau)\right)$ $\displaystyle+\Sigma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}^{*}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A*}(\tau)+\Sigma_{k,s;k^{\prime},s^{\prime};1}^{*}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A*}(\tau)\right)\bigg{]}\ .$ (47) $\displaystyle\Xi^{B}_{q,r;k,s}(\tau)=\sum_{q^{\prime},k^{\prime},r^{\prime},s^{\prime}}\bigg{[}\Gamma_{q,r;q^{\prime},r^{\prime};2}\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}\Xi^{A}_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)+\Sigma_{k,s;k^{\prime},s^{\prime};1}\Lambda^{A}_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}(\tau)\right)$ (48) $\displaystyle-\Sigma_{q,r;q^{\prime},r^{\prime};2}\times\left(\Gamma_{k,s;k^{\prime},s^{\prime};1}\Lambda_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A*}(\tau)-\Sigma_{k,s;k^{\prime},s^{\prime};1}\Xi_{q^{\prime},r^{\prime};k^{\prime},s^{\prime}}^{A*}(\tau)\right)\bigg{]}\ .$ For $\tau<\tau_{A}$ $\Lambda^{A}(\tau),\Xi^{A}(\tau)$ are trivial, and vice- versa, for $\tau>\tau_{B}$, $\Lambda^{B}(\tau),\Xi^{B}(\tau)$ are trivial. Choosing, for instance, the $B$ representation, one would have a trivial expression for $\Lambda^{B}(\tau),\Xi^{B}(\tau)$ and one can make use of eqs. (IV.4), (48) to obtain $\Lambda^{B}(\tau_{0}),\Xi^{B}(\tau_{0})$ in terms of $\Lambda^{A}(\tau_{0}),\Xi^{A}(\tau_{0})$, which are also trivial. Then one can plug $\Lambda^{B}(\tau,\tau_{0})$ and $\Xi^{B}(\tau,\tau_{0})$ in equation (17) to obtain $P^{e\rightarrow\mu}_{k,s\rightarrow q,r}(\tau)$ for $\tau\geq\tau_{B}$ and the reference hypersurface $\tau_{0}\leq\tau_{A}$. #### IV.4.1 Scwharzschild black hole As a realization of this scheme, consider the (static) Schwarzschild metric $\displaystyle ds^{2}=\left(1-\frac{2GM}{r}\right)dt^{2}-\left(1-\frac{2GM}{r}\right)^{-1}dr^{2}-r^{2}d\Omega,$ (49) and two sequences of spacelike hypersurfaces Note6 $\\{\Sigma_{n}^{+}\\}_{n\in\mathbb{N}}$, $\\{\Sigma_{n}^{-}\\}_{n\in\mathbb{N}}$ approaching, respectively, the future $\mathcal{I}^{+}$ and the past null infinity $\mathcal{I}^{-}$ as $n\rightarrow\infty$. We require that $\Sigma_{n}^{+}$ and $\Sigma_{n}^{-}$ be Cauchy surfaces respectively for the causal past $J^{-}(\mathcal{I}^{+})$ of $\mathcal{I}^{+}$ and the causal future $J^{+}(\mathcal{I}^{-})$of $\mathcal{I}^{-}$ for each $n$. For $n$ large enough, these surfaces span an approximately flat portion of the Schwarzschild spacetime. On the surfaces $\Sigma_{n}^{-}$, as $n$ approaches infinity, we expand the massive fields in terms of the incoming solutions, with frequency defined with respect to the schwarzschild time $t$ and with definite angular momentum, $\displaystyle\zeta^{IN}_{\omega,\kappa_{j},m_{j};i}(t,r,\theta,\phi)\propto e^{-i\omega t}\,,\qquad\qquad\xi^{IN}_{\omega,\kappa_{j},m_{j};i}(t,r,\theta,\phi)\propto e^{i\omega t}.$ (50) These modes reduce to the flat space solutions (37) and (38) as $\mathcal{I}^{-}$ is approached. Omitting the irrelevant angular and spin quantum numbers, as $n\rightarrow\infty$, we get $\displaystyle\Lambda^{IN}_{\omega;\omega^{\prime}}(\Sigma^{-}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta m^{2}}}|U_{\omega,\omega^{\prime}}|e^{i\varphi^{-}(\omega,n)}$ (51) $\displaystyle\Xi^{IN}_{\omega;\omega^{\prime}}(\Sigma^{-}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta m^{2}}}|U_{\omega,\omega^{\prime}}|e^{i\rho^{-}(\omega,n)}$ (52) with $|U_{\omega,\omega^{\prime}}|$ and $|V_{\omega,\omega^{\prime}}|$ flat space spherical mixing coefficients as defined in (35) and (36), and $\varphi^{-}(\omega,n),\rho^{-}(\omega,n)$ phase factors depending on $\omega$ and $n$. A similar reasoning can be carried on for the _outgoing_ modes emerging at $\mathcal{I}^{+}$, $\zeta^{OUT}_{\omega,\kappa_{j},m_{j};i}(t,r,\theta,\phi)$ , $\xi^{OUT}_{\omega,\kappa_{j},m_{j};i}(t,r,\theta,\phi)$, so to yield as $n\rightarrow\infty$ $\displaystyle\Lambda^{OUT}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta m^{2}}}|U_{\omega,\omega^{\prime}}|e^{i\varphi^{+}(\omega,n)}$ (53) $\displaystyle\Xi^{OUT}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta m^{2}}}|V_{\omega,\omega^{\prime}}|e^{i\rho^{+}(\omega,n)}\,.$ (54) Because of the Black Hole, the $IN$ and $OUT$ modes do not coincide. Fermion creation by the Schwarzschild black hole has been studied, via the tunneling method, in Kerner . There it has been shown that the Hawking temperature $T_{H}=\frac{1}{8\pi GM}$ is recovered for the emission of spin–$\frac{1}{2}$ particles. We then infer that the $IN$ and $OUT$ modes are related by a thermal Bogoliubov transformation at the Hawking temperature $T_{H}$, corrected for fermions: $\displaystyle\zeta^{OUT}_{\omega,\kappa_{j},m_{j};i}=\sqrt{\frac{e^{\frac{\omega}{k_{B}T_{H}}}}{e^{\frac{\omega}{k_{B}T_{H}}}+1}}\zeta^{IN}_{\omega,\kappa_{j},m_{j};i}+\sqrt{\frac{1}{e^{\frac{\omega}{k_{B}T_{H}}}+1}}\xi^{IN}_{\omega,\kappa_{j},m_{j};i}$ (55) $\displaystyle\xi^{OUT}_{\omega,\kappa_{j},m_{j};i}=\sqrt{\frac{e^{\frac{\omega}{k_{B}T_{H}}}}{e^{\frac{\omega}{k_{B}T_{H}}}+1}}\xi^{IN}_{\omega,\kappa_{j},m_{j};i}-\sqrt{\frac{1}{e^{\frac{\omega}{k_{B}T_{H}}}+1}}\zeta^{IN}_{\omega,\kappa_{j},m_{j};i}\ ,$ (56) It is understood that these equations hold as long as the spacetime is stationary (eternal black hole). For a body that collapses to a Schwarzschild black hole, as considered in the original paper by Hawking Hawking , we expect a slightly modified version of the Bogoliubov transformations, comprising non–diagonal thermal coefficients $\Gamma_{\omega,\omega^{\prime};i}$ and $\Sigma_{\omega,\omega^{\prime};i}$ with $\omega\neq\omega^{\prime}$. We pick the ingoing representation to calculate the probabilities (of course, the outgoing representation yields the same result, as the Bogoliubov transformations are diagonal), and employ eqs. (IV.4, 48) to obtain, with $F_{H}(\omega)=\frac{1}{e^{\frac{\omega}{k_{B}T_{H}}}+1}$, $\displaystyle\Lambda^{IN}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})$ $\displaystyle=$ $\displaystyle\bigg{\\{}\sqrt{\left[1-F_{H}(\omega)\right]\left[1-F_{H}(\omega^{\prime})\right]}\Lambda^{OUT}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})-\ \ \sqrt{F_{H}(\omega)\left[1-F_{H}(\omega^{\prime})\right]}\Xi^{OUT}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})$ (57) $\displaystyle+$ $\displaystyle\sqrt{F_{H}(\omega^{\prime})\left[1-F_{H}(\omega)\right]}\left(\Xi^{OUT}_{\omega;\omega^{\prime}}\right)^{*}(\Sigma^{+}_{n})+\sqrt{F_{H}(\omega)F_{H}(\omega^{\prime})}\left(\Lambda^{OUT}_{\omega;\omega^{\prime}}\right)^{*}(\Sigma^{+}_{n})\bigg{\\}}\,,$ Considered that $\Lambda^{OUT}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta m^{2}}}|U_{\omega,\omega^{\prime}}|e^{i\varphi^{+}(\omega,n)}$ and $\Xi^{+}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})\rightarrow\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta m^{2}}}|V_{\omega,\omega^{\prime}}|e^{i\rho^{+}(\omega,n)}$ as $n\rightarrow\infty$, we obtain $\displaystyle\Lambda^{IN}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})$ $\displaystyle=$ $\displaystyle\bigg{\\{}\sqrt{\left[1-F_{H}(\omega)\right]\left[1-F_{H}(\omega^{\prime})\right]}|U_{\omega;\omega^{\prime}}|e^{i\varphi^{+}(\omega,n)}-\sqrt{F_{H}(\omega)\left[1-F_{H}(\omega^{\prime})\right]}|V_{\omega;\omega^{\prime}}|e^{i\rho^{+}(\omega,n)}$ (58) $\displaystyle+$ $\displaystyle\sqrt{F_{H}(\omega^{\prime})\left[1-F_{H}(\omega)\right]}|V_{\omega;\omega^{\prime}}|e^{-i\rho^{+}(\omega,n)}+\sqrt{F_{H}(\omega)F_{H}(\omega^{\prime})}|U_{\omega,\omega^{\prime}}|e^{-i\varphi^{+}(\omega,n)}\bigg{\\}}\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta m^{2}}}$ and $\displaystyle\Xi^{IN}_{\omega;\omega^{\prime}}(\Sigma^{+}_{n})$ $\displaystyle=$ $\displaystyle\bigg{\\{}\sqrt{\left[1-F_{H}(\omega)\right]\left[1-F_{H}(\omega^{\prime})\right]}|V_{\omega,\omega^{\prime}}|e^{i\rho^{+}(\omega,n)}+\sqrt{F_{H}(\omega)\left[1-F_{H}(\omega^{\prime})\right]}|U_{\omega,\omega^{\prime}}|e^{i\varphi^{+}(\omega,n)}$ (59) $\displaystyle-\sqrt{F_{H}(\omega^{\prime})\left[1-F_{H}(\omega)\right]}|U_{\omega,\omega^{\prime}}|e^{-i\varphi^{+}(\omega,n)}+\sqrt{F_{H}(\omega)F_{H}(\omega^{\prime})}|V_{\omega,\omega^{\prime}}|e^{-i\rho^{+}(\omega,n)}\bigg{\\}}\delta_{\omega^{\prime},\sqrt{\omega^{2}+\Delta m^{2}}}\ .$ Choosing as reference hypersurface $\Sigma^{-}_{m}$ for large $m$, we can now compute the probabilities (17) for a neutrino propagating from $\Sigma^{-}_{m}$ to $\Sigma^{+}_{n}$, i.e. from $\mathcal{I}^{-}$ to $\mathcal{I}^{+}$ in the limit $m,n\rightarrow\infty$. We find, for $m,n\rightarrow\infty$ $\displaystyle P^{e\rightarrow\mu}_{\omega}(m,n)$ $\displaystyle\approx$ $\displaystyle 2\cos^{2}\theta\sin^{2}\theta\ \bigg{(}1-\sqrt{\left[1-F_{H}(\omega)\right]\left[1-F_{H}(\omega^{\prime})\right]}\left[|U_{\omega;\omega^{\prime}}|^{2}\cos(\Delta^{-}_{\omega;m,n})+|V_{\omega;\omega^{\prime}}|^{2}\cos(\Phi^{-}_{\omega;m,n})\right]$ (60) $\displaystyle+$ $\displaystyle\sqrt{F_{H}(\omega)\left[1-F_{H}(\omega^{\prime})\right]}|U_{\omega;\omega^{\prime}}||V_{\omega;\omega^{\prime}}|\left[\cos(\Theta^{-}_{\omega;m,n})-\cos(\Psi^{-}_{\omega;m,n})\right]$ $\displaystyle+$ $\displaystyle\sqrt{F_{H}(\omega^{\prime})\left[1-F_{H}(\omega)\right]}|U_{\omega;\omega^{\prime}}||V_{\omega;\omega^{\prime}}|\left[\cos(\Psi^{+}_{\omega;m,n})-\cos(\Theta^{+}_{\omega;m,n})\right]$ $\displaystyle-$ $\displaystyle\sqrt{F_{H}(\omega)F_{H}(\omega^{\prime})}\left[|U_{\omega;\omega^{\prime}}|^{2}\cos(\Delta^{+}_{\omega;m,n})+|V_{\omega;\omega^{\prime}}|^{2}\cos(\Phi^{+}_{\omega;m,n})\right]\bigg{)}\ .$ with $\Delta^{\pm}_{\omega;m,n}=\varphi^{+}(\omega,n)\pm\varphi^{-}(\omega,m)$ , $\Phi^{\pm}_{\omega;m,n}=\rho^{+}(\omega,n)\pm\rho^{-}(\omega,m)$, $\Psi^{\pm}_{\omega;m,n}=\rho^{+}(\omega,n)\pm\varphi^{-}(\omega,m)$ and $\Theta^{\pm}_{\omega;m,n}=\varphi^{+}(\omega,n)\pm\rho^{-}(\omega,m)$ . In particular, for large energies of the mass fields $\omega,\omega^{\prime}$, $|V_{\omega,\omega^{\prime}}|\rightarrow 0$ and $|U_{\omega,\omega^{\prime}}|\rightarrow 1$, thus $\displaystyle P^{e\rightarrow\mu}_{\omega}(m,n)\approx 2\cos^{2}\theta\sin^{2}\theta\ \bigg{(}1-\sqrt{\left[1-F_{H}(\omega)\right]\left[1-F_{H}(\omega^{\prime})\right]}\cos(\Delta^{-}_{\omega,\kappa_{j};m,n})-\sqrt{F_{H}(\omega)F_{H}(\omega^{\prime})}\cos(\Delta^{+}_{\omega,\kappa_{j};m,n})\bigg{)}\ ,$ (61) where it is understood that $\omega^{{}^{\prime}2}=\omega^{2}+\Delta m^{2}$. If the limit $T_{H}\rightarrow 0$ is taken in equation (61), one obtains, apart from a phase factor, the usual Pontecorvo oscillation formulae. To compute the oscillation formulae on a Schwarzschild background for an arbitrary propagation, since exact analytical solutions are unavailable, one has to resort to approximate solutions to the Dirac equation. In all the applications considered, the oscillation formulae do not depend on the helicity $s$ of neutrinos. However, when additional complications due to frame–dragging and non–conservation of angular momentum arise (see e.g. Lambiase1 ; Lambiase2 ), the formulae for left-handed and right-handed neutrinos can differ. ### IV.5 Quantum mechanical limit It is a general feature of equations (17) that when all the quantum field theoretical effects are negligible, the oscillation formulae are modified only for a phase factor. Indeed, when one can neglect $\Xi_{k,s}$ in equation (17), one immediately has $|\Lambda^{*}_{k,s}(\tau)|=1$, ($|\Lambda_{\omega,\omega^{\prime}}(\tau)|=1$ respectively, in the non–diagonal case) for each $\tau$ from equation (III.1). Then the product $\Lambda_{k,s}(\tau_{0})^{*}\Lambda_{k,s}(\tau)$, ($\Lambda_{\omega,\omega^{\prime}}(\tau_{0})^{*}\Lambda_{\omega,\omega^{\prime}}(\tau)$ respectively) is just a phase $e^{i\varphi(\tau_{0},\tau)}$ and the net effect is a phase shift with respect to the Pontecorvo oscillation formulae, consistently with previous results obtained in a heuristic treatmentCardall . Of course, the explicit value of the phase $\varphi(\tau_{0},\tau)$ depends on the metric and the surfaces $\tau$ considered, as well as on the mode expansion chosen for the mass fields. When the gravitational fields are weak enough, the phase can be computed by means of geometrical optics considerations Cardall ; Lambiase1 . ## V Conclusions We have developed a quantum field theoretical approach to the vacuum neutrino oscillations in curved space, discussing the transition probabilities, and their behaviour under changes of mass representation. We have analyzed the non–trivial interplay between quantum field mixing and field quantization in curved space, and have found that the former has a remarkably richer structure when compared to its flat space counterpart. In particular, the formalism has to be versatile enough to deal with the existence of infinite unitarily inequivalent representations of the mass fields, which is to say that no preferred notion of particle does generally exist. In the spirit of general covariance, we have determined the effect on flavor fields of a shift in the expansion of the mass fields, and established under which conditions the resulting transition probabilities are the same. We have then computed the oscillation formulae in three example metrics, including two FRLW spacetimes and the static Schwarzschild black hole. In the latter we have found that the Hawking radiation affects, although very slightly, the oscillations for neutrinos propagating from the asymptotic past infinity to the asymptotic future infinity. As a general result, it is found that when all the quantum field theoretical effects on neutrino mixing can be neglected, the gravitational background only affects the phase of the oscillations, constistently with previous analyses. ## Acknowledgements Partial financial support from MIUR and INFN is acknowledged. A.C. and G.L. also acknowledge the COST Action CA1511 Cosmology and Astrophysics Network for Theoretical Advances and Training Actions (CANTATA). ## References * (1) L. M. Brown, Physics Today 31, 9, 23 (1978). * (2) Q.R. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett. 87 (7): 071301 (2001). * (3) Y. Fukuda et al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 81 (8), 1562 - 1567 (1998). * (4) S. M. Bilenky and B. Pontecorvo, Phys. Rep. 41, 225 (1978). * (5) S. M. 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D 54, 1587 (1996). * (16) C. Y. Cardall and G. M. Fuller, Phys. Rev. D 55, 7960 (1997). * (17) N. Birrell and P. Davies, Quantum Fields in Curved Space (Cambridge Monographs on Mathematical Physics)., Cambridge: Cambridge University Press, doi:10.1017/CBO9780511622632 (1982). * (18) R. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago Lectures in Physics), The University of Chicago Press, ISBN: 9780226870274 (1994). * (19) Here the index $k$ refers to a generic set of quantum numbers labelling the modes. The sums $\sum_{k}$ are understood as integrals if $k$ is a continuous index. * (20) M. Blasone, A. Capolupo, G. Vitiello, Phys. Rev. D 66, 025033 (2002) and references therein; A. Capolupo, I. De Martino, G. Lambiase and A. Stabile, Phys. Lett. B 790, 427 (2019); K. Fujii, C. Habe and T. Yabuki, Phys. Rev. D 59, 113003 (1999); Phys. Rev. D 64, 013011 (2001); K.C. Hannabuss and D.C. Latimer, J. Phys. A 33, 1369 (2000); J. Phys. A 36, L69 (2003); C.R Ji and Y. Mishchenko, Phys. Rev. D 65, 096015 (2002); Ann. Phys. 315, 488 (2005); M. Blasone, A. Capolupo, O. Romei and G. Vitiello, Phys. Rev. D 63, 125015 (2001); A. Capolupo, C. R. Ji , Y. Mishchenko and G. Vitiello, Phys. Lett. B 594, 135 (2004); M. Blasone, A. Capolupo, F. Terranova, G. Vitiello, Phys. Rev. D 72, 013003 (2005). * (21) A. Capolupo, Adv. High Energy Phys. 2016, 8089142, 10 (2016); Adv. High Energy Phys. 2018, 9840351 (2018); A. Capolupo, S. Capozziello and G. Vitiello, Phys. Lett. A 373, 601 (2009); Phys. Lett. A 363, 53 (2007); Int. J. Mod. Phys. A 23, 4979 (2008); M. Blasone, A. Capolupo, S. Capozziello, G. Vitiello, Nucl. Instrum. Meth. A 588, 272 (2008); M. Blasone, A. Capolupo, G. Vitiello, Prog. Part. Nucl. Phys. 64, 451 (2010); M. Blasone, A. Capolupo, S. Capozziello, S. Carloni and G. Vitiello, Phys. Lett. A 323, 182 (2004); A. Capolupo, M. Di Mauro, A. Iorio, Phys. Lett. A 375, 3415 (2011). * (22) A. Capolupo, S. M. Giampaolo, G. Lambiase and A. Quaranta, arXiv:1912.03332 [hep-ph]. * (23) The helicity index can always be decoupled by choosing eigenstates of the helicity operator. * (24) This does _not_ imply that $Q_{e}$ and $Q_{\mu}$ are conserved separately. * (25) L. Biedenharn, J. Louck, P. Carruthers, Angular Momentum in Quantum Physics: Theory and Application (Encyclopedia of Mathematics and its Applications), Cambridge: Cambridge University Press., doi:10.1017/CBO9780511759888 (1984). * (26) B. Thaller, Advanced Visual Quantum Mechanics, Springer–Verlag New York, doi:10.1007/b138654 (2005). * (27) M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications Inc. New York, ISBN: 9780486612720 (1965). * (28) In this case the label $\omega$ refers to the energies of the reference mass field, that is $\psi_{1}$ for $\rho=e$ and $\psi_{2}$ for $\rho=\mu$. This _must not_ be interpreted as the energy of neutrinos with flavor $\rho$, since by definition they have no definite energy. Indeed, the non–diagonality of (34) shows that it is impossible to classify flavor neutrinos with a single energy $\omega$. * (29) A. O. Barut, H. Duru, Phys. Rev. D 36, 3705 (1987). * (30) We use here two discrete families of surfaces, but also continuous families would do the trick, as long as the same limiting process is involved ($\Sigma^{\pm}\rightarrow\mathcal{I}^{\pm}$). * (31) R. Kerner, R. B. Mann, Class.Quant.Grav. 25, 095014 (2008). * (32) S. W. Hawking, Particle creation by black holes, Comm. Math. Phys. 43 , no. 3, 199–220 (1975). * (33) G. Lambiase, G. Papini, Raffaele Punzi, G. Scarpetta, Phys.Rev. D 71, 073011 (2005). * (34) G. Lambiase, Mon.Not.Roy.Astron.Soc. 362, 867-871 (2005).

2024-09-04T02:54:56.045374

2003.00532

arxiv-papers

language=Shell, columns=flexible, basicstyle=, literate=$$1 language=mlir, basicstyle=, columns=flexible, mathescape # High Performance Code Generation in MLIR: An Early Case Study with GEMM Uday Bondhugula Dept of CSA, Indian Institute of Science & PolyMage Labs Bengaluru, Karnataka, India. <EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract This article is primarily meant to present an early case study on using MLIR [1], a new compiler intermediate representation infrastructure, for high- performance code generation. Aspects of MLIR covered in particular include memrefs, the affine dialect, and polyhedral utilities and pass infrastructure surrounding those. This article is also aimed at showing the role compiler infrastructure could play in generating code that is competitive with highly tuned manually developed libraries, albeit in a more modular, reusable, and automatable way. _K_ eywords MLIR $\cdot$ GEMM $\cdot$ polyhedral $\cdot$ code generation ## 1 Introduction, Motivation, and Setup It is well-known that the computationally demanding routines driving the state-of-the-art in domains such as dense linear algebra and deep learning are all based on carefully hand-optimized and tuned libraries developed with significant engineering effort and insights over time. The techniques and the level of attention necessary to optimize for near-peak performance on a target architecture makes the process often inaccessible to those without a deep knowledge of code optimization and low-level interactions with hardware. In many cases, the best performing code comes from the hardware vendors themselves. Yet, fortunately, there are published works such as those of Goto and Van de Geijn [2] that have described in great detail how such close to peak performance could be obtained. Subsequent works [3] made the process more modular, reusable, and accessible to a wider audience, having translated to an open-source project FLAME/BLIS. This article alludes to the fact that this process could potentially be made even more modular, automatable and systematic — by hosting it on top of a real compiler IR that has the necessary abstractions and infrastructure. This completely avoids the need to write any code by hand — be it C or inline assembly. The IR infrastructure that will be used here is of course, MLIR [1, 4] and we will try to recreate the OpenBLAS/BLIS’ optimization approach in a compiler-oriented manner using MLIR. MLIR is a new intermediate representation designed to provide a unified, modular, and extensible infrastructure to progressively lower dataflow compute graphs, through loop nests potentially, to high-performance target-specific code. While MLIR shares similarities with traditional CFG-based three-address SSA representations (including LLVM IR and Swift intermediate language), it also introduces notions from the polyhedral compiler framework as first class concepts to allow powerful analysis and transformation in the presence of loop nests and multi-dimensional arrays. It is thus a useful infrastructure suitable for developing new compilers, whether general-purpose or domain-specific, and whether targeting general-purpose multicores or specialized accelerator chips. ### 1.1 Setup We are going to be using an Intel Skylake-based high-end desktop/workstation processor for all experimentation. The processor is an Intel(R) Core(TM) i7-8700K CPU @ 3.70GHz, which is actually based on Coffee Lake, a process refinement of Skylake, and thus with the same core/performance characteristics. Note that although this is based on the Skylake microarchitecture, it is not a SkyLake-X: as such, its vectors are not AVX-512 but just AVX-2 (256-bit). It has a 32 KiB L1 data cache and a 256 KiB L2 unified cache per core, and a 12 MiB shared L3 cache (2 MiB / core). Before we start, we are going to compute the machine peak on this processor. This CPU supports AVX-2 and has two FMA units that can operate on 256-bit wide vectors. As such, one can perform $2*4*2$ double-precision floating-point multiply and adds per cycle, which at the max turbo boost frequency of 4.7 GHz translates to 75.2 GFLOPS per core. (Frequency scaling can impact how this figure is calculated, but we will ignore that for now.) ### 1.2 Running example: DGEMM Matrix-matrix multiplication is often an excellent routine for a tutorial or exercise in code optimization because a number of standard practices could be demonstrated using it. It is also one of the most important ones for many domains and thus often the first thing one builds an optimized implementation for when rolling out a new architecture. ### 1.3 Starters Let us take a 2088x2048 double precision matrix-matrix multiplication, and benchmark it in various ways. We choose 2088 here since it is divisible by both 4 and 3: this makes it easier to read some of the snippets given that we will be exploring register tiling by sizes that are multiples either of 2, 3, or 4. #### 1.3.1 Compilers with -O3, Pluto Let us see what current compilers do if this is written as a naive nest in C. These results can be reproduced with the setup in Pluto from under examples/matmul/. (We use -ffast-math along with -O3 to be fair given the other things we are going to compare this with.) ⬇ $ clang -v clang version 8.0.0 (Fedora 8.0.0-3.fc30) Target: x86_64-unknown-linux-gnu $ clang -O3 -ffast-math -DTIME matmul.c -o matmul.clang -lm $ ./matmul.clang 36.484855s 0.47 GFLOPS $ gcc –version gcc (GCC) 9.2.1 20190827 (Red Hat 9.2.1-1) $ gcc -O3 -ffast-math -DTIME matmul.c -o matmul.gcc -lm $ ./matmul.gcc 3.788421s 4.53 GFLOPS Disappointingly, clang and GCC are at 0.6% and 6% of the machine peak respectively! But general-purpose compilers are not expected or meant to get anywhere to the peak. Programmers instead typically use highly tuned libraries. We are going to get to that shortly, but a quick note on why Clang performs poorly here. Clang only performs innermost loop vectorization, which is really a bad choice here (due to poor locality) when that’s the only thing being done. Even a textbook loop interchange here from $ijk\rightarrow ikj$ improves Clang performance by about 10x (because now the right loop would be at the innermost level for both locality and auto-vectorization). Clang does not perform any unroll-and-jam here either. While on this let us also see what a polyhedral tool Pluto, a source to source translator, does on this. The below run shows that it is at about 25% of the machine peak, which is better but also pretty unsatisfying. ⬇ $ make tiled $ ./tiled 0.889177s 19.32 GFLOPS #### 1.3.2 Libraries Now, let us see what the fastest libraries in the world do on matmul. Again, the setup in Pluto allows experimenting with OpenBLAS, BLIS, or MKL quickly. ⬇ $ make mkl blis openblas $ ./blis 63.29 GFLOPS $ ./openblas 0.259527s 67.49 GFLOPS $ ./mkl 0.273492s 67.70 GFLOPS MKL and OpenBLAS are nearly at 92% of the peak (68/75.2), while BLIS is close at 85% of the peak. We will keep these in mind as we try out how to build high-performance versions with MLIR. Note: we haven’t been very rigorous with the above experimentation in terms of performing warmup runs, taking an average across repetitions etc., but our goal is to just get a reasonably accurate picture for now. ## 2 Using MLIR to Optimize GEMM There is currently no C/C++ or another language/programming model frontend that emits MLIR. The way to get something in MLIR run on CPUs is through mlir- cpu-runner which can take MLIR as input and JIT compile and execute it. As part of this process, optimized MLIR is converted to LLVM IR, which then goes through its optimization pipeline to generate target code, all of this through mlir-cpu-runner. ### 2.1 Affine Ops Here is how a simple canonical matrix-matrix multiplication looks like in MLIR – the structure is close to its form in C. ⬇ // C += A * B. func @matmul(%A: memref<2048x2048xf64>, %B: memref<2048x2048xf64>, %C: memref<2048x2048xf64>) { affine.for %arg3 = 0 to 2048 { affine.for %arg4 = 0 to 2048 { affine.for %arg5 = 0 to 2048 { %a = affine.load %A[%arg3, %arg5] : memref<2048x2048xf64> %b = affine.load %B[%arg5, %arg4] : memref<2048x2048xf64> %ci = affine.load %C[%arg3, %arg4] : memref<2048x2048xf64> %p = mulf %a, %b : f64 %co = addf %ci, %p : f64 affine.store %co, %C[%arg3, %arg4] : memref<2048x2048xf64> } } } return } func @main() { %A = alloc() : memref<2048x2048xf64> %B = alloc() : memref<2048x2048xf64> %C = alloc() : memref<2048x2048xf64> %cf1 = constant 1.00000e+00 : f64 linalg.fill(%A, %cf1) : memref<2048x2048xf64>, f64 linalg.fill(%B, %cf1) : memref<2048x2048xf64>, f64 linalg.fill(%C, %cf1) : memref<2048x2048xf64>, f64 call @matmul(%A, %B, %C) : (memref<2048x2048xf64>, memref<2048x2048xf64>, memref<2048x2048xf64>) -> () call @print_memref_2d_f64(%C): (memref<2048x2048xf64>) -> () return } func @print_memref_2d_f64(memref<2048x2048xf64>) affine.for and affine.load/store are ops from the Affine dialect [5] used above. The IR above also uses a helper op from the LinAlg dialect to initialize matrices. The rest like (addf, mulf, alloc) are all ops from MLIR’s standard dialect. ### 2.2 MemRefs A memref [6] in MLIR is the in-memory representation of a tensor in MLIR. Depending on its layout map, it could refer to a potentially non-contiguous sub-region of the underlying buffer. All memrefs in the above snippet have the default identity layout map $(d0,d1)\rightarrow(d0,d1)$, which corresponds to a row major layout in memory. 2088 x 2048 is the shape of memref where each of its elements is an f64 (double). There is other information such as an affine layout map (a mapping for its logical coordinate space to physical memory) that is elided when these are identity maps (the default). We will look at an example of this a little later (Section 2.9). The only way to access the elements of a memref is through load and store operations. ### 2.3 A high-level domain-specific op that can expand MLIR is extensible in several ways, and one of them is in the ease of adding/creating new ops at the level of abstraction we desire. One way of classifying ops or IR in general in MLIR is into high-level, mid-level, and low-level, although there is not a clear distinction and lowering can be progressive and you can have a mix. High-level ops operate on tensor and memref types themselves, i.e., their inputs (operands) and results are of that type. affine.for and affine.if are mid-level ops – these have nested blocks (which are themselves a list of ops) and a particular structure to them. Low- level ops are at the same level as instructions in LLVM, and these typically correspond to either one or a flat sequence of instructions matching a target (three address code style in other words). If one is building a DSL and knows that they need or have a matmul, one could just define an op that does the matmul taking memrefs as operands. For the purpose of this article, we will create a hop.matmul op. An operation in MLIR has inputs, results, and attributes (ones with regions have region arguments as well). ⬇ hop.matmul %A, %B, %C {some_attr = 96, another_attr = 4} : (memref<2088x2048xf64>, memref<2048x2048xf64>, memref<2088x2048xf64>) hop.matmul above is an operation that operates on three memrefs, %A, %B, and %C, which are the LHS, the RHS, and the output corresponding to a matrix- matrix multiplication. This op has zero results: since these are memrefs, the underlying memory can be updated, but the memref itself is an SSA value and thus can be defined only once. In this case, %C will be both read and written by this heavy op. some_attr is an attribute of the op, which like with LLVM metadata is meant to be a constant of one of the numerous types available including some polyhedral types like affine maps and integer sets. Such polyhedral attributes could be used to encode powerful information and we will see an example of this later. For the purpose of this article, we create an -hopt pass that is able to expand out hop.matmul into the naive 3-d loop nest we showed further above. Let us just execute this with MLIR without any optimization to see where we are starting our journey from. ⬇ // This is where we start. No optimizations performed in MLIR besides canonicalization. $ mlir-opt -hopt -convert-linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -entry-point-result=void -shared- libs=lib/libmlir_runner_utils.so Compilation time: 0.015995s 0.558177 GFLOPS This is pretty close to the execution time of ‘clang -O3’ that we saw above, which makes sense, but this is equally slow (less than 1% of the machine peak)! But we have not really made use of or run any of the optimization infrastructure in MLIR. The -hopt here just lowers this op into the canonical 3-d loop nest we showed further above, which is then lowered to LLVM and goes through the same LLVM -O3 pipeline that clang went through. ### 2.4 Tiling for caches: OpenBLAS/BLIS tiling as a polyhedral schedule Tiling and blocking has been studied for several decades, and it is now known how matrix-matrix multiplication should be tiled for high performance on widely used architectures. We are going to use the same cache tiling strategy used in OpenBLAS and BLIS. [This scheme](https://dl.acm.org/citation.cfm?id=1356053) is a very clever one that carefully tiles for multiple levels of the cache in a way that exploits reuse for each of the matrices involved at one level or another: the objective is to make sure the vector FMA/add mul pipeline remains full and one does not wait for loads. It does not blindly tile the nest multiple times to exploit reuse for all matrices at L1, L2, and L3, as would most polyhedral tools do. There are various ways of describing OpenBLAS’s/BLIS’ tiling scheme and they are explained in excellent detail with illustration in the papers by Van Zee and van de Geijn on BLIS [3], and later with analysis and modeling by Low et al. 2016 [7]. But here, we will describe the tiling transformation compactly via polyhedral schedules. For an introduction to polyhedral schedules, see here [8] or the material on polyhedral.info. In particular, the tiling and corresponding intra-tile loop orders used in the OpenBLAS/BLIS approach can be expressed as the following polyhedral schedule (a multi-dimensional affine map / transformation): $(i,j,k)\rightarrow(j/N_{C},k/K_{C},i/M_{C},j/N_{R},i/M_{R},k\ \textrm{ mod }\ K_{C},j\ \textrm{ mod }\ N_{R},i\textrm{ mod }M_{R}),$ where ‘/’ is a floor division. The innermost two loops after applying the above schedule are meant to be fully unrolled leading to an $M_{R}$ x $N_{R}$ loop body. Ignoring the last level of tiling (for the L3 cache), this becomes: $(i,j,k)\rightarrow(k/K_{C},i/M_{C},j/N_{R},i/M_{R},k\textrm{ mod }K_{C},j\textrm{ mod }N_{R},i\textrm{ mod }M_{R}).$ Figure 1 from the BLIS works is a great illustration of its optimization strategy: Figure 1: Tiling strategy of OpenBLAS/BLIS. Figure courtesy: Field Van Zee and Robert van de Geijn The parameters ($M_{C}$, $K_{C}$) are chosen such that a tile of the LHS matrix (%A) of size $M_{C}$ x $K_{C}$ is reused in the L2 cache, an RHS matrix tile of size $K_{C}$ x $N_{R}$ is reused in the L1 cache, while a register tile of the output matrix $M_{R}$ x $N_{R}$ is reused in just the registers. There are several other considerations in choosing these parameters if one is interested in analytical modeling; those are described in the work of Low et al. [7]. The last three dimensions of the schedule are multiplying a panel of %A of size $M_{R}$ x $K_{C}$ with a panel of %B of size $K_{C}$ x $N_{R}$. Note that %C is being both read and written, and intuitively, it makes sense to exploit reuse for it in the registers while running a large enough k intra- tile loop at the innermost level. In the schedule above, the innermost two dimensions (j, i) are fully unrolled, i.e., you get an unrolled body of size $M_{R}$ x $N_{R}$ times the original one. For readers seeking more detail in general on this and on programming for high performance, please see course notes [9] from the BLIS team members. ### 2.5 Tiling in MLIR There are two ways of achieving this tiling in MLIR: one by calling [mlir::tile] (which is also what the loop tiling pass in MLIR uses), and then performing the desired loop interchange via mlir::interchangeLoops The other is by implementing a higher order polyhedral (HOP) approach based on domains and schedules. We use this latter approach here since MLIR’s affine analysis machinery does not yet have the necessary simplification to get rid of certain redundant bounds resulting from tiled code generation in advanced cases. The HOP approach depends on an external library, [ISL](http://isl.gforge.inria.fr), and we implement this as part of the -hopt pass. We express the tiling schedule as an MLIR affine map (in fact, any affine schedule could be used), perform the code generation via ISL, and convert the ISL AST back to MLIR. We will now use $M_{C}$, $K_{C}$, $M_{R}$, $N_{R}$ as attributes. on the hop.matmul op. ⬇ hop.matmul %A, %B, %C { M_C = 64 : i32, K_C = 256 : i32, M_R = 4 : i32, N_R = 8 : i32} : (memref<2088x2048xf64>, memref<2048x2048xf64>, memref<2048x2048xf64>) We have used $K_{C}$ = 256, $M_{C}$ = 64, $M_{R}$ = 4, $N_{R}$ = 8 as a starting point: these can be analyzed to be good values given the cache size constraints described earlier. This is functionally equivalent to using the following schedule with HOP and fully unrolling the last two dimensions, d1, d0, which will be of trip counts 8 and 4 respectively): ⬇ hop.matmul %A, %B, %C { schedule = (d0, d1, d2) -> (d2 floordiv 256, d0 floordiv 64, d1 floordiv 8, d0 floordiv 4, d2, d1, d0) } : (memref<2088x2048xf64>, memref<2048x2048xf64>, memref<2088x2048xf64>) In order to just evaluate cache tiling for now, we will just use the following schedule (drop the tiling for registers): ⬇ hop.matmul %A, %B, %C { schedule = (d0, d1, d2) -> (d2 floordiv 256, d0 floordiv 64, d1, d0, d2) } : (memref<2088x2048xf64>, memref<2048x2048xf64>, memref<2088x2048xf64>) Now, with the above tiling, we get this loop nest: ⬇ … #map8 = (d0) -> (d0 * 16) #map9 = (d0) -> (522, d0 * 16 + 16) … func @matmul(%A: memref<2088x2048xf64>, %B: memref<2048x2048xf64>, %C: memref<2088x2048xf64>) { %c0 = constant 0 : index affine.for %arg3 = 0 to 8 { affine.for %arg4 = 0 to 33 { affine.for %arg5 = 0 to 2048 { affine.for %arg6 = #map7(%arg4) to min #map8(%arg4) { %0 = alloca() : memref<1xf64> %1 = affine.load %C[%arg6, %arg5] : memref<2088x2048xf64> affine.store %1, %0[%c0] : memref<1xf64> affine.for %arg7 = 0 to 256 { %3 = affine.load %A[%arg6, %arg3 * 256 + %arg7] : memref<2088x2048xf64> %4 = affine.load %B[%arg3 * 256 + %arg7, %arg5] : memref<2048x2048xf64> %5 = affine.load %0[0] : memref<1xf64> %6 = mulf %3, %4 : f64 %7 = addf %5, %6 : f64 affine.store %7, %0[0] : memref<1xf64> } %2 = affine.load %0[%c0] : memref<1xf64> affine.store %2, %C[%arg6, %arg5] : memref<2088x2048xf64> } } } } return } Note that the invariant load on %B has been hoisted out. When we execute this: ⬇ // With tiling. $ mlir-opt -hopt -convert-linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -entry-point-result=void -shared- libs=lib/libmlir_runner_utils.so Compilation time: 0.0443082s 1.6012 GFLOPS This is nearly a 3x improvement in performance, but still nowhere close to the machine peak or the performance of MKL, OpenBLAS or BLIS! ### 2.6 Explicit Copying or Packing Whenever a code with multi-dimensional arrays exhibits reuse, explicit copying or packing is a commonly used technique in HPC, which first copies or packs accessed data into contiguous buffers and indexes those buffers for computation. Such copying reduces or nearly eliminates conflict misses, TLB misses, and improves hardware prefetching performance (since the access will lead to fewer prefetch streams). More importantly, in conjunction with tiling, it provides the intended benefits of tiling. On the one hand, tiling is performed so that reuse is exploited in multiple directions when the data accessed fits in a higher level of the memory hierarchy, on the other hand, the data accessed for a tile is no longer contiguous in the original matrix/tensor leading to conflict misses, TLB misses, and more prefetch streams — taking away a lot of the gain even if there is high reuse. There is also a choice in terms of which depth / where one would like to introduce the copies, but this is quite clear for the tiling strategy we are using. The packing optimization is something that one would really wish a compiler performs automatically. MLIR has a pass -affine-data-copy-generate and a utility mlir::affineDataCopyGenerate that can perform explicit copying or packing. The utility can also be called on a specific memref to perform the packing at a specific loop depth, and with a number of other options (including DMA support). With the tiling approach we are using here, packing is performed for the LHS matrix %A right under the second loop ($i\textrm{ floordiv }K_{C}$ dimension in the schedule). For %B (RHS matrix), the copying is performed right under j floordiv $N_{R}$ (the third dimension). Figure 2 illustrates the packing needed. Figure 2: The one in green is the LHS to be packed into a buffer that will reside in L2, and the RHS in blue will reside in L1. (Figure courtesy: Field Van Zee and Robert van de Geijn). To perform the packing automatically with MLIR, instead of calling the -affine-data-copy-generate pass, we will just use the underlying utility [mlir::affineDataCopyGenerate] to place copies at the right depths. For the purpose of this article, we enabled this under the -hopt-copy option. With -hopt-copy, now the nest looks like: ⬇ affine.for %arg3 = 0 to 8 { affine.for %arg4 = 0 to 32 { %0 = alloc() : memref<64x256xf64> // Packing %A into a 64x256 buffer. affine.for %arg5 = #map6(%arg4) to #map7(%arg4) { affine.for %arg6 = #map4(%arg3) to #map5(%arg3) { %1 = affine.load %arg0[%arg5, %arg6] : memref<2048x2048xf64> affine.store %1, %0[%arg4 * -64 + %arg5, %arg3 * -256 + %arg6] : memref<64x256xf64> } } affine.for %arg5 = 0 to 256 { %1 = alloc() : memref<256x8xf64> // Packing %B into a 256x8 buffer. affine.for %arg6 = #map4(%arg3) to #map5(%arg3) { affine.for %arg7 = #map9(%arg5) to #map10(%arg5) { %2 = affine.load %arg1[%arg6, %arg7] : memref<2048x2048xf64> affine.store %2, %1[%arg3 * -256 + %arg6, %arg5 * -8 + %arg7] : memref<256x8xf64> } } affine.for %arg6 = #map16(%arg4) to #map17(%arg4) { // This is multiplying a packed 64x256 LHS panel with a packed 256x8 RHS panel. affine.for %arg7 = 0 to 256 { affine.for %arg8 = 0 to 8 { affine.for %arg9 = 0 to 4 { %2 = affine.load %0[%arg4 * -64 + %arg6 * 4 + %arg9, %arg7] : memref<64x256xf64> %3 = affine.load %1[%arg7, %arg8] : memref<256x8xf64> %4 = affine.load %arg2[%arg6 * 4 + %arg9, %arg5 * 8 + %arg8] : memref<2048x2048xf64> %5 = mulf %2, %3 : f64 %6 = addf %4, %5 : f64 affine.store %6, %arg2[%arg6 * 4 + %arg9, %arg5 * 8 + %arg8] : memref<2048x2048xf64> } } } } dealloc %1 : memref<256x8xf64> } dealloc %0 : memref<64x256xf64> } } ⬇ // With tiling and packing. $ mlir-opt -hopt -hopt-copy -convert-linalg-to-loops -lower-affine -convert- std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.0506358s 2.38624 GFLOPS This is nearly a 1.5x improvement in performance, but in absolute terms still pretty bad. On a general note, the benefit of any optimization often depends on how well optimized the code is on other aspects before the optimization was applied – even more so in this case. It’s important to be mindful of this in order to not draw incorrect conclusions on the relative benefits of each individual transformation at this time. For example, the benefits of cache tiling on a code where unroll-and-jam or register tiling has already been performed is significantly lower than on a version where no optimizations were performed. We will look at an example a little later. ### 2.7 Unroll and Jam We will now use the complete schedule that has the $M_{R}$ and $N_{R}$ register tiling so that an unroll-and-jam of the innermost two loops by $N_{R}$ and $M_{R}$ respectively could be achieved, but we actually didn’t enable that yet. We use the option -hopt-unroll to turn this on and off. We also run scalar replacement in MLIR post unroll-and-jam. Besides turning memref locations being reduced into scalars (single element memrefs) and hoisting them out, it also eliminates redundant loads, and hoists invariant loads out of loops. While at unroll-and-jam, we will also add one more parameter to the list, $K_{U}$, which will be the unroll factor for the $K_{C}$ loop (intra-tile loop corresponding to k reduction dimension). We use $K_{U}$ = 4 to unroll the k loop (which will end up as the innermost loop post all register tiling). So, we have: ⬇ hop.matmul %A, %B, %C { M_C = 64 : i32, K_C = 256 : i32, M_R = 4 : i32, N_R = 8 : i32, K_U = 4 : i32} : (memref<2088x2048xf64>, memref<2048x2048xf64>, memref<2088x2048xf64>) ⬇ // With tiling, packing, and unroll-and-jam/unroll. $ mlir-opt -hopt -hopt-copy -hopt-unroll -hopt-scalrep -convert-linalg-to- loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.11306s 23.2737 GFLOPS A code is only as fast as its weakest link. While the cache tiling really optimized reuse for the LHS and RHS, the strategy used by OpenBLAS and BLIS exploits reuse for the output matrix only in registers (not in L1 nor L2). As such, without unroll-and-jam aka register tiling, brakes had been applied. So, this last step opened the floodgates here yielding a 10x improvement and getting us to 23 GFLOPS. Let us go back a step and check what would happen if we disabled packing while performing unroll-and-jam. ⬇ // With tiling, unroll-and-jam/unroll, but no packing. $ mlir-opt -hopt -hopt-copy=false -hopt-unroll -hopt-scalrep -convert-linalg- to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.0568249s \\\ 11.5595 GFLOPS We lose over 2x of the performance if we do not perform packing here! And if we had not performed the innermost loop unrolling (i.e., set $K_{U}$ = 1), we get: ⬇ // With tiling, packing, and unroll-and-jam/unroll, but K_U = 1\. $ mlir-opt -hopt -hopt-copy -hopt-unroll -hopt-scalrep -convert-linalg-to- loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.0881901s 20.249 GFLOPS All of these improvements are significant from where we started but they are all finally not enough at all on an absolute scale — we are still at about 34% of the peak even after having reproduced a good part of the OpenBLAS/BLIS recipe! Clearly, there is something orthogonal that all of this is missing. It looks like we completely missed vectorization! The highly tuned library kernels that we have set as a target use a careful selection and schedule of vector instructions in inline assembly. Let us see whether our code is even being vectorized reasonably under the hood by LLVM. It’s pretty straightforward to check this. Instead of looking at the pass output remarks, we will just look at the generated assembly. The ’-dump- object-file -object-filename’ options of mlir-cpu-runner are useful for this purpose. ⬇ $ mlir-opt -hopt -hopt-copy -hopt-unroll -hopt-scalrep -convert-linalg-to- loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so -dump-object-file -object-filename=hopt.o $ llvm-objdump -d hopt.o … 14e5: c4 e2 a9 b9 e2 vfmadd231sd %xmm2, %xmm10, %xmm4 1856: c4 42 e5 a8 f8 vfmadd213pd %ymm8, %ymm3, %ymm15 185b: c4 e3 79 05 a9 30 ff ff ff 01 vpermilpd $1, -208(%rcx), %xmm5 1865: c4 41 7d 28 e6 vmovapd %ymm14, %ymm12 186a: c4 63 fd 01 f5 55 vpermpd $85, %ymm5, %ymm14 1870: c4 62 7d 19 c5 vbroadcastsd %xmm5, %ymm8 1875: c4 62 e5 a8 c2 vfmadd213pd %ymm2, %ymm3, %ymm8 187a: c4 42 e5 a8 f5 vfmadd213pd %ymm13, %ymm3, %ymm14 187f: c5 fd 10 b9 40 ff ff ff vmovupd -192(%rcx), %ymm7 1887: c4 e2 7d 19 b1 40 ff ff ff vbroadcastsd -192(%rcx), %ymm6 1890: c4 e3 fd 01 d7 ff vpermpd $255, %ymm7, %ymm2 1896: c4 e2 e5 a8 d0 vfmadd213pd %ymm0, %ymm3, %ymm2 189b: c5 fd 29 93 00 01 00 00 vmovapd %ymm2, 256(%rbx) 18a3: c4 e3 fd 01 ef 55 vpermpd $85, %ymm7, %ymm5 18a9: c4 63 fd 01 ef aa vpermpd $170, %ymm7, %ymm13 18af: c4 62 e5 a8 ab 80 03 00 00 vfmadd213pd 896(%rbx), %ymm3, %ymm13 18b8: c4 e2 e5 a8 e9 vfmadd213pd %ymm1, %ymm3, %ymm5 18bd: c4 c2 e5 a8 f4 vfmadd213pd %ymm12, %ymm3, %ymm6 18c2: c5 fb 10 99 60 ff ff ff vmovsd -160(%rcx), %xmm3 18ca: c5 f9 28 93 20 01 00 00 vmovapd 288(%rbx), %xmm2 18d2: c4 62 e9 b9 cb vfmadd231sd %xmm3, %xmm2, %xmm9 18d7: c4 c1 7b 10 bc f7 f0 ef ff ff vmovsd -4112(%r15,%rsi,8), %xmm7 18e1: c4 62 e1 b9 df vfmadd231sd %xmm7, %xmm3, %xmm11 18e6: c4 c1 7b 10 84 f7 f0 f7 ff ff vmovsd -2064(%r15,%rsi,8), %xmm0 18f0: c4 62 e1 b9 d0 vfmadd231sd %xmm0, %xmm3, %xmm10 18f5: c4 c1 7b 10 4c f7 f0 vmovsd -16(%r15,%rsi,8), %xmm1 18fc: c4 e2 f1 a9 dc vfmadd213sd %xmm4, %xmm1, %xmm3 1901: c5 c1 14 e2 vunpcklpd %xmm2, %xmm7, %xmm4 1905: c5 f1 14 c0 vunpcklpd %xmm0, %xmm1, %xmm0 1909: c4 e3 7d 18 cc 01 vinsertf128 $1, %xmm4, %ymm0, %ymm1 190f: c4 62 7d 19 a1 68 ff ff ff vbroadcastsd -152(%rcx), %ymm12 1918: c4 42 f5 a8 e7 vfmadd213pd %ymm15, %ymm1, %ymm12 191d: c4 e3 79 05 a1 70 ff ff ff 01 vpermilpd $1, -144(%rcx), %xmm4 1927: c4 e3 fd 01 c4 55 vpermpd $85, %ymm4, %ymm0 192d: c4 e2 7d 19 fc vbroadcastsd %xmm4, %ymm7 1932: c4 c2 f5 a8 c6 vfmadd213pd %ymm14, %ymm1, %ymm0 1937: c4 c2 f5 a8 f8 vfmadd213pd %ymm8, %ymm1, %ymm7 … This is really not the code we would like to see in the innermost loop! Although the code is vectorized, it is using XMM registers (128-bit) for a good part instead of YMM ones (256-bit). More importantly, the FMAs are regularly accompanied by loads around them and neither should this have been necessary nor is it any good for performance! A good inner loop here is expected to only have one load of the LHS and RHS every $N_{R}$ and $M_{R}$ FMA ops, roughly speaking, because that’s the amount of reuse we should be getting. In addition, there should be no loads or stores for the output matrix in the innermost loop, which happens to be the case here. And finally, it all should have been %ymm. It is also important to note that vectorization is often nearly useless unless the code has been optimized for locality, (i.e., we should be getting data from caches at least), and it is in fact still not as useful unless we are not reading from registers most of the time. We will not dig and analyze any further here to figure out what went wrong with clang/LLVM and why (this is actually due to a sub-optimal vectorization choice)! Instead, we will see how we could get the vectorization done in MLIR itself where we have all the information and opportunity. ### 2.8 Vectorization MLIR supports vector types and vector types as element types for memrefs. Unlike in LLVM, vector types in MLIR can be multi-dimensional. However, we only need 1-d vector types here. Loop vectorization in MLIR would just lead to IR that turns memrefs of f64 into memrefs of vector of f64 besides transforming loops/loop bodies. A casting op that changes the elemental type of a memref, and the splat op are also needed here. In more complex cases, the view/subview ops are also needed. But for this benchmark, the vectorization needed is quite straightforward, and is a simple outer loop vectorization along the j loop. The existing “super vectorizer” in MLIR is really not functional or complete for the auto-vectorization we need. For this article, we build and use a new loop vectorizer (enabled by -hopt-vect or via -affine-vectorize when running separately). We introduce a new memref_shape_cast op which is needed to change the elemental type on a memref. Here’s how the vectorized MLIR looks like if we started with the naive matmul nest. ⬇ // mlir-opt -hopt-vect func @matmul(%A: memref<2048x2048xf64>, %B: memref<2048x2048xf64>, %C: memref<2048x2048xf64>) { %0 = memref_shape_cast %B : memref<2048x2048xf64> to memref<2048x512xvector<4xf64>> %1 = memref_shape_cast %C : memref<2048x2048xf64> to memref<2048x512xvector<4xf64>> affine.for %arg3 = 0 to 2048 { affine.for %arg4 = 0 to 512 { affine.for %arg5 = 0 to 2048 { %2 = affine.load %A[%arg3, %arg5] : memref<2048x2048xf64> %3 = splat %2 : vector<4xf64> %4 = affine.load %0[%arg5, %arg4] : memref<2048x512xvector<4xf64>> %5 = mulf %3, %4 : vector<4xf64> %6 = affine.load %1[%arg3, %arg4] : memref<2048x512xvector<4xf64>> %7 = addf %5, %6 : vector<4xf64> affine.store %7, %1[%arg3, %arg4] : memref<2048x512xvector<4xf64>> } } } return } The above IR is generated by vectorizing along the $j$ loop, which is a data parallel loop as opposed to a reduction dimension). The memref_shape_cast op casts a memref to one of another shape as long as the product of the dimension sizes all the way up to any aggregate elemental types remain the same. memref_shape_cast actually does not exist in MLIR trunk. Also, we do not discuss the case of trip counts not being a multiple of vector size here since some of the infrastructure around it in MLIR is still being built. Let us look at the performance with vectorization and with nothing else enabled. ⬇ // Just vectorization. Compilation time: 0.016561s 1.53043 GFLOPS #### 2.8.1 Rewind and measure with vectorization Let us rewind and see how the vectorized code performs step by step, with tiling, with packing, and with register tiling / unroll-and-jam. ⬇ // Just vectorization. Compilation time: 0.016561s 1.53043 GFLOPS # Vectorization with tiling Compilation time: 0.0209861s 7.05307 GFLOPS As mentioned earlier, simply vectorizing without worrying about memory bandwidth gets us nowhere. We have now broken that barrier, and can go further here. ⬇ // Vectorization, tiling, and packing/copying $ mlir-opt -hopt -hopt-vect -hopt-copy -convert-linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -entry-point-result=void -reps=5 -shared-libs=lib/libmlir_runner_utils.so,/usr/local/lib/libblis.so Compilation time: 0.0409529s 11.2309 GFLOPS // Vectorization, tiling, packing, and unroll-and-jam, unrolling with MLIR scalar replacement. $ mlir-opt -hopt -hopt-vect -hopt-copy -hopt-unroll -hopt-scalrep -convert- linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -reps=5 -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.0383081s 49.8336 GFLOPS As observed earlier, once again, the last step has opened the floodgates here yielding a 4.5x improvement and getting us to 49 GFLOPS – only 23% away from BLIS and about 27% away from MKL or OpenBLAS. Let us see how the MLIR looks like right after vectorization, tiling, copying, and unroll-and-jam (we will disable the $K_{C}$ loop unroll, i.e., set $K_{U}$ = 1 to make it easier to read; the unroll-jamming of $i,j$ is still shown). ⬇ affine.for %arg3 = 0 to 8 { affine.for %arg4 = 0 to 33 { %2 = alloc() : memref<64x256xf64> affine.for %arg5 = #map6(%arg4) to min #map7(%arg4) { affine.for %arg6 = #map4(%arg3) to #map5(%arg3) { %3 = affine.load %arg0[%arg5, %arg6] : memref<2088x2048xf64> affine.store %3, %2[%arg4 * -64 + %arg5, %arg3 * -256 + %arg6] : memref<64x256xf64> } } affine.for %arg5 = 0 to 256 { %3 = alloc() : memref<256x2xvector<4xf64>> affine.for %arg6 = #map4(%arg3) to #map5(%arg3) { affine.for %arg7 = #map9(%arg5) to #map10(%arg5) { %4 = affine.load %0[%arg6, %arg7] : memref<2048x512xvector<4xf64>> affine.store %4, %3[%arg3 * -256 + %arg6, %arg5 * -2 + %arg7] : memref<256x2xvector<4xf64>> } } affine.for %arg6 = #map26(%arg4) to min #map27(%arg4) { %4 = alloca() : memref<1xvector<4xf64>> %5 = affine.load %1[%arg6 * 4, %arg5 * 2] : memref<2088x512xvector<4xf64>> affine.store %5, %4[0] : memref<1xvector<4xf64>> %6 = alloca() : memref<1xvector<4xf64>> %7 = affine.load %1[%arg6 * 4 + 1, %arg5 * 2] : memref<2088x512xvector<4xf64>> affine.store %7, %6[0] : memref<1xvector<4xf64>> %8 = alloca() : memref<1xvector<4xf64>> %9 = affine.load %1[%arg6 * 4 + 2, %arg5 * 2] : memref<2088x512xvector<4xf64>> affine.store %9, %8[0] : memref<1xvector<4xf64>> %10 = alloca() : memref<1xvector<4xf64>> %11 = affine.load %1[%arg6 * 4 + 3, %arg5 * 2] : memref<2088x512xvector<4xf64>> affine.store %11, %10[0] : memref<1xvector<4xf64>> %12 = alloca() : memref<1xvector<4xf64>> %13 = affine.load %1[%arg6 * 4, %arg5 * 2 + 1] : memref<2088x512xvector<4xf64>> affine.store %13, %12[0] : memref<1xvector<4xf64>> %14 = alloca() : memref<1xvector<4xf64>> %15 = affine.load %1[%arg6 * 4 + 1, %arg5 * 2 + 1] : memref<2088x512xvector<4xf64>> affine.store %15, %14[0] : memref<1xvector<4xf64>> %16 = alloca() : memref<1xvector<4xf64>> %17 = affine.load %1[%arg6 * 4 + 2, %arg5 * 2 + 1] : memref<2088x512xvector<4xf64>> affine.store %17, %16[0] : memref<1xvector<4xf64>> %18 = alloca() : memref<1xvector<4xf64>> %19 = affine.load %1[%arg6 * 4 + 3, %arg5 * 2 + 1] : memref<2088x512xvector<4xf64>> affine.store %19, %18[0] : memref<1xvector<4xf64>> affine.for %arg7 = 0 to 256 { // Unroll-and-jammed body (M_R = 4, N_R = 8 (but vectorized and hence 8/4 effectively). %28 = affine.load %2[%arg4 * -64 + %arg6 * 4, %arg7] : memref<64x256xf64> %29 = splat %28 : vector<4xf64> %30 = affine.load %3[%arg7, 0] : memref<256x2xvector<4xf64>> %31 = mulf %29, %30 : vector<4xf64> %32 = affine.load %4[0] : memref<1xvector<4xf64>> %33 = addf %31, %32 : vector<4xf64> affine.store %33, %4[0] : memref<1xvector<4xf64>> %34 = affine.load %2[%arg4 * -64 + %arg6 * 4 + 1, %arg7] : memref<64x256xf64> %35 = splat %34 : vector<4xf64> %36 = mulf %35, %30 : vector<4xf64> %37 = affine.load %6[0] : memref<1xvector<4xf64>> %38 = addf %36, %37 : vector<4xf64> affine.store %38, %6[0] : memref<1xvector<4xf64>> %39 = affine.load %2[%arg4 * -64 + %arg6 * 4 + 2, %arg7] : memref<64x256xf64> %40 = splat %39 : vector<4xf64> %41 = mulf %40, %30 : vector<4xf64> %42 = affine.load %8[0] : memref<1xvector<4xf64>> %43 = addf %41, %42 : vector<4xf64> affine.store %43, %8[0] : memref<1xvector<4xf64>> %44 = affine.load %2[%arg4 * -64 + %arg6 * 4 + 3, %arg7] : memref<64x256xf64> %45 = splat %44 : vector<4xf64> %46 = mulf %45, %30 : vector<4xf64> %47 = affine.load %10[0] : memref<1xvector<4xf64>> %48 = addf %46, %47 : vector<4xf64> affine.store %48, %10[0] : memref<1xvector<4xf64>> %49 = affine.load %3[%arg7, 1] : memref<256x2xvector<4xf64>> %50 = mulf %29, %49 : vector<4xf64> %51 = affine.load %12[0] : memref<1xvector<4xf64>> %52 = addf %50, %51 : vector<4xf64> affine.store %52, %12[0] : memref<1xvector<4xf64>> %53 = mulf %35, %49 : vector<4xf64> %54 = affine.load %14[0] : memref<1xvector<4xf64>> %55 = addf %53, %54 : vector<4xf64> affine.store %55, %14[0] : memref<1xvector<4xf64>> %56 = mulf %40, %49 : vector<4xf64> %57 = affine.load %16[0] : memref<1xvector<4xf64>> %58 = addf %56, %57 : vector<4xf64> affine.store %58, %16[0] : memref<1xvector<4xf64>> %59 = mulf %45, %49 : vector<4xf64> %60 = affine.load %18[0] : memref<1xvector<4xf64>> %61 = addf %59, %60 : vector<4xf64> affine.store %61, %18[0] : memref<1xvector<4xf64>> } %20 = affine.load %18[0] : memref<1xvector<4xf64>> affine.store %20, %1[%arg6 * 4 + 3, %arg5 * 2 + 1] : memref<2088x512xvector<4xf64>> %21 = affine.load %16[0] : memref<1xvector<4xf64>> affine.store %21, %1[%arg6 * 4 + 2, %arg5 * 2 + 1] : memref<2088x512xvector<4xf64>> %22 = affine.load %14[0] : memref<1xvector<4xf64>> affine.store %22, %1[%arg6 * 4 + 1, %arg5 * 2 + 1] : memref<2088x512xvector<4xf64>> %23 = affine.load %12[0] : memref<1xvector<4xf64>> affine.store %23, %1[%arg6 * 4, %arg5 * 2 + 1] : memref<2088x512xvector<4xf64>> %24 = affine.load %10[0] : memref<1xvector<4xf64>> affine.store %24, %1[%arg6 * 4 + 3, %arg5 * 2] : memref<2088x512xvector<4xf64>> %25 = affine.load %8[0] : memref<1xvector<4xf64>> affine.store %25, %1[%arg6 * 4 + 2, %arg5 * 2] : memref<2088x512xvector<4xf64>> %26 = affine.load %6[0] : memref<1xvector<4xf64>> affine.store %26, %1[%arg6 * 4 + 1, %arg5 * 2] : memref<2088x512xvector<4xf64>> %27 = affine.load %4[0] : memref<1xvector<4xf64>> affine.store %27, %1[%arg6 * 4, %arg5 * 2] : memref<2088x512xvector<4xf64>> } dealloc %3 : memref<256x2xvector<4xf64>> } dealloc %2 : memref<64x256xf64> } } The memref<1 x f64> values (single element memrefs) that we see below are a result of the scalar replacement pass (-affine-scalrep) that runs after the unroll-and-jam. We see eight of these summing up to 32 elements of the output matrix, and these are held in registers when the innermost loop (k) is iterating. LLVM’s passes later turn these into virtual registers. Both BLIS and OpenBLAS use inline assembly microkernels and other hand written components that are composed in a modular/reusable way. On the other hand, we have been trying to generate all our code all the way down, including using LLVM’s instruction selection, scheduling, and register allocation. Note that we have got here through the hop.matmul with the values of $M_{C}$, $K_{C}$, $M_{R}$, $N_{R}$, $K_{U}$ indicated below: ⬇ hop.matmul %A, %B, %C { M_C = 64 : i32, K_C = 256 : i32, M_R = 4, N_R = 8 : i32, K_U = 4 : i32 } : (memref<2088x2048xf64>, memref<2048x2048xf64>, memref<2088x2048xf64>) Before we look at adjusting these for maximum mileage, we will take a detour to memref layouts. ### 2.9 A Quick Detour into Affine Map Layouts We are going to take a quick detour here to understand how a memref’s layout map [6] works. A memref type has three pieces of information: its shape, a affine layout map, and a memory space. ⬇ memref<64x256xf64, (d0, d1) -> (d0, d1), 0> This is a memref of shape 64x256, and with the identity map (d0, d1) -> (d0, d1) which corresponds to a row-major layout in memory. 64 rows and 256 columns, and each of its elements is a float. An element %A[%i, %j] will map to element ($256*d0+d1$) in the buffer – in this case all elements are contiguous. ’0’ is the memory space the memref lives in. On architectures that have different kinds of explicitly addressable memory, a different memory space id is assigned to each. Whenever the layout map is identity, its printing is elided; similarly, in the case of memory space 0. Now, let us look at the memref corresponding to the buffer for the LHS matrix (%A), which is of type memref<64x256xf64>. Figure 3: LHS buffer layout. This block is being multiplied with the RHS panel of type memref<256x2xvector<2xf64>>. The figure above shows how the elements of A block get traversed: along columns of height $M_{R}$ all the way up to $K_{C}$ columns, and then turn around for the next panel, until one has processed all $M_{C}$/$M_{R}$ panels. Note that the elements are not being accessed contiguously. But since the block of A is L2 cache resident and is just streamed through L1, we still get both spatial reuse for it in L1 as well as L2, i.e., it is just that the reuse in space for A does not happen immediately but once you’ve traversed the $M_{R}$ height. And we know per this whole approach that it still stays in cache. However, there is no harm (and it can only potentially get better) if we ensure that A is accessed contiguously. For eg., it can only help with prefetching into L1. Hence, we would like to pick a layout different from the default row-major for memref<64x256xf64>, and MLIR’s affine layout maps make it convenient to express the desired layout here in a compact and powerful way. If we simply change the memref type to: memref<64x256xf64, (d0, d1) $\rightarrow$ (d0 floordiv $M_{R}$, d1, d0 mod $M_{R}$)> it yields the desired layout. The rest of the IR does not need a single change! The mapping specifies that a $(d0,d1)$ in the logical access space should be mapped to (d0 flooridv $M_{R}$, d1, d0 mod $M_{R}$) in a physical (contiguous) buffer of size 16x256x4. When mlir::normalizeMemRef runs, it will turn this memref into: memref<16x256x4xf64> And an access Abuf[%i, %j] is remapped to Abuf[%i floordiv 4, %j, %i mod 4]. This is an example of a tiled layout, akin to how iteration space tiling is performed – stripmine and interchange – but on the data space. As another example, here’s a memref that accesses a non-contiguous sub-space of its underlying buffer. memref<64x256xf64, (d0, d1) -> (d0, d1 floordiv 6, d1 mod 6)> Note that we would have a “padding” worth 2 elements (256 % 6) at the end of each row that can never be accessed via this memref while staying in bounds. Another way to write this is: memref<64x256xf64, $(d0,d1)\rightarrow(d0*258+d1)$> A memref with access strides say 128, 2 (for major, minor respectively) in a larger underlying buffer can be expressed for example as: memref<126x100xf32, $(d0,d1)\rightarrow(d0*128+d1*2)$>. More examples can be found here. The copy options supplied to mlir::affineDataCopyGenerate allows one to choose a custom data layout for the buffer (being copied into/from). One can choose any layout map as long as it is injective: any injective layout will lead to semantically correct code. We will see that this layout is only needed for an experiment later (Section 2.12). We will now get back to adjusting parameters to go full throttle. ### 2.10 Tweaking $M_{C}$, $K_{C}$, $M_{R}$, $N_{R}$ to maximize reuse Using a different setting of these parameters is as simple as changing the attributes on the hop.matmul op. Note that although one can tune these parameters on BLIS, its ASM kernel has to be among those that shipped — these are long hand-written / hand-unrolled ones. However, with MLIR we are in a position to also play around with the register tile sizes just as easily. Let us pick a different set of values of $M_{C}$, $K_{C}$, $M_{R}$, and $N_{R}$ for better utilization of registers and L2 cache capacity as far %A’s buffer goes. We’ll set $M_{R}$ to 3 and $N_{R}$ to 16 since this uses 12 vector registers for %C instead of the under utilization of 8 earlier (there are a total of 16 %ymm registers on x86-64). ⬇ hop.matmul %A, %B, %C { M_C = 180 : i32, K_C = 480 : i32, M_R = 3, N_R = 16 : i32, K_U = 4 : i32 } : (memref<2088x2048xf64>, memref<2048x2048xf64>, memref<2088x2048xf64>) When one chooses $M_{R}$ = 3 and $N_{R}$ = 16, it leads to 3 * 16 /4 = 12 256-bit vector registers for the output values, 3 values for the LHS, and 1 for the RHS, which amounts to 12 + 3 + 1 = 16 registers — exactly the number of VEX encodable registers for AVX-2. The BLIS provided kernels for Haswell otherwise include $6*8$, $8*6$, $4*12$, $12*4$ – they too use ($2*6$ =) 12 registers for the output, but these options differ in how much register reuse is relatively exploited along the two dimensions and how big the L1 resident buffer for the RHS is. Let us execute the $M_{R}$ = 3, $N_{R}$ = 16 configuration. ⬇ // M_C = 180 : i32, K_C = 480 : i32, M_R = 3, N_R = 16 : i32, K_U = 4 : i32 $ mlir-opt -hopt -hopt-vect -hopt-copy -hopt-unroll -hopt-scalrep -convert- linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -reps=3 -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.039474s 61.939 GFLOPS We immediately see a 24% boost in performance! A big strength of a pure compiler approach here is that one automatically generates everything – all the way down to the innermost loop body. So, we have the opportunity to explore more than what’s possible with a fixed set of manually pre-optimized kernels. This is all under the assumption that the compiler generated code is competitive or could be made competitive. Let us now explore the impact of just the $M_{R}$, $N_{R}$ values around the best configuration we have found. ⬇ // Let us see the impact of just the M_R, N_R values. // M_C = 72 : i32, K_C = 256 : i32, M_R = 4, N_R = 8 : i32, K_U = 4 : i32 $ mlir-opt -hopt -hopt-vect -hopt-copy -hopt-unroll -hopt-scalrep -convert- linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -reps=3 -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.04356s 49.7282 GFLOPS // M_R = 6, N_R = 8 (this is what BLIS’ micro kernel uses for Haswell). $ mlir-opt -hopt -hopt-vect -hopt-copy -hopt-unroll -hopt-scalrep -convert- linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -reps=3 -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.0479391s 40.4135 GFLOPS // The best conf so far. // M_C = 180 : i32, K_C = 480 : i32, M_R = 3, N_R = 16 : i32, K_U = 4 : i32. $ mlir-opt -hopt -hopt-vect -hopt-copy -hopt-unroll -hopt-scalrep -convert- linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -reps=3 -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.039474s 61.843 GFLOPS Let us now peek at the generated assembly of the innermost loop to see how good it is. A necessary thing is that the innermost block should have no load/store’s on the output (everything should be in registers) and we should have a continuous sequence of VFMA instructions on 256-bit AVX registers which are named %ymm[0-15]. ⬇ # => This Inner Loop Header: Depth=5 vbroadcastsd -7704(%rbx), %ymm12 vmovapd -480(%r13), %ymm14 vfmadd231pd %ymm12, %ymm14, %ymm0 # ymm0 = (ymm14 * ymm12) + ymm0 vbroadcastsd -3864(%rbx), %ymm13 vfmadd231pd %ymm13, %ymm14, %ymm1 # ymm1 = (ymm14 * ymm13) + ymm1 vbroadcastsd -24(%rbx), %ymm15 vfmadd213pd %ymm10, %ymm15, %ymm14 # ymm14 = (ymm15 * ymm14) + ymm10 vmovapd -448(%r13), %ymm10 vfmadd231pd %ymm10, %ymm12, %ymm2 # ymm2 = (ymm12 * ymm10) + ymm2 vfmadd231pd %ymm10, %ymm13, %ymm3 # ymm3 = (ymm13 * ymm10) + ymm3 vfmadd231pd %ymm10, %ymm15, %ymm4 # ymm4 = (ymm15 * ymm10) + ymm4 vmovapd -416(%r13), %ymm10 vfmadd231pd %ymm10, %ymm12, %ymm5 # ymm5 = (ymm12 * ymm10) + ymm5 vfmadd231pd %ymm10, %ymm13, %ymm6 # ymm6 = (ymm13 * ymm10) + ymm6 vfmadd231pd %ymm10, %ymm15, %ymm7 # ymm7 = (ymm15 * ymm10) + ymm7 vmovapd -384(%r13), %ymm10 vfmadd213pd %ymm9, %ymm10, %ymm12 # ymm12 = (ymm10 * ymm12) + ymm9 vfmadd213pd %ymm11, %ymm10, %ymm13 # ymm13 = (ymm10 * ymm13) + ymm11 vfmadd231pd %ymm10, %ymm15, %ymm8 # ymm8 = (ymm15 * ymm10) + ymm8 vbroadcastsd -7696(%rbx), %ymm9 vmovapd -352(%r13), %ymm11 vfmadd231pd %ymm9, %ymm11, %ymm0 # ymm0 = (ymm11 * ymm9) + ymm0 vbroadcastsd -3856(%rbx), %ymm10 vfmadd231pd %ymm10, %ymm11, %ymm1 # ymm1 = (ymm11 * ymm10) + ymm1 vbroadcastsd -16(%rbx), %ymm15 vfmadd213pd %ymm14, %ymm15, %ymm11 # ymm11 = (ymm15 * ymm11) + ymm14 vmovapd -320(%r13), %ymm14 vfmadd231pd %ymm14, %ymm9, %ymm2 # ymm2 = (ymm9 * ymm14) + ymm2 vfmadd231pd %ymm14, %ymm10, %ymm3 # ymm3 = (ymm10 * ymm14) + ymm3 vfmadd231pd %ymm14, %ymm15, %ymm4 # ymm4 = (ymm15 * ymm14) + ymm4 vmovapd -288(%r13), %ymm14 vfmadd231pd %ymm14, %ymm9, %ymm5 # ymm5 = (ymm9 * ymm14) + ymm5 vfmadd231pd %ymm14, %ymm10, %ymm6 # ymm6 = (ymm10 * ymm14) + ymm6 vfmadd231pd %ymm14, %ymm15, %ymm7 # ymm7 = (ymm15 * ymm14) + ymm7 vmovapd -256(%r13), %ymm14 vfmadd213pd %ymm12, %ymm14, %ymm9 # ymm9 = (ymm14 * ymm9) + ymm12 vfmadd213pd %ymm13, %ymm14, %ymm10 # ymm10 = (ymm14 * ymm10) + ymm13 vfmadd231pd %ymm14, %ymm15, %ymm8 # ymm8 = (ymm15 * ymm14) + ymm8 vbroadcastsd -7688(%rbx), %ymm12 vmovapd -224(%r13), %ymm14 … This looks as good as we may expect! The output matrix values are always in registers (no spilling). LHS and RHS values have the intended amount of reuse in registers. The LHS is loaded in once via broadcast/splat (vbroadcastsd), and reused along the $j$ register tile dimension, while the RHS is moved in via aligned vector loads and reused along the $i$ register tile dimension. ⬇ // Let us look at the benefit of packing in isolation on the best code we have. $ mlir-opt -hopt -hopt-vect -hopt-copy -hopt-unroll -hopt-scalrep -convert- linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -reps=3 -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.039474s 61.843 GFLOPS ⬇ // Without packing (but with everything else). $ mlir-opt -hopt -hopt-vect -hopt-copy=false -hopt-unroll -hopt-scalrep -convert-linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so,/usr/local/lib/libblis.so Compilation time: 0.030535s 22.5257 GFLOPS // A 3x drop in performance just due to the lack of packing, while it was just // 1.5x on a code that was not fully optimized. ⬇ // Without unroll-and-jam (but with packing and everything else) $ mlir-opt -hopt -hopt-vect -hopt-copy -hopt-unroll=false -hopt-scalrep -linalg-lower-to-loops -linalg-convert-to-llvm -convert-linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -reps=5 -e main -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so,/usr/local/lib/libblis.so Compilation time: 0.0424139s 15.5651 GFLOPS ⬇ // Without packing and without unroll-and-jam (but with everything else) $ mlir-opt -hopt -hopt-vect -hopt-copy=false -hopt-unroll=false -hopt-scalrep -linalg-lower-to-loops -linalg-convert-to-llvm -convert-linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -reps=5 -e main -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so,/usr/local/lib/libblis.so Compilation time: 0.0228369s 10.785 GFLOPS As for $M_{C}$, $K_{C}$, note that we are using large enough values that the $M_{C}$ x $K_{C}$ tile (675 KB for $M_{C}$ = 180, $K_{C}$ = 480) of the LHS matrix (A) actually fits in the L3 cache, as opposed to in the L2 cache, which was the plan with the BLIS/OpenBLAS tiling approach. Since we haven’t really done the additional level of tiling ($N_{C})$ per the OpenBLAS/BLIS strategy, we notice that we get more mileage here out of using a larger tile for the LHS in L3 instead of a smaller tile in L2 (the L3 was sort of unoccupied for us). Note that a larger $M_{C}$ in particular amortizes the cost of the RHS’s transfer into L1 and exploits more L1 reuse on that tile. The additional level of tiling is conceptually straightforward given where we are now, and this could be evaluated along with better values for $M_{C}$, $K_{C}$ for improved L2 and L3 reuse. We skip that for this article. Let us now look back at our journey up until this best performing result, and the role each optimization played. ### 2.11 Within 9% of OpenBLAS and MKL Let us plot the overall picture so far. We have now nearly matched BLIS performance (61 vs 63 GFLOPS), and are only 9away from MKL and OpenBLAS’s performance! Note that BLIS’ performance could also likely improved by tuning and playing around $M_{C}$ and $K_{C}$ as opposed to choosing the configuration it preset ($M_{C}$ = 72, $K_{C}$ = 256), but as we observed here, it is the $M_{R}$, $N_{R}$ values that have a greater impact here as far as the MLIR + LLVM approach goes. Secondly, further tuning of BLIS is likely to still be around the OpenBLAS performance (based on published results). #### 2.11.1 Vectorization with copying, unroll-and-jam, unrolling but no scalar replacement On this note, let us look at what happens if we disabled MLIR’s scalar replacement but enabled unroll-and-jam, sort of leaving it to LLVM to perform the replacement. ⬇ $ mlir-opt -hopt -hopt-vect -hopt-scalrep=false -convert-linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -entry-point-result=void -reps=5 -shared-libs=lib/libmlir_runner_utils.so,/usr/local/lib/libblis.so Compilation time: 0.038455s 10.7969 GFLOPS Relying on LLVM to do the scalar replacement does not work! We will not again go into why. Although the passes in LLVM could be improved here, this is an optimization that MLIR is supposed to perform well since it has to do with multidimensional subscripts and loops. ### 2.12 Map to BLIS micro-kernel: “Outer MLIR, Inner BLIS” We might wonder at this point as to what would happen if the MLIR approach just borrowed the BLIS micro-kernel for its inner three loops (the ones running with trip counts $K_{C}$, $M_{R}$, $N_{R}$). How much better performance would it lead to for the best MLIR version? Given that we were able to get close to BLIS using MLIR + LLVM all the way, a no difference in performance would be great news for those who have worked on the x86/x86-64 LLVM backends and most of the low level infrastructure surrounding it. We will do a simple experiment here. One could imagine employing BLIS’ micro- kernel (multiplication of two panels: $M_{R}$ x $K_{C}$ and $K_{C}$ x $N_{R}$) within MLIR itself, i.e., the outer loops, tiling, explicit copying etc. are all automated with MLIR the way we described above, but the carefully crafted inline assembly kernel of BLIS is called for the inner part. This is also an approach of high interest to several research groups working on HPC compilers today because many believe that (1) the performance of the hand-written inner kernel cannot be attained through compiler generated code, (2) one shouldn’t be subject to the vagaries of a compiler when generating such code where even the last ounce of performance should be extracted. Both points are debatable, and the issues involved are addressable. Over here, our experiment is also interesting because it tells us whether the remaining difference in performance is coming from the carefully tailored instruction selection and schedule used in BLIS’ micro-kernel, which the compiler (having been tasked to deal with the entire world of programs) is unable to nail down optimally. Seeing a big difference when we swap out our inner loops with the BLIS kernel could be disappointing because it would mean we will have to now get into LLVM to see how we could improve things as one option. On the other hand, the lack of any significant improvement would be great news for us because we could then focus on the "macro kernel", code and loops that we have all possible control over, to go closer to the peak. It would also be a tribute to all those who have worked on LLVM backends and surrounding infrastructure. Figure 4: The BLIS micro-kernel (figure courtesy: Field Van Zee and Robert van de Geijn). The BLIS micro-kernel being used here is bli_dgemm_haswell_asm_6x8; this corresponds to $M_{R}$ = 6 and $N_{R}$ = 8. We’ll thus run a pure MLIR code generated version with $M_{C}$ = 174, $K_{C}$ = 512, $M_{R}$ = 6, $N_{R}$ = 8, and then the same one with the innermost 3-d nest mapped to BLIS’ microkernel. This keeps the comparison meaningful. (We have adjusted 180 and 480 slightly to 174 and 512 respectively to ensure that all cache tiles are full tiles to keep things more readable). We also need to change the layout of the LHS buffer to the packed layout shown in green - this was discussed in detail earlier when we described how memref layouts work (Section 2.9). Let us look at the structure of MLIR here with tiling and packing, but without vectorization or any unroll-jamming enabled since the inner portion is going to be swapped out. ⬇ affine.for %arg1 = 0 to 4 { affine.for %arg2 = 0 to 12 { %9 = alloc() : memref<29x512x6xf64> affine.for %arg3 = #map4(%arg2) to #map5(%arg2) { affine.for %arg4 = #map2(%arg1) to #map3(%arg1) { %10 = affine.load %0[%arg3, %arg4] : memref<2088x2048xf64> affine.store %10, %9[%arg2 * -29 + %arg3 floordiv 6, %arg1 * -512 + %arg4, %arg3 mod 6] : memref<29x512x6xf64> } } affine.for %arg3 = 0 to 256 { %10 = alloc() : memref<512x8xf64> affine.for %arg4 = #map2(%arg1) to #map3(%arg1) { affine.for %arg5 = #map7(%arg3) to #map8(%arg3) { %11 = affine.load %1[%arg4, %arg5] : memref<2048x2048xf64> affine.store %11, %10[%arg1 * -512 + %arg4, %arg3 * -8 + %arg5] : memref<512x8xf64> } } affine.for %arg4 = #map15(%arg2) to #map16(%arg2) { affine.for %arg5 = 0 to 512 { affine.for %arg6 = 0 to 8 { %11 = affine.load %10[%arg5, %arg6] : memref<512x8xf64> affine.for %arg7 = 0 to 6 { %12 = affine.load %9[%arg2 * -29 + %arg4 + %arg7 floordiv 6, %arg5, %arg7 mod 6] : memref<29x512x6xf64> %13 = affine.load %2[%arg4 * 6 + %arg7, %arg3 * 8 + %arg6] : memref<2088x2048xf64> %14 = mulf %12, %11 : f64 %15 = addf %14, %13 : f64 affine.store %15, %2[%arg4 * 6 + %arg7, %arg3 * 8 + %arg6] : memref<2088x2048xf64> } } } } dealloc %10 : memref<512x8xf64> } dealloc %9 : memref<29x512x6xf64> } } Note that the buffer of shape <174x512xf64> after being assigned the tiled layout and post memref normalization has been converted to the shape <29x512x6xf64>, and it has elements exactly in the order the BLIS micro-kernel wants. One can notice that the blocks and panels corresponding to the buffers here being multiplied (LHS buffer: <29x512x6xf64>, RHS buffer: <512x8xf64>, the output is just 2048x2048xf64 since we will reuse it in registers). For the purpose of this article, we developed a -hopt-blis pass option that actually maps the right part to the BLIS micro-kernel. For a 2088x2048 matrix, this is the generated IR: ⬇ affine.for %arg1 = 0 to 4 { affine.for %arg2 = 0 to 12 { %9 = alloc() {alignment = 32 : i32} : memref<29x512x6xf64> affine.for %arg3 = #map4(%arg2) to #map5(%arg2) { affine.for %arg4 = #map2(%arg1) to #map3(%arg1) { %10 = affine.load %0[%arg3, %arg4] : memref<2088x2048xf64> affine.store %10, %9[%arg2 * -29 + %arg3 floordiv 6, %arg1 * -512 + %arg4, %arg3 mod 6] : memref<29x512x6xf64> } } affine.for %arg3 = 0 to 256 { %10 = alloc() {alignment = 32 : i32} : memref<512x8xf64> affine.for %arg4 = #map2(%arg1) to #map3(%arg1) { affine.for %arg5 = #map7(%arg3) to #map8(%arg3) { %11 = affine.load %1[%arg4, %arg5] : memref<2048x2048xf64> affine.store %11, %10[%arg1 * -512 + %arg4, %arg3 * -8 + %arg5] : memref<512x8xf64> } } call @hopt_dgemm_blis_kernel(%9, %10, %2, %c512, %arg2, %arg3) : (memref<29x512x6xf64>, memref<512x8xf64>, memref<2088x2048xf64>, index, index, index) -> () dealloc %10 : memref<512x8xf64> } dealloc %9 : memref<29x512x6xf64> } } The hopt_dgemm_blis_kernel function is added to mlir_runtime_utils, and just wraps around bli_dgemm_haswell_asm_6x8. The allocs above have alignments since the BLIS kernel uses 256-bit aligned loads on these buffers. So, this experimentation was done with additional support in the MLIR’s std to llvm dialect conversion to use aligned_alloc. We have these alignments set even for the pure MLIR generated code since the loads and stores on vector types will have the vector size as the default ABI alignment, leading to aligned load/stores during LLVM code generation; so the alloc’s need to be aligned to vector size boundaries. ⬇ // MLIR with BLIS micro-kernel: M_R = 6, N_R = 8\. $ mlir-opt -hopt -hopt-blis -convert-linalg-to-loops -lower-affine -convert- std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so,/usr/local/lib/libblis.so -dump-object-file -object-filename=hopt.o Compilation time: 0.0281591s 61.421 GFLOPS ⬇ // MLIR with pure codegen M_R = 6, N_R = 8\. $ mlir-opt -hopt -convert-linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -reps=5 -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.0475061s 40.2426 GFLOPS ⬇ // MLIR with pure codegen M_R = 4, N_R = 8\. $ mlir-opt -hopt -convert-linalg-to-loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -reps=5 -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.0415299s 50.7075 GFLOPS ⬇ // Recall with M_R = 3, N_R = 16\. $ mlir-opt -hopt -hopt-copy -hopt-unroll -hopt-scalrep -convert-linalg-to- loops -lower-affine -convert-std-to-llvm dgemm.mlir | mlir-cpu-runner -O3 -e main -reps=3 -entry-point-result=void -shared-libs=lib/libmlir_runner_utils.so Compilation time: 0.039474s 61.939 GFLOPS There is nearly no difference when comparing the hybrid MLIR and BLIS micro kernel with the best MLIR version we had. For a given $M_{R}$ x $N_{R}$ register tile with schedule we have been using, the register requirement (assuming the instruction scheduler is not doing any non-trivial reordering) would be $M_{R}*N_{R}/4+M_{R}$ \+ 1 (the division by four is for the vector width). With $M_{R}=3$, $N_{R}=16$, this requirement would exactly be 16, which is the number of %ymm registers we have! Using $M_{R}$ = 6, $N_{R}$ = 8 with our code (as opposed to with the BLIS micro-kernel), the requirement goes up to 6⁢2 + 6 + 1 = 19 registers, and leads to a spill. The assembly dump confirms this (notice the vmovupd stores in between the FMAs, which didn’t exist earlier). Also, the intuitively sub-optimal sizes of $M_{R}$ = 4, $N_{R}$ = 8 provide better performance than the tighter $M_{R}$ = 6, $M_{R}$ = 8, indicating the cliff associated with spilling. ⬇ c4 62 95 b8 f0 vfmadd231pd %ymm0, %ymm13, %ymm14 1360: c4 a2 7d 19 84 ef e8 c3 ff ff vbroadcastsd -15384(%rdi,%r13,8), %ymm0 136a: c5 fd 11 84 24 80 01 00 00 vmovupd %ymm0, 384(%rsp) 1373: c4 e2 95 b8 c8 vfmadd231pd %ymm0, %ymm13, %ymm1 1378: c4 a2 7d 19 84 ef e8 d2 ff ff vbroadcastsd -11544(%rdi,%r13,8), %ymm0 1382: c5 fd 11 84 24 a0 01 00 00 vmovupd %ymm0, 416(%rsp) 138b: c4 e2 95 b8 d0 vfmadd231pd %ymm0, %ymm13, %ymm2 1390: c4 a2 7d 19 84 ef e8 e1 ff ff vbroadcastsd -7704(%rdi,%r13,8), %ymm0 139a: c4 e2 95 b8 d8 vfmadd231pd %ymm0, %ymm13, %ymm3 139f: c4 22 7d 19 a4 ef e8 f0 ff ff vbroadcastsd -3864(%rdi,%r13,8), %ymm12 13a9: c5 7d 11 a4 24 e0 01 00 00 vmovupd %ymm12, 480(%rsp) 13b2: c4 c2 95 b8 e4 vfmadd231pd %ymm12, %ymm13, %ymm4 However, $M_{R}$ = 6, $N_{R}$ = 8 could be made to fit in the register budget by using a different permutation of instructions for the innermost loop body, i.e., with a register tile whose intra-tile loops have been permuted before the unrolling was performed. In such a case, the register requirement would be: $M_{R}$ * $N_{R}$/4 + $N_{R}$/4 + 1 = 15! The schedule to be used to realize that would be: $(d0,d1,d2)\rightarrow(d2\textrm{ floordiv }480,d0\textrm{ floordiv }330,d1\textrm{ floordiv }8,d0\textrm{ floordiv }6,d2,d0,d1).$ Notice the flip $(\dots,d0,d1)$ instead of $(\dots,d1,d0)$. This also means that the LLVM backend is not going to automatically do a complex permutation on the innermost body that changes live ranges, reducing register requirements to fit within the budget and thereby avoiding a spill. Using the above schedule leads to no spill and the code performs at about 5% slower though than $M_{R}$ = 3, $N_{R}$ = 16. It is thus also important to get the right permutation for the unrolled register tile (at the time of selecting the loop transformation) — so that the matrix values with longer reuse distances for that intra register tile permutation (LHS values in the case of (d1, d0) and RHS values in the case of (d0, d1)) do not cause a spill. Alternatively, this problem could be viewed as one that performs the unroll-and-jam for $i$, $j$ in the right order. The register requirement in the context of this "known" computation and its interaction with the intra-tile loop permutation and register tile sizes ($M_{R}$, $N_{R}$) are pretty easy to capture at a high level to make sure even a simple register allocator does the right thing. Overall, we conclude here that the kind of code we could get automatically is clearly on par or close to what was achieved with expert written assembly. The best performing $M_{R}$x$N_{R}$ of 3x16 is on par with the MLIR-BLIS configuration we explored. The 6x8 version, if made to work without spilling, will likely be on par with 3x16. This basically means the 4-9% difference between the pure MLIR one and the BLIS/OpenBLAS ones likely stems from things around and outside the micro-kernel. This part requires more experimentation before any firm conclusions can be made. On a separate note, one can see that MLIR does make it easier to map to external hand-optimized kernels, and combine the best of both worlds if that’s necessary for peak performance. Also, it is straightforward to perform a systematic exploration around interesting values by generating ops with different parameter attributes. However, each choice requires a compile and execute cycle since we are not generating any target code parametric in the identified parameters. Figure 5: DGEMM performance. ### 2.13 SGEMM performance We can similarly now benchmark SGEMM performance quickly. All that we need to change on the op’s operands is an s/f64/f32! In addition, we will just double the register tile size $N_{R}$ from 16 to 32 since two times the number of f32 elements can be held in a vector register (8 x f32 instead of 4 x f64). We would thus still be using 3x4 = 12 vector registers for C. In addition, we will also double $M_{C}$ to 348 for the same reason. We thus use this op to generate SGEMM. ⬇ hop.matmul %A, %B, %C { M_C = 348 : i32, K_C = 512 : i32, M_R = 3, N_R = 32 : i32, K_U = 4 : i32 } : (memref<2088x2048xf32>, memref<2048x2048xf32>, memref<2088x2048xf32>) Evaluating this, we find that the performance of purely MLIR generated code is within 2% of MKL performance here!, and in fact marginally better than OpenBLAS and BLIS. The outer MLIR + inner BLIS version here delivers the expected performance, nearly on par with pure MLIR here. Figure 6: SGEMM performance. ### 2.14 What about the remaining 9%? At this point, we would like to see what stands in the way of bridging the remaining 9% gap in getting to MKL/OpenBLAS’s performance. In particular, the hand-written kernels in OpenBLAS, BLIS also use prefetching – we are not yet generating any of these instructions. In summary, we have been able to generate all code starting from just this op: ⬇ hop.matmul %A, %B, %C { M_C = 180 : i32, K_C = 480 : i32, M_R = 3, N_R = 16 : i32, K_U = 4 : i32 } : (memref<2088x2048xf64>, memref<2048x2048xf64>, memref<2088x2048xf64>) ### 2.15 Other questions There may be a few questions here. 1. 1. While the code was automatically generated, the transformation sequence was specified from within an MLIR pass. What does the sequence look like - just C++ IR building and calls to utilities/passes? How productive is it and the subsequent exploration? 2. 2. What about problem sizes that are not a perfect multiple of register tile sizes ($M_{R}$, $N_{R}$)? While MLIR’s unroll-and-jam works with cleanup code generated, there are unresolved issues in interactions downstream (for performance). There is literature in the polyhedral body of works on separation of full and partial tiles, and this is largely an implementation issue. Note that cache tile sizes need not perfectly divide problem sizes - this already works in MLIR in conjunction with packing and other transformations presented here. For eg. the best $M_{C}$ size we used (180) does not divide 2048. 3. 3. We have primarily used the polyhedral passes in MLIR here since the generated code at each stage always stays affine. The LinAlg dialect does not have the utilities to automatically analyze and generate packing code for example. Nearly all passes and utilities used here such as unroll-and-jam, scalar replacement, and memref normalization work on affine dialect ops. 4. 4. What about all the other options around input matrices such as strides and transpose? These all are cleanly expressed via memrefs’ affine layout map discussed above (Section 2.9). We haven’t considered the $\alpha$ and $\beta$ arguments for DGEMM though (it is really $C=\alpha\\*A\\*B+\beta\\*C$) - we have actually assumed $\alpha$ = $\beta$ = 1. In fact, MLIR’s pattern rewrites can fold and optimize away code/nests when $\alpha$ or $\beta$ are 0 or 1. Overall, these aspects are something to keep in mind before making a complete/exact comparison. 5. 5. How do we determine good or the optimal parameter values? The work of Low et al. is on analytical modeling to derive good parameter values (the derived formulae are pretty straightforward to just plug values from hardware resources into). Besides such analytical modeling, there is a lot of room to play here depending on how powerful the generative infrastructure is, and to what extent we would like to generalize this approach beyond the domain considered here. ## 3 Related Work Most of the related work on OpenBLAS [2] and BLIS [3] was already described inline. The work by Gareev et al. [10] is the only approach I am aware of that tried the BLIS approach from inside a compiler (Polly/LLVM there). From their results, they appeared to have had reached about 70% of MKL/OpenBLAS performance. With a low-level IR such as LLVM, and with Polly and LLVM infrastructure decoupled to some extent, one’s expressive power and mileage is expected to vary. There has been a large amount of work on empirical search driven library generation for GEMM [11] (ATLAS), and works like that of Yotov et al. [12] have plugged in analytical models to drive such optimization with comparable performance. The work of Low et al. [7] makes a similar attempt with BLIS. However, all of these frameworks were centered around generation of C code along with inner inline assembly kernels. ## 4 Reproducing these Results ##### Software setup: Fedora Linux 30 running kernel 5.3.6-200.fc30.x86_64, BLIS version 0.6.0-40-gf4f5170f, MKL version 2019.4.243, OpenBLAS 0.3.7-1.fc30.x86_64, and Pluto git 0.11.4-903-g7f21ab57. cpupower was used to set the frequency governor to ‘performance’. A good part of the experiments presented in this article can be reproduced with MLIR trunk. There are major features though that are pending upstream integration (memref_shape_cast op, alloca op, scalar replacement, a new vectorization pass/utility, and support for a few packing options), but these are available in the $hop$ branch here, which is the recommended way of reproducing results presented herein as is. Please see this README to run most of the experiments. The hop branch however is not intended to be continuously kept in sync with changing upstream MLIR development — so that the examples, IR and results presented herein remain valid. On the other hand, the MLIRX [13] repository provides all of the features presented here while being maintained to be in sync with upstream MLIR development. If the reader’s objective is to build upon the augmented IR infrastructure as opposed to just trying to reproduce things, MLIRX is recommended. ## References * [1] Chris Lattner, Mehdi Amini, Uday Bondhugula, Albert Cohen, Andy Davis, Jacques Pienaar, River Riddle, Tatiana Shpeisman, Nicolas Vasilache, and Oleksandr Zinenko. MLIR: A compiler infrastructure for the end of Moore’s law, 2020. * [2] Kazushige Goto and Robert A. van de Geijn. Anatomy of high-performance matrix multiplication. ACM Trans. Math. Softw., 34(3), 2008. * [3] Field G. Van Zee and Robert A. van de Geijn. BLIS: A framework for rapidly instantiating BLAS functionality. ACM Trans. Math. Softw., 41(3), June 2015. * [4] MLIR: A multi-level intermediate representation. https://mlir.llvm.org. * [5] Affine dialect in MLIR. https://github.com/llvm/llvm-project/tree/master/mlir/docs/Dialects/Affine.md. * [6] MemRefs in MLIR. https://github.com/llvm/llvm-project/tree/master/mlir/docs/LangRef.md#memref-type. * [7] Tze Meng Low, Francisco D. Igual, Tyler M. Smith, and Enrique S. Quintana-Orti. Analytical modeling is enough for high-performance BLIS. ACM Trans. Math. Softw., 43(2), August 2016. * [8] Uday Bondhugula. An introduction to the polyhedral framework, 2016. https://www.csa.iisc.ac.in/$\sim$udayb/slides/uday-polyhedral-opt.pdf. * [9] Robert van de Geijn, Margaret Myers, and Devangi Parikh. LAFF-On programming for high performance: ulaff.net, 2019. http://www.cs.utexas.edu/users/flame/laff/pfhp/. * [10] Roman Gareev, Tobias Grosser, and Michael Kruse. High-performance generalized tensor operations: A compiler-oriented approach. ACM Trans. Archit. Code Optim., 15(3), 2018. * [11] R. Clint Whaley and Jack Dongarra. Automatically tuned linear algebra software. In SuperComputing 1998: High Performance Networking and Computing, 1998. http://math-atlas.sourceforge.net/. * [12] Kamen Yotov, Xiaoming Li, Gang Ren, María Jesús Garzarán, David A. Padua, Keshav Pingali, and Paul Stodghill. Is search really necessary to generate high-performance blas? Proceedings of the IEEE, 93(2):358–386, 2005. * [13] PolyMage Labs. MLIRX, 2020. https://github.com/polymage-labs/mlirx.

2024-09-04T02:54:56.060692

2003.00536

arxiv-papers

# Into the UV: A precise transmission spectrum of HAT-P-41b using Hubble’s WFC3/UVIS G280 grism H.R. Wakeford School of Physics, University of Bristol, HH Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA D.K. Sing Department of Earth & Planetary Sciences, Johns Hopkins University, Baltimore, MD, USA Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD, USA K.B. Stevenson JHU Applied Physics Laboratory, 11100 Johns Hopkins Rd, Laurel, MD 20723, USA N.K. Lewis Department of Astronomy and Carl Sagan Institute, Cornell University, 122 Sciences Drive, Ithaca, NY 14853, USA N. Pirzkal Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA T.J. Wilson University of Exeter, Physics Building, Stocker Road, Exeter, Devon, Ex4 4QL, UK Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA J. Goyal Department of Astronomy and Carl Sagan Institute, Cornell University, 122 Sciences Drive, Ithaca, NY 14853, USA T. Kataria NASA Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA T. Mikal-Evans Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue, 37-241, Cambridge, MA 02139, USA N. Nikolov Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD, USA J. Spake Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA (Accepted to AJ February 29, 2020) ###### Abstract The ultraviolet-visible wavelength range holds critical spectral diagnostics for the chemistry and physics at work in planetary atmospheres. To date, exoplanet time-series atmospheric characterization studies have relied on several combinations of modes on Hubble’s STIS/COS instruments to access this wavelength regime. Here for the first time, we apply the Hubble WFC3/UVIS G280 grism mode to obtain exoplanet spectroscopy from 200-800 nm in a single observation. We test the G280 grism mode on the hot Jupiter HAT-P-41b over two consecutive transits to determine its viability for exoplanet atmospheric characterization. We obtain a broadband transit depth precision of 29–33 ppm and a precision of on average 200 ppm in 10 nm spectroscopic bins. Spectral information from the G280 grism can be extracted from both the positive and negative first order spectra, resulting in a 60% increase in the measurable flux. Additionally, the first HST orbit can be fully utilized in the time- series analysis. We present detailed extraction and reduction methods for use by future investigations with this mode, testing multiple techniques. We find the results fully consistent with STIS measurements of HAT-P-41b from 310–800 nm, with the G280 results representing a more observationally efficient and precise spectrum. HAT-P-41b’s transmission spectrum is best fit with a model with Teq=2091 K, high metallicity, and significant scattering and cloud opacity. With these first of their kind observations, we demonstrate that WFC3/UVIS G280 is a powerful new tool to obtain UV-optical spectra of exoplanet atmospheres, adding to the UV legacy of Hubble and complementing future observations with the James Webb Space Telescope. ††software: POET pipeline (Stevenson et al., 2012b; Cubillos et al., 2013), IRAF (Tody, 1986, 1993), IDL Astronomy user’s library (Landsman, 1995), NumPy (Oliphant, 2006–), SciPy (Virtanen et al., 2019), MatPlotLib (Caswell et al., 2019), AstroPy (Astropy Collaboration et al., 2018), Photutils (Bradley et al., 2019) ## 1 Introduction The characterization of planetary atmospheres in the solar system and beyond has long leveraged the ultra-violet (UV) to near-infrared (NIR) spectroscopic capabilities of the Hubble Space Telescope (HST). Observations with HST have been critical in the exploration of the chemical composition, climate, and aerosol properties of exoplanet atmospheres (see Kreidberg et al., 2018, and references therein). With the help of HST we now know that clouds and hazes are likely present in all types of exoplanetary atmospheres (e.g. Marley et al., 2013; Helling, 2019; Wakeford et al., 2019), but we currently lack information related to their abundances, physical properties and extent throughout the atmosphere. We also know that exoplanets exhibit extended upper atmospheres with evidence for atmospheric escape (e.g. Ehrenreich et al., 2014; Bourrier et al., 2018; Sing et al., 2019), but struggle to connect physical processes in the lower and upper portions of exoplanet atmospheres. The UV through optical (200–800 nm) spectra of planets hold rich information about the chemistry and physics at work across a broad range of atmospheric pressures. In the solar system, UV and near-UV spectroscopy has been critical in identifying and measuring the abundances of a variety of hydrocarbon and sulfur-bearing species, produced via photochemical mechanisms, as well as oxygen and ozone and more. For exoplanets, UV to near-UV spectroscopy has been especially useful for constraining aerosol properties and exploring atmospheric chemistry in hot ($>$1000 K) atmospheres (e.g. Sing et al., 2016; Evans et al., 2018). To date, only a handful of exoplanets have been probed in the critical 200–400 nm wavelength range that crosses the optical to UV boundary. Results from these studies have been mixed, limited by the wavelength coverage and sensitivity of the workhorse instrument for such studies, HST’s Space Telescope Imaging Spectrograph (STIS) G430L and E230M gratings. It is important to remember that none of HST’s instruments or modes were specifically designed to support exoplanet observations. It has only been through the development of new observational strategies, such as spatial scanning (McCullough & MacKenty, 2012; McCullough et al., 2014), and data reduction techniques that the potential for HST to probe exoplanet atmospheres has been achieved. In general, slitless spectroscopic observing modes have been preferred for high-precision time-series observations of exoplanets that transit, pass in front of, their host stars because they typically offer more throughput and temporal stability. The slitless spectroscopy capabilities HST’s Wide Field Camera 3 (WFC3) have been heavily used by the exoplanet community at infrared wavelengths (750–1800 nm) with the G102 and G141 grisms. However, HST’s WFC3 UV/Visible (UVIS) channel also offers slitless spectroscopy in the UV through visible (200–800 nm) wavelength range that has yet to be leveraged for exoplanet observations. In fact, this mode has only been employed in a handful of scientific investigations, first used as part of HST WFC3 early release science programs in cycle 16 (2006), however, none of the G280 work was published from this study. Here we detail for the first time the observations, spectral extraction, and analysis processes taken to apply Hubble’s WFC3/UVIS G280 spectroscopic grism to transiting exoplanet observations. We first introduce the challenges in using the UVIS G280 grism in §2. In §3 we detail the observations and spectral extraction procedures used. We then detail the broadband time-series analysis using two systematic reduction techniques in §4. We use Spitzer transit observations to refine system parameters and update the orbital ephemeris in §4.1 and 4.2. We outline the spectroscopic analysis in §5 and discuss the results in §6 including searching for evidence of atmospheric escape and comparisons to STIS data. We then conclude with a summary of our results and the potential of WFC3/UVIS G280 for future exoplanet investigations. ## 2 Introduction to the UVIS G280 grism The WFC3 instrument on HST is fitted with two channels, UVIS and IR. Across these two channels are three slitless spectroscopic grisms: G280 in UVIS and G102 and G141 in the IR channel. The IR grisms have been extensively applied to exoplanet atmospheric studies with increasing success at the advent of spatial scanning (McCullough & MacKenty, 2012), where HST slews in the cross- dispersion direction to spread the target light over a column of pixels (e.g., Deming et al. 2013; Kreidberg et al. 2014; Wakeford et al. 2013, 2016; de Wit et al. 2016). However, the UVIS G280 grism has not had such usage despite large throughput in the near-UV (NUV) and wide coverage from 200–800 nm. More commonly, studies that cover 300–900 nm are conducted with multiple observations using HST’s STIS G430L and G750L low resolution gratings from 300–550 nm and 500–900 nm respectively (e.g., Nikolov et al. 2014; Sing et al. 2016; Lothringer et al. 2018) despite their comparatively low throughput (Fig. 1). The UVIS grism, however, comes with several quirks that make it difficult to observe with and challenging to analyze. A number of these challenges will also affect observations with James Webb Space Telescope’s (JWST) spectroscopic instrument modes. Therefore, WFC3/UVIS G280 is a current working example of the challenges that will be faced with JWST. Here we detail each of the challenges associated with WFC3’s UVIS grism and also the advantages it has over other instrument modes in the NUV to optical wavelengths. ### 2.1 Challenges We detail some of the challenges encountered with this dataset and those expected in the general use of this instrument mode for exoplanet time-series characterisation. \- Curved spectral trace: The trace for spectral order with the G280 grism is strongly curved at shorter wavelengths. The trace is best fit with a 6th order polynomial function detailed by Pirzkal et al. (2017) and section 3. This curvature causes it to be offset in the cross-dispersion direction from the 0th order position, meaning subarray sizes need to be carefully chosen. Unlike the IR grisms, the spectra should not be spatially scanned as this would result in overlapping wavelength regions along the scan direction. The curved spectral trace also introduces a non-linear wavelength solution, meaning each pixel has a slightly different wavelength range than the surrounding pixels in that column. The wavelength solution is therefore extracted relative to the fitted trace position on the detector with a 6th order polynomial. \- Multiple overlapping orders: Additional spectral orders, both positive and negative, overlap with the first order spectra at wavelengths greater than 550 nm. In many cases these additional orders will be much dimmer than the first order spectrum and not impact the observations. However, for stars bright in the NUV such as O,B,A stars the additional spectral orders may impact the spectral extraction. In the presented case, the second order spectrum is $\approx$ 65$\times$ dimmer than the primary spectrum in both positive and negative orders. This would negligibly contribute to the measured transit depths causing the measured planetary radius ($R_{m}$) to be $\approx$ 99.24% of the true planetary radius ($R$) following, $\frac{R_{m}}{R}=\sqrt{\frac{1}{1+\frac{1}{65}}}=0.9924.$ (1) \- Geometric distortion: Using the grism filters causes the spectra to be offset spatially in the detector relative to their direct image X and Y coordinates. For the UVIS array the offset varies as a function of the position due to geometric distortion (Rothberg et al., 2011). The relationship between the coordinates in x and y pixel position also needs to be taken into account when planning the observations in X and Y arcsecond coordinates (see WFC3 data handbook for conversion functions 111WFC3 Data Handbook Appendix C.2). \- Orientation constraints: The spectral trace of the positive and negative orders extend across over 500 pixels each depending on the brightness of the target. In a crowded field or where a target is part of a visual binary system, tight orient constraints need to be placed on the observations to prevent contamination from nearby stars. This is often mitigated in WFC3/IR grism observations using spatial scans where the spectra can be extracted by differencing the individual non-destructive reads within the final science frame. However, as WFC3/UVIS grism observations can only be conducted in stare mode, up-the-ramp sampling cannot be used to recover overlapping spectra. \- Cosmic rays: The large wavelength coverage that extends significantly into the blue wavelengths increases the number of detected comic rays compared to the IR detectors. \- JWST challenges: For JWST a number of the instrument modes that will be utilized for exoplanet timeseries data exhibit curved spectral traces, overlapping spectral orders, and contamination constraints from additional sources on the sky. NIRISS SOSS mode is most similar to the G280 grism with both strongly curved spectral traces and overlapping spectral orders. It is also expected that NIRSpec Bright Object Time Series observations will also have a slightly curved trace. For all slitless modes on JWST used for exoplanet time series observations contamination overlap will need to be carefully considered and orientation constrained carefully sampled. ### 2.2 Advantages While we have detailed many challenges there are also significant advantages to this instrument over other modes in the NUV and optical. We detail these here. \- Wide wavelength coverage: observations are conducted over the whole wavelength range 200–800 nm in a single exposure. Low resolution spectra across this wide wavelength range can address the two main exoplanet science points revealed by HST observations; cloud opacities and atmospheric escape. The G280 grism can measure both the lower atmosphere sensitive to aerosol scattering, while large atmospheric escape signatures can be detectable in narrow bands around strong Fe and Mg signatures at $<$300 nm. Figure 1: Throughput curves for HST instruments and modes commonly used for exoplanet time-series observations. Solid lines are the WFC3-UVIS G280 grism +1 and -1 orders; the dark dot-dashed line is the combined transmission of both orders. Dashed lines are STIS G430L and G750L gratings. -..- line shows the COS G230L. Dotted lines are WFC3-IR G102 and G141 grisms. \- Multiple spectral orders: both the positive and negative orders are measured in each exposure. The UVIS CCD is split into two chips (1 & 2), with each chip of 2051$\times$2048 pixels easily encompassing both spectral orders which each cover $\approx$500 pixels in the dispersion direction. In the presented observations we use chip 2 as it has been shown to be more stable than chip 1. We therefore also recommend the use of chip 2 for future studies. \- Throughput: WFC3/UVIS has the highest throughput amongst all HST instruments in the wavelength range from its lower cut off at 200 nm to the upper end at $\sim$800 nm. The throughput of UVIS G280 in the NUV is on average 25 times that of STIS E230M between 200–400 nm, and roughly four times that of STIS G430L at 350 nm. UVIS G280 also has the advantage of being able to measure both positive and negative spectral orders that have a combined throughput greater than STIS G430L across the whole wavelength range (see Fig. 1). \- New calibration program: Prior to these observations there have been three instrument science reports (Kuntschner et al., 2009; Rothberg et al., 2011; Pirzkal et al., 2017) and no scientific papers using this grism. As demand for time-series observations with this grism have increased there are now new calibration programs being implemented to better characterize the detector and improve the trace fitting for all spectral orders. Calibration of the instrument and mode are important to understand the structure of the CCD, on- sky changes in the PSF, and wavelength dispersion across the detector - especially under the requirements of long term stability for exoplanet investigations that span multiple HST orbits. Overall the WFC3/UVIS G280 grism has many challenges that are difficult but not impossible to overcome, and a significant advantage over other instrument modes in this wavelength range. In the following sections we detail the observations taken and the measurements made with the tools to overcome these challenges. ## 3 UVIS G280 Observations We used HST’s WFC3/UVIS channel with the G280 spectroscopic slitless grism to observe the spectrum of the transiting exoplanet host star HAT-P-41 from 200–800 nm (GO-15288, PIs D.K. Sing & N.K. Lewis). Unlike the WFC3/IR G102 and G141 grisms, the UVIS G280 grism produces a spectrum that is strongly curved, with overlapping spectral orders at longer wavelengths, and a dimmer ($\sim$60%) -1 order spectrum compared to the +1 order. We designed an observation strategy that would cover both +1 and -1 orders simultaneously to examine this difference in flux and test the usability of the G280 grism for time-series exoplanet studies. We observed the target HAT-P-41, in the constellation of Altair, over two visits, each consisting of five HST orbits, to measure the transit of the hot Jupiter exoplanet HAT-P-41b. The two visits were separated by a single planetary orbital period (visit 1: 2018 August 1st; visit 2: 2018 August 4th, period = 2.694047 days), significantly reducing the potential impact of any stellar variations on the transits of this quiet F6 star. Each visit consists of 54 exposures, over 5 HST orbits, with exposure times of 190 seconds. We used a 2100 $\times$ 800 sub-array, with a POS TARG Y offset of -50” to center the spectrum on chip 2. The sub-array is cut out of the full 2051 $\times$ 4096 pixel CCD which contains chip 1 and 2, where chip 1 and chip 2 are separated by 1.2”. Our target star, HAT-P-41, has a nearby companion separated by 3.615”, equivalent to $\approx$ 91.5 pixels on the detector. The nearby companion resulted in a number of tight orientation constraints on the observation. However, our sub-array is large enough to capture both full +1 and -1 spectral orders around the 0th order trace. The maximum flux obtained in a single pixel in the spectral trace is $\approx$ 36,000 e-, keeping it well within the saturation and non-linearity limit of the detector, which is approximately 67,000–72,000 e- (Gilliland et al., 2010). ### 3.1 Spectral Extraction The spectral traces for both visits and both +1 and -1 orders were extracted using calibration files provided by the WFC3 team. A complete extraction and reduction of the provided data requires the following steps: a) cosmic ray removal b) background subtraction c) aperture determination, and d) trace fitting We then use the WFC3 UVIS calibration files to compute the wavelength solution for each spectral order. We also performed spectral extraction with IRAF and custom IDL routines as a second check on the extraction procedure as this is the first published analysis of G280 grism data for time-series spectroscopy (see 3.1.1 for details). ##### Cosmic ray removal We use the “flt” files from the Calwfc3 pipeline to analyze each exposure. Cosmic rays were then rejected by examining the time series for each pixel, and flagging and replacing 3.5-$\sigma$ outliers in an iterative process. We also applied a further spatial cosmic ray cleaning step by rejecting cosmic rays through Laplacian Edge Detection (van Dokkum, 2001). We do a final cosmic ray cleaning on the extracted 1D stellar spectra by comparing them to the median combined spectra and replacing outliers which deviated by more than 3.5 $\sigma$. Where cosmic rays are flagged temporally we replace the pixel value with a median of the time sampled pixel, where cosmic rays are spatially flagged a median of the surrounding pixels in the same frame is used. Figure 2: Modal background count for each exposure in visit 1 (solid) and visit 2 (dashed) across the whole sub-array. The dotted vertical lines indicate the start of a new HST orbit. The background is higher at the start of each HST orbit with a bi-model distribution, perhaps due to stray Earthshine or orbital effects on the telescope. ##### Background subtraction We use the local background estimation, similar to WISE (see section 4.4 c Cutri et al. 2012 222WISE All-sky release explanatory supplement, Section 4.4 c), by computing the pixel mode for each image and subtracting that from each pixel. The mode, or most common binned histogram value, tends to be robust against the bright tail of the distribution that is caused by other stars and cosmic-ray (or hot) pixels in the exposure. We compared this to the mean and median sigma clipped pixel values and found good agreement, giving weight to the mode being resistant to outliers. In each visit the first exposure of each orbit has much higher background than the other exposures with a slightly bi- modal distribution around the peak of the histogram (see Fig. 2), perhaps due to stray Earthshine or orbital effects on the telescope. We remove the first exposure of each orbit in both visits in the lightcurve analysis. Figure 3 shows the visual difference between the original “flt” images and a cleaned-background-subtracted exposure. We save the cleaned and background subtracted images as fits files to be used for the trace fitting routines. Figure 3: HST WFC3 UVIS/G280 spectral image. Top: “flt” file processed and available on the MAST archive. Bottom: cleaned file with cosmic rays and hot pixels removed, and flat fielding applied. In this comparison you can clearly see the difference between the original and cleaned data demonstrating the requirement for accurate and precise treatment of detector artifacts and cosmic ray hits. ##### Trace fitting To extract the target spectrum using the provided calibration files for UVIS G280333G280 UVIS grism files, the sub-array image needs to be re-embedded into the full frame (Rothberg et al., 2011). This can be done using the embedsub routine in wfc3tools444https://github.com/spacetelescope/wfc3tools. This routine also requires the “spt” files be downloaded from the MAST database and contained within the same folder as the cleaned fits files generated from the previous steps. Figure 4: HST WFC3 UVIS/G280 spectral image. Top: visit 1 +1 spectral order (left) and -1 spectral order (right). Bottom: visit 2 +1 spectral order (left) and -1 spectral order (right). All images are background subtracted and cosmic rays have been removed. The dotted black line shows the calculated trace center, with the extent of the +-12 pixel aperture shown in orange dashed lines. At lower flux values the spectral trace does not fit quite as well but the full flux is captured inside the selected aperture. Color shows flux, truncated at 25 e-s-1. Direct images of the target were taken with the F300X filter at the start of each visit to provide an accurate location of our target on the detector. Visits 1 and 2 were positioned on the detector within 1 pixel of each other with x, y centroid positions of [2040.8756, 1063.8825] and [2041.0399, 1062.9073] respectively. Using the description of the spectral trace of the G280 UVIS grism from Pirzkal et al. (2017), we computed the expected location of the trace in each exposure of our G280 datasets. In summary, Pirzkal et al. (2017) compute the trace location as a function of the x-pixel on the detector and a high order 2D polynomial is fit across the trace. The best fit trace is defined by a 6th order polynomial function with a linear 2D field dependence. The reference column for the polynomial fit is chosen to be close to the inflection point of the trace to ensure the best fit to both the highly curved spectrum at short wavelengths and the near-linear trace at longer wavelengths. The polynomial function reproduces the position of both the +1 and -1 spectral orders to within a fraction of a pixel from 200–800 nm. Figure 4 shows the central trace position for both visits and computed for the +1 and -1 spectral orders. The trace fits are currently best calibrated to the +1 order, however, the authors note that there is a new WFC3/UVIS G280 calibration program that will fully characterize the -1 and additional spectral orders. At longer wavelengths, toward the tail end of the spectral trace, fringing effects come into play that divert the spectra from the fit polynomial trace (see Fig. 5). A simple extraction of the spectrum contained in each dataset was created by adding up the observed count rates in pixels above and below the computed location of the trace. We tested apertures ranging from $\pm$5 pixels around the central trace to $\pm$50 pixels. To determine the best aperture we minimized the standard deviation of the residuals for out-of-transit counts. We find that the optimal aperture is $\pm$12 pixels (see Fig. 4), to account for the slightly extended wings of the trace (Kuntschner et al., 2009). Both the +1 and -1 spectra orders were processed in this manner. The overlapping spectral orders are expected to impact the spectrum in the long wavelengths approximately beyond 400 nm. However, these observations were not ideal to show the impact of overlapping spectral orders as the brightness of the star in the shorter wavelengths is too dim, $\approx$65$\times$ dimmer than the first order trace. We discuss potential corrections to this in more detail in § 5. ##### Wavelength solution The wavelength solution is calculated from the trace position using the equation detailed in Pirzkal et al. (2017) which is calibrated from 190 to 800 nm. The extracted wavelength solution is good to +/- 0.7 nm which is roughly half of a UVIS resolution element. We measure the mean spectral dispersion in the first order which varies from $\sim$1.1–1.6 nm per pixel over the full spectral range 200–800 nm. We plot the stellar spectra for both visits and first order spectra in Fig. 5, showing the 16-84 percentile range of each spectrum with remarkable agreement between visits, demonstrating the stability of the instrument. Beyond 800 nm the target spectrum shows extreme fringing effects and is not calibrated, thus we remove it from this analysis. It is also clear to see that the -1 order is significantly dimmer across the whole wavelength range with a large impact on the short wavelengths, short of 250 nm, where the flux drops to near-zero. Figure 5: The 16–84 percentile range of each visit and orders spectral trace. The +1 orders and -1 orders overlap closely, making it difficult to tell the two visits apart and demonstrating the stability of the instrument and the star. The -1 orders are $\sim$50% dimmer than the +1 orders, with little to no flux short of 250 nm. Above 800 nm fringing patterns can clearly be seen in the stellar spectra and we do not use these wavelengths for the lightcurve analysis. #### 3.1.1 IRAF APALL Spectral Extraction We also performed spectral extraction with IRAF and custom IDL routines. The images were first background subtracted and cosmic rays were removed in the same way as detailed above. We then used IRAF’s APALL routine to extract the spectra for each image in the time series, finding an 8th order legendre polynomial was optimal for the spectral trace extraction as measured by the trace root mean square residuals. We note that with IRAF, the fixed aperture center varies smoothly to follow changes in the position of the spectrum across the dispersion axis and partial pixels are used at the ends. We extracted the spectra with a wide range of aperture sizes, finding a 24 pixel aperture was optimal. Similar to the UVIS calibration pipeline routines, the extracted spectra still exhibited a few cosmic rays not cleaned in previous processes, we also then perform the 1D stellar spectra cosmic ray removal step. Using IRAF APALL we were unable to replicate the wavelength solution calculation and therefore used the one calculated following Pirzkal et al. (2017) that required the trace fitting following the UVIS calibration pipeline. Both spectral extraction techniques produce near identical stellar spectra and transmission spectra. However, in the following sections we adopt and present the analysis based on the spectra extracted using the UVIS calibration pipeline as it is widely accessible, publicly available extraction method that does not rely on proprietary custom routines, and has a fully consistent wavelength solution. ## 4 Broadband white-light analysis Prior to measuring the transmission spectrum of HAT-P-41b, we first analyze the broadband white lightcurve from 200–800 nm. In this section we detail the analysis of the broadband whitelight transit depth measured in the UVIS G280 transits for each visit and spectral order based on two different systematic treatment methods - instrument systematic marginalization (Wakeford et al., 2016) and jitter decorrelation (Sing et al., 2019). ##### Instrument systematic marginalization uses a pseudo-stochastic grid of corrective systematic models to measure the desired lightcurve parameters, namely the transit depth, via an evidence-based weight assigned by the data to each potential systematic model. We run a grid of 50 systematic models in the form of an extended polynomial; $S(\mathbf{x})=t_{1}\phi_{t}\times\sum^{n}_{i=1}p_{i}\phi^{i}_{HST}\times\sum^{n}_{j=1}l_{j}\delta^{j}_{\lambda}+1$ (2) where $\phi_{t}$ is the planetary phase representing a linear slope over the whole visit, $\phi_{HST}$ is the HST orbital phase accounting for “HST thermal breathing” effects, and $\delta_{\lambda}$ is the positional shift in the wavelength direction on the detector over the visit. Each of these parameters have scaling factors with the linear slope defined by $t_{1}$, and “HST breathing” and positional shifts fit up to a 4th order polynomial function defined by $p_{1-n}$ and $l_{1-n}$, respectively. Each of the scaling parameters are then either fit as free parameters to activate the systematic model or fixed to zero. The whole grid of 50 systematic models used in this analysis can be found in Table 2 of Wakeford et al. (2016) note the table is 0 indexed. We approximate the evidence (marginal likelihood) of each systematic model fit to the data using the Akaike Information Criterion (AIC). We then calculate the evidence-based weight ($W_{q}$) across all 50 systematic models and use the information from all models to marginalize over the desired parameter ($\alpha_{q}$). $\alpha_{m}=\sum^{N_{q}}_{q=0}(W_{q}\times\alpha_{q})$ (3) Equation (15) of Wakeford et al. (2016), where $N_{q}$ is the number of models fit, and $\alpha_{m}$ is the resulting marginalized parameter. The uncertainty is then calculated in a similar way based on the weights (see Equation (16) of Wakeford et al. 2016). Figure 6: Spectral position changes over the course of each visit, measured by cross-correlating to a template spectrum (points, Cc), and by fitting background sources on the full exposure image (shaded regions, stars). Each are shown relative to the final exposure for comparison. The spectral shifts are accounted for in the systematic treatment of each lightcurve. Figure 7: Mean flux in Spitzer’s 3.6 $\mu$m channel. Plotting on a logarithmic scale reveals HAT-P-41’s faint, nearby companion at pixel position (12,15). We limit our photometry aperture size to 2.25 pixels to minimize contamination from the companion. Bad pixels are masked in white. ##### Jitter decorrelation uses HST’s Pointing Control System to detrend photometric time-series data. Based on the results of (Sing et al., 2019), we include optical state vectors traditionally used for STIS (Sing et al., 2011) as well as several jitter vectors. The full systematics model, $S(\mathbf{x})$, used to detrend the lightcurve is written as, $S(\mathbf{x})=p_{1}\phi_{t}+\sum^{4}_{i=1}p_{i+1}\phi^{i}_{HST}+p_{6}\delta_{\lambda}\\\ +p_{7}X_{psf}+p_{8}Y_{psf}+p_{9}RA+p_{10}DEC\\\ +p_{11}Vn_{roll}+p_{12}Vt_{roll}+1,$ (4) where $\phi_{t}$ is a linear baseline time trend, $\phi_{HST}$ is the 96 minute HST orbital phase, $X_{psf}$ and $Y_{psf}$ are the detector positions of the PSF as measured by the spectral trace, $\delta_{\lambda}$ is the wavelength shift of each spectra as measured by cross-correlation, $V2\\_roll$ and $V3\\_roll$ are roll of the telescope along the V2 and V3 axis, $RA$ and $DEC$ are the right ascension and declination of the aperture reference, and $p_{1..12}$ are the fit systematic parameter coefficients. The first portion of this function was found to be the best functional form of the additional systematic features and corresponds to one of the models used in the marginalization grid. This function is then fit for all transit lightcurves in this form and is not marginalized over to determine the optimal functional form in each lightcurve. The full jitter decorrelation set results in up to twelve total terms used to describe the instrument systematics of the dataset in question. However, in practice not all of these parameters are needed. For each visit and each of the two orders, we used the AIC and measured red noise, $\sigma_{\rm r}$, to determine the optimal optical state vectors to include from the full set without over-fitting the data and minimizing the red noise. Both systematic marginalization and jitter decorrelation require a measurement of the spectral positional changes on the detector across the duration of the observation ($\delta_{\lambda}$). To calculate the shift, we cross-correlate the 1D stellar spectra to a template spectrum and measure the displacement across the whole wavelength range. To demonstrate that this accurately represents the physical shift on the detector, we measured the position for three background sources distributed across the exposure image. We selected the most Gaussian-like sources from the full image and used a 2D-Gaussian fit to their 0th order spectrum in each exposure of each visit. In this case we cannot use the 0th order of the target or its stellar companion to measure this shift as they are both saturated on the detector. Figure 6 shows $\delta_{\lambda}$ for visits 1 and 2 measured using the cross-correlation method (Cc) and the range of positional values measured from the three background sources (stars). The form of the positional shifts are very similar with the vertical breaks showing where the telescope is reset after each HST orbit. The magnitude of the positional shifts is on the sub-pixel scale and is easily accounted for with either of the systematic treatments detailed. Using the 2D-Gaussian fit to the background sources, we find that positional shifts in the y-direction are negligible and do not improve the fit to the data. Due to the phase coverage of HST observations, resulting from Earth occultation events, we are unable to accurately fit for the inclination, a/R∗, and orbital period of the system. Unfortunately, HAT-P-41b was not observed by the Transiting Exoplanet Survey Satellite (TESS) which would have allowed us to easily constrain the system parameters. To fit for these vital parameters we instead use two transit observations from the Spitzer Space Telescope IRAC instrument to obtain accurate system parameters for the inclination and a/R∗ of HAT-P-41b, detailed in §4.1. In §4.2 we present the measured center of transit times for these and previous transit observations of HAT-P-41b to determine the period of the planet, and in §4.3 we present the results of the UVIS G280 broadband lightcurves for the two visits and for each spectroscopic order using both systematic treatments. Figure 8: Transit light curves of HAT-P-41b using Spitzer’s 3.6 $\mu$m (left) and 4.5 $\mu$m (right) channels. We bin the data for plotting purposes only. The 3.6 $\mu$m residuals demonstrate a small amount of correlated noise at timescales shorter than the transit duration. ### 4.1 Spitzer Data Analysis Spitzer program 13044 (PI: Deming) acquired transit observations of HAT-P-41b at 3.6 and 4.5 $\mu$m on 2017 January 18 and 2017 February 3, respectively. The IRAC instrument (Fazio et al., 2004) acquired 32$\times$32 pixel subarray frames at 2 second intervals in batches of 64. Each observation acquired a total of 21,632 frames over a span of $\sim$12 hours. Using the POET pipeline (Stevenson et al., 2012a; Cubillos et al., 2013), we apply a double-iteration, $4\sigma$ outlier rejection routine, 2D Gaussian centroiding, and $5\times$ interpolated aperture photometry over a range of aperture sizes. We convert times to BJDTDB using the JPL Horizons interface. We find that the best aperture size (as defined by the lowest standard deviation of the normalized residuals) is 3.0 pixels; however, at this size there is noticeable contamination from the nearby binary companion. This is evidenced by the correlation between aperture size and transit depth (significant at $3.3\sigma$). HAT-P-41’s stellar companion is located $\sim 3$ pixels away, in the wings of the primary star’s point response function. This is shown in Figure 7, where we depict the mean flux at 3.6 $\mu$m on a logarithmic scale. We find that the impact of the stellar companion on the measured transit depth is minimal ($<1\sigma$) for apertures $\leq 2.25$ pixels and, thus, adopt this value for our final analyses. We note that the transit time, inclination, and semi-major axis parameters do not vary with our choice of aperture size. To derive our best-fit values (see Tables 1 and 2), we fit both Spitzer channels simultaneously using the transit model described by Mandel & Agol (2002), a linear trend in time, and a BLISS map (Stevenson et al., 2012a) to account for intrapixel sensitivity variations. We estimate uncertainties using the Differential-Evolution Markov Chain Monte Carlo technique (ter Braak & Vrugt, 2008) and test for convergence using the Gelmin-Rubin statistic (Gelman & Rubin, 1992) by ensuring that the potential scale reduction factor is within 1% of unity. Figure 8 shows Spitzer’s normalized light curves and residuals. The best-fit 3.6 and 4.5 µm transit depths are $0.992\,{\pm}\,0.008$ % and $1.028\,{\pm}\,0.013$ %, respectively. Figure 9: Observed minus calculated (O-C) diagram of measured HAT-P-41b transit times. The dashed line shows the 1-sigma uncertainty. Table 1: Star and Planet parameters used in the lightcurve fitting process for this analysis. Parameter | Value | Reference ---|---|--- Star | | V (mag) | 11.087 | Hartman et al. (2012) M∗ (M⊙) | 1.418 | Hartman et al. (2012) R∗ (R⊙) | 1.786 | Morrell & Naylor (2019) Teff (K) | 6340 | Morrell & Naylor (2019) [Fe/H] (dex) | 0.21 | Hartman et al. (2012) log(g) | 4.14 | Hartman et al. (2012) Planet | | Mp (MJ) | 0.795 | Bonomo et al. (2017) Rp (RJ) | 1.685 | Hartman et al. (2012) Period (days) | 2.69404861 $\pm$0.00000092 | This work T0 (days) | 2456600.29325$\pm$0.00050 | This work inclination (∘) | 89.17 $\pm$ 0.62 | This work a/R∗ | 5.55 $\pm$ 0.04 | This work ecc | 0.0 | Bonomo et al. (2017) ### 4.2 Updated Orbital Ephemeris We used previous and current data to calculate an up-to-date orbital period for HAT-P-41b, including the ephemeris from the discovery (Hartman et al., 2012), as well as HST and Spitzer transit data (see Table 2). The HST data includes the WFC3/UVIS transits where the +1 and -1 orders were treated independently (see §4.3), as well as WFC3/IR and STIS transits from the Hubble PanCET program (GO-14767, PIs D.K. Sing & M. Lopez-Moralez, Sheppard 2020 in prep - private communication). We converted all of the available transit times to BJDTDB using the tools from (Eastman et al., 2010). These times were fit with a linear function of the period $P$ and transit epoch $E$, $T(E)=T_{0}+EP.$ (5) The resulting ephemeris is given in Table 2, with the linear function giving a reasonable fit to the data (see Fig. 9), with a $\chi^{2}$ value of 14.47 for 9 degrees of freedom (DOF). Table 2: Center of transit times used in Fig.9 to calculate the period of the planetary orbit as well as the resulting best-fit orbital ephemeris. All times have been converted to BJDTBD. Instrument | Mode | Epoch | Note ---|---|---|--- | | (BJDTDB) (days) | | | 2454983.86247 $\pm$ 0.00107 | Hartman et al. (2012) HST WFC3-IR | G141 | 2457677.912139 $\pm$ 0.0008 | Spitzer IRAC | CH1 | 2457772.20477 $\pm$ 0.00021 | Spitzer IRAC | CH2 | 2457788.36879 $\pm$ 0.00027 | HST STIS | G430L | 2458001.197547 $\pm$ 0.001151 | visit 1 HST STIS | G430L | 2458246.357040 $\pm$ 0.000339 | visit 2 HST STIS | G750L | 2458281.379682 $\pm$ 0.000363 | HST WFC3-UVIS | G280 | 2458332.566558 $\pm$ 0.000656 | Visit 1, +1 order HST WFC3-UVIS | G280 | 2458332.564321 $\pm$ 0.001366 | Visit 1, -1 order HST WFC3-UVIS | G280 | 2458335.260623 $\pm$ 0.000303 | Visit 2, +1 order HST WFC3-UVIS | G280 | 2458335.259912 $\pm$ 0.000290 | Visit 2, -1 order Period $P$ (days) | | $T_{0}$ (BJDTDB) (days) | 2.69404861$\pm$0.000000918 | | 2456600.293253 $\pm$ 0.000504 | ### 4.3 UVIS G280 Broadband Lightcurve Results Figure 10: Top: broadband lightcurves. We show the raw extracted lightcurve for the visit 1 +1 spectral order to demonstrate the stability of the 1st HST orbit in the time series (light grey). The systematic corrected and normalized white lightcurves for each visit and spectroscopic order (colored labeled points) with the best fit transit model. Each point represents a single exposure. Each lightcurve is offset for clarity. Middle: residuals from each ligthcurve fit using the systematic marginalization method. Bottom: residuals for each lightcurve fit using the jitter decorrelation method. We measure the combined transit depth of HAT-P-41b to be $(R_{p}/R_{*})^{2}$ = 1.0406 $\pm$ 0.0029 % (SDNR = 221 ppm) and 1.0330 $\pm$ 0.0033 % (SNDR = 281 ppm), for each method respectively. Figure 11: The evidence-based weight for each systematic model used in instrument systematic marginalization for each visit and order for the broadband lightcurve analysis. The table of systematic models relating to each number can be found in Wakeford et al. (2016). We measure the broadband transit depth for UVIS G280 by summing the flux from 200–800 nm and correcting for systematics via systematic marginalization and jitter decorrelation independently for both visits and both spectral orders. We measure a combined transit depth of all four transit timeseries measurements of $(R_{p}/R_{*})^{2}$ = 1.0406 $\pm$ 0.0029 % and 1.0330 $\pm$ 0.0033 %, with an average standard deviation on the residuals of 221 ppm and 281 ppm, using the systematic marginalization and jitter decorrelation methods respectively. There is a 1.7$\sigma$ difference between the two methods, likely due to the small differences between the uncertainties on each exposure for each analysis method that can be seen by comparing the bottom two panels of Fig. 10. In each analysis we use the same extracted stellar spectra, the same limb-darkening coefficients derived using the 3D stellar models presented in Magic et al. (2015), and the same system parameters shown in Table 1. We show the four transit lightcurves (2 visits + 2 orders) corrected in Fig. 10. The lightcurves shown have been corrected using the most favored model applied in systematic marginalization, with the underlying models derived from the same most-likely systematic model. For both data analysis methods, systematic marginalization and jitter decorrelation, the transit model is fit iteratively with the systematic model to measure the transit depth. We note that the lightcurves in Fig.10 only represents a portion of the information obtained through marginalization as all the information from corrected data using other weighted systematic models also go into the final marginalized transit depth measurement (contribution weights can be seen in Fig. 11). Using jitter decorrelation, we derive a single solution for the lightcurve corrections and transit depth for each visit and spectral order. The individual lightcurves from jitter decorrelation are indistinguishable by eye compared to the systematic marginalization ones presented here. For a more direct comparison we show the residuals of both systematic analyses at the bottom of Fig. 10 with their related uncertainties, both achieving near photon noise precision. Figure 12: Intensity plot of the spectroscopic lightcurve residuals for each wavelength bin using the systematic marginalization method. The color bar shows the residuals amplitude for all intensity plots. For the -1 orders we do not compute the transmission below 250 nm as the flux is too low to produce convergent results in the systematic analysis. While jitter decorelation uses a fixed systematic model plus the jitter files directly from the telescope as a main decorelation factor, systematic marginalization derives its information from evidence obtained from an array of independent systematic models. Systematic marginalization therefore accounts for the unknown factors affecting the lightcurves by weighting them according to the reduced data rather than the telescopes fine guidance sensors. Using systematic marginalization we find that each transit and spectral order favors slightly different combinations of systematic corrections. For visit 1 both orders predominantly favor models with a quadratic correction to $\delta_{\lambda}$, while both orders of visit 2 favor a 3rd order $\phi_{HST}$ correction with additional correction for $\delta_{\lambda}$. Given the similarity in the $\delta_{\lambda}$ trend for each visit and spectral order, as shown in Fig. 6, the more favored correction of the HST breathing in visit 2 suggests that this movement on the detector is likely connected with the thermal effects of the telescope and thus the corrections themselves are interchangeable in this specific case where the structure of the systematic is similar. For each lightcurve there is a marginal preference to correct for a linear trend in time across the whole visit; however, it is slightly more significant in visit 1. This linear trend across the whole visit has been noted in several other HST timeseries observations (e.g., Deming et al., 2013; Sing et al., 2015; Kreidberg et al., 2014; Wakeford et al., 2016), and is thus likely related to the observatory as a whole rather than a specific instrument. For each visit and order we show the weighting assigned to each systematic model in the systematic marginalization reduction for the broadband analysis in Fig.11, these model weights are later applied to the spectroscopic lightcurves. The weights shown correspond to the systematic models shown in Table 2 of Wakeford et al. (2016). The structure of this grid is such that it first loops through polynomials correcting for $\delta_{\lambda}$, followed by added corrections for $\phi_{HST}$ with the second half of the grid (25-49) adding in corrections for $\phi_{t}$. The overall structure of the computed weights shows that the corrections for $\delta_{\lambda}$ are the dominant factor given causing the loop every four models. Figure 13: The individual and combined transmission spectra using both systematic marginalization and jitter decorrelation. The two visits and +1/-1 spectral orders are shown as colored shaded regions representing the range of the uncertainties for each spectrum. The final transmission spectrum combining the results of all four are shown as joined black points with errorbars. ## 5 Spectroscopic Analysis To measure the transit depth as a function of wavelength and produce an atmospheric transmission spectrum for HAT-P-41b, we divide the stellar flux into 10 nm bins ($\sim$5 detector resolution elements) from 200 – 800 nm. We note that it is possible to sample the transmission spectrum at a higher resolution ($>$2 resolution elements) in the most optimal portions of the spectrum where the flux is high; however, we use uniform bins across the whole wavelength range for consistency and accurate comparison. We analyze each individual spectroscopic lightcurve in the same way, as described in §4 for the broadband lightcurve, using both systematic marginalization and jitter decorrelation methods. In jitter decorrelation, the systematic correction model is unchanged between wavelength bins, thus assuming all systematics are wavelength independent. Using systematic marginalization, we account for any wavelength dependent systematics by running the full grid of systematic models in each spectroscopic lightcurve. We then use the evidence based weights for each of those models measured in the broadband lightcurve (see Fig 11) to marginalize over the measured values for each model in each lightcurve. By fixing the systematic model weighting to those derived from the broad-band analysis, the uncertainty is then more representative of the dominant wavelength independent systematics while incorporating the scatter measured across wavelength dependent systematics being fit to the data. Figure 14: Direct comparison of the final combined transmission spectrum for each systematic treatment: jitter decorrelation (dark squares) and systematic marginalization (light circles). The horizontal dashed lines show the measure broadband depth using each method. Figure 15: The standard deviation between the four individual transmission spectra in each wavelength bin for systematic marginalization (pink) and jitter decorrelation (purple). Each visit and +1/-1 spectral orders were analyzed separately using the parameters detailed in Table 1 fixing the period, inclination, and a/R∗, and using the center of transit times listed in Table 2. Using both jitter decorrelation and systematic marginalization independently, we find consistent results across both visits and spectral orders. Both methods reach photon noise precision in each of the channels determined by calculating the white and red noise associated with the fits (see Pont et al. 2006), and finding a beta value of 1 consistent with no correlated noise. We show the residuals from each of the spectroscopic lightcurves for the systematic marginalization analysis in Fig. 12 as an intensity residual map to show any global structure in the fit. From the residuals it is clear that the -1 order lightcurves are noisier than the +1 orders. There is also an increase in the scatter at the edges of the wavelength regime, with shorter wavelengths dominating the overall noise range associated with the pure count rates measured from the stellar spectrum in each of the bins (see Fig. 5). In Fig. 13, we present the transmission spectrum measured using both methods for each visit and each +/- first order spectrum with the combined transmission spectrum overlaid. We show a direct comparison between the combined transmission spectrum measured using the two systematic treatments in Fig. 14, with 90% of the points overlapping at the 1-$\sigma$ uncertainty level. A direct comparison between the two methods is best demonstrated by looking at the standard deviation and uncertainty in the transit depth measured across the four transits analyzed (see Fig. 15). It is again evident in the standard deviations and uncertainties that the lower counts measured in the near-UV wavelengths ($<$300 nm) introduce larger scatter and uncertainty to the transit depths. The standard deviation in the short wavelengths indicates that that derived transit depths in each lightcurve are more similar within the uncertainties using systematic marginalization compared to the Jitter decorrelation method. However, there is added scatter with the marginalization method at longer wavelengths. Both methods have similar uncertainty profiles indicating the ability to analyse these data with multiple methods. The unique contribution of the UV points to the transmission spectrum of an exoplanet atmosphere in combination with the optical from a single observation with this low-resolution grism cannot be overstated. ## 6 Discussion We present HST’s WFC3/UVIS G280 grism as a reliable observational mode to measure the transmission spectrum of exoplanet atmospheres from 200–800 nm, critically reaching down to near-UV and optical wavelengths not accessible to JWST. This wavelength range is important to understand and measure cloud opacity sources and their scattering profiles that are defined by the particle sizes (e.g., Lecavelier des Etangs et al. 2008; Wakeford & Sing 2015; Wakeford et al. 2017), escaping atmospheres (e.g., Ehrenreich et al. 2014; Sing et al. 2019), and absorption from Na. To test this new mode, we measured the atmosphere of the hot Jupiter HAT-P-41b over the course of two consecutive transits with the WFC3/UVIS G280 grism. We obtained the positive and negative first order spectra of the target star in each observation and extracted the stellar flux following the methods outlined by the UVIS calibration pipelines (Kuntschner et al., 2009; Rothberg et al., 2011; Pirzkal et al., 2017). We analysed the transit data for each visit and spectral order using two well established techniques, instrument systematic marginalization (Wakeford et al., 2016) and jitter decorrelation (Sing et al., 2019). Both analysis techniques produced statistically similar transmission spectra for the atmosphere of HAT-P-41b. We obtain a precision of 29–33 ppm on the broadband transit depth from 200–800 nm, and an average precision of $\approx$200 ppm in 10 nm spectroscopic bins. ##### Comparison to STIS Observations Figure 16: Transmission spectrum of HAT-P-41b measured with WFC3/UVIS G280 grism using systematic marginalization combining two HST observations (pink), compared to STIS G430L combined spectra from two HST observations (dark green) and one observation with the HST STIS G750L grating (Sheppard, 2020 in prep - private communication). The WFC3/UVIS G280 grism is able to efficiently measure the atmosphere of a transiting exoplanet from 200–800 nm to high precision, matching and exceeding that of STIS. Figure 17: Transmission model fit using the planetary specific grid with rainout condensation by Goyal et al. (2018). Both the jitter decorrelated and systematic marginalization G280 spectra were fit independently with the Spitzer data to the full grid of HAT-P-41b models. Both datasets found the same best fit model with Teq = 2091 K, [M/H] = +2.0, C/O = 0.7, 1100$\times$scattering, cloud = 0.2. We compare the transmission spectrum measured of HAT-P-41b with WFC3/UVIS G280 grism to that measured using STIS G430L and G750L gratings. We find that the combination of the two HST observations in the G280 UVIS grism results in resolution and precision exceeding that of STIS, which required the combination of three HST observations to cover the whole wavelength range compared to two for UVIS. Figure 16 shows the transmission spectrum derived using systematic marginalization from two transits with UVIS G280 compared to the transmission spectrum from three transits with STIS G430L and G750L presented by Sheppard (2020 in prep - private communication). Assessing the overall use of UVIS G280 over the STIS gratings, there are a number of trade offs to consider. As G280 cannot be scanned and the throughput is much higher it will likely be more difficult to observe bright (Vmag $<$ 7) targets, especially considering the impact of overlapping spectral orders that will make it difficult to extract individual spectral bins at this resolution. Therefore, bright targets will be more efficiently observed with STIS/G430L in particular. Additionally, although UVIS G280 can efficiently measure a wide wavelength range in a single observation it does not extend to wavelengths spanning the potassium absorption line that can only be accurately captured with the STIS G750L grating. However, the extended wavelength coverage into the UV compared to the G430L grism and the comparable resolution means that a potential Na line can be resolved just as easily with UVIS as with STIS but with potentially higher precisions in UVIS. The measured UVIS spectrum far exceeds the resolution and precision over the comparative wavelengths than can be achieved by STIS/G750L (see Fig. 16). This direct comparison for the same planet demonstrates that the UVIS G280 grism can easily exceed the precision and resolution of STIS in an equivalent number of observations, while being more efficient and requiring less observing time. UVIS G280 also has the advantage of spanning the whole wavelength range in one shot, dramatically reducing the potential impact of stellar activity and systematics which can cause offsets between datasets from different instrument modes. In summary, for targets with Vmag $\geqslant$ 7 the UVIS G280 grism shows reduced systematics, higher resolution, precision, and wavelength coverage with more efficient observing compared to STIS G430L and G750L gratings. ##### Searching for Evidence of Atmospheric Escape The UVIS G280 grism has ideal wavelength coverage to search for signatures of atmospheric escape of the Fe II at 240 nm and 260 nm, and the prominent Mg II doublet at 279.63 nm. A single resolution element for the G280 grism is $\sim$2 nm which encompasses the whole Mg II doublet absorption line, thus limiting us to strong, low resolution detections. At a single resolution element of the detector, the scatter becomes large and we were unable to converge on a solution to fit the lightcurve systematics. We therefore conducted an analysis of the HAT-P-41b transit data in 4 nm bins (2 resolution elements) across the 230 – 290 nm, with individual moving analyses in 10 nm steps to search for excess absorption from escaping ions. In this analysis, we find little significant evidence for additional absorption by Fe II and Mg II in the atmosphere. In a single 4 nm bin centred at 280 nm we measure additional 0.2% absorption compared to the average transit depth which could potentially correspond to Mg II. However, this absorption is not seen in bins centered 10 nm either side of 280 nm that encompass the peak of the absorption. The scatter is on the order of 0.3% across the whole sampled range. We conducted our search predominantly using the positive spectral orders for each visit as the throughput and flux levels are high enough for the precision needed at these wavelengths. However, for strong signatures such as those seen in WASP-121b (Sing et al., 2019) or KELT-9b (Hoeijmakers et al., 2018), which also orbit bright stars, the absorption signature will likely also be measurable in the negative order spectra as well. We conclude that there is no evidence of significant Fe II and Mg II escaping from the atmosphere of HAT-P-41b based on the precision of these measurements. However, we cannot currently conclude where this places HAT-P-41b in the comparative phase space as more measurements with this mode or similar to that shown in Sing et al. (2019) will be required over a wide temperature phase space to examine the likelihood of detection. ##### Planetary Specific Model Comparison We ran each of the transmission spectra including the measured Spitzer transit depths through the planetary specific forward model grid for HAT-P-41b using rainout condensation presented by Goyal et al. (2018, 2019). In each case, the model fits have the same number of degrees of freedom with the only additional fitting parameter being the absolute altitude of the model. For each UVIS G280 spectrum, we trim the first and last two data points that are likely most affected by low flux and fringing, respectively, and append on the Spitzer transit depths. Each transmission spectrum independently favors the same atmospheric model that has: Teq = 2091 K, atmospheric metallicity [M/H] = +2.0, C/O = 0.7, 1100$\times$scattering profile, and uniform cloud opacity = 0.2 (see Fig. 17). We find a $\chi^{2}_{\nu}$ = 1.45 and 1.72 when fitting the most favored model to the jitter decorrelated and marginalized transmission spectrum, respectively. The model shows prominent TiO/VO features in the near-UV fitting the UVIS G280 data well in the optical with a wavelength dependent slope associated with a scattering opacity source composed of small sub-micron particles. This model predicts a muted H2O feature in the near-IR that would be detectable with WFC3’s G102 and G141 grisms. The Spitzer IR is dominated by CO2 that would add additional constraints on the atmospheric metallicity (Moses et al., 2011) and can be validated by JWST NIRSpec observations. ## 7 Conclusions We present HST’s WFC3/UVIS G280 grism as a new and ideal instrument mode for exoplanet time-series characterisation. This is the first time scientific analysis of any observation with this instrument mode has been published. As such, we provide a detailed breakdown of the challenges and advantages of the instrument, detailed instructions on the spectral extraction with reference to data files and programs provided through UVIS calibration files, and a comparative study of two well established systematic reduction methods. To test the UVIS G280 grism for time-series data, we observed the transit of the hot Jupiter HAT-P-41b over two consecutive transit events. This allowed us to measure the overall stability of the instrument, the precision, and resolution without additional concerns associated with potential stellar activity. We obtained both positive and negative first order spectra from each observations providing four different datasets from 200–800 nm. We analysed each dataset separately before combining the information to produce the final atmospheric transmission spectrum of HAT-P-41b. We applied two different extraction and systematic analysis techniques to the data and find them to be statistically similar across the whole transmission spectrum demonstrating the robust and consistent nature of the instrument critical for accurate exoplanet transmission spectral studies. We measure the complete transmission spectrum of the hot Jupiter HAT-P-41b from 200–800 nm in 10 nm bins and at 3.6 and 4.5 $\mu$m with Spitzer’s IRAC instrument. In the broadband UVIS lightcurves, we reach a precision of 29-33 ppm, with an average of $\approx$200 ppm in 10 nm wide spectroscopic channels. The transmission spectrum shows evidence of TiO/VO in the near-UV to optical with significant absorption from CO2 in the Spitzer 4.5 $\mu$m channel. We fit a grid of forward models specifically derived for HAT-P-41b to the transmission spectrum from multiple reduction pipelines and find constant results with a Teq = 2091 K, [M/H] = +2.0, C/O = 0.7, scattering $\times$1100, and cloud opacity = 0.2 for rainout condensation (see Goyal et al. 2018, 2019). Additional measurements in the near-IR will further aid the interpretation of this planets atmospheric transmission and will be detailed in future publications. We demonstrate that Hubble’s WFC3 UVIS G280 grism is superior to the combination of STIS G430L and G750L gratings for time-series observations in terms of efficiency, precision, and resolution from 300–800 nm for exoplanet time-series observations. Notably the UVIS G280 grism also allows access to wavelengths as short as 200 nm with the potential to measure the escaping atmosphere of giant exoplanets via Fe II and Mg II absorption lines and a broad of range of other atmospheric processes. The wavelength coverage offered by the UVIS G280 grism (200–800 nm) provides a perfect complement to the spectroscopic capabilities of the James Webb Space Telescope (600–14000 nm), which together can probe the full extent of atmospheric processes in exoplanets that closely orbit their host star. ## Acknowledgements We thank D. Deming for use of data from his Spitzer program 13044 that provided the two transits used to obtain accurate system parameters. Thanks to S. Morrell for discussions on the stellar parameters with updates from Gaia. We acknowledge private communication from N. Nikolov and K. Sheppard for the STIS data analysis from the Hubble PanCET program (Sheppard, 2020 in prep - private communication). This work is based on observations made with the NASA/ESA Hubble Space Telescope, https://archive.stsci.edu/hst/search.php (catalog HST-GO-15288), that were obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. A portion of this work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. H.R. Wakeford acknowledges support from the Giacconi Prize Fellowship at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. Author contributions: H.R. 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K. 2019, Research Notes of the American Astronomical Society, 3, 7, doi: 10.3847/2515-5172/aafc63 Table 3: Transmission spectrum of HAT-P-41b based on the combined spectrum of +1 and -1 spectral orders over two transt events. for both systematic marginalization and jitter decorrelation | Systematic marginalization | Jitter Decorrelation | Limb-darkening coefficients∗ ---|---|---|--- Wavelength | Transit depth | Uncertainty | Transit depth | Uncertainty | c1 | c2 | c3 | c4 (nm) | (%) | (%) | (%) | (%) | | | | 200-800 | 1.04056 | 0.00293 | 1.03303 | 0.00331 | | | | 205 | 1.07731 | 0.14700 | 0.85089 | 0.11217 | 0.27642 | -0.27861 | 0.17437 | 0.81199 215 | 1.04749 | 0.09769 | 1.04495 | 0.06266 | 0.44951 | -1.31326 | 2.39714 | -0.53880 225 | 1.11641 | 0.06163 | 0.97186 | 0.04573 | 0.26022 | -0.34716 | 0.79494 | 0.26854 235 | 0.95137 | 0.08046 | 1.02503 | 0.05374 | 0.44189 | -0.57725 | 1.16157 | -0.07411 245 | 1.09086 | 0.03881 | 0.94739 | 0.04950 | 0.54045 | -1.11394 | 2.37372 | -0.82917 255 | 0.93706 | 0.08189 | 0.98717 | 0.05082 | 0.48574 | -0.52757 | 1.52725 | -0.54102 265 | 1.05554 | 0.07504 | 1.01539 | 0.03792 | 0.33614 | -0.12161 | 1.23718 | -0.49873 275 | 0.92522 | 0.06059 | 1.01556 | 0.04516 | 0.51787 | -0.22060 | 0.71649 | -0.07533 285 | 0.95923 | 0.04902 | 0.97523 | 0.03401 | 0.47085 | -0.29824 | 1.30007 | -0.53323 295 | 1.00307 | 0.02041 | 1.00242 | 0.02578 | 0.43969 | -0.17729 | 1.45827 | -0.79410 305 | 1.06577 | 0.01350 | 1.03449 | 0.02395 | 0.40399 | 0.12345 | 0.91689 | -0.53110 315 | 0.99512 | 0.02188 | 0.97180 | 0.02567 | 0.34789 | 0.38909 | 0.43002 | -0.26221 325 | 1.05841 | 0.01795 | 1.01076 | 0.01979 | 0.41584 | 0.42379 | 0.33271 | -0.27752 335 | 0.98227 | 0.01837 | 1.00338 | 0.01746 | 0.32808 | 0.83687 | -0.47180 | 0.19469 345 | 0.99562 | 0.01689 | 0.98657 | 0.02122 | 0.50090 | 0.47375 | -0.04764 | -0.04352 355 | 1.03245 | 0.01882 | 0.99715 | 0.02232 | 0.49485 | 0.49152 | -0.08977 | -0.02466 365 | 1.00155 | 0.01805 | 1.00064 | 0.01927 | 0.49130 | 0.55355 | -0.23093 | 0.04401 375 | 1.00250 | 0.01921 | 1.01414 | 0.02044 | 0.51901 | 0.65211 | -0.42396 | 0.11740 385 | 1.04525 | 0.01217 | 0.99727 | 0.01692 | 0.44683 | 0.40462 | 0.26623 | -0.24303 395 | 0.93250 | 0.03636 | 1.03279 | 0.02120 | 0.48246 | 0.16780 | 0.44794 | -0.21622 405 | 1.03092 | 0.01341 | 1.02334 | 0.01455 | 0.48142 | 0.45253 | 0.03247 | -0.08316 415 | 1.01568 | 0.01890 | 1.00675 | 0.01593 | 0.43310 | 0.43124 | 0.12149 | -0.10687 425 | 1.03996 | 0.01103 | 1.01964 | 0.01239 | 0.47865 | 0.31563 | 0.19400 | -0.11728 435 | 1.02160 | 0.00993 | 1.03207 | 0.01504 | 0.50686 | 0.42439 | -0.02042 | -0.06239 445 | 1.02758 | 0.01111 | 1.01087 | 0.01519 | 0.62723 | 0.00768 | 0.49756 | -0.26869 455 | 1.03755 | 0.01292 | 1.03748 | 0.01523 | 0.66796 | -0.03296 | 0.39110 | -0.16936 465 | 1.03632 | 0.00862 | 1.04188 | 0.01242 | 0.64089 | 0.10690 | 0.17586 | -0.07282 475 | 1.03320 | 0.01313 | 1.01325 | 0.01207 | 0.68519 | 0.04928 | 0.14144 | -0.03468 485 | 1.02089 | 0.01256 | 1.02408 | 0.01507 | 0.79901 | -0.19234 | 0.36598 | -0.16162 495 | 1.04370 | 0.01210 | 1.02236 | 0.01338 | 0.77433 | -0.20489 | 0.40845 | -0.15622 505 | 1.03363 | 0.01274 | 1.00958 | 0.01437 | 0.71140 | -0.05044 | 0.22921 | -0.08213 515 | 1.06672 | 0.01122 | 1.03961 | 0.01399 | 0.64107 | -0.00460 | 0.31643 | -0.15507 525 | 1.02409 | 0.01215 | 1.01568 | 0.01591 | 0.73764 | -0.16794 | 0.39393 | -0.17123 535 | 1.03226 | 0.00823 | 1.03075 | 0.01383 | 0.79821 | -0.32921 | 0.54626 | -0.23226 545 | 1.05347 | 0.01105 | 1.04898 | 0.01551 | 0.82715 | -0.38564 | 0.54019 | -0.20221 555 | 1.04829 | 0.01132 | 1.06219 | 0.01434 | 0.81141 | -0.35516 | 0.49708 | -0.18347 565 | 1.04683 | 0.01666 | 1.02657 | 0.01682 | 0.82890 | -0.41148 | 0.54895 | -0.20500 575 | 1.03509 | 0.01478 | 0.98730 | 0.01769 | 0.84535 | -0.43068 | 0.52234 | -0.18103 585 | 1.07381 | 0.01539 | 1.05872 | 0.01698 | 0.83910 | -0.44512 | 0.56000 | -0.20974 595 | 1.04651 | 0.01203 | 1.04453 | 0.01652 | 0.88137 | -0.54670 | 0.63637 | -0.22889 605 | 1.04134 | 0.01189 | 1.03169 | 0.01478 | 0.88844 | -0.57180 | 0.65069 | -0.23386 615 | 1.03707 | 0.01348 | 1.00632 | 0.01922 | 0.81257 | -0.38094 | 0.42040 | -0.13042 625 | 1.01444 | 0.01252 | 0.98434 | 0.01837 | 0.87418 | -0.59117 | 0.70278 | -0.27158 635 | 1.01662 | 0.01658 | 1.02845 | 0.01979 | 0.87714 | -0.58646 | 0.66610 | -0.24723 645 | 1.02919 | 0.01724 | 1.00875 | 0.01949 | 0.86284 | -0.56322 | 0.65497 | -0.25552 655 | 1.06727 | 0.01847 | 1.05593 | 0.02030 | 0.95153 | -0.74127 | 0.72558 | -0.26379 665 | 1.00227 | 0.01846 | 1.02700 | 0.02128 | 0.90248 | -0.67572 | 0.74030 | -0.27779 675 | 1.03065 | 0.01782 | 1.03617 | 0.02138 | 0.87733 | -0.62328 | 0.65601 | -0.22774 685 | 1.01072 | 0.01601 | 0.94646 | 0.02119 | 0.88769 | -0.66447 | 0.70310 | -0.25098 695 | 1.01402 | 0.02043 | 1.00378 | 0.02471 | 0.89369 | -0.70395 | 0.75795 | -0.27823 705 | 1.07414 | 0.01838 | 1.07324 | 0.02316 | 0.88686 | -0.68657 | 0.72303 | -0.26083 715 | 1.03237 | 0.02539 | 0.99668 | 0.02136 | 0.87414 | -0.65906 | 0.67614 | -0.23359 725 | 1.04279 | 0.02215 | 1.03922 | 0.02722 | 0.89617 | -0.72378 | 0.74812 | -0.26705 735 | 1.02738 | 0.01973 | 1.03978 | 0.02539 | 0.89467 | -0.73795 | 0.77170 | -0.28323 745 | 1.04238 | 0.01927 | 1.01427 | 0.03020 | 0.86509 | -0.64972 | 0.65973 | -0.23664 755 | 1.04771 | 0.02398 | 1.02146 | 0.02712 | 0.89132 | -0.75184 | 0.79302 | -0.29984 765 | 1.06149 | 0.01834 | 0.98611 | 0.03130 | 0.88774 | -0.75638 | 0.78905 | -0.29437 775 | 1.05751 | 0.02014 | 1.09123 | 0.02938 | 0.88279 | -0.73219 | 0.73173 | -0.25886 785 | 1.10846 | 0.02307 | 1.10165 | 0.03805 | 0.88973 | -0.77932 | 0.81639 | -0.31113 * Based on using the +1 order throughput curve and 3D stellar models from Magic et al. (2015) .

2024-09-04T02:54:56.081725

2003.00544

arxiv-papers

# Exploiting Ergonomic Priors in Human-to-Robot Task Transfer Jeevan Manavalan1∗, Prabhakar Ray & Matthew Howard 1Jeevan Manavalan, Prabhakar Ray and Matthew J. Howard are with the Centre for Robotics Research, Department of Engineering, King’s College London, London, UK. <EMAIL_ADDRESS> ###### Abstract In recent years, there has been a booming shift in the development of versatile, autonomous robots by introducing means to intuitively teach robots task-oriented behaviour by demonstration. In this paper, a method based on programming by demonstration is proposed to learn null space policies from constrained motion data. The main advantage to using this is generalisation of a task by retargeting a systems redundancy as well as the capability to fully replace an entire system with another of varying link number and lengths while still accurately repeating a task subject to the same constraints. The effectiveness of the method has been demonstrated in a 3-link simulation and a real world experiment using a human subject as the demonstrator and is verified through task reproduction on a 7DoF physical robot. In simulation, the method works accurately with even as little as five data points producing errors less than $\mathbf{10^{-14}}$. The approach is shown to outperform the current state-of-the-art approach in a simulated 3DoF robot manipulator control problem where motions are reproduced using learnt constraints. Retargeting of a systems null space component is also demonstrated in a task where controlling how redundancy is resolved allows for obstacle avoidance. Finally, the approach is verified in a real world experiment using demonstrations from a human subject where the learnt task space trajectory is transferred onto a 7DoF physical robot of a different embodiment. ## I Introduction In recent years, there has been a booming shift in the development of versatile, autonomous robots capable of performing increasingly complex tasks. Such robots are expected to enhance the capabilities of ordinary people to introduce automation into their lives by means of intuitively teaching robots task-oriented behaviour by demonstration [1, 2]. When humans perform task-oriented movement, it is often the case that there is a high level of redundancy, with the number of degrees of freedom (DoF) available to execute the task usually much higher than those required [3]. For instance, in the task of opening a drawer (see Figure 1), the primary objective is to _manipulate the drawer from the closed to the open position_ , however, there is redundancy in the several possible ways this can be achieved. For example, a human performing this task can adopt different elbow postures, such as flaring it out at various degrees, while still managing to move the drawer (Figure 1(a)). Having this flexibility is beneficial as it not only allows for multiple ways of achieving the task, but can also enhance efficiency or robustness, thereby enhancing overall performance. Humans tend to take advantage of this flexibility in predictable, stereotypical ways, commonly, seeking to minimise discomfort or energy expenditure [4], [5]. For instance, in human drawer opening, despite the variety of postures that can be taken, it is typical to keep ones wrist straight, to avoid uncomfortable joint flexion, and ones elbow down, to avoid working unnecessarily against gravity. Indeed, such features are codified in human ergonomics literature, to the point that they shape work environments and policy on safe working practice [6, 7, 8]. \begin{overpic}[width=216.81pt,height=75.8804pt]{images/drawerFlaring4.jpeg}\put(0.0,75.0){\ref{f:drawingPosesA}}\end{overpic} \begin{overpic}[width=216.81pt,height=75.8804pt]{images/drawerConfiguration3.jpeg}\put(70.0,75.0){\ref{f:drawingPosesB}}\end{overpic} Figure 1: (a) Redundancy in elbow postures when opening a drawer. In absence of other constraints, humans tend to avoid using non-ergonomic postures (shown in blue). (b) A comfortable pose for a human is different to one maximises manipulability in a robot with different joint limits. Similarly, a multi-DoF robotic system imitating the human can adopt different joint configurations that are consistent with maintaining the end-effector on the drawer handle, including those that closely match the human’s posture. Therefore, the simplest way to have a robot learn is to match it to the human’s posture as closely as possible when executing the task. However, such an approach neglects the differences in embodiment between human and robot that may lead to sub-optimal performance of the robot [9, 10]. For instance, maintaining a posture in which ones wrist joint is kept straight (Figure 1(b)), while comfortable for the human, may represent a singular posture for the robot that can lead to dangerous unstable movements. Moreover, for a robot with geared, non-backdriveable joints, it may cost little energy to maintain the elbow in a flared posture, whereas moving the arm to a more human-like, elbow-down posture may actually expend energy unnecessarily. Such cases suggest a more nuanced approach to human-to-robot behaviour transfer is required, that takes explicit account of the stereotypical features of human movement, and the desirability, or otherwise, to reproduce them in a robotic imitator. To this end, this paper investigates how stereotypical features of demonstrators’ posture control can be used to decompose observed behaviour into task-oriented and redundant components of motion. Specifically, it presents a new method for programming by demonstration whereby explicit use of the underlying null space control policy—as determined by the stereotypical or ergonomic features—is used to learn the task and null spaces involved in the behaviour and their underlying constraints [11, 12]. The latter allows the original behaviour to be retargeted to an imitator robot that has a different kinematic embodiment, by optimising movement according to the robot’s own structure without causing any interference with the task goal [13]. Numerical and physical evaluations are reported in which the proposed approach is applied to learn constraints in a toy experiment where performance on varying data lengths and noise are evaluated, a simulated 3DoF experiment where the approach is compared to the state-of-the-art approach, a demonstration on the benefits of retargeting the system to resolve redundancy for obstacle avoidance, task reproduction from a demonstrator system to an imitator of a different embodiment and finally a real world experiment where demonstrations from a human are used to learn and reproduce the task space motions with a Sawyer, a 7DoF physical robot. The results indicate that learning in this way outperforms several state-of-the-art methods [14, 15, 16, 17] in its ability to accurately learn the decomposition from relatively little data where in a comparative study the constraint is learnt from one trajectory of length $2\,s$ (100 data points), with minimal assumptions made on the form of the data. ## II Background & Related Work ### II-A Stereotypical Movement In the natural motion of people, it can generally be assumed that the observed movement will be optimised for efficiency according to their embodiment. In the context of humans teaching robots task-oriented movements, this means that postures adopted by a person in the course of a demonstration are likely to not only meet the requirements of the task, but also show stereotypical traits that reflect the demonstrator’s embodiment. For instance, human demonstrators will typically adopt postures that avoid working against gravity, or limb flexion or extension away from the resting posture of the limb. The study of postural preferences and stereotypical movement features in humans has long been studied in the field of _human movement science_ and, is particularly well documented in the domain of _human ergonomics_ [6, 7]. Note that, these stereotypical features are _secondary to the task_ —that is, they will tend to be promoted to seek comfort and minimise fatigue in movement—but are _subject to any applicable task constraints_. This means that they can be inhibited if the task demands it. For example, in drawer opening, the default rest posture of the shoulder is not maintained: since the hand must be lifted to the drawer handle for the task. Furthermore, if maintaining the elbow-down posture during opening _conflicts with the task_ (e.g., would result in a collision with a obstacle), then task space extends to the elbow elevation and overrides the default behaviour. This flexibility is a hallmark of human behaviour that is not currently captured by existing imitation learning approaches and poses an ongoing challenge. Traditional imitation learning approaches tend to treat behaviours as monolithic control policies, and so do not lend themselves well to task- priorised behaviours [18, 1, 19]. ### II-B Task Prioritised Behaviour To better capture this task-prioritised view of behaviour, several studies have recently focused on modelling demonstrations hierarchically, whereby movement is decomposed into the task space—the DoF required for the primary task—and a null space (i.e., the remaining DoF). This draws on several well- established hierarchical control schemes, such as Liegeois’ redundant kinemetic control scheme [20], or Khatib’s Operational Space Formulation [21]. In this view, actions $\mathbf{u}\in\mathbb{R}^{\mathcal{{Q}}}$ are assumed to take the form $\mathbf{u}(\mathbf{x})=\underbrace{\mathbf{A}^{\dagger}(\mathbf{x})\mathbf{b}(\mathbf{x})}_{\mathbf{v}}+\underbrace{\mathbf{N}(\mathbf{x})\boldsymbol{\pi}(\mathbf{x})}_{\mathbf{w}}$ (1) where $\mathbf{x}\in\mathbb{R}^{\mathcal{{P}}}$ represents state (usually represented either in end-effector or joint space) and $\mathbf{A}(\mathbf{x})\in\mathbb{R}^{\mathcal{{S}}\times\mathcal{{Q}}}$ is a matrix describing a system of $\mathcal{{S}}$-dimensional constraints $\mathbf{A}(\mathbf{x})\mathbf{u}(\mathbf{x})=\mathbf{b}(\mathbf{x})$ (2) and $\mathbf{N}(\mathbf{x}):=\mathbf{I}-\mathbf{A}(\mathbf{x})^{\dagger}\mathbf{A}(\mathbf{x})\in\mathbb{R}^{\mathcal{{Q}}\times\mathcal{{Q}}}$ (3) is the null space projection matrix that projects the policy $\boldsymbol{\pi}(\mathbf{x})$ onto the null space of $\mathbf{A}$. Here, $\mathbf{I}\in\mathbb{R}^{\mathcal{{Q}}\times\mathcal{{Q}}}$ denotes the identity matrix and $\mathbf{A}^{\dagger}=\mathbf{A}^{\top}(\mathbf{A}\mathbf{A}^{\top})^{-1}$ is the Moore-Penrose pseudo-inverse of $\mathbf{A}$. In this view, $\mathbf{b}(\mathbf{x})\in\mathbb{R}^{\mathcal{{S}}}$ represents the _task space policy_ describing the primary task to be accomplished, and the lumped term $\mathbf{v}=\mathbf{A}^{\dagger}(\mathbf{x})\mathbf{b}(\mathbf{x})$ represents that policy projected into the configuration space. $\boldsymbol{\pi}(\mathbf{x})$ represents the _null space policy_ , that encapsulates any actions in the configuration space secondary to the task. Note that, it is typically the case that $\mathbf{b}$ is unknown (since this is the task that should be learnt by demonstration), and $\mathbf{A}$ (and therefore $\mathbf{N}$) is also not explicitly known (since this describes the space in which the unknown task is defined). The key insight of this paper is that in many cases, _prior knowledge of $\boldsymbol{\pi}$ may be assumed_, since it commonly represents the _stereotypical features of secondary movements_. Furthermore, as shown in §III, knowledge of $\boldsymbol{\pi}$ enables efficient estimation of the other quantities in (1) ($\mathbf{v}$ and $\mathbf{N}$) that can in turn be used to separate out the task-oriented part of the demonstrations, and thereby replace the secondary components with a control policy tailored to the imitator’s embodiment. ### II-C Learning the Decomposition Several prior studies have examined the possibility of robot learning by demonstration using the representation (1)-(3), and in particular the possibility of learning $\mathbf{v}$ or $\mathbf{A}$ under the assumption that only $\mathbf{x}$ and $\mathbf{u}$ are observable. However, as noted in §II-A, in many cases such an assumption is _overly stringent_ and can result in degraded estimation performance. Depending on the assumptions made on its representation (see §III-C), one of several learning methods can be used to estimate $\mathbf{A}$ [14, 15, 17]. However, all of these methods, rely on the ability to separate the lumped task space term $\mathbf{v}$ (or, equivalently, the null space term $\mathbf{w}$) from the demonstrations, and learning performance is highly dependant on the quality of the separation. These studies rely on the same approach, first proposed by Towell et al. [12], that uses variations in the task space policy $\mathbf{b}$ and consistency in the null space policy $\boldsymbol{\pi}$ to form an estimate of the separation. This has several limitations in practice. First, for the separation to work, it can be difficult to ensure that the data is ‘rich’ enough in terms of the variations seen in the task space policy $\mathbf{b}$. Second, if working with data which contains different several distinct task-spaces, it is important to separate the data into subgroups and learn within each subset individually. However, this diminishes the learning quality as it tends to make less data available within each subgroup. These requirements can hamper the methods’ efficacy as increasingly complex systems and constraints are considered. The approach proposed here does not have such prerequisites—instead, it exploits prior knowledge of the control policy $\boldsymbol{\pi}$, a component that can often be estimated through an understanding of stereotypical behaviour or consideration of human ergonomics. ## III Method In this section, a new method is defined for estimating the null space projection matrix $\mathbf{N}$ in redundant systems where some prior knowledge of the redundancy resolution strategy is assumed available. ### III-A Data The proposed method works on data given as $\mathcal{{N}}$ pairs of observed states $\mathbf{x}_{n}$ and actions $\mathbf{u}_{n}$ collected from task- oriented movement demonstrations. It is assumed that (i) observations are in the form presented in (1), (ii) $\mathbf{A}$, $\mathbf{N}$ and $\mathbf{b}$ are not explicitly known for any given observation and (iii) $\boldsymbol{\pi}$is _known_ (or a good estimate is available) . As noted in §II, assumption (iii) is reasonable depending on several factors, including the task at hand and the environment. In most circumstances, healthy human demonstrators will tend to perform tasks in a way that _promotes comfort_. In the experiments reported here, this tendency is captured by assuming that the secondary control policy is a point attractor $\boldsymbol{\pi}(\mathbf{x})=\beta(\mathbf{x}^{*}-\mathbf{x})$ (4) where $\beta$ is a gain matrix and the point of attraction $\mathbf{x}^{*}$ is a posture that scores highly according to standard ergonomic assessment procedures such as Rapid Upper Limb Assessment [8]. While many ergonomic measures are applicable depending on the experiment, RULA is selected as it is quick and simple to classify static postures, focuses on the upper body which is in line with the drawer handling experiments, and provides a constant joint posture range for optimal scoring [22], [23]. ### III-B Learning the Null Space Projection Matrix The proposed method works by exploiting the orthogonality between the task and null space parts in (1). Specifically, noting that $\mathbf{v}^{\top}\mathbf{w}=\mathbf{w}^{\top}\mathbf{v}=0$, (1) can be written $\mathbf{w}^{\top}\mathbf{u}=\mathbf{w}^{\top}\mathbf{v}+\mathbf{w}^{\top}\mathbf{w}=\mathbf{w}^{\top}\mathbf{w}$ (5) yielding the identity $\mathbf{w}^{\top}(\mathbf{u}-\mathbf{w})=0.$ (6) An estimate of $\mathbf{N}$, and therefore the null space component $\mathbf{w}$, can be formed by choosing $\tilde{\mathbf{w}}=\tilde{\mathbf{N}}\boldsymbol{\pi}$ consistent with this identity by minimising111For brevity, here, and throughout the paper, the notation $\mathbf{a}_{n}$ is used to denote the quantity $\mathbf{a}$ evaluated on the $n$th sample. For example, if $\mathbf{a}$ is a vector quantity computed from the state $\mathbf{x}$, then $\mathbf{a}_{n}=\mathbf{a}(\mathbf{x}_{n})$. $E[\tilde{\mathbf{N}}]=\sum^{\mathcal{{N}}}_{n=1}||\boldsymbol{\pi}_{n}^{\top}\tilde{\mathbf{N}}_{n}(\mathbf{u}_{n}-\boldsymbol{\pi}_{n})||.$ (7) This minimisation problem can be solved using various nonlinear optimisation tools, in this case Matlab’s fmincon is used with the interior-point algorithm, which is a nonlinear optimisation solver for constrained multivariable functions. ### III-C Representation of the Constraints In order to efficiently learn $\tilde{\mathbf{N}}$, a suitable representation needs to be selected. The approach chosen here follows that first proposed by Lin et al. [11, 17] that represents $\tilde{\mathbf{N}}$ in terms of an underlying constraint matrix $\tilde{\mathbf{A}}$ according to (3). This has been shown to be effective both for unstructured problems (i.e., where the form of the constraint matrix $\mathbf{A}$ is completely unknown) and for situations where some features (i.e., candidate rows) of $\mathbf{A}$ are available. #### III-C1 Unit Vector Representation of $\mathbf{A}$ Following [17], if the form of $\mathbf{A}$ is completely unknown, it can be represented using a (potentially, state-dependent) set of $\mathcal{{S}}$ orthonormal vectors $\tilde{\mathbf{A}}=\left[\boldsymbol{\alpha}_{1}^{\top}\,\boldsymbol{\alpha}_{2}^{\top}\,\cdots\,\boldsymbol{\alpha}_{\mathcal{{S}}}^{\top}\right]^{\top}$ (8) where $\boldsymbol{\alpha}_{s}=(a_{s,1},a_{s,2},...,a_{s,\mathcal{{Q}}})$ corresponds to the $s$th constraint in the observations. The latter can be constructed iteratively by selecting vectors orthonormal to one another where the $s$th vector has the form $\displaystyle a_{s,1}$ $\displaystyle=\cos\theta_{1}$ $\displaystyle a_{s,2}$ $\displaystyle=\sin\theta_{1}\cos\theta_{2}$ $\displaystyle a_{s,3}$ $\displaystyle=\sin\theta_{1}\sin\theta_{2}\cos\theta_{3}$ $\displaystyle\vdots$ $\displaystyle a_{s,\mathcal{{Q}}-1}$ $\displaystyle=\prod_{q=1}^{\mathcal{{Q}}-2}\sin\theta_{q}\cos\theta_{\mathcal{{Q}}-1}$ $\displaystyle a_{s,\mathcal{{Q}}}$ $\displaystyle=\prod_{q=1}^{\mathcal{{Q}}-1}\sin\theta_{q}.$ (9) The resultant matrix is represented by a total of $\mathcal{{P}}=\mathcal{{S}}(2\mathcal{{Q}}-\mathcal{{S}}-1)/2$ parameters $\boldsymbol{\theta}=(\theta_{1},\theta_{2},\cdots,\theta_{\mathcal{{P}}})^{\top}$. #### III-C2 Representation of $\mathbf{A}$ with Candidate Rows In the case that prior information about the form constraint matrix is available, this can be incorporated into the estimate using the approach first proposed by [11]. Here, a suitable representation of the constraint matrix is $\tilde{\mathbf{A}}=\tilde{\mathbf{\Lambda}}\mathbf{\Phi}$ (10) where $\tilde{\mathbf{\Lambda}}\in\mathbb{R}^{\mathcal{{S}}\times\mathcal{{P}}}$ is a _selection matrix_ (to be estimated during learning) and $\mathbf{\Phi}\in\mathbb{R}^{\mathcal{{P}}\times\mathcal{{Q}}}$ is a (possibly, state-dependent) feature matrix. The rows of the latter can take generic forms such as a series of polynomials, or can contain candidate constraints if there is prior knowledge of those potentially affecting the system. For instance, one may choose $\mathbf{\Phi}=\mathbf{J}(\mathbf{x})$, the Jacobian of the manipulator, so that $\tilde{\mathbf{A}}=\tilde{\mathbf{\Lambda}}\mathbf{J}(\mathbf{x})$ encodes constraints on the motion of specific degrees of freedom in the end-effector space. ### III-D Estimating the Components of the Behaviour Once $\tilde{\mathbf{N}}$ is estimated, the decomposition of the behaviour into task-oriented and null space parts is straightforward. The null space space component is given as $\tilde{\mathbf{w}}=\tilde{\mathbf{N}}\boldsymbol{\pi}$ (11) and the task space part is computed as $\tilde{\mathbf{v}}=\mathbf{u}-\tilde{\mathbf{w}}.$ (12) Note that, given the estimate $\tilde{\mathbf{A}}$, $\tilde{\mathbf{N}}$ can be computed using (3), and an estimate of the task space policy $\tilde{\mathbf{b}}$ can be obtained using (2). ### III-E Substituting the Non-task oriented Behaviour As noted in §I, in many cases the redundancy resolution strategy seen in demonstrators’ task-oriented behaviour may be ill-suited to the robot imitator. The proposed method provides a simple means of _retargeting the behaviour to the robotic system_ , while maintaining the task-oriented parts. Specifically, this is achieved by replacing the controls (1) with $\mathbf{u}=\tilde{\mathbf{A}}^{\dagger}\tilde{\mathbf{b}}+\tilde{\mathbf{N}}\boldsymbol{\psi}$ (13) where $\boldsymbol{\psi}$ is a (possibly, state-dependent) redundancy resolution policy for the robot. For instance, $\boldsymbol{\psi}$ could be chosen so as to avoid robot-specific joint limits or singularities [13]. Alternatively, if the task space trajectory is predictable, $\tilde{\mu}$ can be used in combination with global optimisation in the null space [24]. ## IV Evaluation In this section, the proposed approach is first examined through a toy experiment, then a more complex 3-link planar system, before evaluating its performance in the context of programming by demonstration with a human demonstrator and a 7DoF physical robot.222The data supporting this research are openly available from King’s College London at http://doi.org/[link to be made available on acceptance]. Further information about the data and conditions of access can be found by emailing<EMAIL_ADDRESS> ### IV-A Toy Problem The aim of the first evaluation is to test the robustness of the proposed method using data from a simple, two-dimensional system with a one-dimensional task space. The set up (based on [11]) is as follows. Constrained motion data is gathered from a two-dimensional system with a one- dimensional constraint $\mathbf{A}=\boldsymbol{\alpha}\in\mathbb{R}^{1\times 2}$. Movement in the task space is defined by the constraint matrix and occurs in the direction of the unit vector $\boldsymbol{\hat{\alpha}}=(\cos{\theta},\sin{\theta})$. This direction is selected from a uniform-random distribution $\theta\sim U[0^{\circ},180^{\circ}]$ at the start of each trial. The task space policy is a linear point attractor $b(\mathbf{x})_{i}=r^{*}-r,i\in\\{1\\}$, where $r$ is the position in the task space and $r^{*}$ is the target point. To simulate varying tasks, the task space targets are selected randomly $r^{*}\sim U[-2,2]$ for each trial. In the below, learning performance is reported for three different secondary control policies $\boldsymbol{\pi}$, namely, 1. 1. A linear policy: $\boldsymbol{\pi}(\mathbf{x})=-\mathbf{L}\bar{\mathbf{x}}$ where $\bar{\mathbf{x}}\coloneqq(\mathbf{x}^{\top},1)^{\top})$ and $\mathbf{L}=((2,4,0)^{\top},(1,3,-1)^{\top})^{\top})$. 2. 2. A limit cycle: $\dot{\rho}=\rho(\rho_{0}-\rho^{2})$ with radius $\rho_{0}=0.75\,m$, angular velocity $\phi=1\,rad/s$, where $\rho$ and $\phi$ are the polar representation of the state, i.e., $\mathbf{x}=(\rho\cos\phi,\rho\sin\phi)^{\top}$. 3. 3. A non-linear (sinusoidal) policy: $\boldsymbol{\pi}=(\cos z_{1}\cos z_{2},-\sin z_{1}\sin z_{2})^{\top}$ where $z_{1}=\pi x_{1}$ and $z_{2}=\pi(x_{2}+\frac{1}{2})$. The training data consists of $150$ data points, drawn uniform randomly across the space $(\mathbf{x})_{i}\sim U[-1,1],i\in\\{1,2\\}$. For testing, a further 150 data points are used, generated through the same procedure as above. The constraint is learnt by finding a $\boldsymbol{\theta}$ which minimises (7). In each trial performance is measured using two metrics. First, the normalised mean squared error (NMSE) in the estimated null space component is evaluated333The notation $\mathbf{C}=\mathbf{A}\oslash\mathbf{B}$ denotes _Hadamard_ (element-wise) _division_ of $\mathbf{A}$ by $\mathbf{B}$, i.e., $(\mathbf{C})_{ij}=(\mathbf{A})_{ij}/(\mathbf{B})_{ij}$. $E_{\tilde{\mathbf{w}}}=\frac{1}{\mathcal{{N}}}\sum_{n=1}^{\mathcal{{N}}}||(\mathbf{w}_{n}-\tilde{\mathbf{w}}_{n})\oslash\boldsymbol{\sigma}_{\mathbf{u}}||^{2}.$ (14) where $\boldsymbol{\sigma}_{\mathbf{u}}\in\mathbb{R}^{\mathcal{{Q}}}$ is a vector containing the element-wise standard deviation of the observations $\mathbf{u}$. Note that, as $\mathbf{w}_{n}=\mathbf{N}_{n}\boldsymbol{\pi}_{n}$ and $\tilde{\mathbf{w}}_{n}=\tilde{\mathbf{N}}_{n}\boldsymbol{\pi}_{n}$, this measure is equal to the normalised projected policy error [17]. Second, the normalised error measure (7) is evaluated $E_{\tilde{\mathbf{N}}}=\frac{1}{\mathcal{{N}}||\sigma_{\mathbf{u}}||^{2}}\sum^{\mathcal{{N}}}_{n=1}||\boldsymbol{\pi}_{n}^{\top}\tilde{\mathbf{N}}_{n}(\mathbf{u}_{n}-\boldsymbol{\pi}_{n})||.$ (15) It indicates the performance of the minimisation function using only known information from given data and is therefore applicable for practical application. 444To evaluate the fitness of $\tilde{\mathbf{v}}$, noting that $\mathbf{v}+\mathbf{w}=\tilde{\mathbf{v}}+\tilde{\mathbf{w}}$ can be written as $\mathbf{v}-\tilde{\mathbf{v}}=-\mathbf{w}+\tilde{\mathbf{w}}$ returns the identity $\mathbf{v}-\tilde{\mathbf{v}}=-(\mathbf{w}-\tilde{\mathbf{w}}$), where the error in both components are opposites. Thus, results for the fitness of $\tilde{\mathbf{w}}$ can also be considered the same for $\tilde{\mathbf{v}}$ and therefore $\tilde{\mathbf{v}}$ is omitted. The experiment is repeated $50$ times. TABLE I: Test data NMSE in $\tilde{\mathbf{w}}$ (mean$\pm$s.d.)$\times 10^{-15}$ and $E_{\tilde{\mathbf{N}}}$ (mean$\pm$s.d.)$\times 10^{-8}$ over $50$ trials for different $\boldsymbol{\pi}$. $\boldsymbol{\pi}$ $E_{\tilde{\mathbf{w}}}$ $E_{\tilde{\mathbf{N}}}$ Linear 1.2147 $\pm$ 2.6458 0.4263 $\pm$ 1.3396 Limit-cycle 0.5462 $\pm$ 0.7043 1.1616 $\pm$ 1.1682 Sinusoidal 0.3020 $\pm$ 0.5741 1.6685 $\pm$ 1.9316 The NMSE in $\tilde{\mathbf{w}}$ and $E_{\tilde{\mathbf{N}}}$ are presented in TABLE I. As can be seen, $\tilde{\mathbf{N}}$ is successfully estimated with errors in $\tilde{\mathbf{w}}$ less than $10^{-14}$ and $\tilde{\mathbf{N}}$ with less than $10^{-7}$ for all of the policies considered. These low errors shows that the constraint matrix can be estimated with very high precision if knowledge of $\boldsymbol{\pi}$ is available. The overall performance of $\tilde{\mathbf{w}}$ is seen to be very roughly around twice as accurate as $E_{\tilde{\mathbf{N}}}$, which shows that the task and null space components can generally be reproduced with greater accuracy than indicated by just evaluating $E_{\tilde{\mathbf{N}}}$. \begin{overpic}[width=433.62pt]{images/allToyResultsV3lowres.png} \put(0.0,0.0){\scriptsize\ref{f:dataToyResults-a}} \put(25.0,0.0){\scriptsize\ref{f:dataToyResults-b}} \put(51.0,0.0){\scriptsize\ref{f:dataToyResults-c}} \put(76.0,0.0){\scriptsize\ref{f:dataToyResults-d}} \end{overpic} Figure 2: NMSE in $\tilde{\mathbf{w}}$ and $E_{\tilde{\mathbf{N}}}$ (mean$\pm$s.d. over $50$ trials) for (a) increasing number of data points for $E_{\tilde{\mathbf{w}}}$, (b) increasing number of data point for $E_{\tilde{\mathbf{N}}}$, (c) increasing noise levels in $\mathbf{u}$ and (d) increasing noise levels in $\boldsymbol{\pi}$. Mean results are plotted as thick lines and their respective standard deviation are the shaded areas of a similar lighter tone. To further characterise the performance of the proposed approach, the experiment is repeated with (i) data sets of varying sizes ($5<\mathcal{{N}}<250$), (ii) varying levels of noise in the training data $\mathbf{u}_{n}$ represented as $N(0,\epsilon\sigma_{\mathbf{u}}^{2})$ additive white Gaussian noise where $0<\epsilon<0.14$ and (iii) varying levels of noise in the estimated $\boldsymbol{\pi}_{n}$ where $0<\epsilon<0.15$ for $50$ trials using the limit cycle policy. The latter case simulates error in the assumed $\boldsymbol{\pi}$ and thereby allows evaluation of the proposed approach in face of an inaccurate estimate of the true underlying redundancy resolution strategy. The results are plotted in Figure 2. As shown in Figure 2(a), the NMSE in $\tilde{\mathbf{w}}$ is less than $10^{-14}$ for both mean and standard deviation with as few as five data points. As the number of data points increases, so does the accuracy for minimising $\tilde{\mathbf{w}}$ where the performance of the method seems to plateau after around 25 data points. This shows that the approach can learn constraints with as few as five data points and for optimal performance with at least around 25 data points. Figure 2(b) shows errors of less $10^{-7}$ for both mean and standard deviation in $\tilde{\mathbf{N}}$. It follows a similar trend to the previous evaluation with respect to accurate learning with as few as five data points and optimal performance after at least around 25 data points. It can also be observed that the learning performance is very roughly half compared to $E_{\tilde{\mathbf{w}}}$ which was also observed in TABLE I. Looking at Figure 2(c), there is a clear trend with a degrading mean accuracy and greater standard deviation as the noise in $\mathbf{u}$ increases. The mean error in $\tilde{\mathbf{w}}$ stays below 0.1 when $\epsilon<=0.14$ and for mean error in $\tilde{\mathbf{N}}$ when $\epsilon<=0.13$. It can also been seen that the error in $\tilde{\mathbf{N}}$ is greater compared to $\tilde{\mathbf{w}}$ in most cases which is in agreement with prior experiments. Looking at Figure 2(d), the accuracy decreases, with greater standard deviation, as the error in the assumed $\boldsymbol{\pi}$ increases. The mean error in $\tilde{\mathbf{w}}$ stays below 0.1 when $\epsilon<0.05$, however when $\epsilon<=0.1$ only 2 mean values are shown to produce an error above 0.1. The mean $E_{\tilde{\mathbf{N}}}$ stays below 0.1 when $\epsilon<=0.08$ and only has a single instance where the mean value is above 0.1 where $\epsilon<=0.1$. Comparing $E_{\tilde{\mathbf{w}}}$ and $E_{\tilde{\mathbf{N}}}$, while both have similar mean performance, the standard deviation of $E_{\tilde{\mathbf{N}}}$ is noticeably smaller. This is expected, as $E_{\tilde{\mathbf{N}}}$ relies on knowledge of the noisy estimate of $\boldsymbol{\pi}$ to obtain $\tilde{\mathbf{N}}$, whereas $E_{\tilde{\mathbf{w}}}$ compares this to the true $\boldsymbol{\pi}$ to present the error within the estimated null space component. ### IV-B Simulated Three Link Planar Arm The aim of the next evaluation is to test the performance of the proposed method on a more complex system with non-linear constraints which simulates a real world system more accurately. The set up is as follows. Constrained motion data is gathered from a kinematic simulation of a three- link planar robot with uniform links of length $10\,cm$. The state and action space refer to the joint angle position and velocities, respectively, i.e., $\mathbf{x}:=\mathbf{q}\in\mathbb{R}^{3}$ and $\mathbf{u}:=\dot{\mathbf{q}}\in\mathbb{R}^{3}$. The task space is described by the coordinates $\mathbf{r}=(r_{x},r_{y},r_{\theta})^{\top}$ referring to the end-effector positions and orientation, respectively. The simulation runs at a rate of $50\,Hz$. Joint space motion of the system is recorded as it performs tasks under different constraints in the end-effector space. Specifically, a task constraint at state $\mathbf{x}$ is described through $\mathbf{A}(\mathbf{x})=\mathbf{\Lambda}\mathbf{J}(\mathbf{x})$ (16) where $\mathbf{J}\in\mathbb{R}^{3\times 3}$ is the manipulator Jacobian, and $\mathbf{\Lambda}\in\mathbb{R}^{3\times 3}$ is a diagonal selection matrix, with elements $\boldsymbol{\lambda}=(\lambda_{x},\lambda_{y},\lambda_{\theta})^{\top}$ along the diagonal, indicating which coordinates should be included in ($\lambda_{i}=1$) or excluded from ($\lambda_{i}=0$) the task space. In the results reported below, the following six selection matrices are considered: (i) $\mathbf{\Lambda}_{x}$where $\boldsymbol{\lambda}=(1,0,0)^{\top}$, (ii) $\mathbf{\Lambda}_{y}$where $\boldsymbol{\lambda}=(0,1,0)^{\top}$, (iii) $\mathbf{\Lambda}_{\theta}$where $\boldsymbol{\lambda}=(0,0,1)^{\top}$, (iv) $\mathbf{\Lambda}_{x,y}$where $\boldsymbol{\lambda}=(1,1,0)^{\top}$, (v) $\mathbf{\Lambda}_{x,\theta}$where $\boldsymbol{\lambda}=(1,0,1)^{\top}$, and (vi) $\mathbf{\Lambda}_{x,\theta}$where $\boldsymbol{\lambda}=(0,1,1)^{\top}$. To simulate demonstrations of reaching behaviour, the robot end-effector starts from a point chosen uniform-randomly $q_{1}\sim U[0^{\circ},10^{\circ}],q_{2}\sim U[90^{\circ},100^{\circ}],q_{3}\sim U[0^{\circ},10^{\circ}]$ to a task space target $\mathbf{r}^{*}$ following a linear point attractor policy $\mathbf{b}(\mathbf{x})=\mathbf{r}^{*}-\mathbf{r}$ (17) where $\mathbf{r}^{*}$ is drawn uniformly from $r^{*}_{x}\sim U[-1,1]$, $r^{*}_{y}\sim U[0,2]$, $r^{*}_{\theta}\sim U[0^{\circ},180^{\circ}]$. As the secondary control policy a simple point attractor of the form $\boldsymbol{\pi}(\mathbf{x})=\beta(\mathbf{x}^{*}-\mathbf{x})$ (18) is used, where $\mathbf{x}^{*}$ is arbitrarily chosen as $x_{1}=10^{\circ}$, $x_{2}=-10^{\circ}$, $x_{3}=10^{\circ}$ and $\beta=1$. For each of the cases (i)–(vi) above, $100$ trajectories are generated each containing $50$ data points, with $50$% of the samples provided for learning and the remainder reserved as unseen testing data. Finally, this whole experiment is repeated $50$ times. TABLE II: Mean $\pm$s.d. $E_{\tilde{\mathbf{w}}}$ and $E_{\tilde{\mathbf{N}}}$ on testing data for different null space policies over $50$ trials. The figures for $E_{\tilde{\mathbf{w}}}$ are (mean$\pm$s.d.)$\times 10^{-11}$ and for $E_{\tilde{\mathbf{N}}}$ are (mean$\pm$s.d.)$\times 10^{-7}$. $\boldsymbol{\pi}$ $E_{\tilde{\mathbf{w}}}$ $E_{\tilde{\mathbf{N}}}$ $\mathbf{\Lambda}_{x}$ 0.0741 $\pm$ 0.0291 1.1527 $\pm$ 2.3654 $\mathbf{\Lambda}_{y}$ 12.1467 $\pm$ 18.6578 11.1195 $\pm$ 9.1619 $\mathbf{\Lambda}_{\theta}$ 0.1496 $\pm$ 0.3732 0.2445 $\pm$ 0.1700 $\mathbf{\Lambda}_{x,y}$ 5.6114 $\pm$ 10.3401 5.1969 $\pm$ 6.5849 $\mathbf{\Lambda}_{x,\theta}$ 0.0139 $\pm$ 0.0320 0.8522 $\pm$ 1.0982 $\mathbf{\Lambda}_{y,\theta}$ 0.0476 $\pm$ 0.1357 0.6719 $\pm$ 0.6424 The NMSE in $\tilde{\mathbf{w}}$ and $E_{\tilde{\mathbf{N}}}$ on the testing data are presented in TABLE II. As shown, the constraints are successfully learnt with $E_{\tilde{\mathbf{w}}}$ less than $10^{-9}$ and $E_{\tilde{\mathbf{N}}}$ less than $10^{-5}$ in all cases. The $E_{\tilde{\mathbf{w}}}$ is roughly half of the $E_{\tilde{\mathbf{N}}}$ which is in agreement with previous experiments. Overall, the constraint matrix can be accurately estimated using data from the observed demonstrations and knowledge of the control policy, without having to explicitly know how the constraints affect the system’s motions. To further evaluate the performance of the proposed method, it is compared to the current state-of-the-art. As discussed in §II-C, while there are many applicable methods to learn the constraint matrix with acceptable performance including [11], they all rely on the method proposed in [12] for separation of the observed actions. Following [12], Figure 3 shows an example of using a learnt constraint to generate a new trajectory. In this experiment, the new trajectory is reproduced using $\tilde{\mathbf{A}}$ which is learnt from a separate training data set, $\mathbf{u}$, $\mathbf{x}$ and $\boldsymbol{\pi}$, where the latter three are given. Firstly, training data consists of one trajectory of length $2\,s$ (100 data points) with a random constraint in $x,y$. Using the state-of-the-art approach in [12], $\mathbf{u}$ is separated into the task and null space components. Now that the null space component is learnt, it is used with $\mathbf{u}$, $\mathbf{x}$ and the approach in [11] to obtain $\tilde{\mathbf{A}}$. On the other hand, the novel approach uses $\mathbf{u}$, $\mathbf{x}$ and $\boldsymbol{\pi}$ to directly obtain $\tilde{\mathbf{A}}$. Now that both approaches have resulted in a learnt constraint, a ground-truth test trajectory is produced subject to the same true constraints in $x,y$ of the training data. Its start pose is $q_{1}=90^{\circ},q_{2}=45^{\circ},q_{3}=-20^{\circ}$ and the $x,y$ position of the end-effector moves towards the target point $(15,10)^{\top}$ which reaches convergence in 4 seconds. To compare the novel and literature approach, both use $\mathbf{x}$ of this ground-truth data to start at the same position. Both produce $\tilde{\mathbf{v}}$ with their respectively learnt $\tilde{\mathbf{A}}$ and use this with $\mathbf{u}$ to estimate $\tilde{\mathbf{b}}$ following (2). The literature approach already has $\tilde{\mathbf{w}}$ which was obtained using [12]. The novel approach uses (3) to obtain $\tilde{\mathbf{N}}$ and uses the known $\boldsymbol{\pi}$ to produce $\tilde{\mathbf{w}}$. Both approaches use this information to iteratively reproduce the ground-truth data and similarly run for 4 seconds which is shown in Figure 3(a) and Figure 3(b). This experiment is repeated for a constraint in $\theta$ shown in Figure 3(c) and Figure 3(d). As can be seen in Figure 3(a) and Figure 3(b), the proposed approach is shown alongside the true policy as well as the current state-of-the-art [12] for comparison. The new method follows the true joint trajectories accurately. The state-of-the-art approach on the other hand takes a different route in both the end effector trajectory as well as joint space leading to a different target. In Figure 3(c) and Figure 3(d), the task space target is set as the orientation of the end-effector which moves towards the target angle of $45^{\circ}$. The novel approach accurately reproduces the movements under this 1DoF constraint unlike the state-of-the-art method in [12] and [11]. \begin{overpic}[width=208.13574pt]{images/allCompare2.png} \put(0.0,45.0){\scriptsize\ref{f:compareLiteratureA}} \put(50.0,45.0){\scriptsize\ref{f:compareLiteratureB}} \put(0.0,0.0){\scriptsize\ref{f:compareLiteratureC}} \put(52.0,0.0){\scriptsize\ref{f:compareLiteratureD}} \end{overpic} Figure 3: Reproducing the ground-truth movement (dotted-black) in both learnt task and null space using the proposed method (solid-red) and the state-of-the-art method [12] (dashed-blue) to learn the null space component and constraint $\mathbf{A}$ to obtain $\mathbf{b}$ [11]. (a) Arm visualisation for example task under constraint space $\mathbf{r}=(x,y)$, (b) Joint angle positions during example movement in task space $\mathbf{r}=(x,y)$, (c) Arm visualisation for example task under constraint space $\mathbf{r}=\theta$ and (d) Joint angle positions during example movement in task space $\mathbf{r}=\theta$. \begin{overpic}[width=205.97214pt]{images/threeToThreeV4Obstacle.png}\put(0.0,0.0){\scriptsize\ref{f:ThreeToThree}}\end{overpic} \begin{overpic}[width=205.97214pt]{images/threeToSevenV2.png}\put(0.0,0.0){\scriptsize\ref{f:ThreeToSeven}}\end{overpic} Figure 4: Retargeting behaviour with an imitator robot (a) with the same embodiment as the demonstrator but located near to an obstacle, and (b) with a different kinematic structure. The demonstrated movement is illustrated in red, while that of the imitators is in blue. The yellow region indicates an obstacle. Figure 5: Sample flow of obtaining natural demonstration data from user. Video of the subject is recorded during the action of repeatedly opening and closing drawers of various heights to different lengths. After collection, the video feed is overlaid with a skeleton using Openpose [25] to obtain the positions of body parts such that joint angles and trajectories can be extracted into Matlab. The shoulder, elbow and wrist joint angles are used to learn the constraints contained within the set of demonstrations. As mentioned in §III-E, the proposed approach allows _retargeting_ of task- oriented behaviour by substituting the demonstrator’s redundancy resolution strategy with one better-suited to the robot. More concretely, consider the scenario where it is desired to reproduce a demonstrated reaching movement (i) with a robot with identical embodiment to the demonstrator, but is located right next to an obstacle (such that there is the risk of collision, see Figure 4(a)), and (ii) with a robot that has a different kinematic structure to the demonstrator (here, different numbers and lengths of links) . In the following, the feasibility of retargeting to these scenarios is assessed. As the reaching movement to be reproduced, a typical trajectory is taken from the training data (given in the absence of any obstacles) described above. Specifically, the example chosen uses $\mathbf{\Lambda}=\mathbf{\Lambda}_{x,y}$, $\mathbf{w}$ derived from the policy (18), $\mathbf{r}^{*}=(-9.12,3.89)^{\top}$ and $\mathbf{q}=(8.67^{\circ},94.18^{\circ},-2.32^{\circ})^{\top}$. This movement is _retargeted_ by using the learnt $\tilde{\mathbf{A}}$ to derive $\tilde{\mathbf{b}}$, and then applying (13) with a replacement null space control policy. In the case of the robot located next to an obstacle, retargeting simply consists of selecting an appropriate null-space control policy $\boldsymbol{\psi}$. Here, $\boldsymbol{\psi}(\mathbf{x})=\beta_{r}(\mathbf{x}^{*}_{r}-\mathbf{x})$ is used, where $\beta_{r}=5$ and $\mathbf{x}^{*}_{r}=(-320^{\circ},100^{\circ},50^{\circ})^{\top}$. The resultant movements _with_ and _without_ retargeting are shown in Figure 4(a) (blue and red figures, respectively). As can be seen, were the arm to directly imitate the demonstration (red figure) a collision would occur (second and third links overlap with the yellow region). This could not only jeopardise the success of the task but also potentially cause damage to the system. In contrast, starting at the same start point, the retargeted controller (blue figure) successfully completes the task (converges to the same target point in end-effector space), however, by resolving its redundancy differently it avoids the collision. In the case of the robot with the different kinematic structure, retargeting is achieved as follows. As noted above, the constraint in this system is represented in the form (10) where the feature matrix is selected as the Jacobian of the demonstrator’s embodiment (i.e., $\mathbf{\Phi}=\mathbf{J}$) and the selection matrix $\tilde{\mathbf{\Lambda}}$ is learnt. Since the rows of $\mathbf{\Phi}$ represent meaningful quantities (here, the relationship between the joint space and the end-effector position and orientation) a correspondence is drawn between these and the equivalent quantities for the new arm (e.g., if the first row of $\mathbf{J}$ relates to the Jacobian for the $r_{x}$ coordinate, the corresponding row of the Jacobian $\mathbf{J}_{r}$ for the imitator is selected), and $\mathbf{A}_{r}$ is constructed accordingly. Substituting into (13) gives the controller $\mathbf{u}=\mathbf{A}_{r}^{\dagger}\tilde{\mathbf{b}}+(\mathbf{I}-{\mathbf{A}_{r}}^{\dagger}\mathbf{A}_{r})\boldsymbol{\psi}.$ (19) In this evaluation, the imitator robot is taken to be a $7$-DoF arm with link lengths $10$, $5$, $5$, $5$, $5$, $5$ and $10\,cm$, and $\boldsymbol{\psi}==\beta_{r}(\mathbf{x}^{*}_{r}-\mathbf{x})$, where $\beta_{r}=1$ and $\mathbf{x}^{*}_{r}=(-10^{\circ},-10^{\circ},-10^{\circ},-10^{\circ},-10^{\circ},-10^{\circ},-10^{\circ})^{\top}$. The start posture is chosen such that the initial end-effector position matches that of the demonstration (i.e., $\mathbf{q}=(0^{\circ},90^{\circ},-90^{\circ},85^{\circ},90^{\circ},-1^{\circ},-81.5^{\circ})^{\top}$). The resultant movement is shown in Figure 4(b). As can be seen, despite the significant difference in embodiment, the task-oriented part of the movement is effectively reproduced, while the imitator-specific null space controller appropriately handles the added redundancy. ### IV-C Real World Human Arm The aim of this final experiment is to test the performance of the proposed approach in the real world using data from a human demonstrator. The set up is as follows. The task chosen for this experiment is to teach a robotic system the skill of opening and closing a three-tiered set of drawers (see Figure 5). To collect data, the demonstrator stands in front of the drawers at approximately an arm’s length distance. The starting state of each drawer is randomised (varying from anywhere between fully closed to completely open). Starting with the top drawer it is moved a random distance in the opening or closing direction. This is repeated for each of the drawers producing a total three trajectories which are used for learning a model. A side view of this is recorded from a single $12$MP phone camera (with a sensor of $1.4\mu$m pixels and aperture of f/$1.7$) placed roughly $1\,m$ away from the demonstrator. The video data is then post-processed to overlay a skeleton using Openpose [25] to estimate the joint lengths and extract the joint angles and velocities during movement. Constrained motion data of movement in the sagittal plane is gathered from three joints of the demonstrator’s arm, by observing flexion and extension of the shoulder, elbow and wrist (as well as abduction and adduction of the wrist depending on the forearms pronation/supination. This yields an average of $11$ frames per trajectory which translates into 11 data points. Now that the data is collected, to set it up for learning, the joint angles of the demonstrator are treated as the state $\mathbf{x}:=\mathbf{q}\in\mathbb{R}^{3}$ and the joint velocities as the action $\mathbf{u}:=\dot{\mathbf{q}}\in\mathbb{R}^{3}$. The task space is described by the end-effector coordinates $\mathbf{r}=(r_{x},r_{y},r_{\theta})^{\top}$ referring to the hand position and orientation, respectively. The task constraints are described in the form (16), where $\mathbf{J}\in\mathbb{R}^{3\times 3}$ is the manipulator Jacobian of the demonstrator. To construct the Jacobian which simulates the demonstrator as a system, the link lengths are calculated from the skeleton in Openpose for each frame. As these can vary from frame to frame depending on obscurities in the demonstrators pose and imprecision of Openpose, the mean of the joint lengths at every frame for the 3 trajectories are used. Moreover, when translating recorded movements from pixels to the $x$ and $y$ axis in Matlab, joint lengths are extremely large where the upper arm measures at around 30 meters. Therefore the scale of these joint lengths are proportionally reduced to around 10cm by dividing each by 300. $\mathbf{\Lambda}\in\mathbb{R}^{3\times 3}$ is the selection matrix specifying the coordinates to be constrained. In this experiment, $\mathbf{\Lambda}_{x,y}$ represents the ground truth, since the end-effector (demonstrator’s hand) must be maintained at the height $y$ of the drawer handle ($y$ changes in each demonstration depending on the which drawer is being manipulated), and $x$ is the task space in which opening or closing action occurs. It is assumed that the control policy in $\mathbf{w}$ is resolved by the subject with comfort in mind and moves towards a target joint pose following (18) with $\mathbf{x}^{*}=(-90^{\circ},90^{\circ},0^{\circ})^{\top}$. This posture is chosen as it lies in the middle of each joints optimal range following RULA, resulting in a high score according to RULA’s standard ergonomic assessment procedure [8] (see §II). Since the human demonstrations are modelled as a system with its respective Jacobian matrix, $\mathbf{u}$, $\mathbf{x}$ and $\boldsymbol{\pi}$, it can be evaluated like any other robotic system presented so far [26], [27]. The experiment is repeated $45$ times yielding a total of $135$ trajectories ($45$ repetitions for each of the 3 drawers). This is done to verify that the performance of learning the constraints is consistent. The true decomposition of the behaviour, (i.e., $\mathbf{v}$ and $\mathbf{w}$) are not known, however they can be estimated using $\mathbf{\Lambda}_{x,y}$. The initial aspect that can be evaluated is the $E_{\tilde{\mathbf{w}}}$, however the variance in $\mathbf{u}$ is quite small and thus $E[\tilde{\mathbf{N}}]$ from (7) is reported which is $6.3329\pm 5.7832$(mean$\pm$s.d.). Looking at the learnt $\mathbf{\Lambda}$, the correct constraints are consistently learnt using the novel approach. To demonstrate this, the learnt constraint is used to produce $\tilde{\mathbf{b}}$ and the task-oriented trajectory is reproduced on the Sawyer, a 7DoF revolute physical robotic system with a maximum reach of 1260mm and precision of $\pm$0.1mm. A closing of a drawer trajectory is selected from the human demonstration data. $\mathbf{J}\in\mathbb{R}^{3\times 7}$ is the manipulator Jacobian where a correspondence is drawn between these and the human arms decomposed Jacobian. The start pose is $\mathbf{q}=(-3.89^{\circ},42.82^{\circ},25.48^{\circ},-76.96^{\circ},-8.23^{\circ},32.82^{\circ},88.93^{\circ})^{\top}$. $\boldsymbol{\psi}==\beta_{r}(\mathbf{x}^{*}_{r}-\mathbf{x})$, where $\beta_{r}=1$ and $\mathbf{x}^{*}_{r}=(-70.29^{\circ},-32.47^{\circ},-15.92^{\circ},53.24^{\circ},-32.55^{\circ},-17.48^{\circ},\newline -68.23^{\circ})^{\top}$. The resulting trajectory is presented in Figure 6. As shown, the Sawyer is able to reproduce the task space component of closing the drawer using its own embodiment and a different $\boldsymbol{\pi}$ to resolve redundancy subject to the same task constraints. Figure 6: 3DoF Human to 7DoF Sawyer Robot task transfer ## V Conclusion In this paper, a method based on programming by demonstration is proposed to learn null space policies from constrained motion data. The main advantage to using this is the retargeting of not only the systems redundancy resolution but also the entire system itself with another of a different embodiment which can repeat a task accurately while being subject to the same constraints. On a lesser note, this proposed approach can be used to learn directly from observed actions without the need to decompose motion data into a task and null space component. The effectiveness of the method has been demonstrated in a simulated toy experiment, a 3-link simulation and a real world experiment using data collected from a human demonstrator which is validated through task-oriented reproduction on a 7DoF physical robot. All experiments are in agreement that the constraints can be learnt from demonstration. In addition, the evaluations show that the method can (i) learn with very little data ($E_{\tilde{\mathbf{w}}}$ below $10^{-14}$ with just five data points) and (ii) handle noise ($E_{\tilde{\mathbf{w}}}$ below $10^{-1}$ with normalised additive white gaussian noise below 0.15) . In a comparative experiment, the approach is shown to outperform the current state-of-the-art approach in a simulated 3DoF robot manipulator control problem where motions are reproduced using the learnt constraints. It is also used to demonstrate retargeting of a systems null space component to resolve redundancy such that an obstacle can be avoided. Moreover, retargeting through the learnt constraints from the simulated 3DoF demonstrator to a 7DoF robot imitator of different embodiment is shown. 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2024-09-04T02:54:56.095409

2003.00565

arxiv-papers

# Proportional Power Sharing Control of Distributed Generators in Microgrids Farzad Aalipour<EMAIL_ADDRESS>Tuhin Das<EMAIL_ADDRESS> ## Abstract This research addresses distributed proportional power sharing of inverter- based Distributed Generators (DGs) in microgrids under variations in maximum power capacity of DGs. A microgrid can include renewable energy resources such as wind turbines, solar panels, fuel cells, etc. The intermittent nature of such energy resources causes variations in their maximum power capacities. Since DGs in microgrids can be regarded as Multi-Agent-Systems (MASs), a consensus algorithm is designed to have the DGs generate their output power in proportion to their maximum capacities under capacity fluctuations. A change in power capacity of a DG triggers the consensus algorithm which uses a communication map at the cyber layer to estimate the corresponding change. During the transient time of reaching a consensus, the delivered power may not match the load power demand. To eliminate this mismatch, a control law is augmented that consists of a finite-time consensus algorithm embedded within the overarching power sharing consensus algorithm. The effectiveness of the distributed controller is assessed through simulation of a microgrid consisting of a realistic model of inverter-based DGs. ## Keywords Consensus, Eigenvalue Perturbation, Distributed Control, Finite-Time Consensus, Inverter-based Microgrid, Proportional Power Sharing, Renewable Energy, Transient Control. ## 1 Introduction Environmentally sustainable electrical energy production depends on renewable energy resources. In this regard, significant amount of researches have been undertaken within the past few decades [1, 2, 3]. Conventionally, control of electric power systems and the main power grid was accomplished through a few central controllers. Through emerging renewable energy plants, intelligent loads located in the demand side and computational advances, distributed energy production and management has become viable. DGs as distributed energy production units, together with local loads which are distinct from the main power, is called a microgrid. Microgrids operate in two different operational modes called grid-connected and islanding. A microgrid is said to work in grid-connected mode when it is connected to the main grid via a tie line at the point of common coupling (PCC) where there exists bidirectional power flow from or into the main grid [4]. In contrast, microgrids in islanding mode generates power for local loads [5]. To deploy small-scale DGs including photovoltaic (PV) cells, wind turbines, fuel cells and energy storage systems (ESSs) in microgrids, power electronics inverters are vital interfaces which connect DGs to the power buses [6]. There have been extensive studies conducted on control of inverter-based microgrids during the past decades [7, 8]. The control strategies can be classified in different categories including frequency and voltage control or power control [9]. Applying these methods also depends on the microgrid’s mode of operation. For instance, in the grid-connected mode the frequency and voltage are imposed by the main grid. However, voltage and frequency control are vital in islanding mode. In this work, power sharing control of microgrids in grid-connected mode is studied [11]. The problem of power sharing has been studied from the aspect of equal power sharing in [12, 13]. Since DGs possess different capacities, the DGs with higher capacities can share more power than the DGs with lower capacities. The power sharing problem becomes challenging under intermittent nature of power resources. Intermittency causes fluctuations in maximum capacity of DGs, which leads to changes in their output power. Thereby, the total power fluctuates, and the load power may not always be maintained. These fluctuations can be addressed by deploying electrical energy storage (EES) or managing the DGs to flexibly address the variations in their capacities. Other approaches proposed to address the power sharing problem in microgrids can be categorized either as proportional power sharing [9, 10, 14], or economic dispatch problem (EDP) [15]. The studies [14] and [16] have proposed techniques for proportional power sharing. Here, proportional power sharing is defined as sharing the load among DGs such that each individual DG shares a fraction of the load in proportion to its maximum capacity. A distributed droop control scheme based on nonlinear state feedback proposed in [9] guarantees that DGs share reactive power proportionally. In [32], through a distributed voltage control, the active and reactive power are shared proportionally, for a microgrid with inductive impedance loads. In addition, [33] formulates the proportional power sharing as a tracking problem and solves it for grid-connected spatially concentrated microgrids. However, [9] and [32] study islanding mode, and none of [9, 32, 33] have covered the power mismatch during transient time of their proposed strategies. On the other hand, EDP is a method to control the power flow among different DGs optimally, where optimality implies minimizing a quadratic performance index assigned to each DG as the cost of their generated power. EDP has been studied through different techniques including the population dynamic method [17], and the lambda iteration [18]. While these methods have been formulated within a centralized control framework, distributed version of EDP can be found in [15, 19]. Motivated by systems with cyber-physical layers, the power sharing control in this study is devised in two layers. The physical layer that consists of DGs, loads, measurement units, etc., is where the power control loop of each DG is established to track the input power command issued from the cyber layer. DGs have their corresponding agents in the cyber layer. Thus, the ideas of MASs can be utilized to establish the DGs’ controllers and their interactions. The agents communicate through a communication network in the cyber layer. The agents can choose different strategies to control the DGs including centralized, decentralized or distributed formats. When the DGs are located in a small region, it is viable to apply centralized controllers. As the number of DGs increases, while geographically scattered in a wide area, applying the centralized controllers faces deficiencies due to some reasons; Firstly, the centralized controller is not reliable due to the dependency of the DGs on a single controller where its malfunctions deteriorate the performance of the microgrid or may result in instability. Besides, in centralized coordinated control, transferring data to a control center and issuing control signals back to DGs require high bandwidth communication, which is not economically efficient, or technically secure, and is prone to failure [20]. On the contrary, distributed control techniques require considerably lower bandwidth which makes the communications among the DGs economically viable. Decentralized controllers are applicable locally, however it does not exploit cooperation of DGs [21]. Therefore, they may not perform efficiently where the global information and cooperation is required. In contrast, a distributed control scheme encompasses the plug and play feature, which it makes it more flexible compared to centralized and decentralized controllers [22]. The centralized control scheme depends on global information while DGs in distributed control exchange information exclusively with the DGs in their neighborhood. In this study, the well-known consensus algorithm is utilized to design a distributed controller for the power sharing control problem. Considering a large number of DGs scattered in a wide area, it takes agents time to transfer all the required signals. Therefore, it is inevitable to have communication delays in the distributed controllers [23]. The ranges of these delays are from tens to hundreds of milliseconds [24]. The delays may result in prolonging the convergence time of consensus algorithm and potentially lead to microgrid instability [16]. The delays can be reduced through increasing the convergence rate of consensus algorithms utilizing approaches including multiplying the weights of the communication graph with a large constant, or through an optimization of the weights [25]. This study considers a microgrid operating in grid-connected mode using proportional power sharing. Proportional power sharing makes the microgrid adaptable to intermittency of power sources. The grid-connected operation enables the microgrid to transmit excess power to the main grid, while relying on it for frequency and voltage control. The contributions of this paper are: 1) A distributed consensus algorithm is designed, by which the DGs are able to estimate the microgrid’s power capacity under perturbations in the power capacity of individual DGs. Convergence rate of the algorithm is studied and bounds on allowable perturbations are derived based on practical constraints. 2) Multiple proportional power sharing strategies are proposed, to meet the demanded power as consensus is reached. The strategies are executed in a distributed manner. They ensure that the microgrid satisfies load power variations dynamically and allow excess power to be transmitted to the main grid. 3) Items (1) and (2) enable proportional power sharing. However, during convergence of the consensus algorithm, a power mismatch occurs between the generated and demanded power. Although the rate of convergence can be increased, the transient power mismatch remains inevitable. To eliminate this mismatch, a fully distributed finite-time consensus algorithm, based on [27], is additionally augmented. 4) A realistic simulation is conducted, using MATLAB’s Simscape toolbox, with a clear primary controller scheme and corresponding parameters. Grid and DG parameters used in simulations are also given. The simulations and the results can be reproduced by readers, allowing further enhancements in future research. The rest of this paper is organized as follows. The preliminary definitions of technical terms are explained in section two. Then, proportional power sharing is defined in the third section. In the fourth section, the consensus algorithm is developed through which the DGs are able to update their information about the total microgrid power capacity following a change in a DG’s capacity. The overarching consensus algorithm and the embedded transient controller are proposed and elaborated in the same section. Fifth section discusses the cyber and physical layers which control the output power of DGs. Next, simulation results are provided in section six to illustrate the effectiveness of the proposed control plan in response to different variations in capacity of a DG. Finally, concluding remarks are provided and references are listed. ## 2 Preliminary Definitions We define the graph $\mathcal{G}$ as the set pair $(\mathcal{\nu},\mathcal{\varepsilon})$ having vertices set $\mathcal{\nu}$ and edge set $\mathcal{\varepsilon}$. Let the number of vertices in $\mathcal{G}$ be $N$, and let the set $\mathcal{\varepsilon}$ consist of the vertices pairs $(i,j)$ for which there exists an edge that connects $j$ to $i$, with $i,j=1,2,\cdots,N$ and $i\neq j$. The intended graph in this study is undirected or bidirectional graph, where the signals flow along edges in both directions, i.e. if $(i,j)\in\mathcal{\varepsilon}$, then $(j,i)\in\mathcal{\varepsilon}$. The adjacency matrix associated with the graph is $A=[a_{ij}]\in R^{N\times N}$ where each element $a_{ij}>0$ if $(i,j)\in\mathcal{\varepsilon}$, otherwise $a_{ij}=0$. As stated above, in the bidirectional graph $\mathcal{G}$, if $(i,j)\in\mathcal{\varepsilon}$, then $(j,i)\in\mathcal{\varepsilon}$, and $a_{ij}=a_{ji}$. Then $A$ is symmetric, i.e. $A=A^{T}$. We define the degree matrix $D=[d_{ii}]\in R^{N\times N}$ as a diagonal matrix as such $d_{ii}=\sum_{j=1}^{N}a_{ij}\vspace{-1.5mm}$ (1) The matrix $L=[l_{ij}]=D-A$ is denoted as the Laplacian Matrix of $\mathcal{G}$. As mentioned above, $A=A^{T}$, and considering $D$ is a diagonal matrix, it follows that $L=L^{T}$. The neighbor set corresponding to each vertex $i$ is defined as $\mathcal{N}_{\;\;i}=\\{j|(i,j)\in\mathcal{\varepsilon}\\}$. Additionally, $\mathcal{N}_{\;\;j}^{+}$ denotes the set of outgoing neighbors of node $j$, i.e., the set of nodes receiving signals form the node $j$, and $\mathcal{N}_{\;\;j}^{-}$ is the set of nodes which sends signals to the node $j$. For the bidirectional graph $\mathcal{G}$, $\mathcal{N}_{\;\;j}^{+}=\mathcal{N}_{\;\;j}^{-}$. A graph is connected if there exists a path between any two distinct vertices [26]. We assume that $\mathcal{G}$ is connected. Next, consider Fig. 1 which shows a sample localized microgrid with four DGs, denoted by DGi, $i=1,2,3,4$. In this figure, the dashed lines show signaling between the cyber layer and physical layer, i.e. the communications between the DGs and their corresponding agents in the cyber layer. The lines with bidirectional arrows represent communications among the corresponding agents of DGi located in the cyber layer. The solid lines are electrical connections. Based on the weights shown in Fig. 1 and the explanations above, the adjacency and degree matrices are defined as, $\displaystyle A\\!\\!=\\!\\!\\!\\!\left[\begin{array}[]{cccc}\\!\\!\\!\\!0&\\!\\!\\!\\!0&\\!\\!\\!\\!a_{13}&\\!\\!\\!\\!0\\!\\!\\!\\!\\!\\!\\\ \\!\\!\\!\\!0&\\!\\!\\!\\!0&\\!\\!\\!\\!a_{23}&\\!\\!\\!\\!a_{24}\\!\\!\\!\\!\\!\\!\\\ \\!\\!\\!\\!a_{31}&\\!\\!\\!\\!a_{32}&\\!\\!\\!\\!0&\\!\\!\\!\\!0\\!\\!\\!\\!\\!\\!\\\ \\!\\!\\!\\!0\\!\\!\\!\\!&\\!\\!\\!\\!a_{42}&\\!\\!\\!\\!0\\!\\!\\!\\!&\\!\\!\\!\\!0\\!\\!\\!\\!\\!\\!\end{array}\right]\\!\\!\\!,D\\!\\!=\\!\\!\\!\\!\left[\begin{array}[]{cccc}\\!\\!\\!\\!\\!a_{13}&\\!\\!\\!\\!\\!\\!\\!\\!0&\\!\\!\\!\\!\\!\\!\\!\\!0&\\!\\!\\!\\!\\!\\!\\!\\!0\\!\\!\\!\\!\\!\\!\\\ \\!\\!\\!\\!\\!0&\\!\\!\\!\\!\\!\\!\\!\\!a_{23}+a_{24}&\\!\\!\\!\\!\\!\\!\\!\\!0&\\!\\!\\!\\!\\!\\!\\!\\!0\\!\\!\\!\\!\\!\\!\\\ \\!\\!\\!\\!\\!0&\\!\\!\\!\\!\\!\\!\\!\\!0&\\!\\!\\!\\!\\!\\!\\!\\!a_{31}+a_{32}&\\!\\!\\!\\!\\!\\!\\!\\!0\\!\\!\\!\\!\\!\\\ \\!\\!\\!\\!\\!0&\\!\\!\\!\\!\\!\\!\\!\\!0&\\!\\!\\!\\!\\!\\!\\!\\!0&\\!\\!\\!\\!\\!\\!\\!\\!a_{42}\\!\\!\\!\\!\\!\end{array}\right]$ (2) Based on the definition of Laplacian matrix, the corresponding Laplacian matrix to the adjacency and diagonal matrices defined in (2) is $\displaystyle L=\left[\begin{array}[]{cccc}\\!\\!a_{13}&\\!\\!0&\\!\\!-a_{13}&\\!\\!0\\\ \\!\\!0&\\!\\!a_{23}+a_{24}&\\!\\!-a_{23}&\\!\\!-a_{24}\\\ \\!\\!-a_{31}&\\!\\!-a_{32}&\\!\\!a_{31}+a_{32}&\\!\\!0\\\ 0&\\!\\!-a_{42}&\\!\\!0&\\!\\!a_{42}\end{array}\right]$ (3) ## 3 Problem Definition We consider a microgrid in the grid-connected mode, where the microgrid’s voltage and frequency are imposed by the main grid, i.e. the microgrid’s frequency and voltage are fixed. Hence, the goal in this mode is to control the output power of the DGs. The cyber-physical systems considered in this paper is similar to the one shown in Fig. 1. The proposed control emerges from consensus control of Multi-Agent Systems (MAS). The control objective is sharing load power in proportion to the maximum power capacity of the DGs, under variations in maximum capacities. Figure 1: Schematic of a microgrid comprising cyber and physical layers We assume that there exists $N$ DGs in a microgrid which are labeled as DGi where $i=1,2,\cdots,N$. The maximum power capacity and instantaneous output power of each DGi are defined as $P_{i,max}$ and $P_{i}$, respectively. Let $P_{L}$ be the load power, which is proportionately shared among the DGs, i.e. $P_{L}=\sum_{i=1}^{N}P_{i},\,\,\,\mbox{s.t.}\,\,\,r=\frac{P_{1}}{P_{1,max}}=\frac{P_{2}}{P_{2,max}}=\cdots=\frac{P_{N}}{P_{N,max}}\vspace{-1mm}$ (4) where $r$ is the proportional power share ratio. Thus, the output power of DGi is $P_{i}=rP_{i,max}$. Let $P_{T}$ be the total power capacity of the microgrid defined as the accumulation of the maximum power capacity of all the DGs in the microgrid. Then, one can conclude that $r=\frac{\sum_{i=1}^{N}P_{i}}{\sum_{i=1}^{N}P_{i,max}}=\frac{P_{L}}{P_{T}}\vspace{-1mm}$ (5) and the output power of DGi is $P_{i}=(P_{L}/P_{T})\,P_{i,max}$. Note that a fluctuation in the maximum capacity of a DG $P_{i,max}$ or a change in $P_{L}$ will cause a change in $r$. The proposed power sharing control will, in response, manage the output power of the DGs flexibly. Throughout this study, the goal is to manage the variations of $P_{i}$s while meeting the requirement $P_{L}=\sum_{i=1}^{N}P_{i}$. It is assumed that all DGs have a knowledge of $P_{L}$ at all times. The power demand $P_{L}$ can vary with time. We next explain two scenarios for which different controllers are designed. At the core of these controllers is a consensus algorithm which is inherently distributed. Recall that an underlying assumption is that the communication graph among the DGs is connected. Before any change happens to the renewable energy resources, we assume all DGs have the knowledge of $P_{T}$ by which they are able to compute $r$ from (5) and thereby generate their appropriate proportional power share $P_{i}=rP_{i,max}$, $i=1,2,\cdots,N$. In the first scenario, assume the maximum capacity of DGk which is $P_{k,max}$, changes. Then $P_{T}$ changes accordingly and all DGs are required to update their value of $P_{T}$ to be able to recalculate the new $r$ based on (5). The only DG that can generate accurate power immediately after a fluctuation happens is the DGk since it is aware of the change in $P_{k,max}$. Let $\delta$ be the change such that $\tilde{P}_{k,max}=P_{k,max}+\delta$, where $\tilde{P}_{k,max}$, is the updated value of $P_{k,max}$. Thus, DGk can compute the updated capacity of the microgrid as $\tilde{P}_{T}$ where $\tilde{P}_{T}=P_{T}+\delta$ and recalculate $r$ and the delivered power $P_{k}$. A consensus algorithm is devised to have other DGs compute the $\tilde{P}_{T}$ and thereby reach the new value of $r$, distributively. In the second scenario, we address the mismatch between load and supplied power before consensus is reached. As was discussed in scenario (1), only DGk can generate an accurate amount of power instantaneously after a fluctuation in DGk. Although the other DGs are able to update $P_{i}$ following a change in $P_{k,max}$, the consensus algorithm takes time to converge, and hence during the transient time $\sum_{i=1}^{N}P_{i}$ would not necessarily be equal to $P_{L}$. The reason is that the other DGs do not have the correct value of $\tilde{P}_{T}$ instantaneously. However, since instantaneous matching of load power is a priority, a control law is augmented with the consensus algorithm to practically remove power mismatch during transients. ## 4 Distributed Microgrid Control ### 4.1 Consensus on Total Power Capacity under Perturbation We consider a scenario where the individual DGs know the power ratio $r$ and generate accurate $P_{i}$ based on proportional power sharing, as shown in (4). Hence, each DG has correct knowledge of $P_{T}$, as per (5). Next, consider a change in $P_{k,max}$ to $\tilde{P}_{k,max}=P_{k,max}+\delta$. Following this change, all agents are required to compute $\tilde{P}_{T}=P_{T}+\delta$, the updated value of $P_{T}$. We define $s_{i}(t)$ as the estimate of $\tilde{P}_{T}$ by DGi. The vector of estimate variables is then, $\mathbf{S}(t)=\left[\begin{array}[]{ccccc}s_{1}&s_{2}&\cdots&s_{N}\end{array}\right]^{T}$, where $N$ is the number of the DGs in the microgrid. As mentioned above, all the DGs know $P_{T}$ before any change happens. Therefore, the initial value of $\mathbf{S}$ is, $\mathbf{S}(0)=P_{T}\textbf{1}$ where $\textbf{1}^{N}=\left[\begin{array}[]{ccccc}1&1&\cdots&1\end{array}\right]^{T}$. Thereafter, we propose the following consensus dynamics in the cyber layer, through which all DGs update their value of $P_{T}$ and converge to $\tilde{P}_{T}$. $\displaystyle\dot{s}_{k}(t)=\\!-h\left(s_{k}(t)-\tilde{P}_{T}\right)-\\!\\!\\!\\!\sum_{j\in\mathcal{N}_{\;\;k}}\\!\\!\\!\\!a_{kj}\left(s_{k}(t)-s_{j}(t)\right)\\!\\!$ (6) $\displaystyle\dot{s}_{i}(t)=\\!-\\!\\!\\!\\!\sum_{j\in\mathcal{N}_{\;\;i}}\\!\\!\\!\\!a_{ij}\left(s_{i}(t)-s_{j}(t)\right),\,\,i=\\!1,2,...,N,\,i\neq k$ where $s_{k}(0)=P_{T}$ and $s_{i}(0)\\!=P_{T}$. In (LABEL:Eq:_Agents_Communication_Dynamics), $a_{ij}>0$ and it denotes the weight of the communication link between agents $i$ and $j$, where $i,j=1,2,\cdots,N$, $i\neq j$, and $h>0$ is a parameter chosen by the $k^{th}$ agent. The parameter $h$ represents a measure of the convergence rate of $s_{i}(t)$ towards $\tilde{P}_{T}$. Since the communication graph is bidirectional, therefore $a_{ij}=a_{ji}$, and this implies that the Laplacian matrix is symmetric, i.e. $L=L^{T}$ (see example in (3). From (LABEL:Eq:_Agents_Communication_Dynamics), the following matrix equation is obtained $\dot{\textbf{S}}=-(L+\Delta)\mathbf{S}+hd_{k}\tilde{P}_{T}$ (7) where $d_{k}=\\!\\!\left[\begin{array}[]{ccccccccc}\sigma_{1,k}\\\ \sigma_{2,k}\\\ \vdots\\\ \sigma_{k,k}\\\ \vdots\\\ \sigma_{N,k}\end{array}\right]\\!\\!\\!=\\!\\!\left[\begin{array}[]{ccccccccc}0\\\ 0\\\ \vdots\\\ 1\\\ \vdots\\\ 0\end{array}\right]\\!\\!\\!\in\\!R^{N\times 1},\Delta=hd_{k}d^{T}_{k}\in\\!R^{N\times N}$ (8) In (8), $\sigma_{i,k}$ is the Kronecker Delta function, implying the $k^{th}$ element of $d_{k}$ is one and rest are zero. Also, $\Delta_{k,k}=h$ and all other elements are zero. We now propose and prove the following Lemma. ###### Lemma 1. The linear dynamic system defined in (7) and (8) is input-to-state stable (ISS), and $S\rightarrow\tilde{P}_{T}\textbf{1}$ given the graph of communication among the agents is connected. ###### Proof. The linear system of (7) and (8) is ISS if $-(L+\Delta)$ is Hurwitz [28]. The input is $\tilde{P}_{nom}$ which is constant and bounded, thus, if $-(L+\Delta)$ is Hurwitz the proof is complete. Since $L=L^{T}$, and by definition $\Delta=\Delta^{T}$, the matrix $-(L+\Delta)$ is symmetric. Hence, it is Hurwitz if $-(L+\Delta)<0$, i.e. negative definite. To prove this, it is required to show that for any vector $u\in R^{N}$, $u^{T}[-(L+\Delta)]u$ is strictly less than zero unless $u=0$. From Eq. 8, $u^{T}[-(L+\Delta)]u=-u^{T}Lu-u^{T}\Delta u=-u^{T}Lu-h{u_{k}}^{2}$ (9) where $h=\Delta_{kk}>0$, and $u_{k}$ is the $k^{th}$ element of the vector $u$. As the communication graph is connected, $L$ is positive semi-definite with a single zero eigenvalue [29, 26], and it is diagonalizable [30], with all real eigenvalues. Let $\lambda_{i}$, $i=1,2,\cdots,N$ be the eigenvalues of $L$ in descending order, $\lambda_{1}\geq\lambda_{2}\geq\cdots>\lambda_{N-1}>\lambda_{N}=0$. The canonical form of $L$ is $L=V\Lambda V^{T}$, where $\Lambda$ is a diagonal matrix consisting of the eigenvalues of $L$, and $V$ is the right eigenvector- matrix, $\Lambda\\!\\!=\\!\\!\left[\begin{array}[]{ccccc}\lambda_{1}&0&&0\\\ 0&\lambda_{2}&&\\\ &&\ddots&\\\ 0&&&\lambda_{N}\end{array}\right]\\!\\!,\,\,V\\!\\!=\left[\begin{array}[]{ccccc}\\!\\!\\!v_{1}&\\!\\!\\!v_{2}&\\!\\!\\!\cdots&\\!\\!\\!v_{N}\\!\\!\\!\end{array}\right]$ (10) Since $v_{N}$ is the eigenvector corresponding to $\lambda_{N}=0$, following the definition of $L$, $v_{N}=c\textbf{1}^{N}$ where $c\neq 0$ is a real value. Substituting $L=V\Lambda V^{T}$ into (9) and taking (10) into account, the following holds: $\displaystyle-u^{T}Lu-h{u_{k}}^{2}=$ $\displaystyle-z^{T}\\!\\!\left[\begin{array}[]{ccccc}\\!\\!\lambda_{1}&\\!\\!\\!\\!0&\\!\\!\\!\\!&\\!\\!\\!\\!0\\!\\!\\\ \\!\\!0&\\!\\!\\!\\!\lambda_{2}&\\!\\!\\!\\!&\\!\\!\\\ \\!\\!&\\!\\!\\!\\!&\\!\\!\\!\\!\ddots&\\!\\!\\\ \\!\\!0\\!\\!\\!\\!&\\!\\!\\!\\!&\\!\\!\\!\\!&\\!\\!\\!\\!\lambda_{N}\\!\\!\end{array}\right]z-h{u_{k}}^{2}$ (11) $\displaystyle=-\sum_{i=1}^{N}\lambda_{i}z_{i}^{2}-h{u_{k}}^{2}$ where $z=V^{T}u$. Since $V^{T}$ is a nonsingular matrix it is invertible, and its inverse matrix is $V$ and $u=Vz$. For any $z=\left[\begin{array}[]{ccccc}z_{1}&z_{2}&\cdots&z_{N}\end{array}\right]^{T}$ and $u_{k}$, the right hand side of (11) is negative, except $z_{i}=0\;\forall\;i=1,2,\cdots,N-1$ and $u_{k}=0$. The remaining condition is $z=\left[\begin{array}[]{ccccc}0&\cdots&0&z_{N}\end{array}\right]^{T}$. As $V$ is not singular, $u=Vz\neq 0$ while $z_{N}\neq 0$. According to (11) and since $v_{N}=c\textbf{1}^{N}$, $u=Vz=\\!\left[\begin{array}[]{ccccc}\\!\\!\\!\\!v_{1}&\\!\\!\\!v_{2}&\\!\\!\\!\cdots&\\!\\!\\!c\textbf{1}\\!\\!\\!\\!\end{array}\right]\left[\begin{array}[]{ccccc}\\!\\!\\!\\!0&\\!\\!\\!\cdots&\\!\\!\\!0&\\!\\!\\!z_{N}\\!\\!\\!\\!\end{array}\right]^{T}\\!\\!=cz_{N}\textbf{1}$ (12) Now that $u=cz_{N}\mathbf{1}$ and $c,z_{N}\in R$ are non-zero, $u_{k}=cz_{N}\neq 0$. However, this is in contradiction with the assumption made before which is $u_{k}=0$. Thus, (11) is negative for any vector $z$ and since $u=Vz$, (9) is negative for any $u\neq 0$ which proves $-(L+\Delta)$ is negative definite. We define $y=\mathbf{S}-\tilde{P}_{T}\textbf{1}$, therefore, $\mathbf{S}=y+\tilde{P}_{T}\textbf{1}$ and $\dot{\mathbf{S}}=\dot{y}$. By substituting $y$ and $\dot{y}$ into (7), we obtain $\dot{y}=\\!-(L+\Delta)y,\,y=\mathbf{S}-\tilde{P}_{T}\textbf{1},\,y(t_{0})\\!=\\!P_{T}\textbf{1}-\tilde{P}_{T}\textbf{1}\\!=\\!-\delta\textbf{1}\\!\\!\\!\\!$ (13) As $-(L+\Delta)$ is Hurwitz the dynamics of (13) is exponentially stable. It means $y\rightarrow 0$ and therefore $\mathbf{S}\rightarrow\tilde{P}_{T}\textbf{1}$. This completes the proof. ∎ We assume that at any time, only one DG can change its power capacity. Thereafter, any subsequent change in power capacity can be done once the consensus algorithm described above has converged. ### 4.2 Observations on Rate of Convergence From Weyl’s theorem on eigenvalue inequalities for sum of two Hermitian matrices [30], the properties of Laplacian matrices in a connected network [29], and Lemma 1, we have $\lambda_{N-1}(-L)\leq\lambda_{N}(-(L+\Delta))<\lambda_{N}(-L)=0$ assuming the eigenvalues of $-L$ are ordered as $\lambda_{1}(-L)\leq\lambda_{2}(-L)\leq\cdots\leq\lambda_{N-1}(-L)\leq\lambda_{N}(-L)=0$, and those of $-(L+\Delta)$ are ordered as $\lambda_{1}(-(L+\Delta))\leq\lambda_{2}(-(L+\Delta))\leq\cdots\leq\lambda_{N-1}(-(L+\Delta))\leq\lambda_{N}(-(L+\Delta))<0$ for all $h>0$. The eigenvalue $\lambda_{N}(-(L+\Delta))$ is the dominant eigenvalue of $-(L+\Delta)$ and hence it determines the rate of convergence of consensus. To explain the effect of $h$ on convergence rate, we provide the following lemma. ###### Lemma 2. The rate of convergence of the consensus algorithm, given by (7) and (8), increases with increasing values of $h>0$. ###### Proof. The characteristic equation of $-L$ can be expressed as $p(\lambda)=0$. The roots of $p(\lambda)$, i.e. $\lambda_{1}(-L)\leq\lambda_{2}(-L)\leq\cdots\leq\lambda_{N-1}(-L)\leq\lambda_{N}(-L)=0$, are all real. It can be verified that the characteristic equation of $-(L+\Delta)$ takes the form $p(\lambda)+hq(\lambda)=0$. Further, $p(\lambda)$ and $q(\lambda)$ are both monic polynomials, with $p(\lambda)$ and $q(\lambda)$ being polynomials of order $N$ and $(N-1)$ respectively. From Lemma 1, we know that the roots of $p(\lambda)+hq(\lambda)$ satisfy $\lambda_{1}(-(L+\Delta))\leq\lambda_{2}(-(L+\Delta))\leq\cdots\leq\lambda_{N-1}(-(L+\Delta))\leq\lambda_{N}(-(L+\Delta))<0$, $\forall\;h>0$. Considering the characteristic equation of $-(L+\Delta)$ in root-locus form, i.e. $1+h\frac{q(\lambda)}{p(\lambda)}=0,\quad h>0,$ we deduce that all roots of $q(\lambda)$ are real and negative. Otherwise, increasing $h$ will eventually cause some eigenvalues of $-(L+\Delta)$ to become complex conjugates and/or unstable, thereby contradicting Lemma 1. This observation is in accordance with the rules of root locus. We further deduce that the largest root of $q(\lambda)$, namely $\lambda_{max}(q(\lambda))$, must satisfy $\lambda_{N-1}(-L)\leq\lambda_{max}(q(\lambda))<\lambda_{N}(-L)=0$ Violating the above condition would also contradict Lemma 1. Since $\lambda_{N}(-L)=0$ and $\lambda_{max}(q(\lambda))<0$ are the largest roots of $p(\lambda)$ and $q(\lambda)$ respectively, there is one branch of root locus that originates from $\lambda_{N}(-L)$ and ends at $\lambda_{max}(q(\lambda))$ for $0\leq h<\infty$. This branch is also the closest to the origin of all $N$ branches of the above root locus (which are all strictly along the negative real axis), and hence contains the locus of the dominant eigenvalue of $-(L+\Delta)$. With increasing $h$, this eigenvalue moves further to the left of the origin, thereby increasing the rate of convergence to the consensus proposed in Lemma 1. This completes the proof. ∎ ### 4.3 Proportional Power Sharing Strategies The consensus algorithm of Section 4.1 enable the DGs to compute the updated capacity of the microgrid under perturbation. In this section, we propose methods by which individual agents command power to the physical layer based on consensus. Subsequent to a capacity variation such as $\delta$ in Section 4.1, three slightly different strategies are proposed through which the DGs meet the load demand $P_{L}$. The first and third strategies are discussed in detail. The second strategy is similar to the first and hence its details are omitted. Assuming at $t=t_{0}$, $P_{k,max}$ changes to $\tilde{P}_{k,max}=P_{k,max}+\delta$, the first strategy to generate $P_{i}$s is $\mbox{Strategy \\!\\!\\! 1:}\left\\{\begin{aligned} &\\!P_{k}=\\!\frac{P_{L}}{s_{k}}\tilde{P}_{k,max}\\\ &\\!P_{i}=\\!\frac{P_{L}}{s_{i}}P_{i,max}\,\,\,\mbox{for}\,\,\;i=1,2,\cdots,N\;i\neq k\\!\\!\\!\\!\end{aligned}\right.$ (14) where $s_{i}$, $i=1,2,\cdots,N$, are the estimates of $\tilde{P}_{T}$, as discussed in Section 4.1, and $\lim_{t\to\infty}s_{i}=\tilde{P}_{T}$ according to Lemma 1. A potential issue may arise when $s_{i}(t)$ crosses or approaches zero for some $t>t_{0}$ such that $P_{i}$ diverges. In this regard, we state and prove the following Lemma: ###### Lemma 3. Considering the LTI system defined in (7), if $|\delta|<{\theta P_{T}}/{(1+\sqrt{N}})$ with $0<\theta<1-(P_{L}/P_{T})$, the following holds $(1\\!-\theta)P_{T}\\!\\!\leq s_{i}(t)\\!\leq\\!\\!(1\\!+\theta)P_{T},\,\,\,\,P_{i}(t)\\!\\!<\\!P_{i,max},\,\,\,\forall\,t\\!>t_{0}\\!\\!\\!\vspace{0.1in}$ (15) The proof of Lemma 3 is given in the Appendix. From Lemma 3, it may appear that as the number of DGs, $N$, increases, there will be a bigger restriction on $\delta$, since $|\delta|<{\theta P_{T}}/{(1+\sqrt{N}})$. However, it can be shown that the above inequality is not restrictive, mainly because as $N$ increases, $P_{T}$ also increases. An analysis of this aspect is given in the Appendix. From Lemma 3, it may also appear that the constraint on $\delta$ is restrictive as $\theta\to 0$, which happens as $P_{L}\to P_{T}$. This restriction is however justified, since $P_{L}\approx P_{T}$ practically implies that the grid is already close to maximum capacity. Hence, further perturbation in DGs may prevent it from meeting the load demand. So far, it is proved that strategy 1 is valid provided changes in $\delta$ satisfy the conditions in Lemma 3. Defining the total instantaneous output power of the microgrid as $P_{O}(t)$, from (14), $P_{O}(t)=\frac{P_{L}}{s_{k}}\tilde{P}_{k,max}+\sum_{i=1,i\neq k}^{N}\frac{P_{L}}{s_{i}}P_{i,max}$ (16) Therefore, $P_{O}(t)=\frac{P_{L}}{s_{k}}\delta+\sum_{i=1}^{N}\frac{P_{L}}{s_{i}}P_{i,max}$ (17) Thus, defining the instantaneous error $E(t)=P_{O}(t)-P_{L}$, we have, $E(t)=P_{O}(t)-P_{L}=\frac{P_{L}}{s_{k}}\delta+\sum_{i=1}^{N}\frac{P_{L}}{s_{i}}P_{i,max}-P_{L}$ (18) At $t=t_{0}$, $s_{i}=P_{T}$ for $i=1,2,\cdots,N$. Thus, $E(t_{0})=P_{L}\Bigl{[}\frac{\delta}{P_{T}}+\sum_{i=1}^{N}\frac{P_{i,max}}{P_{T}}-1\Bigr{]}$ (19) Since $\sum_{i=1}^{N}\frac{P_{i,max}}{P_{T}}=1$, therefore $E(t_{0})=P_{L}\frac{\delta}{P_{T}}$ (20) Equation (20) shows that $E(t_{0})\neq 0$, and since $E(t)$ is continuous, it implies that a perturbation $\delta$ causes a transient mismatch between the delivered power $P_{O}(t)$ and the load $P_{L}$. The error $E(t)\to 0$ at steady-state, as proven in Lemma 1. Therefore, Strategy 1 given in (14), causes a temporary mismatch of power following a perturbation. This issue is addressed in Section 4.4. The second strategy, which is slightly different from the first one, is as follows: $\mbox{ Strategy \\!\\!\\! 2:}\left\\{\begin{aligned} &\\!P_{k}=\\!\frac{P_{L}}{\tilde{P}_{T}}\tilde{P}_{k,max}\\\ &\\!P_{i}=\\!\frac{P_{L}}{s_{i}}P_{i,max}\,\,\,\mbox{for}\,\,\;i=1,2,\cdots,N\;i\neq k\\!\\!\\!\\!\\!\end{aligned}\right.$ (21) As before, the total instantaneous output power of the microgrid $P_{O}(t)$ is $P_{O}(t)=\frac{P_{L}}{P_{T}+\delta}(\tilde{P}_{k,max})+\sum_{i=1,i\neq k}^{N}\frac{P_{L}}{s_{i}}P_{i,max}$ (22) We again evaluate the error $E(t)=P_{O}(t)-P_{L}$ for $t\geq t_{0}$, yielding $E(t)=\frac{P_{L}}{P_{T}+\delta}(P_{k,max}+\delta)+\sum_{i=1,i\neq k}^{N}\frac{P_{L}}{s_{i}}P_{i,max}-P_{L}$ (23) Upon simplifying, we obtain $E(t)=P_{L}\Bigl{[}\frac{(P_{k,max}+\delta)}{P_{T}+\delta}+\sum_{i=1,i\neq k}^{N}\frac{P_{i,max}}{s_{i}}-1\Bigr{]}$ Since at $t=t_{0}$, $s_{i}=P_{T}$ for $i=1,2,\cdots,N$, and $\sum_{i=1,i\neq k}^{N}P_{i,max}=P_{T}-P_{k,max}$, $E(t_{0})=P_{L}\frac{\delta(P_{T}-P_{k,max})}{P_{T}(P_{T}+\delta)}$ (24) Equation (24) shows that $E(t_{0})\neq 0$, and since $E(t)$ is continuous, it implies that similar to Strategy 1, a perturbation $\delta$ causes a transient mismatch between the delivered power $P_{O}(t)$ and the load $P_{L}$ in Strategy 2. The error $E(t)\to 0$ at steady-state, as proven in Lemma 1. The last candidate strategy is proposed as $\mbox{ Strategy \\!\\!\\! 3:}\left\\{\begin{aligned} &\\!P_{k}=\\!\frac{P_{L}}{s_{k}}({P_{k,max}}+s_{k}-P_{T})\\\ &\\!P_{i}=\\!\frac{P_{L}}{s_{i}}P_{i,max}\,\,\,\mbox{for}\,\,\;i=1,2,\cdots,N\;i\neq k\\!\\!\\!\\!\\!\end{aligned}\right.$ (25) The Strategy 3 allows DGs to update their output power more smoothly compared to the first two strategies. In this case, $P_{O}(t)=\frac{P_{L}}{s_{k}}(P_{k,max}+s_{k}-P_{T})+\sum_{i=1,i\neq k}^{N}\frac{P_{L}}{s_{i}}P_{i,max}$ (26) Therefore, $E(t)=P_{L}\Bigl{[}-\frac{P_{T}}{s_{k}}+\sum_{i=1}^{N}\frac{P_{i,max}}{s_{i}}\Bigl{]}$ (27) Since $s_{i}=P_{T}$ for all $i=1,2,\cdots,N$, at $t=t_{0}$, $E(t_{0})=0$. However, $E(t)$ still undergoes transient fluctuations. Based on (27), $\frac{\dot{E}(t)}{P_{L}}=\frac{P_{T}\dot{s}_{k}(t)}{s_{k}^{2}(t)}-\sum_{i=1}^{N}\frac{P_{i,max}\dot{s}_{i}(t)}{s_{i}^{2}(t)}$ (28) Equation (28) can be further simplified using (7) as follows, $\displaystyle\frac{\dot{E}(t)}{P_{L}}=$ $\displaystyle\frac{P_{T}\dot{s}_{k}}{s_{k}^{2}}-$ (29) $\displaystyle\left[\begin{array}[]{ccccccc}\frac{P_{1,max}}{s_{1}^{2}}&\frac{P_{2,max}}{s_{2}^{2}}&\cdots&\frac{P_{N,max}}{s_{N}^{2}}\end{array}\right]$ $\displaystyle\hskip 42.67912pt\times[-(L+\Delta)S+hd_{k}(P_{T}+\delta)]$ Since at $t=t_{0}$, $s_{i}=P_{T}$ for all $i=1,2,\cdots,N$, $\mathbf{S}(t_{0})=P_{T}\mathbf{1}$ and (29) becomes $\displaystyle\frac{\dot{E}(t_{0})}{P_{L}}=$ $\displaystyle\frac{P_{T}\dot{s}_{k}(t_{0})}{P_{T}^{2}}-$ (30) $\displaystyle\left[\begin{array}[]{ccccccc}\\!\\!\frac{P_{1,max}}{P_{T}^{2}}&\\!\\!\\!\frac{P_{2,max}}{P_{T}^{2}}&\\!\\!\\!\cdots&\\!\\!\\!\frac{P_{N,max}}{P_{T}^{2}}\end{array}\right]$ $\displaystyle\hskip 22.76219pt\times\left[-L\mathbf{1}P_{T}-hd_{k}P_{T}+hd_{k}P_{T}+hd_{k}\delta\right]$ Simplifying (30) yields $\frac{\dot{E}(t_{0})}{P_{L}}=\frac{P_{T}\dot{s}_{k}(t_{0})}{P_{T}^{2}}-\frac{hP_{k,max}\delta}{P_{T}^{2}}$ (31) Referring to (7), at $t=t_{0}$, the term $\dot{s}_{k}(t_{0})$ in (31) is $\dot{s}_{k}(t_{0})=h(P_{T}+\delta)-hP_{T}=h\delta$ (32) Therefore, from (31) and (32), $\frac{\dot{E}(t_{0})}{P_{L}}=h\delta\frac{P_{T}-P_{k,max}}{P_{T}^{2}}$ (33) where from (LABEL:Eq:_Agents_Communication_Dynamics), $h$ is a positive scalar chosen by $k^{th}$ agent. Thus, although $E(t_{0})=0$, $\dot{E}(t_{0})\neq 0$. Therefore, as $E(t)$ is continuous, similar to Strategies 1 and 2, a change $\delta$ results in a transient mismatch between $P_{O}$ and $P_{L}$. It is shown that the three strategies proposed above match the load power $P_{L}$ at steady-state while producing transient deviations. This transient issue is resolved in the next section, where a strategy is proposed to practically maintain $P_{O}=P_{L}$ at any time. ### 4.4 Proportional Power Sharing with Transient Power Match Upon a perturbation in $P_{k,max}$, which results in a change in $P_{T}$, the agents estimate $\tilde{P}_{T}=P_{T}+\delta$ through (7). Among the DGs, only DGk has a knowledge of $\tilde{P}_{T}=P_{T}+\delta$. The other DGs in the microgrid converge to $\tilde{P}_{T}$ through consensus only at steady-state. This leads to the transient power mismatch discussed in Section 4.3. To remove this transient mismatch, we propose a strategy where DGk modulates its power delivery as follows, while the other DGs maintain the same strategy as in Section 4.3: $\displaystyle P_{k}=\frac{P_{L}}{s_{k}}P^{\prime}_{k,max}$ (34) $\displaystyle P_{i}=\frac{P_{L}}{s_{i}}P_{i,max}\qquad\quad\mbox{for}\quad i=1,2,\cdots,N\quad i\neq k$ where $P^{\prime}_{k,max}$ is an auxiliary dynamic variable required to modulate the instantaneous power of DGk. Hence, at $t=t_{0}$, $P^{\prime}_{k,max}(t_{0})=P_{k,max}(t_{0})$, and it is required to converge to $(P_{k,max}+\delta)$ while $s_{i}$ converges via consensus. With the goal of maintaining $P_{O}(t)=P_{L}$ for all $t>t_{0}$, we must have $P_{L}=P_{O}(t)=\frac{P_{L}}{s_{k}}P^{\prime}_{k,max}+\sum_{i=1,i\neq k}^{N}\frac{P_{L}}{s_{i}}P_{i,max}$ (35) Thus, $\frac{P^{\prime}_{k,max}}{s_{k}}+\sum_{i=1,i\neq k}^{N}\frac{P_{i,max}}{s_{i}}-1=0$ (36) Therefore, $P^{\prime}_{k,max}$ is $P^{\prime}_{k,max}(t)=s_{k}(t)\Bigl{[}1-\sum_{i=1,i\neq k}^{N}\frac{P_{i,max}}{s_{i}(t)}\Bigr{]}$ (37) The algorithm for updating $P^{\prime}_{k,max}$ and $s_{i}$ for $i=1,2,\cdots,N$ in (34) is as follows: $\displaystyle P^{\prime}_{k,max}(t)=s_{k}(t)\Bigl{[}1-\sum_{i=1,i\neq k}^{N}\frac{P_{i,max}}{s_{i}(t)}\Bigr{]}$ (38a) $\displaystyle\dot{\textbf{S}}=-(L+\Delta)\mathbf{S}+hd_{k}\tilde{P}_{T}\quad\,\,\,\mbox{where}\,\,\,\,\,\,\,\mathbf{S}(t_{0})=P_{T}\mathbf{1}$ (38b) Based on (34) and (38), we state and prove the following lemma: ###### Lemma 4. The dynamic system of (38) is stable, i.e. the terms $P^{\prime}_{k,max}$, $\frac{P_{i,max}}{s_{i}}$ and S remain bounded if $|\delta|<{\theta P_{T}}/{(1+\sqrt{N}})$, where $0<\theta<1-(P_{L}/P_{T})$. Furthermore, $P^{\prime}_{k,max}\rightarrow\tilde{P}_{k,max}$ and $\mathbf{S}\rightarrow\tilde{P}_{T}\mathbf{1}$, while the instantaneous delivered power satisfies (35) for all $t\geq t_{0}$. ###### Proof. Since (38b) is equivalent to (7), per Lemma 1, the dynamic system of (38b) is ISS. Therefore S is bounded. Additionally, as (38b) and 7) have the same initial conditions, i.e. $\mathbf{S}(t_{0})=P_{T}\mathbf{1}$, thus $\mathbf{S}\rightarrow\tilde{P}_{T}\mathbf{1}$. Following $|\delta|<{\theta P_{T}}/{(1+\sqrt{N}})$, from Lemma 3 we have $(1-\theta)P_{T}\leq s_{i}(t)\leq(1+\theta)P_{T}$ with $0<\theta<1-(P_{L}/P_{T})$. Thus, $\frac{P_{i,max}}{(1+\theta)P_{T}}\leq\frac{P_{i,max}}{s_{i}}\leq\frac{P_{i,max}}{(1-\theta)P_{T}}$ (39) Therefore, $\frac{P_{i,max}}{s_{i}}$ is bounded for all $i=1,2,\cdots,N$. It demonstrates that (38a) represents a viable way to update $P^{\prime}_{k,max}$. By plugging $P^{\prime}_{k,max}$ from (38a) into (34), $P_{O}(t)$ simplifies to $P_{O}(t)=\frac{P_{L}}{s_{k}}s_{k}\Bigl{[}1\\!-\\!\\!\\!\\!\sum_{i=1,i\neq k}^{N}\\!\\!\\!\\!\frac{P_{i,max}}{s_{i}}\Bigr{]}\\!+\\!\\!\\!\\!\\!\\!\sum_{i=1,i\neq k}^{N}\\!\\!\frac{P_{L}}{s_{i}}P_{i,max}=P_{L}$ (40) for all $t\geq t_{0}$. Since $\mathbf{S}$ converges to $\tilde{P}_{T}\mathbf{1}$, from (38a) we therefore deduce $P^{\prime}_{k,max}(t)\rightarrow\tilde{P}_{T}\Bigl{[}1-\sum_{i=1,i\neq k}^{N}\frac{P_{i,max}}{\tilde{P}_{T}}\Bigr{]}=\tilde{P}_{k,max}$ (41) This completes the proof. ∎ The controller designed in (38a) and (38b) maintains $P_{O}(t)=P_{L}$ following a variation in the power capacity of a DG, namely $P_{k,max}$. However, to compute the term $\sum_{i=1,i\neq k}^{N}\frac{P_{i,max}}{s_{i}}$ (42) in $P^{\prime}_{k,max}$, as given in (38a), the $k^{th}$ agent requires additional information. The following approach is proposed to enable the $k^{th}$ agent to attain this information distributively. This approach is based on the distributed finite-time average consensus studied in [27]. According to [27], each agent $i$, shares $\frac{P_{i,max}}{s_{i}(t)}$ to its outgoing neighbors $\mathcal{N}_{\;\;i}^{+}$ where, following Section 2, $\mathcal{N}_{\;\;i}^{+}$ stands for the set of nodes which receives signals from node $i$. Accordingly, based on what follows the agents are able to distributively compute the instantaneous average of all $\frac{P_{i,max}}{s_{i}(t)}$ where $i=1,2,\cdots,N$, i.e. $C_{a}(t)=\frac{\sum_{i=1}^{N}\frac{P_{i,max}}{s_{i}(t)}}{N}$ (43) Then, the $k^{th}$ agent can compute (42) via $\sum_{i=1,i\neq k}^{N}\frac{P_{i,max}}{s_{i}}=N\,C_{a}(t)-\frac{P_{k,max}}{s_{k}}$ (44) One example of applying this distributed finite-time average consensus is presented in [31]. Similar to [31], the steps of executing the finite-time algorithm is as following: $\displaystyle\overline{g}_{i}(m+1)$ $\displaystyle=p_{ii}\overline{g}_{i}(m)+\sum_{j\in\mathcal{N}_{\;\;i}^{-}}p_{ij}\overline{g}_{j}(m)$ (45) $\displaystyle g_{i}(m+1)$ $\displaystyle=p_{ii}g_{i}(m)+\sum_{j\in\mathcal{N}_{\;\;i}^{-}}p_{ij}g_{j}(m)$ where $\overline{g}_{i}(0)=\frac{P_{i,max}}{s_{i}}$ and $g_{i}(0)=1$ for $i=1,2,\cdots,N$. Additionally, $p_{ij}=\frac{1}{1+|\mathcal{N}_{\;\;j}^{+}|}$ for $i\in\mathcal{N}_{\;\;j}^{+}\cup\\{j\\}$, otherwise is zero. Let us define the vectors $\displaystyle\overline{g}_{i,2m}^{T}\\!\\!=\\!\\![\overline{g}_{i}(1)\\!-\\!\overline{g}_{i}(0),\overline{g}_{i}(2)\\!-\\!\overline{g}_{i}(1),\cdots,\overline{g}_{i}(2m+1)\\!-\\!\overline{g}_{i}(2m)]$ (46) $\displaystyle g_{i,2m}^{T}\\!\\!=\\!\\![g_{i}(1)\\!-\\!g_{i}(0),g_{i}(2)\\!-\\!g_{i}(1),\cdots,g_{i}(2m+1)\\!-\\!g_{i}(2m)]$ and the following Hankel matrices $\small{\Gamma\\{\overline{g}_{i,2m}^{T}\\}\triangleq\begin{bmatrix}\overline{g}_{i,2m}(1)&\cdots&\overline{g}_{i,2m}(m+1)\\\ \overline{g}_{i,2m}(2)&\cdots&\overline{g}_{i,2m}(m+2)\\\ \vdots&\ddots&\vdots\\\ \overline{g}_{i,2m}(m+1)&\cdots&\overline{g}_{i,2m}(2m+1)\end{bmatrix}}\normalsize$ (47) and $\small{\Gamma\\{g_{i,2m}^{T}\\}\triangleq\begin{bmatrix}g_{i,2m}(1)&\cdots&g_{i,2m}(m+1)\\\ g_{i,2m}(2)&\cdots&g_{i,2m}(m+2)\\\ \vdots&\ddots&\vdots\\\ g_{i,2m}(m+1)&\cdots&g_{i,2m}(2m+1)\end{bmatrix}}\normalsize$ (48) Each agent $i$ runs the steps in (45) for $2N+1$ times and keeps the values $\overline{g}_{i}(m)$ and $g_{i}(m)$ for $m=1,2,\cdots,2N+1$. Having $\overline{g}_{i}(m)$ stored for the $2N+1$, each agent $i$ establishes the vectors $\overline{g}_{i,2m}^{T}$ and $g_{i,2m}^{T}$ defined in (46) starting from $m=0$. At the same time, all individual agents construct their Hankel matrices $\Gamma\\{\overline{g}_{i,2m}^{T}\\}$ and $\Gamma\\{g_{i,2m}^{T}\\}$ defined in (47) and (48), respectively. Additionally, they calculate the ranks of the Hankel matrices for each $m$ and repeat the same procedure for the next $m+1$ until for a specific $m$ either $\Gamma\\{\overline{g}_{i,2m}^{T}\\}$ or $\Gamma\\{g_{i,2m}^{T}\\}$ becomes a defective matrix. Assume $\Gamma\\{\overline{g}_{i,2M_{i}}^{T}\\}$ or $\Gamma\\{g_{i,2M_{i}}^{T}\\}$ is the first matrix which loses its full rank where $\beta_{i}=[\beta_{i,0},\cdots,\beta_{i,M_{i}-1},1]^{T}$ is its corresponding kernel. Having the kernel $\beta_{i}$, the $i^{th}$ agent computes the average of all $\overline{g}_{i}(0)=\frac{P_{i,max}}{s_{i}}$ for $i=1,2,\cdots,N$ defined as $C_{a}$ in (43) through the following $C_{a}(t)=\frac{1}{N}\sum_{i=1}^{N}\overline{g_{i}}(0)=\frac{\left[\overline{g}_{i}(0),\overline{g}_{i}(1),\cdots,\overline{g}_{i}(M_{i})\right]\beta_{i}}{[g_{i}(0),g_{i}(1),\cdots,g_{i}(M_{i})]\beta_{i}}$ (49) Thereby, the $k^{th}$ agent achieves $C_{a}(t)$, distributively. At this step, the $k^{th}$ agent obtains the term in (42) via (44). By plugging (42) back to (37) the $k^{th}$ agent is able to compute $P^{\prime}_{k,max}$. To implement the proposed strategy practically, the mentioned procedures are required to be discretized firstly, since in practice the signals and algorithms update, digitally. We define an index $w$, starting from $w=0$, that represents the discrete instants at which the overall consensus algorithm is executed. At $w=0$, the $i^{th}$ agent, $i=1,2,\cdots,N$, has the value of $s_{i}(w)=P_{T}$. Therefore, they can compute $P_{i,max}/s_{i}(w)$, individually. Following the finite-time algorithm, they implement the procedure in (45)-(49). Now that all the agents, including the $k^{th}$ agent, obtains $C_{a}(w)$ from (49), then using $C_{a}(w)$, the $k^{th}$ agent computes $P^{\prime}_{k,max}(w)=s_{k}(w)\Bigl{[}1-\sum_{i=1,i\neq k}^{N}\frac{P_{i,max}}{s_{i}(w)}\Bigr{]}$ (50) The command signals to DGs are as follows $\displaystyle P_{k}(w)=\frac{P_{L}(w)}{s_{k}(w)}P^{\prime}_{k,max}(w)$ (51) $\displaystyle P_{i}(w)=\frac{P_{L}(w)}{s_{i}(w)}P_{i,max}\qquad\quad\mbox{for}\quad i=1,2,\cdots,N\quad i\neq k$ Thereafter, the agents compute $s_{i}(w+1)$ through $\textbf{S}(w+1)=\textbf{S}(w)+dt[-(L+\Delta)\mathbf{S}(w)+hd_{k}\tilde{P}_{T}]$ (52) Set $w=w+1$, and repeat the above strategy with the updated value of the $s_{i}(w)$ defined in (52). (a) (b) (c) Figure 2: Physical layer control scheme Figure 3: Power circuit diagram of a DG Figure 4: Simulated microgrid bus system We end this section with the following two observations: ###### Remark 1. The consensus algorithms proposed are independent from the load demand $P_{L}$ and its changes, referring to 4.1 and 4.4. ###### Remark 2. All agents in the cyber layer have access to the instantaneous value of $P_{L}$. The power generation command for strategies 1, 2 and 3 introduced in 4.3 and the control scheme in 4.4 incorporate $P_{L}$ and $s_{i}(t)$ in the commanded power. Therefore, during consensus the power demand is met, irrespective of whether $P_{L}$ changes or not. ## 5 Controller Layout of Physical Layer The proposed power control methods for DGs introduced in this study are required to be implemented on both cyber and physical layer of microgrids. The physical layer which includes DGs is where controllers are designed to control the output power of DGs. In this study, the problem of proportional power sharing is addressed in the grid-connected mode, hence the frequency and voltage of DGs are imposed by the main grid. Therefore, frequency and voltage control methods, such as droop control, are not considered in this study. Furthermore, the reactive power control in the grid-connected mode is not studied for the practical reason of availability of reactive power in the main grid. Therefore, the required reactive power of the microgrid can be maintained from the main grid. The desired active power command of each DGi, i.e $P_{i}^{*}$ for $i=1,2,\cdots,N$ is calculated by its corresponding agent, i.e $i^{th}$ agent in the cyber layer. Then, this signal of $P_{i}^{*}$ is sent to the power control block of DGi located in the physical layer. The power control block of DGs is represented in Fig. 2a. This block receives the voltage $V_{abc}$ and current $I_{abc}$ from the voltage and current measurement units installed on the output of each DG, as shown in Fig. 3. Figure 3 also shows that each DG is connected to the main grid via a dedicated transformer to match the voltage between the DG and the main grid, as the output voltage of the main grid is significantly higher than the output voltage of DGs. To control the generated power of a DGi, i.e, $P_{i}$, it is required to control its output current since $V_{abc}$ and the frequency of the microgrid are fixed by the main grid. To achieve this, the desired active power command $P_{i}^{*}$ issued from $i^{th}$ agent is also considered as the other input in Fig. 2a. Using the Phase-Locked-Loop (PLL) block, the signals in Fig. 2a are converted to their equivalent values in the $dq0$ reference frame, i.e. $V_{dq}$ and $I_{dq}$. Next, the outputs of the $V_{dq}$ and $I_{dq}$ are fed as inputs to Fig. 2b. The parameters $C_{1}$, $C_{2}$ and the $PI$ controllers coefficients together with the upper and lower bounds of the saturation blocks in Fig. 2b are all defined in Section 6. The outputs of the Fig. 2b, regarded as the imaginary and real values of a complex number, are the inputs of the Fig. 2c. These inputs are converted to the amplitude and phase angle of the same complex value. The amplitude and the phase signals, together with the voltage angle $\omega\,t$, obtained from the PLL in the Fig. 2a, constitute the three phase signal fed to the PWM in Fig. 2c. Finally, each PWM sends the switching signals to the three level inverter of its corresponding DG which is illustrated in Fig. 3. Figure 5: Communication graph of the simulated $DG$s (a) (b) (c) (d) (e) (f) Figure 6: Output powers $P_{i}$, $i=1,2,\cdots,6$, under a variation in $P_{1,max}$ are depicted in the figures LABEL:sub@fig:_DG1_output_power, LABEL:sub@fig:_DG2_output_power, LABEL:sub@fig:_DG3_output_power, LABEL:sub@fig:_DG4_output_power, LABEL:sub@fig:_DG5_output_power and LABEL:sub@fig:_DG6_output_power, respectively. Figure 7: Microgrid total output power $P_{O}$ obtained from the proposed control method of Section 4.4 (a) (b) Figure 8: Consensus trajectories of agents on $\tilde{P}_{T}$ from (38b) and trajectories of $r_{i}$, $i=1,2,\cdots,6$, from (34). (LABEL:sub@fig:_Consensus_Convergence) Consensus trajectories on $\tilde{P}_{T}$ (LABEL:sub@fig:_Zoomed_in_Consensus_Convergence) The signals $r_{i}$ Figure 9: Individual ${P_{i,max}}/{s_{i}(t)}$ and the average ${P_{i,max}}/{s_{i}(t)}$ at each time step, as defined in (42) and (43) ## 6 Simulations In this section, the performance of the proposed control methods explained in Section 4.4 is evaluated through the simulation of a microgrid consisting of six inverter-based DGs shown in Fig. 4. The model layout in Fig. 4 is inspired from Matlab-based example available in [35] and the studies in [32, 34]. Then, the performances of the strategies 1 and 3, provided in (25) and (14) respectively, are juxtaposed with the performance of the controller in Section 4.4. The simulations are accomplished using the Simscape toolbox of Matlab. The simulated DGs are numbered from 1 to 6 and are connected to the main grid in parallel as depicted in Fig. 4. Each DG has a corresponding agent in the cyber layer where the updated value of the desired output power is computed by the agents using the information obtained through their bidirectional communication structure, as shown in Fig. 5. Note that the communication graph of the DGs in Fig. 5 is connected per its definition in Section 2. As the communication graph in Fig. 5 is a bidirectional graph, per Section 2, the adjacency matrix of the graph is symmetric. The adjacency and degree matrices are chosen as $A=\\!\\!\left[\begin{array}[]{ccccccc}\\!\\!\\!\\!0&\\!\\!\\!6&\\!\\!\\!0&\\!\\!\\!6&\\!\\!\\!6&\\!\\!\\!0\\!\\!\\!\\!\\\ \\!\\!\\!\\!6&\\!\\!\\!0&\\!\\!\\!0&\\!\\!\\!6&\\!\\!\\!0&\\!\\!\\!0\\!\\!\\!\\!\\\ \\!\\!\\!\\!0&\\!\\!\\!0&\\!\\!\\!0&\\!\\!\\!6&\\!\\!\\!6&\\!\\!\\!0\\!\\!\\!\\!\\\ \\!\\!\\!\\!6&\\!\\!\\!6&\\!\\!\\!6&\\!\\!\\!0&\\!\\!\\!6&\\!\\!\\!6\\!\\!\\!\\!\\\ \\!\\!\\!\\!6&\\!\\!\\!0&\\!\\!\\!6&\\!\\!\\!6&\\!\\!\\!0&\\!\\!\\!6\\!\\!\\!\\!\\\ \\!\\!\\!\\!0&\\!\\!\\!0&\\!\\!\\!0&\\!\\!\\!6&\\!\\!\\!6&\\!\\!\\!0\\!\\!\end{array}\right]\\!\\!\\!,\,\,D=\\!\\!\left[\begin{array}[]{ccccccc}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end{array}\right]\\!\\!$ (53) From Section 2, the correlated Laplacian matrix $L=A-D$ is defined as $L=\left[\begin{array}[]{ccccccc}18&-6&0&-6&-6&0\\\ -6&12&0&-6&0&0\\\ 0&0&12&-6&-6&0\\\ -6&-6&-6&30&-6&-6\\\ -6&0&-6&-6&24&-6\\\ 0&0&0&-6&-6&12\end{array}\right]$ (54) For the simulations, we set $h=10$ in (7), (38b) and (52). Let the maximum capacity of the DGs $P_{i,max}$ for $i=1,2\cdots,6$ be $\begin{split}P_{1,max}=600\,kw\;\;P_{2,max}=450\,kw\;\;P_{3,max}=300\,kw\\\ P_{4,max}=150\,kw\;\;P_{5,max}=750\,kw\;\;P_{6,max}=150\,kw\end{split}$ (55) Thus, the maximum capacity of the whole microgrid is $P_{T}=\sum_{i=1}^{6}P_{i,max}=2400\,kw$, and assume the load demand is $P_{L}=1600\,kw$. According to Section 3, the agents have knowledge of $P_{L}$ at all times and $P_{T}$ at initial time. Therefore, each agent is able to compute the proportional power share ratio $r=\frac{P_{L}}{P_{T}}$ defined in (5), independently, which is $\frac{2}{3}$, initially. Therefore, the output power of each DGi for $i=1,2,\cdots,6$, based on the proportional power sharing, must be, $\displaystyle P_{1}=400\,kW\quad P_{2}=300\,kW\quad P_{3}=200\,kW$ (56) $\displaystyle P_{4}=100\,kW\quad P_{5}=500\,kW\quad P_{6}=100\,kW$ During $t=[0,3]sec$, the power capacity of each DG remains unchanged and hence, each of the DGs generates its active power share as calculated in (56). At $t=3sec$ the power capacity of DG1 undergoes a step change, thereby $P_{T}$ has an increment of $300\,kW$, and at $t=9\,sec$ we introduce a decrement of $600kW$ to the capacity of DG1. The simulated microgrid consists of several components such as inverters, output filters of inverters, transformers, PWM, PI controllers, line impedance, loads, DC resources, measurement units, PLL and abc/dq0 converters. To emulate the main grid a dispatchable generator is considered in the simulation as shown in Fig. 4. The parameters of the transformer that connects the main grid to the distribution system and those of the transformers which connect the DGs to the distribution system are given in Table 1. The parameters of the distribution system are provided in Table 2. The PI controllers depicted in Fig. 2b are identical. The PI controllers of all DGs are also identical, meaning they all have the same $P$ and $I$ gains, chosen as $k_{P}=0.3$ and $k_{I}=30$, respectively. The energy resource of each DG is simulated as a DC power source, then by utilizing an inverter, the DC current converts to the AC current, as shown in Fig. 3. In the same figure, to remove the harmonics from the output power of the inverter, an output filter is applied and then connected to the main grid via a transformer, as illustrated in Fig. 3. The output filter consists of RL and RC branches. The resistive and inductive elements of each RL component are set as R${}_{1}=5.4946\times 10^{-4}\Omega$ and L$=1.4575\times 10^{-4}H$, respectively. The RC components for the output filters of each individual DGi arranged in the delta format have $P_{i}(W)$ and reactive power $Q_{i}(kVar)$ as given in Table 3. In Fig. 2b, $C_{1}=0.0039$, $C_{2}=0.21$, and the upper and lower limit of the saturation blocks are $+1.5$ and $-1.5$, respectively. The power control of each DG established in the physical layer is depicted in Fig. 2. Table 1: Parameters of the Transformers Transformer Connected to DGs --- Parameter | Value Nominal Power (kVA) | 100 Nominal Frequency (Hz) | 60 winding 1 | $V_{1,rms,ph-ph,kV}$ | 25 $R_{1}(pu)$ | 0.0012 $L_{1}(pu)$ | 0.03 winding 2 | $V_{1,rms,ph-ph,V}$ | 270 $R_{2}(pu)$ | 0.0012 $L_{2}(pu)$ | 0.03 Magnetization resistance Rm (pu) | 200 Magnetization inductance Lm (pu) | 200 Transformer Connected to Dispatchable Generator --- Prameter | Value Nominal Power (kVA) | 47000 Nominal Frequency (Hz) | 60 winding 1 | $V_{1,rms,ph-ph,kV}$ | 1200 $R_{1}(pu)$ | 0.0026 $L_{1}(pu)$ | 0.08 winding 2 | $V_{2,rms,ph-ph,kV}$ | 25 $R_{2}(pu)$ | 0.0026 $L_{2}(pu)$ | 0.08 Magnetization Resistance Rm (pu) | 500 Magnetization Inductance Lm (pu) | 500 Table 2: Parameters of the Grid Parameter | Value ---|--- Load 1 Nominal Voltage ($kV_{ph-ph}$) | 25 Load 1 Active Power P (kW) | 250 Load 2 Active Power P (kW) | 2000 Load 3 Power S (kVA) | 30000+j2000 Line 1 $Z_{1}$ Positive and Zero Sequence | Length (km) | 8 $R(\Omega/km)$ | [0.1153 0.413] $L(H/km)$ | [1.05e-3 3.32e-3] $C(F/km)$ | [11.33e-009 5.01e-009] Line 2 $Z_{2}$ Positive and Zero Sequence | Length (km) | 14 $R(\Omega/km)$ | [0.1153 0.413] $L(H/km)$ | [1.05e-3 3.32e-3] $C(F/km)$ | [11.33e-009 5.01e-009] Nominal Frequency (Hz) | 60 Table 3: Active and reactive powers of RC components of each output filter DG Number | Active Power(W) | Reactive Power(kVar) ---|---|--- 1 | 400 | 20 2 | 200 | 10 3 | 600 | 30 4 | 500 | 25 5 | 300 | 15 6 | 400 | 20 (a) (b) Figure 10: (LABEL:sub@fig:_DG1_Different_Methods) Output power $P_{1}$ according to Strategies 1, 3 and the proposed controller of Section 4.4 and (LABEL:sub@fig:_Output_Sum_Diff_Method) Their corresponding microgrid total power $P_{O}$ Starting from $t=3\,sec$, $P_{1,max}$ increases from $600\,kW$ to $900\,kW$. Therefore, the microgrid maximum power capacity increases from $2400\,kW$ to $2700\,kW$. Based on the approach explained in Section 4.4, the finite-time algorithm in (45)-(49) is embedded in the consensus algorithm (38b) to have DGs apply the proposed control law in (34) and (38), distributively. The results of the simulations are shown in Fig. 6 where $P_{1}$ increases and $P_{i}$, $i=2,3,4,5,6$, decreases. During $t=[3,9]\,sec$, the microgrid output power $P_{O}$ remains almost equal to $P_{L}=1600\,kW$, as shown in Fig. 7. The slight difference between $P_{O}$ and $P_{L}$ is due to resistive losses in the DGs due to the resistor elements shown in Fig. 3. Considering (38b), the six estimation variables $s_{i}$ for all $i=1,2,\cdots,6$ are updated through the information exchange until they reach a consensus concerning $\tilde{P}_{T}=2700kW$ as demonstrated in Fig. 8a. The ratios of $r_{i}=\frac{P_{L}}{s_{i,max}}$, for $i=1,2,\cdots,6$, are shown in Fig. 8b, where before reaching a consensus during $t=(3,8)\,sec$, these ratios are different, however they become almost identical during $t=[8,9]\,sec$. Figure 6 illustrates that after the consensus algorithm converges, the steady state values of the output powers of DGs are $P_{1}=533.33\,kW$, $P_{2}=266.66\,kW$, $P_{3}=177.77\,kW$, $P_{4}=88.88\,kW$, $P_{5}=444.44\,kW$ and $P_{6}=88.88\,kW$. Recalling the finite-time average consensus algorithm is embedded in the consensus algorithm, at each time step of evolution of consensus algorithm, the finite-time algorithm is applied. Through this approach, the agents compute the average of $\frac{P_{i,max}}{s_{i}(t)}$ for $i=1,2,\cdots,6$ in a distributed way where its corresponding result is illustrated in Fig. 9. The next variation of the microgrid maximum power capacity occurs at $t=9s$ where $P_{1,max}$ decreases for $-600kW$. Therefore, starting from $t=9\,sec$, the current capacity of the microgrid which is $P_{T}=2700\,kW$ changes to $\tilde{P}_{T}=2100kW$. Then, similar to the same procedure adopted in reaction to a change in a microgrid capacity, the control method in (34) and (38) is triggered. Hence, during $t=[9,18]sec$, according to Fig. 8a, the agents have reached to another consensus on the maximum power capacity of the microgrid which is $2100kW$. Figure 6a demonstrates that $P_{1}$ becomes $228.57\,kW$ after the convergence during $[9,18]sec$. Figure 6 also shows that the output power of the other DGs have increased due to the microgrid capacity reduction. The power sharing ratio $r_{i}$ for $i=1,2,\cdots,6$ are shown in Fig. 8b. The figure illustrates that, during the transient duration of $(9,15)\,sec$, $r_{i}$ ratios are not equal. On the contrary, they converge to steady state conditions in $[14,18]\,sec$. Furthermore, from Fig. 7, $P_{O}$ remains practically equal to $P_{L}=1600\,kW$. In Fig. 10a, the results of the proposed control algorithm in Section 4.4 is compared with the results of the strategies defined as (14) and (25) in Section 4.3. From this figure, it is clear that the output power of $P_{1}$ obtained from the proposed control algorithm Section 4.4 differs from the other two ones during the transient duration. However, after the transient durations of $(3,8)\,sec$ and $(9,15)\,sec$ the output power of $P_{1}$ from all three methods are the same. Figure 10b demonstrates that the approaches of (14) and (25) are ineffective to address the load demand. They produce significant deviation in $P_{O}$ from $P_{L}$ during transient. On the other hand, upon applying the method of Section 4.4, the deviation drastically reduces, both for increase and decrease in maximum power capacity of DG1. ## 7 Conclusion In this research, the problem of distributed proportional power sharing is studied for microgrids that operate in the grid-connected mode. Firstly, a consensus algorithm is designed through which, under a variation in the maximum power capacity of a DG, all DGs in the microgrid estimate the updated microgrid capacity. Utilizing the estimations, they generate their output powers in a distributed manner. Stability and convergence of the consensus algorithm are proven. While the consensus algorithm operates in the cyber layer, power commands are sent to the DGs at the physical layer using multiple strategies, discussed in the research. In this regards, practical issues such as ensuring power commands are within acceptable bounds during the transient time of the consensus method, are addressed. However, the consensus algorithm along with the aforementioned strategies does not guarantee maintaining load power during the transient time. Therefore, a modified strategy is proposed to guarantee a match between demanded and delivered power during transient, while the DGs reach a new consensus following a perturbation in grid capacity. The distributed controller is tested in a simulated microgrid. The microgrid is modeled in Matlab/Simulink using the Simscape toolbox. A complete description of the model along with parameters values used for simulation, are given. Simulation results confirm the effectiveness of the proposed strategy. ## Appendix Proof of Lemma 3 ###### Proof. From (13), we note that $y_{i}=s_{i}-\tilde{P}_{T}$. Since Lemma 1 shows that $-(L+\Delta)$ is Hurwitz, therefore from (13) we have, $y(t)=e^{-(L+\Delta)t}y(t_{0})\;\\!\\!\Rightarrow\\!\\!\;\|y(t)\|\\!\\!\leq\|e^{-(L+\Delta)t}\|\|y(t_{0})\|$ (57) As explained in Lemma 1, $-(L+\Delta)$ is diagonalizable and all of its eigenvalues are negative and real. Assuming $\lambda_{1}<0$ is the largest eigenvalue of $-(L+\Delta)$ and since $y(t_{0})=-\delta\mathbf{1}$, we have, $\|y(t)\|\leq e^{\lambda_{1}t}\|y(t_{0})\|=e^{\lambda_{1}t}\sqrt{N}|\delta|\leq\sqrt{N}|\delta|$ (58) Hence, $\|y\|$ is bounded. Since $|y_{i}|\leq\|y(t)\|$, therefore $|y_{i}(t)|\leq\sqrt{N}|\delta|\quad\forall\;i=1,2,\cdots,N$ (59) If $|\delta|<{\theta P_{T}}/{(1+\sqrt{N}})$, then it follows that $|y_{i}(t)|\leq\sqrt{N}|\delta|\leq\sqrt{N}{\theta P_{T}}/{(1+\sqrt{N})}$ (60) and since $y_{i}=s_{i}-\tilde{P}_{T}$, therefore we have $\displaystyle\tilde{P}_{T}-\sqrt{N}{\theta P_{T}}/$ $\displaystyle{(1+\sqrt{N}})\leq s_{i}(t)\leq$ (61) $\displaystyle\tilde{P}_{T}+\sqrt{N}{\theta P_{T}}/{(1+\sqrt{N}})$ Since $\tilde{P}_{T}=P_{T}+\delta$, and from the assumption $|\delta|<{\theta P}/{(1+\sqrt{N}})$, we have $\displaystyle P_{T}-$ $\displaystyle{\theta P_{T}}/{(1+\sqrt{N}})-\sqrt{N}{\theta P_{T}}/{(1+\sqrt{N}})\leq s_{i}(t)$ (62) $\displaystyle\leq P_{T}+{\theta P_{T}}/{(1+\sqrt{N}})+\sqrt{N}{\theta P_{T}}/{(1+\sqrt{N}})$ Thus, $(1-\theta)P_{T}\leq s_{i}(t)\leq(1+\theta)P_{T}$ (63) Since for all $t>t_{0}$, the output power of each DGi should satisfy $P_{i}(t)=\frac{P_{L}}{s_{i}(t)}P_{i,max}<P_{i,max}$, it is required that $s_{i}(t)>P_{L}$ for all $t>t_{0}$. For guaranteeing $s_{i}(t)>P_{L}$, from (63), we can impose $(1-\theta)P_{T}>P_{L}$. Therefore, under the dynamics of $\mathbf{S}$ in (7), $P_{T}>P_{L}/(1-\theta)$ or $1-(P_{L}/P_{T})>\theta$ ensures that $P_{i}(t)<P_{i,max}$. This completes the proof. ∎ Observation on $\delta$: Lemma 3 gives the condition $|\delta|<{\theta P_{T}}/{(1+\sqrt{N}})$ to prevent unfavorable transients in $s_{i}(t)$. To demonstrate that this condition is not restrictive as $N$ increases, we consider a change in $N$ to $N+1$ and a corresponding change from $P_{T}$ to $P_{T}+P_{N+1,max}$. Further, we impose $\frac{\theta(P_{T}+P_{N+1,max})}{1+\sqrt{N+1}}>\frac{\theta P_{T}}{1+\sqrt{N}}$ (64) to derive the condition under which $|\delta|$ will increase as we increase $N$ to $N+1$. From (64), we have, $P_{N+1,max}>\Bigl{[}\frac{\sqrt{N+1}-\sqrt{N}}{1+\sqrt{N}}\Bigr{]}P_{T}$ (65) From (65), it can be observed that $P_{N+1,max}$ can be only a small fraction of $P_{T}$ to allow $|\delta|$ to increase rather than decrease. For instance, if $N=3$, then $P_{N+1,max}>0.098P_{T}$, and if $N=8$, then $P_{N+1,max}>0.045P_{T}$ which are small fractions of $P_{T}$. In addition, comparing the right hand side of (65) with the average of $P_{T}$, $P_{T,avg}=P_{T}/N$, we obtain the minimum ratio of $P_{N+1,max}/P_{T,avg}$ as following $N\Big{[}\frac{\sqrt{N+1}-\sqrt{N}}{1+\sqrt{N}}\Big{]}$ (66) Equation (66) is strictly less than $\frac{1}{2}$ and it converges to $\frac{1}{2}$ for large values of $N$. This proves that $P_{N+1,max}$ is required to be $P_{N+1,max}\geq(1/2)P_{avg}$ at the worst cases to satisfy the condition on $|\delta|$. Therefore, the condition on $|\delta|$ is not restrictive. ## References * [1] J. Rocabert, A. Luna, F. Blaabjerg, and P. 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2024-09-04T02:54:56.112109

2003.00579

arxiv-papers

# Application of the ASBM-SA closure in a turbulent flow over a hump in the presence of separation control. C. F. Panagiotou F. S. Stylianou E. Gravanis E. Akylas S. C. Kassinos Laboratory Of Environmental Engineering (GAIA), Nireas-International Water Research Centre, University of Cyprus, Nicosia, Cyprus Department of Civil & Environmental Engineering, University of Cyprus, Nicosia, Cyprus Computational Sciences Laboratory (UCY-CompSci), Nireas-International Water Research Centre, University of Cyprus, Nicosia, Cyprus Department of Mechanical & Manufacturing Engineering, University of Cyprus, Nicosia, Cyprus Department of Civil Engineering & Geomatics, Cyprus University of Technology, Limassol, Cyprus Eratosthenes Centre of Excellence, Cyprus University of Technology, Limassol, Cyprus ###### Abstract We demonstrate the coupling between the Algebraic Structure-Based Model (ASBM) and the one-equation Spalart–Allmaras (SA) model, which provides an easy route to bringing structure information in engineering turbulence closures. The estimation ability of the hybrid model was tested for a flow over a hump model with no-flow control and steady suction. ASBM-SA model produced satisfactory predictions for the streamwise Reynolds stress component, while a qualitative agreement with the experiments was achieved for the transverse component. Regarding the shear stress component, ASBM-SA closure provides improved predictions compared to SA in the entire domain. ## 1 Introduction The class of RANS models most often used in engineering applications is that of Eddy Viscosity Models (EVM). One of the most popular EVM is the Spalart–Allmaras (SA) one-equation model [1]. The SA model is often favored by practicing engineers because it exhibits superior robustness, low CPU time requirements and substantially lower sensitivity to grid resolution compared to two-equation models. On the other hand, one has to recognize that, despite its computational and implementational attractiveness, the eddy viscosity assumption is also the source of some of the most important performance limitations. For example, like other EVM, the SA model fails to capture important flow features, such as turbulence anisotropy or the effects of mean or system rotation. A common feature of the classical closure approaches described so far is the assumption that all key information about the turbulence is contained in the scales of the turbulence and in the turbulence stress tensor. However, one should consider that the turbulent stresses contain information only about the componentality of the turbulence, i.e. about the directions in which the turbulence fluctuations associated with large-scale eddies are most energetic. Thus, traditional closures do not take into account the morphology of the energy-containing eddies. Yet, eddies tend to organize spatially the fluctuating motion in their vicinity. In doing so, they eliminate gradients of fluctuation fields in some directions (those in which the spatial extent of the structure is significant) and enhance gradients in other directions (those in which the extent of the structure is small). Thus, associated with each eddy are local axes of dependence and independence that determine the dimensionality of the local turbulence structure. This structure dimensionality information is complementary to the componentality information contained in the Reynolds stresses, and as Kassinos & Reynolds [2] and Kassinos et al. [3] have shown, it is dynamically important. A detailed description of the complete set of the turbulence structure tensors is given in several works [3, 4, 5]. The significant effect that the morphology of the large-scale structures has on the evolution of turbulent statistics [6] has motivated the development of structure-based models. These models can be classified into two categories: differential and algebraic. The first category involves solving a set of transport equations to evaluate structure tensors. Simplified models have been proposed that are applicable at the homogeneous limit [7, 8, 9], while a more sophisticated differential model was proposed by Poroseva et al. [10] that was tested in a turbulent flow passing through a cylindrical pipe that rotates around its longitudinal axis. Furthermore, structure-based models were recently constructed to evaluate scalar transport, ranging from simple passive scalars [5, 11] to strongly stably stratified flows [12]. The second category refers to algebraic approaches, which are based on assumptions that lead to constitutive equations. The Algebraic Structure-Based Model (ASBM) [13, 14] is an engineering structure-based turbulence model that follows this second approach. It is a fully realizable two-equation structure- aware model that provides the full Reynolds stress tensor. Panagiotou & Kassinos [15] presented a successful coupling between the ASBM with the one- equation SA model. Their intention was to combine the numerical robustness and stability of the SA model along with the deeper physical content of the ASBM. The performance of the hybrid model, called ASBM-SA, was evaluated in several standard benchmark cases, ranging from simple fully-developed channel flows to a flow over a steep hill, achieving an overall good agreement between model and experimental predictions. Hence, the aim of this study is to evaluate the performance of the ASBM-SA closure to a more complex case, in particular the case of turbulent flow over a two-dimensional hill with and without separation control. ## 2 Coupling between ASBM and the SA model The main limitation of SA (and any other one-equation model) is that it does not provide of a complete set of turbulence scales. On the other hand, the ASBM closure relies on the availability of suitable turbulence scales and this was the key stumbling block in trying to couple the ASBM and SA closures. The Bradshaw hypothesis [16] has been used as the starting point for transforming turbulence scales $\kappa-\epsilon$ closures, where $\kappa$ and $\epsilon$ are the turbulent kinetic energy and energy dissipation rate respectively, to one equation models. Here, our objective is to use the same phenomenology in order to extract these scales from the SA one-equation closure. A complete description of this formulation and on how the coupling between the two closures has been achieved can be found in Panagiotou & Kassinos [15]. ## 3 Results and discussion ### 3.1 Outline In this study we have considered the case of no-flow control, as well as the case where active-control is applied via steady-suction. During the CFDVAL2004 Workshop [17], these two cases were investigated in depth to assess the performance of popular engineering turbulence models in strongly separated flows subjected to favorable/adverse pressure gradients. Experiments have been conducted in the NASA Langley Transonic Cryogenic Tunnel by Greenblatt et al. [18]. The shape of the hump is that of a “Modified Glauert-Goldschmied” hill, similar to the one used by Seifert and Pack [19]. The experiments are nominally two-dimensional (2D), despite the presence of three dimensional (3D) effects near the side end-plates. The scenarios involved both uncontrolled and controlled flow (steady suction) for Reynolds numbers ($Re$) ranging from 0.37 up to 1.1 million, corresponding to Mach numbers ($M$) ranging from 0.04 up to 0.12. One no-flow control case and one active-control case were selected for the extraction of detailed experimental measurements. Figure 1 shows the geometry of the whole domain, including a detailed view of the flow control slot. The chord length of the hump is denoted as $c$, the height of the domain $H$ is $90\%$ the chord length, while the maximum height of the hump is approximately $0.13\ c$. The slot is located near $x/c\approx 0.65$, where the slot width $h$ is $0.00187\ c$. Detailed information regarding the geometry, computational grids and the relevant experiments can be found in [20]. One of the conclusions reached during the CFDVAL2004 Workshop is that blockage effects stemming from the presence of side plates need to be accounted for in simulations, otherwise the computed pressure coefficients levels exhibit significant discrepancy relative to the experiments, especially over the hump. Thus, the top tunnel surface around hump location is modified so as to reflect the change in the tunnel cross-sectional area due to the presence of the side- plates, as described in [20]. Figure 1: Sketch of the geometry, with a modification along the top-surface such as to account for the side-plate effects, as described in [20]. ### 3.2 Validation cases #### 3.2.1 Turbulent boundary layer As a first step, we performed steady computations of a spatially developing boundary layer flow over a flat plate for flow conditions that correspond to the experiments of Greenblatt et al. [18]. Profiles of the converged solution at a specific streamwise location were extracted and then used as inlet boundary conditions for the cases involving the 2D hump. The desired Reynolds number is $Re_{\delta}\approx 68200$ based on the freestream velocity $U_{\infty}$, where subscript $\infty$ denotes freestream values, and the boundary layer thickness $\delta\approx 0.074\,c$. At the inlet, Dirichlet boundary conditions are imposed for the mean streamwise velocity $U_{x}$ and the pseudo-viscosity $\tilde{\nu}$ variables, such as ${\tilde{\nu}_{\infty}}/{\nu}\approx 3$ and $U_{\infty}=0.1\ M$, yielding a freestream Reynolds number $Re_{\infty}=929,000$ based on the chord length $c$ of the hump, air viscosity $\nu$ and the freestream velocity $U_{\infty}$. At the outlet, a penalty condition is imposed to prevent the occurrence of reflectional effects while ensuring mass conservation. A slip condition was imposed at the top surface, a no-slip condition at the bottom wall surface and periodic conditions along the spanwise direction. In order to obtain grid- independent solutions, three different meshes of increasing resolution were considered. For each mesh, geometric functions were used to define the normal distribution of the nodes, while uniform spacing has been adopted along the streamwise direction. Grid 1 contains a non-uniform mesh of size 120 x 90 x 1 along the streamwise, wall-normal and spanwise directions respectively. The corresponding size for Grid 2 is 130 x 120 x 1 and for Grid 3 is 140 x 150 x 1. The finest grid yields a value of $y^{+}$ around 0.5 for the wall-adjacent cell at the location of the extracted data. Figures 2a-b show predictions using the SA closure for the streamwise mean velocity and pseudo-viscosity respectively. In Figure 3 we show a comparison between the predictions of the SA model using Grid 3 as the baseline grid, and the experimental data for the streamwise mean velocity $U_{x}$, yielding a good agreement. (a) (b) Figure 2: Grid-convergence analysis for a spatially developing turbulent boundary layer at $Re_{\delta}\approx 68,200$. SA model predictions for (a) the streamwise mean velocity and (b) the pseudo-viscosity. Comparison is made among three different grids: Grid 1 ($\solid$) ; Grid 2 ($\dashed$) ; Grid 3 ($\chndotdotdot$). Figure 3: SA model predictions (lines) for the streamwise mean velocity. Comparison is made to the experiments (symbols) of Greenblatt et al. [18]. #### 3.2.2 No-flow control The case of flow over a hump having the shape of “Modified Glauert” hill is considered next. This case was originally conceived for testing the ability of active control to reduce the size of the existing recirculation bubble. However, from a turbulence modeling perspective, even the uncontrolled case is interesting due to the presence of strong separation, which proves to be challenging to turbulence engineering models. Thus, in our numerical experiments, we considered first the uncontrolled case of flow. Simulations have been performed using SA and ASBM-SA models, which are compared to the experimental work of Greenblatt et al. [18] At the inlet surface, profiles for the variables are obtained from the SA solution for the turbulent boundary layer corresponding to $Re_{\delta}\approx 68200$ as described in the previous subsection. At the floor surface, as well as at the wall surfaces inside the cavity, solid wall (no-slip) boundary conditions were applied. A penalty condition is imposed at the outlet surface to ensure that mass flow exits the domain properly, while slip conditions are used at the top surface and periodic conditions for the spanwise direction. Two grids were considered to conduct a grid-sensitivity analysis. The coarser grid contains approximately 103,000 grid points, whereas the finer grid posses approximately 160,000 grid points. Mesh details are shown in Figure 4, together with a zoomed view of the cavity region. (a) (b) (c) Figure 4: Unstructured computational grid with details of the slot region. Figures 5a-b show SA model predictions for the wall-static pressure coefficient $c_{p}=(p-p_{0})/\frac{1}{2}\rho U^{2}_{\infty}$ and skin-friction coefficient $c_{f}=\tau_{w}/\frac{1}{2}\rho U^{2}_{\infty}$ respectively. Based on these results, the coarser grid is shown to be fine enough for the current case. (a) (b) Figure 5: Effect of grid on SA model predictions for the uncontrolled case, for (a) the wall-static pressure coefficient and (b) the skin-friction coefficient. Two grids are shown: coarse grid ($\solid$) ; fine grid ($\dashed$). Due to the algebraic nature of the ASBM closure, numerical difficulties are encountered for the cases where strongly separated flows are considered, as the present ones. During previous works, a filtering scheme was applied in order to smoothen the profiles, improving that way the numerical stability of the solution. However, use of this scheme in the current case led to a mismatch between SA and ASBM-SA predictions for the skin-friction coefficients in regions where a good agreement was expected, such as upstream of the leading edge of the hill and downstream of the recirculation region. As a result, we separated the domain into two zones, one prior the leading edge where the filtering scheme is not active, and the region downstream the leading edge where the scheme is switched on (Figure 6). Figure LABEL:fig:cfcp_zf shows a comparison between SA and ASBM-SA predictions using both approaches for the filtering scheme, revealing the significant role that the filtering details play on the skin-friction distribution all along the bottom surface. In contrast, the pressure coefficient remains unaffected by the choice of filtering scheme. Figure 6: Separation zones showing where the filtering scheme is active (on) or not (off). Figure 8 displays the evolution of the maximum mean streamwise velocity residual. The residual is divided by its initial value, denoted by subscript 0, and is defined by $\text{ Residual }=\max\bigg{[}\frac{V\times\Delta U_{x}/\Delta t}{(V\times\Delta U_{x}\Delta t)_{0}}\bigg{]}\,,\hskip 8.5359pt\Delta U_{x}=U^{n+1}_{x}-U^{n}_{x}\,,$ (1) where $n$ refers to the $n$-th iteration, $\Delta t$ to the time step and $V$ to the volume of the corresponding cell. For both SA and ASBM-SA models, a drop of at least 5 orders of magnitude for the residuals compared to the initial field is achieved, which is believed to be sufficient to provide time- converged solutions. Figure 8: Time history of the streamwise mean velocity residual for the uncontrolled case. The sudden jump in the residual levels indicates the point where the ASBM coupling is switched on. In order to both accelerate our simulations and overcome some stability issues related to the ASBM-SA computations inside the cavity in the case of no- suction, additional computations using similar meshes but without the presence of the cavity were conducted. For these cases, we considered a solid-wall condition along the slot exit. Figures 9a-b show SA predictions for the mean velocity streamlines at the vicinity of the hump in the presence and absence of the cavity respectively, demonstrating the trivial discrepancies between the two approaches. Figure 10 shows the corresponding comparison for the wall- static pressure coefficient, which again reveals the negligible effect on the results of the cavity absence when the flow control is inactive (no suction). (a) (b) Figure 9: SA model predictions for the streamlines of the mean flow approaching the hump (a) in the presence of the cavity and (b) in the absence of the cavity. Figure 10: SA model predictions in the presence ($\solid$) or absence ($\dashed$) of the cavity for the wall-static pressure coefficient. In the following figures, all turbulent and mean quantities (except $c_{f}$, $c_{p}$) are normalized based on the chord length $c$ and the reference inlet freestream velocity $U_{\infty}$. Setting the leading edge of the hump as the origin of the streamwise distance ($x/c=0$), profiles are extracted at three different stations inside the recirculation bubble as measured by the experiments ($x/c=0.66,\ 0.8,\ 1.0$) and one station at the recovery region ($x/c=1.2$), denoted as stations $A,\ B,\ C,\ D$ respectively (Figure 11). Figure 11: Geometrical and mesh details in the absence of cavity. Data is extracted at four stations, denoted as A, B, C and D. The leading-edge point (LE) and the re-attachment point (R) are also shown. Figure 12 shows the predictions of the ASBM-SA and SA closures for the variation of the pressure ($c_{p}$) and skin friction ($c_{f}$) coefficients along the wall surface. Comparison is made to experimental measurements. The ASBM-SA model captures accurately the peak magnitude of the pressure coefficient around $x/c\approx 0.57$. For $x/c$ ranging from -1 to 1.1, that is from the inlet up to about the re-attachment point, the ASBM-SA provides slightly improved $c_{p}$ predictions when compared to SA. Right after re- attachment though, the ASBM-SA predicts a slightly delayed recovery of $c_{p}$ as compared to the experiments (and the SA closure). The two models produce comparable agreement with the experiments for the skin-friction coefficient. ASBM-SA provides an improvement right after the upstream edge of the hill ($x/c\approx 0.2$), while SA predicts correctly the magnitude near the sharp geometry change that occurs around $x/c\approx 0.65$. According to Table 1, both models overpredict the recirculation bubble, with ASBM having the tendency to delay the re-attachment of the flow further downstream, an observation which is in agreement with previous cases in which separated flows over 2D hills were considered, such as the “Witch of Agnesi” hump, as described in detail in [15]. Figure 13 shows results for the streamwise mean velocity $U_{x}$ at the four stations. As shown in Figures 13a-b, at the first two stations ASBM-SA provides slightly improved predictions relative to the SA model, while SA is in better agreement with the experimental data at the next two stations, mostly due to the greater delay of re-attachment in the ASBM-SA predictions. (a) (b) Figure 12: SA ($\solid$) and ASBM-SA ($\dashed$) model predictions for the no- flow control case for (a) the wall static-pressure coefficient and (b) the wall skin-friction coefficient. Comparison is made to experimental values (symbols) of Greenblatt et al. [18]. (a) station A, $x/c=0.66$ (b) station B, $x/c=0.8$ (c) station C, $x/c=1.0$ (d) station D, $x/c=1.2$ Figure 13: Turbulent flow over the “Glauert-Goldschmied” 2D hill for the no- flow control case. Model predictions for the streamwise mean velocity $U_{x}$ at various $x$-stations for SA ($\solid$) and ASBM-SA ($\dashed$) closures. Comparison is made to experimental values of Greenblatt et al. [18]. Next, we consider the performance of the ASBM-SA closure for the turbulent intensities and the fluctuating shear stress with respect to experimental results. The SA model is included only in the comparison for the fluctuating shear stress component, since it cannot provide predictions for the turbulent intensities. Figure 14b displays the streamwise Reynolds stress component $R_{xx}$. At station A, ASBM-SA yields reasonable predictions, being able to capture the near-wall peak magnitude. At the remaining three stations ($x/c=0.8,1.0,1.2$), ASBM-SA is able to capture satisfactorily the peak magnitude. We note that the wiggles near $y/c=0.2$ at station A (Figure 14a) originate from the algebraic expressions for the estimation of the turbulent kinetic energy (not shown here). Following term by term the algebraic procedure for the calculation of kinetic energy, we found that this issue is most likely related to the local mean velocity gradients. These wiggles are also present in previous works [15], for which similar findings were deduced. We also point out that the location $y/c=0.2$ at which the wiggles appear is close to the interface between two grid blocks. Overall, we believe that this is a localized effect that does not affect the quality of the solution. (a) station A, $x/c=0.66$ (b) station B, $x/c=0.8$ (c) station C, $x/c=1.0$ (d) station D, $x/c=1.2$ Figure 14: Turbulent flow over the “Glauert-Goldschmied” 2D hill for the no- flow control case. ASBM-SA model predictions (lines) for the streamwise Reynolds stress component $R_{xx}$ at various $x$-stations are shown. Comparison is made to experimental values (symbols) of Greenblatt et al. [18]. , Figure LABEL:fig:ryy_hill depicts the corresponding predictions for the transverse Reynolds stress component $R_{yy}$. ASBM-SA closure strongly overpredicts the near-wall magnitude at station A, while a fair agreement with the experiments is achieved at the remaining stations. Figure LABEL:fig:rxy_hill shows SA and ASBM-SA predictions for the fluctuating shear stress component. At the first station, ASBM-SA exhibits a similar behavior as for the transverse Reynolds stress component. At the remaining three stations, ASBM-SA provides noticable improvement relative to the SA model, in both the near-wall and freestream regions. This improvement is evident in the whole range of the recirculation bubble, suggesting a satisfactory response of the hybrid model to the strong anisotropic effects that characterize this region. #### 3.2.3 Control via steady suction As a first step, we define the steady mass transfer momentum coefficient $c_{\mu}=\frac{\rho hU^{2}_{jet}}{1/2c\rho U^{2}_{o}}\,,$ (2) where $U_{jet}$ denotes the jet velocity. For the current case, $c_{\mu}$ is set equal to $0.241\,\%$, corresponding to a constant mass flow rate of $\dot{m}=0.01518\ kg/s$ being sucked through the slot, in order to match the experimental conditions of Greenblatt et al. [18]. Figure 17 shows ASBM-SA model predictions for the wall-normal spacing along the floor surface for the uncontrolled and controlled cases, needed to ensure that our solutions are obtained in sufficiently resolved grids. As expected, a sharp increase in $y^{+}$ levels occurs at the location of the slot exit where no-wall is present. Figure 17: ASBM-SA model predictions for the streamwise variation at the bottom surface for the normal spacing at the wall normalized in wall-units. Results are shown for both the no-flow control ($\solid$) and the steady- suction $(\dashed)$ cases. Results for the wall-static pressure coefficient are shown in Figure 18, where the predictions of the SA and ASBM-SA models are compared to the experimental data of Greenblatt et al. [18]. The ASBM-SA closure manages to capture accurately the magnitude and the location of the first sharp change right after the slot, located around $x/c\approx 0.67$, followed by a small recovery delay, as compared to the SA model, till the trailing edge of the hill ($x/c=1$) where the two models coincide again. (a) Figure 18: SA ($\solid$) and ASBM-SA ($\dashed$) model predictions for the steady-suction case for the wall static-pressure coefficient. Comparison is made to experimental values of Greenblatt et al. [18]. Figure 19 displays results for the streamwise mean velocity $U_{x}$ at the four stations. As shown, SA provides slightly better agreement with the experiments than the ASBM-SA model. Figure 20 shows the corresponding comparison for the transverse mean velocity $U_{y}$. In general, SA predictions are in better agreement with the experimental data compared to ASBM-SA model. (a) station A, $x/c=0.66$ (b) station B, $x/c=0.8$ (c) station C, $x/c=1.0$ (d) station D, $x/c=1.2$ Figure 19: Turbulent flow over the “Glauert-Goldschmied” hill for the steady- suction case. Model predictions for the streamwise mean velocity $U_{x}$ at various $x$-stations for SA and ASBM-SA closures. Comparison is made to experimental values of Greenblatt et al. [18]. (a) station A, $x/c=0.66$ (b) station B, $x/c=0.8$ (c) station C, $x/c=1.0$ (d) station D, $x/c=1.2$ Figure 20: Turbulent flow over the “Glauert-Goldschmied” hill for the steady- suction case. Model predictions for the transverse mean velocity $U_{y}$ at various $x$-stations for SA and ASBM-SA closures. Comparison is made to experimental values of Greenblatt et al. [18]. Table 1 shows details regarding the recirculation region. SA predicts more accurately the re-attachment point for both cases, providing an indication why SA closure obtains better results than the hybrid model for the mean statistics. Case | Model | sep.loc. | sep.loc. | Error | reatt.loc. | reatt.loc. | Error ---|---|---|---|---|---|---|--- | | experiment | CFD | ($\%$) | experiment | CFD | ($\%$) no-flow control | SA | $\approx 0.67$ | 0.663 | 1.0 | $1.11\pm 0.003$ | 1.235 | 11.3 no-flow control | ASBM-SA | $\approx 0.67$ | 0.656 | 2.1 | $1.11\pm 0.003$ | 1.330 | 19.8 steady suction | SA | $\approx 0.68$ | 0.676 | 0.6 | $0.94\pm 0.005$ | 1.113 | 18.4 steady suction | ASBM-SA | $\approx 0.68$ | 0.665 | 2.2 | $0.94\pm 0.005$ | 1.180 | 25.5 Table 1: Details of SA and ASBM-SA model predictions regarding the recirculation bubble for each case. Comparison is made to the experimental work of Greenblatt et al [18]. Figure 21 shows the corresponding comparison for the streamwise Reynolds stress component $R_{xx}$. At three of the four stations, ASBM-SA correctly predicts the near-wall peak magnitude and the freestream values, yielding a fair agreement with the experiments. (a) station A, $x/c=0.66$ (b) station B, $x/c=0.8$ (c) station C, $x/c=1.0$ (d) station D, $x/c=1.2$ Figure 21: Turbulent flow over the “Glauert-Goldschmied” hill for the steady- suction case. Model predictions for the streamwise Reynolds stress component $R_{xx}$ at various $x$-stations for SA and ASBM-SA closures. Comparison is made to experimental values of Greenblatt et al. [18]. In Figure LABEL:fig:ryy_on_hill, the agreement is qualitative between the ASBM-SA predictions and the experimental measurements for the transverse Reynolds stress component $R_{yy}$. Combining these results with the analogous ones for $R_{xx}$ reveals the sensitivity of the algebraic model to the anisotropic nature of the flow. Results for the fluctuating shear stress component $R_{xy}$ are shown in Figure LABEL:fig:rxy_on_hill. As shown, the hybrid ASBM-SA model is able to provide significantly improved predictions compared to the SA closure in the whole range of the recirculation region. Overall, ASBM-SA provides a satisfactory agreement with experiments. The active control effect on the recirculation bubble is visualized in Figure 24. ASBM-SA model predictions for the streamlines of the mean velocity for both cases are shown, revealing a noticeable reduction of the bubble size. (a) (b) Figure 24: ASBM-SA model predictions for the streamlines of the mean velocity for (a) the uncontrolled case and (b) the controlled case . ## 4 Summary and Conclusions The ASBM-SA closure has been tested for the case of a flow over a two- dimensional smooth hill in the shape of a “Modified Glauert-Goldschmied” hump both in the presence and absence of separation control. For both cases considered, the ASBM-SA model produced satisfactory predictions for the streamwise Reynolds stress component $R_{xx}$, while a qualitative agreement with the experiments was achieved for the transverse component $R_{yy}$. ASBM- SA closure provided improved predictions compared to SA for the shear stress at all stations. Regarding the mean quantities, the predictions of both closures are comparable, providing fair agreement with the experiments. Overall, the hybrid model managed to capture satisfactory the traits of these highly anisotropic flows, while maintaining the high robustness of the SA model, at a good convergence rate. Use of separated zones for the activation of the filtering scheme resulted to smoother mean velocity profiles in the recirculation region, and more meaningful skin-friction profiles upstream and downstream the separated region. As part of future work we intend to ascertain the performance of the hybrid model over challenging two-dimensional flows, such as turbulent flows around a wall-mounted cube and plane wall jets, while an extension to three-dimensional smooth hills will be attempted. Regarding the refinement issues encountered in the current and previous works, further consideration is needed to understand why ASBM-SA delays further the re-attachment compared to SA for all validation cases considered until now, even though it gives better predictions for the shear stress over the entire region of the recirculation. We have already started developing more advanced filtering schemes, suitable for highly deformed meshes, since we believe that the choice of filtering scheme plays a role on the delay of flow’s re-attachment, and contributes to yielding larger recirculation bubbles. ## References * [1] P. Spalart and S. Allmaras. A one-equation turbulence model for aerodynamic flows. Recherche Aerospatiale, 1:5–21, 1994. doi: 10.2514/6.1992-439. * [2] S.C. Kassinos and W.C. Reynolds. A structure-based model for the rapid distortion of homogeneous turbulence. PhD Thesis, Thermosciences Division Department of Mechanical Engineering Stanford, California 94305, 1994. * [3] S.C. Kassinos, W.C. Reynolds, and M.M. Rogers. One-point turbulence structure tensors. J. Fluid Mech., 428:213–248, 2001. doi: 10.1017/S0022112000002615. * [4] F.S. Stylianou, R. Pecnik, and S.C. Kassinos. A general framework for computing the turbulence structure tensors. Comput. Fluids, 106:54–66, 2015. doi: 10.1016/j.compfluid.2014.09.042. * [5] C.F. Panagiotou and S.C. Kassinos. A structure-based model for the transport of passive scalars in homogeneous turbulent flows. Int. J. Heat Fluid Fl., 57:109–129, 2016. doi: 10.1016/j.ijheatfluidflow.2015.11.008. * [6] B. Shraiman and E. Sigga. Scalar turbulence. Nature, 405:639–646, 2000. doi: 10.1038/35015000. * [7] S.C. Kassinos and W.C. Reynolds. A particle representation model for the deformation of homogeneous turbulence. pages 31–51. Annual Research Briefs, Center for Turbulence Research, 1996\. * [8] S.C. Kassinos and E. Akylas. Advances in particle representation modeling of homogeneous turbulence. From the linear PRM version to the interacting viscoelastic IPRM, volume 18. ERCOFTAC Series, 2012. doi: 10.1007/978-94-007-2506-5_6. * [9] C.F. Panagiotou and S.C. Kassinos. A differential structure-based model based on stochastic evolution equations for the scalar turbulence parameters. European Conference on Computational Fluid Dynamics (ECFD 6), Barcelona, Spain, 2014. * [10] S.V. Poroseva, S.C. Kassinos, C.A. Langer, and W.C. Reynolds. Structure-based turbulence model: Application to a rotating pipe flow. Phys. Fluids, 14(4):1523–1532, 2002. doi: 10.1063/1.1458008. * [11] C.F. Panagiotou, F.S. Stylianou, and S.C. Kassinos. Structure-based transient models for scalar dissipation rate in homogeneous turbulence. Int. J. Heat Fluid Fl., 82:108557, 2020. * [12] C.F. Panagiotou and S.C. Kassinos. A structure-based model for transport in stably stratified homogeneous turbulent flows. Int. J. Heat Fluid Fl., 65:309–322, 2017. doi: 10.1016/j.ijheatfluidflow.2016.12.005. * [13] S.C. Kassinos, C.A. Langer, G. Kalitzin, and G. Iaccarino. A simplified structure-based model using standard turbulence scale equations: Computation of rotating wall-bounded flows. Int. J. Heat Fluid Fl., 27:653–660, 2006. doi: 10.1016/j.ijheatfluidflow.2006.02.018. * [14] C.A. Langer and W.C. Reynolds. A new algebraic structure-based turbulence model for rotating wall-bounded flows. PhD Thesis, Thermosciences Division Department of Mechanical Engineering Stanford, California 94305, 2003. * [15] C.F. Panagiotou and S.C. Kassinos. The ASBM-SA turbulence closure: Taking advantage of structure-based modeling in current engineering CFD codes. Int. J. Heat Fluid Fl., 52:111–128, 2015. * [16] P. Bradshaw, D.H. Ferriss, and N.P. Atwell. Calculation of boundary layer development using the turbulent energy equation. J. Fluid Mech., 28(3):593–616, 1967. doi: 0.1017/S0022112067002319. * [17] C. Rumsey, T.B. Gatski, L. Sellers, V. Vatsa, and S. Viken. Summary of the 2004 CFD validation workshop on synthetic jets and turbulent separation control. In AIAA 2004-2217, 2004. doi: 10.2514/6.2004-2217. * [18] D. Greenblatt, K.B. Paschal, N.W. Schaeffler, A.E. Washburn, J. Harris, and C. Yao. A separation control CFD validation test case. Part 1: Baseline and steady suction. In 2nd AIAA Flow Control Turbulence, 2004. doi: 10.2514/6.2004-2220. * [19] A. Seifert and L.G. Pack. Active flow separation control on wall-mounted hump at high Reynolds numbers. AIAA J., 40(7):1363–1372, 2002. doi: 10.2514/2.1796. * [20] T.B. Gatski and C. Rumsey. Langley Research Center Workshop: CFD validation of synthetic jets and turbulent separation control, (Cited 1 March 2020).

2024-09-04T02:54:56.124095

2003.00584

arxiv-papers

# The Use of Change Point Detection to Identify Software Performance Regressions in a Continuous Integration System David Daly 0000-0001-9678-3721 MongoDB Inc<EMAIL_ADDRESS>, William Brown Columbia University<EMAIL_ADDRESS>, Henrik Ingo 0000-0003-1571-5108 MongoDB Inc<EMAIL_ADDRESS>, Jim O’Leary 0000-0002-3923-5742 MongoDB Inc<EMAIL_ADDRESS>and David Bradford 0000-0003-2282-6535 MongoDB Inc<EMAIL_ADDRESS> (2020) ###### Abstract. We describe our process for automatic detection of performance changes for a software product in the presence of noise. A large collection of tests run periodically as changes to our software product are committed to our source repository, and we would like to identify the commits responsible for performance regressions. Previously, we relied on manual inspection of time series graphs to identify significant changes. That was later replaced with a threshold-based detection system, but neither system was sufficient for finding changes in performance in a timely manner. This work describes our recent implementation of a change point detection system built upon the E-Divisive means (Matteson and James, 2014) algorithm. The algorithm produces a list of change points representing significant changes from a given history of performance results. A human reviews the list of change points for actionable changes, which are then triaged for further inspection. Using change point detection has had a dramatic impact on our ability to detect performance changes. Quantitatively, it has dramatically dropped our false positive rate for performance changes, while qualitatively it has made the entire performance evaluation process easier, more productive (ex. catching smaller regressions), and more timely. change points, performance, testing, continuous integration ††journalyear: 2020††copyright: rightsretained††conference: Proceedings of the 2020 ACM/SPEC International Conference on Performance Engineering; April 20–24, 2020; Edmonton, AB, Canada††doi: 10.1145/3358960.3375791††ccs: Software and its engineering Software performance††ccs: Information systems Database performance evaluation††ccs: Mathematics of computing Time series analysis ## 1\. Introduction We work in a software and services company and need to understand the performance of the software we develop and sell. Our continuous integration system runs thousands of benchmarks periodically (most commonly every 2 hours or every 24 hours), which each produce one or more scalar values as a result. It is a challenge to analyze all of those results and historical data to determine whether the test should be considered passed or failed. It is inherent to this type of benchmarking that the results contain some level of noise. In most of our tests the worst case run to run variation level is currently less than $10\%$, but some sensitive tests can fluctuate as much as $20\%$ or more over the course of a small numbers of runs. In this paper, we detail the results of our experience in deploying automated change point detection software for identifying performance changes in an evolving code base. ### 1.1. Existing performance results and analysis Our performance testing infrastructure is built upon our continuous integration (CI) system: Evergreen (Erf, 2016; Eve, [n.d.]). Evergreen tracks every commit to our source repository, compiles the software on a number of different build targets, and runs correctness tests. Evergreen also runs our performance tests. The performance tests take longer to run than the correctness tests, so we run them less frequently, but otherwise the performance and correctness tests are the same from the perspective of Evergreen. With any performance test, we must * • Setup a system under test * • Run a workload against the system under test * • Report the results from the test * • Decide (and alert) if the performance changed * • Visualize the results This paper focuses on the last two bullets111Previous work focused on the first two bullets and getting reproducible results (Ingo and Daly, 2019). When we first setup our performance tests in our CI system, we had a visualization of the performance over time, but no automated alerting. We had a person (called the “Build Baron”) looking through the graphs for regressions, and opening Jira tickets222Atlassian’s Jira is our ticketing system. for any changes requiring further investigation. The Build Baron would be looking at performance trend graphs like those shown in Figure 1 and Figure 2. Figure 1 shows a test with some basic noise, but no real changes in performance, while Figure 2 has two very clear changes in performance. The cases in Figures 1 and 2 should be easy for the Build Baron to triage, but many other cases are not, such as the two graphs in Figure 3 and the graph in Figure 4, which have smaller changes and larger run to run variation. We quickly realized that having humans triaging performance changes solely on the trend graphs was not a tenable solution: humans looking through all the graphs lose focus quickly and the problem would get worse as we added more tests and system configurations. Therefore we added an automated detection system. Figure 1. Performance trend graph with noise and no discernible changes. Figure 2. Example of a clear drop in performance on August 8th and subsequent performance recovery on August 20th. Figure 3. Performance trend graphs with noise. The second graph (insert_ttl) has a change in performance on June 13th smaller than the noise threshold of the first graph. The amount of noise also changes over time for the insert_ttl graph. Figure 4. Performance trend graph with a small drop in performance on August 5th. It would be easy for a person to miss that regression when looking through many graphs. The automated detection system was designed to catch * • Sudden drops in performance from a specific software change * • Multiple small changes over time causing a drift down in performance It did this in a very simple manner by doing three comparisons: 1. (1) Compare to the immediately preceding run. Did it change more than $X\%$333Where a human then looked at flagged failures and $X$ is a parameter of the system. We used $10\%$ by default? 2. (2) Compare to a run from a week ago. Did it change more than $X\%$? 3. (3) Compare to a run for the last stable release of the software. Did it change more than $X\%$? This system was a vast improvement over just staring at graphs. It was automated and computers do not get bored or lose focus. However, we soon realized we had a large number of false positives with which to deal (up to 99% as discussed in Section 5 depending on how you count). All the tests have some inherent noise in them. However, different tests and different test configurations have different levels and types of noise. Using a static threshold (e.g., $10\%$) led to a lot of alerts for changes on noisy tests. At the same time it would either miss real changes that were less than the threshold, or detect the smaller changes at some later time (when the change plus the noise crossed the threshold). For example, in Figure 3 the insert_ttl test shows a clear, but very small change around June 13th, while the map_reduce workload is steady with run to run variation (noise). There is no way we could adjust a common threshold for those two tests that would both catch the change in insert_ttl and not constantly flag the map_reduce test, or constantly flag the insert_ttl 1 thread line after July 19th. While the system was noisy and painful, it was still useful for us. Over time we added fixes and band aids to the system to reduce the number of false positives (e.g., adjust the thresholds separately for different tests). Even with all those fixes to the previous system, the system: * • Produced a large number of false positive alerts that had to be processed * • Missed smaller regressions (false negatives) * • Flagged failures on commits unrelated to the regression, requiring humans to trace back to find when a regression actually happened * • Only flagged regressions, not improvements * • Consumed a huge amount of time from the team to process all the results We wanted to improve the process to something fundamentally better. We started by trying to clearly state the problem. ## 2\. Problem Definition The software we are testing is updated in discrete commits to our source repository. Our problem is to: ###### Problem 1. Detect which commits change the performance of the software (as measured by our performance tests) in the presence of the noise from the testing infrastructure. When the problem is phrased like that, it is clear that this is a signal processing problem. The noise of the system makes the performance results a stochastic process. As such, we reviewed the literature on signal processing444See Section 7 for references on signal processing and change point detection. In particular, we decided our problem was a change point detection problem. From Matteson and James (Matteson and James, 2014): “Change point analysis is the process of detecting distributional changes within time- ordered observations.” Our system can be represented as the time series: (1) $S_{t}=P_{t}+N_{t}$ Where $S_{t}$ is the measured signal, $P_{t}$ is a constant value of performance, and $N_{t}$ is a random variable representing the noise. It is not clear a priori that the noise component is independent, identically distributed (IID) or not. In fact, we have examples in which the noise is correlated with time. The noise is really composed of at least three components: (2) $N=N_{p}+N_{s}+N_{w}$ * • $N_{p}$: The noise from the test platform, including CPU, network, OS, etc. * • $N_{s}$: The noise from the software itself. This can be from the compiler as well as other non-determinism or randomness in the software. * • $N_{w}$: The noise from the workload generator. The workload generator is also software and may vary in the load it applies to the system. Reducing $N_{p}$ and $N_{w}$ was the focus of previous work (Ingo and Daly, 2019). However, as computers and computer networks are inherently very complicated, non-deterministic machines, including things like caches and predictors, we can only hope to reduce those noise components, not remove them. As such, we have to learn to live with noise. ### 2.1. E-Divisive Means After reviewing the literature on change point detection, we decided to use the E-Divisive means algorithm (Matteson and James, 2014). E-Divisive means has the following useful properties for us: * • Does not require any distributional assumptions other than a finite mean * • Can be recursively applied to identify multiple change points * • Does not require training data and works out of the box on a time series * • Optimal sets of change points can be efficiently computed via dynamic programming and permutation sampling Additionally, the main focus of our work is on retroactive identification of change points, not prediction. Many time series algorithms are intended for forecasting, which is not relevant in the context of commits being pushed by developers. The algorithm works by hierarchically selecting distributional change points that divide the time series into clusters. For a given cluster, a candidate change point is chosen by computing a statistic $\widehat{Q}$ at every point within the cluster and selecting the point with the largest value. This statistic $\widehat{Q}$ is the discrete analog of the divergence between continuous multivariate distributions. After the first change point is determined, the algorithm is then reapplied to each cluster created by the existing change point. This process repeats until the largest candidate $\widehat{Q}$ value is below a threshold of significance. #### 2.1.1. $\widehat{Q}$ Statistic Computation To compute $\widehat{Q}$ for a series of n + m points bisected at the nth point, let: (3) $\mathbf{X}_{n}=\\{X_{i}:i=1,...,n\\}$ (4) $\mathbf{Y}_{m}=\\{Y_{j}:j=1,...,m\\}$ Represent the first n points and the last m points, respectively. An empirical divergence metric can be computed as follows: (5) $\begin{split}\widehat{\mathcal{E}}(\mathbf{X}_{n},\mathbf{Y}_{m};\alpha)=&\frac{2}{mn}\sum_{i=1}^{n}\sum_{j=1}^{m}{|X_{i}-Y_{i}|}^{\alpha}\\\ &-\binom{n}{2}^{-1}\sum_{1\leq i<k\leq n}{|X_{i}-X_{k}|}^{\alpha}\\\ &-\binom{m}{2}^{-1}\sum_{1\leq j<k\leq m}{|Y_{j}-Y_{k}|}^{\alpha}\end{split}$ The $\alpha$ parameter can be any value between 0 and 2. We use 1 for simplicity555Matteson and James (edvisive2014) observed similar results with $\alpha$ of 0.5 and 1.5, and presented their simulation studies with $\alpha$ of 1.. We then attain $\widehat{Q}$ by weighting the previous result by the size of our clusters: (6) $\widehat{Q}(\mathbf{X}_{n},\mathbf{Y}_{m};\alpha)=\frac{mn}{m+n}\widehat{\mathcal{E}}(\mathbf{X}_{n},\mathbf{Y}_{m};\alpha)$ #### 2.1.2. Termination The procedure for termination in the paper involves numerous random permutations of the values in each cluster to determine the statistical significance. Intuitively, if rearranging the order of data in a cluster does not significantly affect the maximum $\widehat{Q}$, there is likely no meaningful change point. ## 3\. Implementation Selecting an algorithm for detecting change points was just the beginning. We then needed to: * • Implement (or find an implementation of the algorithm) * • Automate processing our data with the algorithm * • Present the data after processing * • Enable users to act on the presented data The first step of implementing the algorithm was straightforward as there is an R package (James and Matteson, 2015) for the algorithm. We also wrote a Python implementation666The Python implementation of e-divisive means is available here: https://github.com/mongodb/signal-processing-algorithms to go with our existing Python code base. Both implementations will iteratively find the most significant change point in the series, and then analyze the clusters on either side of the change point, until reaching the stopping criterion. Even though we used our Python implementation, we benefited from the existence of the R implementation. We were able to compare the results from the two implementations to gain confidence in the correctness of our implementation. It also gave us a reference point for the performance of the algorithm. It took a little more work to transform our existing data into a form that could be consumed by the algorithm. We store results in a MongoDB database, with all the results for a given task build stored together. We unbundled the data so we could easily generate a vector of results for each individual combination of (system variant, task, test, test configurations). We then saved the computed change points in the database in a separate collection. Each change point has the same identifying fields as the results data, plus data from the algorithm itself, such as the $\widehat{Q}$ and the order of the change point (was it the first or $N^{th}$ change point for the series when recursively identifying change points). After identifying the change points for a series, each series is partitioned into clusters in which the results are unchanging. We call those clusters “stable regions”. We calculate common statistics for each stable region, and include that in the entry for each change point. Statistics include the min, max, median, mean, variance for the given measure, as well as how many data points are in the stable region. The algorithm is run at the completion of every set of performance tests. The result data is uploaded and transformed before the algorithm is run. We replace any existing change points with the newly generated change points. Rerunning the analysis allows the change points to be updated as more data becomes available. The data is statistical in nature, so having more data allows better analysis. As such, it is expected that on occasion a previously computed change point will “move” to a new commit or even completely go away as more data becomes available. Additionally, the change point algorithm will never find that newest result to be a change point. If a regression was introduced, the algorithm will only detect a change point after several more test results are collected. A three to five day delay is common for our tests. This contrasts with processing results for correctness tests in which the pass/fail result is determined at run time and will not change at a later time. ### 3.1. Display We then displayed the data in two distinct ways. First, we updated the trend graphs discussed in Section 1.1. We added annotations to show the change points as well as any Jira tickets that match the test and revision. Figures 5 and 6 show two trend graphs annotated with both change points and Jira tickets. The green diamonds are the Jira tickets, and the highlighted segments are change points. You can clearly see two tickets matching two green highlighted sections in Figure 5, while there is a small drop with change point and ticket highlighted in Figure 6. That figure is particularly interesting as the change in mean is on the same scale as the level of the noise before and after the change, and a version of that graph without annotations appeared earlier in Figure 4. As we will discuss in Section 3.2, it is a goal of our process that each change point is matched to a Jira ticket or hidden from view if it is clear noise. Figure 5. Trend graph for index build background test. Notice two change points (green highlights) and two JIRA tickets (diamonds). Figure 6. Trend graph for the wildcard index test with annotations for Jira tickets and change points. This figure can be compared to Figure 4 which does not have the annotations. We added necessary meta data fields to our Jira tickets so that they can be mapped to their originating Evergreen project, task, test and git revisions. We track both “First failing revision” and “Fix revision”. Additionally, we needed to query this data every time we drew the displays. For performance reasons, we setup a regular batch job to query Jira and load the data into our database. Note that we do not run the performance tests on every commit to the source repository. That adds complexity to the visualization above. A change point is identified for commits that were built and run. The algorithm cannot determine which commit caused a performance change if the commits have not run. Instead, we can identify the range of commits between the change point and the previous test result as suspect commits. When displaying these results and annotations, we must be able to match up change point, Jira ticket, and builds that have actually run, even if they point to close but different commits. Those annotations are great for examining how one test performs over time, and identifying the location of significant changes. It is not useful though for tracking all tests, triaging the most important regressions, and grouping tests that change at the same time. For that, we built a new page for viewing the change points. This new page shows all the change points on one page, grouped by commit and is shown in Figures 7 and 8. Change points are initially “unprocessed” and through triage become “processed”. Figure 7. Triage view of unprocessed change points. Figure 8. View of processed change points888Our Best Buy tests are based off of the open data set from Best Buy (bes, [n.d.]). Figure 7 shows the list of software revisions (identified by their git hashes) associated with “unprocessed” change points. All the change points for a single git hash are grouped together. The display has the git hash, test name, task name, date of the build, and the hazard level of the change. Hazard level is computed as the log of the ratio of mean before to the mean after the change point. The hazard level has proven extremely useful in focusing triage on the major regressions and it has the property that two changes that cancel out (e.g., $33\%$ drop followed by a $50\%$ rise) have the same magnitude. The display has the ability to filter or sort on any of the fields. There are two buttons along the top to acknowledge or hide the change points. Hiding or acknowledging makes a copy of the changed point with the addition of field to indicate that the change point has been processed. The process of acknowledging or hiding a change point also removes it from the unprocessed change point page. It will also be removed from this page if there is already a Jira ticket for the same test at that commit. The list is short in the figure because the remaining change points had already been “processed”, with only two less interesting change points remaining. Once a commit is acknowledged or hidden, it shows up on the processed tab as shown in Figure 8. There we have shown one change point with a significant improvement in performance. We will isolate that change to a single commit, and open a Jira ticket to document it. Once that ticket is opened, the change point will also be removed from this view. ### 3.2. Triage We have a rotating role of a dedicated person to check for performance changes. We call this person the “Build Baron”. The Build Baron uses all the views of the change points shown in Figures 7 and 8 along with the trend graphs as shown in Figures 5 and 6. Ideally, the Build Baron reviews every change point, deciding if it is a real change worthy of investigation, if it is insignificant, or if it is just noise. The Build Baron starts with the list of the unprocessed change points as shown in Figure 7. By default we have the list set to filter out some things (lower priority, canary999Canary tests are tests run to detect changes to the test system itself. Changes in performance for canary tests indicate a problem with the test itself. See (Ingo and Daly, 2019) for more on our use of canary tests. tests, or some results that are merely informational). The Build Baron goes down this list investigating each git hash in the list. Each git hash can have multiple change points (multiples tests, variants, etc). The test column includes links to the task page with the matching trend graph for the Build Baron to decide if the change point is worth further investigation. They may determine from inspection that there is a clear change. If so, they will try to isolate the offending commit. Recall that we do not run every commit, so once a performance change is detected, we need to run the tests on the missing commits to determine the precise commit that introduced the performance change. The Build Baron will schedule the builds appropriately, and “acknowledge” the change point by pressing the “acknowledge” button on the change point page. The Build Baron will need to wait for the tests to finish. When the tests finish, the system will rerun the change point detection algorithm in order to update the change point to the correct commit. If the change point represents a new issue, the Build Baron will open a Jira ticket for the change and assign it to the appropriate team to investigate. If there is an existing Jira ticket (e.g., this is the fix to a previous regression) the Build Baron will update the existing Jira ticket. If the performance change is a significant enough regression, we may also revert the commit. Not all change points lead to Jira tickets. Occasionally we will have spurious drops in performance, with the performance changing for one build and then returning to previous performance. Drops such as these may go away if we rerun the task. If the performance change is large enough it can lead to a change point being created. We start by checking these against our canary workloads. As part of previous noise reduction work we have a number of canary workloads. The canary workloads tell us nothing about the software we are testing, but a lot about the test system. The results should be constant. If we detect a statistically significant change for a canary or otherwise suspect a spurious drop, we treat all the results from that test run as suspect and rerun the test. Alternatively, the change point may be due to noise. The algorithm is very good at accounting for IID noise, but not for correlated noise. Unfortunately, we have some correlated noise in our system – some from time varying behavior on our cloud providers, other due to things as finicky as code alignment issues in compilation and its impact on processor caches. Either case usually can be confirmed visually, and the Build Baron will “Hide” the change point by using the Hide button. That will remove the change point from the change point page and change the color of the change point on the trend graph to blue. If we have a sustained change in performance due to cloud changes, we open a Jira ticket for that in order to document it. That way, anyone looking at the trend graphs will be able to see why the performance changed. All change points that are processed (hidden or acknowledged) and all change points that match a Jira ticket are removed from the change point list. The list is meant for triage and those points have already been triaged. We have the history of the points however and can use it to compile effectiveness data about our system. Since we group change points by git hash in the display, we may have the case in which we want to acknowledge some of the change points in the group and hide others. This is fully supported. ## 4\. Optimizations The computational complexity of a naive implementation of E-Divisive means is $\mathcal{O}(\kappa T^{3})$, where $\kappa$ is the number of change points and $T$ the number of observations in the series. Matteson and James (Matteson and James, 2014) point out that it is possible to compute the series $\widehat{Q}$ in $\mathcal{O}(T^{2})$ (per change point) as each value in the series can be derived from the previous with linear time. We implemented this optimization in our POC. The major generations of the implementation were: * • Naive $\mathcal{O}(\kappa T^{3})$ implementation * • $\mathcal{O}(\kappa T^{2})$ implementation (row/column differences) * • Use NumPy arrays * • Use native C implementation (Python module) We also tested compiling the first 2 versions with Cython, but this yielded only about $10\%$ improvements and was discarded as an option. With a test data series of 173 values, using an Intel(R) Core(TM) i7-6560U CPU @ 2.20GHz, single threaded executions yielded the following results: Implementation | Time (s) | Ratio ---|---|--- Naive | 0.622551 | 6103.44 Row/column differences | 0.011567 | 113.40 NumPy | 0.001282 | 12.57 Native | 0.000102 | 1.00 At this modest problem size, the native C implementation is over 6,000x faster than our original naive implementation, 113x faster than a non-naive implementation, and 13x faster than our best Python only implementation. At the end of a CI task there are 20-60 new results. The historical time series for those is fetched and the algorithm is used to re-compute change points. The computation is easy to parallelize as there are multiple series to be processed. Using the native C implementation and parallelizing the computation, we are able to re-compute all the change points in under 10 seconds. The majority of that time is spent waiting for network communication to the database, with the E-Divisive mean computation taking only milliseconds. The computation performance is a combination of the performance enhancements discussed above, and a conscious decision to limit our analysis to results from N recent months, by using up to 500 data points101010This is approximate. If more than 500 data points are available, we will select data points going back to an already computed change beyond that range. As such, the computation will use more than 500 points.. We do this since we are interested in catching regressions in new commits and new releases. Due to this, our time series typically have 100 to 500 values and 1 to 10 change points, and the computation time is minimal. In our original project plan we were concerned about the $\mathcal{O}(\kappa T^{3})$ complexity and had envisioned running change point detection in a separate asynchronous job so as to not impact test execution time. Due to the above optimizations we are able to run the change point detection immediately after each test execution in a simpler process. When the Build Baron is running additional tests to isolate the exact commit that changed performance, having the change point detection immediately refreshed reduces the time to isolate the change. Even though the computation time is minimal in production, we still care about the execution time. Occasionally we need (or want) to recompute all the change points in our system (e.g., we updated something related to the computation or we fixed a bug). Based on the current performance of the algorithm, we believe we could recompute all the change points across our system in under an hour. Previously we had a need to regenerate change points for a section of the data and we tried to regenerate the points over the weekend. The job was still running Monday morning when we checked on it. When we tried to regenerate the data more recently for the same subset, it took approximately 30 minutes. Finally, we may want to extend the length of history we use in the computation in the future, for example if we add a project that runs much more frequently. ## 5\. Impact The point of this work, as discussed in Section 2, was to more efficiently detect which commits change the performance of our software. It has succeeded. Triaging performance changes has gone from being a full time job that one person could not completely keep up with, to one that the Build Baron can process all of the change points and have time left over. There is a morale improvement, as the Build Baron is working on real performance changes that make a difference to the company. Additionally, we are able to detect smaller changes in performance than we could previously. We are also tracking performance improvements now in addition to performance regressions. This has had a few impacts: * • We have caught correctness bugs (e.g., important check suddenly skipped). * • We have been able to provide positive feedback to developers for making things faster. Someone notices the impact of their work now. * • We can share improvements with marketing (Walker-Morgan, 2019). While we clearly and strongly know that we have qualitatively improved our system, we have also quantitatively improved it. We have two sets of data: First, we performed a proof of concept (POC) before fully implementing the system. During the POC we looked at false negative and false positive rates, as well as general feasibility related issues. Since then we have implemented the system in full, including collecting the labeling data and the information in Jira tickets. The combination of that data allows us also to measure the effectiveness of the system in production. ### 5.1. Learnings from the POC For the POC we implemented the calculation of change points as a batch job. We then compared the changes points to the results from our existing system and in Jira tickets. We focused on a 5 month period of results. At that time our automated system used two level of Jira tickets. The first level was automatically generated by the analysis code. The Build Baron then would go through that list. The Build Baron would move representative failures to the second Jira project, and link the rest. For the 5 month period we had 2393 tickets in the first project, and 160 in the second. Of the 160 tickets in the second project, 24 of them were “useful.”111111For this use we considered a ticket useful if it reflected a change in performance of the software. So, on average 100 tickets in the first project reduced down to 1 useful ticket in the second project, with a human going through all of those. In other words, the old system generated a huge number of Jira tickets, of which the vast majority were false positives, while also suffering from false negatives. The change point algorithm found all the regressions we were tracking in Jira tickets. The change point algorithm may have missed performance changes, but if it did those were performance changes we were already missing. From a false negative rate, the new system was strictly better than the old system. We also looked at the change points that did not match existing tickets. We started by sorting on the $\widehat{Q}$ value. The first three that we investigated led to two new Jira tickets. The first element was one ticket, while the second and third items shared a ticket. The second item corresponded to a drop in performance and and the third item was a later fix for the earlier drop. Since we did not have complete reference data for all the performance data, we instead sampled the change points to estimate a false positive rate. We considered only the change points that did not match existing Jira tickets, as those are the potential false positives. We looked at 10 of those change points. Of those: * • 2 were clear changes in performance we had not caught previously. * • 4 were clearly noise. * • 4 were not clear cut. They included * – One clear change in distribution (the test got noisier) * – Two cases in which the distribution changed slightly. We think this was likely correlated noise in our system. * – One large outlier value got the system to create a change point. Of the cases that were noise we used the order of change point statistic to search for more significant regressions. In all cases there was a point in which change points went from noise or maybe regressions, to clear cut changes. Depending on how you count the four “not clear cut” cases, we had a false positive rate between 40% and 80%, with the possibility to tune the system. Given the previous system required looking at 100 things for each useful one, this was a huge improvement and we knew we wanted to move forward. ### 5.2. Learnings from Production After the POC we put the change point detection algorithm into production and have been collecting data. Looking at a 3 month period from summer of 2019 for our primary performance project we have: * • 12,321 distinct change points * • Corresponding to 178 unique git hash values * • 126 of those 178 correspond to a Jira ticket * • 122 of those 178 were acknowledged or hidden * • there were 79 unique Jira tickets for those change points Therefore, the Build Baron had 178 things to look at during this period, and more than 2/3 of all of them were issues we felt were worth tracking and possibly working on. We went from 100 notifications leading to 1 tracked issue, to 9 notifications leading to 6 worth follow-up and 4 tracked issues. Given that we want the process to be slightly biased toward false positives, this result is close to optimal. The process still is not perfect though. We also have: * • Acknowledged change points that do not match a raw change point * • Acknowledged change points without a matching Jira ticket * • Hidden change points associated with Jira tickets The two cases of acknowledged change points not matching a Jira ticket or a raw change point appear to come from a few cases related to triaging: aggressive triaging and mistaken triaging. The change points are grouped by git hash. It is easy to check some of the change points for the git hash and observe a distinct performance change, and aggressively acknowledge all the points. Alternatively, sometimes someone hits the wrong button and acknowledges when they probably should have hidden a point. We observe both of those cases when inspecting points for these two cases. We are adding more auditing of how we label change points. Hidden change points that do not match a raw change point mainly come from a combination of marginal change points aggressively triaged, in which the change point algorithm is able to exclude once it has additional data. ## 6\. Open Problems This work is part of a larger effort to understand the performance of our software, and to know when the fundamental performance changes. This builds on work to * • Build tools for making performance workloads * • Automate running of performance tests * • Include external benchmarks in our system * • Reduce the noise in our test bed There are a number of areas we continue to work on in this space * • Better workload generation tools * • More flexible result types * • Better signaling of performance changes * • Determine if a “patch” has changed performance * • Identify, exclude, and rerun test runs in which “canaries” have failed * • Use the data from processing change points to make the algorithm more effective Better workload generation tools is about making it easier for our engineers to craft interesting workloads, while the more flexible result types deal with handling latencies, percentiles, throughputs, etc. The next three items show the limits of the change point detection algorithm. E-Divisive means is great at detecting changes in distribution, but it needs multiple data points to compute that distribution. It also has a bias towards the center of the longest clusters. As such, it cannot tell you anything about the most recent test run, and it will not detect two distinct changes if they are too close together, such as in Figure 9. In Figure 9 a performance impacting code change went in and was reverted soon afterwards. There are not enough commits for the algorithm to detect that two distinct changes happened. Figure 9. Example of a drop in performance followed quickly by a fix. The regression and fix are too close together for the E-Divisive means algorithm to identify both changes. The bias towards the center of clusters is a result of the coefficient $mn/(m+n)$ in Equation 6. The coefficient was an integral assumption in the consistency proof in (Matteson and James, 2014). Since $m+n=T$, where $T$ is the number of observations in the series, we get $mn/(m+n)=m(T-m)/T=m-m2/T$. The first and last values of this coefficient are therefore $1-1/T$ when either $m$ or $n$ equals $1$. All other values of this coefficient are greater than one (e.g. $2-4/T$ for $m=2$) and the coefficient achieves the maximum value of $T/4$ when $m=n=T/2$. Even if it would lead to possibly invalidating the mathematical proof behind E-Divisive, in practice it is tempting to explore whether these limitations could be removed. The algorithm has a parameter for the minimum cluster size of observations: N. The tests used later in (Matteson and James, 2014) used minimum cluster sizes N=30 and N=10, while we have operated with N=3 to have faster notification. With N=3 we require at least 3 data points after the change (including the change) to detect the change. The combined effect of these limitations is that in our primary performance project we typically get alerts of a new regression 5-6 days after it was committed. Ultimately change point detection is not suitable for determining whether a single new test result is a regression or not. We can use outlier detection to answer that question. The change point detection algorithm can identify a stable region, and then we can use outlier detection to determine if the patch build is statistically the same as the stable region or not. This will allow us to react early to very clear regressions, while change point detection can catch smaller regressions after a delay. We also have a use case in which developers can test (patch test) their code changes (patches) before pushing them to the source repository. We can use the same outlier detection to detect performance changes in patch tests. Finally, with our canary tests the goal is reversed: Rather than ignoring outliers, we want to be alerted about outliers so that we can rerun the tests or quarantine the results. As mentioned in Section 3.2, we run a number of canary tests to detect changes to the test system itself. If we have a sudden change in the canary workloads we would like to flag the results and possibly rerun the tests. The detection of performance chances in the canary workloads, flagging the results, and rerunning suspect executions are all amenable to automation. Finally, we are collecting a large corpus of training data related to the change point algorithm, through the acknowledged and hidden change points. This will allow us to compare the existing algorithm to other change point detection algorithms. Additionally, we expect we can use this data to either manually adjust the algorithm, or train more advanced machine learning algorithms. ## 7\. Related Work Previous work has focused on reducing the noise in our test environment (Ingo and Daly, 2019). Our test environment is built on the Evergreen (Erf, 2016) CI system. There are many other continuous integration systems, such as Jenkins (Smart, 2011), BuildBot (Ablett et al., 2007), Travis CI (Tra, [n.d.]), Gitlab CI (Git, [n.d.]), and Circle CI (Cir, [n.d.]). The other CI systems have various supports for graphing performance results. For instance, Jenkins has a performance plugin (Jen, [n.d.]) to support running common benchmarking tools, reporting results, and plotting trend graphs. The Chromium project has implemented its own CI system called LUCI (luc, [n.d.]) and implemented regression detection (Chr, [n.d.]) within it. Each test requires setting explicit threshold values for alerting a regression. As such, it is very similar in concept to our previous analysis system. Google has a patent filing (Vallone, 2018) for an algorithm called “Window deviation analyzer” (WDA). It describes identifying windows of data in a time series, computing median and mean for each window, and estimating if the most recent sample differs more than some threshold from those values. Like our system, Google’s WDA will fail a test some number of commits after the commit that introduced a regression. Unlike our system, WDA requires an explicit threshold and it is does not identify a specific offending commit (it identifies a window of commits within which a regression has occurred). For algorithms for dealing with noise in time series data, we found the NIST/Sematech e-handbook on Statistical Methods (NIST/SEMATECH, 2012) a good overview, particularly chapter 6 “Process and Product Monitoring and Control”. We looked into using techniques such as Moving Averages or Box Jenkins (Box et al., 2015) for the time series data. That would involve building a model of the process and detecting when samples diverge from the existing model. Additionally, see (Aminikhanghahi and Cook, 2017) for an extensive survey on change point detection techniques in time series data, covering supervised methods such as multi-class, binary, and virtual classifiers, and unsupervised methods such as likelihood ratio, subspace models, and probabilistic models. There are frequentist/Bayesian and parametric/non-parametric methods algorithms for both the supervised and unsupervised classes. ## 8\. Conclusion In this paper we describe how we detect performance changes in our software using our continuous integration system and change point detection. We run a large collection of tests periodically as changes to our software product are committed to our source repository. We would like to know when the performance changes for those tests. Originally we used humans looking and graphs, which was later replaced with a threshold based automatic detection system, but neither system was sufficient for finding changes in performance in a timely manner. This paper describes our move to using change point detection (specifically E-Divisive means) to detect and notify when the performance changes. In addition to implementing the change point algorithm, we also had to integrate it into our existing performance testing system and visualize the results so that engineers could triage and use the information. We describe what was required to get change point detection working in our system and how we triage and process the results. We also describe our efforts to speed up the algorithm so that we could use it synchronously within our workflow. The net impact of this work was large for us: Quantitatively, it dramatically dropped our false positive rate for performance changes, while qualitatively it made the entire process easier, more productive (ex. catching smaller regressions), and more timely. There is more work to be done to continue to improve our ability to detect when the performance of our software changes. We discuss a number of open problems for performance testing in general and for using change point detection in particular. ## References * (1) * bes ([n.d.]) [n.d.]. Best Buy APIs open data set. https://bestbuyapis.github.io/api-documentation/#overview * Chr ([n.d.]) [n.d.]. (Chromium) Regression Detection for Performance Tests. wiki. https://www.chromium.org/chromium-os/testing/perf-regression-detection * Cir ([n.d.]) [n.d.]. Circle CI. https://circleci.com/docs/ * Eve ([n.d.]) [n.d.]. Evergreen Continuous Integration. https://github.com/evergreen-ci/evergreen/wiki * Git ([n.d.]) [n.d.]. Gitlab CI/CD. https://docs.gitlab.com/ee/ci/ * Jen ([n.d.]) [n.d.]. Jenkins Performance Plugin. wiki. https://wiki.jenkins.io/display/JENKINS/Performance+Plugin * luc ([n.d.]) [n.d.]. (LUCI) A Tour of Continuous Integration UI. https://chromium.googlesource.com/chromium/src.git/+/master/docs/tour_of_luci_ui.md * Tra ([n.d.]) [n.d.]. Travis CI. https://docs.travis-ci.com * Ablett et al. (2007) Ruth Ablett, Ehud Sharlin, Frank Maurer, Jorg Denzinger, and Craig Schock. 2007. Buildbot: Robotic monitoring of agile software development teams. In _RO-MAN 2007-The 16th IEEE International Symposium on Robot and Human Interactive Communication_. IEEE, 931–936. * Aminikhanghahi and Cook (2017) Samaneh Aminikhanghahi and Diane J Cook. 2017. A survey of methods for time series change point detection. _Knowledge and information systems_ 51, 2 (2017), 339–367. * Box et al. (2015) George EP Box, Gwilym M Jenkins, Gregory C Reinsel, and Greta M Ljung. 2015. _Time series analysis: forecasting and control_. John Wiley & Sons. * Erf (2016) Kyle Erf. 2016. Evergreen Continuous Integration: Why We Reinvented The Wheel. Blog Post. https://engineering.mongodb.com/post/evergreen-continuous-integration-why-we-reinvented-the-wheel * Ingo and Daly (2019) Henrik Ingo and David Daly. 2019. Reducing variability in performance tests on EC2: Setup and Key Results. Blog Post. https://engineering.mongodb.com/post/reducing-variability-in-performance-tests-on-ec2-setup-and-key-results * James and Matteson (2015) Nicholas James and David Matteson. 2015. ecp: An R Package for Nonparametric Multiple Change Point Analysis of Multivariate Data. _Journal of Statistical Software, Articles_ 62, 7 (2015), 1–25. https://doi.org/10.18637/jss.v062.i07 * Matteson and James (2014) David S. Matteson and Nicholas A. James. 2014. A Nonparametric Approach for Multiple Change Point Analysis of Multivariate Data. _J. Amer. Statist. Assoc._ 109, 505 (2014), 334–345. https://doi.org/10.1080/01621459.2013.849605 arXiv:https://doi.org/10.1080/01621459.2013.849605 * NIST/SEMATECH (2012) NIST/SEMATECH. 2012\. e-Handbook of Statistical Methods. http://www.itl.nist.gov/div898/handbook/ note. * Smart (2011) John Ferguson Smart. 2011\. _Jenkins: The Definitive Guide: Continuous Integration for the Masses_. ” O’Reilly Media, Inc.”. * Vallone (2018) Anthony Vallone. 2018\. Window Deviation Analyzer. Patent Publication 2018/0150373 A1, Filed Nov. 28, 2016, Published May 31, 2018. * Walker-Morgan (2019) DJ Walker-Morgan. 2019\. MongoDB 4.2 Performance Boosts. Blog. https://www.mongodb.com/blog/post/mongodb-42-performance-boosts

2024-09-04T02:54:56.141510

2003.00653

arxiv-papers

# Adversarial Attacks and Defenses on Graphs: A Review, A Tool and Empirical Studies Wei Jin† Yaxin Li† Han Xu† Yiqi Wang† Shuiwang Ji‡ Charu Aggarwal§ Jiliang Tang† † ‡ § Data Science and Engineering Lab, Michigan State University Texas A&M University IBM T. J. Watson Research Center <EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract Deep neural networks (DNNs) have achieved significant performance in various tasks. However, recent studies have shown that DNNs can be easily fooled by small perturbation on the input, called adversarial attacks. As the extensions of DNNs to graphs, Graph Neural Networks (GNNs) have been demonstrated to inherit this vulnerability. Adversary can mislead GNNs to give wrong predictions by modifying the graph structure such as manipulating a few edges. This vulnerability has arisen tremendous concerns for adapting GNNs in safety- critical applications and has attracted increasing research attention in recent years. Thus, it is necessary and timely to provide a comprehensive overview of existing graph adversarial attacks and the countermeasures. In this survey, we categorize existing attacks and defenses, and review the corresponding state-of-the-art methods. Furthermore, we have developed a repository with representative algorithms111https://github.com/DSE- MSU/DeepRobust. The repository enables us to conduct empirical studies to deepen our understandings on attacks and defenses on graphs. ## 1 Introduction Graphs can be used as the representation of a large number of systems across various areas such as social science (social networks), natural science (physical systems, and protein-protein interaction networks), and knowledge graphs [2, 26]. With their prevalence, it is of great significance to learn effective representations for graphs that can facilitate various downstream tasks such as node classification, graph classification, link prediction and recommendation [2, 26]. Graph Neural Networks (GNNs), which generalize traditional deep neural networks (DNNs) to graphs, pave one effective way to learn representations for graphs [26, 73, 3, 84]. The power of GNNs relies on their capability of capturing the graph structure simultaneously with the node features. Instead of only considering the instances (nodes with their features) independently, GNNs also leverage the relationships between them. Specifically, GNNs follow a message-passing scheme [24], where the nodes aggregate and transform the information from their neighbors in each layer. By stacking multiple GNN layers, information can be propagated further through the graph structure and we can embed nodes into low-dimensional representations. The obtained node representations can then be fed into any differentiable prediction layer so that the whole model can be trained in an end-to-end fashion. Due to their strong representation learning capability, GNNs have gained practical significance in various applications ranging from data mining [34, 63], natural language processing [48], and computer vision [37] to healthcare and biology [45]. As new generalizations of traditional DNNs to graphs, GNNs inherit both advantages and disadvantages of traditional DNNs. Similar to traditional DNNs, GNNs are also powerful in learning representations of graphs and have permeated numerous areas of science and technology. Traditional DNNs are easily fooled by adversarial attacks [25, 75, 40, 39]. In other words, the adversary can insert slight perturbation during either the training or test phases, and the DNN models will totally fail. It is evident that GNNs also inherit this drawback [89, 17, 91]. The attacker can generate graph adversarial perturbations by manipulating the graph structure or node features to fool the GNN models. As illustrated in Figure 1, originally node $7$ was classified by the GNN model as a green node; after node $7$ creates a new connection with node $3$ and modifies its own features, the GNN model misclassifies it as a blue node. Such vulnerability of GNNs has arisen tremendous concerns on applying them in safety-critical applications such as financial system and risk management. For example, in a credit scoring system, fraudsters can fake connections with several high-credit customers to evade the fraud detection models; and spammers can easily create fake followers to increase the chance of fake news being recommended and spread. Therefore, there is an urgent need to investigate graph adversarial attacks and their countermeasures. Figure 1: An example of adversarial attack on graph data. The goal of the GNN is to predict the color of the nodes. Here node 7 is the target node. Attacker aims to change the prediction of GNN on node 7 by modifying the edges and features. Pushing this research has a great potential to facilitate the successful adoption of GNNs in a broader range of fields, which encourages increasing attention on graph adversarial attacks and defenses in recent years. Thus, it is necessary and timely to provide a comprehensive and systematic overview on existing algorithms. Meanwhile, it is of great importance to deepen our understandings on graph adversarial attacks via empirical studies. These understandings can not only provide knowledge about the behaviors of attacks but also offer insights for us to design defense strategies. These motivate this survey with the following key purposes: * • We categorize existing attack methods from various perspectives in Section 3 and review representative algorithms in Section 4. * • We classify existing countermeasures according to their defense strategies and give a review on representative algorithms for each category in Section 5. * • We perform empirical studies based on the repository we developed that provide comprehensive understandings on graph attacks and defenses in Section 6. * • We discuss some promising future research directions in Section 7. ## 2 Preliminaries and Definitions Before presenting the review and empirical studies, we first introduce concepts, notations and definitions in this section. ### 2.1 Learning on Graph Data In this survey, we use $G=(V,E)$ to denote the structure of a graph where $V=\\{v_{1},\dots,v_{N}\\}$ is the set of $N$ nodes and $E=\\{e_{1},\dots,e_{K}\\}$ is the edge set. We use matrix ${\bf A}\in\\{0,1\\}^{{N}\times{N}}$ to denote the adjacency matrix of $G$, where each entry ${\bf A}_{ij}=1$ means nodes $v_{i}$ and $v_{j}$ are connected in $G$. Furthermore, we use ${\bf X}\in\mathbb{R}^{{N}\times{D}}$ to denote the node attribute matrix where $D$ is the dimension of the node feature vectors. Thus, graph data can be denoted as $G=({\bf A},{\bf X})$. There are a lot of learning tasks on graphs and in this work, we focus on the classification problems on graphs. Furthermore, we use $f_{\theta}$ with parameters $\theta$ to denote the learning models in this survey. Node-Level Classification. For node-level classification, each node in the graph $G$ belongs to a class in the label set $Y$. The graph model aims to learn a neural network, based on labeled nodes (training nodes), denoted as $V_{L}$, to predict the class of unlabeled nodes (test nodes). The training objective function can be formulated as: $\min_{\theta}~{}\mathcal{L}_{\text{train}}(f_{\theta}(G))=\sum_{v_{i}\in{V}_{L}}\ell\left(f_{\theta}({\bf X},{\bf A})_{i},y_{i}\right),$ (1) where $f_{\theta}({\bf X},{\bf A})_{i}$ and $y_{i}$ are the predicted and the true label of node $v_{i}$ and $\ell(\cdot,\cdot)$ is a loss function such as cross entropy. Graph-Level Classification. For graph-level classification, each individual graph has a class in the label set $Y$. We use $\mathcal{G}$ to denote a set of graphs, and $\mathcal{G}_{L}$ is the labeled set (training set) of $\mathcal{G}$. The goal of graph-level classification is to learn a mapping function $f_{\theta}:\mathcal{G}\rightarrow{Y}$ to predict the labels of unlabeled graphs. Similar to node-level classification, the objective function can be formulated as $\min_{\theta}~{}\mathcal{L}_{\text{train}}(\mathcal{G})=\sum_{G_{i}\in\mathcal{G}_{L}}\ell\left(f_{\theta}(G_{i}),y_{i}\right),$ (2) where $G_{i}$ is the labeled graph with ground truth $y_{i}$ and $f_{\theta}(G_{i})$ is the prediction of the graph $G_{i}$. ### 2.2 A General Form of Graph Adversarial Attack Based on the objectives in Section 2.1, we can define a general form of the objective for adversarial attacks, which aims to misguide the victim model by minimizing some attack loss. Thus, the problem of node-level graph adversarial attacks can be stated as: Given $G=({\bf A},{\bf X})$ and victim nodes subset $V_{t}\subseteq{V}$. Let $y_{u}$ denote the class for node $u$ (predicted or using ground truth). The goal of the attacker is to find a perturbed graph $\hat{G}=({\bf\hat{A},\hat{X}})$ that minimize the following attack objective $\mathcal{L}_{\text{atk}}$, $\displaystyle{\min~{}\mathcal{L}_{\text{atk}}(f_{\theta}({\hat{G}}))=\sum_{u\in V_{t}}\ell_{\text{atk}}\left(f_{\theta^{*}}(\hat{G})_{u},y_{u}\right)}$ (3) $\displaystyle s.t.,\quad{\theta^{*}=\arg\min_{\theta}\mathcal{L}_{\text{train}}\left(f_{\theta}\left({G^{\prime}}\right)\right)},$ where $\ell_{\text{atk}}$ is a loss function for attacking and one option is to choose $\ell_{\text{atk}}=-\ell$ and $G^{\prime}$ can either be $G$ or $\hat{G}$. Note that $\hat{G}$ is chosen from a constrained domain $\Phi(G)$. Given a fixed perturbation budget $\Delta$, a typical $\Phi(G)$ can be implemented as, $\|\hat{\bf A}-{\bf A}\|_{0}+\|\hat{\bf X}-{\bf X}\|_{0}\leq\Delta.$ (4) Table 1: Commonly used notations Notations | Description | Notations | Description ---|---|---|--- $G$ | Graph | $u$ | Target node $\hat{G}$ | Perturbed graph | $y_{u}$ | Label of node $u$ $V$ | The set of nodes | $f_{\theta}$ | Neural network model $V_{L}$ | The set of labeled nodes | $\mathcal{L}$ | Loss function $E$ | The set of edges | $l(\cdot,\cdot)$ | Pair-wise loss function $\mathbb{A}$ | Adjacency matrix | $\|\cdot\|_{0}$ | $\ell_{0}$ norm $\mathbb{\hat{A}}$ | Perturbed adjacency matrix | $\Delta$ | Perturbation budget $\mathbb{X}$ | Node attribute matrix | ${\bf Z}$ | Predicted probability $\mathbb{\hat{X}}$ | Perturbed node attribute matrix | $\mathbb{h}_{u}$ | Hidden representation of node $u$ $D$ | Dimension of node features | $e_{ij}$ | Edge between node $v_{i}$ and $v_{j}$ We omit the definition of graph-level adversarial attacks since (1) the graph- level adversarial attacks can be defined similarly and (2) the majority of the adversarial attacks and defenses focus on node-level. Though adversarial attacks have been extensively studied in the image domain, we still need dedicated efforts for graphs due to unique challenges – (1) graph structure is discrete; (2) the nodes in the graph are not independent; and (3) it is difficult to measure whether the perturbation on the graph is imperceptible or not. ### 2.3 Notations With the aforementioned definitions, we list all the notations which will be used in the following sections in Table 1. ## 3 Taxonomy of Graph Adversarial Attacks In this section, we briefly introduce the main taxonomy of adversarial attacks on graph structured data. Attack algorithms can be categorized into different types based on different goals, resources, knowledge and capacity of attackers. We try to give a clear overview on the main components of graph adversarial attacks. ### 3.1 Attacker’s Capacity The adversarial attacks can happen at two phases, i.e., the model training and model testing. It depends on the attacker’s capacity to insert adversarial perturbation: * • Evasion Attack: Attacking happens after the GNN model is trained or in the test phase. The model is fixed, and the attacker cannot change the model parameter or structure. The attacker performs evasion attack when $G^{\prime}=G$ in Eq. (3). * • Poisoning Attack: Attacking happens before the GNN model is trained. The attacker can add “poisons” into the model training data, letting trained model have malfunctions. It is the case when $G^{\prime}=\hat{G}$ in Eq. (3). ### 3.2 Perturbation Type The attacker can insert adversarial perturbations from different aspects. The perturbations can be categorized as modifying node features, adding/deleting edges, and adding fake nodes. Attackers should also keep the perturbation unnoticeable, otherwise it would be easily detected. * • Modifying Feature: Attackers can slightly change the node features while maintaining the graph structure. * • Adding or Deleting Edges: Attackers can add or delete edges for the existing nodes under certain budget of total actions. * • Injecting Nodes: Attackers can insert fake nodes to the graph, and link it with some benign nodes in the graph. ### 3.3 Attacker’s Goal According to the goals of attacks, we can divide the attacks for node-level classification into the following two categories: * • Targeted Attack: There is a small set of test nodes. The attacker aims to let the trained model misclassify these test samples. It is the case when $V_{t}\subset{V}$ in Eq. (3). We can further divide targeted attacks into (1) direct attack where the attacker directly modifies the features or edges of the target nodes and (2) influencer attack where the attacker can only manipulate other nodes to influence the targets. * • Untargeted Attack: The attacker aims to insert poisons to let the trained model have bad overall performance on all test data. It is the case when $V_{t}={V}$ in Eq. (3). Noth that for graph-level classification, there also exist targeted and untargeted attacks. Targeted attack aims to induce the model to give a specific label to a given graph sample, while untargeted attack only wants the model to perform incorrectly. ### 3.4 Attacker’s Knowledge Attacker’s knowledge means how much information an attacker knows about the model that he aims to attack. Usually, there are three settings: * • White-box Attack: All information about the model parameters, training input (e.g, adjacency matrix and attribute matrix) and the labels are given to the attacker. * • Gray-box Attack: The attacker only has limited knowledge about the victim model. For example, the attacker cannot access the model parameters but can access the training labels. Then it can utilize the training data to train surrogate models to estimate the information from victim model. * • Black-box Attack: The attacker does not have access to the model’s parameters or training labels. It can access the adjacency matrix and attribute matrix, and do black-box query for output scores or labels. ### 3.5 Victim Models In this part we are going to summarize the victim models that have been proven to be susceptible to adversarial examples. Graph Neural Networks. Graph neural networks are powerful tools in learning representation of graphs [56, 73]. One of the most successful GNN variants is Graph Convolutional Networks (GCN) [34]. GCN learns the representation for each node by keeping aggregating and transforming the information from its neighbor nodes. Though GNNs can achieve high performance in various tasks, studies have demonstrated that GNNs including GCN are vulnerable to adversarial attacks [89, 56]. Besides, it is evident from recent works [89, 87, 44] that other graph neural networks including column network (CLN) [52], graph attention network (GAT) [63] and JKNet [77] have the same issue. Other Graph Learning Algorithms. In addition to graph neural networks, adversary may attack some other important algorithms for graphs such as network embeddings including LINE [59] and Deepwalk [51], graph-based semi- supervised learning (G-SSL) [88], and knowledge graph embedding [8, 42]. Table 2: Categorization of representative attack methods Attack Methods | | Attack --- Knowledge | Targeted or --- Non-targeted | Evasion or --- Poisoning Perturbation Type | Application | Victim Model PGD, Min-max [76] | White-box | Untargeted | Both | Add/Delete edges | Node Classification | GNN | IG-FGSM [72] --- IG-JSMA [72] White-box | Both | Evasion | | Add/Delete edges --- Modify features Node Classification | GNN Wang et al. [64] | | White-box --- Gray-box Targeted | Poisoning | Add/Delete edges | Node Classification | GNN Nettack [89] | Gray-box | Targeted | Both | | Add/Delete edges --- Modify features Node Classification | GNN Metattack [91] | Gray-box | Untargeted | Poisoning | Add/Delete edges | Node Classification | GNN NIPA [57] | Gray-box | Untargeted | Poisoning | Inject nodes | Node Classification | GNN RL-S2V [17] | Black-box | Targeted | Evasion | Add/Delete edges | | Graph Classification --- Node Classification GNN ReWatt [46] | Black-box | Untargeted | Evasion | Add/Delete edges | Graph Classification | GNN Liu et al. [43] | | White-box --- Gray-box Untargted | Poisoning | | Flip label --- Modify features | Classification --- Regression G-SSL FGA [13] | White-box | Targeted | Both | Add/Delete edges | | Node Classification --- Community Detection | Network --- Emebdding GF-Attack [9] | Black-box | Targeted | Evasion | Add/Delete edges | Node Classification | | Network --- Emebdding Bojchevski et al. [5] | Black-box | Both | Poisoning | Add/Delete edges | | Node Classification --- Community Detection | Network --- Emebdding Zhang et al. [81] | White-box | Targeted | Poisoning | Add/Delete facts | Plausibility Prediction | | Knowledge --- Graph Embedding CD-Attack [38] | Black-box | Targeted | Poisoning | Add/Delete edges | Community Detection | | Community --- Detection Algorithm ## 4 Graph Adversarial Attacks In this section, we review representative algorithms for graph adversarial attacks. Following the categorizations in the previous section, we first divide these algorithms into white-box, gray-box and black-box and then for algorithms in each category, we further group them into targeted and untargeted attacks. Note that without specific mention, the following attack methods focus on node-level classification. An overall categorization of representative attack methods is shown in Table 2. In addition, some open source implementations of representative algorithms are listed in Table 9. ### 4.1 White-box Attacks In white-box attack setting, the adversary has access to any information about the victim model such as model parameters, training data, labels, and predictions. Although in most of the real world cases we do not have the access to such information, we can still assess the vulnerability of the victim models under the worst situation. Typically, white-box attacks use the gradient information from the victim model to guide the generation of attacks [13, 76, 72, 10]. #### 4.1.1 Targeted Attack Targeted attack aims to mislead the victim model to make wrong predictions on some target samples. A lot of studies follow the white-box targeted attack setting with a wide range of real-world applications. FGA [13] extracts the link gradient information from GCN, and then greedily selects the pair of nodes with maximum absolute gradient to modify the graph iteratively. Genetic algorithm based Q-Attack is proposed to attack a number of community detection algorithms [10]. Iterative gradient attack (IGA) based on the gradient information in the trained graph auto-encoder, which is introduced to attack link prediction [11]. Furthermore, the vulnerability of knowledge graph embedding is investigated in [81] and the plausibility of arbitrary facts in knowledge graph can be effectively manipulated by the attacker. Recommender systems based on GNNs are also vulnerable to adversarial attacks, which is shown in [86]. In addition, there are great efforts on attacking node-level classification. Traditional attacks in the image domain always use models’ gradients to find adversarial examples. However, due to the discrete property of graph data, directly calculating gradients of models could fail. To solve this issue, the work [72] suggests to use integrated gradient [58] to better search for adversarial edges and feature perturbations. During the attacking process, the attacker iteratively chooses the edge or feature which has the strongest effect to the adversarial objective. By this way, it can cause the victim model to misclassify target nodes with a higher successful rate. The work [79] assumes there is a set of “bad actor” nodes in a graph. When they flip the edges with any target nodes in a graph, it will cause the GNN model to have a wrong prediction on the target node. These “bad actor” nodes are critical to the safety of GNN models. For example, Wikipedia has hoax articles which have few and random connections to real articles. Manipulating the connections of these hoax articles will cause the system to make wrong prediction of the categories of real articles. #### 4.1.2 Untargeted Attack Currently there are not many studies on untargeted white-box attack, and topology attack [76] is one representative algorithm. It first constructs a binary symmetric perturbation matrix ${\bf S}\in\\{0,1\\}^{n}$ where ${\bf S}_{ij}=1$ indicates to flip the edge between $i$ and $j$ and ${\bf S}_{ij}=0$ means no modification on ${\bf A}_{ij}$. Thus, the goal of the attacker is to find ${\bf S}$ that minimizes the predefined attack loss given a finite budget of edge perturbations $\Delta$, i.e., $\|{\bf S}\|_{0}\leq\Delta$. It considers two different attack scenarios: attacking pre-trained GNN with fixed parameters $\theta$ and attacking a re-trainable GNN $f_{\theta}$. For attacking a fixed $f_{\theta}$, the problem can be formulated as, $\displaystyle\min\limits_{{\bf{S}}\in{\\{0,1\\}^{n}}}~{}\mathcal{L}_{\text{atk}}(f_{\theta}(\bf{S,{A}},\bf{X}))$ $\displaystyle\quad\text{s.t.}~{}\|{\bf S}\|_{0}\leq{\Delta}.$ (5) It utilizes the Projected Gradient Descent (PGD) algorithm in [47] to search the optimal $\bf S$. Note that the work [47] is also one popular attack algorithm in the image domain. For the re-trainable GNNs, parameter $\theta$ will be retrained after adversarial manipulation. Thus the attack problem is formulated as a min-max form where the inner maximization is to update $\theta$ by maximizing the attack loss and can be solved by gradient ascent while the outer minimization can be solved by PGD. Another work on untargeted white-box attack aims to attack multi-network mining by perturbing the networks (graphs) [85]. Specifically, in each iteration it measures the influence of network elements (edges and node attributes) to the mining results and attack one network element with the highest influence value. ### 4.2 Gray-box Attacks White-box attacks assume that attackers can calculate gradient through model parameters, which is not always practical in real-world scenarios. Gray-box attacks are proposed to generate attacks with limited knowledge on the victim model [89, 91, 57]. Usually they first train a surrogate model with the labeled training data to approximate the information of the victim model and then generate perturbations to attack the surrogate model. It is noted that these models need the access to the labels of training data, thus they are not black-box attacks that will be introduced in the following subsection. #### 4.2.1 Targeted Attack The early work on targeted gray-box attacks is for graph clustering [15]. It demonstrates that injecting noise to a domain name system (DNS) query graph can degrade the performance of graph embedding models. Different from [15], the work [89] proposes an attack method called Nettack to generate structure and feature attacks, aiming at solving Eq. (3). Besides, they argue that only limiting the perturbation budgets cannot always make the perturbation “unnoticeable”. They suggest the perturbed graphs should also maintain important graph properties, including degree distribution and feature co- occurrence. Therefore, Nettack first selects possible perturbation candidates not violating degree distribution and feature co-occurrence of the original graph. Then it greedily chooses the perturbation that has the largest score to modify the graph, where the score is defined as, $\max_{i\neq y}\ln\left({\bf Z}_{u,i}\left(G^{\prime}\right)\right)-\ln\left({\bf Z}_{u,y}\left(G^{\prime}\right)\right),$ (6) where ${\bf Z}_{u,i}$ is the probability of node $u$ to be the class $i$ predicted by the surrogate model. Thus, the goal of the attacker is to maximize the difference in the log-probabilities of the target node $u$. By doing this repeatedly until reaching the perturbation budge $\Delta$, it can get the final modified graph. Furthermore, it suggests that such graph attack can also transfer from model to model, just as the attacks in the image domain [25]. The authors also conduct influencer attacks where they can only manipulate the nodes except the target. It turns out that influencer attacks lead to a lower decrease in performance compared with directly modifying target node given the same perturbation budget. However, Nettack cannot handle large-scale graphs due to its high time complexity. The work [90] employs statistical analysis to investigate the patterns exhibited by Nettack perturbations. According to the observations on perturbation patterns, a scalable attack method Fasttack is proposed. It first ranks the possible perturbations by their impact on the victim model and chooses the top perturbations to generate attacks. In [67], AFGSM is proposed to derive an approximate closed-form solution with a lower time cost for the attack model and sequentially inject fake nodes into the graph. Another line of works focus on generating backdoor attacks for graphs [83, 74]. Backdoor attack aims to find some ”trigger” (or hidden pattern) that would produce misclassified results when added to an input [66, 14]. To generate backdoor attacks for graphs, the attacker injects triggers (e.g. carefully-crafted subgraphs) in the training samples, and aims to fool GNNs into misclassifying those samples with triggers while maintaining the performance on other testing samples. Note that graph backdoor attacks can be applied in both node-level and graph-level classification [74]. #### 4.2.2 Untargeted Attack Although following the same way of training a surrogate model as Nettack, Metattack [91] is a kind of untargeted poisoning attack. It tackles the bi- level problem in Eq. (3) by using meta-gradient. Basically, it treats the graph structure matrix as a hyper-parameter and the gradient of the attacker loss with respect to it can be obtained by: $\displaystyle\nabla_{G}^{\text{meta }}=\nabla_{G}\mathcal{L}_{\text{atk }}\left(f_{\theta^{*}}(G)\right).$ (7) Note that $\nabla_{G}\mathcal{L}_{\text{atk }}\left(f_{\theta^{*}}(G)\right)$ is actually a function with respect to both $G$ and $\theta$. If $\theta^{*}$ is obtained by some differential operations, we can compute $\nabla_{G}^{\text{meta }}$ as follows, $\nabla_{G}^{\text{meta }}=\nabla_{f}\mathcal{L}_{\text{atk }}\left(f_{\theta^{*}}(G)\right)\cdot\left[\nabla_{G}f_{\theta^{*}}(G)+\nabla_{\theta^{*}}f_{\theta^{*}}(G)\cdot\nabla_{G}\theta^{*}\right]$ (8) where $\theta^{*}$ is often obtained by gradient descent in fixed iterations $T$. At iteration $t+1$, the gradient of $\theta_{t+1}$ with respect to $G$ can be formulated as, $\nabla_{G}\theta_{t+1}=\nabla_{G}\theta_{t}-\alpha\nabla_{G}\nabla_{\theta_{t}}\mathcal{L}_{\text{train }}\left(f_{\theta_{t}}(G)\right),$ (9) where $\alpha$ denotes learning rate of the gradient descent operation. By unrolling the training procedure from $\theta_{T}$ back to $\theta_{0}$, we can get $\nabla_{G}\theta_{T}$ and then $\nabla_{G}^{\text{meta }}$. A greedy approach is applied to select the perturbation based on the meta gradient. Specifically, given the meta-gradient for a node pair $(u,v)$, it defines a score $S(u,v)=\nabla_{{\bf A}_{uv}}^{\operatorname{meta}}\cdot\left(-2\cdot{\bf A}_{uv}+1\right)$ and greedily picks the perturbation which has the highest score but satisfies the unnoticeable constraints as in Nettack. Instead of modifying the connectivity of existing nodes, a novel reinforment learning method for node injection poisoning attacks (NIPA) [57] is proposed to inject fake nodes into graph data . Specifically, NIPA first injects singleton $n$ nodes into the original graph. Then in each action $a_{t}$, the attacker first chooses an injected node to connect with another node in the graph and then assigns a label to the injected node. By doing this sequentially, the final graph is statistically similar to the original graph but can degrade the overall model performance. ### 4.3 Black-box Attacks Different from gray-box attacks, black-box attacks [17, 46, 57, 5, 9, 44] are more challenging since the attacker can only access the adjacency matrix, attribute matrix and output of the victim model. The access of parameters, labels and predicted probability is prohibited. #### 4.3.1 Targeted Attack As mentioned earlier, training a surrogate model requires access to the labels of training data, which is not always practical. We hope to find a way that we only need to do black-box query on the victim model [17] or attack the victim in an unsupervised fashion [5, 9]. To do black-box query on the victim model, reinforcement learning is introduced. RL-S2V [17] is the first work to employ reinforcement learning technique to generate adversarial attacks on graph data under the black-box setting. They model the attack procedure as a Markov Decision Process (MDP) and the attacker is allowed to modify $m$ edges to change the predicted label of the target node $u$. They study both node-level (targeted) and graph-level (untargeted) attacks. For node-level attack, they define the MDP as follows, * • State. The state $s_{t}$ is represented by the tuple $\left(G^{(t)},u\right)$ where $G^{(t)}$ is the modified graph at time step $t$. * • Action. A single action at time step $t$ is denoted as $a_{t}$. For each action $a_{t}$, the attacker can choose to add or remove an edge from the graph. Furthermore, a hierarchical structure is applied to decompose the action space. * • Reward. Since the goal of the attacker is to change the classification result of the target node $u$, RL-S2V gives non-zero reward $r$ to the attacker at the end of the MDP: $r\left(s_{m},a_{m}\right)=\left\\{\begin{array}[]{ll}{1}&{\text{ if }f_{\theta}\left(G^{(m)},u\right)\neq y},\\\ {-1}&{\text{ if }f_{\theta}\left(G^{(m)},u\right)=y.}\end{array}\right.$ (10) In the intermediate steps, the attacker receives no reward, i.e., $\forall t=1,2,\dots,m-1,r\left(s_{t},a_{t}\right)=0$. * • Termination. The process terminates when the attacker finishes modifying $m$ edges. Since they define the MDP of graph-level attack in the similar way, we omit the details. Further, the Q-learning algorithm [50] is adopted to solve the MDP and guide the attacker to modify the graph. Instead of attacking node-level classification, the work [5] shows a way to attack the family of node embedding models in the black-box setting. Inspired by the observation that DeepWalk can be formulated in matrix factorization form [53], they maximize the unsupervised DeepWalk loss with matrix perturbation theory by performing $\Delta$ edge flips. It is further demonstrated that the perturbed structure is transferable to other models like GCN and Label Propagation. However, this method only considers the structure information. GF-Attack [9] is proposed to incorporate the feature information into the attack model. Specifically, they formulate the connection between the graph embedding method and general graph signal process with graph filter and construct the attacker based on the graph filter and attribute matrix. GF- Attack can also be transferred to other network embedding models and achieves better performance than the method in [5]. #### 4.3.2 Untargeted Attack It is argued that the perturbation constraining only the number of modified edges may not be unnoticeable enough. A novel framework ReWatt [46] is proposed to solve this problem and perform untargeted graph-level attack. Employing a reinforcement learning framework, ReWatt adopts the rewiring operation instead of simply adding/deleting an edge in one single modification to make perturbation more unnoticeable. One rewiring operation involves three nodes $v_{1},v_{2}$ and $v_{3}$, where ReWatt removes the existing edge between $v_{1}$ and $v_{2}$ and connects $v_{1}$ and $v_{3}$. ReWatt also constrains $v_{3}$ to be the 2-hop neighbor of $v_{1}$ to make perturbation smaller. Such rewiring operation does not change the number of nodes and edges in the graph and it is further proved that such rewiring operation affects algebraic connectivity and effective graph resistance, both of which are important graph properties based on graph Laplacian, in a smaller way than adding/deleting edges. To perform graph adversarial attacks in a more realistic way, the work [44] proposes a more restricted black-box setting where the attackers only have access to a subset of nodes and can only attack a small number of them. Under this setting, the attacker is required to generate attacks in two steps: 1) selecting a small subset of nodes to attack under the limits of node access; 2) modifying node features or edges under the perturbation budget. By exploiting the structural inductive biases of the GNN models [77, 36] as the information source for attacking, it proposes a practical greedy method of adversarial attacks for node-level classification tasks and effectively degrades the performance of GNNs. ## 5 Countermeasures Against Graph Adversarial Attacks In previous sections, we have shown that graph neural networks can be easily fooled by unnoticeable perturbation on graph data. The vulnerability of graph neural networks poses great challenges to apply them in safety-critical applications. In order to defend the graph neural networks against these attacks, different countermeasure strategies have been proposed. The existing methods can be categorized into the following types: (1) adversarial training, (2) adversarial perturbation detection, (3) certifiable robustness, (4) graph purification, and (5) attention mechanism. ### 5.1 Adversarial Training Adversarial training is a widely used countermeasure for adversarial attacks in image data [25]. The main idea of adversarial training is to inject adversarial examples into the training set such that the trained model can correctly classify the future adversarial examples. Similarly, we can also adopt this strategy to defend graph adversarial attacks as follows, $\min_{\theta}\max_{\delta_{\bf A}\in\mathcal{P}_{\bf A}\atop\delta_{\bf X}\in\mathcal{P}_{\bf X}}\mathcal{L}_{\text{train}}\left(f_{\theta}({\bf A}+\delta_{\bf A},{\bf X}+\delta_{\bf X})\right),$ (11) where $\delta_{\bf A}$, $\delta_{\bf X}$ denote the perturbation on ${\bf A},{\bf X}$, respectively; $\mathcal{P}_{\bf A}$ and $\mathcal{P}_{\bf X}$ stand for the domains of imperceptible perturbation. The min-max optimization problem in Eq (11) indicates that adversarial training involves two processes: (1) generating perturbations that maximize the prediction loss and (2) updating model parameters that minimize the prediction loss. By alternating the above two processes iteratively, we can train a robust model against to adversarial attacks. Since there are two inputs, i.e., adjacency matrix ${\bf A}$ and attribute matrix ${\bf X}$, adversarial training can be done on them separately. To generate perturbations on the adjacency matrix, it is proposed to randomly drop edges during adversarial training [17]. Though such simple strategy cannot lead to very significant improvement in classification accuracy (1% increase), it shows some effectiveness with such cheap adversarial training. Furthermore, projected gradient descent is used to generate perturbations on the discrete input structure, instead of randomly dropping edges [76]. On the other hand, an adversarial training strategy with dynamic regularization is proposed to perturb the input features [22]. Specifically, it includes the divergence between the prediction of the target example and its connected examples into the objective of adversarial training, aiming to attack and reconstruct graph smoothness. Furthermore, batch virtual adversarial training [19] is proposed to promote the smoothness of GNNs and make GNNs more robust against adversarial perturbations. Several other variants of adversarial training on the input layer are introduced in [12, 18, 69, 20]. The aforementioned adversarial training strategies face two main shortcomings: (1) they generate perturbations on ${\bf A}$ and ${\bf X}$ separately; and (2) it is not easy to perturb the graph structure due to its discreteness. To overcome the shortcomings, instead of generating perturbation on the input, a latent adversarial training method injects perturbations on the first hidden layer [32]: $\min_{\theta}\max_{\delta\in\mathcal{P}}\mathcal{L}_{\text{train}}\left(f_{\theta}(G;{\bf H}^{(1)}+\delta)\right),$ (12) where ${\bf H}^{(1)}$ denotes the representation matrix of the first hidden layer and $\delta\in\mathcal{P}$ is some perturbation on ${\bf H}$. It is noted that the hidden representation is continuous and it incorporates the information from both graph structure and node attributes. ### 5.2 Adversarial Perturbation Detection To resist graph adversarial attacks during the test phase, there is one main strategy called adversary detection. These detection models protect the GNN models by exploring the intrinsic difference between adversarial edges/nodes and the clean edges/nodes [78, 28]. The work [78] is the first work to propose detection approaches to find adversarial examples on graph data. It introduces four methods to distinguish adversarial edges or nodes from the clean ones including (1) link prediction (2) sub-graph link prediction (3) graph generation models and (4) outlier detection. These methods have shown some help to correctly detect adversarial perturbations. The work [28] introduces a method to randomly draw subsets of nodes, and relies on graph-aware criteria to judiciously filter out contaminated nodes and edges before employing a semi-supervised learning (SSL) module. The proposed model can be used to detect different anomaly generation models, as well as adversarial attacks. ### 5.3 Certifiable Robustness Previously introduced adversarial training strategies are heuristic and only show experimental benefits. However, we still do not know whether there exist adversarial examples even when current attacks fail. Therefore, there are works [92, 6, 30, 7, 93, 61, 65] considering to seriously reason the safety of graph neural networks which try to certify the GNN’s robustness. As we know, GNN’s prediction on one node $v_{t}$ always depends on its neighbor nodes. In [92], they ask the question: which nodes in a graph are safe under the risk of any admissible perturbations of its neighboring nodes’ attributes. To answer this question, for each node $v$ and its corresponding label $y_{v}$, they try to exactly calculate an upper bound $U(v)$ of the maximized margin loss : $U(v)\geq\max_{G^{\prime}\in\mathcal{G}}\Big{(}\max_{i\neq y}{\bf Z}_{v,i}\left(G^{\prime}\right)-{\bf Z}_{v,y}\left(G^{\prime}\right)\Big{)},$ (13) where $\mathcal{G}$ denotes the set of all allowed graph perturbations (in [92] only attribute perturbation). This upper bound $U$ is called the Certificate of node $v$. During the certification, for $v$, if $U(v)\leq 0$, any perturbation cannot result the model to give a larger score to a wrong class $(i\neq y_{v})$ than the class $y_{v}$, so there is no adversarial attack in $\mathcal{G}$ that can change the model’s prediction. During the test phase, they calculate the certificate for all test nodes, thus they can know how many nodes in a graph is absolutely safe under attributes perturbation. Moreover, this certificate is trainable, directly minimizing the certificates will help more nodes become safe. However, the work [92] only considers the perturbations on node attributes. Analyzing certifiable robustness from a different perspective, in [6], it deals with the case when the attacker only manipulates the graph structure. It derives the robustness certificates (similar to Eq. (13)) as a linear function of personalized PageRank [29], which makes the optimization tractable. In [93], it also tries to certify robustness of graph neural networks under graph structural perturbations. It successfully solves the certification problem using a jointly constrained bilinear programming method. In [65], it borrows the idea from randomized smoothing [16] to achieve certifiable robustness for graphs under structural perturbation. Meanwhile, the sparsity of graph data during certification is considered in [7]. It improves the efficiency and accuracy of the certifications against attacks on both graph features and structures. Immunization methods are introduced to improve graphs’ certifiable robustness [61]. Besides, there are also works studying certifiable robustness on GNN’s other applications such as community detection [30]. ### 5.4 Graph Purification Both adversarial training or certifiable defense methods only target on resisting evasion attacks, which means that the attack happens during the test time. However, graph purification defense methods mainly focus on defending poisoning attacks. Since the poisoning attacks insert poisons into the training graph, purification methods aim to purify the poisoned graph and learn robust graph neural network models based on it. There are two approaches to realize graph purification: pre-processing [72, 21] and graph learning [33, 35]. #### 5.4.1 Pre-processing Pre-processing methods first purify the perturbed graph data and then train the GNN model on the purified graph. In this way, the GNN model is trained on a clean graph. The work [72] proposes a purification method based on two empirical observations of the attack methods: (1) Attackers usually prefer adding edges over removing edges or modifying features and (2) Attackers tend to connect dissimilar nodes. As a result, they propose a defense method by eliminating the edges whose two end nodes have small Jaccard Similarity [54]. Because these two nodes are different and it is not likely they are connected in reality, the edge between them may be adversarial. The experimental results demonstrate the effectiveness and efficiency of the proposed defense method. However, this method can only work when the node features are available. In [21], it is observed that Nettack [89] generates the perturbations which mainly changes the small singular values of the graph adjacency matrix. Thus it proposes to purify the perturbed adjacency matrix by using truncated SVD to get its low-rank approximation. it further shows that only keeping the top $10$ singular values of the adjacency matrix is able to defend Nettack and improve the performance of GNNs. #### 5.4.2 Graph Learning Pre-procesing might not be an ideal choice for purifying the graph, as it is independent of GNN’s training process and could mistakenly remove normal edges. An alternative purification strategy is graph learning, which targets at removing adversarial patterns and obtaining clean graph structure. Traditional graph learning methods [23, 31] do not directly deal with adversarial attacks. Hence, to make GNNs more robust, it is of great importance to leverage the characteristics of adversarial attacks to guide the graph learning process. In [33], it has demonstrated that adversarial attacks can essentially violate some important graph properties, i.e, low-rank, sparsity and feature smoothness. Then it explores these graph properties to design robust graph neural networks, Pro-GNN, which jointly learns clean graph structure and robust GNN parameters. Specifically, Pro-GNN alternatively updates graph structure by preserving the aforementioned properties and trains GNN parameters on the updated graph structure. In [80], variational graph autoencoder [35] is employed to reconstruct graph structure, which can also reduce the effects of adversarial perturbations. ### 5.5 Attention Mechanism Different from the purification methods which try to exclude adversarial perturbations, attention-based defense methods aim to train a robust GNN model by penalizing model’s weights on adversarial edges or nodes [87, 60, 82]. Basically, these methods learn an attention mechanism to distinguish adversarial edges and nodes from the clean ones, and then make the adversarial perturbations contribute less to the aggregation process of the GNN training. The work [87] assumes that adversarial nodes may have high prediction uncertainty, since adversary tends to connect the node with nodes from other communities. In order to penalize the influence from these uncertain nodes, they propose a defense method named RGCN to model the $l$-th layer hidden representation $\boldsymbol{h}_{i}^{(l)}$ of nodes as Gaussian distribution with mean value $\boldsymbol{\mu}_{\mathrm{i}}^{(l)}$ and variance $\boldsymbol{\sigma}_{i}^{(l)}$, $\boldsymbol{h}_{i}^{(l)}\sim N\left(\boldsymbol{\mu}_{\mathrm{i}}^{(l)},\operatorname{diag}\left(\boldsymbol{\sigma}_{i}^{(l)}\right)\right),$ (14) where the uncertainty can be reflected in the variance $\boldsymbol{\sigma}_{i}^{(l)}$. When aggregating the information from neighbor nodes, it applies an attention mechanism to penalize the nodes with high variance, $\boldsymbol{\alpha}_{i}^{(l)}=\exp\left(-\gamma\boldsymbol{\sigma}_{i}^{(l)}\right),$ (15) where $\boldsymbol{\alpha}^{(l)}_{i}$ is the attention score assigned to node $i$ and $\gamma$ is a hyper-parameter. Furthermore, it is verified that the attacked nodes do have higher variances than normal nodes and the proposed attention mechanism does help mitigate the impact brought by adversarial attacks. From a different perspective, GNNGuard is proposed to employ the theory of network homophily [49] to assign higher scores to edges connecting similar nodes while pruning edges between unrelated node [82]. In [60], it suggests that to improve the robustness of one target GNN model, it is beneficial to include the information from other clean graphs, which share the similar topological distributions and node attributes with the target graph. For example, Facebook and Twitter have social network graph data that share similar domains; Yelp and Foursquare have similar co-review graph data. Thus, it first generates adversarial edges $E_{P}$ on the clean graphs, which serve as the supervision of known perturbation. With this supervision knowledge, it further designs the following loss function to reduce the attention scores of adversarial edges: $\mathcal{L}_{dist}=-\min\left(\eta,\underset{e_{ij}\in{E}\backslash{E_{P}}}{\mathbb{E}}\boldsymbol{\alpha}_{ij}^{(l)}-\underset{e_{ij}\in{E_{P}}}{\mathbb{E}}\boldsymbol{\alpha}_{ij}^{(l)}\right),$ (16) where $\mathbb{E}$ denotes the expectation, $E\backslash{E_{P}}$ represents normal edges in the graph, $\boldsymbol{\alpha}^{(l)}_{ij}$ is the attention score assigned to edge $e_{ij}$ and $\eta$ is a hyper-parameter controlling the margin between the expectation of two distributions. It then adopts meta- optimization to train a model initialization and fine-tunes it on the target poisoned graph to get a robust GNN model. ## 6 A Repository for Graph Attacks and Defenses In this section, we give a brief introduction to the repository we have developed for adversarial attacks and defenses, DeepRobust222https://github.com/DSE-MSU/DeepRobust [41]. To facilitate the research on adversarial attacks and defenses, DeepRobust includes the majority of representative attack and defense algorithms for both graph data and image data. The repository can enable researchers to deepen our understandings on attacks and defenses via empirical studies. Specifically, for graph adversarial learning, DeepRobust provides 7 datasets: four citation graphs including Cora [55], Cora-ML [4], Citeseer [4] and Pubmed [55], one co-author graph ACM [68], one blog graph Polblogs [1] and one social graph BlogCatalog [27]. On top of that, DeepRobust covers the following attacks and defenses: (1) 5 targeted attack algorithms, i.e., FGA [13], Nettack [89], RL-S2V [17], integrated gradient attack [72] and random attack [89]; (2) 3 untargeted algorithms, i.e., Metattack [91], Topology attack [76] and DICE [70]; (3) one victim model, i.e. GCN [34]; and (4) 5 defense algorithms, i.e., adversarial training [25], GCN-Jaccard [72], GCN-SVD [21], RGCN [87] and Pro-GNN [33]. DeepRobust is an easy-to-use platform for researchers who are working on adversarial attacks and defenses. With DeepRobust, users can generate graph adversarial attacks by training an attack model through the attacking APIs or loading the pre-attacked graphs provided by the repository. Robust models can be trained on the perturbed graph with the defense APIs. Once we obtain the perturbed graph and the trained defense model, they can be fed into the evaluation API to compete against each other, and the model performance under the corresponding attack will be reported. In the next section, we use DeepRobust for empirical studies on graph adversarial attacks. ## 7 Empirical Studies With the help of DeepRobust, we are able to conduct empirical studies on graph adversarial attacks and discover their important patterns. Next we first introduce the experimental settings and then present the empirical results and findings. ### 7.1 Experimental Setup Different attack and defense methods have been designed under different settings. We perform the experiments with one of the most popular settings – the untargeted poisoning setting. Correspondingly we choose representative attack and defense methods that have been designed for this setting. Three representative attack methods are adopted to generate perturbations including DICE [70], Metattack [91] and Topology attack [76]. It is noted that DICE is a white-box attack which randomly connects nodes with different labels or drops edges between nodes sharing the same label. To evaluate the performance of different defense methods under adversarial attacks, we compare the robustness of the natural trained GCN [34] and four defense methods on those attacked graphs, i.e., GCN [34], GCN-Jaccard [72], GCN-SVD [21], RGCN [87] and GAT [62]. Following [91], we use three datasets: Cora, Citeseer and Polblogs. For each dataset, we randomly choose 10% of nodes for training, 10% of nodes for validation and the remaining 80% for test. We repeat each experiment for 5 times and report the average performance. On Cora and Citeseer datasets, the most destructive variant CE-min-max [76] is adopted to implement Topology attack. But CE-min-max cannot converge on Polblogs dataset, we adopt another variant called CE-PGD [76] on this dataset. ### 7.2 Analysis on Attacked Graph One way to understand the behaviors of attacking methods is to compare the properties of the clean graph and the attacked graph. In this subsection, we perform this analysis from both global and local perspectives. Global Measure. We have collected five global properties from both clean graphs and perturbed graphs generated by the three attacks on the three datasets. These properties include the number of added edges, the number of deleted edges, the number of edges, the rank of the adjacent matrix, and clustering coefficient. We only show the results of Metattack in Table LABEL:tab:pro-meta. Results for Topology attack and DICE can be found in Appendix A.1. Note that we vary the perturbation rates from $0$ to $25\%$ with a step of $5\%$ and $0\%$ perturbation denotes the original clean graph. It can be observed from the table: * • Attackers favor adding edges over deleting edges. * • Attacks are likely to increase the rank of the adjacency matrix. * • Attacks are likely to reduce the connectivity of a graph. The clustering coefficients of a perturbed graph decrease with the increase of the perturbation rate. Table 3: Properties of attacked graphs under Metattack. Note that $r$ denotes perturbation rate and $0\%$ perturbation indicates the original clean graph. Dataset | $r$(%) | edge+ | edge- | edges | ranks | | clustering --- coefficients Cora | 0 | 0 | 0 | 5069 | 2192 | 0.2376 5 | 226 | 27 | 5268 | 2263 | 0.2228 10 | 408 | 98 | 5380 | 2278 | 0.2132 15 | 604 | 156 | 5518 | 2300 | 0.2071 20 | 788 | 245 | 5633 | 2305 | 0.1983 25 | 981 | 287 | 5763 | 2321 | 0.1943 Citeseer | 0 | 0 | 0 | 3668 | 1778 | 0.1711 5 | 181 | 2 | 3847 | 1850 | 0.1616 1 | 341 | 25 | 3985 | 1874 | 0.1565 15 | 485 | 65 | 4089 | 1890 | 0.1523 20 | 614 | 119 | 4164 | 1902 | 0.1483 25 | 743 | 174 | 4236 | 1888 | 0.1467 Polblogs | 0 | 0 | 0 | 16714 | 1060 | 0.3203 5 | 732 | 103 | 17343 | 1133 | 0.2719 10 | 1347 | 324 | 17737 | 1170 | 0.2825 15 | 1915 | 592 | 18038 | 1193 | 0.2851 20 | 2304 | 1038 | 17980 | 1193 | 0.2877 25 | 2500 | 1678 | 17536 | 1197 | 0.2723 Local Measure. We have also studied two local properties including the feature similarity and label equality between two nodes connected by three kinds of edges: the newly added edges, the deleted edges and the normal edges which have not been changed by the attack methods. Since features are binary in our datasets, we use jaccard similarity as the measure for feature similarity. For label equality, we report the ratio if two nodes share the same label or have different labels. The feature similarity and label equality results are demonstrated in Figures 2 and 3, respectively. We show the results for Metattack with $5\%$ perturbations. Results for Topology attack and DICE can be found in Appendix A.2. Note that we do not have feature similarity results on Polblogs since this dataset does not have node features. We can make the following observations from the figures. * • Attackers tend to connect nodes with dissimilar features and different labels. As shown in Figure 2 and Figure 3, most of the added edges connect nodes with very dissimilar features and different labels. * • Attackers tend to remove edges from nodes which share similar features and same label. As shown in Figure 2 and Figure 3, most of the deleted edges originally connect nodes sharing similar features and same labels. (a) Cora (b) Citeseer Figure 2: Node feature similarity for Metattack. (a) Cora (b) Citeseer (c) Polblogs Figure 3: Label equality for Metattack. ### 7.3 Attack and Defense Performance In this subsection, we study how the attack methods perform and whether the defense methods can help resist to attacks. Similarly, we vary the perturbation rates from $0$ to $25\%$ with a step of $5\%$. The results are demonstrated in Table 4. We show the performance for Metattack. Results for Topology attack and DICE are shown in Appendix A.3. Note that we do not have the performance for Jaccard defense model in Polblogs since this mode requires node features and Polblogs does not provide node features. According to the results, we have the following observations: * • With the increase of the perturbations, the performance of GCN dramatically deceases. This result suggests that Metattack can lead to a significant reduce of accuracy on the GCN model. * • When the perturbations are small, we observe small performance reduction for defense methods which suggests their effectiveness. However, when the graphs are heavily poisoned, their performance also reduces significantly which indicates that efforts are needed to defend heavily poisoning attacks. Table 4: Performance (accuracy) under Metattack Dataset | $r$ (%) | 0 | 5 | 10 | 15 | 20 | 25 ---|---|---|---|---|---|---|--- Cora | GCN | 83.10 | 76.69 | 65.58 | 54.88 | 48.66 | 38.44 Jaccard111 | 82.39 | 81.02 | 77.28 | 72.74 | 69.16 | 64.56 SVD222 | 77.97 | 75.67 | 70.51 | 64.34 | 55.89 | 45.92 RGCN | 84.81 | 81.32 | 72.12 | 60.25 | 49.75 | 37.76 GAT | 81.69 | 74.75 | 61.69 | 52.56 | 45.30 | 38.52 Citeseer | GCN | 74.53 | 72.59 | 63.96 | 61.66 | 50.58 | 44.32 Jaccard111 | 74.82 | 73.60 | 73.50 | 72.80 | 72.97 | 72.53 SVD222 | 70.32 | 71.30 | 67.58 | 63.86 | 56.91 | 45.28 RGCN | 74.41 | 72.68 | 71.15 | 69.38 | 67.93 | 67.24 GAT | 74.23 | 72.01 | 67.12 | 57.70 | 47.97 | 38.70 Polblogs | GCN | 95.80 | 73.93 | 72.07 | 67.69 | 62.29 | 52.97 SVD222 | 94.99 | 82.64 | 71.27 | 66.09 | 61.37 | 52.82 RGCN | 95.60 | 72.01 | 67.12 | 57.70 | 47.97 | 38.70 GAT | 95.40 | 84.83 | 77.03 | 69.94 | 53.62 | 53.76 * 1 Jaccard: GCN-Jaccard defense model. * 2 SVD: GCN-SVD defense model. ## 8 Conclusion & Future Directions In this survey, we give a comprehensive overview of an emerging research field, adversarial attacks and defenses on graph data. We investigate the taxonomy of graph adversarial attacks, and review representative adversarial attacks and the corresponding countermeasures. Furthermore, we conduct empirical study to show how different defense methods behave under different attacks, as well as the changes in important graph properties by the attacks. Via this comprehensive study, we have gained deep understandings on this area that enables us to discuss some promising research directions. * • Imperceptible perturbation measure. Different from image data, humans cannot easily tell whether a perturbation on graph is imperceptible or not. The $\ell_{0}$ norm constraint on perturbation is definitely not enough. Currently only very few existing work study this problem, thus finding concise perturbation evaluation measure is of great urgency. * • Different graph data. Existing works mainly focus on static graphs with node attributes. 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Adversarial attacks on graph neural networks via meta learning. arXiv preprint arXiv:1902.08412, 2019. * [92] D. Zügner and S. Günnemann. Certifiable robustness and robust training for graph convolutional networks. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 246–256, 2019. * [93] D. Zügner and S. Günnemann. Certifiable robustness of graph convolutional networks under structure perturbations. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 1656–1665, 2020. ## Appendix A Additional Results ### A.1 Global Measures for Topology Attack and DICE The global measures for Topology attack are shown in Table LABEL:tab:pro-top; the global measures for DICE are shown in Table LABEL:tab:pro-dice. Table 5: Properties of attacked graphs under Topology attack. Dataset | $r$(%) | edges+ | edges- | edges | ranks | | clustering --- coefficients Cora | 0 | 0 | 0 | 5069 | 2192 | 0.2376 5 | 255 | 0 | 5324 | 2292 | 0.2308 10 | 508 | 0 | 5577 | 2369 | 0.2185 15 | 762 | 0 | 5831 | 2417 | 0.2029 20 | 1015 | 0 | 6084 | 2442 | 0.1875 25 | 1269 | 0 | 6338 | 2456 | 0.1736 Citeseer | 0 | 0 | 0 | 3668 | 1778 | 0.1711 5 | 185 | 0 | 3853 | 1914 | 0.1666 10 | 368 | 0 | 4036 | 2003 | 0.1568 15 | 552 | 0 | 4220 | 2058 | 0.1429 20 | 735 | 0 | 4403 | 2077 | 0.1306 25 | 918 | 0 | 4586 | 2087 | 0.1188 Polblogs | 0 | 0 | 0 | 16714 | 1060 | 0.3203 5 | 716 | 96 | 17334 | 1213 | 0.2659 10 | 1532 | 128 | 18118 | 1220 | 0.2513 15 | 2320 | 146 | 18887 | 1221 | 0.2408 20 | 3149 | 155 | 19708 | 1221 | 0.2317 25 | 3958 | 163 | 20509 | 1221 | 0.2238 Table 6: Properties of attacked graphs under DICE attack. Dataset | $r$(%) | edge+ | edge- | edges | ranks | | clustering --- coefficients Cora | 0 | 0 | 0 | 5069 | 2192 | 0.2376 5 | 125 | 128 | 5066 | 2208 | 0.2165 10 | 251 | 255 | 5065 | 2241 | 0.1963 15 | 377 | 383 | 5063 | 2256 | 0.1768 20 | 504 | 509 | 5063 | 2262 | 0.1588 25 | 625 | 642 | 5053 | 2271 | 0.1436 Citeseer | 0 | 0 | 0 | 3668 | 1798 | 0.1711 5 | 91 | 92 | 3667 | 1829 | 0.1574 10 | 183 | 183 | 3668 | 1843 | 0.1404 15 | 276 | 274 | 3670 | 1858 | 0.1279 20 | 368 | 365 | 3672 | 1872 | 0.1170 25 | 462 | 455 | 36755 | 1871 | 0.1068 Polblogs | 0 | 0 | 0 | 16714 | 1060 | 0.3203 5 | 420 | 415 | 16719 | 1151 | 0.2742 10 | 846 | 825 | 16736 | 1191 | 0.2341 15 | 1273 | 1234 | 16752 | 1206 | 0.2077 20 | 1690 | 1652 | 16752 | 1214 | 0.1862 25 | 2114 | 2064 | 16765 | 1216 | 0.1675 ### A.2 Local Measures for Topology Attack and DICE The node feature similarity and label equality for Topology attack are shown in Figure 4 and Figure 6, respectively; the node feature similarity and label equality for DICE are shown in Figure 5 and Figure 7, respectively. (a) Cora (b) Citeseer Figure 4: Node feature similarity for Topology attack. (a) Cora (b) Citeseer Figure 5: Node feature similarity for DICE attack. (a) Cora (b) Citeseer (c) Polblogs Figure 6: Label equality for Topology attack. (a) Cora (b) Citeseer (c) Polblogs Figure 7: Label equality for DICE attack. ### A.3 Attack and Defense Performance for Topology Attack and DICE The attack and defense performance for DICE is shown in Table 7; the performance for DICE is shown in Table 8. Table 7: Performance (accuracy) under Topology attack. Dataset | $r$ (%) | 0 | 5 | 10 | 15 | 20 | 25 ---|---|---|---|---|---|---|--- Cora | GCN | 83.10 | 71.82 | 68.96 | 66.77 | 64.21 | 62.52 Jaccard111 | 82.39 | 73.05 | 72.62 | 71.84 | 71.41 | 70.85 SVD222 | 77.97 | 78.17 | 75.92 | 73.69 | 72.03 | 70.11 RGCN | 84.81 | 72.68 | 71.15 | 69.38 | 67.92 | 67.23 GAT | 81.69 | 71.03 | 68.80 | 65.66 | 64.29 | 62.58 Citeseer | GCN | 74.53 | 79.29 | 75.47 | 72.89 | 70.12 | 68.49 Jaccard111 | 74.82 | 79.07 | 76.76 | 74.29 | 71.87 | 69.55 SVD222 | 70.32 | 78.17 | 75.92 | 73.69 | 72.03 | 70.11 RGCN | 74.41 | 78.13 | 75.93 | 73.93 | 72.32 | 70.60 GAT | 74.23 | 77.52 | 74.09 | 71.90 | 69.62 | 66.99 Polblogs | GCN | 95.80 | 72.04 | 65.87 | 63.35 | 61.06 | 58.49 SVD222 | 94.99 | 71.90 | 65.42 | 63.01 | 60.74 | 58.26 RGCN | 95.60 | 71.27 | 65.30 | 62.76 | 60.25 | 57.89 GAT | 95.40 | 72.56 | 65.97 | 63.35 | 60.94 | 58.77 * 1 Jaccard: GCN-Jaccard defense model. * 2 SVD: GCN-SVD defense model. Table 8: Performance (accuracy) under DICE attack. Dataset | $r$ (%) | 0 | 5 | 10 | 15 | 20 | 25 ---|---|---|---|---|---|---|--- Cora | GCN | 83.10 | 81.56 | 80.28 | 78.64 | 77.10 | 75.17 Jaccard111 | 82.39 | 80.91 | 79.80 | 79.23 | 77.81 | 76.35 SVD222 | 77.97 | 75.24 | 73.33 | 70.92 | 69.47 | 67.29 RGCN | 84.81 | 83.53 | 82.56 | 80.70 | 79.30 | 77.89 GAT | 81.69 | 78.86 | 77.50 | 74.56 | 70.54 | 69.25 Citeseer | GCN | 74.53 | 74.41 | 73.61 | 71.98 | 71.47 | 69.92 Jaccard111 | 74.82 | 74.73 | 74.11 | 72.88 | 72.36 | 71.32 SVD222 | 70.32 | 70.68 | 69.89 | 68.59 | 67.09 | 66.65 RGCN | 74.41 | 74.46 | 73.93 | 72.37 | 71.61 | 70.85 GAT | 74.23 | 73.70 | 72.71 | 70.52 | 69.27 | 67.78 Polblogs | GCN | 95.80 | 89.90 | 85.87 | 82.41 | 79.47 | 78.02 SVD222 | 94.99 | 90.88 | 87.79 | 85.09 | 83.64 | 81.25 RGCN | 95.60 | 89.86 | 86.11 | 82.25 | 78.81 | 77.18 GAT | 95.40 | 90.74 | 86.22 | 82.41 | 78.83 | 76.77 * 1 Jaccard: GCN-Jaccard defense model. * 2 SVD: GCN-SVD defense model. ## Appendix B Open Source Code We list some open source implementations of representative algorithms in Table 9. Table 9: A summary of open-source implementations | Methods | Framework | Github Link ---|---|---|--- Attack | PGD, Min-max [76] | | tensorflow --- pytorch | https://github.com/KaidiXu/GCN_ADV_Train --- https://github.com/DSE-MSU/DeepRobust DICE [71] | python | https://github.com/DSE-MSU/DeepRobust FGA [13] | pytorch | https://github.com/DSE-MSU/DeepRobust IG-FGSM [72] | pytorch | https://github.com/DSE-MSU/DeepRobust Nettack [89] | tensorflow | https://github.com/danielzuegner/nettack Metattack [91] | | tensorflow --- pytorch | https://github.com/danielzuegner/gnn-meta-attack --- https://github.com/ChandlerBang/pytorch-gnn-meta-attack RL-S2V [17] | pytorch | https://github.com/Hanjun-Dai/graph_adversarial_attack Bojchevski et al. [5] | tensorflow | https://github.com/abojchevski/node_embedding_attack GF-Attack [9] | tensoflow | https://github.com/SwiftieH/GFAttack Defense | RGCN [87] | | tensorflow --- pytorch | https://github.com/thumanlab/nrlweb/blob/master/static/ --- assets/download/RGCN.zip https://github.com/DSE-MSU/DeepRobust GCN-Jaccard [72] | pytorch | https://github.com/DSE-MSU/DeepRobust GCN-SVD [21] | pytorch | https://github.com/DSE-MSU/DeepRobust Pro-GNN [33] | pytorch | https://github.com/ChandlerBang/Pro-GNN | Adversarial --- Training [76] | tensorflow --- pytorch | https://github.com/KaidiXu/GCN_ADV_Train --- https://github.com/DSE-MSU/DeepRobust PA-GNN [60] | tensorflow | https://github.com/tangxianfeng/PA-GNN Graph-Cert [6] | python | https://github.com/abojchevski/graph_cert DefenseVGAE [80] | pytorch | https://github.com/zhangao520/defense-vgae Bojchevski et al. [7] | pytorch | https://github.com/abojchevski/sparse_smoothing Zügner et al. [93] | python | https://github.com/danielzuegner/robust-gcn-structure

2024-09-04T02:54:56.170815

2003.00761

arxiv-papers

# $5$-rank of ambiguous class groups of quintic Kummer extensions Fouad ELMOUHIB Mohamed TALBI Abdelmalek AZIZI ###### Abstract Let $k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_{5})$, where $n$ is a positive integer $5^{th}$ power-free, whose $5-$class group denoted $C_{k,5}$ is isomorphic to $\mathbb{Z}/5\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$. Let $k_{0}\,=\,\mathbb{Q}(\zeta_{5})$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $\zeta_{5}$. Let $C_{k,5}^{(\sigma)}$ the group of the ambiguous classes under the action of $Gal(k/k_{0})$ = $\langle\sigma\rangle$. The aim of this paper is to determine all integers $n$ such that the group of ambiguous classes $C_{k,5}^{(\sigma)}$ has rank $1$ or $2$. ## 1 Introduction One of the most important problems in number theory is the determination of the structure of class group of a number field, particularly its rank. The case of quadratic fields, Gauss’s genus theory, determines the rank of $2$-class group. In a series of papers ( see [References], [References], [References]), Frank Gerth III proved several results on the $3$-class groups of pure cubic extension of $\mathbb{Q}$ and cyclic cubic extension of $\mathbb{Q}$. Recently, S.Aouissi, M.C.Ismaili, M.Talbi, A.Azizi in [References] had classified all fields $\mathbb{Q}(\sqrt[3]{n},\zeta_{3})$ whose $3-$class group is of type $(9,3)$. Let $k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_{5})$, a number of researchers have studied the $5$-class group $C_{k,5}$. M.Kulkarni, D.Majumdar and B.Sury in [References] proved some results that are seen as generalisation of Gerth’s work to the case of any odd prime, and they give more details in case of $5$-class group of $k$. In [References], C.Parry gives a formula between class numbers of pure quintic field $\mathbb{Q}(\sqrt[5]{n})$ and its normal closure $k$. In References H.Kobayashi proved that if the radicand $n$ has a prime factor $p$ congruent to $-1$ modulo $5$, then the class number of the pure quintic field $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$ is a multiple of $5$, and the class number of $k$ is multiple of $25$. Let $n>1$ be a $5^{th}$ power-free integer and $k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_{5})$ be a quintic Kummer extension of the cyclotomic field $k_{0}\,=\,\mathbb{Q}(\zeta_{5})$. By $C_{k,5}^{(\sigma)}$ we denote the $5$-group of ambiguous ideal classes under the action of $Gal(k/k_{0})$ = $\langle\sigma\rangle$, i.e $C_{k,5}^{(\sigma)}\,=\,\\{\mathcal{A}\,\in\,C_{k,5}|\,\mathcal{A}^{\sigma}\,=\,\mathcal{A}\\}$. Let $k^{\ast}\,=\,(k/k_{0})^{\ast}$ be the maximal abelian extension of $k_{0}$ contained in the Hilbert $5$-class field $k_{5}^{(1)}$ of $k$, which is called the relative $5$-genus field of $k/k_{0}$. We consider the problem of finding the radicands $n$ of all pure quintic fields $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$, for which the Galois group $\operatorname{Gal}(k^{\ast}/k)$ is non-trivial. The present work gives the complete solution of this problem by characterizing all quintic Kummer extensions $k/k_{0}$ with $5$-group of ambiguous ideal classes $C_{k,5}^{(\sigma)}$ of order $5$ or $25$. This paper can be viewed as the continuation of the work of M.Kulkarni, D.Majumdar and B.Sury in [References]. In fact, we shall prove the following Main Theorem: ###### Theorem 1.1. Let $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n>1$ is a $5^{th}$ power-free integer, and $k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_{5})$ be its normal closure. We assume that the $5-$class group $C_{k,5}$ is of type $(5,5)$. (1) If rank $(C_{k,5}^{(\sigma)})\,=\,1$, then the integer $n$ can be written in one of the following forms: $n\,=\,\left\\{\begin{array}[]{ll}5^{e}q_{1}^{2}q_{2}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)&\text{ with }\quad q_{1}\,\text{ or }\,q_{2}$ $\not\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\ 5^{e}p\not\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,25)&\text{ with }\quad p\,\not\equiv\,-1\,(\mathrm{mod}\,25)\\\ 5^{e}q_{1}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)&\text{ with }\quad q_{1}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\ p^{e}q_{1}\,\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,25)&\text{ with }\quad p\,\not\equiv\,-1\,(\mathrm{mod}\,25),\,q_{1}\,\not\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\ p^{e}\,\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,25)&\text{ with }\quad p\,\equiv\,-1\,(\mathrm{mod}\,25)\\\ q_{1}^{e_{1}}q_{2}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)&\text{ with }\quad q_{i}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)\par\end{array}\right.$ (1) where $p\,\equiv\,-1\,(\mathrm{mod}\,5)$ and $q_{1},q_{2}\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$ are primes and $e,e_{1}$ are integers in $\\{1,2,3,4\\}$. (2) If rank $(C_{k,5}^{(\sigma)})\,=\,2$, then the integer $n$ can be written in one of the following forms: $n\,=\,\left\\{\begin{array}[]{ll}5^{e}l\not\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,25)&\quad\text{ with }\quad l\,\not\equiv\,1\,(\mathrm{mod}\,25),\\\ l^{e_{1}}q_{1}\,\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,25)&\quad\text{ with }\quad l\,\equiv\,1\,(\mathrm{mod}\,5),\,q_{1}\,\equiv\,\pm 2,\pm 7,\pm 3\,(\mathrm{mod}\,25)\\\ l^{e_{1}}\,\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,25)&\quad\text{ with }\quad l\,\equiv\,1\,(\mathrm{mod}\,25),\\\ \end{array}\right.$ (2) where $l\,\equiv\,1\,(\mathrm{mod}\,5)$ and $q_{1}\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$ are primes and $e,e_{1}$ are integers in $\\{1,2,3,4\\}$. This result will be underpinned by numerical examples obtained with the computational number theory system PARI/GP [References] in section 3. Notations. Throughout this paper, we use the following notations: * • The lower case letter $p$,$q$ and $l$ will denote a prime numbers such that, $p\,\equiv\,-1\,(\mathrm{mod}\,5)$, $q\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$ and $l\,\equiv\,1\,(\mathrm{mod}\,5)$. * • $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$: a pure quintic field, where $n\neq 1$ is a $5^{th}$ power-free positive integer. * • $k_{0}\,=\,\mathbb{Q}(\zeta_{5})$, the cyclotomic field, where $\zeta_{5}\,=\,e^{\frac{2i\pi}{5}}$ a primitive $5^{th}$ root of unity. * • $k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_{5})$: the normal closure of $\Gamma$, a quintic Kummer extension of $k_{0}$. * • $\Gamma^{{}^{\prime}},\,\Gamma^{{}^{\prime\prime}},\,\Gamma^{{}^{\prime\prime\prime}},\,\Gamma^{{}^{\prime\prime\prime\prime}},\,$ the four conjugates quintic fields of $\Gamma$, contained in $k$. * • $\langle\tau\rangle\,=\,\operatorname{Gal}(k/\Gamma)$ such that $\tau^{4}\,=\,id,\,\tau^{3}(\zeta_{5})\,=\,\zeta_{5}^{3},\,\tau^{2}(\zeta_{5})\,=\,\zeta_{5}^{4},\,\tau(\zeta_{5})\,=\,\zeta_{5}^{2}$ and $\tau(\sqrt[5]{n})\,=\,\sqrt[5]{n}$. * • $\langle\sigma\rangle\,=\,\operatorname{Gal}(k/k_{0})$ such that $\sigma^{5}\,=\,id,\ \sigma(\zeta_{5})\,=\,\zeta_{5}$ and $\sigma(\sqrt[5]{n})\,=\,\zeta_{5}\sqrt[5]{n},\,\sigma^{2}(\sqrt[5]{n})\,=\,\zeta_{5}^{2}\sqrt[5]{n},\,\\\ \\\ \sigma^{3}(\sqrt[5]{n})\,=\,\zeta_{5}^{3}\sqrt[5]{n},\,\sigma^{4}(\sqrt[5]{n})\,=\,\zeta_{5}^{4}\sqrt[5]{n}$. * • $\lambda\,=\,1-\zeta_{5}$ is prime element above $5$ of $k_{0}$. * • $q^{\ast}\,=\,0,\,1$ or $2$ according to whether $\zeta_{5}$ and $1+\zeta_{5}$ is not norm or is norm of an element of $k^{*}\,=\,k\setminus\\{0\\}$. * • $d$: the number of prime ideals of $k_{0}$ ramified in $k$. * • For a number field $L$, denote by: * – $\mathcal{O}_{L}$: the ring of integers of $L$; * – $E_{L}$: the group of units of $L$; * – $C_{L}$, $h_{L}$, $C_{L,5}$: the class group, class number, and $5$-class group of $L$. * – $L_{5}^{(1)},L^{\ast}$: the Hilbert $5$-class field of $L$, and the absolute genus field of $L$. $\mathbf{k}$$\mathbf{\Gamma}$ $\mathbf{\Gamma^{\prime}}$ $\mathbf{\Gamma^{\prime\prime}}$ $\mathbf{\Gamma^{\prime\prime\prime}}$ $\mathbf{\Gamma^{\prime\prime\prime\prime}}$ $\mathbf{k_{0}}$$\mathbb{Q}$ Figure 15445 ## 2 Proof of Main Theorem ###### Theorem 2.1. (Decompositon in cyclotomic fields) Let $m$ a positive integer and $p$ a prime number. Suppose $p$ do not divides $m$, and let $f$ be the smallest positive integer such that $p^{f}\,\equiv\,1\,(\mathrm{mod}\,m)$. Then $p$ splits into $\phi(m)/f$ distinct primes in $\mathbb{Q}(\zeta_{m})$ each of which has a residue class degree $f$. In particular, p splits completely if and only if $p\,\equiv\,1\,(\mathrm{mod}\,m)$ ###### Proof. see [References] page 14. ∎ ###### Corollary 2.1. Let $p$ a prime integer, we have : If $p\,=\,5$, then $\lambda\,=\,1-\zeta_{5}$ is the unique prime over 5 in $\mathbb{Q}(\zeta_{5})$. If $l\,\equiv\,1\,(\mathrm{mod}\,5)$, then $l$ splits completely in $\mathbb{Q}(\zeta_{5})$ as $l\,=\,\pi_{1}\pi_{2}\pi_{3}\pi_{4}$, with $\pi_{i}$ are primes in $\mathbb{Q}(\zeta_{5})$ If $q\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$, then $q$ is inert in $\mathbb{Q}(\zeta_{5})$. If $p\,\equiv\,-1\,(\mathrm{mod}\,5)$, then $p$ splits in $\mathbb{Q}(\zeta_{5})$ as $p\,=\,\pi_{1}\pi_{2}$, with $\pi_{i}$ are primes in $\mathbb{Q}(\zeta_{5})$. Before proving the main theorem, we give a proof of the existance of a unique prime $l\,\equiv\,1\,(\mathrm{mod}\,5)$ divides the radicand $n$ in the case of rank $(C_{k,5}^{(\sigma)})\,=\,2$ ###### Theorem 2.2. If rank $C_{k,5}^{(\sigma)}=2$, so $C_{k,5}\,=\,C_{k,5}^{(\sigma)}$, and there is unique prime $l\,\equiv\,1\,(\mathrm{mod}\,5)$ that divides the radicand $n$. Furthermore, we have $(k/k_{0})^{*}\,=\,k_{5}^{(1)}$. ###### Proof. If rank $(C_{k,5}^{(\sigma)})=2$ so the order of $C_{k,5}^{(\sigma)}$ is at least $25$, since $C_{k,5}^{(\sigma)}\subseteq C_{k,5}$ and $|C_{k,5}|\,=\,25$ because $C_{k,5}$ is of type $(5,5)$ we have $C_{k,5}\,=\,C_{k,5}^{(\sigma)}$ it means that all ideal classes are ambiguous. By class fields theory $C_{k,5}^{1-\sigma}$ correspond to $(k/k_{0})^{*}$ and $Gal(k_{5}^{(1)}/k)\cong C_{k,5}$. Since $C_{k,5}^{(\sigma)}\,=\,C_{k,5}$, we get $C_{k,5}^{1-\sigma}\,=\,\\{1\\}$, hence $(k/k_{0})^{*}\,=\,k_{5}^{(1)}$, and by [References, proposition 5.8] we know explicitly in this case the Hilbert $5-$class field of $k$. We assume now that there is no prime $l\,\equiv\,1\,(\mathrm{mod}\,5)$ divides $n$. We can write $n$ as $n\,=\,5^{e}q_{1}^{f_{1}}...q_{r}^{f_{r}}p_{1}^{g_{1}}....p_{s}^{g_{s}}$ with $q_{i}\,\equiv\,\pm 2\,(mod\,5)$ and $p_{j}\,\equiv\,-1\,(mod\,5)$, $f_{i}\,=\,1,2,3,4$ for $1\leq i\leq r$ and $g_{j}\,=\,1,2,3,4$ for $1\leq j\leq s$, and $e\,=\,0,1,2,3,4$, by Corollary 2.1 each $q_{i}$ is inert in $k_{0}$, and $q_{i}$ is ramified in $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$, also by Corollary 2.1 $p_{j}$ splits in $k_{0}$ as $p_{j}\,=\,\pi_{1}\pi_{2}$, where $\pi_{1},\pi_{2}$ are primes in $k_{0}$, and $p_{j}$ is ramified in $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$, so the prime ideals ramified in $k/k_{0}$ are those above $q_{i}$ and $\pi_{j}$ and the ideal above $\lambda$ with $\lambda\,=\,1-\zeta_{5}$ if 5 is ramified in $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$. If $\lambda$ is ramified in $k/k_{0}$, we note $\mathcal{I}$ the prime ideal in $k$ above $\lambda$, and for $1\leq i\leq r$ ,$\mathcal{Q}_{i}$ the prime ideal above $q_{i}$ in $k$, and for $1\leq j\leq s$ ,$\mathcal{P}_{j}$ the prime ideal above $\pi_{j}$ in $k$, with $\pi_{j}$ is prime of $k_{0}$ above $p_{j}$. We have evidently $\mathcal{I}^{5}\,=\,(\lambda)$, $\mathcal{Q}_{i}^{5}\,=\,(q_{i})$, $\mathcal{P}_{j}^{5}\,=\,(\pi_{j})$ in $k$. we note by $C_{k,s}^{(\sigma)}$ the group of strong ambiguous adeal classes. We have to treat two cases: (i) $C_{k,s}^{(\sigma)}\,\neq\,C_{k,5}^{(\sigma)}\,=\,C_{k,5}$: Let $C_{k,5}^{+}\,=\,\\{\mathcal{A}\in\,C_{k,5}|\mathcal{A}^{\tau^{2}}\,=\,\mathcal{A}\\}$ and $C_{k,5}^{-}\,=\,\\{\mathcal{A}\in\,C_{k,5}|\mathcal{A}^{\tau^{2}}\,=\,\mathcal{A}^{-1}\\}$ be a nontrivial subgoups of $C_{k,5}$. We have $(C_{k,5}^{(\sigma)})^{+}\,=\,C_{k,5}^{+}$, by [References, Lemma 6.2] $C_{k,5}^{+}\,\simeq\,C_{\Gamma,5}$ i.e $C_{k,5}^{+}$ can be generated by $5-$class comes from $\Gamma$ ($|C_{k,5}^{+}|\,=\,5$). The strong ambiguous classes are those of primes ramified in $k/k_{0}$, namly $[\mathcal{Q}_{i}]$ for $1\leq i\leq r$, $[\mathcal{P}_{j}]$ for $1\leq j\leq s$ and $[\mathcal{I}]$ if $\lambda$ is ramified in $k/k_{0}$. Its easy to see that: $[\mathcal{Q}_{i}^{\tau^{2}}]\,=\,[\mathcal{Q}_{i}]^{\tau^{2}}\,=\,[\mathcal{Q}_{i}]$ and $[\mathcal{P}_{j}^{\tau^{2}}]\,=\,[\mathcal{P}_{j}]^{\tau^{2}}\,=\,[\mathcal{P}_{j}]$ and $[\mathcal{I}^{\tau^{2}}]\,=\,[\mathcal{I}]^{\tau^{2}}\,=\,[\mathcal{I}]$, we now that $C_{k,5}/C_{k,s}^{(\sigma)}$ is generated by image in $C_{k,5}/C_{k,s}^{(\sigma)}$ of element in $C_{k,5}^{+}$. Since $C_{k,s}^{(\sigma)}$ is generated by $[\mathcal{Q}_{i}],[\mathcal{P}_{j}]$ and $[\mathcal{I}]$ if $\lambda$ is ramified in $k/k_{0}$, all elements of $C_{k,5}$ will be fixed by $\tau^{2}$, in particular whose of $C_{k,5}^{-}$, therefore $\forall\mathcal{A}\in C_{k,5}^{-}$, $\mathcal{A}^{\tau^{2}}\,=\,\mathcal{A}^{-1}\,=\,\mathcal{A}$ i.e $\mathcal{A}^{2}\,=\,1$ i.e $\mathcal{A}^{4}\,=\,1$, hence $\mathcal{A}\,=\,1$ because $\mathcal{A}$ is $5-$class, so we get $C_{k,5}^{-}\,=\,{1}$, and this contradict the fact that $|C_{k,5}^{-}|=5$. (ii) $C_{k,s}^{(\sigma)}\,=\,C_{k,5}^{(\sigma)}\,=\,C_{k,5}$: In this case $C_{k,5}$ will be generated by $[\mathcal{Q}_{i}]$, $[\mathcal{P}_{j}]$ and $[\mathcal{I}]$ if $\lambda$ is ramifed in $k/k_{0}$, and as in (i) all the classes are fixed by $\tau^{2}$, which give the same contradiction. Thus we proved the existence of a prime $l$ divides $n$ such that $l\,\equiv\,1\,(\mathrm{mod}\,5)$. According to [References,section 5.1], we have rank $C_{k,5}^{(\sigma)}\,=\,d-3+q^{*}$ where $d$ is the number of prime ramified in $k/k_{0}$ and $q^{*}$ is an index has as value 0,1 or 2. Assuming that there is two prime $l_{1}$ and $l_{2}$ divides $n$ such that $l_{i}\,\equiv\,1\,(\mathrm{mod}\,5)$, $(i=1,2)$, then $d\geq 8$ and rank $C_{k,5}^{(\sigma)}$ is at least $5$, that is impossible. Thus if rank $C_{k,5}^{(\sigma)}\,=\,2$ its exsite unique prime $l\,\equiv\,1\,(\mathrm{mod}\,5)$ divides $n$. ∎ ### 2.1 Proof of Theoreme 1.1 Let $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n\geq 2$ is a $5^{th}$ power-free integer, $k\,=\,\Gamma(\zeta_{5})$ be its normal closure, and $C_{k,5}$ be the $5$-class group of $k$. Let $C_{k,5}^{(\sigma)}$ be the ambigous ideal classes group under the action of $Gal(k/k_{0})\,=\,\langle\sigma\rangle$. Since $k_{0}$ has class number $1$, $C_{k,5}^{(\sigma)}$ is un elementary abelian $5$-group, so rank $(C_{k,5}^{(\sigma)})\,=\,1\,\mathrm{or}\,2$. According to [References,section 5.1], the rank of $C_{k,5}^{(\sigma)}$ is given as follows: rank $(C_{k,5}^{(\sigma)})\,=\,d-3+q^{*}$ where $d$ is the number of prime ideals of $k_{0}$ ramified in $k$, and $q^{\ast}\,=\,0,\,1$ or $2$ according to whether $\zeta_{5}$ and $1+\zeta_{5}$ is not norm or is norm of an element of $k^{*}\,=\,k\setminus\\{0\\}$ as follows: $q^{*}$ = $\begin{cases}2&\text{if }\,\zeta,1+\zeta\in N_{k/k_{0}}(k^{*}),\\\ 1&\text{if }\,\zeta^{i}(1+\zeta)^{j}\in N_{k/k_{0}}(k^{*})\,\text{ for some i and j },\\\ 0&\text{if }\,\zeta^{i}(1+\zeta)^{j}\notin N_{k/k_{0}}(k^{*})\,\text{for}\hskip 5.69054pt0\leq i,j\leq 4\text{ and}\hskip 5.69054pti+j\neq 0.\\\ \end{cases}$ We can writ $n$ as $n\,=\,\mu\lambda^{e}\pi_{1}^{e_{1}}....\pi_{g}^{e_{g}}$, where $\mu$ is a unit in $\mathcal{O}_{k_{0}}$, $\lambda=1-\zeta_{5}$, $\pi_{1},,,,\pi_{g}$ are primes in $k_{0}$ and $e\in\\{0,1,2,3,4\\}$, $e_{i}\in\\{1,2,3,4\\}$ for $1\leq i\leq g$. According to [References, Lemma 5.1] we have, $\zeta_{5}\,\in N_{k/k_{0}}(k^{*})\,\Longleftrightarrow\,N_{k_{0}/\mathbb{Q}}((\pi_{i}))\,\,\equiv\,1\,(\mathrm{mod}\,25)$ for all i, and from [References, proposition 8.2] if $\pi$ is a prime of $\mathcal{O}_{k_{0}}$ over a prime $p\,\in\,\mathbb{Z}$ we have $N_{k_{0}/\mathbb{Q}}((\pi))\,=\,p^{f}$, with $f$ is the least positif integer such that $p^{f}\,\,\equiv\,\,1\,(\mathrm{mod},\,5)$, so we can get that $\zeta_{5}$ is norm of an element of $k^{*}\,=\,k\setminus\\{0\\}$ if and only if $p^{f}\,\,\equiv\,\,1\,(\mathrm{mod},\,25)$ for all prime $p\,\neq\,5$ dividing $n$: 1. $-$ If $l\,\,\equiv\,\,1\,(\mathrm{mod},\,5)$, by corollary 2.1 $l\,=\,\pi_{1}\pi_{2}\pi_{3}\pi_{4}$, we have $N_{k_{0}/\mathbb{Q}}((\pi_{i}))\,=\,l$, so to have $N_{k_{0}/\mathbb{Q}}((\pi_{i}))\,\,\equiv\,\,1\,(\mathrm{mod},\,25)$ the prime $l$ must verify $l\,\,\equiv\,\,1\,(\mathrm{mod}\,25)$. 2. $-$ If $q\,\,\equiv\,\,\pm 2\,(\mathrm{mod},\,5)$ we have $q$ is inert in $k_{0}$, so $N_{k_{0}/\mathbb{Q}}((q))\,=\,q^{4}$, so to have $N_{k_{0}/\mathbb{Q}}((q))\,\,\equiv\,\,1\,(\mathrm{mod},\,25)$ the prime $q$ must verify $q\,\equiv\,\pm 7\,(\mathrm{mod}\,25)$. 3. $-$ If $p\,\,\equiv\,\,-1\,(\mathrm{mod}\,5)$,by corollary 2.1 $p\,=\,\pi_{1}\pi_{2}$ we have $N_{k_{0}/\mathbb{Q}}((\pi))\,=\,p^{2}$, so to have $N_{k_{0}/\mathbb{Q}}((\pi))\,\,\equiv\,\,1\,(\mathrm{mod}\,25)$ the prime $p$ must verify $p\,\,\equiv\,\,-1\,(\mathrm{mod}\,25)$. (1) If rank $(C_{k,5}^{(\sigma)})\,=\,1$, we get that $d+q^{*}\,=\,4$, so there are three possible cases as follows: * • Case 1: $q^{*}=0\,\,\mathrm{and}\,\,d=4$, * • Case 2: $q^{*}=1\,\,\mathrm{and}\,\,d=3$, * • Case 3: $q^{*}=2\,\,\mathrm{and}\,\,d=2$, We will successively treat the three cases to prove the first point of the main theorem. * • Case 1: we have $q^{*}=0$ and $d=4$, so the number of prime ideals which are ramified in $k/k_{0}$ should be $4$. According to the proof of theorem 2.2, if $n$ is divisible by a prime $l\,\equiv\,1\,(\mathrm{mod}\,5)$ then the prime $l$ is unique. \- If $l\,\equiv\,1\,(\mathrm{mod}\,5)$ divides $n$, then by Corollary 2.1, $l\,=\,\pi_{1}\pi_{2}\pi_{3}\pi_{4}$ where $\pi_{i}$ are primes of $k_{0}$. The prime $l$ is ramified in $\Gamma$, because disk$(\Gamma/\mathbb{Q})\,=\,5^{5}n^{4}$ and $l$ divides this discriminent, then $\pi_{1},\pi_{2},\pi_{3}$ and $\pi_{4}$ are ramified in $k$. Hence we have $d=4$, so $l$ is the unique prime divides $n$, because if $n$ is dividing by other prime we obtain $d>4$, which is impossible in the three cases. So $n\,=\,l^{e_{1}}$ with $l\,\equiv\,1\,(\mathrm{mod}\,5)$ and $e_{1}\in\\{1,2,3,4\\}$. According to [References, Lemma 5.1] we have $(\lambda)$ ramifies in $k/k_{0}\,\Longleftrightarrow\,n\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$, with $\lambda\,=\,1-\zeta_{5}$, so in the case $n\,=\,l^{e_{1}}$ where $l\,\equiv\,1\,(\mathrm{mod}\,5)$ we should have $n\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$, so the only $l$ verifiy $n\,=\,l^{e_{1}}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ is $l\,\equiv\,1\,(\mathrm{mod}\,25)$, and for this prime we have $q^{*}\,\geq\,1$, because $\zeta_{5}$ is norme, which is impossible in this case. We note that if $n\,=\,l^{e_{1}}$ with $l\,\equiv\,1\,(\mathrm{mod}\,25)$ and $q*\,=\,2$, there is no fields $k$ of type $(5,5)$, because we have rank$(C_{k,5}^{(\sigma)})\,=\,3$, and if $q^{*}\,=\,1$ we have rank$(C_{k,5}^{(\sigma)})\,=\,2$, which will treat in the second point of the proof. -If no prime $l\,\equiv\,1\,(\mathrm{mod}\,5)$ divides $n$, we have two forms of $n$: 1. $(i)$ $n\,=\,5^{e}p^{e_{1}}q_{1}^{e_{2}}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\equiv\,-1\,(\mathrm{mod}\,5),\,\,q_{1}\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$ and $e,e_{1},e_{2}\in\\{1,2,3,4\\}$. By Corollary 2.1, $p\,=\,\pi_{1}\pi_{2}$, where $\pi_{1},\,\pi_{2}$ are primes in $k_{0}$ and $q_{1}$ is inert in $k_{0}$, the prime $p$ is ramified in $\Gamma$, then $\pi_{1},\,\pi_{2}$ are ramified in $k$. We have $q_{1}$ is ramified in $\Gamma$. Since $n\,\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$, then $\lambda\,=\,1-\zeta_{5}$ is ramified in $k$, so we get $d=4$. To verifiy that $n\,=\,5^{e}p^{e_{1}}q_{1}^{e_{2}}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ we can choose $e_{1}\,=\,2$ and $e_{2}\,=\,1$ because $\mathbb{Q}(\sqrt[5]{ab^{2}c^{3}d^{4}})\,=\,\mathbb{Q}(\sqrt[5]{a^{2}b^{4}cd^{3}})\,=\,\mathbb{Q}(\sqrt[5]{a^{3}bc^{4}d^{2}})\,=\,\mathbb{Q}(\sqrt[5]{a^{4}b^{3}c^{2}d})$, so $n\,=\,5^{e}p^{2}q_{1}$ with $e\in\\{1,2,3,4\\}$. In one hand if $e\,=\,2,3,4$ we have $n\,\equiv\,0\,(\mathrm{mod}\,25)$, in the other hand if $n\,=\,5p^{2}q_{1}$ we have $p^{2}q_{1}\,=\,5\alpha+2$ or $p^{2}q_{1}\,=\,5\alpha^{\prime}+3$ with $\alpha,\alpha^{\prime}\in\mathbb{Z}$, so $5p^{2}q_{1}\,\equiv\,10\,(\mathrm{mod}\,25)$ or $5p^{2}q_{1}\,\equiv\,15\,(\mathrm{mod}\,25)$, therefore we conclude that $n\,\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$. According to [References, Theorem 5.18], if $p\,\equiv\,-1\,(\mathrm{mod}\,25)$ and $q_{1}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)$, rank $(C_{k,5})$ is at least $3$ which contradict the fact that $C_{k,5}$ is of type $(5,5)$, and by the proof of [References, Theorem 5.13, Theorem 5.15], for the other congruence cases of $p$ and $q_{1}$ we have $q^{*}\,=\,1$ which is impossible in this case. 2. $(ii)$ $n\,=\,p^{e}q_{1}^{e_{1}}q_{2}^{e_{2}}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\equiv\,-1\,(\mathrm{mod}\,5),\,\,q_{1}\,\equiv\,2\,(\mathrm{mod}\,5),\,\,\\\ q_{2}\,\equiv\,3\,(\mathrm{mod}\,5)$ and $e,e_{1},e_{2}\in\\{1,2,3,4\\}$. As $\mathbb{Q}(\sqrt[5]{ab^{2}c^{3}d^{4}})\,=\,\mathbb{Q}(\sqrt[5]{a^{2}b^{4}cd^{3}})\,=\,\mathbb{Q}(\sqrt[5]{a^{3}bc^{4}d^{2}})\,=\,\mathbb{Q}(\sqrt[5]{a^{4}b^{3}c^{2}d})$ we can choose $e_{1}\,=\,2,e_{2}\,=\,1$ i.e $n\,=\,p^{e}q_{1}^{2}q_{2}$ with $e\in\\{1,2,3,4\\}$. By Corollary 2.1 $p\,=\,\pi_{1}\pi_{2}$ where $\pi_{1},\,\pi_{2}$ are primes in $k_{0}$ and $q_{1},\,q_{2}$ are inert in $k_{0}$. We have $p,\,q_{1}$ and $q_{2}$ are ramified in $\Gamma$,so $\pi_{1},\,\pi_{2},\,q_{1}$ and $q_{2}$ are ramified in $k$ then $d=4$. The condition $n\,=\,p^{e}q_{1}^{2}q_{2}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ is not verified for all $p\,\equiv\,-1\,(\mathrm{mod}\,5),\,\,q_{1}\,\equiv\,2\,(\mathrm{mod}\,5)$ and $q_{2}\,\equiv\,3\,(\mathrm{mod}\,5)$, so we combine all the cases of congruence and we obtain that $p\,\equiv\,-1\,(\mathrm{mod}\,25),\,\,q_{1}\,\equiv\,12\,(\mathrm{mod}\,25),\,\,q_{2}\,\equiv\,3\,(\mathrm{mod}\,25)$. By [References, Lemma 5.1], since $N_{k_{0}/\mathbb{Q}}(\pi_{i})\,\equiv\,1\,(\mathrm{mod}\,25),\,N_{k_{0}/\mathbb{Q}}(q_{1})\,\not\equiv\,1\,(\mathrm{mod}\,25),N_{k_{0}/\mathbb{Q}}(q_{2})\,\not\equiv\,1\,(\mathrm{mod}\,25)$, we have $q^{*}\,=\,1$ which is impossible in this case. We deduce that in the case 1, there is no radicand $n$ who verifiy rank$(C_{k,5}^{(\sigma)})\,=\,1$. * • Case 2: we have $q^{*}=1$ and $d=3$, so the number of prime ideals which are ramified in $k/k_{0}$ should be $3$. According to case 1, n is not divisible by any prime $l\,\equiv\,1\,(\mathrm{mod}\,5)$ in this case. We can writ $n$ as $n\,=\,\mu\lambda^{e}\pi_{1}^{e_{1}}....\pi_{g}^{e_{g}}$, where $\mu$ is a unit in $\mathcal{O}_{k_{0}}$, $\lambda=1-\zeta_{5}$, $\pi_{1},,,,\pi_{g}$ are primes in $k_{0}$ and $e\in\\{0,1,2,3,4\\}$, $e_{i}\in\\{1,2,3,4\\}$ for $1\leq i\leq g$. By [References, proposition 5.2] $d\,=\,g$ or $g+1$ according to whether $n\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ or $n\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$, to obtain $d=3$, $n$ must be written in $\mathcal{O}_{k_{0}}$ as: $n\,=\,\pi_{1}^{e_{1}}\pi_{2}^{e_{2}}\pi_{3}^{e_{3}}$ or $n\,=\,\lambda^{e}\pi_{1}^{e_{1}}\pi_{2}^{e_{2}}$, therefore we have three forms of $n$: 1. $(i)$ $n\,=\,5^{e}p^{e_{1}}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\equiv\,-1\,(\mathrm{mod}\,5)$ and $e,e_{1}\in\\{1,2,3,4\\}$. As $\mathbb{Q}(\sqrt[5]{ab^{2}c^{3}d^{4}})\,=\,\mathbb{Q}(\sqrt[5]{a^{2}b^{4}cd^{3}})\,=\,\mathbb{Q}(\sqrt[5]{a^{3}bc^{4}d^{2}})\,=\,\mathbb{Q}(\sqrt[5]{a^{4}b^{3}c^{2}d})$ we can choose $e_{1}\,=\,1$ i.e $n\,=\,5^{e}p$ with $e\in\\{1,2,3,4\\}$. By Corollary 2.1 $p\,=\,\pi_{1}\pi_{2}$ with $\pi_{1},\,\pi_{2}$ are primes in $k_{0}$, we have $p$ is ramified in $\Gamma$, so $\pi_{1},\,\pi_{2}$ are ramified in $k$, and since $n\,\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$, according to [References, Lemma 5.1], $\lambda\,=\,1-\zeta_{5}$ is ramified in $k$ so we obtain $d=3$. The condition $n\,\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ is verified for all $p\,\equiv\,-1\,(\mathrm{mod}\,5)$ because, if $e\,=\,2,3,4$ we have $n\,=\,5^{e}p\,\equiv\,0\,(\mathrm{mod}\,25)$, if $e=1$ i.e $n\,=\,5p$ we have $p\,=\,5\alpha+4$ that implie $5p\,=\,25\alpha+20$ with $\alpha\in\mathbb{Z}$, so $n\,=\,5p\,\equiv\,20\,(\mathrm{mod}\,25)$. According to the proof of [References, theorem 5.15] if $p\,\equiv\,-1(\mathrm{mod}\,25)$ we have $q^{*}\,=\,2$, and if $p\,\not\equiv\,-1(\mathrm{mod}\,25)$ we have $q^{*}\,=\,1$, we conclude that $n\,=\,5^{e}p\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\not\equiv\,-1(\mathrm{mod}\,25)$. We note that the computational number theory system PARI [References], show that if $n\,=\,5^{e}p\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\not\equiv\,-1(\mathrm{mod}\,25)$, the field $k$ is not always of type $(5,5)$. 2. $(ii)$ $n\,=\,5^{e}q_{1}^{e_{1}}q_{2}^{e_{2}}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $q_{1}\,\equiv\,2\,(\mathrm{mod}\,5),\,\,q_{2}\,\equiv\,3\,(\mathrm{mod}\,5)$ and $e,e_{1},e_{2}\in\\{1,2,3,4\\}$. As $\mathbb{Q}(\sqrt[5]{ab^{2}c^{3}d^{4}})\,=\,\mathbb{Q}(\sqrt[5]{a^{2}b^{4}cd^{3}})\,=\,\mathbb{Q}(\sqrt[5]{a^{3}bc^{4}d^{2}})\,=\,\mathbb{Q}(\sqrt[5]{a^{4}b^{3}c^{2}d})$, we can choose $e_{1}\,=\,2$ and $e_{2}\,=\,1$ i.e $n\,=\,5^{e}q_{1}^{2}q_{2}$ with $e\in\\{1,2,3,4\\}$. By Corollary 2.1, $q_{1}$ and $q_{2}$ are inert in $k_{0}$ , and $q_{1}$, $q_{2}$ are ramified in $\Gamma$, then $q_{1},\,q_{2}$ are ramified in $k$. Since $n\,\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$, then $\lambda\,=\,1-\zeta_{5}$ is ramified in $k$, so we get $d=3$.The condition $n\,\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ is verified for all $q_{1}\,\equiv\,2\,(\mathrm{mod}\,5),\,\,q_{2}\,\equiv\,3\,(\mathrm{mod}\,5)$, if $e\,=\,2,3,4$ we have $n\,=\,5^{e}q_{1}^{2}q_{2}\,\,\equiv\,0\,(\mathrm{mod}\,25)$, if $n\,=\,5q_{1}^{2}q_{2}$ we have $q_{1}^{2}q_{2}\,=\,5\alpha+2$ with $\alpha,\in\mathbb{Z}$, so $5q_{1}^{2}q_{2}\,\equiv\,10\,(\mathrm{mod}\,25)$. If $q_{1}\,\equiv\,7\,(\mathrm{mod}\,25)$ and $q_{2}\,\equiv\,-7\,(\mathrm{mod}\,25)$ we have $q^{*}\,=\,2$, and if $q_{1}\,\not\equiv\,7\,(\mathrm{mod}\,25)$ or $q_{2}\,\not\equiv\,-7\,(\mathrm{mod}\,25)$, according to the proof of [References, theorem 5.13] we have $q^{*}\,=\,1$, but for this form of the radicand $n$ the computational number theory system PARI [References] show that $C_{k,5}\,\simeq\,\mathbb{Z}/5\mathbb{Z}$. 3. $(iii)$ $n\,=\,p^{e}q_{1}^{e_{1}}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\equiv\,-1\,(\mathrm{mod}\,5),\,\,q_{1}\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$ and $e,e_{1}\in\\{1,2,3,4\\}$. As $\mathbb{Q}(\sqrt[5]{ab^{2}c^{3}d^{4}})\,=\,\mathbb{Q}(\sqrt[5]{a^{2}b^{4}cd^{3}})\,=\,\mathbb{Q}(\sqrt[5]{a^{3}bc^{4}d^{2}})\,=\,\mathbb{Q}(\sqrt[5]{a^{4}b^{3}c^{2}d})$ we can choose $e_{1}\,=\,1$ i.e $n\,=\,p^{e}q_{1}$ with $e\in\\{1,2,3,4\\}$. By Corollary 2.1 $p\,=\,\pi_{1}\pi_{2}$ where $\pi_{1},\,\pi_{2}$ are primes in $k_{0}$ and $q_{2}$ is inert in $k_{0}$. We have $p$ is ramified in $\Gamma$,so $\pi_{1},\,\pi_{2}$ are ramified in $k$. $q_{2}$ is ramified in $\Gamma$ too, we obtain $d=3$. The condition $n\,=\,p^{e}q_{1}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ is not verified for all $p\,\equiv\,-1\,(\mathrm{mod}\,5),\,\,q_{1}\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$, so we combine all the cases of congruence and we obtain that $p\,\not\equiv\,-1\,(\mathrm{mod}\,25)$ and $q_{1}\,\not\equiv\,\pm 7\,(\mathrm{mod}\,25)$. According to [References, theorem 5.18] If $p\,\equiv\,-1\,(\mathrm{mod}\,25)$ and $q_{1}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)$ we have rank $C_{k,5}\geq 3$ which is impossible in our cases. If $p\,\not\equiv\,-1\,(\mathrm{mod}\,25)$ and $q_{1}\,\not\equiv\,\pm 7\,(\mathrm{mod}\,25)$, according to [References, theorem 5.13] we have $q^{*}\,=\,1$. Using the computational number theory system PARI/GP [References], if $n\,=\,p^{e}q_{1}$ with $p\,\not\equiv\,-1\,(\mathrm{mod}\,25)$ and $q_{1}\,\not\equiv\,\pm 7\,(\mathrm{mod}\,25)$, the field $k$ is not always of type $(5,5)$. We summarize all forms of integer $n$ in the case 2, for which $k$ is of type $(5,5)$ and rank $(C_{k,5}^{(\sigma)})\,=\,1$ as follows: $n=\left\\{\begin{array}[]{ll}5^{e}p\not\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,25)&\text{ with }p\,\not\equiv\,-1\,(\mathrm{mod}\,25),\\\ p^{e}q_{1}\,\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,25)&\text{ with }p\,\not\equiv\,-1\,(\mathrm{mod}\,25)\text{ and }q_{1}\,\not\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\ \end{array}\right.$ (3) * • Case 3: we have $q^{*}=2$ and $d=2$, so the number of prime ideals which are ramified in $k/k_{0}$ should be $2$. Let $n\,=\,\mu\lambda^{e}\pi_{1}^{e_{1}}....\pi_{g}^{e_{g}}$, where $\mu$ is a unit in $\mathcal{O}_{k_{0}}$, $\lambda=1-\zeta_{5}$, $\pi_{1},,,,\pi_{g}$ are primes in $k_{0}$ and $e\in\\{0,1,2,3,4\\}$, $e_{i}\in\\{1,2,3,4\\}$ for $1\leq i\leq g$. $d\,=\,g$ or $g+1$ according to whether $n\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ or $n\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$, to obtain $d=2$, $n$ must be written in $\mathcal{O}_{k_{0}}$ as: $n\,=\,\pi_{1}^{e_{1}}\pi_{2}^{e_{2}}$ or $n\,=\,\lambda^{e}\pi_{1}^{e_{1}}$, therefore we have three forms of $n$: 1. $(i)$ $n\,=\,5^{e}q_{1}^{e_{1}}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $q_{1}\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$ and $e,e_{1}\in\\{1,2,3,4\\}$. As $\mathbb{Q}(\sqrt[5]{ab^{2}c^{3}d^{4}})\,=\,\mathbb{Q}(\sqrt[5]{a^{2}b^{4}cd^{3}})\,=\,\mathbb{Q}(\sqrt[5]{a^{3}bc^{4}d^{2}})\,=\,\mathbb{Q}(\sqrt[5]{a^{4}b^{3}c^{2}d})$ we can choose $e_{1}\,=\,1$ i.e $n\,=\,5^{e}q_{1}$ with $e\in\\{1,2,3,4\\}$. Since $q^{*}=2$ so we have $q_{1}\,\,\equiv\,\,\pm 7\,(\mathrm{mod}\,25)$. By Corollary 2.1 $q_{1}$ is inert in $k_{0}$, we have $q_{1}$ is ramified in $\Gamma$, so $q_{1}$ is ramified too in $k$, and since $n\,\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$, according to [References, Lemma 5.1], $\lambda\,=\,1-\zeta_{5}$ is ramified in $k$ so we obtain $d=2$. The condition $n\,\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ is verified for all $q_{1}\,\,\equiv\,\,\pm 7\,(\mathrm{mod}\,5)$, because if $e\,=\,2,3,4$ we have $n\,=\,5^{e}q_{1}\,\equiv\,0\,(\mathrm{mod}\,25)$, if $e=1$ i.e $n\,=\,5q_{1}$ we have $n\,=\,5q_{1}\,\equiv\,\pm 10\,(\mathrm{mod}\,25)$, we conclude that $n\,=\,5^{e}q_{1}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $q_{1}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)$, but for this form of the radicand $n$ the computational number theory system PARI/GP [References] show that $C_{k,5}\,\simeq\,\mathbb{Z}/5\mathbb{Z}$. 2. $(ii)$ $n\,=\,q_{1}^{e_{1}}q_{2}^{e_{2}}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $q_{1}\,\equiv\,3\,(\mathrm{mod}\,5),\,\,q_{2}\,\equiv\,2\,(\mathrm{mod}\,5)$ and $e_{1},e_{2}\in\\{1,2,3,4\\}$. As $\mathbb{Q}(\sqrt[5]{ab^{2}c^{3}d^{4}})\,=\,\mathbb{Q}(\sqrt[5]{a^{2}b^{4}cd^{3}})\,=\,\mathbb{Q}(\sqrt[5]{a^{3}bc^{4}d^{2}})\,=\,\mathbb{Q}(\sqrt[5]{a^{4}b^{3}c^{2}d})$ we can choose $e_{2}\,=\,1$ i.e $n\,=\,q_{1}^{e_{1}}q_{2}$ with $e_{1}\in\\{1,2,3,4\\}$. Since $q^{*}=2$, we have $\zeta_{5}\,\in\,N_{k/k_{0}}(k^{*})$, we get that $q_{1}\,\,\equiv\,\,-7\,(\mathrm{mod}\,25)$ and $q_{2}\,\,\equiv\,\,7\,(\mathrm{mod}\,25)$. By Corollary 2.1 $q_{1}$ and $q_{2}$ are inert in $k_{0}$, and we have $q_{1},\,q_{2}$ are ramified in $\Gamma$,so $q_{1},\,q_{2}$ are ramified in $k$, so we obtain $d=2$. The condition $n\,=\,q_{1}^{e_{1}}q_{2}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ is verified, because we have $n\,=\,q_{1}^{e_{1}}q_{2}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)$ for all $e_{1}$, so we conclude that $n\,=\,q_{1}^{e_{1}}q_{2}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $q_{1}\,\equiv\,-7\,(\mathrm{mod}\,25),\,\,q_{2}\,\equiv\,7\,(\mathrm{mod}\,25)$, but for this form of the radicand $n$ the computational number theory system PARI/GP [References] show that $C_{k,5}\,\simeq\,\mathbb{Z}/5\mathbb{Z}$ 3. $(iii)$ $n\,=\,p^{e}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\equiv\,-1\,(\mathrm{mod}\,5)$ and $e\in\\{1,2,3,4\\}$. Since $q^{*}=2$, we have $\zeta_{5}\,\in\,N_{k/k_{0}}(k^{*})$, we get that $p\,\,\equiv\,\,-1\,(\mathrm{mod}\,25)$. By Corollary 2.1, $p\,=\,\pi_{1}\pi_{2}$ where $\pi_{1},\,\pi_{2}$ are primes of $k_{0}$. The prime $p$ is ramified in $\Gamma$, then $\pi_{1},\pi_{2}$ are ramified in $k$. hence we have $d=2$. The condition $n\,=\,p^{e}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ is verified for all $p\,\,\equiv\,\,-1\,(\mathrm{mod}\,25)$, we conclude that $n\,=\,p^{e}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\equiv\,-1\,(\mathrm{mod}\,25)$. Using the computational number theory system PARI/GP [References], if $n\,=\,p^{e}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\equiv\,-1\,(\mathrm{mod}\,25)$, the field $k$ is not always of type $(5,5)$. We deduce that in the case 3, there is one form of $n$ for which the fields $k$ is of type $(5,5)$ and rank$(C_{k,5}^{(\sigma)})\,=\,1$ as follows: $n\,=\,p^{e}\,\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,25)\text{ with }p\,\equiv\,-1\,(\mathrm{mod}\,25)$ (2) If rank $(C_{k,5}^{(\sigma)})\,=\,2$, so $C_{k,5}\,=\,C_{k,5}^{(\sigma)}$. According to [References,section 5.1], the rank of $C_{k,5}^{(\sigma)}$ is given as follows: rank $(C_{k,5}^{(\sigma)})\,=\,d-3+q^{*}$ where $d$ et $q^{*}$ are defined previously. Since rank $(C_{k,5}^{(\sigma)})\,=\,2$ and $q^{*}\,=\,0,1\,\mathrm{or}\,2$, there are three possible cases as follows: * • Case 1: $q^{*}=0\,\,\mathrm{and}\,\,d=5$, * • Case 2: $q^{*}=1\,\,\mathrm{and}\,\,d=4$, * • Case 3: $q^{*}=2\,\,\mathrm{and}\,\,d=3$, We will treat the three cases to prove the forms of the radicand $n$. By theorem 2.2, $n$ must be divisible by one prime $l\,\equiv\,1\,(\mathrm{mod}\,5)$ in all cases, and since rank $(C_{k,5}^{(\sigma)})\,=\,2$, the invariant $q^{*}$ should be $0$ or $1$, because if $q^{*}\,=\,2$ and $l\,\equiv\,1\,(\mathrm{mod}\,5)$ divides $n$, we get that the invariant $d$ is at least $4$, so we obtain that rank $(C_{k,5}^{(\sigma)})$ is at least $3$. * • Case 1: we have $q^{*}=0$ and $d=5$, so the number of prime ideals which are ramified in $k/k_{0}$ should be $5$. The radicand $n$ must be divisible by one prime $l\,\equiv\,1\,(\mathrm{mod}\,5)$. We can writ $n\,\in\,\mathcal{O}_{k_{0}}$ as $n\,=\,\mu\lambda^{e}\pi_{1}^{e_{1}}....\pi_{g}^{e_{g}}$, where $\mu$ is a unit in $\mathcal{O}_{k_{0}}$, $\lambda=1-\zeta_{5}$, $\pi_{1},,,,\pi_{g}$ are primes in $k_{0}$ and $e\in\\{0,1,2,3,4\\}$, $e_{i}\in\\{1,2,3,4\\}$ for $1\leq i\leq g$. $d\,=\,g$ or $g+1$ according to whether $n\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ or $n\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$. To obtain $d=5$, $n$ must be written in $\mathcal{O}_{k_{0}}$ as: $n\,=\,\pi_{1}^{e_{1}}\pi_{2}^{e_{2}}\pi_{3}^{e_{3}}\pi_{4}^{e_{4}}\pi_{5}^{e_{5}}$ or $n\,=\,\lambda^{e}\pi_{1}^{e_{1}}\pi_{2}^{e_{2}}\pi_{3}^{e_{3}}\pi_{4}^{e_{4}}$, therefore we have two forms of $n$: 1. $(i)$ $n\,=\,5^{e}l^{e_{1}}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $l\,\equiv\,1\,(\mathrm{mod}\,5)$ and $e,e_{1}\in\\{1,2,3,4\\}$. As $\mathbb{Q}(\sqrt[5]{ab^{2}c^{3}d^{4}})\,=\,\mathbb{Q}(\sqrt[5]{a^{2}b^{4}cd^{3}})\,=\,\mathbb{Q}(\sqrt[5]{a^{3}bc^{4}d^{2}})\,=\,\mathbb{Q}(\sqrt[5]{a^{4}b^{3}c^{2}d})$ we can choose $e_{1}\,=\,1$ i.e $n\,=\,5^{e}l$ with $e\in\\{1,2,3,4\\}$. By Corollary 2.1 $l\,=\,\pi_{1}\pi_{2}\pi_{3}\pi_{4}$ with $\pi_{i}$ are primes in $k_{0}$, we have $l$ is ramified in $\Gamma$, so $\pi_{1},\,\pi_{2},\pi_{3}$ and $\pi_{4}$ are ramified in $k$, and since $n\,\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$, according to [References, Lemma 5.1], $\lambda\,=\,1-\zeta_{5}$ is ramified in $k$ so we obtain $d=5$. The condition $n\,\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ is verified for all $l\,\equiv\,1\,(\mathrm{mod}\,5)$, because if $e\,=\,2,3,4$ we have $n\,=\,5^{e}l\,\equiv\,0\,(\mathrm{mod}\,25)$, if $e=1$ i.e $n\,=\,5l$ we have $l\,=\,5\alpha+1$ that implie $5l\,=\,25\alpha+5$ with $\alpha\in\mathbb{Z}$, so $n\,=\,5l\,\equiv\,5\,(\mathrm{mod}\,25)$. If $l\,\equiv\,1(\mathrm{mod}\,25)$ we have $\zeta_{5}\,\in\,N_{k/k_{0}}(k^{*})$, so $q^{*}\,\geq\,1$ which in impossible in this case. We conclude that $n\,=\,5^{e}l\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $l\,\not\equiv\,1(\mathrm{mod}\,25)$. Using the computational number theory system PARI/GP [References], if $n\,=\,5^{e}l\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $l\,\not\equiv\,1(\mathrm{mod}\,25)$, the field $k$ is not always of type $(5,5)$. 2. $(ii)$ $n\,=\,l^{e}q_{1}^{e_{1}}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $l\,\equiv\,1\,(\mathrm{mod}\,5),\,\,q_{1}\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$ and $e,e_{1}\in\\{1,2,3,4\\}$. As $\mathbb{Q}(\sqrt[5]{ab^{2}c^{3}d^{4}})\,=\,\mathbb{Q}(\sqrt[5]{a^{2}b^{4}cd^{3}})\,=\,\mathbb{Q}(\sqrt[5]{a^{3}bc^{4}d^{2}})\,=\,\mathbb{Q}(\sqrt[5]{a^{4}b^{3}c^{2}d})$ we can choose $e_{1}\,=\,1$ i.e $n\,=\,l^{e}q_{1}$ with $e\in\\{1,2,3,4\\}$. By Corollary 2.1 $l\,=\,\pi_{1}\pi_{2}\pi_{3}\pi_{4}$ where $\pi_{i}$ are primes in $k_{0}$ and $q_{1}$ is inert in $k_{0}$. We know that $l$ is ramified in $\Gamma$,so $\pi_{1},\,\pi_{2},\pi_{3}$ and $\pi_{4}$ are ramified in $k$. $q_{1}$ is ramified in $\Gamma$ too, we obtain $d=5$. The condition $n\,=\,l^{e}q_{1}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ is not verified for all $l\,\equiv\,1\,(\mathrm{mod}\,5),\,\,q_{1}\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$, so we combine all the cases of congruence and we obtain that $l\,\equiv\,1\,(\mathrm{mod}\,5)$ and $q_{1}\,\equiv\,\pm 2,\pm 3\,\pm 7\,(\mathrm{mod}\,25)$. Using the computational number theory system PARI/GP [References], if $n\,=\,l^{e}q_{1}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$, with $l\,\equiv\,1\,(\mathrm{mod}\,5)$, and $q_{1}\,\equiv\,\pm 2,\pm 3\,\pm 7\,(\mathrm{mod}\,25)$, the field $k$ is not always of type $(5,5)$ We summarize all forms of integer $n$ in this case as follows: $n=\left\\{\begin{array}[]{ll}5^{e}l\not\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,25)&\text{ with }l\,\not\equiv\,1\,(\mathrm{mod}\,25),\\\ l^{e}q_{1}\,\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,25)&\text{ with }l\,\equiv\,1\,(\mathrm{mod}\,5)\,q_{1}\,\equiv\,\pm 2,\pm 3\,\pm 7\,(\mathrm{mod}\,25)\\\ \end{array}\right.$ (4) * • Case 2: We have $q^{*}\,=\,1$ and $d=4$, so the number of prime ideals which are ramified in $k/k_{0}$ should be $4$. The radicand $n$ must be divisible by one prime $l\,\equiv\,1\,(\mathrm{mod}\,5)$, and according to Corollary 2.1 $l$ splits in $k_{0}$ as $l\,=\,\pi_{1}\pi_{2}\pi_{3}\pi_{4}$ with $\pi_{i}$ are primes in $k_{0}$. Since $l$ is ramified in $\Gamma$, so $\pi_{1},\,\pi_{2},\pi_{3}$ and $\pi_{4}$ are ramified in $k$, hence if $n$ is devisible by another prime than $l$ the number of primes which are ramified in $k/k_{0}$ surpass $4$, therefore we have unique form of $n$ in this case, its $n\,=\,l^{e}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $l\,\equiv\,1\,(\mathrm{mod}\,5)$ and $e\in\\{1,2,3,4\\}$. The condition $n\,\equiv\,\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ is verified only for $l\,\,\equiv\,1,(\mathrm{mod}\,25)$, and we have $q^{*}\,=\,1$. In conclusion we get $n\,=\,l^{e}$ with $l\,\,\equiv\,1,(\mathrm{mod}\,25)$. Using the computational number theory system PARI/GP [References], if $n\,=\,l^{e}\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $l\,\equiv\,1(\mathrm{mod}\,25)$, the field $k$ is not always of type $(5,5)$ * • case 3: We have $q^{*}\,=\,2$ and $d=3$, so the number of prime ideals which are ramified in $k/k_{0}$ should be $3$. The radicand $n$ must be divisible by one prime $l\,\equiv\,1\,(\mathrm{mod}\,5)$, and according to Corollary 2.1 $l\,=\,\pi_{1}\pi_{2}\pi_{3}\pi_{4}$ with $\pi_{i}$ are primes in $k_{0}$. Since $p$ is ramified in $\Gamma$, so $\pi_{1},\,\pi_{2},\pi_{3}$ and $\pi_{4}$ are ramified in $k$, so we deduce that the number of primes ramified in $k/k_{0}$ is at least $4$, so the case 3 does not exist. ## 3 Numerical examples Let $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a positive integer, $5^{th}$ power-free, and let $k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_{5})$ its normal closure. We assume that $C_{k,5}$ is of type $(5,5)$. Using the system PARI/GP [References], we illustrate our main result Theorem 1.1. ### 3.1 rank $(C_{k,5}^{(\sigma)})\,=\,1$ Table 1: $n\,=\,p^{e}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\equiv\,-1\,(\mathrm{mod}\,25)$ $p$ | $n\,=\,p^{e}$ | $p\,(\mathrm{mod}\,5)$ | $p\,(\mathrm{mod}\,25)$ | $h_{k,5}$ | $C_{k,5}$ | rank $(C_{k,5}^{(\sigma)})$ ---|---|---|---|---|---|--- 149 | 22201 = $149^{2}$ | -1 | -1 | 25 | $(5,5)$ | 1 199 | 7880599 = $199^{3}$ | -1 | -1 | 25 | $(5,5)$ | 1 349 | 42508549 = $349^{3}$ | -1 | -1 | 25 | $(5,5)$ | 1 449 | $449$ | -1 | -1 | 25 | $(5,5)$ | 1 559 | $559$ | -1 | -1 | 25 | $(5,5)$ | 1 1249 | $1249$ | -1 | -1 | 25 | $(5,5)$ | 1 1499 | $1499$ | -1 | -1 | 25 | $(5,5)$ | 1 1949 | $1949$ | -1 | -1 | 25 | $(5,5)$ | 1 1999 | $1999$ | -1 | -1 | 25 | $(5,5)$ | 1 2099 | $449$ | -1 | -1 | 25 | $(5,5)$ | 1 Table 2: $n\,=\,q_{1}^{e_{1}}q_{2}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $q_{i}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)$ $q_{1}$ $q_{1}\,(\mathrm{mod}\,5)$ $q_{1}\,(\mathrm{mod}\,25)$ $q_{2}$ $q_{2}\,(\mathrm{mod}\,5)$ $q_{2}\,(\mathrm{mod}\,25)$ $n\,=\,q_{1}^{e_{1}}q_{2}$ $h_{k,5}$ $C_{k,5}$ rank $(C_{k,5}^{(\sigma)})$ 7 2 7 43 3 -7 2107 = $7^{2}\times 43$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 7 2 7 193 3 -7 1351 = $7\times 193$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 7 2 7 293 3 -7 2051 = $7\times 293$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 107 2 7 43 3 -7 492307 = $107^{2}\times 43$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 107 2 7 193 3 -7 20651 = $107\times 193$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 107 2 7 293 3 -7 31351 = $107\times 293$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 107 2 7 443 3 -7 47401 = $107\times 443$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 157 2 7 43 3 -7 6751 = $157\times 43$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 157 2 7 193 3 -7 30301 = $157\times 193$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 157 2 7 443 3 -7 69551 = $157\times 443$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 257 2 7 193 3 -7 49601 = $257\times 293$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 257 2 7 293 3 -7 75301 = $257\times 293$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 307 2 7 193 3 -7 59251 = $307\times 193$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 457 2 7 43 3 -7 19651 = $457\times 43$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 457 2 7 443 3 -7 202451 = $457\times 443$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 557 2 7 43 3 -7 23251 = $557\times 43$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 607 2 7 43 3 -7 26101 = $607\times 43$ 5 $\mathbb{Z}/5\mathbb{Z}$ 1 Table 3 : $n\,=\,5^{e}q_{1}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $q_{1}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)$ $q_{1}$ | $n\,=\,5^{e}q_{1}$ | $q_{1}\,(\mathrm{mod}\,5)$ | $q_{1}\,(\mathrm{mod}\,25)$ | $h_{k,5}$ | $C_{k,5}$ | rank $(C_{k,5}^{(\sigma)})$ ---|---|---|---|---|---|--- 7 | 175 = $5^{2}\times 7$ | 2 | 7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 107 | 535 = $5\times 107$ | 2 | 7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 157 | 19625 = $5^{3}\times 157$ | 2 | 7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 257 | 6425 = $5^{2}\times 257$ | 2 | 7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 307 | 38375 = $5^{3}\times 307$ | 2 | 7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 457 | 2285 = $5\times 457$ | 2 | 7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 557 | 2785 = $5\times 557$ | 2 | 7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 607 | 3053 = $5\times 607$ | 2 | 7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 757 | 3785 = $5\times 457$ | 2 | 7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 857 | 4285 = $5\times 457$ | 2 | 7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 907 | 4535 = $5\times 907$ | 2 | 7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 43 | 1075 = $5^{2}\times 43$ | 3 | -7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 193 | 120625 = $5^{4}\times 193$ | 3 | -7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 293 | 120625 = $5^{4}\times 193$ | 3 | -7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 443 | 11075 = $5^{2}\times 443$ | 3 | -7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 643 | 3215 = $5\times 643$ | 3 | -7 | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 Table 4: $n\,=\,5^{e}q_{1}^{2}q_{2}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $q_{1}$ or $q_{2}$ $\not\equiv\,\pm 7\,(\mathrm{mod}\,25)$ $q_{1}$ | $q_{1}\,(\mathrm{mod}\,5)$ | $q_{1}\,(\mathrm{mod}\,25)$ | $q_{2}$ | $q_{2}\,(\mathrm{mod}\,5)$ | $q_{2}\,(\mathrm{mod}\,25)$ | $n\,=\,5^{e}q_{1}^{2}q_{2}$ | $h_{k,5}$ | $C_{k,5}$ | rank $(C_{k,5}^{(\sigma)})$ ---|---|---|---|---|---|---|---|---|--- 2 | 2 | 2 | 3 | 3 | 3 | 60 = $5\times 2^{2}\times 3$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 2 | 2 | 2 | 13 | 3 | 13 | 260 = $5\times 2^{2}\times 13$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 2 | 2 | 2 | 53 | 3 | 3 | 1060 = $5\times 2^{2}\times 53$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 2 | 2 | 2 | 23 | 3 | -2 | 460 = $5\times 2^{2}\times 23$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 7 | 2 | 7 | 3 | 3 | 3 | 735 = $5\times 7^{2}\times 3$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 17 | 2 | 17 | 3 | 3 | 3 | 108375 = $5^{3}\times 17^{2}\times 3$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 17 | 2 | 17 | 23 | 3 | -2 | 33235 = $5\times 17^{2}\times 23$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 37 | 2 | 17 | 3 | 3 | 3 | 20535 = $5\times 37^{2}\times 13$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 37 | 2 | 17 | 13 | 3 | 13 | 88985 = $5\times 37^{2}\times 13$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 47 | 2 | -3 | 3 | 3 | 3 | 33135 = $5\times 47^{2}\times 3$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 47 | 2 | -3 | 13 | 3 | 13 | 143585 = $5\times 47^{2}\times 13$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 47 | 2 | -3 | 23 | 3 | -2 | 254035 = $5\times 47^{2}\times 23$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 47 | 2 | -3 | 43 | 3 | -7 | 474935 = $5\times 47^{2}\times 43$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 107 | 2 | 7 | 23 | 3 | -2 | 1316635 = $5\times 2^{2}\times 3$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 67 | 2 | 17 | 3 | 3 | 3 | 67335 = $5\times 67^{2}\times 3$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 67 | 2 | 17 | 53 | 3 | 3 | 1189585 = $5\times 67^{2}\times 53$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 97 | 2 | -3 | 43 | 3 | -7 | 2022935 = $5\times 97^{2}\times 43$ | 5 | $\mathbb{Z}/5\mathbb{Z}$ | 1 Table 5: $n\,=\,p^{e}q_{1}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\not\equiv\,-1\,(\mathrm{mod}\,25)$, $q_{1}\,\not\equiv\,\pm 7\,(\mathrm{mod}\,25)$ $p$ $p\,(\mathrm{mod}\,5)$ $p\,(\mathrm{mod}\,25)$ $q_{1}$ $q_{1}\,(\mathrm{mod}\,5)$ $q_{1}\,(\mathrm{mod}\,25)$ $n\,=\,p^{e}q_{1}$ $h_{k,5}$ $C_{k,5}$ rank $(C_{k,5}^{(\sigma)})$ 59 -1 9 2 2 2 118 = $59\times 2$ 25 $(5,5)$ 1 19 -1 19 3 3 3 57 = $19\times 3$ 25 $(5,5)$ 1 59 -1 9 23 3 -2 1357 = $59\times 23$ 25 $(5,5)$ 1 359 -1 9 2 2 2 718 = $359\times 2$ 25 $(5,5)$ 1 409 -1 9 2 2 2 816 = $409\times 2$ 25 $(5,5)$ 1 59 -1 9 127 2 2 7493 = $59\times 127$ 25 $(5,5)$ 1 109 -1 9 23 3 -2 2507 = $109\times 23$ 25 $(5,5)$ 1 509 -1 9 2 2 2 1018 = $509\times 2$ 25 $(5,5)$ 1 709 -1 9 2 2 2 1418 = $709\times 2$ 25 $(5,5)$ 1 19 -1 19 53 3 3 1007 = $19\times 53$ 25 $(5,5)$ 1 Table 6: $n\,=\,5^{e}p\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $p\,\not\equiv\,-1(\mathrm{mod}\,25)$ $p$ | $n\,=\,5^{e}p$ | $p\,(\mathrm{mod}\,5)$ | $p\,(\mathrm{mod}\,25)$ | $h_{k,5}$ | $C_{k,5}$ | rank $(C_{k,5}^{(\sigma)})$ ---|---|---|---|---|---|--- 19 | 475 = $5^{2}\times 19$ | -1 | 19 | 25 | (5,5) | 1 29 | 145 = $5\times 29$ | -1 | 4 | 25 | (5,5) | 1 59 | 7375 = $5^{3}\times 59$ | -1 | 9 | 25 | (5,5) | 1 89 | 55625 = $5^{4}\times 89$ | -1 | 14 | 25 | (5,5) | 1 109 | 2725 = $5^{2}\times 109$ | -1 | 9 | 25 | (5,5) | 1 229 | 28625 = $5^{3}\times 229$ | -1 | 4 | 25 | (5,5) | 1 239 | 1195 = $5\times 239$ | -1 | 14 | 25 | (5,5) | 1 269 | 6725 = $5^{2}\times 19$ | -1 | 19 | 25 | (5,5) | 1 379 | 168125 = $5^{4}\times 379$ | -1 | 4 | 25 | (5,5) | 1 389 | 1945 = $5^{2}\times 389$ | -1 | 14 | 25 | (5,5) | 1 ### 3.2 rank $(C_{k,5}^{(\sigma)})\,=\,2$ Table 1: $n\,=\,5^{e}l\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $l\,\not\equiv\,1(\mathrm{mod}\,25)$ $l$ | $n\,=\,5^{e}l$ | $l\,(\mathrm{mod}\,5)$ | $l\,(\mathrm{mod}\,25)$ | $h_{k,5}$ | $C_{k,5}$ | rank $(C_{k,5}^{(\sigma)})$ ---|---|---|---|---|---|--- 11 | 55 = $5\times 11$ | 1 | 11 | 25 | (5,5) | 2 41 | 5125 = $5^{3}\times 41$ | 1 | -9 | 25 | (5,5) | 2 61 | 5125 = $5^{4}\times 61$ | 1 | 11 | 25 | (5,5) | 2 71 | 1775 = $5^{2}\times 71$ | 1 | -4 | 25 | (5,5) | 2 131 | 655 = $5\times 131$ | 1 | 6 | 25 | (5,5) | 2 181 | 113125 = $5^{4}\times 181$ | 1 | 6 | 25 | (5,5) | 2 241 | 30125 = $5^{3}\times 241$ | 1 | -9 | 25 | (5,5) | 2 311 | 1555 = $5\times 311$ | 1 | 11 | 25 | (5,5) | 2 331 | 8275 = $5^{2}\times 331$ | 1 | 6 | 25 | (5,5) | 2 431 | 2155 = $5\times 431$ | 1 | 6 | 25 | (5,5) | 2 Table 2: $n\,=\,l^{e}q_{1}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $l\,\equiv\,1\,(\mathrm{mod}\,5)$ and $q_{1}\,\equiv\,\pm 2,\pm 3\,\pm 7\,(\mathrm{mod}\,25)$ $l$ $l\,(\mathrm{mod}\,5)$ $l\,(\mathrm{mod}\,25)$ $q_{1}$ $q_{1}\,(\mathrm{mod}\,5)$ $q_{1}\,(\mathrm{mod}\,25)$ $n\,=\,l^{e}q_{1}$ $h_{k,5}$ $C_{k,5}$ rank $(C_{k,5}^{(\sigma)})$ 31 1 6 2 2 2 $31\times 2$ 25 $(5,5)$ 2 131 1 6 23 3 -2 $131^{3}\times 23$ 25 $(5,5)$ 2 181 1 6 47 2 -3 $181\times 47$ 25 $(5,5)$ 2 11 1 11 3 3 3 $11\times 3$ 25 $(5,5)$ 2 41 1 16 23 3 -2 $41\times 23$ 25 $(5,5)$ 2 191 1 16 2 2 2 $191\times 2$ 25 $(5,5)$ 2 41 1 16 47 2 -3 $41^{2}\times 47$ 25 $(5,5)$ 2 311 1 11 2 2 2 $311^{4}\times 2$ 25 $(5,5)$ 2 Table 3: $n\,=\,l^{e}\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)$ with $l\,\equiv\,1(\mathrm{mod}\,25)$ $l$ $n\,=\,l^{e}$ $l\,(\mathrm{mod}\,5)$ $l\,(\mathrm{mod}\,25)$ $h_{k,5}$ $C_{k,5}$ rank $(C_{k,5}^{(\sigma)})$ 151 $151$ 1 1 25 (5,5) 2 251 $251^{2}$ 1 1 25 (5,5) 2 601 $601^{3}$ 1 1 25 (5,5) 2 1051 $1051^{4}$ 1 1 25 (5,5) 2 1301 $1301$ 1 1 25 (5,5) 2 1451 $1451^{2}$ 1 1 25 (5,5) 2 1801 $1801^{3}$ 1 1 25 (5,5) 2 1901 $1901^{4}$ 1 1 25 (5,5) 2 2111 2111 1 1 25 (5,5) 2 2131 $2131^{2}$ 1 1 25 (5,5) 2 ## 4 Conjecture In this article, we have classified some pure quintic fields $\mathbb{Q}(\sqrt[5]{n})$, more precisely, we focused on the ones whose normal closures $\mathbb{Q}(\sqrt[5]{n},\zeta_{5})$ possesses a $5$-class groups of type $(5,5)$, by treating the rank of the ambiguous classes, that can be characterized by the radicand $n$. As to provide numerical examples, we use the system PARI/GP References. Thus, we have noticed that the done calculations for some $n$ forms, show that $5$-class groupe $C_{k,5}$ of the field $k$, is isomorphic to $\mathbb{Z}/5\mathbb{Z}$, which allows us to give this conjecture as follows: ###### Conjecture 4.1. Let $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a positive integer, $5^{th}$ power-free. Let $k\,=\Gamma(\zeta_{5})$ be the normal closure of $\Gamma$. Denote by $C_{k,5}$ the 5-class group of $k$, $q_{1},q_{2}\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$ are primes and $e\in\\{1,2,3,4\\}$. If the radicand $n$ take one form as follows: $n\,=\,\begin{cases}q_{1}^{e_{1}}q_{2}\,\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)&\text{ with }\quad q_{i}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\ 5^{e}q_{1}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)&\text{ with }\quad q_{1}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\ 5^{e}q_{1}^{2}q_{2}\not\equiv\,\pm 1\pm 7\,(\mathrm{mod}\,25)&\text{ with }\quad q_{1}\,\text{ or }\,q_{2}$ $\not\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\ \end{cases}$ (5) Then $C_{k,5}$ is a cyclic groupe of order $5$. ## References * [1] S. Aouissi, M. Talbi, M. C. Ismaili and A. Azizi. _Fields $\mathbb{Q}(\sqrt[3]{d},\zeta_{3})$ whose $3$-class group is of type $(9,3)$_, Int.J.Number Theory (2019) * [2] F. Gerth III, _On $3$-class groups of cyclic cubic extensions of certain number fields_, J. Number Theory 8 (1976), No. 1, 84–98. * [3] F. Gerth III, _On $3$-class groups of pure cubic fields,_ J. Reine Angew. Math. 278/279 (1975), 52–62. * [4] F. Gerth III, _On $3$-class groups of certain pure cubic fields_,Bull. Austral. Math. Soc. 72 (2005), 471–476. * [5] David Grant. _A proof of quintic reciprocity using the arithmetic of $y^{2}=x^{5}+\frac{1}{4}$_. ACTA ARITHMETICA (1996). * [6] G.Gras, _Sur les $l$-classes d’idéux dans les extensions cycliques relatives de degré premier impaire $l$_. _Annales de l’institut Fourier_ , (1973). * [7] E.Hecke, _Algebraic Number Theory_ , GTM 77, Springer-Verlag 1981. * [8] M.Ishida, _The genus Fields of Algebraic Number Fields_. Lecture notes in Mathematics Vol 555, Springer-Verlag (1976). * [9] K. Iwasawa, _A note on the group of units of an algebraic number field_ , J. Math. Pures Appl. (9) 35 (1956), 189–192. * [10] K.Ireland and M.Rosen, _A Classical Introduction to modern Number Theory_. Graduate Texts in Mathematics 84, Springer-Verlag (1982). * [11] G.J Janus, _Algebraic Number Fields_. Academic Press, New York-London (1973). * [12] M.Kulkarni,D. Majumdar, B.Sury _$l$ -class groups of cyclic extension of prime degree $l$_, J. Ramanujan Math. Soc. 30, No.4 (2015), 413-454. * [13] H. Kobayashi, _Class numbers of pure quintic fields_ , Journal of Number Theory 160 (2016) 463-477. * [14] C. Parry, _Class number relations in pure quintic felds_ , Symposia Mathematica. 15 (1975), 475-485. * [15] Lawrence C. Washington, _Introduction to Cyclotomic Fields_ , Springer-Verlag New York Inc (1982). * [16] The PARI Group, PARI/GP, Version 2.4.9, Bordeaux, 2017, http://pari.math.u-bordeaux.fr. Fouad ELMOUHIB Department of Mathematics and Computer Sciences, Mohammed 1st University, Oujda - Morocco, <EMAIL_ADDRESS> Mohamed TALBI Regional Center of Professions of Education and Training in the Oriental, Oujda - Morocco, <EMAIL_ADDRESS> Abdelmalek AZIZI Department of Mathematics and Computer Sciences, Mohammed 1st University, Oujda - Morocco, <EMAIL_ADDRESS>

2024-09-04T02:54:56.183203

2003.00764

arxiv-papers

# Weak deflection angle of a dirty black hole Reggie C. Pantig<EMAIL_ADDRESS>Physics Department, De La Salle University-Manila, 2401 Taft Ave., 1004 Manila Philippines Emmanuel T. Rodulfo<EMAIL_ADDRESS> ###### Abstract In this paper, we present the weak deflection angle in a Schwarzschild black hole of mass $m$ surrounded by the dark matter of mass $M$ and thickness $\Delta r_{s}$. The Gauss-Bonnet theorem, formulated for asymptotic spacetimes, is found to be ill-behaved in the third-order of $1/\Delta r_{s}$ for very large $\Delta r_{s}$. Using the finite-distance for the radial locations of the source and the receiver, we derived the expression for the weak deflection angle up to the third-order of $1/\Delta r_{s}$ using Ishihara (et al.) method. The result showed that the required dark matter thickness is $\sim 2\sqrt{3mM}$ for the deviations in the weak deflection angle to occur. Such thickness requirement is better by a factor of 2 as compared to the deviations in the shadow radius ($\sim\sqrt{3mM}$). It implies that the use of the weak deflection angle in detecting dark matter effects in one’s galaxy is better than using any deviations in the shadow radius. ## I Introduction One of the fruitful achievements of the human mind is the general theory of relativity by Albert Einstein, which relates the phenomena of gravitation to the geometry of spacetime. One consequence of the theory is the existence of black holes that have long remained to be a theoretical construct, until the release of the first image of the black hole shadow at the center of the M87 galaxy on April 10, 2019 Akiyama _et al._ (2019). At present, dark matter is another entity that remains so mysterious and elusive. The $\Lambda$CDM model of cosmology suggests that the content of our universe is made up of 27$\%$ dark matter, which constitutes 85$\%$ of the total mass Jarosik _et al._ (2011). Using Earth-based laboratories, several scientists attempted to detect dark matter by direct means, but they gave inconsistent results with each other. (see Bernabei _et al._ (2010) and Aprile _et al._ (2010)). Even long before these experiments, indirect search through dark matter annihilation also revealed null results Desai _et al._ (2004); Peirani _et al._ (2004). It proves that dark matter detection is more difficult compared to gravitational wave detection Abbott _et al._ (2016). Emerging theoretical efforts have shown alternative means of possible dark matter detection using the changes in the silhouette of a black hole. The spacetime of pure dark matter and black hole were combined rigorously by Xu in Ref. Xu _et al._ (2018), and this formalism of using dark matter density profiles to black hole spacetime is applied immediately to our own and M87 galaxy Hou _et al._ (2018a); Jusufi _et al._ (2019). Several studies are also present that analyze the black hole shadow and deflection angle under the influence of dark matter modeled as a perfect fluid Hou _et al._ (2018b); Haroon _et al._ (2019). However, Konoplya in Ref. Konoplya (2019) considered these black hole metrics as highly model-dependent. Using a less model- dependent and agnostic view on dark matter, he estimated the condition for dark matter to have some notable effect on the shadow radius. Not only the study of shadow from various black hole models has gained much attention from many researchers, but also the study of gravitational lensing (GL). GL has proved useful in probing many astrophysical objects. Long ago, it is used to probe the coevolution of supermassive black holes (SMBH) and galaxies Peng _et al._ (2006). It is also used to probe exotic entities like dark matter Trimble (1987); Metcalf and Madau (2001); Metcalf and Zhao (2002) that permeates a whole galaxy, and even galaxy clusters Hoekstra and Jain (2008); Ellis (2010). Astrophysical black holes, which can also be approximately described by the Schwarzschild metric for useful estimates, are studied extensively in terms of lensing and the relativistic images that it produces Virbhadra and Ellis (2000); Virbhadra (2009). Perhaps the most popular way to obtain the weak deflection angle is by using the Gauss-Bonnet theorem (GBT) introduced in Ref. Gibbons and Werner (2008). Since then, the GBT has been proven very useful in calculating the weak deflection angle of various black hole models that demonstrate asymptotic behavior Övgün _et al._ (2018a, b); Övgün (2018); Övgün _et al._ (2019a); Övgün (2019a); Jusufi and Övgün (2018); Javed _et al._ (2019). Other notable studies also existed that includes dark matter, phantom matter, and quintessential energy, Övgün (2019b); Övgün _et al._ (2019b); Haroon _et al._ (2019); De Leon and Vega (2019) on their analysis of the weak deflection angle. Gravitational lensing by exotic matter or energy, which possibly deviate from some well-known equation of state, is also explored in Kitamura _et al._ (2014); Nakajima _et al._ (2014); Izumi _et al._ (2013). Calculation of the weak deflection angle for non-asymptotic spacetimes seems problematic using the GBT as far as its asymptotic form is concerned. Recently, an improved analysis in Ref. Ishihara _et al._ (2016) made the calculation possible by considering the finite distances of the source and receiver. However, the correspondence between the GBT and finite-distance involves the brute force of evaluating integrals, which contain the orbit equation. This method, whether the spacetime is asymptotic or not, axially- symmetric or not Ono _et al._ (2019); Li and Övgün (2020); Zhang _et al._ (2019), is also useful in strong gravitational lensing regime Ishihara _et al._ (2017); Azreg-Aïnou _et al._ (2017). It is interesting to investigate the effect of dark matter configuration used in Ref. Konoplya (2019) on a different black hole phenomenon: weak gravitational lensing. In particular, this paper seeks to derive the expression for the weak deflection angle caused by the main property of dark matter - its mass. Although that the deviation in the shadow radius already gives us the idea that null geodesics are affected, it is interesting to find out whether if the deviation in the black hole’s weak deflection angle offers a better condition for dark matter detection. We organize the paper as follows: Sect. II introduces the dirty Schwarzschild metric alongside with the description of the simple dark matter model in consideration. In Sect. III, we present three possible cases that show the different positions of the source and receiver relative to the dark matter distribution and utilize the GBT. In Sect. IV, we proceed to calculate the weak deflection angle by assuming some finite-distance of the source and the receiver using the Ishihara (et al.) method. Lastly, in Sect. V, we summarize the results and indicate some possible future research direction. The metric signature in this study is +2, and we use $G=c=1$. ## II Dirty Schwarzschild black hole The term ”dirty” black hole has its roots in Ref. Visser (1992, 1993); Macedo _et al._ (2016); Leung _et al._ (1997); Krauss _et al._ (1996) which describes a black hole surrounded by some astrophysical environment. These dirty black holes can be categorized as follows: (a) those that came from a specific Lagrangian, or a certain field theory which results to a metric as the Einstein field equation is solved (for examples, see Refs. Shapere _et al._ (1991); Dowker _et al._ (1992); Gibbons and ichi Maeda (1988); Allen _et al._ (1990); Galt’sov and Ershov (1989); Lahiri (1992)), (b) generic black hole metrics with sufficient generality Nielsen and Birnholz (2019) that came from some hypothetical configuration (or derived from empirical data) of astrophysical environment, and (c) dirty black holes which came from solutions to a non-Einsteinian theory such as pseudo-complex general relativity Moffat (1979); Hess and Greiner (2009); Mann and Moffat (1982). Here, we give a brief overview and describe the dirty Schwarzschild black hole used in Ref. Konoplya (2019), where the author studied dark matter effects on the photonsphere and black hole shadow. The astrophysical environment in consideration is a spherical shell of dark matter described only by its mass M, inner radius $r_{s}$, and thickness $\Delta r_{s}$ (the subscript $s$ denotes shell). Further, the dark matter mass $M$ is treated as an additional effective mass to the black hole while maintaining its non-interaction with the electromagnetic field. Since the physical parameters of the dark matter shell are hypothetical, such a dirty black hole falls to the second category described above. The generic metric produced still has sufficient generality because the black hole itself came from the vacuum solution of the Einstein equation. One can then assume a piecewise function to impose three domains Konoplya (2019): $\mathcal{M}(r)=\begin{cases}m,&r<r_{s};\\\ m+MG(r),&r_{s}\leq r\leq r_{s}+\Delta r_{s};\\\ m+M,&r>r_{s}+\Delta r_{s}\end{cases}$ (1) where $G(r)=\left(3-2\frac{r-r_{s}}{\Delta r_{s}}\right)\left(\frac{r-r_{s}}{\Delta r_{s}}\right)^{2}.$ (2) The expression for $G(r)$ is chosen so that $\mathcal{M}(r)$ and $\mathcal{M}^{\prime}(r)$ are continuous (see Fig. 1). Note also the possibility of $M<0$. Figure 1: Example of a choice for mass function. Here, $M=10m$, $r_{s}=2m$, $\Delta r_{s}=20m$, $m=1$. The Schwarzschild metric is one of the famous, yet simplest, vacuum solution of the Einstein field equations. Consider surrounding it with a spherical shell of dark matter whose parameters are described by Eq. (2). Then we have $d{s}^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right)$ (3) where the metric function $f(r)$ is now given by $f(r)=1-\frac{2\mathcal{M}(r)}{r}.$ (4) The general scenario is depicted in Fig. 2 where $r_{s}\neq r_{h}$, which makes the piecewise function in Eq. (1) very clear. Considering the first domain and if $r$ is between $r_{h}$ and $r_{s}$, the dark matter outside $r$ has no bearing to any black hole phenomena that we want to analyze. The situation is completely equivalent as if there’s no dark matter surrounding the black hole. Suppose that $r$ describes the photonsphere radius $r_{ph}$, then we simply have the trivial expression $r_{ph}=3m$. Figure 2: A representation of a black hole surrounded by a thin shell of dark matter where $r_{s}\neq r_{h}$. The third domain of Eq. (1) seems an innocent-looking condition where the parameters $r_{s}$ and $\Delta r_{s}$ does not exist. The domain can be interpreted in two ways: (1) an isolated black hole surrounded by some dark matter shell, and (2) a black hole at the center of a galaxy (combined mass is $m$) where both are surrounded by dark matter halo. Taking the example of deriving the photonsphere radius, the former case gives $r_{ph}=3(m+M)$ as the null geodesic has no way to cross the outer radius of the dark matter shell. The study of the deflection angle is more applicable to the latter case, but the third domain’s applicability rests on the assumption that the null geodesic never crosses the region of dark matter halo. For this to happen, the location of the source and the receiver must be tremendously remote from the lensing galaxy. It is also easy to see why the first and third domains should satisfy the Einstein equation. For the second domain, the spacetime where the $r$-coordinate is located is now different due to the introduction of dark matter mass $M$ and its physical dimensions. Since the configuration is hypothetical, we don’t know the specific field theory, which may lead to the stress-energy tensor that produces such expression of the metric if the Einstein equation is solved. It is also true for black hole spacetime fused with dark matter models whose density profiles came from observational data Xu _et al._ (2018); Hou _et al._ (2018a); Jusufi _et al._ (2019), hence they can be considered as dirty black holes. It is possible, however, to satisfy the Einstein equation. The key is to express the Einstein tensor in terms of appropriately chosen orthogonal bases along with the stress-energy tensor Azreg-Aïnou (2014). Assuming that the new spacetime expressed through the second domain of Eq. (1) satisfies the Einstein equation, one can proceed with the analysis of black hole properties. It is found out that it gives a non-trivial expression for the photonsphere radius inside the dark matter shell, and is also true even with the approximation $\Delta r_{s}>>r_{s}$ Konoplya (2019). Nevertheless, it gave relevant insights to the effect of dark matter on the photonsphere radius, and as a consequence, to the deviations in the shadow radius. Note that being in the piecewise relation, the equation for the second domain cannot be applied for values of $r$ beyond its lower and upper bounds. An observer inside the dark matter shell has a different reality condition compared to being outside it, or as the first domain applies. Looking at Eq. (2) again, the function allows $r_{s}$ to have values smaller than $r=2m$. For simplicity, one can set $r_{s}=2m$ and assume that the dark matter shell is static. By being static, it means that dark matter is not affected by the radial pull of the black hole. Nevertheless, it amplifies the change in the black hole’s geometry, which then manifests to the dynamics of null and time-like particles. The idea of an astrophysical environment that surrounds a black hole is common in several studies Konoplya _et al._ (2019); Cardoso and Pani (2017). The reasons include (1) verify whether the deviation observed is due to the new physics happening near the horizon, or due to some effect of the astrophysical environment; and (2) directly examine and study the influence of astrophysical environment to the geometry of black hole. Only recently that the mass function in Eq. (1) has been used to make some estimates of dark matter effects to the radius of the black hole shadow Konoplya (2019). ## III Deflection angle in a dirty black hole using the Gauss-Bonnet theorem Consider $D$ as a freely orientable two-dimensional curved surface described by the Gaussian curvature $K$, and $dS$ is its infinitesimal area element (See Fig. 3). It contains several boundaries that are differentiable curves, denoted by $\partial D_{a}$ ($a=1,2,...,N$), with geodesic curvature $\kappa_{g}$. Such boundary also contains the line element $\ell$ where its sign remains consistent with the surface orientation. Let $\theta_{a}$ represents the exterior or jump angles along the vertices of the surface. Then the Gauss-Bonnet theorem states that Ishihara _et al._ (2016, 2017); Do Carmo (2016); Klingenberg (2013) $\iint_{D}KdS+\sum\limits_{a=1}^{N}\int_{\partial D_{a}}\kappa_{g}d\ell+\sum\limits_{a=1}^{N}\theta_{a}=2\pi.$ (5) Figure 3: Schematic diagram of a curved surface for Gauss-Bonnet theorem. The interior angles are denoted by $\varepsilon$ while the exterior (or ”jump”) angles are $\theta$. In the weak lensing regime, the Gauss-Bonnet theorem proves to be a powerful tool in calculating the weak deflection angle $\hat{\alpha}$ as long as the spacetime is asymptotically flat. If the spacetime is static, and spherically symmetric (SSS), $\kappa_{g}=0$ along the boundary curves and the second term in Eq. (5) vanishes. Assuming asymptotic flatness of the spacetime being considered, the weak deflection angle can be derived as Gibbons and Werner (2008); Javed _et al._ (2019); Ishihara _et al._ (2016, 2017); Werner (2012); Ono _et al._ (2017) $\hat{\alpha}=-\iint_{{}_{R}^{\infty}\square_{S}^{\infty}}KdS$ (6) where $K$ is the Gaussian optical curvature that is integrated over the quadrilateral ${}_{R}^{\infty}\square_{S}^{\infty}$, $dS$ is the surface area element. These quantities that are important in GBT, are defined as follows: $K=\frac{R_{r\phi r\phi}}{\gamma},$ (7) $dS=\sqrt{\gamma}drd\phi$ (8) where $\gamma$ denotes the determinant of the spatial metric $\gamma_{ij}$ as $i$ and $j$ run from 1 to 3. The spatial metric for an SSS spacetime can be found by considering the null condition $ds^{2}=0$, and solving for $dt$ yields Övgün _et al._ (2018a) $dt=\sqrt{\gamma_{ij}dx^{i}dx^{j}}.$ (9) Rewriting Eq. (3) as $d{s}^{2}=A(r)dt^{2}+B(r)dr^{2}+C(r)d\theta^{2}+D(r,\theta)d\phi^{2},$ (10) the definition of $\gamma_{ij}$ is given by $\gamma_{ij}dx^{i}dx^{j}=\frac{1}{A(r)}\left(B(r)dr^{2}+C(r)d\theta^{2}+D(r,\theta)d\phi^{2}\right).$ (11) Dealing only in equatorial orbits, the Gaussian curvature $K$, being related to the two-dimensional Riemann tensor, can be written in terms of affine connection as Haroon _et al._ (2019); Övgün _et al._ (2019a) $K=\frac{1}{\sqrt{\gamma}}\left[\frac{\partial}{\partial\phi}\left(\frac{\sqrt{\gamma}}{\gamma_{rr}}\Gamma_{rr}^{\phi}\right)-\frac{\partial}{\partial r}\left(\frac{\sqrt{\gamma}}{\gamma_{rr}}\Gamma_{r\phi}^{\phi}\right)\right]$ (12) and using Eq. (10), $K$ can be written as Ono _et al._ (2017) $K=-\sqrt{\frac{A(r)^{2}}{B(r)D(r)}}\frac{\partial}{\partial r}\left[\frac{1}{2}\sqrt{\frac{A(r)^{2}}{B(r)D(r)}}\frac{\partial}{\partial r}\left(\frac{D(r)}{A(r)}\right)\right].$ (13) With the area surface element $dS$ in Eq. (8), we can write Eq. (6) as $\hat{\alpha}=\int_{0}^{\pi}\int_{r_{o}}^{\infty}\mathcal{K}drd\phi,$ (14) where the Gaussian curvature term De Leon and Vega (2019) reads $\mathcal{K}=-\frac{\partial}{\partial r}\left[\frac{1}{2}\sqrt{\frac{A(r)^{2}}{B(r)D(r)}}\frac{\partial}{\partial r}\left(\frac{D(r)}{A(r)}\right)\right].$ (15) Also, $r_{o}$ denotes the radial distance of the photon’s closest approach to the lensing black hole, and can be obtained through the orbit equation. For an SSS metric such as the Schwarzschild metric, the photon’s orbit equation in the equatorial plane reads $\left(\frac{dr}{d\phi}\right)^{2}=\frac{D(r)(D(r)-A(r)b^{2})}{A(r)B(r)b^{2}}$ (16) where $b$ is the impact parameter. For convenience and as usually done in celestial mechanics, we can set $u=\frac{1}{r}$. Thus, the above can be expressed as $\left(\frac{du}{d\phi}\right)^{2}\equiv F(u)=\frac{u^{4}D(u)(D(u)-A(u)b^{2})}{A(u)B(u)b^{2}}.$ (17) The closest approach $u_{o}$ can be found by iteratively solving $u$ in Eq. (17) and imposing the boundary condition $\frac{du}{d\phi}\big{|}_{\phi=\frac{\pi}{2}}=0$. For example, for the Schwarzschild metric, $u$ is given by $u=\frac{\sin\phi}{b}+\frac{m}{b^{2}}(1+\cos^{2}\phi).$ (18) With $u_{o}$, we can now recast Eq. (14) as $\hat{\alpha}=\int_{0}^{\pi}\int_{0}^{u_{o}}-\frac{\mathcal{K}}{u^{2}}dud\phi.$ (19) ### III.1 Case 1: $\mathcal{M}(r)=m$ Let $u_{S}$ and $u_{R}$ be the reciprocal of the source’s and receiver’s radial distance from the lensing object. The use of Eq. (14) assumes that these radial distances are so far away that the spacetime becomes approximately Minkowskian. For the first domain in Eq. (1) to hold, the inner radius $r_{s}$ of the dark matter shell should always larger than the source’s and receiver’s radial distance (See Fig. 4). With the metric function taking the form $f(r)=1-\frac{2m}{r},$ (20) the Gaussian curvature and the area element in the equatorial plane reads $\displaystyle K$ $\displaystyle=-\frac{2m}{r^{3}}+\frac{3m^{2}}{r^{4}}$ $\displaystyle dS$ $\displaystyle=r+3m+\mathcal{O}(m^{2})$ (21) respectively. Using Eq. (19), we find $\displaystyle\hat{\alpha}$ $\displaystyle=\int_{0}^{\pi}\int_{0}^{\frac{\sin(\phi)}{b}+\frac{m}{b^{2}}(1+\cos^{2}\phi)}-\frac{\mathcal{K}}{u^{2}}dud\phi$ $\displaystyle=\int_{0}^{\pi}\int_{0}^{\frac{\sin(\phi)}{b}+\frac{m}{b^{2}}(1+\cos^{2}\phi)}2m+3m^{2}u+\mathcal{O}\left(m^{3}\right)dud\phi$ $\displaystyle=\frac{4m}{b}+\frac{15\pi m^{2}}{4b^{2}}+\mathcal{O}\left(m^{3}\right)$ (22) which is the weak deflection angle of a Schwarzschild black hole up to the second-order in $m$. Figure 4: Quadrilateral embedded in curved spacetime while enclosing dark matter shell. ### III.2 Case 2: $\mathcal{M}(r)=m+M$ Following the discussion in Sect. II about the possible interpretation of this case, Fig. 5 shows the use of the third domain of Eq. (1). As we deal only with a non-rotating black hole, the metric function is trivial: $f(r)=1-\frac{2(m+M)}{r}.$ (23) Using Eq. (14), we find $\displaystyle\hat{\alpha}$ $\displaystyle=\int_{0}^{\pi}\int_{0}^{\frac{\sin(\phi)}{b}+\frac{m+M}{b^{2}}(1+\cos^{2}\phi)}-\frac{\mathcal{K}}{u^{2}}dud\phi$ $\displaystyle=\int_{0}^{\pi}\int_{0}^{\frac{\sin(\phi)}{b}+\frac{m+M}{b^{2}}(1+\cos^{2}\phi)}2(m+M)$ $\displaystyle+3(m+M)^{2}u+\mathcal{O}\left[(m+M)^{3}\right]dud\phi$ $\displaystyle=\frac{4(m+M)}{b}+\frac{15\pi(m+M)^{2}}{4b^{2}}+\mathcal{O}\left[(m+M)^{3}\right]$ (24) The result in Eq. (III.2) shows clearly how the dark matter mass acts as an additional effective mass to the black hole - it increases the value of the weak deflection angle. It clearly shows how $M$ adds up to the spacetime distortion that affects the path of the null geodesic. It also means that the photon’s path never enters and leaves the dark matter shell as it travels from the source to the receiver. Suppose that $M=0$ and we increase $m$, the null geodesic would only shift to its new orbital equilibrium while being outside the vicinity of the event horizon. Figure 5: Geometrical configuration of the quadrilateral when S and R are beyond the outermost radius of dark matter shell. ### III.3 Case 3: $\mathcal{M}(r)=m+MG(r)$ In this scenario, the null geodesic should remain inside the dark matter shell for the second domain in Eq. (1) to hold. Such restriction is to determine how the weak deflection angle should behave, knowing that the dark matter beneath or above the null geodesic can affect it. This scenario is the same as the restriction imposed in Ref. Konoplya (2019) for the photonsphere radius. Suppose that $\Delta r_{s}$ is fixed where its thickness is not yet comparable to dark matter halos and accommodates both the source and the receiver inside the shell. In calculating the weak deflection angle using Eq. (19), the asymptotic condition must apply, but the problem is that the source and the receiver locations exceed that of the shell’s outer radius. Hence, given Eq. (1), such a situation invalidates the use of Eq. (2). Therefore, there is a necessity for the approximation of $\Delta r_{s}$ that it must be very large. Such approximation directs the analysis to the scenario involving the source and the receiver inside a very large dark matter halo. This means there is an assumption that $\Delta r_{s}>>\frac{1}{u_{S}}$ (and $u_{R}$). See Fig. 6. Figure 6: Far approximation of S and R (i.e. $u_{S}<<1$ and $u_{S}<<1$) always implies the approximation that $\Delta r_{s}>>\frac{1}{u_{S}}$ (or $u_{R}$) for the 2nd domain in Eq. (1) to hold. The orbit equation in this case is $\displaystyle F(u)$ $\displaystyle=\frac{1}{b^{2}}+u^{2}(2mu-1)$ $\displaystyle+\frac{2M}{\Delta r_{s}^{2}}(r_{s}u-1)^{2}\left(\frac{2r_{s}u}{\Delta r_{s}}-\frac{2}{\Delta r_{s}}+3u\right).$ (25) Note that there is no coupling between the black hole mass $m$ and dark matter mass $M$ in Eq. (III.3), unlike the coupling between the spin parameter $a$ and $m$ in Kerr spacetime. The iterative solution of Eq. (III.3) to obtain $u_{o}$ is then $u=\frac{\sin\phi}{b}+\frac{m}{b^{2}}(1+\cos^{2}\phi)+\frac{3Mr_{s}^{2}}{b^{2}\Delta r_{s}^{2}}.$ (26) Hence, the integrand of Eq. (19), up to the third-order of $1/\Delta r_{s}$, is expressed by $\displaystyle\hat{\alpha}$ $\displaystyle=\int_{0}^{\pi}\int_{0}^{u_{o}}-\frac{\mathcal{K}}{u^{2}}dud\phi=\int_{0}^{\pi}\int_{0}^{u_{o}}\bigg{[}2m+\frac{6Mr_{s}^{2}}{\Delta r_{s}^{2}}$ $\displaystyle-\frac{6mMr_{s}(2-3r_{s}u)}{\Delta r_{s}^{2}}-\frac{4M}{\Delta r_{s}^{3}}\left(\frac{1}{u^{3}}-r_{s}^{3}\right)$ $\displaystyle-\frac{12mMr_{s}^{2}(1-r_{s}u)}{\Delta r_{s}^{3}}+\mathcal{O}\left(m^{2},\frac{1}{\Delta r_{s}^{4}}\right)\bigg{]}dud\phi.$ (27) The result shows divergence in the 4th term if one attempts to evaluate the integral. It means that in the third-order of $1/\Delta r_{s}$, if one insists to be more precise in their calculations, the spacetime is not asymptotically flat and GBT cannot be used. The same case also occur in other non-asymptotic spacetime such as those that involved the cosmological constant such as the Kottler spacetime Kottler (1918). However, the cosmological constant has no adjustable parameters where an approximation can be made possible. The condition $\Delta r_{s}>>\frac{1}{u}$ gives the approximate expression if $\Delta r_{s}$ is very large, at least comparable to the known size of dark matter halos. Therefore, the manifestation of dark matter effects to the weak deflection angle can be seen to begin with the second-order of $1/\Delta r_{s}$. Fortunately, there is asymptotic flatness and proceeding with the GBT calculation, we find $\displaystyle\hat{\alpha}$ $\displaystyle=\int_{0}^{\pi}\int_{0}^{u_{o}}-\frac{\mathcal{K}}{u^{2}}dud\phi=\int_{0}^{\pi}\int_{0}^{u_{o}}\bigg{[}2m+\frac{6Mr_{s}^{2}}{\Delta r_{s}^{2}}$ $\displaystyle-\frac{6mMr_{s}(2-3r_{s}u)}{\Delta r_{s}^{2}}\bigg{]}dud\phi=\frac{4m}{b}+\frac{12Mr_{s}^{2}}{b\Delta r_{s}^{2}}$ $\displaystyle-\frac{24mMr_{s}}{b\Delta r_{s}^{2}}+\frac{15\pi m^{2}}{4b^{2}}+\frac{39\pi mMr_{s}^{2}}{2b^{2}\Delta r_{s}^{2}}.$ (28) showing higher-order terms. It is now a matter of question whether the third term in Eq. (III.3) can also be neglected due to the nature of dark matter model used in this study. To find out, we will use Ishihara (et al.) method Ishihara _et al._ (2016) to purposely show that we can compute the weak deflection angle up to the third-order in $1/\Delta r_{s}$, and compare the result to Eq. (III.3) at least in second-order of $1/\Delta r_{s}$. The formalism developed is very useful in dealing with Kottler spacetime and Schwarzschild-like solutions in the Weyl conformal gravity Mannheim and Kazanas (1989) because it assumes that $u_{S}$ and $u_{R}$ are at finite distance from the black hole. The computation using this method will be the subject of the next section. ## IV Deflection angle using finite distance In this section, we use the method in Ref. Ishihara _et al._ (2016) to calculate the weak deflection angle of a black hole under the influence of dark matter, as demonstrated in Fig. 6. From the GBT, the generalized correspondence between the deflection angle and the surface integral of the Gaussian curvature reads $\hat{\alpha}=\phi_{RS}+\Psi_{R}-\Psi_{S}$ $=\int_{u_{R}}^{u_{o}}\frac{1}{\sqrt{F(u)}}du+\int_{u_{S}}^{u_{o}}\frac{1}{\sqrt{F(u)}}du+\Psi_{R}-\Psi_{S}$ (29) where $F(u)$ and $u_{o}$ are given by Eq. (III.3) and Eq. (26) respectively. The angles $\Psi_{S}$ and $\Psi_{R}$ in Eq. (29) are determined through the inner product of the unit basis vector in the spacetime considered, and the unit vector with respect to the lensing object. The unit basis vector $e^{i}$, along the equatorial plane, is given by $e^{i}=\left(\frac{dr}{dt},0,\frac{d\phi}{dt}\right)=\frac{d\phi}{dt}\left(\frac{dr}{d\phi},0,1\right)$ (30) while the unit radial vector, which is along the radial direction from the lens is $R^{i}=\left(\frac{1}{\sqrt{\gamma_{rr}}},0,0\right).$ (31) Hence, the inner product suggests the definition $\cos\Psi\equiv\gamma_{ij}e^{i}R^{j}$ $\cos\Psi=\sqrt{\gamma_{rr}}\frac{A(r)b}{D(r)}\frac{dr}{d\phi}.$ (32) Using $F(u)$ in Eq. (17), it is easy to see that $\sin\Psi=\sqrt{\frac{A(r)}{D(r)}}b$ (33) where it clear that it is favorable to use $\sin\Psi$ than using $\cos\Psi$. Taylor expansion of $\Delta r_{s}$ as it is very large resulted to an angle $\Psi$ calculated as $\Psi=\arcsin(bu\sqrt{1-2mu})$ $-\frac{3bM}{\Delta r_{s}^{2}}\frac{(r_{s}u-1)^{2}}{\sqrt{1-2mu}\sqrt{b^{2}u^{2}(2mu-1)+1}}$ $-\frac{2bM}{\Delta r_{s}^{3}}\frac{(r_{s}u-1)^{3}}{u\sqrt{1-2mu}\sqrt{b^{2}u^{2}(2mu-1)+1}}+\mathcal{O}\left(\frac{1}{\Delta r_{s}^{4}}\right).$ (34) Continuing, we find that $\Psi_{R}-\Psi_{S}=(\Psi_{R}^{\text{Schw}}-\Psi_{S}^{\text{Schw}})-\frac{3bM}{\Delta r_{s}^{2}}\left[\frac{(r_{s}u_{R}-1)^{2}}{\sqrt{1-b^{2}u_{R}^{2}}}+\frac{(r_{s}u_{S}-1)^{2}}{\sqrt{1-b^{2}u_{S}^{2}}}\right]$ $-\frac{2bM}{\Delta r_{s}^{3}}\left[\frac{(r_{s}u_{S}-1)^{3}}{u_{R}\sqrt{1-b^{2}u_{R}^{2}}}+\frac{(r_{s}u_{S}-1)^{3}}{u_{S}\sqrt{1-b^{2}u_{S}^{2}}}\right]$ $+\frac{3bmM}{\Delta r_{s}^{2}}\left[\frac{u_{R}\left(2b^{2}u_{R}^{2}-1\right)(r_{s}u_{R}-1)^{2}}{\left(1-b^{2}u_{R}^{2}\right)^{3/2}}+\frac{u_{S}\left(2b^{2}u_{S}^{2}-1\right)(r_{s}u_{S}-1)^{2}}{\left(1-b^{2}u_{S}^{2}\right)^{3/2}}\right]$ $+\frac{2bmM}{\Delta r_{s}^{3}}\left[\frac{\left(2b^{2}u_{R}^{2}-1\right)(r_{s}u_{R}-1)^{3}}{\left(1-b^{2}u_{R}^{2}\right)^{3/2}}+\frac{\left(2b^{2}u_{S}^{2}-1\right)(r_{s}u_{S}-1)^{3}}{\left(1-b^{2}u_{S}^{2}\right)^{3/2}}\right]+\mathcal{O}\left(\frac{1}{\Delta r_{s}^{4}}\right)$ (35) where $(\Psi_{R}^{\text{Schw}}-\Psi_{S}^{\text{Schw}})=[\arcsin(u_{R}b)+\arcsin(u_{S}b)-\pi]$ $-bm\left[\frac{u_{R}^{2}}{\sqrt{1-b^{2}u_{R}^{2}}}+\frac{u_{S}^{2}}{\sqrt{1-b^{2}u_{S}^{2}}}\right].$ (36) Note the above expression contains a divergent term in the third-order of $1/\Delta r_{s}$. Hence, the spacetime caused by the combination of black hole and dark matter with very large $\Delta r_{s}$ is non-asymptotic and the limit $u_{S}\rightarrow 0$ and $u_{R}\rightarrow 0$ is not allowed. We now proceed to calculate the $\phi_{RS}$ part by evaluating the integrals in Eq. (29). The function $F(u)$ in Eq. (III.3) results to an integrand of the form $\frac{1}{\sqrt{F(u)}}=\frac{b}{\sqrt{1-b^{2}u^{2}}}-\frac{b^{3}u^{3}m}{\left(1-b^{2}u^{2}\right)^{3/2}}$ $-\frac{3Mb^{3}}{\Delta r_{s}^{2}}\frac{u(r_{s}u-1)^{2}}{\left(1-b^{2}u^{2}\right)^{3/2}}+\frac{9b^{5}mMu^{4}}{\Delta r_{s}^{2}}\frac{(r_{s}u-1)^{2}}{\left(1-b^{2}u^{2}\right)^{5/2}}$ $-\frac{2Mb^{3}}{\Delta r_{s}^{3}}\frac{(r_{s}u-1)^{3}}{\left(1-b^{2}u^{2}\right)^{3/2}}+\frac{6b^{5}mMu^{3}}{\Delta r_{s}^{3}}\frac{(r_{s}u-1)^{3}}{\left(1-b^{2}u^{2}\right)^{5/2}}+\mathcal{O}\left(\frac{1}{\Delta r_{s}^{4}}\right).$ (37) The evaluation of the integral gives a cumbersome expression and reveals no divergence especially in the third-order of $1/\Delta r_{s}$ and hence, can be safely omitted for brevity. Due to Eq. (29), the expression for the source and receiver are essentially the same, and that is $\int\frac{1}{\sqrt{F(u)}}du=\arcsin bu+\frac{m}{b}\frac{\left(b^{2}u^{2}-2\right)}{\sqrt{1-b^{2}u^{2}}}$ $+\frac{9M}{\Delta r_{s}^{2}}\left[\left(m-\frac{2r_{s}}{3}\right)+\frac{5mr_{s}^{2}}{2b^{2}}\right]\arcsin(bu)-\frac{3M}{b\Delta r_{s}^{2}}\frac{\left[b^{2}\left(-r_{s}^{2}u^{2}-2r_{s}u+1\right)+2r_{s}^{2}\right]}{\sqrt{1-b^{2}u^{2}}}$ $-\frac{3mMr_{s}}{2b\Delta r_{s}^{2}}\frac{\left(48r_{s}b^{2}u^{2}+6b^{2}u+15r_{s}^{2}u-32r_{s}\right)}{\left(1-b^{2}u^{2}\right)^{3/2}}+\mathcal{O}\left(\frac{1}{\Delta r_{s}^{3}}\right)+C$ (38) where C is a constant. The expression for $\phi_{RS}$ includes the sum of two evaluated integrals: $\phi_{RS}=\phi_{RS}^{\text{Schw}}-\frac{9M}{\Delta r_{s}^{2}}\left[\left(m-\frac{2r_{s}}{3}\right)+\frac{5mr_{s}^{2}}{2b^{2}}\right]\left[\arcsin(bu_{R})+\arcsin(bu_{S})\right]$ $+\frac{3M}{b\Delta r_{s}^{2}}\left\\{\frac{\left[b^{2}\left(-r_{s}^{2}u_{R}^{2}-2r_{s}u_{R}+1\right)+2r_{s}^{2}\right]}{\sqrt{1-b^{2}u_{R}^{2}}}+\frac{\left[b^{2}\left(-r_{s}^{2}u_{S}^{2}-2r_{s}u_{S}+1\right)+2r_{s}^{2}\right]}{\sqrt{1-b^{2}u_{S}^{2}}}\right\\}$ $\displaystyle+\frac{3mMr_{s}}{2b\Delta r_{s}^{2}}\bigl{[}\frac{\left(48r_{s}b^{2}u_{R}^{2}+6b^{2}u_{R}+15r_{s}^{2}u_{R}-32r_{s}\right)}{\left(1-b^{2}u_{R}^{2}\right)^{3/2}}$ $\displaystyle+\frac{\left(48r_{s}b^{2}u_{S}^{2}+6b^{2}u_{S}+15r_{s}^{2}u_{S}-32r_{s}\right)}{\left(1-b^{2}u_{S}^{2}\right)^{3/2}}\bigr{]}+\mathcal{O}\left(\frac{1}{\Delta r_{s}^{3}}\right)$ (39) where we introduced $\phi_{RS}^{\text{Schw}}=\pi-\arcsin(u_{R}b)-\arcsin(u_{S}b)-\frac{m}{b}\left[\frac{\left(b^{2}u_{R}^{2}-2\right)}{\sqrt{1-b^{2}u_{R}^{2}}}+\frac{\left(b^{2}u_{S}^{2}-2\right)}{\sqrt{1-b^{2}u_{S}^{2}}}\right].$ (40) We can finally calculate the deflection angle $\hat{\alpha}$ using Eq. (29). Combining Eq. (35) and Eq. (IV), we find $\hat{\alpha}\approx\frac{2m}{b}\left[\sqrt{1-b^{2}u_{R}^{2}}+\sqrt{1-b^{2}u_{S}^{2}}\right]$ $-\frac{9M}{\Delta r_{s}^{2}}\left[\left(m-\frac{2r_{s}}{3}\right)+\frac{5mr_{s}^{2}}{2b^{2}}\right]\left[\arcsin(bu_{R})+\arcsin(bu_{S})\right]$ $+\frac{6Mr_{s}^{2}}{b\Delta r_{s}^{2}}\left[\sqrt{1-b^{2}u_{R}^{2}}+\sqrt{1-b^{2}u_{S}^{2}}\right]-\frac{2bM}{\Delta r_{s}^{3}}\left[\frac{(r_{s}u_{R}-1)^{3}}{u_{R}\sqrt{1-b^{2}u_{R}^{2}}}+\frac{(r_{s}u_{S}-1)^{3}}{u_{S}\sqrt{1-b^{2}u_{S}^{2}}}\right]-$ $\frac{3mM}{2\Delta r_{s}^{2}}\left\\{\frac{\left[4b^{2}u_{R}+r_{s}(52b^{2}u_{R}^{2}+15r_{s}u_{R}-32)\right]}{b\left(1-b^{2}u_{R}^{2}\right)^{3/2}}+\frac{\left[4b^{2}u_{S}+r_{s}(52b^{2}u_{S}^{2}+15r_{s}u_{S}-32)\right]}{b\left(1-b^{2}u_{S}^{2}\right)^{3/2}}\right\\}$ $+\frac{2bmM}{\Delta r_{s}^{3}}\left[\frac{\left(2b^{2}u_{R}^{2}-1\right)(r_{s}u_{R}-1)^{3}}{\left(1-b^{2}u_{R}^{2}\right)^{3/2}}+\frac{\left(2b^{2}u_{S}^{2}-1\right)(r_{s}u_{S}-1)^{3}}{\left(1-b^{2}u_{S}^{2}\right)^{3/2}}\right]+\mathcal{O}\left(\frac{1}{\Delta r_{s}^{4}}\right).$ (41) The next procedure is to impose the limit $u_{S}\rightarrow 0$ and $u_{S}\rightarrow 0$ in Eq. (41) which is not problematic in asymptotic spacetimes. However, this limit is not applicable or allowed in Eq. (41) since $\hat{\alpha}$ will apparently diverge. Hence, the reciprocal of $u_{S}$ and $u_{R}$ can be interpreted as finite distances but with very small values. For the far source and receiver, it is then safe to impose $u_{S}<<1$ and $u_{R}<<1$ Ishihara _et al._ (2016); Haroon _et al._ (2019). Eq. (41) then becomes $\hat{\alpha}\approx\frac{4m}{b}+\frac{12Mr_{s}^{2}}{b\Delta r_{s}^{2}}-\frac{96mMr_{s}}{b\Delta r_{s}^{2}}$ $+\frac{2Mb}{\Delta r_{s}^{3}}\left[\left(\frac{1}{u_{R}}+\frac{1}{u_{S}}\right)+2m\right]+\mathcal{O}\left(\frac{1}{\Delta r_{s}^{4}}\right).$ (42) Comparing this result to Eq. (III.3), we see that the first two terms are identical, except the third term, and the existence of terms in the third- order of $1/\Delta r_{s}$ which cannot be derived from the GBT given by Eq. (14). Dark matter mass, being an additional effective mass, makes the $mM$ term to be interpreted as a higher-order term in mass and hence, can safely be neglected. The significant contribution to the weak deflection angle only arises from the first two terms. Moreover, the source and the receiver are inside the dark matter halo, hence $\Delta r_{s}>>1/u_{S}$ and $\Delta r_{s}>>1/u_{R}$ is possible. Hence, we are left with $\hat{\alpha}\approx\frac{4m}{b}+\frac{12Mr_{s}^{2}}{b\Delta r_{s}^{2}}$ (43) and we see that dark matter has an effect to increase the value of the weak deflection angle. See Fig. (7). Moreover, the abnormality of such increase in the weak deflection angle due to the very small $\Delta r_{s}$ might constrain the value of dark matter mass $M$. The weak deflection angle is asymptotic to the Schwarzschild case as $\Delta r_{s}$ increases, as expected. Note that if the mass $M$ is negative, which might represent an exotic matter with negative kinetic term, the weak deflection angle decreases. Figure 7: Weak deflection angle as $\Delta r_{s}$ varies. Here, $r_{s}=2m$ which coincides with the event horizon, and $b=1000m$. We recall the estimate in Ref. Konoplya (2019) that for notable dark matter effects to occur in the shadow radius, the effective radius of the dark matter distribution must be in the order of $\Delta r_{s}\sim\sqrt{3mM}$. This result is found by the same analysis using Fig. 6. For a given value of $M$, it revealed a very small value of $\Delta r_{s}$, which implies that dark matter must be concentrated near the black hole to change the shadow radius considerably. Further analysis also revealed that when the dark matter density is very low ($\Delta r_{s}$ is very large), the effect of dark matter outside the photon’s orbit can be safely neglected. Hence, the dark matter mass beneath the photon’s orbit is the main contributor to any deviations. Such a previous conclusion synchronizes with the results in Eq. (III.3) and Eq. (42). Since dark matter acts as an additional effective mass to the black hole, we can interpret the $mM$ term as a higher-order term in mass, and their coupling can be ignored. Surprisingly, Eq. (43) reveals that dark matter effects will occur in the weak deflection angle (as deviation) when $\Delta r_{s}=2\sqrt{3mM}$. Since there is an increase in the thickness requirement, dark matter detection via deviations in the weak deflection angle of a black hole is better compared when a deviation in shadow radius is used. Unfortunately, however, such an increase is still irrelevant, at least for the technological capabilities we have today. Consider an estimate for dark matter mass in our galaxy which is $M\approx 1.0\times 10^{12}M_{\odot}$ Battaglia _et al._ (2005) while the mass of the central black hole is around $m\approx 4.3\times 10^{6}M_{\odot}$. It gives a required dark matter thickness of $\Delta r_{s}\approx 6.13\times 10^{12}$ m $\approx 2\times 10^{-4}$ pc to see any changes in the weak deflection angle. Such value is lower in many orders of magnitude even if we compare it to the core radius of the dark matter halo present in our galaxy ($r_{o}\approx 15.7-17.46$ kpc) de Oliveira _et al._ (2015). ## V Conclusion In this paper, we present an analytic formula for the weak deflection angle using a simple dark matter model that only incorporates its basic features, such as mass and physical parameters. It is shown that the GBT, with its form given by Eq. (19), can be used only to the second-order of $1/\Delta r_{s}$, but is ill-behaved in the third-order of $1/\Delta r_{s}$. 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2024-09-04T02:54:56.195532

2003.00768

arxiv-papers

# Convolutional Sparse Support Estimator Network (CSEN) From energy efficient support estimation to learning-aided Compressive Sensing Mehmet Yamaç Tampere University, Faculty of Information Technology and Communication Sciences, Tampere, Finland Mete Ahishali Tampere University, Faculty of Information Technology and Communication Sciences, Tampere, Finland Serkan Kiranyaz Department of Electrical Engineering, Qatar University, Qatar Moncef Gabbouj Tampere University, Faculty of Information Technology and Communication Sciences, Tampere, Finland ###### Abstract Support estimation (SE) of a sparse signal refers to finding the location indices of the non-zero elements in a sparse representation. Most of the traditional approaches dealing with SE problem are iterative algorithms based on greedy methods or optimization techniques. Indeed, a vast majority of them use sparse signal recovery techniques to obtain support sets instead of directly mapping the non-zero locations from denser measurements (e.g., Compressively Sensed Measurements). This study proposes a novel approach for learning such a mapping from a training set. To accomplish this objective, the Convolutional Support Estimator Networks (CSENs), each with a compact configuration, are designed. The proposed CSEN can be a crucial tool for the following scenarios: (i) Real-time and low-cost support estimation can be applied in any mobile and low-power edge device for anomaly localization, simultaneous face recognition, etc. (ii) CSEN’s output can directly be used as ”prior information” which improves the performance of sparse signal recovery algorithms. The results over the benchmark datasets show that state-of-the-art performance levels can be achieved by the proposed approach with a significantly reduced computational complexity. ###### Index Terms: Support Recovery, Sparse Signal Representation, Learned Compressive Sensing. ## I Introduction Sparse Representation or Sparse Coding (SC) denotes representing a signal as a linear combination of only a small subset of a pre-defined set of waveforms. Compressive Sensing (CS) [1, 2] can be seen as a special form of SC while a signal, $\mathbf{s}\in\mathbb{R}^{d}$ which has a sparse representation, $\mathbf{x}\in\mathbb{R}^{n}$ in a dictionary or basis $\mathbf{\Phi}\in\mathbb{R}^{d\times n}$, can be acquired in a compressed manner using a linear dimensional reductional matrix, $\mathbf{A}\in\mathbb{R}^{m\times d}$. Therefore, this signal can also be represented in a sparse manner in the dictionary, $\mathbf{D}\in\mathbb{R}^{m\times n}$, (that can be called equivalent dictionary [3], where $m<<n$, and typically assumed to be full-row rank), which is the matrix multiplication of the measurement matrix, $\mathbf{A}$ and pre-defined dictionary, $\mathbf{\Phi}$, i.e., $\mathbf{D}=\mathbf{A}\mathbf{\Phi}$. In SC literature, signal synthesis refers to producing a signal, $\mathbf{y}=\mathbf{Dx}\in\mathbb{R}^{m}$, using a sparse code, $\mathbf{x}\in\mathbb{R}^{n}$ and a pre-specified dictionary, $\mathbf{D}$. On the other hand, signal analysis deals with finding the sparse codes, $\mathbf{x}$ from the given measurements, $\mathbf{y}$, with respect to the dictionary $\mathbf{D}$ [4]. Sparse Support Estimation or simply Support Estimation (SE) [5, 6, 7], refers to finding the location indices of non-zero elements in SCs. In other words, it is the localization of the smallest subset of the atoms, which are the basis waveforms in the dictionary, whose linear combination represents the given signal well enough. On the other hand, sparse Signal Recovery (SR) refers to finding the values of these non-zero elements of SCs. SE and SR are intimately linked in such a way that the SE of a sparse signal is first performed, then an SR will be trivial using the ordinary Least Squares optimization. Actually, this is the main principle of most greedy algorithms [8, 9] Figure 1: The proposed CSEN with two potential applications: a) (bottom-left) Sparse Support Estimation b) (top-middle) Learned aided CS-sparse signal reconstruction with CSEN vs. (top-right) traditional recovery methods; i) OMP [8] and ii) $\ell_{1}$-minimization. The literature that purely targets SE is relatively short compared to extensive studies on sparse signal recovery [10]. Many existing works, first apply a coarse SR using existing SR methods, and then SE can be easily performed if SE is the main objective. Indeed, there are many applications where computing the support set is more important than computing the magnitudes of SCs. For instance, in an SR based classification (SRC) [11], such as face recognition [12], the training samples are stacked in the dictionary in such a way that a subset of the columns consists of the samples of a specific class. As another example, in cognitive radio systems, only a small ratio of all spectrum is occupied for a given time interval. Therefore, finding the occupied spectrum (i.e., the support set) is the primary concern [13, 14]. Similarly, in a ground-penetrating radar imaging system, finding the location of the target is more important than predicting the actual signal magnitudes [15]. In this study, a novel Convolutional Support Estimator Network (CSEN) is proposed with two primary objectives as in Figure 1. First, this approach enables learning-based non-iterative support estimation with minimal computational complexity. To accomplish this, we use two compact Convolutional Neural Network (CNN) configurations, both of which are designed without the dense layers [16]. The proposed CSENs are trained to optimize the support estimations. To our knowledge, this is the first study that proposes a learning-based approach for non-iterative support estimation. Hence, in order to perform comparative evaluations we train the following state-of-the-art CS signal reconstruction deep neural networks as the support estimators: 1) ReconNet [17] that originally works on the spatial domain and, 2) the Learned AMP (LAMP) [18], that is the deep version of AMP [19], which is the state-of- the-art optimization scheme working on sparse domain. An extensive set of experiments over three benchmark datasets has demonstrated that the proposed CSEN approach outperforms both deep counterparts, especially dealing with a structural sparse signal. In the first experimental setup, we simulate a CS system making data acquisition from the MNIST data set in different measurement rates. Moreover, the proposed SE system is shown to improve the SE performance when compared to its deep counterparts, especially in low measurement rates and imperfect sparsity (in the case of CS of approximate sparse signal or noisy environment). Furthermore, CSEN is tested on a well- known support recovery problem, where face recognition is performed based on sparse codes [11]. We use two benchmark datasets, Yale-B [20], and CelebA [21], in our experiments. Comparative evaluations performed against the two state-of-the-art dictionary-based (representation-based) face recognition methods in the literature, SR based face recognition [11], and collaborative learning [22], have demonstrated that the proposed CSEN approach outperformed both methods. As for the second objective, we focus on an alternative usage of CSENs. Instead of using them as support estimators, which naturally requires the hard-thresholding of the network outputs, these outputs can be directly used as prior information about the sparse signals. It is a well-known fact that having prior information about the non-zero locations such as the probability map, $p(x)$ (or simply $\mathbf{p}$), on the support set, could improve the conventional signal recovery algorithms [23]. However, in many cases, it is not clear how to obtain such prior information in advance. The most common usage of such a system appears in dynamical sparse recovery [24], where previous support estimations can be used as priors for the next estimation. In this study, we have demonstrated that CSEN outputs can be a better alternative for the prior information of the non-zero locations. Therefore, CSEN is now used as a learning-aided CS reconstruction scheme, where the prior information comes directly from the CSEN outputs. A wide range of experiments shows that this approach has great potential to improve SR performance of traditional approaches for sparse signal recovery problems. As mentioned above, we used CS imaging simulation, but this time signal reconstruction error is compared with state-of-the-art conventional SR approaches. Figure 1 illustrates a representative graph of two different applications of CSENs; a) performing SE from CS measurement vector, $\mathbf{y}$, and b) the output of CSEN is used as the side information, $\mathbf{p}$, which gives the estimated probability of being non-zero for each index. In this simple illustration we assume that the hand-writing signal ’$2$’ is sparse in spatial domain such that $\mathbf{\Phi}=\mathbf{I}$; therefore, $\mathbf{D}=\mathbf{A}\mathbf{I}=\mathbf{A}$ and $\mathbf{B}$ is a denoiser matrix such as $\mathbf{D}^{T}$, or $\left(\mathbf{D}^{T}\mathbf{D}+\lambda\mathbf{I}\right)^{-1}\mathbf{D}^{T}$ where $\lambda$ is the regularization parameter. The rest of the paper is organized as follows. In Section II, we start by giving mathematical notation that is used in this article. A brief overview of sparse representation and compressive sensing theory, with emphasis on state- of-art sparse signal recovery and support estimation techniques, will be given in Section III. In the same section, we also introduce case studies of support estimation that are chosen for this work. Then we discuss the limitations of existing support estimator techniques. In Section IV, we will present the proposed learned based SE scheme and the two compact CSEN models. Experimental evaluations of the study will also be given at the end of this section, which we can divide into three main categories according to the case studies: (i) Basic support estimation performance evaluation on MNIST dataset that is performed to compare CSENs with the aforementioned state-of-art deep networks. (ii) SE based face recognition performance evolution of proposed SE with an emphasis on how CSEN based SE has the ability to improve the classical representation-based approaches. (iii) Performance comparison of classical compressing sensing reconstruction techniques and proposed learned-aided SR in terms of both speed and reconstruction accuracy. Having theoretical and experimental analysis, in Section VI we will present a more detailed discussion on how the proposed scheme differs from the state-of-the-art SR and SE techniques, pros and cons, and possible usage scenarios with an emphasis on the flexibility of proposed CSEN in different scenarios. Finally, the conclusions are drawn in Section VII. ## II Notations In this work, we define the $\ell_{p}$ norm of any vector $\mathbf{x}\in\mathbb{R}^{n}$ as $\left\|\mathbf{x}\right\|_{\ell_{p}^{n}}=\left(\sum_{i=1}^{n}\left|x_{i}\right|^{p}\right)^{1/p}$ for $p\geq 1$. The $\ell_{0}$-norm of the vector $\mathbf{x}\in\mathbb{R}^{n}$ is given as $\left\|\mathbf{x}\right\|_{\ell_{0}^{n}}=\lim_{p\to 0}\sum_{i=1}^{n}\left|x_{i}\right|^{p}=\\#\\{j:x_{j}\neq 0\\}$ and the $\ell_{\infty}$ is defined as $\left\|\mathbf{x}\right\|_{\ell_{\infty}^{n}}=\max_{i=1,...,n}\left(\left|x_{i}\right|\right)$. A signal $\mathbf{s}$ can be defined as a strictly $k$-sparse signal if it can be represented with less than $k+1$ non-zero coefficients in a proper basis $\mathbf{\Phi}$, i.e., $\left\|\mathbf{x}\right\|_{0}\leq k$, where $\mathbf{s}=\mathbf{\Phi}\ \mathbf{x}$. We also define a sparse support set or simply support set, $\Lambda\subset\\{1,2,3,...,n\\}$, as the set of indices that represents the non-zero coefficients, i.e., $\Lambda:=\left\\{i:x_{i}=0\right\\}$. The complement of support set, $\Lambda$, with respect to $\\{1,2,3,...,n\\}$ is given as $\Lambda^{c}=\\{1,2,3,...,n\\}\setminus\Lambda$. In this manner, $\mathbf{x}_{\Lambda}\in\mathbb{R}^{\left|\Lambda\right|}$ is a vector consisting of non-zero elements of $\mathbf{x}\in\mathbb{R}^{n}$, where $\left|\Lambda\right|$ refers to the number of the non-zero coefficients . Similarly, $\mathbf{M}_{\Lambda}\in\mathbb{R}^{m\times\left|\Lambda\right|}$ denotes a matrix that consists of the columns of a matrix $\mathbf{M}\in\mathbb{R}^{m\times n}$ indexed by support ${\Lambda}$. ## III Related Work CS theory claims that a signal $\mathbf{s}$ can be sensed using far fewer linear measurements $m$ than Nyquist/Shannon based traditional methods use, $d$, i.e., $\mathbf{y}=\mathbf{A}\mathbf{s}=\mathbf{A}\mathbf{\Phi}\mathbf{x}=\mathbf{D}\mathbf{x},$ (1) where $\mathbf{A}\in\mathbb{R}^{m\times d}$ is the measurement matrix and $\mathbf{D}\in\mathbb{R}^{m\times n}$ is called the equivalent dictionary. It can be demonstrated that sparse representation, $\min_{\mathbf{x}}~{}\left\|\mathbf{x}\right\|_{0}~{}\text{subject to}~{}\mathbf{D}\mathbf{x}=\mathbf{y}$ (2) is unique if $m\geq 2k$ [25] and $\left\|\mathbf{x}\right\|_{0}\leq k$. In brief, the uniqueness of the sparse representation in Eq. (2) shows that any $k$-sparse signal pair can still be distinguished in the equivalent dictionary, $\mathbf{D}$. However, the problem in Eq. (2) is that this is a non-convex problem and known to be NP-hard. The most common approach is the relaxation of the $\ell_{0}$ norm to the closest convex norm, which is $\ell_{1}$-norm, $\min_{\mathbf{x}}\left\|\mathbf{x}\right\|_{1}~{}s.t.~{}\mathbf{x}\in\mho\left(\mathbf{y}\right)$ (3) where $\mho\left(\mathbf{y}\right)=\left\\{\mathbf{x}:\mathbf{D}\mathbf{x}=\mathbf{y}\right\\}$, which is known as Basis Pursuit [26]. The surprising result of the CS theory is that even if the exact recovery of the signal, $\mathbf{s}$ was not possible by using the minimum norm solution, a tractable solution is possible using (3), when $\mathbf{D}$ satisfies some properties such as Restricted Isometry Property [27] and $m>k(\log(n/k))$. However, the signal of interest, $\mathbf{x}$, is not perfectly $k$-sparse but approximately sparse in most of the cases. In addition, CS measurements, most probably, are corrupted by an additive noise during data acquisition, quantization, etc. As a result, we handle $\mathbf{y}=\mathbf{D}\mathbf{x}+\mathbf{z}$, where $\mathbf{z}$ is additive noise. In this case, the constraint can be relaxed by setting $\mho\left(\mathbf{y}\right)=\left\\{\mathbf{x}:\left\|\mathbf{D}\mathbf{x}-\mathbf{y}\right\|_{2}\leq\epsilon\right\\}$ which is known as Basis Pursuit Denoising [28] or Dantzig Selector [29] if we set $\mho\left(\mathbf{y}\right)=\left\\{\mathbf{x}:\left\|\mathbf{D}^{T}\left(\mathbf{y}-\mathbf{D}\mathbf{x}\right)\right\|_{\infty}\leq\lambda\right\\}$. In the noisy case, even exact recovery of sparse signal is not possible, stable recovery is well studied in the literature for Basis Pursuit Denoising [30] and Dantzig Selector [31, 32]. We mean by stable recovery is that a stable solution $\hat{\mathbf{x}}$ obeys $\left\|\mathbf{x}-\hat{\mathbf{x}}\right\|\leq\kappa\left\|\mathbf{z}\right\|$ where the $\kappa$ is small constant. Another related formulation is $\min_{\mathbf{x}}\left\\{\left\|\mathbf{D}\mathbf{x}-\mathbf{y}\right\|_{2}^{2}+\lambda\left\|\mathbf{x}\right\|_{1}\right\\}$ (4) which is known as Lasso [33] formulation, which is also known to produce stable solution in noisy case and exact solution in noise free case [34]. Figure 2: Most common model for a practical support estimator. ### III-A Generic Sparse Support Estimation (SE) In many application scenarios, detecting the indices of the non-zero coefficients’ location, $\Lambda$, is more important than computing these coefficients. To list a few, in a sparse anomaly (either from CS [35] or uniform sampled measurements) detection problem [36], where a group of users initiates a flooding attack to a communication network (specifically for a VoIP network), detecting the malicious user group (sub-set of all users) is more critical. Other, CS-based active user detection in the downlink of a CDMA system [37], and for the uplink of a NOMA [38, 39] system. Such systems are believed to play an important role in 5G communication technology. As discussed in Section I, other examples may be listed as sparse representation based classifications [11, 12] and radar imaging [15, 40]. Mathematically speaking, for the linear measurement model given in Eq. (1) and with additive noise, $\mathbf{y}=\mathbf{Dx}+\mathbf{z}$, we define the following support estimator $\mathcal{E}(.,.)$, $\hat{\Lambda}=\mathcal{E}\left(\mathbf{y},\mathbf{D}\right)$ (5) where $\hat{\Lambda}$ is the estimated support. For the noise-free case and $\mathbf{x}$ is exactly $k$-sparse, the exact $\Lambda$ recovery performance of an algorithm coincides with the sparse signal recovery performance. This an expected outcome since the unique representation is satisfied when $m>2k$. In the noisy case, even if the exact signal recovery is not possible, it is still possible to recover the support set exactly. In the literature, several studies have proposed to provide information-theoretical (i.e., the optimal decoder, $\mathcal{E}$’s performance) guarantee conditions for exact [41, 5, 42, 10], and partial support estimation [7, 43, 10]. However, in most of the practical applications, a tractable signal recovery method is applied first to find an estimation $\hat{\mathbf{x}}$ of the sparse signal $\mathbf{x}$, then a component-wise thresholding is applied to $\hat{\mathbf{x}}$ to compute the estimated support as illustrated in Figure 2. A common approach is to follow an iterative sparse signal recovery method from the CS literature. For instance, it is proven in [44] that if $min_{i\in\Lambda}\left|x_{i}\right|>8\sigma\sqrt{2*log(n)}$, then one can recover the support set exactly using Lasso with $\lambda=2\sqrt{2*log(n)}$, where $\sigma^{2}$ is variance of the measurement noise. This theorem is valid in the case that the equivalent dictionary satisfies the mutual coherence property defined in [44]. One may clearly deduce from their results that accurate support estimation is possible via Lasso if the non-zero coefficients’ magnitudes are above a certain level determined by the noise. Similarly, the conditions of exact support recovery under noise using OMP are given in [45], and partial support recovery performance bounds of AMP are in [46]. Along with these SR algorithms in CS literature, which are iterative methods, traditional linear decoders such as Maximum Correlation (MC) [47], $\hat{\mathbf{x}}^{MC}=\mathbf{D}^{T}\mathbf{y}$ and LMSEE [46], $\hat{\mathbf{x}}^{LMMSE}=\left(\mathbf{D}^{T}\mathbf{D}+\sigma_{z}^{2}\mathbf{I}_{n\times n}\right)^{-1}\mathbf{D}^{T}\mathbf{y}$ are also used in many applications. The theoretical performance bounds of these methods are also given in [46]. ### III-B Case study of SE: Representation based classification Consider an image from a particular class is queried: It can be expected from the estimated SCs, $\hat{\mathbf{x}}$ to have significant (non-zero) entries which are located in a specific location so that the corresponding columns in the dictionary matrix, $\mathbf{D}$, are the samples from the actual class of the image. This problem is also known as the representation based classification, which is a typical example where the support set location is the main information we are seeking. In [11], $\ell_{1}$-minimization is used to obtain such a sparse code to determine the identity of face images. However, in reality such an ideal decomposition is not accomplished in general because face images show a high correlation among different classes. This is why, instead of using the estimated sparse codes, $\hat{\mathbf{x}}$ obtained by an SR technique such as Eq. (4), the authors propose a four steps solution: (i) Normalization: Normalize all the atoms in $\mathbf{D}$ and $\mathbf{y}$ to have unit $\ell_{2}$-norm, (ii) SR: $\hat{\mathbf{x}}=\arg\min_{\mathbf{x}}\left\|\mathbf{x}\right\|_{1}\text{s.t}\left\|\mathbf{y}-\mathbf{D}\mathbf{x}\right\|_{2}$, (iii) Residual finding: $\mathbf{e_{i}}=\left\|\mathbf{y}-\mathbf{D_{i}}\mathbf{\hat{x}_{i}}\right\|_{2}$, where $\mathbf{\hat{x}_{i}}$ is the estimated coefficients corresponding the class $i$, (iv) Class determination: $\text{Class}\left(\mathbf{y}\right)=\arg\min\left(\mathbf{e_{i}}\right)$. This technique and its similar variants have been reported to perform well not only in face recognition but many other classification problems [48, 49]. Later, the authors of [22] propose to change the second step, from $\ell_{1}$-minimization to the classical $\ell_{2}$-minimization; $\mathbf{\hat{x}}=\arg\min_{\mathbf{x}}\left\\{\left\|\mathbf{y}-\mathbf{D}\mathbf{x}\right\|_{2}^{2} +\lambda\left\|\mathbf{x}\right\|_{2}^{2} \right\\}$, which has a closed-form solution, $\hat{\mathbf{x}}=\left(\mathbf{D}^{T} \mathbf{D}+\lambda\mathbf{I}_{n\times n} \right)^{-1}\mathbf{D}^{T} \mathbf{y}$. This collaborative representation based classification (CRC) was reported to achieve a comparable classification performance for different classification problems. For face recognition problems, in particular, the authors reported that high classification accuracies were obtained especially for high measurement rates (MRs). Figure 3: Proposed model for an efficient support estimator. ### III-C Sparse signal reconstruction with side information of support set Consider the case where SE is not the main concern but SR is. In case side information is available about the support set, an improvement to $\ell_{1}$-minimization can be achieved in sparse signal recovery as follows: $\min_{\mathbf{x}}\left\\{\left\|\mathbf{D}\mathbf{x}-\mathbf{y}\right\|_{2}^{2}+\lambda\left\|\mathbf{w\odot x}\right\|_{1}\right\\}$ (6) where $\odot$ is element-wise multiplication operator and $\mathbf{w}$ is the predefined cost that imposes the prior information about each element’s values. In the concept of modified CS [50] and CS with prior information literature, the cost function, $\mathbf{w}$ generally appears in the form of $w_{i}=\frac{1}{p_{i}+\epsilon}$, where $\epsilon>0$ is a predefined constant and $p_{i}$ is the $i^{th}$ element of the vector $\mathbf{p}$ which is a type of a measure such as prior likelihood [23] of the support set, which could represent the probability of $(i)^{th}$ element being non-zero. ### III-D Limitations of existing Support Estimators Both SE and SR algorithms guarantee to perform well if the equivalent dictionary $\mathbf{D}$ satisfies certain properties such as mutual incoherence [51]. However, in many practical scenarios, $\mathbf{D}$ fails to satisfy these properties, e.g., in face recognition problem, the atoms of $\mathbf{D}$, vectorized faces, are highly correlated. The second limitation of traditional sparse recovery algorithms is that they are iterative methods and computationally costly. Therefore, the support estimators relying on these sparse recovery algorithms may not be feasible, especially in real-time applications. The third limitation of state-of-the-art SR techniques such as $\ell_{1}$-minimization is that there is a lower limit for MR (see phase transition [52]); below this limit, the SR algorithms start to fail completely. This limit generally depends on the wellness of $\mathbf{D}$ (defined by properties such as mutual incoherence [51] ). Therefore, SE techniques that build upon an SR algorithm tend to fail if $\mathbf{D}$ does not satisfy the required properties, e.g., if the atoms of $\mathbf{D}$ are highly correlated. On the other hand, when it comes to SR techniques leveraging SE as prior information, despite the fact that a good improvement can be achieved using such prior information, most of the works assume that the information is available in advance; however, they do not mention how to obtain such a $\mathbf{p}$. ## IV Convolutional Support Estimator Network Recent advance in deep neural networks [53, 18] enables a non-iterative solution for the sparse signal recovery. It is often reported that they produce a solution $\mathbf{\hat{x}}$, which is closer to $\mathbf{x}$ than the ones obtained by an iterative approach. They can still work under those measurement rates where classical CS recovery algorithms fail. Nevertheless, a crucial disadvantage is that they require a large number of training samples to achieve a high generalization capability. Second, their complex configuration with millions of parameters causes certain computational complexity issues such as speed and memory problems, especially when they are used in edge devices with limited power, speed and memory. If one may wish to find only support $\Lambda$ instead of the sign and amplitude of $\mathbf{x}$, a traditional Machine Learning approach would be sufficient. In this study, we propose a support estimator, $\mathcal{E}(.)$, that can be performed by a compact CSEN network. Another crucial objective is to have the ability to learn from a minimal training set with a limited number of labeled data. A typical application where this approach can benefit from is face recognition via sparse representations, where only a few samples of each identity are available. Let us define a binary mask $\mathbf{v}\in\left\\{0,1\right\\}^{n}$, as follows $\displaystyle v_{i}=$ $\displaystyle 1\hfill$ if $i\in\Lambda$ (7a) $\displaystyle v_{i}=$ $\displaystyle 0$ else . (7b) Consequently, the problem of finding an estimation $\hat{\mathbf{v}}$ of this binary mask will be equivalent to producing a support estimation $\hat{\Lambda}$, i.e., $\hat{\Lambda}=\left\\{i\in\left\\{1,2,..,n\right\\}:\hat{v}_{i}=1\right\\}$. To accomplish this objective, first, the CSEN network with input and output, $\mathcal{P}\left(\mathbf{y},\mathbf{D} \right):\mathbb{R}^{n} \mapsto\left[0,1\right]^{n}$, produces a vector $\mathbf{p}$ that gives the information about the probability of each index to be in support set such that $p_{i}\in\left[0,1\right]$. Then, the final support estimator, $\mathcal{E}(\mathbf{y},\mathbf{D})$ will produce a support estimation such that $\hat{\Lambda}=\left\\{ i\in \left\\{ 1,2,..,n\right\\}:p_{i}>\tau \right\\}$, by thresholding $\mathbf{p}$ with $\tau$ where $\tau$ is a fixed threshold. As shown in Figure 3, the proposed SE approach is different from the conventional SR based methods, which directly threshold $\hat{\mathbf{x}}$ for support estimation. Moreover, the input-output pair is different. The proposed CSEN learns over $\left(y^{train},v^{train}\right)$ to compute $\mathbf{p}$ while the conventional SR methods work with $\left(y^{train},x^{train}\right)$ to first make the sparse signal estimation, and then compute support estimation by thresholding it. As evident in Figure 1, the application of direct signal recovery may cause noisy estimation of the support codes while the proposed CSEN has the advantage of learning the pattern of the support codes and, therefore, can predict their most-likely location with proper training. Figure 4: Type-I Convolutional Support Estimator Network (CSEN1). Figure 5: Type-II Convolutional Support Estimator Network (CSEN2). In this study, the proposed CSEN models consist of only convolutional layers in the type of fully convolutional networks [16] that are trained by optimizing the support estimations. Since the SE problem involves one-to-one mapping, other network types such as Multi-Layer Perceptrons (MLPs) can also be used as in [18]. However, this brings two limitations compared to CSENs: high computational complexity and over-fitting due to the limited training data and number of parameters in the network. In Section V, it will be shown that such an approach yields a poor generalization and is not robust to noise. When a CSEN is trained, it learns the following transformation: $\hat{\mathbf{v}}\leftarrow \mathcal{P}\left(\mathbf{\tilde{x}}\right)$, where $\hat{\mathbf{v}}$ is the estimation of binary mask representing the estimated support for the signal $\mathbf{x}$, and the proxy $\mathbf{\tilde{x}=By}$ with $\mathbf{B=D^{T}}$, or $\left(\mathbf{D}^{T}\mathbf{D}+\lambda\mathbf{I}\right)^{-1}\mathbf{D}^{T}$, i.e., the Maximum Correlation and LMMSE formula in [46]; and hence, $\mathbf{x},\mathbf{\tilde{x}}\in\mathbb{R}^{N}$. First, the proxy $\mathbf{\tilde{x}}$ is reshaped to 2-D plane (e.g., original size of the image or pre-defined search grid). Correspondingly, the proxy $\mathbf{\tilde{x}}$ is convolved with $\mathbf{W_{1}}$, the weight kernels connecting the input layer to the next layer with $N$ filters to form the input of the next layer with the summation of weight biases $\mathbf{b_{1}}$ as follows: $\mathbf{F_{1}}=\\{S(ReLu(b_{1}^{i}+\mathbf{w}_{1}^{i}*\tilde{\mathbf{x}}))\\}_{i=1}^{N},$ (8) where $S(.)$ is the down- or up-sampling operation and $ReLu(x)=max(0,x)$. In more general form, the $k^{th}$ feature map of layer $l$ can be expressed as, $\mathbf{f_{l}^{k}}=\textsc{S}(\textsc{ReLu}(b_{l}^{k}+\sum_{i=1}^{N_{l-1}}\textsc{conv2D}(\mathbf{w}_{l}^{ik},\mathbf{f}_{l-1}^{i},^{\prime}\textsc{ZeroPad}^{\prime}))).$ (9) The trainable parameters of the network would be: $\mathbf{\Theta_{CSEN}}=\big{\\{}\\{\mathbf{w}_{1}^{i},b_{1}^{i}\\}_{i=1}^{N_{1}},\\{\mathbf{w}_{2}^{i},b_{2}^{i}\\}_{i=1}^{N_{2}},...\\{\mathbf{w}_{L}^{i},b_{L}^{i}\\}_{i=1}^{N_{L}}\big{\\}}$ for a L layer CSEN. In the proposed approach, the Mean-Square Error (MSE) is computed between its binary mask, $\mathbf{v}$, and CSEN’s actual output, $\mathcal{P}_{\Theta}\left(\mathbf{x}\right)_{p}$ as follows: $E(\mathbf{x})=\sum_{p}\left(\mathcal{P}_{\Theta}\left(\mathbf{x}\right)_{p}-v_{p}\right)^{2}$ (10) where $v_{p}$ is the $p^{th}$ pixel of $\mathbf{v}$. The CSEN network is trained using samples in the train data, $D_{train}=\big{\\{}(\tilde{\mathbf{x}}^{(1)},\mathbf{v}^{(1)}),(\tilde{\mathbf{x}}^{(2)},\mathbf{v}^{(2)}),...(\tilde{\mathbf{x}}^{(s)},\mathbf{v}^{(s)})\big{\\}}$. Please note that, even if we use MSE as the loss function in the original CSEN design, depending on the application, any other regularization function (e.g., $\ell_{1}$-norm, mixed norm, etc.) can be added to this cost function. As an example, we present a strategy to approximate the loss function which is group $\ell_{1}$-norm in addition to MSE. ## V Results In order to evaluate the effect of different network configurations, in this study, we use two different CSEN configurations and perform a comprehensive analysis over each of them. Generally, each convolutional layer has a dimension reduction capability with pooling functions. However, the first proposed network architecture consists of only convolutional layers with ReLu activation functions to preserve the sparse signal (e.g., image) dimensions at the output layer. In this configuration (CSEN1), we propose to use three convolutional layers with $48$ and $24$ hidden neurons and $3\times 3$ filter size as given in Figure 4. CSEN2 is a slight modification of CSEN1 configuration, as shown in Figure 5 by using up- and down-sampling layers. Although this modification increases the number of parameters, in return, it yields substantial performance improvement over MNIST. While the SE performance analysis over MNIST has done using CSEN1 and CSEN2, only CSEN1 results are reported since CSEN2 produces similar recognition rates ($\sim$ 0.001 difference) for face recognition. In any case, both network configurations are compact compared to the deep CNNs that have been proposed recently. For example, the study in [17] proposes ReconNet for SR, which consists of six convolutional layers with $32$ neurons or more in each layer. Since there is no competing method for SE that is similar to the proposed method, we use the ReconNet [17] in this study on the SE problem by directly giving $\mathbf{\tilde{x}}$ as the input, and removing the denoiser block at the end for comparative evaluations. Finally, we apply thresholding over the output of ReconNet to generate SE i.e., $\hat{\Lambda}_{\text{R}}=\left\\{ i\in \left\\{ 1,2,..,n\right\\}:\mathcal{P}_{\text{R}}\left(\mathbf{\tilde{x}}\right)>\tau \right\\}$, where $\mathcal{P}_{\text{R}}(.)$ is ReconNet with fully convolutional layers. ReconNet is originally a CS recovery algorithm working directly on spatial domain, i.e., $\mathbf{\hat{s}}\leftarrow\mathcal{P}\left(\mathbf{y}\right)$ instead of solving them in the sparsifying dictionary, i.e., $\mathbf{\hat{s}}=\mathbf{\Phi}\mathbf{\hat{x}}$ where $\mathbf{\hat{x}}\leftarrow\mathcal{P}\left(\mathbf{y}\right)$. Therefore, ReconNet serves as a deep CSEN approach against which the performance of the two compact CSENs will be compared. Moreover, we also train the state-of-the- art deep SR solution, LAMP, in order to use it over SE problem. For LAMP method, it is possible to predefine the number of layers in advance. For a fair comparison, we have tested the algorithm for three different setups: $2$, $3$, and $4$ layers design using their provided implementation. Next, in the experiments of face recognition based on SR, we consider both speed and recognition accuracy of the algorithms as it is performed only for $\ell_{1}$-minimization toolbox in [54]. Thus, in order to perform comparative evaluations, the proposed CSEN approach is evaluated against most of the conventional state-of-the-art SR techniques along with ReconNet. Finally, CSEN2 is applied as a pre-processing step for the CS-recovery to obtain $\mathbf{w}$ in the cost function as illustrated in Figure 1. Figure 6: Histogram of $\rho_{i}$’s obtained from the 10k samples (test set). The vectorized gray scale images, $\mathbf{x}_{i}$ in MNIST dataset are already sparse in the spatial domain (in canonical basis, i.e., $\Phi=I$) with $\left\|\mathbf{x}_{i}\right\|\leq k_{i}$. The experiments in this study have been carried out on a workstation that has four Nvidia® TITAN-X GPU cards and Intel ® Xeon(R) CPU E5-2637 v4 at 3.50GHz with 128 GB memory. Tensorflow library [55] is used with Python. ADAM optimizer [56] is utilized during the training with the proposed default values of the learning parameters: learning rate, $\alpha=0.001$, and moment updates $\beta_{1}=0.9$, $\beta_{2}=0.999$ with only 100 and 30 Back- Propagation iterations for MNIST and face recognition experiments, respectively. ### V-A Experiment I: Support Estimation from CS measurements For the experiments in this section, MNIST dataset is used. This dataset contains 70000 samples (50K/10K/10K as the sizes of the train/validation/test sets) of the handwritten digits (0 to 9). Each image in the dataset is a $28\times 28$ pixel resolution with intensity values ranging from 0 (black, background) to 1 (white, foreground). Since the background covers more area than the foreground, each image can be considered as a sparse signal. Mathematically speaking, we may assume that the $i^{th}$ vectorized, $\mathbf{x}_{i}\in\mathbb{R}^{n=784}$ can be considered as the $k_{i}$-sparse signal. The sparsity rates of each sample is calculated as $\rho_{i}=\frac{k_{i}}{n}$, and its histogram is given in Figure 6. We have designed an experimental setup where these sparse signals (sparse in canonical basis) $\mathbf{x}_{i}$’s are compressively sensed, $\mathbf{y_{i}}=\mathbf{A}\mathbf{{x_{i}}}=\mathbf{D}\mathbf{x_{i}}$ (11) where $\mathbf{D}=\mathbf{A}\in\mathbb{R}^{m\times n}$ since $\mathbf{\Phi}=\mathbf{I}$. We calculate the measurement rate as $\mathbf{MR}=\frac{m}{n}$. Therefore, the problem is SE from each CS measurement, i.e., finding $\hat{\Lambda}_{i}$ from each $\mathbf{y_{i}}$ in the test dataset. Figure 7: F1 Measure graph of CSEN and Lamp configurations in different noise level at MR = 0.25. TABLE I: Support Recovery Performance of Algorithms from the noise-free measurements. MR | 0.25 | 0.1 | 0.05 | 0.25 | 0.1 | 0.05 | 0.25 | 0.1 | 0.05 | 0.25 | 0.1 | 0.05 ---|---|---|---|---|---|---|---|---|---|---|---|--- | F1 Measure | Precision | Recall | CE CSEN | 0.91 | 0.85 | 0.80 | 0.90 | 0.84 | 0.77 | 0.92 | 0.87 | 0.84 | 0.03 | 0.06 | 0.08 CSEN2 | 0.94 | 0.89 | 0.84 | 0.93 | 0.88 | 0.82 | 0.94 | 0.90 | 0.87 | 0.02 | 0.04 | 0.06 ReconNet | 0.90 | 0.85 | 0.79 | 0.89 | 0.82 | 0.76 | 0.90 | 0.87 | 0.83 | 0.05 | 0.06 | 0.09 LAMP (2) | 0.92 | 0.89 | 0.82 | 0.94 | 0.90 | 0.82 | 0.89 | 0.87 | 0.83 | 0.05 | 0.05 | 0.08 LAMP (3) | 0.93 | 0.89 | 0.82 | 0.95 | 0.90 | 0.82 | 0.91 | 0.88 | 0.82 | 0.03 | 0.05 | 0.08 LAMP (4) | 0.93 | 0.90 | 0.83 | 0.95 | 0.92 | 0.82 | 0.92 | 0.89 | 0.83 | 0.03 | 0.04 | 0.08 For this dataset, the measurement rate, (MR) is varied from $0.05$ to $0.25$ in order to investigate the effect of MR on the SE performance. The measurement matrix is then chosen as the ”Gaussian”, the matrix whose elements $A_{i,j}$ are i.i.d. drawn from $\mathcal{N}\left(0,\frac{1}{m}\right)$. It is worth mentioning that the approximate message passing (AMP) algorithm is a well-optimized method for the Gaussian measurement matrix, and LAMP is a learned version of this algorithm. Therefore, they are reported to be state- of-the-art if the measurement matrix is Gaussian but they do not even guarantee the converge for other types of measurement matrices. On the other hand, the comparative performance evaluations against LAMP and deep CS-sparse methods are presented in Table I, and the results clearly indicate that the proposed method achieves the best SE performance in terms of F1 measure for MR = 0.25 and 0.05 and comparable for MR = 0.1. The results presented in Table I indicate that despite its deep and complex configuration, compact CSENs achieve superior performance levels compared to ReconNet. TABLE II: Support Recovery Performance of Algorithms under 10 dB measurement noise. MR | 0.25 | 0.1 | 0.05 | 0.25 | 0.1 | 0.05 | 0.25 | 0.1 | 0.05 | 0.25 | 0.1 | 0.05 ---|---|---|---|---|---|---|---|---|---|---|---|--- | F1 Measure | Precision | Recall | CE CSEN | 0.89 | 0.82 | 0.77 | 0.89 | 0.82 | 0.75 | 0.89 | 0.82 | 0.79 | 0.04 | 0.07 | 0.09 CSEN2 | 0.92 | 0.86 | 0.80 | 0.92 | 0.86 | 0.80 | 0.92 | 0.86 | 0.82 | 0.03 | 0.06 | 0.08 ReconNet | 0.89 | 0.83 | 0.78 | 0.89 | 0.81 | 0.74 | 0.89 | 0.85 | 0.81 | 0.04 | 0.07 | 0.09 LAMP (2) | 0.87 | 0.85 | 0.79 | 0.90 | 0.86 | 0.78 | 0.84 | 0.83 | 0.80 | 0.08 | 0.08 | 0.10 LAMP (3) | 0.87 | 0.84 | 0.77 | 0.91 | 0.87 | 0.78 | 0.84 | 0.81 | 0.77 | 0.06 | 0.08 | 0.12 LAMP (4) | 0.86 | 0.85 | 0.77 | 0.87 | 0.87 | 0.78 | 0.85 | 0.82 | 0.77 | 0.08 | 0.07 | 0.12 Furthermore, comparative evaluations are performed when the measurements are exposed to noise in the test set, i.e., $\mathbf{y}_{i}=\mathbf{D}\mathbf{x}_{i}+\mathbf{z}_{i}$, where $\mathbf{z}_{i}$ is an additive white Gaussian noise. The results presented in Figure 7 show that SE performances of the LAMP method are adversely affected by increased measurement noise. Their performance gets even worse when the number of layers is increased (i.e., see results for LAMP (2) to LAMP (4)). CSEN2, on the other hand, achieves the highest F1 measure for all noise levels. ### V-B Experiment II: Face Recognition based on Sparse Representation Figure 8: Reconstruction accuracy vs. process time comparison of Algorithms in Yale-B database. As explained in Section III-B, the dictionary-based (representation based) classification could be seen as a SE problem. Therefore, CSEN presents an alternative and better approach to both CRC and SRC solutions. In this manner, the proposed CSEN approach is evaluated against both CRC and the state-of-the- art SRC techniques recently proposed. The algorithms are chosen by considering both their speed and performance on the SR problem, since the speed-accuracy performance of SRC directly depends on the performance of the sparse signal recovery algorithm [54], and there is no unique winner to achieve the top performance level for all databases. The proposed method is, of course, not limited to face recognition but can be applied in any other representation based classification problem. For the sake of brevity, in this study, we focus only on the face recognition problem. Figure 9: Reconstruction accuracy vs. process time comparison of Algorithms in CelebA database. In dictionary-based classification designs, the samples of a specific class are stacked in the dictionary as atoms with pre-defined indices, e.g., the atoms belonging to a particular class can be located in concatenate manner. Consequently, in sparse representation based classification, instead of using $\ell_{1}$-minimization in Eq. (4), group $\ell_{1}$-minimization can be introduced as follows, $\min_{\mathbf{x}}\left\\{\left\|\mathbf{D}\mathbf{x}-\mathbf{y}\right\|_{2}^{2}+\lambda\sum_{i=1}^{c}\left\|\mathbf{x_{Gi}}\right\|_{2}\right\\}.$ (12) where $\mathbf{x_{Gi}}$ is the group of coefficients corresponds to class $i$. Hence, the MSE cost function in Eq. (10) can be modified accordingly: $E(\mathbf{x})=\sum_{p}(\mathcal{P}_{\Theta}\left(\mathbf{x}\right)_{p}-v_{p})^{2}+\lambda\sum_{i=1}^{c}\left\|\mathcal{P}_{\Theta}\left(\mathbf{x}\right)_{Gi}\right\|_{2}.$ (13) To approximate this, a simple average pooling can be applied after the last layer of CSEN which is then followed by SoftMax function to produce class probabilities. Therefore, the modified cost function with Cross-Entropy loss at the output would be: $E(\mathbf{x})=-\sum_{i}^{C}t_{i}log\left(\mathcal{P}_{\Theta}(\mathbf{x})\right)$ where $t_{i}$ and $\mathcal{P}_{\Theta}(\mathbf{x})$ are the real and the predicted values by CSEN, respectively, for class $i\in C$. By this way, the modified network can directly produce the predicted class labels as the output. In the experiments, we have used Yale-B [20] and CelebA [21] databases. In Yale-B dataset, there are $2414$ face images with 38 identities; and a sub-set of CelebA is chosen with 5600 images and 200 identities. The face recognition experiments are repeated 5 times with samples randomly selected to build the dictionary, train, and test sets with 32, 16, 16, and 8, 12, 8 samples each for Yale-B and CelebA, respectively, and 25% of training data is separated as validation set. The selected sub-set of CelebA dataset is also different between each repeated run. For Yale-B database, we use vectorized images in the dictionary. Earlier studies reported that both SRC and CRC techniques achieve a high recognition accuracy such as 97 - 98%, especially for high MR rate scenarios ($m/d>0.25$ for $\mathbf{A}\in\mathbb{R}^{m\times d}$). On the other hand, for CalebA both CRC and SRC solution tends to fail when we use raw atoms in the dictionary without extracting descriptive features. This is why in this study, we propose to use a more representative dictionary. Instead of using raw images, the atoms consist of more descriptive features extracted by a neural network-based face feature extractor in the library [57]. The proposed method is compared against CRC and SRC techniques with the following 7 state-of-the-art SR solver: ADDM [58], Dalm [54], OMP [54], Homotopy [59], GPSR [60], L1LS [61], $\ell_{1}$-magic [62], and Palm [54]. Overall, when we perform experiments in two facial image databases, Yale-B and CalebA for different MRs, the CSEN based classification proves to be very stable; and in all MRs, it gives the highest or comparable recognition accuracy to the highest ones for all experiments as presented in Figure 8 and 9. Furthermore, it is significantly superior in terms of computational speed when compared with SRC solutions. To be able to use the same CSEN designs introduced in Section IV, we re-order the positions of the atoms, i.e., in the representative sparse codes corresponding non-zero coefficients remain next to each other in 2-D plane. A simplified illustration on the comparison of conventional dictionary design and the proposed design for sparse representation based classification is shown in Figure 10. Defined sparse code sizes and their representations in 2-D grid for Yale-B and CelebA datasets are also given in Table III. Figure 10: The graphical representation of proposed dictionary design vs. conventional design for face recognition problem. TABLE III: Utilized face recognition benchmark datasets are given with their corresponding mask size and number of samples in dictionary, training, and testing per class. Dataset | | Dictionary --- Samples Train Samples | Test Samples | SC Size in 2D-Plane Yale-B | 32 | 16 | 16 | 16 x 76 CelebA | 8 | 12 | 8 | 8 x 200 ### V-C Experiment III: Learning-aided Compressive Sensing Figure 11: Examples from MNIST that are compressively sensed, and then reconstructed at MR=$0.25$. As the experimental setup, we randomly choose, sparse signals, $\mathbf{x}$ in the MNIST database and use the Gaussian measurement matrix, $\mathbf{A}$ to simulate the CS, i.e., $\mathbf{y}=\mathbf{A}\mathbf{x}$. Then, we recover the sparse signal from $\mathbf{y}$ by using the aforementioned state-of-the-art SR tools and the proposed weighted $\ell_{1}$-minimization Eq. (6), where the weights $\mathbf{w}$ are obtained using CSEN output such that $\mathbf{w}=\frac{1}{\mathbf{p}+\epsilon}$. The two examples where signals are compressively sensed with $MR=0.25$ and their estimated versions by different SR methods are shown in Figure 11. It is clear that the proposed approach recovers the sparse signal with the best quality while the other state-of-art SR techniques perform poorly. Figure 12 shows an illustration of how proposed compressive sensing reconstruction scheme differs from traditional compressive sensing recovery setup. Using the output of CSEN as prior information not only provide more accurate signal recovery, but also faster convergence of iterative sparse signal recovery such as $\ell_{1}$-minimization. Figure 12: (Top) Proposed Compressive Sensing Reconstruction. (Bottom) Traditional $\ell_{1}$-minimization based CS-recovery. Furthermore, we draw the estimated phase transition of the algorithms in Figure 13 using an experimental setup whose procedure is explained in [19]. Briefly summarizing the procedure, a grid of (MR, $\rho$) is generated for each algorithm, with 20 independent realization of the problem: according to their sparsity ratios, $\rho$, randomly chosen sparse signal $\mathbf{x}$, among $10000$ MNIST test images, are compressively sensed with independent realization of measurement matrices. Then, they are recovered using the competing algorithms, and each realization is considered a success for the specific algorithm if $\frac{\left\|\mathbf{x}-\mathbf{{\hat{x}}}\right\|_{2}}{\left\|\mathbf{x}\right\|}\leq\text{tol}$ (14) where tol is a predefined parameter, we choose $\text{tol}=10^{-1}$ in our experiments. For a specific algorithm, we draw the phase transition in the border where a $50\%$ success rate is achieved. The procedure is similar to [19], with the exception that they repeated the experiment only once, while we repeat it 100 times for each method, except L1LS due to its infeasibly high computational cost (it took almost two weeks with an ordinary computer). With an accurate SR algorithm, we expect the transition border to be close to the left-top corner in the phase transition graph because it is a good indicator that the algorithm performs well in low MRs and with high sparsity ratio, $\rho$. From the Figure, one can easily deduce that the proposed CS- reconstruction approach clearly outperforms all competing state-of-the-art SR reconstruction methods. Figure 13: Phase Transition of the Algorithms. ## VI Discussion ### VI-A Sparse Modelling vs Structural Sparse Modelling The first generation CS-recovery or sparse representation methods only use the information that the signal, we encounter in real life, is sparse in a proper domain or dictionary. These models do not utilize any further assumptions about the sparse signal, $\mathbf{x}$, in signal recovery or support estimation. Therefore, they only impose sparsity to the signal to have support set with elements in arbitrary location i.e., $\min\left\|\mathbf{x}\right\|_{0}~{}\text{s.t.}~{}\mathbf{Dx}=\mathbf{y}$. However, most sparse signals we face in practical applications exhibit a kind of structure. In second-generation sparse representation models, researchers realized that in addition to arbitrary sparsity any prior information about the sparse code can be used in modeling more advance recovery schemes [63, 64]. For instance, the indices of the non-zero wavelet coefficients of an image mostly exhibit grouping effect [65]. This kind of group sparsity patterns can be imposed by designing the optimization problem involving mixed norm minimization problems [66] instead of simple $\ell_{1}$-norm. On the other hand, more complex sparsity structures require more complex model design. This work proposes an alternative solution to the hand-crafted model-based sparse signal recovery approaches, to be able to learn the pattern inside sparse code (or structural sparse signals), $\mathbf{x}$ by a Machine Learning technique. This proof of the concept work in which the performance is tested over 3 real datasets, MNIST, Yale and CelabA, validates the possibility of such learning and deserves further investigation in different sparse representation problems. ### VI-B Unrolling Deep Models vs CSEN The most common approaches to reconstruct sparse signals, $\mathbf{x}$ from the given measurements, $\mathbf{y}$ with a fixed dictionary $\mathbf{D}$ can be listed as follows: i) Convex Relaxation (or $\ell_{1}$ minimization) such as Basis Pursuit [26]: $\min_{\mathbf{x}}\left\|\mathbf{x}\right\|~{}\text{s.t.}~{}\mathbf{y}=\mathbf{Dx}$ or Basis Pursuit Denoising (BPDN) [26]: $\min_{\mathbf{x}}\left\|\mathbf{x}\right\|~{}\text{s.t.}~{}\left\|\mathbf{y}-\mathbf{Dx}\right\|\leq\epsilon$, where $\epsilon$ is a small constant. ii) Greedy Algorithms such as matching pursuit (MP) [67], Orthogonal Matching Pursuit (OMP) [8], Compressive Sampling Matched Pursuit (CoSaMP) [9]. iii) Bayesian Framework [68], etc. These conventional algorithms dealing with sparse inverse problems work in an iterative manner; for instance, most convex relaxation techniques such as BPDN, minimize the data fidelity term (e.g., $\ell_{2}$-norm) and sparsifying term (e.g., $\ell_{1}$-norm) in an alternating manner in each iteration. Therefore, these schemes suffer from computational complexity and not suitable for real-time applications. Along with the traditional approaches listed above, Deep Learning methods used in this domain have recently become very popular: $\mathbf{\hat{x}}\leftarrow\mathcal{P}\left(\mathbf{y}\right)$, where $\mathcal{P}$ is a learned mapping from $m$ dimensional compressed domain to $n$ dimensional sparse domain. These techniques are built on the idea that the performance of existing convex relaxation can further be improved by reducing the number of iterations and enhancing the reconstruction accuracy. The key idea is that both the possible denoiser matrices, $\mathbf{B}$ (responsible from dealing with data fidelity term) such as $\mathbf{D}^{T}$, or $\left(\mathbf{D}^{T}\mathbf{D}+\lambda\mathbf{I}\right)^{-1}\mathbf{D}^{T}$ where $\lambda$ is the regularization parameter, and the thresholding values (responsible from sparsifying) can be learned from the training data using a deep network generally with dense layers. For instance, the first example of this type is Learned-ISTA (LISTA) [53] which is built upon iterative soft- thresholding algorithm (ISTA) [69]. These category of methods, also called unrolled deep models, design networks in iterative manner are powerfull tools for sparse signal recovery. However, in many practical applications, we may either no need to estimate the sparse signal itself or not have a large amount of training data for deep unrolling networks. In that manner, CSEN provides a third approach, by directly estimating the support set via a compact design which requires less computational power, memory and training set. It exhibits very good performance, especially in the problems that include sparse representation with sparse codes having structural patterns. The other advantage of the compact design with convolutional layers is that it more stable against noise compared to unrolled deep models which include dense layers. ### VI-C Proxy signal vs measurement vector as input to CSEN The proposed support estimation scheme utilizes proxy $\mathbf{\tilde{x}}=\mathbf{By}$ as input to convolutional layers. Making inference directly on proxy using ML approach has been recently reported by several studies. For example, the study in [70, 71] proposed to perform reconstruction-free image classification on proxy, and the study in [72] performed signal reconstruction using proxy as an input to a deep fully convolutional network. Furthermore, proxy $\mathbf{\tilde{x}}$ can be learnt by fully-connected dense layers as presented in [70]. However, this brings additional complexity and training the network may cause over-fitting with limited number of training data. As in [70], they had to adapt by first training the fully-connected layers or try to freeze the other layers during the training. On the other hand, choosing the denoiser matrix, $\mathbf{B}$, is another design problem. For example, in [70, 71] the authors use $\mathbf{B=D^{T}}$ as denoiser to obtain proxy. We reported the results in this paper for denoiser matrix, $\mathbf{B}=\left(\mathbf{D}^{T}\mathbf{D}+\lambda\mathbf{I}\right)^{-1}\mathbf{D}^{T}$, because it gives slightly more stable performance over $\mathbf{B}$. ## VII Conclusions Sparse support estimators that work based on traditional sparse signal recovery techniques suffer from computational complexity and noise. Moreover, they tend to fail at low MRs completely. The proposed CSENs can be considered as reconstruction-free and non-iterative support estimators. Of course, despite their high computational complexity, recent state-of-the-art deep signal reconstruction algorithms may be a cure to sparse recovery methods. However, they are still redundant if SR is not the main concern. In addition, such deep networks often require a large amount of training data that is not available in many practical applications. To address these drawbacks and limitations, in this study, we introduce a novel learning-based support estimators, which have compact network designs. The highlights of the proposed system are as follows: i) Signal reconstruction-free support estimation where sparse estimation can be done in a feed-forward manner, non-iteratively at a low cost. ii) Compact network designs enabling efficient learning even from a small-size training set. iii) The proposed solution is generic; it could be used in any support estimation task such as SE based classification. ## References * [1] D. L. Donoho _et al._ , “Compressed sensing,” _IEEE Transactions on information theory_ , vol. 52, no. 4, pp. 1289–1306, 2006. * [2] E. J. Candès _et al._ , “Compressive sampling,” in _Proceedings of the International Congress of Mathematicians_ , vol. 3, 2006, pp. 1433–1452. * [3] G. Li, Z. Zhu, D. Yang, L. Chang, and H. Bai, “On projection matrix optimization for compressive sensing systems,” _IEEE Transactions on Signal Processing_ , vol. 61, no. 11, pp. 2887–2898, 2013. * [4] M. 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2024-09-04T02:54:56.210907

2003.00769

arxiv-papers

# SOL-KiT - fully implicit code for kinetic simulation of parallel electron transport in the tokamak Scrape-Off Layer S. Mijin A. Antony F. Militello R.J. Kingham Blackett Lab., Plasma Physics Group, Imperial College, London SW7 2AZ, UK CCFE, Culham Science Centre, Abingdon, Oxon OX14 3DB, UK ###### Abstract Here we present a new code for modelling electron kinetics in the tokamak Scrape-Off Layer (SOL). SOL-KiT (Scrape-Off Layer Kinetic Transport) is a fully implicit 1D code with kinetic (or fluid) electrons, fluid (or stationary) ions, and diffusive neutrals. The code is designed for fundamental exploration of non-local physics in the SOL and utilizes an arbitrary degree Legendre polynomial decomposition of the electron distribution function, treating both electron-ion and electron-atom collisions. We present a novel method for ensuring particle and energy conservation in inelastic and superelastic collisions, as well as the first full treatment of the logical boundary condition in the Legendre polynomial formalism. To our knowledge, SOL-KiT is the first fully implicit arbitrary degree harmonic kinetic code, offering a conservative and self-consistent approach to fluid-kinetic comparison with its integrated fluid electron mode. In this paper we give the model equations and their discretizations, as well as showing the results of a number of verification/benchmarking simulations. ###### keywords: kinetic , non-local , electron , atomic , SOL , implicit ††journal: Computer Physics Communications *[ allpages, angle=45, scale=10, xpos=-50, ypos=50 ] PREPRINT PROGRAM SUMMARY Program Title: SOL-KiT Licensing provisions: GNU GPLv3 Programming language: Fortran 90 Nature of problem: Fluid models of parallel transport in the Scrape-Off Layer (SOL) fail to account for the fact that the electron disctribution function is often far from a Maxwellian, and kinetic effects have been linked to discrepencies between experiment and fluid modelling[1]. A kinetic treatment of electrons in the SOL requires detailed accounting of collisional processes, especially those with neutral particles, as well as a proper implementation of the logical boundary condition at the material surface[2]. Furthermore, the ability to identify differences between fluid and kinetic modelling using self-consistent comparison is desirable. Solution method: Electrons are modelled either as a fluid, or kinetically, maintaining self-consistency between models. All equations are solved using finite difference and the implicit Euler method, with fixed-point iteration. The kinetic approach is based on solving the Vlasov-Fokker-Planck-Boltzmann equation, decomposed in Legendre polynomials. Equations for the harmonics (or electron fluid equations) are solved alongside fluid equations for the ions, Ampère-Maxwell’s law for the electric field, as well as a diffusive-reactive Collisional-Radiative model for the evolution of hydrogenic atomic states. Each individual operator is built into a matrix, combined with all other operators, and the matrix equation arising from the Euler method is solved using the PETSc library, with MPI parallelization. Additional comments: This article presents the physical and numerical outline of the code, and is accompanied by html documentation, a small test suite based on benchmarking runs presented below, as well as instructions and means for compiling and executing SOL-KiT. Special focus in the article is given to the novel numerical and model aspects in the greater context of the developed software. ## References * [1] A. Chankin, D. Coster, On the locality of parallel transport of heat carrying electrons in the SOL, J. Nucl. Mater. 463 (2015) * [2] R. J. Procassini, C. K. Birdsall, B. I. Cohen, Particle simulations of collisional transport in a high recycling, diverted tokamak scrape-off layer, Nucl. Fusion 30 (11) (1990) ## 1 Introduction The heat flow onto the plasma facing components of both present day and future magnetically confined fusion (MCF) devices is of considerable importance [1, 2], as it will greatly affect the lifetime of the material. This is true in both steady state operation and during transients (such as ELMs - Edge Localalized Modes). Understanding the heat flux in the Scrape-Off Layer (SOL) is thus of key importance for the design and operation of future fusion devices. Classic fluid modelling of the parallel (to the magnetic field lines) energy transport in the edge region of MCF devices has relied on the fluid closure of Braginskii [3], or otherwise on various flux limiter approaches [4]. However, it is now well known that there exist discrepancies between experiments and the widely used fluid codes. These discrepencies can be, at least partly, attributed to the effect of non-local transport in the SOL [5]. In this paper, the term “non-local” is used to describe behaviour that stems from strong departure of the electron distribution function from a Maxwellian, in particular due to the fact that electron-ion mean-free paths in situations of interest are comparable to or greater than the temperature gradient scale lengths. In the following text we focus mainly on aspects of the divertor SOL. The main feature of the divertor configuration is that the location of the primary plasma-surface interaction is relatively far away from the hot core [6] to specifically designed target plates. We distinguish between the “upstream”, closer to the core, and the “downstream”, where the plasma near the divertor targets is considerably cooler, and the ionization degree can be well below 100%, rendering plasma-neutral interaction important. As such, a large temperature gradient is present, and plasma collisionality (measured with the electron-ion collision mean free path $\lambda_{ei}$) varies greatly along the magnetic field lines in the SOL. Of critical importance is the ratio of the mean free path to the temperature gradient scale length $L_{\nabla T}=(\nabla_{||}T/T)^{-1}$. Once the ratio $\lambda_{ei}/L_{\nabla T}$ is no longer much less than unity, the classical transport results are no longer valid [4]. However, another important concept in the understanding of energy transport in the SOL is that of the high energy heat-carrying electrons (HCE), which become marginally collisionless before the bulk of the distribution [7, 8], and will remain so in most SOL situations. In other words, even if the ratio $\lambda_{ei}/L_{\nabla T}$ might still imply the correctness of classical transport coefficients, the HCE could be collisionless, and thus modify the transport by producing non-Maxwellian distribution functions. A further complication in the understanding of the SOL, as mentioned above, is the importance of electron-neutral interactions. This is especially true during detachment[9], when the ionization degree drops considerably, and a neutral cloud is formed between the divertor targets and the upstream plasma. As this regime of operation offers better protection to the divertor plate materials, it becomes important to understand the interplay of kinetic/non- local effects already present in the SOL with detachment. ### 1.1 SOL kinetic modelling In order to properly capture non-local effects in the SOL it is necessary to treat the plasma using a kinetic approach. Broadly speaking, the two main approaches in the kinetic modelling of the SOL are the Particle-in-Cell (PIC) and finite difference methods solving the Vlasov-Fokker-Planck equation sometimes combined with the Boltzmann collision integral. A representative PIC code for SOL simulations is BIT1[10, 11], used in a variety of simulation scenarios corresponding to present day machine conditions. The PIC method is highly parallelizable, and naturally accommodates the addition of many different collision types (e.g. tungsten impurities[12, 13]). While detailed simulations covering many aspects of SOL transport are possible, two issues make PIC codes complicated to operate.. These are the usually long run times (compared to finite difference methods and fluid codes), and the fact that the number of particles simulated in a PIC code can never approach reality, requiring smoothing techniques[14], and potentially not resolving the high energy tails of distributions with enough accuracy. Finite difference methods do not suffer from the noise problems of PIC codes, as they solve for the distribution function directly. A number of finite difference codes with different approaches have been utilized in the modeling of the SOL, with a few examples mentioned here. An early example of a completely kinetic code (treating every species kinetically) was the ALLA code[15]. Another code, utilizing a similar method to what is presented in this paper, albeit with an explicit algorithm, is the FPI code[16, 17, 18], where electrons are treated kinetically while others species are stationary. More recently, the code KIPP[8, 19, 20], with kinetic electrons, has been coupled with the 2D fluid code SOLPS, providing the latter with kinetically calculated transport coefficients. ### 1.2 Motivation to develop SOL-KiT Due to the great mass difference between electrons and other species within a hydrogen plasma, electrons mainly suffer pitch-angle scattering collisions when colliding with those heavier particles. Eigenfunctions of such collision operators are spherical harmonics, and an expansion of the electron distribution function in spherical harmonics becomes natural[21]. This approach has been used in the modeling of Scrape-Off-Layer transport to a limited extent[16, 17, 18], but has been used both in codes dealing with laser-plasma interactions (KALOS[22]/OSHUN[23], and IMPACT[24]), as well as electron swarm transport models[25, 26]. The expansion has been used to efficiently model both Coulomb and electron-neutral collisions, and has proven itself to be a powerful tool in treating plasmas of various collisionality. With this in mind, a marriage of the Vlasov-Fokker-Planck (VFP) approach in laser plasmas and the Boltzmann approach of electron swarms in neutral gases seems to be potentially a highly applicable model for the Scrape-Off Layer plasma, where collisionality changes, and neutral-plasma interactions carry a great deal of importance. As is typical with solutions of differential equations, the boundary conditions tend to define the system behaviour and dictate the approach in the numerical solution. In modeling the SOL, it becomes necessary to incorporate the effect of the plasma sheath formed at the boundary, i.e. at the divertor target. While the traditional approach of Procassini et al.[27] has been used in many kinetic codes, when utilizing the spherical harmonic expansion it becomes necessary to formulate the well known boundary condition in terms of the expansion basis. We present this formulation, and give its implementation in SOL-KiT. The divertor target plate acts as a sink of particles, and as such generates flows towards the target. Since the divertor boundary condition is formulated in the lab frame, it is then necessary to treat the ion flow in the lab frame as well. As a consequence, the electron-ion collision operator must be extended to account for the moving ions. This is a different strategy for incorporating ion motion in the electron VFP equation, than used elsewhere. There the Vlasov terms are transformed instead into the local rest frame of the ions[28]. We present a simplified treatment of this lab frame operator in the next section, together with fluid ion equations. In order to be able to resolve the low velocity cells for higher harmonics (avoiding the CFL condition) it then becomes necessary to treat the electron kinetic equation implicitly. Another consequence of the divertor boundary conditions is that it also acts as a source of neutral particles. The inclusion of electron-neutral collisions on a nonuniform velocity grid poses particle and energy conservation problems. We present a method of mapping inelastic collisions on a nonuniform velocity grid which conserves both energy and particles, as well as obeying a numerically consistant detailed balance condition when calculating superelastic collision cross-sections. In order to self-consistently model the interaction of atomic states and the electrons, we include a diffusive- reactive Collisional Radiative model[29] for the evolution of hydrogenic atomic states. Finally, in order to provide a modular, self-consistent one-to-one comparison of kinetic and fluid modelling, the code also includes fluid equations for the electrons, which can be solved instead of the kinetic model, while making sure that all of the physical processes are the same, and that the atomic data is used consistently between the two models. To the authors’ knowledge, this is the first fully implicit arbitrary Legendre polynomial/Spherical Harmonic code that has an inbuilt sheath boundary condition, inelastic electron-neutral collisions, and a self-consistent fluid mode for clean comparisons. In the following sections the equations of SOL-KiT and their numerical implementation will be presented, starting with the analytical aspects of the model in section 2, before moving on to the model’s numerical implementation in section 3. Finally, details of performed benchmarking runs will be given in section 4. We discuss the various aspects of the code in section 5. ## 2 Physical model In this section we will introduce the equations being solved in the SOL-KiT model, giving a condensed overview of the physics before expanding on individual operators. While the code is capable of handling both fixed and periodic boundary conditions, since the most involved cases utilize the Scrape-Off Layer domain, we start with describing it. The domain is 1D, and is assumed to be along a (straightened) field line. The field line is taken as the $x$-axis, around which the domain is symmetric. The point $x=0$ is taken to be the symmetry plane, representing the ”upstream” of the SOL. A sketch of the simplified SOL domain is given in Figure 1. Figure 1: The SOL simulation domain (not to scale): the $x$-axis is the principle axis of the system, with $x=0$ being the upstream symmetry plane; the right boundary of the system is at the divertor target, which acts as a sink for the plasma, as well as a source of neutrals via the recycling flux $\Gamma_{REC}$ The equations solved by SOL-KiT are the following: * 1. Electron equations - either fluid (density, parallel velocity, temperature) or kinetic * 2. Ion fluid equations - density and parallel velocity (assuming either $T_{i}=0$ or $T_{i}=T_{e}$) * 3. Diffusive-reactive Collisional Radiative Model for atomic states * 4. Ampère-Maxwell law for the evolution of the electric field For the electrons we either solve the kinetic equation, which is the main mode of the code, or we can solve local fluid equations, obtained by taking moments of the kinetic model, ensuring maximum correspondance between the kinetic and fluid modes. This in turn allows for easy comparison between the fluid and kinetic model, further highlighting kinetic effects. The 1D kinetic equation solved for the electrons is the Vlasov-Fokker-Planck-Boltzmann equation, given by $\frac{\partial f(x,\vec{v},t)}{\partial t}+v_{x}\frac{\partial f(x,\vec{v},t)}{\partial x}-\frac{e}{m_{e}}E\frac{\partial f(x,\vec{v},t)}{\partial v_{x}}=C[f,...],$ (1) where $E$ is the electric field (assuming azimuthal symmetry and straightening out the magnetic field we ignore magnetic effects). The RHS contains all of the collision and source operators. Details on collision operators and the Legendre polynomial decomposition of the electron distribution function are given in Section 2.1. In the electron fluid mode, the continuity, momentum, and temperature equations are solved instead. The continuity equation is given by $\frac{\partial n_{e}}{\partial t}+\frac{\partial(n_{e}u_{e})}{\partial x}=S,$ (2) while the momentum equation is $\frac{\partial u_{e}}{\partial t}=-u_{e}\frac{\partial u_{e}}{\partial x}-\frac{e}{m_{e}}E+\frac{R_{ei}+R_{en}}{m_{e}n_{e}}-\frac{S}{n_{e}}u_{e}-\frac{1}{m_{e}n_{e}}\frac{\partial(n_{e}kT_{e})}{\partial x},$ (3) where $n_{e}$,$u_{e}$, and $T_{e}$ are the electron density, flow velocity, and temperature, respectively. $S=S_{ion}+S_{rec}$ is the particle source including ionization and recombination, and $R_{ei}=R_{T}+R_{u}$ is the classical Braginskii[3] friction $R_{u}=-\frac{m_{e}n_{e}}{\tau_{e}}0.51(u_{e}-u_{i}),$ (4) $R_{T}=-0.71n_{e}\frac{\partial(kT_{e})}{\partial x},$ (5) where the $\tau_{e}$ is the Braginskii collision time[3]. $R_{en}$ is the total friction from all electron-neutral collisions, assuming a slowly (compared to electron thermal speed) drifting Maxwellian distribution for electrons. The electron temperature equation is $\frac{\partial kT_{e}}{\partial t}=-u_{e}\frac{\partial kT_{e}}{\partial x}+\frac{2}{3}\left[\frac{Q}{n_{e}}-kT_{e}\frac{\partial u_{e}}{\partial x}-\frac{1}{n_{e}}\frac{\partial q_{e}}{\partial x}-\frac{S}{n_{e}}\left(\frac{3}{2}kT_{e}-\frac{m_{e}u_{e}^{2}}{2}\right)-\frac{u_{e}(R_{ei}+R_{en})}{m_{e}n_{e}}\right],$ (6) where $q_{e}=q_{T}+q_{u}$ is again the classical Braginskii[3] heat flux $q_{T}=-\kappa_{e}\frac{\partial(kT_{e})}{\partial x},$ (7) $q_{u}=0.71n_{e}kT_{e}(u_{e}-u_{i}),$ (8) where $\kappa$ is taken to be either the Spitzer-Härm result or the Lorentz result [30, 31], depending on whether we treat electron-electron momentum transfer collisions. $Q=Q_{ext}+Q_{en}$ is the combination of the external heating energy source (see 2.1.4.), as well as any inelastic collision energy transfer between the electrons and the atoms. The ion equations are analogous, with a few differences. Firstly, the ion continuity equation can be solved, or quasi-neutrality can be enforced artificially by setting $Zn_{i}=n_{e}$. Secondly, there is no ion temperature equation in the current version of the code. Instead, ion temperature is set to either zero, or to the electron temperature. The equations are $\frac{\partial n_{i}}{\partial t}+\frac{\partial(n_{i}u_{i})}{\partial x}=S,$ (9) $\frac{\partial u_{i}}{\partial t}=-u_{i}\frac{\partial u_{i}}{\partial x}+\frac{Ze}{m_{i}}E+\frac{R_{ie}+R_{CX}}{m_{i}n_{i}}-\frac{S}{n_{i}}u_{i}-\frac{1}{m_{i}n_{i}}\frac{\partial(n_{i}kT_{i})}{\partial x},$ (10) where $R_{ie}$ is obtained by requiring total momentum in electron-ion collisions to be conserved, ie. $R_{ie}=-R_{ei}$. $R_{CX}$ is a simple charge exchange friction term, given by $R_{CX}=-n_{i}m_{i}u_{i}|u_{i}|\sum_{b}n_{b}\sigma_{CX,b},$ (11) where the sum is over neutral atomic states, and both the ions and neutrals are approximated as cold, with the simplified constant hydrogenic charge exchange cross sections given by approximate low energy values obtained from Janev[32] $\sigma_{CX,1}=3\times 10^{-19}m^{2},\quad\sigma_{CX,2}=2^{4}\times 10^{-19}m^{2},\quad\sigma_{CX,3}=3^{4}\times 7\times 10^{-20}m^{2},\quad\sigma_{CX,b\geq 4}=b^{4}\times 6\times 10^{-20}m^{2}.$ To calculate the electric field, we solve Ampère-Maxwell’s law, which contains only the displacement current $\frac{\partial E}{\partial t}=-\frac{1}{\epsilon_{0}}(j_{e}+Zen_{i}u_{i}),$ (12) where $j_{e}$ is either given as a moment of the electron distribution function, or simply as $j_{e}=-en_{e}u_{e}$ in the electron fluid case. Finally, the atomic state distribution of the neutrals must be tracked. This is done using a diffusive-reactive Collisional Radiative model (CRM) to obtain the evolution of the neutral state densities $n_{b}$ (where $b$ here denotes the principal quantum number of the state) $\displaystyle\frac{\partial n_{b}}{\partial t}$ $\displaystyle=\frac{\partial}{\partial x}\left(D_{b}\frac{\partial n_{b}}{\partial x}\right)+\sum_{b^{\prime}<b}\left[K_{b^{\prime}\rightarrow b}^{e}n_{b^{\prime}}-A_{b\rightarrow b^{\prime}}n_{b}-K_{b\rightarrow b^{\prime}}^{e}n_{b}\right]$ $\displaystyle+\sum_{b^{\prime}>b}\left[K_{b^{\prime}\rightarrow b}^{e}n_{b^{\prime}}+A_{b^{\prime}\rightarrow b}n_{b^{\prime}}-K_{b\rightarrow b^{\prime}}^{e}n_{b}\right]-K_{b}^{ion}n_{b}+\alpha_{b}n_{e}^{2}n_{i}+\beta_{b}n_{e}n_{i},$ (13) where the ionization and (de-)excitation rates $K$, as well as three-body recombination rates $\alpha$ are calculated using moments of the distribution function (see 2.1.3.). The inelastic cross-sections and radiative de- excitation/recombination rates $A$ and $\beta$ are all taken from Janev[32] and NIST[33]. Radiative de-excitation is included only up to state number $b=20$, due to lack of available data. Since higher excited states are primarily collisionally coupled in most situations of interest this should not cause significant discrepancies. The classical 1D diffusion coefficient is simply $D_{b}=\frac{v_{tn}}{2[(n_{i}+n_{1})\sigma_{el}+\sigma_{CX,b}n_{i}]}$ (14) where $v_{tn}$ is the thermal speed of neutrals, $\sigma_{el}$ is the elastic collisions cross-section (see electron-neutral elastic collision operator in 2.1.3.), and $n_{1}$ is the ground state density. When charge exchange is used, it is the dominant term in the diffusion coefficient, but the elastic collision diffusion is included for cases when charge exchange is turned off. Since gas temperature and elastic cross-section are free parameters in SOL- KiT, this operator can be tuned. Ideally, however, a self-consistent neutral dynamics model should be implemented, and this is a planned extension of the model. The boundary condition at the divertor target is the logical boundary condition [27] when electrons are treated kinetically, while the ions are always assumed to reach sound speed (as per the Bohm criterion). Details of the kinetic boundary condition are given below. In both the fluid and kinetic model, the flow into the sheath is ambipolar $\Gamma_{e}=\Gamma_{i}$. When electrons are treated as a fluid the sheath heat transmission coefficient[6] in $q_{sh}=\gamma kT_{e}\Gamma_{e}$ is taken to be $\gamma_{e}=2-0.5\ln(2\pi(1+T_{i}/T_{e})m_{e}/m_{i})$. Finally, the atomic neutrals are recycled with a recyling flux $\Gamma_{REC}=-R\Gamma_{i}$, where $R\leq 1$ is the recycling coefficient. This is simply imposed by setting $D_{1}\partial n_{1}/\partial x$ in (13) at the boundary to $\Gamma_{REC}$, whereas it would otherwise be zero. ### 2.1 Electron kinetic equation in Legendre formalism Spherical harmonics are an orthonormal basis set in the solid angle space $\left(\theta,\varphi\right)\in\left[0,\pi\right)\times\left[0,2\pi\right)$, and can be written as a suitably normalized product of associated Legendre polynomials $P_{l}^{m}(\cos\theta)$ and the complex phase $e^{im\varphi}$. The spherical harmonic convention used in SOL-KiT is same as the ones in KALOS[22]/OSHUN[23]. As such, the traditional Cartesian coordinate system for velocity $(v_{x},v_{y},v_{z})$ will be rotated so that the angles $\theta$ and $\varphi$ are defined as part of a spherical coordinate system: $v=\sqrt{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}},\quad\theta=\arccos(v_{x}/v),\quad\varphi=\arctan(v_{z}/v_{y}).$ (15) If then we decompose the distribution function $f(v,\theta,\phi)$ in spherical harmonics we get: $f(v,\theta,\varphi)=\sum_{l=0}^{\infty}{\sum_{m=-l}^{l}{f_{l}^{m}(v)P_{l}^{|m|}(\cos\theta)e^{im\varphi}}}$ (16) where the complex expansion coefficients $f_{l}^{m}$ satisfy $(f_{l}^{m})^{*}=f_{l}^{-m}$. This allows writing the kinetic equation for electrons as a set of equations for the amplitudes $f_{l}^{m}$, whose physical significance becomes obvious when moments of the distribution function are expressed using the expansion. If $\phi$ is a scalar function of $v$ then $\int\phi f(\vec{v})d\vec{v}=4\pi\int_{0}^{\infty}\phi f_{0}^{0}(v)v^{2}dv$ (17) while if $\vec{a}$ is a vector function of $v$ $\int\vec{a}f(\vec{v})d\vec{v}=\frac{4\pi}{3}\int_{0}^{\infty}||a||\begin{pmatrix}f_{1}^{0}\\\ 2Re(f_{1}^{1})\\\ -2Im(f_{1}^{1})\end{pmatrix}v^{2}dv$ (18) and similarly for higher order tensors[21, 23]. As can be seen from this, not only does the spherical harmonic expansion have useful properties in relation to collisions, it also provides a physically meaningful decomposition for evaluating transport quantities. Since the current model is 1D, and azimuthal symmetry around the $x$-axis is assumed, we do not treat magnetic field effects, and the spherical harmonic decomposition reduces to a Legendre polynomial decomposition ($m=0$ always). Accordingly, in the rest of this paper, harmonics will be labeled only by their $l$-number, i.e. $f_{l}(v)$. Following the Legendre decomposition of the distribution function as outlined above, the 1D kinetic equation can be written as a set of equations $\frac{\partial f_{l}(x,v,t)}{\partial t}=A_{l}+E_{l}+C_{l},$ (19) where now all of the operators are moved to the RHS and are functions of $l$ in addition to whatever arguments they naturally have. These will be examined in detail below. #### 2.1.1 Vlasov terms The terms on the LHS of equation (1) are usually referred to as the Vlasov terms. The two Vlasov terms in equation (19) are the spatial advection term $A_{l}$ (corresponding to second LHS term in (1)), and the velocity space advection term due to the electric field in the $x$-direction $E_{l}$ (corresponding to third LHS term in (1)). Firstly, the spatial advection term (advection in the $x$-direction), for a given harmonic $l$ is $A_{l}=-\frac{l}{2l-1}v\frac{\partial f_{l-1}}{\partial x}-\frac{l+1}{2l+3}v\frac{\partial f_{l+1}}{\partial x}.$ (20) Spatial advection couples harmonics with different $l$ numbers. The physical significance of this coupling is most easily seen in the coupling between $f_{0}$ and $f_{1}$. The moments of $f_{0}$ are the density and total energy, while $f_{1}$ is associated with flows (of particles and energy). Thus gradients in $f_{0}$ (density and temperature) drive advection of $f_{1}$ (flows), and vice-versa. The velocity space advection term due to the electric field couples harmonics as well, albeit through velocity space gradients in $f_{l}$. As only the $x$ component of the electric field is treated, in the remainder of the paper it will simply be written as $E$ for brevity. The velocity space advection operator is given by[22] $E_{l}=\frac{e}{m}E\left[\frac{l}{2l-1}G_{l-1}+\frac{l+1}{2l+3}H_{l+1}\right]$ (21) where $G_{l}(v)=v^{l}\frac{\partial v^{-l}f_{l}}{\partial v},$ (22) $H_{l}(v)=\frac{1}{v^{l+1}}\frac{\partial v^{l+1}f_{l}}{\partial v}.$ (23) As is evident from equation (21), the electric field couples harmonics through the $G_{l}$ and $H_{l}$ functions, which contain velocity space gradients of the coupled harmonics. Thus, Vlasov terms provide coupling of harmonics through either spatial gradients or the electric field. #### 2.1.2 Coulomb collision terms Let us consider the effect of Coulomb collisions on the distribution function $f$ of particles of mass $m$ and charge $q=ze$ colliding with particles of mass $M=\mu m$ and charge $Q=Ze$ with distribution $F$. We follow the formalism of Shakorfsky et al.[21], starting from the Rosenbluth coefficient formulation of the Fokker-Planck collision operator for Coulomb collisions $\frac{1}{\Gamma_{zZ}}\frac{\delta f}{\delta t}=\frac{4\pi}{\mu}Ff+\frac{\mu-1}{\mu+1}\nabla\mathcal{H}(F)\cdot\nabla f+\frac{\nabla\nabla\mathcal{G}(F):\nabla\nabla f}{2},$ (24) where $\nabla=\partial/\partial\vec{v}$ and $\Gamma_{zZ}=(zZe^{2})^{2}\ln\Lambda/(4\pi(m\epsilon_{0})^{2})$, with $\ln\Lambda$ denoting the Coulomb logarithm. The Rosenbluth drag and diffusion coefficients are respectively $\mathcal{H}$ and $\mathcal{G}$. We separate the distribution functions into their isotropic and anisotropic components $F=F_{0}+F_{a}$, $f=f_{0}+f_{a}$. The key assumption going forward is that the anisotropic component is small compared to the isotropic one, so that it becomes possible to linearize equation (24). Expanding the distribution function and the Rosenbluth coefficients in harmonics and using the integrals[21] $I_{j}(F_{l})=\frac{4\pi}{v^{j}}\int_{0}^{v}F_{l}(u)u^{j+2}du,\quad J_{j}(F_{l})=\frac{4\pi}{v^{j}}\int_{v}^{\infty}F_{l}(u)u^{j+2}du,$ (25) one can derive the expressions for the harmonic components of the Fokker- Planck collision integral for all $l$. For $l=0$ this is $\frac{1}{\Gamma_{zZ}}\frac{\delta f_{0}}{\delta t}=\frac{1}{3v^{2}}\frac{\partial}{\partial v}\left[\frac{3}{\mu}f_{0}I_{0}(F_{0})+v\left(I_{2}(F_{0})+J_{-1}(F_{0})\right)\frac{\partial f_{0}}{\partial v}\right].$ (26) For $l>0$ the following is obtained[23] $\displaystyle\frac{1}{\Gamma_{zZ}}\frac{\partial f_{l}}{\partial t}$ $\displaystyle=\frac{4\pi}{\mu}\left[F_{0}f_{l}+f_{0}F_{l}\right]$ $\displaystyle-\frac{(\mu-1)}{\mu v^{2}}\left\\{\frac{\partial f_{0}}{\partial v}\left[\frac{l+1}{2l+1}I_{l}(F_{l})-\frac{l}{2l+1}J_{-1-l}(F_{l})\right]+I_{0}(F_{0})\frac{\partial f_{l}}{\partial v}\right\\}$ $\displaystyle+\frac{I_{2}(F_{0})+J_{-1}(F_{0})}{3v}\frac{\partial^{2}f_{l}}{\partial v^{2}}+\frac{-I_{2}(F_{0})+2J_{-1}(F_{0})+3I_{0}(F_{0})}{3v^{2}}\frac{\partial f_{l}}{\partial v}$ $\displaystyle-\frac{l(l+1)}{2}\times\frac{-I_{2}(F_{0})+2J_{-1}(F_{0})+3I_{0}(F_{0})}{3v^{3}}f_{l}$ $\displaystyle+\frac{1}{2v}\frac{\partial^{2}f_{0}}{\partial v^{2}}\left[C_{1}I_{l+2}(F_{l})+C_{1}J_{-1-l}(F_{l})+C_{2}I_{l}(F_{l})+C_{2}J_{1-l}(F_{l})\right]$ $\displaystyle+\frac{1}{v^{2}}\frac{\partial f_{0}}{\partial v}\left[C_{3}I_{l+2}(F_{l})+C_{4}J_{-1-l}(F_{l})+C_{5}I_{l}(F_{l})+C_{6}J_{1-l}(F_{l})\right],$ (27) where the $C$ coefficients are functions of $l$ $\displaystyle C_{1}$ $\displaystyle=\frac{(l+1)(l+2)}{(2l+1)(2l+3)},\quad C_{2}=-\frac{(l-1)l}{(2l+1)(2l-1)},\quad C_{3}=-\frac{(l+1)l/2+l+1}{(2l+1)(2l+3)},$ $\displaystyle C_{4}$ $\displaystyle=\frac{-(l+1)l/2+l+2}{(2l+1)(2l+3)},\quad C_{5}=\frac{(l+1)l/2+l-1}{(2l+1)(2l-1)},\quad C_{6}=-\frac{(l+1)l/2-l}{(2l+1)(2l-1)}.$ For electron-electron collisions $\mu=1$. The effect of e-e collisions on the isotropic part of the distribution function is given by $\frac{1}{\Gamma_{ee}}\left(\frac{\delta f_{0}}{\delta t}\right)_{e-e}=\frac{1}{v^{2}}\frac{\partial}{\partial v}\left[C(f_{0})f_{0}+D(f_{0})\frac{\partial f_{0}}{\partial v}\right],$ (28) where the drag and diffusion coefficients are defined as $C(f_{0})=4\pi\int_{0}^{v}f_{0}(u)u^{2}du,$ (29) $D(f_{0})=4\pi\int_{0}^{v}u^{2}\left[\int_{u}^{\infty}f_{0}(u^{\prime})u^{\prime}du^{\prime}\right]du.$ (30) Note that $D$ is not in the form one would expect from equation (26). Instead, it is written in an analytically equivalent form (see section 3.5.). The electron-electron collision operator for $l=0$ is important for the proper relaxation of the electron distribution function to a Maxwellian. For higher harmonics, from (27) we get $\begin{split}\frac{1}{\Gamma_{ee}}\left(\frac{\delta f_{l}}{\delta t}\right)_{e-e}&=8\pi f_{0}f_{l}+\frac{I_{2}(f_{0})+J_{-1}(f_{0})}{3v}\frac{\partial^{2}f_{l}}{\partial v^{2}}\\\ &+\frac{-I_{2}(f_{0})+2J_{-1}(f_{0})+3I_{0}(f_{0})}{3v^{2}}\frac{\partial f_{l}}{\partial v}\\\ &-\frac{l(l+1)}{2}\times\frac{-I_{2}(f_{0})+2J_{-1}(f_{0})+3I_{0}(f_{0})}{3v^{3}}f_{l}\\\ &+\frac{1}{2v}\frac{\partial^{2}f_{0}}{\partial v^{2}}\left[C_{1}I_{l+2}(f_{l})+C_{1}J_{-l-1}(f_{l})+C_{2}I_{l}(f_{l})+C_{2}J_{1-l}(f_{l})\right]\\\ &+\frac{1}{v^{2}}\frac{\partial f_{0}}{\partial v}\left[C_{3}I_{l+2}(f_{l})+C_{4}J_{-l-1}(f_{l})+C_{5}I_{l}(f_{l})+C_{6}J_{1-l}(f_{l})\right].\end{split}$ (31) As the ions are either assumed cold or with the same temperature as the electrons, in the current version of SOL-KiT the effect of electron-ion collisions on $f_{0}$ is not included. For higher harmonics we distinguish two cases of electron-ion collisions. The first is the classical stationary ion case, where $F_{0}=n_{i}\delta(v)/(4\pi v^{2})$. It can easily be shown that equation (27) reduces to the following eigenfunction form $\left(\frac{\delta f_{l}}{\delta t}\right)_{e-i}=-\frac{l(l+1)}{2}\frac{\Gamma_{ei}n_{i}}{v^{3}}f_{l}.$ (32) Here it can be seen that this is purely angular scattering, which dampens higher harmonics, helping us truncate the expansion. However, when ions are not stationary, but are moving at some velocity much smaller than the electron thermal velocity (the case we expect in the SOL), it is necessary to modify the electron-ion collision operator. This is done by first getting rid of all the terms proportional to the inverse mass ratio in equation (27). To calculate the $I$ and $J$ integrals for a cold ion stream we let the ion distribution function be $F(\vec{v})=n_{i}\delta(\vec{v}-\vec{u_{i}}),$ (33) Recasting the Dirac delta into spherical coordinates assuming azimuthal symmetry and expanding in Legendre polynomials gives us $F_{l}(v)=\frac{n_{i}(2l+1)}{4\pi v^{2}}\delta(v-u_{i}).$ (34) Substituting these harmonics into the equations for the $I$ and $J$ integrals gives us $I_{j}(F_{l})=(2l+1)n_{i}\frac{u_{i}^{j}}{v^{j}}\Theta(v-u_{i}),$ (35) $J_{j}(F_{l})=(2l+1)n_{i}\frac{u_{i}^{j}}{v^{j}}\Theta(u_{i}-v),$ (36) where $\Theta$ denotes the Heaviside step function. It is now trivial to see that all but the $I_{0}$ integrals vanish when $u_{i}=0$, recovering the stationary ion collision integral. It can also be easily shown that for small enough $u_{i}$, the collision integral for $f_{1}$ reduces to[21] $\frac{1}{\Gamma_{zZ}}\frac{\partial f_{1}}{\partial t}=-\frac{n_{i}}{v^{3}}\left(f_{1}+u_{i}\frac{\partial f_{0}}{\partial v}\right).$ (37) If $f_{0}$ is taken to be Maxwellian, this gives a stationary solution to the $f_{1}$ collision operator to be that $f_{1}$ which yields a slowly drifting Maxwellian with drift velocity $u_{i}$. Taking a closer look at the obtained collision integral, we see that electrons slower than the ions are being carried around by them, as one would expect. This modifies all harmonics for small $v$, so we lose the convenient property of clean truncation of the distribution function by electron-ion collisions. However, in realistic simulation cases, this happens only for a small handful of velocity space cells, while higher harmonics are normally dampened in the rest of velocity space. This is to be expected, as our electrons see an ion Dirac delta and are collisionally driven towards it. #### 2.1.3 Boltzmann collision terms We start from the general form of the Boltzmann collision integral for the effect of collisions on the distribution function of species $s$ colliding with species $s^{\prime}$ $C[f_{s},f_{s^{\prime}}](v)=\int d\vec{v_{2}}d\Omega|\vec{v}-\vec{v_{2}}|\sigma(|\vec{v}-\vec{v_{2}}|,\Omega)[f_{s}(\vec{v^{\prime}})f_{s^{\prime}}(\vec{v_{2}^{\prime}})-f_{s}(\vec{v})f_{s^{\prime}}(\vec{v_{2}})],$ (38) where primed velocities denote values before a collision, and $\sigma$ is the appropriate differential cross-section. The following results are all derived under the assumption of a small mass ratio and stationary (slow compared to the electrons) neutral particles (atoms)[21, 25, 26]. Furthermore, all differential cross-sections are assumed to be azimuthally symmetric, i.e. are only a function of energy/velocity of the impacting particle (electron), and the deflection angle (here $\chi$). This just means that the cross-sections do not depend on the orientation of the neutral particle. ##### Elastic electron-neutral collisions The first term in the small mass expansion of the electron-neutral elastic collision integral cancels for $l=0$. It is therefore necessary to include higher order effects of collisional energy transfer. This way, one can allow for long term relaxation of the electron distribution function to a Maxwellian with temperature equal to the gas temperature $T_{g}$. If $M$ is the neutral mass, and $n_{b}$ the gas density, the collision integral takes the form $\left(\frac{\delta f_{0}}{\delta t}\right)_{e-n_{b}}=\frac{m_{e}}{M+m_{e}}\frac{1}{v^{2}}\frac{\partial}{\partial v}\left[n_{b}v^{4}\left(\int d\Omega(1-\cos\chi)\sigma_{b}^{el}(\chi,v)\right)\left(f_{0}+\frac{kT_{g}}{m_{e}v}\frac{\partial f_{0}}{\partial v}\right)\right]$ (39) As is the case with electron-ion Coulomb collisions, due to a great mass difference, the $l>0$ integral is considerably simplified, and can be written as $\left(\frac{\delta f_{l>0}}{\delta t}\right)_{e-n_{b}}=-n_{b}v\left[\int d\Omega(1-P_{l}(\cos(\chi)))\sigma_{b}^{el}(\chi,v)\right]f_{l}.$ (40) where $P_{l}$ is simply the $l$-th Legendre polynomial. While implemented and tested in the current version of SOL-KiT, these two processes are rarely used, because of the lack of proper elastic collision cross-section data. Currently the cross-section for elastic collisions of an electron with a hydrogen atom in a given state is obtained using the classical expression for orbit size. Namely, this gives for the integral elastic collision cross-section of electrons with hydrogen atoms in the $b$-th state $\sigma_{b}^{el,TOT}=\pi a_{0}^{2}b^{4}$ (41) where $a_{0}$ is the Bohr radius. ##### Inelastic electron-neutral collisions We start with inelastic collisions where the total number of particles of each species is conserved (e.g. excitation). A standard procedure exists[21, 25] for an inelastic collision for which the pre-collision and post-collision velocities are related as $\frac{m_{e}v^{\prime 2}}{2}=\frac{m_{e}v^{2}}{2}+\epsilon$ (42) where $\epsilon$ is the inelastic energy loss. Defining $\alpha=v^{\prime}/v=(1+2\epsilon/mv^{2})^{1/2}$, one can write the collision integral as $\left(\frac{\delta f_{l}}{\delta t}\right)^{ex}_{b\rightarrow b^{\prime}}=-n_{b}v\left[\sigma^{TOT}_{b\rightarrow b^{\prime}}(v)f_{l}(v)-f_{l}(\alpha v)\alpha^{2}\left(\sigma^{TOT}_{b\rightarrow b^{\prime}}(\alpha v)-\sigma^{(l)}_{b\rightarrow b^{\prime}}(\alpha v)\right)\right].$ (43) Here $\sigma^{TOT}=\int d\Omega\sigma(\chi,v)$ is the integral cross section, while $\sigma^{(l)}(v)=\int d\Omega(1-P_{l}(\cos\chi))\sigma(\chi,v).$ Using equation (43) for $l=0$ one can easilly show that particle number is conserved. On the other hand, collisional processes such as ionization do not conserve the total number of particles. The main difficulty in treating ionization is the fact that it is, ultimately, a 3-body process, and ideally one would like to know the triply differential cross section for such a process. However, such an approach is not only complicated, but cross-section data (to the author’s knowledge) are not systematically available. Because of this, the approach taken here will be the simplest possible[15], where all electrons produced in ionization are put in the lowest velocity cell, while the original electron experiences a standard energy loss/deflection as in the case of excitation. The collisional operator for ionization takes the following form $\left(\frac{\delta f_{l}}{\delta t}\right)_{b}^{ion}=\left(\frac{\delta f_{l}}{\delta t}\right)^{ex}(\sigma_{b}^{ion})+n_{b}K^{ion}_{b}\frac{\delta(v)}{4\pi v^{2}}\delta_{l,0}$ (44) where $\left(\frac{\delta f_{l}}{\delta t}\right)^{ex}(\sigma_{b}^{ion})$ is equation 43, but with $\sigma^{ex}$ replaced with $\sigma^{ion}$, and with the collisional ionization rate coefficient defined (unconventionally) as $K^{ion}_{b}=4\pi\int dvv^{3}f_{0}(v)\sigma^{TOT,ion}_{b}(v).$ (45) Particle sources are then computed using $S_{ion}=\sum_{b}K^{ion}_{b}n_{b}$, and similarly for recombination, while the inelastic collision contribution to $Q$ is similarly calculated by taking the product of each transition rate and the associated transition energy. Of course, one operator of the above kinds exists for each possible process of the given kind, i.e. one operator for each excitation process, and one for each ionization process, taking in the appropriate neutral state densities and cross-sections for the given process. The rate coefficients are the same ones used in equation (13). Inverse processes (deexcitation and 3-body recombination) are treated using the principle of detailed balance [34, 35] to obtain cross-sections. For deexcitation, this gives $\sigma_{deex}(i,j,v^{\prime})=\frac{g_{j}}{g_{i}}\frac{v^{2}}{v^{\prime 2}}\sigma_{ex}(j,i,v)$ (46) where $i$ and $j$ are atomic states ($j<i$), and $g_{i}$ and $g_{j}$ their statistical weights. For hydrogen these are simply $g_{n}=2n^{2}$. Equation (42) defines the velocities, but we use a negative $\epsilon$. For 3-body recombination, using the statistical weights of a free electron gas we get for the cross-section $\sigma_{3b-recomb}(i,v^{\prime})\frac{1}{n_{e}}=\frac{g_{i}}{2g_{1}^{+}}\left(\frac{h^{2}}{2\pi m_{e}kT_{e}}\right)^{3/2}\times\frac{v^{2}}{v^{\prime 2}}\sigma_{ion}(i,v),$ (47) where $h$ is the Planck constant, $n_{e}$ and $T_{e}$ are the electron density and temperature, respectively, and $g_{1}^{+}$ is the ion ground state statistical weight (for hydrogen simply $g_{1}^{+}=1$). To calculate the electron fluid mode $R_{en}$ in equation (3) we use terms of the form $R_{en}^{ion}=\sum_{b}\frac{4\pi}{3}\int_{0}^{\infty}\left(\frac{\delta f_{1}}{\delta t}\right)_{b}^{ion}v^{3}dv,$ where $f_{1}(v)=-u_{e}\partial f_{0}/\partial v$ (with Maxwellian $f_{0}$), and similarly for other neutral processes. #### 2.1.4 Electron heating operator The implemented diffusive heating operator has the form $\left(\frac{\partial f_{0}}{\partial t}\right)_{heating}=\Theta(L_{h}-x)D(x,t)\frac{1}{3v^{2}}\frac{\partial}{\partial v}v^{2}\frac{\partial f_{0}}{\partial v},$ (48) where $\Theta(L_{h}-x)$ is the step function designating the heating region. It is easy to check that this operator conserves particle number if $\partial f/\partial v=0$ on the system boundaries. If we assume a spatially uniform heating, it is easy to show that $D(t)=\frac{W_{h}(t)}{m_{e}\int_{0}^{L_{h}}n_{e}(x,t)dx},$ (49) where $W_{h}(t)$ is the heat flux entering the SOL over length $L_{h}$. This is related to the fluid model heating $Q_{ext}$ via $Q_{ext}=W_{h}/L_{h}$. #### 2.1.5 Particle source operator In order to treat upstream density perturbations, the following electron and ion particle sources (in kinetic mode only) are implemented $\left(\frac{\partial f_{0}}{\partial t}\right)_{source}=\Theta(L_{s}-x)F_{R}(x,t)\left(\frac{m_{e}}{2\pi kT_{source}}\right)^{3/2}e^{\frac{-mv^{2}}{2kT_{source}}},$ (50) where $F_{R}$ is the source rate coefficient $F_{R}=\frac{\Gamma_{in}}{L_{s}},$ (51) with $\Gamma_{in}$ being the effective upstream flux. The particles are injected over a length of $L_{s}$ and with temperature $T_{source}$, which can be the background temperature. The ion particle source is simply $\left(\frac{\partial n_{i}}{\partial t}\right)_{source}=F_{R}.$ (52) #### 2.1.6 Divertor target boundary condition with Legendre polynomials The boundary condition at the divertor target is calculated using the standard logical boundary condition [27], setting the ion and electron fluxes to be equal at the sheath entrance. The logical boundary condition assumes that all electrons with a parallel velocity above some $v_{c}$ moving towards the target are lost, while all others are reflected. This translates to having a sharp cut-off in the electron distribution function. The challenge when formulating this condition in a Legendre polynomial formalism is the extreme anisotropy that results from it, which would require a high number of harmonics to resolve to a satisfactory level. Fortunately, this number is usually not prohibitively high (see 4.4.). The harmonic content of the ”cut- off” distribution $f_{cl}$ can be written as a linear combination of known harmonics $f_{cl}(v)=\sum_{l^{\prime}}P_{ll^{\prime}}f_{l^{\prime}}(v).$ (53) For details on the calculation of the transformation matrix $P_{ll^{\prime}}$ see Appendix A. Knowing the form of the distribution function, one can solve the ambipolarity condition $\frac{4\pi}{3}\int_{0}^{\infty}v^{3}f_{c1}dv=n_{i,sh}u_{i,sh},$ (54) where $n_{i,sh}$ is the extrapolated density at the sheath boundary (see 3.7. below), and $u_{i,sh}$ is the ion velocity at the boundary, given by the Bohm condition $u_{i}\geq c_{s}=\sqrt{\frac{k(T_{e}+T_{i})}{m_{i}}},$ (55) where $T_{e}$ is the electron temperature in the last simulation cell, and $T_{i}$ is the ion temperature. Solving the ambipolarity condition gives the value of $v_{c}$, and with it the value of the sheath potential drop $\Delta\Phi=m_{e}v_{c}^{2}/(2e)$. The electron distribution harmonics with the correct cut-off velocity can then be used as the dynamically updated boundary condition. ## 3 Numerical Methods In this section we present numerical details of the SOL-KiT algorithm, starting with the definitions of the normalization scheme and the grids used. An overview of discretization schemes for the various operators in the code follows. ### 3.1 Normalization The temperature is normalized to some reference value (in eV), while the reference density is assumed to be given in $m^{-3}$. These normalization constants will be refered to as $T_{0}$ and $n_{0}$, respectively. The velocity is normalized to the electron thermal speed $v_{th}=({2T_{0}[J]}/{m_{e}})^{1/2}$, time is normalized to the $90^{\circ}$ electron-ion collision time $t_{0}={v_{th}^{3}}/{(\Gamma_{ei}^{0}n_{0}\ln\Lambda_{ei}(T_{0},n_{0})/Z)}$, and the length to the thermal electron-ion collision mean free path $x_{0}=v_{th}t_{0}$. Here $T_{0}[J]$ denotes the normalization temperature converted from eV to Joules, with $\Gamma_{ei}^{0}=Z^{2}\Gamma_{ee}^{0}=Z^{2}\frac{e^{4}}{4\pi(m_{e}\epsilon_{0})^{2}},$ and where $\ln\Lambda_{ei}(T_{0},n_{0})$ is the Coulomb logarithm for electron-ion collisions calculated for the normalization temperature and density (taken from [36]). All normalized quantities are $\displaystyle\tilde{v}$ $\displaystyle=\frac{v}{v_{th}},\quad\tilde{t}=\frac{t}{t_{0}},\quad\tilde{x}=\frac{x}{x_{0}},$ $\displaystyle\tilde{f_{l}}$ $\displaystyle=\frac{f_{l}}{n_{0}v_{th}^{-3}},\quad\tilde{E}=\frac{Eet_{0}}{m_{e}v_{th}},\quad\tilde{q}=\frac{q}{m_{e}n_{0}v_{th}^{3}},$ $\displaystyle\tilde{T}_{e,i,g}$ $\displaystyle=\frac{T_{e,i,g}}{T_{0}},\quad\tilde{n}_{e,i,b}=\frac{n_{e,i,b}}{n_{0}},\quad\tilde{u}_{e,i}=\frac{u_{e,i}}{v_{th}},$ $\displaystyle\tilde{\epsilon}$ $\displaystyle=\frac{\epsilon}{T_{0}},\quad\tilde{\sigma}=\frac{\sigma}{\sigma_{0}},$ where $\sigma_{0}=a_{0}^{2}\pi$ (where $a_{0}$ is the Bohr radius) . In the following sections the normalized quantities will be written without the tilde in order to lighten the notation. ### 3.2 Grids The velocity grid is a uniform or geometric grid of cells with (starting) width $\Delta v_{1}$ and width multiplier $c_{v}$, with $N_{v}$ cell centres distributed as $\displaystyle v_{1}$ $\displaystyle=\frac{\Delta v_{1}}{2},\quad\Delta v_{n}=c_{v}\Delta v_{n-1},\quad v_{n}=v_{n-1}+\frac{1}{2}\left(\Delta v_{n}+\Delta v_{n-1}\right),$ while the spatial grid is staggered, i.e. consists of cell centres and boundaries. $N_{c}$ is the number of cells (cell centres), while $N_{x}$ is used to denote the total number of spatial points (cells and boundaries), which depends on the boundary conditions of the grid (while $N_{c}$ is an input parameter). In the following text, spatial points with an odd index ($x_{1}$, $x_{3}$, etc.) will denote cell centres and those with an even index will denote cell boundaries. The values of these points are determined by $\displaystyle x_{1}=0,\quad x_{k}=x_{k-1}+\frac{\Delta x_{m}^{c}}{2},$ where $m=k$ if $k$ is odd, or $m=k-1$ if $k$ is even. $\Delta x_{m}^{c}$ denotes the cell width of the cell whose centre is at $m$. For a uniform grid, this is constant, while for a “logarithmic” grid it is an exponential function that starts at a prescribed width for the first cell $dx$, and drops to a prescribed width of the last cell $\Delta x_{L}$ (with a fixed number of cells $N_{c}$). Variables are positioned on the staggered grid in the following manner: * 1. In cell centres: * (a) $f_{l}$ for even $l$ * (b) Number densities: $n_{e}$, $n_{i}$, and $n_{b}$ * (c) Electron fluid temperatures $T_{e}$ * 2. On cell boundaries: * (a) $f_{l}$ for odd $l$ * (b) $E$-field * (c) Ion and electron fluid velocities, $u_{i}$ and $u_{e}$ Variables not evolved in cell centres are linearly interpolated from neighbouring cell boundaries, and vice versa. ### 3.3 Timesteps, nonlinear iteration, and vectorization of variables SOL-KiT uses either uniform timesteps of prescribed length $\Delta t$, or rescales $\Delta t$ with $\min(T_{e}(x)^{3/2}/n_{TOT}(x))$ (where $n_{TOT}$ is the total density of heavy particles). This is a conservative estimate for the shortest Coulomb collision time. In the following text the timestep will be assumed uniform, but generalization to adaptive timesteps is straightforward. Timestepping is done using a first order implicit backwards Euler method. Let $F$ be the vector containing all the evolved quantities, and $M(F)$ the evolution matrix. Then $\frac{F^{i+1}-F^{i}}{\Delta t}=M(F^{i^{*}})F^{i+1},$ (56) or $F^{i+1}=(I-\Delta tM(F^{i^{*}}))^{-1}F^{i},$ (57) Within a given timestep, we use fixed point iteration to solve the non-linear system (57), with $M(F^{i^{*}})$ being evaluated with the solution at the previous iteration. For the first iteration,$F^{i^{*}}=F^{i}$. (57) is iterated until convergence is established is established within a prescribed tolerance. Implicit variables (those that appear at time $i+1$ on RHS of equation (56)) in the most important nonlinear terms are given in Tables 1-3. Table 1: Fluid continuity and velocity equation implicit variables in nonlinear terms (including fluid contribution to Ampère-Maxwell law) Term | $\frac{\partial(nu)}{\partial x}$ | $S_{ion}{\textsuperscript{a}}$ | $S_{rec}{\textsuperscript{a}}$ | $-u\frac{\partial F}{\partial x}$ | $\frac{\partial(nkT)}{\partial x}$ | $-\frac{u}{n}S$ | $R_{u}$ | $R_{en}$ | $\frac{\partial E}{\partial t}$ ---|---|---|---|---|---|---|---|---|--- Implicit variable | $u$ | $n_{b}$ | $n_{e}$b | $F$ | $n$ | $u$ | $u$ | $n_{b}$ or $n_{e}$ | $u$ * a Collisional-radiative model uses the same implicit variable as in sources. * b If using kinetic model this is replaced by $4\pi\int v^{2}f_{0}dv$. Table 2: Fluid temperature equation implicit variables in nonlinear terms; $S$ in fifth term refers to same variable as in $S_{ion}$ and $S_{rec}$ of Table 1 Term | $-u\frac{\partial T}{\partial x}$ | $-T\frac{\partial u}{\partial x}$ | $q_{T}$ | $q_{u}$ | $-\frac{S}{n}\left[\frac{3}{2}kT-u^{2}\right]$ | $\frac{u}{n}R_{T}$ | $\frac{u}{n}R_{u}$ | $\frac{u}{n}R_{en}$ ---|---|---|---|---|---|---|---|--- Implicit variable | $T$ | $u$ | $T$ | $u$ | $S$ | $T$ | $u$ | $n_{b}$ or $n_{e}$ Table 3: Electron kinetic equation implicit variables in nonlinear terms; Coulomb collision terms for $f_{0}$ written out in more detail to avoid confusion (see below) Term | $E\frac{\partial f}{\partial v}$ | $\left(\frac{\delta f_{0}}{\delta t}\right)_{e-e}$ | $\left(\frac{\delta f_{l>0}}{\delta t}\right)_{e-i}^{stationary}$ | $\left(\frac{\delta f_{l>0}}{\delta t}\right)_{e-i}^{moving}$ | $\left(\frac{\delta f_{l}}{\delta t}\right)_{e-n}$ ---|---|---|---|---|--- Implicit variable | $E$ | $C^{i^{*}},D^{i^{*}},f_{0}^{i+1},\partial f_{0}^{i+1}/\partial v$ | $f_{l}$ | $f_{l}$, and $n_{i}$ in $I(F_{l})$,$J(F_{l})$ | $f_{l}$ The structure of the variable vector (with all variables present) is the following: $F=\begin{pmatrix}F_{loc}(x_{1})\\\ F_{loc}(x_{2})\\\ \vdots\\\ F_{loc}(x_{N_{x}})\end{pmatrix},$ (58) where $F_{loc}(x_{k})$ is the (spatially) local subvector for quantites at point $x_{k}$ given as $F_{loc}(x_{k})=\begin{pmatrix}f_{l_{max}}(x_{k})\\\ f_{l_{max-1}}(x_{k})\\\ \vdots\\\ f_{0}(x_{k})\\\ n_{1}(x_{k})\\\ n_{2}(x_{k})\\\ \vdots\\\ n_{N_{n}}(x_{k})\\\ E(x_{k})\\\ n_{e}(x_{k})\\\ u_{e}(x_{k})\\\ T_{e}(x_{k})\\\ n_{i}(x_{k})\\\ u_{i}(x_{k})\end{pmatrix},$ (59) where $N_{n}$ is the total number of neutral states tracked, $l_{max}$ is the highest resolved harmonic, and $f_{l}(x_{k})$ is the $l$-th harmonics subvector $f_{l}(x_{k})=\begin{pmatrix}f_{l}(x_{k},v_{1})\\\ f_{l}(x_{k},v_{2})\\\ \vdots\\\ f_{l}(x_{k},v_{N_{v}})\end{pmatrix}.$ (60) Note that when running in kinetic mode, the vector does not contain electron fluid quantities, while when the code is running with fluid electrons the distribution function harmonics are not evolved, but are updated after each timestep to be a slowly drifting Maxwellian with current temperature, density, and electron fluid velocity. From knowing the input parameters $N_{c}$ (and its derived parameters $2N_{c}-1\leq N_{x}\leq 2(N_{c}+2)-1$), $l_{max}$, $N_{n}$, and $N_{v}$ we can calculate the total vector length (for a kinetic run) as $N_{total}=N_{x}\left((l_{max}+1)N_{v}+N_{n}+3\right).$ (61) For a representative system with $l_{max}=5$, $N_{v}=80$, $N_{n}=30$, and $N_{x}=128$ this gives $N_{total}=55424$. To solve the above matrix system, we use the MPI and PETSc libraries [37, 38, 39]. Domain decomposition is done in the spatial dimension with as close to even distibution of grid points between processors as possible. ### 3.4 Velocity and spatial derivative discretization The velocity space derivatives appearing in the various kinetic operators are all implemented using a central difference scheme: $\displaystyle\frac{\partial^{2}F}{\partial v^{2}}(x_{k},v_{n})$ $\displaystyle=\frac{1}{\Delta v_{n}}\left[\frac{F(x_{k},v_{n+1})-F(x_{k},v_{n})}{v_{n+1}-v_{n}}-\frac{F(x_{k},v_{n})-F(x_{k},v_{n-1})}{v_{n}-v_{n-1}}\right],$ $\displaystyle\frac{\partial F}{\partial v}(x_{k},v_{n})$ $\displaystyle=\frac{F(x_{k},v_{n+1})-F(x_{k},v_{n-1})}{v_{n+1}-v_{n-1}},$ where $F$ is any velocity space function. Spatial derivatives are mostly discretized using an analogous central difference scheme to the above velocity space one. The only exceptions are the advection terms in the momentum and temperature equations (eq. (3),(6),(10)), where an upwind scheme can be used instead of the central difference. The upwind scheme used is simply $\frac{\partial F}{\partial t}(x_{k})=-u(x_{k})\frac{\partial F}{\partial x}(x_{k}),$ (62) where if $u(x_{k})\geq 0$ $\frac{\partial F}{\partial x}(x_{k})=\frac{F(x_{k})-F(x_{k-1})}{x_{k}-x_{k-1}},$ (63) and if $u(x_{k})<0$ $\frac{\partial F}{\partial x}(x_{k})=\frac{F(x_{k+1})-F(x_{k})}{x_{k+1}-x_{k}},$ (64) where if $F$ is not evolved on $x_{k\pm 1}$ we use use $x_{k\pm 2}$ instead (this is the case for temperature advection in (6)). Finally, the diffusion type derivatives are given by $\displaystyle\frac{\partial}{\partial v}\left(A\frac{\partial F}{\partial v}\right)(x_{k},v_{n})$ $\displaystyle=\frac{1}{\Delta v_{n}}[A(x_{k},v_{n+1/2})\frac{F(x_{k},v_{n+1})-F(x_{k},v_{n})}{v_{n+1}-v_{n}}$ $\displaystyle-A(x_{k},v_{n-1/2})\frac{F(x_{k},v_{n})-F(x_{k},v_{n-1})}{v_{n}-v_{n-1}}],$ $\displaystyle\frac{\partial}{\partial x}\left(A\frac{\partial F}{\partial x}\right)(x_{k})$ $\displaystyle=\frac{1}{x_{k+2}-x_{k-2}}[A(x_{k+1})\frac{F(x_{k+2})-F(x_{k})}{x_{k+2}-x_{k}}$ $\displaystyle-A(x_{k-1})\frac{F(x_{k})-F(x_{k-2})}{x_{k}-x_{k-2}}],$ where the velocity grid boundaries $v_{n+1/2}$ are given as $v_{n+1/2}=\sum_{l=1}^{n}\Delta v_{l}$. The only simple derivatives that do not obey the above are in the velocity advection terms, where in equations (22) and (23) we use the conservative form $\frac{\partial F}{\partial v}(x_{k},v_{n})=\frac{F(x_{k},v_{n+1/2})-F(x_{k},v_{n-1/2})}{\Delta v_{n}},$ where $F(x_{k},v_{n+1/2})$ is obtained through linear interpolation. The following velocity space boundary conditions are assumed for the distribution function harmonics $f_{0}(x_{k},0)=\frac{f_{0}(x_{k},v_{1})-f_{0}(x_{k},v_{2})v_{1}^{2}/v_{2}^{2}}{1-v_{1}^{2}/v_{2}^{2}},$ (65) $f_{l>0}(x_{k},0)=0,$ (66) i.e. $f_{0}$ at $v=0$ is quadratically extrapolated (this is used whenever $f_{0}$ at $v=0$ is required) and higher harmonics are set to $0$. At the velocity space boundary after $v_{N_{v}}$ all $f_{l}$ and their derivatives are assumed to be zero. ### 3.5 Coulomb collision operators Here, Coulomb collision operator discretization will briefly be presented to supplement the material in previous sections. The method used for the electron-electron collisions for $l=0$ is the Chang-Cooper-Langdon[40, 41, 42] method. The implementation in SOL-KiT is very similar to that in IMPACT[24], and as such we will leave out some of the details. We write the collision operator as a divergence of fluxes $F$ $C_{ee0}^{i+1}(x_{k},v_{n})=\frac{A_{ee}^{0}\ln\Lambda_{ee}(T_{e}^{i}(x_{k}),n_{e}^{i}(x_{k}))}{v_{n}^{2}}\frac{F^{i+1}(x_{k},v_{n+1/2})-F^{i+1}(x_{k},v_{n-1/2})}{\Delta v_{n}},$ (67) where $A_{ee}^{0}=\Gamma_{ee}^{0}n_{0}t_{0}/v_{t}^{3}=1/(Z\ln\Lambda_{ei}(T_{0},n_{0}))$, and the flux is given by $\displaystyle F^{i+1}(x_{k},v_{n+1/2})$ $\displaystyle=C^{i^{*}}(x_{k},v_{n+1/2})f_{0}^{i+1}(x_{k},v_{n+1/2})$ $\displaystyle+D^{i^{*}}(x_{k},v_{n+1/2})\frac{f_{0}^{i+1}(x_{k},v_{n+1})-f_{0}^{i+1}(x_{k},v_{n})}{v_{n+1}-v_{n}}.$ (68) $f_{0}^{i+1}(x_{k},v_{n+1/2})$ is then calculated using a special weighted interpolation, which ensures relaxation to a Maxwellian $\displaystyle f_{0}^{i+1}(x_{k},v_{n+1/2})$ $\displaystyle=(1-\delta^{i^{*}}(x_{k},v_{n+1/2}))f_{0}^{i+1}(x_{k},v_{n+1})$ $\displaystyle+\delta^{i^{*}}(x_{k},v_{n+1/2})f_{0}^{i+1}(x_{k},v_{n}),$ (69) $\displaystyle\delta^{i^{*}}(x_{k},v_{n+1/2})$ $\displaystyle=\frac{1}{W^{i^{*}}(x_{k},v_{n+1/2})}-\frac{1}{\exp[W^{i^{*}}(x_{k},v_{n+1/2})]-1},$ (70) $\displaystyle W^{i^{*}}(x_{k},v_{n+1/2})$ $\displaystyle=(v_{n+1}-v_{n})\frac{C^{i^{*}}(x_{k},v_{n+1/2})}{D^{i^{*}}(x_{k},v_{n+1/2})}.$ (71) The friction coefficient is $C^{i^{*}}(x_{k},v_{n+1/2})=4\pi\sum_{l=1}^{n}f_{0}^{i^{*}}(x_{k},v_{l})v_{l}^{2}\Delta v_{l}.$ (72) As previously noted in the Section 2.1., we chose to write the diffusion coefficient in a way different from what is usually in the literature. This allows discretization that conserves energy as well $D^{i^{*}}(x_{k},v_{n+1/2})=\frac{4\pi}{v^{*}_{n+1/2}}\sum_{l=1}^{n}v_{l}^{2}\left[\sum_{m=l}^{N_{v}-1}f_{0}^{i^{*}}(x_{k},v_{m+1/2})v_{m+1/2}(v_{m+1}-v_{m})\right]\Delta v_{l},$ (73) where $v^{*}_{n+1/2}=(v_{n}+v_{n+1})/2$. Boundary conditions for the $C$ and $D$ coefficients are taken in such a way to conserve particle density (see [24]). The resulting submatrix from this operator is tridiagonal. Thus we have an iterative method which conserves particles and energy (up to nonlinear iteration tolerance), and which relaxes the distribution to a Maxwellian in the absence of other driving forces. The electron-electron collision terms for higher $l$ (equation (31) have the same normalization constant as that for $l=0$. Here we use a discretization method similar to that in OSHUN [23], and will hence again, for the sake of brevity go over only the most important elements. The first three terms in (31) produce a tridiagonal matrix when discretized, while the remaining terms produce an upper and lower triangular submatrix due to the the $I$ and $J$ integrals, which are discretized as $I_{j}[f_{l}](x_{k},v_{n})=4\pi\begin{pmatrix}\left(\frac{v_{1}}{v_{n}}\right)^{j}v_{1}^{2}\Delta v_{1}\\\ \left(\frac{v_{2}}{v_{n}}\right)^{j}v_{1}^{2}\Delta v_{2}\\\ \vdots\\\ \left(\frac{v_{n-1}}{v_{n}}\right)^{j}v_{n-1}^{2}\Delta v_{n-1}\\\ \frac{1}{2}v_{n}^{2}\Delta v_{n}\\\ 0\\\ \vdots\\\ 0\end{pmatrix}^{T}\cdot\begin{pmatrix}f_{l}(x_{k},v_{1})\\\ \vdots\\\ f_{l}(x_{k},v_{N})\end{pmatrix},$ (74) $J_{j}[f_{l}](x_{k},v_{n})=4\pi\begin{pmatrix}0\\\ \vdots\\\ 0\\\ \frac{1}{2}v_{n}^{2}\Delta v_{n}\\\ \left(\frac{v_{n+1}}{v_{n}}\right)^{j}v_{n+1}^{2}\Delta v_{n+1}\\\ \vdots\\\ \left(\frac{v_{N_{v}-1}}{v_{n}}\right)^{j}v_{N_{v}-1}^{2}\Delta v_{N_{v}-1}\\\ \left(\frac{v_{N_{v}}}{v_{n}}\right)^{j}v_{N_{v}}^{2}\Delta v_{N_{v}}\end{pmatrix}^{T}\cdot\begin{pmatrix}f_{l}(x_{k},v_{1})\\\ \vdots\\\ f_{l}(x_{k},v_{N})\end{pmatrix}.$ (75) For $l=1$ the discretized e-e collision operator does not numerically conserve momentum, unfortunately, as the discertized form loses the partial integration properties that would analytically conserve momentum. However, the numerically lost momentum is transfered to the ions, and total momentum is thus conserved. The electron-ion operator for stationary cold ions is trivial, and is discretized straightforwardly, while the moving ion operator is discretized similarly to the e-e operator for higher $l$, with the exception of having mainly a tridiagonal component (with terms containing $F_{0}$), and a part (terms containing $F_{l}$) where ion density is the implicit variable (if working on a cell boundary, it is implicitly interpolated) . For the second part we need the previously presented $I(F_{l})$ and $J(F_{l})$ integrals for cold ions ((35) and (36)). For a given lagged ion velocity $u_{i}^{i^{*}}(x_{k})$ and density $n_{i}^{i+1}(x_{k})$ these integrals in discrete form are $I_{j}^{i+1}[F_{l}](x_{k},v_{n})=(2l+1)n_{i}^{i+1}(x_{k})\frac{u_{i}^{i^{*}}(x_{k})^{j}}{v_{n}^{j}}\Theta(v_{k}-u_{i}^{i^{*}}(x_{k})),$ (76) $J_{j}^{i+1}[F_{l}](x_{k},v_{n})=(2l+1)n_{i}^{i+1}(x_{k})\frac{u_{i}^{i^{*}}(x_{k})^{j}}{v_{n}^{j}}\Theta(u_{i}^{i^{*}}(x_{k})-v_{k}).$ (77) ### 3.6 Electron-neutral collision numerics In this section we briefly go over the basic properties of the elastic electron-neutral collision operator before moving on to the important conservative discretization of inelastic electron-neutral collisions. ##### Elastic collisions The discretization of elastic collisions borrows greatly from the Coulomb collision operators. Equation (39) is discretized in a way similar to the Chang-Cooper-Langdon scheme. However, since the integral is linear in $f_{0}$, there is no need for special interpolation ($f_{0}$ and $\sigma$ are simply linearly interpolated), and just the flux formalism was used, with the $C$ and $D$ coefficients (here left unnormalized) being $C(v_{n+1/2})=n_{b}v_{n+1/2}^{4}\sigma^{el}(v_{n+1/2}),$ (78) $D(v_{n+1/2})=n_{b}v_{n+1/2}^{3}\sigma^{el}(v_{n+1/2})\frac{kT_{g}}{m_{e}},$ (79) where $\sigma^{el}(v_{n})=\int d\Omega(1-\cos\chi)\sigma^{el}(\chi,v_{n}).$ (80) Since no differential cross section data is available, this is simply set to the constant cross-section (41). The above discretization produces an operator that conserves particle number and relaxes the electron $f_{0}$ to a Maxwellian with temperature $T_{g}$ (within finite grid effects). For higher $l$, equation (40) is discretized in a straightforward way. #### 3.6.1 Conservative discretization scheme for inelastic collisions In order to streamline the arguments in the next sections, let us introduce the following notation. If we label inelastic processes with transition energy $\epsilon$ as $\Pi$, we can label the corresponding superelastic (inverse) processes as $\Pi^{-1}$ (with transition energy $-\epsilon$). Electron energy and particle conservation are governed by the $l=0$ harmonic, which we will denote simply as $f$ in the following derivation. Then the value of $f$ in the $n$-th velocity cell centre can be written as $f_{n}$, while the total cross- section notation for any given process will be labeled as $\sigma_{n}$. Equation (43) contains two terms, a loss/emission term and a gain/absorption term. While the first term is defined on the used velocity grid, the second term requires evaluation of the distribution function on (most likely) non- existant points $\alpha(v)v$ (where $\alpha=(1\pm 2\epsilon/mv^{2})^{1/2}$). A straightforward interpolation here fails to produce a discretization that conserves particles and energy. In order to develop such a scheme we start by writing the collision operator in the emission/absorption form $C_{n}=-E_{n}+A_{n},$ (81) where $E_{n}=n_{neut}v_{n}f_{n}\sigma_{n},$ (82) and $A_{n}=\sum_{m}W_{nm}E_{m},$ (83) where $W_{nm}$ are weights determining the contribution of the emission from cell $m$ to the absorption in cell $n$. For particle conservation, we want the number of particles emitted by a single cell $m$ \- $4\pi E_{m}v_{m}^{2}\Delta v_{m}$ to be absorbed exactly by some other cells $n$ \- $4\pi W_{nm}E_{m}v_{n}^{2}\Delta v_{n}$. After tidying up, the resulting particle conservation condition is $v_{m}^{2}\Delta v_{m}=\sum_{n}W_{nm}v_{n}^{2}\Delta v_{n}.$ (84) For energy conservation, we note that all energy being emitted via collisions from one cell needs to either be lost to the energy of internal atomic states or be absorbed by some cells $n$. In a similar way to the above, one can cancel distribution functions and cross-sections to obtain the energy conservation condition $v_{m}^{2}\Delta v_{m}\left(v_{m}^{2}\mp\epsilon\right)=\sum_{n}W_{nm}v_{n}^{4}\Delta v_{n}.$ (85) Finally, it is useful to write down the numerical version of detailed balance for the collisional cross-section of the inverse process. Here we show the result for de-excitation ($\Pi=ex$, $\Pi^{-1}=deex$), with the same procedure applicable to recombination. Detailed balance implies that for a Maxwellian distribution of electrons and for a Boltzmann distribution of excited states the rates of excitation and de-excitation become equal. This can be written as $n_{l}\sum_{m}f_{m}\sigma_{m}^{ex}v_{m}^{3}\Delta v_{m}=n_{u}\sum_{n}f_{n}\sigma_{n}^{deex}v_{n}^{3}\Delta v_{n},$ (86) where $n_{l}$ and $n_{u}$ denote the densities of the lower and upper excited states, respectively. Using a Maxwellian for $f$, relating $n_{u}$ and $n_{l}$ via the Boltzmann distribution, and utilizing equation (84) we obtain $\sigma_{n}^{deex}=\frac{g_{l}}{g_{u}}e^{\epsilon/T}\sum_{m}W_{nm}\sigma_{m}^{ex}\frac{v_{m}}{v_{n}}e^{-\left(v_{m}^{2}-v_{n}^{2}\right)/T}.$ (87) Note that this depends on the temperature and the excitation weights, contrary to the analytical version from equation (46). This is due to the discrete nature of the grid, where energy differences between absorbing and emitting cells do not have to be equal to the analytical value $\epsilon$. #### 3.6.2 Two-absorber mapping Note that in the above conservation condition we haven’t specified the summation range for $n$. The simplest way to do this is the following. For each emitter $m$ we choose exactly two (consecutive) absorber cells, $n_{1}$ and $n_{2}$ such that $\sqrt{v_{m}^{2}\mp\epsilon}$ lies between $v_{n_{1}}$ and $v_{n_{2}}$. We will refer to these two points as the ideal absorber pair. This way we do not need any further instructions on partitioning the emitted particles and energy, and can proceed to calculate weights. Since for any given $m$ there are only two absorbing cells, we can denote the weights simply as $W_{1}^{m}$ and $W_{2}^{m}$, and solving the conservation conditions we get $W_{1}^{m}=\frac{v_{m}^{2}\Delta v_{m}}{v_{n_{1}(m)}^{2}\Delta v_{n_{1}(m)}}\left(1-\frac{v_{m}^{2}-v_{n_{1}(m)}^{2}\mp\epsilon}{v_{n_{2}(m)}^{2}-v_{n_{1}(m)}^{2}}\right),$ (88) $W_{2}^{m}=\frac{v_{m}^{2}\Delta v_{m}}{v_{n_{2}(m)}^{2}\Delta v_{n_{2}(m)}}\frac{v_{m}^{2}-v_{n_{1}(m)}^{2}\mp\epsilon}{v_{n_{2}(m)}^{2}-v_{n_{1}(m)}^{2}}.$ (89) Figure 2 shows an illustration of this mapping for an excitation collision with both particle and energy transfer obeying conservation conditions. Figure 2: Two-absorber mapping for an excitation collision; particles and energy emitted by higher energy cell distributed among the two absorbers, while a portion of energy is lost to internal energy states of collision target (atom) - red line. Blue and green lines denote emission to first and second cell of absorber pair, respectively. The black star shows the location of the analytical absorption point. To establish that this scheme can reduce to the analytical result, we note two properties we expect in the analytic limit. Firstly $n_{1}\rightarrow n_{2}=n$, i.e. each emitting point has one and only one absorbing point. This can be done by letting one of the weights tend to $0$. For the sake of this illustrative argument, let that be $W_{1}^{m}$, from which we see that the second fraction in (89) tends to unity. Then, we note that the first fraction will tend to $\alpha(v_{n})^{2}\Delta v_{m}/\Delta v_{n}$ (since $v_{m}=\alpha(v_{n})v_{n}$ in analytical limit). Taking the differential of (42) points us toward a grid that satisfies $\Delta v_{m}/\Delta v_{n}\rightarrow 1/\alpha(v_{n})$ which finally yields the limits $W_{1}^{m}\rightarrow 0,\quad W_{2}^{m}\rightarrow\alpha(v_{n}).$ Noting that we have chosen to write the gain term as a sum of weighted loss terms, another $\alpha$ factor can be extracted from the single emitter cell velocity in the absorption term, and the analytical $\alpha^{2}$ form from (43) is recovered. At first glance, this mapping looks good, we have found pairs of absorbers and weighted their absorption terms to conserve particles and energy, but after a closer look at eq. (87) a potential problem reveals itself. Suppose that for some $n$ and every $m$ the weights $W_{nm}=0$ in an excitation process. In this case $\sigma_{n}^{deex}=0$ for the corresponding deexcitation process. In other words, if cell $n$ does not absorb in process $\Pi$, it will not emit in $\Pi^{-1}$, and gaps are left on our velocity grids where cells that should emit do not. This is not physical, and the mapping needs to be refined. #### 3.6.3 Two-absorber mapping with pair partition In the previous section we have associated every cell (emitter) $m$ with an ideal absorber pair $(n_{1}(m),n_{2}(m))$. We define the absorber list $\mathcal{A}^{\Pi}_{m}$ as a list of absorber pairs of point $m$ in process $\Pi$. The two-absorber mapping implies $\mathcal{A}^{\Pi}_{m}=\left\\{(n_{1}(m),n_{2}(m))\right\\}$, where the only pair is the ideal absorber pair. As noted above, this produces gaps in the the velocity grid where cells do not emit in the inverse process $\Pi^{-1}$. In order to fill out the aforementioned gaps, we must potentially include absorbers other than those in the ideal pair. This can be done by looking at all cells $n$ that aren’t in the ideal absorber pair, but whose $\sqrt{v_{n}^{2}\pm\epsilon}$ (note the change in sign!) falls into cell $m$. We then refer to $m$ as the ideal emitter for cells $n$. We then move to generate absorber pairs that would include the non-ideal absorbers $n$, while satisfying the original constraint, namely that $\sqrt{v_{m}^{2}\mp\epsilon}$ is between the points of each new pair. In SOL- KiT this is done by always pairing non-ideal absorbers with one of the ideal absorbers. If we denote $P_{m}\geq 1$ as the number of (both ideal and non- ideal) absorber pairs of cell $m$, the absorber list becomes $\mathcal{A}^{\Pi}_{m}=\left\\{(n_{1}^{p}(m),n_{2}^{p}(m)),p=1,...,P_{m}\right\\}$, where we label the first and second cell in pair $p$ as $n_{1}^{p}(m)$ and $n_{2}^{p}(m)$, respectively. This allows us to use the previous solution, but we require a way to partition the emitted energy/particles among pairs. We do this by defining a total energy width of the entire absorber list and normalizing the energy widths of each cell within the list to the total energy width of the list: $\beta_{TOT}^{m}=\sum_{n\in\mathcal{A}^{\Pi}_{m}}2v_{n}\Delta v_{n},\quad\beta_{n}=2v_{n}\Delta v_{n}/\beta_{TOT}^{m}.$ Then we define $\delta_{n}$ as the number of pairs in $\mathcal{A}^{\Pi}_{m}$ which contain point $n$. We can then define $\gamma^{p}=\frac{\beta_{n_{1}^{p}}}{\delta_{n_{1}^{p}}}+\frac{\beta_{n_{2}^{p}}}{\delta_{n_{2}^{p}}}$ (90) to be the fraction of the emission given to pair $p$. It is trivial to check that the sum of all $\gamma^{p}$ for a given absorber list is equal to unity, as one would expect. An example with an absorber list with three distinct cells is given in Figure 3. Figure 3: Pair partition calculation example for an excitation collision; emitter cell has a list of three absorbers, grouped into two pairs, with emitted energy and particles partitioned according to factors $\gamma^{1}$ and $\gamma^{2}$ Then we can go through the same process as the one for the two-absorber case, except the final results for $W_{1}^{m}$ and $W_{2}^{m}$ will now be functions of the pair $p$ for which they have been calculated, and will have an extra $\gamma^{p}$ factor multiplying the previous simple results (see below). To then calculate the total absorption term for a given cell $n$, it should be summed over each pair the cell belongs to, i.e. $A_{n}=\sum_{m}\sum_{p}W_{nm}^{p}E_{m}.$ (91) This way we have both ensured particle and energy conservation, as well as a reasonably physical numerical detailed balance condition. The weights in this case are given by $W_{n_{1}^{p}(m)m}=\gamma_{m}^{p}\frac{v_{m}^{2}\Delta v_{m}}{v_{n_{1}^{p}(m)}^{2}\Delta v_{n_{1}^{p}(m)}}\left(1-\frac{v_{m}^{2}-v_{n_{1}^{p}(m)}^{2}\mp\epsilon}{v_{n_{2}^{p}(m)}^{2}-v_{n_{1}^{p}(m)}^{2}}\right),$ (92) $W_{n_{2}^{p}(m)m}=\gamma_{m}^{p}\frac{v_{m}^{2}\Delta v_{m}}{v_{n_{2}^{p}(m)}^{2}\Delta v_{n_{2}^{p}(m)}}\frac{v_{m}^{2}-v_{n_{1}^{p}(m)}^{2}\mp\epsilon}{v_{n_{2}^{p}(m)}^{2}-v_{n_{1}^{p}(m)}^{2}}.$ (93) Using the above method for every process produces both a transition mapping and weights for each one. These depend solely on the grid and the inelastic processes being considered, and as such do not change during a simulation. The final discretized (unnormalized) form of the inelastic collision integral for particle conserving collisions is then $\displaystyle\left(\frac{\delta f_{l}}{\delta t}\right)^{inel,i+1}_{b\rightarrow b^{\prime}}(x_{k},v_{n})$ $\displaystyle=-v_{n}n_{b}^{i^{*}}(x_{k})[\sigma^{TOT}_{b\rightarrow b^{\prime}}(v_{n})f_{l}^{i+1}(x_{k},v_{n})$ $\displaystyle-\sum_{m}\sum_{p}W_{nm}^{p}(\sigma^{TOT}_{b\rightarrow b^{\prime}}(v_{m})-\sigma^{(l)}_{b\rightarrow b^{\prime}}(v_{m}))f_{l}^{i+1}(x_{k},v_{m})].$ (94) ### 3.7 Divertor target boundary condition discretization In order to implement the boundary condition at the divertor target (not explicitly present on the spatial grid), as was implied in the Section 2.1.6., we require knowledge of the forward going electron distribution function. We reconstruct it by extrapolating harmonics from cells leading up to the boundary $f_{\text{odd }l}^{\text{forward,target}}=\frac{n_{e}^{2}(x_{N_{x}})}{n_{e}^{2}(x_{N_{x}-1})}f_{\text{odd }l}^{i^{*}}(x_{N_{x}-1}),\quad f_{\text{even }l}^{\text{forward,target}}=\frac{n_{e}(x_{N_{x}})}{n_{e}(x_{N_{x}-1})}f_{\text{even }l}^{i^{*}}(x_{N_{x}}),$ (95) where we scale the harmonics by the ratio of the electron densities at spatial cells $N_{x}$ and $N_{x}-1$. The choice to perform this sort of extrapolation, and not a linear one, comes from the danger of large density gradients (which are often present at the divertor target) to produce negative extrapolated densities when extrapolated linearly. Knowing the forward going distribution allows us to calculate the cut-off distribution using equation (53), while equation (54) imposes $v_{c}$. However, due to the discretization of velocity space, this will not be one of the resolved cell centres, and we are required to interpolate in order to calculate the precise cut-off. A diagram of the interpolation region is given in Figure 4. Figure 4: The interpolation region on the velocity grid; cell $K$ contains the cut-off, and cells $K$ and $K-1$ are replaced with interpolated cells with centres at $v_{i1}$,$v_{i2}$ and widths $\Delta v_{1}$,$\Delta v_{2}$ The interpolation replaces cell $K$ containing the cut-off and its preceding cell $K-1$ with two new cells with centres $v_{i1}=\frac{v_{K+1/2}+v_{c}}{2},\quad v_{i2}=\frac{v_{K-3/2}+v_{c}}{2},$ and with widths $\Delta v_{1}=v_{K+1/2}-v_{c},\quad\Delta v_{2}=v_{c}-v_{K-3/2}.$ The electron flux to the boundary (given in equation (54)) is then calculated using this updated grid, while linearly interpolating the $f_{1}$ component of the cut-off distribution in the two new cells. The electron flux can then be matched to the ion flux using a bisection or false position method (the latter being implemented in the current version of SOL-KiT) to find $v_{c}$ with a desired accuracy. ## 4 Benchmarking SOL-KiT operators A number of verification tests have been performed using SOL-KiT, aimed at checking the properties of various implemented operators. The test details are presented in the following sections, while an overview of the tests and their scope is given in Table 4. In all tests we consider hydrogenic species, ie. $Z=1$. Table 4: List of performed test runs and sets of runs with their target operators. Run/Set | Targeted operators ---|--- Runs 1-4 | e-i and e-e collisions, Vlasov, Maxwell Runs 5-6 | fluid ion and electron operators Set 1 | e-i and e-e collisions, Vlasov, Maxwell, high $l$ Runs 7-8 | kinetic e-n inel. collisions, CRM Run 9 | fluid e-n inel. collisions, CRM, detailed balance Run 10 | e-i and e-e collisions, Vlasov, Maxwell, kinetic e-n inel. collisions, CRM Run 11 | fluid ion operators - advection Set 2 | fluid ion operators - charge-exchange friction Set 3 | divertor boundary condition - analytic limit Set 4 | divertor boundary condition, high $l$ ### 4.1 Heat conduction tests In order to test a number of operators we performed both local tests with different scenarios, as well as non-local perturbation tests. ##### Local heat conduction tests Two types of tests were used to verify the local limit of heat conduction in SOL-KiT. The first type is a small perturbation test on a periodic grid, with the initial temperature given by $T(x)=T_{0}+T_{1}\sin\left(2\pi\frac{x}{L}\right),$ (96) where $L$ is the total length of the system, given by $L=N_{c}\Delta x$. The density was set to $n_{0}=10^{19}m^{-3}$, and the rest of the simulation parameters are $T_{0}=100\text{eV},\quad T_{1}=0.05\text{eV},$ $N_{c}=64,\quad\Delta x=150\ x_{0},\quad N_{v}=120,$ with $\Delta v$,$\Delta t$, and the total number of timesteps being able to vary between runs. The total time simulated was always set to $30\ t_{0}$, i.e. to 30 electron-ion collision times. In all runs the full set of Coulomb collision operators was active, and the highest harmonic number was kept at $l_{max}=1$. The calculated conduction heat flux was compared to the Spitzer- Härm (SH) heat flux $q_{SH}$ (equation (7)), with the heat conductivity for $Z=1$ in SOL-KiT normalized units being $\kappa=0.6\sqrt{\pi}.$ The ratio of the calculated heat flux and the reference SH flux at the end of each run would be averaged along the simulation domain, and these are the results presented below. Three pure kinetic runs (with initial $f_{1}$ set to 0, and no fluid ion operators active) were performed. The reference run (Run 1) with $\Delta t=0.1\ t_{0}$, $\Delta v=0.1\ v_{th}$, $N_{t}=300$ produces an average heat flux of $q=(0.988296\pm 1\times 10^{-6})q_{SH}$, i.e. reproduces the reference results with less than 1.2% error. Run 2 used a smaller timestep $\Delta t=0.05\ t_{0}$, $N_{t}=600$, but the ratio obtained was the same as in Run 1. Run 3, on the other hand, tested the velocity resolution dependence by setting $\Delta v=0.05\ v_{th}$, and the obtained heat flux was $q=(0.997488\pm 1\times 10^{-6})q_{SH}$, reducing the relative error below 0.3%. At this point we believe the relative error is smaller than the precision of the reference SH flux value, and should thus be taken with a grain of salt. The next set of small perturbation tests aimed to compare the results of kinetic simulations to those performed using the fluid mode of SOL-KiT. For this purpose, since the fluid model assumes the reference SH heat flux described above from the very start of the simulation, we initialized $f_{1}$ in the kinetic simulation (Run 4) to a local solution which would give the same heat flux as the one in the fluid run (Run 5). Other than this, the parameters of Run 4 were identical to those of Run 1, and the final heat flux ratio was the same as the one there. Run 5 was performed with the same parameters as Run 4, but instead of solving the kinetic equation for the electrons, fluid equations for both electrons and ions were solved. The total changes in the temperature value at its maximum for both runs were compared. It was found that the relative diference in the changes was less than 0.3%, showing good agreement between the kinetic and fluid model in the small perturbation test. The final heat conduction test run (Run 6) was done for the fluid model, where the plasma was initialized on a logarithmic spatial grid ($\Delta x=8.5\ x_{0}$ and $\Delta x_{L}=0.5\ x_{0}$) with the Two-Point Model[6] profile $T(x)=\left(T_{u}^{7/2}+\frac{x}{L}(T_{u}^{7/2}-T_{d}^{7/2}\right)^{2/7},\quad n(x)=n_{u}T_{u}/T(x),$ where $n_{u}=0.368\times 10^{19}m^{-3}$ and $T_{u}=18\text{eV}$ are the upstream density and temperature, while $T_{d}=5\text{eV}$ is the downstream temperature. Boundary conditions were fixed. We expect that this profile should not evolve, and after 30000 collision times (for a reference plasma of $n_{0}=2.5\times 10^{19}m^{-3}$ and $T_{0}=10\text{eV}$), the largest deviation from the initial profile was 0.12 %, which is due to linear interpolation close to the downstream boundary, with most points having less than 0.01% deviation. ##### Non-local heat conduction tests In order to test how SOL-KiT handles a more non-local regime, we have performed a set of runs (Set 1) with a setup similar to Run 1 and related runs, featuring a sinusoidal temperature perturbation. For completeness, we give the common run parameters for Set 1 $T_{0}=100\text{eV},\quad T_{1}=0.1\text{eV},\quad n_{0}=10^{19}m^{-3},$ $N_{c}=63,\quad N_{v}=120,\quad\Delta v_{1}=0.0307,\quad c_{v}=1.01.$ The various runs in the set were performed with different system lengths and with varying number of resolved harmonics $l_{max}=1,3,5,7$. In Figure 5 we plot the $\kappa/\kappa^{(B)}=q/q_{SH}$ for each of the runs, comparing it with values obtained with the code KIPP[8, 19, 20] and reported by Brodrick et al.[43]. The ratio $q/q_{SH}$ was obtained by running each run in the set until the ratio equilibriated. The different values in Figure 5 are plotted as a function of $k\lambda_{ei}^{(B)}$, where $\lambda_{ei}^{(B)}=3(\pi/2)^{1/2}x_{0}/4$ is the Braginskii electron-ion collisional mean freepath, as used by Brodrick et al., and $k=2\pi/L$ is the perturbation wavenumber. KIPP results were fitted using the following function based on equation (25) in [43] $\frac{\kappa}{\kappa^{(B)}}=\left[1+\left(\frac{1}{a(k\lambda_{ei}^{(B)})^{2}}+\frac{1}{bk\lambda_{ei}^{(B)}}\right)^{-1}\right]^{-1},$ (97) where the values obtained for the parameters $a$ and $b$ are $51.409789$ and $4.4682314$, respectively. We show that the increase in number of harmonics leads to an increase in agreement between SOL-KiT and KIPP. Already at $l=5$ SOL-KiT results appear to be very close to the fit values, with the increase to $l=7$ having a negligible effect on the result. We also note that the diffusive approximation ($l=1$) appears to break down around $k\lambda_{ei}^{(B)}=0.075$, while $l=3$ seems to hold up further into the non-local regime. Figure 5: Comparison of SOL-KiT and KIPP results for the non-local value of heat conductivity for different number of resolved harmonics. KIPP results have been fitted with a version of the fitting function presented in [43], here see equation 97. ### 4.2 Collisional-Radiative model and inelastic collision tests (a) Uniform grid - Run 7 (b) Geometric grid - Run 8 Figure 6: Evolution of total density in the two inelastic collision discretization test runs. The discretization scheme for inelastic collisions presented in this paper is designed with three properties in mind - particle and energy conservation, as well as numerically consistent detailed balance. In this section we present several tests aimed at confirming the desired properties. ##### Conservation property tests We start with two runs differing only in the utilized velocity space grid. The common run parameters are $n_{0}=10^{19}m^{-3},\quad n_{e}=n_{0},\quad n_{1}=0.1n_{0},\quad N_{n}=30,$ $\Delta t=0.5t_{0},\quad N_{t}=30000,\quad N_{v}=120,\quad T_{0}=5\text{eV},\quad T_{e}=T_{0},$ where $N_{n}$ is the total number of resolved hydrogen states. Both runs included only the inelastic electron-atom processes, and were performed in quasi-0D (low number of spatial cells). The first of the two runs (Run 7) used a uniform grid with $\Delta v=0.05v_{th}$, while the second (Run 8) used a geometric grid with $\Delta v_{1}=0.01$ and $c_{v}=1.025$. Figure 6 shows the average heavy particle density ($n_{i}+\sum n_{b}$) in the two runs. As can be seen the total error is of the order of $10^{-14}$, which is consistent with round-off errors. Similar results are obtained for the relative deviation of the total energy density $E_{TOT}=3n_{e}kT_{e}/2+n_{e}\epsilon_{ion}+\sum n_{b}\epsilon_{b}$ from initial value, where $\epsilon_{b}$ is the $1\rightarrow b$ transition energy. This result is shown in Figure 7, where we see that the geometric grid (Run 8) is performing somewhat better in the later part of the simulation, likely due to a finer velocity grid near the origin. Figure 7: Relative change in total energy density in both electron motion and atomic states for the two inelastic collision discretization test runs (Runs 7-8). ##### Detailed balance test In order to test the numerical detailed balance condition for the inverse processes we treat the electrons as a fluid, thus forcing the distribution function to a Maxwellian. This simulation (Run 9) was performed with the same grid, normalization, and initial conditions as the uniform grid run discussed above. The only difference was the timestep parameters being $N_{t}=1000$ and $\Delta t=100t_{0}$. By the end of the simulation the atomic states settled close to a Saha-Boltzmann equilibrium, as would be expected from the detailed balance condition. The ionization degree relative error computed against the analytical solution for hydrogen was $\delta X=1.134\times 10^{-8}$, while the total density and energy errors at the end of the simulation were $\approx 6\times 10^{-14}$ and $\approx 5\times 10^{-14}$, respectively. (a) Electron temperature (b) Electron density Figure 8: Evolution of temperature and density in reproduction of results from Allais et al. [18] (Run 10) - corresponds to Figures 1 and 2 from original paper. Note that in all three runs discussed above we disabled all radiative processes, as those would mask any energy conservation processes in the first two runs, and would not allow for equilibrum in the detailed balance test. ##### Integrated test Finally, an integrated test (Run 10) was performed in order to test the full interplay of electon and neutral processes. We have attempted to replicate as closely as possible the simulation performed by Allais et al.[18] using the code FPI. Parameters used in this run were $T_{0}=10\text{eV},\quad n_{0}=2.5\times 10^{19}m^{-3},\quad N_{n}=30,\quad n_{TOT}=n_{0},$ $N_{v}=80,\quad\Delta v_{1}=0.05v_{th},\quad c_{v}=1.025,\quad l_{max}=1$ $N_{c}=64,\quad\Delta x=5.536x_{0},\quad N_{t}=16600,\quad\Delta t=0.1t_{0},$ amounting to a domain of length $L=20.31\text{m}$ and total simulation time $t_{TOT}=49.96\mu\text{s}$. At $x=0$ the temperature is initialized to 25 eV, and the plasma is 100% ionized. From $x=2.5$m to $x=4.06$m the temperature drops exponentially to 1eV, while the ionization degree drops to 10%. All inelastic collisions were enabled, as well as radiative processes, while ions and neutrals were left stationary as in the original paper. Boundary conditions were fixed to initial values. Figure 8 shows the evolution of electron temperature and density corresponding to Fig. 1. and Fig. 2. in [18]. Qualitative behaviour is recovered, while discrepancies of less than 10% are most likely caused by potential differences in the intialization, as well as likely different spatial and velocity space resolutions (not reported in [18]). Another potential cause of discrepency is the use different databases in SOL-KiT compared to FPI. ### 4.3 Ion flow tests ##### Acoustic wave test To test the ion advection, we performed the following isothermal ion acoustic wave test (Run 11). Parameters of the simulation were $T_{0}=100\text{eV},\quad n_{0}=10^{19}m^{-3},\quad N_{t}=6000,\quad dt=0.01t_{0},\quad l_{max}=1$ $N_{v}=120,\quad\Delta v=0.1v_{th},\quad N_{c}=128,\quad dx=0.0078125x_{0},$ with the density and ion velocity initialized on a periodic grid of length $L=x_{0}$ as $n(x)=n_{0}+0.01n_{0}\sin\left(2\pi\frac{x}{L}\right),\quad u_{i}(x)=\frac{v_{th}}{6000}\sin\left(2\pi\frac{x}{L}\right).$ Electron-electron collisions for $l=0$ and electron-ion collisions were turned on in the simulation. The density and ion velocity are presented at several points during the evolution of the acoustic wave in Figure 9. The sound speed was evaluated by fitting a sine function to the ion velocity profiles at different times, giving the value $c_{s}=(1.001\pm 0.013)c_{s}^{analytical}$. Estimation of the error was performed conservatively, by taking the greatest deviation from the computed mean sound speed. Nonetheless, the obtained agreement is satisfactory even with the conservative error estimate. (a) Electron/ion density (b) Ion velocity Figure 9: Evolution of density and ion velocity during the acoustic wave propagation test from Run 11. ##### Charge-exchange friction test To test the charge-exchange cross-section implementation, a set of runs (Set 2) with the following parameters was performed $T_{0}=10\text{eV},\quad n_{0}=2.5\times 10^{20}m^{-3},\quad N_{n}=1,\quad n_{TOT}=n_{0},\quad n_{e,i}=n_{0},$ with the electrons and ions decoupled by turning the $E$-field and electron- ion collisions off. The only operator being tested was the charge-exchange friction term in the ion equation. The total simulation time was kept at 500 e-i reference collision times. The ion velocity was initialized at $u_{i}=v_{th}$, and its value was compared to the analytical solution $u(t)=u(0)/(1+u(0)n_{i}m_{i}n_{1}\sigma_{CX,1}t)$ as the ions were slowed down by friction. The relative error of the evolved ion velocity is plotted in Figure 10 for different timestep lengths. It should be noted that the initial value of the ion velocity is unphysically high, and was used only for stress testing the operator, with velocities towards the end of the simulation being more in line with values observed in the SOL. Figure 10: Relative difference between analytical and computed values during charge-exchange friction test for various timestep lengths - Set 2. ### 4.4 Divertor boundary condition tests ##### Analytic limit convergence test The first condition a cut-off logical boundary condition would need to satisfy is the analytical Maxwellian limit for the sheath properties. In order to test this aspect of the operator, the cut-off procedure and calculation of the sheath heat transmission factor and potential drop were performed without evolving the initially Maxwellian distribution. A set of single step $\Delta t=0$ simulations (Set 3) with different velocity space resolutions (constant $v_{max}=6v_{th}$) was used to evaluate convergence of the cut-off calculation without evolving the distribution. The results of this test are shown in Figure 11, where we see that both the sheath heat transmission coefficient and the potential are within a few percent of the analytical values[6] ( $\gamma_{e}=2-0.5\ln(2\pi(1+T_{i}/T_{e})m_{e}/m_{i})$ and $\Delta\Phi=\gamma_{e}-2$) even for the coarsest grid used. Figure 11: Relative error of computed sheath heat transmission coefficient and potential drop as a function of velocity grid resolution - Set 3. Distribution was set to Maxwellian and wasn’t evolved in order to compare to the analytical results (see text). ##### High harmonic convergence test In order to test convergence of the sheath properties in a driven system when the number of harmonics is varied a set of runs (Set 4) was performed with the following parameters $T_{0}=10\text{eV},\quad n_{0}=2.5\times 10^{19}m^{-3},$ $N_{v}=80,\quad\Delta v_{1}=0.05v_{th},\quad c_{v}=1.01,$ with a short system $L=1x_{0}$ and $N_{c}=2$. Electron-electron collisions were included only for $l=0$ while electron-ion collisions were on for all $l>0$. A Maxwellian with $T_{e}=T_{0}$ and $n_{e}=n_{0}$ was imposed as a fixed boundary condition at $x=0$. This provided a scenario where the boundary condition operator was sufficiently stressed without entering a strictly collisionless regime. Simulations were run until the sheath properties equilibriated, and the results of the test are shown in Figure 12. Simulations were performed up to $l=25$, and the relative changes in the sheath potential drop and heat transmission coefficient were tracked. Very quickly the change drops below 1%, before nonmonotonic behaviour is observed around $l=9$. While convergence appears slow, this is to be expected with such a sharply cut-off distribution being expanded in Legendre polynomials. Fortunately, even in this highly stressed scenario a relatively low number of harmonics $l\leq 5$ seems to be enough to capture the physically important behaviour. Figure 12: Relative change in sheath properties when increasing the number of harmonics in the simulation. Non-monotonic behaviour was observed around $l=9$, causing the drop in relative change - Set 4. ## 5 Discussion In the previous section we have presented tests that show SOL-KiT agreeing with established analytical results for phenomena of interests, namely the heat conduction and logical sheath boundary condition. The Epperlein-Short test (Set 1) presented in Section 4.1. as well as Set 4 in Section 4.4. both show that the code is capable of handling non-local behaviour, as well as demonstrating that suitable convergence in harmonic number $l$ can be achieved. The novel discretization scheme for the Boltzman collision integral for inelastic collisions on a velocity grid has been shown to conserve particles and energy up to round-off error. This is to be expected as with $N_{n}=30$ hydrogen states, the number of transitions treated, not including radiative transitions, is 930. Given the stiff nature of the Collisional-Radiative model matrix, as well as the corresponding terms in the electron kinetic equations and the large number of transitions, we expected relative errors of order $\approx 10^{-13}$. Runs 7-8 in the previous section agree with this estimate, with the geometric grid notably having better conservation properties. This is likely due to the fact that the high excited state transitions are better resolved with a finer grid near the origin. Detailed balance has also been shown to be observed within a high accuracy, owing to the numerically adapted detailed balance cross-sections derived in Section 3.6.1. At the sheath boundary, we implement the logical boundary condition [27]. While this is a standard boundary condition in SOL kinetic codes, this is the first time it has been implemented in a harmonic decomposition code. A reasonable concern in this case would be that the sheath properties would be hard to pinpoint with accuracy, given that the sharp cut-off is poorly approximated with Legendre polynomials. While we do notice the effects of the Gibbs phenomenon in the reconstructed distribution function when high $l$ low collisionality systems are treated, we show in Set 4 that a relatively low number of harmonics is sufficient to provide an estimate of the relevant sheath properties. Furthermore, the Maxwellian analytic limit of the sheath properties requires only resolving up to $l=1$. This further supports the notion that, while important in general, high harmonic approximations of the cut-off are not necessary for the transport calculations at the sheath boundary. SOL-KiT is a fully implicit code, and is thus not limited by the CFL condition. It is, therefore, able to resolve the lowest velocity cells for higher harmonics, unlike explicit codes [23]. This is one of the key features allowing the implementation of both lab frame ion flow, as well as inelastic electron-neutral collisions, with the former being the main beneficiary of the implicit scheme. The reason for this is that the effect of flowing ions in the lab frame is visible primarily in the low velocity cells, where electrons are highly collisional and are dragged with the ions. A scheme that cannot resolve those cells must resort to calculations in the ion centre of mass frame[44, 28]. Given the complicated sheath boundary condition necessary in the simulation of the SOL, the lab frame is the natural choice. The effect of this choice is that the complexity normally arising in various LHS operators (in the Vlasov part) of the electron kinetic equation is transferred into the electron-ion collision operator. In this work the ions are treated as a cold beam for the sake of this collisional computation. While the addition of ion temperature effects would in theory increase the fidelity of the operator, it would strongly depend on the velocity grid resolution around the ion flow velocity. It is likely that the effect would be negligible as the ion thermal velocity is much smaller than the electron thermal velocity, and that the main collisional effect of the ion thermal population is well captured by the delta function of a cold beam. However, refinement of the electron-ion collision operator is a potential path in the development of the code. Momentum is not explicitly conserved in the electron-electron collision operator for $l>0$ as implemented in the code. As suggested by Tzoufras et al. [23], we ensure total momentum conservation by transfering the momentum lost in electron-electron collisions to the ions. This produces a spurious electric field in the regions of high flow speeds. However, this electric field is negligible compared to the physical electric fields occuring in realistic SOL simulations, as the region of non-zero ion flow tends to be in front of the divertor target, where the pre-sheath electric field is generated through pressure gradients. The way we calculate the electric field also ensures that quasineutrality is observed up to displacement current effects. This allows SOL-KiT to treat collisionless phenomena on the Debye length scale. However, as the logical sheath boundary condition does not require resolving the Debye sheath, this regime has not been fully explored. Performance of the code was tracked in several of the presented runs. Specifications of the used workstation include an Intel® Xeon(R) CPU E5-2630 v3 @ 2.40GHz with a maximum of 16 threads, and 15.6GB of RAM. In all runs performed so far it appears that the CPU requirement always outweighs the memory needs of the code. Execution times on the workstation during tests varied from under 6 minutes for the local heat conductivity runs, to 18.5h for the reproduction in Run 10. Most runs were performed with the full 16 threads, excepts for the several quasi-0D tests. These runs (such as Runs 7-8) took just under 7h to complete on a single core. Run 9, with both electrons and ions treated as a fluid took slightly over 3h. Since fluid runs have both a simpler matrix, as well as no requirement to resolve the collisional times, they tend to run much faster and for longer physical simulation times. The two main bottlenecks in the current version of the code have been identified as the electron-electron collisions for $l>0$ as well as the electron-neutral collision operators. As was noted above, the e-e collisions for $l>0$ produce dense submatrices for each harmonic, both increasing the matrix computation times as well as increasing the difficulty of solving the matrix system. The e-n collisions similarly produce almost dense matrices, but the main bottleneck there is the large number of transitions, translating into hundreds of effective collision operator computations per timestep. In order to speed this up, the detailed balance cross-sections can be updated only once per timestep, as opposed to during every nonlinear iteration. Another way of speeding the neutral physics up would be grouping the higher excited states into effective states, thus reducing the total number of collision integral matrices computed every timestep. While SOL-KiT is implicit, in practice the timestep is limited by the capabilities of the solver. We currently use the Bi-CGSTAB iterative matrix solver with Block Jacobi preconditioning in the PETSc package. This solver tends to struggle when the timestep is many ($\approx 50$) times the electron- ion collisional time, as well as when higher harmonics are included due to the added stiffness of the matrix due to the fast evolution of $f_{l}$ for high $l$’s. However, in most situations we are aiming to resolve the collisional phenomena, and the timestep-limiting effect of higher harmonics becomes evident at a number of harmonics already high enough where the dense matrix effects already cause the majority of the slowdown. SOL-KiT is currently parallelized using MPI, with domain decomposition along the $x$-axis. Basic optimization was performed on the code, however, many avenues of potential improvement remain. Besides the already mentioned grouping of excited neutral states, parallelization in $l$ number is also being considered, though the degree of speedup attainable would depend heavily on the inter-processor communication. Different preconditioners and solvers could also be employed depending on the problem, though this would require more in-depth tests of the code’s performance. One of the code’s design goals was to enable consistent and convenient one-to- one comparisons between a fluid and kinetic model of electron transport. This was accomplished by ensuring that the physics implementated in the kinetic and fluid modes use the same data (e.g. cross-sections etc.). A possible extension of this approach would be the inclusion of various non-local fluid models as comparison options. The easiest to implement would possibly be the flux limiter method [4], and more complicated models like the SNB model and others[43] could be implemented self-consistently with the current framework. The elastic electron-neutral collision operator currently implemented assumes a simple cross-section (see 2.1.3.). As we believe the cross-section could be improved with a more detailed model based on experimental data, while tested, this operator is not in regular use. Improvements of the neutral model are being planned, including potential additions of molecules, the implementation of a fluid as opposed to a diffusive neutral model, and the option to include high $Z$ impurities. To accompany the improved neutral model, the next step in SOL-KiT’s development will be the addition of an ion temperature equation, as well as an electron-ion collision operator for $l=0$. This will allow us to probe more varied SOL regimes, where the ion and electron temperatures are decoupled. Compared to PIC codes, the finite difference approach taken with SOL-KiT provides a speed-up in computation, as well as avoiding the noise issues present in PIC codes. The closest comparison to SOL-KiT, however, is the code FPI [16, 17, 18]. While a form of the logical boundary condition appears to have been implemented in FPI [16, 17], to our knowledge no in-depth discussion of its Legendre formalism form is available. Similarly, electron-neutral inelastic collisions have been implemented and reported in both ALLA[15] and FPI, but the conservative properties of the operators were not explored in detail. In this paper we report in detail both of these numerical aspects in the context of a Legendre decomposition model. Furthermore, the combination of a fully implicit algorithm together with the lab frame ion treatment and the aforementioned numerical facets has been realized for the first time in SOL- KiT. The addition of a self-consistent fluid mode for the electrons adds another aspect to the code, with work on coupling the two modes under way. ## 6 Conclusion We have presented the model and the numerics behind the newly developed fully implicit arbitrary harmonic electron kinetic code SOL-KiT. The code is designed for the exploration of kinetic effects in the Scrape-Off Layer, and includes a self-consistent fluid electron mode for one-to-one comparisons. Novel conservative implemenentation of the electron-neutral collision operators for inelastic collisions and the Legendre polynomial version of the logical sheath boundary condition are presented and tested, showing good agreement with expected properties. We have shown that the code can resolve highly non-local effects utilizing high harmonics, and have demonstrated that some of the more demanding operators converge well with the increase in the number of harmonics. The next steps in SOL-KiT development have been laid out, focusing on both improving performance, as well as adding new physics to extend the applicability of the code. ## Acknowledgements This work was supported through the Imperial President’s PhD Scholarship scheme. This work was partially funded by the RCUK Energy Programme [grant number: EP/P012450/1]. Some of the simulation results were obtained with the use of the Imperial College Research Computing Service [45] ## Appendix A Divertor target boundary condition transformation matrix derivation As stated above, we label the cut-off velocity $v_{c}$ and take that all electrons with parallel velocity greater than $v_{c}$ are lost to the sheath, and all other electrons are reflected. This informs the form of the distribution function at the sheath entrance in the following way. If we know the distribution function for positive $v_{x}$ (moving towards the divertor target) and can expand it into Legendre polynomials (assuming azimuthal symmetry, of course) $f(v_{x}>0,v_{\perp})=\sum_{l}f_{l}(v)P_{l}(\cos\theta),\quad\cos\theta>0.$ Now, if we were to reflect all electrons back, the total (reflection included) distribution function would be $f_{R}(v,\theta)=\sum_{l}f_{l}(v)P_{l}(|\cos\theta|).$ Finally, we take away all of the electrons that would have been lost to the sheath and thus should not be in the distribution function. This is simply all electrons with $v\cos\theta=v_{x}<-v_{c}$. Using a Heaviside step function, this can be expressed as $f_{c}(v,\theta)=f_{R}(v,\theta)\Theta(v\cos\theta+v_{c}).$ (98) From here we can use Legendre polynomial orthogonality to extract the $l$-th harmonic of the “cut-off” distribution $f_{cl}(v)=\frac{2l+1}{2}\int_{-1}^{1}f_{R}(v,\theta)\Theta(v\cos\theta+v_{c})P_{l}(\cos\theta)d(\cos\theta).$ (99) Using the decomposition of $f_{R}$ we write equation (99) as $f_{cl}(v)=\frac{2l+1}{2}\sum_{l^{\prime}}f_{l^{\prime}}(v)\int_{-1}^{1}P_{l^{\prime}}(|\cos\theta|)P_{l}(\cos\theta)\Theta(v\cos\theta+v_{c})d(\cos\theta).$ (100) Here it’s helpful to visualize the two-dimensional velocity space of the cut- off. This is shown in Fig 13. As can be seen, the integral in (100) has different limits depending on the value of the velocity $v$. This can be written as (taking $x=\cos\theta$) $f_{cl}(v)=\frac{2l+1}{2}\sum_{l^{\prime}}f_{l^{\prime}}(v)\int_{x_{min}}^{1}P_{l^{\prime}}(|x|)P_{l}(x)dx,$ (101) where $x_{min}=\max(-1,\cos\theta_{max}(v))$, with $\cos\theta_{max}=-v_{c}/v$. Introducing the transformation matrix $P_{ll^{\prime}}$ ($v$ dependence implied) $P_{ll^{\prime}}=\frac{2l+1}{2}\int_{x_{min}}^{1}P_{l^{\prime}}(|x|)P_{l}(x)dx,$ (102) we get a compact expression for the sheath edge electron distribution harmonics $f_{cl}(v)=\sum_{l^{\prime}}P_{ll^{\prime}}f_{l^{\prime}}(v).$ (103) Figure 13: The distribution function (anisotropy exaggerated) after applying reflection and cut-off. Highlighted is the integration limit $\theta_{max}(v).$ Now the problem is reduced to computing the matrix $P_{ll^{\prime}}$. First, let us dispose of the absolute value in the argument of $P_{l^{\prime}}$ by separating the integral into positive and negative $x$ intervals $P_{ll^{\prime}}=\frac{2l+1}{2}\left((-1)^{l^{\prime}}\int_{x_{min}}^{0}P_{l^{\prime}}(x)P_{l}(x)dx+\int_{0}^{1}P_{l^{\prime}}(x)P_{l}(x)dx\right),$ (104) where we have used the parity property of Legendre polynomials $P_{l}(-x)=(-1)^{l}P_{l}(x)$. In order to reduce the integrals in (104) to forms available in the literature we expand the integration range in the first integral up to $1$ and after grouping terms up, we get $P_{ll^{\prime}}=P^{x}_{ll^{\prime}}+P^{0}_{ll^{\prime}},$ (105) where $P^{x}_{ll^{\prime}}=\frac{2l+1}{2}(-1)^{l^{\prime}}\int_{x_{min}}^{1}P_{l^{\prime}}(x)P_{l}(x)dx,$ (106) $P^{0}_{ll^{\prime}}=\frac{2l+1}{2}(1-(-1)^{l^{\prime}})\int_{0}^{1}P_{l^{\prime}}(x)P_{l}(x)dx.$ (107) The integral in (106) can be found in the literature (see for example [46] for general case) and well known recurrence formulae for the derivatives of Legendre polynomials can be used to get $P^{x}_{ll^{\prime}}=\begin{cases}(-1)^{l^{\prime}}\delta_{ll^{\prime}}\quad\text{if $x_{min}=-1$}\\\ F_{ll^{\prime}}\quad\text{if $x_{min}>-1$ and $l\neq l^{\prime}$}\\\ F_{ll}\quad\text{if $x_{min}>-1$ and $l=l^{\prime}$}\end{cases}$ where $\begin{split}F_{ll^{\prime}}&=(-1)^{l^{\prime}}\frac{2l+1}{2}\Big{[}-\frac{x_{min}}{l+l^{\prime}+1}P_{l}(x_{min})P_{l^{\prime}}(x_{min})\\\ &+\frac{1}{(l-l^{\prime})(l+l^{\prime}+1)}\left(lP_{l-1}(x_{min})P_{l^{\prime}}(x_{min})-l^{\prime}P_{l}(x_{min})P_{l^{\prime}-1}(x_{min})\right)\Big{]},\end{split}$ and $F_{ll}=(-1)^{l}\left[\frac{1}{2}-\left(\frac{x_{min}}{2}[P_{l}(x_{min})]^{2}+\sum_{k=1}^{l-1}P_{k}(x_{min})(x_{min}P_{k}(x_{min})-P_{k+1}(x_{min}))\right)\right].$ For $P^{0}_{ll^{\prime}}$ we get $P^{0}_{ll^{\prime}}=\begin{cases}0\quad\text{if $l^{\prime}$ even}\\\ \delta_{ll^{\prime}}\quad\text{if $l^{\prime},l$ odd and $l=l^{\prime}$}\\\ 0\quad\text{if $l^{\prime},l$ odd and $l\neq l^{\prime}$}\\\ \frac{2l+1}{(l-l^{\prime})(l+l^{\prime}+1)}\left(lP_{l-1}(0)P_{l^{\prime}}(0)-l^{\prime}P_{l}(0)P_{l^{\prime}-1}(0)\right)\quad\text{if $l^{\prime}$ odd and $l$ even}\end{cases}$ It should be noted that for $v<v_{c}$ the above equations give $P_{ll^{\prime}}=0$ for all odd $l$, which is the consequence of reflection symmetry, from which one can also see that there is no effective flux contribution for $v<v_{c}$ . ## References * [1] B. 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2024-09-04T02:54:56.230711

2003.00786

arxiv-papers

# Riemann Solitons and Almost Riemann Solitons on Almost Kenmotsu Manifolds V. Venkatesha H. Aruna Kumara and Devaraja Mallesha Naik Department of Mathematics Kuvempu University Shankaraghatta Karnataka 577 451 India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract. The aim of this article is to study the Riemann soliton and gradient almost Riemann soliton on certain class of almost Kenmotsu manifolds. Also some suitable examples of Kenmotsu and $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifolds are constructed to justify our results. ###### Key words and phrases: Riemann soliton, Ricci soliton, almost Kenmotsu manifold, Einstein manifold. ###### 1991 Mathematics Subject Classification: 53C25, 53C15, 53D15 ## 1\. Introduction Hamilton [9] introduced the concept of Ricci flow. The idea of Ricci flow generalized to the concept of Riemann flow (see [16, 17]). As an analog of Ricci soliton, Hiric̆a and Udrişte [10] introduced and studied Riemann soliton. This is appears as a self-similar solution of the Riemann flow [16]: $\displaystyle\frac{\partial}{\partial t}G(t)=-2R(g(t)),\quad t\in[0,I],$ where $G=\frac{1}{2}g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g$, $R$ is the Riemann curvature tensor associated to the metric $g$ and $\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}$ is Kulkarni-Nomizu product. These extensions are natural, since some results in the Riemann flow resembles the case of Ricci flow (for detail, see [17]). For instance, the Riemann flow satisfies the short time existence and uniqueness [11]. Further the authors in [14] characterize the Riemann soliton in terms of infinitesimal harmonic transformation. For $(0,2)$-tensors $A$ and $B$, thier Kulkarni- Nomizu product $A\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}B$ is given by $\displaystyle(A\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}B)(X_{1},X_{2},X_{3},$ $\displaystyle X_{4})=A(X_{1},X_{3})B(X_{2},X_{4})+A(X_{2},X_{4})B(X_{1},X_{3})$ $\displaystyle-A(X_{1},X_{4})B(X_{2},X_{3})-A(X_{2},X_{3})B(X_{1},X_{4}).$ (1.1) Riemann soliton is a smooth manifold $M$ together with Riemannian metric $g$ that satisfies $\displaystyle 2R+\lambda(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g)+(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}\pounds_{V}g)=0,$ (1.2) where $V$ is a vector field called as a potential vector field, $\pounds_{V}$ denotes the Lie-derivative and $\lambda$ is a constant. The Riemann soliton also corresponds as a fixed point of the Riemann flow, and they can be viewed as a dynamical system, on the space of Riemannian metric modulo diffeomorphism. Note that notion of Riemann soliton generalizes the space of constant sectional curvature (i.e., $R=kg\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g$, for some constant $k$). A Riemann soliton is called shrinking when $\lambda<0$, steady when $\lambda=0$ and expanding $\lambda>0$. If the vector field $V$ is the gradient of a potential function $u$, then we get the notion of gradient Riemann soliton. In such a case, the equation (1.2) transform into $\displaystyle R+\frac{1}{2}\lambda(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g)+(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}Hess~{}u)=0,$ (1.3) where $Hess~{}u$ denotes the Hessian of the smooth function $u$. In [10], it is proved that a Riemann soliton on a compact Riemannian manifold is a gradient. A Riemann soliton is called trivial when potential vector field $V$ vanishes identically, and this case manifold is of constant sectional curvature. If $\lambda$ appearing in equation (1.2) and (1.3) is a smooth function, then $g$ is called almost Riemann soliton and almost gradient Riemann soliton respectively. In [10], Hiric̆a and Udrişte studied Riemann soliton and gradient Riemann soliton within the framework of Sasakian manifold. In their study, it is proved that if Sasakian manifold admits a Riemann soliton whose soliton vector field $V$ is pointwise collinear with $\xi$ (or a gradient Riemann soliton and potential function is harmonic), then it is Sasakian space form. In this study, the authors exposes an open problem as classify gradient Riemann soliton for arbitrary potential function on Sasakian manifold. To fulfill this classification, the present authors in [4] consider a Riemann soliton on contact manifold, and obtained several intersting results. These results on contact geometry intrigues us to study the Riemann soliton on other almost contact metric manifolds. In this paper, we classify certain class of almost Kenmotsu manifold which admits a Riemann soliton and almost gradient Riemann solion. ## 2\. Preliminaries Let $(M,g)$ be a $(2n+1)$-dimensional smooth Riemannian manifold. On this manifold if there exists a (1,1)-type tensor field $\varphi$, a global vector field $\xi$ and 1-form $\eta$ such that $\displaystyle\varphi^{2}X=-X+\eta(X)\xi,\quad\eta(\xi)=1,$ (2.1) $\displaystyle g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y),$ (2.2) for any vector field $X,Y$ on $M$, then we say that $(\varphi,\xi,\eta,g)$ is an almost contact metric structure and $M$ is an almost contact metric manifold [1]. Generally, $\xi$ and $\eta$ are called Reeb or characterstic vector field and almost contact 1-form respectively. We remark that an almost contact metric structure on a Riemannian manifold $M$ may be regarded as a reduction of the structure group $M$ to $U(n)\times 1$. For such a manifold, we define a fundamental 2-form $\Phi$ by $\Phi(X,Y)=g(X,\varphi Y)$. It is well known that the normality of an almost contact structure is expressed by vanishing of the tensor $N_{\varphi}=[\varphi,\varphi]+2d\eta\otimes\xi$, where $[\varphi,\varphi]$ is the Nijenhuis tensor of $\varphi$ (see [1]). Almost Kenmotsu manifold [8] is an almost contact metric manifold such that $d\eta=0$ and $d\Phi=2\eta\wedge\Phi$. A normal almost Kenmotsu manifold is called Kenmotsu manifold, and this normality condition is expressed as $\displaystyle(\nabla_{X}\varphi)Y=g(\varphi X,Y)\xi-\eta(Y)\varphi X,$ for any vector field $X,Y$ on $M$, where $\nabla$ denotes the Levi-Civita connection of the Riemannian metric $g$. On an almost Kenmotsu manifold, we set $\ell=R(\cdot,\xi)\xi$, $2h=\pounds_{\xi}\varphi$ and $h^{\prime}=h\circ\varphi$, where $R$ denotes the curvature tensor of $M$ and $\pounds$ is the Lie-derivative. These mentioned tensor fields plays important role in almost Kenmotsu manifold, and it is easily seen that both are symmetric and satisfies the following equations $\displaystyle\nabla_{X}\xi=X-\eta(X)\xi+h^{\prime}X,$ (2.3) $\displaystyle h\xi=\ell\xi=0,\quad trh=trh^{\prime}=0,\quad h\varphi+\varphi h=0,$ (2.4) $\displaystyle tr\ell=S(\xi,\xi)=g(Q\xi,\xi)=-2n-trh^{2},$ (2.5) where $S$ is the Ricci curvature tensor, $Q$ is the Ricci operator with respect to $g$ anb $tr$ is the trace operator. ###### Definition 2.1. On almost contact metric manifold $M$, a vector field $V$ is said to be infinitesimal contact transformation if $\pounds_{V}\eta=\sigma\eta$, for some function $\sigma$. In particular, we called $V$ is a strict infinitesimal contact transformation if $\pounds_{V}\eta=0$. Now, we recall some formulas which are helpfull to prove our main results. Using the symmetry of $\pounds_{V}\nabla$ in the commutation formula (see Yano [22]): $\displaystyle(\pounds_{V}$ $\displaystyle\nabla_{X}g-\nabla_{X}\pounds_{V}g-\nabla_{[V,X]}g)(Y,Z)$ $\displaystyle=-g((\pounds_{V}\nabla)(X,Y),Z)-g((\pounds_{V}\nabla)(X,Z),Y),$ we derive $\displaystyle 2g((\pounds_{V}\nabla)(X,Y),Z)=(\nabla_{X}\pounds_{V}g)(Y,Z)+(\nabla_{Y}\pounds_{V}g)(Z,X)-(\nabla_{Z}\pounds_{V}g)(X,Y).$ (2.6) The following well-known commutation equations is also known to us from Yano [22]: $\displaystyle(\pounds_{V}R)(X,Y)Z=(\nabla_{X}\pounds_{V}\nabla)(Y,Z)-(\nabla_{Y}\pounds_{V}\nabla)(X,Z),$ (2.7) $\displaystyle\pounds_{V}\nabla_{X}Y-\nabla_{X}\pounds_{V}Y-\nabla_{[V,X]}Y=(\pounds_{V}\nabla)(X,Y).$ (2.8) ## 3\. Riemann soliton and almost gradient Riemann soliton on Kenmotsu manifolds Dileo and Pastore [2] proved that the almost contact metric structure of an almost Kenmotsu manifold is normal if and only if the foliations of the distribution $\mathcal{D}$ are Kählerian and the $(1,1)$-type tensor field $h$ vanishes. In particular, we have immediately that almost Kenmotsu manifold is a Kenmotsu manifold if and only if $h=0$. The following formulae are holds for a Kenmotsu manifold [12] $\displaystyle\nabla_{X}\xi=X-\eta(X)\xi,$ (3.1) $\displaystyle R(X,Y)\xi=\eta(X)Y-\eta(Y)X,$ (3.2) $\displaystyle Q\xi=-2n\xi.$ (3.3) Differentiating (3.3) along an arbitrary vector field $X$ and recalling (3.1) we deduce $\displaystyle(\nabla_{X}Q)\xi=-QX-2nX.$ (3.4) In this section, we aim to investigate the existence of geometry of Riemann soliton and an almost gradient Riemann soliton on Kenmotsu manifold. First, we prove the following result. ###### Lemma 3.1. Let $M$ be a $(2n+1)$-dimensional Kenmotsu manifold. If $(g,V)$ is a Riemann soliton with soliton vector $V$ has a constant divergence, then the potential vector field and Ricci tensor satisfy $\displaystyle divV=2n(1-\lambda),$ (3.5) $\displaystyle(\pounds_{V}S)(X,\xi)=\frac{1}{2n-1}\\{-(Yr)+(\xi r)\eta(Y)\\},$ (3.6) where $div$ denotes the divergence operator. ###### Proof. As a result of (1), the Riemann soliton equation (1.2) can be expressed as $\displaystyle 2R(X,Y,$ $\displaystyle Z,W)+2\lambda\\{g(X,W)g(Y,Z)-g(X,Z)g(Y,W)\\}$ $\displaystyle+\\{g(X,W)(\pounds_{V}g)(Y,Z)+g(Y,Z)(\pounds_{V}g)(X,W)$ $\displaystyle-g(X,Z)(\pounds_{V}g)(Y,W)-g(Y,W)(\pounds_{V}g)(X,Z)\\}=0.$ (3.7) Contractiong (3.7) over $X$ and $W$, we obtain $\displaystyle(\pounds_{V}g)(Y,Z)+\frac{2}{2n-1}S(Y,Z)+\frac{2}{2n-1}(2n\lambda+divV)g(Y,Z)=0.$ (3.8) Taking covariant derivative of the above relation gives $\displaystyle(\nabla_{X}\pounds_{V}g)(Y,Z)+\frac{2}{2n-1}(\nabla_{X}S)(Y,Z)=0,$ (3.9) where we applied $V$ has a constant divergence. Inserting the foregoing equation into (2.6) we obtain $\displaystyle g((\pounds_{V}\nabla)(X,Y),Z)=\frac{1}{2n-1}\\{(\nabla_{Z}S)(X,Y)-(\nabla_{X}S)(Y,Z)-(\nabla_{Y}S)(Z,X)\\}.$ (3.10) The present authors in [18] and Ghosh in [7] independently gave the complete proof of the following formula which is valid for any Kenmotsu manifold $\displaystyle(\nabla_{\xi}Q)X=-2QX-4nX.$ (3.11) Replacing $Y$ by $\xi$ in (3.10), applying (3.4) and (3.11) we get $\displaystyle(\pounds_{V}\nabla)(X,\xi)=\frac{2}{2n-1}QX+\frac{4n}{2n-1}X.$ (3.12) Differentiating (3.12) covariantly along $Y$ and utilizing (3.1), we find $\displaystyle(\nabla_{Y}\pounds_{V}\nabla)(X,\xi)+(\pounds_{V}\nabla)(X,Y)-\frac{2}{2n-1}\eta(Y)\\{QX+2nX\\}=\frac{2}{2n-1}(\nabla_{Y}Q)X.$ Making use of this in the well known expression (2.7), we have $\displaystyle(\pounds_{V}R)(X,Y)\xi=$ $\displaystyle\frac{2}{2n-1}\\{\eta(X)QY-\eta(Y)QX+(\nabla_{X}Q)Y-(\nabla_{Y}Q)X\\}$ $\displaystyle+\frac{4n}{2n-1}\\{\eta(X)Y-\eta(Y)X\\}.$ (3.13) We insert $Y=\xi$ in the above relation and utilizing (3.4) and (3.11) to achieve $(\pounds_{V}R)(X,\xi)\xi=0$. On the other hand, operating $\pounds_{V}$ to the formula $R(X,\xi)\xi=-X+\eta(X)\xi$ (follows from (3.2)) yields $\displaystyle(\pounds_{V}R)(X,\xi)\xi+g(X,\pounds_{V}\xi)\xi-2\eta(\pounds_{V}\xi)X=\\{(\pounds_{V}\eta)X\\}\xi,$ which by virtue of $(\pounds_{V}R)(X,\xi)\xi=0$ becomes $\displaystyle g(X,\pounds_{V}\xi)\xi-2\eta(\pounds_{V}\xi)X=\\{(\pounds_{V}\eta)X\\}\xi.$ (3.14) With the aid of (3.3), the equation (3.8) takes the form $\displaystyle(\pounds_{V}g)(X,\xi)=-\frac{2}{2n-1}\\{2n\lambda+divV-2n\\}\eta(X).$ (3.15) Now, Lie-differentiating $\eta(X)=g(X,\xi)$ and $g(\xi,\xi)=1$ along $V$ and taking into account (3.15) provides $\displaystyle(\pounds_{V}\eta)X-g(X,\pounds_{V}\xi)+\frac{2}{2n-1}\\{2n\lambda+divV-2n\\}\eta(X)=0,$ $\displaystyle\eta(\pounds_{V}\xi)=\frac{1}{2n-1}\\{2n\lambda+divV-2n\\}.$ Utilizing these equations in (3.14) yields $(2n\lambda+divV-2n)\\{X-\eta(X)\xi\\}=0$. Tracing this provides (3.5). Now contracting the equation (3.13) provides $\displaystyle(\pounds_{V}S)(Y,\xi)=\frac{1}{2n-1}\\{-(Yr)-2(r+2n(2n+1))\\}\eta(Y),$ (3.16) where we applied the well-known formula $divQ=\frac{1}{2}grad\,\,r$ and $tr\nabla Q=grad\,\,r$. Taking trace of (3.11) provides $(\xi r)=-2(r+2n(2n+1))$. By virtue of this and (3.16) gives (3.6). This completes the proof. ∎ ###### Theorem 3.2. Let $M$ be an $\eta$-Einstein Kenmotsu manifold of dimension higher than $3$. If $(g,V)$ is a Riemann soliton with $divV=constant$, then $M$ is Einstein. ###### Proof. We know that a Kenmotsu manifold $M$ is $\eta$-Einstein if and only if $\displaystyle S(X,Y)=\left(\frac{r}{2n}+1\right)g(X,Y)-\left(\frac{r}{2n}+2n+1\right)\eta(X)\eta(Y).$ (3.17) With the aid of the above equation, it has been proved by present authors in ([18], Lemma 3.4) that on $\eta$-Einstein Kenmotsu manifold of $dim>3$ there holds $\displaystyle Dr=(\xi r)\xi.$ Making use of this in (3.6), we have $(\pounds_{V}S)(X,\xi)=0$. Now, applying $\pounds_{V}$ to (3.3), recalling (3.17), (3.15), (3.5) and $\eta(\pounds_{V}\xi)=0$, we obtain $\displaystyle(r+2n(2n+1))\pounds_{V}\xi=0.$ (3.18) Suppose that $r\neq 2n(2n+1)$ in some open set $\mathcal{O}$ of M. Then on $\mathcal{O}$, $\pounds_{V}\xi=0=\pounds_{V}\eta$. Replacing $Y$ by $\xi$ in (2.8) and using $\pounds_{V}\xi=0=\pounds_{V}\eta$, we have $\displaystyle(\pounds_{V}\nabla)(X,\xi)=$ $\displaystyle\pounds_{V}X-\eta(X)\pounds_{V}\xi-\\{(\pounds_{V}\eta)X\\}\xi$ $\displaystyle-\eta(\pounds_{V}X)\xi-\pounds_{V}X+\eta(\pounds_{V}X)\xi=0.$ By virtue of this in (3.12), we have $QX=-2nX$. Taking trace of this gives $r=-2n(2n+1)$ on $\mathcal{O}$, which is a contradiction on $\mathcal{O}$. Thus, equation (3.18) gives $r=-2n(2n+1)$ and therefore we can conclude from $\eta$-Einstein condition (3.17) that $M$ is Einstein. ∎ It is known that any $3$-dimensional Kenmotsu manifold is $\eta$-Einstein. As a result of Theorem 3.2, it is interesting to study Riemann soliton in Kenmotsu $3$-manifold and here, we prove the following outcome. ###### Theorem 3.3. Let be a 3-dimensional Kenmotsu manifold. If $(g,V)$ represents a Riemann soliton with soliton vector $V$ has a constant divergence, then $M$ is of constant negative curvature $-1$. ###### Proof. It is well known that the Riemannian curvature tensor of $3$-dimensional Riemannian manifold is given by $\displaystyle R(X,Y)Z=$ $\displaystyle g(Y,Z)QX-g(X,Z)QY+g(QY,Z)X-g(QX,Z)Y$ $\displaystyle-\frac{1}{2}\\{g(Y,Z)X-g(X,Z)Y\\}.$ (3.19) Replacing $\xi$ in place of $Y$ and $Z$ in above equation, employing (3.2) and (3.3) gives $\displaystyle QX=\left(\frac{r}{2}+1\right)X-\left(\frac{r}{2}+3\right)\eta(X)\xi,$ which is equivalent to $\displaystyle S(X,Y)=\left(\frac{r}{2}+1\right)g(X,Y)-\left(\frac{r}{2}+3\right)\eta(X)\eta(Y).$ (3.20) Applying $\pounds_{V}$ to (3.3), recalling (3.20) yields $\displaystyle(\pounds_{V}S)(X,\xi)+\left(\frac{r}{2}+1\right)g(X,\pounds_{V}\xi)-\left(\frac{r}{2}+3\right)\eta(X)\eta(\pounds_{V}\xi)=-2(\pounds_{V}\eta)(X).$ This together with (3.6) provides $\displaystyle-Y(r)+\xi(r)\eta(Y)+\left(\frac{r}{2}+1\right)g(X,\pounds_{V}\xi)$ $\displaystyle-\left(\frac{r}{2}+3\right)\eta(X)\eta(\pounds_{V}\xi)+2(\pounds_{V}\eta)(X)=0.$ (3.21) By virtue of (3.5) and (3.15), it follows from (3.21) that $\displaystyle(r+6)g(X,\pounds_{V}\xi)=2\\{Y(r)-\xi(r)\eta(Y)\\},$ (3.22) where we used Theorem 4.1 of Wang [20]. Now suppose that $r=-6$. Then from (3.20) we have $QX=-2X$, and substituting this in (3.19) one gets $\displaystyle R(X,Y)Z=-\\{g(Y,Z)X-g(X,Z)Y\\},$ which means $M$ is of constant curvature $-1$. Now we suppose that $r\neq-6$ in some open set $\mathcal{O}$ of $M$. Then on $\mathcal{O}$, the relation (3.22) can be written as $\displaystyle\pounds_{V}\xi=f\\{Dr-\xi(r)\xi\\},$ (3.23) where $f=\frac{2}{r+6}$. Taking $Y$ by $\xi$ in (2.8) and using (3.1), (3.23) and (3.12), it follows that $\displaystyle f\\{$ $\displaystyle X(r)\eta(Y)+Y(r)\eta(X)-2\xi(r)\eta(X)\eta(Y)+g(\nabla_{X}Dr,Y)-X(\xi(r))\eta(Y)\\}$ $\displaystyle+X(f)\\{Y(r)-\xi(r)\eta(Y)\\}+\left\\{f~{}\xi(r)-\frac{1}{2}\xi(r)\right\\}g(\varphi X,\varphi Y)=0.$ Antisymmetrizing the foregoing equation and keeeping in mind the Poincare lemma: $g(\nabla_{X}Dr,Y)=g(\nabla_{Y}Dr,X)$, we find $\displaystyle f\\{Y(\xi(r))\eta(X)-X(\xi(r))\eta(Y)\\}+X(f)\\{Y(r)-\xi(r)\eta(Y)\\}$ $\displaystyle-Y(f)\\{X(r)-\xi(r)\eta(X)\\}=0.$ Putting $Y=\xi$ in the above equation and using Theorem 4.1 of Wang [20] one get $\displaystyle(2f-\xi(f))\\{X(r)-\xi(r)\eta(X)\\},$ which implies $\displaystyle(2f-\xi(f))\\{Dr-\xi(r)\xi\\}.$ (3.24) In the following first case we show that $Dr=\xi(r)\xi$, one get $g(Dr,\xi)=-2(r+6)$. Since $f=\frac{2}{r+6}$ we obtain $\xi(f)=2f$. In the second case we show $\xi(f)=2f$ implies $Dr=\xi(r)\xi$. Thus for any point $p\in\mathcal{O}$, we have $(Dr)_{p}=\xi_{p}(r)\xi_{p}$ if and only if $\xi_{p}(f)=2f$. Hence from (3.24), we have either $\xi(f)=2f$ or $Dr=\xi(r)\xi$. Case 1: First we assume $\displaystyle Dr=\xi(r)\xi,$ (3.25) and so (3.23) becomes $\pounds_{V}\xi=0$ on the open set $\mathcal{O}$ of $M$. Setting $Y$ by $\xi$ in (2.8) and using $\pounds_{V}\xi=0$, we have $(\pounds_{V}\nabla)(X,\xi)=0$. This together with (3.12) provides $QX=-2X$. Contracting this with respect to $X$ gives $r=-6$ which yields a contradiction. Case 2: Suppose if $\xi(f)=2f$, then using $f=\frac{2}{r+6}$ we obtain $\xi(r)=-2(r+6)$ which means $\displaystyle g(Dr,\xi)=-2(r+6).$ As $r\neq-6$ on $\mathcal{O}$, the above equation implies $Dr=f\xi$, for some smooth function $f$. In fact, we have $Dr=\xi(r)\xi$ and so following the Case 1 we arrive the contradiction. This completes the proof. ∎ Now, we consider a Kenmotsu metric as a gradient almost Riemann soliton and first, we need the following result to prove our main theorem. ###### Lemma 3.4. For gradient almost Riemann soliton, the following formula is valid $\displaystyle R(X,Y)Du=$ $\displaystyle\frac{1}{2n-1}\\{(\nabla_{Y}Q)X-(\nabla_{X}Q)Y\\}$ $\displaystyle+\frac{1}{2n-1}\\{Y(2n\lambda+\Delta u)X-X(2n\lambda+\Delta u)Y\\},$ (3.26) where $\Delta u=divDu$, $\Delta$ is the Laplacian operator. ###### Proof. Contracting gradient almost Riemann soliton equation (1.3), we get $\displaystyle Hess~{}u+\frac{1}{2n-1}S+\frac{1}{2n-1}(2n\lambda+\Delta u)g=0.$ Note that the above equation may exhibited as $\displaystyle\nabla_{Y}Du=-\frac{1}{2n-1}QY-\frac{1}{2n-1}(2n\lambda+\Delta u)Y.$ (3.27) By straightforward computations, using the well known expression of the curvature tensor: $\displaystyle R(X,Y)=\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]},$ and the repeated use of equation (3.27) gives the desired result. ∎ ###### Theorem 3.5. If metric of a Kenmotsu manifold $M$ of dimeninsion $(2n+1)$ represents a gradient almost Riemann soliton, then either $M$ is Einstein or the soliton vector field $V$ is pointwise collinear with the characteristic vector field $\xi$ on an open set $\mathcal{O}$ of $M$. In the first case, if $M$ is complete, then $M$ is locally isometric to a hyperbolic space $\mathbb{H}^{2n+1}$, and the function $2n\lambda+\Delta u$, upto an additive constant, can be expressed as a linear combination of $cosh\,t$ and $sinh\,t$. ###### Proof. Replacing $Y$ by $\xi$ in (3.26) and employing (3.4) and (3.11), we find $\displaystyle R(X,\xi)Du=-\frac{1}{2n-1}\\{QX+2nX\\}+\frac{1}{2n-1}\\{\xi(2n\lambda+\Delta u)X-X(2n\lambda+\Delta u)\xi\\}.$ (3.28) From (3.2), we have $R(X,\xi)Y=g(X,Y)\xi-\eta(Y)X$. Employing this into (3.28) provides $\displaystyle g(X,Du+\frac{1}{2n-1}D(2n\lambda+\Delta u))$ $\displaystyle\xi-\\{\xi(u)+\frac{1}{2n-1}\xi(2n\lambda+\Delta u)\\}X$ $\displaystyle=-\frac{1}{2n-1}\\{QX+2nX\\}.$ (3.29) Inner product of the previous equation with $\xi$ and using (3.3) provides $X(u+\frac{1}{2n-1}(2n\lambda+\Delta u))=\xi(u+\frac{1}{2n-1}(2n\lambda+\Delta u))\eta(X)$, from which we have $\displaystyle d(u+\frac{1}{2n-1}(2n\lambda+\Delta u))=\xi(u+\frac{1}{2n-1}(2n\lambda+\Delta u))\eta,$ (3.30) where $d$ is the exterior derivative. This means that $u+\frac{1}{2n-1}(2n\lambda+\Delta u)$ is invariant along the distribution $\mathcal{D}$, i.e., $X(u+\frac{1}{2n-1}(2n\lambda+\Delta u))=0$ for any vector field $X\in\mathcal{D}$. Utilizing (3.30) into (3.29), we obtain $\displaystyle\\{\xi(u)+\frac{1}{2n-1}\xi(2n\lambda+\Delta u)\\}\eta(X)$ $\displaystyle\xi-\\{\xi(u)+\frac{1}{2n-1}\xi(2n\lambda+\Delta u)\\}X$ $\displaystyle=-\frac{1}{2n-1}\\{QX+2nX\\}.$ (3.31) Contracting the above equation, one immediately obtain $\displaystyle\xi(u+\frac{1}{2n-1}(2n\lambda+\Delta u))=\frac{1}{2n-1}\\{\frac{r}{2n}+2n+1\\}.$ (3.32) Making use of last equation in (3.31) one can obtain $\eta$-Einstein condition (3.17). Now, we contract equation (3.26) over $X$ to deduce $\displaystyle S(Y,Du)=\frac{1}{2(2n-1)}Y(r)+\frac{2n}{2n-1}Y(2n\lambda+\Delta u).$ In the contrast of above equation with (3.17) we obtain $\displaystyle(r+2n)Y(u)-($ $\displaystyle r+2n(2n+1))\xi(u)\eta(Y)-\frac{1}{2(2n-1)}Y(r)$ $\displaystyle-\frac{4n^{2}}{2n-1}Y(2n\lambda+\Delta u)=0.$ (3.33) Inserting $Y=\xi$ in (3.33) and recalling (3.32) one can get $\xi(r)=-2\\{r+2n(2n+1)\\}$. Action of $d$ on (3.30), we get $dr\wedge\eta=0$, where we used $d^{2}=0$ and $d\eta=0$. Hence by virtue of $\xi(r)=-2\\{r+2n(2n+1)\\}$ we have $\displaystyle Dr=-2\\{r+2n(2n+1)\\}\xi.$ (3.34) Suppose that $X$ in (3.33) is orthogonal to $\xi$. Taking into account $(u+\frac{1}{2n-1}(2n\lambda+\Delta u))$ being a constant along $\mathcal{D}$ and utilizing (3.30) and (3.34), then we get $(r+2n(2n+1))X(u)=0$ for any $X\in\mathcal{D}$. This implies that $\displaystyle(r+2n(2n+1))(Du-\xi(u)\xi)=0.$ (3.35) If $r=-2n(2n+1)$, then the equation (3.17) shows that $QX=-2nX$, and hence $M$ is Einstein. Since $r=-2n(2n+1)$, it follows from (3.32) that $\xi(u)=-\frac{1}{2n-1}\xi(2n\lambda+\Delta u)$ and therefore $Du=-\frac{1}{2n-1}D(2n\lambda+\Delta u)\xi$. Thus, equation (3.27) can be exhibited as $\displaystyle\nabla_{X}D(2n\lambda+\Delta u)=((2n\lambda+\Delta u)-2n)X.$ (3.36) According to Theorem 2 of Tashiro [15] we obtain $M$ is locally isometric to the hyperbolic space $\mathbb{H}^{2n+1}$, when $M$ is complete. Since $\nabla_{\xi}\xi=0$ and $g(\xi,D(2n\lambda+\Delta u))=\xi(2n\lambda+\Delta u)$, we deduce from (3.36) that $\xi(\xi(2n\lambda+\Delta u))=(2n\lambda+\Delta u)-2n$. But, it is known [12] that a $(2n+1)$-dimensional Kenmotsu manifold is locally isometric to the warped product $I\times_{ce^{t}}N^{2n}$, where $N$ is a Kähler manifold and $I$ is an open interval. Employing $\xi=\frac{\partial}{\partial t}$ (where $t$ denotes the coordinate on $I$) into (3.36) we obtain $\displaystyle\frac{d^{2}(2n\lambda+\Delta u)}{dt}=(2n\lambda+\Delta u)-2n.$ The solution of this can be exhibited as $(2n\lambda+\Delta u)=A\,cosh\,\,t+B\,sinh\,\,t+2n$, where $A$, $B$ constants on $M$. Now, suppose that $r\neq-2n(2n+1)$ in some open set $\mathcal{O}$ of $M$, then from (3.35) we have $Du=\xi(u)\xi$, and this completes the proof. ∎ ###### Remark 3.6. Theorem 1 in [6], Theorem 1 in [5] and Theorem 3 in [7] of Ghosh is a direct corollary of Theorem 3.2, Theorem 3.3 and Theorem 3.5 respectively. Now we construct an example of Riemann soliton on 3-diemnsional Kenmotsu manifold, and verify our results. ###### Example. Let us indicate the canonical coordinates on $\mathbb{R}^{3}$ by $(x,y,z)$, and take the 3-dimensional manifold $M\subset\mathbb{R}^{3}$ defined by $\displaystyle M=\\{(x,y,z)\in\mathbb{R}^{3}|z\neq 0\\}.$ We may easily verify that putting $\displaystyle\varphi\left(\frac{\partial}{\partial x}\right)=\frac{\partial}{\partial y},\quad\varphi\left(\frac{\partial}{\partial y}\right)=-\frac{\partial}{\partial x},\quad\varphi\left(\frac{\partial}{\partial z}\right)=0,$ $\displaystyle\xi=\frac{\partial}{\partial z},\quad\eta=dz,\quad g=\frac{1}{2}exp(2z)(dx^{2}+dy^{2})+dz^{2},$ $(\varphi,\xi,\eta,g)$ is an almost contact metric structure on $M$. The matrix representation of $g$ with respect to $\frac{\partial}{\partial x}$, $\frac{\partial}{\partial y}$ and $\frac{\partial}{\partial z}$ is $\displaystyle(g_{ij})=\begin{pmatrix}\frac{1}{2}exp(2z)&0&0\\\ 0&\frac{1}{2}exp(2z)&0\\\ 0&0&1\end{pmatrix}.$ Using the Koszul formula we have $\displaystyle\begin{gathered}\nabla_{\frac{\partial}{\partial x}}\frac{\partial}{\partial x}=-\frac{1}{2}exp(2z)\frac{\partial}{\partial z},\quad\nabla_{\frac{\partial}{\partial x}}\frac{\partial}{\partial y}=0,\quad\nabla_{\frac{\partial}{\partial x}}\frac{\partial}{\partial z}=\frac{\partial}{\partial x},\\\ \nabla_{\frac{\partial}{\partial y}}\frac{\partial}{\partial x}=0,\quad\nabla_{\frac{\partial}{\partial y}}\frac{\partial}{\partial y}=-\frac{1}{2}exp(2z)\frac{\partial}{\partial z},\quad\nabla_{\frac{\partial}{\partial y}}\frac{\partial}{\partial z}=\frac{\partial}{\partial y},\\\ \nabla_{\frac{\partial}{\partial z}}\frac{\partial}{\partial x}=\frac{\partial}{\partial x},\quad\nabla_{\frac{\partial}{\partial z}}\frac{\partial}{\partial y}=\frac{\partial}{\partial y},\quad\nabla_{\frac{\partial}{\partial x}}\frac{\partial}{\partial z}=0,\end{gathered}$ (3.40) where $\nabla$ denotes the Levi-Civita connection of the Riemannian metric $g$. It is not hard to verify that the condition (3.1) for Kenmotsu manifold is satisfied. Hence, the manifold under consideration is a Kenmotsu manifold. By a straightforward calculation we have $\displaystyle R\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)\frac{\partial}{\partial z}=0,\quad R\left(\frac{\partial}{\partial x},\frac{\partial}{\partial z}\right)\frac{\partial}{\partial y}=0,\quad R\left(\frac{\partial}{\partial x},\frac{\partial}{\partial z}\right)\frac{\partial}{\partial z}=-\frac{\partial}{\partial x},$ $\displaystyle R\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)\frac{\partial}{\partial x}=\frac{1}{2}exp(2z)\frac{\partial}{\partial y},\quad R\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)\frac{\partial}{\partial y}=-\frac{1}{2}exp(2z)\frac{\partial}{\partial x},$ $\displaystyle R\left(\frac{\partial}{\partial x},\frac{\partial}{\partial z}\right)\frac{\partial}{\partial x}=\frac{1}{2}exp(2z)\frac{\partial}{\partial z},\quad R\left(\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\frac{\partial}{\partial x}=0,$ $\displaystyle R\left(\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\frac{\partial}{\partial y}=\frac{1}{2}exp(2z)\frac{\partial}{\partial z},\quad R\left(\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\frac{\partial}{\partial z}=-\frac{\partial}{\partial y},$ (3.41) and applying these, we obtain $\displaystyle S\left(\frac{\partial}{\partial x},\frac{\partial}{\partial x}\right)=-exp(2z),\quad S\left(\frac{\partial}{\partial y},\frac{\partial}{\partial y}\right)=-exp(2z),\quad S\left(\frac{\partial}{\partial z},\frac{\partial}{\partial z}\right)=-2.$ Thus, the Ricci tensor satisfy $S(X,Y)=-2ng(X,Y)$ for any $X,Y\in\Gamma(TM)$, that is, $M$ is Einstein. From (3.41), we can easily shows that $\displaystyle R(X,Y)Z=-\\{g(Y,Z)X-g(X,Z)Y\\},$ (3.42) for any $X,Y,Z\in\Gamma(TM)$. Next consider a vector field $\displaystyle V=a\left(y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}\right),$ (3.43) where $a\neq 0$ is a constant. One can easily verify that $V$ has constant divergence. As a result of (3.40), one can easily justify that $\displaystyle(\pounds_{V}g)(X,Y)=0,$ (3.44) for any $X,Y\in\Gamma(TM)$. Unifying (3.44) and (3.42), we obtain that $g$ is a Riemann soliton, that is, (1.2) holds true with $V$ as in (3.43) and $\lambda=1$. Further (3.42) shows that $M$ is of constant negative curvature $-1$ and this verifies the Theorem 3.3. ## 4\. Riemann solitons and almost gradient Riemann solitons on $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifolds with $\kappa<-1$ If the Reeb or characterstic vector field $\xi$ of an almost Kenmotsu manifold $M$ belonging to the $(\kappa,\mu)^{\prime}$-nullity distribution (see [3]), that is, $\displaystyle R(X,Y)\xi=\kappa\\{\eta(Y)X-\eta(X)Y\\}+\mu\\{\eta(Y)h^{\prime}X-\eta(X)h^{\prime}Y\\},$ (4.1) for some constants $\kappa$ and $\mu$, then $M$ is called $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold. Classification of almost Kenmotsu manifold with $\xi$ belonging to the $(\kappa,\mu)^{\prime}$-nulliy distribution have done by many geometers. For more detail, we refer to [3, 13, 19, 20]. On $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold, the following relation hold $\displaystyle h^{\prime 2}=(\kappa+1)\varphi^{2},$ $\displaystyle\quad\text{or}\quad h^{2}=(\kappa+1)\varphi^{2},$ (4.2) $\displaystyle Q\xi=2n\kappa\xi.$ (4.3) Let $X\in Ker\eta$ be an eigenvector field of $h^{\prime}$ orthogonal to $\xi$ with the corresponding eigenvalue $\theta$. As a result of (4.2), we get $\theta^{2}=-(\kappa+1)$ and hence we have $\kappa\leq-1$. From (4.2) we remark that the tensor field $h^{\prime}=0$ (equivalently, $h=0$) if and only if $\kappa=-1$, and $\kappa<-1$ if and only if $h^{\prime}\neq 0$. As stated by Dileo and Pastore [3] in Proposition 4.1 that, on a $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold with $\kappa<-1$, we have $\mu=-2$. In this section, we paln to investigate the geometry of a Riemann soliton and gradient almost Riemann soliton on non-Kenmotsu $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold. First, we reminisce the following result for our later use. ###### Lemma 4.1. In a $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold $M$ with $\kappa<-1$, the Ricci operator satisfies $\displaystyle QX=-2nX+2n(\kappa+1)\eta(X)\xi-2nh^{\prime},$ (4.4) where $h^{\prime}\neq 0$. Moreover, the scalar curvature $r$ of $M$ is $2n(\kappa-2n)$. Now, we prove the following outcome. ###### Theorem 4.2. Let $M$ be a non-Kenmotsu $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold. If metric of $M$ is a Riemann soliton with $divV=constant$, then either the manifold is locally isometric to the Reimannian product $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$ or the potential vector field $V$ is strict infinitesimal contact transformation. ###### Proof. Contracting the equation (1.2) over $X$ and $W$, one can get $\displaystyle(\pounds_{V}g)(Y,Z)+\frac{2}{2n-1}S(Y,Z)+\frac{2}{2n-1}(2n\lambda+divV)g(Y,Z)=0.$ As a result of Lemma 4.1, the above equation transform into $\displaystyle(\pounds_{V}g)(Y,$ $\displaystyle Z)+\frac{2}{2n-1}\\{(2n\lambda+divV-2n)g(Y,Z)$ $\displaystyle+2n(\kappa+1)\eta(Y)\eta(Z)-2ng(h^{\prime}Y,Z)\\}=0.$ (4.5) Differentiating (4.5) covariantly along $X$, recalling (2.3) and $V$ has a constant divergence, we have $\displaystyle(\nabla_{X}\pounds_{V}g)(Y,Z)=-$ $\displaystyle\frac{4n}{2n-1}(\kappa+1)\\{g(X+h^{\prime}X,Y)\eta(Z)+g(X+h^{\prime}X,Z)\eta(Y)$ $\displaystyle-2\eta(X)\eta(Y)\eta(Z)\\}+\frac{4n}{2n-1}g((\nabla_{X}h^{\prime})Y,Z).$ (4.6) In [3] Dileo and Pastore proved that on $M$ there holds $\displaystyle g((\nabla_{X}h^{\prime})Y,Z)=g$ $\displaystyle((\kappa+1)X-h^{\prime}X,Y)\eta(Z)+\eta(Y)g((\kappa+1)X-h^{\prime}X,Z)$ $\displaystyle-2(\kappa+1)\eta(X)\eta(Y)\eta(Z).$ (4.7) Making use of (4.6) into (2.6), we entails that $\displaystyle(\pounds_{V}\nabla)(X,Y)=-\frac{4n}{2n-1}(\kappa+2)g(h^{\prime}X,Y)\xi,$ (4.8) where we applied (4.7). Taking covariant differentiation of (4.8) we get $\displaystyle(\nabla_{Y}\pounds_{V}\nabla)(X,Z)=$ $\displaystyle-\frac{4n}{2n-1}(\kappa+2)g((\nabla_{Y}h^{\prime})Y,Z)\xi$ $\displaystyle-\frac{4n}{2n-1}(\kappa+2)g(h^{\prime}X,Z)(Y-\eta(Y)\xi+h^{\prime}Y).$ Putting the foregoing equation into (2.7) yields $\displaystyle(\pounds_{V}$ $\displaystyle R)(X,Y)Z=\frac{4n}{2n-1}\\{g((\nabla_{Y}h^{\prime})X,Z)\xi-g((\nabla_{X}h^{\prime})Y,Z)\xi$ $\displaystyle+g(h^{\prime}X,Z)(Y-\eta(Y)\xi+h^{\prime}Y)-g(h^{\prime}Y,Z)(X-\eta(X)\xi+h^{\prime}X)\\}.$ (4.9) Contracting the equation (4.9) with respect to $X$ and recalling (4.7) we easily obtain $\displaystyle(\pounds_{V}S)(Y,Z)=-\frac{8n^{2}}{2n-1}(\kappa+2)g(h^{\prime}Y,Z).$ (4.10) From (4.4), the Ricci tensor can be written as $\displaystyle S(Y,Z)=-2ng(Y,Z)+2n(\kappa+1)\eta(Y)\eta(Z)-2ng(h^{\prime}Y,Z).$ Lie-derivative of this equation along potential vector field $V$, utilization of (2.3) and (4.5) yields $\displaystyle(\pounds_{V}S)(Y,Z)$ $\displaystyle=\frac{4n}{2n-1}\\{(2n\lambda+divV+2n\kappa)g(Y,Z)-2(\kappa+1)(2n\lambda+divV$ $\displaystyle+2n\kappa)\eta(Y)\eta(Z)+(2n\lambda+divV-4n)g(h^{\prime}Y,Z)\\}+2n(\kappa+1)$ $\displaystyle\\{\eta(Y)g(\pounds_{V}\xi,Z)+\eta(Z)g(\pounds_{V}\xi,Y)\\}-2ng((\pounds_{V}h^{\prime})Y,Z).$ (4.11) Putting (4.11) together with (4.10), we get $\displaystyle g((\pounds_{V}h^{\prime})Y,Z)=$ $\displaystyle\frac{2}{2n-1}\\{(2n\lambda+divV+2n\kappa)g(Y,Z)-2(\kappa+1)(2n\lambda+divV$ $\displaystyle+2n\kappa)\eta(Y)\eta(Z)+(2n(\kappa+2)+2n\lambda+divV-4n)g(h^{\prime}Y,Z)\\}$ $\displaystyle+(\kappa+1)\\{\eta(Y)g(\pounds_{V}\xi,Z)+\eta(Z)g(\pounds_{V}\xi,Y)\\}.$ (4.12) Note that by switching $Y=Z=\xi$ in (4.5) we obtain $\eta(\pounds_{V}\xi)-\frac{1}{2n-1}(2n\lambda+divV+2n\kappa)=0$. Applying this equation and substituting $Y=Z=\xi$ in (4.12) one can get $(2n\lambda+divV+2n\kappa)=0$ by reason of $\kappa<-1$ and eventually we have $\displaystyle(\pounds_{V}h^{\prime})Y=(\kappa+1)\\{\eta(Y)\pounds_{V}\xi+g(\pounds_{V}\xi,Y)\xi\\}.$ Inserting $Y$ by $\xi$ in the foregoing equation and utilization of $(2n\lambda+\Delta u+2n\kappa)=0$ gives $\displaystyle h^{\prime}\pounds_{V}\xi=-(\kappa+1)\pounds_{V}\xi.$ (4.13) In view of (4.2), $(2n\lambda+divV+2n\kappa)=0$ and $\kappa<-1$, the action of $h^{\prime}$ on the above equation gives $h^{\prime}\pounds_{V}\xi=\pounds_{V}\xi$. This together with (4.13) yields $\displaystyle(\kappa+2)\pounds_{V}\xi=0.$ Suppose $\kappa=-2$, then it follows from Proposition 4.1 and Corollary 4.2 of Dielo and Pastore [3] that a non-Kenmotsu $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold is locally isometric to the Riemannian product $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$. If $\kappa\neq-2$, then we have $\pounds_{V}\xi=0$. As a result of $2n\lambda+divV+2n\kappa=0$ and $(\pounds_{V}g)(X,\xi)=0$ we obtain $\displaystyle(\pounds_{V}\eta)X=(\pounds_{V}g)(X,\xi)+g(X,\pounds_{V}\xi)=0.$ This means potential vector field $V$ is strict infinitesimal contact transformation. This finishes the proof. ∎ As a result of Lemma 4.1, the relation (3.27) and (3.26), we obtain the following fruitful result. ###### Theorem 4.3. Let $M$ be a non-Kenmotsu $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifold which admits a gradient almost Riemann soliton. Then, the soliton is expanding with $\lambda=\frac{6n-2}{2n-1}$ and $M$ is locally isometric to the Riemannian product $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$. Moreover, the potential vector field is tangential to the Euclidean factor $\mathbb{R}^{n}$. ###### Proof. With the help of (4.4) we have $\displaystyle(\nabla_{Y}Q)X-(\nabla_{X}Q)Y=$ $\displaystyle-2n((\nabla_{Y}h^{\prime})X-(\nabla_{X}h^{\prime})Y)$ $\displaystyle-2n(\kappa+1)(\eta(Y)(X+h^{\prime}X)-\eta(X)(Y+h^{\prime}Y)).$ Utilization of the above equation in (3.26) gives $\displaystyle R(X,Y)Du=$ $\displaystyle-\frac{2n}{2n-1}\\{((\nabla_{Y}h^{\prime})X-(\nabla_{X}h^{\prime})Y)$ $\displaystyle+(\kappa+1)(\eta(Y)(X+h^{\prime}X)-\eta(X)(Y+h^{\prime}Y))\\}$ $\displaystyle+\frac{1}{2n-1}\\{Y(2n\lambda+\Delta u)X-X(2n\lambda+\Delta u)Y\\}$ (4.14) Using $\xi$ in place of $X$ in (4.14) we get an equation, then taking inner product of the resulting equation with $\xi$ gives $\displaystyle g(R(\xi,Y)Du,\xi)=\frac{1}{2n-1}\\{Y(2n\lambda+\Delta u)-\xi(2n\lambda+\Delta u)\eta(Y)\\}.$ (4.15) As a result of (4.1), we get $\displaystyle g(R(\xi,Y)\xi,Du)=\kappa\\{\xi(u)\eta(Y)-Y(u)\\}+2g(h^{\prime}Du,Y),$ (4.16) where we used $\mu=-2$. Comparing (4.15) with (4.16) yields that $\displaystyle\frac{1}{2n-1}D(2n\lambda+\Delta u)=\kappa Du-\kappa\xi(u)\xi-2h^{\prime}Du+\frac{1}{2n-1}\xi(2n\lambda+\Delta u)\xi.$ (4.17) According to Corollary 4.1 of Delio and Pastore [3], and Lemma 3.4 of Wang and Liu [21], we have that $tr(\nabla_{X}h^{\prime})=0$ and $(divh^{\prime})X=2n(\kappa+1)\eta(X)$. Applying these equation and contracting (4.14) over $X$ gives that $\displaystyle S(Y,Du)=\frac{2n}{2n-1}Y(2n\lambda+\Delta u).$ As a result of (4.4), the above equation transform into $\displaystyle\frac{1}{2n-1}D(2n\lambda+\Delta u)=-Du+(\kappa+1)\xi(u)\xi-h^{\prime}Du.$ (4.18) By virtue of (4.17) and (4.18) we obtain $\displaystyle(\kappa+1)Du-(2\kappa+1)\xi(u)\xi-h^{\prime}Du+\frac{1}{2n-1}\xi(2n\lambda+\Delta u)=0.$ Remembering the assumption $\kappa<-1$ and equation (4.2), the action of $h^{\prime}$ on above equation gives that $\displaystyle h^{\prime}Du=-Du+\xi(u)\xi.$ (4.19) Again making use of (4.2), the action $h^{\prime}$ on (4.19) yields $(\kappa+1)(Du-\xi(u)\xi)=h^{\prime}Du$. This together with (4.19) gives that $\displaystyle(\kappa+2)(Du-\xi(u)\xi)=0.$ (4.20) Thus, we have either $\kappa=-2$ or $Du=\xi(u)\xi$. Now, we prove that the second case can not occur. In fact, if we assume that the second case is true, that is, $Du=\xi(u)\xi$, then covariant derivative of this along $X$ gives $\displaystyle\nabla_{X}Du=X(\xi(u))\xi+\xi(u)\\{X-\eta(X)\xi+h^{\prime}X\\}.$ With the aid of above equation, (3.27) becomes $\displaystyle\frac{1}{2n-1}QX=-(\xi(u)+\frac{1}{2n-1}(2n\lambda+\Delta u))X+(\xi(u)\eta(X)-X(\xi(u)))\xi-\xi(u)h^{\prime}X.$ Utilization of (4.4) in the foregoing equation furnishes $\displaystyle\left(\xi(u)+\frac{1}{2n-1}(2n\lambda+\Delta u-2n)\right)X+\left(\xi(u)-\frac{2n}{2n-1}\right)h^{\prime}X$ $\displaystyle+\left(\frac{2n}{2n-1}(\kappa+1)\eta(X)+X(\xi(u))-\xi(u)\eta(X)\right)\xi=0.$ (4.21) Contracting (4.21) with respect to $X$ and recalling (2.4) we obtain $\displaystyle 2n(\kappa+1)\left(\xi(u)-\frac{2n}{2n-1}\right)=0,$ and this exhibits that $\xi(u)=\frac{2n}{2n-1}$ because of $\kappa<-1$. Putting $\xi(u)=\frac{2n}{2n-1}$ into (4.21) we obatin $\displaystyle(2n\lambda+\Delta u)X+2n\kappa\eta(X)\xi=0.$ (4.22) Considering $X$ in (4.22) is orthogonal to $\xi$, we have $2n\lambda+\Delta u=0$. Thus, (4.22) becomes $2n\kappa\eta(X)\xi=0$. It follows that $\kappa=0$, this contradicts our assumption $\kappa<-1$. Therefore, we obtain from (4.20) that $\kappa=-2$. In view of $\kappa=-2=\mu$, on the basis of results of Dileo and Pastore [3] we obtain that $M$ is locally isometric to the Riemannian product $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$. Employing $\kappa=-2$ and (4.19) in (4.17) we obtain $\displaystyle D(2n\lambda+\Delta u)=\xi(2n\lambda+\Delta u).$ (4.23) Utilizing (4.23) and (4.19) in (4.18) provides $\displaystyle\xi\left(\frac{1}{2n-1}(2n\lambda+\Delta u)+2u\right)=0.$ (4.24) In the framework of (4.4), (3.27) and (4.24) we acquire that $\displaystyle 2n\lambda+\Delta u=4n+\frac{1}{2}\xi(\xi(2n\lambda+\Delta u)).$ (4.25) From (4.2) we have $h^{\prime 2}=-\varphi^{2}$. Now, we shall indicate by $[1]^{\prime}$ and $[-1]^{\prime}$ the distribution of the eigen vectors of $h^{\prime}$ orthogonal to $\xi$ with eigen values $1$ and $-1$ respectively, and also we may consider a local orthonormal $\varphi$-frame $\\{E_{i},\varphi E_{i},\xi\\}_{i=1}^{n}$ with $E_{i}\in[1]^{\prime}$ and $\varphi E_{i}\in[-1]^{\prime}$. As a result of (4.19), we easily find that $Du$ has no components on the distribution $[1]^{\prime}$. Thus, we write $Du=\sum_{i=n}^{n}\gamma_{i}\varphi E_{i}+\xi(u)\xi$, where $\\{\gamma_{i}\\}_{i=1}^{n}$ are smooth function on $M$. Applying this and (4.24) in (3.27) we obtain that $\displaystyle\frac{1}{2n-1}QX=$ $\displaystyle\frac{1}{2n-1}\left(\frac{1}{2}\xi(2n\lambda+\Delta u)-(2n\lambda+\Delta u)\right)X-\sum_{i=1}^{n}X(\gamma_{i})\varphi E_{i}$ $\displaystyle-\sum_{i=1}^{n}\gamma_{i}\nabla_{X}\varphi E_{i}+\frac{1}{2(2n-1)}\xi(2n\lambda+\Delta u)h^{\prime}X$ $\displaystyle+\frac{1}{2(2n-1)}\left(X(\xi(2n\lambda+\Delta u))-\xi(2n\lambda+\Delta u)\eta(X)\right)\xi.$ By virtue of (4.4), the above equation transform into $\displaystyle\frac{1}{2n-1}$ $\displaystyle\left(\frac{1}{2}\xi(2n\lambda+\Delta u)-(2n\lambda+\Delta u)+2n\right)X-\sum_{i=1}^{n}X(\gamma_{i})\varphi E_{i}-\sum_{i=1}^{n}\gamma_{i}\nabla_{X}\varphi E_{i}$ $\displaystyle+\frac{1}{2n-1}\left(\frac{1}{2}\xi(2n\lambda+\Delta u)+2n\right)h^{\prime}X+\frac{1}{2(2n-1)}\\{X(\xi(2n\lambda+\Delta u))$ $\displaystyle-\xi(2n\lambda+\Delta u)\eta(X)+4n\eta(X)\\}\xi=0.$ (4.26) According to the proof of Proposition 4.1 of [3], we have that $\nabla_{E_{j}}\varphi E_{i}\in[-1]^{\prime}$ for any $E_{i}\in[1]^{\prime}$, $1\leq j\leq n$. Consequently, using $E_{j}\in[1]^{\prime}$ in place of $X$ in (4.26) provides $\displaystyle(2n\lambda+\Delta u)=\xi(2n\lambda+\Delta u)+4n.$ (4.27) Unifying (4.25) with (4.27) yields that $2n\lambda+\Delta u=4n$. Now, contracting (3.27) over $X$ gives $\displaystyle\Delta u=-\frac{4n^{2}}{2n-1},$ where we applied $\kappa=-2$ and $r=2n(\kappa-2n)$. Combining the above equation with $2n\lambda+\Delta u=4n$ yields that $\lambda=\frac{6n-2}{2n-1}>0$, and this means that the gradient almost Riemann soliton is expanding. Eventually, utilization of $2n\lambda+\Delta u=4n$ in (4.24) gives $\xi(u)=0$ and thus from (4.19) we have $h^{\prime}Du=-Du$. According to Theorem 4.2 of [3], the factor $\mathbb{R}^{n}$ in the product space $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^{n}$ is the integral submanifold of the distribution $[-1]^{\prime}$. Therefore, $h^{\prime}Du=-Du$ implies that the gradient of the potential function $Du$ is tangential to the Euclidean factor $\mathbb{R}^{n}$. ∎ ###### Remark 4.4. Theorem 3.1 of Wang [20] is a direct corollary of the above Theorem 4.3. Before closing this section, we present an example of 3-dimensional $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifolds admitting a Riemann soliton. ###### Example. We consider a 3-dimensional manifold $\displaystyle M=\\{(x,y,z)\in\mathbb{R}^{3},z\neq 0\\},$ and linear independently vector fields $\displaystyle e_{1}=-\frac{\partial}{\partial x}+2y\frac{\partial}{\partial y}-\frac{\partial}{\partial z},\quad e_{2}=\frac{\partial}{\partial y},\quad e_{3}=\frac{\partial}{\partial z}.$ One can easily check that $\displaystyle[e_{1},e_{2}]=-2e_{2},\quad[e_{2},e_{3}]=0,\quad[e_{1},e_{3}]=0.$ On $M$ we define a (1,1)-tensor field $\varphi$ by $\varphi(e_{1})=0$, $\varphi(e_{2})=e_{3}$ and $\varphi(e_{3})=-e_{2}$, and we define a Riemannian metric $g$ such that $g(e_{i},e_{j})=\delta_{ij}$, $1\leq i,j\leq 3$. We denote by $\xi=e_{1}$ and $\eta$ its dual 1-form with respect to the metric $g$. On the other hand, it is not hard to justify that $M$ with the structure $(\varphi,\xi,\eta,g)$ is an almost Kenmotsu structure, which is not Kenmotsu, since the operator $h^{\prime}$ does not vanish. In fact, it is given by $\displaystyle h^{\prime}(e_{1})=0,\quad h^{\prime}(e_{2})=e_{2},\quad h^{\prime}(e_{3})=-e_{3}.$ Utilization of Koszul formula gives $\displaystyle\begin{gathered}\nabla_{e_{1}}e_{1}=0,\quad\nabla_{e_{1}}e_{2}=0,\quad\nabla_{e_{1}}e_{3}=0,\\\ \nabla_{e_{2}}e_{1}=2e_{2},\quad\nabla_{e_{2}}e_{2}=-2e_{1},\quad\nabla_{e_{2}}e_{3}=0,\\\ \nabla_{e_{3}}e_{1}=0,\quad\nabla_{e_{3}}e_{2}=0,\quad\nabla_{e_{3}}e_{3}=0,\end{gathered}$ (4.31) where $\nabla$ denotes the Levi-Civita connection of the Riemannian metric $g$. By a straightforward calculation we have $\displaystyle\begin{gathered}R(e_{1},e_{2})e_{1}=4e_{2},\quad R(e_{1},e_{2})e_{2}=-4e_{1},\quad R(e_{1},e_{2})e_{3}=0,\\\ R(e_{1},e_{3})e_{1}=0,\quad R(e_{1},e_{3})e_{2}=0,\quad R(e_{1},e_{3})e_{3}=0,\\\ R(e_{2},e_{3})e_{1}=0,\quad R(e_{2},e_{3})e_{2}=0,\quad R(e_{2},e_{3})e_{3}=0.\end{gathered}$ (4.35) With the help of the expressions of the curvature tensor, we conclude that the characteristic vector field $\xi$ belongs to the $(\kappa,\mu)^{\prime}$-nullity distribution with $\kappa=-2$ and $\mu=-2$. Let us consider a vector field $\displaystyle V=e^{-2x}\frac{\partial}{\partial y}+4(x-z)\frac{\partial}{\partial z}.$ (4.36) One can easily check that $divV=-4$, a constant. As a result of (4.31), one can get $\displaystyle\begin{gathered}(\pounds_{V}g)(e_{1},e_{1})=0,\quad(\pounds_{V}g)(e_{2},e_{2})=0,\quad(\pounds_{V}g)(e_{3},e_{3})=-8,\\\ (\pounds_{V}g)(e_{1},e_{2})=0,\quad(\pounds_{V}g)(e_{1},e_{3})=0,\quad(\pounds_{V}g)(e_{2},e_{3})=0\end{gathered}$ (4.39) In view of (4.39) and (4.35), one can easily verify that $\displaystyle 2R(e_{i},e_{j},e_{k},e_{l})+4(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}g)(e_{i},e_{j},e_{k},e_{l})+(g\mathbin{\bigcirc\mspace{-15.0mu}\wedge\mspace{3.0mu}}\pounds_{V}g)(e_{i},e_{j},e_{k},e_{l})=0,$ for $1\leq i,j,k,l\leq 3$. Thus $g$ is a Riemann soliton with the soliton vector field $V$ as given in (4.36) and $\lambda=4$. According to Dileo and Pastore [3], we obtain that $(-2,-2)^{\prime}$-almost Kenmotsu manifold $M$ is locally isometric to the product $\mathbb{H}^{2}(-4)\times\mathbb{R}$. This verifies Theorem 4.2. ## References * [1] D.E. Blair, Riemannian geometry of contact and symplectic manifolds. In: Progress in Mathematics, 203. Birkhäuser, Boston (2010). * [2] G. Dileo, A. M. Pastore, Almost Kenmotsu manifolds and local symmetry, Bull. Belg. Math. Soc. Simon Stevin. 14(2) (2007), 343-354. * [3] G. Dileo, A. M. Pastore, Almost Kenmotsu manifolds and nullity distributions, J. Geom. 93(1-2) (2009), 46-61. * [4] M.N. Devaraja, H.A. Kumara, V. Venkatesha, Riemann soliton within the framework of contact geometry, Quaestiones Mathematicae, (2020) DOI: 10.2989/16073606.2020.1732495 * [5] A. Ghosh, Kenmotsu 3-metric as a Ricci soliton, Chaos Solitons & Fractals, 44 (2011), 647-650. * [6] A. Ghosh, An $\eta$-Einstein Kenmotsu metric as a Ricci soliton, Publ. Math. Debrecen, 82 (2013), 691-598. * [7] A. Ghosh, Ricci soliton and Ricci almost soliton within the framework of Kenmotsu manifold, Carpathian Math. Publ. 11(1) (2019), 59-69. * [8] D. Janssens, L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J. 4(1) (1981), 1-27. * [9] R.S, Hamilton, The Ricci flow on surfaces. Mathematics and general relativity. Contemp. Math., Amer. Math. Soc. 71 (1988), 237-262. * [10] I.E. Hiric̆a, C. Udrişste, Ricci and Riemann solitons, Balkan J. Geom. Applications. 21(2) (2016), 35-44. * [11] I.E. Hiric̆a, C. Udrişste, Basic evolution PDE’s in Riemannian Geometry, Balkan J.Geom.Appl. 17(1) (2012), 30-40. * [12] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tôhoku Math. J. 24(1) (1972), 93-103. * [13] D.G. Prakasha, P. Veeresha, Venkatesha, The Fischer–Marsden conjecture on non-Kenmotsu $(\kappa,\mu)^{\prime}$-almost Kenmotsu manifolds, J. Geom. 110 (1) (2019), https://doi.org/10.1007/s00022-018-0457-8. * [14] S.E. Stepanov, I.I. Tsyganok, The theory of infinitesimal harmonic transformations and its applications to the global geometry of Riemann solitons, Balk. J. Geom. Appl. 24 (2019), 113–121. * [15] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251-275. * [16] C. Udrişte, Riemann flow and Riemann wave, Ann. Univ. Vest, Timisoara. Ser. Mat.-Inf. 48(1-2) (2010), 265-274. * [17] C. Udrişte, Riemann flow and Riemann wave via bialternate product Riemannian metric. preprint, arXiv.org/math.DG/1112.4279v4 (2012). * [18] Venkatesha, D. M. Naik, H. A. Kumara, $*$-Ricci soliton and gradient almost $*$-Ricci soliton on Kenmotsu manifolds, Mathemtica Slovoca (accepted). * [19] V. Venkatesha, H. A. Kumara, Gradient $\rho$-Einstein soliton on almost Kenmotsu manifolds, Ann. Univ. Ferrara. 65(2) (2019), 375-388. * [20] Y. Wang, Gradient almost Ricci soliton on two classes of almost Kenmotsu manifolds, J. Korean Math. Soc. 53(5) (2016), 1101-1114. * [21] Y. Wang, X. Liu, Locally symmetric CR-integrable almost Kenmotsu manifolds, Mediterr. J. Math. 12(1) (2015), 159-171. * [22] K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970.

2024-09-04T02:54:56.253122

2003.00809

arxiv-papers

# Vision based body gesture meta features for Affective Computing Indigo Jay Dennis Orton (June, 2019) ###### Abstract ##### Title: Vision based body gesture meta features for Affective Computing Early detection of psychological distress is key to effective treatment. Automatic detection of distress, such as depression, is an active area of research. Current approaches utilise vocal, facial, and bodily modalities. Of these, the bodily modality is the least investigated, partially due to the difficulty in extracting bodily representations from videos, and partially due to the lack of viable datasets. Existing body modality approaches use automatic categorization of expressions to represent body language as a series of specific expressions, much like words within natural language. In this dissertation I present a new type of feature, within the body modality, that represents meta information of gestures, such as speed, and use it to predict a non-clinical depression label. This differs to existing work by representing overall behaviour as a small set of aggregated meta features derived from a person’s movement. In my method I extract pose estimation from videos, detect gestures within body parts, extract meta information from individual gestures, and finally aggregate these features to generate a small feature vector for use in prediction tasks. No existing, publicly available, dataset for distress analysis contains source videos of participants or pose data extracted from the source videos. As such, I introduce a new dataset of 65 video recordings of interviews with self- evaluated distress, personality, and demographic labels. This dataset enables the development of features utilising the whole body in distress detection tasks. I evaluate my newly introduced meta-features for predicting depression, anxiety, perceived stress, somatic stress, five standard personality measures, and gender. A linear regression based classifier using these features achieves a 82.70% F1 score for predicting depression within my novel dataset. My results suggest this feature type has value for distress prediction and, more broadly, has potential within the affective computing domain, as they appear to be useful aggregate representations of human behaviour. Word count: 14,586 ###### keywords: computer-vision affective-computing Hughes Hall Computer Science ###### Acknowledgements. First and foremost I thank my supervisor Dr. Marwa Mahmoud, her guidance has been invaluable and her collaboration in the development of the dataset has been integral. Thank you to my peers who have contributed immensely to my learning through countless rambling and broad ranging discussions. Their breadth of interests and expertise is always engaging. Finally, thank you to my family for their endless support and encouragement. ## Chapter 1 Introduction Mental health is an increasingly prominent area of discussion and research in society, it is a main contributor to overall disease burden globally [41]. Awareness of its effects and symptoms, destigmatization of its results, consideration of its causes, and approaches to its treatment and diagnosis, in the case of illness, are all rapidly evolving. According to the UK Mental Health Foundation [41], depression is the “predominant mental health problem worldwide”. Within the UK in 2014, 19.7% of people over the age of 16 showed symptoms of depression or anxiety. Early detection is paramount to long term health, 10% of children aged 5–16 years have a clinically diagnosable mental health issue, of children who have experienced mental health problems 70% do not have their issues detected or addressed early enough [41]. #### Automatic Distress Detection - Why and How Automatic distress detection enables large scale early screening. Early screening can be used to enable mitigation of distress at an earlier stage that it might otherwise be identified and also prioritization of services. For example, Crisis Text Line [12] (CTL) provides a free mental health counseling service via text messages. The communication medium means consumers can message without a counselor being exclusively available. To better support those consumers most in need (e.g. those most at risk of suicide), CTL analyses text messages in real time to triage consumers for counselor attention [15]. While the CTL example is a specific use case with a single modality, text, the broader area of distress detection uses many modalities including facial, eye, head, vocal, bodily, and speech. Existing work has investigated usage of psychology coding systems such as FACS levels of activity in eye and head movement [1, 55, 16], orientation of eye gaze and head [46], fundamental frequencies and biomarkers of vocal recordings [43, 2], automatic categorization of body expressions [36, 34, 51], and natural language artifacts from speech [13]. Moreover, many methods incorporate multiple modalities and achieve significantly better results. One of the difficulties in this area is the lack of available and relevant datasets. Distress focused datasets are not readily shared given the sensitive nature of the subject. Those that are shared rarely include source data, but rather provide processed outputs, again due to the sensitive of the subject. Finally, each modality has different source data requirements and datasets are generally designed with a specific method in mind. All of this creates a barrier to the development of novel features and thus research within this area is concentrated on the modalities available within the few public datasets. #### Body Modality - A Research Gap The body modality includes the whole body from the neck down, sometimes the head is also included. It is the least researched of the modalities. This is, partially, due to the dataset barrier. No public distress dataset, that I am aware of, provides body modality data, or the source video recordings to extract such data. The majority of relevant body modality based automatic distress detection work is based on private clinical datasets. ##### Existing Approaches Within these body modality methods there are two primary approaches: body expression categorization and hand-crafted distress behaviour descriptors. The first approach uses unsupervised learning to cluster spatio-temporal descriptors of movement to define expression codebooks and then uses these codebooks to predict depression severity [36, 34, 51]. The second approach defines hand-crafted descriptors of behaviour psychology literature has shown to correlate to distress [23], such as self-adaptors and fidgeting [46, 40], and to model the intensity of such behaviours to indicate distress severity. ##### Thesis Contributions My core research question is: is meta information from gestures predictive of psychological distress? In this dissertation I present novel generic body gesture meta features for automatic distress detection. These features are based on the movement, frequency, and duration of body gestures. Per-gesture these features are cheap to compute and relatively simplistic, aggregated across all gestures they aim to represent general behavioural characteristics of a subject. I evaluate the use of these features for detecting depression (non-clinical) and find that they provide useful predictive information within linear models. I then evaluate the generalisability of these features by applying them to classification tasks for other distress and personality labels. As I noted earlier, a core barrier to the development of new features within the body modality is the lack of available distress datasets that expose the required source data. As such I introduce a novel audio-visual dataset gathered for this dissertation in collaboration with Dr. Marwa Mahmoud. The dataset contains audio-visual recordings of semi-structured interviews and labels based on established psychology self-evaluation questionnaires for distress and personality measures. ### Dissertation structure Chapter 2 contains a literature review covering related distress detection research and relevant technology for working with the body modality specifically. Chapter 3 introduces a novel dataset for depression detection. I present my method in Chapter 4. I evaluate my features’ validity in Chapter 5. Finally, I conclude and outline potential future work in Chapter 6. ## Chapter 2 Literature Review This review is structured in two parts: automatic depression detection and body modality technology. In the first component I review existing approaches to automatic depression detection and related fields, and the variety of modalities used by these approaches. This defines the research space this dissertation occupies. The second component outlines the current methods for working with the body modality and the technology involved in these methods, such as pose estimation. This provides the basis for the core data I use within my method. ### 2.1 Automatic Detection of Distress Distress is expressed through all modalities. Many approaches have been developed to automatically detect distress using behavioural cues, these include both mono-modal and multi-modal approaches. For example, face analysis methods are one of the common, and powerful, techniques enabled by the development of tools for extracting accurate positional information, such as facial landmarks, and tools for automatic interpretation based on existing psychology approaches such as FACS [21]. I review uses of the primary modalities, cross-modal considerations, multi- modal approaches, and the expanding use of deep learning within automatic distress detection. The face, head, and body modalities are the most relevant, though I briefly provide examples of text and vocal modal usage. #### Text Modality Dang et al. [13] use linguistic attributes, auxiliary speech behaviour, and word affect features to predict depression severity and emotional labels within the DAIC dataset [26]. Linguistic attribute features include total number of words, unique words, pronouns, lexical proficiency, among others. Auxiliary speech behaviour cover parallel actions, such as laughing or sighing, and meta information such as word repeats and average phrase length. Word features are determined by a collection of annotated corpora that assign $n$-grams categorizations or ratings relating to their affective semantic. For example, assigning an emotion type (anger, disgust, joy, etc) to a word or rating words from 0 to 10 on affective attributes such as arousal, valence, dominance, and pleasure. These three feature types are used to predict depression measures. #### Audio Modality The audio modality can be a strong predictive source as non-verbal features of speech can be predictive of distress irrespective of the content of a person’s speech [43]. Features from audio data commonly include prosody, jitter, intensity, loudness, fundamental frequency, energy, Harmonic-to-Noise-Ratio (HNR), among others. Ozdas et al. [43] present a method for analysing fluctuations in the fundamental frequency of a person’s voice to assess their risk of suicide. Alghowinem et al. [2] explore the use of a broad selection of vocal features, extracted using the “openSMILE” toolkit [22] to detect depression. Dibeklioglu et al. [16] use vocal prosody to detect depression. In their investigation of psychomotor retardation caused by depressive states, Syed et al. [55] use low-level descriptors to model turbulence in subject speech patterns. By profiling the turbulence of depressed and non-depressed participants with a depression dataset they develop a model for predicting depression severity based on the level of turbulence. #### Facial Modality Joshi et al. [36] present a “bag of facial dynamics” depression detection method based on the same expression clustering as their “bag of body dynamics” method described in more depth below. For this facial method the space-time interest points (STIPs) are generated for face aligned versions of the source videos. While Joshi et al. use a categorization approach, Dibeklioglu et al. [16] use generic features representing facial movement dynamics for depression detection. This method involves generating statistical derivations from movement features such as velocity, acceleration, and facial displacement over a time period and then modeling their effect. Whilst Dibeklioglu et al. take a generic approach to facial movement, Syed et al. [55] attempt to model behaviour discussed in psychology literature, using features representing psychomotor retardation to predict depression severity. Psychomotor retardation has been shown to be linked to depression [50]. In particular, Syed et al. generate features to capture craniofacial movements that represent psychomotor retardation, and thus indicate depression. To capture the target movements they design features that represent muscular tightening, a depressed subject is expected to have impaired muscle movements. They model three types of movement: head movement, mouth movement, and eyelid movement. These movements are represented by temporal deltas to define the amount of movement in the region. From these localised movement deltas the authors aim to represent specific actions such as blinking or contorting of the mouth. Relatively simplistic features derived from these actions, such as blink rate, can be indicative of depression [3, 19]. The nature of human behaviour is that these kinds of simplistic features can contribute useful information to distress detection models. Moreover, modeling specific facial actions has been examined as well. For example, Scherer et al. [46] use smile features such as intensity and duration, along with other modalities, to detect depression. Yang et al. [58] present a novel facial descriptor, a “Histogram of Displacement Range (HDR)”, which describes the amount of movement of facial landmarks. The histogram counts the number of occurrences of a displacement within a certain range of movement. Where Syed et al. represented the amount of movement of certain facial features to measure psychomotor retardation, Yang et al. represent the number of times the face is distorted, so to speak, by landmarks moving a certain amount. ###### Eye Sub-Modality While Syed et al. [55] explored the use of eye lid features, eye gaze features have also been shown to be effective in predicting distress. This modality has become viable as eye tracking technology has progressed sufficiently to enable accurate processing of eye features. Alghowinem et al. [1] use eye gaze/activity to perform binary classification of depression in cross-cultural datasets. They extract iris and eyelid movements to extract features such as blink rate, duration of closed eyes, and statistical “functionals” (i.e. simple derivations) of the amount of activity. However, activity is not the only indicator, Scherer et al. [46] use average eye gaze vertical orientation (among other features), in the span $[-60,60]$ degrees. #### Head Modality Joshi et al. [36] present a “histogram of head movements” depression detection method that models movement of a person’s head over time. They use three facial landmarks, the corner of each eye and the tip of the nose, to compute the orientation of the subject’s head. The histogram uses orientation bins of width 10 within the range $[-90,90]$ degrees for windows of time within a video. These windowed histograms are then averaged over the full length of the video. The resulting average histogram is a descriptor of the amount of movement within the video by representing the variety of angles the head orients to. This method achieves comparable performance to their “bag of facial dynamics” method. A number of the methods using eye activity features also incorporate head activity in their models. Alghowinem et al. [1] model head activity similarly to their modeling of eye activity. As with eye activity, they extract statistical derivations of movement and angular shift. They also include the duration of the head at different orientations, the rate of change of orientation, and the total number of orientation changes. Scherer et al. [46] use head vertical orientation, similar to their eye gaze orientation feature, as a feature for predicting depression. They use the average pitch of the head within a 3D head orientation model. Dibeklioglu et al. [16] also model head movement dynamics in a similar fashion to their facial movement dynamics features. Similar to Alghowinem et al. they extract statistical derivations of movement velocity, amplitude, and acceleration as head movement features. The head movement statistical derivation features presented by Alghowinem et al. and Dibeklioglu et al. are similar to the features I introduce in this dissertation, in that they represent meta movement information of the modality, rather than categorizing movement. Though, of course, Alghowinem et al. also incorporate categorized movement via their orientation features. #### Body Modality The body modality is the least researched of the modalities reviewed. This is due to a number of factors including the difficulty of dataset creation and the available technology for extracting raw body data. Contrast this with the relative ease of working with the other modalities and it is no surprise they received more attention. However, much of the relevant body modality based automatic distress detection research appeared in the early 2010s as some private clinical datasets were gathered and the parallel technological ecosystem expanded to support generation of body modality features. ##### The Case for Further Investigation De Gelder [14] presents the case for further research of bodily expressions within affective neuroscience. Though a parallel field, the core argument is much the same for affective computing’s investigation of bodily expressions. Specifically, at the time of writing (2009) de Gelder asserts that 95% of “social and affective neuroscience” focuses on faces and that the remaining 5% is mostly split between vocal, musical, and environmental modalities with a very few number of papers investigating the body modality. This was similar to the state of automatic distress detection in the early 2010s, though the vocal modality has a more prominent position in the literature and is more evenly balanced with the facial modality. ###### Affectation control and robust automatic detection Non-verbal features provide discriminative information regardless of a person’s conscious communication, this is particularly important for automatic distress detection. Different modalities can be consciously controlled to varying degrees. For example, facial expressions are more easily controlled than bodily expressions [14]. By including more modalities, and representations of those modalities, automatic distress detection could become more robust to conscious affectation modification of modalities. Further to robustness, one of the advantages of the face and body modalities is their ability to detect micro-expressions. Micro-expressions are instinctual reactions to some stimulus that can be predictive of emotional and distress state [20, 28]. They are significantly harder to control than general expressions. ##### Expression Categorization Much of the body modality research has approached the problem as a transfer of the methods from the facial modality by modeling body movements as expressions in much the same way facial expressions are [33, 36, 34]. Differences in these methods have been centred around: what tracklets form the basis of the body data [36], the use of deep learning vs manual descriptor definition [46], and process for generating categories of expressions [51]. Joshi et al. [36] demonstrate the discriminative power of bodily expressions for predicting clinically based depression measures. They use STIPs from recordings of a participant’s upper body and generate a “Bag of Body Dynamics (BoB)” based on codebooks of expression representations. STIPs are generated for a video, then Histograms of Gradient (HoG) and Optical Flow (HoF) are computed spatio-temporally around the STIPs, these histograms are then clustered within a sample, the cluster centres form a representation of the movements occurring in the video. The cluster centres from all videos in a training set are clustered again to generate the codebook of expressions. A BoB feature vector is generated for each sample by counting the number of cluster centres within the sample that fit within each codebook expression. Finally, these BoB feature vectors are used by an SVM to detect depression. Joshi et al. [34] extend on this approach by combining a “holistic body analysis”, similar to the BoB method, this method uses STIPs for whole body motion analysis and adds relative body part features. These features represent the movement of the head and limbs relative to the participant’s trunk, represented as polar histograms. Applying the same essential method as Joshi et al., Song et al. [51] present a method for learning a codebook of facial and bodily micro-expressions. They identify micro-expressions by extracting STIPs over very short time intervals (e.g. a few hundred milliseconds), then, as Joshi et al. do, they compute local spatio-temporal features around the STIPs and learn a codebook of expressions based on these local features. Finally, for each sample they generate a Bag-of-Words style feature vector based on the codified micro- expressions present in the sample. ##### Distress Behaviour Descriptors Psychology literature describes specific behaviours that are correlated with psychological distress and disorders. For example, Fairbanks et al. [23] find self-adaptors and fidgeting behaviour to be correlated to psychological disorders. Based on this work, Scherer et al. [46] evaluate the use of these behaviours for automatic distress detection. Specifically, they manually annotate their dataset for hand self-adaptor behaviours and fidgets, including hand tapping, stroking, grooming, playing with hands or hair, and similar behaviours. To identify whether regions are relevant to these behaviours they also annotate these behaviours with categories such as head, hands, arms, and torso, and then extract statistical information such as the average duration of self-adaptors in each region. They also annotate leg fidgeting behaviour such as leg shaking and foot tapping, again they use statistical derivations of these behaviours as features for detection. Whilst Scherer et al. manually annotated self-adaptors and fidgets, Mahmoud et al. [40] present an automatic detector of fidgeting, and similar behaviours, based on a novel rhythmic motion descriptor. They extract SURF interest point tracklets from colour and depth data and then apply their novel rhythmic measure to check similarity among cyclic motion across tracklets. Rhythmic motion is then localised based on Kinect skeletal regions and classified as one of four classes: Non-Rhythmic, Hands, Legs, and Rocking. #### Multi-Modal Fusion Combining modalities for prediction has proven effective when combining a variety of modalities [58, 35, 16]. There are four primary types of fusion: feature fusion such as feature vector concatenation, decision fusion such as majority vote, hybrid fusion which uses both, and deep learning fusion which merges inner representations of features within a deep learning architecture. The deep learning fusion method differs from feature fusion as the features are provided to separate input layers and only merged after inner layers, but before decision layers. Song et al. [51] combine micro-expressions from facial and bodily modalities with sample-level audio features. They evaluate three methods, early fusion by concatenating audio features to features from each visual frame, early fusion using a CCA [30] kernel, and late fusion based on voting, where the per-frame predictions from the visual modalities are averaged over the sample and then combined with the audio prediction. Dibeklioglu et al. [16] fuse facial, head, and vocal modalities using feature concatenation. They extend on this by performing feature selection on the concatenated vector, rather than the source vectors, using the Min-Redundancy Max-Relevance algorithm [45]. Alghowinem et al. [1] perform hybrid modality fusion, both combining feature vectors from modalities and performing a majority vote on individual modality classification predictions. The vote fusion is based on three classifiers, two mono-modal classifiers and one feature fusion classifier. Huang et al. [31] train long-short-term memory (LSTM) models on facial, vocal, and text modalities and then use a decision level fusion, via a SVR, to predict the final regression values. This paper differs from many deep learning approaches as it uses the decision level fusion, rather than having the deep learning models find patterns across feature types. ###### Temporal contextualisation Gong & Poellabauer [25] present another approach to usage of multiple modalities where the text modality provides contextualisation for features from the audio-visual modalities. They apply a topic modeling method for depression detection using vocal and facial modalities, where features are grouped based on the topic being responded to within an interview. The authors suggest that without topic modeling the features are averaged over too large a time period such that all temporal information is lost. By segmenting the samples they aim to retain some of the temporal information. Arbitrary segmentation would not necessarily be useful, thus their use of logical segmentation based on topic. #### Deep Learning Much of the recent work in distress detection leverages advances in deep learning, especially advances related to recurrent architectures, such as LSTMs, which can model sequence data well. In distress detection most modalities provide sequence data, audio-visual streams, natural language, or sequence descriptors of the data (such as FACS AUs). Chen et al. [9] present a method utilizing text, vocal, and facial modalities for emotion recognition. They explore the use of existing vocal features such as fundamental frequency analysis, auto-learnt features based on pre-trained CNNs to extract vocal and facial features, and word embedding features for the text. The auto-learnt facial features are derived from existing facial appearance data already extracted from the raw videos (i.e. they do not have their CNNs process raw frames to extract features). They then experiment with SVR and LSTM models to evaluate the temporal value of the LSTM model. They find that fused auto-learnt features from all modalities in combination with the LSTM model provides the greatest performance. Yang et al. [59] present an interesting multi-level fusion method that incorporates text, vocal, and facial modalities. They design a Deep Convolutional Neural Network (DCNN) to Deep Neural Network (DNN) regression model that is trained, separately, for audio and video modalities. They also trained their regression models separately for depressed and non-depressed participants, resulting in four separate DCNN - DNN models. They use the openSMILE toolkit for their audio features, this is common among many of the vocal modality methods (e.g. Alghowinem et al. from above), and FACS AUs for their visual features. They derive a temporally relevant feature vector from the set of all AUs by calculating the change in AUs over time. Their text modality model uses Paragraph Vectors in combination with SVMs and random forests and predicts a classification task rather than a regression task. Finally, they fuse, using DNNs, the audio and visual model predictions per training set, i.e. the depressed participant trained models are fused and the non-depressed models are fused. They then use another DNN to fuse the two fused regression predictions (i.e. depressed and non-depressed) and the classification prediction from the text modality. While they use DNNs to fuse decisions at multiple levels, this is a decision-level fusion method, not a deep learning fusion method, as they fuse the regression predictions from each model rather than the inner layer outputs. Yang et al. [58] present a second paper which utilises the same structure of DCNNs and DNNs, with two significant changes: firstly, the video features are changed to a new global descriptor they present, the “Histogram of Displacement Range (HDR)” which describes the amount of movement of facial landmarks, and secondly, the text modality now uses the same DCNN - DNN architecture to perform regression based on text data. Having changed the output of the text model the final fusion is a regression fusion using the same method as the audio visual models were fused in the first paper. Finally, no method, that I am aware of, uses deep learning end-to-end for automatic distress detection such that features are learnt from raw data for predicting distress. All methods apply deep learning on top of audio-visual descriptors and hand-crafted features. One of the core difficulties is the relative sparsity of existing data and thus the restricted ability to learn interesting features. Therefore, continued development of features and approaches to behavioural representation is valuable and able to contribute to methods that use deep learning. ### 2.2 Body Gestures Body gestures present a number of challenges including deriving features directly from pixels, extracting positional data, detecting gestures within the data, and designing representative features based on gestures. This section focuses on previous work on body gesture and pose representation, which forms the basis of my feature definition and generation. There are three primary approaches to body gesture representation within the affective computing literature: traditional computer vision feature detection algorithms such as STIPs [36, 34], pose estimation [8] from standard video recordings, and use of specialised 3D capture equipment such as Kinects [40]. #### Recording Type - 2D vs 3D The first two approaches use standard 2D video recordings to extract the base data for calculating features. The third approach uses 3D video recordings to enable more detailed analysis of body movement via depth. The most common 3D capture equipment used within related work is Kinects. Kinects have an added benefit that they provide skeletal key points (i.e. joint locations) within a recording, thus enabling more accurate representation of body data. #### Extracting Body Representations Given a video recording, 2D or 3D, with or without skeletal information, the next phase is feature extraction. Feature extraction approaches fall into two primary categories, which apply to both 2D and 3D videos: generic video feature representations and body modality specific representations. ##### Generic Interest Points The first approach does not target specific body areas or movement, instead it assumes that the only subject within the video is the subject the model is concerned with, i.e. the participant, and extracts generic video features from the recording to represent the body and gestures. Examples of these generic features are Space-Time Interest Points (STIPs), SURF, Histogram of Gradients (HOGs) and Optical Flow (HOFs), among others. These features can be extracted from colour and depth recordings such that they are applicable to both 2D and 3D recordings. While some approaches apply these generic features (or rather, derivations of these features) directly to prediction tasks [36, 51] (examples are discussed in Section 2.1), others aim to incorporate heuristics and information based on body parts. For example, interest points can be segmented based on location within the frame to heuristically distinguish body parts (e.g. head and arms) [34]. Another example is Mahmoud et al. [40] who use SURF keypoints from colour and depth images across a video to define tracklets for their rhythmic motion measure. Their dataset is based on Kinect captured videos to provide depth, this also provides skeletal regions such as feet and head. They use these Kinect skeletal regions to localise their keypoints’ motion. ##### Body Modality Specific The second approach extracts body modality specific interest points (e.g. skeletal models) to calculate features from. There are two primary methods for extracting these interest points: joint estimations from specialised capture equipment such as Kinects and pose estimation from existing video recordings. Such skeletal models have gained popularity in the past few years for action recognition tasks [54, 17, 10, 48, 56, 18]. In this dissertation I use pose estimation to extract body interest points (e.g. joints) from each frame in a two-dimensional video. In the past three to four years there has been substantial work on these pose estimation systems [57, 7, 49, 60, 39]. The current state-of-the-art is OpenPose by Cao et al. [8]. OpenPose uses skeletal hierarchy and part affinity fields to estimate pose interest points (i.e. joints) relative to each other and determine both part and orientation based on the direction of the proposed limb. The state- of-the-art model, at the time of writing, generates 25 interest points identifying the location of all major joints and more detailed information regarding head and feet. OpenPose can also be used to extract facial landmarks and detailed hand models. ### 2.3 Summary I have reviewed methods for automatic depression detection and the variety of modalities and models used within the field. I have also discussed, in broad terms, methods used for working with body data within related fields, from variation in capture equipment to difference in core interest points. The first component of the review outlined the gap within the existing use of the body modality that my proposed methodology addresses. While the second component outlined methods for working with the body modality, and specifically technology that I use to implement my methodology. ## Chapter 3 Dataset To develop whole body features I need a dataset containing video recordings and distress labels. However, due to the sensitive nature of this data no existing dataset makes both the source video and distress labels publicly available. Though some datasets make video derived data available, such as facial landmarks, none include derived data for the participant’s whole body. Given this I introduce a new dataset containing audio-visual recordings of interviews and labeled with distress and personality measures. This is not a clinical dataset, the interviews are performed by a computer science researcher and the labels are based on established psychology self- evaluation questionnaires. #### Collaborators Dr. Marwa Mahmoud from the Department of Computer Science at the University of Cambridge was the interviewer. Dr. Gabriela Pavarini from the Department of Psychiatry at the Univeristy of Oxford provided advice regarding the interview procedure for collection. ### 3.1 Existing Related Datasets There are a number of relevant existing datasets, however none satisfy the requirements of my research. Namely that they: are available, include psychological distress labels, and include source videos. I provide an overview of four existing datasets, two clinical datasets, a distress focused dataset using facial, auditory, and speech modalities, and one affective computing body gesture dataset. Each of these datasets, and their related work, provide useful insight for my data collection and model development research. #### 3.1.1 Clinical Joshi et al. describe a clinical dataset [33] for depression analysis that contains recordings of participants’ upper bodies and faces during a structured interview. It includes clinical assessment labels for each participant, including depression severity. This dataset was collected at the Black Dog Institute [6], a clinical research institute in Australia. Joshi et al. [34] describe a clinical dataset collected at the University of Pittsburgh containing full body recordings of participants in interviews, it uses Major Depressive Disorder (MDD) [4] diagnosis and Hamilton Rating Scale of Depression (HRSD) [29] as labels. The authors validate the use of full body features for depression prediction within this dataset. Given the clinical nature of these datasets they are not publicly available. #### 3.1.2 Audio-Visual Distress Gratch et al. introduce the Distress Analysis Interview Corpus (DAIC) dataset [26] containing audio-visual recordings of interviews with participants with varying levels of distress. Participants are assessed for depression, post traumatic stress disorder (PTSD), and anxiety based on self-evaluation questionnaires. The dataset provides audio data (and derivations such as transcripts) and extracted facial landmark data, though they do not provide source video data nor pose data. Source video recordings are rarely shared in distress focused datasets due to the sensitive of the data. The dataset contains four participant interview structures: face-to-face interviews with a human, teleconference interviews with a human interviewer, “Wizard-of-Oz” interviews with a virtual agent controlled by an unseen interviewer, and automated interviews with a fully autonomous virtual agent. This dataset informs my dataset collection via the labels it contains and their interview structures. The DAIC method has participants complete a set of questionnaires, then participants are interviewed and recorded, and finally the participant completes another set of questionnaires. Within the interview the interviewer asks a few neutral questions to build rapport, then asks questions related to the participant’s distress symptoms, and finally asks some neutral questions so the participant can relax before the interview finishes. #### 3.1.3 Body Gestures Palazzi et al. introduce a audio-visual dataset of dynamic conversations between different ethnicities annotated with prejudice scores. Videos in this dataset contain two participants interacting in an empty confined space. Participants move around the space throughout the session, enabling analysis of body language affected during conversation. This dataset highlights, and focuses on, the effect attributes of a counterpart, such as race, gender, and age, have on a person’s behaviour. In this dataset counterparts are explicitly non-uniform. Whereas, in my dataset the interviewer (i.e. counterpart for a participant) is the same for all interviewees. This is useful in controlling for the counterpart variable in human behaviour, supporting the isolation of correlations between distress and behaviour. Datasets such as this one are not useful for my current research question, however, as they provide source videos they present opportunities for future work investigating my features generalisability to other domains. ### 3.2 Method #### 3.2.1 Design This dataset is designed to enable investigation of the body modality for use in automatic detection of distress for early screening. This is a non-clinical dataset. Its source data is audio-visual recordings of conversational interviews. The recordings capture the whole body of participants to enable features based on a whole body modality. These interviews involve questions related to distress to elicit emotive responses from participants, however the responses to these questions are irrelevant to the core data. The interviews use a conversational style to best enable naturalistic gestures from participants. Labels are scored results from established self-evaluation questionnaires for assessing distress and personality traits, as well as demographic labels such as gender. The distress questionnaires are: the PHQ-8 [38, 37] for depression, GAD-7 [52] for anxiety, SSS-8 [24] for somatic symptoms, and the PSS [11] for perceived stress. Personality traits are measured using the Big Five Inventory [32]. #### 3.2.2 Participants ##### Recruitment I advertised for participants via University of Cambridge email lists, student social media groups, classified sections of websites, such as Gumtree [27], specific to the Cambridge area, and paper fliers posted around the University of Cambridge. Participants were directed to a website111helpresearch.org describing the study along with a participant registration form. The registration form captured demographic data and two self-evaluation psychological distress questionnaires. Demographic data captured includes gender, age, ethnicity, and nationality. Gender and age were required while ethnicity and nationality were not. The two psychological distress questionnaires were the PHQ-8 [38, 37] and GAD-7 [52]. ##### Selection In total 106 people registered to participate and 35 were invited to the face to face session. The participant population is balanced with regards to distress levels and gender222 Non-binary/other was given as an option in the registration form. A number of people registered with this option. However, none of those people met the distress level criteria and were thus not selected for an interview.. Distress level balancing aims to include participants at the extents of the distress spectrum such that there is a distinct difference between the high and low distress populations. Participant distress level is selected based on PHQ-8 and GAD-7 questionnaire responses such that participants are balanced between high (i.e. major or severe) and low (i.e. mild) distress. Of the invited participants, there are 18 with high distress and 17 with low distress. ##### Compensation Participants are compensated for their time with a £15 voucher. #### 3.2.3 Face to Face Session During the face to face session participants sign a research consent form outlining the interview process, complete a battery of five established psychology questionnaires evaluating distress levels and personality traits, are interviewed by a researcher, and finally sign a debrief research consent form that outlines the full purpose of the study. Participants are not aware of the focus of the research (i.e. body modality analysis) before the interview such their affectations are natural. To achieve the conversational interview dynamic the interviewer asks general questions regarding the participant’s life and further encourages the participant to elaborate. For example, the interviewer asks “can you tell me about one time in your life you were particularly happy?” and then asks follow up questions regarding the example the participant provides. The interview style and structure is inspired by the DAIC dataset. In developing the interview structure and questions I also drew on training documents provided by Peer2Peer Cambridge [44], an organisation dedicated to peer support for mental health which trains students in general counseling. So as to avoid influencing participants’ behaviour the interviewer remains as neutral as possible during the interview, while still responding naturally such that the participant is comfortable in engaging in the questions. Furthermore, to ensure neutrality the interviewer is explicitly not aware of the distress level of participants before the interview and has no prior relationships with any participant. ##### Technical Faults 16 interviews are interrupted due to technical faults333The camera disconnected.. Recordings that are interrupted are treated as multiple individual samples within the dataset (though they remain connected by their participant ID). ### 3.3 Preliminary Analysis The dataset contains a total of 35 interviewed participants with a total video duration of 7 hours 50 minutes and 8 seconds. Each participant provides responses to 5 questionnaires, including 2 responses to both the PHQ-8 and GAD-7 questionnaires as participants completed both during registration and the face-to-face session. Though significantly more people registered for participation, I include only those interviewed in this analysis. #### 3.3.1 Validation Criteria There are three primary criteria the dataset should satisfy with regards to label results: 1. 1. The psychological measures statistically match previous work and published norms444 Published norms are the standard values for questionnaire results as defined by the psychology literature. These aim to be representative of the general population. They thus provide a benchmark for other work (generally within psychology) to check smaller populations’ results against. (i.e. the distribution within the participant population is similar to that of the general population). 2. 2. There are no confounding correlations. For example, gender correlating highly to depression would indicate a poorly balanced dataset and would be confounding for depression analysis. 3. 3. The labels are well balanced to enable machine learning. As the common measure of similar distress detection research is depression I focus primarily on it for this validation. General statistics regarding the questionnaire and demographic results within the dataset are provided in Table 3.1. Covariance is presented as normalized covariance values, also known as the correlation coefficient. Label | Possible range | Max | Min | Mean | Median | Std. | Depression covariance ---|---|---|---|---|---|---|--- Distress | | | | | | | Depression | 0–24 | 19 | 0 | 7.43 | 8 | 5.87 | - Anxiety | 0–21 | 19 | 0 | 7.00 | 8 | 5.53 | 86.15% Perceived stress | 0–40 | 30 | 1 | 18.17 | 18 | 8.03 | 84.00% Somatic symptoms | 0–32 | 27 | 1 | 9.06 | 7 | 6.94 | 74.16% Personality | | | | | | | Extraversion | 0–32 | 31 | 3 | 16.37 | 17 | 6.33 | -30.49% Agreeableness | 0–36 | 34 | 12 | 25.67 | 26 | 5.60 | -42.21% Openness | 0–40 | 39 | 7 | 27.29 | 28 | 6.77 | 4.29% Neuroticism | 0–32 | 31 | 1 | 16.86 | 18 | 8.60 | 80.00% Conscientiousness | 0–36 | 36 | 10 | 21.46 | 21 | 6.87 | -46.41% Demographic | | | | | | | Gender | - | - | - | - | - | - | 9.47% Age | - | 52 | 18 | 25.40 | 22 | 9.1 | -11.09% Table 3.1: General statistics regarding questionnaire and demographic results within the dataset. The “Depression covariance” column is most important as it demonstrates the independence of variables with regards to depression (for example, it shows that age and gender are not confounding of depression). ##### Published Norms A comparison of the mean values for distress and personality measures between my dataset and the published norms is presented in Table 3.2. While there are differences, the measures are generally in line with the published norms. The dataset has slightly higher mean distress scores, though a substantially higher mean perceived stress score. Depression, extraversion, and neuroticism measures are particularly close to their published norms. While the dataset mean for agreeableness and openness are substantially greater than the published norms (over 10% over the technical range for those measures). Label | Dataset mean | Norm mean | Source ---|---|---|--- Distress | | | Depression | 7.43 | 6.63 | Ory et al. [42] Anxiety | 7.00 | 5.57 | Spitzer et al. [52] Perceived stress | 18.17 | 12.76 | Cohen et al. [11] Somatic symptoms | 9.06 | 12.92 | Gierk et al. [24] Personality | | | Extraversion | 16.37 | 16.36 | Srivastava et al. [53] Agreeableness | 25.67 | 18.64 | Srivastava et al. [53] Openness | 27.29 | 19.61 | Srivastava et al. [53] Neuroticism | 16.86 | 16.08 | Srivastava et al. [53] Conscientiousness | 21.46 | 18.14 | Srivastava et al. [53] Table 3.2: Comparison of the mean questionnaire values within my dataset to the published norms. This shows that the population distribution, with regards to these distress and personality measures, is generally in line with the broader population. ##### Confounding Correlations While the other distress measures (anxiety, perceived stress, and somatic stress) are strongly correlated with depression, the personality measures have below 50% covariance with the exception of neuroticism which has an 80% covariance. Furthermore, the demographic measures, gender and age, are negligibly correlated, with 9.47% and -11.09% covariance, respectively. This suggests that the labels are not confounding of each other. ##### Label Balance There are 17 participants below the mean depression result (7.43) and 18 participants above. The mean depression score of the group below the overall mean is 2.18 while the score for those above is 12.39. Ideally for machine learning the dataset’s distribution would include more participants at the severe depression end of the spectrum, though the present distribution still places the below group firmly in the “mild” category and the above group in the “major depression” category. There are 18 male and 17 female participants. As the gender covariance shows, the split on the depression measure and the split on gender are not the same participants (gender is balanced across the distress spectrum). #### 3.3.2 Difference from Registration Participants complete the PHQ-8 and GAD-7 questionnaires during registration and during the interview process. These questionnaires are temporal, specifically, they relate to the participant’s mental state in the past two weeks. Given this, some difference between registration and interview results is expected. With the exception of a small number of outliers, participants were generally consistent in self-evaluation between registration and interview. PHQ-8 responses have a mean difference of 0.89 while GAD-7 responses have a mean difference of 0.63. This supports the selection of participants based on temporal self-evaluation questionnaire results. #### 3.3.3 Interview Meta Statistics There is a total of 7 hours 50 minutes and 8 seconds of participant interview recordings, with a mean interview duration of 13 minutes and 25 seconds. The standard deviation of interview duration is 3 minutes and 20 seconds and the median interview duration is 13 minutes and 8 seconds. Depression score and interview duration are not correlated, with a covariance of 6.95%. Furthermore, interview duration is not correlated with any questionnaire result (i.e. distress or personality measure), all absolute covariance values are below 25%, which provides confidence in the reliability of the data. ### 3.4 Summary I have introduced a new audio-visual dataset containing recordings of conversational interviews between participants and a researcher, and annotated with established psychology self-evaluation questionnaires for depression, anxiety, somatic symptoms, perceived stress, and personality traits. This dataset involves 35 participants and 65 recordings (due to recording interruptions) with a total video duration of 7 hours 50 minutes and 8 seconds. There are a number of relevant existing datasets including clinical datasets which contain body gestures but are inaccessible beyond their home institute, distress datasets that contain facial expressions and speech modalities but no body gestures or source videos, and video datasets containing body gestures but lacking distress labels. While these datasets inform my dataset design and collection, no dataset I am aware of satisfies the criteria for research on body gesture modalities for predicting distress. An analysis of the questionnaire results in the dataset show they are aligned with psychology literature published norms, they are not confounded by factors such as gender or age, and have a useful and balanced distribution across the distress spectrum. ## Chapter 4 Method Figure 4.1: High level pipeline description of feature extraction process. Having collected the dataset, I now describe my methodology for extracting gesture meta features and applying them to automatic distress detection. To this end I define four core stages of my methodology, from video to prediction (as shown in Figure 4.1): 1. 1. Pose estimation \- extract pose estimation data from video recordings. 2. 2. Data preparation \- When preparing the data for learning I clean and smooth the extracted pose data. Pose data extracted from 2D video frames has a not- insignificant level of noise. 3. 3. Feature extraction \- Detect gestures, extract per-gesture features, and aggregate features across all gestures. I define gestures as sustained movement of some body part over a minimum period of time (this is elaborated on in Section 4.3.2). 4. 4. Classifier training \- a classifier is then trained to predict a target label (e.g. depression) based on these aggregate features. Classifier choice is explained in the evaluation, Chapter 5. ### 4.1 Pose Estimation Figure 4.2: Example of OpenPose estimation output of the participant interview position. Subject is not a study participant. Pose points are indicated by the green dots. Pose estimation extracts per-frame approximations of skeletal joint locations. This enables more accurate gesture analysis than direct pixel based approaches (such as STIPs per Joshi et al.’s method [36]). I process each video to extract per-frame skeletal pose data using OpenPose [8] by Cao et al. (OpenPose is discussed in more detail in Section 2.2) as it is the current state-of-the-art in pose estimation. I use the BODY_25 pose model provided with OpenPose111 OpenPose models provided at https://github.com/CMU-Perceptual-Computing-Lab/openpose.. As the name suggests, this model generates 25 pose points corresponding to a subject’s joints. OpenPose also extracts more detailed hand data that provides joint estimations for each joint in the hand. Figure 4.2 presents an example of extracted pose points. ### 4.2 Data Preparation I perform three data preparation steps: filtering, recovery, and smoothing. Filtering smooths dataset-level noise, recovery fixes outlier noise (where outliers are detection failures for specific pose points, but not the whole body), and smoothing reduces detection noise. Extracted pose estimation data has two primary forms of noise: individual frames, or short segments of frames, where detection of a person, or part of a person, is lost, and detection points moving around slightly even if the person is static. Manual review of a selection of detection loss cases shows no consistent cause (for both complete detection loss and partial loss). It appears to be the deep learning model failing inexplicably on some frames. #### 4.2.1 Filtering The absence of, or low representation of, gestures is relevant information for prediction tasks (simply put, if more activity is relevant then less activity must be too). However, the absence of gestures can also be due to a lack of opportunity to express gestures, such as when a sample is too short. These short samples lead to dataset-level sample noise that can hinder predictive models. Samples shorter than 1 minute are excluded as it is difficult to provide enough opportunity for gesture dynamics in less than 1 minute of video. 12 out of 65 samples within the dataset are shorter than 1 minute. #### 4.2.2 Detection Recovery Pose estimation within my dataset contains frames where certain pose points (e.g. an arm or a leg) are not detected. In these cases OpenPose returns $0$ for each pose point not detected. Manual review of a number of instances shows that the joint does not moved much, if at all, during the lost frames. However, the pose point “moving” to position $0$ causes significant noise in feature calculation. Therefore, I perform detection “recovery” to infer the position of the pose point in the missing frames, thus providing a smoother pose point movement. I recover the position by linearly interpolating the pose point’s position between the two closest detected frames temporally surrounding the lost frame(s). It is worth noting that this pose point detection failure is different to full detection failure where the whole participant is not detected within a frame. I do not attempt to recover such full failure cases as the failure is more serious and cause is ambiguous. I do not want to introduce stray data by “recovering” significantly incorrect data. Partial failures suggest a simple failure of OpenPose to extend through its body hierarchy. Since other pose points are still detected I am more confident that it is not a “legitimate” failure to do with participant position. Instead, full failure cases are treated as “separators” within a sample. Gesture detection occurs on either side but not across such separators. #### 4.2.3 Detection Smoothing To extract more accurate and relevant features I smooth the pose data by removing high frequency movement within pose points. Such high frequency movement of pose points is caused by OpenPose’s detection noise (i.e. the exact pose point might move back and forth by a few pixels each frame while its target joint is static). Thus smoothing is not smoothing human movement, but rather smoothing pose extraction noise. To smooth the data I apply a fourier transform filter to each dimension for each pose point. The smoothing steps are: 1. 1. Separate the data into $x$ and $y$ positions and smooth them (i.e. apply the following steps) independently. 2. 2. I convert the position sequence data using a fourier transform with a window length of $64$. 3. 3. Set the medium and high frequency values (all frequencies above the first five) to $0$. 4. 4. Invert the fourier transform on the updated fourier values to reconstruct the smoothed pose data. 5. 5. Concatenate the smoothed windows. ### 4.3 Feature Extraction I extract two types of features: aggregate features across the whole body and localised features for specific body parts, which I term “body localisations”. The localisations are: head, hands, legs, and feet. As the participants are seated the body trunk does not move substantially such that a gesture might be detected. The whole-body features include aggregations of the body localisation features as well as features incorporating the whole body. By including localised and non-localised features I can model the information provided by individual body parts and also the overall behaviour. ##### Gesture definition I define a gesture as a period of sustained movement within a body localisation. Multiple body localisations moving at the same time are treated as multiple individual, overlapping, gestures. A gesture is represented as a range of frames in which the target body localisation has sustained movement. For example, if a participant waves their hand while talking it would be a hand gesture. If they were to cross their legs it would register as a gesture in both the leg and feet localisations. ##### Whole Body Features * • Average frame movement \- the per-frame average movement of every tracked pose point (i.e. whole body). This is the only feature that is not based on detected gestures. * • Proportion of total movement occurring during a gesture \- the proportion of total movement (i.e. the whole body) that occurred while some body localisation was affecting a gesture. * • Average gesture surprise \- gesture surprise is calculated per-gesture as the elapsed proportional time since the previous gesture in the same localisation, or the start of the sample (proportional to length of the sample) for the first gesture. This overall feature averages the surprise value calculated for every gesture across all tracked localisations. I use the term “surprise” as the feature targets the effect on a gesture level basis, rather than the sample level. This is not a measure of how much of a sample no gesture is occurring as it is normalised on both the sample length and the number of gestures222 To illustrate further: if 2 gestures occurred within a sample such that 80% of the sample duration had no gesture occurring, the average gesture surprise would be $\dfrac{80}{2}=40$. Whereas, if there were 100 gestures, still with 80% of the sample with no gesture occurring, the average surprise be 0.8%, even though both samples had the same proportion without any gesture occurring. This matches the intuition that each gesture within 100 evenly spaced gestures would be unsurprising as they were regularly occurring, whereas the 2 evenly spaced gestures would be surprising because nothing was happening in between. . * • Average gesture movement standard deviation \- the standard deviation of per- frame movement within a gesture is averaged across all detected gestures. This is intended to indicate the consistency of movement intensity through a gesture. * • Number of gestures \- total number of detected gestures across all tracked localisations. ##### Localised Features Whole body and localised features are concatenated in the same feature vector. Localised features are included in the final feature vector for each localisation included in the vector. * • Average length of gestures \- the average number of frames per gesture. * • Number of gestures \- the total number of gestures, irrespective of gesture length. * • Average per-frame gesture movement \- the average movement across all gestures. * • Total movement in gestures \- the total amount of movement affected by the detected gestures. * • Average gesture surprise \- the average surprise across all gestures. ##### Normalisation All features are normalised such that the length of the sample does not affect the results. I normalise sum based features (e.g. gesture length, gesture count, total movement, etc) against the total number of frames in the sample and against the total number of gestures for gesture average values. For example, gesture surprise is normalised against the total number of frames and normalised a second time against the total number of gestures. ##### Absent Features If a feature has no inputs (such as when no gesture was detected within a body localisation) its value is set to $-1$ to enable models to incorporate the absence of movement in their predictions. #### 4.3.1 Body Localisation Gestures in different body localisations provide distinct information. Aggregating gestures from different localisations provides a general representation of this information, however, having features localised to specific body localisations provides further information, without significantly increasing the dimensionality. I define four localisations, hands, head, legs, and feet, based on specific pose estimation points. ###### Hands I use the finger tip points (including thumb) detected by OpenPose as the gesture detection points. This means wrist based gestures (e.g. rolling of a hand) are detected. Each hand is processed separately, that is, I detect gestures and calculate individual gestures independently in each hand, these gestures are then aggregated into a single body localisation feature vector. This makes the final features side agnostic. This ensures differences in dominant hand between participants will not affect the result. ###### Head While OpenPose includes face detection, providing detailed facial landmarks, I use the general head position points provided by the standard pose detection component. ###### Legs I represent legs using the knee pose points from OpenPose. As with hands, I process gestures in each leg independently and then aggregate to a single feature vector. ###### Feet Each foot is comprised of four pose points within OpenPose. Aggregation is the same as hands and legs. ###### Trunk I do not include a “trunk” localisation as there is minimal movement in the trunk, given the seated nature of the dataset interviews. Though some participants may lean forwards and backwards, these movements are not represented well within my data as the camera faces the participants directly such that forwards and backwards leaning would be towards the camera, thus requiring depth perception which is not included in my data. Side to side leaning is restricted by the arms of the participant’s chair. As such the localisations that are relatively free to move, those other than the trunk, are intuitively the most likely areas to provide predictive information. #### 4.3.2 Gestures ##### Gesture Detection To detect gestures within a set of pose points (e.g. finger tips, knees, feet, head, etc) I scan the activity of the target points for sustained periods of movement. The gesture detection step takes cleaned per-frame pose estimations and outputs a collection of ranges of non-overlapping frames that contain gestures within the localisation. First, the per-frame absolute movement delta is calculated for each pose point. The movement is then averaged across all localisation pose points. Movement deltas are $L^{2}$ distances. Formally, $\begin{split}M_{p,t}&=P_{p,t}-P_{p,t-1}\\\ F_{t}&=\dfrac{1}{|P|}\sum_{p\in P}M_{p,t}\end{split}$ (4.1) where $M_{p,t}$ is the amount of movement for pose point $p$ at time $t$, $P_{p,t}$ is the position value of pose point $p$ at time $t$, and $F_{t}$ is the averaged per-frame movement across all points. Second, I average the movement of each frame within a window such that a small number of frames do not have a disproportionate effect on the detection. That is, $\begin{split}W_{i}=\dfrac{1}{l}\sum_{t=i\times l}^{t<i\times(l+1)}F_{t}\end{split}$ (4.2) where $W_{i}$ is the windowed average at window index $i$, $l$ is the length of the window, and $F_{t}$ is the average movement at frame $t$, from Equation 4.1. In this dissertation I use $l=10$, i.e. a second of movement is represented by 3 windows, this is experimentally chosen. Third, the detector iterates through the averaged windows until a window with an average movement above a threshold is found. The first frame of this initial window is considered the beginning of the gesture. The gesture continues until $n$ consecutive windows (I use 3, i.e. 30 frames, as an approximate of a second) are found below the movement threshold. The last frame of the final window above the movement threshold is considered the end of the gesture. This is provided more formally in Algorithm 1. Algorithm 1 Gesture detection 1:$n\leftarrow\textit{number of consecutive windows}\text{ below threshold for end}$ 2:$m\leftarrow\text{threshold of window }\textit{movement}$ 3:$l\leftarrow\text{minimum }\textit{number of windows}\text{ for a gesture}$ 4:$gestures\leftarrow EmptyList$ 5:$start\leftarrow NULL$ 6:$belowThresholdCount\leftarrow 0$ 7: 8:for each window movement $W$ at index $i$ do 9: if $W\geq m$ then 10: // Start the gesture on the first window that exceeds the movement threshold. 11: if $start\equiv NULL$ then 12: $start\leftarrow i$ 13: $belowThresholdCount\leftarrow 0$ 14: 15: else if $start\neq NULL$ then 16: $belowThresholdCount\leftarrow belowThresholdCount+1$ 17: // A gesture is completed after $n$ consecutive windows below the movement threshold. 18: if $belowThresholdCount\equiv n$ then 19: // The end of a gesture is the final window that exceeds the threshold. 20: $end\leftarrow i-n$ 21: if $end-start\geq l$ then 22: $gestures$ append $[start,\ end]$ 23: // Reset to find the next gesture. 24: $start\leftarrow NULL$ 25: $belowThresholdCount\leftarrow 0$ 26: 27: 28:// Close the final gesture. 29:if $start\neq NULL$ then 30: $end\leftarrow finalIndex$ 31: if $end-start\geq l$ then 32: $gestures$ append $[start,\ end]$ Having detected gestures in each body localisation I extract the features described above from each gesture and aggregate them to form the final feature vector. ### 4.4 Feature Space Search I have described a collection of novel features whose individual and mutual predictive value is as yet unknown. Some features are potentially unhelpful to predictive models. Therefore, distinguishing useful features from unhelpful features is a key operation to enable development of accurate models, and thus validate these features. To this end, I perform an exhaustive search of the feature space to identify the combination of features with the best performance. Alghowinem et al. [1] demonstrate the benefits of a variable feature set, achieving accuracy improvements of up to 10%. Their classification method is based on SVMs and the dimensionality reduction enabled by feature filtering is significant. They use a statistical T-threshold based approach to feature selection. However, Dibeklioglu et al. [16] argue that basing selection on optimization of mutual information will achieve better results than individual information based selection. I follow this mutual information approach and thus feature selection is based on the results achievable given a combination of features, rather than features’ individual relevance. I define few enough features such that a brute force feature combination space search (i.e. test every permutation) is viable. As each feature can be included or excluded the space has $2^{n}$ permutations, where $n$ is the number of features being searched. I iterate over every permutation and perform three fold cross validation using the combination of features, the permutation with the greatest average cross validation F1 score is taken as the best permutation, which enables testing and evaluating the effectiveness of the proposed features. ### 4.5 Summary In this chapter I have described my methodology for extracting gesture meta features from videos. The four core stages are: pose estimation, data preparation, feature extraction, and finally classifier training. I use OpenPose to extract pose data per-frame as it is the state-of-the-art in pose estimation. I then filter samples and perform two operations to reduce noise within the remaining samples: recovery of partial detection failures and smoothing of high frequency detection noise. Features are extracted by detecting individual gestures, calculating per-gesture features such as speed, and aggregating per-gesture features within their localisations and over the whole body. Classifier choice is an evaluation detail and discussed in the next chapter. ## Chapter 5 Evaluation Having described my gesture meta features, I now evaluate their predictive capability on the dataset introduced in Chapter 3. Section 5.1 describes implementation details of my evaluations. Section 5.2 defines a simple baseline using a single feature. In Section 5.3 I evaluate the features’ potential using the exhaustive feature search described in Section 4.4. Section 5.4 investigates the influence of different body localisations on the features’ effectiveness. I then demonstrate the features’ generalisability beyond the primary depression detection task in Section 5.5. In Section 5.6 I compare the features’ automatic depression detection performance with an existing body-modality method and a straightforward face-modality method. In Section 5.7 I experiment with a multi-modal classifier that combines the gesture meta features with face-modality features. Finally, I summarize the results in Section 5.8. ### 5.1 Implementation details Before presenting evaluation results, I outline the details of the evaluation setup. #### Evaluation I perform evaluations using a three fold cross validation. The training and test samples are split in a participant-independent manner and use stratified folding to balance them with regards to labels. Cross validating with more folds leads to fewer test samples per fold. Given the small size of the dataset, this can lead to more erratic performance (i.e. more extremes in cross validation results). I assess results based on the average cross validation F1 score and the standard deviation between fold F1 results. Average F1 provides an objective measure of model quality. Whilst the standard deviation provides an indicator of the consistency of the model. #### Classifier Models I evaluate four types of classifiers on all tasks: * • A linear regression based classifier (denoted as lin) using a classification threshold of 0.5. * • A logistic regression based classifier (denoted as log) using the L-BFGS solver. * • A linear kernel SVM (denoted as svm) with balanced class weighting (without balancing the class weightings the classifier consistently chooses to predict a single class on every fold). * • A random forest (denoted as rf) with 40 trees, feature bootstrapping, a minimum of 3 samples per leaf, a maximum depth of 5, balanced class weighting, and exposing 80% of features per node. These parameters were chosen experimentally. #### Labels I only evaluate binary classification tasks (i.e. high vs. low depression score) within this dissertation. However, the dataset contains continuous values for distress and personality measures, thus a constant threshold is required for each label (similar to the participation selection criteria discussed in Section 3.2.2). These thresholds are chosen such that the resulting binary classes are as balanced as possible. Given the small size of the dataset, balancing the classes is important to classifier training. Per- label thresholds are reported in Table 5.1. Label | Threshold | # Participants above | # Participants below ---|---|---|--- Distress | | | Depression | 7 | 18 | 17 Anxiety | 7 | 18 | 17 Perceived stress | 17 | 18 | 17 Somatic stress | 6 | 19 | 16 Personality | | | Neuroticism | 17 | 18 | 17 Extraversion | 16 | 18 | 17 Agreeableness | 25 | 20 | 15 Conscientiousness | 20 | 20 | 15 Openness | 27 | 19 | 16 Table 5.1: Binary classification thresholds for distress and personality labels. A larger dataset, or an expansion of this dataset, would enable regression models. Regression models would also naturally provide a multi-class classification solution as the original questionnaire scoring categories could be applied to the regressed predictions. Though I evaluate the features’ predictive capability against multiple labels, my primary focus is on depression detection, as this is the area that most of the related work discusses. An evaluation the features’ capability on other labels is presented in Section 5.5. #### Localisations Though I evaluate four localisation types: head, hands, legs, and feet, the feet localisation has a negative effect on performance, shown in Section 5.4. As such, all other sections use the three useful localisations: head, hands, and legs, in their evaluations. #### Feature Reporting Notation To concisely report features used by different classification models I describe a brief notation for enumerating features. The notation defines tokens for localisations, feature type, and how the lin model interprets the feature. The structure is “[localisation]-[feature type][linear polarity]”. Localisation and feature type token mappings are provided in Table 5.2. Localisation | Token ---|--- Overall | O Hands | Hn Head | He Legs | L Feet | F | Feature | Token ---|---|--- Overall | Average frame movement | FM Proportion of total movement occurring during a gesture | GM Average gesture surprise | GS Average gesture movement standard deviation | GD Number of gestures | GC Localised | Average length of gesture | GL Average per-frame gesture movement | GA Total movement in gestures | GT Average gesture surprise | GS Number of gestures | GC Table 5.2: Feature notation tokens. I define an indicator of how a model interprets a feature based on its contribution to a positive classification. A greater value (e.g. more activity) contributing to a positive classification is denoted by “+”. Conversely a greater value contributing to a negative classification is denoted by “$\neg$”. A value which has a negligible effect (defined as a linear model applying a near-zero coefficient) is denoted by “/”. Finally, if the lin classifiers for each cross-validation fold are inconsistent in usage of the feature the “?” indicator is used. For example, F-GS$\neg$ would denote that a greater amount of surprise in feet gestures indicates a negative classification. ### 5.2 Baseline As a baseline I evaluate the use of the only non-gesture based feature, average per frame movement (O-FM). Results are given in Table 5.3. Model | F1 avg | F1 std ---|---|--- lin | 34.43% | 11.45% log | 34.43% | 11.45% svm | 33.82% | 10.73% rf | 64.29% | 5.83% Table 5.3: F1 aggregate scores for models using the baseline feature on the depression detection task. The rf model achieves the best performance. In this evaluation only one feature is used, this means that the feature is available to every decision node in the random forest, enabling strong overfitting. While the rf model achieves the best results, it is inconsistent across its folds. Its 3 F1 scores are: 70.06%, 60.00%, and 47.06%. However, it does suggest that the movement feature alone is valuable for prediction. Though, the worse than random results achieved by the other models suggest that movement is not a linear indicator. It is possible, indeed likely, that the rf model’s result is reflective an overfitting of the dataset. ### 5.3 Feature Search I evaluate the effect of the feature space search to identify the best feature combination. #### All Features Table 5.4 presents results for each model when provided with the full feature vector unmodified. Given all features the lin, log, and svm models all improve on their baselines, while the rf model is worse than its baseline. The reduction in performance by the rf model can be attributed to the overfitting ability of random forests. This ability is especially prevalent with single feature problems as the feature must be made available to every node within the forest, thus enabling direct fitting of multiple decision trees to the dataset. The lin model achieves significantly better-than-random performance, indicating that the features have some linear predictive capability. Model | F1 avg | F1 std ---|---|--- lin | 66.81% | 8.89% log | 56.55% | 5.12% svm | 62.70% | 9.19% rf | 41.88% | 6.04% Table 5.4: Classifier F1 performance for detecting depression within my dataset given all features. There are two primary interest points in this table: 1) the lin classifier performs best given all features, suggesting that the features provide some linear information, and 2) the rf classifier performs significantly worse than the one feature baseline in Table 5.3, further suggesting the rf classifier is overfitting the one feature baseline. #### Searched Features I perform an exhaustive feature space search to identify the best combination of features (determined by the average cross-validation F1 score). This provides two important outcomes: the best accuracy possible for a given model using these features within my dataset and the most relevant features to predict depression within my dataset. A good accuracy from the first outcome validates the features I have developed. The second outcome then provides a basis for further analysis of these features and their relation to depression within my dataset. This search requires a model fast enough to train that the full space can be searched in a viable amount of time and, preferably, a model that is interpretable. For these reasons I use the lin model to search the feature space, and then evaluate the best feature combination with the other classifier types. Figure 5.1: Comparison of baseline results to feature searched results. This shows that performing a full feature search enables the lin classifier to outperform the other classifiers and feature combinations. Also of interest is the reduction in performance by the rf model when provided with more features, suggesting that it may be overfitting when provided with the single baseline feature. The specific feature combination resulting from the search is discussed below in Chosen Features. Model | F1 avg | F1 std ---|---|--- lin | 82.70% | 8.95% log | 54.17% | 5.89% svm | 56.55% | 5.12% rf | 53.18% | 14.96% Table 5.5: Performance of the best feature combination as determined by an exhaustive feature space search using the lin classifier to detect depression. This demonstrates the power, and the linearity, of the gesture meta features as the lin classifier is able to achieve a high F1 score. ##### Feature Search Improves Best Performance The best performance when applying the feature search, again from the lin classifier, is significantly better, 82.70%, than the classifier’s baseline of 34.43% and all features baseline of 66.81%. This demonstrates the importance of reducing dimensionality and feature confusion, especially when using relatively direct methods such as linear regression. Results are provided in Table 5.5, a comparison of these results to the baseline results is presented in Figure 5.1. ##### Chosen Features The full feature combination is: {O-FM?, O-GM+, O-GC?, Hn-GC?, Hn-GT$\neg$, Hn-GS?, He-GL?, He-GC?, He-GA+, He-GT$\neg$, He-GS?, L-GL$\neg$, L-GC?, L-GT+}. Analysing this feature set, we can derive some intuition as to information indicative of depression. The overall (O-*) features suggest that the number of gestures (O-GC) and the amount of movement within those gestures (O-GM) is relevant to depression. The O-GM+ token suggests that more movement within gestures relative to all other movement is indicative of depression. The localised features suggest that the length of gestures (*-GL) has a correlation with depression, however, this correlation differs between localisations. The head localisation is ambiguous as to whether shorter or longer gestures (He-GL?) is indicative of depression. Whilst longer gestures in the legs localisation (L-GL$\neg$) is indicative of less depression. Within this model, less total movement of the hands (Hn-GT$\neg$) is indicative of distress. ##### Negative Performance Impact on Other Models and Overfitting The identified feature set is chosen using the lin model, so it is unsurprising it has a greater improvement than any other model. While the log classifier’s performance does not change much, the svm and rf classifiers have reduced performance compared to their all features and one feature baselines, respectively. There are two aspects to consider here: the value of pre-model filtering to each model and the potential for each model to overfit. Focusing on the rf classifier as it has a more distinct reduction in performance; random forests have inbuilt feature discretion, so the pre-model filtering of features does not have as great a complexity reducing effect as it does on the other models. My hyper-parameters give the rf model a relatively large selection of features (80%) per node, thus it should generally filter, to some extent, those naturally unhelpful features. Random forests, as decision tree ensemble methods, have a propensity for overfitting data. By reducing the number of features available I reduce the available surface for overfitting. Moreover, when only using one feature, as in the baseline, every decision node in the random forest has access to the feature, thus enabling particularly strong overfitting of a small dataset. #### Dimensionality Compression I do not perform any dimensionality compression operations in my presented evaluations. However, I have experimented with Principle Component Analysis (PCA) both independently and in combination with a feature search. Neither approach achieved especially interesting results, all were worse than when not applying PCA. Given this, and the already relatively low number of dimensions, I do not see it as a critical path of investigation for this dissertation. ### 5.4 Body Localisations Not all localisations are necessarily beneficial. Identifying which localisations are helpful is made more difficult by the localisations’ interactions within a classifier and the effect they have on the overall features. Though a localisation may generate features that are chosen using feature search, they may reduce overall accuracy by obfuscating predictive information in the overall features. Given this, I experiment with localisations included individually and in varying combinations. I also provide an example of a localisation, feet, that negatively effects performance when included, even when all other localisations are also included (and are thus providing the same predictive information). A comparison of the best F1 scores for localisation combinations is presented in Figure 5.2. Figure 5.2: Comparison of the best F1 average scores from localisation combinations using the lin classifier using all features. The most interesting results are the bottom four (vertically) localisation results: head - legs, hands - head - legs, feet - head - legs, and hands - feet - head - legs. Specifically, the feet localisation impairs the performance of the localisation combinations when included. This trend is also seen in feet - legs and feet - head. ##### Localisation Inclusion Clearly not all of the features generated per localisation provide useful information. Inclusion of more localisations, and thus a larger feature space, does not guarantee a better optimal result is available within the space. As the overall features (those aggregated across all localisations) are effected by each localisation, the quality of the feature space can be degraded with the inclusion of localisations. For example, this occurs regularly in Figure 5.2 when including the feet localisation. In particular, the best base configuration, head - legs using the lin classifier, achieves a 70.88% F1 score, when the feet localisation is included this drops to 49.84%. I have not identified any psychology literature, or clear intuition, as to why the feet localisation hinders performance. I see three probable explanations: 1) some literature does support this and I have simply not identified the literature, 2) this is accurately representing that feet movement meta information does not distinctively change with distress, but no literature has explicitly investigated this, and 3) this is simply a attribute of the dataset that is not reflective of any broader trend, either due to the dataset size or the nature of the interview dynamic. ##### Best Base Performance Configuration Though the head - legs configuration achieves the best performance when all features are used, it does not achieve the best performance when features are chosen based on an exhaustive search. While my primary configuration, hands - head - legs, achieves 82.70% F1 average, the head - legs configuration achieves 80.53%, results are presented in Table 5.6. Model | F1 avg | F1 std ---|---|--- lin | 80.53% | 4.04% log | 59.52% | 8.91% svm | 65.40% | 8.40% rf | 51.32% | 5.76% Table 5.6: Performance of models when using features chosen via exhaustive search with source features from the head - legs configuration. This configuration achieves close to the best performance of the standard configuration (Table 5.5). The lin classifier also has more consistent performance across cross validation folds than it does on the standard configuration, with a standard deviation of 4.04% compared to 8.95%. However, these results do not clearly define which configuration is generally better as the differences are quite minor. ### 5.5 Generalisability I have demonstrated the gesture meta features’ predictive value with regards to depression detection. I now evaluate their predictive capability for other labels including participants’ gender, personality measures, anxiety, perceived stress, and somatic stress. I apply the best feature combination identified in Section 5.3 to each of the labels, presented in Table 5.7. I also perform a feature space search for each label, using the same process as Section 5.3, to provide greater insight into the effect of features and their reliability across labels, presented in Table 5.8. A comparison is shown in Figure 5.3. Consistent identification of features across uncorrelated labels reinforces the hypothesis that they provide predictive information beyond a specific label and dataset (i.e. are less likely to be overfitting). Figure 5.3: Comparison of F1 average scores when using the optimal feature combination for the depression label vs. the optimal feature combination for each label. Each label has a significant performance improvement when using its optimal feature combination. Label | lin | log | svm | rf ---|---|---|---|--- | F1 avg | F1 std | F1 avg | F1 std | F1 avg | F1 std | F1 avg | F1 std Distress | | | | | | | | Depression | 82.70% | 8.95% | 54.17% | 5.89% | 56.55% | 5.12% | 53.18% | 14.96% Anxiety | 47.18% | 23.21% | 38.33% | 19.20% | 30.18% | 4.02% | 53.26% | 16.58% Perceived stress | 47.18% | 23.21% | 38.33% | 19.20% | 30.18% | 4.02% | 53.26% | 16.58% Somatic stress | 42.74% | 30.29% | 51.68% | 4.01% | 44.44% | 6.29% | 54.89% | 8.37% Personality | | | | | | | | Neuroticism | 62.08% | 0.76% | 31.71% | 10.89% | 33.36% | 12.59% | 38.30% | 8.30% Extraversion | 51.04% | 14.06% | 67.95% | 17.33% | 65.14% | 12.98% | 52.94% | 20.83% Agreeableness | 69.48% | 5.97% | 73.61% | 2.13% | 67.98% | 3.92% | 56.96% | 20.79% Conscientiousness | 71.28% | 6.53% | 72.95% | 6.93% | 78.77% | 6.24% | 79.19% | 6.11% Openness | 49.64% | 14.94% | 65.08% | 5.94% | 64.46% | 5.31% | 61.47% | 8.64% Demographic | | | | | | | | Gender | 34.26% | 18.18% | 52.96% | 2.28% | 40.63% | 18.42% | 63.14% | 5.29% Table 5.7: Performance of models using the feature combination identified for the depression label (Section 5.3) for predicting a variety of labels. The best results per-label are bolded. These results suggest that the depression chosen feature combination does not generalise particularly well. These results are also surprising as the distress labels (anxiety, perceived stress, and somatic stress), which are correlated to the depression label, perform poorly, whilst uncorrelated labels (such as agreeableness, openness, and gender) perform better. This may be due to overfitting of feature profiles to labels (i.e. the optimal features for the label) or truly distinct feature profiles between the correlated distress labels, though the former appears more probable. Label | lin | log | svm | rf ---|---|---|---|--- | F1 avg | F1 std | F1 avg | F1 std | F1 avg | F1 std | F1 avg | F1 std Distress | | | | | | | | Depression | 82.70% | 8.95% | 54.17% | 5.89% | 56.55% | 5.12% | 53.18% | 14.96% Anxiety | 88.46% | 9.53% | 46.14% | 9.16% | 54.33% | 12.32% | 52.94% | 12.85% Perceived stress | 88.46% | 9.53% | 46.14% | 9.16% | 54.33% | 12.32% | 52.94% | 12.85% Somatic stress | 86.37% | 6.45% | 58.44% | 8.85% | 68.08% | 16.97% | 49.48% | 11.38% Personality | | | | | | | | Neuroticism | 76.39% | 8.56% | 38.85% | 5.15% | 48.25% | 4.47% | 40.78% | 13.59% Extraversion | 84.04% | 10.24% | 74.61% | 10.34% | 73.39% | 12.75% | 58.59% | 31.33% Agreeableness | 82.70% | 5.77% | 69.14% | 6.67% | 50.43% | 14.85% | 67.97% | 1.85% Conscientiousness | 88.72% | 4.25% | 73.14% | 3.13% | 69.90% | 7.84% | 78.91% | 2.80% Openness | 85.01% | 6.62% | 64.32% | 5.06% | 68.77% | 8.84% | 60.32% | 7.36% Demographic | | | | | | | | Gender | 81.69% | 5.32% | 68.11% | 10.22% | 76.71% | 5.10% | 64.81% | 11.42% Table 5.8: Performance of models for predicting a variety of labels when using features identified via a feature space search specific to the label. This demonstrates the performance improvement, compared to Table 5.7, achieved via label specific feature combinations. The best results for each label are bolded. #### Results The depression feature combination does not generalise well to the other distress measures for the lin model, though it achieves 62–71% F1 scores for neuroticism, agreeableness, and conscientiousness. Interestingly, the other labels’ best results using the depression combination are above 60% F1, with the exception of the other distress measures, which only achieve a best of 53–54% F1. This is surprising as the distress labels are strongly correlated to the depression label. However, when features are chosen on a per-label basis the results improve significantly. All labels111 Though perceived stress and anxiety measures have a covariance of 82.98% within my dataset, once reduced to binary classifications they are equivalent (i.e. 100% correlation). Thus results for both labels are the same. achieve 76%+ (all but one are above 80%) average F1 score with their best classifier. The best classifier for all labels is the lin classifier, which is to be expected given it is the search classifier and previous evaluation has shown the features provide useful linear information. ##### Fitting features to labels There are two potential explanations for the substantial improvement in performance when using label specific feature combinations: each label could legitimately have a unique feature profile through which its distinctive attributes are expressed or the labels could be experiencing a level of overfitting due to the small size of the dataset. While it is likely to be a combination of the two reasons, labels that are relatively uncorrelated with depression are still able to achieve good performance using the depression chosen features, suggesting it is not just overfitting. For example, agreeableness, extraversion, and conscientiousness all have covariances of less than 50% with depression, yet they achieve 72.61%, 67.95%, and 79.19% average F1 scores, respectively, with the depression features. Openness which has a 4.29% covariance with depression achieves 65.08% with the depression chosen features. Furthermore, as Table 5.9 shows, the openness feature combination shares only 6 out of its 9 features with the depression combination, whilst it also excludes 8 out of 14 of the depression combination’s features. Therefore, it can be reasonably suggested that the features do generalise, acknowledging that fitting the features directly to a label achieves the best performance, as would be expected with all prediction tasks. ##### Cross classifier generalisability Within the depression feature combination results, the lin, log, and rf models all achieve the best results on at least 2 labels each. The svm classifier performs close to the best on many labels, such as conscientiousness where it achieves 78.77% compared to the best result of 79.19%. The svm model performs better when classifying conscientiousness, extraversion, and agreeableness, using the depression feature combination, than it does when predicting the depression label. It is also better at transferring the depression feature combination to other labels than the lin model that the feature combination was chosen with. Its performance improves on most labels when using the label targeted feature sets, such as on gender where it achieves 76.71% and somatic stress with 68.08% F1, whilst with the depression feature combination its F1 result was less than 50% for both. Interestingly, whilst the svm model performed particularly well on agreeableness using the depression feature combination, it performed worse when using the label targeted feature combination. ##### Personality Within the results for the depression feature combination the conscientiousness label achieves the best average F1 score across all classifiers, 75.55%, with an average standard deviation of 6.45%. Unlike the distress evaluation questionnaires, the BFI (personality) questionnaire is designed such that responses should remain relatively consistent over time. Future work could investigate whether the features are able to consistently predict personality measures over time for a participant. Do the features remain predictive of personality as a person’s temporal distress measures change? ##### Chosen Features Table 5.9 presents the feature sets resulting from the per-label feature searches. There are some consistencies and intuitive inversions within the feature sets. For example, the head localised average gesture movement (He-GA) is relevant for many of the labels, though in some it is inconsistent whether greater or lesser is indicative of the label (instances of inconsistencies are denoted as He-GA? within Table 5.9). The feature usage is inverted between neuroticism, where a faster average speed is indicative of a positive classification, and extraversion and agreeableness, where a slower average speed is indicative of positive classification. Furthermore, as stated above, features that are chosen by uncorrelated labels support the hypothesis that these features are providing useful information. For example, of the 7 features the gender label chooses, 5 are also chosen by the anxiety label, including hand total gesture movement (Hn-GT), hand gesture surprise (Hn-GS), and leg gesture surprise (L-GS). Whilst gender and anxiety have a covariance of 7.23% within the dataset. Label | # Feat. | Localizations ---|---|--- | | Overall | Hands | Head | Legs Distress | | | | | Depression | 14 | FM?, GM+, GC? | GC?, GT$\neg$, GS? | GL?, GC?, GA+, GT$\neg$, GS? | GL$\neg$, GC?, GT+ Anxiety | 13 | FM+, GM+, GD$\neg$, GC$\neg$ | GT?, GS+ | GL+, GA?, GT$\neg$ | GL$\neg$, GA+, GT+, GS+ Perceived stress | 13 | FM+, GM+, GD$\neg$, GC$\neg$ | GT?, GS+ | GL+, GA?, GT$\neg$ | GL$\neg$, GA+, GT+, GS+ Somatic stress | 9 | GD$\neg$ | GL?, GT?, GS+ | GC$\neg$, GA+ | GL$\neg$, GC$\neg$, GT+ Personality | | | | | Neuroticism | 14 | GM+, GS?, GC$\neg$ | GC+, GA$\neg$ | GL?, GC+, GA+, GT$\neg$ | GL$\neg$, GC+, GA?, GT+, GS+ Extraversion | 10 | FM+, GC+ | GL$\neg$, GC+, GA+ | GC?, GA$\neg$ | GL+, GT$\neg$, GS$\neg$ Agreeableness | 9 | GS+, GC+ | GC$\neg$, GA$\neg$ | GA$\neg$, GS+ | GC$\neg$, GA?, GT? Conscientiousness | 8 | FM?, GM$\neg$, GS? | GL+, GS? | GC+ | GL?, GC? Openness | 9 | GM+, GD+ | GL$\neg$, GC? | GC$\neg$, GA?, GS? | GA?, GT? Demographic | | | | | Gender | 7 | GS+, GD$\neg$ | GC+, GT+, GS$\neg$ | GA+ | GS$\neg$ Table 5.9: Features chosen for each label when features are searched specifically for the label. “Positive classification” for the gender label is female (i.e. He-GA+ indicates a higher average speed of head gestures is indicative of the participant being female). Refer to Table 5.2 for the full notation mapping. ##### Movement I represent four measures of movement: average movement per-frame (i.e. speed), standard deviation of movement (i.e. consistency of speed), total movement both over an entire sample and over individual gestures, and proportion of total movement that occurs during gestures (i.e. how much/little movement is there outside of gestures). The amount and speed of movement is intuitively correlated to distress and personality. This intuition is validated as these movement representations are included in a variety of feature sets resulting from feature space searches, across multiple labels. Indeed, Table 5.9 shows that the speed of the head during a gesture (i.e. He- GA) is correlated with positive classifications for labels including neuroticism, depression, somatic stress, while it is inversely correlated with other classifications including extraversion and agreeableness. ##### Surprise One of the less initially intuitive features is “gesture surprise”. This feature represents the distance between a gesture and the previous gesture (or beginning of the sample) as a proportion of the total length of the sample. This per-gesture value is then averaged across all gestures, this means it is not a measure of the proportion of the sample when no gesture occurs (as discussed in Section 4.3). The intuition is to represent whether the participant’s gestures occur regularly or after a period of stillness (i.e. how “surprising” is the gesture). Gesture surprise is included in every feature combination resulting from a feature search of a label, shown in Table 5.9, suggesting it provides useful information. ##### Linearity The features are, in general, linearly correlated with different labels, as shown by the effectiveness of the lin classifier, which has been the focus of the feature development and evaluation. Using this classifier provides interpretability of the features and integrity that the features are providing real, useful, information, not simply a platform for overfitting. However, not all features are consistently linear with regards to certain labels. Some features are learnt as correlated and inversely correlated on the same label in different cross validation rounds. This inconsistency denoted with the usage indicator “?”. For example, while the hands localised gesture surprise feature (Hn-GS) is linearly correlated with somatic stress, anxiety, perceived stress, and gender, it is inconsistently correlated with conscientiousness and depression. These inconsistencies within feature correlation are the exception, not the rule. Indeed, all of the features that experience inconsistency on a label are shown to be linearly correlated to another label, or linearly correlated within another localisation. ### 5.6 Comparison with Related Work I evaluate two comparison methods: a linear SVM model based on Bag-of-Words features of FACS [21] Action Units (AUs), and the Bag-of-Body Dynamics (BoB) method presented by Joshi et al. [36]. The first method provides a comparison of modality (facial-modality) within my dataset and a cross-dataset benchmark with the closest related dataset, DAIC. The second method provides a comparison to an existing body-modality method that uses an automatic expression categorization approach to predict depression. #### AU SVM This method uses basic facial expression analysis to predict a binary depression label. I implement a simple linear kernel SVM based on a Bag-of- Words (BoW) feature vector of FACS AUs to predict the binary depression label in my dataset and, separately, a binary depression label within DAIC. As with my dataset, I apply a threshold to DAIC’s PHQ-8 score labels such that the dataset is as balanced as possible. The resulting threshold is 5 (i.e. 6 and above is considered “depressed”), this results in 68 of 139 samples classed as “depressed”. This is a lower threshold than used in my dataset (which is 7). Within the BoW feature vector, binary AUs are counted for each frame they are detected in, whilst intensity measured AUs (i.e. continuous value measures) are summed across all frames. The sums and counts are then normalised against the number of frames in which the face was successfully detected. The DAIC dataset provides per-frame predictions for 20 AUs. I use OpenFace [5] to extract per-frame AUs from my dataset. OpenFace predicts 35 AUs per-frame, substantially more than provided by DAIC. While this method does not use the gesture features, I still only evaluate it on the samples that pass the gesture filtering step (Section 4.2.1) so that the evaluation is consistent with previous results. #### Bag-of-Body Dynamics Joshi et al. [36] use a BoW approach to predict depression in clinically assessed participants from their BlackDog dataset. The BoW feature vector is comprised of categorized body expressions based on K-Means clustering of histograms surrounding STIPs within a video of the subject. Their method is: 1. 1. Extract Space-Time Interest Points (STIPs). 2. 2. Calculate histograms of gradients (HoG) and optic flow (HoF) around the STIPs. 3. 3. Cluster the histograms per-sample using K-Means, the resulting cluster centres define the sample’s “key interest points” (KIPs). 4. 4. Cluster KIPs across all samples to define a BoB codebook, also using K-Means. 5. 5. Generate a BoB feature vector for each sample by fitting its KIPs to the codebook. 6. 6. Finally, apply a non-linear SVM to the resulting feature vectors to predict depression. They use a radial basis function kernel (RBF) for their SVM. ###### Differences in Method In the original paper Joshi et al. experiment with multiple KIP counts, codebook sizes, and perform an extensive grid search for their SVM parameters. I use a single KIP count and codebook size, 1,000 and 500, which they also experiment with. While I perform a grid search on the RBF parameters, it may not be as extensive as their search. I also exclude samples that generate fewer STIPs than the KIP count. This results in 45 out of 65 samples being included. #### Results The results for each model are presented in Table 5.10, a comparison of methods applied to my dataset is shown in Figure 5.4. I compare the best depression detection model from my previous evaluations (i.e. the feature combination identified in Section 5.3 with the lin classifier), and a lin classifier using all gesture meta features, with two BoB based SVMs and a FACS AUs based linear SVM for predicting depression. The lin model performs best, achieving an 82.70% F1 average with the optimal feature combination and 66.81% with all features, compared to the next best at 63.71% from the FACS AUs SVM model and 61.10% from the BoB RBF SVM model. Results for all models are based on the same three fold cross-validation mean F1 as previously. Figure 5.4: Comparison of method F1 scores for predicting the depression label in my dataset. This shows the lin classifier using the optimal depression feature combination identified in Section 5.3 (lin (opt.)) outperforming the comparison methods. The lin classifier using all of the gesture meta features (lin (all)) also outperforms the comparison methods, though not by as much. There are two caveats to this comparison: the features for the lin (opt.) classifier have been fitted to the label and the BoB results are for a reduced dataset (45 samples compared to 53 for the lin and AUs models) as not all of the samples produced enough STIPs for the methods to operate appropriately. The FACS AUs SVM model performs better on my dataset than the DAIC dataset. This is almost certainly due to the difference in quantity and quality of available FACS AU features. The DAIC dataset provides 20 AU predictions per frame while my dataset provides 35 AU predictions. Model | F1 avg | F1 std ---|---|--- lin | Optimal feature combination | 82.70% | 8.95% All features baseline | 66.81% | 8.89% BoB | RBF kernel | 61.10% | 6.09% Linear kernel | 60.13% | 9.24% FACS AUs | My dataset | 63.71% | 8.02% DAIC dataset | 56.53% | 3.10% Table 5.10: Comparison of my lin model, FACS AUs based SVM models, and the BoB SVM models for predicting depression as defined by the PHQ-8 questionnaire, on my dataset. The FACS AUs DAIC model is the exception, its results are for the DAIC dataset. The lin model using the optimal feature combination identified in Section 5.3 achieves the best results. The lin model using all features (i.e. the all feature baseline from Section 5.3) also beats the comparison methods. ### 5.7 Multi-Modal Given the success of the FACS AUs SVM classifier in Section 5.6, I also evaluate a multi-modal method that fuses my gesture meta features and FACS AUs. I perform multiple multi-modal experiments: 1. 1. Feature vector fusion including all features, evaluated across all four classifier types. 2. 2. Feature vector fusion with a feature search on my gesture meta features, though all AU features are retained in every search iteration. 1. (a) Search using the lin classifier. 2. (b) Search using the svm classifier, as this was used successfully with the AUs features alone. 3. (c) Search using a radial-basis function kernel SVM classifier which achieves comparable results to the linear kernel SVM classifier on the AUs features alone. 3. 3. Hybrid fusion inspired by Alghowinem et al. [1]. This involved feature fusion and decision level fusion via a majority vote of three classifiers: one classifier with only meta features, one with only AUs features, and one with the fused feature vector. I experimented with all meta features and the feature searched meta features. I used the best classifier type for the individual classifiers, i.e. lin for gesture meta features, svm for AUs features, and then also svm for fused features. #### Results The best result was by the feature search method with an svm classifier (2.b), achieving 81.94% F1 average. The feature search with lin classifier (2.a) was second best with an 80.93% F1 average. However, these are worse than the best depression detection score of 82.70% when using the gesture meta features alone with the lin model. Moreover, the hybrid fusion approaches achieved F1s in the mid-70s, so rather than correcting errors by the meta features, it averaged the success rate between the meta classifier and the AUs classifier, resulting in a worse F1. #### Potential Future Fusion Approaches These approaches to fusion are somewhat simplistic. More sophisticated approaches, such as deep learning fusion, may achieve better results than the meta features alone. An interesting route for future work is to use recurrent deep learning to fuse temporally aligned meta features with AUs, rather than fusing them post-aggregation. ### 5.8 Summary In this chapter I have evaluated the introduced gesture meta features and found they provide linear predictive information. To achieve the best performance I perform an exhaustive search of the feature space to identify the best feature combination. This demonstrates the potential of the features, however, it also introduces a risk of overfitting. This risk is mitigated by two factors: the best performance is achieved by a linear regression based classifier (i.e. a classifier not prone to overfitting) and many features are present in optimal feature combinations for multiple uncorrelated labels. I compared my method to a basic facial-modality method and an existing body- modality method based on STIPs (i.e. generic analysis of video data). My method outperforms both when using the lin classifier, both when using all possible features and when using the optimal feature combination for the depression label. Finally, I perform a multi-modal experiment utilising the facial-modality comparison method and my novel gesture meta features. Despite testing multiple approaches to fusion, no method I evaluated beat the mono-modal gesture meta features classifiers. However, all related work that I am aware of achieves better performance when incorporating multiple modals, as such it is likely that further investigation of multi-modal approaches incorporating the gesture meta features will identify a multi-modal approach that does improve upon my mono-modal results. ## Chapter 6 Conclusion I have presented a novel set of generic body modality features based on meta information of gestures. To develop and evaluate these features I also introduced a novel non-clinical audio-visual dataset containing recordings of semi-structured interviews along with distress and personality labels. I evaluated these features for detecting depression as a binary classification task based on PHQ-8 scores. A linear regression based classifier achieved a 82.70% average F1 score, suggesting these features are both useful for depression detection and provide linear information. I further evaluated the features’ generalisability via binary classification tasks for other distress labels, such as anxiety, personality measures, and gender. All generalisation labels achieved better than 80% average F1 scores, with the exception of one label, personality neuroticism, which achieved 76.39%. These are novel features as previous works within the body modality for automatic distress detection are based on either unsupervised definitions of expressions or hand-crafted descriptors of distress indicative behaviour. These features are similar to some of the work within the eye and head modalities which are based on modeling generic activity levels within these modalities, among other features. Finally, these features exist within a broader ecosystem of affect based features. The best results for automatic distress detection, and similar tasks, is achieved by integrating multiple modalities and multiple representations of modalities. Future work may apply these features to new tasks, extend the feature type by extracting similar meta features, and develop methods for combining them with other modalities to amplify the information they provide. ### Future Work ### Dataset Future work could expand the introduced dataset and increase its quality via manual annotations. For example, annotating time frames based on the topic of conversation could enable sample segmentation methods such as Gong & Poellabauer [25]. This could improve the feature modeling and classifier stages, another route of future work is gesture definition and detection. For example, manual annotation of gestures would enable auto-learnt gesture detectors. ### Regression Tasks All labels I evaluate, with the exception of gender, are scales based on self- evaluation questionnaires relating to distress or personality. The distress scales define multiple levels of severity while the personality scales do not. Both types of labels are prime for regression prediction tasks. Moreover, distress classification prediction could be provided by the defined severity levels once a regression had been performed. However, this requires a larger dataset (initial experiments with regression on the dataset support this assertion). ### Features Applying the presented features within larger datasets could further illuminate the properties of certain features, either confirming their inconsistency, or solidifying the linearity in one direction. Such properties should then be tested with regards to the psychology literature. #### Trunk I do not evaluate a “trunk” localisation (i.e. hips to shoulders), though it may prove useful in future work. In my dataset participants are seated during the whole interview, thus the two-dimensional range of motion in the trunk (i.e. up/down and side to side) is small. As I do not include depth recording the greatest range of motion, forward and backward, is difficult to incorporate. Future work that either includes depth sensing or has participants in a wider variety of scenarios, where they may be standing or more physically engaged, may find some use in a trunk localisation. #### Cross Localisation Co-occurrence Co-occurrence of gestures across localisations is not considered in this dissertation, though it is an area for future work. Applying deep learning to co-occurrence modeling to identify relevant patterns would be particularly interesting. #### Improved Surprise In future work this feature could be extended to be more indicative of “surprise”, rather than simply an average of distance since last gesture. For example, a gesture that occurs within a regular pattern of gestures, regardless of their distance, might be considered to have low surprise, while a gesture interrupting that pattern by occurring in the middle of the pattern’s silent period would be very surprising. This more sophisticated measure of surprise could account for naturalistic behaviour such as rhythmic movement/gestures. These types of extensions of features are exciting for two reasons: firstly, they are interpretable, they do not come from a black box model, and thus can be supported by psychology literature. Secondly, they are immediately measurable and their design is based on traditional statistical techniques such as repetition modeling and can therefore be iterated on more readily than deep learning based features. #### Generalisation Future work, some of which has been discussed here, could extend these features to achieve better results in more diverse datasets. A greater variety of scenarios where the participant is more constantly physically engaged such that movement is more constant, e.g walking or a standing conversation, would challenge the design of the presented gesture meta features. Indeed, the current design would have trouble as it relies on the default state being a lack of movement. #### Deep Learning for Feature Generation As I discussed in Chapter 2, deep learning has been applied to the depression detection task, but still relies on hand-crafted features and descriptors. As with many fields it is likely that deep learning methods will, in the next few years, achieve state-of-the-art results, far exceeding the potential of more traditional approaches. However, to reach this point the deep learning models need large enough datasets to train on. Assuming no especially large dataset is developed specifically for automatic depression detection, one potential for future work is the use of transfer learning (as Chen et al. [9] explore for vocal and facial features) for body expressions. For example, CNNs could be trained on general body expression datasets to learn inner representations of body language. The output of inner layers could then be used in depression detection tasks (Razavian et al. [47] present a transfer learning approach to image recognition tasks and achieve very good results on niche tasks using a generic pre-trained image CNN). ### Classifiers and Deep Learning In this dissertation I have used two traditional statistical models, linear and logistic regression, and two more advanced machine learning methods, support vector machines and random forests. However, I have purposefully avoided more complex machine learning methods such as neural networks, and especially deep learning. These more sophisticated methods have achieved significant improvements on the state-of-the-art in many domains, however, they suffer from a lack of interpretability and a propensity to overfit data. As my primary focus has been validating gesture meta features I define interpretability as a core requirement and, given the size of my dataset, avoiding methods that overfit is important. However, having validated these features, deep learning presents opportunities with regards to learning more sophisticated aggregation functions (this is slightly different to the feature generating deep learning discussed above). I aggregate three core features of gestures: their movement, duration, and “surprise”. I perform aggregation via averaging and standard deviations. However, given a larger dataset, a deep learning model could foreseeably learn both more sophisticated aggregation functions and core meta-features. It is important to emphasize here the burden on dataset size that exists when attempting to learn valuable features using deep learning. #### Multi-Modal Approaches with Deep Learning Deep learning can learn useful modality integration functions. While I experimented with a multi-modal approach in Section 5.7 it did not achieve better results than my mono-modal method. 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2024-09-04T02:54:56.300581

2003.00837

arxiv-papers

# On Parameter Tuning in Meta-learning for Computer Vision Farid Ghareh Mohammadi1 M. Hadi Amini2 and Hamid R. Arabnia1 1: Department of Computer Science Franklin College of Arts and Sciences University of Georgia Athens Georgia 30602 2: School of Computing and Information Sciences College of Engineering and Computing Sustainability Optimization and Learning for InterDependent networks laboratory (solid lab) Florida International University Miami FL 33199 Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract Learning to learn plays a pivotal role in meta-learning (MTL) to obtain an optimal learning model. In this paper, we investigate image recognition for unseen categories of a given dataset with limited training information. We deploy a zero-shot learning (ZSL) algorithm to achieve this goal. We also explore the effect of parameter tuning on performance of semantic auto-encoder (SAE). We further address the parameter tuning problem for meta-learning, especially focusing on zero-shot learning. By combining different embedded parameters, we improved the accuracy of tuned-SAE. Advantages and disadvantages of parameter tuning and its application in image classification are also explored. Keywords: Advanced Machine Learning, Data Science, Meta-Learning, Zero-Shot Learning, Few-Shot Learning, Optimized Learning, Parameter Tuning ## 1 Introduction Motivation: Computer vision algorithms are essential to enable modern functionalities in future smart cities [1] [2], including face recognition [3] and automatic license plate recognition. Image classification stands as the primary challenging problem in computer vision [4] [5] [6] [7]. Computer vision provides important applications such as visual and infrared sensor data-based obstacle detection for the visually impaired [8]. Within computer vision, deep learning is an advanced learning tools based on using a training dataset to obtain a high performance on a testing dataset [9] [10]. Furthermore, auto-encoders have become recent challenging work in computer vision [11]; therefore, we focus on semantic auto-encoder in this paper. The ability to learn new tasks efficiently by leveraging prior experience from related tasks plays a main role in the world of artificial intelligence, especially learning. Meta-Learning (MTL) is the most advanced machine learning algorithm because it acquires the potential ability to learn as efficiently as possible from prior information. MTL first was presented as early as 1987 by Schmidhuber [12], and recent state-of-the-art studies [13] [14] [15] illustrate how MTL perfectly learns from a limited training dataset. In this paper, we investigate a task of image recognition to classify given images of unseen categories in the testing dataset for which we had a lack of examples in the training dataset. We choose zero-shot learning (ZSL) for the process of unseen image recognition [16] to overcome this limitation properly. Zero-shot learning is known as attribute-based learning, defined as the process of learning to recognise unseen objects. ZSL emphasizes learning of a new distribution of seen classes, given meta description of the seen categories, and seeks correlations with existing seen categories in a training dataset. This means that ZSL no longer needs to have any seen samples of unseen classes before evaluating a performance of predicting unseen classes. In recent years, ZSL [17] [18] [15] [13] [19] has been an active, challenging and hot research topic in advanced computer vision, machine learning, and medical data analysis [20]. Drug discovery [20], image classification [18] [13] and meta-sense [21] are examples of such research studies. Furthermore, ZSL has penetrated other domains like human action recognition, networks [18], etc. In [18], researchers provided comprehensive information about ZSL applications in computer and mobile networks. ZSL is a promising MTL algorithm that behaves like human cognitive activity. Here, we discuss emerging problems with ZSL. First, semantic space representations, which include attributes (’A’) and word vector (’W’), are critical for understanding and learning from the seen categories to apply on unseen categories, but these representations appear challenging to ascertain a high performance [17]. Human can only understand visual objects and attributes based on image explanations to recognise new images. However, the explanations do not yield distinguishable results [22], even traditional machine learning algorithms will not have promising performances [23]. Second, ZSL provides a mapping model selection to work with the unseen classes. The most important feature of ZSL consists of learning a compatible visual-semantic or attribute- based function and its ability to semantically represent objects. All in all, the more complex functions we have, the higher the risk of over-fitting, which yields poor results applied on the unseen classes. Whereas, simple linear function currently yields poor classification accuracy on the seen classes and does not adapt accurately on the unseen classes. Contribution: In this paper, we address the two aforementioned emerging problems in ZSL and present one promising solution. The main contribution of this paper is to provide a linear optimal function using tuned parameters to reach the most promising classification result that outperforms the most state-of-the-art work. We illustrate a detailed procedure of the work for meta-Mapping which is inspired from semantic auto-encoder (SAE) in algorithm 1. In the given meta-mapping, we extend the work presented by Kodirov _et al_ [24] Algorithm Semantic Auto-Encoder (SAE) [24] is an advanced linear mapping auto encoder that aims to find an optimal mapping function ($\mathscr{W}$) to recognise and classify unseen classes. Algorithm 1 illustrates a comprehensive procedure of SAE. First, in this paper, we develop SAE properly, and in the second step, we optimise the algorithm leveraging the tuning of a few embedded parameters of SAE. Note that in addition to the meta-learning for computer vision application, parameter tuning plays a pivotal role in ensuring convergence of several algorithms [25, 26]. Algorithm 1 Implementation of optimized zero-shot learning (SAE) [24] 1:A batch set of training input $(\mathscr{X,Y})$ and $training_{size}$ 2:The best mapping matrix $(\mathscr{W})$ for zero-shot learning 3:Tuning embedded parameters $\left.\begin{array}[]{@{}r@{}r@{}r@{}}\\\ {}\hfil\end{array}\color[rgb]{1,0,0}\right\\}\color[rgb]{1,0,0}\begin{tabular}[]{l}We contribute here\end{tabular}$ 4:Begin Training 5:for t=0 $\cdots training_{size}$ do 6: Learn $\mathscr{W}$:$\mathcal{Y}$ $\Leftarrow$ $\mathscr{W}$ $\mathscr{X}$ 7: $Err_{dst.}$=$\mid\mid$ $\mathscr{X}$ \- $\mathscr{W}$$\mathscr{W}^{\prime}$ $\mathscr{X}$ $\mid\mid_{F}$ $\left.\begin{array}[]{@{}r@{}r@{}r@{}}\\\ {}\hfil\\\ {}\hfil\\\ {}\hfil\\\ {}\hfil\end{array}\color[rgb]{1,0,0}\right\\}\color[rgb]{1,0,0}\begin{tabular}[]{l}We learn $\mathscr{W}$\\\ until $Err_{Dst.}\leadsto 0$\end{tabular}$ 8: Optimize $(\mathscr{W})$ :$\mathscr{W}$ =$\frac{\mathscr{C}}{\mathscr{A}+\mathscr{B}}$ 9: Return $\mathscr{W}$ and$\mathscr{W^{T}}$ 10:end for 11:End Training 12:Begin Testing 13:Compute acc. on Unseen Classes 14:for t=0 $\cdots test_{size}$ do 15: $\Delta=\mid\mid Pred-Ground_{T}\mid\mid$ $\left.\begin{array}[]{@{}r@{}r@{}r@{}}\\\ {}\hfil\\\ {}\hfil\\\ {}\hfil\\\ {}\hfil\\\ {}\hfil\\\ {}\hfil\end{array}\color[rgb]{1,0,0}\right\\}\color[rgb]{1,0,0}\begin{tabular}[]{l}We test\\\ unseen classes\\\ maximize the\\\ performance of SAE\end{tabular}$ 16: Minimise $\Delta$ 17: if ($\Delta$ is minimum) then 18: $\text{Performance}+=\frac{1}{test_{size}}$ 19: end if 20:end for 21:End Testing Organization: The rest of this paper is organized as follows. First we review related works regarding the use of ZSL for recognition of unseen classes. In Section 3, we state preliminary know-ledge of meta-learning before expressing a suggested meta-mapping (ZSL) algorithm. We then present experimental results and discussion, followed by conclusions. ## 2 Related Works In this section, we study related works on the use of zero-shot learning and presented previous examination on the evaluating unseen classes. While the number of new zero-shot learning methods increasing yearly, the criteria to examine all methods are not the same [19]. This makes our evaluation the methods particularly difficult. We present meta-learning to overcome the traditional machine learning limitation, which is when the learned model fails to predict testing data when the class is not already trained in training phases. Meta-learning, specially zero-shot learning overcomes this limitation by recognizing instances of unseen classes, which may not have been trained during training. In this section, we discuss related ZSL research studies. Table 1: Comparing related datasets ( SS-D refers to dimension of semantic space dimension) Features Datasets | AwA [16] | CUB [27] | ImNet-2 [28] ---|---|---|--- images | 30,475 | 11,788 | 218,000 total classes | 50 | 200 | 1360 seen classes | 40 | 150 | 1000 unseen classes | 10 | 50 | 360 SS-D | 85 | 312 | 1000 Lampert _et al_ [16] proposed an attribute-based prediction method for ZSL towards object recognition. They introduced a direct attribute prediction (DAP) method to do classification that learns from seen classes (y) and is tested on unseen classes ($y^{\prime}$). The authors stated that leveraging attributes enables us to have highly accurate and cost effective knowledge transfer between seen classes in a training dataset and unseen classes in a testing dataset. Romera and Torr _et al_ [29] presented embarrassingly simple zero-shot learning (ESZSL) and evaluated this with DAP, which is a baseline algorithm. Then, they updated compatibility of the learning algorithm by adding a regularization term. Zhang and Saligrama [30] [31] introduced a new ZSL based on semantic similarity embedding (SSE) and joint latent similarity embedding (JLSE) respectively. These researchers [31] formulated zero-shot recognition (ZSR) as a binary classification problem. They studied a framework leveraging dictionary learning to entirely learn the parameters of the model, that finally led to a class-free classifier. Akata _et al_ [32] presented an image classification technique called, structured joint embedding (SJE). Meanwhile, Changpinyo _et al_ [33] introduced a synthesized classifier (SYNC) method using combined semantic descriptions of $(A+W)$ to provide a higher performance on image recognition in comparison with DAP. From there, Bucher _et al_ [34] proposed a method, which leveraged adding visual features into attribute space, and then learned a metric to minimize the inconsistency by maximizing adaptability of the semantic embedding. Shigeto _et al_ [35] found that a least square regularised mapping function does not yield a good result for the hubness problem. Thus, they proposed regression and CCA-based approaches to ZSL to compute reverse regression, which means embedding class prototypes into the visual feature space. SAE [24] proposed by Kodirov _et al_ a semantic auto-encoder to regularize the learned model by mapping the image feature to semantic space. Although Xina _et al_ [36] proposed a feature generating framework for $any- shot$ learning called, f-VAEGAN-D2, there is room for improvement. A lot of work have done for small datasets [16] [30] [31] [29] [16] [33] [32] [30] [31] [34] [35] [24], however, just few methods are proposed for large datasets [37] [28]. Hybrid embedding system: Norouzi _et al_ [37] proposed a hybrid image embedding system, referred to as convex combination of semantic embeddings (ConSE), to deal with n-way image classification that lies in between independent classifier learning and compatibility learning frameworks. ConSE takes images and maps them into the semantic embedding space using convex combination of the class label embedding vectors. Furthermore, Fu and Sigal [28] presented a learning called semi-supervised vocabulary-informed learning (SS-Voc). ## 3 Meta-learning for Computer Vision: Preliminaries and Algorithms ### 3.1 Preliminaries In order to cover all available classes, we need machine learning and evolutionary algorithms try to solve high dimensionality problems of large datasets such as curse of dimensionality (CoD) [5] [7]. But, machine learning still cannot learn all samples when there are few instances per class. Meta- learning alleviates this problem by providing an advanced learning process. Meta-learning has three important promises for computer vision problems, specifically image classification and image recognition: 1) few-shot learning (FSL), 2) one-shot learning (OSL), and 3) zero-shot learning (ZSL). The crux of FSL is to learn a meta-learner to understand categories with boundaries without more than few examples of each category. FSL or k-shot learning takes k samples for each category in the training phase. Algorithm 2, which is combination of FSL and OSL, presents semantic pseudocode of model agnostic meta-learning (MAML) which was proposed by Finn [13]. This algorithm added one more step, which better illustrates how MTL can overcome traditional machine learning limitation using a meta-learner to optimize the learner with gradient decent optimization. Algorithm 2 Implementation of Meta-learning (MAML) [13] 1:A batch set of input targets (I,Y) and $Batch_{size}$ 2:The best meta-learner $F(\theta^{\prime})$ for few shot learning and an optimal mapping matrix $(\mathscr{W})$ to zero-shot learning 3:Begin 4:while work is not done do 5: for t=0 $\cdots Batch_{size}$ do 6: Learn from training batch set $\left.\begin{array}[]{@{}r@{}r@{}r@{}}\\\ {}\hfil\\\ {}\hfil\\\ {}\hfil\end{array}\color[rgb]{1,0,0}\right\\}\color[rgb]{1,0,0}\begin{tabular}[]{l}We learn\\\ the learner\end{tabular}$ 7: Learn new $(\theta)$ 8: Update the $(F(\theta))$ 9: end for 10: $\theta^{\prime}$=$\theta^{\prime}$+$\nabla\mathscr{L(F(\theta))}$ $\left.\begin{array}[]{@{}r@{}r@{}r@{}}\\\ {}\hfil\\\ {}\hfil\end{array}\color[rgb]{1,0,0}\right\\}\color[rgb]{1,0,0}\begin{tabular}[]{l}We optimize the meta-learner\\\ until $\nabla\mathscr{L(F(\theta))}\leadsto 0$.\end{tabular}$ 11: Update $\mathscr{F(\theta^{\prime})}\backsim\theta^{\prime}$ 12:end while 13:End ### 3.2 Preliminaries for Zero Shot Learning Notations Let’s suppose $(\mathscr{D})$ stands for a training dataset $(\mathscr{D})$=( $\mathscr{X}$, $\mathscr{Y}$) with seen classes $E$=$\\{1,2,\cdots,n\\}$. Consider $\mathscr{X}$ involving $\mathscr{d}$ dimensions of semantic space with required data, we map $\mathscr{X}$ into k-dimensional latent space with a mapping matrix $\mathscr{W}$. We name this latent representation, ${S}$. Then, we map this latent representation back to feature space $\mathscr{\hat{X}}$ using the transpose of $\mathscr{W}$, which is $\mathscr{W^{T}}$. In line 6 of algorithm 1, the authors used a well-known Sylvester equation to calculate an optimum mapping matrix using $\mathscr{A}$, $\mathscr{B}$ and $\mathscr{C}$ which stands for $SS^{T}$, $\lambda XX^{T}$ and $(1+\lambda)SX^{T}$, respectively. We use this $\mathscr{W}$ to recognise unseen classes in a testing dataset, $D^{t}$=( $X^{t}$, $Y^{t}$), with unseen classes $Z$=$\\{n+1,n+2,\cdots,m\\}$. ## 4 Experimental Result We investigate extensively to understand the benefits of a few factors in SAE algorithm [24]. SAE only has one parameter called $\lambda$ which is set differently for separate datasets. However, SAE has a few embedded parameters including HITK, Dist and sorting mode, which are required to be optimally tuned. We tune the embedded parameters with different ranges and values as follows: HITK ranges between 1 and the total number of unseen classes per dataset, Dist includes a kernel algorithm to calculate similarity between mapped unseen class instances and learned seen class instances, sorting mode occurs in either ascending or descending order. It is worth mentioning that we mostly give HITK the values of 1$\cdots$7 and 10. Examining all possible combinations is quite expensive, therefore, we present a set of results per different value of HITK for comparison. We find out that only HITK, one of embedded parameters, has direct sensible, publishable and positive effect on result of tuned-SAE performance. ### 4.1 Semantic Space To accurately calculate performance of tuned-SAE, we describe the semantic space (SS) and dimension of semantic space (SS-D). We directly emphasise on semantic space since the idea behind zero-shot learning relies on this. The way we describe the inputs becomes important in the training phase to ZSL, especially SAE. In previous research studies, scientists applied two different types of semantic space in their work. One is attribute ($A$), one is word vector ($W$). All the work we present in tables 2 and 3 mostly used attribute (A), except two research studies. SJE [32] worked with a combined semantic space (A+W) that yielded a result, 76.3, which was better than the basic attribute-based method (DAP). Moreover, SS-Voc[28] leveraged both (A/W) but not at the same time, and the performed results, 78.3/68.9, illustrate that their approach had been well-computed. However, SAE [24] only used attribute- based semantic space to calculate the performance of recognising unseen classess. It is noteworthy to say that, only word-vector(’W’) as SS has been used for large datasets in the compared work illustrated in table 3 [24]. ### 4.2 Pre-defined Parameters In [24], Kodirov _et al_ compared their proposed method, SAE, with more than 10 highly qualified methods using small datasets, which include AwA [16], CUB [27], $aP\&Y$ [38] and SUN [39]. The authors improved the accuracy of recognising unseen images at least $\%6$ in comparison with $SS-voc$ and at most $\%20$ in comparison with basic attibute-based learning method, DAP. Further, the researchers used two large datasets: $ImNet-1$ [28] and $ImNet-2$ [28]. Their image recognition errors for large datasets are beyond $\%60$. ### 4.3 Effect of Parameter Tuning on Accuracy of Meta-learning for Computer Vision To illustrate in detail the effect of parameter tuning on accuracy of meta- learning for computer vision, first, we discuss the used datasets, provide an ablation study, define an evaluation metric and present the state-of-the-art works and provide comparative evaluation in the following sections. ### 4.4 Dataset We choose two small, but popular, and one large benchmark dataset for ZSL in this study: AwA (Animals with Attributes) [16] consists of more than 30,000 images with respect to 50 different classes of animals; CUB-200-2011 Birds (CUB) [27] consists of 218 instances, 1000 seen classes and 360 unseen classes; ImNet-2 [28] provides 1000 classes for seen classes and 360 classes for unseen classes, where seen and unseen classes are extracted from ILSVRC2012, ILSVRC2010, respectively. Table 1 illustrates details information of these datasets either training and testing. Table 2: Comparing the related methods with our contribution for small datasets methods Datasets | AwA | CUB ---|---|--- DAP [16] | 60.1 | - ESZSL [29] | 75.3 | 48.7 SSE [30] | 76.3 | 30.4 JLSE [31] | 80.5 | 41.8 SJE [32](A+W) | 76.3 | 50.1 SynC [33] | 72.9 | 54.4 MLZSC [34] | 72.9 | 54.4 PRZSL[35] | 80.4 | 52.4 f-VAEGAN-D2 (IND) [36] | 71.1 | 61.0 f-VAEGAN-D2 (TRAN)[36] | 89.8 | 71.1 SS-Voc[28](A/W) | 78.3/68.9 | - SAE (W) [24] | 84.7 | 61.4 $SAE(W^{T})$ [24] | 84.0 | 60.9 Our contribution | | $SAE(W)-1$ | 84.7 | 61.4 $SAE(W^{T})-1$ | 84.0 | 60.9 $SAE(W)-2$ | 74.6 | 78.0 $SAE(W^{T})-2$ | 88.9 | 97.4 $SAE(W)-3$ | 83.7 | 85.13 $SAE(W^{T})-3$ | 94.4 | 97.4 $SAE(W)-4$ | 91.0 | 89.9 $SAE(W^{T})-4$ | 97.5 | 97.4 $SAE(W)-5$ | 96.4 | 92.4 $SAE(W^{T})-5$ | 99.1 | 97.4 $SAE(W)-6$ | 99.7 | 94.1 $SAE(W^{T})-6$ | 99.7 | 97.4 $SAE(W)-7$ | 99.5 | 95.3 $SAE(W^{T})-7$ | 99.8 | 97.4 $SAE(W)-10$ | 100 | 97.4 $SAE(W^{T})-10$ | 100 | 97.4 Table 3: Comparing the related methods with our contribution for a large dataset methods Datasets | ImNet-2 | ImNet-2 ---|---|--- ConSE[37] | 15.5 | 15.5 SS-Voc[28] | 16.8 | 16.8 $SAE(W)$ [24] | 26.3 | 26.3 $SAE(W^{T})$ [24] | 27.2 | 27.2 Our solution | | with parameter tuning | $\lambda$= 5 | $\lambda$= 6 $SAE(W)-1$ | 12.2 | 12.2 $SAE(W^{T})-1$ | 12.9 | 13.0 $SAE(W)-2$ | 17.6 | 17.6 $SAE(W^{T})-2$ | 18.3 | 18.4 $SAE(W)-3$ | 21.2 | 21.1 $SAE(W^{T})-3$ | 22.1 | 22.1 $SAE(W)-4$ | 24.0 | 23.9 $SAE(W^{T})-4$ | 24.9 | 24.9 $SAE(W)-5$ | 26.3 | 26.3 $SAE(W^{T})-5$ | 27.2 | 27.3 $SAE(W)-6$ | 28.3 | 28.3 $SAE(W^{T})-6$ | 29.2 | 29.3 $SAE(W)-7$ | 30.1 | 30.1 $SAE(W^{T})-7$ | 31.1 | 31.2 $SAE(W)-10$ | 34.8 | 34.7 $SAE(W^{T})-10$ | 35.6 | 35.7 ### 4.5 Ablation Study In this paper, we work jointly with semantic auto-encoder (SAE), which is an advanced supervised clustering algorithm for the specific purpose of zero-shot learning. The main strength of this paper is tuning SAE, which comes from examining the output by updating parameters. ZSL usually uses a complex projection function, but SAE leverages a linear projection, according to algorithm 1. ### 4.6 Evaluation Metric We compute performance of the tuned-SAE based on the loss function $\mid\mid Pred-Ground_{T}\mid\mid$, which is also presented in [40] for a metric learning function and supervised clustering. ### 4.7 Competitors We compare our method, tuned-SAE, with state-of-the-art method [36] and other work are compared in [24]. All compared research studies have used zero-shot learning (supervised learning) [31] [16] and semi-supervised learning[28]. ### 4.8 Comparative Evaluation We make the following observations according to the results in tables 2 and 3: (1) Our tuned-SAE model obtained the best results on both small and large datasets. (2) On the small datasets, the gap between tuned-SAE′s results and the strongest competitors are varied due to different results of SAE. Note that our tuned model is a linear projection function, while most of the compared models use complex nonlinear projection functions, and some of them use more than one semantic space like SJE [32] and SS-voc [28]. (3) Although tuned-SAE performed well on the large-scale dataset ($ImNet-2$), our model did not improve the performance more than $\%7$ in comparison with SAE, but tuned- SAE yields a promising result in comparison with other methods. (4) The performance of tuned-SAE on the small datasets is far better than the large dataset. (5) Last but not least, by increasing HITK value from 2 to 10, our performance directly increases. ## 5 Discussion Tuning plays a main role in all algorithms, especially machine learning algorithms. It provides a comprehensive situation for the algorithms to learn from a training dataset, so the learned model yields a high performance on a testing dataset. However, machine learning algorithms may not do well for unseen classes, although they obtained optimised parameters. In this paper, we address this problem, study semantic auto-encoder (SAE) and develop tuned-SAE in order to gain a better performance on unsupervised clustering, which is one of the approaches for zero-shot learning. The results in table 2 depict that tuned-SAE leads to a better performance than SAE and other related methods. To compare table 2 with table 3, we find that tuned-SAE performs far better for small datasets than large datasets. This paper proves that tuning is a big advantage in zero-shot learning for image recognition. However, it does not work well for large datsets. Table 3 shows that with different $\lambda$ values we have have different results for $SAE(W)$ and $SAE(W^{T})$, such that $SAE(W^{T})$ plays as a decoder and maps data from semantic space to feature space to compute performance of the work. ## 6 Conclusion Having an optimal learning model is key in the world of machine learning. However, learning from unseen classes is a critical issue in traditional machine learning. In this paper, we address this problem and investigate advanced learning processes to enable learning from seen classes to predict unseen classes accurately. We aim to focus on SAE as a semantic auto-encoder, which enables us to find an optimal mapping function between semantic space and feature space, such that it also works for unseen semantic space and classes. In this paper, we tune embedded SAE’s parameters in a way that SAE yields better results than the original parameters presented in [24]. 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2024-09-04T02:54:56.310899

2003.00853

arxiv-papers

# Sensitivity-Enhanced Fourier Transform Mid-Infrared Spectroscopy Using a Supercontinuum Laser Source Ivan Zorin Research Center for Non-Destructive Testing Science Park 2, Altenberger Str.69 4040 Linz, Austria <EMAIL_ADDRESS> &Jakob Kilgus Research Center for Non-Destructive Testing Science Park 2, Altenberger Str.69 4040 Linz, Austria &Kristina Duswald Research Center for Non-Destructive Testing Science Park 2, Altenberger Str.69 4040 Linz, Austria &Bernhard Lendl Institute for Chemical Technologies and Analytics TU Wien, Getreidemarkt 9 1060 Vienna, Austria &Bettina Heise Research Center for Non-Destructive Testing Science Park 2, Altenberger Str.69 4040 Linz, Austria &Markus Brandstetter Research Center for Non-Destructive Testing Science Park 2, Altenberger Str.69 4040 Linz, Austria <EMAIL_ADDRESS> Corresponding<EMAIL_ADDRESS> ###### Abstract Fourier transform infrared (FTIR) spectrometers have been the dominant technology in the field of mid-infrared (MIR)spectroscopy for decades. Supercontinuum laser sources operating in the MIR spectral region now offer the potential to enrich the field of FTIR spectroscopy due to their distinctive properties, such as high-brightness, broadband spectral coverage and enhanced stability. In our contribution, we introduce this advanced light source as a replacement for conventional thermal emitters. Furthermore, an approach to efficient coupling of pulsed MIR supercontinuum sources to FTIR spectrometers is proposed and considered in detail. The experimental part is devoted to pulse-to-pulse energy fluctuations of the applied supercontinuum laser, performance of the system, as well as the noise and long-term stability. Comparative measurements performed with a conventional FTIR instrument equipped with a thermal emitter illustrate that similar noise levels can be achieved with the supercontinuum-based system. The analytical performance of the supercontinuum-based FTIR spectrometer was tested for a concentration series of aqueous formaldehyde solutions in a liquid flow cell (500 µmm path length) and compared with the conventional FTIR (130 µm path length). The results show a four-times-enhanced detection limit due to the extended path length enabled by the high brightness of the laser. In conclusion, FTIR spectrometers equipped with novel broadband MIR supercontinuum lasers could outperform traditional systems providing superior performance, e.g., interaction path lengths formerly unattainable, while maintaining low noise levels known from highly stable thermal emitters. _K_ eywords Mid-infrared spectroscopy $\cdot$ MIR $\cdot$ supercontinuum laser source $\cdot$ Fourier transform infrared spectroscopy $\cdot$ FTIR ## 1 Introduction Fourier-transform infrared (FTIR) spectroscopy has been a well established and widely used tool for chemical characterization in various application scenarios for decades. FTIR spectrometers are still the gold standard in the mid-infrared (MIR) spectral range exhibiting reasonable acquisition times, high sensitivity and spectral resolution[1]. In their typical configuration, these spectrometers employ thermal light sources emitting black-body radiation perfectly fitted for many applications. Nevertheless, thermal emitters impose several limitations caused by their inherent properties, such as spatial incoherence, low power and omnidirectionality. Recently, a highly interesting new type of broadband source has been emerging into the MIR spectral region, namely the supercontinuum laser source. MIR supercontinuum lasers are nowadays operating in the same or broader wavelength range as thermal sources[2, 3]. In contrast to the latter, they exhibit drastically higher brightness, spatial coherence, and stability [4, 5] making them a promising tool for spectroscopy. Furthermore, supercontinuum sources are an attractive alternative to the already established MIR quantum cascade lasers (QCL)[6, 7]. The noise characteristics of supercontinuum generation, which had been a deterrent in their early days, were significantly improved since then[8, 9, 5]. Successful demonstrations of MIR supercontinuum source-based setups in a wide variety of noise sensitive applications [10, 11, 12, 13, 14, 15] prove their practical suitability. Furthermore, it is expected that the trend of noise reduction will continue e.g. by implementing the supercontinuum generation in an all-normal-dispersion regime, which is insensitive to the input pump noise delivering the enhanced shot-to-shot spectral coherence[16, 17, 18, 19, 20]. Meanwhile, with respect to the achieved average power, intensities up to 21.8 W were recently achieved in the wavelength range of 1.9 µm - 3.8 µm[21]. The broad spectral range covered by these sources is continuously extending [22, 23, 24, 25, 26, 27] even beyond the fingerprint region revealing attractive potentials for spectroscopic measurements. This ultra-broadband coverage, including both near-infrared (NIR) and also the MIR range, is unique and only offered by supercontinuum lasers, while QCLs inherently offer a narrow-band wavelength tunability. Due to the strict requirements on precise control of the mirror position, FTIR spectrometers are designed to operate with continuous-wave (CW) sources as the modulated signal is sampled at a defined frequency. This scheme introduces the limitations for operating FTIR spectrometers with pulsed sources without taking care of the sampling and pulse repetition rates. For this reason, the latter is usually set much higher to go into quasi-CW mode. This approach has been evaluated by e.g. the combination of an FTIR and a NIR supercontinuum source radiating at repetition rates of 80 MHz and 100 MHz already resulting in an improved signal-to-noise ratio (SNR) and detection limits as well as extended interaction path lengths[28, 29, 30]. In the much more attractive MIR spectral range, currently available supercontinuum sources typically operate at repetition rates from tens of kHz and up to several MHz with a pulse duration in the sub-nanosecond range. Due to the asynchronization of the emission and sampling performed by FTIR spectrometers, additional noise and spectral distortions are introduced resulting in insufficient stability of the signal for FTIR measurements[31, 32]. CW supercontinuum may be a key to solve the problem, however, the technical complexity and disadvantages (fibers with several tens of kilometers length, high power requirements, poor broadening) [2, 33] make them challenging especially in the MIR spectral range. From an analytical point of view, CW operation of such bright sources would introduce substantial thermal load, which may cause damage of the sample under investigation. In this contribution, we prove the principle suitability of pulsed MIR supercontinuum lasers as a light source in an FTIR instrument. We define and evaluate practical problems for coupling with an FTIR spectrometer that is highly sensitive to intensity variations and report a successful method to overcome the sampling problem. In the experimental part, we examine pulse-to- pulse intensity fluctuations over the supercontinuum emission spectrum, the noise of the experimental setup and its stability. Finally, the analytical performance of the proposed solution is investigated and compared to conventional state-of-the-art instrumentation. ## 2 Materials and methods ### 2.1 Mid-infrared supercontinuum laser source In our study, a novel and commercially available supercontinuum source (NKT Photonics, SuperK MIR) with a spectral coverage from 1.1 µm to 4.4 µm was applied. The ultra-broadband spectrum is generated due to a sequence of non- linear processes initiated in a ZBLAN (ZrF4-BaF2-LaF3-AlF3-NaF) fiber[2], pumped in the spectral region of around 2 µm by a pre-broadened and amplified seed laser (1.55 µm). The repetition rate of the source is adjustable in the range of 2 MHz to 3 MHz (2.5 MHz default). The supercontinuum pulse width is shorter than 1 ns yielding in a duty cycle of less than 0.5% causing the spectrum distortion and irreproducibility of FTIR measurements when directly applied without additional measures[31]. The output beam of the supercontinuum source is a single Gaussian mode with a beam quality parameter $\mathrm{M^{2}\leq 1.1}$ and a diameter of around 4 mm with a beam divergence of $2$ mrad. The total average power of the source was measured as 475 mW, whereof 200 mW are radiated in MIR spectral range. Characterization and verification of the pulse-to-pulse energy fluctuations were performed using a monochromator (Gilden Photonics, GM500) and a high- speed Mercury Cadmium Telluride (MCT) detector (PVM series, Vigo, rise time $\leq 0.7$ ns and 230 MHz bandwidth). ### 2.2 Experimental setup The experimental setup consisted of a commercial FTIR instrument (Bruker Optics, Vertex 70) and the pulsed supercontinuum laser as depicted in Fig. 1. Function GeneratorLock-In AmplifierSCLFCOpticsM1M2Ref. LaserBSApertureDetectorRD3 MHzLockedBPFROEFTIR Spectrometer Figure 1: Scheme of the experimental setup: FTIR spectrometer (Bruker Vertex 70, simplified scheme), SC - supercontinuum laser source, BPF - band-pass spectral filter, LFC - liquid flow cell, ROE - spectrometer built-in read-out electronics and mirror control, M1 and M2 - interferometer mirrors and BS - beam splitter; RD - detector used to produce reference interferogram. In order to effectively exploit the dynamic range of the detector and avoid oversaturation by NIR spectral components (e.g. strong and high power seed laser line at 1.55 µm), a suitable bandpass filter (BPF, Thorlabs FB3500-500, 3100 cm-1 \- 2650 cm-1) was employed thereby determining the investigated spectral range; the average power of the transmitted band (i.e. incident on the sample) was measured equal to 17 mW (power meter, Coherent, LM-10 HTD), which did not lead to observable heating of the sample. It should be noted that the measurements over the entire MIR part of the emission spectrum are restricted by the dynamic range of the detector, however, they are realizable applying edgepass (1.65 µm cut-on wavelength) and neutral density filters or introducing smaller diameters of the aperture. The collimated radiation was transmitted through the sample inserted in a liquid flow cell (PIKE Technologies, 4 mm CaF2 windows) before it entered the FTIR spectrometer via the external optical input. The FTIR spectrometer with a typical configuration based on a Michelson configuration (BS - beam splitter, M1 and M2 - movable and fixed mirrors, respectively) is schematically shown in Fig. 1. The interferogram was recorded with a Mercury Cadmium Telluride detector (variable gap Vigo PCI-4TE-12, detectivity within the spectral range D$\geq~{}2.0\times~{}10^{9}$ $\mathrm{cm\cdot\sqrt{Hz}\cdot W^{-1}}$). The built-in spectrometer control and read-out unit (ROE) utilized for the signal acquisition was used to control mirror scan velocity (i.e sampling frequency) and size of the output aperture. In order to achieve the equidistant time sampling, the reference monochromatic laser is used to produce a quasi-sine interferogram [34] (e.g. see Fig. 2(c)) to be recorded by the reference detector (RD). Any asynchronization of the signal acquisition and pulse generation would result in strong noise and low intensity of the recorded interferogram (Fig. 3), since the sampling would frequently be performed in the absence of any pulse, thereby being equivalent to the measurement of the detector noise. Figure 2 shows a zoomed-in part of a simulated interferogram and serves purely for illustration purposes defining the sampling problem and a possible solution by using an external integrator. The pulses presented in Fig. 2(a) were chosen according to the parameters of the supercontinuum radiation exploited in our study. A reference signal (150 kHz, Fig. 2(c)) that defines the sampling frequency was simulated in order to temporarily scale the graph and show the infeasibility of the traditional approach to sample the signal produced by any low-duty cycle source. Time (µs)Intensity (a.u.)a)b)c) Figure 2: Problem of the signal sampling for low-duty cycle sources applied in the FTIR spectrometer and approach to overcome the asynchronization using an external integrator: (a) Interferogram for the pulsed light source, (b) Integrated CW signal of the detector, i.e. lock-in amplifier (LIA) output, (c) Reference interferogram produced by the monochromatic source, defines the sampling frequency (red). In the system proposed in Fig. 1, the lock-in amplifier (LIA, Stanford Research Systems, SR844) was used as an integrator to suppress the pulsed nature of the signal. Therefore, the output of the MCT detector was transformed into a signal similar to that obtained in the traditional configuration with a CW source (i.e. to one shown Fig. 2(c)). According to the Nyquist-Kotelnikov-Shannon theorem, the measurement approach is feasible only for integration times of the LIA shorter than half of the period of sampling, otherwise, the interforogram is distorted or undetectable. Nevertheless, very short time constants that are achievable with modern LIAs (e.g. 1 µs) result in a higher spectral noise due to the averaging of fewer pulses. The LIA that was used in this work has a fixed and discrete integration time range (100 µs, 300 µs, 1 ms and longer). The sampling rate, i.e. the mirror velocity of the FTIR spectrometer, could be adjusted within the range from 1 kHz to 60 kHz. Therefore, the requirements defined by the sampling theorem were satisfied by selecting the proper parameters of the LIA and the FTIR spectrometer. With respect to the trade-off between the speed and sensitivity (100 µs and 300 µs time constants correspond to 5 kHz and 1.7 kHz sampling rates), the following measurements were executed at the longer integration time, i.e. 300 µs, and at 1 kHz sampling frequency. The external function generator was utilized to trigger the supercontinuum source at the highest available frequency (3 MHz, according to the maximum repetition rate) and to synchronize the laser with the LIA. Consequently, around 900 pulses were integrated for each sampling point. a)b)c)d)SNR=$\mathbf{3.74}$SNR=$\mathbf{7320}$ Figure 3: Raw interferograms and reconstructed emission spectra of the supercontinuum source (band-pass filter inserted) obtained using the traditional sampling approach without external integrator (a,b) and applying the lock-in amplifier (c,d). The intensity values are not normalized and correspond to the amplitude of the raw signals. Figure 3 illustrates raw signals and spectra obtained by the supercontinuum- based FTIR spectrometer in two configurations: a direct coupling and a coupling using the LIA locked at the repetition rate of the source. The optical parameters and conditions were set the same for both measurements (aperture 8 mm, 500 µm path length liquid flow cell, distilled water). SNR values were traditionally derived as the ratio between maximum signal levels and calculated standard deviations of the zero absorbance lines (also known as 100% lines). It should be noted that the peak intensity when using the LIA- based integration scheme showed a 250 times higher magnitude, which is proportional to the pulse sampling probability, i.e. $1/\mathrm{D_{c}}$, where Dc is the duty cycle. ### 2.3 Samples Formaldehyde (CH2O or methanal) is a pollutant that is widely distributed in the environment. Due to its high water solubility [35] and a wide variety of natural sources such as e.g. oxidation of organic matters[36] and industrial effluents, formaldehyde and formaldehyde monohydrate (methanediol) can e.g. be frequently found in food, natural- and drinking water[37]. The evidence of mutagenicity and risks of carcinogenicity of formaldehyde are well investigated and verified[38]. Histopathological short and long-term effects in the mucosa of rats are documented at the concentration threshold of 260 ppm[39], while a tolerable concentration of 2.6 ppm with an uncertainty factor of 100 is defined by the World Health Organization (WHO)[37]. During the metabolism following oral exposure, formaldehyde is oxidized to formic acid that also introduces specific toxic effects on human health[40, 41]. The detection of low concentrations of formaldehyde dissolved in water is a practically interesting problem for cost-efficient spectroscopy since it is usually done by sophisticated and expensive chromatographic analysis[42]. In our study, a concentration series of formaldehyde aqueous solutions has been prepared for spectroscopic analysis by dilution from 37% stock solution (Carl Roth GmbH). ## 3 Results and discussions The measured emission spectra of the thermal and supercontinuum source are depicted in Fig. 4(a), edgepass filter (1.65 µm cut-on wavelength) and neutral density filter were used to avoid oversaturation of the detector. (a) Emission spectra of the sources (b) Absorption spectra of formaldehyde and water Figure 4: (a) Emission spectra of the supercontinuum laser source and the thermal source with respect to (b) absorption of water (recorded with the thermal emitter, 90 µm liquid flow cell) and formaldehyde (spectrum taken from database[43]). The supercontinuum spectrum was recorded using a neutral- density filter (optical density of 1), an aperture size was set to 0.25 mm; the emission spectrum of the thermal source was measured with an aperture size of 8 mm in the absence of any filters; both recorded emission spectra reveal strong absorption of CO2 around 2350 cm-1. Intensity values are not normalized, since different optical parameters were used to avoid oversaturation of the MCT detector: the aperture was set to 8 mm diameter for the thermal emitter, while for the high brightness supercontinuum laser the aperture was set to the minimum available aperture of 0.25 mm with an additionally inserted neutral density filter (10% transmission). In order to determine the analytical performance (i.e. limit of detection) of the supercontinuum-based FTIR, an aqueous dilution series was measured. Formaldehyde, i.e more accurately its mixture with the hydrated form methanediol (CH2(OH)2)[44] was chosen as the target analyte. Formaldehyde exhibits strong absorption due to the stretching vibration of C-H, in a spectral range where water absorption shows a relative minimum (see Fig. 4(b), spectrum of formaldehyde taken from a database[43]). Hence, the bandpass filter with a quasi-Gaussian profile was used to select the spectral range (3100 cm-1 \- 2650 cm-1); the default spectral resolution was set to 4 cm-1 for the following experimental verification and noise analysis. ### 3.1 Noise and long-term stability In order to characterize and compare the noise of the supercontinuum-based system (Fig. 5) and conventional FTIR with the CW thermal emitter, zero absorbance lines were obtained within the spectral range of interest (3100 cm-1 \- 2650 cm-1). (a) 100% zero absorbance lines, 130 µm liquid flow cell (b) Allan-Werle variance SCTERMS=0.00078RMS=0.00067 Figure 5: (a) Noise performance of the experimental setup using a 130 µm liquid flow cell filled with blank water: the supercontinuum source (marked as SC) in comparison to the thermal emitter (TE) expressed in the form of 100% (zero absorbance) lines and (b) long-term stability of the LIA based setup illustrated by the Allan Werle variance. Since the spectroscopic analysis could not be performed directly for the same path length due to the different intensity levels of the thermal and supercontinuum source and the limited dynamic range of the detector[45], a direct comparison of the same sample at the same optical path length was not feasible. Therefore, a neutral density filter (optical density of 1) was used to reduce the output power of the laser. Figure 5(a) depicts the recorded 100% lines measured through a 130 µm liquid flow cell filled with blank water. Four consecutive spectra were averaged for the calculation of each zero absorption line. Root-mean-square (RMS) values for both sources exhibit comparable noise levels (0.67$\times$10-3 and 0.78$\times$10-3 standard errors for thermal and supercontinuum correspondingly), while the raw signal level of the quasi- Gaussian peak transmitted through BPF shows a factor of 25 higher magnitude in the case of the supercontinuum laser. The spectral region beyond 2800 cm-1 is not used in the calculations due to the total attenuation imposed by water and thus insufficient light transmitted in this range. Additionally, long-term measurements using the same 130 µm liquid flow cell (blank water) were carried out over around 28 hours (10.000 spectra) to specify the stability of the system, the prevalent types of noise sources and to estimate detection limits that could be achieved by averaging. The series of error values calculated for zero absorbance lines (within the same spectral range of 2800 cm-1 \- 2650 cm-1) were used to derive the Allan-Werle variance[46] as depicted in Fig. 5(b). The dependence of the signal deviation on the integration time illustrated by the Allan-Werle variance demonstrates the stability of the supercontinuum-based FTIR in a long-term perspective. Analysing the plot, white noise can be observed as a dominant noise floor in the system, as indicated by the decreasing tendency with a constant slope. This noise is common and most likely inherited by the nonlinear processes (initiated by amplification of the pump shot noise) [8, 9] that occurred during the spectral broadening. A weak long-term drift was observed only after 4$\times$104 seconds and could be caused by temperature changes. The evolution of the variance reveals that the SNR of the system and the corresponding limit of detection could be enhanced by 4 orders of magnitude applying the signal averaging. However, such long averaging is not expedient so that the spectroscopic measurements in the following sections are performed with an averaging over 10 spectra only (around 70 seconds). ### 3.2 Pulse-to-pulse energy fluctuations To specify and complete the noise characterization of the supercontinuum source, the pulse-to-pulse behaviour over the emission spectrum has been investigated (Fig. 6). Figure 6: Normalized pulse-to-pulse energy fluctuations over the emission spectrum of the supercontinuum laser source (indicated for reference). A Czerny-Turner monochromator was used to access the pulse-to-pulse energy fluctuations within a narrow spectral band (3 nm resolution, equidistant in the wavelength space). The high-speed MCT detector and oscilloscope (400 MHz, 5 GS/s) were used to record pulse waveforms. The measurements were performed by sweeping over the MIR part of the spectrum with 6 nm step size, while for each position 1000 pulses were analysed. The normalized pulse energy fluctuations, depicted in Fig. 6, were calculated as a standard deviation of the pulse areas (time-integrals, represent a pulse energy) normalized by their mean. An average pulse-to-pulse energy fluctuation of around 6.4% could be observed within the spectral range of interest. For the Gaussian noise, the standard error for the averaged measurement is proportional to $1/\sqrt{N}$, where $N$ is the number of samples. Thereby, the evaluated noise is verified, since the measurements coincide quite well with the results demonstrated in the previous section: the obtained dependence yields the RMS of around 0.0010, derived for $N=3600$, while the measurements of the 100% lines, where 4 spectra were averaged (900 pulses integrated for each spectrum), give a RMS of 0.00078. ### 3.3 Quantitative measurements The performance and analytical usability of the supercontinuum-based FTIR spectrometer were verified for the quantification of aqueous formaldehyde solutions (calibration curves shown in Fig. 7). (a) Thermal source (130 um liquid flow cell) (b) Supercontinuum source (500 um liquid flow cell) Figure 7: Calibration curves of a formaldehyde (methanediol) dilution series obtained by integrating the absorbance within the spectral range of 2720 cm-1 \- 2620 cm-1, covering parts of the C-H stretching band (a) standard FTIR instrument with CW thermal emitter and 130 µm optical path length flow cell; (b) supercontinuum-based FTIR and 500 µm flow cell. The spectroscopic measurements were performed using both the CW thermal emitter (conventional system without external integrator) and the supercontinuum laser (experimental setup). The spacers of the liquid flow cell (130 µm and 500 µm) were selected correspondingly with respect to the available intensity levels, while the aperture size was set to 8 mm for both configurations. Despite the different path lengths, the raw signal recorded by FTIR in the case of the supercontinuum source exhibited a 50 times higher magnitude due to the distinctive properties of the radiation and the applied pulse integration approach; for each measurement 10 spectra were averaged. The calculated absorbance spectra of the standard solutions were integrated within the spectral range of 2720 cm-1 \- 2620 cm-1, where the sufficient intensity is obtained for both light sources. Figure 7 depicts the obtained linear calibration curves. The inset in Fig. 7(b) presents absorbance spectra for the lowest concentrations of the dilution series. It should be noted that the spectral band, within which the spectroscopic analysis is performed, covers the range near the maximuma of the absorption band where it coincides with relatively weak water absorption. The linear fitting model delivered $R^{2}$ values of 0.9992 for the thermal emitter and 0.9989 for the supercontinuum source respectively. Hence, corresponding slopes of 0.035 (thermal source) and 0.162 (supercontinuum) and standard errors (SE) yield the limits of detection (LOD) for both configurations calculated according to the IUPAC definition:[47] a LOD of 580 ppm was achieved using conventional FTIR, while a superior LOD of 140 ppm has been demonstrated by the experimental system applying the supercontinuum source. ## 4 Conclusions and Outlook Novel commercially available noise reduced mid-infrared (MIR) supercontinuum sources have recently evolved to a state where they can reasonably be applied for various spectroscopic analytical tasks[10, 11, 12, 13, 14]. However, they are widely underestimated among spectroscopists, which is, for instance, illustrated by the fact that they are still basically unnoticed at the relevant conferences. The main reason might be a specification lag, i.e. high noise and instability, of their first versions. Therefore, one of the main goals of this contribution was to demonstrate that in fact MIR supercontinuum laser sources have achieved a level of maturity to be competitive with the state-of-the-art equipment. In order to support this idea, we demonstrated their applicability for the gold standard in MIR spectroscopy, Fourier- transform infrared spectroscopy (FTIR) that is strongly sensitive to intensity fluctuations. The basic idea here was to replace the broadband but weak and spatially incoherent thermal emitters by a broadband and high brightness spatially coherent source. In the experimental part, a simple solution to overcome the sampling problem for the direct coupling of pulsed supercontinuum and FTIR has been proposed and realized. Utilizing the prospected experimental system, we achieved a factor of 200 greater amplitude of the raw interferogram and enhanced signal- to-noise ratio. The setup and the supercontinuum laser source were characterized with respect to noise and stability of the measurements. A satisfying long-term stability over around 28 hours and a noise level, which is comparable to the conventional thermal source, were demonstrated. The standard errors of the 100% zero absorbance lines (0.67$\times$10-3 and 0.78$\times$10-3 for thermal and supercontinuum sources respectively) were calculated for the measurements of blank water within a 130 µm liquid flow cell. The superior brightness of the supercontinuum source that is provided by the directionality of the radiation and the high output power allowed us to increase the interaction path length in a transmission measurement and to demonstrate the enhanced performance of the supercontinuum-based system. A spectroscopic analysis of an aqueous formaldehyde dilution series has been performed. The absorbance spectra (measured for 500 µm liquid flow cell) yielded a limit-of-detection (LOD) of 140 ppm. In total, a 4-times enhanced detectivity over a conventional FTIR system with the thermal source was demonstrated, while the investigation of path lengths over 130 µm were not expedient with the thermal emitter due to the almost total absorption by water. The next generation of supercontinuum sources being developed is based on chalcogenide fibers and has already reached a milestone in terms of spectral coverage, starting in the NIR spectral region and spanning up to 14-16 µm emission wavelength[3, 48, 49, 27, 26]. Thus, they cover almost the entire MIR spectral range, which makes them a highly promising technology to significantly push the field of MIR spectroscopy in the nearest future. Considering the gained results and current developments[50, 19], we believe that the proposed solution offers new potentials for enhancing the currently applied methods in this field. The demonstrated approach to couple pulsed supercontinuum sources appears to be universal and could be applied for any low-duty cycle source. Meanwhile, a purpose-oriented adaptation of the presented signal acquisition scheme or investigation of other types of integration devices, e.g. boxcar, could be considered in order to assemble a fully-integrated FTIR system, since a price reduction of supercontinuum sources is expected. ## 5 Funding Financial support was provided by the Marie Sklodowska-Curie Action SUPUVIR of the European Union’s H2020-MSCA-ITN-2016 Programme under REA grant agreement no. 722380; and the strategic economic and research program ”Innovative Upper Austria 2020” of the province of Upper Austria. ## References * [1] H. Günzler and H.U. Gremlich. IR Spectroscopy: An Introduction. Wiley, 2002. * [2] John Dudley and Roy Taylor. Supercontinuum Generation in Optical Fibers. 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2024-09-04T02:54:56.332938

2003.00866

arxiv-papers

# Wireless AI: Enabling an AI-Governed Data Life Cycle Dinh C. Nguyen, Peng Cheng, Ming Ding, David Lopez-Perez, Pubudu N. Pathirana, Jun Li, Aruna Seneviratne, Yonghui Li, and H. Vincent Poor Dinh C. Nguyen and Pubudu N. Pathirana are with School of Engineering, Deakin University, Australia (e-mails: {cdnguyen, pubudu.pathirana}@deakin.edu.au).Peng Cheng and Yonghui Li are with the School of Electrical and Information Engineering, the University of Sydney, Australia, (e-mails: {peng.cheng, yonghui.li}@sydney.edu.au). Ming Ding is with Data61, CSIRO, Australia (email: [email protected]). David Lopez-Perez is with Nokia Bell Labs (e-mail: [email protected]).Jun Li is with School of Electrical and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: jun.li @njust.edu.cn).Aruna Seneviratne is with School of Electrical Engineering and Telecommunications, University of New South Wales (UNSW), NSW, Australia (email: [email protected]).H. Vincent Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: [email protected]). ###### Abstract Recent years have seen rapid deployment of mobile computing and Internet of Things (IoT) networks, which can be mostly attributed to the increasing communication and sensing capabilities of wireless systems. Big data analysis, pervasive computing, and eventually artificial intelligence (AI) are envisaged to be deployed on top of IoT and create a new world featured by data-driven AI. In this context, a novel paradigm of merging AI and wireless communications, called Wireless AI that pushes AI frontiers to the network edge, is widely regarded as a key enabler for future intelligent network evolution. To this end, we present a comprehensive survey of the latest studies in wireless AI from the data-driven perspective. Specifically, we first propose a novel Wireless AI architecture that covers five key data- driven AI themes in wireless networks, including Sensing AI, Network Device AI, Access AI, User Device AI and Data-provenance AI. Then, for each data- driven AI theme, we present an overview on the use of AI approaches to solve the emerging data-related problems and show how AI can empower wireless network functionalities. Particularly, compared to the other related survey papers, we provide an in-depth discussion on the Wireless AI applications in various data-driven domains wherein AI proves extremely useful for wireless network design and optimization. Finally, research challenges and future visions are also discussed to spur further research in this promising area. ###### Index Terms: Wireless Networks, Artificial Intelligence, Deep Learning, Machine Learning, Data-driven AI. ## I Introduction Digitization and automation have been widely recognized as the next wave of technological revolution, which is envisaged to create a connected world and fundamentally transform industry, business, public services, and our daily life. In this context, recent years have seen rapid advancements in the deployment of Internet of Things (IoT) networks, which can be mostly attributed to the increasing communication and sensing capabilities combined with the falling prices of IoT devices [1], [2]. According to the Cisco forecast, by 2030 there will be more than 500 billion IoT devices connected to the Internet [3]. In the meanwhile, a vast array of multimedia services are blooming rapidly, while the data traffic is growing at an astonishing pace. Annual global data traffic is predicted to increase threefold over the next 5 years, reaching 4.8 ZB per year by 2022 [4]. Fueled by soaring mobile data and increasing device communication, wireless communication systems are evolving toward next generation (5G) wireless networks, by incorporating massive networking, communication, and computing resources. Especially, future wireless networks should provide many innovative vertical services to satisfy the diverse service requirements, ranging from residence, work, to social communications. The key requirements include the support of up to 1000 times higher data volumes with ultra-low latency (1ms or less than 1ms), and massive device connectivity supporting 10-100x number of connected devices with ultra- high reliability of 99.999% [5], [6]. Such stringent service requirements associated with the increasing complexity of future wireless communications make traditional methods to network design, deployment, operation, and optimization no longer adequate. Past and present wireless communications, regulated by mathematical models, are mostly derived from extant conventional communication theories. Communication system design significantly depends on initial network conditions and/or theoretical assumptions to characterize real environments. However, these techniques are unlikely to handle complex scenarios with many imperfections and network nonlinearities [7]. The future wireless communication networks, where the quality of services (QoS) is far beyond the capabilities and applicability of current modeling and design approaches, will require robust and intelligent solutions to adapt to the high network dynamicity for different services in different scenarios [8]. Figure 1: An illustration of the Wireless AI paradigm. Among the existing technologies, artificial intelligence (AI) is one of the most promising drivers for wireless communications to meet various technical challenges. AI techniques such as Machine Learning (ML), Deep Learning (DL) and Deep Reinforcement Learning (DRL), well known from computer science disciplines, are beginning to emerge in wireless communications with recent successes in complicated decision making, wireless network management, resource optimization, and in-depth knowledge discovery for complex wireless networking environments [9]. It has been proved that AI techniques can achieve outstanding performances for wireless communication applications thanks to their online learning and optimization capabilities in various complex and dynamic network settings. AI also lowers the complexity of algorithm computations enabled by data learning and interactions with the environment, and hence accelerates the convergence in finding sub-optimal solutions compared to conventional techniques [10]. By bringing the huge volumes of data to AI, the wireless network ecosystem will create many novel application scenarios and fuel the continuous booming of AI. Therefore, the integration of AI techniques into the design of intelligent network architectures becomes a promising trend for effectively addressing technical challenges in wireless communication systems. Indeed, the marriage of wireless communications and AI opens the door to an entirely new research area, namely ”Wireless AI” [11], [12]. Instead of heavily relying on datacentres, i.e. cloud servers, to run AI applications, Wireless AI takes advantage of resources at the network edge to gain AI insights in wireless networks. More specifically, AI functions can be deployed over every corner of the wireless IoT networks, including in data generators (i.e. IoT devices), data receivers (i.e. mobile users, edge devices) and during on the wireless data transmissions (i.e. wireless links). This would make the wireless networks fully self-automating all operations, controls and maintenance functions with limited human involvement. Wireless AI thus solves effectively the heterogeneity and dynamicity of future wireless networks by leveraging its intelligent, self-adaptive, and auto-configurable capabilities. Notably, Wireless AI has attracted much attention from both the academia and industry. For example, according to Ericsson, wireless AI will have a significant role in reshaping next-generation wireless cellular networks, from intelligent service deployment to intelligent policy control, intelligent resource management, intelligent monitoring, and intelligent prediction [13]. Major enterprises, such as Apple and Google, have been designing and developing many wireless AI-empowered applications for mobile users (i.e. Face ID and Siri apps as Apple products [14] or Google Assistant tool of Google [15]). These effort have boosted a wide spectrum of wireless AI applications, from data collection, data processing, massive IoT communication to smart mobile services. Therefore, Wireless AI is potentially a game changer not only for wireless communication networks, but also for emerging intelligent services, which will impact virtually every facet of our lives in the coming years. ### I-A Wireless AI Architecture In this paper, we propose a novel Wireless AI architecture for wireless networks as shown in Fig. 1. This new paradigm is positioned to create cognition, intelligence, automation, and agility for future wireless communication system design. As shown in Fig. 1, there are mainly four entities involved in this architecture. * • Firstly, our environment, society, and people are monitored and sensed by a variety of data generators, e.g. sensors and wearable devices to transform the observations and measurements of our physical world into digital via smart IoT devices. * • Secondly, such data will be transferred via wireless or wire-line links to wireless data receivers or directly to the Internet. Typical wireless data receivers include 4G/5G networks, Wi-Fi, LoRa, etc. * • Thirdly, such data will be collected by data custodians via the Internet using cloud services, private cloud or enterprise cloud. A data custodian usually maintains a platform that organizes, stores and provides private and secure data access to data consumers. Note that a data custodian can also assume the role of a data consumer. * • Fourthly, data consumers such as government, individuals, analysts, etc., will access the data so that they can conduct big data mining and harvest useful information to analyze and predict our physical world. Besides, wireless service operators such as base station controllers can also exploit such data to optimize their network for seamless service delivery across data utilities in wireless networks. It is important to note that in Fig. 1 there are two shortcuts originated from the physical world as follows, * • The first shortcut goes from the physical world to the data receivers, skipping the data generators. This framework is also known as wireless sensing because the data receivers directly see the physical world using radio frequency spectrum. Typical applications include people counting, human activity detection and recognition, etc. * • The second shortcut goes from the physical world to the data custodians, skipping both the data generators and the data receivers. This framework allows the data custodians to directly collect data using survey, report, and cross-organization data sharing, etc. Typical applications include national population censuses, tax return lodging, authorized medical record sharing, etc. Conventionally, AI functions sit in the cloud, or in powerful data center owned by data custodians, or in data consumers. In some cases, AI functions may even exist in an edge cloud, close to data receivers, performing edge computing tasks. Generally speaking, those AI functions are mainly designed for big data analysis and most of them follow a traditional paradigm of centralized AI, i.e. bringing data to computation functions for processing. Recently, many concerns begin to rise regarding this centralized AI paradigm, such as user privacy, power consumption, latency, cost, availability of wireless links, etc. Among them is the user data privacy bottleneck caused by the centralized network architecture where IoT devices mainly rely on a third party, i.e. a cloud service provider. Although this model can provide convenient computing and processing services, the third party can obtain illegally personal information, leading to serious information leakages and network security issues. For example, due to the Facebook data privacy scandal in Year 2018, privacy has become an obstacle for many applications involving personal data, such as living habits, healthcare data, travel routes, etc. Hence, enabling useful data collection/analysis without revealing individuals sensitive information has become an urgent task for the research community. Moreover, the traditional centralized AI paradigm shows critical limitations in optimizing wireless networks [19], [20]. In the 5G era with ubiquitous mobile devices and multi-channel communications, the complexity of wireless networks is extremely high. The solution of using only a centralized AI controller to manage the whole network is obviously inefficient to optimize network operations and ensure high-quality service delivery. In this context, pushing AI to the network edge, i.e. bringing computation functions to data, close to data generators and receivers, has been recognized as a promising solution to protect privacy, reduce latency, save power and cost, and increase reliability by less-communication. In this paper, we identify four categories of wireless AI at the network edge and discuss their technologies, recent developments and future trends. More specifically, the four categories of wireless AI are shown in Fig. 1 and briefly explained as follows. * • Sensing AI at the Data Receivers: AI supports well sensing tasks such as people and object detection over wireless networks. By analyzing Wi-Fi CSI characteristics, AI-based people counting and sensing systems can be designed to train the CSI values and generate the model so that people detection can be implemented. For example, convolutional neural network (CNN) is particularly useful in learning human images and recognizing humans among the crowd at different scales [73], [75]. Besides, AI can be used for human motion and activity recognition using wireless signal properties [90]. In fact, the relation between the CSI value and the dynamicity of human motion has been exploited to build the learning model using AI algorithms [100]. * • User and Network Device AI at the Data Generators and Receivers: At the data generators such as sensors and mobile devices, AI functions have been embedded into end devices to create a new paradigm: on-device AI. This would open up new opportunities for simplifying AI implementations in wireless networks by eliminating the need of external servers, i.e. clouds. For example, deep neural network (DNN) inference now can be run on mobile devices such as Android phones for local client applications such as image classification [197], object recognition [199]. At the data receivers like edge servers, AI has been applied in the network edge to learn and detect signals, by embedding an AI-based intelligent algorithm on the base stations or access points. Specially, content catching at the network edge is now much easier thanks to the prediction capability of AI. Moreover, AI also has the potential in edge computing management such as edge resource scheduling [111] and orchestration management [116]. * • Access AI in the Data Transfer Process: AI can be applied to facilitate the data transmission from three main layers, including physical (PHY) layer, media access control (MAC) layer, and network layer. For example, in PHY layer, AI has been applied to facilitate CSI acquisition and estimation [125] that are vitally important for improving data transfer process in terms of channel allocation, traffic congestion prediction. Further, AI also supports channel allocation, power control and interference mitigation functions in MAC layer. In fact, the learning capability of AI is particularly useful in estimating traffic channel distribution to realize flexible channel allocation [146] and channel power management [152], aiming at minimizing channel interference in wireless networks. Specially, AI is able to empower spectrum sharing, data offloading, multi-connectivity, and resource management services from the network layer perspective. For example, AI, especially deep learning has the potential to estimate dynamically a spectrum resource management policy in the network by online adaptive learning to achieve a reliable and efficient spectrum sharing among network users [177], [178]. * • Data-provenance AI in the Data Generation Process: In the data generation process, AI can provide various useful services, such as AI data compression, data privacy, and data security. As an example, AI has been applied for supporting sensor data aggregation and fusion, by classifying useful information from the collected data samples [224]. Also, the classification of AI helps monitor the data aggregation process, aiming to analyze the characteristics of all data and detect data attacks [234]. In addition to that, AI has been an ideal option to recognize legitimate users from attackers by training the channel data and performing classification [275]. Meanwhile, DRL has proved extremely useful in building attack detection mechanisms [244], [245] that can learn from the user behaviours and estimate the abnormal events for appropriate security actions. ### I-B Comparisons to Other Related Works and Our Contributions TABLE I: Existing surveys on AI and wireless networks. Related Surveys | Topic | Key contributions ---|---|--- [16] | Edge computing for AI | This work mainly focused on the survey on the use of edge computing for supporting AI, including edge computing for AI training, edge computing for AI model inference. [17] | Edge computing for ML | This work focused on the analysis on the edge ML architectures and presented how edge computing can support ML algorithms. [18] | AI for heterogeneous networks | This study provided a very brief review on the use of AI techniques for heterogeneous networks. They analyzed each AI approaches and then showed the general potentials in Hetnet, but no clear use cases were given. [19] | DL for edge computing | This work discusses the roles of DL in edge computing-based applications and analyses the benefits of edge computing in supporting DL deployment. [20] | DL for mobile networks | This survey focused on the discussion on the integration of DL and mobile networks. The authors focused on the survey related to the following aspects: DL for Mobile Data Analysis, DL for User Mobility Analysis, DL for User Localization, DL for Wireless Sensor Networks, and DL for Network Security. [21] | DL for IoT data analytics | This survey focused on the discussion on the integration of DL and IoT data analytics. The main topics includes the analysis on the use of DL for IoT devices, and DL for fog/cloud computing. [22] | DL for network traffic control | This survey only focused on the discussion on the use of DL for network traffic control. [23] | DL for wireless networks | This survey paid attention to performing in-depth quantitative analysis of several applications of DL for the design of wireless networks. [24] | DRL for communication networks | This survey focused on the discussion on the integration of DRL and communication networks. They focused on reviewing the benefits of DRL in solving communication issues, including network access and control, content catching and offloading, network security and privacy, resource sharing, and data collection. [25] | ML for IoT wireless communications | This work presents a survey of ML applications to address some key problems in IoT wireless communications such as interference management, spectrum sensing and the future use on the IoT beyond communication. [26] | ML for wireless networks | The authors mainly concentrated on the use of ML for some domains in wireless networks, including resource management, networking, mobility management, and localization. [27] | ML for wireless networks | The authors mainly concentrated on the analysis on the roles of AI in wireless networks from the physical layer and the application layer. [28] | ML for optical networks | The authors mainly concentrated on the use of ML in optical networks. Our paper | Data-driven AI for wireless networks | We propose a novel Wireless AI architecture for wireless communication networks with five key data-driven AI domains, including Sensing AI, Network Device AI, Access AI, User Device AI and Data-provenance AI. Then, we focus on analyzing the role of AI in each each domain from the data perspective. Specially, we provide an in-depth survey on the Wireless AI applications with various related use cases. Driven by the recent advances of AI in wireless networks, some efforts have been made to review related works and provide useful guidelines. Specifically, the work in [16] presented a survey on the basics of AI and conducted a discussion on the use of edge computing for supporting AI, including edge computing for AI training, and edge computing for AI model inference. Similarly, the authors in [17] analyzed the edge ML architectures and presented how edge computing can support ML algorithms. Furthermore, the potential of AI in heterogeneous networks (Hetnet) was investigated in [18] with a focus on a review of AI approaches for wireless networks, but did not pay attention to AI-based applications. Meanwhile, DL which has been an active research area in AI, is integrated in wireless systems for various purposes. For example, the authors in [19] discussed the roles of DL in edge computing- based applications and then analyses the benefits of edge computing in supporting DL deployment. The study in [20] presented the discussion on the integration of DL and mobile networks. The authors focused on the survey on various aspects, including DL for mobile data analysis, user mobility analysis, user localization, wireless sensor networks, and network security. The authors in [21], [22] studied the potentials of DL in enabling IoT data analytics (including surveys on the use of DL for IoT devices, and the performance of DL-based algorithms for fog/cloud computing) and network traffic control, while the authors in [23] mainly concentrated on the quantitative analysis of DL applications for the design of wireless networks. DRL, an emerging AI technique recent years, has been studied for wireless networks in [24]. This survey focused on the discussion on the integration of DRL and communication networks, with a holistic review on the benefits of DRL in solving communication issues, including network access and control, content catching and offloading, network security and privacy, resource sharing, and data collection. Meanwhile, the benefits and potentials of ML in wireless communications and networks were surveyed in recent works [25], [26], [27], [28] in various use cases, such as resource management, IoT data networks, mobility management and optical networking. The main contributions of related AI survey works are summarized in TABLE I. Despite the significant progress of surveying wireless AI applications, some limitations have remained to be considered in current works. * • Most existing works mainly focused on each of the specific AI techniques (ML/DL or DRL) for wireless networks. * • Most existing surveys on ML/DL only focused on some certain network scenarios in wireless communication from either data transmitters [16], [17] or data receivers [18], [19] perspectives. * • There is no any existing survey working on the use of AI for wireless communication networks from the data-driven perspective, i.e. data sensing AI, data access AI, data device AI, and data-provenance AI. Considering these limitations and the ongoing research activities, we provide a more comprehensive survey, which incorporates the recent achievements in the applications of diverse AI techniques in a wide range of network scenarios. To this end, our survey provides the following contributions: 1. 1. We provide an overview of the state-of-the-art AI techniques used in wireless communication networks. 2. 2. We propose a novel Wireless AI architecture that covers five key data-driven AI themes in wireless networks, including Sensing AI, Network Device AI, Access AI, User Device AI and Data-provenance AI. 3. 3. For each data-driven AI theme, we present an overview on the use of AI approaches to solve the emerging data-related problems and show how AI can empower wireless network functionalities. Particularly, we provide an in-depth discussion on the Wireless AI applications in various data-driven use cases. The key lessons learned from the survey of each domain are also summarized. 4. 4. We provide an elaborative discussion on research challenges in wireless AI and point out some potential future directions related to AI applications in wireless networking problems. ### I-C Organization of the Survey Paper The structure of this survey is organized as follows. Section II presents an overview of state-of-the-art AI techniques including ML, DL, and DRL. Based on the proposed wireless AI architecture outlined in the introduction section, we start to analyze the wireless AI applications in communication networks with five key domains: Sensing AI, Network Device AI, Access AI, User Device AI and Data-provenance AI. More specifically, we present a state-of-the-art review on the existing literature works in the Sensing AI area in Section III. We classify into three main specific domains, namely people and objection detection, localization, and motion and activity recognition. Section IV presents a state-of-the-art survey on the Network Device AI domain, highlighting the development of AI for supporting wireless services on edge servers (i.e. base stations). Next, Section V provides an extensive review for the Access AI domains, with a focus on the discussion on the roles of AI in PHY layer, MAC layer, and network layer. We then analyze the recent developments of the User Device AI domain in Section VI, by highlighting the development of AI for supporting wireless services on user devices. Meanwhile, Section VII presents a discussion on the recent use of Data-provenance AI applications in the data generation process in various domains, namely AI data compression, data clustering, data privacy, data security. Lastly, we point out the potential research challenges and future research directions in Section VIII. Finally, Section IX concludes the paper. A list of key acronyms used throughout the paper is presented in TABLE II. TABLE II: List of key acronyms. Acronyms | Definitions ---|--- AI | Artificial Intelligence ML | Machine Learning DL | Deep Learning DRL | Deep Reinforcement Learning SVM | Support vector machine SVR | Support vector regression k-NN | k-neural networks DNN | Deep Neural Network CNN | Convolutional Neural Network LSTM | Long-short Term Memory RNN | Recurrent Neural Networks DQN | Deep Q-network FL | Federated learning PCA | Principal component analysis MDp | Markov decision processes MAC | Media Access Control IoT | Internet of Things MEC | Mobile edge computing UAV | Unmanned aerial vehicle MIMO | Multiple-input and multiple-output NOMA | Non-orthogonal multiple access OFDM | Orthogonal Frequency-Division Multiplexing CSI | Channel state information V2V | Vehicle-to-vehicle D2D | Device-to-Device M2M | Machine-to-Machine MTD | Machine type device BS | base station MCS | Mobile Crowd Sensing QoS | Quality of Service SINR | Signal-to-interference-plus-noise ratio ## II AI technology: State of the Art Reviewing the state-of-the-art studies related to the wireless AI topic, we find that there are three key AI techniques used for intelligent wireless communication applications, including ML, DL and DRL. The literature on AI is very extensive, so the detailed AI survey is beyond this section. The readers are suggested to refer to a number of fundamental books and tutorials [29], [30], [31], [32], [33]. Here, we focus on reviewing the recent AI developments with discussions on some important AI approaches that have been used in the wireless domains. ### II-A Machine Learning This sub-section covers analysis on some of the most researched and applied ML algorithms to wireless communications, including supervised learning, semi- supervised learning, unsupervised learning, and reinforcement learning. #### II-A1 Supervised learning Supervised learning, as the name implies, aims to learn a mapping function representing the relation of the input and the output under the control of a supervisor with a labelled data set. In this way, the data model can be constructed so that each new data input can be learned by the trained model for making decisions or predictions [34]. Supervised learning is a very broad ML domain. Here we pay attention to some techniques that have been employed in wireless AI systems, including support vector machine, K-nearest neighbors, decision trees. \- Support vector machine (SVM): The SVM utilizes a part of dataset as support vectors, which can be considered as training samples distributed close to the decision surface. The main idea behind SVM is to use a linear classifier that maps an input dataset into a higher dimensional vector. The aim of its mapping concept is to separate between the different classes by maximizing their distance. In SVMs, a nonlinear optimization problem is formulated to solve the convex objective function, aiming at determining the parameters of SVMs. It is also noting that SVMs leverage kernel functions, i.e. polynomial or Gaussian kernels, for feature mapping. Such functions are able to separate data in the input space by measuring the similarity between two data points. In this way, the inner product of the input point can be mapped into a higher dimensional space so that data can be separated [35]. \- K-nearest neighbors (k-NN): Another supervised learning technique is K-NN that is a lazy learning algorithm which means the model is computed during classification. In practical applications, k-NN is used mainly for regression and classification given an unknown data distribution. It utilizes the local neighbourhood to classify a new sample by setting the parameter K as the number of nearest neighbours. Then, the distance is computed between the test point and each training point using the distance metrics such as Euclidean or Chebyshev [36]. \- Decision trees: Decision trees aim to build a training model that can predict the value of target variables through decision rules. Here, a tree- based architecture is used to represent the decision trees where each leaf node is a class model, while the training is set as the root. From the dataset samples, the learning process finds the outcome at every leaf and the best class can be portioned [37]. #### II-A2 Unsupervised Learning Unsupervised learning is a ML technique that learns a model to estimate the output without the need of labelled dataset. One of the most important unsupervised learning algorithms used in wireless networks is K-means that aims to find the clusters from the set of unlabelled data. The algorithm implementation is simple with two required parameters, including original data set and the target number of clusters. Another popular unsupervised learning algorithm is Expectation-Maximization (EM) that is used to estimate the values of latent variables (which cannot be observed directly) under the known variable probability distribution. The importance of EM algorithms is shown via some use cases, such as estimation for parameters in Hidden Markov Models, finding the missing data in the data sample space, or support for clustering tasks [38]. #### II-A3 Semi-supervised learning As the intermediate technique between supervised learning and unsupervised learning, semi-supervised learning uses both labelled and unlabelled samples. The key objective is to learn a better prediction rule using the unlabelled data than that based on the data labelling in supervised learning that is always time-consuming and easy to bias on the model. Semi-supervised learning is mainly used for classification tasks in computer visions, image processing [39]. Figure 2: Future AI functions in wireless networks. #### II-A4 Reinforcement Learning In Reinforcement Learning (RL), there is no need to define the target outcomes. This method is based on the concept of environment interaction which means that an agent directly interacts with the surrounding environment, sense the states and select appropriate actions [40]. As a result, the agent can get a reward after an action is taken. The aim of the agent is to accumulate the long-term reward as much as possible that corresponds the objective in wireless networks, such as system throughput optimization. The most popular RL technique is Q-learning that uses the Q functions to adjust its policy to seek an optimal solution via trial and error without the need of prior environment knowledge. This concept can be applied to wireless scenarios where the system model is hard to find due to the lack of mathematical formulation, such as real time network optimization [41], mobile computation offloading [42]. ### II-B Deep Learning Deep Learning (DL) includes supervised or unsupervised learning techniques that uses multiple neural network layers in a deep architecture [43]. The neural network consists of neurons connected via weighted connections among layers. A basic deep neural network (DNN) contains an input layer for receiving data samples, a single hidden layer for training and an output layer for generating training outcomes. Here, the number of hidden layers reflects the depth of the DNN architecture. To generate the desired output, supervised or unsupervised learning techniques are used with the labeled or unlabeled data samples, associated with the adjustment of weight values among perceptrons. DL is known for AI applications such as speech recognition, computer vision, image processing, object detection [44]. In the literature of wireless network topics, some of the most popular DL architectures are adopted including Convolutional Neural Networks (CNN), Recurrent Neural Networks (RNN), Long-short Term Memory (LSTM) Networks. #### II-B1 Convolutional Neural Network (CNN) CNN is a type of neural network used mainly for image processing and recognition with large pixel datasets [45]. Basically, the CNN architecture is similar to a deep neural network, but adding two new convolutional and pooling layers in between hidden layers. The convolution layer is regarded as the core component of a CNN, responsible to process the input samples based on a convolution operation to extract the features during the sample compression thanks to a set of filters. Another block of a CNN is the pooling layers that aim to make the spatial size of the representation smaller for reducing the computing overheads and mitigating overfitting. In this fashion, the output of previous layer can be combined with a perceptron of the next layer. Lastly, the extracted features are further processed in the final layer until reaching the output layer. Deep CNNs have been proved with many successes in wireless communication applications, such as massive MIMO [46], modulation classification in wireless networks [47]. #### II-B2 Recurrent Neural Network (RNN) RNNs can be regarded as a deep generative architecture since they can be considered as several non-linear neuron layers between the input layer and the output layer in an unfolded fashion [48]. The length of RNN is highly correlated to the length of data sequence and thus RNNs are well suitable for modelling the data sequence with text data or audio time series. This is based on the RNN training procedure where the training process is performed from the interactions between multilayer perceptrons in a feedback loop. That means the hidden layer is chained with the others via a loop to create a chained training structure. This operation is based on the memory capability of each neuron to store information of computation from the previous data input. Some RNN-based wireless applications can be fault detection in wireless sensor networks [49], wireless channel modelling [50]. #### II-B3 Long Short Term Memory (LSTM) LSTM is an extended version of RNN. Unlike the conventional RNN, LSTM operates based on the gate concept [51]. It keep information of the recurrent network in the neuron or gated cell that has different types of gates such as read, write and erasure gate. Different from digital logic gate in computers, these gates are analog and formulated using element-wise multiplication by sigmoids ranging from 0 to 1. Analog gates are able to be differentiable that is well suitable for backpropagation tasks. These gates are operated similar to the neurons of neural networks. The cells perform data learning via an iterative process to adjust the weights via gradient descent. RNN has been applied in several wireless scenarios such as orthogonal frequency division multiplexing (OFDM) [52], IoT data analysis [53]. ### II-C Deep Reinforcement Learning Reinforcement Learning is a strong AI technique, which allows a learning agent to adjust its policy to seek an optimal solution via trial and error without the need of prior environment knowledge. However, when the dimension of state and action space explodes, the RL-based solutions can also become inefficient. Deep Reinforcement Learning (DRL) methods such as deep Q-network (DQN) have been introduced and demonstrated their superiority for solving complex problems [54]. Specially, instead of storing the variables in the table as the RL technique, DRL leverages a deep neural network as the approximator to store high-space actions and states. DRL is mainly used for model optimization, classification, and estimation where a decision making problem is formulated by using an agent and interactive environment. At present, DRL in wireless is a very active research area where some popular DRL techniques such as deep Q-learning, double DRL, duelling DRL have been used widely for specific applications such as data offloading [55], [56], channel allocation [57]. ### II-D Visions of Wireless AI The aforementioned AI techniques would have the great potential in supporting wireless networks from two perspectives: signal processing and data processing. Here, signal processing aims to increase the data throughput, while data processing focuses on producing data utility for specific applications. In fact, AI functions are particularly useful for signal processing tasks through signal learning and estimation. In general, signal processing takes advantages of signal parameters to recognize and classify types of signal in involved applications. Some AI functions for signal processing have been recently explored, such as channel interfere signal detection, frequency and bandwidth estimation, modulation recognition [20], [23]. AI is able to facilitate feature selection, an important domain in signal processing, for extracting instantaneous time features, statistical features or transform features. In practice, feature selection from the received signals is particularly challenging due to the lack of prior information of signal parameters and the uncertainty of signal time series. Feature engineering could be left out by using AI architectures to recognize effectively important signal features such as amplitude, phase, and frequency that are important for many signal-related domains, i.e. spectrum sharing [127], channel coding [81] or edge signal detection [195]. Moreover, AI functions are also particularly useful for data processing. In the era of IoT, data exhibits unique characteristics, such as large-scale streaming, heterogeneity, time and space correlation. How to obtain hidden information and knowledge from big IoT data is a challenge. AI can come as a promising tool for recognizing and extracting meaningful patterns from enormous raw input data, aiming to create a core utility for big data processing in wireless applications such as object detection, big data analysis or security. For example, big data collected from ubiquitous sensors can be learned using DL for human/object recognition [259]. In such scenarios, AI is able to extract useful features which represent most the target object, and then classified using DL architectures, i.e. CNN or DNN. The classification capability of AI is also beneficial to data processing tasks for network security enhancements. Recent works [284], [285] show that a DL network is able to detect intrusions and threats in IoT datasets so that modified or unauthorized data can be recognized and eleminated for better security of data-driven applications. The migration of computation power from the core to the edge has already begun. Wireless AI will further lead such an exodus beyond the edge, illuminating a future with pervasive and end-to-end AI. In the next sections, we conduct a detailed survey on AI functions in wireless networks following a migration trail of computation functions from the edge to the source data. Such AI functions include Sensing AI, Network Device AI, Access AI, User Device AI and Data-provenance AI as shown in Fig. 2. ## III Sensing AI As shown in the Wireless AI architecture in Fig. 2, the first Wireless AI function of the data life cycle is Sensing AI that is responsible to sense and collect data from the physical environments using sensors and wearable devices, followed by smart data analytics using AI techniques. In this section, we present a state-of-the-art review on the existing literature studies working around the Sensing AI function for wireless networks as shown in Fig. 3. Here, Sensing AI focuses on three main domains, namely people and objection detection, localization, and motion and activity recognition. ### III-A People and object detection #### III-A1 People detection Here we focus on discussing the use of AI in two people detection domains, including people counting and mobile crowd sensing. Figure 3: Sensing AI function. 1.1) People counting: Problem formulation: People counting provides essential information for various pervasive applications and services, e.g., crowd control for big events, smart guiding in events, indoor monitoring. It is important to support crowd analytics for human crowd size estimation and public space design. Drawbacks of conventional methods: Most of the existing work uses computer vision, infrared sensors and devices attached on people such as RFID and smart phones. However, infrared sensors deployed at the doorway count the entering or exiting people based on the assumption that people are moving through one by one under certain time interval. Vision-based approaches count people by image object detection, while its performance can be weakened by object overlapping, poor lighting conditions and dead zones [58]. Unique advantages of using wireless AI techniques: The use of AI would enable smart people counting with light-weight and better accurate human information analytics [59]. Recent works: We here summarize some representative liturature studies working on AI techniques for people counting, mainly using linear regression and SVR methods, AI-based crowdsourcing methods, and DL methods. * • Linear regression and SVR methods: In order to improve the accuracy of the people counting, ML methods such as linear regression and SVR have been proposed for estimating the number of people using the existing Wi-Fi access points in indoor environments [60]. The experiment result indicates that SVR yields the better accuracy in estimating people numbers. SVR is also applied for pedestrian counting from captured video images [61]. To improve the efficiency of feature extraction, a Histogram of Oriented Gradient technique is designed that can capture more meaningful information from the raw images for better human movement recognition. SVR is also used to learn the estimation function modelled from the extracted features for people counting tasks [62]. Radar sensor BumbleBee is employed to detect the human movement images, and features extracted from the images are classified into three domains, i.e. time, frequency, and joint time-frequency domains. Here, an experiment is set up with 60 sessions of data collected in room settings with 43 people hired to do a trial. K-fold cross validation is used to evaluate the feasibility of the proposed scheme, showing a high correlation and low error with baseline methods. * • Crowdsourcing methods: Another AI-based approach for people counting is proposed in [63] that implements a people counting model called Wi-Fi-counter using smartphone and Wi-Fi. The Wi-Fi-based counter scheme employs a crowdsourcing technique to gather the information of people and Wi-Fi signals. However, the useful information between of them that is necessary to estimate people count may be hidden. Then a five-layer neural network architecture is built, including two main phases: the offline phase for computing the eigenvalues from Wi-Fi signals at the access point and the online phase for collecting Wi-Fi data that is then used for training the model. The performance of Wi-Fi-counter model is mainly evaluated via counting accuracy and system power usage overheads. Meanwhile, the people counting project in [64] investigates the efficiency of Random Forest technique. The number of people at train stations is estimated from images captured from CCTV videos. Instead of using features extracted randomly that is time-consuming, each captured image is divided into sub-windows for fast human recognition during the learning. * • DL methods: In general, ML has been applied and achieved certain successes in people counting over wireless networks. However, in fact the performance in terms of accuracy, efficient large-scale training and the ability to solve complex counting tasks with variety of data samples still need to be improved [65]. DL can be an answer for facilitating future people counting applications. For example, the authors in [66] work in developing a new crowd counting model, call Wicount, using Wi-Fi devices. The dynamicity of human motions makes CSI values changeable over time. Based on this concept, the CSI characteristics are captured, including amplitude and phase information. A critical challenge is to find a mathematical function which represents the relationship between CSI values and people count due to the varying number of people. To solve this issue, a DL solution is proposed to train the CSI values and generate the model so that people counting can be implemented. A cross- scene crowd counting framework is presented in [67]. The key focus is on mapping the images captured by camera sensors and people counts. To solve the issue in people counting in the case of surveillance crowd scenes unseen in the training, a CNN architecture is proposed to train the samples iteratively and a data-driven technique is used to fine-tune the trained CNN model. 1.2) Mobile Crowd Sensing: Problem formulation: Mobile Crowd Sensing (MCS) is a large-scale sensing paradigm empowered by the sensing capability of mobile IoT devices, such as smartphones, wearable devices. MCS enables mobile devices to sense and collect data from sensors and share with cloud/edge servers via communication protocols for further processing. The high mobility of mobile devices makes MCS flexible to provide different ubiquitous services, such as environment monitoring, human crowd monitoring, traffic planning [68]. Drawbacks of conventional methods: Existing approaches like dynamic programming have tried to assume MCS as a static model where the motion of mobile devices are assumed to follow in a predefined trajectory in the static environement [68]. In fact, MCS tasks should be always considered in time- varying environment and the uncertainty of mobile devices, which make current MCS solutions inefficient. Unique advantages of using wireless AI techniques: AI helps to realize the potential of MCS and overcome challenges in crowd sensing applications in terms of high MCS system dynamicity, sensing network complexity and mobile data diversity. Recent works: Several ML and DL approaches have been used to faciliate MCS applications. * • MCS using ML approaches: For example, the work in [69] considers a robust crowd sensing framework with mobile devices. The key objective is to solve the uncertainty of sensing data from the mobile user participants and optimize the sensing revenue under a constrained system budget. In practical scenarios, a sensing task owner often selects randomly the member who can gather the most information among all participants for maximizing the profits. However, it may be infeasible in the context where data quality is uncertain which makes crowdsensing challenging. To solve this uncertainty issue, an online learning approach is applied to obtain the information from sensing data thanks to a selection step without knowing the prior statistics of the system. The authors in [70] suggest to use ML for feature selection from similar metrics in mobile sensing data (i.e. Wi-Fi signal strength and acceleration data). To detect human movement patterns, two techniques: individual following and group leadership are proposed that use an underlying ML algorithm for identifying features that can classify human groups. Meanwhile, unsupervised ML is investigated in [71] to estimate the number of people from audio data on mobile devices. An application called Crowd++ is implemented on Android phones to collect over 1200 minutes of audio from 120 participants. To estimate the number of target object, namely active speakers in the crowd, a ML process is taken with three steps: speech detection, feature extraction and counting. * • MCS using DL approaches: Recently, DL has been investigated to facilitate MCS ecosystems [72]. How to achieve high-quality data collection in terms of optimizing collected data with lowest energy consumption is important for MCS applications in smart cities where mobile terminals equipped with sensors to gather ubiquitous data. Therefore, a DRL approach is proposed to create a movement and sensing policy for mobile terminals so that the efficiency of data collection is maximized. Here, each mobile terminal acts as an agent to interact with the MCS environment to find the best trajectory for data sensing and collection. Another DL approach is introduced in [73] where a joint data validation and edge computing scheme is proposed for a robust MCS. Stemming from the fact that the current MCS models suffer from several limitations, lack of incentive mechanism for attracting more members for data sensing, lack of validation of collected data, and network congestion due to high transmit data volumes. Thus, DL based on a CNN network for pattern recognition is adopted to detect unimportant information and extract useful features for lightweight data sensing. Further, edge computing is necessary to provide a low-latency data processing for MCS, associated with an incentive mechanism to motivate more users in collecting data. DRL is also applied in [74] for mobile crowd sensing in smart cities based on unmanned aerial vehicle (UAVs). To maximize the data collection rate, a joint adaptive energy allocation and task routing scheme is considered with charging stations which provide a solution for replenishing the energy of UAVs. In this context, how to balance the efficiency of data collection and energy preservation for UAVs is crucial. Unlike the traditional approaches that formulate MCS as a constrained optimization problem, here DRL-based DL is leveraged for UAV trajectory modelling by observations, training and action decision making in an iterative manner. #### III-A2 Object detection In this subsection, we study the roles of AI in object detection. Problem formulation: As one of the key computer vision areas, object detection plays an important role in realizing the knowledge and understanding of images and videos, and is used in various applications, such as image/video classification, object analytics or object motion tracking. Drawbacks of conventional methods: Traditional object detection approaches mostly use a three-step model, with the adoption of informative region selection, feature selection, and classification using mathematical models [75]. For example, a multi-scale sliding window is often used to scan the image to detect target objects, but it is computationally expensive and time- costing. Further, due to the high similarity of visual features, current solutions may be unlikely to represent exactly the detection models and mis- classify objects. Unique advantages of using wireless AI techniques: Recent years, many research have made attempts to use AI for object detection thanks to its higher prediction accuracy but low computational complexity. Recent works: Object detection (i.e. salient object detection) employ deep neural networks to recognize the distinctive regions in an image. The work in [76] uses a fully CNN architecture for salient object detection. A new encoder-decoder network is created where a global perception module is constructed to produce a low-resolution saliency map of the object. Another work in [77] presents a recurrent fully convolutional network for better accurate saliency detection. This is enabled by an iterative correction process during learning, which refines better saliency maps and brings a better performance in terms of higher prediction accuracy. The authors in [78] focus on designing a cascaded partial decoder mechanism which helps to remove shallower features and refine the features of deeper layers in the CNN network for obtaining precise saliency maps. Based on that, the proposed scheme can achieve higher prediction accuracy but lower algorithm running time. To reduce the training complexity and improve the evaluation speed in traditional neural network in salient object detection, a simplified CNN network is introduced in [79]. Instead of minimizing the boundary length as the existing Mumford-Shah approaches have implemented, the focus of this work is to minimize the intersection over union among the pixels of the images. This non-reliance on super-pixels makes the method fully convolutional and thus enhances the estimation speed. Besides, a hybrid contrast-oriented CNN architecture is investigated in [80] to solve the issues of the computation latency and redundancies in current CNN networks. The fully convolutional networks are devised in the first stream for fast learning of the image to map accurately it with dense saliency prediction. Also, a segment-level spatial pooling (SP) stream complementary is integrated for sparse saliency inference and modelling visual contrast among super-pixels. ### III-B Localization Figure 4: DL for CSI-based human activity classification. Problem formulation: Localization aims to determine the geographic coordinates of network nodes and objects [81]. It is important to various applications, such as object tracking, human recognition or network resource optimization. Drawbacks of conventional methods: In many scenarios, it is infeasible to use global positioning system (GPS) hardware to specify where a node is located, especially in indoor environments. Distance of nodes can be calculated using mathematical techniques such as RSSI, TOA, and TDOA [82]. However, the location of nodes may be changeable over time due to movements. Unique advantages of using wireless AI techniques: AI comes as a promising solution to estimate effectively locations of objects/nodes due to some benefits. First, AI can approximate the relative locations of nodes to absolute locations without the need of physical measurements. Second, in the complex dynamic network, AI can help to divide into sub-classes using classifiers, which reduces the complexity of localization problem for better node estimation. Here, we summarize some recent research results on the use of AI in localization tasks. Recent works: Some important works are highlighed with the exploitation of AI for localization tasks, by using ML and DL approaches. * • ML-based localization: The work in [83] presents a sensor node localization scheme using ML classifiers. Data is collected with time-stamp from four capacitive sensors with a realistic room setting. A variety of ML classifiers such as Bayes network, random forest, and SVM are used to evaluate the performance of indoor localization, showing that random forest performs best overall. The authors in [84] propose a non-linear semi supervised noise minimization algorithm for sensor node localization via iterative learning. The collected labelled and unlabelled data are expressed as a weighted graph. The localization of wireless sensor node can be considered as semi supervised learning. Another work in [85] implements a neural network on Android phones for indoor localization. The system is a two-layer model, where the first layer is a neural network coupled with random forest, while the second one is a pedometer for accelerating the data sampling process. Different from [85], the work in [86] focuses on using ML for supporting floor localization in a multi-access point-based floor setting. To eliminate the redundancy information during the data collection, principal component analysis (PCA) is employed in the pre-processing step. Then, an extreme learning machine approach is taken to for model learning to build a classification function which is used for estimating the floor location. Recently, the use of ML for coarse localization in human activity recognition tasks is also investigated [87]. In this work, data as micro-Doppler signatures that are collected from radar sensors is classified using SVM and k-nearest neighbors (kNN) classifiers. To further improve the accuracy of classification, a CNN architecture is designed with respect to angles of object movement, and numbers of radars. * • DL-based localization: The potential of DL in supporting localization applications has been proven via recent successes. The work in [88] explores a solution for device-free wireless localization in wireless networks. The discriminative features are first extracted from wireless signals and learned by a three-layer deep neural network. Using the extracted features from the DL model, a ML-based softmax regression algorithm is then employed to estimate the node location in indoor lab and apartment settings. Meanwhile, a deep extreme learning machine technique is presented in [89] for fingerprint localization based on signal strength of wireless sensor networks. Instead of using random weight generation, neural network-based autoencoder is used for sample training in the encoding and decoding manner. This allows to extract better features and then localize the node better. DL is also considered in [90] to support the estimation of target locations in IoT networks. More specific, a CNN model is developed that learns the amplitude of channel frequency from the Wi-Fi CSI signals collected by a gateway. The model is tested in real room environments and shows high localization precision. CSI data is also exploited in [91] to estimate the object location. From the CSI packets, random forest is used for selecting key features, including variance, median, and maximum, and minimum amplitudes of each frequency band. These features are fed to a CNN model using Keras to train the model for location estimation. Recently, DRL has been applied to facilitate localization tasks [92]. To formulate the localization problem as a MDP, which is the key structure of DRL approaches, a continuous wireless localization process is derived. The localization task is divided into time slots where the agent inputs the location information at the previous time slot and outputs the new location in the next slot. Therefore, the computation of the location at each step only depends on the location at the previous step, which makes it appropriate to formulate a MDP problem. Then, a deep Q-learning algorithm is built to solve the proposed problem, aiming to minimize the error of localization estimations. ### III-C Motion and Activity Recognition Problem formulation: In social networks, human motion data may be very large and activity parterns are changable over time. How to deal with a large-scale datasets for motion and activity recognition tasks is a challenge. Drawbacks of conventional methods: Many traditional techniques have assumed to have prior knowledge of the CSI or environment information for motion recognition [93], [94], but it is hard to achieve in practice. Moreover, the samples collected from human motion activities are often very large associated with the complexity of multiple motion patterns, which make current approaches such as dynamic programming inefficient to model exactly the motion estimation task so that the problem can be solved effectively. Unique advantages of using wireless AI techniques: Thanks to high classification accuracy and efficient online learning based on scalable big datasets, AI has been employed for human motion and activity recognition based on wireless signal properties for better accuracy of activity recognition. Recent works: In [93], the authors study the CSI retrieved from the host computer to estimate and classify the four body motions: normal gait, abnormal gait (ataxia) [95]. The main idea is to analyze the relation between the CSI value and the change of human motion. Then, a system setting with an access point and network interface for CSI detection is considered so that gait abnormality and hand tremors can be estimated using a ML algorithm. The authors in [96] develop a motion detection system using Wi-Fi CSI data. The system relies on the variance of CSI to detect the human motion that is empowered by a moving filter in supervised ML algorithm. Compared to traditional motion recognition approaches using wearable sensors, radars that require hardware settings and management efforts, motion detection based on Wi-Fi signals is much more flexible, cheaper due to using the available infrastructure. Moreover, a multiple access point-based CSI aggregation architecture is introduced in [97], aiming to estimate human motion. A large-scale Wi-Fi CSI datasets from the data collection process are fed to a multi-layer CNN model for training and analyzing the relations between CSI strengths and levels of human motion. In comparison to SVM-based training, CNN provides a better accuracy in activity recognition. As an investigation on the influence of human motions on Wi-Fi signals, the work in [98] suggest a DL network for image processing. From the measures on multiple channels, CSI values are transformed to a radio image, which is then extracted to generate features via a DNN learning process and trained for motion classification with ML. A DL model for CSI-based activity recognition can be seen in Fig. 4. As an further effort for improving the feature selection from CSI datasets in human activity recognition, an enhanced learning model is suggested in [99]. In fact, the data collected from Wi-Fi signals contains redundancy information that may not be related to human motion, and thus needs to be remoted for better estimation. Motivated by this, a background reduction module is designed to filter unrelated motion information from the original CSI data. Then, correlation features are extracted from the filtered CSI data. To this end, a DL-based RNN is employed to extract the deeper features via offline training. Specially, LSTM is integrated with RNN to achieve better recognition accuracy with lower computational complexity. Moreover, a Wi-Fi CSI collection framework based on IoT devices is introduced in [100] for recognizing human activity. To learn effectively the features extracted from CSI datasets, a joint DL model of autoencoder, CNN and LSTM is designed. Here, autoencoder aims to eliminate the noise from CSI data, CNN extracts the important features from the output of autoencoder, while LSTM extracts inherent dependencies in the features. Compared to existing feature selection techniques, the proposed scheme can achieve a much lower algorithm running latency with better recognition accuracy. ### III-D Lessons learned The main lessons acquired from the survey on the Sensing AI domain are highlighted as the following. * • AI has been applied and achieved certain successes in people and object detection over wireless networks. By analyzing Wi-Fi CSI characteristics, AI- based people counting and sensing systems can be built to train the CSI values and generate the model so that people detection can be implemented. We have also observed that CNN is particularly useful in learning human images and recognizing humans among the crowd at different scales. Besides, many recent researches have made attempts to use DL for object detection, with a focus on two domains: salient object detection and text detection. We also find that most studies leverage deep CNN as the classifier for better accurate object recognition thanks to its fully convolutional networks [79], [101], [102]. * • Moreover, AI helps estimate effectively locations of objects/nodes due to some benefits. First, AI can approximate the relative locations of nodes to absolute locations without the need of physical measurements. Second, in the complex dynamic network, AI can help to divide into sub-class using classifiers, which reduces the complexity of localization problem for better node estimation. Many research efforts have been made to apply both ML and DL in supporting localization applications in mobile IoT networks [89], [92]. * • Lastly, the use of AI for human motion and activity recognition using wireless signal properties has been realized. The relation between the CSI value and the change of human motion has been exploited to build the learning model using AI algorithms. Especially, to learn effectively the features extracted from CSI datasets for human detection tasks, a joint DL model of autoencoder, CNN and LSTM is designed [100]. Here, autoencoder aims to eliminate the noise from CSI data, CNN extracts the important features from the output of autoencoder, while LSTM extracts inherent dependencies in the features. This triple AI combination is promising to improve recognition accuracy and reduce the training latency, which is important for wireless AI deployments in realistic scenarios. In summary, we list Sensing AI use cases in the taxonomy TABLE III to summarize the key contributions of each reference work. TABLE III: Taxonomy of Sensing AI use cases. Category | Ref. | Use case | AI techniques applied | Main contributions ---|---|---|---|--- People and object detection | [60] | People counting | SVR | A scheme for estimating the number of people using the existing Wi-Fi access points. [62] | People counting | SVR | An estimation function for people counting using extracted features. [63] | People counting | Five-layer neural networks | A people counting model, called Wi-Fi-counter using smartphone and Wi-Fi. [69] | Crowd sensing | Online ML | A robust crowd sensing framework with mobile devices. [73] | Crowd sensing | CNN | A pattern recognition model for detecting unimportant information and extract useful features for lightweight crowd sensing. [77] | Salient object detection | Recurrent convolutional network | A recurrent fully convolutional network for better accurate saliency detection. [79] | Salient object detection | CNN | A simplified CNN network for salient object detection. Localization | [83] | Sensor node localization | Bayes network, random forest, and SVM | A sensor node localization scheme using ML classifiers. [84] | Sensor node localization | Semi supervised learning | A non-linear semi supervised noise minimization algorithm for sensor node localization via iterative learning. [85] | Indoor localization | Random forest | A neural network on Android phones for indoor localization. [88] | Wireless localization | Three-layer DNNs | A solution for device-free wireless localization in wireless networks. AI at Devices | [93] | Human body motion classification | Online ML | A study of CSI estimation retrieved from the host computer to estimate and classify the four body motions. [96] | Motion detection | Supervised ML | A motion detection system using Wi-Fi CSI data. [97] | Human motion detection | Multi-layer CNN | A multiple access point-based CSI aggregation architecture, aiming to estimate human motion. [98] | Human motion detection | DNNs | An investigation on the influence of human motions on Wi-Fi signals. [100] | Human activity recognition | CNN | A Wi-Fi CSI collection framework based on IoT devices for recognizing human activity. ## IV Network Device AI The next Wireless AI function of the data life cycle in Fig. 2 is Network Device AI where AI has been applied in the network edge to learn and detect signals, by embedding an AI-based intelligent algorithm on the base stations or access points. We here focus on two main applications of the Network Device AI function, including content catching and edge computing management as shown in Fig. 5. Figure 5: Network Device AI function. ### IV-A Content caching Problem formulation: Caching contents at base stations/access points at the edge of the network has emerged as a promising approach to reduce traffic and enhance the quality of experience (QoE) [103]. Drawbacks of conventional methods: Most existing studies assume that the distribution of content catching popularity is known, but in practice, it is complex and non-stationary because of high content dymanics [103]. Unique advantages of using wireless AI techniques: Consider the complexity and dynamicity of contents, recent studies have investigated the integration of AI at BSs to facilitate content caching policies. Recent works: Many solutions have been proposed to support content catching, using ML and DL approaches. * • ML-based edge content catching: For example, the authors in [104] consider optimization of caching efficiency at small cell base stations. They focus on minimizing the backhaul load, formulate it as an optimization problem and leverage a K-means clustering algorithm to identify spatiotemporal patterns from the content requests at the BS. As a result, the similar content preferences are grouped into different classes so that the BS can cache the most important content with high accuracy and low caching latency. Another work that uses ML for edge caching, aiming to to estimate content popularity and design a caching scheme is proposed in [105]. The caching entities with cabled backhaul or wireless backhaul are deployed at the BSs, i.e. macro BSs or small BSs, to enable a flexible and low-latency method of content distribution, compared to with caching on remote cloud servers. Because of the fact that the computation resource at the BS can be insufficient for processing cached data and implementing the learning process, a power- efficient learning approach is developed using unsupervised learning (i.e. K-means clustering), an improved DNN network. The simulation results verify that compared to conventional optimization algorithms, the learning-based scheme can achieve better cached data prediction rate with higher accuracy and lower energy consumption. RL has recently been applied to edge-based content caching architectures. With an assumption that content popularity and user preference are not available at MEC servers, a multi-agent RL is an appropriate option for learning and training the caching decisions [106]. Each MEC server as an agent learns on how to coordinate its individual caching actions to maximize the optimal caching reward (i.e. the downloading latency metric). Due to the large state space, the conventional Q-table may not enough for action exploration, a combinatorial upper confidence bound solution is integrated to mitigate the RL algorithm complexity. * • DL-based edge content catching: Unlike aforementioned solutions, the work in [107] suggests a DRL approach for content caching at the edge nodes, i.e. at the base stations, to solve the policy control problem. Indeed, content delivery is always complex with a huge number of contents and different cache sizes in realistic scenarios, which pose critical challenges on estimating content popularity distribution and determining contents for storage in caches. DRL which is well known in solving complex control problems is suitable for this caching decisions at the BS. User requests to the BS are considered as the system state and caching decisions (cache or not) are action variables in the RL formulation, while cache hit rate is chosen as the reward to represent the goal of the proposed scheme. Through the simulation, DRL has proven its efficiency with better cache hit rate in various system settings (i.e. edge cache capability). For content caching in fog computing-empowered IoT networks, an actor–critic deep RL architecture is provided in [108] and deployed at the edge router with the objective of reducing caching latency. Instead of deriving a mathematical formulation which is hard to obtain in practice, the authors propose a model-free learning scheme, where the agent continuously interacts with the fog caching environment to sense the states (i.e. numbers of contents and user requests). Since the edge router prefers to cache those contents with better popularity, the content transmission delay is formulated as a reward function. By this way, the content with higher popularity is more likely to have lower time cost, which motivates the agent to take the action of caching for long-term reward. ### IV-B Edge Computing Management Problem formulation: With the assistance of mobile edge computing, mobile devices can rely on edge services for their computation and processing tasks. Due to the constrained resource of edge servers and the unprecedented mobile data traffic, resource scheduling and orchestration management are required to achieve robust edge management and ensure long-term service delivery for ubiquitous mobile users. Drawbacks of conventional methods: Most of current strategies rely on mixed integer nonlinear programming or dynamic programming to find the edge computing management policies, but it is unlikely to apply to complex wireless edge networks with unpredictable data traffic and user demands [16], [17]. Further, when the dimension of the network increases due to the increase of mobile users, the traditional techniques are unable to solve the increasingly computational issue and then hard to scale well to meet QoS of all users. Unique advantages of using wireless AI techniques: AI would be a natural option to achieve a scalable and low-complex edge computing management thanks to its DL and large-scale data optimization. In this sub-section, we investigate the use of AI for edge computing management with some representative recent works in two use case domains: edge resource scheduling and orchestration management. Recent works: Many studies have leveraged AI techniques to support edge computing management via two important services: edge resource scheduling and orchestration management. * • Edge resource scheduling: AI has been applied widely to facilitate edge resource scheduling. The authors in [109] present a DL-based resource scheduling scheme for hybrid MEC networks, including base stations, vehicles and UAVs connecting with edge cloud. A DNN network is proposed with a scheduling layer aimed to perform edge computing resource allocation and user association. To provide an adaptive policy for resource allocation at the MEC nodes in mobile edge networks with multi-users, a deep Q-learning scheme is introduced in [110]. In the multi-user MEC network, it is necessary to have a proper policy for the MEC server so that it can adjust part of edge resource and allocate fairly to all users to avoid interference and reduce latency of users for queuing to be served. Therefore, a smart resource allocation scheme using deep Q-learning is developed and deployed directly on the MEC server so that it can allocate effectively resource for offloaded tasks of network users under different data task arrival rates. Moreover, a multi-edge resource scheduling scheme is provided in [111]. An edge computing use case for a protest crowd incident management model is considered where ubiquitous data including images and videos are collected and offloaded to the nearby MEC server for execution. Here, how to allocate edge resource to the offloaded tasks and adjust edge computation with varying user demands to avoid network congestion is a critical challenge. Motivated by this, an edge computing prediction solution using ML is proposed, from the real data (transmission and delay records) in wireless network experiments. The ML algorithm can estimate network costs (i.e. user data offloading cost) so that efficient edge resource scheduling can be achieved. To further improve the efficiency of online edge computing scheduling, another solution in [112] suggests a DRL-based schem. The authors focus on the mobility-aware service migration and energy control problem in mobile edge computing. The model includes three main entities: a network hypervisor for aggregating the states of input information such as server workloads, bandwidth consumption, spectrum allocation; a RL-based controller for producing the control actions based on the input information, and action executor for obtaining the control decisions from the RL controller to execute corresponding actions. As a case study, the authors investigate the proposed DRL scheme with Deep Q-learning for an edge service migration example, showing that DRL can achieve superior performance, compared to the greedy scheme in terms of lower system cost [113]. * • Edge orchestration management: In addition to edge resource scheduling, the potential of AI has been investigated in edge orchestration management. As an example, the work in [114] considers an orchestration model of networking, caching and computing resources with MEC. The authors concentrate on the resource allocation problem that is formulated as an optimization problem with respect to the gains of networking, caching and computing. The triple model is highly complex that is hard to be solved by traditional optimization algorithms. Therefore, a deep Q-learning approach is proposed to achieve the high convergence performance with the best system utility. Meanwhile, in [115], a joint scheme of virtual edge orchestration with virtualized network functions and data flow scheduling is considered. Motivated by the fact that real-world networks may be hard to be modelled due to the complex and dynamic nature of wireless networks, a model-free DRL algorithm using deep deterministic policy gradients is proposed, aiming to minimize system delay and operation costs (i.e. execution time). Specially, an edge-Service Function Chain (SFC) orchestration scheme based on blockchain [116] is introduced in [117] for hybrid cloud-edge resource scheduling. Blockchain is adopted to create secure transmission between users and edge service providers, while the SFC orchestration model is formulated as a time-slotted chain to adapt with the dynamicity of IoTs. Based on that, a joint problem of orchestration control and service migration is derived and solved by a DRL approach using deep Q-learning. The numerical simulation results show that the learning-based method can achieve high performance with low latency of orchestration algorithm, and system cost savings. ### IV-C Lessons learned The deployment of AI functions at edge servers (i.e. base stations) instead of remote clouds has emerged as a new direction for flexile network designs with low latency and privacy protection. AI has been applied in the network edge learn and detect signals, by embedding an AI-based intelligent algorithm on the base stations or access points [118]. Specially, content catching at the network edge is now much easier thanks to the prediction capability of AI. Due to the huge number of contents and different cache sizes in realistic network scenarios, how to accurately estimate content popularity to achieve a high catching rate at the edge is highly challenging. AI comes as a natural choice to provide smart caching policies for implementing content catching with respect to latency and content loss constraints [105], [119]. Moreover, the potential of AI in edge computing management has been also investigated in mainly two use-case domains: edge resource scheduling [109] and orchestration management [114]. We list Network Device AI use cases in the taxonomy TABLE IV to summarize the contributions of each reference work. TABLE IV: Taxonomy of Network Device AI use cases. Category | Ref. | Use case | AI techniques applied | Main contributions ---|---|---|---|--- Network Device AI | [104] | Catching optimization | K-means | A scheme for optimization of caching efficiency at small cell base stations. [106] | Caching decisions | Multi-agent RL | A scheme for coordinating individual caching actions to maximize the optimal caching reward in MEC. [110] | Edge resource allocation | Deep Q-learning | A scheme for resource allocation at the MEC nodes in mobile edge networks with multi-users. [111] | Multi-edge resource scheduling | Online ML | A protest crowd incident management model for multi-edge resource scheduling. [112] | Edge computing management | DRL | A DRL-enabled edge computing management framework with MEC. [113] | Edge orchestration management | Deep Q-learning | An orchestration model of networking, caching and computing resources with MEC. [115] | Virtual edge orchestration | Model-free DRL | A joint scheme for virtual edge orchestration with virtualized network functions and data flow scheduling. ## V Access AI The third Wireless AI function of the data life cycle in Fig. 2 is Access AI where AI can be applied to facilitate the data transmission from three main layers, including PHY layer, MAC layer, and network layer. Here, we focus on discussions on the roles of AI in such layers as shown in Fig. 6. Figure 6: Access AI function. ### V-A PHY Layer AI AI has been applied in physical layer communications and shown impressive performances in recent years [120], [121], [122]. Here, we focus on discussing the roles of AI in three important domains, namely CSI acquisition, precoding, and coding/decoding. #### V-A1 CSI acquisition In this sub-section, we analyze the application of AI in CSI acquisition. Problem formulation: The massive multiple-input multiple-output (MIMO) system is widely regarded as a major technology for fifth-generation wireless communication systems. By equipping a base station (BS) with hundreds or even thousands of antennas in a centralized or distributed manner, such a system can substantially reduce multiuser interference and provide a multifold increase in cell throughput. This potential benefit is mainly obtained by exploiting CSI at BSs. In current frequency division duplexity (FDD) MIMO systems (e.g., long-term evolution Release-8), the downlink CSI is acquired at the user equipment (UE) during the training period and returns to the BS through feedback links. Drawbacks of conventional methods: Vector quantization or codebook-based approaches are usually adopted to reduce feedback overhead. However, the feedback quantities resulting from these approaches need to be scaled linearly with the number of transmit antennas and are prohibitive in a massive MIMO regime. Unique advantages of using wireless AI techniques: Recently, AI techniques such as ML and DL have been adopted as a promising tool to facilitate CSI acquisition in wireless communications. For example, AI potentially lowers the complexity and latency for CSI acquisition tasks such as CSI feedback and CSI sensing. Particularly, it is shown that a deep neural network with online learning is able to facilitate the parameter tuning and expedite the convergence rate, which improves the estimation of CSI without occupation of uplink bandwidth resource [121]. Recent works: There are a number of studies working on ML and DL techniques for CSI acquisition. * • ML-based CSI acquisition: The work in [123] introduces a ML-based CSI acquisition architecture for massive MIMO with frequency division duplex (FDD). The learning algorithm is divided into three steps: offline regression model training, online CSI estimation, and online CSI prediction by using linear regression (LR) and support vector regression (SVR). The proposed ML solution enables each user to estimate the downlink channel separately and then feed back the estimated low dimension channel to the base station based on the simple codebook method, which reduces the overhead of uplink feedback. Another work in [124] is also related to CSI estimation. It provides a ML- based time division duplex (TDD) scheme in which CSI can be estimated via a ML-based predictor instead of conventional pilot-based channel estimator. Here, two ML-based structures are built to improve the CSI prediction, namely, CNN combined with autoregressive (AR) predictor and autoregressive network with exogenous inputs, which can improve the prediction accuracy, and alleviate overheads of channel estimation. * • DL-based CSI acquisition: DL has been also widely used in CSI acquisition tasks. As an example, the authors in [125] introduce CsiNet, a novel DL- enabled CSI sensing and recovery mechanism that learns to effectively use channel structure from training samples. In CsiNet, the recent and popular CNNs are explored for the encoder (CSI sensing) and decoder (recovery network) so that they can exploit spatial local correlation by enforcing a local connectivity pattern among the neurons of adjacent layers. To compress CSI in massive MIMO system, a multiple-rate compressive sensing neural network framework is proposed in [126]. The key focus is on the investigation of CSI reconstruction accuracy and feasibility at the user equipments (UE) for the CSI feedback problem by using a DL-based network architecture, called CsiNet+. This framework not only enhances reconstruction accuracy but also decreases storage space at the UE, thus enhancing the system feasibility. The authors in [127] also propose a convolutional DL architecture, called DeepCMC, for efficient compression of the channel gain matrix to alleviate the significant CSI feedback load in massive MIMO systems. The proposed scheme is composed of fully convolutional layers followed by quantization and entropy coding blocks for better CSI estimation under low feedback rates. In the similar direction, the study in [128] presents a new architecture of DL in CSI feedback compression, which also takes advantage of the memory characteristic of RNN for feature extraction, compression and decompression. To reduce the size of the model, a depthwise separable convolutions is adopted in feature recovery for better recovery quality at different compression ratios (CRs). For reducing CSI feedback overhead due to significant amount of BS antennas in massive MIMO systems, DL is used in [129] decoupled with superimposed coding. In particular, a multi-task neural network with subnet- by-subnet training method is employed to facilitate the parameter tuning and expedite the convergence rate, which improves the estimation of CSI without occupation of uplink bandwidth resource. #### V-A2 Precoding In this sub-section, we discuss the roles of AI in supporting precoding. Problem formulation: Hybrid precoding is a significant issue in millimeter wave (mmWave) massive MIMO system. To cope with a large number of antennas, hybrid precoding is required where signals are processed from both analog and digital precoders. Typically, the estimation of MIMO channels is neccessary to establish the hybrid precoding matrices where AI has emerged as a promising tool to support precoding. Drawbacks of conventional methods: Due to the large-scale antennas as well as the high resolution analog-to-digital converters (ADCs)/digital-to analog converters (DACs), current digital precoding frameworks [130], [131] suffer from high power consumption and hardware costs. Besides, the radio frequency (RF) chains is considerable that may make traditional solutions such as one- bit ADCs processing [132] inefficient to achieve a desired throughput rate. Unique advantages of using wireless AI techniques: Recently, AI has attracted much attention in precoding system designs thanks to its low complexity, online learning for better estimation of hybrid precoding matrices, and its adaptability to time-varying mmWave channels. Recent works: There are many AI techniques used to facilitate precoding tasks in the literature. * • ML approaches: For example, ML has been adopted in [133] to build a hybrid precoding architecture with two main steps. First, an optimal hybrid precoder is designed with a hybrid precoding algorithm enabled by an enhanced cross- entropy approach in ML. The second step is to design the optimal hybrid combiner with an approximate optimization method, aiming to obtain the optimal precoding matrix and combining matrix to maximize the spectrum efficiency of mmWave massive MIMO system with low computation complexity [134]. To support hybrid precoding in mmWave MIMO-OFDM systems, a ML approach using cluster analysis is introduced in [135]. The key purpose is to group dynamic subarray with the help of PCA that is used to extract frequency-flat RF precoder from the principle component of the optimal frequency-selective precoders. Similarly, the study in [136] incorporates an unadjusted probability and smoothing constant to across-entropy method in ML to perform a hybrid precoding implementation to solve the issue of energy loss in mmWave massive MIMO systems with multiple RF chains. * • DL approaches: Meanwhile, the authors in [137] investigate the use of DL, namely neural networks called an auto-precoder, to design the hybrid precoding metrics by using the prior observations of the channel. Based on a learning procedure, the auto-precoder can learn to optimize the channel sensing vectors, which are then used to predict the hybrid beamforming vectors without negligible training overheads. The work in [138] applies DNNs to obtain a decentralized robust precoding framework in multi-user MIMO settings, aiming to cope with the continuous decision space and the required fine granularity of the precoding. To this end, the authors develop a decentralized DNNs architecture, where each TX is responsible to approximate a decision function using a DNN, and then all the DNNs are jointly trained to enforce a common behaviour for all cooperative TXs. To reduce the large computational complexity and local-minimum problems faced by the traditional design of hybrid precoders for multi-user MIMO scenarios, the work in [139] considers a DL scheme using a CNN for a new mmWaves hybrid precoding architecture. More specific, the precoder performs an offline training process by using the channel matrix of users as the input to train the CNN, which generates the output labelled as the hybrid precoder weights. After that, based on the trained model, the CNN-MIMO can predict the hybrid precoders by feeding the network with the user channel matrix. A significant advantage of the CNN architecture is a capability to handling the imperfections of the input data and the dynamicity of the network, which contributes a stable and better performance in terms of high throughput rates [140]. * • DRL approaches: Recently, DRL has been appeared as a new research direction to further improve the performance of current ML/DL approaches in precoding design. The work in [141] also focuses on precoding issues, but it uses DRL for hybrid precoding in MIMO under uncertain environment and unknown interference. The benefit of DRL is the ability to learn from the environment using an agent to force a policy for optimal actions and then obtain rewards without a specific model. Using this theory, the authors develop a game theoretic DRL-based algorithm to learn and update intelligently a dynamic codebook process for digital precoding. The simulation results show that DRL can effectively maximize multi-hop signal coverage and data rate for MIMO systems. Similarly, the study in [142] also analyzes a mmWave hybrid precoding model. This work introduces a mmWave point-to-point massive MIMO system where the hybrid beamforming design is considered using DRL. In this scheme, channel information is considered as the system state, precoder matrix elements are actions, while spectral efficiency and bit error rate are regarded jointly as a reward function. Based on that, a mmWave hybrid precoding problem is formulated as a markov decision process (MDP) that is solved by a deep Q-learning algorithm. The authors concluded that the DRL adoption can achieve high performance in terms of high spectral efficiency and minimal error rate, compared to its algorithm counterparts. #### V-A3 Coding/decoding In this sub-section, we present a discussion on the use of AI in supporting coding/decoding in wireless networks. Problem formulation: Channel coding and decoding are significant components in modern wireless communications. Error correcting codes for channel coding are utilized to achieve reliable communications at rates of the Shannon channel capability. For instance, low-density parity-check (LDPC) codes are able to exhibit near Shannon capability with decoding algorithms such as belief propagation (BP) [143]. Drawbacks of conventional methods: Many coding and decoding solutions have been proposed in the literature but they still remain some limitations. For example, LDPC codes may not achieve desired performance due to channel fading and coloured noise that also introduces high coding complexity [143]. Another approach is whitening that aims to transform coloured noise to white noise, but it requires matrix multiplication that is known to highly complex for long-length codes. Drawbacks of conventional methods: The advances of AI open up new opportunities to address coding/decoding issues. Instead of relying on a pre- defined channel model, AI is able to learn the network channel via learning with low computation complexity, better coding accuracy and improved decoding performances. Recent works: The work in [144] uses CNN to develop a smart receiver architecture for channel decoding. Stemming from the fact that the difficulty of accurate channel noise estimation with presence of decoding error, neural layers are exploited to estimate channel noise. In this process, training is particularly important to eliminate estimation errors and capture useful features for the BP decoder. The continuous interaction between the BP decoder and CNN would improve decoding SNR with much lower complexity and better decoding accuracy, as confirmed from simulation results. The authors in [145] concern about the stability of channel coding in terms of block lengths and number of information bits. This can be achieved by using neural networks as replacement of an existing polar decoder. In this case, the large polar encoding graph is separated into sub-graphs so that the training of codeword lengths is improved for better decoding performances with low computation latency. An improvement of BP algorithms for decoding block codes using AI is shown in [146]. A RNN decoder is designed to replace the conventional BP decoder which desires to enhance the decoding efficiency on parity check matrices. The ability of AI in assisting coding and decoding is also demonstrated in [147] that employs parameterized RNN decoder for decoding. The main objective is to train weights of the decoder using a dataset that is constructed from noise collections of a codeword. This training process is performed using stochastic gradient descent, showing that the bit error rate is reduced significantly without causing computational complexity. ### V-B MAC layer AI In this sub-section, we survey the recent AI-based solutions for MAC layer in three main domains, including channel allocation, power control and interference mitigation. #### V-B1 Channel allocation In this sub-section, we present a discussion on the use of AI in channel allocation. Problem formulation: Channel allocation is a major concern in densely deployed wireless networks wherein there has a large number of base stations/ access points but limited available channels. Channel allocation is possible to lower network congestion, improve network throughput and enhance user experience. Drawbacks of conventional methods: In existing studies [148], [149], the major challenge lies in accurately modelling the channel characteristics to make effectively channel decisions for long-term system performance maximization under the complex network settings. Unique advantages of using wireless AI techniques: AI with its learning and prediction capabilities shows great prospects in solving wireless channel allocation, i.e. better solving high dimensions of states, improving channel network learning for optimal channel allocation. Recent works: Some ML and DL approaches are applied to empower channel allocation applications. * • ML approaches: An AI-based solution is proposed in [148] for channel allocation in Wireless Networked Control Systems (WNCSs) where each subsystem uses a timer to access the channel. To formulate a channel allocation for subsystems, an online learning is considered so that the subsystems can learn the channel parameters (i.e. Gilbert-Elliott (GE) channel) to adjust their timers according to channel variation for gaining optimal channel resource. To support channel assignment for multi-channel communication in IoT networks, the work in [149] introduces a ML-based algorithm running on IoT devices. Through the learning process, each IoT device estimates the channel with higher resources (i.e. bandwidth) to make decisions for selecting the appropriate channel in the dynamic environment. The authors in [150] also employ ML to implement a channel allocation strategy for Wi-Fi devices in the context of channel interference. The learning allows each device to estimate the channel states and detect the behaviour of neighbour devices for minimizing channel usage overlapping. Another solution for channel allocation is [151] where a RL-based ML approach is designed for optimizing channel allocation in wireless sensor networks. The RL agent is used to interact with the environment (i.e. sensor networks) and select the channel with high frequency for sensor nodes, aiming to reduce the error rate during the transmission. * • DL approaches: Another approach in [152] solves the channel allocation issue by taking advantage of pilot symbols prior known by both transmitters and receivers in OFDM systems. In fact, the estimation of sub-channels relies on the number of pilots and pilot positions. Motivated by this, the authors propose a autoencoder-based scheme with DNN to find the pilot position. The autoencoder is first trained to encode the input (i.e. the true channel) into representation code to reconstruct it. Then, the pilots are allocated at nonzero indices of the sparse vector so that the channel allocation can be done only at the indices corresponding to the pilots. Besides, a conjunction of DRL and Graph Convolutional Networks is considered in [153] for a channel allocation scheme in wireless local area networks (WLANs). The high density makes the use of DRL suitable for solving high dimensions of states coming from the large number of mobile users and coordinators (i.e. access points). For channel allocation in multi-beam satellite (MBS) systems, the study in [154] suggests a DRL-based solution to achieve a long-term optimal channel resource allocation under the complex interactions between satellite and user terminals. #### V-B2 Power control In this sub-section, we present a discussion on the use of AI in power control. Problem formulation: The energy-efficient power control is one of the most important issues for the sustainable growth of future wireless communication networks. How to ensure smooth communications of devices while efficient power control on the channels is critical. This topic is of paramount importance for practical communication scenarios such as MIMO systems, OFDM networks and D2D communications [155]. Drawbacks of conventional methods: Most traditional approaches follow a predefined channel of wireless communication to formulate the power control/allocation problem [156], [157]. However, these schemes only work well in the static wireless networks with predictable channel conditions (i.e. user demands, transmit data sizes), while it is hard to apply in the networks without a specific channel model. Unique advantages of using wireless AI techniques: Recently, AI has been investigated to solve power control issues in complex and dynamic wireless communications with low computation complexity and better throughput for power transmisison and power allocation. Recent works: * • ML approaches: The work in [156] presents a power control scheme using ML for D2D networks. The major objective is to dynamically allocate power to the network of D2D users and cellular users for avoiding channel interference. This can be done by using a Q-learning algorithm which learns a power allocation policy such that the successful data transmission of each user is ensured with a certain level of power. In the complex wireless communication networks with nonstationary wireless channels, the control of transmit power for transmitters is a challenging task. The authors in [157] propose a learning-based scheme with the objective of optimizing transmit power over the wireless channel under a certain power budget. The distribution of power allocation to channels is estimated through a learning process so that the use of power serving data transmissions is efficiently predicted and optimized in a long run. * • DL approaches: Meanwhile, the study in [158] investigates the potential of DL in addressing power control issues in wireless networks. As an experimental implementation, the authors employ a DNN architecture to perform a power control in large wireless systems without the need of a complex channel estimation procedure. The DNN network is trained using CSI values on all network links as inputs to capture the interference pattern over the different links. With the aid of learning, the proposed scheme can adapt well when the size of the network increases. Similar to this work, a solution for power allocation is proposed in [159] that uses ANNs to estimate power allocation in wireless interference networks. Here, the distribution of wireless channels is used as the input for the deep ANNs architecture to perform training, which then generates an optimization model to specify how much power should be allocated to each channel. In [160], a LSTM which is a DL-based RNN architecture is recommended for building a channel prediction mechanism in wireless body are networks. Based on the constructed model, a predictor is derived that estimates the power consumption on each channel in order to adjust the suitable level of power over the channels. [161] also leverage DL to propose a dynamic power control scheme for NOMA in wireless catching systems. They concentrate on minimizing the transmission delay by formulating it as an optimization problem with respect to transmit deadline between base station and users as well as total power constraint. Then, a DNN architecture is built to learn in an iterative manner, evaluate and find an optimal solution in a fashion the transmission time is minimized and optimal power allocation is achieved under a power budget. * • DRL approaches: DRL also shows the high potential in solving power control problems. A multi-agent DRL-based transmit power control scheme is introduced in [162] for wireless communications. Different from the existing power control frameworks which require accurate information of channel gains of all users, the proposed model only needs certain some channels with high power values. As a result, the complexity of the learning algorithm is low regardless of the size of the network, which shows a good scalability of the proposed scheme. DRL is also employed in [163] to develop a joint scheme of resource allocation and power control for mobile edge computing (MEC)-based D2D networks. A model-free reinforcement learning algorithm is designed to update the parameters and rules of resource allocation and power control via a policy gradient method. In the training, each D2D transmitter acts as an agent to interact with the environment and take two appropriate actions: channel selection and power selection, aiming to find a near-optimal solution to maximize these two objective values. Another possible solution for the power control problem is [164] that leverages DRL to estimate optimal power allocation policies in wireless interference channels. The state space includes the number of channels and the levels of power over the channels, while the action space is defined as a conjunction of channel selection and power allocation policies. The learning process is conducted in an iterative manner to find a reward of transmission rate, sum-rate of all users, and fairness among users. #### V-B3 Interference management In this sub-section, we present a discussion on the use of AI in interference management. Problem formulation: In the ultra-dense networks, interference stems from the usage of the same frequency resource among neighbouring users or network cells. In OFDMA networks, for example, inter-cell interference can be caused by two or more neighbour cells using the same frequency band, which can degrade the overall performance of wireless networks. Drawbacks of conventional methods: Usually, different frequency reuse factors (FRF) or partial-frequency reuse approaches are employed to coordinate and manage interference among users. However, these strategies may be inefficiently in future wireless networks with variable network traffic and dynamic user patterns, leading to under-usage or lack of the spectrum in user/cell networks. Unique advantages of using wireless AI techniques: By using AI, it is possible that an adaptive and online approach is devised to enable minimum network interference for efficient cellular deployments in terms of spectral allocation efficiency and as well as QoS fulfilment. Recent works: Many AI-based studies have been done to solve interference management issues. * • ML approaches: In [165], the authors consider uplink interference management in multi-user LTE cellular networks with the aid of data-driven ML. A strategy for power control is necessary to adjust the transmit power on the transmission channels for interference balance among users under the settings of cell-specific power control parameters in LTE networks. Then, a stochastic leaning-enabled gradient algorithm is developed to determine optimal power control parameters in a fashion interference among user channels is maintained in an acceptable level for improving the data rate in LTE networks. The work in [166] solves the issue of inter-cell interference which is caused by two or more neighbour cells using the same frequency band in OFDMA networks through a decentralized RL approach. Different from the previous study, the authors focus on the spectrum assignment for interference control among cells. To formulate a learning problem, each cell is regarded as an agent which determines optimally an amount of spectrum resource based on obtained context information (i.e. average signal to interference ratio (SIR)/chunk in the cell and user throughput). Concretely, the proposed scheme demonstrates its high performance in cellular deployments in terms of spectral allocation efficiency and SINR enhancement as well as QoS fulfilment. * • DL approaches: To investigate the potential of DL in interference management for wireless communications, the authors in [167] suggest a DNN architecture to develop a real-time optimization scheme for time sensitive signal processing over interference-limited channels in multi-user networks. By introducing a learning process, the relationship between the input and the output of conventional signal processing algorithms can be approximated and then the DNN-based signal processing algorithm can be implemented effectively in terms of low computation complexity and better system throughput (high spectrum efficiency and low user interference levels). Moreover, a joint DL-enabled scheme of interference coordination and beam management (BM-IC) is presented in [168] for dense mmWave networks. The ultimate objective is to minimize the interference and enhance the network sum-rate with respect to beam directions, beamwidths, and transmit power resource allocations. To do so, a beamforming training is required to establish directional links in IEEE 802.11ay. Then, the BM-IC is formulated as a joint optimization problem that is approximated by a DNN network through an iterative learning procedure. The authors in [169] also consider an interference classification approach in satellite systems where DL is employed to provide an interference classifier for classification of some cellular standards, including LTE, UMTS and GSM. DRL is also investigated for dealing interference management issues [170] that aims at minimizing a joint metric of wireless transmit latency and interference levels in UAVs-enabled cellular networks. To realize this, a cooperative game is derived in which UAVs are players to learn its trajectory, power allocation in the context of insufficient information of network settings, such as user service demands, UAV locations. To deal with this dynamic game, DRL comes as an ideal candidate to allow UAVs to estimate its path and adjust power resource allocation so that the interference among UAVs is minimized. * • DRL approaches: The interference management problem is also considered in [171] that concentrates on in the context of vehicle-to-vehicle (V2V) communications. In fact, in the complex vehicular networks, a large number of V2V links results in a high interference level, which degrades the overall performance of the involved network, such as high latency and less transmission reliability. By introducing an RL concept where each V2V link acts as an agent and spectrum usage and power allocation are determined using CSI and mutual information among channels, a DRL architecture is built to provide a unified network observation model for adjusting resource allocation among vehicles to avoid high interference on V2V links. Lastly, another solution for interference management is presented in [172] from the QoS perspective. To evaluate QoS of massive MIMO cognitive radio networks in terms of transmission rate and interference, a user selection approach is proposed in using a deep Q-learning. Two key cognitive radio scenarios are taken into consideration, including radio networks with available and unavailable CSI knowledge at the secondary base station. While the former case can be considered with a conventional programming tool to formulate and solve the optimization problem of power allocation, the later case is realized by using a Deep Q-learning scheme. The base station is regarded as an agent which senses the cognitive radio environment and makes decisions on how much power resource should be allocated to each secondary user so that interference in the network is minimal [173]. ### V-C Network layer AI In this section, we discuss the roles of AI in supporting spectrum sharing, data offloading, multi-connectivity, and network resource management. #### V-C1 Spectrum sharing In this sub-section, we discuss the roles of AI in supporting spectrum sharing. Problem formulation: The ultra-dense networks in the emerging 5G era promises to support one million devices per $km^{2}$ in the future. However, with the full swing of IoTs reaching every corner, the anticipated device density could even surpass this target, particularly in large-scale cellular networks. To address this problem, consensus has been reached that spectrum sharing by various users and devices will play a key role, monitoring frequency-band use and dynamically determining spectrum sharing. With the available spectrum identified, secondary users (SUs) will share the opportunities efficiently and fairly, through an appropriate spectrum sharing strategy. Essentially, a distributed self-organized multi-agent system is highly desirable, where each SU acts in a self-organised manner without any information exchange with its peers. Since the complete state space of the network is typically very large, and not available online to SUs, the direct computation of optimal policies becomes intractable. Drawbacks of conventional methods: Most existing studies on the multi-agent case use game theory [174], matching theory [175], and graph colouring [176] to obtain structured solutions. But these model-dependent solutions make many impractical assumptions, such as the knowledge of signal-to-interference-plus- noise ratio (SINR), transmission power, and price from base stations. Moreover, when the number of users is larger than that of channels, model- dependent methods (graph colouring in particular) will become infeasible. Therefore, a fundamentally new approach to spectrum sharing needs to be developed, handling the large state space and enabling autonomous operation among a large number of SUs. Unique advantages of using wireless AI techniques: AI has emerged as a highly efficient solution to overcome these challenges in spectrum sharing. Here, we focus on analyzing the applications of AI in spectrum sharing with recent successful works. Many AI-empowered solutions with ML, DL and DRL techniques have been proposed to spectrum sharing issues in wireless communications. Recent works: The work in [177] presents an inter-operator spectrum sharing (IOSS) model that takes advantage of the benefits of spectral proximity between users and BSs. By executing a Q-learning scheme at each BSs, the network operators can achieve efficient spectrum sharing among users with high quality of experience and spectral resource utilization. To be clear, the network operators can share their licenced spectrum with others via an orthogonal spectrum sharing based business model. The BSs can make decisions to utilize the IOSS mode based on the amount of spectral resources requested from the neighbours, while inferences among users are taken to formulate the action (i.e. spectrum sharing parameters) to satisfy user demands in different load scenarios. To this end, a Q-learning algorithm is derived so that the BS can adjust dynamically the spectrum amount based on the states and actions for sharing among different types of users, including users with high service demands and users with less resource requirements. Database-based spectrum sharing is of the important solutions for wireless IoT communications [178]. In fact, the spectrum sharing policies of a spectrum can be aggregated as a database so that the SUs can refer for avoiding spectrum interference with primary users, especially in the primary exclusive region (PER). Therefore, updating the PER once spectrum interference occurs is necessary to make sure that the smooth spectrum exchange is ensured. To do that, a supervised learning algorithm is derived to update the PER and learn a sharing policy over time for allocating optimally spectrum over the network, aiming to increase the efficiency of spectrum usage among users. To support spectrum sharing in vehicular networks, the work in [179] considers a multi-agent RL scheme where multiple vehicle-to-vehicle (V2V) links reuse the frequency spectrum occupied by vehicle-to-infrastructure (V2I) links. Here, the V2V links interconnect neighbouring vehicles, while the V2I network links each vehicle to the nearby BS. The process of spectrum sharing among V2V links and V2I connections is formulated as a multi-agent problem with respect to spectrum sub-band selection. The key objective is to optimize the capability of V2I links for high bandwidth content delivery, which is enabled by a reward design and centralized learning procedure. Then, a multi-agent deep RL is proposed where each V2V link acts as an agent to learn from the vehicular environment for gaining high spectrum usage. The implementation results indicate the advantage of the learning approach in terms of better system capacity of V2I links and enhanced load delivery rate of V2V links with spectrum savings. #### V-C2 Data offloading In this sub-section, we discuss the roles of AI in supporting data offloading. Problem formulation: Data offloading refers to the mechanism where the resource-limited nodes offload part or full of their data to the resourceful nodes (i.e. edge/cloud servers). Data offloading potentially mitigates the burden on resource-constrained nodes in terms of computation and storage savings. Worthy of mention, this solution would benefit both end users for better service processing and service providers (i.e. edge/cloud services) for network congestion control and better user service delivery. For example, the advances of mobile cloud/edge technologies open up opportunities to solve the computation challenges on mobile devices by providing offloading services. In this way, mobile users can offload their computation-extensive tasks (i.e. data and programming codes) to resourceful cloud/edge servers for efficient execution. This solution potentially reduces computation pressure on devices, enhances service qualities to satisfy the ever-increasing computation demands of modern mobile devices. Drawbacks of conventional methods: Some literature works listed in [180], [181] show that the traditional offloading strategies mainly leverage Lyapunov or convex optimization methods to deal with the data offloading problem. However, such traditional optimization algorithms is only suitable for low- complexity tasks and often needs the information of system statistics that may not be available in practice. Besides, how to adapt the offloading model to the varying environments (i.e. varying user demands, channel availability) is a critical challenge to be solved. Unique advantages of using wireless AI techniques: In the data offloading process, AI plays an important role in supporting computation services, channel estimations and device power control for optimal offloading. Recent works: Many AI-based studies have been done to solve data offloading issues. * • ML approaches: For example, the authors in [182] apply ML associated with big data to a traffic offloading in heterogeneous ultra-dense networks. They investigate the potential of ML in overcome challenges such as latency, energy consumption in offloading applications such as D2D offloading and V2V offloading. An AI-enhanced offloading scheme is proposed in [180] for industrial IoT where mobile devices can offload their data tasks to nearby edge servers or remote cloud servers for computation. The main performance indicator considered in offloading is service accuracy (i.e. accuracy of computation results). A compressed neural network is deployed on edge servers to learn model parameters from data across edge servers, instead of relying on a cloud server for model updates. Then, the predictive model is trained based on a specific requirement of service delay and accuracy of computation tasks, congestion status of the edge-IoT network, and available computing edge resource. The learning-based offloading framework is able to estimate how much data should be offloaded for optimal service accuracy in real-time network settings. A hierarchical ML approach is also proposed in [181] for an industrial IoT network with mobile edge computing (MEC), where each user device makes decision to offload to MEC servers with respect to transmission error rate, QoS level, and communication resource constraints for data processing delay minimization. * • DL approaches: A DL-based solution for supporting offloading in wireless networks is evaluated in [183]. To be clear, a vehicular traffic offloading scheme is considered to deal with the issue of computation burden on resource- limited vehicles by using MEC services. Transmission mode selection and MEC server selection are taken into consideration for building an optimization problem, which is then solved effectively by a deep Q-learning algorithm. Moreover, a cooperative offloading architecture for multi-access MEC is given in [184]. In the complex environment with multiple MEC servers and varying user offloading requirements, how to offload tasks to support mobile users while preserve network resource and improve data catching on MEC servers is challenging. Inspired by these, a joint offloading scheme of MEC and D2D offloading is formulated and offloading decisions are formulated as a multi- label classification problem. To solve the proposed problem, a deep supervised learning-based algorithm is then derived which aims to minimize the overhead of data offloading. Considering the limitation of battery and computation capability of IoT devices, the study in [185] suggests an offloading scheme which enable IoT devices to submit their heavy data tasks to nearby MEC servers or cloud servers. A challenge here is how to choose which computation platform (MEC or cloud) to serve IoT users for the best offloading latency efficiency. To tackle this problem, a DL mechanism is proposed to learn the offloading behaviours of users and estimate offloading traffic to reduce computing latency with respect to task size, data volume and MEC capability. * • DRL approaches: Moreover, to learn a computation offloading process for wireless MEC networks, a DRL approach is described in [186] as shown in Fig. 7. The key problem to be solved is how to optimize offloading decisions (i.e. offloading tasks to MEC servers or locally executing at mobile devices) and achieve network power resource preservation with respect to network channel dynamicity. Motivated by this, a joint optimization problem is derived that is then solved effectively by a deep Q-learning empowered by a DNN network, showing that DRL not only provides flexible offloading decision policies but also adjusts adaptively network resource to satisfy the computation demands of all network users. DRL is aslo useful in vehicular data offloading for Internet of vehicle networks [187]. To achieve an optimal offloading decision for tasks with data dependence, a RL agent can be neccessary to sense the vehicular networks to collect necessary information, including user mobility and resource demands, and then train offline the collected data on the MEC nodes. Besides, thanks to the online training process, the vehicular service transactions can well adapt with the changes of vehicular environments for efficient offloading. In addition, to assist computation offloading in 5G vehicular IoT applications, an intelligent offloading scheme using DRL is proposed in [188]. The main focus is on optimizing offloading costs (i.e. latency, power usage) by taking spectrum allocation into consideration. Unlike the traditional DRL approach, this work considers a multi-agent DRL architecture where each vehicular user acts as an agent to cooperatively make decisions based on data task information and CSI feedback. Figure 7: Deep reinforcement learning for data offloading. #### V-C3 Multi-connectivity Due to the increasing number of base stations and mobile users in 5G ultra- dense networks, the demands of multi-connectivity with high reliability and spectrum efficiency is growing. In such as a context, AI has come as a promising solution to cope with the enormous state space caused by high network complexity and well support interconnection among users [189]. Communication techniques such as duel connectivity can realize the data transmission between base station and mobile users such that each station can select a codeword from a code book to establish an analog beam for assisting downlink communication with users. ML approaches such as SVM can support classification for codeword selection by a learning process where the inputs include user transmit power and CSI data. By using trained model, users can select effectively channels for uplink transmission and base stations can select codeword for downlink transmission with low complexity, which achieves fast and reliable ultra-dense network communications. To solve the mobility management issue caused by frequent handovers in ultra-dense small cells, a LSTM-based DL algorithm is proposed in [190]. This aims to learn the historical mobility patterns of users to estimate future user movement, which is important for hanover estimation for regulate user connectivity with base stations. Further, to improve the scalability of wireless communications in 5G vehicular networks, a traffic workload prediction scheme using neural networks is considered in [191]. The model is able to learn connectivity behaviour based on data rate, numbers of users and QoS requirements in order to make decisions for adjusting scalability for efficient vehicular communications in terms of low latency and high service request rates. #### V-C4 Network resource management In this sub-section, we present how AI can enable smart network resource management. Problem formulation: The current cellular technology and wireless networks may not satisfy the strides of wireless network demands. Resource management becomes more complex to maintain the QoE and achieve the expected system utility. The development of advanced technologies in 5G such as software defined networking (SDN) [192] and network slicing makes network resource management much more challenging that thus urgently requires new innovative and intelligent solutions. Drawbacks of conventional methods: Many solutions have been proposed to solve network resource management issues, such as generic algorithms or dynamic programming approaches, but they have high complexity in terms of computation and optimization [25]. Further, these conventional approaches have been demonstrated to be inefficient under presence of dynamicity of network resources and user demands. Drawbacks of conventional methods: AI can achieve smart and reliable network resource management thanks to its online learning and optimization capability. An important feature of AI is its smart making decision ability (i.e. RL approaches) to adaptively allocate network resources and control intelligently resource so that the network stability is guaranteed. Recent works: The work in [193] presents an AI-based network resource management scheme for SDN-empowered 5G vehicular networks. LSTM and CNN are used to classify and detect the resource demands at the control plane. To mimic the resource usage behaviour, learning is necessary to train the virtualization manager placed on the SDN controller. It is able to adjust the resource (i.e. bandwidth) based on the user demand and QoS requirements so that network fairness among users can be achieved. In the line of discussion, DL has been used in [194] to manage SDN resources in vehicular networks. To do that, a flow management scheme is designed and integrated in the SDN controller so that resource utilization is optimized. Due to the different priority levels and data requests of different vehicular users, a dynamic resource allocation algorithm is vitally important to avoid over resource usage as well as resource scarcity at any users. CNN is deployed on the network controller to manage the flow control for multiple users, aiming to optimize resource usage. Based on the network information such as bandwidth, available resource, user requests, the network controller can learn the new resource usage patterns for better allocation. The authors in [195] focus on the use of DRL for resource management in network slicing. Here, a deep Q-network controller tries to seek a bandwidth sharing solution to achieve a fair and optimal bandwidth resource with respect to available resource and dynamic user demands without knowing the specific network model. Another work in [196] also considers DRL as a solution for resource management in network slices. In the presence of uncertain network slice demands, designing an autonomous controller with the ability to interact and explore the environment to make decisions for resource allocation is desired. Deep Q network can fulfil this requirement by using an agent that estimate the state-action distribution where slice demands are regarded as states while allocated resources are seen as actions. The implementation results show that the combination of a reward-clipping scheme and duelling deep network improves the bandwidth allocation under different slicing scenarios. ### V-D Lessons learned The main lessons acquired from the survey on the Access AI domain are highlighted as the following. * • AI well support the designs of three key network layers: PHY layer, MAC layer, and Network layer. For PHY layer, AI has been applied to facilitate CSI acquisition tasks in wireless communications such as CSI estimation [124], learning for CSI sensing and recovery [125], and CSI feedback compression [128]. AI provides promising solutions by its learning and prediction capabilities to simplify the precoding design under uncertain environments and unknown network traffic patterns [141]. Further, AI also facilitates channel coding/decoding for wireless systems [146]. * • Many AI-based solutions have been proposed for MAC layer, mainly in three domains, namely channel allocation, power control and interference mitigation. In the multi-channel communications with less CSI and high channel variation, AI especially DL is able to estimate dynamically the channel allocation by establishing the adaptive channel assignment policy with low computational complexity. AI has been also investigated to solve power control issues in complex and dynamic wireless communications with recent successes [156], [157], [161]. Specially, some emerging research results demonstrate that AI is particularly useful for interference management in small cell networks or mmWave networks [168]. * • AI also gain enormous interests in network layer designs from four fundamental domains, spectrum sharing, data offloading, multi-connectivity, and network resource management. For instance, many AI-based approaches have been introduced to solve the critical spectrum sharing issues in wireless communications, ranging from wireless spectrum sharing [177] to IoT spectrum exchange [178]. Meanwhile, the learning capability of AI is also helpful to estimate data traffic and channel distribution to realize flexible data offloading [180], multi-connectivity [190], and resource management [193] in wireless networks. In summary, we list Access AI use cases in the taxonomy TABLE V to summarize the contributions of each reference work. TABLE V: Taxonomy of Access AI use cases. Category | Ref. | Use case | AI techniques applied | Main contributions ---|---|---|---|--- PHY layer AI | [123] | CSI acquisition | Support vector regression-SVR | A ML-based CSI acquisition architecture for massive multiple input multipleoutput (MIMO) with frequency division duplex (FDD). [124] | CSI estimation | CNN | A ML-based time division duplex (TDD) scheme in which CSI can be estimated via a ML-based predictor. [126] | CSI compression | DL (CsiNet+) | A multiple-rate compressive sensing neural network framework for CSI compression in MIMO. [127] | CSI compression | Convolutional DL | A convolutional DL architecture, called DeepCMC, for efficient compression of the channel gain matrix. [137] | Hybrid channel precoding | Neural networks | Designing the hybrid precoding metrics by using the prior observations of the channel with neural networks. [138] | Channel precoding | DNNs | A decentralized robust precoding framework in multi-user MIMO settings. [139] | mmWaves hybrid precoding | CNN | A DL scheme using a convolutional neural network (CNN) for a new mmWaves hybrid precoding architecture. MAC layer AI | [148] | Distributed channel access | Online ML | A distributed channel access scheme for Wireless Networked Control Systems. [149] | Channel assignment | Online ML | A DL-based scheme for channel assignment for multi-channel communication in IoT networks. [151] | Channel allocation | RL | A RL based ML approach is designed for optimizing channel allocation in wireless sensor networks. [156] | Power control | Q-learning | A power control scheme using ML for D2D networks. [157] | Transmit power optimization | ML | A learning-based scheme for optimizing transmit power over the wireless channel under a certain power budget. [165] | Interference management | Stochastic gradient leaning | A scheme for interference management in multi-user LTE cellular networks with the aid of data-driven ML. [166] | Intercell interference control | Distributed RL | A scheme for intercell interference in OFDMA networks. Network layer AI | [177] | Spectrum sharing | Q-learning | An inter-operator spectrum sharing (IOSS) model that takes advantage of the benefits of spectral proximity between users and BSs. [178] | Spectrum sharing | Supervised learning | A Database-based spectrum sharing model for PU-SU networks. [179] | Spectrum sharing | RL | A scheme for spectrum sharing in vehicular networks with V2V and V2I links. [180] | Data offloading | Compressed neural networks | An AI-enhanced offloading scheme for industrial IoT. [183] | Vehicular traffic offloading | Deep Q-learning | A vehicular traffic offloading scheme for vehicles. [186] | Computation offloading | DRL | A computation offloading scheme for wireless MEC networks. [190] | Multi-connectivity | LSTM-based DL | A DL scheme for mobility management caused by frequent handovers in ultra-dense small cells [191] | Multi-connectivity | Neural networks | A model for supporting connectivity in 5G vehicular networks [193] | Network resource management | LSTM and CNN | A DL-based scheme for 5G wireless networks. ## VI User Device AI The next Wireless AI function of the data life cycle in Fig. 2 is User Device AI where AI have been embedded into end devices to create a new on-device AI paradigm. In this section, we study the User Device AI function for wireless networks by analyzing the integration of AI in mobile user devices as shown in Fig. 8. ### VI-A Embedded AI functions Figure 8: User Device AI function. In this sub-section, we present how AI is integrated into mobile devices to develop on-develop AI functions. Problem formulation: With the rapid growth of smart mobile devices and improved embedded hardware, there is interest in implementing AI on the device [197], for both ML and DL functions. In the future networks with higher requirements in terms of low-latency data processing and reliable intelligence, pushing AI functions to mobile devices with privacy enhancement would be a notable choice to wireless network design. Drawbacks of conventional methods: Traditionally, AI functions are placed on remote cloud servers, which results in long latency for AI processing due to long-distance transmissions. Further, the reliance on a central server faces a serious scalable problem when processing data from distributed mobile devices. The use of external servers to run AI functions is also vulnerable to security and data privacy issues due to the third parties. Unique advantages of using wireless AI techniques: AI can be run directly on mobile devices to provide smart and online learning functions for mobile networks at the edge. The key benefits of embedded AI functions consist of low-latency learning and improve the intelligence of mobile devices, which can open up new intesting on-device applications, such as mobile smart object detection, mobile human recognition. Recent works: Some important works are highlighed with the integration of AI on mobile devices, including embedded ML functions and embedded DL functions. * • Embedded ML functions: The work in [198] introduces nnstreamer, an AI software system running on mobile devices or edge devices. This software is able to execute directly real-time ML functions, such as neural networks, with complex mobile data streams by simplifying ML implementations on mobile devices without relying on cloud servers. The feasibility of nnstreamer has been investigated via real-world applications, such as event detection through neural networks with sensor stream samples as the input data for training and classification. In [199], a GPU-based binary neural networks (BNN) engine for Android devices is designed, that optimizes both software and hardware to be appropriate for running on resource-constrained devices, i.e. Android phones. More specific, the authors exploit the computation capability of BNN on mobile devices, decoupled with parallel optimizations with the OpenCL library toolkit for enabling real-time and reliable neural network deployments on Android devices. Another research effort in buiding AI on devices is in [200] that applies ML to Android malware detection on Android smart devices. They build an Application Program Interface (API) call function and integrate a ML classification algorithm to detect abnormal mobile operations or malicious attack on the device. As a trial, some typical ML classifiers are considered, including SVM, decision tree, and bagging to characterize the designed API scheme, showing that ML can assist to recognize malware, showing that the ML- based bagging classifier achieves the best performance in terms of higher malicious application detection rate. Moreover, a ML-based API module is designed in [201] to recognize malware on Android phones. There are 412 samples including 205 benign app and 207 malware app used in the test for classification by integrating with SVM and tree random forest classifiers. The trial results show that ML classifiers can achieve high malware detection rates, with over 90% for cross validation examination. Similarly, a ML-based malware detection scheme called Talos is also presented in [202] on Android operating systems. Talos only relies on on-device ML to identify and detect malicious apps on Android devices without the need of external computing services for running the ML algorithm. * • Embedded DL functions: Software and hardware support for performing DL on mobile devices is now evolving extremely fast. In [203], the authors run DNNs on over 10,000 mobile devices and more than 50 different mobile system-on- chips. The most complex task that can be run with the help of TensorFlow Mobile or TensorFlow Lite is image classification with MobileNet or Inception CNNs at the moment. The work in [204] proposes Minerva as a highly automated co-design approach across the algorithm, architecture, and circuit levels to optimize DNN hardware accelerators. Minerva enables highly accurate, ultra-low power DNN accelerators (in the range of tens of milliwatts), making it feasible to deploy DNNs in power-constrained IoT and mobile devices. Another example of mobile AI function is in [205] that develops a MobileNet app running lightweight CNN on mobile devices for mobile localization applications. Images captured by phone cameras can be recognized by CNN to specify the centroid of the object (i.e. the human hand), which is useful to various applications in human motion detections. CNN is also applied to build classification functions for preserve user privacy on embedded devices [206]. With the popularity of Android apps on the market, malicious apps can attack mobile operating systems to exploit personal information, which breaches mobile data over the Internet. By using a CNN-based training model, the mobile processor can recognize attack activities by integrating the trained model into the malicious apps. Therefore, all behaviour and data streams on mobile websites can be identified and detected. CNN inference is becoming necessary for improving user experience with recent applications, such as information extraction from mobile data streams, virtual reality. The authors in [207] present a hardware design that considers to integrate CNN inference into embedded AMR devices. They implement a Kernel- levels scheme which represents the four-layer CNN across heterogeneous cores. Different cores would process different layers based on the consecutive images in a stream. The improvement of overall throughput of the CNN architecture stems from the computation resources of multi-cores and on-chip data memory. In addition to CNN, DNN has been explored for mobile learning [208]. In this work, an on-chip DL model is designed for edge AI devices. Due to the overhead of computation on mobile CPUs, an improved DNN inference engine is considered with half precision support and optimized neural network layers to be suitable for resource-limited devices. This model potentially enhances the GPU throughput by 20 times and saves 80% memory usage. As a different AI platform, a collaborative AI scheme is proposed in [209] where cloud servers are used to support AI training, while AI model is executed on the mobile devices. All features of an AI model (i.e. DNN) are offloaded to cloud for processing. A loss function is created that calculates the compressibility of intermediate features in the model, and it is then integrated with the overall loss function. This solution would accelerate the learning process for multi-tasks, and mitigate the workload offloaded to cloud through compression. In particular, CNN can be integrated in edge devices to create an on-device CNN architecture [210]. CNN layers can learn the features collected from the data samples on wearable sensor devices. The edge board such as SparkFun Apollo3 chip is used to run the DL algorithm. The novelty of this work is to use batch normalization for recognition tasks with CNN on mobile devices. To optimize the performance of on-chip DL-based system, a memory footprint optimization technique is also involved and compared with conventional statistical ML technique, showing a lower execution delay for running mobile DL models. Recently, an on-board AI model is designed for mobile devices for sensing and prediction tasks [211]. To do this, a CNN architecture is considered, with two convolutional blocks, two linear blocks with a sigmoid block to perform learning and classification on embedded devices. A use-case using CNN for detecting the seed germination dynamics in agriculture is taken with seed images as training data inputs, showing the high accuracy in seed recognition with power savings for running DL algorithm. Figure 9: Federated learning for mobile wireless networks. ### VI-B Federated learning at user side In this sub-section, we present the recent development of federated learning (FL) in wireless networks. Conceptually, FL requires a cross-layer implementation involving user devices, network devices, and the access procedure. However, since almost all of the AI-related computations are carried out by user devices in FL, we put FL under the category of this section ”User device AI”. Problem formulation: In the large-scale mobile networks, it is in-efficient to send the AI learning model to remote cloud or centralized edge servers for training due to high communication costs and AI data privacy issues. With widely distributed and heterogeneous user data located outside the cloud nowadays, centralized AI would be no longer a good option to run AI in wireless networks. FL can preserve privacy while training a AI model based on data locally hosted by multiple clients. Rather than sending the raw data to a centralized data-center for training, FL allows for training a shared model on the server by aggregating locally computed updates black [212]. Drawbacks of conventional methods: The traditional AI models rely on a cloud or edge server to run AI functions, which put high pressure on such a centralized data center in terms of high data storage and processing demands. Further, offloading to remote servers to run AI function may result in data loss due to the user reluctancy of providing sensitive data, which in return degrades data privacy. Unique advantages of using wireless AI techniques: FL allows to train AI models in a distributed manner where local data can be trained at mobile devices for low-latency learning and data privacy protection. It also reduce the flow of data traffic to avoid network congestion. Recent works: In [213], the authors integrate the DRL techniques and FL framework with mobile edge systems, for optimizing, caching in mobile edge computing. The work in [214] presents a collaborative FL network for smart IoT users. The FL includes two steps, including model training on the smartphones using data collected from IoT devices and model sharing over the blockchain network [215]. To encourage more users to join the FL, an incentive mechanism is designed to award coins to users, which in return enhances the robustness of the mobile FL network. To support low-latency vehicle-to-vehicle (V2V) communications, the authors in [216] employ FL that enables to interconnect vehicular users in a distributed manner. In this context, each user transmits its local model information such as vehicle queue lengths to the roadside unit (RSU) where a centralized learning module is available for perform learning for maximum likelihood estimation. FL is also useful to DRL training in the offloading process of the MEC-based IoT networks [217]. In the offloading scheme, each IoT device takes actions to make offloading decisions based on system states, i.e. device transmit power, CSI conditions. In such a context, FL allows each IoT user to use his data to train the DRL agent locally without exposing data to the edge server, which preserves the privacy of personal information. Each trained model on individual IoT user is then aggregated to build a common DRL model for. By using FL, the IoT data is trained locally that eliminates the channel congestion due to a large amount of data transmitted over the network. In addition to training support, FL is able to preserve the privacy of DL schemes, i.e. in the neural network training process [218]. Here, the training model consists of centralized learning on the edge server and FL at the users. Specially, in FL, both users and servers collaborate to train a common neural network. Each user keeps local training parameters and synchronize with the cloud server to achieve an optimal training performance. By implementing the FL, user privacy can be ensured due to the local training. Moreover, a FL scheme is introduced in [219] to support training the collaborative model over a network of agents of users, service providers and agents. These entities can incorporate to predict the information sharing using a privacy-preserved centralized model. Here, the centralized model is stored in the cloud server, but learning parameter settings or updates are implemented by each local agent. The cloud server then averages the updates to produce the shared model. All training data is kept on the local user agent, and no updates are stored on the cloud server for privacy protection. Another solution introduced in [220] uses FL for building a self-learning distributed architecture that aims to monitor security of IoT devices (i.e. detecting compromised devices). In this case, FL is used to aggregate detection profiles of the IoT network where the mobile gateways can train data locally without sharing with remote servers. This would not only mitigate the latency overhead of the training process, but also ensure IoT data privacy by using available resource for model training. Besides, to support low-latency communication in mobile wireless networks, a federated edge computing scheme is proposed in [221] as depicted in Fig. 9. User data can be trained locally using local datasets and model updates can be transmitted over the multi- access channel to the edge server for aggregation of local models. The process is repeated until the global model is converged. Further, an on-device FL model for wireless network is suggested by [222]. The main focus is on a design for communication latency mitigation using a Newton method and a scheme for reducing error during the FL model update based on difference-of-convex functions programming. Different from these works, the authors in [223] pay attention to privacy preservation for mobile FL. The FL model includes key generation centre, cloud server and participants (users). Each user has private dataset and performs training locally, then extracts the model updates (gradient parameters). These updates would be exchanged with the cloud server to achieve a global training. Here, to preserve privacy for FL, a distributed Gaussian scheme is integrated that allows to encrypt local data to achieve a secure aggregation protocol. ### VI-C Lessons learned With the rapid growth of smart mobile devices and improved hardware technology, AI functions have been embedded into mobile devices to create a new paradigm: on-device AI. This would open up new opportunities in simplifying AI implementation in wireless networks by eliminating the need of external servers, i.e. clouds. In fact, the feasibility of on-device AI model has been demonstrated via recent successes with designs of mobile ML functions [198] and mobile DL functions [203]. For example, DNN inference now can be run on mobile devices such as Android phones for local client applications such as image classification [205], object recognition [210]. Developing deep AI models on mobile devices is predicted to revolutionize the way future intelligent networks operates and provides user services, especially in the booming era of the Internet of Things. Moreover, AI can be implemented by the cooperation of distributed mobile devices, which gives birth to FL paradigms. Rather than sending the raw data to a centralized data-center for training, FL allows for training a shared model on the server by aggregating locally computed updates [212]. FL can support training the collaborative model over a network of agents of users, service providers and agents with low training latency, traffic congestion mitigation and privacy preservation [217]. We list User Device AI use cases in the taxonomy TABLE VI to summarize the contributions of each reference work. TABLE VI: Taxonomy of User Device AI use cases. Category | Ref. | Use case | AI techniques applied | Main contributions ---|---|---|---|--- User Device AI | [198] | Embedded ML functions | Neural networks | An AI software system running on mobile devices or edge devices. [199] | Mobile ML engine | Bsinary neural networks | A GPU-based binary neural networks (BNN) engine for Android devices. [200] | Malware detection | SVM, decision tree | An Android malware detection on Android smart devices. [201] | Malware detection | SVM, tree random forest | A ML-based API module to recognize malware on Android phones. [203] | Image classification | DNNs | A scheme for image classification using deep neural networks (DNNs) on mobile devices. [205] | Mobile localization | CNN | A MobileNet app, which run lightweight CNN on mobile devices for mobile localization applications. [206] | Mobile privacy protection | CNN | A scheme for building classification functions for preserve user privacy on embedded devices. [208] | DNN inference engine | DNN | An on-chip DL model for edge AI devices. [210] | Mobile activity recognition | CNN | An on-device CNN architecture for support human activity recognition. [211] | Mobile sensing and prediction | CNN | An on-board AI model for mobile devices for sensing and prediction tasks [214] | Collaborative federated learning | FL | A collaborative federated learning network for smart IoT users. [216] | Vehicular communications | FL | A FL model for supporting low-latency vehicle-to-vehicle (V2V) communications. [219] | Collaborative learning | FL | A FL scheme for training the collaborative model over a network of agents of users, service providers and agents. [220] | Security monitoring | FL | A self-learning distributed architecture to monitor security of IoT devices. [221] | Low-latency communication | FL | A federated edge computing for supporting low-latency communication in mobile wireless networks. [222] | Communication latency mitigation | FL | An on-device FL model for wireless networks. ## VII Data-provenance AI The final Wireless AI function of the data life cycle in Fig. 2 is Data- provenance AI where AI can provide various useful services, such as AI data compression, data privacy, and data security. We will focus on discussing the roles of AI for such important services that are shown in Fig. 10. Figure 10: Data-provenance AI function. ### VII-A AI Data compression We here analyze how AI can help data compression tasks in wireless networks. Problem formulation: In the era of big data, i.e. big sensor data, sensor data needs to be compressed to mitigate network traffic and save resources of wireless networks. There are two important data compression tasks in wireless networks, including data aggregation and data fusion. Here, data aggregation is a process that aims to minimize the data packets from different sources (i.e. sensor nodes) to alleviate the transmit workload. Due to the resource constraints of wireless sensor devices, data aggregation is of paramount importance for achieving efficient data collection while preserving energy resources on wireless networks. Meanwhile, in the data fusion, data from ubiquitous sensors can be fused to analyze and create a global view on the target subject. For example, in a smart home setting where sensors in each room collect data that represents the location information. Drawbacks of conventional methods: Many data compression algorithms have been proposed in [23], [25], but they remain some unsolved issues. For example, a wireless sensor network often consists of heterogeneous sensor nodes with multi-media data associated with different statistical properties, which makes traditional data compression approaches using mathematical formulations inefficient. Further, in the future networks, the number of sensor nodes would increase rapidly with higher data complexity and high data volume, how to compress data with high scalability and robustness without loss of valuable data is a challenge. Unique advantages of using wireless AI techniques: The major benefits of AI for data compression tasks in wireless networks are intelligence and scalability. Thanks to the online learning and prediction, AI can extract the useful information to compress the ubiquitous sensor data and identify the most important features that are necessary for later data processing. Moreover, AI can provide scalable data learning, data processing and data computation for a large dataset. The use of DNN would overcome the limitations of current ML approaches to map and learn the big data collected from sensors. Recent works: Some important works are highlighed with the exploitation of AI for data compression tasks, including sensor data aggregation and sensor data fusion. * • Sensor data aggregation: The work in [224] presents a sensory data aggregation scheme for wireless sensor clusters. The main focus is on classifying the collected data using a supervised intrusion detection system. Here a hybrid ML system including a misuse detection and an anomaly detection system are designed to monitor the data aggregation process and detect data attacks. To investigate the feasibility of the proposed approach, a knowledge discovery in data mining dataset [225] is employed as the dataset for training and testing, showing that ML can help achieve above 99% detection rate and 99.8% accuracy. A new tree-based data aggregation framework is introduced in [226] for sensor networks. Sensory data can be collected by an aggregator and then a part of data is transmitted to the sink associated with a function that represents the characteristics of all data, while the remaining data is processed in the sink for transmit energy savings. The classification of the aggregated data is implemented by a Bayesian belief network, which can achieve 91% accuracy via numerical simulations. The efficiency of data aggregation in wireless networks can be further improved with DL approaches [227]. In this study, the focus is on privacy preservation for sensor data on the link between gateways and servers. Here, a DL autoencoder is used to ensure security for the sensor communication channel by reconstructing the obfuscated data from the original data, which can hidden data from unauthorized parties. In some practical scenarios, aggregators may not be willing to collect data due to the lack of motivation (i.e. less aggregation revenues). To solve this issue, the work in [228] implements a payment-privacy preservation scheme that enables aggregators to collect securely data while paying (i.e. rewards) to them. This process can formulated as a MDP problem and since the transition probabilities of payment is unknown to the aggregators, a model-free DRL approach using deep Q-learning is proposed to learn the model via trial and error with an agent. The reward here correlates with the optimal payment so that the agent explores the data aggregation network to find an optimal policy for maximizing revenues. * • Sensor data fusion: The study in [229] presents a multi-sensor data fusion that serves human movement classification based on ML. A network of heterogeneous sensors of a tri-axial accelerometer, a micro-Doppler radar and a camera is used to collect data that is then fused and classified by a ML algorithm with SVM and KNN. In [230], a sensor fusion architecture is proposed for object recognition. Parameters or information measured from sensors can be fused with those collected from other sensors to create a global view on the target subject, which helps evaluate it more comprehensively. To facilitate the fusion process, a processing model based on unsupervised ML is integrated that helps classify the features from the fused datasets. Experiments use camera images as the input data for training and classification, showing that ML can recognize the object well in different parameter settings. Another solution for sensor fusion in wireless sensor networks is presented in [231]. In particular, a system setting with various rooms is considered, where sensors in each room collect data that represents the location information. Then, data collected from different sensors is fused and a ML-based K-Spectral Centroid algorithm is adopted to cluster the data from different rooms together. In practice, heterogeneous sensory data fusion may be challenging due to the insufficient information of collected data sources and the difficulty to represent the multimodal data, which can make data estimation infeasible. To solve this issue, a new multimodal data fusion for sensor networks is proposed in [232]. More specific, a deep multimodal encoder based on neural networks is designed to perform learning from data collected from sensors to capture the correlated features, which addresses the missing data issue. Importantly, an unsupervised learning algorithm is derived to compress data by learning joint features among the set of features in the raw dataset. Meanwhile, a Bayesian network is employed in [233] for sensor data fusion. The complex non-Gaussian sample dynamics in sensor signal patterns are fused and classified based on an offline Bayesian non-parametric Dirichlet Process model, which achieves a higher accuracy performance compared with traditional Bayes network and SVM classifiers. ### VII-B AI Data Privacy In this sub-section, we present how AI can help improve data privacy in wireless networks. Problem formulation: At the network edge, data privacy protection is the most important service that should be provided. The dynamicity of the data collection, data transmisison and data distribution in future networks would put at risks of user’s information leakage. For example, the data collection can expose sensor’s information to third parties which makes privacy (i.e. privacy of user address, user name) vulnerable. Therefore, protecting data privacy is of paramount importance for network design and operations. By ensuring data privacy, a wireless network can appeal to a larger number of users which increases the utility of such a system. Drawbacks of conventional methods: Many existing approaches have been proposed to solve data privacy issues, mainly using encryption protocols [23], [26]. Although such techniques can keep the content of user’s information private from the others and third parties, it is hard to apply to scalable wireless networks with large data. Further, they often apply a specific encryption protocol to a group of multi-media data collected from sensors and devices, which is infeasible to future large-scale networks. Unique advantages of using wireless AI techniques: AI potentially provides privacy functions for sensed data via classification and detection to recognize potential data threats. AI is also useful to user location privacy. In fact, AI can also support protection of user location privacy by building decision-making policies for users to adjust their location information and selecting privacy location preference. Recent works: From the perspective of data generation process, we consider two AI data privacy domains, including data sensing privacy and user location privacy. * • Data sensing privacy: The work in [234] concerns about privacy preservation for mobile data sensing. Differential privacy is used to protect the individual information during the data collection process. To authenticate the user identity based on their own data, a ML-based classifier approach using two-layer neural networks is designed. It classifies the privacy sensitivity levels (0-9) and evaluates the privacy vulnerability with leakage possibility. To overcome the limitations of existing encryption solutions in terms of computation complexity and the lack of ability to solving large-scale domains, a multi-functional data sensing scheme with awareness of different privacy is proposed in [235]. The key objective is to provide flexible sensing functions and aggregation properties to serve different demands from different users. Besides, it is important to preserve the collected sensor data against adversarial attacks for better user information protection. In such contexts, the aggregation queries from sensors are collected as the dataset for training that is implemented by a ML algorithm, while the multifunctional aggregation can be performed by a Laplace differential privacy technique. Then, learning helps to estimate the aggregation results without revealing user privacy. As an example, a distributed privacy-preserved ML scheme for crowd sensing is presented in [236] that combines sensing, learning and privacy protection via a classifier design on mobile devices. The behind ML algorithm is a stochastic gradient descent model that is implemented on each local device. This solution not only distributes learning workload over the network of devices, but also keeps data private due to no need to transfer data on the network. Meanwhile, the work in [237] solves the data collection issue with privacy awareness using K-mean clustering. Indeed, to ensure data privacy, a technique of adding Laplace noise to the data compression process. Also, aggregated data is distributed over the mobile device network where each device holds a differentially-private sketch of the network datasets. In the multimedia IoT networks, the sensor nodes can sense data from the environment and transmit collected data to cloud servers for further processing. A critical issue here is the data sensing and transmission can reveal private data, i.e. location information. It is thus important to protect data privacy during the data sensing process. The work in [238] proposes a cluster approach that divides sensor devices into different groups. Each cluster controller ensures the privacy of its sensor group, associated with a neural network that classifies the collected data and extracts useful features for data learning. Such data processing can be helped by using MEC services that not only provides low-latency computation but also enhances privacy for physical sensor devices [16]. A DL scheme for privacy protection and computation services for physical sensors can be seen in Fig. 11. * • User location privacy: An example is in [239] that introduces a privacy preservation scheme to help users to manage their location information. The key objective is to enable users to obtain the privacy preferences based on their choice factors such as QoS, trust levels. Based on that, a k-learning model is built that can make decisions for users in preserving location privacy. The authors in [240] are interested in privacy in mobile social networks. They first perform a real experiment to collect user location information in a campus network and investigate the possibility of location privacy leakage. Then, a privacy management scheme is designed to provide privacy control for mobile users. This model is empowered by a ML-based decision tree algorithm that learns the user privacy preference and classifies location contexts in the social ecosystem. At present, the location of mobile devices is mostly identified by GPS technology, but this platform remains inherent limitations in terms of energy overheads and reliance on the third party (i.e. service providers), which poses user information at privacy risks. A new method presented in [241] can achieve a user privacy location guarantee with energy savings. By using the features of current location that represents the privacy location preference, a ML approach is applied to learn the privacy model, aiming to classify privacy preferences depending on whether the user want or not want to share its location information based on their activities. It is noting that capturing features only relies on sensors available on smart phones without using GPS that is power saving and eliminates the need of third party. To preserve user location privacy in crowd sensing, a learning scheme is presented in [242] to model the correlations between sensitive and non- sensitive scenarios according to the relations with private locations. The main aim is to find a strategy so that the system utility is maximized while user location is protected against malicious attacks. The work in [243] jointly takes context factors, i.e. occasion and time, and individual factors into considerations to establish a privacy location preservation protocol in social networks. A ML-based regression model is involved to evaluate the privacy levels based on such features, showing that the occasion factor dominates in determining the privacy degree during the location sharing. Figure 11: DL for MEC privacy preservation. ### VII-C AI Data Security In addition to privacy protection, AI also has the great potential for enhancing security for wireless networks. We here present how AI can help to protect data security. Problem formulation: Data security is the top research concern in wireless network design. A wireless network system is unable to work well and robust under the presence of security threats and attacks. The openness and dynamicity of wireless networks makes them prone to external data threats, while unauthorized users can access the resource of the network. Drawbacks of conventional methods: Wireless networks have gained increasing influences on modern life that makes cyber security an significant research area. Despite installed defensive security layers, the wireless networks inherit cyber attacks from the IT environments. Many security models have been proposed, mainly using middle boxes such as Firewall, Antivirus and Intrusion Detection Systems (IDS) [244]. These solutions aims to control the traffic flow to a network based on predefined security rules, or scan the system for detecting malicious activities. However, many traditional security mechanisms still suffer from a high false alarm rate, ignoring actual serious threats while generating alerts for non-threatening situations. Another issue of the existing security systems is the impossibility of detecting unknown attacks. In the context of increasingly complex mobile networks, the human-device environments change over time, sophisticated attacks emerge constantly. How to develop a flexible, smart and reliable security solution to cope with various attacks and threats for future wireless networks is of paramount importance for network operators. Unique advantages of using wireless AI techniques: AI has been emerged as a very promising tool to construct intelligent security models. Thanks to the ability of learning from massive datasets, AI can recognize and detect attack variants and security threats, including unknown ones in an automatic and dynamic fashion without relying on domain knowledge [245]. Also, by integrating with on-device learning algorithms, it is possible for AI to recognize the unique features of a certain user (i.e. through AI classifiers), aiming for the user classification against threats. Recent works: We here introduce some important security services that AI can offer, including user authentication and attack detection. * • AI for user authentication: Some recent works are interested in exploiting AI for user authentication in wireless networks. An example in [246] introduces a user authentication model using millimetre sensors. Each radar sensor is equipped with four receive antennas and two transmit antennas. Signals that re reflected from a part of human body are analyzed and extracted to produce a feature dataset that is then fed to a ML-based random forest classifier. Through training and learning, the classifier can recognize the biometric properties of any individuals. In [247], a user authentication scheme is proposed based on CSI measurements working for both stationary and mobile users. For the former case, the authors build a user profile with considerations of spoofing attacks, while for the later case, the correlations of CSI measures are taken to implement user authentication checking. Specially, an authenticator is created that runs a ML algorithm to learn from the CSI profiles and determine whether the CSI value measured in the coming packet matches with the one created in the profile storage. If they match, the authentication is successful, otherwise that user is denied for further processing. The Wi-Fi signals can be exploited to build datasets of learning for user authentication. The authors in [248] try the first time for a new user authentication system using human gait biometrics from Wi-Fi signals. Here, CSI value samples are collected from Wi-Fi devices. The core component of the scheme is a 23-layer DNN network that is modelled to extract the important salient gait features collected from gait biometrics in the CSI measurement dataset. DNN would act as a classifier to extract unique features of a user and identify that user among the mobile user set. A user authentication scheme for Wi-Fi networks is also suggested in [249] which can be installed on mobile devices to detect malicious access and prevent spoofing attacks. By integrating with on-device learning algorithms, it is possible to recognize the unique features of a certain user (i.e. through ML classifiers) where visible light can be an indicator input for the user classification. As a different viewpoint of user authentication, the work in [250] analyzes CSI data for user authentication in wireless networks. First, a CNN classifier is built to learn the extracted features from CSI dataset such that pooling layers are utilized to simplify the computation process for faster learning. To analyze CSI features, a RNN network is then designed, including some recurrent layers and a fully connected layer where user recognition is implemented. The combination of CNN and RNN models leads to a new scheme, called CRNN, aiming to use the capability of information representation of CNN and information modelling of RNN. Thus, the accuracy for user authentication is improved that is also confirmed through computer simulations. It is noted that in wireless networks, the authentication between two users on the physical network layer based on traditional approaches is challenging due to the time variance of channels and mobility of users. * • AI for attack detection: The work in [251] introduces a scalable intrusion detection system (IDS) for classifying and recognizing unknown and unpredictable cyberattacks. The IDS dataset collected from a large-scale network-level and host-level events is processed and learned by a DNN architecture that can classify the network behaviours and intrusions. Compared to commercial IDSs that mainly rely on physical measurements or defined thresholds on feature sets to realize the intrusions, learning can assist flexible IDS without any further system configurations, and work well under varying network conditions such as network traffic, user behaviour, data patterns. Another solution in [252] formulates an intrusion detection system using RNN. The main focus is on evaluating the detection rate and false positive rate. The intrusion detection can be expressed equivalently to a multi-class classification problem for recognizing whether the traffic behaviour in the network is normal or not. The key motivation here is to enhance the accuracy of the classifiers so that intrusive network behaviours can be detected and classified efficiently. Motivated by this, a RNN-based classifier is leveraged to extract abnormal traffic events from the network features using the well-known NSL-KDD dataset [253]. Specially, the attacks can be divided into four main types, including DoS (Denial of Service attacks), R2L (Root to Local attacks), U2R (User to Root attack), and Probe (Probing attacks). The results of deep training and learning show a much better classification accuracy of RNN in comparison with ML techniques. In addition to that, wireless networks also have to solve spoofing attacks where an attack claims to be a legitimate node to gain malicious access or perform denial of service attacks. Conventional security solutions, such as digital signatures, cannot fully address the spoofing issues, especially in the dynamic environment with unknown channel parameters. Recently, AI gains interest in spoofing detection tasks. As an example, a solution proposed in [254] uses ML to detect and identify spoofing attacks in massive MIMO 5G networks. By collecting the virtual path gain of massive MIMO channel, a virtual channel space is created to detect the anomaly among the network users. In particular, a learning-based algorithm is built to estimate the anomalies and recognize spoofing attacks without knowing the knowledge of channel/signal information. Further, the work in [255] also leverages the potential of ML to detect spoofing attacks in wireless sensor networks. By analyzing the correlation of signal strength on the mobile radio channel, the learning algorithm receives signal strength samples as the input data for abnormal behaviour analysis. To enhance the detection accuracy, the collected samples can be categorized into small windows that may include the attacking nodes and legitimate nodes. Then, a collaborative learning scheme with k-NN and k-means is applied to identify the spoofing attacks in the dense networks. Meanwhile, the authors in [256] use RL to implement a spoofing detection model from the PHY layer perspective for wireless networks. The key concept is to exploit the CSI of radio packets to detect spoofing vulnerabilities. The spoofing authentication process is formulated as a zero-sum game including a receiver and spoofers. Here, the receiver specifies the test threshold in PHY spoofing detection, while spoofers select attack frequency to optimize its utility with respect to Bayesian risks. Specially, the threshold selection is achieved by trial and error process using Q-learning without knowing CSI such as channel variations. ### VII-D Lessons learned The main lessons acquired from the survey on the Data-provenance AI domain are highlighted as the following. * • Recent studies also raise the discussion on the AI adoption for data-driven applications during the data generation process. They mainly concentrate on four specific domains, namely AI data compression, data clustering, data privacy, data security. As an example, AI has been applied for supporting sensor data aggregation and fusion, by classifying useful information from the collected data samples [224]. Also, the classification of AI helps monitor the data aggregation process, aiming to analyze the characteristics of all data and detect data attacks [226]. * • At the network edge, usage privacy and data security protection are the most important services that should be provided. From the perspective of data generation process, the applications of AI in data privacy are reflected in two key domains, including data sensing privacy and user location privacy. Currently, AI-based data classification algorithms can be implemented on each local device. This solution not only distributes learning workload over the network of devices, but also keeps data private due to no need to transfer data on the network [236]. Interestingly, recent studies in [239], [240] show that ML can help learn the user privacy preference and classify location contexts in the social ecosystem. * • In addition to privacy protection, AI also has the great potential for enhancing security for wireless networks. In the literature, AI mainly provide some important security services, including user authentication and access control. In practical wireless networks, the authentication between two users on the physical network layer based on traditional approaches is challenging due to the time variance of channels and mobility of users. AI may be an ideal option to recognize legitimate users from attackers by training the channel data and perform classification. Meanwhile, DRL has proved extremely useful in building access control mechanisms [257], [258] that can learn from the user behaviours and estimate the access events for appropriate security actions. In summary, we list Data-provenance AI use cases in the taxonomy TABLE VII to summarize the contributions of each reference work. TABLE VII: Taxonomy of Data-provenance AI use cases. Category | Ref. | Use case | AI techniques applied | Main contributions ---|---|---|---|--- AI Data compression | [224] | Sensory data aggregation | Supervised ML | A sensory data aggregation scheme for wireless sensor clusters. [226] | Sensory data aggregation | Bayesian belief network | A new tree-based data aggregation framework for sensor networks. [227] | Data aggregation | DL autoencoder | A scheme for data aggregation tasks with a focus on privacy preservation. [229] | Multi-sensor data fusion | SVM and KNN | A multi-sensor data fusion that serves human movement classification based on ML. AI Data Privacy | [234] | Data privacy preservation | Neural networks | A ML scheme for privacy preservation in mobile data sensing. [235] | Different privacy | Neural networks | A multifunctional data sensing scheme with awareness of different privacy. [237] | Privacy awareness | K-mean clustering | A model for the data collection issue with privacy awareness using K-mean clustering. [238] | Data privacy | Neural networks | A scheme for protecting data privacy during the data sensing process. [239] | User location privacy | k-learning | A privacy preservation scheme that helps users to manage their location information. [243] | Privacy location preservation | Regression learning | A privacy location preservation protocol in social networks. AI Data Security | [246] | User authentication | Random forest | A user authentication model using millimetre sensors. [248] | User authentication | Multi-layer DNN | A new user authentication system using human gait biometrics from Wi-Fi signals. [250] | User authentication | CNN | A scheme for analyzing CSI data for user authentication in wireless networks. [251] | Intrusion detection | DNN | A scalable intrusion detection system (IDS) for classifying and recognizing unknown and unpredictable cyberattacks. [252] | Intrusion detection | RNN | An intrusion detection system for evaluating the detection rate and false positive rate. [254] | Spoofing detection | Supervised learning | A ML approach to detect and identify spoofing attacks in massive MIMO 5G networks. [255] | Spoofing detection | k-NN and k-means | A learning model for detecting spoofing attacks in wireless sensor networks. [256] | Spoofing detection | RL | A spoofing detection model from the PHY layer perspective for wireless networks. ## VIII Research Challenges and Future Directions Different approaches surveyed in this paper clearly show that AI plays a key role in the evolution of wireless communication networks with various applications and services. However, the throughout review on the use of AI in wireless networks also reveals several critical research challenges and open issues that should be considered carefully during the system design. Further, some future directions are also highlighted. ### VIII-A Research challenges We analyze some important challenges in wireless AI from three main aspects: security threats, AI limitations, and complexity of AI wireless communications. #### VIII-A1 Security threats Security is a major concern in wireless AI systems. Data collected from sensors and IoT devices are trained to build the model that takes actions for the involved wireless applications, such as decision making for channel allocation, user selection, or spectrum sharing. Adversaries can inject false data or adversarial sample inputs which makes AI learning invalid. Attackers can also attempt to manipulate the collected data, tamper with the model and modify the outputsc [259]. For instance, in reinforcement learning, the learning environment may be tampered with an attacker so that the agent senses incorrectly the information input data. Another concern can stem from the manipulation on hardware implementation by modifying hardware settings or re- configuring system learning parameters. The work in [260] investigated the multi-factor adversarial attack issues in DNN with input perturbation and DNN model-reshaping risks which lead to wrong data classification and then damages the reliability of the DNN system. Evaluating security threats on AI models and providing creative solutions to prevent attack risks and protect AI operations is vitally important to AI-based wireless applications. #### VIII-A2 Limitations of AI While AI shows impressive results in wireless communication applications, its models still remain performance limitations. For example, the imperfection bottleneck in the training process of DNN can be exploited by adversarial attacks to tamper with input data that leads to misclassification [261]. Further, an investigation in [262] reveals the false confidence of DNN in estimating images. By generating fooling samples (i.e. images) in a way imperceptible to humans, a DNN can mislabel the image and thus recognize wrongly the object. Another limitation is the high cost of training the AI model. Indeed, it is not uncommon to take some days for the AI training on mobile devices or even edge devices with limited computation resources. A more challenge is the scarcity of real datasets available for wireless communications necessary to for the AI training, i.e. in supervised learning [263]. In many wireless scenarios where only a small dataset is not sufficient for the training which degrades the performance of AI model in terms of low classification or prediction accuracy, and high model overfitting, and thus the confidence of AI model may not be guaranteed. This drawback would be an obstacle for deployment of AI in real-world wireless applications since the feasibility of AI should be demonstrated in testbeds or experiments in the real environments. #### VIII-A3 Complexity of AI wireless communication networks The complexity may stem from wireless data and network architecture. In fact, in AI wireless communication networks, the data is produced from both application traffic and information of devices and objects. Considering the complex wireless network with various data resources, it is challenging to extract data needed for the AI training of a specific task. In some cognitive network scenarios, how to label the massive learning data for training AI models is also highly challenging. From the network architecture perspective, the traditional protocol of wireless communication is based on a multi-layer network architecture. However, currently it is difficult to determine which and how many layer should be configured to run AI functions and deploy AI technologies in devices. Therefore, it is essential to consider intelligent protocol architectures with the simplified and adaptive capability according to AI functions and requirements [264]. ### VIII-B Future directions We discuss some of the future research directions on wireless AI motivated from the recent success in the field. #### VIII-B1 Specialized AI architecture for wireless networks The AI architecture has significant impacts on the overall performance of the wireless AI systems, while the current AI mode design is still simple. Many AI architectures proposed in [21], [22], [24], [26] are mostly relied on conventional algorithms with learning iterations, data compression, decoding techniques. The performance of wireless AI algorithms can be reduced thanks to recent AI advances with sub-block partition solutions for neural networks [146] or data augmentation for DL [265]. However, these AI approaches suffer from dimensionality problems when applying to wireless applications due to the high complexity of the wireless systems, and data training would be inefficient when the network dimension is increased with the increase of mobile devices and traffic volumes. Therefore, developing adaptive AI architectures specialized in wireless communication scenarios is the key for empowering future intelligent applications. As suggested by [266], by adapting with the dynamicity of wireless communications, AI can model effectively the wireless systems and discover the knowledge of data generated from network traffic and distributed devices. Besides, motivated by [24], [26], AI functions should be implemented and deployed in the wireless network in a distributed manner. This architecture not only influences the wireless data transmission infrastructure, but also modernize the way that wireless communications are organized and managed [267]. Furthermore, with the ambition of deploying AI on mobile devices/edge devices to extend the network intelligence to local clients, developing specialized hardware/chipsets for AI is vitally important. Recent advances in hardware designs for high-bandwidth CPUs and specialized AI accelerators open up new opportunities to the development of AI devices at the edge of the network [268], [269]. Motivated by this, recently Facebook bring ML inference to the mobile devices and edge platforms [270] thanks to specialized AI functions with high data training speed and low energy costs. The work in [271] also introduces a DL inference runtime model running on embedded mobile devices. This scheme is able to support data training and MIMO modelling using a hardware controller empowered by mobile CPUs with low latency and power consumption with bandwidth savings. #### VIII-B2 From simulation to implementation Reviewing the literature, most of the AI algorithms conceived for wireless communication networks are in their simulation stages. Thus, the researchers and practitioners needs to further improve AI functions before they are realized in real-world wireless networks. We here propose some solutions that can be considered in practical wireless AI design. First, real-world datasets collected from physical communication systems must be made available for ready AI function implementation. The sets of AI data generated from simulation environments such as deepMIMO dataset [272] can contain the important network features to perform AI functions, but they are unable to reflect exactly the characteristics of a real wireless system. Second, as discussed in the previous section, designing specialized hardware/chipsets for AI is extremely important to deploy AI applications on edge/mobile devices. These new hardware architectures should provide functions specializing AI to support on-device training, modelling and classification at the edge of the network. Third, the multidimensional assessment for wireless AI applications in various scenarios is also vitally necessary to realize the feasibility of the AI adoption in practice. By comparing with simulation performances, the designers can adjust the model parameters and architecture settings so that AI can well fit the real channel statistical characteristics. #### VIII-B3 AI for 5G and beyond The heterogeneous nature of 5G and beyond wireless networks features multi- access ultra-dense networks with high data transmission capability and service provisions. AI has been regarded as the key enabler to drive the 5G networks by providing a wide range of services, such as intelligent resource allocation, network traffic prediction, smart system management. For example, the integration of AI enables network operators to realize intelligent services, such as autonomous data collection across heterogeneous access networks, dynamic network slicing to provide ubiquitous local applications for mobile users, and service provisioning management [272]. Wireless AI also acts as AI-as-a-service that can realize the potentials of Massive MIMO, a key 5G technology. For example, AI can be used to predict the user distribution in MIMO 5G cells, provide MIMO channel estimation, and optimize massive MIMO beamforming [273]. With the rapid development of edge 5G services, AI would potentially transform the way edge devices perform 5G functionalities. In fact, the network management and computation optimization of mobile edge computing (MEC) would be simplified by using intelligent AI software through learning from the historical data. Lightweight AI engines at the 5G devices can perform real-time mobile computing and decision making without the reliance of remote cloud servers. Beyond the fifth-generation (B5G) networks, or so-called 6G, will appear to provide superior performance to 5G and meet the increasingly service requirements of future wireless networks in the 2030s. The 6G networks are envisioned to provide new human-centric values with transformative services, such as quantum communications, wearable computing, autonomous driving, virtuality, space-air-ground networks [274], [275], and the Internet of Nano- Things [276]. AI technologies are expected to play an indispensable role in facilitating end-to-end design and optimization of the data processing in 6G wireless networks [277]. The future cellular networks will be much more complex and dynamic, and the number of parameters to be controlled will increase rapidly. In such scenarios, AI can provide smart solutions by its self-learning and online optimization capabilities to support advanced data sensing, data communication, and signal processing [278]. These benefits that AI bring about would accelerate the deployment of future 6G networks and enable newly innovative wireless services. The application of AI in wireless is still at its infancy and will quickly mature in the coming years for providing smarter wireless network services. The network topology model and wireless communication architecture will be more complex with the dynamicity of data traffic and mobile devices. AI is expected to play a key role in realizing the intelligence of the future wireless services and allow for a shift from human-based network management to automatic and autonomous network operations. Therefore, the integration of AI in wireless networks has reshaped and transformed the current service provisions with high performances. This detailed survey is expected to pave a way for new innovative researches and solutions for empowering the next generation of wireless AI. ## IX Conclusions In this paper, we have presented a state-of-the-art survey on the emerging Wireless AI in communication networks. We have first proposed a novel Wireless AI architecture from the data-driven perspective with a focus on five main themes in wireless networks, namely Sensing AI, Network Device AI, Access AI, User Device AI and Data-provenance AI. Then, we have focused on analyzing the use of AI approaches in each data-driven AI theme by surveying the recent research efforts in the literature. For each domain, we have performed a holistic discussion on the Wireless AI applications in different network scenarios, and the key lessons learned from the survey of each domain have been derived. 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2024-09-04T02:54:56.365775

2003.00886

arxiv-papers

11institutetext: IEOR, IIT Bombay, India 11email: {indrojit<EMAIL_ADDRESS> # Financial replicator dynamics: emergence of systemic-risk-averting strategies Indrajit Saha Veeraruna Kavitha ###### Abstract We consider a random financial network with a large number of agents. The agents connect through credit instruments borrowed from each other or through direct lending, and these create the liabilities. The settlement of the debts of various agents at the end of the contract period can be expressed as solutions of random fixed point equations. Our first step is to derive these solutions (asymptotically), using a recent result on random fixed point equations. We consider a large population in which the agents adapt one of the two available strategies, risky or risk-free investments, with an aim to maximize their expected returns (or surplus). We aim to study the emerging strategies when different types of replicator dynamics capture inter-agent interactions. We theoretically reduced the analysis of the complex system to that of an appropriate ordinary differential equation (ODE). We proved that the equilibrium strategies converge almost surely to that of an attractor of the ODE. We also derived the conditions under which a mixed evolutionary stable strategy (ESS) emerges; in these scenarios the replicator dynamics converges to an equilibrium at which the expected returns of both the populations are equal. Further the average dynamics (choices based on large observation sample) always averts systemic risk events (events with large fraction of defaults). We verified through Monte Carlo simulations that the equilibrium suggested by the ODE method indeed represents the limit of the dynamics. ###### Keywords: Evolutionary stable strategy (ESS), Replicator dynamics, Ordinary differential equation, Random graph, Systemic risk, Financial network. ## 1 Introduction We consider a financial network with large number of agents. These agents are interconnected to each other through financial commitments (e.g., borrowing -lending etc.). In addition they make investments in either risk-free (risk neutral) or risky derivatives. In such a system the agents not only face random economic shocks (received via significantly smaller returns of their risky investments), they are also affected by the percolation of the shocks faced by their neighbours (creditors), neighbours of their neighbours etc. In the recent years from $2007-2008$ onwards, there is a surge of activity to study the financial and systemic level risks caused by such a percolation of shocks ([6, 5, 4, 1]). Systemic risk is the study of the risks related to financial networks, when individual or entity level shocks can trigger severe instability at system level that can collapse the entire economy (e.g., [6, 5, 4]). In this set of papers, the author study the kind of topology (or graph structure) that is more stable towards the percolation of shocks in financial network, where stability is measured in terms of the total number of defaults in the network. In contrast to many existing studies in literature related to systemic risk, we consider heterogeneous agents and we consider evolutionary framework. In our consideration, there are two groups of agents existing simultaneously in the network; one group of agents invest in risk-free instruments, while the other group considers risky investments. The second group borrows money from the other members of the network to gather more funds towards the risky investments (with much higher expected returns). These investments are subjected to large (but rare) economic shocks, which can potentially percolate throughout the network and can even affect the ‘risk-free’ agents; the extent of percolation depends upon relative sizes of the two groups. We consider that new agents join such a network after each round of investment; they chose their investment type (risky or risk-free) based on their observations of the returns (the surplus of the agents after paying back their liabilities) of a random sample of agents that invested in previous round. The relative sizes of the two groups changes, the network structure changes, which influences the (economic shock-influenced) returns of the agents in the next round, which in turn influences the decision of the new agents for the round after. Thus the system evolves after each round. We study this evolution process using the well known evolutionary game theoretic tools. In a financial network perspective, this type of work is new to the best of our knowledge. We found few papers that consider evolutionary approach in other aspects related to finance; in [8], the authors study the financial safety net (a series of the arrangement of the firms to maintain financial stability), and analyze the evolution of the bank strategies (to take insurance or not); recently in [7] authors consider an evolutionary game theoretic model with three types of players, i) momentum traders ii) contrarian traders iii) fundamentalists and studied the evolution of the relative populations. As already mentioned, these papers relate to very different aspects in comparison with our work. Evolutionary stable strategies Traditionally evolutionary game models have been studied in the literature to study animal behaviour. The key ingredients of the evolutionary game models are a) a large number of players, b)the dynamics and c) the pay-off function (e.g., see the pioneering work [13]). Replicator dynamics deals with evolution of strategies, reward based learning in dynamic evolutionary games. Typically it is shown that these dynamics converge to a stable equilibrium point called Evolutionary Stable Strategy (ESS), which can be seen as a refinement of a strict Nash Equilibrium ([13]); a strategy prevailing in a large population is called evolutionary stable if any small fraction of mutants playing a different strategy get wiped out eventually. Formally, in a 2-player symmetric game, a pure strategy $\hat{s}$ is said to be evolutionary stable if 1. 1. $(\hat{s},\hat{s})$ is a Nash Equilibrium; i.e., $u(\hat{s},\hat{s})\geq u(s^{{}^{\prime}},\hat{s})$ $\forall s^{{}^{\prime}}$ and 2. 2. If $(\hat{s},\hat{s})$ is not a strict NE (i.e., $\exists$ some $s^{{}^{\prime}}\neq\hat{s}$ such that $u(\hat{s},\hat{s})=u(s^{{}^{\prime}},\hat{s})$), then $u(\hat{s},s^{{}^{\prime}})>u(s^{{}^{\prime}},s^{{}^{\prime}})$. We study the possible emergence of evolutionary stable strategies, when people chose either a risky or a risk-free strategy; the main difference being that the returns of either group are influenced by the percolation of shocks. The returns of the portfolios depend further upon the percolation of shocks due to layered structure of financial connections, and not just on the returns of the investments, i.e., not just on economic shocks. Our main conclusions are two fold; a) when agents consider large sample of data for observation and learning, the replicator dynamics can settle to a mixed ESS, at which the expected returns of the two the groups are balanced; b) in many other scenarios, through theoretical as well as simulation based study, we observed that the replicator dynamics converges to one of the two strategies, i.e., to a pure ESS (after completely wiping out the other group). The analysis of these complex networks (in each round) necessitated the study of random fixed point equations (defined sample path-wise in large dimensional spaces), which represent the clearing vectors of all the agents ([1, 6, 5, 4] etc). The study is made possible because of the recent result in [1], which provided an asymptotically accurate one dimensional equivalent solution. ## 2 Large Population Finance Network We consider random graphs, where the edges represent the financial connection between the two nodes. Any two nodes are connected with probability $p_{ss}>0$ independent of the others, but the weights on the edges depend on (the number of) neighbors. This graph represents a large financial network where borrowing and lending are represented by the edges and the weights over them. The modeller may not have access to the exact connections of the network, but random graph model is a good approach to analyse such a complex system. In particular we consider the graphs that satisfy the assumptions of [1]. The agents are repeatedly investing in some financial projects. In each round of investment, the agents borrow/lend to/from some random subset of the agents of the network. Some of them may invest the remaining in a risk-free investment (which has a constant rate of interest $r_{s}$). While the others invest the rest of their money in risky investments which have random returns; we consider a binomial model in which returns are high (rate $u$) with high probability $\delta$ and can have large shocks (rate $d$), but with small probability ($1-\delta$); it is clear that $d<r_{s}<u$. We thus have two types of agents, we call the group that invests in risk-free projects as ‘risk-free’ group ($G_{1}$), the rest are being referred to as ‘risky’ group ($G_{2}$) New agents join the network in each round of investment. They chose their investment type, either risk-free or risky, for the first time based on the previous experience of the network and continue the same choice for all future rounds of investment. The new agents learn from network experience (returns of agents of the previous round of investments) and chose a suitable investment type, that can potentially give them good returns. The new agents either learn from the experience of a random sample (returns of two random agents) of the network or learn from a large number of agents. In the former case, their choice of investment type depends upon the returns of the random sample in the previous round. While in the latter case the decision can also depend on the average utility of each group of the agents, obtained after observing large number of samples. Two strategies: As mentioned before, there are two strategies available in the financial market. Risk-free agents of $G_{1}$ use strategy 1; these agents lend some amount of their initial wealth to other agents (of $G_{2}$) that are willing to borrow, while the rest is invested in a government security, for example, bonds, government project etc. Risky agents of $G_{2}$ are adapting strategy $2$, wherein they borrow funds from the other agents and invest in risky security, for example, derivative markets, stocks, corporate loans etc.. These agents also lend to other agents of $G_{2}.$ Let $\epsilon_{t}$ be the fraction of the agents in $G_{1}$ group and let $n(t)$ be the total number of agents in round $t$. Thus the total number of agents (during round $t$) in group 1 equals $n_{1}(t):=|G_{1}|=n(t)\epsilon_{t}$ and $n_{2}(t):=|G_{2}|=n(t)(1-\epsilon_{t})$. We consider that one new agent is added in each round 111this approach can easily be generalized to several other types of dynamics and we briefly discuss a few of them towards the end., and thus size of the graph/network is increasing. The agents are homogeneous, i.e., they reserve the same wealth $w>0$ for investments (at the initial investment period) of each round. Each round is composed of two time periods, the agents invest during the initial investment period and they obtain their returns after some given time gap. The two time period model is borrowed from [5, 4, 1] etc. The new agents make their choice for the next (and the future) round(s), based on their observations of these returns of the previous round. Initial investment phases: During the initial investment phases (of any round $t$), any agent $i\in G_{1}$ lends to any agent $j\in G_{2}$ with probability $p_{ss}$ and it lends (same) amount222This normalization, (after choosing the required parameters, like $w$, appropriately) is done to derive simpler final expressions. $w/(n(t)p_{ss})$ to each of the approachers based on the number that approached it for loan; let $I_{ij}$ be the indicator of this lending event. Note that for large $n(t)$, the number of approachers of $G_{2}$ approximately equals $n(t)(1-\epsilon_{t})p_{ss}$, and, thus any agent of $G_{1}$ lends approximately $w(1-\epsilon_{t})$ fraction to agents of $G_{2}$. The agents of $G_{1}$ invest the rest $w\epsilon_{t}$ in risk-free investment (returns with fixed rate of interest $r_{s}$). Let $\tilde{w}$ be the accumulated wealth333These amounts could be random and different from agent to agent, but with large networks (by law of large numbers) one can approximate these to be constants. of any agent of $G_{2}$ out of which a positive fraction $\alpha$ is invested towards the other banks of $G_{2}$ and $(1-\alpha)$ portion is invested in risky security. Thus the accumulated wealth of a typical $G_{2}$ agent is governed by the following equation, $\tilde{w}=\underbrace{w+w\epsilon}_{\text{Initial wealth + Borrowed from $G_{1}$}}+\underbrace{\tilde{w}\alpha}_{\text{Lend/borrow $G_{2}$}}\mbox{ and thus }\tilde{w}=\frac{w(1+\epsilon)}{(1-\alpha)}.$ (1) Thus the total investment towards the risky venture equals $\tilde{w}(1-\alpha)=w(1+\epsilon)$. The $G_{2}$ agents have to settle their liabilities at the end of the return/contract period (in each round) and this would depend upon their returns from the risky investments. Thus the total liability of any agent of $G_{2}$ is $y=(w\epsilon+\tilde{w}\alpha)(1+r_{b})$, where $r_{b}$ is the borrowing rate444For simplicity of explanation, we are considering constant terms to represent all these quantities, in reality they would be i.i.d. quantities which are further independent of other rounds and the asymptotic analysis would go through as in [1].; by simplifying $y=\frac{w(\epsilon+\alpha)(1+r_{b})}{(1-\alpha)}.$ Similarly, any agent of $G_{2}$ lends the following amount to each of its approachers (of $G_{2}$): $\frac{\alpha{\tilde{w}}}{n(1-\epsilon_{t})p_{ss}}=\frac{\alpha w(1+\epsilon)}{n(1-\epsilon_{t})p_{ss}(1-\alpha)}.$ (2) Figure 1: Apportioning of $G_{1}$ Figure 2: Apportioning of $G_{2}$ Return and Settling phases, Clearing Vectors: We fix the round $t$ and avoid notation $t$ for simpler notations. The agents of $G_{2}$ have to clear their liabilities during this phase in every round. Recall the agents of $G_{2}$ invested $w(1+\epsilon)$ amount in risky-investments and the corresponding random returns (after economic shocks) are: $K_{i}=\begin{cases}w(1+\epsilon)(1+u)=:k_{u},&\text{w.p. (with probability) }\ \delta\\\ w(1+\epsilon)(1+d)=:k_{d},&\text{otherwise}\end{cases}$ (3) This is the well known binomial model, in which the upward moment occurs with probability $\delta$ and downward moment with $(1-\delta).$ The agents have to return $y$ (after the interest rate $r_{b}$) amount to their creditors, however may not be able to manage the same because of the above economic shocks. In case of default, the agents return the maximum possible; let $X_{i}$ be the amount cleared by the $i^{th}$ agent of group $G_{2}$. Here we consider a standard bankruptcy rule, limited liability and pro-rata basis repayment of the debt contract (see [5, 4]), where the amounts returned are proportional to their liability ratios. Thus node $j$ of $G_{2}$ pays back $X_{i}L_{ji}(1+r_{b})/y$ towards node $i$, where $L_{ji}$ the amount borrowed (liability) during initial investment phases equals (see the details of previous subsection and equation (2)): $L_{ji}=\begin{cases}I_{ji}\frac{w}{np_{ss}},&\text{if}\ i\in G_{1}\\\ I_{ji}\frac{\alpha w(1+\epsilon)}{np_{ss}(1-\alpha)(1-\epsilon)},&\text{if}\ i\in G_{2}.\end{cases}$ (4) Thus the maximum amount cleared by any agent $j\in{\cal G}_{2}$, $X_{j}$, is given by the following fixed point equation in terms of the clearing vector $\\{X_{i}\\}_{i\in G_{2}}$ composed of clearing values of all the agents (see [5, 4] etc): $X_{i}=\min\left\\{\bigg{(}K_{i}+\sum_{j\in G_{2}}X_{j}\frac{L_{ji}}{y}-v\bigg{)}^{+},\ y\right\\},$ (5) with the following details: the term $K_{i}$ is the return of the risky investment, the term $\sum_{j\in G_{2}}X_{j}\ L_{ji}/y$ equals the claims form the other agents (those borrowed from agent $i$) and $v$ denotes the taxes to pay. In other words, agent $i$ will pay back the (maximum possible) amount $K_{i}+\sum_{j\in G_{2}}X_{j}\frac{L_{ji}}{y}-v$ in case of a default, and in the other event, will exactly pay back the liability amount $y$. Surplus of any agent is defined as the amount obtained from various investments, after clearing all the liabilities. This represents the utility of the agent in the given round. The surplus of the agent $i\in G_{2}$: $R^{2}_{i}=\left(K_{i}+\sum_{j\in G_{2}}X_{j}\frac{L_{ji}}{y}-v-y\right)^{+},$ (6) while that of agent $i\in G_{1}$ is given by: $R^{1}_{i}=\left(w\epsilon(1+r_{s})+\sum_{j\in G_{2}}X_{j}\frac{L_{ji}}{y}-v\right)^{+}.$ (7) In the above, the first term is the return from the risk free investment. The second term equals the returns or claims form $G_{2}$ agents (whom they lent) and $v$ denotes the amount of taxes. ## 3 Asymptotic approximation of the large networks We thus have dynamic graphs whose size increases with each round. In this section, we obtain appropriate asymptotic analysis of these graphs/systems, with an aim to derive the pay-off of each group after each round. Towards this, we derive the (approximate) closed form expression of the equations (6) and (7), which are nothing but the per-agent returns after the settlement of the liabilities. The returns of the agents depend upon how other agents settle their liabilities to their connections/creditors. Thus our first step is to derive the solution of the clearing vector fixed point equations (5). Observe that the clearing vector $\\{X_{j}\\}_{j\in G_{2}}$ is the solution of the vector- valued random fixed point equations (5) in $n$-dimensional space (where $n$ is the size of the network), defined sample-path wise. Clearing vectors using results of [1]: Our financial framework can be analysed using the results of [1], as the details of the model match555Observe that $\alpha(1+\epsilon)/(\alpha+\epsilon)<1$. the assumptions of the paper. By [1, Theorem 1], the aggregate claims converge almost surely to constant values (as the network size increases to infinity): $\displaystyle\mbox{(claims of agents of $G_{1}$)},$ $\displaystyle\displaystyle\sum_{j\in G_{2}}X_{j}\frac{L_{ji}}{y}\to$ $\displaystyle\frac{(1-\alpha)(1-\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}\mbox{ a.s.},\mbox{ and }$ $\displaystyle\mbox{(claims of agents of $G_{2}$)},$ $\displaystyle\displaystyle\sum_{j\in G_{2}}X_{j}\frac{L_{ji}}{y}\to$ $\displaystyle\frac{\alpha(1+\epsilon)}{(\alpha+\epsilon)}{\bar{x}}^{\infty}\mbox{ a.s.},$ where the common expected clearing value ${\bar{x}}^{\infty}$ satisfies the following fixed point equation in one-dimension: ${\bar{x}}^{\infty}=E\bigg{(}\min\left\\{K_{i}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v,y\right\\}\bigg{)}^{+}.$ (8) Further by the same Theorem, the clearing vectors converge almost surely to (asymptotically independent) random vectors: $X_{i}\to\bigg{(}\min\left\\{K_{i}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v,y\right\\}\bigg{)}^{+}\mbox{, for each }i\in G_{2}.$ (9) By virtue of the above results, the random returns given by equations (6) and (7), converge almost surely: $\displaystyle R^{1}_{i}$ $\displaystyle\to$ $\displaystyle\left(w\epsilon(1+r_{s})+\frac{(1-\alpha)(1-\epsilon)}{(\alpha+\epsilon)}{\bar{x}}^{\infty}-v\right)^{+},\mbox{ for each }i\in G_{1}$ (10) $\displaystyle R^{2}_{i}$ $\displaystyle\to$ $\displaystyle\left(K_{i}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v-y\right)^{+},\mbox{ for each }i\in G_{2}.$ (11) Probability of Default is defined as the fraction of agents of $G_{2}$ that failed to pay back their full liability, i.e., $P_{d}:=P({X_{i}}<y)$. For large networks (when the initial network size $n_{0}$ itself is sufficiently large), one can use the above approximate expressions and using the same we obtain the default probabilities and the aggregate clearing vectors in the following (proof in Appendix). ###### Lemma 1 The asymptotic average clearing vector and the default probability of $G_{2}$ is given by : $({\bar{x}}^{\infty},P_{d})=\begin{cases}(y,\hskip 68.2866pt0)&\text{if }\ \ c_{\epsilon}>\frac{y-\underline{w}}{y}\\\ \left(\frac{\delta y+(1-\delta)\underline{w}}{1-(1-\delta)c_{\epsilon}},\hskip 11.38109pt1-\delta\right)&\text{if}\ \ \frac{y-\overline{w}}{y-(1-\delta)(\overline{w}-\underline{w})}<c_{\epsilon}<\frac{y-\underline{w}}{y}\\\ \left(\frac{k_{d}(1-\delta)+k_{u}\delta}{1-c_{\epsilon}},\ \ \ \ \ \ 1\right)&\text{if }\ \ c_{\epsilon}<\frac{y-\overline{w}}{y-(1-\delta)(\overline{w}-\underline{w})}\end{cases}$ (12) where, $c_{\epsilon}=\frac{\alpha+\alpha\epsilon}{\alpha+\epsilon}$, $E(W)=\delta k_{u}+(1-\delta)k_{d}$ , $\underline{w}=k_{d}-v$ and $\overline{w}=k_{u}-v$. $\blacksquare$ Expected Surplus: By virtue of the Theorem developed in [1, Theorem 1] we have a significantly simplified limit system, whose performance is derived in the above Lemma. We observe that this approximation is sufficiently close (numerical simulations illustrate good approximations), and assume the following as the pay-offs of each group after each round of the investments: $\displaystyle\phi_{1}(\epsilon):=E(R^{1}_{i})=\bigg{(}w\epsilon(1+r_{s})+\frac{(1-\alpha)(1-\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v\bigg{)}^{+},\mbox{ for any agent of }G_{1}$ $\displaystyle\phi_{2}(\epsilon):=E(R^{2}_{i})=E\bigg{(}K_{i}+\frac{(1+\epsilon)\alpha}{\alpha+\epsilon}{\bar{x}}^{\infty}-v-y\bigg{)}^{+},$ (13) $\displaystyle=\bigg{(}k_{u}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v-y\bigg{)}^{+}\delta+\bigg{(}k_{d}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v-y\bigg{)}^{+}(1-\delta),$ for any agent of $G_{2}.$ Observe here that the aggregate limits are almost sure constants, hence the expected surplus of all the agents of the same group are equal, while the random returns of the same group are i.i.d. (independent and identically distributed). ## 4 Analysis of Replicator dynamics In every round of investments, we have a new network that represents the liability structure of all the agents of that round formed by the investment choices of the agents, and, in the previous two sections we computed the (asymptotically approximate) expected returns/utilities of each agent of the network. As already mentioned in Section 2, new agents join the network in each round, and chose their strategies depending upon their observations of these expected returns of the previous round. These kind of dynamics is well described in literature by name replicator dynamics (e.g.,[12, 7, 2] etc). The main purpose of such a study is to derive asymptotic analysis and answer some or all of the following questions: will the dynamics converge, i.e., would the relative fractions of various populations settle as the number of rounds increase? will some of the strategies disappear eventually? if more than one population type survives what would be the asymptotic fractions? etc. These kind of analysis are common in other types of networks (e.g., wireless networks (e.g., [12]), biological networks ([2])), but are relatively less studied in the context of financial networks (e.g., [7]). We are interested in knowing the asymptotic outcome of these kind of dynamics (if there exists one) and study the influence of various network parameters on the outcome. We begin with precise description of the two types of dynamics considered in this paper. ### 4.1 Average Dynamics The new agent contacts two random (uniformly sampled) agents of the previous round. If both the contacted agents belong to the same group, the new agent adapts the strategy of that group. When it contacts agents from both the groups it investigates more before making a choice; the new agent observes significant portion of the network, in that, it obtains a good estimate of the average utility of agents belonging to both the groups. It adapts the strategy of the group with maximum (estimated) average utility. Say it observes the average of each group with an error that is normally distributed with mean equal to the expected return of the group and variance proportional to the size of the group, i.e., it observes (here ${\cal N}(0,\sigma^{2})$ is a zero mean Gaussian random variable with variance $\sigma^{2}$) ${\hat{\phi}}_{i}(\epsilon)=\phi_{i}(\epsilon)+{\cal N}_{i}\mbox{ with }{\cal N}_{1}\sim{\cal N}\left(0,\frac{1}{{\bar{c}}\epsilon}\right)\mbox{ and }{\cal N}_{2}\sim{\cal N}\left(0,\frac{1}{{\bar{c}}(1-\epsilon)}\right),$ for some ${\bar{c}}$ large. Observe by this modeling that: the expected values of the observations are given by $(\phi_{1}(\epsilon),\ \phi_{2}(\epsilon))$ and are determined by the relative proportions of the two populations, while the variance of any group reduces as its proportion increases to 1 and increases as the proportion reduces to zero. We also assume that the estimation errors $\\{{\cal N}_{1},{\cal N}_{2}\\}$ (conditioned on the relative fraction, $\epsilon$) corresponding to the two groups are independent. Then the probability that the new agent chooses strategy 1 is given by $Prob({\hat{\phi}}_{1}(\epsilon)-{\hat{\phi}}_{2}(\epsilon)>0)=Prob({\cal N}_{2}-{\cal N}_{1}\leq\phi_{1}(\epsilon)-\phi_{2}(\epsilon)),$ which by (conditional) independence of Gaussian random variables equals666 because $\frac{1}{\epsilon}+\frac{1}{1-\epsilon}=\frac{1}{\epsilon(1-\epsilon)}$ $\displaystyle g(\epsilon):=\int_{-\infty}^{\left(\phi_{1}(\epsilon)-\phi_{2}(\epsilon)\right)\sqrt{{\bar{c}}\epsilon(1-\epsilon)}}e^{-x^{2}/2}\frac{dx}{\sqrt{2\pi}}.$ (14) Let $(n_{1}(t),n_{2}(t))$ respectively represent the sizes of $G_{1}$ and $G_{2}$ population after round $t$ and note that $\epsilon_{t}=\frac{n_{1}(t)}{n_{1}(t)+n_{2}(t)}$. Then the system dynamics is given by the following ($g(\cdot)$ given by (14)): $\displaystyle(n_{1}(t+1),\ n_{2}(t+1))$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{llll}\big{(}n_{1}(t)+1,&n_{2}(t)\big{)}&\text{w.p.}\ \ \epsilon_{t}^{2}+2\epsilon_{t}(1-\epsilon_{t})g(\epsilon_{t})\\\ \big{(}n_{1}(t),&n_{2}(t)+1\big{)}&\text{w.p.}\ \ (1-\epsilon_{t})^{2}+2\epsilon_{t}(1-\epsilon_{t})(1-g(\epsilon_{t})).\\\ \end{array}\right.$ (17) (18) It is clear that (with $\epsilon_{0}$ and $n_{0}$ representing the initial quantities), $\displaystyle\epsilon_{t+1}$ $\displaystyle=$ $\displaystyle\frac{n_{1}(t+1)}{t+n_{0}+1}=\frac{(t+n_{0})\epsilon_{t}+Y_{t+1}}{t+n_{0}+1}=\epsilon_{t}+\frac{1}{t+n_{0}+1}\left(Y_{t+1}-\epsilon_{t}\right)\mbox{ where }$ $\displaystyle Y_{t+1}$ $\displaystyle=$ $\displaystyle\begin{cases}1&\text{wp}\ \ \epsilon_{t}^{2}+2\epsilon_{t}(1-\epsilon_{t})g(\epsilon_{t})\\\ 0&\text{wp}\ \ (1-\epsilon_{t})^{2}+2\epsilon_{t}(1-\epsilon_{t})(1-g(\epsilon_{t}))\mbox{, for all }t\geq 1.\end{cases}$ One can rewrite the update equations as $\displaystyle\epsilon_{t+1}$ $\displaystyle=$ $\displaystyle\epsilon_{t}+\frac{1}{t+n_{0}+1}\left(h(\epsilon_{t})+M_{t+1}\right),\mbox{ with, }\ \ M_{t+1}\ :=\ Y_{t+1}-\epsilon_{t}-h(\epsilon_{t}),\mbox{ where, }$ $\displaystyle h(\epsilon)$ $\displaystyle:=$ $\displaystyle E\big{[}Y_{t+1}-\epsilon_{t}|\epsilon_{t}=\epsilon\big{]}=\epsilon(1-\epsilon)\left(2g(\epsilon)-1\right)\mbox{ for any }0\leq\epsilon\leq 1.$ and observe that (with ${\cal F}_{t}$ the natural filtration of the process till $t$) $\displaystyle E[M_{t+1}|{\cal F}_{t}]$ $\displaystyle=$ $\displaystyle E[M_{t+1}|\epsilon_{t}]=0\mbox{ and }\ E[M_{t+1}^{2}|{\cal F}_{t}]\ \leq\ C\mbox{ for some constant }C<\infty.$ Further observe that $\hskip 19.91692pt0\leq\epsilon_{t}\leq 1\mbox{ for all }t\mbox{ and all sample paths}.$ Thus our algorithm satisfies assumptions777The assumptions require that the process is defined for the entire real line. One can easily achieve this by letting $h(\epsilon)=0$ for all $\epsilon\notin[0,1]$, which still ensures required Lipschitz continuity and by extending $M_{t+1}=0$ for all $\epsilon_{t}\notin[0,1]$. A.1 to A.4 of [11] and hence we have using [11, Theorem 2] that ###### Theorem 1 The sequence $\\{\epsilon_{t}\\}$ generated by average dynamics (18) converges almost surely (a.s.) to a (possibly sample path dependent) compact connected internally chain transitive invariant set of ODE: $\dot{\epsilon_{t}}=h(\epsilon_{t}).\hskip 56.9055pt\mbox{ {\hfill$\blacksquare$} }$ (19) The dynamics start with initial condition $\epsilon_{0}\in(0,1)$ and clearly would remain inside the interval $[0,1]$, i.e., $\epsilon_{t}\in[0,1]$ for all $t$ (and almost surely). Thus we consider the invariant sets of ODE (19) within this interval for some interesting case studies in the following (Proof in Appendix). ###### Corollary 1 Define ${\bar{r}}_{r}:=u\delta+d(1-\delta)$. And assume $w(1+d)>v$ and observe that $u>r_{b}\geq r_{s}>d.$ Assume $\epsilon_{0}\in(0,1).$ Given the rest of the parameters of the problem, there exists a ${\bar{\delta}}<1$ (depends upon the instance of the problem) such that the following statements are valid for all $\delta\geq{\bar{\delta}}$: 1. (a) If ${\bar{r}}_{r}>r_{b}>r_{s}$ then $\phi_{2}(\epsilon)>\phi_{1}(\epsilon)$ for all $\epsilon$, and $\epsilon_{t}\to 0$ almost surely. 2. (b) If $\phi_{1}(\epsilon)>\phi_{2}(\epsilon)$ for all $\epsilon$ then $\epsilon_{t}\to 1$ almost surely. 3. (c) When $r_{b}>{\bar{r}}_{r}>r_{s}$, and case (b) is negated there exists a unique zero $\epsilon^{*}$ of the equation $\phi_{1}(\epsilon)-\phi_{2}(\epsilon)=0$ and $\hskip 28.45274pt\epsilon_{t}\to\epsilon^{*}\mbox{ almost surely; further for $\delta\approx 1$, }\epsilon^{*}\approx\frac{r_{b}-{\bar{r}}_{r}}{{\bar{r}}_{r}-r_{s}}.\hskip 42.67912pt\mbox{{\hfill$\blacksquare$} }$ From (13) and Lemma 1, it is easy to verify that all the limit points are evolutionary stable strategies (ESS). Thus the replicator dynamics either settles to a pure strategy ESS or mixed ESS (in part (c) of the corollary), depending upon the parameters of the network; after a large number of rounds, either the fraction of agents following one of the strategies converges to one or zero or the system reaches a mixed ESS which balances the expected returns of the two groups. In many scenarios, the expected rate of return of the risky investments is much higher than the rate of interest related to lending/borrowing, i.e., ${\bar{r}}_{r}>r_{b}$. Further the assumptions of the corollary are satisfied by more or less all the scenarios (due to standard no-arbitrage assumptions) and because the shocks are usually rare (i.e., $\delta$ is close to 1). Hence by the above corollary, in majority of scenarios, the average dynamics converges to a pure strategy with all ‘risky’ agents (i.e., $\epsilon_{t}\to 0$). The group $G_{1}$ gets wiped out and almost all agents invest in risky ventures, as the expected rate of returns is more even in spite of large economic shocks. One can observe a converse or a mixed ESS when the magnitude of the shocks is large ($d$ too small) or when the shocks are too often to make ${\bar{r}}_{r}<r_{b}$. ### 4.2 Random dynamics When the new agent contacts two random agents of different groups, its choice depends directly upon the returns of the two contacted agents. The rest of the details remain the same as in average dynamics. In other words, the new agents observe less, their investment choice is solely based on the (previous round) returns of the two contacted agents. In this case the dynamics are governed by the following (see (10)-(11)): $\displaystyle(n_{1}(t+1),n_{2}(t+1))$ $\displaystyle=$ $\displaystyle\begin{cases}\left(n_{1}(t)+1,\hskip 28.45274ptn_{2}(t)\right)&\text{wp}\ \ \epsilon_{t}^{2}\\\ \left(n_{1}(t),\hskip 45.5244ptn_{2}(t)+1\right)&\text{wp}\ \ (1-\epsilon_{t})^{2}\\\ \left(n_{1}(t)+G(\epsilon_{t}),\ \ n_{2}(t)+(1-G(\epsilon_{t}))\right)&\text{else, with}\\\ \end{cases}$ $\displaystyle G(\epsilon_{t})$ $\displaystyle=$ $\displaystyle 1_{\\{R^{1}\geq R^{2}\\}}$ $\displaystyle=$ $\displaystyle 1_{\left\\{\left(w\epsilon(1+r_{s})+\frac{(1-\alpha)(1-\epsilon)}{(\alpha+\epsilon)}{\bar{x}}^{\infty}-v\right)^{+}\geq\left(K_{i}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}{\bar{x}}^{\infty}-v-y\right)^{+}\right\\}},$ where ${\bar{x}}^{\infty}={\bar{x}}^{\infty}(\epsilon_{t})$ is given by Lemma 1. Here we assume people prefer risk-free strategy under equality, one can easily consider the other variants. Once again this can be rewritten as $\displaystyle\epsilon_{t+1}=\epsilon_{t}+\frac{Z_{t+1}-\epsilon_{t}}{t+n_{0}+1}\mbox{ with }\ \ Z_{t+1}=\begin{cases}1&\text{wp}\ \ \epsilon_{t}^{2}\\\ 0&\text{wp}\ \ (1-\epsilon_{t})^{2}\\\ G(\epsilon_{t})&\text{else}.\end{cases}$ (21) As in previous section the above algorithm satisfies assumptions888In the current paper, we consider scenarios in which $h_{R}(\cdot)$ is Lipschitz continuous, basically under the conditions of Corollary 2. A.1 to A.4 of [11] and once again using [11, Theorem 2], we have: ###### Theorem 2 The sequence $\\{\epsilon_{t}\\}$ generated by average dynamics (4.2) converges almost surely (a.s.) to a (possibly sample path dependent) compact connected internally chain transitive invariant set of ODE: $\dot{\epsilon}(t)=h_{R}(\epsilon(t)),\ h_{R}(\epsilon):=E_{\epsilon}\big{[}Z_{t+1}-\epsilon_{t}|\epsilon_{t}=\epsilon\big{]}=\epsilon(1-\epsilon)(2E[G(\epsilon)]-1).\mbox{ {\hfill$\blacksquare$} }$ (22) One can derive the analysis of this dynamics in a similar way as in average dynamics, however there is an important difference between the two dynamics; we can never have random dynamics converges to an intermediate attractor, like the attractor in part (c) of Corollary 1 (unique $\epsilon^{*}$ satisfying $\phi_{1}=\phi_{2}$). This is because $E_{\epsilon}[G]=P(R^{1}(\epsilon)>R^{2}(\epsilon))$ equals $0$, $1-\delta$ or 1 and never $1/2$ (unless $\delta=1/2$, which is not a realistic case). Nevertheless, we consider the invariant sets (corresponding to pure ESS) within $[0,1]$ for some cases (Proof in Appendix): ###### Corollary 2 Assume $\epsilon_{0}\in(0,1).$ Given the rest of the parameters of the problem, there exists a $1/2<\delta<1$ (depends upon the instance of the problem) such that the following statements are valid: 1. (a) If $E_{\epsilon}[G]=0$ for all $\epsilon$ or $1-\delta$ for all $\epsilon$, then $\epsilon_{t}\to 0$ almost surely. 2. (b) If $E_{\epsilon}[G]=1$ for all $\epsilon$, then $\epsilon_{t}\to 1$ almost surely. 3. (c) When $w(1+d)>v$ and $u>r_{b}\geq r_{s}>d$, there exists a ${\bar{\delta}}<1$ such that for all $\delta\geq{\bar{\delta}}$, the default probability $P_{d}\leq(1-\delta)$ and $E[G]=1-\delta$ and this is true for all $\epsilon$. Hence by part (a), $\epsilon_{t}\to 0$ almost surely. $\blacksquare$ Remarks: Thus from part (c), under the conditions of Corollary 1, the random dynamics always converges to all ‘risky’ agents (pure ESS), while the average dynamics, as given by Corollary 1, either converges to pure or mixed ESS further based on system parameters (mainly various rates of return). From this partial analysis (corollaries are for large enough $\delta$) it appears that one can never have mixed ESS with random dynamics, and this is a big contrast to the average dynamics; when agents observe sparsely the network eventually settles to one of the two strategies, and if they observe more samples there is a possibility of emergence of mixed ESS that balances the two returns. We observe similar things, even for $\delta$ as small as $0.8$ in numerical simulations (Table 4). We are keen to understand this aspect in more details as a part of the future work. To summarize we have a financial network which grows with new additions, in which the new agents adapt one of the two available strategies based on the returns of the agents that they observed/interacted with. Our asymptotic analysis of [1] was instrumental in deriving these results. This is just an initial study of the topic. One can think of other varieties of dynamics, some of which could be a part of our future work. The existing agents may change their strategies depending upon their returns and observations. The agents might leave the network if they have reduced returns repeatedly. The network may adjust itself without new additions etc. ## 5 Numerical observations We performed numerical simulations to validate our theory. We included Monte- Carlo (MC) simulation based dynamics in which the clearing vectors are also computed by directly solving the fixed point equations, for any given sample path of shocks. Our theoretical observation well matches the MC based limits. In Tables 4, 4 and 4 we tabulated the limits of the average dynamics for various scenarios, and the observations match the results of Corollary 1. The configuration used for Table 4 is: $n_{0}=2000,\epsilon_{0}=0.75,r_{s}=0.18,r_{b}=0.19,w=100,v=46,\alpha=0.1$, while that used for Table 4 is: $n_{0}=2000,\epsilon_{0}=0.5,r_{s}=0.17,r_{b}=0.19,w=100,v=40,\alpha=0.1$. For both these tables risky expected rate of returns ${\bar{r}}_{r}$ is smaller than $r_{b}$ and the dynamics converges either to ‘all risky’ agents configuration or to a mixed ESS. In Table 4, the risky expected rate of returns ${\bar{r}}_{r}=.1250$ which is greater than $r_{b}$ and $r_{s}$, thus the dynamics converges to all risky-agents, as indicated by Corollary 1. $u$ | $d$ | $\delta$ | $\phi_{1}$ | $\phi_{2}$ | $\epsilon^{*}$ ---|---|---|---|---|--- 0.2 | -0.05 | 0.8 | 72 | 0 | 1 0.2 | -0.1 | 0.8 | 72 | 0 | 1 0.2 | -0.15 | 0.8 | 72 | 0 | 1 0.2 | -0.2 | 0.8 | 72 | 0 | 1 0.2 | -0.25 | 0.8 | 72 | 0 | 1 Table 1: When the shocks are too large along with larger taxes ($v=46$), the average dynamics converges to a configuration with all ‘risk-free agents’! $u$ | $d$ | $\delta$ | $\phi_{1}$ | $\phi_{2}$ | $\epsilon^{*}$ ---|---|---|---|---|--- 0.2 | -0.1 | 0.95 | 78.33 | 78.27 | 0.3326 0.2 | -0.11 | 0.95 | 78.24 | 78.31 | 0.3791 0.2 | -0.12 | 0.95 | 78.14 | 78.14 | 0.4288 0.2 | -0.13 | 0.95 | 78.04 | 78.04 | 0.4820 0.2 | -0.14 | 0.95 | 77.92 | 77.92 | 0.5385 Table 2: Average dynamics converges to mixed ESS, at which both populations survive with $\phi_{1}=\phi_{2}$, $u$ | $d$ | $\delta$ | $\phi_{1}$ | $\phi_{2}$ | $\epsilon^{*}$ ---|---|---|---|---|--- 0.15 | -0.1 | 0.9 | 0 | 82.12 | 0 0.16 | -0.1 | 0.9 | 0 | 83.24 | 0 0.17 | -0.1 | 0.9 | 0 | 84.29 | 0 0.18 | -0.1 | 0.9 | 0 | 85.19 | 0 Table 3: Average Dynamics converges to all ‘risky-agents’; Configuration: $n_{0}=2000,\epsilon_{0}=.5,r_{s}=0.10,r_{b}=0.12,w=100,v=30,\alpha=0.5$ Config | $\epsilon^{*}$(Theory) | $\epsilon^{\star}$(Monte Carlo) ---|---|--- ($d,\delta,v$) | Avg | Rndm | Avg | Rndm 0.10, 0.95, 40 | 0 | 0 | .0016 | 0.0011 -0.10, 0.95, 40 | 0.33 | 0 | .3214 | 0.0004 -0.15, 0.95, 40 | 0.6 | 0 | .5988 | 0.0014 0.10, 0.80, 46 | 1 | 0 | .9896 | 0.0065 Table 4: Average and Random dynamics, Comparison of MC results with theory Configuration: $n_{0}=2000,u=0.2,r_{s}=0.17,r_{b}=0.19,w=100,\alpha=0.1$ In Table 4 we considered random dynamics as well as average dynamics. In addition, we provided the Monte-Carlo based estimates. There is a good match between the MC estimates and the theory. Further we have the following observations: a) random dynamics always converge to a configuration with all ‘risky’ agents, as given by Corollary 2; b) when ${\bar{r}}_{r}>r_{b}$, the average dynamics also converges to $\epsilon^{*}=0$ as suggested by Corollary 1; and c) when ${\bar{r}}_{r}<r_{b}$, the average dynamics converges to mixed ESS or to a configuration with all ‘risk-free’ agents, again as given by Corollary 1. As the ‘risk increases’, i.e., as the amount of taxes increase and or as the expected rate of return of risky investments ${\bar{r}}_{r}$ decreases, one can observe that the average dynamics converges to all ‘risk-free’ agents (last row of Table 4) thus averting systemic risk event (when there are large number of defaults, $P_{d}$). While the random dynamics fails to do the same. As predicted by theory (the configurations satisfying part (b) of Corollary 2), random dynamics might also succeed in averting the systemic risk event, when the expected number of defaults is one for all $\epsilon>0.$ It is trivial to verify that the configuration with $w(1+u)<v$, is one such example. Thus, average dynamics is much more robust towards averting systemic risk events. ## 6 Conclusions We consider a financial network with a large number of agents. The agents are interconnected via liability graphs. There are two types of agents, one group lends to others and invests the rest in risk-free projects, while the second group borrows/lends and invests the rest in risky ventures. Our study is focused on analysing the emergence of these groups, when the new agents adapt their strategies for the next investment round based on the returns of the previous round. We considered two types of dynamics; in average dynamics the new agents observe large sample of data before deciding their strategy, and in random dynamics the decision is based on a small random sample. We have the following important observations: a) when the expected rate of return of the risky investments is higher (either when the shocks are rare or when the shocks are not too large) than the risk-free rate, then ‘risk-free’ group wipes out eventually, almost all agents go for risky ventures; this is true for both types of dynamics; b) when the expected rate of risky investments is smaller, a mixed ESS can emerge with average dynamics while the random dynamics always converges to all risky agents; at mixed ESS the expected returns of both the groups are equal; more interestingly, when the risky-expected rate is too small, the average dynamics converges to a configuration with all risk-free agents. In other words, in scenarios with possibility of a systemic risk event, i.e., when there is a possibility of the complete-system collapse (all agents default), the average dynamics manages to wipe out completely the risky agents; the random dynamics can fail to do the same. Thus when agents make their choices rationally and after observing sufficient sample of the returns of the previous round of investments, there is a possibility to avoid systemic risk events. These are some initial results and we would like to investigate further in future to make more affirmative statements in this direction. ## References * [1] Veeraruna Kavitha, Indrajit Saha, and Sandeep Juneja. ”Random Fixed Points, Limits and Systemic risk.” In 2018 IEEE Conference on Decision and Control (CDC), pp. 5813-5819. IEEE, 2018. * [2] Miekisz, Jacek. ”Evolutionary game theory and population dynamics.” Multiscale Problems in the Life Sciences. Springer, Berlin, Heidelberg, 2008. 269-316. * [3] Benveniste, Albert, Michel Métivier, and Pierre Priouret. Adaptive algorithms and stochastic approximations. Vol. 22. Springer Science & Business Media, 2012. * [4] Larry Eisenberg and Thomas H Noe. Systemic risk in financial systems. Management Science, 2001. * [5] Daron Acemoglu, Asuman Ozdaglar, and Alireza Tahbaz-Salehi. Systemic risk and stability in financial networks. The american economic review, 2015. * [6] Franklin Allen and Douglas Gale. Financial contagion. Journal of political economy, 2000. * [7] Li, Honggang, Chensheng Wu, and Mingyu Yuan. ”An evolutionary game model of financial markets with heterogeneous players.” Procedia Computer Science 17 (2013): 958-964. * [8] Yang, Ke, Kun Yue, Hong Wu, Jin Li, and Weiyi Liu. ”Evolutionary Analysis and Computing of the Financial Safety Net.” In International Workshop on Multi-disciplinary Trends in Artificial Intelligence, pp. 255-267. Springer, Cham, 2016. * [9] Brock, William A., and Cars H. Hommes. ”Heterogeneous beliefs and routes to chaos in a simple asset pricing model.” Journal of Economic dynamics and Control 22, no. 8-9 (1998): 1235-1274. * [10] Friedman, Daniel. ”Towards evolutionary game models of financial markets.” (2001): 177-185. * [11] Borkar, Vivek S. Stochastic approximation: a dynamical systems viewpoint. Vol. 48. Springer, 2009. * [12] Tembine, Hamidou, Eitan Altman, Rachid El-Azouzi, and Yezekael Hayel. ”Evolutionary games in wireless networks.” IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 40, no. 3 (2009): 634-646. * [13] Smith, J. Maynard, and George R. Price. ”The logic of animal conflict.” Nature 246, no. 5427 (1973): 15-18. ## Appendix Proof of Lemma 1 : We consider the following: Case 1: First consider the case when downward shock can be absorbed , in this case the clearing vector ${\bar{x}}^{\infty}=y\delta+y(1-\delta)=y$, default probability is $P_{d}=0$. The region is true if the following condition is meet i.e., if $K_{d}-v+yc_{\epsilon}>y\implies c_{\epsilon}>\frac{y-\underline{w}}{y}$ Case 2: Consider the case with banks receive shock will default and the corresponding average clearing vector ${\bar{x}}^{\infty}=y\delta+(\underline{w}+c_{\epsilon}{\bar{x}}^{\infty})(1-\delta)$ which simplifies to : ${\bar{x}}^{\infty}=\frac{y\delta+\underline{w}(1-\delta)}{1-c_{\epsilon}(1-\delta).}$ This region lasts if the following conditions hold to be true $K_{d}-v+c_{\epsilon}{\bar{x}}^{\infty}<y,\mbox{ and }K_{u}-v+c_{\epsilon}{\bar{x}}^{2\infty}>y.$ Substituting ${\bar{x}}^{\infty}=\frac{y\delta+\underline{w}(1-\delta)}{1-c_{\epsilon}(1-\delta)}$ we have, $\hskip 159.33542pt\frac{y-\overline{w}}{y-(1-\delta)(\overline{w}-\underline{w})}<c_{\epsilon}<\frac{y-\underline{w}}{y}.$ Case 3: In this we first calculate ${\bar{x}}^{\infty}$ which is obtained by solving following fixed point equation: $\displaystyle{\bar{x}}^{\infty}$ $\displaystyle=$ $\displaystyle(K_{d}-v+c_{\epsilon}{\bar{x}}^{\infty})(1-\delta)+(K_{u}-v+c_{\epsilon}{\bar{x}}^{\infty})\delta\ \ \implies{\bar{x}}^{\infty}\ =\ \frac{EW}{1-c_{\epsilon}}.$ In this case the default probability is $P_{d}=1$. The regime satisfies if the following hold $\displaystyle K_{u}-v+c_{\epsilon}\frac{EW}{1-c_{\epsilon}}<y\ \ \implies c_{\epsilon}<\frac{y-\overline{w}}{y-(1-\delta)(\overline{w}-\underline{w})}.\hskip 14.22636pt\mbox{ {\hfill$\blacksquare$} }$ Proof of Corollary 1: First consider the system with $\delta=1$, i.e., system without shocks. From Lemma 1, $P_{D}\leq(1-\delta)$ for all $\epsilon$ because (with $\delta=1$) $y\left(c_{\epsilon}-\frac{y-{\bar{w}}}{y}\right)=w(1+\epsilon)(1+u)-v-w\epsilon(1+r_{b})=w(1+u)-v+w\epsilon(u-r_{b})\geq w(1+u)-v,$ for all $\epsilon$ (the lower bound independent of $\epsilon$). Under these assumptions, there exists ${\bar{\delta}}<1$ by continuity of the involved functions such that $y\left(c_{\epsilon}-\frac{y-{\bar{w}}}{y-(1-\delta)({\bar{w}}-{\underline{w}})}\right)>0\mbox{ for all }\delta\geq{\bar{\delta}}\mbox{ and for all }\epsilon.$ Thus from Lemma 1 ${\bar{x}}^{\infty}=y$ or ${\bar{x}}^{\infty}=\frac{\delta y+(1-\delta)\underline{w}}{1-(1-\delta)c_{\epsilon}}$ for all such $\delta\geq{\bar{\delta}}$. We would repeat a similar trick again, so assume initially ${\bar{x}}^{\infty}=y$ for all $\epsilon$ and consider $\delta\geq{\bar{\delta}}$. With this assumption we will have: $\displaystyle R^{1}(\epsilon)$ $\displaystyle=$ $\displaystyle\left(w\epsilon(1+r_{s})+\frac{(1-\alpha)(1-\epsilon)}{(\alpha+\epsilon)}y-v\right)^{+}$ $\displaystyle=$ $\displaystyle\left(w\epsilon(1+r_{s})+w(1-\epsilon)(1+r_{b})-v\right)^{+}$ $\displaystyle=$ $\displaystyle\left(w(1+r_{b})-v+w\epsilon(r_{s}-r_{b})\right)\mbox{, under the given hypothesis, and }$ $\displaystyle R^{2}(\epsilon)$ $\displaystyle=$ $\displaystyle\left(K_{i}+\frac{\alpha(1+\epsilon)}{\alpha+\epsilon}y-v-y\right)^{+}=\left(K_{i}-w\epsilon(1+r_{b})-v\right)^{+}$ (24) $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{llll}R^{2}_{u}&\mbox{ w.p. }\delta&\mbox{ where }R^{2}_{u}:=w(1+u)-v+w\epsilon(u-r_{b})\\\ \left(R^{2}_{d}\right)^{+}&\mbox{ w.p. }1-\delta&\mbox{ where }R^{2}_{d}:=w(1+d)-v+w\epsilon(d-r_{b}).\end{array}\right.$ (27) Note that $R^{2}_{u}\geq w(1+u)-v>0$ (for any $\epsilon$) under the given hypothesis. Proof of part (a): When ${\bar{r}}_{r}>r_{b}$, from (13), it is clear that (inequality only when $R^{2}_{d}$ is negative) $\displaystyle\phi_{2}(\epsilon)-\phi_{1}(\epsilon)\geq R^{2}_{u}\delta+R^{2}_{d}(1-\delta)-\phi_{1}(\epsilon)=w({\bar{r}}_{r}-r_{b})+w\epsilon({\bar{r}}_{r}-r_{s})>0.$ Thus in this case $\phi_{2}>\phi_{1}$ for all $\epsilon$ and hence $g(\epsilon)<1/2\mbox{ and }2g(\epsilon)-1<0\mbox{ for all $0<\epsilon<1$. }$ Therefore with Lyaponuv function $V_{0}(\epsilon)=\epsilon/(1-\epsilon)$ on defined on neighbourhood $[0,1)$ of $0$ (in relative topology on $[0,1]$) we observe that $\frac{dV_{0}}{d\epsilon}h(\epsilon)=\frac{\epsilon}{1-\epsilon}(2g(\epsilon)-1)<0\mbox{ for all }0<\epsilon<1\mbox{ and equals }0\mbox{ for }\epsilon=0.$ Further $V_{0}(\epsilon)\to\infty$ as $\epsilon\to 1$, the boundary point of $[0,1)$. Thus $\epsilon^{*}=0$ is the asymptotically stable attractor of ODE (19) (see [11, Appendix, pp.148]) and hence the result follows by Theorem 2. For all $\delta\geq{\bar{\delta}}$, from Lemma 1, we have the following $\displaystyle\sup_{\epsilon}|y-{\bar{x}}^{\infty}|=\sup_{\epsilon}(1-\delta)\bigg{|}\frac{y-c_{\epsilon}-\underline{w}}{1-(1-\delta)c_{\epsilon}}\bigg{|}<\frac{1-\delta}{\delta}\eta$ (28) for some $\eta>0$ , which decreases to 0 as $\delta\to 1$. ( The last inequality is due to $c_{\epsilon}<1$ and then taking supremum over $\epsilon$). By continuity of the above upper bound with respect to $\delta$ and the subsequent functions considered in the above parts of the proof, there exists a ${\bar{\delta}}<1$ (further big if required) such that all the above arguments are true for all $\delta>{\bar{\delta}}$. Proof of part (b): The proof follows in similar way, now using Lyaponuv function $V_{1}(\epsilon)=(1-\epsilon)/\epsilon$ on neighbourhood $(0,1]$ of 1, and by observing that $g(\epsilon)>1/2$ for all $\epsilon<1$ and hence $\frac{dV_{1}}{d\epsilon}h(\epsilon)=-\frac{1-\epsilon}{\epsilon}(2g(\epsilon)-1)<0\mbox{ for all }0<\epsilon<1\mbox{ and equals }0\mbox{ for }\epsilon=1.$ Proof of part (c): It is clear that $\phi_{1}(\epsilon)=R^{1}(\epsilon)$ decreases linearly as $\epsilon$ increases: $\phi_{1}(\epsilon)=w(1+r_{b})-v+w\epsilon(r_{s}-r_{b}).$ For $\epsilon$ in the neighbourhood of $0$, $\phi_{2}(\epsilon)>0$ and is decreasing linearly with slope ${\bar{r}}_{r}-r_{b}$, because $R^{2}_{d}(0)=w(1+d)-v>0$ and thus for such $\epsilon$ $\phi_{2}(\epsilon)=w(1+{\bar{r}}_{r})-v+w\epsilon({\bar{r}}_{r}-r_{b}).$ From (24), $R^{2}_{d}({\epsilon})$ is decreasing with increase in $\epsilon.$ There is a possibility of an ${\bar{\epsilon}}$ that satisfies $R^{2}_{d}({\bar{\epsilon}})=0$, in which case $\phi_{2}$ increases linearly with slope $\delta w(u-r_{b})$, i.e., $\phi_{2}(\epsilon)=\delta\left[w(1+u)-v+w\epsilon(u-r_{b})\right]\mbox{ for all }\epsilon\geq{\bar{\epsilon}}.$ When ${\bar{r}}_{r}<r_{b}$ we have, $\hskip 68.2866pt\phi_{1}(0)=w(1+r_{b})-v>w(1+{\bar{r}}_{r})-v=\phi_{2}(0).$ By hypothesis $\phi_{1}(\epsilon)<\phi_{2}(\epsilon)$ for some $\epsilon$, hence by intermediate value theorem there exists at least one $\epsilon^{*}$ that satisfies $\phi_{1}(\epsilon^{*})=\phi_{2}(\epsilon^{*}).$ Further the zero is unique because $\phi_{2}$ is either linear or piece-wise linear (with different slops), while $\phi_{1}$ is linear. Consider Lyaponuv function $V_{*}(\epsilon):=(\epsilon-\epsilon^{*})^{2}/(\epsilon(1-\epsilon))$ on neighbourhood $(0,1)$ of $\epsilon^{*}$, note $V_{*}(\epsilon)\to\infty$ as $\epsilon\to 0$ or $\epsilon\to 1$ and observe by (piecewise) linearity of the functions we will have $\displaystyle\phi_{1}(\epsilon)$ $\displaystyle>$ $\displaystyle\phi_{2}(\epsilon)\mbox{ and thus }(2g(\epsilon)-1)>0\mbox{ for all }0<\epsilon<\epsilon^{*}\mbox{ and }$ $\displaystyle\phi_{2}(\epsilon)$ $\displaystyle>$ $\displaystyle\phi_{1}(\epsilon)\mbox{ and thus }(2g(\epsilon)-1)<0\mbox{ for all }1>\epsilon>\epsilon^{*}.$ Thus we have999When $\epsilon<1/2$ and $\epsilon<\epsilon^{*}$ then clearly $\frac{(\epsilon-\epsilon^{*})(2\epsilon-1)}{\epsilon(1-\epsilon)}>0$. When $\epsilon>1/2$ we have $(2\epsilon-1)/\epsilon<1/2$ and with $\epsilon<\epsilon^{*}$ we have $\epsilon^{*}-\epsilon<1-\epsilon$ and thus $2+\frac{(\epsilon-\epsilon^{*})(2\epsilon-1)}{\epsilon(1-\epsilon)}\geq 3/2>0\mbox{ for all }\epsilon<\epsilon^{*}.$ In a similar way $\epsilon>\epsilon^{*}$, then we will have that the above term is again positive. , $\frac{dV_{*}}{d\epsilon}=2\frac{\epsilon-\epsilon^{*}}{\epsilon(1-\epsilon)}+\frac{(\epsilon-\epsilon^{*})^{2}(2\epsilon-1)}{\epsilon^{2}(1-\epsilon)^{2}}\mbox{ and hence }$ $\frac{dV_{*}}{d\epsilon}h(\epsilon)=(\epsilon-\epsilon^{*})\left(2+\frac{(\epsilon-\epsilon^{*})(2\epsilon-1)}{\epsilon(1-\epsilon)}\right)(2g(\epsilon)-1)<0\mbox{ for all }\epsilon\notin\\{0,1,\epsilon^{*}\\}.$ Thus $\epsilon^{*}$ is the asymptotically stable attractor of ODE (19) and hence the result follows by Theorem 2. The result can be extended for $\delta<1$ as in case (a) and the rest of the details follow by direct verification (at $\delta=1$), i.e., by showing that $\phi_{1}(\epsilon^{*})=\phi_{2}(\epsilon^{*})$ at $\delta=1$ and the equality is satisfied approximately in the neighbourhood of $\delta=1$. $\blacksquare$ Proof of Corollary 2: For part (a), $h_{R}(\epsilon)=-c_{G}\epsilon(1-\epsilon)$, where the constant $c_{G}=1$ (or respectively $c_{G}=2\delta-1$). Using Lyanponuv function of part (a) of Corollary 1, the proof follows in exactly the same lines. For part (b), $h_{R}(\epsilon)=\epsilon(1-\epsilon)$, and proof follows as in part (b) of Corollary 1. For part (c), first observe (using equations (Appendix)-(24) of proof of Corollary 1) $\displaystyle R^{2}_{u}(\epsilon)-R^{1}(\epsilon)$ $\displaystyle\geq$ $\displaystyle w(1+u)+w\epsilon(u-r_{s})-y+{\bar{x}}^{\infty}\bigg{(}\frac{2\alpha+\epsilon-1}{\alpha+\epsilon}\bigg{)}$ $\displaystyle=$ $\displaystyle w(1+u)+w\epsilon(u-r_{s})+({\bar{x}}^{\infty}-y)-{\bar{x}}^{\infty}\bigg{(}\frac{1-\alpha}{\alpha+\epsilon}\bigg{)}$ $\displaystyle=$ $\displaystyle w(u-r_{b})+w\epsilon(u-r_{s})+({\bar{x}}^{\infty}-y)\bigg{(}1-\frac{1-\alpha}{\alpha+\epsilon}\bigg{)}>0.$ The last inequality is trivially true for $\delta=1$ (and so ${\bar{x}}^{\infty}=y$) for the given hypothesis, and then by continuity as in proof of Corollary 1, one can consider $\bar{\delta}<1$ such that for all $\delta\geq\bar{\delta}$, the term $({\bar{x}}^{\infty}-y)\big{(}1-\frac{1-\alpha}{\alpha+\epsilon}\big{)}$ (uniformly over $\epsilon$) can be made arbitrarily small. When $P_{d}=0$, i.e., ${\bar{x}}^{\infty}=y$ for some $\epsilon$, then $R^{2}_{d}(\epsilon)-R^{1}(\epsilon)=w(d-r_{b})+w\epsilon(d-r_{s})<0$ for all such $\epsilon$. When $P_{d}\neq 0$, then $R^{2}_{d}=0\leq R^{1}$. Thus in either case $R^{2}_{d}(\epsilon)\leq R^{1}(\epsilon)$ for all $\epsilon$. By virtue of the above arguments we have $P_{D}\leq(1-\delta)$ and $E[G]=1-\delta$ and this is true for all $\epsilon$, for all $\delta\geq{\bar{\delta}}$. The rest of the proof follows from part(a). $\blacksquare$

2024-09-04T02:54:56.377861

2003.00899

arxiv-papers

# Demonstrating Rosa: the fairness solution for any Data Analytic pipeline Kate Wilkinson illumr Ltd., London, United Kingdom George Čevora illumr Ltd., London, United Kingdom <EMAIL_ADDRESS> ###### Abstract Most datasets of interest to the analytics industry are impacted by various forms of human bias. The outcomes of Data Analytics [DA] or Machine Learning [ML] on such data are therefore prone to replicating the bias. As a result, a large number of biased decision-making systems based on DA/ML have recently attracted attention. In this paper we introduce Rosa, a free, web-based tool to easily de-bias datasets with respect to a chosen characteristic. Rosa is based on the principles of Fair Adversarial Networks, developed by illumr Ltd., and can therefore remove interactive, non-linear, and non-binary bias. Rosa is stand- alone pre-processing step / API, meaning it can be used easily with any DA/ML pipeline. We test the efficacy of Rosa in removing bias from data-driven decision making systems by performing standard DA tasks on five real-world datasets, selected for their relevance to current DA problems, and also their high potential for bias. We use simple ML models to model a characteristic of analytical interest, and compare the level of bias in the model output both with and without Rosa as a pre-processing step. We find that in all cases there is a substantial decrease in bias of the data- driven decision making systems when the data is pre-processed with Rosa. A topic that has recently received much attention both in the media and in academia is the bias in decisions made based on the output of Machine Learning [ML] algorithms [22][16][10]. This bias is especially problematic when the result is unfair discrimination between groups on the basis of a protected characteristic. In UK Law, protected characteristics are defined as age, disability, gender reassignment, marriage and civil partnership, pregnancy and maternity, race, religion or belief, sex and sexual orientation. The Equality Act, 2010 [3], made it illegal to discriminate based on any of these characteristics. Discrimination in the output of ML algorithms most often stems from biased human judgements made in the past. These biased judgements can make up the dataset which an algorithm is trained on, or they can shape the composition of the dataset so that it does not represent the true population. If an ML algorithm is trained to predict biased human judgements, then the bias will be replicated in decisions made based on the output of the algorithm. One widely publicised example occurred in 2015, when Amazon revealed that its experimental recruitment algorithm was heavily biased against women [9]. The dataset used to train the algorithm comprised successful resumes submitted to the company over the last 10 years. The technology industry is historically and currently male-dominated, and therefore the proportion of female resumes in the dataset was far less than the true proportion of female applicants that would be suitable for the job today. It is understandable that an algorithm trained to assess the suitability of applicants would find a link between successful applicants and gender. If decisions are made based on the assessments of this algorithm, then an unfair proportion of female applicants will be hired, propagating the bias further. Almost every dataset has some degree of bias, and not using biased data at all is certainly worse than making decisions without any data. This calls for methods of debiasing data with respect to a given characteristic, so that we can still use potentially biased datasets without fear of creating algorithms that discriminate unfairly. Unfortunately, removing bias from a dataset or algorithm is not straightforward. Simply removing the column that contains information about the protected characteristic does not necessarily remove information about that characteristic from the dataset. Datasets often contain variables which are proxies for other variables. For example, postcode may be a proxy for race, or education may be a proxy for gender. As discussed in Čevora 2019 [6], the most common methods of bias removal do not work particularly well, are based on subjective notions of fairness and/or are very difficult to implement. Fair Adversarial Networks [FANs] are an alternative technique for removing multivariate, non-binary and/or non-linear bias from a dataset, whilst being trivial to implement [6]. The FAN methodology encodes a novel, biased dataset which can subsequently be used with a range of analytical tools, free of bias. FANs therefore directly tackle the problem of biased datasets in ML. In December 2019, illumr Ltd. released a tool called Rosa, which uses the principles of FANs to debias novel datasets using a web-based interface. A free, demo version of Rosa is available online (at illumr’s website), where any user can upload a small dataset and debias it with respect to a chosen characteristic. In particular, Rosa aims to be accessible to data analysts who may not have much experience with data preparation or advanced Machine Learning techniques. In this paper, we demonstrate the effect of Rosa on 5, real-world datasets. The datasets are selected for their high potential to be used in real-world decision making, and also their high potential for inherent bias. To test the effectiveness of Rosa, we perform a simple data analytical task using each dataset, and examine the degree of bias in the output. We then repeat the task using datasets which have been processed by Rosa, and investigate whether the bias has decreased. We find significant/total reductions in bias on all datasets after processing by Rosa. The models that are chosen for the analytical tasks are basic, out-of-the box models, in order to imitate the scenario in which an inexperienced analyst may wish to use Rosa. ## 1 Data Analytic Examples ### 1.1 Criminal Recidivism Criminal sanctions in most countries do not only depend solely on the criminal act that has been committed, but also on personal circumstances (e.g. having children) and the level of threat the individual poses to society. Of a particular interest to us is estimation of the likelihood of recidivism for a criminal defendant, as it is becoming increasingly common to delegate this task to automated decision-making systems. These systems can have a huge impact on people’s lives, as individuals deemed unlikely to recidivate may be released on bail or assigned lower sentences than those deemed otherwise. Correctional Offender Management Profiling for Alternative Sanctions [COMPAS] is a technology used by the Department of Justice of the United States [DoJ] to assess the likelihood of recidivism of criminal defendants and guide bail decisions. Larson and colleagues [12] found significant racial bias in these assessments, with African-Americans being assigned significantly higher recidivism risk scores, independent of whether they recidivated or not. This means COMPAS is much more likely to wrongly label African-Americans as potential recidivists than White Americans. Conversely White Americans are much more likely to be wrongly labelled as non-recidivists. The DoJ unfortunately defends this pattern of discrimination as being fair [11]. Their argument relies on the fact that more African-Americans recidivate according to DoJ statistics. Shockingly, Flores and colleagues, writing in defense of the DoJ, fail to acknowledge the significant evidence for racial bias within the criminal justice system itself, with Black populations suffering from higher rates of stop, search, and arrest, despite research to suggest that their crime rate is relatively similar to other populations [17, 7, 5, 8, 18, 19]. As a result, crime datasets often imply a much higher rate of criminality for Black populations than is likely to be the case. #### 1.1.1 Data and Methods In this section we use a dataset complied by ProPublica that includes the criminal profiles of 11,757 defendants from Broward County, Florida, along with an indication of whether they recidivated during a two-year follow-up study. The dataset is freely available to download at Github. Before starting the analysis we discarded non-categorical or numerical data, and data relating to the COMPAS risk-assessment scores allocated to the offenders. We kept data from only two race categories, _Caucasian_ and _African-American_. To replicate a recidivism model such as COMPAS we used a basic logistic regression model from Python’s sklearn package to predict likelihood of recidivism while excluding the race variable from the data, and looked for bias in the predictions of our model. We then processed the data with Rosa and repeated the exact same modelling, checking to see whether the bias had reduced. #### 1.1.2 Results As shown in Figures 1 and 2 the distribution of model estimates is clearly discriminatory against African-Americans when Rosa is not used. This bias essentially disappeared when the data was pre-processed using Rosa. | ---|--- | Figure 1: Without data pre-processing using Rosa (left-hand plots), African-Americans are labelled as having a higher probability of recidivism than their Caucasian counterparts by our model, independent of whether they recidivated or not. Using Rosa as a pre-processing step (right-hand plot), this difference has been almost entirely removed. Note that without considering the estimates for recidivists and non-recidivists separately, any difference in the score distribution for African-Americans and Caucasians could be explained by a different rate of recidivism between the racial categories, as opposed to an unfair pattern of misclassification. | Original data | Rosa pre-processed ---|---|--- | non-recid | recid | non-recid | recid $\mu$ African-American | 0.37 | 0\. 46 | 0.35 | 0.37 $\mu$ Caucasian | 0.27 | 0.33 | 0.35 | 0.36 $\mu$ Diff | 0.10 | 0.13 | 0.00 | 0.01 $\sigma$ African-American | 0.13 | 0.14 | 0.06 | 0.06 $\sigma$ Caucasian | 0.11 | 0.13 | 0.05 | 0.06 $\sigma$ Average | 0.12 | 0.14 | 0.06 | 0.06 $\mu$ Diff / $\sigma$ Average | 0.85 | 0.92 | 0.00 | 0.17 Figure 2: Caucasian individuals receive significantly lower recidivism likelihood estimates from our recidivism model when Rosa was not used than their African-American counterparts, irrespective of whether they later recidivate or not. The amount of bias in the model has been quantified by dividing the difference in mean estimate for African-Americans and Caucasians by the average standard deviation of the two distributions. The higher the value the greater the bias. Using Rosa has significantly decreased the level of bias in the estimates both for recidivists and non-recidivists. The accuracy of the regression model prior to debiasing by Rosa was 67%, and post-debiasing it was 63%. This drop, while significant is likely due to the bias inherent in the dataset - which was also used to evaluate accuracy. Unfortunately the real-world nature of the examples presented in this paper do not allow unbiased evaluation. #### 1.1.3 Discussion We have performed a simple DA task to model the chance of recidivism of a criminal defendant. Similar systems are used across USA to determine a defendant’s suitability for alternative sanctions and can therefore have a significant impact on the individual’s life. Without correcting for bias our model discriminated against African-Americans, who were always considered of higher recidivism risk whether they recidivated or not. These results are in line with an analysis of COMPAS - a system developed by Equivant and used by DoJ that has been demonstrated to perform similar discrimination [12]. Replicating the exact same DA pipeline that resulted in a discriminatory model, but applying Rosa as a data pre-processing step, resulted in model that did not discriminate. It is hardly surprising that the initial model was biased against African- Americans. The dataset was likely affected by the issues discussed in Section 1.1, such as a heightened rate of stop and search for African-Americans compared to Caucasians, leading to an over-representation of African-Americans (and black people in general) in criminal datasets relative to their rate of criminality. For example, arrests for marijuana possession are 3.7 times higher for the black population in the US than the white population, despite similar reported levels of usage [5]. For the COMPAS dataset, ‘Marijuana’ was one of the most frequently occurring words in the text describing the charge for each defendant. Dealing with such systematic bias in criminal datasets is essential as data-driven decision making becomes more widespread. Here we have demonstrated that Rosa is a suitable tool for such a task. ### 1.2 Absenteeism at work DA has an ever-increasing role in Human Resource Management [HR], with employers aiming to only hire/retain the highest performing employees. Absenteeism is an important aspect of HR for many organisations and therefore its estimation is an interesting feature that can be used to screen applicants during the hiring process. While employers should be considering absenteeism during the hiring process, it is known to be related to age [14] which is a protected characteristic. Achieving lower absenteeism in a workforce by age- skewed hiring would therefore be illegal. #### 1.2.1 Data and Methods We have performed a simple analysis to model absenteeism in the _Absenteeism at work_ dataset available from the UCI Machine Learning Repository. This dataset holds personal information on a set of employees from a courier company in Brazil, along with their total absenteeism time over three years. Before commencing with the modelling exercise we removed data on the type/season of absence, and replaced the ‘Absenteeism Time in Hours’ variable with a derived feature: ‘Absenteeism in Upper Quartile’, which indicates whether an employee has absence in the upper quartile of all employee absences. We also replaced the age (in years) with a categorical variable, indicating whether employees were under 35, 35 to 45 or over 45. This simplified the identification of bias as we could compare the model estimates of absenteeism directly between the three age groups, rather than in a continuous fashion. We used the Logistic Regression model from Python’s sklearn package to predict whether employees had absenteeism in the upper quartile of all the employees in the dataset while excluding the age variable from the data, and analysed for bias with respect to age. We then used the same model to predict absenteeism with data that had been debiased by Rosa, and checked whether the bias has reduced. #### 1.2.2 Results | ---|--- | Figure 3: For both the upper quartile and lower three-quartile absentees, those under 35 received lower estimates of absenteeism than those over 45. The right-hand plots show the model estimates using data pre-processed by Rosa, where most of the difference in model estimates for the over 45 and under 35 groups has been removed. In particular, before Rosa was applied to the dataset, the older lower three-quartile absentees received higher estimates of absenteeism than the younger upper quartile absentees, indicating a high degree of bias with respect to age. This pattern disappeared when Rosa was used to pre-process the data. | Original data | Rosa pre-processed ---|---|--- | upper quart | lower quarts | upper quart | lower quarts $\mu$ Under 35 | 0.33 | 0.29 | 0.40 | 0.37 $\mu$ Over 45 | 0.52 | 0.37 | 0.41 | 0.35 $\mu$ Diff | 0.19 | 0.08 | 0.01 | 0.02 $\sigma$ Under 35 | 0.05 | 0.10 | 0.17 | 0.15 $\sigma$ Over 45 | 0.16 | 0.14 | 0.11 | 0.08 $\sigma$ Average | 0.11 | 0.12 | 0.12 | 0.12 $\mu$ Diff / $\sigma$ Average | 1.81 | 0.69 | 0.13 | 0.20 Figure 4: Younger individuals receive significantly lower estimates of absenteeism than their older counterparts when we investigate the upper and lower three-quartiles of absentees separately in a model without any compensation of age related bias. The amount of bias in the model has been quantified by dividing the difference in mean estimate for those over 45 and those under 35 by the average standard deviation of the two distributions. The higher the value the greater the bias. Using Rosa has significantly decreased the level of bias in the estimates both for those with upper quartile absenteeism and those without. There was a decrease in prediction accuracy of the model from 72% to 65%, although the unbiased prediction accuracy cannot be directly compared with the prediction accuracy of a biased model. This is because predicting certain outcomes with and without bias are fundamentally different tasks, and therefore success on these separate tasks cannot be compared directly. #### 1.2.3 Discussion We demonstrated that a simple model to predict employee absenteeism shows significant bias with respect to age. This is concerning as it is becoming increasingly common for automated assessments to form a part of hiring processes, and predicting absenteeism is of much interest to employers. While it is acceptable to discriminate based on likely absenteeism alone, we have seen that this correlates with age in a simple model, which would likely result in age-based discrimination in hiring, which is illegal. A particular pattern of discrimination appeared in the analysis of the original dataset: the older lower three-quartile absentees received higher estimates of absenteeism than the younger upper-quartile absentees, indicating a large degree of discrimination with respect to age. This pattern disappeared when Rosa was used to pre-process the data. Using the exact same DA pipeline to predict absenteeism after pre-processing by Rosa, the model estimates showed near zero bias, meaning that the dataset would be safe to use in an automated hiring procedure without risk of age discrimination. ### 1.3 Heart Disease Coronary heart disease [CHD] is the leading cause of death in women [2]. At certain ages, CHD mortality rates are up to 5x higher in men than women [4]. However, this risk is highly age dependent and means that over a lifetime, men and women have a relatively similar risk. Despite this, the higher risk for men during middle-age leads to the common misconception that CHD is a ‘men’s disease’, in turn leading to a lack of research and data collected on CHD in women. Although the major risk factors for CHD in healthy women are the same as those identified for healthy men in epidemiological studies, the relative strength of certain risk factors may depend on gender [13][20]. This means that a model which is trained to predict the risk of heart disease using data mostly from men may perform less well on women. This can also lead to the misdiagnosis of heart disease in women, as symptoms that indicate the presence of heart disease in women are often considered atypical [15]. Misdiagnosis is a serious issue, as the longer a patient goes without appropriate treatment, the greater the risk of mortality. A study by the University of Leeds found that women who had a final diagnosis of a STEMI-type heart attack had a 59 per cent greater chance of a misdiagnosis compared with men [21]. #### 1.3.1 Data and Methods We used a simple DA model to predict the presence of heart disease from a dataset of various blood measurements from patients in Cleveland, Ohio. The _Heart Disease Dataset_ is available at Kaggle. The original dataset has a scale for type of heart disease, with 0 indicating the absence of heart disease and 1 to 3 indicating some degree of heart disease. To simplify modelling, we converted these categories into a binary variable indicating whether a patient has any degree of heart disease or not. We used the Logistic Regression model from Python’s sklearn package to predict the presence of heart disease for individuals in the dataset while excluding the gender variable from the data, and checked for bias with respect to gender in the model estimates. We then debiased the data with respect to gender, and repeated the exact same analytical process to see whether the bias had decreased. #### 1.3.2 Results As shown in Figure 5 and 6, the model estimates of heart disease for men are clearly higher than for women without data pre-processing with Rosa, regardless of whether the patient had heart disease or not. Using data that has been pre-processed by Rosa, this bias is significantly reduced. | ---|--- | Figure 5: Higher model estimates of heart disease were assigned to men compared to women, both for patients with and without heart disease before the data was pre-processed by Rosa. Using the same model with data pre-processed by Rosa, the model assigns equal estimates to men and women with and without heart disease. | Original data | Rosa pre-processed ---|---|--- | Healthy | Heart Disease | Healthy | Heart Disease $\mu$ Women | 0.37 | 0.64 | 0.44 | 0.54 $\mu$ Men | 0.44 | 0.67 | 0.44 | 0.54 $\mu$ Diff | 0.07 | 0.03 | 0.00 | 0.00 $\sigma$ Women | 0.24 | 0.18 | 0.11 | 0.12 $\sigma$ Men | 0.20 | 0.16 | 0.11 | 0.11 $\sigma$ Average | 0.22 | 0.16 | 0.11 | 0.11 $\mu$ Diff / $\sigma$ Average | 0.30 | 0.20 | 0.04 | 0.05 Figure 6: Women received lower estimates of heart disease from our regression model compared to men, independent of whether they had heart disease or not. This means that women are more likely to be incorrectly given a negative diagnosis when they actually have heart disease, compared to men. The amount of bias in the model has been quantified by dividing the difference in mean estimate for men and women with and without heart disease by the average standard deviation of the two distributions. The higher the value the greater the bias. Using Rosa has significantly decreased the level of bias in the estimates both for those with heart disease and those without. The accuracy of the model in diagnosing heart disease using biased data was 0.74, and post-Rosa it was 0.67, although it is not reasonable to directly compare the accuracy of a biased model with the accuracy of an unbiased model. This is because predicting certain outcomes with and without bias are fundamentally different tasks. #### 1.3.3 Discussion We found that a simple model to predict heart disease was biased with respect to gender. This fits well with research on the under-/mis-diagnosis of heart disease in women compared to men [21]. The dataset may suffer from two problems which could lead to biased predictions: 1) the dataset is likely to contain an over-representation of men compared to the true population of heart disease sufferers, as women are less likely be diagnosed correctly; and 2) signals in the data which may lead to good predictions for men do not necessarily apply to women. After de-biasing with Rosa, the predictions made using the heart disease dataset had significantly reduced bias. This has significant implications, as it means that existing data collected from clinical trials on men can be used to predict risk of heart disease without disadvantaging women. ### 1.4 Predicting the Economic Needs Index of Schools As the financial situation of a school can have a large impact on the success of its students, it is important to identify and direct resources to financially disadvantaged schools. In New York City, elite schools admit students based on the results of a single test: the Specialized High Schools Admissions Test, or SHSAT. Unfortunately, due to the lack of resources available to schools in deprived areas, and also the link between race and socio-economic status, there is little diversity in those admitted to these elite high schools. This is a problem that replicates across many cities and many countries. In order to counter this problem, more resources must be directed to under- performing schools, allowing students from deprived areas to catch-up with their less deprived counterparts. A key variable indicating whether students at a particular school are likely to benefit from additional resources is the economic needs index of the school. The economic needs index of the school is the average economic needs index of its pupils, and is based on factors such as whether the student is eligible for assistance from the state and whether they have lived in temporary housing [1]. It is important that the economic needs index can be estimated accurately, however, as race correlates with socio-economic status, there is opportunity for inadvertent racial discrimination in estimates (as we will demonstrate). #### 1.4.1 Data and Methods PASSNYC is a not-for-profit organisation that uses public data to identify promising students in New York’s under-performing schools. They direct resources to these schools in order to increase the diversity of students taking SHSAT. We used a dataset on schools in New York City, compiled by PASSNYC, to predict the economic needs index of students. The _PASSNYC dataset_ is available at Kaggle. We converted the ‘Percent Black / Hispanic’ column to a binary variable which indicated whether the school had a majority of black students or not, in order to ease the identification of bias. We used the Ridge regression model from Python’s sklearn package to predict the economic needs index of each school while excluding the race variable from the data, and looked for racial bias in the predictions. We then debiased the data with respect to race, and made another set of predictions using the debiased data to see whether the bias had decreased when the data was pre- processed using Rosa. #### 1.4.2 Results | ---|--- Figure 7: Before data pre-processing by Rosa, schools with a majority of black students receive much higher estimates of economic needs index than schools with a majority of white students, with very little overlap. Using the same model with data pre-processed by Rosa, the estimates are far less polarised. | True Values | Pre-Rosa | Post-Rosa ---|---|---|--- $\mu$ Majority Black | 0.76 | 0.75 | 0.69 $\mu$ Majority White | 0.42 | 0.44 | 0.57 $\mu$ Diff | 0.34 | 0.31 | 0.12 $\sigma$ Majority Black | 0.13 | 0.08 | 0.11 $\sigma$ Majority White | 0.19 | 0.06 | 0.08 $\sigma$ Average | 0.16 | 0.07 | 0.095 $\mu$ Diff / $\sigma$ Average | 1.93 | 4.43 | 1.26 Figure 8: Prior to data de-biasing using Rosa, the model overestimates the true difference between the economic needs index of majority black schools and majority white schools. The amount of bias in the model has been quantified by dividing the difference in mean estimate for majority black and majority white schools by the average standard deviation of the two distributions. The higher the value the greater the bias. Using Rosa has significantly decreased the level of bias in the estimates both for those with heart disease and those without. It is clear that the pre-Rosa model overestimates this difference compared to the true value, and after data pre-processing with Rosa the difference is far closer to the true value. The $R^{2}$ for predictions prior to debiasing was 0.57, and post-debiasing it was 0.35, although we cannot properly assess model accuracy on biased data. #### 1.4.3 Discussion We found that without using Rosa our model overestimated the economic needs index of majority black schools, and underestimated the economic needs of majority white schools. This is problematic as it means that white students from disadvantaged backgrounds may not receive the same level of support as similarly disadvantaged black students. After debiasing the dataset with Rosa, the difference in mean economic needs index estimate assigned by our model was much closer to the true value than before de-biasing. This means that organisations like PASSNYC can better allocate resources by predicting the economic need of students without racial bias. ### 1.5 Communities and Crime Being able to predict the rate of crime in different communities is of interest to law enforcement bodies, as it can allow them to better distribute resources such as police officers. However, there is a significant body of research to suggest that Black people are unfairly represented in criminal datasets due to the racial bias of those responsible for enacting the law [8, 18, 19]. In certain circumstances, it has even been found that the true rate of crime for Black persons is likely equal to that of White persons, despite large differences in the data collected by law enforcement bodies [5, 17]. This is highly problematic, as any model used to predict crime rate based on such datasets is likely to overestimate the strength of the true relationship between race and crime, leading to biased predictions. In this example it might lead to an unnecessary direction of police resources to communities with a large Black population. In turn, this is likely to lead to a greater rate of arrest of Black individuals (as there will be more police officers in areas with a large Black population), further strengthening the bias in crime datasets. #### 1.5.1 Data and Methods We used a dataset on communities and crime within the United States (combining data from the 1990 US Census, law enforcement data from the 1990 US LEMAS survey, and crime data from the 1995 FBI UCR) to predict the rate of violent crime in different communities. The _Communities and Crime dataset_ is available from the UCI Machine Learning Repository. We discarded 22 of the most sparsely populated columns, and converted the ‘RacePercentBlack’ column to a binary label indicating whether the black proportion in a given community is in the upper quartile across all communities in the dataset. We removed all other columns that contained information about the racial profile of each community. We used the Linear Regression model in Python’s sklearn package to predict the rate of violent crime in each community while excluding the race variable from the data, and looked for bias with respect to race. We then debiased the data with respect to race, and made another set of predictions using the exact same DA pipeline to see whether the bias had truly been removed. #### 1.5.2 Results | ---|--- Figure 9: Before the data was pre-processed by Rosa, our model assigns much higher estimates of violent crime rate to majority Black communities compared to majority White communities. After the data has been pre-processed by Rosa, there is less discrepancy between model estimates for majority Black and majority White communities. | True Values | Pre-Rosa | Post-Rosa ---|---|---|--- $\mu$ Black Upper Quart | 0.49 | 0.51 | 0.39 $\mu$ Black Lower Quarts | 0.16 | 0.15 | 0.20 $\mu$ Diff | 0.33 | 0.36 | 0.19 $\sigma$ Black Upper Quart | 0.27 | 0.17 | 0.29 $\sigma$ White Lower Quarts | 0.16 | 0.13 | 0.17 $\sigma$ Average | 0.215 | 0.15 | 0.23 $\mu$ Diff / $\sigma$ Average | 1.53 | 2.40 | 0.83 Figure 10: There is a large difference in the rate of violent crime between majority Black and majority White communities in the _Communities and Crime dataset_ , despite evidence to suggest that the true difference should be minimal. This is reflected in the estimates from the model before data pre- processing with Rosa. The amount of bias in the model has been quantified by dividing the difference in mean estimate for upper-quartile black and lower three-quartile black communities by the average standard deviation of the two distributions. The higher the value the greater the bias. The pre-Rosa model has even greater bias than the original dataset. After data pre-processing with Rosa, our model estimates have less bias than the original dataset. The $R^{2}$ for predictions prior to debiasing was 0.69, and post-debiasing it was 0.52, although we cannot properly assess model accuracy on biased data. #### 1.5.3 Discussion There is evidence to suggest that there is only a small difference in the rate of crime for Black persons and White populations, however, most criminal datasets currently have a large over-representation of Black persons due to the well documented biases of those enacting the law. This is problematic, as police resources may be incorrectly distributed based on this data, further strengthening the existing bias. We found that our simple model overestimated violent crime rate in communities with a high proportion of Black residents. After pre-processing the data using Rosa, there was much less difference in the model estimates for majority Black and majority White communities. Although the difference in model estimates for communities with a large proportion of Black residents and those with a large proportion of White residents after pre-processing with Rosa was smaller than the true difference in the dataset, extensive research suggests the dataset itself is biased, and therefore it cannot be used as a benchmark. We must instead aim for a level of bias below the true dataset bias in order to prevent the further propagation of racial bias through the criminal justice system. This was achieved by using Rosa. ## 2 Discussion Rosa is a free, web-based tool, which uses the principles of Fair Adversarial Networks [FANs] to remove bias from a dataset with respect to a certain characteristic or characteristics. In this paper we demonstrated the bias removing capabilities of Rosa on five datasets which were related to real and current issues in the world of Data Analytics [DA] and Machine Learning [ML]. These datasets contained racial, gender or age bias that made the results of our analysis clearly biased as well; however the bias was successfully removed/significantly decreased each time we have used Rosa as a pre- processing step in our analysis. The main advantage of Rosa is its wide applicability and simplicity for the end user. Rosa is stand-alone and does not require any integration into the ML pipeline that a dataset is intended for. It is therefore compatible with a wide range of ML and data analysis techniques. Less technically minded users may choose to use a graphical user interface such as the one available at rosa.illumr.com to pre-process their data for further use in software such as MS Excel or IBM SPSS. A data scientist comfortable with scripting, on the other hand, can use the Rosa API directly. In such cases debiasing data becomes a single line of code. It should be noted that although we have demonstrated the efficacy of Rosa at removing bias with respect to protected characteristics [3], Rosa can remove any type of bias. For example, it might be desirable to remove regional bias when evaluating performance of an enterprise’s regional offices. In such cases performance of employees in different parts of the country may not be comparable without removing the bias of the region. There were slight decreases in prediction accuracy across all models after data debiasing. However, because the datasets themselves were all biased in some respect, we cannot directly compare model accuracy before and after debiasing. The model trained on the biased dataset was only tested on biased data - we do not know what its performance would be on fair data. Tests on synthetic datasets with artificially injected bias indicate that the accuracy increases when using Rosa (running the DA pipeline with biased data but evaluating the performance on data before artificial biasing). This result is, however, reliant on the assumptions about how human biases work and therefore do not provide a definite proof of increase in accuracy when using Rosa. The nature of the problem that Rosa addresses makes it impossible to prove an increase in accuracy in the real-world. The stochastic nature of both the simple DA models and the FANs behind Rosa mean that the results of similar experiments will vary every time, and for a thorough analysis should be repeated multiple times. In our analyses for this paper, we present a randomly selected run of the DA pipeline to keep the matter simple. We have observed only very marginal variance in the output with repeated runs. As demonstrated in this paper, Rosa is capable of removing many different types of bias, with no change in approach taken by the end user other than selecting the right characteristic. This ease-of-use is unrivalled by alternative bias removal methods. ## References * [1] Student economic need index. Available at: https://data.cccnewyork.org/data/bar/1371/student-economic-need-index#1371/a/1/1622/62, 2019\. * [2] II Abubakar, T Tillmann, and A Banerjee. Global, regional, and national age-sex specific all-cause and cause-specific mortality for 240 causes of death, 1990-2013: a systematic analysis for the global burden of disease study 2013. Lancet, 385(9963):117–171, 2015. * [3] Equality Act. c. 15. Retrieved from the UK National Archives website: http://www. legislation. gov. uk/ukpga/2010/15/contents, 2010. * [4] Sophie H Bots, Sanne A E Peters, and Mark Woodward. Sex differences in coronary heart disease and stroke mortality: a global assessment of the effect of ageing between 1980 and 2010. BMJ Global Health, 2(2), 2017. * [5] WC Bunting, Lynda Garcia, and Ezekiel Edwards. The war on marijuana in black and white. American Civil Liberties Union, June, 2013. * [6] George Cevora. Fair adversarial networks. 2020\. * [7] Dean A Dabney, Laura Dugan, Volkan Topalli, and Richard C Hollinger. The impact of implicit stereotyping on offender profiling: Unexpected results from an observational study of shoplifting. Criminal Justice and Behavior, 33(5):646–674, 2006. * [8] Dean A Dabney, Richard C Hollinger, and Laura Dugan. Who actually steals? a study of covertly observed shoplifters. Justice Quarterly, 21(4):693–728, 2004. * [9] Jeffrey Dastin. Amazon scraps secret ai recruiting tool that showed bias against women. Reuters, 2018. * [10] Virginia Eubanks. Automating inequality: How high-tech tools profile, police, and punish the poor. St. Martin’s Press, 2018. * [11] Anthony W Flores, Kristin Bechtel, and Christopher T Lowenkamp. False positives, false negatives, and false analyses: A rejoinder to machine bias: There’s software used across the country to predict future criminals. and it’s biased against blacks. Fed. Probation, 80:38, 2016. * [12] Jeff Larson, Surya Mattu, Lauren Kirchner, and Julia Angwin. How we analyzed the compas recidivism algorithm. ProPublica (5 2016), 9, 2016. * [13] JoAnn E Manson, Heather Tosteson, Paul M Ridker, Suzanne Satterfield, Patricia Hebert, Gerald T O’Connor, Julie E Buring, and Charles H Hennekens. The primary prevention of myocardial infarction. New England journal of medicine, 326(21):1406–1416, 1992. * [14] Joseph J Martocchio. Age-related differences in employee absenteeism: a meta-analysis. Psychology and Aging, 4(4):409, 1989. * [15] Jean C McSweeney, Leanne L Lefler, and Beth F Crowder. What’s wrong with me? women’s coronary heart disease diagnostic experiences. Progress in Cardiovascular Nursing, 20(2):48–57, 2005. * [16] Cathy O’Neil. Weapons of math destruction: How big data increases inequality and threatens democracy. Broadway Books, 2016. * [17] Alex R Piquero and Robert W Brame. Assessing the race–crime and ethnicity–crime relationship in a sample of serious adolescent delinquents. Crime & Delinquency, 54(3):390–422, 2008. * [18] George E Schreer, Saundra Smith, and Kirsten Thomas. “shopping while black”: Examining racial discrimination in a retail setting 1. Journal of Applied Social Psychology, 39(6):1432–1444, 2009. * [19] Andy Shallice and Paul Gordon. Black People, White Justice?: Race and the Criminal Justice System. Runnymede Trust London, 1990. * [20] Eric Vittinghoff, Michael G Shlipak, Paul D Varosy, Curt D Furberg, Christine C Ireland, Steven S Khan, Roger Blumenthal, Elizabeth Barrett-Connor, Stephen Hulley, et al. Risk factors and secondary prevention in women with heart disease: the heart and estrogen/progestin replacement study. Annals of internal medicine, 138(2):81–89, 2003. * [21] Jianhua Wu, Chris P Gale, Marlous Hall, Tatendashe B Dondo, Elizabeth Metcalfe, Ged Oliver, Phil D Batin, Harry Hemingway, Adam Timmis, and Robert M West. Editor’s choice-impact of initial hospital diagnosis on mortality for acute myocardial infarction: A national cohort study. European Heart Journal: Acute Cardiovascular Care, 7(2):139–148, 2018. * [22] James Zou and Londa Schiebinger. Ai can be sexist and racist—it’s time to make it fair, 2018.

2024-09-04T02:54:56.388535

2003.00913

arxiv-papers

# Experimental Tuning of Transport Regimes in Hyperuniform Disordered Photonic Materials Geoffroy J. Aubry<EMAIL_ADDRESS>Département de Physique, Université de Fribourg, Switzerland Institut de Physique de Nice, Université Côte d’Azur/CNRS, France Luis S. Froufe-Pérez Département de Physique, Université de Fribourg, Switzerland Ulrich Kuhl Institut de Physique de Nice, Université Côte d’Azur/CNRS, France Olivier Legrand Institut de Physique de Nice, Université Côte d’Azur/CNRS, France Frank Scheffold Département de Physique, Université de Fribourg, Switzerland Fabrice Mortessagne Institut de Physique de Nice, Université Côte d’Azur/CNRS, France (August 27, 2024) ###### Abstract We present wave transport experiments in hyperuniform disordered arrays of cylinders with high dielectric permittivity. Using microwaves, we show that the same material can display transparency, photon diffusion, Anderson localization, or a full band gap, depending on the frequency $\nu$ of the electromagnetic wave. Interestingly, we find a second weaker band gap, which appears to be related to the second peak of the structure factor. Our results emphasize the importance of spatial correlations on different length scales for the formation of photonic band gaps. In analogy to electronic semiconductors, dielectric materials in a periodic [1, 2, 3, 4], quasiperiodic [5], or amorphous configuration [6, 7, 8, 9, 10] can all display full band gaps. For the latter materials, due to the absence of long range order, the band gap has been associated with local resonances of the scatterers or correlated scattering clusters, which is reminiscent of the tight-binding model in electronic semiconductors [11]. In contrast to electrons, however, there exist no bound photon states making this analogy questionable. Other proposals have linked the opening of a gap directly to the suppression of density fluctuations on large length scales, known as stealthy hyperuniformity (SHU) [7]. While the precise origin of a band gap in an amorphous dielectric material is yet unknown, the transport properties inside the gap are well understood [3, 12, 9, 10]. In both periodic and nonperiodic band gap materials, an incident light wave enters by a finite distance $L_{\mathrm{B}}$, called the Bragg length, and is then totally reflected. For a slab of thickness $L$, the wave can tunnel through the material with a probability $T\sim e^{-L/L_{\mathrm{B}}}$. However, outside the gap, the transport properties differ strongly. Photonic crystals either reflect, diffract into Bragg peaks, or they are transparent, which is a direct consequence of long-range order and the corresponding sharp Bragg maxima in the structure factor $S(\vec{k})$. The situation is entirely different for amorphous materials, which scatter light strongly over a broad range of $\vec{k}$. Recent numerical work has revealed that this leads to a rich transport phase diagram for amorphous band gap materials—with regions of transparency, Anderson localization, and light diffusion—not present in ordered materials [10]. In contrast to disordered photonic crystals, discussed for example in the celebrated article by Sajeev John in 1987 [2], the diffuse scattering and localization observed outside the gap is not a consequence of imperfections, but an inherent feature of the amorphous material [9]. Introduced in 2004, stealthy hyperuniformity provides an elegant way to construct such idealized disordered materials with finely tunable correlations encoded by the degree of stealthiness $\chi$, ranging from $0\to 0.5$ before the onset of crystallization [13, *Uche2004]. Thirty years after John’s seminal work on the interplay between photonic band gap formation and strong localization in disordered dielectric lattices [2], a controlled experimental study of the optical transport properties in between ordered and disordered states of matter is still lacking [15]. Here, we present experimental results obtained for a 2D system composed of high index dielectric cylinders in air [16] placed according to SHU point patterns [7]. To probe the different transport regimes experimentally, we conduct measurements in the microwave regime since the frequency span in this regime is much larger than in the optical one. Furthermore, our microwave setup provides a more versatile platform compared to optics. Our samples consist of about $N\simeq 200$ cylindrical scatterers (dielectric permittivity $\varepsilon\simeq 36$, radius $r=3$ mm, height $h=5$ mm; the Mie scattering efficiency of such a cylinder is shown in the Supplemental Material, Fig. S1) placed in an aluminum 2D cavity ($50\times 50\times 0.5$ cm3) on a SHU point pattern (on a square of size of approximately $25\times 25$ cm2) generated by simulating an annealing relaxation scheme [9] (see Fig. 1(a)). Figure 1: (a) Setup for 2D microwave scattering and transport experiments. The dielectric cylinders are placed in between two conducting aluminum plates. To reveal the interior of the sample the top plate has been removed. We place absorbing foam (LS-14 from Emerson&Cuming) around the sample. A fixed antenna (1, black arrow) is positioned at the center of the cavity, $(x,y)=(0,0)$. The mobile antenna (2, red arrow) enters the cavity through small holes arranged on a $(x,y)$ grid in the top plate. (b) Transmitted power $\left|S_{12}(\nu)\right|^{2}$ for different configurations ($\chi$) and for a given distance $d=\sqrt{x^{2}+y^{2}}$ between (1) and (2). We perform measurements on five different configurations $\chi=0.15,0.25,0.30,0.40$, and a triangular lattice. For all the samples studied, we kept the number density constant ($\rho\simeq 0.32$ cm-2). The point patterns and the structure factors of the samples are shown in the Supplemental Material Fig. S2. The cavity can be considered as two dimensional for the microwave frequencies $\nu<10$ GHz studied. Under this condition, only the first transverse magnetic mode, TM0, exists in air: the electric field is perpendicular to the plane, and the field amplitude is uniform over the cavity height [17]. We mimic an infinite 2D system by placing absorbing carbon loaded polyurethane foam between the sample and the metallic walls of the cavity. We raster the cavity with a mobile antenna that is inserted by a robotic arm through holes drilled into the upper plate with a diameter 2 mm, on a $5\times 5$ mm2 grid unit cell. Considering the sample size, and the fact that we are not able to penetrate the cavity at the holes above the scatterers, we end up with about $\sim 2700$ measured positions. At each grid point $(x,y)$, we measure the complex transmission spectrum $S_{12}(\nu)$ between a fixed antenna (1) placed at the center of the cavity and the mobile antenna (2) using a vector network analyzer. Figure 1(b) shows examples of measured spectra $|S_{12}(\nu)|^{2}$ between the central position $1$ and probe position $2$ for different $\chi$ values and for a given distance $d$ between the antennas. The small transmission values of order $10^{-6}$ or less are because the receiving antenna is weakly coupled to the cavity. The measured spectra consist of a superposition of peaks which are associated to the resonances of the system. We extract their frequency, complex amplitude and width using harmonic inversion as described in Ref. [18, 19]. We then cluster the resonances measured on all the lattice points in order to reveal all the eigenmodes present in the system without being spoiled by false resonances induced by noise (See Supplemental Material III.1 [20, 21, 22]). Figure 2: Experimental density of states (DOS). Histogram of states per $0.15$ GHz frequency interval for different configurations: $\chi$ between 0.15 and 0.40, and for the triangular lattice. The hatched areas are a guide to the eye to illustrate the measured band gap widths as a function of $\chi$. In Fig. 2, we plot a histogram of the frequencies of the eigenmodes, which is directly proportional to the density of states (DOS). We compare the results for SHU point patterns with different values of $\chi$, to the results obtained for a triangular lattice. As shown in earlier numerical work, the triangular lattice is the champion photonic crystal structure in 2D, with a gap slightly larger than disordered hyperuniform structures [9]. Our experimental data confirms the two first TM photonic band gaps predicted for the triangular lattice [3]. We also find frequency windows without states for the SHU disordered systems. Surprisingly, not only the first but also the second band gap is present in the $\chi=0.4$ sample. To our knowledge, second and higher order band gaps have so far neither been predicted nor observed in disordered systems. This finding is in contradiction to previous claims about the origin of band gaps in disordered photonic materials [6, 23, 24]. To corroborate additional evidence for this interesting observation, we performed band structure calculations, using the same parameters as in the experiment (see Supplemental Material §. IV [25]). These numerical data confirm the existence of a second-order band gap for $\chi\geq 0.4$. Both the first and the second gap approximately match the maxima of $S(k)$ of the triangular lattice and of the SHU structures, supporting earlier proposals that short- range spatial correlations play a key role for the opening of band-gaps in amorphous photonic materials [9]. Experimentally, we observe a narrow photonic band gap even for our most disordered sample ($\chi=0.15$). Our numerical data for a large ensemble of system realizations, however, suggest that the band gap closes for $\chi\lesssim 0.3$ and reduces to a pseudogap with a small but finite density of states. Naturally, variations between different realizations of hyperuniform materials become more pronounced for smaller values of $\chi$ (see Supplemental Material Fig. S5) and moreover the number of states per frequency bin is small for a finite sized system. This can lead to the situation that the central frequency and width of the band gaps depend on the precise realization of the point pattern, which is a distinct feature of disordered materials not found in crystals. For larger values of $\chi$ these variations are suppressed, and the gap becomes more robust against statistical fluctuations. We now consider the optical properties of our material outside the gap [10]. The amplitude of the peaks observed in Fig. 1(b), and clustered to reveal the eigenmodes, differs from one position to the other and from this we obtain an electric field amplitude map $E_{\nu}(x,y)$ of an eigenmode [26] (see Supplemental Material § III [20, 21, 22, 27]). These eigenmodes maps, shown in the first line of Fig. 3, reveal the striking variations in optical transport properties across the spectral range covered by our experiment. Figure 3: Electromagnetic field distribution of the eigenmodes and wave transport in the time domain for a sample with $\chi=0.30$ ($\nu_{\mathrm{c}}=2.88$GHz). (a-e): Signed amplitudes of selected eigenmodes at different characteristic frequencies. (a) cavity mode, (b) diffusive mode, (c) dielectric localized mode, (d) air localized mode and (e) diffusive mode. (f-j): Maps of the electric field for wave transport at different times $t_{1},t_{2},t_{3}$ and for different central frequencies $f_{0}$. The wave—a Gaussian pulse centered at $f_{0}$ and having a width of 0.5 GHz in the frequency domain—is emitted at the center of the maps, and its temporal representation is shown in the last line ($\Re[\tilde{F}_{f_{0},\Delta\nu}(t)]$ is the real part of the Fourier transform of the Gaussian band pass filter). The colored vertical lines indicate the time of each frame shown $t_{1},t_{2},t_{3}$. Entire videos are included in the Supplemental Material, Videos S6. The color scale is adjusted for each individual panels. At low frequencies, we observe simple square cavity modes as if the medium was homogeneous, which is a remarkable result given the fact that at $\nu\sim 2$GHz, the system size $L=25$ cm is almost two orders of magnitude larger than the Boltzmann mean free path $\ell_{\mathrm{s}}(\nu)$ of the cylinder ensemble (see Supplemental Material Fig. S1), with $\ell_{\mathrm{s}}(\nu)=[\sigma_{\mathrm{s}}(\nu)\rho]^{-1}$ given by the total scattering cross section $\sigma_{\mathrm{s}}(\nu)$ and the number density $\rho$. An alternative way to study wave propagation in the SHU material is to monitor the wave emitted by the central antenna as it propagates through the medium in the time domain. By calculating the real part of the Fourier-transform of $S_{12}(\nu)\times F_{f_{0},\Delta\nu}(\nu)$ (with $F_{f_{0},\Delta\nu}$ a band pass filter of bandwidth $\Delta\nu$ centered around $f_{0}$) at all points on the lattice, we reconstruct movies of the propagating electromagnetic fields as a function of time for the selected bandwidth $\Delta\nu$. Individual frames of the movies are shown in Figs. 3(f-j) (details on the numerical procedure and the entire movies are included in the Supplemental Material § V). Figure 3(f) shows that at low frequencies a circular wave propagates from the central antenna into the medium again signaling transparency. Note that the disordered pattern observed at $t_{3}$ in Fig. 3(f) is due to the nonperfectly absorbing foams placed around the sample which reflect part of the signal (for more details, see Supplemental Material Videos S6-1 and S6-2). From the velocity of the circular wave in the medium we can derive the effective refractive index of the samples and find $n_{\mathrm{eff}}\sim 1.8$. Equally, counting the nodal lines of the modes (Fig. 3(a)) and relating them to their frequencies, we obtain values of the effective refractive index of the metamaterial in the range $n_{\mathrm{eff}}=1.7\pm 0.3$. The uncertainty is due to the fact that, for disordered systems, the cavity size is not well defined and moreover, we observe a slight increase of $n_{\mathrm{eff}}$ from $\nu=1\to 3$ GHz. For comparison, the Maxwell-Garnett effective refractive index, which in 2D corresponds to the square root of the surface averaged permittivity, is $n_{\text{MG}}=2.05$. Torquato and coworkers named their designer materials “stealthy” hyperuniform because they predicted them to be fully transparent below a threshold frequency $\nu<\nu_{\mathrm{c}}$ [28]. The latter is equivalent to saying that $L/\ell^{\star}\to 0$ (with $\ell^{\star}$ the transport mean free path), while $L/\ell_{\mathrm{s}}$ remains finite and can even be larger than one. In this first-order or single-scattering approximation $\nu_{\mathrm{c}}=\frac{c}{n_{\mathrm{eff}}}\sqrt{\frac{\rho\chi}{\pi}}$ [10]. For our system parameters, the theoretical $\nu_{\mathrm{c}}$ range from $\simeq 2.2$ GHz ($\chi=0.15$) to $\simeq 3.0$ GHz ($\chi=0.4$) based on an effective refractive index of $n_{\mathrm{eff}}\sim 1.8$. Leseur _et al._ [29] demonstrated recently that stealthy transparency is also robust against recurrent multiple scattering. They establish a stricter criterion for transparency, $L/\ell_{\mathrm{s}}\ll k\ell_{\mathrm{s}}$, in a dense SHU disordered material composed of dipolar point scatterers. While transparency is retained under this condition it also implies that the transition at $\nu_{\mathrm{c}}$ is not sharp but system size dependent. From a theoretical evaluation of $\sigma_{\mathrm{s}}(\nu)$ for our $\varepsilon=36$ cylinders in air, however, we find that only for $\nu<1$ GHz the condition $L/\ell_{\mathrm{s}}<k\ell_{\mathrm{s}}$ is met (see Supplemental Material Fig. S1 [30]). The experimental results, however, suggest that the condition set by Leseur _et al._ [29] is too restrictive and transparency remains a robust feature for $\nu<\nu_{\mathrm{c}}$ in our dense, high index SHU materials, even for $k\ell_{\mathrm{s}}\lesssim 1$ (see also Supplemental Material Fig. S7). For frequencies $\nu>\nu_{\mathrm{c}}$ transparency is clearly lost and we observe scattering and wave diffusion. The modes become disordered, Fig. 3(b), and the propagating wavefronts in the time domain are highly distorted signaling mean free paths smaller than the system size, Fig. 3(g). A closer inspection of the propagating wave fronts, Supplemental Material Fig. S7, illustrates how the onset of scattering and wave diffusion is shifted to higher frequencies $\nu_{\mathrm{c}}(\chi)\propto\sqrt{\chi}$ as the system becomes more and more stealthy. At frequencies close to the first band gap, we observe spatially localized modes as shown in Figs. 3(c) and (d) [16, 31, 32]. In the time domain, we find that, at longer times, the wave stays localized near the central antenna, as shown in the panels framed red in Figs. 3(h,i) and in the corresponding Supplemental Material videos S6-4 and S6-6. Figure 4: Thouless conductance for different degrees of stealthy hyperuniformity $\chi$ between 0.15 and 0.40. The curves are shifted by a factor 10 for clarity. The hatched areas show the width of the experimentally observed band gaps for each value of $\chi$ using the same colors. We note that the modes below the band gap are localized on the dielectric cylinders, Fig. 3(c), and the modes above the band gap are localized in air, Fig. 3(d). For frequencies in between the first and the second band gap we again observe diffusive modes, Fig. 3(e), as well as extended waves at later times, Fig. 3(j). For frequencies in the band gaps we find no modes, all positions are phase coherent and there is no propagation. Next, we calculate the Thouless conductance $g_{\mathrm{Th}}=\delta\nu/\Delta\nu$, which is a fundamental localization parameter [33, 34, 35]. Thouless argued that in the Anderson localization regime, the dimensionless ratio $g_{\mathrm{Th}}=\delta\nu/\Delta\nu$ falls below unity. In this case, the spectral widths $\delta\nu$ of the modes are smaller than their spacing $\Delta\nu$, and the modes are isolated [33]. In the opposite limit, for $g_{\mathrm{Th}}\geq 1$ modes overlap and waves can propagate. By calculating the average width of the modes in each frequency bin, Fig. 2, we extract the mean Thouless conductance for each frequency bin as shown in Fig. 4. We have marked the data points directly at the band edges by open circles in Fig. 4. Note that, due to the discretization, their values can be affected by the zeroes of the DOS in the gap. Inside the band gap there are no modes and $\left<g_{\mathrm{Th}}\right>$ is not defined. We find values of $\left<g_{\mathrm{Th}}\right>\sim 1$ everywhere except in the vicinity of the gap where $\left<g_{\mathrm{Th}}\right>$ drops by up to two orders of magnitude, signaling localization. This result is consistent with both the finite spatial extension of the modes we observe experimentally, see Figs. 3(c,d), and the localization of the propagating wave in the same frequency domain, Fig. 3(h,i). In the low-frequency regime, the Thouless conductance is close to one, and wave transport expands over the whole system size. In conclusion, we show experimentally that disordered dielectric structures display different characteristic transport regimes such as transparency, photon diffusion, Anderson localization, as well as first and even second order band gaps. We rationalize our findings by analyzing the mode structure and the propagation of waves in the time domain. We find evidence that transparency is robust against recurrent multiple scattering, and that the stealthy materials we study retain their low-frequency transparency even for the unusually strong refractive index mismatch between our scatterers and air $\sqrt{\varepsilon/\varepsilon_{\text{air}}}=6$. Our results lend support to recent numerical predictions and shed new light on the interplay between disorder and correlations [10]. We believe this will have significant consequences for the design of photonic materials, such as two-dimensional nanostructured materials for light harvesting in solar cells [36] or light guiding in all-optical circuit applications [37]. ### Acknowledgments G.A., L.S.F., and F.S. acknowledge funding by the Swiss National Science Foundation through Project No. 169074 and No. 188494, and through the National Center of Competence in Research Bio-Inspired Materials. We would like to thank Paul Chaikin and Juanjo Saenz for discussions. ## References * Yablonovitch [1987] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58, 2059 (1987). * John [1987] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58, 2486 (1987). * Joannopoulos _et al._ [2008] J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, _Photonic Crystals: Molding the Flow of Light_, 2nd ed. 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Florescu, Hyperuniform disordered waveguides and devices for near infrared silicon photonics, Scientific Reports 9, 20338 (2019). ## Supplementary Material This document contains the scattering properties of a single rod, details on the structures of the point patterns, the band structure calculation, details on the time domain propagation videos and all the technical information on the data analysis. The seven videos (permanently stored on the Zenodo repository: https://doi.org/10.5281/zenodo.3978032) show how the electromagnetic wave propagates in the cavity for different frequency ranges (see Supplemental Material Fig. S6 for the description of the videos). ## I Boltzmann scattering mean free path. In Fig. S1 we show the scattering efficiency $Q$ of an individual cylinder in TM polarization calculated using Mie theory [30] (upper panel). In the lower panel, we show how the corresponding Boltzmann scattering mean free path $\ell_{\mathrm{sca}}(\nu)=[\sigma_{\mathrm{sca}}(\nu)\rho]^{-1}$ (with $\sigma_{\mathrm{sca}}(\nu)=2rQ_{\mathrm{sca}}$ the total scattering cross section) compares with $L$, the size of the system, and $\lambda_{0}$, the wavelength in vacuum of the wave. Figure S1: Upper panel: Scattering efficiency $Q_{\mathrm{sca}}$ of individual cylinders in TM polarization (solid blue line), and the three first terms in the Mie expansion (dashed lines). Lower panel: optical density $L/\ell_{\mathrm{s}}$ in the independent scattering approximation using the Boltzmann scattering mean free path and the sample size $L$. Also shown is $k_{0}\ell_{\mathrm{s}}$ with $k_{0}=2\pi/\lambda$ and the wavelength in vacuum $\lambda_{0}$ ($k_{0}=2\pi/\lambda_{0}$). ## II Point patterns and their structure factors. Figure S2(a) shows the point patterns of the samples studied in this study, and Fig. S2(b) the corresponding average structure factors $\displaystyle S(\mathbf{k})={\frac{1}{N}}\sum_{j=1}^{N}\sum_{l=1}^{N}\mathrm{e}^{-i\mathbf{k}\cdot(\mathbf{R}_{j}-\mathbf{R}_{l})},$ (S1) over 1000 samples generated as the ones used in this study, where $R_{j}$ are the positions of the $N$ points, and $\mathbf{k}$ is the wavevector. Figure S2: (a) Point patterns of the studied samples. (b) Radially averaged structure factors $S(k)$ of the studied samples as a function of $ka$, where $a=1/\sqrt{\rho}$ and $\rho$ denotes the number density of scatterers. The structure factors are averaged over 1000 different realizations of about 200 points. The grey vertical lines indicate the peaks of the radially averaged triangular lattice structure factor (Bragg peaks). ## III Visualization of the eigenmodes of the disordered cavity ### III.1 Clustering of the resonances into modes The measured spectra consist of a superposition of peaks (see Main Text Fig. 1(b)). which are associated to the resonances of the system. We determine the frequencies $\nu^{i}$, widths $\gamma^{i}$ and complex amplitudes $A^{i}$ of each resonance $i=1,\dots,N$ using the harmonic inversion method described in ref. [18, 19]. Ideally, resonances belonging to the same mode should all have the same frequency. In practice, the presence of the mobile antenna at every point $(x,y)$ shifts the resonant frequency by a small amount depending on the intensity of the electromagnetic field at the specific mobile antenna position [20], see Fig. S3. Figure S3: For each position $(x,y)$, a spectrum is measured and the frequencies are extracted using harmonic inversion: these are the points plotted in this figure for two different frequency ranges. The points are then clusterized: each color corresponds to a cluster found by the algorithm. The upper panel corresponds to a typical situation in the stealth regime where the intensity is almost uniform over the sample (small frequency shifts). The lower panel corresponds to the case of localized modes with large intensities corresponding to large frequency shifts. Note that we minimize this perturbation due to the mobile antenna by having it extending into the cavity by only 1 mm whereas the height of the cavity is 5 mm. This has the consequence that it is weakly coupled to the field, and explains the low transmission values as seen in Main Text Fig. 1(b). We identify all data points belonging to a certain cluster by using a density- based clustering algorithm [22] fulfilling the condition that two points having the same coordinate $(x,y)$ cannot be in the same cluster. To associate each resonant signal at position $(x,y)$ to a specific mode, we apply a semi- supervised clustering algorithm. This allows us to identify every single mode of the disordered cavity, associated with discrete resonance frequencies, as long as the mode amplitude is large enough to be detected by the vector network analyzer [21, 22]. More precisely, we use a slightly modified version of the C-DBSCAN algorithm published in Ref. [22]. In our version, step 2 of the algorithm [22] either labels the points in the KD-tree leaf as noise ratio (if the density is too small), or we create a local cluster for each point in the leaf. Depending on the frequency range, we run our modified version of C-DBSCAN either in the $(x,y,\nu)$, $(x,y,\nu,\gamma)$ or $(x,y,\nu,\gamma,\ln A)$ space to reach the best clustering results. An example of the result is shown in Fig. S3 where the different clusters, or modes, found by the algorithm are plotted using different colors. ### III.2 Electric field amplitude maps In the first line of Main Text Fig. 3, we plot the signed amplitude $E_{\nu}^{\pm}(x,y)=\operatorname{sgn}\left(\textrm{Re}[\tilde{S}_{12}]\right)|\tilde{S}_{12}|$, where $\tilde{S}_{12}$ is the transmission deduced from $S_{12}$ after the ad hoc rotation of the global phase making the real and imaginary parts statistically independent [27]. This allows to represent both the real and imaginary parts of the eigenmodes on the same map. ## IV Numerical simulations of the DOS Figure S4 shows the normalized density of states (nDOS) of the stealthy hyperuniform samples obtained numerically for a large statistical ensemble of point patterns and using periodic boundary conditions. Figure S4: Normalized density of states (nDOS) obtained by taking the average over the band structure calculated numerically for 500 system realizations at each value of $\chi$. The properties of the dielectric cylinders and their density are identical to those of the system studied in the experiment. The nDOS was calculated using the MIT Photonic Bands [25] software using the supercell method [3] as described earlier in ref. [9]. This dataset was obtained by calculating 500 different samples for each $\chi$-value (between 0.1 and 0.5, every 0.05). Figure S5 shows the average and the standard deviation of the gap central frequency and width found for the samples used in Fig. S4. Figure S5: Spread of the first gap central frequency and width found in the numerical results used to obtain Fig. S4. The error bars correspond to the standard deviations, the scattered points to the 500 individual systems per $\chi$-value used to compute the statistics. The dashed lines correspond to the results obtained for the triangular lattice. The right panel shows the histograms for the $\chi=0.30$ samples. The statistical variations are large at low and intermediate $\chi$-values (between 0.10 and 0.35). At large $\chi$-values ($\geq 0.4$), the standard deviation vanishes: the gap central frequencies and widths are similar from sample to sample. ## V Time domain propagation videos We obtain time domain propagation signals from the real part of the Fourier transform of the complex transmission spectra multiplied by a chosen bandpass filter centered at $f_{0}$ with a standard deviation $\Delta\nu$. We use a Gaussian bandpass filter to avoid window effects in the Fourier transform. The excitation in the time domain is therefore a Gaussian pulse with a temporal spread inversely proportional to $1/\Delta\nu$ of the Gaussian bandpass filter. Videos S6-1, 2 and 3 show the propagation of the wave in the low frequency regime (well below the gap frequency $\nu_{\mathrm{G}}\simeq 5$ GHz. 1. 1. Stealth regime (Gaussian bandpass filter, $f_{0}=1.75$ GHz, $\Delta\nu=0.25$ GHz) 2. 2. Stealth regime (Gaussian bandpass filter, $f_{0}=2.25$ GHz, $\Delta\nu=0.25$ GHz) 3. 3. Wave diffusion (Gaussian bandpass filter, $f_{0}=3.5$ GHz, $\Delta\nu=0.25$ GHz) 4. 4. Dielectric Anderson localized modes just below the band gap (Gaussian bandpass filter, $\Delta\nu=0.25$ GHz) 5. 5. Square filter in the band gaps 6. 6. Air Anderson localized modes just above the band gap (Gaussian bandpass filter, $\Delta\nu=0.25$ GHz) 7. 7. Wave diffusion (Gaussian bandpass filter, $f_{0}=6.5$ GHz, $\Delta\nu=0.25$ GHz) Figure S6: Videos description. The videos are permanently stored on the Zenodo repository: https://doi.org/10.5281/zenodo.3978032. We observe that for frequencies $\nu<\nu_{\mathrm{c}}$ and at early times, the spherical wave structure is well preserved, indicating the absence of scattering. This boundary between the stealth regime and the diffusive regime is also shown in more detail in Fig. S7. Figure S7: Maps of the electric field amplitude for the propagation of a pulse of spectral width $\Delta\nu=0.125$ GHz at different central frequencies $f_{0}$ (for details see text and Main Text Fig. 3), and first half of the Gaussian pulse used for the excitation. The frames shown in the figure are taken at the time marked by the blue vertical line. The panels in the green polygon indicate frequencies below $\nu_{\mathrm{c}}(\chi)$. The radius of the dashed circles indicate the place where a wave emitted at the time marked by the red vertical line should be at the time marked by the blue vertical line, for a homogeneous medium with $n_{\mathrm{eff}}=1.8$. The color scale is adjusted for each individual panels. The panels in the green shaded polygon indicate that the Gaussian pulse central frequency $f_{0}$ is below the critical stealth frequency $\nu_{\mathrm{c}}=\frac{c}{n_{\mathrm{eff}}}\sqrt{\frac{\rho\chi}{\pi}}$, and above $\nu_{\mathrm{c}}$ elsewhere. By eye, we see a clear correlation between the wave front smoothness and the transition from the stealth regime to the diffusive regime for frequencies $\nu>\nu_{\mathrm{c}}$. Since $\nu_{\mathrm{c}}\propto\sqrt{\chi}$ the transition is shifted to higher frequencies when increasing the degree of stealthiness $\chi$. Note that the wave distortion at later times (in the videos) is explained by reflections of the wave on the non-ideal absorbing foam walls. Video S6-4 (respectively S6-6) shows the electromagnetic field for a Gaussian pulse centered 0.25 GHz below (resp. above) the band gap and having a width $\Delta\nu=0.25$ GHz. Video S6-7 shows the propagation of the wave in the high frequency regime, well above the first band gap. As in the low frequency regime for frequencies above $\nu_{\mathrm{c}}$, we observe a strong scattering and wave diffusion. Finally, video S6-5 shows the electromagnetic field in the band gap. For this video, the bandpass filter was chosen to be a square filter fitting exactly the band gaps as extracted from Main Text Fig. 2. This explains the windowing effect seen in the input signal.

2024-09-04T02:54:56.399821

2003.00924

arxiv-papers

# MOZARD: Multi-Modal Localization for Autonomous Vehicles in Urban Outdoor Environments Lukas Schaupp1, Patrick Pfreundschuh3, Mathias Bürki1,2, Cesar Cadena1, Roland Siegwart1, and Juan Nieto1 1Autonomous Systems Lab, ETH Zürich<EMAIL_ADDRESS> 2Sevensense Robotics AG<EMAIL_ADDRESS>3ETH Zurich, <EMAIL_ADDRESS> ###### Abstract Visually poor scenarios are one of the main sources of failure in visual localization systems in outdoor environments. To address this challenge, we present MOZARD, a multi-modal localization system for urban outdoor environments using vision and LiDAR. By extending our preexisting key-point based visual multi-session local localization approach with the use of semantic data, an improved localization recall can be achieved across vastly different appearance conditions. In particular we focus on the use of curbstone information because of their broad distribution and reliability within urban environments. We present thorough experimental evaluations on several driving kilometers in challenging urban outdoor environments, analyze the recall and accuracy of our localization system and demonstrate in a case study possible failure cases of each subsystem. We demonstrate that MOZARD is able to bridge scenarios where our previous work VIZARD fails, hence yielding an increased recall performance, while a similar localization accuracy of 0.2$m$ is achieved. ## I Introduction Due to increasing traffic in urban environments and changing customer demands, self-driving vehicles are one of the most discussed and promising technologies in the car and robotics industry. Still no system was presented yet, that allows to localize robustly under all light, weather and environmental conditions. However, precise localization is a vital feature for each autonomous driving task, since a wrong pose estimate may lead to accidents. Especially in urban environments, safety margins on the position of the car are small due to crowded traffic and other traffic participants (e.g. pedestrians, cyclists). Because of multi-path effects or satellite blockage, GPS sensors cannot be used reliably under those urban conditions. Thus, other sensors have to be used for localization. For this purpose, mainly LiDARs and cameras have been used in the last years. Appearance changes in urban environments challenge visual localization approaches. However, such driving scenarios contain persistent structures even under those appearance changes. Curbstones are one such feature. Curbstones are used to protect pedestrians from cars and to separate the sidewalk from the street. As they delimit the street, they also offer information of the area where the car is allowed to be placed in. Detection of their position relative to the car can thus allow to localize inside the lane. In contrast to other geometrical shapes such as poles and road markings, curbstone measurements are found more frequently in urban environments and yield a reliable, continuous lateral constraint for pose refinement. Due to their shape and their contrasting color with respect to the pavement in many cases, they can be detected both in camera images as well as in LiDAR pointclouds. Figure 1: We aim at accurately localizing the UP-Drive vehicle in a map of features extracted from vision and LiDAR data depicted on the right side. Our proposed algorithm can be separated into two distinct steps. We extract keypoint-based features from camera and additional 3D geometrical curbstone information from a semantic vision-LiDAR pipeline. The features extracted from images of the surround-view camera system (top-left corner) are matched against $3D$ landmarks in the map while our raw curbstone measurements (bottom-left corner) are matched to their corresponding landmarks. Inlier matches, centered on the estimated $6DoF$ pose of the vehicle in the map, are illustrated as dark yellow lines on the right side. Purple indicates our pre- generated curbstone map data represented by splines. During runtime we downsample the nearest splines spatially to match with the current raw curbstone measurements indicated by the red and green color. Therefore, our pipeline named MOZARD extends our previous visual localization system - VIZARD[1] with the use of additional geometrical features from LiDAR data, for the self-driving cars used in the UP-Drive project111The UP-Drive project is a research endeavor funded by the European Commission, aiming at advancing research and development towards fully autonomous cars in urban environment. See www.up-drive.eu.. In a thorough evaluation of our proposed localization system using our long-term outdoor dataset collection, we investigate key performance metrics such as localization accuracy and recall and demonstrate in a case study possible failure scenarios. We see the following aspects as the main contributions of this paper: * • A semantic extension of our key-point-based localization pipeline based upon the extraction of curbstone information is presented, that allows to bridge sparse key-point based feature scenarios in visual localization. * • In a thorough evaluation on the long-term dataset collection UP-Drive, we demonstrate a reliable localization performance across different appearance conditions in urban outdoor environments. We compare our results to our vision-based localization pipeline and demonstrate significant performance increases. * • A computational performance analysis showing that our proposed algorithm exhibits real-time capabilities and better scalability. ## II related work Since our localization system is a multi-modal semantic extension of our previous work we will concentrate the related work on frameworks that exploit semantic features using either one modality or fusing multiple modalities in different ways. Therefore these works can be subdivided into each of their specific sensor setup. For general related work on our prior visual key-point based system we refer to our previous work [1]. #### Vision-only A recent example of using semantic features for the purpose of localization and mapping is Lu et al. [2] using a monocular camera for road-mark detection whereas other studies use traffic signs [3], line segments from multi-view cameras [4] or poles [5] [6] for feature matching. On the detection of curbstones, traditional image based curbstone detection mostly uses the vanishing point and color distribution to detect the corresponding pixels [7], [8]. Recent work such from Enzweiler et al. [9] and Panev et al. [10] also demonstrate a learning based approach to detect curbs in images. In contrast to the vision based approaches, we concentrate on the multi-modal aspect as provided by Goga et al. [11]. #### LiDAR-only LiDAR based methods use assumptions about the shape of the semantic features like curbs, poles and planes by evaluating the difference in elevation [12, 13], slope [12] or curvature [14, 15]. Authors such as Schaefer et al. [16] detect and extract 3D poles from the scenery which are then being used for map tracking. In regards to the usage of curbstones, most applications use geographical map data [17, 18] or road networks [19] as a reference to localize with detected curbstones. Unlike these works, our approach does not rely on external pre-generated data for curbstone map construction. #### Vision and LiDAR Recent work such from Kampker et al. [20] use a camera to extract pole like landmarks and a LIDAR for cylinder shapes for the task of self-localization. Kummerle et al. [21] demonstrate that basic geometric primitives can be extracted using vision and LiDAR to obtain road markings, poles and facades which can then be used for localization and mapping for the purpose of self- localization on various weather conditions. While these approaches use both modalities for mapping and localization separately, there has been recent research into cross-modality. Xiao et al. [22] uses a LiDAR to build an HD map and extract 3D semantic features from the map. Then a monocular camera is used with a deep learning based approach to match these semantic features with the ones from the camera. In contrast to the graph-based SLAM formulation used in our approach, the mentioned approaches are filter-based, with Extended Kalman Filters [23] or Monte Carlo Localization [18, 20], and they are evaluated at low speed and/or on short maps of a few hundred meters [22]. In addition they do not use raw curbstone measurements as a feature for localization and mapping [21, 22, 16, 17, 18]. Our work is evaluated on a long-term map with a length of over 5$km$ using urban driving speed of around 50$km/h$. ## III Methodology Figure 2: The key-point based map-tracking module extracts 2D features from current camera images, and matches them with 3D map landmarks locally in image space using a pose prior $\hat{T}^{t}_{MB}$ while our semantic map-tracking module matches point-cloud based curbstone measurements between our map and immediate input. The state estimation module fuses the visual 2D-3D and geometrical 3D-3D matches with the current wheel-odometry measurement to obtain a current vehicle pose estimate $T^{t}_{MB}$. A schematic overview of MOZARD can be found in Figure 2. Since our work extends the VIZARD framework, we refer to the general methodology from Buerki et al. [1]. We assume that our visual localization pipeline already created a map by tracking and triangulating local 2D features extracted along a trajectory. ### III-A Curbstone Detection For our curbstone detection we employ the work from Goga et al. [11]. Goga et al., fuse a vision-based segmentation CNN with LiDAR data. In a post- processing step they extract, refine and filter semantic curb ROIs to obtain new curb measurements. In the following we use their curbstone detection as input into our pipeline. ### III-B Map Extension The curbstones are added to a map that was built using the VIZARD pipeline [1]. This map is called the base map in the following. Curbstone points detected in a specific LiDAR pointcloud frame will be called curbstone observation. The detection pipeline finds curbstone pointclouds in the vehicle frame $\mathcal{F}_{B}$. We find the closest vertex in time within the base map and allocate the curbstone pointcloud. This is performed for all curbstone detections along the trajectory. From the base map, the respective transformations from the map coordinate frame $\mathcal{F}_{M}$ to the body frame at time $t$ can be looked up. Using $T_{MB}^{t}$ at each vertex in the base map that contains a curbstone observation, a curbstone map in $\mathcal{F}_{M}$ can then be created. ### III-C Curbstone Map-Tracking Figure 3: For our curbstone parameterization pipeline we first obtain raw curbstone measurements. After a voxelization processing step, we cluster the curbstones into segments. Finally, a cubic b-spline is fitted to each segment. In case the fitting algorithm fails, we store the raw points. The curbstone tracking module is the core component of the curbstone localization pipeline. It performs an alignment between the map curbstones and the input curbstones to estimate the vehicle pose. A sanity check is performed to detect wrong alignments. If it is fulfilled, a pose constraint is added to the graph. The integration into the VIZARD system is shown in Figure 2. The single steps are explained in detail in the following section. #### III-C1 Reference Curbstone Retrieval To retrieve the map curbstones, a prior estimate $\hat{T}^{t}_{MB}$ is used. In a fixed radius $r_{lookup}$ around the estimated position $\hat{P}^{t}$, we then search for the closest vertex in Euclidean distance in the base map that contains curbstones. Furthermore, a criterion on the maximum yaw angle between the prior pose estimate and the base map vertex pose is used to prevent wrong associations. #### III-C2 Pointcloud Registration Given the map and input curbstones a pointcloud registration is performed. As the registration algorithm NDT (Normal Distribution Transform) [24] is used from the implementation of the Point Cloud Library [25]. Additionally, outlier points are removed using a fixed ratio since some artifacts might only be included in one of both pointclouds, due to occlusion or unsimilar detection. The pointcloud registration estimates a transformation $T_{align}$ that aligns the input cloud to the map cloud. #### III-C3 Sanity Check In some regions, the input or map pointcloud can consist of very few points. Matching in those scenarios can be ambiguous and lead to wrong associations. Thus, matching is only performed if both pointclouds exceed a minimum amount of points. Since urban street scenarios change frequently, e.g due to constructions or parked cars, the input and map pointcloud can diverge heavily. In those cases, pointcloud registration might fail, ending up in wrong alignments and thereby wrong pose estimates. Therefore, a sanity check has to be performed, to detect wrong pose estimates. To do so, a matching score can be calculated, that can be used as an indicator if the alignment was successful. Magnusson et al. [24] proposed a matching score for NDT. It corresponds to the likelihood that the aligned input points lie on the reference scan. A more detailed explanation can be found in his work [24]. The alignment is considered valid, if the mean likelihood over all input points is higher than a threshold $P_{min}$. #### III-C4 Pose Constraint The new pose estimate is calculated as: $T^{t}_{MB_{estimate}}=\hat{T}^{t}_{MB}*T_{align}$ If the sanity check is successful, the pose constraint is added to the pose graph using a fixed covariance. For our experiments, the covariance was determined empirically. ### III-D Curbstone Parameterization Since curbstone maps can scale quickly given the multitude of possibly redundant observations, a memory overhead is induced. To reduce this memory footprint, we perform a curbstone parameterization. Since the map contains several artifacts like intersections or roundabouts, a curve parameterization was preferred over a polyline. Curbstones are not continuous throughout the whole map, as they often end at intersections. Thus, it naturally makes sense to split the map into single connected regions. In a first step, the raw curbstone pointcloud is subsampled. A clustering is then performed on the subsampled points, to find connected segments of a maximum length. The length- to-width ratio of each segment is then calculated. If a certain threshold is fulfilled, a Cubic B-Spline is fitted to the segment. By doing so, only the control points of the spline have to be saved, instead of all raw curbstone points. If the threshold is not fulfilled, the raw points are saved. The steps are explained in detail in the following and shown in Figure 3. #### III-D1 Subsampling The high point density of the raw pointcloud can result in high runtimes of the clustering as well as in overfitting of the spline to noise in the points. Thus, a spatial subsampling using a voxel grid with a leaf size of $30cm$ is performed. A pointcloud of the means of the points inside each voxel is then used for clustering. #### III-D2 Clustering The clustering is performed in a two-step fashion. First, a Euclidean clustering using a tolerance of more than $2m$ is performed to find large segments. Since the curvature can vary along long segments, fitting a single spline to it can be problematic, as different levels of detail are needed along the segment. An example is a curb going around a corner: While low curvature is desired in the straight sections, high curvature is needed in the area of the corner to properly describe the curb. Thus, the coarse cluster is split into smaller sub-clusters with a maximum expansion of 20$m$ before validating each sub-cluster by using an SVD Decomposition. #### III-D3 Cubic B-Spline Fitting Spline fitting usually refers to fitting a spline that goes through each single input point. However, due to the noisy nature of the curbstone segment, a best fit given a fixed amount of control points is preferred in this case instead of fitting every single point. To achieve this, the approach proposed by Wang et al. [26] to fit an open cubic B-Spline is used. The number of control points is calculated proportionally to the approximate length of the segment, using $0.25\ points/m$, but a minimum of $4$. For segments with a large width (indicating an intersection, road curve or round-about) a fixed amount of $20$ points is used to allow for a proper representation. To validate our fitted spline, we define our goodness score $GS$ as follows: $GS=\frac{\\#Spline\ Inliers}{\\#Spline\ Points}*\frac{\\#Point\ Inliers}{\\#Points}$ whereas $SplineInliers$ is the number of sampled spline point close to a raw point and $PointInliers$ the number of raw points close to points sampled from a spline. Naively using all sub-segment points for the spline fitting can lead to an overfitting of the curve. Thus, the best set of points is found in a RANSAC-like manner. In each iteration, one third of the sub-segment points is sampled randomly. The spline is then fitted to the sampled points. Eventually, the spline with the highest score is chosen. #### III-D4 Spline Sampling To be able to perform matching with the input cloud, points are spatially uniformly sampled from the splines during runtime. Those sampled points are then used as the map pointcloud. ## IV Evaluation In the following section, the performance of the proposed pipeline is evaluated and compared against the VIZARD pipeline as a benchmark. Long-term experiments in an urban scenario are performed on varying weather and appearance conditions. A special focus is set on how curbstone map tracking influences localization accuracy and recall. Example cases are presented, where localization gaps in the VIZARD pipeline could be bridged using curbstone localization. The sensor set-up of the UP-Drive vehicle and the datasets used in the experiments are described in the next section. ### IV-A The UP-Drive Platform For the collection of the datasets, the UP-Drive vehicle was used. Its sensor setup consists of four fish-eye cameras, resulting in a surround view of the car. Gray-scale images with a resolution of $640x400px$ are recorded at $30Hz$. Five Velodyne LiDARs are mounted on top of the car. Curbstones are obtained from the approach as described by Goga et al. [11]. Additionally, a low-cost IMU and wheel tick encoders are used to provide odometry measurements. A consumer-grade GPS sensor is used to gain an initial position estimate and near-by map poses are used to generate an initial orientation estimate. ### IV-B UP-Drive Dataset Collection The UP-Drive dataset collection was recorded between December 2017 and November 2019 in Wolfsburg, Germany, at the Volkswagen factory and its surrounding area and aggregate a total driving distance of multiple $100$ kilometers. The environment is urban, with common artifacts such as busy streets, buses, zebra crossings and pedestrians. Since the data was collected over several months, seasonal appearance changes as well as multiple weather and day-time conditions are present. For this work, our dataset selection is dependent on the availability of curbstone measurements, which result from the curbstone detection pipeline from Goga et al. [11]. Since the curbstone detection module was only enabled in some of our datasets, our evaluation dataset collection consists of $5$ sessions which totals to $10$ drives from August 2019 to November 2019. Each session contains two partially overlapping routes in opposite directions and consists of the same amount of sunny and cloudy/rainy conditions captured throughout a day. Recordings in rainy conditions are categorized as Cloudy, since there is little difference in performance on rainy datasets as opposed to in dry conditions. ### IV-C Metrics #### IV-C1 Localization Recall The fraction of the total travelled distance in which a successful localization was achieved is calculated as the localization recall $r$[%]. While using only the visual pipeline, a localization attempt at time $t$ is accepted as successful if a minimum of $10$ inlier landmark observations is present after pose optimization. When using the combined pipeline, a localization is counted as successful, if the condition above is fulfilled or if a viable curbstone alignment (see section III-C) could be performed. #### IV-C2 Localization Accuracy As no ground-truth for the described dataset exists, the poses estimated by an RTK GPS sensor are used instead as a reference. RTK GPS altitude estimates are not reliable, thus the error in $z$ can not be calculated reliably. Therefore, we focus on the planar $\mathbf{p^{e}_{xy}}$ and lateral translation error $\mathbf{p^{e}_{y}}$ as well as on the orientation error $\mathbf{\theta^{e}_{xyz}}$. | $\mathbf{r}_{mt}$[%] | | $\mathbf{\bar{p}^{e}_{xy}}$, $\mathbf{\bar{p}^{e}_{y}}$ | | $\mathbf{\bar{\theta}^{e}_{xyz}}$ ---|---|---|---|---|--- | $10$-$08$ | $10$-$25$ | $11$-$08$ | $11$-$20$ | | $10$-$08$ | $10$-$25$ | $11$-$08$ | $11$-$20$ | | $10$-$08$ | $10$-$25$ | $11$-$08$ | $11$-$20$ $MOZARD$ \- $Map$ | | | | | | | | | | | | | | $($08-21$)$ | 100.0 | 99.94 | 99.06 | 99.82 | | 0.08 [0.34],0.04 [0.21] | 0.07 [0.26], 0.03 [0.13] | 0.09 [0.37], 0.04 [0.2] | 0.13 [0.37], 0.05 [0.2] | | 0.64 [0.75] | 0.74 [0.76] | 1.04 [1.33] | 1.09 [1.4] $($08-21; 10-08$)$ | - | 100.0 | 100.0 | 100.0 | | - | 0.06 [0.15], 0.03 [0.08] | 0.07 [0.24], 0.03 [0.13] | 0.1 [0.29], 0.04 [0.14] | | - | 0.66 [0.65] | 1.15 [1.4] | 1.2 [1.4] $($08-21; 10-08; 10-25$)$ | - | - | 100.0 | 100.0 | | - | - | 0.07 [0.22], 0.03[0.11] | 0.09 [0.27], 0.03 [0.12] | | - | - | 1.21 [1.43] | 1.23 [1.42] $($08-21; 10-08; 10-25; 11-08$)$ | - | - | - | 100.0 | | - | - | - | 0.07 [0.18], 0.02 [0.1] | | - | - | - | 1.13 [1.38] $VIZARD$ \- $Map$ | | | | | | | | | | | | | | $($08-21$)$ | 100.0 | 98.2 | 97.94 | 91.76 | | 0.08 [0.28], 0.04 [0.16] | 0.07 [0.26], 0.03 [0.13] | 0.09 [0.29], 0.04 [0.17] | 0.13 [0.37], 0.05 [0.21] | | 0.64 [0.76] | 0.74 [0.76] | 1.1 [0.16] | 1.07 [1.28] $($08-21; 10-08$)$ | - | 100.0 | 99.9 | 97.89 | | - | 0.06 [0.13], 0.02 [0.07] | 0.07 [0.24], 0.03 [0.13] | 0.1 [0.29], 0.04 [0.14] | | - | 0.55 [0.67] | 1.15 [1.4] | 1.19 [1.37] $($08-21; 10-08; 10-25$)$ | - | - | 100.0 | 99.18 | | - | - | 0.07 [0.22], 0.03 [0.11] | 0.09 [0.25], 0.03 [0.13] | | - | - | 1.21 [1.43] | 1.23 [1.41] $($08-21; 10-08; 10-25; 11-08$)$ | - | - | - | 99.7 | | - | - | - | 0.07 [0.18], 0.02 [0.1] | | - | - | - | 1.13 [1.38] TABLE I: The localization performance on the UP-Drive dataset, showing localization recall, and the median planar $\mathbf{\bar{p}^{e}_{xy}}$, lateral $\mathbf{\bar{p}^{e}_{y}}$ and orientation ($\mathbf{\bar{\theta}^{e}_{xyz}}$) accuracy. The $90$ percentile is shown in square brackets. Numbering in round brackets defines the timestamp of the sessions used for mapping. E.g. (08-21) represents August, 21th. ### IV-D Localization Accuracy and Recall In order to fully rely on MOZARD to control the car in the UP-Drive project, a high localization recall with an accuracy below 0.5$m$ is paramount, as only short driving segments with no localization may be bridged with wheel-odometry before the car may deviate from its designated lane. Curbstones are not available for the whole trajectory, but for around 89% of the distance of the map. We compare localization recall and accuracy of our localization system to our prior work on visual localization - VIZARD [1]. Note, however, that our prior work relied on the use of cameras for localization, as in contrast to the former, the latter is now able to use LiDAR and vision. To demonstrate that curbstones provide useful additional information, we construct and expand a map iteratively using multiple datasets. Our first map is constructed from two datasets (one session) from August 2019. We then evaluate this map against multiple sessions from different months and add these sessions to our (multi- session) map in a iterative fashion. We present the resulting key evaluation metrics (localization recall $\mathbf{r}_{mt}$[%] and localization accuracy) in Table I over all sessions. By including this comparison, we aim at highlighting the gain in localization recall attainable by using MOZARD while keeping a consistent median translation and orientation error. As shown in Table I, MOZARD is able to attain close to a $100\%$ recall performance on all $4$ sessions on the UP-Drive dataset, while VIZARD performance increase correlates with the addition of sessions to its base map due to the change in visual appearance. We further note that in both cases the planar median localization accuracy are below 15$cm$, while the median lateral error is below 10$cm$. The median orientation errors are on average less than 1 $degree$. For MOZARD the 90th percentile shows an increase which is likely to be due to the higher uncertainty in precision of curbstone measurements. ### IV-E Runtime On our live car platform Goga et al. [11] demonstrated that their curbstone detection pipeline deployed on 2 Nvidia GTX 1080 takes around 20$ms$ for the CNN image segmentation to complete on all $4$ cameras. An additional 32$ms$ are needed for the fusion of 5 LiDARs to run on an Intel i7-3770K CPU. Our curbstone alignment module takes an average of approximately 25$ms$, while the map tracking module (with vision) can take from 27$ms$ with a single session map up to 48$ms$ on our largest multi-session map (see Table I) and has been evaluated on an Intel Xeon E3-1505M CPU. This would allow MOZARD to run with around 10$Hz$ on a single machine on a single session map. Table II summarizes our findings. Module | Average Runtime [ms] ---|--- Curbstone Detection | 52 Curbstone Tracking | 25 Map Tracking (VIZARD) | 27-48 Total | 104-125 TABLE II: Runtime of each component of MOZARD . Curbstone Detection and Curbstone Tracking with average runtime over all evaluated datasets on a single session map, while Map Tracking shows average runtime for running on a single session map and on the largest multi-session map. ### IV-F Case Study We provide further insights into our pipeline by showing specific failure examples for each component. Sample images of a section where the visual localization fails on the evaluated datasets are depicted in Figure 4. Due to occlusion and the absence of surrounding building structures, barely any stable visual cues are found in this section, preventing the visual localization system from matching a sufficient amount of landmarks from the map. This example demonstrates the current limitations of VIZARD, while our MOZARD pipeline is able to handle these sparse keypoint-based scenarios. Unfortunately there are also scenarios where a lack of keypoints and curbstones exists or our curbstone alignment fails - hence conditions where both pipeline are likely to fail as depicted in the right image of Figure 4. A further extension of our current framework to other geometric shapes such as poles, road markings could provide additional useful information that would allow us to further increase our localization performance. Note that we used a single session map for the evaluation of this case study and VIZARD is able to bridge some of these scenarios if enough datasets are provided during the mapping process. Figure 4: On the left, a sample image of a trajectory segment that fails to localize due to occlusion. A lack of keypoints renders it unfeasible to match a sufficient number of map landmarks. On the middle, the projected curbstone information is depicted in the camera frame in red - enabling a continued localization although visual localization failed. On the right, a sample image is depicted where our curbstone and vision pipeline fail. In this case curbstones are actually detected but alignment fails due to our constraints. ## V conclusions We presented MOZARD, a geometric extension to our visual localization system for urban outdoor environments. Through our evaluation on $8$ datasets, including several kilometers of real-world driving conditions, we demonstrated the benefits of using curbstone information for localization and mapping. Our datasets used in the experiments contain challenging appearance conditions such as seasonal changes, wet road surfaces and sun reflections. A comparison with our prior work demonstrated that we can achieve a higher recall performance while using less datasets during the mapping process, as the pipeline would fail due to sparse keypoint scenarios. Our run-time analysis shows that our approach demonstrates real time capabilities. Although the curbstone detection stack of MOZARD takes in average more computing time than VIZARD, it is to note that an object segmentation/detection algorithm on a self-driving car has to be deployed for environmental perception independent of whether a localization takes place or not. Even taking in account the total computational time, our approach still runs at $10Hz$ while needing up to four times less data while achieving the same localization performance. We also showed specific cases where both of our pipelines would fail due to occlusions and/or curbstone misalignment giving suggestions for future work such as the extension of our approach to poles and road markings. 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2024-09-04T02:54:56.411613

2003.00931

arxiv-papers

# Unmixedness of some weighted oriented graphs Lourdes Cruz<EMAIL_ADDRESS>Departamento de Matemáticas Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional Apartado Postal 14–740, Ciudad de México 07000 México Yuriko Pitones<EMAIL_ADDRESS>Departamento de Matemáticas Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional Apartado Postal 14–740, Ciudad de México 07000 México Enrique Reyes<EMAIL_ADDRESS>Departamento de Matemáticas Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional Apartado Postal 14–740, Ciudad de México 07000 México ( ) ###### Abstract Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph whose edge ideal is $I(D)$. In this paper, we characterize the unmixed property of $I(D)$ for each one of the following cases: $G$ is an $SCQ$ graph; $G$ is a chordal graph; $G$ is a simplicial graph; $G$ is a perfect graph; $G$ has no $4$\- or $5$-cycles; $G$ is a graph without $3$\- and $5$-cycles; and ${\rm girth}(G)\geqslant 5$. Keywords: Weighted oriented graphs, edge ideal, unmixed ideal, $SCQ$ graph, perfect graph. ## 1 Introduction A weighted oriented graph $D$ is a triplet $(G,\mathcal{O},w)$, where $G$ is a simple graph whose vertex set is $V(G)$; $\mathcal{O}$ is an edge orientation of $G$ (an assignment of a direction to each edge of $G$); and $w$ is a function $w:V(G)\to\mathbb{N}$. In this case, $G$ is called the underlying graph of $D$. The vertex set of $D$ is $V(D):=V(G)$, the edge set of $D$, denoted by $E(D)$, is the set of oriented edges of the oriented graph $(G,\mathcal{O})$. The _weight_ of $x\in V(D)$ is $w(x)$ and we denote the set $\\{x\in V(D)\mid w(x)>1\\}$ by $V^{+}$. If $R=K[x_{1},\ldots,x_{n}]$ is a polynomial ring over a field $K$, then the edge ideal of $D$ is $I(D)=(x_{i}x_{j}^{w(x_{j})}\mid(x_{i},x_{j})\in E(D))$ where $V(D)=\\{x_{1},\ldots,x_{n}\\}$. These ideals (introduced in [10]) generalize the usual edge ideals of graphs (see [14]), since if $w(x)=1$ for each $x\in V(D)$, then $I(D)$ is the edge ideal of $G$, i.e. $I(D)=I(G)$. An interest in $I(D)$ comes from coding theory in some studies of Reed–Muller typed codes, (see [1, 9]). Furthermore, some algebraic and combinatorial invariants and properties of $I(D)$ have been studied in some papers [7, 8, 10, 11, 15]. In particular, in [10] is given a characterization of the irredundant irreducible decomposition of $I(D)$. This characterization permits studying when $I(D)$ is unmixed, using the strong vertex covers (Definition 2.10 and Theorem 2.12). The unmixed property of $I(D)$ have been studied when $G$ is one of the following graphs: cycles in [10]; graphs with whiskers in [7, 10]; bipartite graphs in [8, 10]; graphs without $3$\- $5$\- and $7$-cycles and König graphs in [11]. In this paper, we study the unmixed property of $I(D)$ for some families of weighted oriented graphs. In Section 2, we give the known results and definitions that we will use in the following sections. In Section 3 (in Theorem 3.10), we characterize when a subset of $V(D)$ is contained in a strong vertex cover. Using this result, we characterize the unmixed property of $I(D)$, when $G$ is a perfect graph (see Theorem 3.11). In Section 4 (in Theorem 4.8), we characterize the unmixed property of $I(D)$ when $G$ is an $SCQ$ graph (see Definition 2.27). These graphs generalize the graph defined in [13] and in the context of this paper, they are important because if $G$ is well–covered such that $G$ is simplicial, $G$ is chordal or $G$ has no some small cycles, then $G$ is an $SCQ$ graph (see Remark 2.29 and Theorem 2.30). In [3], using the $SCQ$ graphs the authors characterize the vertex decomposable property of $G$ when each $5$-cycle of $G$ has at least $4$ chords. Also, in Section 4, we characterize the unmixed property of $I(D)$ when $G$ is Köning, $G$ is simplicial or $G$ is chordal (see Corollaries 4.2 and 4.9). In Section 5, we characterize the unmixed property of $I(D)$, when $G$ has no $3$\- or $5$-cycles; or $G$ has no $4$\- or $5$-cycles; or $G$ has girth greater than 4 (see Theorems 5.4, 5.10 and 5.13). Finally, in Section 6, we give some examples. Our results generalize the results about the unmixed property of $I(D)$ given in [7, 8, 10, 11], since if $G$ is well-covered and $G$ is one of the following graphs: cycles, graphs with whiskers, bipartite graphs, Köning graphs, or graphs without $3$-, $5$\- and $7$-cycles, then $G$ is an $SCQ$ graph. ## 2 Preliminaries In this Section, we give some definitions and well-known results that we will use in the following sections. Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph, recall that $V^{+}=\\{x\in V(D)\mid w(x)>1\\}$ and $I(D)=\big{(}x_{i}x_{j}^{w(x_{j})}\mid(x_{i},x_{j})\in E(D)\big{)}$. ###### Definition 2.1 Let $x$ be a vertex of $D$, the sets $N_{D}^{+}(x):=\\{y\mid(x,y)\in E(D)\\}\quad{\rm and}\quad N_{D}^{-}(x):=\\{y\mid(y,x)\in E(D)\\}$ are called the out-neighbourhood and the in-neighbourhood of $x$, respectively. The neighbourhood of $x$ is the set $N_{D}(x):=N_{D}^{+}(x)\cup N_{D}^{-}(x)$. Furthermore, $N_{D}[x]:=N_{D}(x)\cup\\{x\\}$. Also, if $A\subseteq V(D)$ then $N_{D}(A):=\\{b\in V(D)\mid b\in N_{D}(a)\ {\rm for\ some}\ a\in A\\}$. ###### Definition 2.2 Let $x$ be a vertex of $D$. If $N_{D}^{+}(x)=\emptyset$, then $x$ is called a sink. On the other hand, $x$ is a source if $N_{D}^{-}(x)=\emptyset$. ###### Remark 2.3 Consider the weighted oriented graph $\tilde{D}=(G,\mathcal{O},\tilde{w})$ with $\tilde{w}(x)=1$ if $x$ is a source and $\tilde{w}(x)=w(x)$ if $x$ is not a source. Hence, $I(\tilde{D})=I(D)$. Therefore, in this paper, we assume that if $x$ is a source, then $w(x)=1$. ###### Definition 2.4 The degree of $x\in V(D)$ is $deg_{G}(x):=|N_{D}(x)|$ and $N_{G}(x):=N_{D}(x)$. ###### Definition 2.5 A vertex cover $\mathcal{C}$ of $D$ (resp. of $G$) is a subset of $V(D)$ (resp. of $V(G)$), such that if $(x,y)\in E(D)$ (resp. $\\{x,y\\}\in E(G)$), then $x\in\mathcal{C}$ or $y\in\mathcal{C}$. A vertex cover $\mathcal{C}$ of $D$ is minimal if each proper subset of $\mathcal{C}$ is not a vertex cover of $D$. ###### Remark 2.6 Let $\mathcal{C}$ be a vertex cover of $D$ and $e\in E(G)$. Then, $\mathcal{C}\cap e\neq\emptyset$. Furthermore, $e\cap(\mathcal{C}\setminus a)\neq\emptyset$ if $a\notin e$, $b\in N_{D}(a)$ and $e=\\{a,b\\}$. Hence, $(\mathcal{C}\setminus a)\cup N_{D}(a)$ is a vertex cover of $D$. ###### Definition 2.7 Let $\mathcal{C}$ be a vertex cover of $D$, we define the following three sets: * • $L_{1}(\mathcal{C}):=\\{x\in\mathcal{C}\mid N_{D}^{+}(x)\cap\mathcal{C}^{c}\neq\emptyset\\}$ where $\mathcal{C}^{c}=V(D)\setminus\mathcal{C}$, * • $L_{2}(\mathcal{C}):=\\{x\in\mathcal{C}\mid\mbox{$x\notin L_{1}(\mathcal{C})$ and $N^{-}_{D}(x)\cap\mathcal{C}^{c}\neq\emptyset$}\\}$, * • $L_{3}(\mathcal{C}):=\mathcal{C}\setminus(L_{1}(\mathcal{C})\cup L_{2}(\mathcal{C}))$. ###### Remark 2.8 If $\mathcal{C}$ is a vertex cover of $G$, $x\in V(G)\setminus\mathcal{C}$ and $y\in N_{G}(x)$, then $e:=\\{x,y\\}\in E(G)$ and $e\cap\mathcal{C}\neq\emptyset$. So, $y\in\mathcal{C}$, since $x\notin\mathcal{C}$. Hence, $N_{G}(x)\subseteq\mathcal{C}$. ###### Remark 2.9 Let $\mathcal{C}$ be a vertex cover of $D$, then $x\in L_{3}(\mathcal{C})$ if and only if $N_{D}[x]\subseteq\mathcal{C}$. Hence, $L_{3}(\mathcal{C})=\emptyset$ if and only if $\mathcal{C}$ is minimal. ###### Definition 2.10 A vertex cover $\mathcal{C}$ of $D$ is strong if for each $x\in L_{3}(\mathcal{C})$ there is $(y,x)\in E(D)$ such that $y\in L_{2}(\mathcal{C})\cup L_{3}(\mathcal{C})=\mathcal{C}\setminus L_{1}(\mathcal{C})$ with $y\in V^{+}$ (i.e. $w(y)>1$). ###### Definition 2.11 An ideal $I$ of a ring $R$ is unmixed if each one of its associated primes has the same height. ###### Theorem 2.12 [10, Theorem 31] The following conditions are equivalent: 1. (1) $I(D)$ is unmixed. 2. (2) Each strong vertex cover of $D$ has the same cardinality. 3. (3) $I(G)$ is unmixed and $L_{3}(\mathcal{C})=\emptyset$ for each strong vertex cover $\mathcal{C}$ of $D$. ###### Definition 2.13 The cover number of $G$ is $\tau(G):={\rm min\ }\\{|\mathcal{C}|\ \mid\mathcal{C}\ {\rm\ is\ a\ vertex\ cover\ of\ }G\\}$. Furthermore, a $\tau$-reduction of $G$ is a collection of pairwise disjoint induced subgraphs $H_{1},\ldots,H_{s}$ of $G$ such that $V(G)=\cup_{i=1}^{s}V(H_{i})$ and $\tau(G)=\sum_{i=1}^{s}\tau(H_{i})$. ###### Remark 2.14 We have $\tau(G)=|\mathcal{C}_{1}|$, for some vertex cover $\mathcal{C}_{1}$. So, $\mathcal{C}_{1}$ is minimal. Thus, by Remark 2.9, $L_{3}(\mathcal{C}_{1})=\emptyset$. Hence, $\mathcal{C}_{1}$ is strong. Now, if $I(D)$ is unmixed, then by (2) in Theorem 2.12, $|\mathcal{C}|=|\mathcal{C}_{1}|=\tau(G)$ for each strong vertex cover $\mathcal{C}$ of $D$. ###### Definition 2.15 A stable set of $G$ is a subset of $V(G)$ containing no edge of $G$. The stable number of $G$, denoted by $\beta(G)$, is $\beta(G):={\rm max\ }\\{|S|\ \mid S{\rm\ is\ a\ stable\ set\ of\ }G\\}$. Furthermore $G$ is well–covered if $|S|=\beta(G)$ for each maximal stable set $S$ of $G$. ###### Remark 2.16 $S$ is a stable set of $G$ if and only if $V(G)\setminus S$ is a vertex cover. Hence, $\tau(G)=|V(G)|-\beta(G)$. ###### Remark 2.17 [11, Remark 2.12] $G$ is well-covered if and only if $I(G)$ is unmixed. ###### Definition 2.18 A collection of pairwise disjoint edges of $G$ is called a matching. A perfect matching is a matching whose union is $V(G)$. On the other hand, $G$ is a König graph if $\tau(G)=\nu(G)$ where $\nu(G)$ is the maximum cardinality of a matching of $G$. ###### Definition 2.19 Let $e$ be an edge of $G$. If $\\{a,a^{\prime}\\}\in E(G)$ for each pair of edges, $\\{a,b\\}$, $\\{a^{\prime},b^{\prime}\\}\in E(G)$ and $e=\\{b,b^{\prime}\\}$, then we say that $e$ has the property (P). On the other hand, we say that a matching $P$ of $G$ has the property (P) if each edge of $P$ has the property (P). ###### Theorem 2.20 [2, Proposition 15] If $G$ is a Köning graph without isolated vertices, then $G$ is well–covered if and only if $G$ has a perfect matching with the property (P). ###### Definition 2.21 $\mathcal{P}=(x_{1},\ldots,x_{n})$ is a walk (resp. an oriented walk) if $\\{x_{i},x_{i+1}\\}\in E(G)$ for $i=1,\ldots,n-1$. In this case, $\mathcal{P}$ is a path (resp. an oriented path) if $x_{1},\ldots,x_{n}$ are different. On the other hand, a walk (resp. an oriented walk), $C=(z_{1},z_{2},\ldots,z_{n},z_{1})$ is a $n$-cycle (resp. an oriented $n$-cycle) if $(z_{1},\ldots,z_{n})$ is a path (resp. is an oriented path). ###### Definition 2.22 Let $A$ be a subset of $V(G)$, then the graph induced by $A$, denoted by $G[A]$, is the subgraph $G_{1}$ of $G$ with $V(G_{1})=A$ and $E(G_{1})=\\{e\in E(G)\mid e\subseteq A\\}$. On the other hand, a subgraph $H$ of $G$ is induced if there is $B\subseteq V(G)$ such that $H=G[B]$. A cycle $C$ of $G$ is induced if $C$ is an induced subgraph of $G$. ###### Definition 2.23 A weighted oriented graph $D^{\prime}=(G^{\prime},\mathcal{O}^{\prime},w^{\prime})$ is a weighted oriented subgraph of $D=(G,\mathcal{O},w)$, if $(G^{\prime},\mathcal{O}^{\prime})$ is an oriented subgraph of $(G,\mathcal{O})$ and $w^{\prime}(x)=w(x)$ for each $x\in V(G^{\prime})$. Furthermore, $D^{\prime}$ is an induced weighted oriented subgraph of $D$ if $G^{\prime}$ is an induced subgraph of $G$. ###### Definition 2.24 A vertex $v$ is called simplicial if the induced subgraph $H=G[N_{G}[v]]$ is a complete graph with $k=|V(H)|-1$, in this case, $H$ is called $k$-simplex (or simplex). The set of simplexes of $G$ is denoted by $S_{G}$. $G$ is a simplicial graph if every vertex of $G$ is a simplicial vertex of $G$ or is adjacent to a simplicial vertex of $G$. ###### Definition 2.25 The minimum length of a cycle (contained) in a graph $G$, is called the girth of $G$. On the other hand, $G$ is a chordal graph if the induced cycles are $3$-cycles. ###### Theorem 2.26 [12, Theorems 1 and 2] If $G$ is a chordal or simplicial graph, then $G$ is well-covered if and only if every vertex of $G$ belongs to exactly one simplex of $G$. ###### Definition 2.27 An induced $5$-cycle $C$ of $G$ is called basic if $C$ does not contain two adjacent vertices of degree three or more in $G$. $G$ is an $SCQ$ graph (or $G\in SCQ$) if $G$ satisfies the following conditions: 1. $(i)$ There is $Q_{G}$ such that $Q_{G}=\emptyset$ or $Q_{G}$ is a matching of $G$ with the property (P). 2. $(ii)$ $\\{V(H)\mid H\in S_{G}\cup C_{G}\cup Q_{G}\\}$ is a partition of $V(G)$, where $C_{G}$ is the set of basic $5$-cycles. In the following three results, we use the graphs of Figure 1. $\mathbf{C_{7}}$$d_{\tiny 1}$$d_{\tiny 2}$$b_{\tiny 2}$$a_{\tiny 2}$$a_{\tiny 1}$$b_{\tiny 1}$$c_{\tiny 1}$$c_{\tiny 2}$$g_{\tiny 2}$$g_{\tiny 1}$$\mathbf{P_{10}}$$C_{1}$$C_{2}$$\mathbf{\tilde{e}_{\tiny 1}}$$\mathbf{\tilde{e}_{\tiny 2}}$$\mathbf{\tilde{e}_{\tiny 3}}$$b_{\tiny 4}$$b_{\tiny 3}$$b_{\tiny 2}$$b_{\tiny 1}$$a_{\tiny 4}$$a_{\tiny 3}$$a_{\tiny 2}$$a_{\tiny 1}$$d_{\tiny 2}$$d_{\tiny 1}$$c_{\tiny 1}$$c_{\tiny 2}$$v$$\mathbf{P_{13}}$$C_{1}$$C_{2}$$\mathbf{\tilde{e}_{\tiny 1}}$$\mathbf{\tilde{e}_{\tiny 2}}$$v$$a_{\tiny 1}$$a_{\tiny 3}$$a_{\tiny 2}$$b_{\tiny 1}$$c_{\tiny 1}$$b_{\tiny 3}$$c_{\tiny 3}$$b_{\tiny 2}$$c_{\tiny 2}$$\mathbf{T_{10}}$$\mathbf{\tilde{e}_{\tiny 1}}$$\mathbf{\tilde{e}_{\tiny 2}}$$\mathbf{\tilde{e}_{\tiny 3}}$$a_{\tiny 1}$$a_{\tiny 2}$$a_{\tiny 7}$$a_{\tiny 6}$$a_{\tiny 3}$$a_{\tiny 4}$$a_{\tiny 5}$$b_{\tiny 1}$$b_{\tiny 7}$$b_{\tiny 2}$$b_{\tiny 6}$$b_{\tiny 3}$$b_{\tiny 5}$$b_{\tiny 4}$$\mathbf{P_{14}}$$\mathbf{Q_{13}}$$a_{\tiny 1}$$a_{\tiny 2}$$d_{\tiny 1}$$d_{\tiny 2}$$h$$g_{\tiny 1}$$g_{\tiny 2}$$h^{\prime}$$b_{\tiny 1}$$b_{\tiny 2}$$v$$c_{\tiny 1}$$c_{\tiny 2}$$C_{1}$$\mathbf{\tilde{e}_{\tiny 1}}$$\mathbf{\tilde{e}_{\tiny 2}}$$\mathbf{\tilde{e}_{\tiny 3}}$ Figure 1: ###### Theorem 2.28 [6, Theorem 1.1] If $G$ is connected without $4$\- and $5$-cycles, then $G$ is well-covered if and only if $G\in\\{C_{7},T_{10}\\}$ or $\\{V(H)\mid H\in S_{G}\\}$ is a partition of $V(G)$. ###### Remark 2.29 Suppose $G$ is well-covered. If $G$ is simplicial, or $G$ is chordal or $G$ is a graph without $4$\- and $5$-cycles and $G\notin\\{C_{7},T_{10}\\}$. Then, by Theorems 2.26 and 2.28, $\\{V(H)\mid H\in S_{G}\\}$ is a partition of $V(G)$. Therefore, $G$ is an $SCQ$ graph with $C_{G}=Q_{G}=\emptyset$. ###### Theorem 2.30 [5, Theorem 2 and Theorem 3] If $G$ is a connected graph without $3$\- and $4$-cycles, then $G$ is well-covered if and only if $G\in\\{K_{1},C_{7},P_{10},P_{13},P_{14},Q_{13}\\}$ or $\\{V(H)\mid H\in S_{G}\cup C_{G}\\}$ is a partition of $V(G)$. ###### Definition 2.31 The complement of $G$, denoted by $\overline{G}$, is the graph with $V(\overline{G})=V(G)$ such that for each pair of different vertices $x$ and $y$ of $D$, we have that $\\{x,y\\}\in E(\overline{G})$ if and only if $\\{x,y\\}\notin E(G)$. ###### Definition 2.32 A $k$-colouring of $G$ is a function $c:V(G)\rightarrow\\{1,2,\ldots,k\\}$ such that $c(u)\neq c(v)$ if $\\{u,v\\}\notin E(G)$. The smallest integer $k$ such that $G$ has a $k$-colouring is called the chromatic number of $G$ and it is denoted by $\chi(G)$. On the other hand, the clique number, denoted by $\omega(G)$ is the size of the largest complete subgraph of $G$. Finally, $G$ is perfect if $\chi(H)=\omega(H)$ for every induced subgraph $H$ of $G$. ###### Remark 2.33 Let $A$ be a subset of $V(G)$, then $A$ is a stable set of $G$ if and only if $\overline{G}[A]$ is a complete subgraph of $\overline{G}$. Hence, $\beta(G)=\omega(\overline{G})$. ###### Theorem 2.34 [4, Theorem 5.5.3] $G$ is perfect if and only if $\overline{G}$ is perfect. ## 3 Strong vertex cover and $\star$-semi-forest Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph. In this Section, we introduce the unicycle oriented subgraphs (Definition 3.2), the root oriented trees (Definition 3.3), and the $\star$-semi-forests of $D$ (Definition 3.4). With this definitions, we characterize when a subset of $V(G)$ is contained in a strong vertex cover (see Theorem 3.10). Using this result, we characterize when $I(D)$ is unmixed if $G$ is a perfect graph (see Definition 2.32 and Theorem 3.11). ###### Proposition 3.1 If $\mathcal{C}$ is a vertex cover of $D$ such that $N_{D}^{+}(A)\subseteq\mathcal{C}$ and $A\subseteq V^{+}$, then there is a strong vertex cover $\mathcal{C}^{\prime}$ of $D$, such that $N_{D}^{+}(A)\subseteq\mathcal{C}^{\prime}\subseteq\mathcal{C}$. Proof. First, we prove that there is a vertex cover $\mathcal{C}^{\prime}$ such that $L_{3}(\mathcal{C}^{\prime})\subseteq N_{D}^{+}(A)\subseteq\mathcal{C}^{\prime}\subseteq\mathcal{C}$. We take $L:=N_{D}^{+}(A)$. If $L_{3}(\mathcal{C})\subseteq L$, then we take $\mathcal{C}^{\prime}=\mathcal{C}$. Now, we suppose there is $a_{1}\in L_{3}(\mathcal{C})\setminus L$, then by Remark 2.9, $N_{D}[a_{1}]\subseteq\mathcal{C}$. Thus, $\mathcal{C}_{1}=\mathcal{C}\setminus\\{a_{1}\\}$ is a vertex cover and $L\subseteq\mathcal{C}_{1}$, since $L\subseteq\mathcal{C}$ and $a_{1}\notin L$. Now, we suppose that there are vertex covers $\mathcal{C}_{0},\ldots,\mathcal{C}_{k}$, such that $L\subseteq\mathcal{C}_{i}=\mathcal{C}_{i-1}\setminus\\{a_{i}\\}$ and $a_{i}\in L_{3}(\mathcal{C}_{i-1})\setminus L$ for $i=1,\ldots,k$ where $\mathcal{C}_{0}=\mathcal{C}$ and we give the following recursively process: If $L_{3}(\mathcal{C}_{k})\subseteq L$, then we take $\mathcal{C}^{\prime}=\mathcal{C}_{k}$. Now, if there is $a_{k+1}\in L_{3}(\mathcal{C}_{k})\setminus L$, then by Remark 2.9, $N_{D}[a_{k+1}]\subseteq\mathcal{C}_{k}$. Consequently, $\mathcal{C}_{k+1}:=\mathcal{C}_{k}\setminus\\{a_{k+1}\\}$ is a vertex cover. Also, $L\subseteq\mathcal{C}_{k+1}$, since $L\subseteq\mathcal{C}_{k}$ and $a_{k+1}\not\in L$. This process is finite, since $|V(D)|$ is finite. Hence, there is $m$ such that $L_{3}(\mathcal{C}_{m})\subseteq L\subseteq\mathcal{C}_{m}\subseteq\mathcal{C}$. Therefore, we take $\mathcal{C}^{\prime}=\mathcal{C}_{m}$. Now, we prove that $\mathcal{C}^{\prime}$ is strong. We take $x\in L_{3}(\mathcal{C}^{\prime})$, then $x\in L=N_{D}^{+}(A)$, since $L_{3}(\mathcal{C}^{\prime})\subseteq L$. Thus, $(y,x)\in E(D)$ for some $y\in A\subseteq V^{+}$. Hence, $y\in\mathcal{C}^{\prime}$, since $x\in L_{3}(\mathcal{C}^{\prime})$. Also, $y\not\in L_{1}(\mathcal{C}^{\prime})$, since $N_{D}^{+}(y)\subseteq N_{D}^{+}(A)\subseteq\mathcal{C}^{\prime}$. Hence, $y\in\big{(}\mathcal{C}^{\prime}\setminus L_{1}(\mathcal{C}^{\prime})\big{)}\cap V^{+}$. Therefore, $\mathcal{C}^{\prime}$ is strong. $\Box$ ###### Definition 3.2 If $B$ is a weighted oriented subgraph of $D$ with exactly one cycle $C$, then $B$ is called unicycle oriented graph when $B$ satisfies the following conditions: 1. $(i)$ $C$ is an oriented cycle in $B$ and there is an oriented path from $C$ to $y$ in $B$, for each $y\in V(B)\setminus V(C)$. 2. $(ii)$ If $x\in V(B)$ with $w(x)=1$, then $deg_{B}(x)=1$. ###### Definition 3.3 A weighted oriented subgraph $T$ of $D$ without cycles, is a root oriented tree (ROT) with parent $v\in V(T)$ when $T$ satisfies the following properties: 1. $(i)$ If $x\in V(T)\setminus\\{v\\}$, there is an oriented path $\mathcal{P}$ in $T$ from $v$ to $x$. 2. $(ii)$ If $x\in V(T)$ with $w(x)=1$, then $deg_{T}(x)=1$ and $x\neq v$ or $V(T)=\\{v\\}$ and $x=v$. ###### Definition 3.4 A weighted oriented subgraph $H$ of $D$ is a $\star$-semi-forest if there are root oriented trees $T_{1},\ldots,T_{r}$ whose parents are $v_{1},\ldots,v_{r}$ and unicycle oriented subgraphs $B_{1},\ldots,B_{s}$ such that $H=\big{(}\cup_{i=1}^{r}\ T_{i}\big{)}\cup\big{(}\cup_{j=1}^{s}\ B_{j}\big{)}$ with the following conditions: 1. $(i)$ $V(T_{1}),\ldots,V(T_{r}),V(B_{1}),\ldots,V(B_{s})$ is a partition of $V(H)$. 2. $(ii)$ There is $W=\\{w_{1},\ldots,w_{r}\\}\subseteq V(D)\setminus V(H)$ such that $w_{i}\in N_{D}(v_{i})$ for $i=1,\ldots,r$ (it is possible that $w_{i}=w_{j}$ for some $1\leqslant i<j\leqslant r$). 3. $(iii)$ There is a partition $W_{1}$, $W_{2}$ of $W$ such that $W_{1}$ is a stable set of $D$, $W_{2}\subseteq V^{+}$ and $(w_{i},v_{i})\in E(D)$ if $w_{i}\in W_{2}$. Also, $N_{D}^{+}(W_{2}\cup\tilde{H})\cap W_{1}=\emptyset$, where $\tilde{H}=\\{x\in V(H)\mid deg_{H}(x)\geqslant 2\\}\cup\\{v_{i}\mid deg_{H}(v_{i})=1\\}.$ ###### Remark 3.5 By Definition 3.2 and Definition 3.3, we have $\tilde{H}\subseteq V^{+}$. Furthermore, if $v_{i}$ is a parent vertex of $T_{i}$, with $deg_{H}(v_{i})\geqslant 1$, then $v_{i}\in\tilde{H}$. ###### Lemma 3.6 If $H$ is a $\star$-semi-forest of $D$, then $V(H)\subseteq N_{D}(W_{1})\cup N_{D}^{+}(W_{2}\cup\tilde{H})$. Proof. We take $x\in V(H)$. Since $H=\big{(}\cup_{i=1}^{r}\ T_{i}\big{)}\cup\big{(}\cup_{j=1}^{s}\ B_{j}\big{)}$, we have two cases: Case 1) $x\in V(B_{j})$ for some $1\leqslant j\leqslant s$. Let $C$ be the oriented cycle of $B_{j}$. If $x\in V(C)$, then there is $y_{1}\in V(C)$ such that $(y_{1},x)\in E(C)$. Furthermore, $deg_{H}(y_{1})\geqslant deg_{C}(y_{1})=2$, then $y_{1}\in\tilde{H}$. Hence, $x\in N_{D}^{+}(y_{1})\subseteq N_{D}^{+}(\tilde{H})$. Now, if $x\in V(B_{j})\setminus V(C)$, then there is an oriented path $\mathcal{P}$ in $B_{j}$ from $C$ to $x$. Thus, there is $y_{2}\in V(\mathcal{P})$ such that $(y_{2},x)\in E(\mathcal{P})$. If $|V(\mathcal{P})|>2$, then $deg_{H}(y_{2})\geqslant deg_{\mathcal{P}}(y_{2})=2$. If $|V(\mathcal{P})|=2$, then $y_{2}\in V(C)$ and $deg_{H}(y_{2})>deg_{C}(y_{2})=2$. Therefore, $y_{2}\in\tilde{H}$ and $x\in N_{D}^{+}(\tilde{H})$. Case 2) $x\in V(T_{i})$ for some $1\leqslant i\leqslant r$. First, assume $x=v_{i}$, then there is $w_{i}\in W$ such that $x\in N_{D}(w_{i})$. Consequently, $x\in N_{D}(W_{1})$ if $w_{i}\in W_{1}$ and, by $(iii)$ of Definition 3.4, $x\in N_{D}^{+}(w_{i})\subseteq N_{D}^{+}(W_{2})$ if $w_{i}\in W_{2}$. Now, we suppose $x\neq v_{i}$, then there is an oriented path $\mathcal{L}$, from $v_{i}$ to $x$. Consequently, there is $y_{3}\in V(\mathcal{L})$ such that $(y_{3},x)\in E(D)$. If $y_{3}\neq v_{i}$, then $deg_{H}(y_{3})\geqslant deg_{\mathcal{L}}(y_{3})=2$. Thus, $y_{3}\in\tilde{H}$ and $x\in N_{D}^{+}(\tilde{H})$. Finally, if $y_{3}=v_{i}$, then $deg_{H}(y_{3})\geqslant 1$. Hence, by Remark 3.5, $y_{3}\in\tilde{H}$ and $x\in N_{D}^{+}(\tilde{H})$. $\Box$ ###### Remark 3.7 Sometimes to stress the relation between $W$ and $H$ in Definition 3.4, $W$ is denoted by $W^{H}$. Similarly, $W_{1}^{H}$ and $W_{2}^{H}$. If $\\{T_{1},\ldots,T_{r}\\}=\emptyset$, then $W^{H}=W_{1}^{H}=W_{2}^{H}=\emptyset$. ###### Lemma 3.8 Let $K$ be a weighted oriented subgraph of $D$. If $H$ is a maximal ROT in $K$ with parent $v$, or $H$ is a maximal unicycle oriented subgraph in $K$ whose cycle is $C$, then there is no $(y,x)\in E(K)$ with $x\in V(K)\setminus V(H)$ and $y\in V^{+}\cap V(H)$. Proof. By contradiction suppose there is $(y,x)\in E(K)$ with $x\in V(K)\setminus V(H)$ and $y\in V^{+}\cap V(H)$. Thus, $H\subsetneq H_{1}:=H\cup\\{(y,x)\\}\subseteq K$. If $H$ is a unicycle oriented subgraph with cycle $C$ (resp. $H$ is a ROT), then there is an oriented path $\mathcal{P}$ from $C$ (resp. from $v$) to $y$. Consequently, $\mathcal{P}\cup\\{(y,x)\\}$ is an oriented path from $C$ (resp. from $v$) to $x$ in $H_{1}$. Furthermore, $H_{1}$ has exactly one cycle (resp. has no cycles), since $deg_{H_{1}}(x)=1$ and $V(H_{1})\setminus V(H)=\\{x\\}$. Now, we take $z\in V(H_{1})$ with $w(z)=1$, then $z=x$ or $z\in V(H)$. We prove $deg_{H_{1}}(z)=1$. If $z=x$, then $deg_{H_{1}}(x)=1$. Now, if $z\in V(H)$, then $z\neq y$, since $y\in V^{+}$. So, $deg_{H_{1}}(z)=deg_{H}(z)$, since $N_{H_{1}}(x)=\\{y\\}$. If $H$ is a ROT with $V(H)=\\{v\\}$, then $y=z=v$. A contradiction, since $w(z)=1$ and $y\in V^{+}$. Consequently, by $(ii)$ in Definitions 3.2 and 3.3, $deg_{H_{1}}(z)=deg_{H}(z)=1$. Hence, $H_{1}$ is a unicycle oriented subgraph with cycle $C$ (resp. is a ROT with parent $v$) of $K$. This is a contradiction, since $H\subsetneq H_{1}\subseteq K$ and $H$ is maximal. $\Box$ ###### Definition 3.9 Let $K$ be a weighted oriented subgraph of $D$ and $H$ a $\star$-semi-forest of $D$. We say $H$ is a generating $\star$-semi-forest of $K$ if $V(K)=V(H)$. ###### Theorem 3.10 Let $K$ be an induced weighted oriented subgraph of $D$. Hence, the following conditions are equivalent: 1. (1) There is a strong vertex cover $\mathcal{C}$ of $D$, such that $V(K)\subseteq\mathcal{C}$. 2. (2) There is a generating $\star$-semi-forest $H$ of $K$. Proof. ${\rm(2)}\Rightarrow{\rm(1)}$ Let $\mathcal{C}_{1}$ be a minimal vertex cover of $D$. By (2), $K$ has a generating $\star$-semi-forest $H$. Now, using the notation of Definition 3.4, we take $\mathcal{C}_{2}=\big{(}\mathcal{C}_{1}\setminus W_{1}\big{)}\cup N_{D}(W_{1})\cup N_{D}^{+}(W_{2}\cup\tilde{H})$. By Remark 2.6, $\mathcal{C}_{2}$ is a vertex cover of $D$. Since $W_{1}$ is a stable set, $N_{D}(W_{1})\cap W_{1}=\emptyset$. Then, $\mathcal{C}_{2}\cap W_{1}=\emptyset$, since $N_{D}^{+}(W_{2}\cup\tilde{H})\cap W_{1}=\emptyset$. By Remark 3.5 and $(iii)$ in Definition 3.4, $\tilde{H}\cup W_{2}\subseteq V^{+}$. So, by Proposition 3.1, there is a strong vertex cover $\mathcal{C}$ of $D$ such that $N_{D}^{+}(W_{2}\cup\tilde{H})\subseteq\mathcal{C}\subseteq\mathcal{C}_{2}$. Consequently, $\mathcal{C}\cap W_{1}=\emptyset$, since $\mathcal{C}_{2}\cap W_{1}=\emptyset$. Thus, $N_{D}(W_{1})\subseteq\mathcal{C}$, since $\mathcal{C}$ is a vertex cover. Then, by Lemma 3.6, $V(H)\subseteq N_{D}(W_{1})\cup N_{D}^{+}(W_{2}\cup\tilde{H})\subseteq\mathcal{C}$. Hence, $V(K)\subseteq\mathcal{C}$, since $H$ is a generating $\star$-semi-forest of $K$. ${\rm(1)}\Rightarrow{\rm(2)}$ We have, $\mathcal{C}$ is a strong vertex cover such that $V(K)\subseteq\mathcal{C}$. If $A:=L_{1}(\mathcal{C})\cap V(K)=\\{v_{1},\ldots,v_{s}\\}$, then there is $w_{i}\in V(D)\setminus\mathcal{C}\subseteq V(D)\setminus V(K)$ such that $(v_{i},w_{i})\in E(D)$. We take the ROT’s $M_{1}=\\{v_{1}\\},\ldots,M_{s}=\\{v_{s}\\}$ and sets $W_{1}^{i}=\\{w_{i}\\}$ and $W_{2}^{i}=\emptyset$ for $i=1,\ldots,s$. Now, we will give a recursive process to obtain a generating $\star$-semi- forest of $K$. For this purpose, suppose we have connected $\star$-semi- forests $M_{s+1},\ldots,M_{l}$ of $K\setminus A$ with subsets $W_{1}^{s+1},\ldots,W_{1}^{l},W_{2}^{s+1},\ldots,W_{2}^{l}\subseteq V(D)\setminus V(K)$ and $V^{s+1},\ldots,V^{l}\subseteq V(K)$ such that for each $s<j\leqslant l$, they satisfies the following conditions: * (a) $V^{j}=\\{v_{j}\\}$ if $M_{j}$ is a ROT with parent $v_{j}$ or $V^{j}$ is the cycle of $M_{j}$ if $M_{j}$ is a unicycle oriented subgraph, * (b) $M_{j}$ is a maximal ROT in $K^{j}:=K\setminus\cup_{i=1}^{j-1}V(M_{i})$ with parent in $V^{j}$ or $M_{j}$ is a maximal unicycle oriented subgraph in $K^{j}$ with cycle $V^{j}$. * (c) $W_{1}^{j}\cap\mathcal{C}=\emptyset$ and $W_{2}^{j}\subseteq\big{(}\mathcal{C}\setminus(L_{1}(\mathcal{C})\cup V(K))\big{)}\cap V^{+}$. Hence, we take $K^{l+1}:=K\setminus\big{(}\cup_{i=1}^{l}V(M_{i})\big{)}$. This process starts with $l=s$; in this case, $K^{s+1}:=K\setminus\big{(}\cup_{i=1}^{s}V(M_{i})\big{)}=K\setminus A$; furthermore, if $A=\emptyset$, then $K^{1}=K$. Continuing with the recursive process, if $K^{l+1}=\emptyset$, then $V(K)=\cup_{i=1}^{l}V(M_{i})$ and we stop the process. Now, if $K^{l+1}\neq\emptyset$, then we will construct a connected $\star$-semi-forest $M_{l+1}$ of $K^{l+1}$ in the following way: Case (1) $L_{2}(\mathcal{C})\cap V(K^{l+1})\neq\emptyset$. Then, there is $z\in L_{2}(\mathcal{C})\cap V(K^{l+1})$. Thus, there is $(z^{\prime},z)\in E(D)$ with $z^{\prime}\notin\mathcal{C}$. We take a maximal ROT $M_{l+1}$ in $K^{l+1}$, whose parent is $z$. Also, we take $V^{l+1}=\\{v_{l+1}\\}=\\{z\\}$, $W_{1}^{l+1}=\\{w_{l+1}\\}=\\{z^{\prime}\\}$ and $W_{2}^{l+1}=\emptyset$. Hence, $M_{l+1}$ satisfies (a), (b) and (c), since $z^{\prime}\notin\mathcal{C}$ and $W_{2}^{l+1}=\emptyset$. Case (2) $L_{2}(\mathcal{C})\cap V(K^{l+1})=\emptyset$. Then, $V(K^{l+1})\subseteq L_{3}(\mathcal{C})$, since $K^{l+1}\subseteq K\setminus A\subseteq\mathcal{C}\setminus L_{1}(\mathcal{C})$. We take $x\in V(K^{l+1})$, then there is $x_{1}\in\big{(}\mathcal{C}\setminus L_{1}(\mathcal{C})\big{)}\cap V^{+}$ such that $(x_{1},x)\in E(D)$, since $\mathcal{C}$ is strong. If $x_{1}\in V(K^{l+1})$, then there is $x_{2}\in\big{(}\mathcal{C}\setminus L_{1}(\mathcal{C})\big{)}\cap V^{+}$ such that $(x_{2},x_{1})\in E(D)$, since $\mathcal{C}$ is strong. Continuing with this process we obtain a maximal path $\mathcal{P}=(x_{r},x_{r-1},\ldots,x_{1},x)$ such that $x_{r-1},\ldots,x_{1},x$ are different in $V(K^{l+1})$ and $x_{1},\ldots,x_{r}\in\big{(}\mathcal{C}\setminus L_{1}(\mathcal{C})\big{)}\cap V^{+}$. Thus, $x_{r}\notin\cup_{j=1}^{s}V(M_{j})$, since $x_{r}\notin L_{1}(\mathcal{C})$. Now, suppose $x_{r}\in V(M_{j})$ for some $s<j\leqslant l$. So, $(x_{r},x_{r-1})\in E(K^{j})$, $x_{r-1}\in V(K^{j})\setminus V(M_{j})$ and $x_{r}\in V^{+}\cap V(M_{j})$. Furthermore, $N_{D}^{+}(x_{r})\cap W_{1}^{j}=\emptyset$, since $x_{r}\in\mathcal{C}\setminus L_{1}(\mathcal{C})$ and $\mathcal{C}\cap W_{1}^{j}=\emptyset$. A contradiction, by Lemma 3.8, since $W^{j}\cap V(K^{j})=\emptyset$. Hence, $x_{r}\notin\cup_{j=1}^{l}V(M_{j})$. Consequently, $x_{r}\notin V(K)$ or $x_{r}\in V(K^{l+1})$. Case (2.a) $x_{r}\notin V(K)$. Then, take a maximal ROT $M_{l+1}$ in $K^{l+1}$ whose parent is $x_{r-1}$. Also, we take $V^{l+1}=\\{v_{l+1}\\}=\\{x_{r-1}\\}$, $W_{1}^{l+1}=\emptyset$; and $W_{2}^{l+1}=\\{w_{l+1}\\}=\\{x_{r}\\}$. Thus, $M_{l+1}$ satisfies (a), (b) and (c), since $W_{1}^{l+1}=\emptyset$, $x_{r}\in\big{(}\mathcal{C}\setminus L_{1}(\mathcal{C})\big{)}\cap V^{+}$ and $x_{r}\notin V(K)$. Case (2.b) $x_{r}\in V(K^{l+1})$. Then, $x_{r}\in L_{3}(\mathcal{C})$, since $V(K^{l+1})\subseteq L_{3}(\mathcal{C})$. Hence, there is $x_{r+1}\in\big{(}\mathcal{C}\setminus L_{1}(\mathcal{C})\big{)}\cap V^{+}$ such that $(x_{r+1},x_{r})\in E(D)$, Then $\tilde{\mathcal{P}}=(x_{r+1},x_{r},\ldots,x_{1},x)$ is an oriented walk. By the maximality of $\mathcal{P}$, we have that $x_{r}\in\\{x_{r-1},\ldots,x_{1},x\\}$. Thus, $\mathcal{P}=(x_{r},\ldots,x_{1},x)$ contains an oriented cycle $C$. We take a maximal unicycle oriented subgraph $M_{l+1}$ of $K^{l+1}$ with cycle $C$, $V^{l+1}=C$ and $W_{1}^{l+1}=W_{2}^{l+1}=\emptyset$. Then, $M_{l+1}$ satisfies (a), (b) and (c). Since $K$ is finite, with this proceeding we obtain $M_{1},\ldots,M_{t}\subseteq K$ such that $V(K)=\cup_{i=1}^{t}\ V(M_{i})$, $W_{1}^{i}\cap\mathcal{C}=\emptyset$ and $W_{2}^{i}\subseteq\big{(}\mathcal{C}\setminus L_{1}(\mathcal{C})\big{)}\cap V^{+}$ for $i=1,\ldots t$. We take $H:=\cup_{i=1}^{t}\ M_{i}$ with $W_{j}=\cup_{i=1}^{t}\ W_{j}^{i}$ for $j=1,2$. So, $V(H)=V(K)$. Also, $W_{1}\cap\mathcal{C}=\emptyset$, then $W_{1}$ is a stable set, since $\mathcal{C}$ is a vertex cover. Furthermore, $W_{2}\subseteq V^{+}$ and $W_{2}\subseteq\mathcal{C}\setminus L_{1}(\mathcal{C})$, then $N_{D}^{+}(W_{2})\subseteq\mathcal{C}$. Then, $N_{D}^{+}(W_{2})\cap W_{1}=\emptyset$, since $\mathcal{C}\cap W_{1}=\emptyset$. If $x\in L_{1}(\mathcal{C})\cap V(K)$, then there is $1\leqslant i\leqslant s$ such that $x=v_{i}$ and $M_{i}=\\{v_{i}\\}$. Consequently, $deg_{H}(x)=deg_{M_{i}}(v_{i})=0$. Thus, $\tilde{H}\cap L_{1}(\mathcal{C})=\emptyset$ implying $N_{D}^{+}(\tilde{H})\subseteq\mathcal{C}$, since $V(H)\subseteq\mathcal{C}$. Hence, $N_{D}^{+}(\tilde{H})\cap W_{1}=\emptyset$, since $W_{1}\cap\mathcal{C}=\emptyset$. Therefore, $H$ is a generating $\star$-semi- forest of $K$. $\Box$ ###### Theorem 3.11 Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph where $G$ is a perfect graph, then $G$ has a $\tau$-reduction $H_{1},\ldots,H_{s}$ in complete subgraphs. Furthermore, $I(D)$ is unmixed if and only if each $H_{i}$ has no generating $\star$-semi-forests. Proof. First, we prove $G$ has a $\tau$-reduction in complete graphs. By Theorem 2.34, $\overline{G}$ is perfect. Thus, $s:=w(\overline{G})=\chi(\overline{G})$. So, there is a $s$-colouring $c:V(\overline{G})\rightarrow\\{1,\ldots,s\\}$. We take $V_{i}:=c^{-1}(i)$ for $i=1,\ldots,s$. Then, $V_{i}$ is a stable set in $\overline{G}$, since $c$ is a $s$-colouring. Hence, by Remark 2.33, $H_{i}:=G[V_{i}]$ is a complete graph in $G$ and $s=\omega(\overline{G})=\beta(G)$. Furthermore, $V_{1},\ldots,V_{s}$ is a partition of $V(\overline{G})=V(G)$, since $c$ is a function. Consequently, $\sum\limits_{i=1}^{s}\tau(H_{i})=\sum\limits_{i=1}^{s}\big{(}|V_{i}|-1\big{)}=\Big{(}\sum\limits_{i=1}^{s}|V_{i}|\Big{)}-s=|V(G)|-\beta(G)=\tau(G)$. Finally, by Remark 2.16, $|V(G)|-\beta(G)=\tau(G)$, then, $H_{1},\ldots,H_{s}$ is a $\tau$-reduction of $G$. Now, we prove that $I(D)$ is unmixed if and only if each $H_{i}$ has no generating $\star$-semi-forests. $\Rightarrow)$ By contradiction, assume $H_{j}$ has a generating $\star$-semi- forest, then by Theorem 3.10 there is a strong vertex $\mathcal{C}$ such that $V_{j}\subseteq\mathcal{C}$. Furthermore, $\mathcal{C}\cap V_{i}$ is a vertex cover of $H_{i}$, then $|\mathcal{C}\cap V_{i}|\geqslant\tau(H_{i})=|V_{i}|-1$ for $i\neq j$. Thus, $|\mathcal{C}|=\sum_{i=1}^{s}|\mathcal{C}\cap V_{i}|\geqslant|V_{j}|+\sum_{\begin{subarray}{c}i=1\\\ i\neq j\end{subarray}}^{s}(|V_{i}|-1)$, since $V_{1},\ldots,V_{s}$ is a partition of $V(G)$. Hence, by Remark 2.16, $|\mathcal{C}|>|V(G)|-s=\tau(G)$, since $s=\beta(G)$. A contradiction, by Remark 2.14, since $I(D)$ is unmixed. $\Leftarrow)$ Let $\mathcal{C}$ be a strong vertex cover, then $\mathcal{C}\cap V_{i}$ is a vertex cover of $H_{i}$. So, $|\mathcal{C}\cap V_{i}|\geqslant\tau(H_{i})=|V_{i}|-1$ for $i=1,\ldots,s$. Furthermore, by Theorem 3.10, $V_{i}\not\subseteq\mathcal{C}$. Consequently, $|\mathcal{C}\cap V_{i}|=|V_{i}|-1$. Thus, $|\mathcal{C}|=\sum_{i=1}^{s}\big{(}|V_{i}|-1\big{)}$, since $V_{1},\ldots,V_{s}$ is a partition of $V(G)$. Therefore, by (2) in Theorem 2.12, $I(D)$ is unmixed. $\Box$ ## 4 Unmixedness of weighted oriented $SCQ$ graphs Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph. If $P$ is a perfect matching of $G$ with the property (P), then in Proposition 4.1, we characterize when $|\mathcal{C}\cap e|=1$, for each strong vertex cover $\mathcal{C}$ of $D$ and each $e\in P$. Using Proposition 4.1 in Corollary 4.2, we characterize when $I(D)$ is unmixed if $G$ is Köning. In Proposition 4.6, we characterize the basic $5$-cycles, $C$ such that $|\mathcal{C}\cap V(C)|=3$ for each strong vertex cover $\mathcal{C}$ of $D$. Furthermore, in Theorem 4.8, we characterize when $I(D)$ is unmixed if $G$ is an $SCQ$ graph (see Definition 2.27). Finally, using this result we characterize the unmixed property of $I(D)$, when $G$ is simplicial or $G$ is chordal (see Corollary 4.9). ###### Proposition 4.1 Let $e$ be an edge of $G$. Hence, the following conditions are equivalent: 1. (1) $|\mathcal{C}\cap e|=1$ for each strong vertex cover $\mathcal{C}$ of $D$. 2. (2) $e$ has the property (P) and $N_{D}(b)\subseteq N_{D}^{+}(a)$ if $(a,b^{\prime})\in E(D)$ with $a\in V^{+}$ and $e=\\{b,b^{\prime}\\}$. Proof. ${\rm(1)}\Rightarrow{\rm(2)}$ First, we show $e$ has the property (P). By contradiction, suppose there are $\\{a,b\\},\\{a^{\prime},b^{\prime}\\}\in E(G)$ such that $\\{a,a^{\prime}\\}\notin E(G)$. This implies, there is a maximal stable set $S$ such that $\\{a,a^{\prime}\\}\subseteq S$. So, $\tilde{\mathcal{C}}=V(G)\setminus S$ is a minimal vertex cover. Consequently, $\tilde{\mathcal{C}}$ is strong. Furthermore, $a,a^{\prime}\notin\tilde{\mathcal{C}}$, then $b,b^{\prime}\in\tilde{\mathcal{C}}$, since $\\{a,b\\},\\{a^{\prime},b^{\prime}\\}\in E(G)$. A contradiction by (1). Now, assume $(a,b^{\prime})\in E(D)$ with $a\in V^{+}$ and $e=\\{b,b^{\prime}\\}$, then we will prove that $N_{D}(b)\subseteq N_{D}^{+}(a)$. By contradiction, suppose there is $c\in N_{D}(b)\setminus N_{D}^{+}(a)$. We take a maximal stable set $S$ such that $b\in S$. Thus, $\mathcal{C}_{1}=V(G)\setminus S$ is a minimal vertex cover such that $b\notin\mathcal{C}_{1}$. By Remark 2.6, $\mathcal{C}=\big{(}\mathcal{C}_{1}\setminus\\{c\\}\big{)}\cup N_{D}(c)\cup N_{D}^{+}(a)$ is a vertex cover. Furthermore, $c\notin\mathcal{C}$, since $c\notin N_{D}^{+}(a)$. By Proposition 3.1, there is a strong vertex cover $\mathcal{C}^{\prime}$ such that $N_{D}^{+}(a)\subseteq\mathcal{C}^{\prime}\subseteq\mathcal{C}$, since $a\in V^{+}$. Also, $b^{\prime}\in N_{D}^{+}(a)\subseteq\mathcal{C}^{\prime}$ and $c\notin\mathcal{C}^{\prime}$, since $(a,b^{\prime})\in E(D)$ and $c\notin\mathcal{C}$. Then, $b\in N_{D}(c)\subseteq\mathcal{C}^{\prime}$. Hence, $\\{b,b^{\prime}\\}\subseteq\mathcal{C}^{\prime}$. This is a contradiction, by (1). ${\rm(2)}\Rightarrow{\rm(1)}$ By contradiction, assume there is a strong vertex cover $\mathcal{C}$ of $D$ such that $|\mathcal{C}\cap e|\neq 1$. So, $|\mathcal{C}\cap e|=2$, since $\mathcal{C}$ is a vertex cover. Hence, by Theorem 3.10, there is a generating $\star$-semi-forest $H$ of $e$. We set $e=\\{z,z^{\prime}\\}$. First, assume $H$ is not connected. Then, using the Definition 3.4, we have $H=M_{1}\cup M_{2}$ where $M_{1}=\\{v_{1}\\}$, $M_{2}=\\{v_{2}\\}$ and $w_{1},w_{2}\in W$ such that $w_{i}\in N_{D}(v_{i})$ for $i=1,2$. Thus, $\\{z,z^{\prime}\\}=\\{v_{1},v_{2}\\}$ and $\\{w_{1},w_{2}\\}\in E(G)$, since $e$ satisfies the property (P). This implies $|W_{1}\cap\\{w_{1},w_{2}\\}|\leqslant 1$, since $W_{1}$ is a stable set. Hence, we can suppose $w_{2}\in W_{2}$, then $w_{2}\in V^{+}$ and $(w_{2},z^{\prime})\in E(D)$. Consequently, by (2), $w_{1}\in N_{D}(z)\subseteq N_{D}^{+}(w_{2})$, then $(w_{2},w_{1})\in E(D)$. Furthermore, by $(iii)$ in Definition 3.4, $N_{D}^{+}(W_{2})\cap W_{1}=\emptyset$, then $w_{1}\in W_{2}$. So, $w_{1}\in V^{+}$ and $(w_{1},z)\in E(D)$. By (1) with $a=w_{1}$, we have $(w_{1},w_{2})\in E(D)$. A contradiction, then $H$ is connected. Thus, $H$ is a ROT with $V(H)=\\{z,z^{\prime}\\}$. We can suppose $v_{1}=z$ and $W^{H}=\\{w_{1}\\}$, then $(z,z^{\prime})\in E(D)$, $w_{1}\in N_{D}(z)$ and $z=v_{1}\in\tilde{H}$, since $deg_{H}(v_{1})=1$. If $w_{1}\in N_{D}^{+}(z)$, then $w_{1}\in W_{1}$, since $z=v_{1}$. A contradiction, since $N_{D}^{+}(\tilde{H})\cap W_{1}=\emptyset$. Then, $w_{1}\notin N_{D}^{+}(z)$. By Remark 3.5, $z=v_{1}\in\tilde{H}\subseteq V^{+}$. Therefore, by (1) (taking $a=b=z$ and $b^{\prime}=z^{\prime}$), we have $N_{D}(z)\subseteq N_{D}^{+}(z)$, since $e=\\{z,z^{\prime}\\}$ and $z^{\prime}\in N_{D}^{+}(z)$. A contradiction, since $w_{1}\in N_{D}(z)\setminus N_{D}^{+}(z)$. $\Box$ ###### Corollary 4.2 [11, Theorem 3.4] Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph, where $G$ is Köning without isolated vertices. Hence, $I(D)$ is unmixed if and only if $D$ satisfies the following two conditions: 1. (a) G has a perfect matching $P$ with the property (P). 2. (b) $N_{D}(b)\subseteq N_{D}^{+}(a)$, when $a\in V^{+}$, $\\{b,b^{\prime}\\}\in P$ and $b^{\prime}\in N_{D}^{+}(a)$. Proof. $\Rightarrow)$ By Theorem 2.12, $I(G)$ is unmixed. Thus, by Remark 2.17 and Theorem 2.20, $G$ has a perfect matching $P$ with the property (P). Consequently, $\nu(G)=|P|$. Also, $\tau(G)=\nu(G)$, since $G$ is Köning. So, $\tau(G)=|P|$. Now, we take a strong vertex cover $\mathcal{C}$ of $D$ and $e\in P$. Then, $|\mathcal{C}\cap e|\geqslant 1$. Furthermore, by Remark 2.14, $|\mathcal{C}|=\tau(G)=|P|$. Hence, $|\mathcal{C}\cap e|=1$, since $\mathcal{C}=\cup_{\tilde{e}\in P}\ \mathcal{C}\cap\tilde{e}$. Therefore, by Proposition 4.1, $D$ satisfies (b). $\Leftarrow)$ We take a strong vertex cover $\mathcal{C}$ of $D$. By Proposition 4.1, $|\mathcal{C}\cap e|=1$ for each $e\in P$, since $D$ satisfies (a) and (b). This implies $|\mathcal{C}|=|P|$, since $P$ is a perfect matching. Therefore, by $(2)$ in Theorem 2.12, $I(D)$ is unmixed. $\Box$ ###### Lemma 4.3 If there is a basic $5$-cycle $C=(z_{1},z_{2},z_{3},z_{4},z_{5},z_{1})$ with $(z_{1},z_{2})$, $(z_{2},z_{3})\in E(D)$, $z_{2}\in V^{+}$ and $C$ satisfies one of the following conditions: 1. (a) $(z_{3},z_{4})\in E(D)$ with $z_{3}\in V^{+}$. 2. (b) $(z_{1},z_{5})$, $(z_{5},z_{4})\in E(D)$ with $z_{5}\in V^{+}$. then there is a strong vertex cover $\tilde{\mathcal{C}}$ such that $|\tilde{\mathcal{C}}\cap V(C)|=4$. Proof. We take $\mathcal{C}=\big{(}\mathcal{C}_{0}\setminus V(C)\big{)}\cup N_{D}(z_{1})\cup N_{D}^{+}(z_{2},x)$ where $\mathcal{C}_{0}$ is a vertex cover and $x=z_{3}$ if $C$ satisfies (a) or $x=z_{5}$ if $C$ satisfies (b). Thus, $x\in V^{+}$. Furthermore, $z_{2},z_{3},z_{5}\in N_{D}(z_{1})\cup N_{D}^{+}(z_{2})$ and $z_{4}\in N_{D}^{+}(z_{3})$ if $C$ satisfies (a) or $z_{4}\in N_{D}^{+}(z_{5})$ if $C$ satisfies (b). Hence, $\\{z_{2},z_{3},z_{4},z_{5}\\}\subseteq N_{D}(z_{1})\cup N_{D}^{+}(z_{2},x)$. Consequently, $\\{z_{2},z_{3},z_{4},z_{5}\\}\subseteq\mathcal{C}$, implying $\mathcal{C}$ is a vertex cover, since $\mathcal{C}_{0}$ is vertex cover and $N_{D}(z_{1})\subseteq\mathcal{C}$. Also, $z_{1}\notin\mathcal{C}$, since $z_{1}\notin N_{D}(z_{1})\cup N_{D}^{+}(z_{2},z_{3})$ and $z_{1}\notin N_{D}^{+}(z_{5})$ if $C$ satisfies (b). By Proposition 3.1, there is a strong vertex cover $\mathcal{C}^{\prime}$ such that $N_{D}^{+}(z_{2},x)\subseteq\mathcal{C}^{\prime}\subseteq\mathcal{C}$, since $\\{z_{2},x\\}\subseteq V^{+}$. So, $z_{1}\notin\mathcal{C}^{\prime}$, since $z_{1}\notin\mathcal{C}$. Then, by Remark 2.8, $N_{D}(z_{1})\subseteq\mathcal{C}^{\prime}$. Hence, $\\{z_{2},z_{3},z_{4},z_{5}\\}\subseteq N_{D}(z_{1})\cup N_{D}^{+}(z_{2},x)\subseteq\mathcal{C}^{\prime}$. Therefore, $|\mathcal{C}^{\prime}\cap V(C)|=4$, since $z_{1}\notin\mathcal{C}^{\prime}$. $\Box$ ###### Definition 4.4 Let $C$ be an induced $5$-cycle, we say that $C$ has the $\star$-property if for each $(a,b)\in E(C)$ where $a\in V^{+}$, then $C=(a^{\prime},a,b,b^{\prime},c,a^{\prime})$ with the following properties: 1. $(\star.1)$ $(a^{\prime},a)\in E(D)$ and $w(a^{\prime})=1$. 2. $(\star.2)$ $N_{D}^{-}(a)\subseteq N_{D}(c)$ and $N_{D}^{-}(a)\cap V^{+}\subseteq N_{D}^{-}(c)$. 3. $(\star.3)$ $N_{D}(b^{\prime})\subseteq N_{D}(a^{\prime})\cup N_{D}^{+}(a)$ and $N_{D}^{-}(b^{\prime})\cap V^{+}\subseteq N_{D}^{-}(a^{\prime})$. ###### Lemma 4.5 Let $C=(a_{1}^{\prime},a_{1},b_{1},b_{1}^{\prime},c_{1},a_{1}^{\prime})$ be a basic $5$-cycle of $D$, such that $(a_{1}^{\prime},a_{1})\in E(D)$, $deg_{D}(a_{1})\geqslant 3$, $deg_{D}(c_{1})\geqslant 3$ and $w(b_{1})=1$. If there is a strong vertex cover $\mathcal{C}$ of $D$, such that $V(C)\subseteq\mathcal{C}$, then $C$ has no the $\star$-property. Proof. By contradiction, suppose $C$ has the $\star$-property and there is a strong vertex cover $\mathcal{C}$, such that $V(C)\subseteq\mathcal{C}$. Then, $deg_{D}(a_{1}^{\prime})=deg_{D}(b_{1}^{\prime})=2$, since $C$ is a basic cycle, $deg_{D}(a_{1})\geqslant 3$ and $deg_{D}(c_{1})\geqslant 3$. Hence, $a_{1}^{\prime},b_{1}^{\prime}\in L_{3}(\mathcal{C})$, since $V(C)\subseteq\mathcal{C}$. Thus, $(c_{1},a_{1}^{\prime})\in E(D)$ and $w(c_{1})\neq 1$, since $a_{1}^{\prime}\in L_{3}(\mathcal{C})$, $deg_{D}(a_{1}^{\prime})=2$, $(a_{1}^{\prime},a_{1})\in E(D)$ and $\mathcal{C}$ is strong. By ($\star.1$) with $(a,b)=(c_{1},a_{1}^{\prime})$, we have that $(b_{1}^{\prime},c_{1})\in E(D)$. Hence, $N_{D}^{-}(b_{1}^{\prime})\subseteq\\{b_{1}\\}$, since $deg_{D}(b_{1}^{\prime})=2$. This is a contradiction, since $b_{1}^{\prime}\in L_{3}(\mathcal{C})$ and $w(b_{1})=1$. $\Box$ ###### Proposition 4.6 Let $C$ be a basic $5$-cycle, then $C$ has the $\star$-property if and only if $|\mathcal{C}\cap V(C)|=3$ for each strong vertex cover $\mathcal{C}$ of $D$. Proof. $\Rightarrow)$ By contradiction, we suppose there is a strong vertex cover $\mathcal{C}$ such that $|\mathcal{C}\cap V(C)|\geqslant 4$. Thus, there is a path $L=(d_{1},d_{2},d_{3},d_{4})\subseteq C$ such that $V(L)\subseteq\mathcal{C}$. Then, $deg_{D}(d_{2})=2$ or $deg_{D}(d_{3})=2$, since $C$ is basic. We can suppose $deg_{D}(d_{2})=2$, then $N_{D}(d_{2})\subseteq\mathcal{C}$. This implies $b_{1}:=d_{2}\in L_{3}(\mathcal{C})$. So, there is $(a_{1},b_{1})\in E(D)$ with $a_{1}\in\big{(}\mathcal{C}\setminus L_{1}(\mathcal{C})\big{)}\cap V^{+}$, since $\mathcal{C}$ is strong. Since, $N_{D}(b_{1})\subseteq C$, we can set $C=(a_{1}^{\prime},a_{1},b_{1},b_{1}^{\prime},c_{1},a_{1}^{\prime})$. Consequently, $\\{a_{1},b_{1}^{\prime}\\}=N_{D}(b_{1})=N_{D}(d_{2})=\\{d_{1},d_{3}\\}\subseteq\mathcal{C}$. By ($\star.1$), $(a_{1}^{\prime},a_{1})\in E(D)$ and $w(a_{1}^{\prime})=1$. If $b_{1}\in V^{+}$, then by Remark 2.3, $b_{1}$ is not a sink. This implies, $(b_{1},b_{1}^{\prime})\in E(D)$. Then, by ($\star.1$) with $(a,b)=(b_{1},b_{1}^{\prime})$, $w(a_{1})=1$. A contradiction, since $a_{1}\in V^{+}$. Hence, $w(b_{1})=1$. We prove $a_{1}^{\prime}\in\mathcal{C}$. By contradiction assume $a_{1}^{\prime}\not\in\mathcal{C}$, then $\\{b_{1},a_{1},c_{1},b_{1}^{\prime}\\}\subseteq\mathcal{C}$, since $|\mathcal{C}\cap V(C)|\geqslant 4$. Suppose $b_{1}^{\prime}\in L_{3}(\mathcal{C})$, then there is $y\in\big{(}N_{D}^{-}(b_{1}^{\prime})\cap V^{+}\big{)}\setminus L_{1}(\mathcal{C})$. Then, by ($\star.3$) with $(a,b)=(a_{1},b_{1})$, $y\in N_{D}^{-}(a_{1}^{\prime})$, i.e. $(y,a_{1}^{\prime})\in E(D)$. Consequently, $y\in L_{1}(\mathcal{C})$, since $a_{1}^{\prime}\notin\mathcal{C}$. This is a contradiction. Hence, $b_{1}^{\prime}\notin L_{3}(\mathcal{C})$, i.e. there is $y^{\prime}\in N_{D}(b_{1}^{\prime})\setminus\mathcal{C}$, since $b_{1}^{\prime}\in\mathcal{C}$. By ($\star.3$), $y^{\prime}\in N_{D}(a_{1}^{\prime})\cup N_{D}^{+}(a_{1})$. Furthermore, $a_{1}^{\prime}\notin\mathcal{C}$, then $N_{D}(a_{1}^{\prime})\subseteq\mathcal{C}$ and $y^{\prime}\notin N_{D}(a_{1}^{\prime})$, since $\mathcal{C}$ is a vertex cover and $y^{\prime}\notin\mathcal{C}$. This implies $y^{\prime}\in N_{D}^{+}(a_{1})$, then $a_{1}\in L_{1}(\mathcal{C})$, since $a_{1}\in\mathcal{C}$ and $y^{\prime}\notin\mathcal{C}$. A contradiction, since $a_{1}\notin L_{1}(\mathcal{C})$. Therefore, $a_{1}^{\prime}\in\mathcal{C}$. Thus, $\\{b_{1},a_{1},a_{1}^{\prime},b_{1}^{\prime}\\}\subseteq\mathcal{C}$. Now, we prove $c_{1}\in\mathcal{C}$, $deg_{D}(a_{1})\geqslant 3$ and $deg_{D}(c_{1})\geqslant 3$. Case (1) $a_{1}\in L_{3}(\mathcal{C})$. Consequently, there is $z\in N_{D}^{-}(a_{1})\cap V^{+}$ such that $z\in\mathcal{C}\setminus L_{1}(\mathcal{C})$. Then, $z\notin V(C)$, since $N_{D}^{-}(a_{1})\cap V(C)=\\{a_{1}^{\prime}\\}$ and $w(a_{1}^{\prime})=1$. By ($\star.2$), $z\in N_{D}^{-}(c_{1})$. Thus, $(z,c_{1})\in E(D)$. Consequently, $c_{1}\in\mathcal{C}$, $deg_{D}(a_{1})\geqslant 3$ and $deg_{D}(c_{1})\geqslant 3$, since $z\in\mathcal{C}\setminus L_{1}(\mathcal{C})$ and $z\in N_{D}(a_{1})\cap N_{D}(c_{1})$. Case (2) $a_{1}\notin L_{3}(\mathcal{C})$. This implies, there is $z^{\prime}\in N_{D}(a_{1})$ such that $z^{\prime}\notin\mathcal{C}$. Then, $z^{\prime}\notin V(C)$, since $N_{D}(a_{1})\cap V(C)=\\{a_{1}^{\prime},b_{1}\\}\subseteq\mathcal{C}$. Consequently, $z^{\prime}\in N_{D}^{-}(a_{1})$, since $a_{1}\in\mathcal{C}\setminus L_{1}(\mathcal{C})$. By ($\star.2$), we have $z^{\prime}\in N_{D}^{-}(a_{1})\subseteq N_{D}(c_{1})$. Hence, $c_{1}\in\mathcal{C}$, $deg_{D}(a_{1})\geqslant 3$ and $deg_{D}(c_{1})\geqslant 3$, since $z^{\prime}\notin\mathcal{C}$ and $z^{\prime}\in N_{D}(a_{1})\cap N_{D}(c_{1})$. This implies, $V(C)\subseteq\mathcal{C}$. A contradiction, by Lemma 4.5, since $C$ has the $\star$-property. $\Leftarrow)$ Assume $C=(a^{\prime},a,b,b^{\prime},c,a^{\prime})$ with $(a,b)\in E(C)$ such that $w(a)\neq 1$. We take a minimal vertex cover $\mathcal{C}$ of $D$. We will prove ($\star.1$), ($\star.2$) and ($\star.3$). $\mathbf{(\star.1)}$ First we will prove $(a^{\prime},a)\in E(D)$. By contradiction, suppose $(a,a^{\prime})\in E(D)$. By Remark 2.3, there is $y\in N_{D}^{-}(a)$, since $a\in V^{+}$. Thus, $y\notin V(C)$ and $deg_{D}(a)\geq 3$. Consequently, $deg_{D}(a^{\prime})=deg_{D}(b)=2$, since $C$ is basic. Also, $deg_{D}(b^{\prime})=2$ or $deg_{D}(c)=2$, since $C$ is basic. We can assume $deg_{D}(c)=2$, then $N_{D}(c)=\\{a^{\prime},b^{\prime}\\}$. So, by Remark 2.6, $\mathcal{C}_{1}=\big{(}\mathcal{C}\setminus\\{y,c\\}\big{)}\cup N_{D}(y,b)\cup N_{D}^{+}(a)$ is a vertex cover, since $\mathcal{C}$ is a vertex cover, $\\{a^{\prime},b^{\prime}\\}\subseteq N_{D}(b)\cup N_{D}^{+}(a)\subseteq\mathcal{C}_{1}$. Since $deg_{D}(c)=2$, we have $c\not\in N_{D}(y)$. Furthermore, $c\notin N_{D}(b)\cup N_{D}^{+}(a)$, since $C$ is induced. Then, $c\not\in\mathcal{C}_{1}$. Also, $N_{D}(b)=\\{b^{\prime},a\\}$, implies $y\not\in\mathcal{C}_{1}$, since $y\not\in N_{D}^{+}(a)$. By Proposition 3.1 there is a strong vertex cover $\mathcal{C}_{1}^{\prime}$ such that $N_{D}^{+}(a)\subseteq\mathcal{C}_{1}^{\prime}\subseteq\mathcal{C}_{1}$, since $a\in V^{+}$. Thus, $c,y\notin\mathcal{C}_{1}^{\prime}$, since $c,y\notin\mathcal{C}_{1}$. By Remark 2.8, $a^{\prime},b^{\prime},a\in N_{D}(c)\cup N_{D}(y)\subseteq\mathcal{C}_{1}^{\prime}$. Furthermore, $b\in N_{D}^{+}(a)\subseteq\mathcal{C}_{1}^{\prime}$. Hence, $|\mathcal{C}_{1}^{\prime}\cap V(C)|=4$. A contradiction. Now, we prove $w(a^{\prime})=1$. By contradiction, assume $w(a^{\prime})\neq 1$. By the last argument, $(c,a^{\prime})\in E(D)$, since $(a^{\prime},a)\in E(D)$ and $a\in V^{+}$. A contradiction, by (a) in Lemma 4.3. $\mathbf{(\star.2)}$ We will prove $N_{D}^{-}(a)\subseteq N_{D}(c)$. By contradiction, suppose there is $y\in N_{D}^{-}(a)\setminus N_{D}(c)$. Also, $N_{D}^{-}(a)\cap V(C)\subseteq\\{a^{\prime}\\}\subseteq N_{D}(c)$, since $b\in N_{D}^{+}(a)$. Hence, $y\notin V(C)$. By Remark 2.6, $\mathcal{C}_{2}=\big{(}\mathcal{C}\setminus\\{y,c\\}\big{)}\cup N_{D}(y,c)\cup N_{D}^{+}(a)$ is a vertex cover. Furthermore, $y,c\notin\mathcal{C}_{2}$, since $y\in N_{D}^{-}(a)\setminus N_{D}(c)$ and $c\notin N_{D}(a,y)$. By Proposition 3.1, there is a strong vertex cover $\mathcal{C}_{2}^{\prime}$ such that $N_{D}^{+}(a)\subseteq\mathcal{C}_{2}^{\prime}\subseteq\mathcal{C}_{2}$, since $a\in V^{+}$. Thus, $y,c\notin\mathcal{C}_{2}^{\prime}$ since $y,c\notin\mathcal{C}_{2}$. By Remark 2.8, $a,a^{\prime},b^{\prime}\in N_{D}(y,c)\subseteq\mathcal{C}_{2}^{\prime}$. Hence, $|\mathcal{C}_{2}^{\prime}\cap V(C)|=4$, since $b\in N_{D}^{+}(a)\subseteq\mathcal{C}_{2}^{\prime}$. A contradiction. Now, we prove $N_{D}^{-}(a)\cap V^{+}\subseteq N_{D}^{-}(c)$. By contradiction, suppose there is $y\in N_{D}^{-}(a)\cap V^{+}\setminus N_{D}^{-}(c)$. By Remark 2.6, $\mathcal{C}_{3}=(\mathcal{C}\setminus\\{c\\})\cup N_{D}(c)\cup N_{D}^{+}(a,y)$ is a vertex cover. Furthermore, $c\notin N_{D}^{+}(a,y)$, then $c\notin\mathcal{C}_{3}$. By Proposition 3.1, there is a strong vertex cover $\mathcal{C}_{3}^{\prime}$ such that $N_{D}^{+}(a,y)\subseteq\mathcal{C}_{3}^{\prime}\subseteq\mathcal{C}_{3}$ since $\\{a,y\\}\subseteq V^{+}$. So, $c\notin\mathcal{C}_{3}^{\prime}$, since $c\notin\mathcal{C}_{3}$. Thus, by Remark 2.8 $a^{\prime},b^{\prime}\in N_{D}(c)\subseteq\mathcal{C}_{3}^{\prime}$. Also, $a,b\in N_{D}^{+}(a,y)\subseteq\mathcal{C}_{3}^{\prime}$. Hence, $|\mathcal{C}^{\prime}_{3}\cap V(C)|=4$, a contradiction. $\mathbf{(\star.3)}$ We prove $N_{D}(b^{\prime})\subseteq N_{D}(a^{\prime})\cup N_{D}^{+}(a)$. By contradiction, we suppose there is $y\in N_{D}(b^{\prime})\setminus\big{(}N_{D}(a^{\prime})\cup N_{D}^{+}(a)\big{)}$. Thus, $y\notin C$, since $N_{D}(b^{\prime})\cap V(C)=\\{c,b\\}\subseteq N_{D}(a^{\prime})\cup N_{D}^{+}(a)$. By Remark 2.6, $\mathcal{C}_{4}=\big{(}\mathcal{C}\setminus\\{y,a^{\prime}\\})\big{)}\cup N_{D}(y,a^{\prime})\cup N_{D}^{+}(a)$. Furthermore, $y\notin\mathcal{C}_{4}$, since $y\notin N_{D}(a^{\prime})\cup N_{D}^{+}(a)$. By ($\star.1$), $(a^{\prime},a)\in E(D)$, then $a^{\prime}\notin\mathcal{C}_{4}$, since $a^{\prime}\notin N_{D}(y)\cup N_{D}^{+}(a)$. By Proposition 3.1, there is a strong vertex cover $\mathcal{C}_{4}^{\prime}$ such that $N_{D}^{+}(a)\subseteq\mathcal{C}_{4}^{\prime}\subseteq\mathcal{C}_{4}$, since $a\in V^{+}$. So, $y,a^{\prime}\notin\mathcal{C}_{4}^{\prime}$, since $y,a^{\prime}\notin\mathcal{C}_{4}$. Thus, by Remark 2.8 $b^{\prime},a,c\in N_{D}(y)\cup N_{D}(a^{\prime})\subseteq\mathcal{C}_{4}^{\prime}$. Also, $b\in N_{D}^{+}(a)\subseteq\mathcal{C}_{4}^{\prime}$. Hence, $|\mathcal{C}_{4}^{\prime}\cap V(C)|=4$, a contradiction. Finally, we prove $N_{D}^{-}(b^{\prime})\cap V^{+}\subseteq N_{D}^{-}(a^{\prime})$. By contradiction, we suppose there is $y\in\big{(}N_{D}^{-}(b^{\prime})\cap V^{+}\big{)}\setminus N_{D}^{-}(a^{\prime})$. By $(\star.1)$, $a^{\prime}\in N_{D}^{-}(a)$. Furthermore, by (a) in Lemma 4.3, $y\neq b$, since $y\in V^{+}$. If $y=c$, then $(c,b^{\prime})\in E(D)$. Thus, by $(\star.1)$, with the edge $(a^{\prime},c)\in E(D)$. A contradiction by (b) in Lemma 4.3, since $c=y\in V^{+}$. Hence, $y\notin V(C)$. By Remark 2.6, $\mathcal{C}_{5}=\big{(}\mathcal{C}\setminus\\{a^{\prime}\\}\big{)}\cup N_{D}(a^{\prime})\cup N_{D}^{+}(y,a)$ is a vertex cover. By Remark 2.6, $a^{\prime}\notin N_{D}^{+}(y,a)$, since $(a^{\prime},a)\in E(D)$ and $y\notin N_{D}^{-}(a^{\prime})$. Consequently, $a^{\prime}\notin\mathcal{C}_{5}$. By Proposition 3.1, there is a strong vertex cover $\mathcal{C}_{5}^{\prime}$ such that $N_{D}^{+}(a,y)\subseteq\mathcal{C}_{5}^{\prime}\subseteq\mathcal{C}_{5}$, since $\\{a,y\\}\subseteq V^{+}$. So, $a^{\prime}\notin\mathcal{C}_{5}^{\prime}$, since $a^{\prime}\notin\mathcal{C}_{5}$. Then, by Remark 2.8 $a,c\in N_{D}(a^{\prime})\subseteq\mathcal{C}_{5}^{\prime}$. Furthermore, $b,b^{\prime}\in N_{D}^{+}(a,y)\subseteq\mathcal{C}_{5}^{\prime}$. Hence, $|\mathcal{C}_{5}^{\prime}\cap V(C)|=4$, a contradiction. $\Box$ ###### Lemma 4.7 Let $\mathcal{C}$ be a vertex cover of $D$ where $G$ is an $SCQ$ graph. Hence, $|\mathcal{C}|=\tau(G)$ if and only if $|\mathcal{C}\cap V(K)|=|V(K)|-1$, $|\mathcal{C}\cap V(C)|=3$ and $|\mathcal{C}\cap e|=1$ for each $K\in S_{G}$, $C\in C_{G}$ and $e\in Q_{G}$, respectively. Proof. We set $\mathcal{C}$ a vertex cover of $D$, $K\in S_{G}$, $C\in C_{G}$ and $e\in Q_{G}$. Then, there are $y\in V(G)$ and $a,a^{\prime}\in V(C)$ such that $K=G[N_{G}[y]]$, $deg_{G}(a)=deg_{G}(a^{\prime})=2$ and $\\{a,a^{\prime}\\}\notin E(G)$. We set $A_{K}:=V(K)\setminus\\{y\\}$ and $B_{C}:=V(C)\setminus\\{a,a^{\prime}\\}$. Also, $\mathcal{C}\cap V(K)$ is a vertex cover of $K$, so $|\mathcal{C}\cap V(K)|\geqslant\tau(K)=|V(K)|-1$. Similarly, $|\mathcal{C}\cap V(C)|\geqslant\tau(C)=3$ and $|\mathcal{C}\cap e|\geqslant\tau(e)=1$. Thus, $|\mathcal{C}|=\sum\limits_{K\in S_{G}}|\mathcal{C}\cap V(K)|+\sum\limits_{C\in C_{G}}|\mathcal{C}\cap V(C)|+\sum\limits_{e\in Q_{G}}|\mathcal{C}\cap e|\geqslant\sum\limits_{K\in S_{G}}\big{(}|V(K)|-1\big{)}+3|C_{G}|+|Q_{G}|,$ (4.1) since $\mathcal{H}=\\{V(H)\mid H\in S_{G}\cup C_{G}\cup Q_{G}\\}$ is a partition of $V(G)$. Now, we take a maximal stable set $S$ contained in $V(Q_{G}):=\\{x\in V(G)\mid x\in e\ {\rm and}\ e\in Q_{G}\\}$. Then, $|S\cap e|\leqslant 1$ for each $e\in Q_{G}$, since $S$ is stable. If $S\cap e=\emptyset$ for some $e=\\{x_{1},x_{2}\\}\in Q_{G}$, then there are $y_{1},y_{2}\in S$ such that $\\{x_{1},y_{1}\\}$, $\\{x_{2},y_{2}\\}\in E(G)$, since $S$ is maximal. But $Q_{G}$ satisfies the property (P), then $\\{y_{1},y_{2}\\}\in E(G)$. A contradiction, since $S$ is stable. Hence, $|S\cap e|=1$ for each $e\in Q_{G}$. Consequently, $|S|=|Q_{G}|$ and $|S^{\prime}|=|Q_{G}|$, where $S^{\prime}=V(Q_{G})\setminus S$. Now, we take $\mathcal{C}(S^{\prime})=\Big{(}\bigcup\limits_{K\in S_{G}}A_{K}\Big{)}\bigcup\Big{(}\bigcup\limits_{C\in C_{G}}B_{C}\Big{)}\bigcup S^{\prime}$. We prove $\mathcal{C}(S^{\prime})$ is a vertex cover of $D$. By contradiction, suppose there is $\hat{e}\in E(G)$ such that $\hat{e}\cap\mathcal{C}(S^{\prime})=\emptyset$. We set $z\in\hat{e}$, then $\hat{e}=\\{z,z^{\prime}\\}$. If $z\in V(\tilde{K})$ for some $\tilde{K}\in S_{G}$, then $\tilde{K}=G[N_{G}[z]]$, since $A_{\tilde{K}}\subseteq\mathcal{C}(S^{\prime})$ and $z\notin\mathcal{C}(S^{\prime})$. So, $z^{\prime}\in N_{G}(z)\subseteq\tilde{K}\setminus\\{z\\}=A_{\tilde{K}}\subseteq\mathcal{C}(S^{\prime})$. A contradiction, since $\hat{e}\cap\mathcal{C}(S^{\prime})=\emptyset$. Now, if $z\in V(\tilde{C})$ for some $\tilde{C}\in C_{G}$, then $z\notin B_{\tilde{C}}$. Thus, $deg_{G}(z)=2$ implying $z^{\prime}\in B_{\tilde{C}}\subseteq\mathcal{C}(S^{\prime})$, since $\\{z,z^{\prime}\\}\in E(G)$. A contradiction. Then, $\hat{e}\subseteq V(Q_{G})$, since $\mathcal{H}$ is a partition of $V(G)$. Also, $\hat{e}\cap S^{\prime}=\emptyset$, this implies $\hat{e}\subseteq V(Q_{G})\setminus S^{\prime}=S$. But $S$ is stable. This is a contradiction. Hence, $\mathcal{C}(S^{\prime})$ is a vertex cover of $D$. Furthermore, $|\mathcal{C}(S^{\prime})|=\sum\limits_{K\in S_{G}}|A_{K}|+\sum\limits_{C\in C_{G}}|B_{C}|+|S^{\prime}|=\sum\limits_{K\in S_{G}}\big{(}|V(K)|-1\big{)}+3|C_{G}|+|Q_{G}|$. Thus, $\tau(G)=\sum_{K\in S_{G}}\big{(}|V(K)|-1\big{)}+3|C_{G}|+|Q_{G}|$. Therefore, by (4.1), $|\mathcal{C}|=\tau(G)$ if and only if $|\mathcal{C}\cap V(K)|=|K|-1$, $|\mathcal{C}\cap V(C)|=3$ and $|\mathcal{C}\cap e|=1$ for each $K\in S_{G}$, $C\in C_{G}$ and $e\in Q_{G}$, respectively. $\Box$ ###### Theorem 4.8 Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph where $G$ is an $SCQ$ graph. Hence, $I(D)$ is unmixed if and only if $D$ satisfies the following conditions: 1. (a) Each basic $5$-cycle of $G$ has the $\star$-property. 2. (b) Each simplex of $D$ has no generating $\star$-semi-forests. 3. (c) $N_{D}(b)\subseteq N_{D}^{+}(a)$ when $a\in V^{+}$, $\\{b,b^{\prime}\\}\in Q_{G}$ and $b^{\prime}\in N_{D}^{+}(a)$. Proof. $\Rightarrow)$ We take a strong vertex cover $\mathcal{C}$ of $D$, then by Remark 2.14, $|\mathcal{C}|=\tau(G)$. Consequently, by Lemma 4.7, $|\mathcal{C}\cap V(K)|=|V(K)|-1$, $|\mathcal{C}\cap V(C)|=3$ and $|\mathcal{C}\cap e|=1$ for each $K\in S_{G}$, $C\in C_{G}$ and $e\in Q_{G}$. Thus, $V(K)\not\subseteq\mathcal{C}$. Consequently, by Theorem 3.10, $D$ satisfies (b). Furthermore, by Propositions 4.6 and 4.1, $D$ satisfies (a) and (c). $\Leftarrow)$ Let $\mathcal{C}$ be a strong vertex cover of $D$. By (a) and Proposition 4.6, we have $|\mathcal{C}\cap V(C)|=3$ for each $C\in C_{G}$. Furthermore, by (b) and Theorem 3.10, $V(K)\not\subseteq\mathcal{C}$ for each $K\in S_{G}$. Consequently, $|V(K)|>|\mathcal{C}\cap V(K)|\geqslant\tau(K)=|V(K)|-1$. So, $|\mathcal{C}\cap V(K)|=|V(K)|-1$. Now, if $e\in Q_{G}$, then $e$ has the property (P), since $Q_{G}$ has the property (P). Thus, by (c) and Proposition 4.1, $|\mathcal{C}\cap e|=1$. Hence, by Lemma 4.7, $|\mathcal{C}|=\tau(G)$. Therefore $I(D)$ is unmixed, by $(2)$ in Theorem 2.12. $\Box$ ###### Corollary 4.9 Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph where $G$ is a simplicial or chordal graph. Hence, $I(D)$ is unmixed if and only if $D$ satisfies the following conditions: 1. (a) Each vertex is in exactly one simplex of $D$. 2. (b) Each simplex of $D$ has not a generating $\star$-semi-forest. Proof. $\Rightarrow)$ By $(3)$ in Theorem 2.12 and Remark 2.17, $G$ is well- covered. Thus, by Theorem 2.26, $G$ satisfies (a). Furthermore, by Remark 2.29, $G$ is an $SCQ$ graph with $C_{G}=Q_{G}=\emptyset$. Hence, by Theorem 4.8, $D$ satifies (b). $\Leftarrow)$ By (a), $\\{V(H)\mid H\in S_{G}\\}$ is a partition of $V(G)$. Hence, $G$ is an $SCQ$ graph with $C_{G}=\emptyset$ and $Q_{G}=\emptyset$. Therefore, by (b) and Theorem 4.8, $I(D)$ is unmixed. $\Box$ ## 5 Unmixedness of weighted oriented graphs without some small cycles Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph. In this Section, we study and characterize the unmixed property of $I(D)$ when $G$ has no $3$\- or $5$\- cycles (Theorem 5.4), or $G$ is a graph without $4$\- or $5$-cycles (Theorem 5.10), or $girth(G)\geqslant 5$ (Theorem 5.13). In other words, in this Section, we characterize the unmixed property of $I(D)$ when $G$ has at most one of the following types of cycles: $3$-cycles, $4$-cycles and $5$-cycles. ###### Proposition 5.1 If for each $(y,x)\in E(D)$ with $y\in V^{+}$, we have that $N_{D}(y^{\prime})\subseteq N_{D}^{+}(y)$ for some $y^{\prime}\in N_{D}(x)\setminus y$, then $L_{3}(\mathcal{C})=\emptyset$ for each strong vertex cover $\mathcal{C}$ of $D$. Proof. By contradiction, suppose there is a strong vertex cover $\mathcal{C}$ of $D$ and $x\in L_{3}(\mathcal{C})$. Hence, there is $y\in\big{(}\mathcal{C}\setminus L_{1}(\mathcal{C})\big{)}\cap V^{+}$ with $(y,x)\in E(D)$. Then, $N_{D}(x)\subseteq\mathcal{C}$ and $N_{D}^{+}(y)\subseteq\mathcal{C}$, since $x\in L_{3}(\mathcal{C})$ and $y\in\mathcal{C}\setminus L_{1}(\mathcal{C})$. By hypothesis, there is a vertex $y^{\prime}\in N_{D}(x)\setminus y\subseteq\mathcal{C}$ such that $N_{D}(y^{\prime})\subseteq N_{D}^{+}(y)\subseteq\mathcal{C}$. Thus, $y^{\prime}\in L_{3}(\mathcal{C})$. Since $\mathcal{C}$ is strong, there is $(y_{1},y^{\prime})\in E(D)$ with $y_{1}\in V^{+}$. So, $y_{1}\in N_{D}(y^{\prime})\subseteq N_{D}^{+}(y)$. On the other hand, $(y_{1},x_{1})\in E(D)$ where $x_{1}:=y^{\prime}$ and $y_{1}\in V^{+}$, then by hypothesis, there is $y_{1}^{\prime}\in N_{D}(x_{1})\setminus y_{1}$ such that $N_{D}(y_{1}^{\prime})\subseteq N_{D}^{+}(y_{1})$. Hence, $y_{1}^{\prime}\in N_{D}(x_{1})=N_{D}(y^{\prime})\subseteq N_{D}^{+}(y)$. Consequently, $y\in N_{D}(y_{1}^{\prime})\subseteq N_{D}^{+}(y_{1})$. A contradiction, since $y_{1}\in N_{D}^{+}(y)$. $\Box$ ###### Corollary 5.2 If $G$ is well-covered and $V^{+}$ is a subset of sinks, then $I(D)$ is unmixed. Proof. If $y\in V^{+}$, then $y$ is a sink. Thus, $(y,x)\notin E(D)$ for each $x\in V(D)$. Hence, by Proposition 5.1, $L_{3}(\mathcal{C})=\emptyset$, for each strong vertex cover $\mathcal{C}$ of $D$. Furthermore, by Remark 2.17, $I(G)$ is unmixed. Therefore $I(D)$ is unmixed, by $(3)$ in Theorem 2.12. $\Box$ ###### Lemma 5.3 Let $(z,y),(y,x)$ be edges of $D$ with $y\in V^{+}$ and $N_{D}(x)=\\{y,x_{1},\ldots,x_{s}\\}$. If there are $z_{i}\in N_{D}(x_{i})\setminus N_{D}^{+}(y)$ such that $\\{z,x,z_{1},\ldots,z_{s}\\}$ is a stable set, then $I(D)$ is mixed. Proof. We take $A:=\\{z,z_{1},\ldots,z_{s}\\}$, then $A\cup\\{x\\}$ is a stable set. We can take a maximal stable set $S$ of $V(G)$, such that $A\cup\\{x\\}\subseteq S$. So, $\tilde{\mathcal{C}}=V(G)\setminus S$ is a minimal vertex cover of $D$. Hence, $\mathcal{C}=\tilde{\mathcal{C}}\cup N_{D}^{+}(y)$ is a vertex cover of $D$. Also $A\cap\mathcal{C}=\emptyset$, since $A\subseteq S$, $z\in N_{D}^{-}(y)$ and $z_{i}\notin N_{D}^{+}(y)$. By Proposition 3.1, there is a strong vertex cover $\mathcal{C}^{\prime}$ of $D$ such that $N_{D}^{+}(y)\subseteq\mathcal{C}^{\prime}\subseteq\mathcal{C}$, since $y\in V^{+}$. Thus, $A\cap\mathcal{C}^{\prime}=\emptyset$, since $A\cap\mathcal{C}=\emptyset$. Then, by Remark 2.8, $N_{D}(A)\subseteq\mathcal{C}^{\prime}$. Furthermore $N_{D}(x)=\\{y,x_{1},\ldots,x_{s}\\}\subseteq N_{D}(A)$. Consequently, $N_{D}(x)\subseteq\mathcal{C}^{\prime}$. Hence, $x\in L_{3}(\mathcal{C}^{\prime})$, since $x\in N_{D}^{+}(y)\subseteq\mathcal{C}^{\prime}$. Therefore, by (3) in Theorem 2.12, $I(D)$ is mixed. $\Box$ ###### Theorem 5.4 Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph such that $G$ has no $3$\- or $5$-cycles. Hence, $I(D)$ is unmixed if and only if $D$ satisfies the following conditions: 1. (a) $G$ is well-covered. 2. (b) If $(y,x)\in E(D)$ with $y\in V^{+}$, then $N_{D}(y^{\prime})\subseteq N_{D}^{+}(y)$ for some $y^{\prime}\in N_{D}(x)\setminus y$. Proof. $\Leftarrow)$ By Proposition 5.1 and (b), we have that $L_{3}(\mathcal{C})=\emptyset$ for each strong vertex cover $\mathcal{C}$ of $D$. Furthermore, by (a) and Remark 2.17, $I(G)$ is unmixed. Therefore, by $(3)$ in Theorem 2.12, $I(D)$ is unmixed. $\Rightarrow)$ By (3) in Theorem 2.12 and Remark 2.17, $D$ satisfies (a). Now, we take $(y,x)\in E(D)$ with $y\in V^{+}$. Then, by Remark 2.3, there is $z\in N_{D}^{-}(y)$. Furthermore $z\notin N_{D}(x)$, since $G$ has no $3$-cycles. We set $N_{D}(x)\setminus y=\\{x_{1},\ldots,x_{s}\\}$. We will prove (b). By contradiction, suppose there is $z_{i}\in N_{D}(x_{i})\setminus N_{D}^{+}(y)$ for each $i=1,\ldots,s$. If $\\{z_{i},z_{j}\\}\in E(G)$ for some $1\leqslant i<j\leqslant s$, then $(x,x_{i},z_{i},z_{j},x_{j},x)$ is a $5$-cycle. But $G$ has no $5$-cycles, then $\\{z_{1},\ldots,z_{s}\\}$ is a stable set. Now, if $\\{x,z_{k}\\}\in E(G)$ or $\\{z,z_{k}\\}\in E(G)$ for some $k\in\\{1,\ldots,s\\}$, then $(x,x_{k},z_{k},x)$ is a $3$-cycle or $(z,y,x,x_{k},z_{k},z)$ is a $5$-cycle. Hence, $\\{x,z,z_{1},\ldots,z_{s}\\}$ is a stable set. A contradiction, by Lemma 5.3, since $I(D)$ is unmixed. $\Box$ In the following results, we use the notation of Figure 1. ###### Remark 5.5 Let $G$ be a graph in $\\{C_{7},T_{10},P_{10},P_{13},P_{14},Q_{13}\\}$. Hence, 1. (a) $G$ does not contain $4$-cycles. Furthermore, if $G$ has a $3$-cycle, then $G=T_{10}$. 2. (b) If $deg_{G}(x)=2$, then $x$ is not in a $3$-cycle of $G$. 3. (c) If $G\neq C_{7}$ and $\tilde{e}=\\{v,u\\}\in E(G)$ with $deg_{D}(v)=deg_{D}(u)=2$, then $\tilde{e}\in\\{\tilde{e}_{1},\tilde{e}_{2},\tilde{e}_{3}\\}$. Also, if $\tilde{e}$ is in a $5$-cycle $C$, then $G\in\\{P_{10},P_{13}\\}$, $\tilde{e}\in\\{\tilde{e}_{1},\tilde{e}_{2}\\}$ and $C\in\\{C_{1},C_{2}\\}$ or $G=Q_{13}$, $\tilde{e}=\tilde{e}_{1}$ and $C=C_{1}$. 4. (d) If $P=(y_{1},y_{2},y_{3})$ is a path in $G$ with $deg_{G}(y_{i})=2$ for $i=1,2,3$, then $G=C_{7}$. Proof. (a) By Theorems 2.28 and 2.30, $G$ has no $4$-cycles. Now, if $G$ has a $3$-cycle then, by Theorem 2.30, $G=T_{10}$. (b) By (a), the unique $3$-cycle is $(c_{1},c_{2},c_{3},c_{1})$ in $T_{10}$ and $deg_{T_{10}}(c_{i})=3$ for $i=1,2,3$. (c) By Figure 1, $\tilde{e}\in\\{\tilde{e}_{1},\tilde{e}_{2},\tilde{e}_{3}\\}$ and $G\in\\{T_{10},P_{10},P_{13},Q_{13}\\}$, since $G\neq C_{7}$. Now, assume $\tilde{e}$ is in a $5$-cycle $C$. By Theorem 2.28, $G\neq T_{10}$. If $G=P_{10}$, then $\tilde{e}_{3}$ is not in a $5$-cycle. Thus, $\tilde{e}\in\\{\tilde{e}_{1},\tilde{e}_{2}\\}$ and $C\in\\{C_{1},C_{2}\\}$. Now, if $G=P_{13}$, then $\tilde{e}\in\\{\tilde{e}_{1},\tilde{e}_{2}\\}$ and $C\in\\{C_{1},C_{2}\\}$. Finally, if $G=Q_{13}$, then $\tilde{e}_{2},\tilde{e}_{3}$ are not in a $5$-cycle. Hence, $\tilde{e}=\tilde{e}_{1}$ and $C=C_{1}$. (d) By contradiction, suppose $G\neq C_{7}$. If $e\in E(P)$, then by (c), $e\in\\{e_{1},e_{2},e_{3}\\}$. But, $e_{i}\cap e_{j}=\emptyset$ for $i\neq j$. A contradiction, since $P$ is a path. $\Box$ ###### Lemma 5.6 Let $G$ be a graph in $\\{C_{7},T_{10},P_{10},P_{13},P_{14},Q_{13}\\}$ with $I(D)$ unmixed. If $(z,y),(y,x)\in E(D)$ with $y\in V^{+}$ and $N_{D}(x)\setminus\\{y\\}=\\{x_{1}\\}$, then $deg_{D}(x_{1})=2$. Proof. By contradiction, suppose $deg_{D}(x_{1})\geqslant 3$. Hence, there are $z_{1},z_{1}^{\prime}\in N_{D}(x_{1})\setminus\\{x\\}$. By hypothesis, $deg_{D}(x)=2$. Then, by (b) in Remark 5.5, $x$ is not in a $3$-cycle. So, $x_{1}\neq z$. Furthermore, by (a) in Remark 5.5, $G$ has no $4$-cycles. Thus, $z\notin\\{z_{1}^{\prime},z_{1}\\}$ and $z_{1},z_{1}^{\prime}\notin N_{D}(y)$. If $z_{1}^{\prime},z_{1}\in N_{D}(z)$, then $(x_{1},z_{1}^{\prime},z,z_{1},x_{1})$ is a $4$-cycle. A contradiction, then we can assume $z_{1}\notin N_{D}(z)$. Consequently, $\\{x,z,z_{1}\\}$ is a stable set, since $x$ is not in a $3$-cycle. A contradiction, by Lemma 5.3, since $I(D)$ is unmixed. $\Box$ ###### Lemma 5.7 If $I(D)$ is unmixed, $G\in\\{C_{7},T_{10},P_{10},P_{13},P_{14},Q_{13}\\}$ and $\tilde{e}=(y,x)\in E(D)$ with $deg_{D}(x)=2$ and $y\in V^{+}$, then $G=P_{10}$ and $\tilde{e}=(d_{i},b_{j})$ with $\\{i,j\\}=\\{1,2\\}$. Proof. By Remark 2.3, there is $z\in N_{D}^{-}(y)$. We set $N_{D}(x)=\\{y,x_{1}\\}$, then by (b) in Remark 5.5, $z\neq x_{1}$. Thus, by Lemma 5.6, $deg_{D}(x_{1})=2$. Now, we set $N_{D}(x_{1})=\\{x,z_{1}\\}$. By (b) in Remark 5.5, $x$ is not in a $3$-cycle. So, $z,z_{1}\notin N_{D}(x)$ and $z_{1}\neq y$. Also, by (a) in Remark 5.5, $G$ has no $4$-cycles. Then, $z_{1}\neq z$ and $z_{1}\notin N_{D}(y)$. If $z\notin N_{D}(z_{1})$, then $\\{x,z,z_{1}\\}$ is a stable set. A contradiction, by Lemma 5.3, since $I(D)$ is unmixed. Hence, $\\{z_{1},z\\}\in E(G)$ and $C:=(z,y,x,x_{1},z_{1},z)$ is a $5$-cycle. Suppose $y^{\prime}\in N_{D}^{-}(y)\setminus\\{z\\}$, then $y^{\prime}\notin N_{D}(z_{1})$, since $(z_{1},z,y,y^{\prime},z_{1})$ is not in a $4$-cycle in $G$. Consequently, $\\{y^{\prime},x,z_{1}\\}$ is a stable set, since $deg_{D}(x)=2$. A contradiction, by Lemma 5.3, since $(y^{\prime},y),(y,x)\in E(D)$, $y\in V^{+}$, $N_{D}(x)=\\{y,x_{1}\\}$ and $z_{1}\in N_{D}(x_{1})\setminus N_{D}^{+}(y)$. Hence, $N_{D}^{-}(y)=\\{z\\}$. Now, by (c) in Remark 5.5 and by symmetry of $P_{10}$ and $P_{13}$, we can assume $\\{x,x_{1}\\}=\tilde{e}_{1}$, $C=C_{1}$ and $G\in\\{P_{10},P_{13},Q_{13}\\}$. First, assume $G=P_{13}$. By symmetry and notation of Figure 1, we can suppose $x_{1}=a_{1}$ and $x=a_{2}$, since $\tilde{e}_{1}=\\{x,x_{1}\\}$. Then, $y=b_{2}$, $z=c_{1}$ and $z_{1}=b_{1}$. Thus, $(b_{2},d_{2})\in E(D)$, since $y=b_{2}$ and $N_{D}^{-}(y)=\\{z\\}=\\{c_{1}\\}$. By Figure 1, $(c_{1},b_{2})=(z,y)\in E(D)$, $(b_{2},d_{2})\in E(D)$, $b_{2}=y\in V^{+}$ and $N_{D}(d_{2})=\\{b_{2},b_{4},v\\}$. Also, $a_{4}\in N_{D}(b_{4})$, $d_{1}\in N_{D}(v)\setminus N_{D}(b_{2})$ and $\\{c_{1},d_{2},a_{4},d_{1}\\}$ is a stable set. A contradiction, by Lemma 5.3, since $I(D)$ is unmixed. Now, suppose $G=Q_{13}$. By the symmetry of we can suppose $x=a_{2}$ and $x_{1}=a_{1}$, then $d_{2}=y\in V^{+}$, $z=h$ and $(h,d_{2})=(z,y)\in E(D)$. So, $(d_{2},c_{2})\in E(D)$, since $N_{D}^{-}(d_{2})=N_{D}^{-}(y)=\\{z\\}=\\{h\\}$. A contradiction, by Lemma 5.3, since $(h,d_{2}),(d_{2},c_{2})\in E(D)$, $d_{2}\in V^{+}$, $N_{D}(c_{2})=\\{d_{2},b_{1}\\}$, $g_{1}\in N_{D}(b_{1})\setminus N_{D}^{+}(d_{2})$ and $\\{h,c_{2},g_{1}\\}$ is a stable set. Hence, $G=P_{10}$ and $\\{x,x_{1}\\}=\tilde{e}_{1}=\\{a_{1},b_{1}\\}$. If $x=a_{1}$ and $x_{1}=b_{1}$, then $g_{1}=y\in V^{+}$, $(d_{1},g_{1})=(z,y)\in E(D)$, since $C=C_{1}$. Furthermore, $(g_{1},c_{1})\in E(D)$, since $N_{D}^{-}(g_{1})=N_{D}^{-}(y)=\\{z\\}=\\{d_{1}\\}$. A contradiction by Lemma 5.3, since $(d_{1},g_{1}),(g_{1},c_{1})\in E(D)$, $g_{1}\in V^{+}$, $N_{D}(c_{1})=\\{g_{1},c_{2}\\}$, $g_{2}\in N_{D}(c_{2})$ and $\\{d_{1},c_{1},g_{2}\\}$ is stable. Therefore, $x=b_{1}$ and $x_{1}=a_{1}$, implying $y=d_{2}$ and $(y,x)=(d_{2},b_{1})$, since $C=C_{1}$. $\Box$ ###### Remark 5.8 Assume $I(D)$ is unmixed, $G\in\\{C_{7},T_{10},Q_{13},P_{13},P_{14}\\}$, $\mathcal{C}$ is a strong vertex cover of $D$ and $y\in\mathcal{C}\cap V^{+}$ such that $N_{G}(y)\setminus\mathcal{C}\subseteq V_{2}:=\\{a\in V(G)\mid deg_{G}(a)=2\\}$. We take $b\in N_{G}(y)\setminus\mathcal{C}$. If $(y,b)\in E(D)$, then by Lemma 5.7, $G=P_{10}$. A contradiction, then $(b,y)\in E(D)$. Consequently, $N_{D}(y)\setminus\mathcal{C}\subseteq N_{D}^{-}(y)$, i.e. $N_{D}^{+}(y)\subseteq\mathcal{C}$. Hence, $y\in\mathcal{C}\setminus L_{1}(\mathcal{C})$. ###### Proposition 5.9 If $I(D)$ is unmixed, with $G\in\\{C_{7},T_{10},Q_{13},P_{13},P_{14}\\}$, then the vertices of $V^{+}$ are sinks. Proof. By contradiction, suppose there is $(y,x)\in E(D)$ with $y\in V^{+}$. Then, by Lemma 5.7, $deg_{D}(x)\geqslant 3$. Thus, $G\neq C_{7}$. By Remark 2.3, $y$ is not a source. So, there is $(z,y)\in E(D)$. We set $V_{2}:=\\{a\in V(G)\mid deg_{G}(a)=2\\}$. By Theorem 2.12, $L_{3}(\tilde{\mathcal{C}})=\emptyset$ for each strong vertex cover $\tilde{\mathcal{C}}$ of $D$, since $I(D)$ is unmixed. Hence, to obtain a contradiction, we will give a vertex cover $\mathcal{C}$ of $D$ such that $L_{3}(\mathcal{C})=\\{x\\}$ and $y\in\mathcal{C}\setminus L_{1}(\mathcal{C})$, since with these conditions $\mathcal{C}$ is strong. We will use the notation of Figure 1. Case (1) If $D=T_{10}$, then $x\in\\{c_{1},c_{2},c_{3},v\\}$, since $deg_{G}(x)\geqslant 3$. Case (1.a) $x\in\\{c_{1},c_{2},c_{3}\\}$. By symmetry of $P_{10}$, we can assume $x=c_{1}$ and $y\in\\{b_{1},c_{2}\\}$. Thus, $\mathcal{C}_{1}=\\{v,a_{2},a_{3},b_{1},c_{1},c_{2},c_{3}\\}$ is a vertex cover with $L_{3}(\mathcal{C}_{1})=\\{x\\}$. If $y=b_{1}$, then $y\in C_{1}$ and $z=a_{1}$, since $N_{D}(b_{1})=\\{a_{1},c_{1}\\}$. So, $N_{D}^{+}(y)=\\{c_{1}\\}\subseteq\mathcal{C}_{1}$. Now, if $y=c_{2}$, then $y\in\mathcal{C}_{1}$. Furthermore, by Lemma 5.7, $(b_{2},c_{2})\in E(D)$, since $c_{2}=y\in V^{+}$, $b_{2}\in V_{2}$ and $I(D)$ is unmixed. Consequently, $N_{D}^{+}(y)\subseteq\\{c_{1},c_{3}\\}\subseteq\mathcal{C}_{1}$. Hence, $y\in\mathcal{C}_{1}\setminus L_{1}(\mathcal{C}_{1})$ and we take $\mathcal{C}=\mathcal{C}_{1}$. Case (1.b) $x=v$. Then, $\mathcal{C}_{2}=\\{v,a_{1},a_{2},a_{3},c_{1},c_{2},c_{3}\\}$ is a vertex cover with $L_{3}(\mathcal{C}_{2})=\\{x\\}$. By symmetry of $P_{10}$, we can suppose $y=a_{1}$. Consequently, $z=b_{1}$ and $N_{D}^{+}(y)=\\{v\\}\subseteq\mathcal{C}_{2}$, since $a_{1}\in V_{2}$. Hence, $y\in\mathcal{C}_{2}\setminus L_{1}(\mathcal{C}_{2})$ and we take $\mathcal{C}=\mathcal{C}_{2}$. Case (2) If $D=P_{14}$, then, by symmetry, we can assume $y=a_{1}$ and $x\in\\{a_{2},b_{1}\\}$. Case (2.a) $x=a_{2}$. Thus, $z\in\\{a_{7},b_{1}\\}$. We take $\mathcal{C}_{3}=\\{a_{1},a_{2},a_{3},a_{4},a_{6},b_{1},b_{2},b_{5},b_{6},b_{7}\\}$ if $z=a_{7}$ or $\mathcal{C}_{3}=\\{a_{1},a_{2},a_{3},a_{5},a_{7},b_{2},b_{3},b_{4},b_{5},b_{6}\\}$ if $z=b_{1}$. So, $\mathcal{C}_{3}$ is a vertex cover of $D$, $y\in\mathcal{C}_{3}$ and $L_{3}(\mathcal{C}_{3})=\\{x\\}$. Furthermore, $N_{D}(y)\setminus\mathcal{C}_{3}=\\{z\\}$ and $z\in N_{D}^{-}(y)$, then $y\in\mathcal{C}_{3}\setminus L_{1}(\mathcal{C}_{3})$ and we take $\mathcal{C}=\mathcal{C}_{3}$. Case (2.b) $x=b_{1}$. By symmetry of $P_{14}$, we can suppose $z=a_{2}$. Then, $\mathcal{C}_{4}=\\{a_{1},a_{3},a_{4},a_{5},a_{7},b_{1},b_{2},b_{3},b_{6},b_{7}\\}$ is a vertex cover of $D$ with $L_{3}(\mathcal{C}_{4})=\\{x\\}$. Also, $N_{D}(y)=\setminus\mathcal{C}_{4}=\\{a_{2}\\}$ and $a_{2}=z\in N_{D}^{-}(y)$. Hence, $y\in\mathcal{C}_{4}\setminus L_{1}(\mathcal{C}_{4})$ and we take $\mathcal{C}=\mathcal{C}_{4}$. Case (3) If $D=P_{13}$, then we can assume $x\in\\{b_{1},c_{2},d_{1}\\}$, since $deg_{G}(x)\geqslant 3$. Case (3.a) $x=c_{2}$. Then, $y\in N_{D}(c_{2})=\\{b_{3},b_{4},c_{1}\\}$. Without loss of generality, we can suppose $y\in\\{b_{3},c_{1}\\}$. If $y=c_{1}$, then $z\in\\{b_{1},b_{2}\\}$. By symmetry, we can assume $z=b_{1}$ so $N_{D}^{+}(y)=N_{D}^{+}(c_{1})\subseteq\\{b_{2},c_{2}\\}$. We take $\mathcal{C}_{5}=\\{a_{1},a_{4},b_{2},b_{3},b_{4},c_{1},c_{2},d_{1},v\\}$. Thus, $\mathcal{C}_{5}$ is a vertex cover of $D$ with $L_{3}(\mathcal{C}_{5})=\\{x\\}$, $y\in\mathcal{C}_{5}$ and $N_{D}^{+}(y)\subseteq\\{b_{2},c_{2}\\}\subseteq\mathcal{C}_{5}$. Now, if $y=b_{3}$, then $b_{3}\in V^{+}\cap\mathcal{C}_{5}$. Hence, by Lemma 5.7, $(a_{3},b_{3})\in E(D)$, since $a_{3}\in V_{2}$ and $I(D)$ is unmixed. Consequently, $N_{D}^{+}(y)\subseteq\\{c_{2},d_{1}\\}\subset\mathcal{C}_{5}$. Therefore, $y\in\mathcal{C}_{5}\setminus L_{1}(\mathcal{C}_{5})$ and we take $\mathcal{C}=\mathcal{C}_{5}$. Case (3.b) $x=b_{1}$. Hence, $\mathcal{C}_{6}=\\{a_{1},a_{3},b_{1},b_{2},b_{3},b_{4},c_{1},d_{1},d_{2}\\}$ is a vertex cover of $D$ with $L_{3}(\mathcal{C}_{6})=\\{x\\}$ and $y\in N_{D}(b_{1})=\\{a_{1},c_{1},d_{1}\\}$. Also, $y\in\mathcal{C}_{6}$. If $y\in\\{a_{1},d_{1}\\}$, then $N_{D}(y)\setminus\mathcal{C}_{6}\subseteq\\{a_{2},v\\}\subseteq V_{2}$. Consequently, by Lemma 5.7, $(b,y)\in E(D)$ for each $b\in N_{D}(y)\setminus\mathcal{C}_{6}$, since $y\in V^{+}$. So, $N_{D}^{+}\subseteq\mathcal{C}_{6}$ implying $y\in\mathcal{C}_{6}\setminus L_{1}(\mathcal{C}_{6})$. Now, if $y=c_{1}$. We can assume $(c_{2},c_{1})\in E(D)$, since in another case we have the case (3.a) with $x=c_{2}$ and $y=c_{1}$. Thus, $y=c_{1}\in\mathcal{C}_{6}\setminus L_{1}(\mathcal{C}_{6})$, since $N_{D}^{+}(c_{1})\subseteq\\{b_{1},b_{2}\\}\subset\mathcal{C}_{6}$. Therefore, we take $\mathcal{C}=\mathcal{C}_{6}$. Case (3.c) $x=d_{1}$. Then, $y\in N_{G}(d_{1})=\\{b_{1},b_{3},v\\}$. By symmetry, we can assume $y\in\\{b_{1},v\\}$. Furthermore, $\mathcal{C}_{7}=\\{a_{2},a_{4},b_{1},b_{2},b_{3},b_{4},c_{1},d_{1},v\\}$ is a vertex cover of $D$ with $L_{3}(\mathcal{C}_{7})=\\{x\\}$ and $y\in\mathcal{C}_{7}$. If $y=b_{1}$, then $N_{D}(y)\setminus\mathcal{C}_{7}\subseteq\\{a_{1}\\}$. Also, by Lemma 5.7, $N_{D}(y)\setminus\mathcal{C}_{7}\subseteq N_{D}^{-}(y)$, since $a_{1}\in V_{2}$ and $y\in V^{+}$. Thus, $y\in\mathcal{C}_{7}\setminus L_{1}(\mathcal{C}_{7})$. Now, if $y=v$, then $z=d_{2}$, since $N_{D}(v)=\\{d_{1},d_{2}\\}$. This implies $N_{D}^{+}(v)=\\{d_{1}\\}\subseteq\mathcal{C}_{7}$. So, $y\in\mathcal{C}_{7}\setminus L_{1}(\mathcal{C}_{7})$ and we take $\mathcal{C}=\mathcal{C}_{7}$. Case (4) $D=Q_{13}$. Hence, $x\in\\{d_{1},d_{2},g_{1},g_{2},h,h^{\prime}\\}$, since $deg_{D}(x)\geqslant 3$. By symmetry, we can suppose $x\in\\{d_{2},g_{2},h,h^{\prime}\\}$. Case (4.a) $x\in\\{d_{2},g_{2}\\}$. We take $\mathcal{C}_{8}=\\{a_{2},c_{1},c_{2},d_{1},d_{2},g_{1},g_{2},h,h^{\prime}\\}$ if $x=d_{2}$ or $\mathcal{C}_{8}=\\{a_{1},b_{2},c_{2},d_{1},d_{2},g_{1},g_{2},h,h^{\prime}\\}$ if $x=g_{2}$. Thus, $\mathcal{C}_{8}$ is a vertex cover of $D$ with $L_{3}(\mathcal{C}_{8})=\\{x\\}$ and $V(G)\setminus\mathcal{C}_{8}=\\{a_{1},b_{1},b_{2},v\\}\cup\\{a_{2},b_{1},c_{1},v\\}\subseteq V_{2}$. Consequently, by Lemma 5.7, $N_{D}(y)\setminus\mathcal{C}_{8}\subseteq N_{D}^{-}(y)$, since $y\in V^{+}$. This implies $N_{D}^{+}(y)\subseteq\mathcal{C}_{8}$. Therefore $y\in\mathcal{C}_{8}\setminus L_{1}(\mathcal{C}_{8})$ and we take $\mathcal{C}=\mathcal{C}_{8}$. Case (4.b) $x\in\\{h,h^{\prime}\\}$. We take $\mathcal{C}_{9}=\\{a_{2},b_{1},b_{2},d_{1},d_{2},g_{1},g_{2},h,v\\}$ if $x=h$ or $\mathcal{C}_{9}=\\{a_{2},c_{1},c_{2},d_{1},d_{2},g_{1},g_{2},h^{\prime},v\\}$ if $x=h^{\prime}$. Thus, $\mathcal{C}_{9}$ is a vertex cover of $D$ with $L_{3}(\mathcal{C}_{9})=\\{x\\}$. Also, $y\in N_{G}(x)\subseteq\\{v,d_{1},d_{2},g_{1},g_{2}\\}$. If $y=v$, then $\\{x,z\\}\subseteq N_{D}(y)=\\{h,h^{\prime}\\}$. Hence, $N_{D}^{+}(y)=\\{x\\}\subseteq\mathcal{C}_{9}$, then $y\in\mathcal{C}_{9}\setminus L_{1}(\mathcal{C}_{9})$. Now, if $y\neq v$, then $y\in\\{d_{1},d_{2}\\}$ when $x=h$ or $y\in\\{g_{1},g_{2}\\}$ when $x=h^{\prime}$. Then, $N_{D}(y)\setminus\mathcal{C}_{9}\subseteq\\{a_{1},c_{1},c_{2}\\}\subseteq V_{2}$ if $x=h$ or $N_{D}(y)\setminus\mathcal{C}_{9}\subseteq\\{b_{1},b_{2}\\}\subseteq V_{2}$ if $x=h^{\prime}$. Consequently, by Lemma 5.7, $N_{D}(y)\setminus\mathcal{C}_{9}\subseteq N_{D}^{-}(y)$, since $y\in V^{+}$. Thus, $N_{D}^{+}(y)\subseteq\mathcal{C}_{9}$ and $y\in\mathcal{C}_{9}\setminus L_{1}(\mathcal{C}_{9})$. Therefore, we take $\mathcal{C}=\mathcal{C}_{9}$. $\Box$ ###### Theorem 5.10 Let $D=(G,\mathcal{O},w)$ be a connected weighted oriented graph without $4$\- and $5$-cycles. Hence, $I(D)$ is unmixed if and only if $D$ satisfies one of the following conditions: 1. (a) $G\in\\{C_{7},T_{10}\\}$ and the vertices of $V^{+}$ are sinks. 2. (b) Each vertex is in exactly one simplex of $D$ and each simplex of $D$ has no generating $\star$-semi-forests. Proof. $\Rightarrow)$ By (3) in Theorem 2.12 and Remark 2.17, $G$ is well- covered. Thus, by Theorem 2.28, $G\in\\{C_{7},T_{10}\\}$ or $\\{V(H)\mid H\in S_{G}\\}$ is a partition of $V(G)$. If $G\in\\{C_{7},T_{10}\\}$, then by Proposition 5.9, $D$ satisfies (a). Now, if $\\{V(H)\mid H\in S_{G}\\}$ is a partition of $V(G)$, then $G$ is an $SCQ$ graph with $C_{G}=Q_{G}=\emptyset$. Hence, by Theorem 4.8, $D$ satisfies (b). $\Leftarrow)$ If $D$ satisfies (a), then by Theorem 2.28, $G$ is well-covered. Consequently, by Corollary 5.2, $I(D)$ is unmixed. Now, if $D$ satisfies (b), then $G$ is an $SCQ$ graph, with $C_{G}=Q_{G}=\emptyset$. Therefore, by (b) and Theorem 4.8, $I(D)$ is unmixed. $\Box$ ###### Corollary 5.11 Let $D$ be a weighted oriented graph without isolated vertices and $girth(G)\geqslant 6$. Hence, $I(D)$ is unmixed if and only if $D$ satisfies one of following properties: 1. (a) $G=C_{7}$ and the vertices of $V^{+}$ are sinks. 2. (b) $G$ has a perfect matching $e_{1}=\\{x_{1},x_{1}^{\prime}\\},\ldots,e_{r}=\\{x_{r},x_{r}^{\prime}\\}$ where $deg_{D}(x_{1}^{\prime})=\cdots=deg_{D}(x_{r}^{\prime})=1$ and $(x_{j}^{\prime},x_{j})\in E(D)$ if $x_{j}\in V^{+}$. Proof. $\Leftarrow)$ If $D$ satisfies (a), then $I(D)$ is unmixed by (a) in Theorem 5.10. Now, assume $D$ satisfies (b). Assume $b^{\prime}\in N_{D}^{+}(a)$ with $a\in V^{+}$ such that $\\{b,b^{\prime}\\}=e_{j}=\\{x_{j},x_{j}^{\prime}\\}$ for some $1\leqslant j\leqslant r$. If $b^{\prime}=x_{j}^{\prime}$, then $x_{j}=a\in V^{+}$, since $deg_{D}(x_{j}^{\prime})=1$. So, $x_{j}^{\prime}=b^{\prime}\in N_{D}^{+}(a)=N_{D}^{+}(x_{j})$. But by hypothesis, $(x_{j}^{\prime},x_{j})\in E(D)$, since $x_{j}\in V^{+}$. A contradiction, then $b^{\prime}=x_{j}$. Thus, $b=x_{j}^{\prime}$ implies $N_{D}(b)=\\{x_{j}\\}=\\{b^{\prime}\\}\subseteq N_{D}^{+}(a)$. Hence, by Proposition 4.1, $|\mathcal{C}\cap e_{j}|=1$ where $\mathcal{C}$ is a strong vertex cover. So, $|\mathcal{C}|=r$. Therefore, by (2) in Theorem 2.12, $I(D)$ is unmixed. $\Rightarrow)$ $G\neq T_{10}$, since $T_{10}$ has $3$-cycles. Hence, by Theorem 5.10, $D$ satisfies (a) or $\\{V(H)\mid H\in S_{G}\\}$ is a partition of $V(G)$. Furthermore, $S_{G}\subseteq E(G)$, since $G$ has not isolated vertices and $girth(G)\geqslant 6$. Thus, we can assume $S_{G}=\\{e_{1},\ldots,e_{r}\\}$ where $e_{i}=\\{x_{i},x_{i}^{\prime}\\}$ and $deg_{D}(x_{i}^{\prime})=1$ for $i=1,\ldots,s$. Also, by Theorem 5.10, each $e_{i}$ has no generating $\star$-semi-forests. So, by Theorem 3.10, $e_{i}\not\subseteq\mathcal{C}$ for each strong vertex cover $\mathcal{C}$. Consequently, $|e_{i}\cap\mathcal{C}|=1$, since $\mathcal{C}$ is a vertex cover. Then, $e_{i}$ satisfies (2) of Proposition 4.1. Now, suppose $(x_{j},x_{j}^{\prime})\in E(D)$ and $x_{j}\in V^{+}$. We take $a=b=x_{j}$ and $b^{\prime}=x_{j}^{\prime}$, then by (2) of Proposition 4.1, $N_{D}(x_{j})=N_{D}(b)\subseteq N_{D}^{+}(a)=N_{D}^{+}(x_{j})$. Hence, $x_{j}$ is a source. A contradiction, by Remark 2.3, since $x_{j}\in V^{+}$. Therefore, $D$ satisfies (b). $\Box$ ###### Proposition 5.12 If $G=P_{10}$, then the following properties are equivalent: 1. (1) $I(D)$ is unmixed. 2. (2) If $y\in V^{+}$ and $y$ is not a sink, then $y=d_{1}$ with $N_{D}^{+}(y)=\\{g_{1},b_{2}\\}$ or $y=d_{2}$ with $N_{D}^{+}(y)=\\{g_{2},b_{1}\\}$. Proof. ${\rm(2)}\Rightarrow{\rm(1)}$ Let $\mathcal{C}$ be a strong vertex cover of $D$. Suppose $x\in L_{3}(\mathcal{C})$. Then, there is $y\in\big{(}\mathcal{C}\setminus L_{1}(\mathcal{C})\big{)}\cap V^{+}$ such that $x\in N_{D}^{+}(y)$. Thus, by (2), $y\in\\{d_{1},d_{2}\\}$ and $x\in N_{D}^{+}(y)\subseteq\\{b_{1},b_{2},g_{1},g_{2}\\}$. Hence, $L_{3}(\mathcal{C})\subseteq\\{b_{1},b_{2},g_{1},g_{2}\\}$. By symmetry of $P_{10}$, we can assume $y=d_{1}$. Then by (2), $x\in N_{D}^{+}(y)=\\{g_{1},b_{2}\\}$. Also, $\\{g_{1},d_{1},b_{2}\\}=N_{D}^{+}[y]\subseteq\mathcal{C}$, since $y\in\mathcal{C}\setminus L_{1}(\mathcal{C})$. But $y=d_{1}\notin L_{3}(\mathcal{C})$, then $N_{D}(y)\not\subseteq\mathcal{C}$. Thus, $d_{2}\notin\mathcal{C}$. So, by Remark 2.8, $\\{b_{1},d_{1},g_{2}\\}=N_{D}(d_{2})\subseteq\mathcal{C}$. Furthermore, $\\{a_{1},a_{2}\\}\cap L_{3}(\mathcal{C})=\emptyset$, since $L_{3}(\mathcal{C})\subseteq\\{b_{1},b_{2},g_{1},g_{2}\\}$. Consequently, $\\{a_{1},a_{2}\\}\cap\mathcal{C}=\emptyset$, since $N_{D}(a_{1},a_{2})=\\{g_{1},b_{1},g_{2},b_{2}\\}\subseteq\mathcal{C}$. But, $x\in\\{g_{1},b_{2}\\}\cap L_{3}(\mathcal{C})$, then $a_{1}\in N_{D}(g_{1})\subseteq\mathcal{C}$ or $a_{2}\in N_{D}(b_{2})\subseteq\mathcal{C}$. Hence, $\\{a_{1},a_{2}\\}\cap\mathcal{C}\neq\emptyset$. A contradiction, then $L_{3}(\mathcal{C})=\emptyset$. Also, by Theorem 2.28 and Remark 2.17, $I(G)$ is unmixed. Therefore, by (3) in Theorem 2.12, $I(D)$ is unmixed. ${\rm(1)}\Rightarrow{\rm(2)}$ We take $V_{2}:=\\{a\in V(G)\mid deg_{G}(a)=2\\}$ and $y\in V^{+}$, such that $y$ is not a sink, then there is $(y,x)\in E(D)$. Also, by Remark 2.3, there is $z\in N_{D}^{-}(y)$. We will prove $y\in\\{d_{1},d_{2}\\}$. If $deg_{D}(x)=2$, then by Lemma 5.7, $y\in\\{d_{1},d_{2}\\}$. Now, we assume $deg_{D}(x)\geqslant 3$, then by symmetry of $P_{10}$, we can suppose $x\in\\{g_{1},d_{1}\\}$. First suppose $x=g_{1}$. Thus, $y\in N_{D}(g_{1})=\\{a_{1},c_{1},d_{1}\\}$. If $y\in\\{a_{1},c_{1}\\}$, then $deg_{D}(y)=2$ and $y\in\mathcal{C}_{1}=\\{a_{1},a_{2},c_{1},d_{1},d_{2},g_{1},g_{2}\\}$ is a vertex cover of $D$ with $L_{3}(\mathcal{C}_{1})=\\{x\\}$. Hence, $N_{D}(y)=\\{x,z\\}$ implies $N_{D}^{+}(y)=\\{x\\}=\\{g_{1}\\}\subseteq\mathcal{C}_{1}$. Consequently, $y\in\mathcal{C}_{1}\setminus L_{1}(\mathcal{C}_{1})$. Hence, $\mathcal{C}_{1}$ is strong, since $L_{3}(\mathcal{C}_{1})=\\{x\\}$. A contradiction, by Theorem 2.12, then $y=d_{1}$. Now, suppose $x=d_{1}$. Then, $\mathcal{C}_{2}=\\{a_{1},b_{2},c_{2},d_{1},d_{2},g_{1},g_{2}\\}$ is a vertex cover of $D$ with $L_{3}(\mathcal{C}_{2})=\\{x\\}$. Also, $y\in N_{D}(x)=\\{g_{1},b_{2},d_{2}\\}$. If $y\in\\{g_{1},b_{2}\\}$, then $N_{D}(y)\setminus\\{d_{1}\\}\subseteq N_{D}(g_{1},b_{2})\setminus\\{d_{1}\\}=\\{a_{1},a_{2},c_{1}\\}\subseteq V_{2}$. Hence, by Lemma 5.7, $N_{D}(y)\setminus\\{d_{1}\\}\subseteq N_{D}^{-}(y)$, since $y\notin\\{b_{1},b_{2}\\}$. Thus, $N_{D}^{+}(y)=\\{d_{1}\\}=\\{x\\}\subseteq\mathcal{C}_{2}$ implying $y\in\mathcal{C}_{2}\setminus L_{1}(\mathcal{C}_{2})$. Consequently, $\mathcal{C}_{2}$ is strong with $L_{3}(\mathcal{C}_{2})\neq\emptyset$. A contradiction, by Theorem 2.12, then $y=d_{2}$. Therefore $y\in\\{d_{1},d_{2}\\}$. By symmetry of $P_{10}$, we can assume $y=d_{1}$. Now, we will prove $N_{D}^{+}(y)=\\{g_{1},b_{2}\\}$. By contradiction, in each one of the following cases, we give a strong vertex cover $\mathcal{C}^{\prime}$ with $L_{3}(\mathcal{C}^{\prime})\neq\emptyset$, since $I(D)$ is unmixed and $z\in N_{D}^{-}(y)$. Case (1) $g_{1}\notin N_{D}^{+}(d_{1})$ and $b_{2}\in N_{D}^{+}(d_{1})$. So, $N_{D}^{+}(d_{1})\subseteq\\{b_{2},d_{2}\\}$ and $\mathcal{C}_{1}^{\prime}=$ $\\{a_{1},a_{2},b_{2},c_{1},c_{2},d_{1},d_{2}\\}$ is a vertex cover of $D$ with $L_{3}(\mathcal{C}_{1}^{\prime})=\\{b_{2}\\}$. Also, $(d_{1},b_{2})\in E(D)$ and $y=d_{1}\in\big{(}\mathcal{C}_{1}^{\prime}\setminus L_{1}(\mathcal{C}_{1}^{\prime})\big{)}\cap V^{+}$, since $b_{2}\in N_{D}^{+}[d_{1}]\subseteq\\{d_{1},b_{2},d_{2}\\}\subset\mathcal{C}_{1}^{\prime}$. Hence, $\mathcal{C}_{1}^{\prime}$ is a strong vertex cover of $D$. Case (2) $b_{2}\notin N_{D}^{+}(d_{1})$ and $g_{1}\in N_{D}^{+}(d_{1})$. Then, $N_{D}^{+}(d_{1})\subseteq\\{g_{1},d_{2}\\}$ and $\mathcal{C}_{2}^{\prime}=$ $\\{a_{1},a_{2},c_{1},d_{1},d_{2},g_{1},g_{2}\\}$ is a vertex cover of $D$ with $L_{3}(\mathcal{C}_{2}^{\prime})=\\{g_{1}\\}$. Furthermore $(d_{1},g_{1})\in E(D)$ and $d_{1}=y\in\big{(}\mathcal{C}_{2}^{\prime}\setminus L_{1}(\mathcal{C}_{2}^{\prime})\big{)}\cap V^{+}$, since $g_{1}\in N_{D}^{+}[d_{1}]\subseteq\\{d_{1},g_{1},d_{2}\\}\subset\mathcal{C}_{2}^{\prime}$. Hence, $\mathcal{C}_{2}^{\prime}$ is a strong vertex cover of $D$. Case (3) $b_{2},g_{1}\notin N_{D}^{+}(d_{1})$. Thus, $z=d_{1}$, $N_{D}^{+}(d_{1})=\\{d_{2}\\}$ and $\mathcal{C}_{3}^{\prime}=$ $\\{a_{2},b_{1},c_{1},d_{1},d_{2},g_{1},g_{2}\\}$ is a vertex cover of $D$ with $L_{6}(\mathcal{C}_{3}^{\prime})=\\{d_{2}\\}$. Also, $(d_{1},d_{2})\in E(D)$ and $d_{1}=y\in\big{(}\mathcal{C}_{3}^{\prime}\setminus L_{1}(\mathcal{C}_{3}^{\prime})\big{)}\cap V^{+}$, since $N_{D}^{+}[d_{1}]=\\{d_{1},d_{2}\\}\subset\mathcal{C}_{3}^{\prime}$. Hence, $\mathcal{C}_{3}^{\prime}$ is a strong vertex cover of $D$. $\Box$ ###### Theorem 5.13 Let $D=(G,\mathcal{O},w)$ be a connected weighted oriented graph, with ${\rm girth}(G)\geqslant 5$. Hence, $I(D)$ is unmixed if and only if $D$ satisfies one of the following properties: 1. (a) $G\in\\{K_{1},C_{7},Q_{13},P_{13},P_{14}\\}$ and the vertices of $V^{+}$ are sinks. 2. (b) $G=P_{10}$, furthermore if $y$ is not a sink in $V^{+}$, then $y=d_{1}$ with $N_{D}^{+}(y)=\\{g_{1},b_{2}\\}$ or $y=d_{2}$ with $N_{D}^{+}(y)=\\{g_{2},b_{1}\\}$. 3. (c) Each vertex is in exactly one simplex of $G$ or in exactly one basic $5$-cycle of $G$. Furthermore, each simplex of $D$ has not a generating $\star$-semi- forest and each basic $5$-cycle of $D$ has the $\star$-property. Proof. $\Rightarrow)$ By (3) in Theorem 2.12 and Remark 2.17, $G$ is well- covered. By Theorem 2.30, $G\in\\{K_{1},C_{7},P_{10},P_{13},P_{14},Q_{13}\\}$ or $\\{V(H)\mid H\in S_{G}\cup C_{G}\\}$ is a partition of $V(G)$. If $\\{V(H)\mid H\in S_{G}\cup C_{G}\\}$ is a partition of $V(G)$, then $G$ is an $SCQ$ graph with $Q_{G}=\emptyset$. Hence, by Theorem 4.8, $D$ satisfies (c). Now, if $G=P_{10}$, then by Proposition 5.12, $D$ satisfies (b). Furthermore, if $G\in\\{C_{7},Q_{13},P_{13},P_{14}\\}$, then by Proposition 5.9, $D$ satisfies (a). Finally, if $G=K_{1}$, then by Remark 2.3, $V^{+}=\emptyset$. $\Leftarrow)$ If $D$ satisfies (b), then by Proposition 5.12, $I(D)$ is unmixed. Now, if $D$ satisfies (c), then $G$ is an $SCQ$ graph with $Q_{G}=\emptyset$. Consequently, by Theorem 4.8, $I(D)$ is unmixed. Finally, if $D$ satisfies (a), then by Theorem 2.30, $G$ is well-covered. Therefore, by Corollary 5.2, $I(D)$ is unmixed. $\Box$ ###### Remark 5.14 A graph is well-covered if and only if each connected component is well- covered. Hence, when $D$ is no connected in Theorem 5.10 (resp. 5.13), $I(D)$ is unmixed if and only if each connected component of $D$ satisfies (a) or (b) (resp. (a), (b) or (c)). ## 6 Examples ###### Example 6.1 Let $D_{1}$ be the following weighted oriented graph. $v_{\small 1}$$1$$w_{\small 1}$$2$$2$$2$$1$$2$$2$$2$$1$$1$$2$$2$$2$$v_{\small 2}$$2$$2$$2$$w_{\small 2}$$2$$1$$2$$1$$2$$2$$2$$2$$1$$2$$w_{\small 5}$$1$$1$$v_{\small 3}$$2$$v_{\small 4}$$2$$2$$2$$w_{\small 4}$$2$$2$$v_{\small 5}$$2$$2$$\mathbf{v_{\small 1}}$$1$$\mathbf{T_{1}}$$w_{\small 1}$$2$$2$$2$$1$$2$$2$$2$$1$$1$$2$$2$$2$$\mathbf{B_{1}}$$\mathbf{v_{\small 2}}$$2$$2$$2$$w_{\small 2}=w_{\small 3}$$2$$1$$2$$1$$2$$\mathbf{T_{2}}$$2$$2$$2$$1$$2$$w_{\small 5}$$1$$1$$\mathbf{v_{\small 3}}$$2$$\mathbf{T_{3}}$$\mathbf{v_{\small 4}}$$2$$2$$2$$w_{\small 4}$$2$$2$$\mathbf{v_{\small 5}}$$2$$2$$\mathbf{T_{4}}$$\mathbf{T_{5}}$ We take the weighted oriented subgraph $K$ of $D_{1}$ induced by $V(D_{1})\setminus\\{w_{1},w_{2},w_{4},w_{5}\\}$. In the following figure, $T_{1},\ldots,T_{5}$ are the ROT’s and $B_{1}$ is the unicycle oriented subgraph such that the parent of $T_{i}$ is $v_{i}$ and $W^{T_{i}}=\\{w_{i}\\}$ for $i=1,\ldots,5$. Furthermore, $H=\cup_{i=1}^{5}\ T_{i}\ \cup B_{1}$ is a $\star$-semi-forest with $W_{1}^{H}=\\{w_{1},w_{5}\\}$ $W_{2}^{H}=\\{w_{2},w_{4}\\}$ and $V(H)=V(K)$. Therefore, $H$ is a generating $\star$-semi-forest of $K$. ###### Example 6.2 Let $D_{2}$ be an oriented weighted graph such as the Figure. Let $K^{4}=H=D_{2}[x_{1},x_{2},x_{3}]$ be an induced weighted oriented subgraph of $D_{2}$ and $H$ a generating $\star$-semi-forest of $K^{4}$ with $\tilde{H}=H=\\{x_{1},x_{2},x_{3}\\}\subseteq V^{+}$ and $W_{1}=W_{2}=\emptyset$. Then, by Theorem 3.10, there is a strong vertex cover $\mathcal{C}$ of $D_{2}$, such that $V(K^{4})\subseteq\mathcal{C}$. For this purpose, first we take a minimal vertex cover $\mathcal{C}_{i}$ of $D$ and define $\mathcal{C}_{i}^{\prime}=(\mathcal{C}_{i}\setminus W_{1})\cup N_{D}(W_{1})\cup N_{D}^{+}(W_{2}\cup\tilde{H})$. Finally, we use the algorithm in the proof of the Proposition 3.1. In this example, $\mathcal{C}_{i}^{\prime}=\mathcal{C}_{i}\cup N_{D}^{+}(\tilde{H})=\mathcal{C}_{i}\cup N_{D}^{+}(\\{x_{1},x_{2},x_{3}\\})=\mathcal{C}_{i}\cup\\{x_{1},x_{2},x_{3},y_{1},y_{2}\\}$. $z_{\tiny 1}$$z_{\tiny 2}$$z_{\tiny 3}$$y_{\tiny 1}$$x_{\tiny 1}$$w>1$$y_{\tiny 2}$$x_{\tiny 2}$$w>1$$y_{\tiny 3}$$x_{\tiny 3}$$w>1$$H$ * • If we take $\mathcal{C}_{1}=\\{x_{1},x_{2},y_{1},y_{2},y_{3}\\}$ as a minimal vertex cover of $D_{2}$, then $\mathcal{C}_{1}^{\prime}=\\{x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}\\}$ and $L_{3}(\mathcal{C}_{1}^{\prime})\setminus N_{D}^{+}(\tilde{H})=\emptyset$. Thus, by Proposition 3.1, $\mathcal{C}=\mathcal{C}_{1}^{\prime}=\\{x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}\\}$ is a strong vertex cover such that $V(K^{4})\subseteq\mathcal{C}$ and it is no minimal. Furthermore, in this case it is enough to know the orientation of $E(K^{4})$, since $L_{3}(\mathcal{C})=\\{x_{1},x_{2},x_{3}\\}$ and $N_{D_{2}}(x_{i})\subseteq\mathcal{C}$ for $i=1,2,3$. * • Now, if we take $\mathcal{C}_{2}=\\{x_{1},x_{2},x_{3},z_{1},z_{2},z_{3}\\}$ as a minimal vertex cover of $D_{2}$, then $\mathcal{C}_{2}^{\prime}=V(D_{2})\setminus\\{y_{3}\\}$ and $L_{3}(\mathcal{C}_{2}^{\prime})\setminus N_{D}^{+}(\tilde{H})=\\{z_{1},z_{2}\\}$. Thus, by the algorithm in the proof of Proposition 3.1, $\mathcal{C}^{\prime}=\mathcal{C}_{4}^{\prime}=\\{x_{1},x_{2},x_{3},y_{1},y_{2},z_{3}\\}$ is a strong vertex cover such that $V(K^{4})\subseteq\mathcal{C}^{\prime}$. Since $\mathcal{C}$ and $\mathcal{C}^{\prime}$ are no minimal with $L_{3}(\mathcal{C})=\\{x_{1},x_{2},x_{3}\\}$ and $L_{3}(\mathcal{C}^{\prime})=\\{x_{1},x_{2}\\}$, then $D_{2}$ is mixed. Furthermore, $V(D_{2})$ has a partition in complete graphs: $K^{i}=D_{2}[y_{i},z_{i}]$ for $i=1,2,3$ and $K^{4}=D_{2}[x_{1},x_{2},x_{3}]$ . ###### Example 6.3 Let $D_{3}=(G,\mathcal{O},w)$ be the following oriented graph. Hence, $w>1$$b^{\prime}$$d_{\small 1}^{\prime}$$b$$d_{\small 2}^{\prime}$$1$$c$$d_{\small 1}$$w>1$$w>1$$a$$d_{\small 2}$$w>1$$a^{\prime}$$1$ * • $G$ has no $3$\- and $4$-cycles. * • The basic $5$-cycle $C=(a^{\prime},a,b,b^{\prime},c,a^{\prime})$ satisfies the $\star$-property. * • $D_{3}[\\{d_{1},d_{1}^{\prime}\\}]$ has not a generating $\star$-semi-forest. But $T_{1}=\tilde{e}=(d_{2},d_{2}^{\prime})$ is a ROT with parent $v_{1}=d_{2}$ and $W=W_{2}=\\{w_{1}=a\\}$. Furthermore, $H=T_{1}$ is a $\star$-semi-forest with $V(H)=V(K)$, where $K:=D_{3}[\\{d_{2},d_{2}^{\prime}\\}]$. Therefore, $H$ is a generating $\star$-semi-forest of $K$ and $I(D_{3})$ is mixed. ###### Example 6.4 Let $D_{4}=(G,\mathcal{O},w)$ be the following oriented weighted graph. Hence, $w>1$$d$$1$$1$$1$$1$$1$$1$$1$$1$$1$ * • $G=P_{10}$, then by Theorem 2.30 and Remark 2.17, $G$ is well-covered and $I(G)$ is unmixed. * • $d$ is not a sink and $d\in V^{+}$. * • By Proposition 5.12, $I(D_{4})$ is unmixed. ## References * [1] C. Carvalho, V. G. Neumann and H. H. López. Projective nested cartesian codes. Bull. Braz. Math. Soc. (N.S.), 48, (2017), no. 2, 283–-302. * [2] I. D. Castrillón, R. Cruz, and E. Reyes. On well-covered, vertex decomposable and Cohen–Macaulay graphs. Electron. J. Combin., 23, (2016), no. 2, Paper 2.39, 17pp. * [3] I. Castrillón and E. Reyes, Pure vertex decomposable simplicial complex associated to graphs whose $5$-cycles are chorded, Bol. Soc. Mat. Mex. 23 (2017), 399–412. * [4] R. Diestel, Graph Theory, Second Edition, Graduate Texts in Mathematics, 173, Springer-Verlag, New York, 2000. * [5] A. Finbow, B. Hartnell and R. J. Nowakowski, A characterization of well-covered graphs of girth 5 or greater, J. of Combinatorial Theory B 57 (1993) 44-68. * [6] A. Finbow, B. Hartnell and R. J. Nowakowski, A characterization of well-covered graphs that contain neither 4- nor 5-cycles, J. Graph Theory 18 (1994) 713-721. * [7] P. Gimenez, J. Martínez-Bernal, A. Simis, R. H. Villarreal, and C. E. Vivares, Symbolic powers of monomial ideals and Cohen–Macaulay vertex-weighted digraphs, Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, Springer, Cham, (2018), 491–510. * [8] H. T. Há, K. N. Lin, S. Morey, E. Reyes and R. H. Villarreal, Edge ideals of oriented graphs, Internat. J. Algebra Comput., 29 (2019), no. 3, 535–559. * [9] J. Martínez-Bernal, Y. Pitones and R. H. Villarreal, Minimum distance functions of graded ideals and Reed-Muller-type codes. J. Pure Appl. Algebra 221 (2017), no. 2, 251–-275. * [10] Y. Pitones, E. Reyes and J. Toledo, Monomial ideals of weighted oriented graphs, Electronic. J. Combin., 26 (2019), no. 3, Paper 3.44, 18 pp. * [11] Y. Pitones, E. Reyes and R. H. Villarreal, Unmixed and Cohen–Macaulay weighted oriented König graphs, (2019), preprint, arxiv:1909.13295. * [12] E. Prisner, J. Topp and P. D. Vestergaard, Well Covered Simplicial, Chordal, and Circular Arc Graphs, J. Graph Theory, 21 (1996) 113-119. * [13] B. Randerath and L. Volkmann, A characterization of well-covered block-cactus graphs, Australas. J. Combin, 9 (1994) 307–-314. * [14] R. H. Villarreal, Monomial Algebras, Second Edition, Monographs ans Research Notes in Mathematics, Chapman and Hall/CRC, 2015. * [15] G. Zhu, L. Xu, H. Wang and Z. Tang, Projective dimensions and regularity of edge ideal of some weighted oriented graphs, Rocky Mountain J. Math., 49 (2019), no. 4, 1391-–1406.

2024-09-04T02:54:56.435743

2003.00964

arxiv-papers

# Almost-Matching-Exactly for Treatment Effect Estimation under Network Interference M. Usaid Awan Department of Economics, Duke University Marco Morucci Department of Political Science, Duke University Vittorio Orlandi Department of Statistical Science, Duke University Sudeepa Roy Department of Computer Science, Duke University Cynthia Rudin Department of Statistical Science, Duke University Department of Computer Science, Duke University Department of Electrical and Computer Engineering, Duke University Alexander Volfovsky Department of Statistical Science, Duke University ###### Abstract We propose a matching method that recovers direct treatment effects from randomized experiments where units are connected in an observed network, and units that share edges can potentially influence each others’ outcomes. Traditional treatment effect estimators for randomized experiments are biased and error prone in this setting. Our method matches units almost exactly on counts of unique subgraphs within their neighborhood graphs. The matches that we construct are interpretable and high-quality. Our method can be extended easily to accommodate additional unit-level covariate information. We show empirically that our method performs better than other existing methodologies for this problem, while producing meaningful, interpretable results. ### 1 INTRODUCTION Randomized experiments are considered to be the gold standard for estimating causal effects of a treatment on an outcome. Typically, in these experiments, the outcome of a unit is assumed to be only affected by the unit’s own treatment status, and not by the treatment assignment of other units (Cox, 1958; Rubin, 1980). However, in many applications – such as measuring effectiveness of an advertisement campaign or a teacher training program – units interact, and ignoring these interactions results in poor causal estimates (Halloran and Struchiner, 1995; Sobel, 2006). We propose a method that leverages the observed network structure of interactions between units to account for treatment interference among them. We study a setting in which a treatment has been uniformly randomized over a set of units connected in a network, and where treatments of connected units can influence each others’ outcomes. The development of methods for this setting is a relatively new field in causal inference methodology, and only few approaches for it have been proposed (e.g., van der Laan, 2014; Aronow et al., 2017; Sussman and Airoldi, 2017). In this paper, we propose a method that leverages matching (Rosenbaum and Rubin, 1983) to recover direct treatment effects from experiments with interference. Our method makes several key contributions to the study of this setting: First, our method explicitly leverages information about the network structure of the experimental sample to adjust for possible interference while estimating direct treatment effects. Second, unlike other methods, matching allows us to nonparametrically estimate treatment effects, without the need to specify parametric models for interference or outcomes. Third, matching produces highly interpretable results, informing analysts as to which features of the input data were used to produce estimates. More specifically, we match on features of graphs that are easy to interpret and visualize. In our setting, units experience interference according to their neighborhood graphs – the graphs defined by the units they are directly connected to. Units with similar neighborhood graphs will experience similar interference. For example, the educational outcome of a student randomly assigned to an extra class depends on whether or not her friends are also assigned to that class, and not just on how many: the specific structure of the student’s friendship circle will influence whether or not study groups are formed, how information is shared, how much attention the treated student will devote to the class, and so on. All of this will impact the overall educational outcomes of interest. Because of this, matching units with similar neighborhood graphs together will enable us to recover direct treatment effects even under interference. We match units’ neighborhood graphs on counts of subgraphs within them, as graphs with similar counts of the same unique subgraphs are naturally likely to be similar. From there, we construct matches on individuals with similar sets of important subgraphs; here, the set of important subgraphs is learned from a training set. We generalize the Almost-Matching-Exactly (AME) framework (Dieng et al., 2019; Wang et al., 2019) to match units on subgraphs in experimental settings. We do this by constructing graph-based features that can explain both the interference pattern in the experiment and predict the underlying social network. We demonstrate that our method performs better than other methods for the same problem in many settings, while generating interpretable matches. The paper will proceed as follows: In Section 2, we make explicit the assumptions underpinning our framework, and outline our matching approach to estimating direct treatment effects. In Sections 3 and 4, we evaluate the effectiveness of our method on simulated and real-world data. Theoretical evaluation of our approach is available in the appendix. #### 1.1 Related Work Work on estimating causal effects under interference between units has three broad themes. First, there has been a growing body of work on the design of novel randomization schemes to perform causal inference under interference (Liu and Hudgens, 2014; Sinclair et al., 2012; Duflo and Saez, 2003; Basse and Airoldi, 2018). Some of this work makes explicit use of observed network structure to randomly assign treatment so as to reduce interference (Ugander et al., 2013; Toulis and Kao, 2013; Eckles et al., 2016, 2017; Jagadeesan et al., 2019). These methodologies are inapplicable to our setting as they require non-uniform treatment assignment, whereas in our setting we wish to correct for interference after randomization. Second, there is work on estimating direct treatment effects in experiments under interference, and after conventional treatment randomization, similar to our setting. Some existing work aims to characterize the behavior of existing estimators under interference (Manski, 2013; Sävje et al., 2017). Other approaches lay out methods based on randomization inference to test a variety of hypotheses under interference and treatment randomization (Rosenbaum, 2007; Aronow, 2012; Athey et al., 2018). Some of these approaches mix randomization inference and outcome models (Bowers et al., 2013). For the explicit problem of recovery of treatment effects under interference, Aronow et al. (2017) provide a general framework to translate different assumptions about interference into inverse- probability estimators, and Sussman and Airoldi (2017) give linearly unbiased, minimum integrated-variance estimators under a series of assumptions about interference. These methods either ignore explicit network structure, or require probabilities under multiple complex sampling designs to be estimated explicitly. Finally, there have been studies of observational inference under network interference (van der Laan, 2014; Liu et al., 2016; Ogburn et al., 2017; Forastiere et al., 2016). However, recovering causal estimates using observational data when units are expected to influence each other requires a structural model of both the nature of interference and contagion among units. ### 2 METHODOLOGY We discuss our problem and approach in this section. #### 2.1 Problem Statement We have a set of $n$ experimental units indexed by $i$. These units are connected in a known graph $G=(V,E)$, where $V(G)=\\{1,\dots,n\\}$ is the set of vertices of $G$, and $E(G)$ is the set of edges of $G$. We disallow self- loops in our graph. We say that $H$ is a subgraph of $G$ if $V(H)\subseteq V(G)$ and $E(H)\subseteq E(G)$. Let $t_{i}\in\\{0,1\\}$ represent the treatment indicator for unit $i$, $\mathbf{t}$ represent the vector of treatment indicators for the entire sample, and $\mathbf{t}_{-i}$ represent the treatment indicators for all units except $i$. Given a treatment vector $\mathbf{t}$ on the entire sample (i.e., for all vertices in $G$), we use $G^{\mathbf{t}}$ to denote the labeled graph, where each vertex $i\in V(G)$ has been labeled with its treatment indicator $t_{i}$. In addition, we use $G_{P}$ to denote a graph induced by the set of vertices $P\subseteq V(G)$ on $G$, such that $V(G_{P})=P$ and $E(G_{P})=\\{(e_{1},e_{2})\in E(G):\;e_{1}\in P,e_{2}\in P\\}$. We use the notation $\mathcal{N}_{i}=\\{j:(i,j)\in E(G)\\}$ to represent the neighborhood of vertex $i$. The labeled neighborhood graph of a unit $i$, $G_{\mathcal{N}_{i}}^{\mathbf{t}}$, is defined as the graph induced by the neighbors of $i$, and labeled according to $\mathbf{t}$. We also define $\mathbf{t}_{\mathcal{N}_{i}}$ to be the vector of treatment indicators corresponding to unit $i$’s neighborhood graph. A unit’s response to the treatment is represented by its random potential outcomes $Y_{i}(\mathbf{t})$ = $Y_{i}(t_{i},\mathbf{t}_{-i})$. Unlike other commonly studied causal inference settings, unit $i$’s potential outcomes are now a function of both the treatment assigned to $i$, and of all other units’ treatments. Observed treatments for unit $i$ and the whole sample are represented by the random variables $T_{i}$ and $\mathbf{T}$ respectively. We assume that the number of treated units is always $n^{(1)}$, i.e., $\sum_{i=1}^{n}T_{i}=n^{(1)}$. A0: Ignorability of Treatment Assignment. We make the canonical assumption that treatments are administered independently of potential outcomes, that is: $Y_{i}(t_{i},\mathbf{t}_{-i})\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10.0mu\perp$}}}\mathbf{T}$, and $0<\Pr(T_{i}=1)<1$ for all units. In practice, we assume that treatment is assigned uniformly at random to units, which is possible only in experimental settings. As stated before, we do not make the canonical Stable Unit Treatment Value Assumption (SUTVA) (Rubin, 1980), which, among other requirements, states that units are exclusively affected by the treatment assigned to them. We do not make this assumption because our units are connected in a network: it could be possible for treatments to spread along the edges of the network and to affect connected units’ outcomes. We do maintain the assumption of comparable treatments across units, which is commonly included in SUTVA. Our causal quantity of interest will be the Average Direct Effect (ADE), which is defined as follows: $\displaystyle ADE$ $\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[Y_{i}(1,\mathbf{0})-Y_{i}(0,\mathbf{0})],$ (1) where $\mathbf{t}_{-i}=\mathbf{0}$ represents the treatment assignment in which no unit other than $i$ is treated. The summand represents the treatment effect on unit $i$ when no other unit is treated, and, therefore, no interference occurs (Halloran and Struchiner, 1995). #### 2.2 Framework We outline the requirements of our framework for direct effect estimation under interference. We denote interference effects on a unit $i$ with the function $f_{i}(\mathbf{t}):\\{0,1\\}^{n}\mapsto\mathbb{R}$, a function that maps each possible treatment allocation for the $n$ units to the amount of interference on unit $i$. We will use several assumptions to restrict the domain of $f$ to a much smaller set (and overload the notation $f_{i}$ accordingly). To characterize $f$, we rely on the typology of interference assumptions introduced by Sussman and Airoldi (2017). The first three assumptions (A0-A2) needed in our framework are common in the interference literature (e.g., Manski, 2013; Toulis and Kao, 2013; Eckles et al., 2016; Athey et al., 2018): A1: Additivity of Main Effects. First, we assume that main treatment effects are additive, i.e., that there is no interaction between units’ treatment indicators. This allows us to write: $\displaystyle Y_{i}(t,\mathbf{t}_{-i})=t\tau_{i}+f_{i}(\mathbf{t}_{-i})+\epsilon_{i}$ (2) where $\tau_{i}$ is the direct treatment effect on unit $i$, and $\epsilon_{i}$ is some baseline effect. A2: Neighborhood Interference. We focus on a specific form of the interference function $f_{i}$ by assuming that the interference experienced by unit $i$ depends only on treatment of its neighbors. That is, if for two treatment allocations $\mathbf{t},\mathbf{t}^{\prime}$ we have $\mathbf{t}_{\mathcal{N}_{i}}=\mathbf{t}_{\mathcal{N}_{i}}^{\prime}$ then $f_{i}(\mathbf{t})=f_{i}(\mathbf{t}^{\prime})$. To make explicit this dependence on the neighborhood subgraph, we will write $f_{i}(\mathbf{t}_{\mathcal{N}_{i}})\equiv f_{i}(\mathbf{t})$. A3: Isomorphic Graph Interference We assume that, if two units $i$ and $j$ have _isomorphic labeled neighborhood graphs_ , then they receive the same amount of interference, denoting isomorphism by $\simeq$, $G_{\mathcal{N}_{i}}^{\mathbf{t}}\simeq G_{\mathcal{N}_{j}}^{\mathbf{t}}\implies f_{i}(\mathbf{t}_{\mathcal{N}_{i}})=f_{j}(\mathbf{t}_{\mathcal{N}_{j}})\equiv f(G_{\mathcal{N}_{i}}^{\mathbf{t}})=f(G_{\mathcal{N}_{j}}^{\mathbf{t}})$. While Assumptions A1 and A2 are standard, A3 is new. This assumption allows us to study interference in a setting where units with similar neighborhood subgraphs experience similar amounts of interference. All our assumptions together induce a specific form for the potential outcomes, namely that they depend on neighborhood structure $G_{\mathcal{N}_{i}}^{\mathbf{t}}$, but not exactly who the neighbors are (information contained in $\mathcal{N}_{i}$) nor treatment assignments for those outside the neighborhood (information contained in $\mathbf{t}_{\mathcal{N}_{i}}$). Namely: ###### Proposition 1. Under assumptions A0-A3, potential outcomes in (2) for all units $i$ can be written as: $\displaystyle Y_{i}(t,\mathbf{t}_{-i})$ $\displaystyle=t\tau_{i}+f(G_{\mathcal{N}_{i}}^{\mathbf{t}})+\epsilon_{i},$ (3) where $\tau_{i}$ is the direct treatment effect on unit $i$, and $\epsilon_{i}$ is some baseline response. In addition, suppose that baseline responses for all units are equal to each other in expectation, i.e., for all $i$, $\mathbb{E}[\epsilon_{i}]=\alpha$. Then under assumptions A0-A3, for neighborhood graph structures $g_{i}$ of unit $i$ and treatment vectors $\mathbf{t}$, the ADE is identified as: $\displaystyle ADE=\frac{1}{n^{(1)}}\sum_{i=1}^{n}\mathbb{E}\bigl{[}T_{i}\times$ $\displaystyle\bigl{(}\mathbb{E}[Y_{i}|G_{\mathcal{N}_{i}}^{\mathbf{T}}\simeq g_{i}^{\mathbf{t}},T_{i}=1]$ $\displaystyle-\mathbb{E}[Y_{i}|G_{\mathcal{N}}^{\mathbf{T}}\simeq g_{i}^{\mathbf{t}},T_{i}=0]\bigr{)}\bigr{]},$ where $G_{\mathcal{N}_{i}}^{\mathbf{T}}$ is the neighborhood graph of $i$ labelled according to the treatment assignment $\mathbf{T}$. The proposition (whose proof is in the appendix) states that the interference received by a unit is a function of each unit’s neighborhood graph. Further, the outcomes can be decomposed additively into this function and the direct treatment effect on $i$. The proposition implies that the ADE is identified by matching each treated unit to one or more control units with an isomorphic neighborhood graph, and computing the direct effect on the treated using these matches. This effect is, in expectation over individual treatment assignments, equal to the ADE. #### 2.3 Subgraph Selection via Almost-Matching-Exactly Given Proposition 1 and the framework established in the previous section, we would ideally like to match treated and control units that have isomorphic neighborhood graphs. This would allow us to better estimate the ADE without suffering interference bias: for a treated unit $i$, if a control unit $j$ can be found such that $G_{\mathcal{N}_{i}}^{\mathbf{t}}\simeq G_{\mathcal{N}_{j}}^{\mathbf{t}}$, then $j$’s outcome will be identical in expectation to $i$’s counterfactual outcome and can be used as a proxy. Unfortunately, the number of non-isomorphic (canonically unique) graphs with a given number of nodes and edges grows incredibly quickly (Harary, 1994) and finding such matches is infeasible for large graphs. We therefore resort to counting all subgraphs that appear in a unit’s neighborhood graph and matching units based on the counts of those subgraphs. However, instead of exactly matching on the counts of those subgraphs, we match treated and control units if they have _similar_ counts, since matching exactly on all subgraph counts implies isomorphic neighborhoods and is also infeasible. Further, absolutely exact matches may not exist in real networks. Constructing inexact matches, in turn, requires a measure of relative graph importance. In Figure 1, for example, there are two control units that the treated unit may be matched to; if triangles contribute more to the interference function, it should be matched to the right; otherwise, if degree and/or two-stars are more important, it should be matched to the left. Of course, these relative importance measures might depend on the problem and we would like to learn them. Figure 1: Inexact matching presupposes an ordering of feature importance; should the the treated ego (black) be matched to a control whose neighborhood graph has the same number of units (left), or same number of triangles (right)? It might be tempting to match directly on $f$, as that would lead to unbiased inference. However, we abstain from doing so for two reasons. Firstly, in practice, the true interference is unknown and we could only match on estimated values of $f$; this suffers from all the problems that afflict matching on estimated propensity scores without appropriate adjustments (Abadie and Imbens, 2016) or parametric approximations (Rubin and Thomas, 1996). Such corrections or approximations do not currently exist for estimated interference functions and their development is an active area of research. Secondly, interpretability is a key component of our framework that would be lost matching on $f$-values; these values are scalar summaries of interference that depends on entire graphs. Estimating $f$ well would also likely require complex and uninterpretable nonparametric methods. In Section K of the appendix, we empirically compare matching units on $f$-values to our subgraph matching method via simulation. The loss of interpretability associated with matching on $f$ does not yield substantial gains in performance, even when using _true_ values of $f$ for matching, which is impossible in practice. Almost-Matching-Exactly (AME) (Wang et al., 2019; Dieng et al., 2019; Awan et al., 2019) provides a framework for the above problem that is explicitly geared towards building interpretable, high-quality matches on discrete covariates, which in our setting are the counts of the treated subgraphs in the neighborhood. AME performs inexact matching while learning importance weights for each covariate from a training set, prioritizing matches on more important covariates. In this way, it neatly addresses the challenge of inexact matching by learning a metric specific to discrete covariates (namely, a weighted Hamming distance). Formally, AME matches units so as to optimize a flexible measure of match quality. For each treated unit $i$, solving the AME problem is equivalent to finding: $\displaystyle\boldsymbol{\theta}^{i^{*}}$ $\displaystyle\in\operatorname*{arg\,max}_{\boldsymbol{\theta}\in\\{0,1\\}^{p}}\boldsymbol{\theta}^{T}\mathbf{w}$ (4) $\displaystyle\text{such that }\exists j:t_{j}=0\text{ and }\mathbf{x}_{j}\circ\boldsymbol{\theta}=\mathbf{x}_{i}\circ\boldsymbol{\theta}$ where $\circ$ denotes the Hadamard product, $\mathbf{w}$ is a vector of weights and $\boldsymbol{x}_{i},\boldsymbol{x}_{j}$ are vectors of binary covariates for units $i$ and $j$ that we might like to match on. In our network interference setting, these are vectors of subgraph counts. The vector $\mathbf{w}$ denotes the importance of each subgraph in causing interference. We will leverage both information on outcomes and networks to construct an estimate for it. We start by enumerating (up to isomorphism) all the $p$ subgraphs $g_{1},\dots,g_{p}$ that appear in any of the $G_{\mathcal{N}_{i}}^{\mathbf{t}},i\in 1,\dots,n$. The covariates for unit $i$ are then given by $S(G_{\mathcal{N}_{i}}^{\mathbf{t}})=(S_{1}(G_{\mathcal{N}_{i}}^{\mathbf{t}}),\dots,S_{p}(G_{\mathcal{N}_{i}}^{\mathbf{t}}))$ where $S_{k}(G_{\mathcal{N}_{i}}^{\mathbf{t}})$ denotes the number of times subgraph $g_{k}$ appears in the subgraphs of $G_{\mathcal{N}_{i}}^{\mathbf{t}}$. These counts are then converted into binary indicators that are one if the count of subgraph $g_{k}$ in each unit’s neighborhood is exactly $x$, for all $x$ observed in the data. Thus, units will be matched exactly if they have identical subgraph counts. We then approximately solve the problem in Equation (4) to find the optimally important set of subgraphs upon which to exactly match each treated unit, such that there is at least one control unit that matches exactly with the treated unit on the chosen subgraph counts. The key idea behind this approach is that we want to match units exactly on subgraph counts that contribute significantly to the interference function, trading off exactly-matching on these important subgraphs with potential mismatches on subgraphs that contribute less to interference. In practice, our implementation enumerates all subgraphs in each unit’s neighborhood and stores the count of each pattern – this is computationally challenging. There is a growing body of work on efficient counting algorithms for pre-specified small patterns (up to 4-5 nodes) but there is little research on fast methods to both enumerate and count all motifs in a graph (e.g., Pinar et al., 2017; Marcus and Shavitt, 2010; Hu et al., 2013). Empirically, we see that this enumeration takes less than 30 seconds for 50 units. ###### The FLAME Algorithm for AME. The Fast Large Almost Matching Exactly (FLAME) algorithm (Wang et al., 2019) approximates the solution to the AME problem. The procedure starts by exactly matching all possible units on all covariates. It then drops one covariate at a time, choosing the drop maximizing the match quality $\mathtt{MQ}$ at that iteration, defined: $\displaystyle\mathtt{MQ}=C\cdot\mathtt{BF}-\widehat{\mathtt{PE}}_{Y}.$ (5) The match quality is the sum of a balancing factor $\mathtt{BF}$ and a predictive error $\widehat{\mathtt{PE}}_{Y}$, with relative weights determined by the hyper-parameter $C$. The balancing factor is defined as the proportion of treated units plus the proportion of control units matched at that iteration. Introducing the balancing factor into the objective has the advantage of encouraging more units to be matched, thereby minimizing variance of estimators (see Wang et al., 2019). In our setting, the second component of the match quality, predictive error, takes the form: $\displaystyle\widehat{\mathtt{PE}}_{Y}=\operatorname*{arg\,min}_{h\in\mathcal{F}_{1}}\sum_{i}^{n}(Y_{i}-h(S(G_{\mathcal{N}_{i}}^{\mathbf{t}})\circ\boldsymbol{\theta},T_{i}))^{2}$ (6) for some class of functions $\mathcal{F}_{1}$. It is computed using a holdout training set and discourages dropping covariates that are useful for predicting the outcome. In this way, FLAME strikes a balance between matching many units and ensuring these matches are of high-quality. By using a holdout set to determine how useful a set of variables is for out-of-sample prediction, FLAME learns a measure of covariate importance via a weighted Hamming distance. Specifically, it learns a vector of importance weights $\mathbf{w}$ for the different subgraph counts that minimizes $\mathbf{w}^{T}\mathbb{I}[S(G_{\mathcal{N}_{i}}^{\mathbf{t}})\neq S(G_{\mathcal{N}_{j}}^{\mathbf{t}})]$, where $\mathbb{I}[S(G_{\mathcal{N}_{i}}^{\mathbf{t}})\neq S(G_{\mathcal{N}_{j}}^{\mathbf{t}})]$ is a vector whose $k^{\textrm{th}}$ entry is 0 if the labeled neighborhood graphs of $i$ and $j$ have the same count of subgraph $k$, and 1 otherwise. To this match quality term, we add a network fit term to give subgraphs more weight that are highly predictive of overall network structure. We fit a logistic regression model in which the edges $(i,j)$ between units $i,j$ are independent given $\mathcal{N}_{i},\mathcal{N}_{j}$, and dependent on the subgraph counts of units $i$ and $j$: $(i,j)\stackrel{{\scriptstyle iid}}{{\sim}}\textrm{Bern}(\textrm{logit}(\beta_{1}^{T}S(G_{\mathcal{N}_{i}}^{\mathbf{t}})+\beta_{2}^{T}S(G_{\mathcal{N}_{j}}^{\mathbf{t}})))$ To the match quality in the original formulation, we then add $\widehat{\mathtt{PE}}_{G}$, defined to be the _AIC_ (Akaike, 1974) of this fitted model, weighted by a hyperparameter $D$. Therefore, at each iteration: $\displaystyle\mathtt{MQ}=C\cdot\mathtt{BF}-\widehat{\mathtt{PE}}_{Y}+D\cdot\widehat{\mathtt{PE}}_{G}.$ Thus, we penalize not only subgraph drops that impede predictive performance or making matches, but also those that make the observed network unlikely. $\widehat{\mathtt{PE}}_{G}$ represents the empirical prediction error of the chosen set of statistics for the observed graph: if $\widehat{\mathtt{PE}}_{G}$ is low, then the chosen subgraphs do a good job of predicting the observed graph. This error is also evaluated at a minimum over another class of prediction functions, $\mathcal{F}_{2}$. This term in the AME objective is justified by Assumption A3: units that have isomorphic labeled neighborhood graphs should experience the same amount of interference, and subgraph counts should be predictive of neighborhood graph structure. Our approach to estimating the ADE is therefore as follows. (1) For each unit $i$, count and label all types of subgraphs in $G_{\mathcal{N}_{i}}^{\mathbf{t}}$. (2) Run FLAME, encouraging large numbers of matches on subgraph counts, while using the covariates that are most important for predicting the outcome and the network. (3) Estimate ADE as $\widehat{ADE}$, by computing the difference in means for each matched group and then averaging across matched groups, weighted by their size. Since our approach is based on FLAME, we call it _FLAME-Networks_. Extensions. FLAME-Networks immediately extends to handling unit-level covariate information for baseline adjustments; we simply concatenate subgraph information and covariate information in the same dataset and then make almost-exact matches. FLAME-Networks will automatically learn the weights of both subgraphs and baseline covariates to make matches that take both into account. Another straightforward extension considers interference not just in the immediate neighborhood of each unit, but up to an arbitrary number of hops away. To extend FLAME-Networks in this way, it is sufficient to enumerate subgraphs in the induced neighborhood graph of each unit $i$ where the vertices considered are those with a path to $i$ that is at most $k$ steps long. Given these counts, our method proceeds exactly as before. ### 3 EXPERIMENTS We begin by evaluating the performance of our estimator in a variety of simulated settings, in which we vary the form of the interference function. We find that our approach performs well in many distinct settings. In Section M of the appendix, we also assess the quality of the matches constructed. We simulate graphs from an Erdős-Rènyi model: $G\sim\textrm{Erd\H{o}s-R\\`{e}nyi}(n,q)$, by which every possible edge between the $n$ units is created independently, with probability $q$. In Sections H and I of the appendix, we also perform experiments on cluster- randomized and real-world networks. Treatments for the whole sample are generated with $\Pr(\mathbf{T}=\mathbf{t})=\binom{n}{n^{(1)}}^{-1}$, where $n^{(1)}$ is the number of treated units. Outcomes are generated according to $Y_{i}(t,\mathbf{t}_{-i})=\mathbf{t}\tau_{i}+f(G_{\mathcal{N}_{i}}^{\mathbf{t}})+\epsilon_{i}$, where $\epsilon\sim N(\mathbf{0},I_{n})$ represents a baseline outcome; $\tau_{i}\sim N(\mathbf{5},I_{n})$ represents a direct treatment effect, and $f$ is the interference function. In Section J of the appendix, we consider a setting in which the errors are heteroscedastic. For the interference function, we use additive combinations of the subgraph components in Table 1 and define $m_{ip},p=1,\dots,7$ to be the counts of feature $p$ in $G_{\mathcal{N}_{i}\cup\\{i\\}}^{\mathbf{t}}$. Lastly, the counts of each component are normalized to have mean 0 and standard deviation 1. $d_{i}$: | The treated degree of unit $i$ ---|--- $\Delta_{i}$: | The number of triangles in $G_{\mathcal{N}_{i}\cup\\{i\\}}^{\mathbf{t}}$ with at least one treated unit. $\bigstar_{i}^{k}$: | The number of $k$-stars in $G_{\mathcal{N}_{i}\cup\\{i\\}}^{\mathbf{t}}$ with at least one treated unit. $\dagger_{i}^{k}$: | The number of units in $G_{\mathcal{N}_{i}\cup\\{i\\}}^{\mathbf{t}}$ with degree $\geq k$ and at least one treated unit among their neighbors. $B_{i}$: | The vertex betweenness of unit $i$. $C_{i}$: | The closeness centrality of unit $i$. Table 1: Interference components used in experiments; see the appendix for more details. We compare our approach with different methods to estimate the ADE under interference: Naïve. The simple difference in means between treatment and control groups assuming no interference. All Eigenvectors. Eigenvectors for the entire adjacency matrix are computed with every treated unit matched to the control unit minimizing the Mahalanobis distance between the eigenvectors, weighing the $k$’th eigenvector by $1/k$. The idea behind this estimator is that the eigendecomposition of the adjacency matrix encodes important information about the network and how interference might spread within it. First Eigenvector. Same as All Eigenvectors except units are matched only on their values of the largest-eigenvalue eigenvector. Stratified Naïve. The stratified naïve estimator as discussed by Sussman and Airoldi (2017). A weighted difference-in-means estimator where units are divided into strata defined by their treated degree (number of treated vertices they are connected to), and assigned weight equal to the number of units within the stratum in the final difference of weighted averages between treated and control groups. SANIA MIVLUE. The minimum integrated variance, linear unbiased estimator under assumptions of symmetrically received interference and additivity of main effects, when the priors on the baseline outcome and direct treatment effect have no correlation between units; proposed by Sussman and Airoldi (2017). FLAME-Networks. Our proposed method. In all simulations, the two components of the PE function are weighted equally, and a ridge regression is used to compute outcome prediction error. #### 3.1 Experiment 1: Additive Interference First we study a setting in which interference is an additive function of the components in Table 1. Outcomes in this experiment have the form: $Y_{i}=\gamma_{1}d_{i}+\gamma_{2}\Delta_{i}+\gamma_{3}\bigstar^{2}_{i}+\gamma_{4}\bigstar^{4}_{i}+\gamma_{5}\dagger_{i}^{3}+\gamma_{6}B_{i}+\gamma_{7}C_{i}+\epsilon_{i}$, with $\epsilon_{i}\sim N(0,1)$. We simulate 50 datasets for each setting, in which the units are in an $ER(50,0.05)$ graph. Table 2 shows values for the $\gamma_{i}$ in each of our experimental settings. Feature | $d_{i}$ | $\Delta_{i}$ | $\bigstar^{2}_{i}$ | $\bigstar^{4}_{i}$ | $\dagger^{3}$ | $B_{i}$ | $C_{i}$ ---|---|---|---|---|---|---|--- Weight | $\gamma_{1}$ | $\gamma_{2}$ | $\gamma_{3}$ | $\gamma_{4}$ | $\gamma_{5}$ | $\gamma_{6}$ | $\gamma_{7}$ Setting 1 | 0 | 10 | 0 | 0 | 0 | 0 | 0 Setting 2 | 10 | 10 | 0 | 0 | 0 | 0 | 0 Setting 3 | 0 | 10 | 1 | 1 | 1 | 1 | -1 Setting 4 | 5 | 1 | 10 | 1 | 1 | 1 | -1 Table 2: Settings for Experiment 1. Results for Experiment 1 are reported in Figure 2. FLAME-Networks outperforms all other methods both in terms of average error, and standard deviation over the simulations. This is likely because FLAME-Networks learns weights for the subgraphs that are proportionate to those we use at each setting, and matches units on subgraphs with larger weights. When the interference function is multiplicative instead of additive, FLAME-Networks performs similarly; results are in the appendix. Figure 2: Results from Experiment 1. Each violin plot represents the distribution over simulations of absolute estimation error. The panels are numbered according to the parameter settings of the simulations. Violin plots are blue if the method had mean error lower than or equal to FLAME-Networks’ and red otherwise. The black line inside each violin is the median error. The dashed line is FLAME-Networks’ mean error. #### 3.2 Experiment 2: Covariate Adjustment A strength of FLAME-Networks is its ability to natively account for covariate information. We analyze a setting in which baseline effects are dependent on an additional discrete-valued covariate, $x$, that is observed alongside the network. Outcomes take the form $Y_{i}=t\tau_{i}+f(G_{\mathcal{N}_{i}}^{\mathbf{t}})+\beta x_{i}+\epsilon_{i}$, where $x_{i}$ is chosen uniformly at random from $\\{1,2,3\\}$ for each unit, and $\beta$ is fixed at 15. This means that our sample is divided into 3 strata defined by the covariate values. We ran FLAME- Networks with a dataset consisting of subgraph counts, plus observed covariates for each unit. For comparison with the other methods, we first regress $Y$ on $x$, and then use the residuals of that regression as outcomes for the other methods. This way, the initial regression will account for the baseline effects of $x$, and the residuals contain only potential interference. The interference function takes the form $f(G_{\mathcal{N}_{i}}^{\mathbf{t}})=d_{i}+\Delta_{i}+B_{i}$, which is what we are trying to learn with the other methods. We simulate the sample network from $ER(50,0.05)$. Results are displayed in Table 3. FLAME-Networks performs, on average, better than all the other methods. Results in Section L of the supplement show that when $\beta$ is increased, none of the methods suffer in performance. While regression adjustment prior to estimation seems to have a positive impact on the performance of other methods in the presence of additional covariates, FLAME-Networks performs best. This is because FLAME-Networks is built to easily handle the inclusion of covariates in its estimation procedure. Method | Median | 25th q | 75th q ---|---|---|--- FLAME-Networks | 0.39 | 0.21 | 0.59 First Eigenvector | 0.47 | 0.40 | 0.83 All Eigenvectors | 0.55 | 0.29 | 0.79 Naive | 0.53 | 0.36 | 0.92 SANIA | 1.93 | 1.75 | 2.25 Stratified | 4.49 | 4.45 | 4.53 Table 3: Results from Experiment 2 with $\beta=5$. Median and 25th and 75th percentile of absolute error over 40 simulated datasets. #### 3.3 Experiment 3: Misspecified Interference We now study the robustness of our estimator in a setting in which one of our key assumptions – A3 – is violated. Specifically, we now allow for treated and control units to receive different amounts of interference, even if they have the same labelled neighborhood graphs. We do this by temporarily eliminating all control-control edges in the network and then counting the features in Table 1 used to assign interference. That is, consider a unit $i$ with a single, untreated neighbor $j$. In our new setting, if degree is a feature of the interference function, then $i$ being treated implies $i$ receives interference from $j$. But if $i$ is untreated, then $i$ would receive no interference from $j$, because its neighbor is also untreated. This crucially implies that FLAME-Networks will be matching–and estimating the ADE from–individuals that do not necessarily receive similar amounts of interference. In this setting, we generate interference according to: $f_{i}=(5-\gamma)d_{i}+\gamma\Delta_{i}$ for $\gamma\in[0,5]$ and assess the performance of FLAME-Networks against that of the SANIA and stratified estimators. Results are shown in Figure 3. We see that, when degree is the only component with weight in the interference function, FLAME-Networks performs better than the stratified estimator, but worse than the SANIA estimator, which leverages aspects of the graph related to degree. However, our performance improves as $\gamma$ increases and the true interference depends more on triangle counts since the triangle counts available to FLAME- Networks represent the actual interference pattern more frequently than the degree counts did. Thus, we see that although violation of our method’s assumptions harms its performance, it still manages at times to outperform estimators that rely too heavily on degree. Figure 3: Results from Experiment 3. These simulations were run on an ER(75, 0.07) graph. The bands around the lines represent 25th and 75th quantiles of the 50 simulations, for each value of $\gamma$. ### 4 APPLICATION In this section, we demonstrate the practical utility of our method. We use data collected by Banerjee et al. (2013) on social networks for 75 villages in Karnataka, India. They are a median distance of 46 km from one another other, motivating the assumption that network interference is experienced solely between individuals from the same village. For each village, we study the effect of 1. lack of education on election participation; 2. lack of education on Self-Help-Group (SHG) participation; and 3. being male on SHG participation. We proxy election participation by ownership of an election card. We compare our estimates – which account for network interference – to naive estimates – which assume no network interference. Data pre-processing is summarized in the appendix. For ADE estimates, we assume the treatment is randomly assigned. We find that lack of education is associated with higher SHG participation, and that males are less likely to participate in SHGs than females (see Figure 4). These results make sense in the context of developing countries where SHGs are mainly utilized by females in low-income families, which generally have lower education levels. We observe that education does not impact election participation; in developing countries, an individual’s decision to participate in an election may be driven by factors such caste, religion, influence of local leaders and closeness of race (Shachar and Nalebuff, 1999; Gleason, 2001). FLAME-Networks matches units in each village by subgraph counts and covariate values to estimate the ADE. Looking at the matched groups, we discover that subgraphs such as 2-stars and triangles were important for matching, implying that second-order connections could be affecting interference in this setting. Further details of the matched groups are in Section F. Figure 4 plots naive and FLAME-Networks ADE estimates. We find a significant difference between our estimates and the naive estimates when estimating the effect of being male on participation in SHGs. The naive estimator overestimates the treatment effect, which is possible when ignoring network interference. It is plausible, in this setting, that interference would heighten these effects for all outcomes. This is because individuals from similar social backgrounds or gender tend to interact more together, and, therefore, are more likely to influence each other’s participation decision, both in elections and in SHGs. Figure 4: Naive and FLAME-Networks ADE estimates and their difference. Red, blue, and green respectively correspond to (treatment, outcome) pairs: (no education, election participation), (no education, SHG participation), and (gender, SHG participation). ### 5 DISCUSSION Conventional estimators for treatment effects in randomized experiments will be biased when there is interference between units. We have introduced FLAME- Networks – a method to recover direct treatment effects in such settings. Our method is based on matching units with similar neighborhood graphs in an almost-exact way, thus producing interpretable, high-quality results. We have shown that FLAME-Networks performs better than existing methods both on simulated data, and we have used real-world data to show how it can be applied in a real setting. Our method extends easily to settings with additional covariate information for units and to taking into account larger neighborhoods for interference. 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Using A0-A3 we can write (2) as: $\displaystyle Y_{i}(t,\mathbf{t}_{-i})$ $\displaystyle=t\tau_{i}+f_{i}(\mathbf{t}_{-i})+\epsilon_{i}$ (By A1) $\displaystyle=t\tau_{i}+f_{i}(\mathbf{t}_{\mathcal{N}_{i}})+\epsilon_{i}$ (By A2) $\displaystyle=t\tau_{i}+f(G_{\mathcal{N}_{i}}^{\mathbf{t}})+\epsilon_{i}$ $\displaystyle\mbox{(By A3)}.$ (7) This proves Equation (3). The first line agrees with the additivity assumption, which permits a contribution $f_{i}(\mathbf{t}_{-i})$ to $Y_{i}(t,\mathbf{t}_{-i})$ arising from anywhere within the rest of the graph. The second line states that this contribution is limited to $i$’s neighborhood. The third line states that the contribution depends only the neighborhood structure and not anything else about the neighbors. To prove identification for the ADE, we must show that the individual treatment effect is identified for a treated unit $i$. We need to show that: $\mathbb{E}[Y_{i}(1,\mathbf{0})-Y_{i}(0,\mathbf{0})]=\mathbb{E}[Y_{i}|T_{i}=1,G_{\mathcal{N}_{i}}^{\mathbf{T}}=g_{i}^{\mathbf{t}}]-\mathbb{E}[Y_{i}|T_{i}=0,G_{\mathcal{N}_{i}}^{\mathbf{T}}=g_{i}^{\mathbf{t}}]$. We have first: $\displaystyle\mathbb{E}[Y_{i}(1,\mathbf{0})-Y_{i}(0,\mathbf{0})]$ $\displaystyle=\mathbb{E}[t\tau_{i}+f(G_{\mathcal{N}_{i}}^{\mathbf{0}})+\epsilon_{i}-f(G_{\mathcal{N}_{i}}^{\mathbf{0}})-\epsilon_{i}]$ $\displaystyle=\mathbb{E}[\epsilon_{i}+\tau_{i}-\epsilon_{i}]=\tau_{i}.$ (8) Second, we have: $\displaystyle\mathbb{E}[Y_{i}|G_{\mathcal{N}_{i}}^{\mathbf{T}}\simeq g_{i}^{\mathbf{t}},T_{i}=1]-\mathbb{E}[Y_{i}|G_{\mathcal{N}_{i}}^{\mathbf{T}}\simeq g_{i}^{\mathbf{t}},T_{i}=0]$ $\displaystyle=$ $\displaystyle\mathbb{E}[Y_{i}(1,\mathbf{T}_{-i})T_{i}+$ $\displaystyle\quad+Y_{i}(0,\mathbf{T}_{-i})(1-T_{i})|G_{\mathcal{N}_{i}}^{\mathbf{T}}\simeq g_{i}^{\mathbf{t}},T_{i}=1]$ $\displaystyle-$ $\displaystyle\mathbb{E}[Y_{i}(1,\mathbf{T}_{-i})T_{i}$ $\displaystyle\quad+Y_{i}(0,\mathbf{T}_{-i})(1-T_{i})|G_{\mathcal{N}_{i}}^{\mathbf{T}}\simeq g_{i}^{\mathbf{t}},T_{i}=0]$ $\displaystyle=$ $\displaystyle\mathbb{E}[\tau_{i}+f(g_{i}^{\mathbf{t}})+\epsilon_{i}|G_{\mathcal{N}_{i}}^{\mathbf{T}}\simeq g_{i}^{\mathbf{t}},T_{i}=1]$ $\displaystyle-\mathbb{E}[f(g_{i}^{\mathbf{t}})+\epsilon_{i}|G_{\mathcal{N}_{j}}^{\mathbf{T}}=g_{i}^{\mathbf{t}},T_{i}=0]$ $\displaystyle=$ $\displaystyle\alpha+f(g_{i}^{\mathbf{t}})+\tau_{i}-\alpha-f(g_{i}^{\mathbf{t}})$ $\displaystyle=$ $\displaystyle\tau_{i}.$ (9) The first equality follows from the definition of $Y_{i}$, the second equality from the result in (3) and A3, and the third equality follows from independence of $T$ and $Y$ given in A0: $\mathbb{E}[\epsilon_{i}|T_{i}]=\mathbb{E}[\epsilon_{i}]=\alpha$ for all $i$. Finally, we can use both of the results above to obtain the ADE: $\displaystyle\frac{1}{n^{(1)}}\sum_{i=1}^{n}\mathbb{E}[T_{i}\times(\mathbb{E}[Y_{i}|G_{\mathcal{N}_{i}}^{\mathbf{T}}\simeq g_{i}^{\mathbf{t}},T_{i}=1]$ $\displaystyle\qquad\qquad\quad-\mathbb{E}[Y_{i}|G_{\mathcal{N}_{i}}^{\mathbf{T}}\simeq g_{i}^{\mathbf{t}},T_{i}=0])]$ $\displaystyle=\frac{1}{n^{(1)}}\sum_{i=1}^{n}\mathbb{E}[T_{i}\tau_{i}]$ (By (9)) $\displaystyle=\frac{1}{n^{(1)}}\sum_{i=1}^{n}\mathbb{E}[T_{i}\times(\mathbb{E}[Y_{i}(1,\mathbf{0})-Y_{i}(0,\mathbf{0})])]$ (By (8)) $\displaystyle=\frac{1}{n^{(1)}}\sum_{i=1}^{n}\Pr(T_{i}=1)(\mathbb{E}[Y_{i}(1,\mathbf{0})-Y_{i}(0,\mathbf{0})])$ $\displaystyle=\frac{1}{n^{(1)}}\sum_{i=1}^{n}\frac{n^{(1)}}{n}(\mathbb{E}[Y_{i}(1,\mathbf{0})-Y_{i}(0,\mathbf{0})])$ $\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[Y_{i}(1,\mathbf{0})-Y_{i}(0,\mathbf{0})],$ where $\Pr(T_{i}=1)=\frac{n^{(1)}}{n}$ by assumption of complete randomization. ∎ ### B Additional Theoretical Results We study the expected error for one AME match on subgraphs under two assumptions: that the true weights for the AME objective (the weighted Hamming distance) are known, and that the candidate units for matching all have independently generated neighborhoods, and none of the units in these neighborhoods are being matched. Additional information on this setting is available in the proof. ###### Proposition 2. (Oracle AME Error With Independent Neighborhoods) Suppose that there are $N$ independently generated graphs, each with $n$ vertices, and all i.i.d. from the same distribution over graphs: $\Pr(G_{1}=g_{1},\dots,G_{N}=g_{N})=\prod_{i=1}^{N}p(g_{i})$. Assume matches are only allowed between units in different graphs. Suppose additionally that $n^{(1)}$ randomly chosen units within each graph are assigned treatment, so that $\Pr(\mathbf{T}_{i}=\mathbf{t}_{i})=\binom{n}{n^{(1)}}^{-1}$. Assume further that interference functions obey the following: $|f(g)-f(h)|\leq K\mathbf{w}^{T}\mathbb{I}[S(g)\neq S(h)]$, where $\mathbf{w}$ is a vector of positive, real-valued, importance weights for the subgraphs counts, such that $\|\mathbf{w}\|_{1}=M_{n}$ for some constant $0<M_{n}<\infty$, and such that the condition above is satisfied for $\mathbf{w}$, and $\mathbb{I}[S(g)\neq S(h)]$ is the Hamming distance: a vector of the same length as $\mathbf{w}$ that is 0 at position $k$ if graphs $g$ and $h$ have the same count of subgraph $k$, and 1 otherwise. Assume that baseline responses have variance $Var(\epsilon_{i})=\sigma^{2}\;\forall i$. Then, for a treatment unit $i$, if $j$ solves the AME problem, i.e., $j\in\operatorname*{arg\,min}\limits_{\begin{subarray}{c}{k=1,\dots,n}\\\ {T_{k}=0}\end{subarray}}\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}})\neq S(G_{\mathcal{N}_{k}}^{\mathbf{T}})]$, under A0-A3: $\displaystyle\mathbb{E}\left[|Y_{i}-Y_{j}-\tau_{i}|\big{|}T_{i}=1,G_{\mathcal{N}_{i}}^{\mathbf{t}}=h_{\mathcal{N}_{i}}^{\mathbf{t}}\right]\leq\sqrt{2}\sigma$ $\displaystyle+K\binom{n-1}{n^{(1)}}^{-1}\times\sum_{g\in\mathcal{G}_{n}}\sum_{\begin{subarray}{c}\mathbf{t}\in\mathcal{T}\\\ t_{j}=0\end{subarray}}\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}})\neq S(g_{\mathcal{N}_{j}}^{\mathbf{t}})]p(g)$ $\displaystyle\times\left(\frac{n^{(1)}}{n}+\frac{n-n^{(1)}}{n}C(g_{\mathcal{N}_{j}}^{\mathbf{t}})\right)^{N-2},$ where $\mathcal{G}_{n}$ is the set of all graphs with $n$ units, and $C(h_{\mathcal{N}_{j}}^{\mathbf{t}})\leq 1$ for all $g$ and $\mathbf{t}$. A proof is available in the following section. The first element in the right hand side of the inequality is the standard deviation of the baseline responses. One summation is over all possible graphs with $n$ units, and the other summation is over possible treatment assignments. The expression inside the summation is the product of three terms. First, the weighted Hamming distance between a graph and the target graph we are trying to match. Second, the probability of observing that graph. Third, an upper bound on the probability that unit $j$ is among the minimizers of the weighted Hamming distance. Note that $\mathbf{w}^{T}\mathbb{I}[S(g_{\mathcal{N}_{j}}^{\mathbf{t}})\neq S(g_{\mathcal{N}_{j}}^{\mathbf{t}})]p(g)$ is bounded for fixed $n$ for all $g$ and $\mathbf{t}$. This implies that the bound converges to $2\sqrt{\sigma}$ as $N\rightarrow\infty$, as long as the size of neighborhood graphs is held fixed, because perfect matching is possible with large amounts of data in this regime. #### B.1 Proof of Proposition 2 We briefly review some notation and assumptions to be used in this proof. For the purposes of theory, we study a simplified setting, in which we have to AME-match a unit $i$ to one unit in a set of candidate units of size $N$ such that: a) all the candidate units belong to disconnected graphs, which we refer to as candidate graphs. b) within each candidate graph there is only one pre- determined candidate unit c) candidate units have neighborhood graphs denoted by $G_{\mathcal{N}_{j}}$. d) all the candidate graphs are drawn independently from the same distribution over graphs: $\Pr(G_{\mathcal{N}_{1}}=g_{1},\dots,G_{\mathcal{N}_{N}}=g_{N})=\prod_{i=1}^{N}p(g_{i})$. The support of $p$ will be $\mathcal{G}_{n}$: the set of all graphs with exactly $n$ units. We use $g_{\mathcal{N}_{i}}$ to denote the subgraph induced over $g$ by the units in the set of neighbors of unit $i$, $\mathcal{N}_{i}\subseteq V(g)$, i.e., $g_{\mathcal{N}_{i}}$ is the graph consisting only of the vertices that share an edge with $i$, in $g$, and of the edges in $g$ that are between these vertices. The ego $i$ is not included in $g_{\mathcal{N}_{i}}$. Assigned treatments are denoted by $\mathbf{T}$, where $\mathbf{T}\in\\{0,1\\}^{n}$, but in this setting treatment assignment is assumed to be independent within the $N$ candidate graphs. Formally, the assumption we make is that $\Pr(\mathbf{T}_{1}=\mathbf{t}_{1},\dots,\mathbf{T}_{N}=\mathbf{t}_{N})=\prod_{i=1}^{N}\binom{n}{n^{(1)}}^{-1}$, i.e., $n^{(1)}$ units are always treated uniformly at random within each of the $N$ candidate graphs. The direct treatment effect for any unit $i$ is given by $\tau_{i}$. We use $\mathbb{I}[S(g)\neq S(h)]$ to indicate the Hamming distance between subgraph counts of graphs $g$ and $h$. This means that $\mathbb{I}[S(g)\neq S(h)]$ is a vector of size $|\mathcal{G}_{n}|$ that will be 1 in the $\ell^{th}$ entry if $g$ and $h$ have the same amount of occurrences of graph $g_{\ell}$ among their subgraphs. Note that this distance is coloring sensitive: two subgraphs that are isomorphic in shape but not labels will belong to different entries in this distance. The matched group of a treated unit $i$, denoted $\mathtt{MG}_{i}$ is the set of all control units that match with $i$. In our setting $j\in\mathtt{MG}_{i}$ if it solves the AME problem, that is $j\in\operatorname*{arg\,min}\limits_{\begin{subarray}{c}k=1\dots,n\\\ T_{k}=0\end{subarray}}\mathbf{w}^{T}\mathbb{I}[S(G_{\mathcal{N}_{k}}^{\mathbf{T}})\neq S(g_{\mathcal{N}_{i}}^{\mathbf{t}})]$. Finally, we assume that both the graph for the unit we want to match and the treatment assignment for that unit’s graph are fixed: $\mathbf{t}_{i}$ is the treatment assignment in the graph of $i$, and $h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}}$ is the neighborhood graph of $i$, where $h$ denotes unit $i$’s graph. All other notation is as in the main paper. ###### Proof. We start by upper-bounding our quantity of interest as follows: $\displaystyle\mathbb{E}[|Y_{i}-Y_{j}-\tau_{i}|\big{|}j\in\mathtt{MG}_{i}]$ $\displaystyle=\mathbb{E}[|Y_{i}(1,\mathbf{t}_{i-i})-Y_{j}(0,\mathbf{T}_{j-j})-\tau_{i}|\big{|}j\in\mathtt{MG}_{i}]$ $\displaystyle=\mathbb{E}[|\tau_{i}+f(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})+\epsilon_{i}-f(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}})-\epsilon_{j}-\tau_{i}|\big{|}j\in\mathtt{MG}_{i}]$ $\displaystyle\leq\mathbb{E}[|f(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})-f(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}})|\big{|}j\in\mathtt{MG}_{i}]$ $\displaystyle\quad+\mathbb{E}[|\epsilon_{i}-\epsilon_{j}|\big{|}j\in\mathtt{MG}_{i}]$ $\displaystyle\leq K\mathbb{E}[\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}})]\big{|}j\in\mathtt{MG}_{i}]$ $\displaystyle\quad+\mathbb{E}[|\epsilon_{i}-\epsilon_{j}|\big{|}j\in\mathtt{MG}_{i}],$ (10) where the notation $\mathbf{T}_{j-j}$ denotes the treatment indicator for candidate graph $j$ for all units except $j$. The first equality follows from A0 since the event $j\in\mathtt{MG}_{i}$ implies that $T_{j}=0$, as only control units are allowed in the matched groups. The second equality follows from Proposition 1. The first inequality is an application of the triangle inequality. The last line follows from the condition on the interference functions. Consider the second term. We can use the Cauchy-Schwarz inequality to construct a simple upper bound on it: $\displaystyle\mathbb{E}[|\epsilon_{i}-\epsilon_{j}|\big{|}j\in\mathtt{MG}_{i}]=\mathbb{E}[|\epsilon_{i}-\epsilon_{j}|]$ $\displaystyle\leq\sqrt{\mathbb{E}[(\epsilon_{i}-\epsilon_{j})^{2}]}$ $\displaystyle=\sqrt{\mathbb{E}[\epsilon_{i}^{2}]+\mathbb{E}[\epsilon_{j}^{2}]-2\mathbb{E}[\epsilon_{i}]\mathbb{E}[\epsilon_{j}]}=\sqrt{2}\sigma$ where the last equality follows for the fact that the $\epsilon_{i}$ have mean 0 and are independent, with $Var(\epsilon_{i})=\sigma^{2}$ for all $i$. Consider now the term $\mathbb{E}[\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}})]|j\in\mathtt{MG}_{i}]$. To upper-bound this, we write it out as follows using the definition of expectation: $\displaystyle\mathbb{E}[\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}})]|j\in\mathtt{MG}_{i}]$ $\displaystyle=\sum_{g\in\mathcal{G}_{n}}\sum_{\mathbf{t}\in\mathcal{T},t_{j}=0}\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(g_{\mathcal{N}_{j}}^{\mathbf{t}_{j}})]$ $\displaystyle\qquad\qquad\times\Pr(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}}=g_{\mathcal{N}_{j}}^{\mathbf{t}},\mathbf{T}_{j}=\mathbf{t}|j\in\mathtt{MG}_{i}).$ We want to find an upper bound on $\Pr(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}}=g_{\mathcal{N}_{j}}^{\mathbf{t}},\mathbf{T}_{j}=\mathbf{t}|j\in\mathtt{MG}_{i})$. We start by writing this quantity out as a product of two probabilities: $\displaystyle\Pr(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}}=g_{\mathcal{N}_{j}}^{\mathbf{t}},\mathbf{T}_{j}=\mathbf{t}|j\in\mathtt{MG}_{i})$ $\displaystyle=\Pr(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}}=g_{\mathcal{N}_{j}}^{\mathbf{t}}|j\in\mathtt{MG}_{i},\mathbf{T}_{j}=\mathbf{t})$ $\displaystyle\times\Pr(\mathbf{T}_{j}=\mathbf{t}|j\in\mathtt{MG}_{i})$ $\displaystyle=\Pr(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}}=g_{\mathcal{N}_{j}}^{\mathbf{t}}|j\in\mathtt{MG}_{i},\mathbf{T}_{j}=\mathbf{t})\binom{n-1}{n^{(1)}}^{-1}.$ Note that $\Pr(\mathbf{T}_{j}=\mathbf{t}|j\in\mathtt{MG}_{i})=\binom{n-1}{n^{(1)}}^{-1}$ because treatment is uniformly randomized with $n^{(1)}$ units always treated in each candidate graph, but $T_{j}=0$ conditionally on $j\in\mathtt{MG}_{i}$. We use Bayes’ rule to write out the first term in the final product as $\Pr(G_{\mathcal{N}_{j}}^{\mathbf{T}}=g_{\mathcal{N}_{j}}^{\mathbf{t}}|j\in\mathtt{MG}_{i},\mathbf{T}_{j}=\mathbf{t})=\frac{\Pr(j\in\mathtt{MG}_{i}|\mathbf{T}_{j}=\mathbf{t},G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}}=g_{\mathcal{N}_{j}}^{\mathbf{t}})\Pr(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}}=g_{\mathcal{N}_{j}}^{\mathbf{t}}|\mathbf{T}_{j}=\mathbf{t})}{\Pr(j\in\mathtt{MG}_{i}|\mathbf{T}_{j}=\mathbf{t})}$. By assumption, if all neighborhood graphs are empty, all units are used for all matched groups, and we are restricting ourselves to assignments in which $T_{j}=0$, therefore, $\Pr(j\in\mathtt{MG}_{i}|\mathbf{T}_{j}=\mathbf{t})=1$. Second, by assumption $\Pr(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}}=g_{\mathcal{N}_{j}}^{\mathbf{t}}|\mathbf{T}_{j}=\mathbf{t})=p(g)$. This is because treatment assignment is independent of the graph. We are left with having to find an expression for the likelihood, this can be written as: $\displaystyle\Pr(j\in\mathtt{MG}_{i}|\mathbf{T}_{j}=\mathbf{t},G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}}=g_{\mathcal{N}_{j}}^{\mathbf{t}})$ $\displaystyle=\Pr(j\in\operatorname*{arg\,min}_{\begin{subarray}{c}k=1,\dots,N,\\\ k\neq i\end{subarray}}\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(G_{\mathcal{N}_{k}}^{\mathbf{T}_{k}})]$ $\displaystyle\qquad\qquad|\mathbf{T}_{j}=\mathbf{t},G_{\mathcal{N}_{j}}^{\mathbf{T}}=g_{\mathcal{N}_{j}}^{\mathbf{t}})$ $\displaystyle=\prod_{\begin{subarray}{c}k=1\\\ k\neq i,j\end{subarray}}^{N}\bigg{[}\Pr(\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(G_{\mathcal{N}_{k}}^{\mathbf{T}_{k}})]\geq$ $\displaystyle\qquad\qquad\>\quad\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(g_{\mathcal{N}_{j}}^{\mathbf{t}})]|T_{k}=0)\Pr(T_{k}=0)$ $\displaystyle\qquad\quad+\Pr(T_{k}=1)\bigg{]}$ $\displaystyle=\prod_{\begin{subarray}{c}k=1\\\ k\neq i,j\end{subarray}}^{N}\bigg{[}\Pr\big{(}\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(G_{\mathcal{N}_{k}}^{\mathbf{T}_{k}})]$ $\displaystyle\qquad\quad\ \ \,\geq\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(g_{\mathcal{N}_{j}}^{\mathbf{t}})]|T_{k}=0\big{)}\frac{n-n^{(1)}}{n}$ $\displaystyle\qquad\quad+\frac{n^{(1)}}{n}\biggl{]}.$ The second equality follows because $k$ can never be in the matched group of unit $i$ if $T_{k}=1$, and, if $T_{k}=0$, then $k$ must be one of the minimizers of the weighted Hamming distance between neighborhood subgraph counts. The probability is a product of densities because of independence of candidate subgraphs. For an arbitrary unit, $k$, we define the following compact notation for the probability that $k$’s weighted Hamming distance from $i$ is larger than the weighted Hamming distance from $j$ to $i$: $\displaystyle\Pr(\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(G_{\mathcal{N}_{k}}^{\mathbf{T}_{k}})]$ $\displaystyle\qquad\geq\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(g_{\mathcal{N}_{j}}^{\mathbf{t}})]|T_{k}=0)$ $\displaystyle=:C_{k}(g_{\mathcal{N}_{j}}^{\mathbf{t}})\leq 1.$ Note that the last inequality follows from the fact that the expression above is a probability. Since graphs and treatment assignments, $G_{k}$ and $\mathbf{T}_{k}$ are the only random variables in the probability denoted by $C_{k}(g_{\mathcal{N}_{j}}^{\mathbf{t}})$, and since they are all independent, and identically distributed, we can say that $C_{1}(g_{\mathcal{N}_{j}}^{\mathbf{t}})=C_{2}(g_{\mathcal{N}_{j}}^{\mathbf{t}})=\dots=C_{N}(g_{\mathcal{N}_{j}}^{\mathbf{t}})=C(g_{\mathcal{N}_{j}}^{\mathbf{t}})$. Because of this we have: $\displaystyle\Pr(j\in\mathtt{MG}_{i}|G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}}=g_{\mathcal{N}_{j}}^{\mathbf{t}},\mathbf{T}_{j}=\mathbf{t})$ $\displaystyle=\left(\frac{n^{(1)}}{n}+\frac{n-n^{(1)}}{n}C(g_{\mathcal{N}_{j}}^{\mathbf{t}})\right)^{N-2}.$ Putting all the elements we have together we get the expression for the first term in the bound: $\displaystyle\mathbb{E}[\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}})]|j\in\mathtt{MG}_{i}]$ $\displaystyle=\sum_{g\in\mathcal{G}_{n}}\sum_{\mathbf{t}\in\mathcal{T}:t_{j}=0}\mathbf{w}^{T}\mathbb{I}[S(g_{\mathcal{N}_{j}}^{\mathbf{t}})\neq S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})]$ $\displaystyle\qquad\qquad\times\Pr(G_{\mathcal{N}_{j}}^{\mathbf{T}}=g_{\mathcal{N}_{j}}^{\mathbf{t}},\mathbf{T}_{\mathcal{N}_{j}}=\mathbf{t}_{\mathcal{N}_{j}}|j\in\mathtt{MG}_{i})$ $\displaystyle=\sum_{g\in\mathcal{G}_{N}}\sum_{\mathbf{t}\in\mathcal{T}:t_{j}=0}\mathbf{w}^{T}\mathbb{I}[S(g_{\mathcal{N}_{j}}^{\mathbf{t}})\neq S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})]$ $\displaystyle\times\binom{n-1}{n^{(1)}}^{-1}p(g)\left(\frac{n^{(1)}}{n}+\frac{n-n^{(1)}}{n}C(g_{\mathcal{N}_{j}}^{\mathbf{t}})\right)^{N-2}.$ ∎ #### B.2 Asymptotic Behavior Here we expand on the asymptotic consequences of Proposition 2: note first, that, by assumption $\|\mathbf{w}\|_{1}=M_{n}$, and that, therefore $\mathbf{w}^{T}\mathbb{I}[S(g)\neq S(h)]\leq M_{n}$ for any graphs $g,h\in\mathcal{G}_{n}$. That is to say, the weighted Hamming distance between any two graphs with $n$ units will be upper-bounded by the sum of the weights. Recall also that $C(g_{\mathcal{N}_{j}}^{\mathbf{t}})\leq 1$ for all $g$ and $\mathbf{t}$ as this quantity is a probability, and let $C_{max}=\max\limits_{\begin{subarray}{c}g\in\mathcal{G}_{n}\\\ \mathbf{t}\in\mathcal{T}\end{subarray}}C(g_{\mathcal{N}_{j}}^{\mathbf{t}})$. We can combine all these bounds with the upper bound in Proposition 2 to write: $\displaystyle\mathbb{E}[\mathbf{w}^{T}\mathbb{I}[S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})\neq S(G_{\mathcal{N}_{j}}^{\mathbf{T}_{j}})]|j\in\mathtt{MG}_{i}]$ $\displaystyle=\sum_{g\in\mathcal{G}_{\mathcal{N}}}\sum_{\mathbf{t}\in\mathcal{T},t_{j}=0}\mathbf{w}^{T}\mathbb{I}[S(g_{\mathcal{N}_{j}}^{\mathbf{t}})\neq S(h_{\mathcal{N}_{i}}^{\mathbf{t}_{i}})]$ $\displaystyle\times\binom{n-1}{n^{(1)}}^{-1}p(g)\left(\frac{n^{(1)}}{n}+\frac{n-n^{(1)}}{n}C(g_{\mathcal{N}_{j}}^{\mathbf{t}})\right)^{N-2}$ $\displaystyle\leq M_{n}\left(\frac{n^{(1)}}{n}+\frac{n-n^{(1)}}{n}C_{max}\right)^{N-2}$ $\displaystyle\times\sum_{g\in\mathcal{G}_{n}}\sum_{\mathbf{t}\in\mathcal{T},t_{j}=0}\binom{n-1}{n^{(1)}}^{-1}p(g)$ $\displaystyle=M_{n}\left(\frac{n^{(1)}}{n}+\frac{n-n^{(1)}}{n}C_{max}\right)^{N-2}.$ The first equality follows from Proposition 2, the first inequality from the bounds previously discussed, and the second equality follows from the fact that the sum in the second to last line is a sum of probability distributions over their entire domain, and therefore is equal to 1. Under the condition that $n$, the number of units in each unit’s candidate graph, stays fixed, and that $C_{max}<1$, then, as $N\rightarrow\infty$, we have $M_{n}\left(\frac{n^{(1)}}{n}+\frac{n-n^{(1)}}{n}C_{max}\right)^{N-2}\rightarrow 0$, because the quantity inside the parentheses is always less than 1. This makes sense, because asymptotically, matches can be made exactly; i.e., units matched in the way described in our theoretical setting have isomorphic neighborhood subgraphs asymptotically. This also has a consequence that the bound in Proposition 2 converges to $\sqrt{2}\sigma$ asymptotically in $N$. This is the variance of the baseline errors and can be lowered by matching the same unit with multiple others. As noted before, for this argument to apply, candidate graphs must remain of fixed size $n$ as they grow in number, so that the quantity $M_{n}$ remains constant: this setting is common in cluster- randomized trials where a growing number of units is clustered into fixed-size clusters of size at most $n$. The asymptotic behavior of our proposed methodology is less clear in settings in which the analyst cannot perform such clustering before randomization and $n$ is allowed to grow with $N$, and is an avenue for potential future research. #### B.3 Heteroskedasticity in The Baseline Effects In a network setting such as the one we study, it is possible that baseline effects of units do not have equal variance. Here we discuss how this setting affects our result in Proposition 2. Here, we maintain that $\mathbb{E}[\epsilon_{i}]=\alpha$ for all $i$, but we assume that $Var(\epsilon_{i})=\sigma_{i}^{2}$, and that $Cov(\epsilon_{i}\epsilon_{j})\neq 0$. Starting from the upper bound on the estimation error given in (10), we can see that the baseline effects only come in in the term: $\mathbb{E}[|\epsilon_{i}-\epsilon_{j}||j\in\mathtt{MG}_{i}]$, we therefore focus our attention on this term, as the rest of this bound does not change when the variance of these terms changes. Note first, that $\mathbb{E}[|\epsilon_{i}-\epsilon_{j}||j\in\mathtt{MG}_{i}]=\mathbb{E}[|\epsilon_{i}-\epsilon_{j}|]$ as the event $j\in\mathtt{MG}_{i}$ is independent of the baseline effects. We can now apply the Cauchy-Schwarz inequality, in the same way as we do in the proof of Proposition 2, to obtain: $\displaystyle\mathbb{E}[|\epsilon_{i}-\epsilon_{j}|]$ $\displaystyle\leq\sqrt{\mathbb{E}[(\epsilon_{i}-\epsilon_{j})^{2}]}$ $\displaystyle=\sqrt{\mathbb{E}[\epsilon_{i}^{2}]+\mathbb{E}[\epsilon_{j}^{2}]-2\mathbb{E}[\epsilon_{i}]\mathbb{E}[\epsilon_{j}]}$ $\displaystyle=\sqrt{\sigma^{2}_{i}+\alpha^{2}+\sigma_{j}^{2}+\alpha^{2}-2\alpha^{2}}$ $\displaystyle=\sqrt{\sigma_{i}^{2}+\sigma_{j}^{2}}.$ Clearly, this is not too different from the homoskedastic setting we study in the proposition: as long as neither of the unit variances is too large for inference, results in the heteroskedastic setting will suffer from similar bias as they would under independent baseline effects with equal variance. Simulations, shown in Section J, also support the above rationale of comparable performance in the heteroskedastic case and demonstrate that FLAME- Networks still outperforms competing methods. ### C Derivation of SANIA MIVLUE Used in Simulations Theorem 6.2 of Sussman and Airoldi, 2017 Suppose potential outcomes satisfy SANIA and that the prior on the parameters (baseline outcome and direct treatment effect) has no correlation between units. If unbiased estimators exist, the MIVLUE weights are: $w_{i}(\mathbf{z})=\frac{C_{i,d_{i}^{\mathbf{z}}}}{\sum_{d=0}^{n-1}C_{i,d}}\cdot\frac{2z_{i}-1}{n\mathrm{P}(\mathbf{z}_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d_{i}^{\mathbf{z}})}$ where $\displaystyle C_{i,d}$ $\displaystyle=$ $\displaystyle\left(\sum_{\mathbf{z}}\mathrm{P}(\mathbf{z})\mathbf{1}\\{d_{i}^{\mathbf{z}}=d\\}\cdot\frac{\Sigma(\mathbf{z})_{ii}}{\mathrm{P}(z_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d)^{2}}\right)^{-1}$ and $C_{i,d}$ is defined to be 0 if the probability in its denominator is 0. #### C.1 Setup and Notation We assume that there is a constant probability $p$ of each unit being treated and that units are independently treated. Let unit $i$ have $d_{i}$ neighbors and a treated degree of $d_{i}^{\mathbf{z}}$. In our setting, $\Sigma(\mathbf{z})_{ii}=\Sigma_{\alpha,ii}+z_{i}\Sigma_{\beta,ii}$ where $\Sigma_{\alpha}$ and $\Sigma_{\beta}$ are the covariance matrices on priors placed on the baseline outcome and the direct treatment effect, respectively. Additionally in our setting, their diagonals are constant and so we let $\sigma^{2}_{\alpha}:=\Sigma_{\alpha,ii}$ and $\sigma^{2}_{\beta}:=\Sigma_{\beta,ii}$. #### C.2 Find $\mathrm{P}(z_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d_{i}^{\mathbf{z}})$ By the setup, the constituent probabilities are independent and the probability of treatment is constant across units and so: $\mathrm{P}(z_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d_{i}^{\mathbf{z}})=[z_{i}p+(1-z_{i})(1-p)]{d_{i}\choose d_{i}^{\mathbf{z}}}p^{d_{i}^{\mathbf{z}}}(1-p)^{d_{i}-d_{i}^{\mathbf{z}}}$ #### C.3 Find $C_{i,d}$ Below, we will consider $\mathbf{z}_{\text{neighbor}(i)}$, the treatment assignment of the neighbors of $i$ (excluding $i$), $z_{i}$, the treatment assignment of unit $i$, and $\mathbf{z}_{\text{rest}(i)}$, the treatment assignment of the remaining units. $\displaystyle C_{i,d}$ $\displaystyle=\left(\sum_{\mathbf{z}}\frac{\mathrm{P}(\mathbf{z})\mathbf{1}\\{d_{i}^{\mathbf{z}}=d\\}\Sigma(\mathbf{z})_{ii}}{\mathrm{P}(z_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d)^{2}}\right)^{-1}$ $\displaystyle=\left(\sum_{\mathbf{z}:d_{i}^{\mathbf{z}}=d}\frac{\mathrm{P}(\mathbf{z}_{\text{neighbor}(i)})\mathrm{P}(z_{i})\mathrm{P}(\mathbf{z}_{\text{rest}(i)})\Sigma(\mathbf{z})_{ii}}{\mathrm{P}(z_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d)^{2}}\right)^{-1}$ $\displaystyle=\left(\frac{\mathrm{P}(\mathbf{z}_{\text{neighbor}(i)})}{\mathrm{P}(z_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d)^{2}}\right)^{-1}$ $\displaystyle\quad\times\left(\sum_{\mathbf{z}:d_{i}^{\mathbf{z}}=d}\Sigma(\mathbf{z})_{ii}\mathrm{P}(z_{i})\mathrm{P}(\mathbf{z}_{\text{rest}(i)})\right)^{-1}$ $\displaystyle=\left(\frac{\mathrm{P}(\mathbf{z}_{\text{neighbor}(i)})}{\mathrm{P}(z_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d)^{2}}\right)^{-1}$ $\displaystyle\quad\times\left(\sum_{\mathbf{z}:d_{i}^{\mathbf{z}}=d}(\sigma^{2}_{\alpha}+z_{i}\sigma^{2}_{\beta})\mathrm{P}(z_{i})\mathrm{P}(\mathbf{z}_{\text{rest}(i)})\right)^{-1}$ $\displaystyle=\left(\frac{\mathrm{P}(\mathbf{z}_{\text{neighbor}(i)})}{\mathrm{P}(z_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d)^{2}}\right)^{-1}$ $\displaystyle\quad\times\left(\sigma^{2}_{\alpha}+\sum_{\mathbf{z}:d_{i}^{\mathbf{z}}=d}(z_{i}\sigma^{2}_{\beta})\mathrm{P}(z_{i})\mathrm{P}(\mathbf{z}_{\text{rest}(i)})\right)^{-1}$ $\displaystyle=\left(\frac{\mathrm{P}(\mathbf{z}_{\text{neighbor}(i)})}{\mathrm{P}(z_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d)^{2}}\right)$ $\displaystyle\quad\times\left(\sigma^{2}{\alpha}+\sum_{\begin{subarray}{c}\mathbf{z}:d_{i}^{\mathbf{z}}=d\\\ z_{i}=1\end{subarray}}(\sigma^{2}_{\beta,})p\mathrm{P}(\mathbf{z}_{\text{rest}(i)})\right)^{-1}$ $\displaystyle=\left(\frac{\mathrm{P}(\mathbf{z}_{\text{neighbor}(i)})\left(\sigma^{2}_{\alpha}+\sigma^{2}_{\beta}\right)}{\mathrm{P}(z_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d)^{2}}\right)^{-1}$ $\displaystyle=\frac{\left([z_{i}p+(1-z_{i})(1-p)]{d_{i}\choose d}p^{d}(1-p)^{d_{i}-d}\right)^{2}}{(\sigma^{2}_{\alpha}+\sigma^{2}_{\beta})p^{d}(1-p)^{d_{i}-d}}$ $\displaystyle=\frac{[z_{i}p+(1-z_{i})(1-p)]^{2}{d_{i}\choose d}^{2}p^{d}(1-p)^{d_{i}-d}}{\sigma^{2}_{\alpha}+\sigma^{2}_{\beta}}$ #### C.4 Find $w_{i}$ Plugging in the expressions we’ve found: $\displaystyle w_{i}(\mathbf{z})=$ $\displaystyle\frac{C_{i,d_{i}^{\mathbf{z}}}}{\sum_{d=0}^{n-1}C_{i,d}}\cdot\frac{2z_{i}-1}{n\mathrm{P}(\mathbf{z}_{i}^{\text{obs}}=z_{i},d_{i}^{\mathbf{z}^{\text{obs}}}=d_{i}^{\mathbf{z}})}$ $\displaystyle=\frac{\frac{[z_{i}p+(1-z_{i})(1-p)]^{2}{d_{i}\choose d_{i}^{\mathbf{z}}}^{2}p^{d_{i}^{\mathbf{z}}}(1-p)^{d_{i}-d_{i}^{\mathbf{z}}}}{(\sigma^{2}_{\alpha}+\sigma^{2}_{\beta})}}{\frac{\sum_{d=0}^{n-1}{d_{i}\choose d}^{2}p^{d}(1-p)^{d_{i}-d}(z_{i}p+(1-z_{i})(1-p))^{2}}{(\sigma^{2}_{\alpha}+\sigma^{2}_{\beta})}}$ $\displaystyle\quad\times\frac{2z_{i}-1}{n[z_{i}p+(1-z_{i})(1-p)]{d_{i}\choose d_{i}^{\mathbf{z}}}p^{d_{i}^{\mathbf{z}}}(1-p)^{d_{i}-d_{i}^{\mathbf{z}}}}$ $\displaystyle=\frac{(z_{i}p+(1-z_{i})(1-p))(2z_{i}-1)}{n\sum_{d=0}^{n-1}{d_{i}\choose d}^{2}p^{d}(1-p)^{d_{i}-d}(z_{i}p+(1-z_{i})(1-p))^{2}}$ $\displaystyle=\frac{(z_{i}p+(1-z_{i})(1-p))(2z_{i}-1)}{n(z_{i}p+(1-z_{i})(1-p))^{2}}$ $\displaystyle\quad\times\frac{1}{\sum_{d=0}^{n-1}{d_{i}\choose d}^{2}p^{d}(1-p)^{d_{i}-d}}$ Note in the first fraction that $(z_{i}p+(1-z_{i})(1-p))(2z_{i}-1)$ equals $p$ when $z_{i}=1$ and $p-1$ when $z_{i}=0$. Also, $(z_{i}p+(1-z_{i})(1-p))^{2}$ equals $p^{2}$ when $z_{i}=1$ and $(1-p)^{2}$ when $z_{i}=0$. Thus, the first term is $1/np$ when $z_{i}=1$ and $-1/n(1-p)$ when $z_{i}=0$ and so the overall expression for the weights is: $w_{i}(\mathbf{z})=\frac{z_{i}/np-(1-z_{i})/(n(1-p))}{\sum_{d=0}^{n-1}{d_{i}\choose d}^{2}p^{d}(1-p)^{d_{i}-d}}$ and the MIVLUE is given by $\sum_{i=1}^{n}w_{i}Y_{i}$. ### D Subgraph Descriptions Here, for graphs without self-loops, we define the interference components used in our simulations: * • Degree: the degree of a node is the number of edges it is a part of. * • Triangle: A graph with three mutually connected nodes (see Figure 5). * • Square: A graph with 4 nodes and 4 edges, such that each node is a part of exactly two distinct edges (see Figure 5). * • $k$-Star: A graph with $k+1$ nodes, the first $k$ of which are all connected to the $(k+1)$st node and no others (see Figure 5). * • Vertex Betweenness: The vertex betweenness of a vertex $v$ is defined as: $\sum_{i\neq v\neq j}\frac{\sigma_{ij}(v)}{\sigma_{ij}}$ where $\sigma_{ij}$ is the number of shortest paths between vertices $i$ and $j$ and $\sigma_{ij}(v)$ is the number of shortest paths between $i$ and $j$ that go through $v$. We use the normalized vertex betweenness which scales the above expression by $2/(n^{2}-3n+2)$ where $n$ is the number of nodes in the graph. * • Closeness Centrality: We use the normalized closeness centrality of a vertex $v$, defined as: $\frac{n-1}{\sum_{i=1}^{n}d(\sigma_{vi})}$ where $d(\sigma_{vi})$ is the length of the shortest path between $v$ and $i$ and $n$ is the number of nodes in the graph. Figure 5: Four types of treated neighborhood subgraphs the colored unit might be a part of: triangle (top right), square (bottom right), 2-star (top left), 4-star (bottom left). ### E Data Pre-processing We estimate the DTE using data from from Banerjee et al. (2013) on social networks for the 75 villages in India. A unit $i$ is defined to be socially connected with unit $j$ if they are connected across at least 3 of the following four types of social connections: (1) $i$ visits $j$’s house, (2) $j$ visits $i$’s house, (3) $i$ and $j$ socialize and are relatives, (4) $i$ and $j$ socialize and are not related to each other. Along with subgraph counts, we also use age and whether or not individual $i$ speaks Telugu as additional covariates to match on. For computational efficiency, we drop units with maximum degree of connection greater than 15, where the cut-off is selected based on computational efficiency. ### F Matched Groups We provide sample matched groups in Table 4. These matched groups were produced by applying FLAME-Networks on the data discussed in Section 4. We report all the covariates used for matching. The first group is comprised of 40-year-old units who do not speak Telugu, and have 2 or 3 triangles, 3 2-stars, and 3 edges in their treated neighborhood graph. These units (given the binning) are matched exactly. The second group is comprised of units who speak Telugu with no triangles, 3 2-stars, 3 edges in their treated neighborhood graph. Note that units in this group are matched approximately, since they are not matched exactly on age. Units | Triangles | 2-Stars | Edges | Telugu | Age | Treated | Outcome ---|---|---|---|---|---|---|--- Matched Group 1 | | | | | | | 1 | 2 or 3 | 3 | 3 | 0 | 40 | 0 | 1 2 | 2 or 3 | 3 | 3 | 0 | 40 | 1 | 0 3 | 2 or 3 | 3 | 3 | 0 | 40 | 1 | 0 4 | 2 or 3 | 3 | 3 | 0 | 40 | 1 | 0 Matched Group 2 | | | | | | | 1 | 0 | 3 | 3 | 1 | 30 | 0 | 0 2 | 0 | 3 | 3 | 1 | 34 | 0 | 0 3 | 0 | 3 | 3 | 1 | 25 | 0 | 1 4 | 0 | 3 | 3 | 1 | 20 | 1 | 0 Table 4: Sample Match Groups. Two sample matched groups generated by FLAME- Networks using data discussed in Section 4. The columns are the covariates used for matching, along with treatment status and outcome. The counts of subgraphs were coarsened into 10 bins defined by deciles of the empirical distribution of counts. The two groups have relatively good match quality overall. Note that the first group matches units exactly (given the binning). However, Matched Group 2 matches units approximately, with exact matches on subgraph counts and whether or not individuals speak Telugu, but inexact matches on age. ### G Additional Experiment: Multiplicative Interference We explore settings in which interference is a nonlinear function of the interference components and their weights. Since matching is nonparametric, it is particularly appropriate for handling non-linearities in interference functions. Outcomes in this experiment have the form $Y_{i}(t,\mathbf{t}_{-i})=t\tau_{i}+\alpha\prod_{p=1}^{P}m_{ip}^{\mathbb{I}[p\text{ is included}]}+\epsilon_{i}$. Table 5 shows which components are included in the outcome function for each setting. We use a small number of parameters in each setting, as their multiplicative interaction suffices to define a complex interference function. The simulations are run on an $ER(50,0.05)$ graph. Setting | $d_{i}$ | $\Delta_{i}$ | $\bigstar_{i}^{4}$ | $B_{i}$ ---|---|---|---|--- 1 | x | x | | 2 | x | | | x 3 | | x | | x 4 | | x | x | Table 5: Parameters included in interference function Experiment 1. The marked components for each setting were the only ones included in those experiments. Results for this experiment are presented in Figure 6. FLAME-Networks performed better than all baseline methods in this setting, both in terms of mean absolute error and, in most cases, in terms of standard deviation over simulations. The stratified and SANIA estimators perform especially poorly, because they cannot handle nonlinear interference settings, unlike FLAME- Networks. Figure 6: Results from experiments with a multiplicative interference function. Each violin plot represents the distribution over simulations of absolute estimation error over for each method. The panels are numbered according to the parameter setting the simulations were ran with. Violin plots are color-coded blue if the method had mean error either equal to or better than FLAME-Networks and red otherwise. The black line inside each violin is the median error. The dashed line is FLAME-Networks’ mean error. ### H Additional Experiment: Graph Cluster Randomization We also explored the performance of FLAME-Networks in settings in which treatment is randomized within multiple clusters, which have few connections between them. More specifically, we simulate a network according to a stochastic block model with 5 clusters. In each cluster, there are 10 units, 5 of which are treated. The probability of edges within clusters is 0.3 and between clusters is 0.05. This results in graphs with few edges between clusters. We then evaluate our method as previously described, simulating the outcome with additive interference and homoskedastic errors. The results in Figure 7 demonstrate that FLAME-Networks outperforms competing methods in this setting as well. Figure 7: Results from experiments with additive interference on graphs in which treatment is randomly assigned within multiple clusters with few edges between them. Each violin plot represents the distribution over simulations of absolute estimation error over for each method. The panels are numbered according to the parameter setting the simulations were ran with. Violin plots are color-coded blue if the method had mean error either equal to or better than FLAME-Networks and red otherwise. The black line inside each violin is the median error. The dashed line is FLAME-Networks’ mean error. ### I Additional Experiment: Real Network To ensure that FLAME-Networks also performs well on real networks, we consider an AddHealth network (Harris, 2013). Specifically, we use the addhealthc3 dataset from the amen R package, with all edges treated as undirected. There are 32 nodes and on every simulation, 16 are randomly selected to be treated. Outcome and additive interference are simulated as previously described. Errors are homoskedastic. The results in Figure 8 demonstrate that FLAME- Networks still outperforms competing methods. Figure 8: Results from experiments on a real, AddHealth network with additive interference. Each violin plot represents the distribution over simulations of absolute estimation error over for each method. The panels are numbered according to the parameter setting the simulations were ran with. Violin plots are color-coded blue if the method had mean error either equal to or better than FLAME-Networks and red otherwise. The black line inside each violin is the median error. The dashed line is FLAME-Networks’ mean error. ### J Additional Experiment: Heteroscedastic Errors We also explored the performance of FLAME-Networks in settings in which the variance of the outcomes is not constant. We simulate a single ER(50, 0.07) graph and randomly treat 25 units. We consider additive interference, as in the body of the text, and all other simulation parameters are the same, expect for that now, each unit $i$ has baseline outcome $\alpha_{i}\stackrel{{\scriptstyle ind}}{{\sim}}N(0,v_{i})$ with $v_{i}\stackrel{{\scriptstyle ind}}{{\sim}}U(0,1)$. We see in Figure 9 that FLAME-Networks outperforms competitors; the fact that it is nonparametric allows it to handle more flexible baseline outcomes and variances. Figure 9: Results from experiments with an additive interference function involving heteroskedasticity in the baseline effects across units. Each violin plot represents the distribution over simulations of absolute estimation error over for each method. The panels are numbered according to the parameter setting the simulations were ran with. Violin plots are color-coded blue if the method had mean error either equal to or better than FLAME-Networks and red otherwise. The black line inside each violin is the median error. The dashed line is FLAME-Networks’ mean error. ### K Additional Experiment: Matching on True Interference Here, we compare FLAME-Networks to approaches that match directly on units’ interference values. FLAME-Networks already has the advantage of interpretably matching on neighborhood graphs that can be visualized and compared, as opposed to uninterpretable scalar values of an interference function. Additionally, to perform well in practice, one would typically need to use equally uninterpretable machine learning methods to estimate units’ interference values well. But even comparing FLAME-Networks to an approach that matches on the true (typically unknown) interference values, we see that our method does well in comparison, because it learns and matches on baseline effects as well as (approximate) interference values. Results using an ER(50, 0.07) graph with 25 units randomly treated, an additive interference function, and homoskedastic errors – as previously described – are shown in Figure 10. ### L Additional Experiment: Covariate Weight In this section, we show that increasing the influence that covariates have on the outcome function harms neither FLAME-Networks nor the competing methods. As in the results shown in the main text, however, the performance of FLAME- Networks is still superior, given that it naturally handles covariate data. The experimental setup is the same as in Experiment 2 in the main text and results are shown in Tables 6 and 7 Method | Median | 25th q | 75th q ---|---|---|--- FLAME-Networks | 0.34 | 0.15 | 0.52 First Eigenvector | 0.41 | 0.24 | 0.49 All Eigenvectors | 0.36 | 0.32 | 0.74 Naive | 0.61 | 0.19 | 0.85 SANIA | 2.31 | 1.78 | 2.75 Stratified | 4.56 | 4.55 | 4.63 Table 6: Additional results from the experimental setup of Experiment 2, but with $\beta=20$. Median and 25th and 75th percentile of absolute error over 10 simulations. – Method | Median | 25th q | 75th q ---|---|---|--- FLAME-Networks | 0.25 | 0.08 | 0.51 First Eigenvector | 0.52 | 0.32 | 0.85 All Eigenvectors | 0.53 | 0.29 | 0.83 Naive | 0.78 | 0.32 | 1.16 SANIA | 1.86 | 1.68 | 2.11 Stratified | 4.78 | 4.74 | 4.80 Table 7: Additional results from the experimental setup of Experiment 2, but with $\beta=25$. Median and 25th and 75th percentile of absolute error over 10 simulations. – Figure 10: Results from experiments comparing FLAME-Networks to matching units on their true interference values. Each violin plot represents the distribution over simulations of absolute estimation error over for each method. The panels are numbered according to the parameter setting the simulations were ran with. Violin plots are color-coded blue if the method had mean error either equal to or better than FLAME-Networks and red otherwise. The black line inside each violin is the median error. The dashed line is FLAME-Networks’ mean error. ### M Match Quality Here, we assess the quality of matches generated by FLAME-Networks versus matching on true interference, and the All Eigenvectors approach. To do so, for FLAME-Networks: for each (control) treated unit, we take the minimal Frobenius norm of the difference between that unit’s neighborhood adjacency matrix and that of all the (treated) control units in its matched group111The Frobenius norm of the difference of the adjacency matrices, up to reordering of the vertices., and average across all units. This gives an average graph distance for a single simulation. And to do so for the true interference matching and All Eigenvectors approaches: for every (control) treated unit, we take the closest (treated) control unit, find the graph distance (as above) between their neighborhood subgraphs, and average across units. This gives an average graph distance for a single simulation. Results from 50 simulations performed on an ER(50, 0.07) graph with additive interference and homoskedastic errors, as described previously, are shown in Figure 11, showing that FLAME-Networks produces more matches between units with similar neighborhood subgraphs than matching on the true interference or the All Eigenvectors method. Figure 11: Results from experiments comparing the average distance between the neighborhood subgraphs of the units matched by different methods. Each violin plot represents the distribution over simulations of graph distance for each method. The panels are numbered according to the parameter setting the simulations were ran with. Violin plots are color-coded blue if the method had mean graph distance either equal to or better than FLAME-Networks and red otherwise. The black line inside each violin is the median graph distance. The dashed line is FLAME-Networks’ mean graph distance.

2024-09-04T02:54:56.452309

2003.00973

arxiv-papers

# Differential Privacy at Risk: Bridging Randomness and Privacy Budget Ashish Dandekar<EMAIL_ADDRESS> Département d’informatique, École Normale Supérieure Paris, France Debabrota Basu<EMAIL_ADDRESS> Data Science and AI Division Department of Computer Science and Engineering Chalmers University of Technology Göteborg, Sweden Stéphane Bressan<EMAIL_ADDRESS> School of Computing National University of Singapore Singapore, Singapore ###### Abstract The calibration of noise for a privacy-preserving mechanism depends on the sensitivity of the query and the prescribed privacy level. A data steward must make the non-trivial choice of a privacy level that balances the requirements of users and the monetary constraints of the business entity. We analyse roles of the sources of randomness, namely the explicit randomness induced by the noise distribution and the implicit randomness induced by the data-generation distribution, that are involved in the design of a privacy- preserving mechanism. The finer analysis enables us to provide stronger privacy guarantees with quantifiable risks. Thus, we propose privacy at risk that is a probabilistic calibration of privacy-preserving mechanisms. We provide a composition theorem that leverages privacy at risk. We instantiate the probabilistic calibration for the Laplace mechanism by providing analytical results. We also propose a cost model that bridges the gap between the privacy level and the compensation budget estimated by a GDPR compliant business entity. The convexity of the proposed cost model leads to a unique fine-tuning of privacy level that minimises the compensation budget. We show its effectiveness by illustrating a realistic scenario that avoids overestimation of the compensation budget by using privacy at risk for the Laplace mechanism. We quantitatively show that composition using the cost optimal privacy at risk provides stronger privacy guarantee than the classical advanced composition. ## 1 Introduction Dwork et al. quantify the privacy level $\varepsilon$ in _$\varepsilon$ -differential privacy_ as an upper bound on the worst-case privacy loss incurred by a privacy-preserving mechanism. Generally, a privacy-preserving mechanism perturbs the results by adding the calibrated amount of random noise to them. The calibration of noise depends on the sensitivity of the query and the specified privacy level. In a real-world setting, a data steward must specify a privacy level that balances the requirements of the users and monetary constraints of the business entity. Garfinkel et al. report the issues in deploying differential privacy as the privacy definition by the US census bureau. They highlight the lack of analytical methods to choose the privacy level. They also report empirical studies that show the loss in utility due to the application of privacy-preserving mechanisms. We address the dilemma of a data steward in two ways. Firstly, we propose a probabilistic quantification of privacy levels. Probabilistic quantification of privacy levels provides a data steward a way to take quantified risks under the desired utility of the data. We refer to the probabilistic quantification as _privacy at risk_. We also derive a composition theorem that leverages privacy at risk. Secondly, we propose a cost model that links the privacy level to a monetary budget. This cost model helps the data steward to choose the privacy level constrained on the estimated budget and vice versa. Convexity of the proposed cost model ensures the existence of a unique privacy at risk that would minimise the budget. We show that the composition with an optimal privacy at risk provides stronger privacy guarantees than the traditional advanced composition (Dwork et al., 2014). In the end, we illustrate a realistic scenario that exemplifies how the data steward can avoid overestimation of the budget by using the proposed cost model by using privacy at risk. The probabilistic quantification of privacy levels depends on two sources of randomness: the _explicit randomness_ induced by the noise distribution and the _implicit randomness_ induced by the data-generation distribution. Often, these two sources are coupled with each other. We require analytical forms of both sources of randomness as well as an analytical representation of the query to derive a privacy guarantee. Computing the probabilistic quantification is generally a challenging task. Although we find multiple probabilistic privacy definitions in the literature (Machanavajjhala et al., 2008; Hall et al., 2012), we are missing analytical quantification bridging the randomness and privacy level of a privacy-preserving mechanism. To the best of our knowledge, we are the first to analytically derive such a probabilistic quantification, namely privacy at risk, for the widely used Laplace mechanism (Dwork et al., 2006b). We also derive a composition theorem with privacy at risk. It is a special case of the advanced composition theorem (Dwork et al., 2014) that deals with a sequential and adaptive use of privacy- preserving mechanisms. We work on a simpler model independent evaluations used in the basic composition theorem (Dwork et al., 2014). The privacy level proposed by the differential privacy framework is too abstract a quantity to be integrated in a business setting. We propose a cost model that maps the privacy level to a monetary budget. The corresponding cost model for the probabilistic quantification of privacy levels is a convex function of the privacy level. Hence, it leads to a unique probabilistic privacy level that minimises the cost. We illustrate a realistic scenario in a GDPR compliant business entity that needs an estimation of the compensation budget that it needs to pay to stakeholders in the unfortunate event of a personal data breach. The illustration shows that the use of probabilistic privacy levels avoids overestimation of the compensation budget without sacrificing utility. In this work, we comparatively evaluate the privacy guarantees using privacy at risk of using Laplace mechanism. We quantitatively compare the composition under the optimal privacy at risk, which is estimated using the cost model, with traditional composition mechanisms - the basic composition and advanced mechanism (Dwork et al., 2014). We observe that it gives stronger privacy guarantees than the ones by the advanced composition without sacrificing on the utility of the mechanism. In conclusion, benefits of the probabilistic quantification i.e. the privacy at risk are twofold. It not only quantifies the privacy level for a given privacy-preserving mechanism but also facilitates decision-making in problems that focus on the privacy-utility trade-off and the compensation budget minimisation. ## 2 Background We consider a universe of datasets $\mathcal{D}$. We explicitly mention when we consider that the datasets are sampled from a data-generation distribution $\mathcal{G}$ with support $\mathcal{D}$. Two datasets of equal cardinality $x$ and $y$ are said to be neighbouring datasets if they differ in one data point. A pair of neighbouring datasets is denoted by $x\sim y$. In this work, we focus on a specific class of queries called numeric queries. A numeric query $f$ is a function that maps a dataset into a real-valued vector, i.e. $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$. For instance, a sum query returns the sum of the values in a dataset. In order to achieve a privacy guarantee, a _privacy-preserving mechanism_ , or mechanism in short, is a randomised algorithm, that adds noise to the query from a given family of distributions. Thus, a privacy-preserving mechanism of a given family, $\mathcal{M}(f,\Theta)$, for the query $f$ and the set of parameters $\Theta$ of the given noise distribution, is a function that maps a dataset into a real vector, i.e. $\mathcal{M}(f,\Theta):\mathcal{D}\rightarrow\mathbb{R}^{k}$. We denote a privacy-preserving mechanism as $\mathcal{M}$, when the query and the parameters are clear from the context. ###### Definition 1 [Differential Privacy (Dwork et al., 2014).] A privacy-preserving mechanism $\mathcal{M}$, equipped with a query $f$ and with parameters $\Theta$, is $(\varepsilon,\delta)$-differentially private if for all $Z\subseteq\textrm{Range}(\mathcal{M})$ and $x,y\in\mathcal{D}$ such that $x\sim y$: $\mathbb{P}(\mathcal{M}(f,\Theta)(x)\in Z)\leq\text{e}^{\varepsilon}\mathbb{P}(\mathcal{M}(f,\Theta)(y)\in Z)+\delta$ $(\varepsilon,0)$-differentially private mechanism is ubiquitously called as $\varepsilon$-differentially private. A privacy-preserving mechanism provides perfect privacy if it yields indistinguishable outputs for all neighbouring input datasets. The privacy level $\varepsilon$ quantifies the privacy guarantee provided by $\varepsilon$-differential privacy. For a given query, the smaller the value of the $\varepsilon$, the qualitatively higher is the privacy. A randomised algorithm that is $\varepsilon$-differentially private is also $\varepsilon^{\prime}$-differential private for any $\varepsilon^{\prime}>\varepsilon$. In order to satisfy $\varepsilon$-differential privacy, the parameters of a privacy-preserving mechanism requires a calculated calibration. The amount of noise required to achieve a specified privacy level depends on the query. If the output of the query does not change drastically for two neighbouring datasets, then small amount of noise is required to achieve a given privacy level. The measure of such fluctuations is called the _sensitivity_ of the query. The parameters of a privacy-preserving mechanism are calibrated using the sensitivity of the query that quantifies the smoothness of a numeric query. ###### Definition 2 [Sensitivity.] The sensitivity of a query $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$ is defined as $\Delta_{f}\triangleq\max_{\begin{subarray}{c}x,y\in\mathcal{D}\\\ x\sim y\end{subarray}}~{}\lVert f(x)-f(y)\rVert_{1}.$ The Laplace mechanism is a privacy-preserving mechanism that adds scaled noise sampled from a calibrated Laplace distribution to the numeric query. ###### Definition 3 [(Papoulis and Pillai, 2002)] The Laplace distribution with mean zero and scale $b>0$ is a probability distribution with probability density function $\mathrm{Lap}(b)\triangleq\frac{1}{2b}\exp{(-\frac{|x|}{b})},$ where $x\in\mathbb{R}$. We write $\mathrm{Lap}(b)$ to denote a random variable $X\sim\mathrm{Lap}(b)$ ###### Definition 4 [Laplace Mechanism (Dwork et al., 2006b).] Given any function $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$ and any $x\in\mathcal{D}$, the Laplace Mechanism is defined as $\mathcal{L}^{\Delta_{f}}_{\varepsilon}(x)\triangleq\mathcal{M}\left(f,~{}\frac{\Delta_{f}}{\varepsilon}\right)(x)=f(x)+(L_{1},...,L_{k}),$ where $L_{i}$ is drawn from $\mathrm{Lap}\left(\frac{\Delta_{f}}{\varepsilon}\right)$ and added to the $i^{\mathrm{th}}$ component of $f(x)$. ###### Theorem 5 (Dwork et al., 2006b) The Laplace mechanism, $\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}$, is $\varepsilon_{0}$-differentially private. ## 3 Privacy at Risk: A Probabilistic Quantification of Randomness The parameters of a privacy-preserving mechanism are calibrated using the privacy level and the sensitivity of the query. A data steward needs to choose appropriate privacy level for practical implementation. Lee and Clifton show that the choice of an actual privacy level by a data steward in regard to her business requirements is a non-trivial task. Recall that the privacy level in the definition of differential privacy corresponds to the worst case privacy loss. Business users are however used to taking and managing risks, if the risks can be quantified. For instance, Jorion defines Value at Risk that is used by risk analysts to quantify the loss in investments for a given portfolio and an acceptable confidence bound. Motivated by the formulation of Value at Risk, we propose to use the use of probabilistic privacy level. It provides us a finer tuning of an $\varepsilon_{0}$-differentially private privacy-preserving mechanism for a specified risk $\gamma$. ###### Definition 6 [Privacy at Risk.] For a given data generating distribution $\mathcal{G}$, a privacy-preserving mechanism $\mathcal{M}$, equipped with a query $f$ and with parameters $\Theta$, satisfies $\varepsilon$-differential privacy with a _privacy at risk_ $0\leq\gamma\leq 1$, if for all $Z\subseteq\textrm{Range}(\mathcal{M})$ and $x,y$ sampled from $\mathcal{G}$ such that $x\sim y$: $\mathbb{P}\left[\left|\ln{\frac{\mathbb{P}(\mathcal{M}(f,\Theta)(x)\in Z)}{\mathbb{P}(\mathcal{M}(f,\Theta)(y)\in Z)}}\right|>\varepsilon\right]\leq\gamma,$ (1) where the outer probability is calculated with respect to the probability space $\mathrm{Range}(\mathcal{M}\circ\mathcal{G})$ obtained by applying the privacy-preserving mechanism $\mathcal{M}$ on the data-generation distribution $\mathcal{G}$. If a privacy-preserving mechanism is $\varepsilon_{0}$-differentially private for a given query $f$ and parameters $\Theta$, for any privacy level $\varepsilon\geq\varepsilon_{0}$, privacy at risk is $0$. Our interest is to quantify the risk $\gamma$ with which $\varepsilon_{0}$-differentially private privacy-preserving mechanism also satisfies a stronger $\varepsilon$-differential privacy, i.e. $\varepsilon<\varepsilon_{0}$. Unifying Probabilistic and Random Differential Privacy. As a natural consequence, Equation 1 unifies the notions of probabilistic differential privacy and random differential privacy by accounting for both sources of randomness in a privacy-preserving mechanism. Machanavajjhala et al. define probabilistic differential privacy that incorporates the explicit randomness of the noise distribution of the privacy-preserving mechanism whereas Hall et al. define random differential privacy that incorporates the implicit randomness of the data-generation distribution. In probabilistic differential privacy, the outer probability is computed over the sample space of $\mathrm{Range}(\mathcal{M})$ and all datasets are equally probable. ### 3.1 Composition theorem Application of $\varepsilon$-differential privacy to many real-world problem suffers from the degradation of privacy guarantee, i.e. privacy level, over the composition. The basic composition theorem (Dwork et al., 2014) dictates that the privacy guarantee degrades linear in the number of evaluations of the mechanism. Advanced composition theorem (Dwork et al., 2014) provides a finer analysis of the privacy loss over multiple evaluations and provides a square root dependence on the the number of evaluations. In this section, we provide the composition theorem for privacy at risk. ###### Definition 7 [Privacy loss random variable.] For a privacy-preserving mechanism $\mathcal{M}:\mathcal{D}\rightarrow R$ and two neighbouring datasets $x,y\in\mathcal{D}$, the privacy loss random variable $\mathcal{C}$ takes a value $r\in R$ $\mathcal{C}\triangleq\ln{\frac{\mathbb{P}(\mathcal{M}(x))}{\mathbb{P}(\mathcal{M}(y))}}$ ###### Lemma 8 If a privacy-preserving mechanism $\mathcal{M}$ satisfies $\varepsilon_{0}$ differential privacy, then $\mathbb{P}[|\mathcal{C}|\leq\varepsilon_{0}]=1$ ###### Theorem 9 For all $\varepsilon_{0},\varepsilon,\gamma,\delta>0$, the class of $\varepsilon_{0}$-differentially private mechanisms, which satisfy $(\varepsilon,\gamma)$-privacy at risk, are $(\varepsilon^{\prime},\delta)$-differential privacy under $n$-fold composition where $\varepsilon^{\prime}=\varepsilon_{0}\sqrt{2n\ln{\frac{1}{\delta}}}+n\mu$ where, $\mu=[\gamma\varepsilon(e^{\varepsilon}-1)+(1-\gamma)\varepsilon_{0}(e^{\varepsilon_{0}}-1)]$ Proof Let, $\mathcal{M}^{1...n}:\mathcal{D}\rightarrow R^{1}\times R^{2}\times...\times R^{n}$ denote the $n$-fold composition of privacy- preserving mechanisms $\\{\mathcal{M}^{i}:\mathcal{D}\rightarrow R^{i}\\}_{i=1}^{n}$. Each $\varepsilon_{0}$-differentially private $\mathcal{M}^{i}$ also satisfies $(\varepsilon,\gamma)$-privacy at risk for some $\varepsilon\leq\varepsilon_{0}$ and appropriately computer $\gamma$. Consider any two neighbouring datasets $x,y\in\mathcal{D}$. Let, $B=\\{(r_{1},...,r_{n})|\bigwedge_{i=1}^{n}\frac{\mathbb{P}(\mathcal{M}^{i}(x)=r_{i})}{\mathbb{P}(\mathcal{M}^{i}(y)=r_{i})}>\text{e}^{\varepsilon}\\}$ Using the technique in (Dwork et al., 2014, Theorem 3.20), it suffices to show that $\mathbb{P}(\mathcal{M}^{1...n}(x)\in B)\leq\delta$. Consider, $\displaystyle\ln{\frac{\mathbb{P}(\mathcal{M}^{1...n}(x)=(r_{1},...,r_{n}))}{\mathbb{P}(\mathcal{M}^{1...n}(y)=(r_{1},...,r_{n}))}}$ $\displaystyle=$ $\displaystyle\ln{\prod_{i=1}^{n}}\frac{\mathbb{P}(\mathcal{M}^{i}(x)=r_{i})}{\mathbb{P}(\mathcal{M}^{i}(y)=r_{i})}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\ln{\frac{\mathbb{P}(\mathcal{M}^{i}(x)=r_{i})}{\mathbb{P}(\mathcal{M}^{i}(y)=r_{i})}}~{}=~{}\sum_{i=1}^{n}\mathcal{C}^{i}$ (2) where $\mathcal{C}^{i}$ in the last line denotes privacy loss random variable related $\mathcal{M}^{i}$. Consider, an $\varepsilon$-differentially private mechanism $\mathcal{M}_{\varepsilon}$ and $\varepsilon_{0}$-differentially private mechanism $\mathcal{M}_{\varepsilon_{0}}$. Let $\mathcal{M}_{\varepsilon_{0}}$ satisfy $(\varepsilon,\gamma)$-privacy at risk for $\varepsilon\leq\varepsilon_{0}$ and appropriately computed $\gamma$. Each $\mathcal{M}^{i}$ can be simulated as the mechanism $\mathcal{M}_{\varepsilon}$ with probability $\gamma$ and the mechanism $\mathcal{M}_{\varepsilon_{0}}$ otherwise. Therefore, privacy loss random variable for each mechanism $\mathcal{M}^{i}$ can be written as $\mathcal{C}^{i}=\gamma\mathcal{C}_{\varepsilon}^{i}+(1-\gamma)\mathcal{C}_{\varepsilon_{0}}^{i}$ where, $\mathcal{C}_{\varepsilon}^{i}$ denotes the privacy loss random variable associated with the mechanism $\mathcal{M}_{\varepsilon}$ and $\mathcal{C}_{\varepsilon_{0}}^{i}$ denotes the privacy loss random variable associated with the mechanism $\mathcal{M}_{\varepsilon_{0}}$. Using (Dwork et al., 2014, Lemma $3.18$), we can bound the mean of every privacy loss random variable as, $\mu\triangleq\mathbb{E}[\mathcal{C}^{i}]\leq[\gamma\varepsilon(e^{\varepsilon}-1)+(1-\gamma)\varepsilon_{0}(e^{\varepsilon_{0}}-1)]$ We have a collection of $n$ independent privacy random variables $\mathcal{C}^{i}$s such that $\mathbb{P}\left[|\mathcal{C}^{i}|\leq\varepsilon_{0}\right]=1$. Using Hoeffding’s bound (Hoeffding, 1994) on the sample mean for any $\beta>0$, $\mathbb{P}\left[\frac{1}{n}\sum_{i}C^{i}\geq\mathbb{E}[\mathcal{C}^{i}]+\beta\right]\leq\exp{\left(-\frac{n\beta^{2}}{2\varepsilon_{0}^{2}}\right)}$ Rearranging the inequality by renaming the upper bound on the probability as $\delta$, we get, $\mathbb{P}\left[\sum_{i}C^{i}\geq n\mu+\varepsilon_{0}\sqrt{2n\ln{\frac{1}{\delta}}}\right]\leq\delta$ Theorem 9 is an analogue, in the privacy at risk setting, of the advanced composition of differential privacy (Dwork et al., 2014, Theorem 3.20) under a constraint of independent evaluations. Note that, if one takes $\gamma=0$, then we obtain the exact same formula as in (Dwork et al., 2014, Theorem 3.20). It provides a sanity check for the consistency of composition using privacy at risk. In fact, if we consider both sources of randomness, the expected value of loss function must be computed by using the law of total expectation. $\mathbb{E}[\mathcal{C}]=\mathbb{E}_{x,y\sim\mathcal{G}}[\mathbb{E}[\mathcal{C}]|x,y]$ Therefore, the exact computation of privacy guarantees after the composition requires access to the data-generation distribution. We assume a uniform data- generation distribution while proving Theorem 9. We can obtain better and finer privacy guarantees accounting for data-generation distribution, which we keep as a future work. ### 3.2 Convexity and Post-processing Since privacy at risk provides a probabilistic privacy guarantee for an $\varepsilon_{0}$-differentially private mechanism, it does not alter the basic properties of differential privacy - convexity and post-processing. We now show that privacy at risk equally adheres to both of these properties. ###### Lemma 10 (Convexity) For a given $\varepsilon_{0}$-differentially private privacy-preserving mechanism, privacy at risk satisfies convexity property. Proof Let $\mathcal{M}$ be a mechanism that satisfies $\varepsilon_{0}$-differential privacy. By the definition of the privacy at risk, it also satisfies $(\varepsilon_{1},\gamma_{1})$-privacy at risk as well as $(\varepsilon_{2},\gamma_{2})$-privacy at risk for some $\varepsilon_{1},\varepsilon_{2}\leq\varepsilon_{0}$ and appropriately computed values of $\gamma_{1}$ and $\gamma_{2}$. Let $\mathcal{M}^{1}$ and $\mathcal{M}^{2}$ denote the hypothetical mechanisms that satisfy $(\varepsilon_{1},\gamma_{1})$-privacy at risk and $(\varepsilon_{2},\gamma_{2})$-privacy at risk respectively. We can write privacy loss random variables as follows: $\displaystyle\mathcal{C}^{1}$ $\displaystyle\leq\gamma_{1}\varepsilon_{1}+(1-\gamma_{1})\varepsilon_{0}$ $\displaystyle\mathcal{C}^{2}$ $\displaystyle\leq\gamma_{2}\varepsilon_{2}+(1-\gamma_{2})\varepsilon_{0}$ where $\mathcal{C}^{1}$ and $\mathcal{C}^{2}$ denote privacy loss random variables for $\mathcal{M}^{1}$ and $\mathcal{M}^{2}$. Let us consider a privacy-preserving mechanism $\mathcal{M}$ that uses $\mathcal{M}^{1}$ with a probability $p$ and $\mathcal{M}^{2}$ with a probability $(1-p)$ for some $p\in[0,1]$. By using the techniques in the proof of Theorem 9, the privacy loss random variable $\mathcal{C}$ for $\mathcal{M}$ can be written as: $\displaystyle\mathcal{C}$ $\displaystyle=p\mathcal{C}^{1}+(1-p)\mathcal{C}^{2}$ $\displaystyle\leq\gamma^{\prime}\varepsilon^{\prime}+(1-\gamma^{\prime})\varepsilon_{0}$ where $\displaystyle\varepsilon^{\prime}$ $\displaystyle=\frac{p\gamma_{1}\varepsilon_{1}+(1-p)\gamma_{2}\varepsilon_{2}}{p\gamma_{1}+(1-p)\gamma_{2}}$ $\displaystyle\gamma^{\prime}$ $\displaystyle=(1-p\gamma_{1}-(1-p)\gamma_{2})$ Thus, $\mathcal{M}$ satisfies $(\varepsilon^{\prime},\gamma^{\prime})$-privacy at risk. It proves that privacy at risk staisfies convexity (Kifer and Lin, 2012, Axiom 2.1.2). ###### Lemma 11 (Post-processing) For a given $\varepsilon_{0}$-differentially private privacy-preserving mechanism, privacy at risk satisfies post-processing property. Proof ${(\varepsilon_{0},0)-DP}$${(\varepsilon_{0},0)-DP}$${(\varepsilon,\gamma)-PaR}$${(\varepsilon,\gamma)-PaR}$$\scriptstyle{\varphi}$ Privacy at risk analyses sources of randomness involved in a privacy- preserving mechanism to provide probabilistic guarantees over the privacy level of differential privacy. Application of a deterministic function as a post-processing does not effect either data-generation distribution or noise- distribution. Thus privacy at risk simply reflects the features of underlying privacy level under post-processing. Since differential privacy is preserved under post-processing (Dwork et al., 2014, Proposition 2.1), so is the privacy at risk. We need to closely observe Lemma 11. Privacy at risk is preserved under post- processing only if the mechanism satisfies a stronger privacy guarantee such as differential privacy. If the mechanism satisfies a variant such probabilistic differential privacy, which does not satisfy post-processing property Machanavajjhala et al. (2008), then privacy at risk will not be preserved under post-processing. ## 4 Privacy at Risk for Laplace Mechanism In this section, we instantiate privacy at risk for the Laplace mechanism for three cases: two cases involving two sources of randomness and third case involving the coupled effect. Three different cases correspond to three different interpretations of the confidence level, represented by the parameter $\gamma$, corresponding to three interpretation of the support of the outer probability in Definition 6. In order to highlight this nuance, we denote the confidence levels corresponding to the three cases and their three sources of randomness as $\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$, respectively. ### 4.1 The Case of Explicit Randomness In this section, we study the effect of the explicit randomness induced by the noise sampled from Laplacian distribution. We provide a probabilistic quantification for fine tuning for the Laplace mechanism. We fine-tune the privacy level for a specified risk under by assuming that the sensitivity of the query is known a priori. For a Laplace mechanism $\mathcal{L}_{\varepsilon_{0}}^{\Delta_{f}}$ calibrated with sensitivity $\Delta_{f}$ and privacy level $\varepsilon_{0}$, we present the analytical formula relating privacy level $\varepsilon$ and the risk $\gamma_{1}$ in Theorem 12. The proof is available in Appendix A. ###### Theorem 12 The risk $\gamma_{1}\in[0,1]$ with which a Laplace Mechanism $\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}$, for a numeric query $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$ satisfies a privacy level $\varepsilon\geq 0$ is given by $\gamma_{1}=\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\varepsilon_{0})},$ (3) where $T$ is a random variable that follows a distribution with the following density function. $P_{T}(t)=\frac{2^{1-k}t^{k-\frac{1}{2}}K_{k-\frac{1}{2}}(t)\varepsilon_{0}}{\sqrt{2\pi}\Gamma(k)\Delta_{f}}$ where $K_{n-\frac{1}{2}}$ is the Bessel function of second kind. Figure 1 shows the plot of the privacy level against risk for different values of $k$ and for a Laplace mechanism $\mathcal{L}^{1.0}_{1.0}$. As the value of $k$ increases, the amount of noise added in the output of numeric query increases. Therefore, for a specified privacy level, the privacy at risk level increases with the value of $k$. The analytical formula representing $\gamma_{1}$ as a function of $\varepsilon$ is bijective. We need to invert it to obtain the privacy level $\varepsilon$ for a privacy at risk $\gamma_{1}$. However the analytical closed form for such an inverse function is not explicit. We use a numerical approach to compute privacy level for a given privacy at risk from the analytical formula of Theorem 12. Figure 1: Figure 2: Figure 3: Privacy level $\varepsilon$ for varying privacy at risk $\gamma_{1}$ for Laplace mechanism $\mathcal{L}_{\varepsilon_{0}}^{1.0}$. In Figure 1, we use $\varepsilon_{0}=1.0$ and different values of $k$. In Figure 2, for $k=1$ and different values of $\varepsilon_{0}$. Figure 4: Number of samples $n$ for varying privacy at risk $\gamma_{2}$ for different error parameter $\rho$. Result for a Real-valued Query. For the case $k=1$, the analytical derivation is fairly straightforward. In this case, we obtain an invertible closed-form of a privacy level for a specified risk. It is presented in Equation 4. $\varepsilon=\ln{\left(\frac{1}{1-\gamma_{1}(1-e^{-\varepsilon_{0}})}\right)}$ (4) Remarks on $\varepsilon_{0}$. For $k=1$, Figure 2 shows the plot of privacy at risk level $\varepsilon$ versus privacy at risk $\gamma_{1}$ for the Laplace mechanism $\mathcal{L}^{1.0}_{\varepsilon_{0}}$. As the value of $\varepsilon_{0}$ increases, the probability of Laplace mechanism generating higher value of noise reduces. Therefore, for a fixed privacy level, privacy at risk increases with the value of $\varepsilon_{0}$. The same observation is made for $k>1$. ### 4.2 The Case of Implicit Randomness In this section, we study the effect of the implicit randomness induced by the data-generation distribution to provide a fine tuning for the Laplace mechanism. We fine-tune the risk for a specified privacy level without assuming that the sensitivity of the query. If one takes into account randomness induced by the data-generation distribution, all pairs of neighbouring datasets are not equally probable. This leads to estimation of sensitivity of a query for a specified data- generation distribution. If we have access to an analytical form of the data- generation distribution and to the query, we could analytically derive the sensitivity distribution for the query. In general, we have access to the datasets, but not the data-generation distribution that generates them. We, therefore, statistically estimate sensitivity by constructing an empirical distribution. We call the sensitivity value obtained for a specified risk from the empirical cumulative distribution of sensitivity the sampled sensitivity (Definition 14). However, the value of sampled sensitivity is simply an estimate of the sensitivity for a specified risk. In order to capture this additional uncertainty introduced by the estimation from the empirical sensitivity distribution rather than the true unknown distribution, we compute a lower bound on the accuracy of this estimation. This lower bound yields a probabilistic lower bound on the specified risk. We refer to it as empirical risk. For a specified absolute risk $\gamma_{2}$, we denote by $\hat{\gamma_{2}}$ corresponding empirical risk. For the Laplace mechanism $\mathcal{L}_{\varepsilon}^{\Delta_{S_{f}}}$ calibrated with sampled sensitivity $\Delta_{S_{f}}$ and privacy level $\varepsilon$, we evaluate the empirical risk $\hat{\gamma_{2}}$. We present the result in Theorem 13. The proof is available in Appendix B. ###### Theorem 13 Analytical bound on the empirical risk, $\hat{\gamma_{2}}$, for Laplace mechanism $\mathcal{L}_{\varepsilon}^{\Delta_{S_{f}}}$ with privacy level $\varepsilon$ and sampled sensitivity $\Delta_{S_{f}}$ for a query $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$ is $\hat{\gamma_{2}}\geq\gamma_{2}(1-2e^{-2\rho^{2}n})$ (5) where $n$ is the number of samples used for estimation of the sampled sensitivity and $\rho$ is the accuracy parameter. $\gamma_{2}$ denotes the specified absolute risk. The error parameter $\rho$ controls the closeness between the empirical cumulative distribution of the sensitivity to the true cumulative distribution of the sensitivity. Lower the value of the error, closer is the empirical cumulative distribution to the true cumulative distribution. Figure 4 shows the plot of number of samples as a function of the privacy at risk and the error parameter. Naturally, we require higher number of samples in order to have lower error rate. The number of samples reduces as the privacy at risk increases. The lower risk demands precision in the estimated sampled sensitivity, which in turn requires larger number of samples. Let, $\mathcal{G}$ denotes the data-generation distribution, either known apriori or constructed by subsampling the available data. We adopt the procedure of (Rubinstein and Aldà, 2017) to sample two neighbouring datasets with $p$ data points each. We sample $p-1$ data points from $\mathcal{G}$ that are common to both of these datasets and later two more data points. From those two points, we allot one data point to each of the two datasets. Let, $S_{f}=\lVert f(x)-f(y)\rVert_{1}$ denotes the sensitivity random variable for a given query $f$, where $x$ and $y$ are two neighbouring datasets sampled from $\mathcal{G}$. Using $n$ pairs of neighbouring datasets sampled from $\mathcal{G}$, we construct the empirical cumulative distribution, $F_{n}$, for the sensitivity random variable. ###### Definition 14 For a given query $f$ and for a specified risk $\gamma_{2}$, sampled sensitivity, $\Delta_{S_{f}}$, is defined as the value of sensitivity random variable that is estimated using its empirical cumulative distribution function, $F_{n}$, constructed using $n$ pairs of neighbouring datasets sampled from the data-generation distribution $\mathcal{G}$. $\Delta_{S_{f}}\triangleq F_{n}^{-1}(\gamma_{2})$ If we knew analytical form of the data generation distribution, we could analytically derive the cumulative distribution function of the sensitivity, $F$, and find the sensitivity of the query as $\Delta_{f}=F^{-1}(1)$. Therefore, in order to have the sampled sensitivity close to the sensitivity of the query, we require the empirical cumulative distributions to be close to the cumulative distribution of the sensitivity. We use this insight to derive the analytical bound in the Theorem 13. Figure 5: Figure 6: Figure 7: Dependence of error and number of samples on the privacy at risk for Laplace mechanism $\mathcal{L}_{1.0}^{\Delta_{S_{f}}}$. For the figure on the left hand side, we fix the number of samples to $10000$. For the Figure 6 we fix the error parameter to $0.01$. ### 4.3 The Case of Explicit and Implicit Randomness In this section, we study the combined effect of both explicit randomness induced by the noise distribution and implicit randomness in the data- generation distribution respectively. We do not assume the knowledge of the sensitivity of the query. We estimate sensitivity using the empirical cumulative distribution of sensitivity. We construct the empirical distribution over the sensitivities using the sampling technique presented in the earlier case. Since we use the sampled sensitivity (Definition 14) to calibrate the Laplace mechanism, we estimate the _empirical risk_ $\hat{\gamma_{3}}$. For Laplace mechanism $\mathcal{L}_{\varepsilon_{0}}^{\Delta_{S_{f}}}$ calibrated with sampled sensitivity $\Delta_{S_{f}}$ and privacy level $\varepsilon_{0}$, we present the analytical bound on the empirical sensitivity $\hat{\gamma_{3}}$ in Theorem 15 with proof in the Appendix C. ###### Theorem 15 Analytical bound on the empirical risk $\hat{\gamma_{3}}\in[0,1]$ to achieve a privacy level $\varepsilon>0$ for Laplace mechanism $\mathcal{L}_{\varepsilon_{0}}^{\Delta_{S_{f}}}$ with sampled sensitivity $\Delta_{S_{f}}$ of a query $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$ is $\hat{\gamma_{3}}\geq\gamma_{3}(1-2e^{-2\rho^{2}n})$ (6) where $n$ is the number of samples used for estimating the sensitivity, $\rho$ is the accuracy parameter. $\gamma_{3}$ denotes the specified absolute risk defined as: $\gamma_{3}=\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\eta\varepsilon_{0})}\cdot\gamma_{2}$ The error parameter $\rho$ controls the closeness between the empirical cumulative distribution of the sensitivity to the true cumulative distribution of the sensitivity. Figure 7 shows the dependence of the error parameter on the number of samples. In Figure 5, we observe that the for a fixed number of samples and a privacy level, the privacy at risk decreases with the value of error parameter. For a fixed number of samples, smaller values of the error parameter reduce the probability of similarity between the empirical cumulative distribution of sensitivity and the true cumulative distribution. Therefore, we observe the reduction in the risk for a fixed privacy level. In Figure 6, we observe that for a fixed value of error parameter and a fixed level of privacy level, the risk increases with the number of samples. For a fixed value of the error parameter, larger values of the sample size increase the probability of similarity between the empirical cumulative distribution of sensitivity and the true cumulative distribution. Therefore, we observe the increase in the risk for a fixed privacy level. Effect of the consideration of implicit and explicit randomness is evident in the analytical expression for $\gamma_{3}$ in Equation 7. Proof is available in Appendix C. The privacy at risk is composed of two factors whereas the second term is a privacy at risk that accounts for inherent randomness. The first term takes into account the implicit randomness of the Laplace distribution along with a coupling coefficient $\eta$. We define $\eta$ as the ratio of the true sensitivity of the query to its sampled sensitivity. $\gamma_{3}\triangleq\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\eta\varepsilon_{0})}\cdot\gamma_{2}$ (7) ## 5 Minimising Compensation Budget for Privacy at Risk Many service providers collect users’ data to enhance user experience. In order to avoid misuse of this data, we require a legal framework that not only limits the use of the collected data but also proposes reparative measures in case of a data leak. General Data Protection Regulation (GDPR)111https://eugdpr.org/ is such a legal framework. Section 82 in GDPR states that any person who suffers from material or non- material damage as a result of a personal data breach has the right to demand compensation from the data processor. Therefore, every GDPR compliant business entity that either holds or processes personal data needs to secure a certain budget in the worst case scenario of the personal data breach. In order to reduce the risk of such an unfortunate event, the business entity may use privacy-preserving mechanisms that provide provable privacy guarantees while publishing their results. In order to calculate the compensation budget for a business entity, we devise a cost model that maps the privacy guarantees provided by differential privacy and privacy at risk to monetary costs. The discussions demonstrate the usefulness of probabilistic quantification of differential privacy in a business setting. ### 5.1 Cost Model for Differential Privacy Let $E$ be the compensation budget that a business entity has to pay to every stakeholder in case of a personal data breach when the data is processed without any provable privacy guarantees. Let $E_{\varepsilon}^{dp}$ be the compensation budget that a business entity has to pay to every stakeholder in case of a personal data breach when the data is processed with privacy guarantees in terms of $\varepsilon$-differential privacy. Privacy level, $\varepsilon$, in $\varepsilon$-differential privacy is the quantifier of indistinguishability of the outputs of a privacy-preserving mechanism when two neighbouring datasets are provided as inputs. When the privacy level is zero, the privacy-preserving mechanism outputs all results with equal probability. The indistinguishability reduces with increase in the privacy level. Thus, privacy level of zero bears the lowest risk of personal data breach and the risk increases with the privacy level. $E_{\varepsilon}^{dp}$ needs to be commensurate to such a risk and, therefore, it needs to satisfy the following constraints. 1. 1. For all $\varepsilon\in\mathbb{R}^{\geq 0}$, $E_{\varepsilon}^{dp}\leq E$. 2. 2. $E_{\varepsilon}^{dp}$ is a monotonically increasing function of $\varepsilon$. 3. 3. As $\varepsilon\rightarrow 0$, $E_{\varepsilon}^{dp}\rightarrow E_{min}$ where $E_{min}$ is the unavoidable cost that business entity might need to pay in case of personal data breach even after the privacy measures are employed. 4. 4. As $\varepsilon\rightarrow\infty$, $E_{\varepsilon}^{dp}\rightarrow E$. There are various functions that satisfy these constraints. For instantiation, we choose to work with a cost model that is convex with respect to $\varepsilon$. The cost model $E_{\varepsilon}^{dp}$ as defined in Equation 8. $E_{\varepsilon}^{dp}\triangleq E_{min}+Ee^{-\frac{c}{\varepsilon}}$ (8) $E_{\varepsilon}^{dp}$ has two parameters, namely $c>0$ and $E_{min}\geq 0$. $c$ controls the rate of change in the cost as the privacy level changes and $E_{min}$ is a privacy level independent bias. For this study, we use a simplified model with $c=1$ and $E_{min}=0$. ### 5.2 Cost Model for Privacy at Risk Let $E_{\varepsilon_{0}}^{par}(\varepsilon,\gamma)$ be the compensation that a business entity has to pay to every stakeholder in case of a personal data breach when the data is processed with an $\varepsilon_{0}$-differentially private privacy-preserving mechanism along with a probabilistic quantification of privacy level. Use of such a quantification allows use to provide a stronger a stronger privacy guarantee viz. $\varepsilon<\varepsilon_{0}$ for a specified privacy at risk at most $\gamma$ for Thus, we calculate $E_{\varepsilon_{0}}^{par}$ using Equation 9. $E_{\varepsilon_{0}}^{par}(\varepsilon,\gamma)\triangleq\gamma E_{\varepsilon}^{dp}+(1-\gamma)E_{\varepsilon_{0}}^{dp}$ (9) #### 5.2.1 Existence of Minimum Compensation Budget We want to find the privacy level, say $\varepsilon_{min}$, that yields the lowest compensation budget. We do that by minimising Equation 9 with respect to $\varepsilon$. ###### Lemma 16 $E_{\varepsilon_{0}}^{par}(\varepsilon,\gamma)$ is a convex function of $\varepsilon$ if $E_{\varepsilon}^{dp}$ is defined by Equation (9). This result also generalises to any other convex cost model satisfying the four conditions. By Lemma 16, there exists a unique $\varepsilon_{min}$ that minimises the compensation budget for a specified parametrisation, say $\varepsilon_{0}$. Since the risk $\gamma$ in Equation 9 is itself a function of privacy level $\varepsilon$, analytical calculation of $\varepsilon_{min}$ is not possible in the most general case. When the output of the query is a real number, we derive the analytic form (Equation 4) to compute the risk under the consideration of explicit randomness. In such a case, $\varepsilon_{min}$ is calculated by differentiating Equation 9 with respect to $\varepsilon$ and equating it to zero. It gives us Equation 10 that we solve using any root finding technique such as Newton-Raphson method (Press, 2007) to compute $\varepsilon_{min}$. $\frac{1}{\varepsilon}-\ln{\left(1-\frac{1-e^{\varepsilon}}{\varepsilon^{2}}\right)}=\frac{1}{\varepsilon_{0}}$ (10) #### 5.2.2 Fine-tuning Privacy at Risk For a fixed budget, say $B$, re-arrangement of Equation 9 gives us an upper bound on the privacy level $\varepsilon$. We use the cost model with $c=1$ and $E_{min}=0$ to derive the upper bound. If we have a maximum permissible expected mean absolute error $T$, we use Equation 12 to obtain a lower bound on the privacy at risk level. Equation 11 illustrates the upper and lower bounds that dictate the permissible range of $\varepsilon$ that a data publisher can promise depending on the budget and the permissible error constraints. $\frac{1}{T}\leq\varepsilon\leq\left[\ln{\left(\frac{\gamma E}{B-(1-\gamma)E_{\varepsilon_{0}}^{dp}}\right)}\right]^{-1}$ (11) Thus, the privacy level is constrained by the effectiveness requirement from below and by the monetary budget from above. (Hsu et al., 2014) calculate upper and lower bound on the privacy level in the differential privacy. They use a different cost model owing to the scenario of research study that compensates its participants for their data and releases the results in a differentially private manner. Their cost model is different than our GDPR inspired modelling. ### 5.3 Illustration Suppose that the health centre in a university that complies to GDPR publishes statistics of its staff health checkup, such as obesity statistics, twice in a year. In January 2018, the health centre publishes that 34 out of 99 faculty members suffer from obesity. In July 2018, the health centre publishes that 35 out of 100 faculty members suffer from obesity. An intruder, perhaps an analyst working for an insurance company, checks the staff listings in January 2018 and July 2018, which are publicly available on website of the university. The intruder does not find any change other than the recruitment of John Doe in April 2018. Thus, with high probability, the intruder deduces that John Doe suffers from obesity. In order to avoid such a privacy breach, the health centre decides to publish the results using the Laplace mechanism. In this case, the Laplace mechanism operates on the count query. Figure 8: Variation in the budget for Laplace mechanism $\mathcal{L}_{\varepsilon_{0}}^{1}$ under privacy at risk considering explicit randomness in the Laplace mechanism for the illustration in Section 5.3. In order to control the amount of noise, the health centre needs to appropriately set the privacy level. Suppose that the health centre decides to use the expected mean absolute error, defined in Equation 12, as the measure of effectiveness for the Laplace mechanism. $\mathbb{E}\left[\lvert\mathcal{L}^{1}_{\varepsilon}(x)-f(x)\rvert\right]=\frac{1}{\varepsilon}$ (12) Equation 12 makes use of the fact that the sensitivity of the count query is one. Suppose that the health centre requires the expected mean absolute error of at most two in order to maintain the quality of the published statistics. In this case, the privacy level has to be at least $0.5$. In order to compute the budget, the health centre requires an estimate of $E$. Moriarty et al. (2012) shows that the incremental cost of premiums for the health insurance with morbid obesity ranges between $\$5467$ to $\$5530$. With reference to this research, the health centre takes $\$5500$ as an estimate of $E$. For the staff size of $100$ and the privacy level $0.5$, the health centre uses Equation 8 in its simplified setting to compute the total budget of $\$74434.40$. Is it possible to reduce this budget without degrading the effectiveness of the Laplace mechanism? We show that it is possible by fine-tuning the Laplace mechanism. Under the consideration of the explicit randomness introduced by the Laplace noise distribution, we show that $\varepsilon_{0}$-differentially private Laplace mechanism also satisfies $\varepsilon$-differential privacy with risk $\gamma$, which is computed using the formula in Theorem 12. Fine- tuning allows us to get a stronger privacy guarantee, $\varepsilon<\varepsilon_{0}$ that requires a smaller budget. In Figure 8, we plot the budget for various privacy levels. We observe that the privacy level $0.274$, which is same as $\varepsilon_{min}$ computed by solving Equation 10, yields the lowest compensation budget of $\$37805.86$. Thus, by using privacy at risk, the health centre is able to save $\$36628.532$ without sacrificing the quality of the published results. Figure 9: $\mathcal{L}_{0.1}^{1}$ satisfies $(0.08,0.80)$-privacy at risk. Figure 10: $\mathcal{L}_{0.5}^{1}$ satisfies $(0.27,0.61)$-privacy at risk. Figure 11: $\mathcal{L}_{1.0}^{1}$ satisfies $(0.42,0.54)$-privacy at risk. Figure 12: Comparing the privacy guarantee obtained by basic composition and advanced composition (Dwork et al., 2014) with the composition obtained using optimal privacy at risk that minimises the cost of Laplace mechanism $\mathcal{L}_{\varepsilon_{0}}^{1}$. For the evaluation, we set $\delta=10^{-5}$. ### 5.4 Cost Model and the Composition of Laplace Mechanisms Convexity of the proposed cost function enables us to estimate the optimal value of the privacy at risk level. We use the optimal privacy value to provide tighter bounds on the composition of Laplace mechanism. In Figure 12, we compare the privacy guarantees obtained by using basic composition theorem (Dwork et al., 2014), advanced composition theorem (Dwork et al., 2014) and the composition theorem for privacy at risk. We comparatively evaluate them for composition of Laplace mechanisms with privacy levels $0.1,0.5$ and $1.0$. We compute the privacy level after composition by setting $\delta$ to $10^{-5}$. We observe that the use of optimal privacy at risk provided significantly stronger privacy guarantees as compared to the conventional composition theorems. Advanced composition theorem is known to provide stronger privacy guarantees for mechanism with smaller $\varepsilon$s. As we observe in Figure 11 and Figure 10, the composition provides strictly stronger privacy guarantees than basic composition, in the cases where the advanced composition fails. ## 6 Balancing Utility and Privacy In this section, we empirically illustrate and discuss the steps that a data steward needs to take and the issues that she needs to consider in order to realise a required privacy at risk level $\varepsilon$ for a confidence level $\gamma$ when seeking to disclose the result of a query. We consider a query that returns the parameter of a ridge regression (Murphy, 2012) for an input dataset. It is a basic and widely used statistical analysis tool. We use the privacy-preserving mechanism presented by Ligett et al. for ridge regression. It is a Laplace mechanism that induces noise in the output parameters of the ridge regression. The authors provide a theoretical upper bound on the sensitivity of the ridge regression, which we refer as sensitivity, in the experiments. ### 6.1 Dataset and Experimental Setup We conduct experiments on a subset of the 2000 US census dataset provided by Minnesota Population Center in its Integrated Public Use Microdata Series (Ruggles et al., 2015). The census dataset consists of 1% sample of the original census data. It spans over 1.23 million households with records of 2.8 million people. The value of several attributes is not necessarily available for every household. We have therefore selected $212,605$ records, corresponding to the household heads, and $6$ attributes, namely, Age, Gender, Race, Marital Status, Education, Income, whose values are available for the $212,605$ records. In order to satisfy the constraint in the derivation of the sensitivity of ridge regression (Ligett et al., 2017), we, without loss of generality, normalise the dataset in the following way. We normalise Income attribute such that the values lie in $[0,1]$. We normalise other attributes such that $l_{2}$ norm of each data point is unity. All experiments are run on Linux machine with 12-core 3.60GHz Intel® Core i7™processor with 64GB memory. Python® 2.7.6 is used as the scripting language. ### 6.2 Result Analysis We train ridge regression model to predict Income using other attributes as predictors. We split the dataset into the training dataset ($80\%$) and testing dataset ($20\%$). We compute the root mean squared error (RMSE) of ridge regression, trained on the training data with regularisation parameter set to $0.01$, on the testing dataset. We use it as the metric of utility loss. Smaller the value of RMSE, smaller the loss in utility. For a given value of privacy at risk level, we compute $50$ runs of an experiment of a differentially private ridge regression and report the means over the $50$ runs of the experiment. Let us now provide illustrative experiments under the three different cases. In every scenario, the data steward is given a privacy at risk level $\varepsilon$ and the confidence level $\gamma$ and wants to disclose the parameters of a ridge regression model that she trains on the census dataset. She needs to calibrate the Laplace mechanism to achieve the privacy at risk required the ridge regression query. The Case of Explicit Randomness (cf. Section 4.1). In this scenario, the data steward knows the sensitivity for the ridge regression. She needs to compute the privacy level, $\varepsilon_{0}$, to calibrate the Laplace mechanism. She uses Equation 3 that links the desired privacy at risk level $\varepsilon$, the confidence level $\gamma_{1}$ and the privacy level of noise $\varepsilon_{0}$. Specifically, for given $\varepsilon$ and $\gamma_{1}$, she computes $\varepsilon_{0}$ by solving the equation: $\gamma_{1}\mathbb{P}(T\leq\varepsilon_{0})-\mathbb{P}(T\leq\varepsilon)=0.$ Since the equation does not give an analytical formula for $\varepsilon_{0}$, the data steward uses a root finding algorithm such as Newton-Raphson method (Press, 2007) to solve the above equation. For instance, if she needs to achieve a privacy at risk level $\varepsilon=0.4$ with confidence level $\gamma_{1}=0.6$, she can substitute these values in the above equation and solve the equation to get the privacy level of noise $\varepsilon_{0}=0.8$. Figure 13: Utility, measured by RMSE (right y-axis), and privacy at risk level $\varepsilon$ for Laplace mechanism (left y-axis) for varying confidence levels $\gamma_{1}$. Figure 14: Empirical cumulative distribution of the sensitivities of ridge regression queries constructed using $15000$ samples of neighboring datasets. Figure 13 shows the variation of privacy at risk level $\varepsilon$ and confidence level $\gamma_{1}$. It also depicts the variation of utility loss for different privacy at risk levels in Figure 13. In accordance to the data steward’s problem, if she needs to achieve a privacy at risk level $\varepsilon=0.4$ with confidence level $\gamma_{1}=0.6$, she obtains the privacy level of noise to be $\varepsilon_{0}=0.8$. Additionally, we observe that the choice of privacy level $0.8$ instead of $0.4$ to calibrate the Laplace mechanism gives lower utility loss for the data steward. This is the benefit drawn from the risk taken under the control of privacy at risk. Thus, she uses privacy level $\varepsilon_{0}$ and the sensitivity of the function to calibrate Laplace mechanism. The Case of Implicit Randomness (cf. Section 4.2). In this scenario, the data steward does not know the sensitivity of ridge regression. She assesses that she can afford to sample at most $n$ times from the population dataset. She understands the effect of the uncertainty introduced by the statistical estimation of the sensitivity. Therefore, she uses the confidence level for empirical privacy at risk $\hat{\gamma_{2}}$. Given the value of $n$, she chooses the value of the accuracy parameter using Figure 4. For instance, if the number of samples that she can draw is $10^{4}$, she chooses the value of the accuracy parameter $\rho=0.01$. Next, she uses Equation 13 to determine the value of probabilistic tolerance, $\alpha$, for the sample size $n$. For instance, if the data steward is not allowed to access more than $15,000$ samples, for the accuracy of $0.01$ the probabilistic tolerance is $0.9$. $\alpha=1-2e^{(-2\rho^{2}n)}$ (13) She constructs an empirical cumulative distribution over the sensitivities as described in Section 4.2. Such an empirical cumulative distribution is shown in Figure 14. Using the computed probabilistic tolerance and desired confidence level $\hat{\gamma_{2}}$, she uses equation in Theorem 13 to determine $\gamma_{2}$. She computes the sampled sensitivity using the empirical distribution function and the confidence level for privacy $\Delta_{S_{f}}$ at risk $\gamma_{2}$. For instance, using the empirical cumulative distribution in Figure 14 she calculates the value of the sampled sensitivity to be approximately $0.001$ for $\gamma_{2}=0.4$ and approximately $0.01$ for $\gamma_{2}=0.85$ Thus, she uses privacy level $\varepsilon$, sets the number of samples to be $n$ and computes the sampled sensitivity $\Delta_{S_{f}}$ to calibrate the Laplace mechanism. The Case of Explicit and Implicit Randomness (cf. Section 4.3). In this scenario, the data steward does not know the sensitivity of ridge regression. She is not allowed to sample more than $n$ times from a population dataset. For a given confidence level $\gamma_{2}$ and the privacy at risk $\varepsilon$, she calibrates the Laplace mechanism using illustration for Section 4.3. The privacy level in this calibration yields utility loss that is more than her requirement. Therefore, she wants to re-calibrate the Laplace mechanism in order to reduce utility loss. For the re-calibration, the data steward uses privacy level of the pre- calibrated Laplace mechanism, i.e. $\varepsilon$, as the privacy at risk level and she provides a new confidence level for empirical privacy at risk $\hat{\gamma_{3}}$. Using Equation 26 and Equation 24, she calculates: $\hat{\gamma_{3}}\mathbb{P}(T\leq\eta\varepsilon_{0})-\alpha\gamma_{2}~{}\mathbb{P}(T\leq\varepsilon)=0$ She solves such an equation for $\varepsilon_{0}$ using the root finding technique such as Newton-Raphson method (Press, 2007). For instance, if she needs to achieve a privacy at risk level $\varepsilon=0.4$ with confidence levels $\hat{\gamma_{3}}=0.9$ and $\gamma_{2}=0.9$, she can substitute these values and the values of tolerance parameter and sampled sensitivity, as used in the previous experiments, in the above equation. Then, solving the equation leads to the privacy level of noise $\varepsilon_{0}=0.8$. Thus, she re-calibrates the Laplace mechanism with privacy level $\varepsilon_{0}$, sets the number of samples to be $n$ and sampled sensitivity $\Delta_{S_{f}}$. ## 7 Related Work Calibration of mechanisms. Researchers have proposed different privacy- preserving mechanisms to make different queries differentially private. These mechanisms can be broadly classified into two categories. In one category, the mechanisms explicitly add calibrated noise, such as Laplace noise in the work of (Dwork et al., 2006c) or Gaussian noise in the work of (Dwork et al., 2014), to the outputs of the query. In the other category, (Chaudhuri et al., 2011; Zhang et al., 2012; Acs et al., 2012; Hall et al., 2013) propose mechanisms that alter the query function so that the modified function satisfies differentially privacy. Privacy-preserving mechanisms in both of these categories perturb the original output of the query and make it difficult for a malicious data analyst to recover the original output of the query. These mechanisms induce randomness using the explicit noise distribution. Calibration of these mechanisms require the knowledge of the sensitivity of the query. Nissim et al. consider the implicit randomness in the data-generation distribution to compute an estimate of the sensitivity. The authors propose the smooth sensitivity function that is an envelope over the local sensitivities for all individual datasets. Local sensitivity of a dataset is the maximum change in the value of the query over all of its neighboring datasets. In general, it is not easy to analytically estimate the smooth sensitivity function for a general query. Rubinstein and Aldà also study the inherent randomness in the data-generation algorithm. They do not use the local sensitivity. We adopt their approach of sampling the sensitivity from the empirical distribution of the sensitivity. They use order statistics to choose a particular value of the sensitivity. We use the risk, which provides a mediation tool for business entities to assess the actual business risks, on the sensitivity distribution to estimate the sensitivity. Refinements of differential privacy. In order to account for both sources of randomness, refinements of $\varepsilon$-differential privacy are proposed in order to bound the probability of occurrence of worst case scenarios. Machanavajjhala et al. propose probabilistic differential privacy that considers upper bounds of the worst case privacy loss for corresponding confidence levels on the noise distribution. Definition of probabilistic differential privacy incorporates the explicit randomness induced by the noise distribution and bounds the probability over the space of noisy outputs to satisfy the $\varepsilon$-differential privacy definition. Dwork and Rothblum propose Concentrated differential privacy that considers the expected values of the privacy loss random variables for the corresponding. Definition of concentrated differential privacy incorporates the explicit randomness induced by the noise distribution but considering only the expected value of privacy loss satisfying $\varepsilon$-differential privacy definition instead of using the confidence levels limits its scope. Hall et al. propose random differential privacy that considers the privacy loss for corresponding confidence levels on the implicit randomness in the data-generation distribution. Definition of random differential privacy incorporates the implicit randomness induced by the data-generation distribution and bounds the probability over the space of datasets generated from the given distribution to satisfy the $\varepsilon$-differential privacy definition. Dwork et al. define approximate differential privacy by adding a constant bias to the privacy guarantee provided by the differential privacy. It is not a probabilistic refinement of the differential privacy. Around the same time of our work, Triastcyn and Faltings independently propose Bayesian differential privacy that takes into account both of the sources of randomness. Despite this similarity, our works differ in multiple dimensions. Firstly, they have shown the reduction of their definition to a variant of Renyi differential privacy that depends on the data-generation distribution. Secondly, they rely on the moment accountant for the composition of the mechanisms. Lastly, they do not provide a finer case-by-case analysis of the source of randomness, which leads to analytical solutions for the privacy guarantee. Kifer and Machanavajjhala define Pufferfish privacy framework, and its variant by Bassily et al., that considers randomness due to data-generation distribution as well as noise distribution. Despite the generality of their approach, the framework relies on the domain expert to define a set of _secrets_ that they want to protect. Composition theorem. Recently proposed technique of the moment accountant (Abadi et al., 2016) has become the state-of-the-art of composing mechanisms in the area of privacy-preserving machine learning. Abadi et al. show that the moment accountant provides much strong privacy guarantees than the conventional composition mechanisms. It works by keeping track of various moments of privacy loss random variable and use the bounds on them to provide privacy guarantees. The moment accountant requires access to data-generation distribution to compute the bounds on the moment. Hence, the privacy guarantees are specific to the dataset. Cost models. (Ghosh and Roth, 2015; Chen et al., 2016) propose game theoretic methods that provide the means to evaluate the monetary cost of differential privacy. Our approach is inspired by the approach in the work of Hsu et al.. They model the cost under a scenario of a research study wherein the participants are reimbursed for their participation. Our cost modelling is driven by the scenario of securing a compensation budget in compliance with GDPR. Our requirement differs from the requirements for the scenario in their work. In our case, there is no monetary incentive for participants to share their data. ## 8 Conclusion and Future Works In this paper, we provide a means to fine-tune the privacy level of a privacy- preserving mechanism by analysing various sources of randomness. Such a fine- tuning leads to probabilistic quantification on privacy levels with quantified risks, which we call as privacy at risk. We also provide composition theorem that leverages privacy at risk. We analytical calculate privacy at risk for Laplace mechanism. We propose a cost model that bridges the gap between the privacy level and the compensation budget estimated by a GDPR compliant business entity. Convexity of the cost function ensures existence of unique privacy at risk that minimises compensation budget. The cost model helps in not only reinforcing the ease of application in a business setting but also providing stronger privacy guarantees on the composition of mechanism. Privacy at risk may be fully analytically computed in cases where the data- generation, or the sensitivity distribution, the noise distribution and the query are analytically known and take convenient forms. 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Federated learning with bayesian differential privacy. _arXiv preprint arXiv:1911.10071_ , 2019. * Zhang et al. (2012) Jun Zhang, Zhenjie Zhang, Xiaokui Xiao, Yin Yang, and Marianne Winslett. Functional mechanism: regression analysis under differential privacy. _Proceedings of the VLDB Endowment_ , 5(11):1364–1375, 2012. ## A Proof of Theorem 12 (Section 4.1) Although a Laplace mechanism $\mathcal{L}^{\Delta_{f}}_{\varepsilon}$ induces higher amount of noise on average than a Laplace mechanism $\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}$ for $\varepsilon<\varepsilon_{0}$, there is a non-zero probability that $\mathcal{L}^{\Delta_{f}}_{\varepsilon}$ induces noise commensurate to $\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}$. This non-zero probability guides us to calculate the privacy at risk $\gamma_{1}$ for the privacy at risk level $\varepsilon$. In order to get an intuition, we illustrate the calculation of the overlap between two Laplace distributions as an estimator of similarity between the two distributions. ###### Definition 17 [Overlap of Distributions, (Papoulis and Pillai, 2002)] The overlap, $O$, between two probability distributions $P_{1},P_{2}$ with support $\mathcal{X}$ is defined as $O=\int_{\mathcal{X}}\min[P_{1}(x),P_{2}(x)]~{}{dx}.$ ###### Lemma 18 The overlap $O$ between two probability distributions, $\mathrm{Lap}(\frac{\Delta_{f}}{\varepsilon_{1}})$ and $\mathrm{Lap}(\frac{\Delta_{f}}{\varepsilon_{2}})$, such that $\varepsilon_{2}\leq\varepsilon_{1}$, is given by $O=1-(\exp{(-\mu\varepsilon_{2}/\Delta_{f})}-\exp{(-\mu\varepsilon_{1}/\Delta_{f})}),$ where $\mu=\frac{\Delta_{f}\ln{(\varepsilon_{1}/\varepsilon_{2})}}{\varepsilon_{1}-\varepsilon_{2}}$. Using the result in Lemma 18, we note that the overlap between two distributions with $\varepsilon_{0}=1$ and $\varepsilon=0.6$ is $0.81$. Thus, $\mathcal{L}^{\Delta_{f}}_{0.6}$ induces noise that is more than $80\%$ times similar to the noise induced by $\mathcal{L}^{\Delta_{f}}_{1.0}$. Therefore, we can loosely say that at least $80\%$ of the times a Laplace Mechanism $\mathcal{L}^{\Delta_{f}}_{1.0}$ will provide the same privacy as a Laplace Mechanism $\mathcal{L}^{\Delta_{f}}_{0.8}$. Although the overlap between Laplace distributions with different scales offers an insight into the relationship between different privacy levels, it does not capture the constraint induced by the sensitivity. For a given query $f$, the amount of noise required to satisfy differential privacy is commensurate to the sensitivity of the query. This calibration puts a constraint on the noise that is required to be induced on a pair of neighbouring datasets. We state this constraint in Lemma 19, which we further use to prove that the Laplace Mechanism $\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}$ satisfies $(\varepsilon,\gamma_{1})$-privacy at risk. ###### Lemma 19 For a Laplace Mechanism $\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}$, the difference in the absolute values of noise induced on a pair of neighbouring datasets is upper bounded by the sensitivity of the query. Proof Suppose that two neighbouring datasets $x$ and $y$ are given input to a numeric query $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$. For any output $z\in\mathbb{R}^{k}$ of the Laplace Mechanism $\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}$, $\displaystyle\sum_{i=1}^{k}\left(|f(y_{i})-z_{i}|-|f(x_{i})-z_{i}|\right)$ $\displaystyle\leq\sum_{i=1}^{k}\left(|f(x_{i})-f(y_{i})|\right)$ $\displaystyle\leq\Delta_{f}.$ We use triangular inequality in the first step and Definition 2 of sensitivity in the second step. We write $\mathrm{Exp}(b)$ to denote a random variable sampled from an _exponential distribution_ with scale $b>0$. We write $\mathrm{Gamma}(k,\theta)$ to denote a random variable sampled from a _gamma distribution_ with shape $k>0$ and scale $\theta>0$. ###### Lemma 20 [(Papoulis and Pillai, 2002)] If a random variable $X$ follows Laplace Distribution with mean zero and scale $b$, $|X|\sim\mathrm{Exp}(b)$. ###### Lemma 21 [(Papoulis and Pillai, 2002)] If $X_{1},...,X_{n}$ are $n$ i.i.d. random variables each following the Exponential Distribution with scale $b$, $\sum_{i=1}^{n}X_{i}\sim\textrm{Gamma}(n,b)$. ###### Lemma 22 If $X_{1}$ and $X_{2}$ are two i.i.d. Gamma$(n,\theta)$ random variables, the probability density function for the random variable $T=|X_{1}-X_{2}|/\theta$ is given by $P_{T}(t;n,\theta)=\frac{2^{2-n}t^{n-\frac{1}{2}}K_{n-\frac{1}{2}}(t)}{\sqrt{2\pi}\Gamma(n)\theta}$ where $K_{n-\frac{1}{2}}$ is the modified Bessel function of second kind. Proof Let $X_{1}$ and $X_{2}$ be two i.i.d. $\mathrm{Gamma}(n,\theta)$ random variables. Characteristic function of a Gamma random variable is given as $\phi_{X_{1}}(z)=\phi_{X_{2}}(z)=(1-\iota z\theta)^{-n}.$ Therefore, $\phi_{X_{1}-X_{2}}(z)=\phi_{X_{1}}(z)\phi_{X_{2}}^{*}(z)=\frac{1}{(1+(z\theta)^{2})^{n}}$ Probability density function for the random variable $X_{1}-X_{2}$ is given by, $\displaystyle P_{X_{1}-X_{2}}(x)$ $\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-\mathrm{i}zx}\phi_{X_{1}-X_{2}}(z)dz$ $\displaystyle=\frac{2^{1-n}{|\frac{x}{\theta}|}^{n-\frac{1}{2}}K_{n-\frac{1}{2}}(|\frac{x}{\theta}|)}{\sqrt{2\pi}\Gamma(n)\theta}$ where $K_{n-\frac{1}{2}}$ is the Bessel function of second kind. Let $T=|\frac{X_{1}-X_{2}}{\theta}|$. Therefore, $P_{T}(t;n,\theta)=\frac{2^{1-n}t^{n-\frac{1}{2}}K_{n-\frac{1}{2}}(t)}{\sqrt{2\pi}\Gamma(n)\theta}$ We use Mathematica (Inc., 2014) to solve the above integral. ###### Lemma 23 If $X_{1}$ and $X_{2}$ are two i.i.d. Gamma$(n,\theta)$ random variables and $|X_{1}-X_{2}|\leq M$, then $T^{\prime}=|X_{1}-X_{2}|/\theta$ follows the distribution with probability density function: $P_{T^{\prime}}(t;n,\theta,M)=\frac{P_{T}(t^{\prime};n,\theta)}{P_{T}(T\leq M)},$ where $P_{T}$ is the probability density function of defined in Lemma 22. ###### Lemma 24 For Laplace Mechanism $\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}$ with query $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$ and for any output $Z\subseteq Range(\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}})$, $\varepsilon\leq\varepsilon_{0}$, $\gamma_{1}\triangleq\mathbb{P}\left[\ln{\left|\frac{\mathbb{P}(\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}(x)\in Z)}{\mathbb{P}(\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}(y)\in Z)}\right|}\leq\varepsilon\right]=\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\varepsilon_{0})},$ where $T$ follows the distribution in Lemma 22, $P_{T}(t;k,\frac{\Delta_{f}}{\varepsilon_{0}})$. Proof Let, $x\in\mathcal{D}$ and $y\in\mathcal{D}$ be two datasets such that $x\sim y$. Let $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$ be some numeric query. Let $\mathbb{P}_{x}(z)$ and $\mathbb{P}_{y}(z)$ denote the probabilities of getting the output $z$ for Laplace mechanisms $\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}(x)$ and $\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}(y)$ respectively. For any point $z\in\mathbb{R}^{k}$ and $\varepsilon\neq 0$, $\displaystyle~{}\frac{\mathbb{P}_{x}(z)}{\mathbb{P}_{y}(z)}$ $\displaystyle=\prod_{i=1}^{k}\frac{\exp{\left(\frac{-\varepsilon_{0}|f(x_{i})-z_{i}|}{\Delta_{f}}\right)}}{\exp{\left(\frac{-\varepsilon_{0}|f(y_{i})-z_{i}|}{\Delta_{f}}\right)}}$ $\displaystyle=\prod_{i=1}^{k}\exp{\left(\frac{\varepsilon_{0}(|f(y_{i})-z_{i}|-|f(x_{i})-z_{i}|)}{\Delta_{f}}\right)}$ $\displaystyle=\exp{\left(\varepsilon\left[\frac{\varepsilon_{0}\sum_{i=1}^{k}(|f(y_{i})-z_{i}|-|f(x_{i})-z_{i}|)}{\varepsilon\Delta_{f}}\right]\right)}.$ (14) By Definition 4, $(f(x)-z),(f(y)-z)\sim\textrm{Lap}(\Delta_{f}/\varepsilon_{0}).$ (15) Application of Lemma 20 and Lemma 21 yields, $\sum_{i=1}^{k}\left(|f(x_{i})-z_{i}|\right)\sim\textrm{Gamma}(k,\Delta_{f}/\varepsilon_{0}).$ (16) Using Equations 15, 16, and Lemma 19, 23, we get $\left(\frac{\varepsilon_{0}}{\Delta_{f}}\sum_{i=1}^{k}\left|\left(|f(y_{i})-z|-|f(x_{i})-z|\right)\right|\right)\\\ \sim P_{T^{\prime}}(t;k,\Delta_{f}/\varepsilon_{0},\Delta_{f}).$ (17) since, $\sum_{i=1}^{k}\left|\left(|f(y_{i})-z|-|f(x_{i})-z|\right)\right|\leq\Delta_{f}$. Therefore, $\mathbb{P}\left(\left[\frac{\varepsilon_{0}}{\Delta_{f}}\sum_{i=1}^{k}\ \left|\left(|f(y_{i})-z|-|f(x_{i})-z|\right)\right|\right]\leq\varepsilon\right)=\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\varepsilon_{0})},$ (18) where $T$ follows the distribution in Lemma 22. We use Mathematica (Inc., 2014) to analytically compute, $\mathbb{P}(T\leq x)\propto\left({}_{1}F_{2}(\frac{1}{2};\frac{3}{2}-k,\frac{3}{2};\frac{x^{2}}{4})\sqrt{\pi}4^{k}x]\right)-\\\ \left(2_{1}F_{2}(k;\frac{1}{2}+k,k+1;\frac{x^{2}}{4})x^{2k}\Gamma(k)\right)$ where ${}_{1}F_{2}$ is the regularised generalised hypergeometric function as defined in (Askey and Daalhuis, 2010). From Equation A and 18, $\mathbb{P}\left[\ln{\left|\frac{\mathbb{P}(\mathcal{L}_{\varepsilon_{0}}^{\Delta_{f}}(x)\in S)}{\mathbb{P}(\mathcal{L}_{\varepsilon_{0}}^{\Delta_{f}}(y)\in S)}\right|}\leq\varepsilon\right]=\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\varepsilon_{0})}.$ This completes the proof of Theorem 12. ###### Corollary 25 Laplace Mechanism $\mathcal{L}^{\Delta_{f}}_{\varepsilon_{0}}$ with $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$ is $(\varepsilon,\delta)$-probabilistically differentially private where $\delta=\begin{cases}1-\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\varepsilon_{0})}&\quad\varepsilon\leq\varepsilon_{0}\\\ 0&\quad\varepsilon>\varepsilon_{0}\end{cases}$ and $T$ follows $\mathrm{BesselK}(k,\Delta_{f}/\varepsilon_{0})$. ## B Proof of Theorem 13 (Section 4.2) Proof Let, $x$ and $y$ be any two neighbouring datasets sampled from the data generating distribution $\mathcal{G}$. Let, $\Delta_{S_{f}}$ be the sampled sensitivity for query $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$. Let, $\mathbb{P}_{x}(z)$ and $\mathbb{P}_{y}(z)$ denote the probabilities of getting the output $z$ for Laplace mechanisms $\mathcal{L}_{\varepsilon}^{\Delta_{S_{f}}}(x)$ and $\mathcal{L}_{\varepsilon}^{\Delta_{S_{f}}}(y)$ respectively. For any point $z\in\mathbb{R}^{k}$ and $\varepsilon\neq 0$, $\displaystyle\frac{\mathbb{P}_{x}(z)}{\mathbb{P}_{y}(z)}$ $\displaystyle=\prod_{i=1}^{k}\frac{\exp{\left(\frac{-\varepsilon|f(x_{i})-z_{i}|}{\Delta_{S_{f}}}\right)}}{\exp{\left(\frac{-\varepsilon|f(y_{i})-z_{i}|}{\Delta_{S_{f}}}\right)}}$ $\displaystyle=\exp{\left(\frac{\varepsilon\sum_{i=1}^{k}(|f(y_{i})-z_{i}|-|f(x_{i})-z_{i}|)}{\Delta_{S_{f}}}\right)}$ $\displaystyle\leq\exp{\left(\frac{\varepsilon\sum_{i=1}^{k}|f(y_{i})-f(x_{i})|}{\Delta_{S_{f}}}\right)}$ $\displaystyle=\exp{\left(\frac{\varepsilon\lVert f(y)-f(x)\rVert_{1}}{\Delta_{S_{f}}}\right)}$ (19) We used triangle inequality in the penultimate step. Using the trick in the work of (Rubinstein and Aldà, 2017), we define following events. Let, $B^{\Delta_{S_{f}}}$ denotes the set of pairs neighbouring dataset sampled from $\mathcal{G}$ for which the sensitivity random variable is upper bounded by $\Delta_{S_{f}}$. Let, $C_{\rho}^{\Delta_{S_{f}}}$ denotes the set of sensitivity random variable values for which $F_{n}$ deviates from the unknown cumulative distribution of $S$, $F$, at most by the accuracy value $\rho$. These events are defined in Equation 20. $\displaystyle B^{\Delta_{S_{f}}}$ $\displaystyle\triangleq\\{x,y\sim\mathcal{G}~{}\textrm{such that}~{}\lVert f(y)-f(x)\rVert_{1}\leq\Delta_{S_{f}}\\}$ $\displaystyle C_{\rho}^{\Delta_{S_{f}}}$ $\displaystyle\triangleq\left\\{\sup_{\Delta}|F_{S}^{n}(\Delta)-F_{S}(\Delta)|\leq\rho\right\\}$ (20) $\displaystyle\mathbb{P}(B^{\Delta_{S_{f}}})$ $\displaystyle=\mathbb{P}(B^{\Delta_{S_{f}}}|C_{\rho}^{\Delta_{S_{f}}})\mathbb{P}(C_{\rho}^{\Delta_{S_{f}}})$ (21) $\displaystyle+\mathbb{P}(B^{\Delta_{S_{f}}}|\overline{C\rho^{\Delta_{S_{f}}}})\mathbb{P}(\overline{C_{\rho}^{\Delta_{S_{f}}}})$ $\displaystyle\geq\mathbb{P}(B^{\Delta_{S_{f}}}|C_{\rho}^{\Delta_{S_{f}}})\mathbb{P}(C_{\rho}^{\Delta_{S_{f}}})$ $\displaystyle=F_{n}(\Delta_{S_{f}})\mathbb{P}(C_{\rho}^{\Delta_{S_{f}}})$ $\displaystyle\geq\gamma_{2}\cdot(1-2e^{-2\rho^{2}n})$ (22) In the last step, we use the definition of the sampled sensitivity to get the value of the first term. The last term is obtained using DKW-inequality, as defined in (Massart et al., 1990), where the $n$ denotes the number of samples used to build empirical distribution of the sensitivity, $F_{n}$. From Equation 19, we understand that if $\lVert f(y)-f(x)\rVert_{1}$ is less than or equals to the sampled sensitivity then the Laplace mechanism $\mathcal{L}_{\varepsilon}^{\Delta_{S_{f}}}$ satisfies $\varepsilon$-differential privacy. Equation 22 provides the lower bound on the probability of the event $\lVert f(y)-f(x)\rVert_{1}\leq\Delta_{S_{f}}$. Thus, combining Equation 19 and Equation 22 completes the proof. ## C Proof of Theorem 15 (Section 4.3) Proof of Theorem 15 builds upon the ideas from the proofs for the rest of the two cases. In addition to the events defined in Equation 20, we define an additional event $A_{\varepsilon_{0}}^{\Delta_{S_{f}}}$, defined in Equation 23, as a set of outputs of Laplace mechanism $\mathcal{L}_{\varepsilon_{0}}^{\Delta_{S_{f}}}$ that satisfy the constraint of $\varepsilon$-differential privacy for a specified privacy at risk level $\varepsilon$. $A_{\varepsilon_{0}}^{\Delta_{S_{f}}}\triangleq\left\\{z\sim\mathcal{L}_{\varepsilon_{0}}^{\Delta_{S_{f}}}~{}:~{}\ln{\left|\frac{\mathcal{L}_{\varepsilon_{0}}^{\Delta_{S_{f}}}(x)}{\mathcal{L}_{\varepsilon_{0}}^{\Delta_{S_{f}}}(y)}\right|}\leq\varepsilon,x,y\sim\mathcal{G}\right\\}$ (23) ###### Corollary 26 $\mathbb{P}(A_{\varepsilon_{0}}^{\Delta_{S_{f}}}|B^{\Delta_{S_{f}}})=\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\eta\varepsilon_{0})}$ where $T$ follows the distribution $P_{T}(t;\Delta_{S_{f}}/\varepsilon_{0})$ in Lemma 22 and $\eta=\frac{\Delta_{f}}{\Delta_{S_{f}}}$. Proof We provide the sketch of the proof. Proof follows from the proof of Lemma 24. For a Laplace mechanism calibrated with the sampled sensitivity $\Delta_{S_{f}}$ and privacy level $\varepsilon_{0}$, Equation 17 translates to, $\left(\frac{\varepsilon_{0}}{\Delta_{S_{f}}}\sum_{i=1}^{k}\left|\left(|f(y_{i})-z|-|f(x_{i})-z|\right)\right|\right)\sim\\\ P_{T^{\prime}}(t;k,\Delta_{S_{f}}/\varepsilon_{0},\Delta_{S_{f}}).$ since, $\sum_{i=1}^{k}\left|\left(|f(y_{i})-z|-|f(x_{i})-z|\right)\right|\leq\Delta_{f}$. Using Lemma 23 and Equation 18, $\mathbb{P}(A_{\varepsilon_{0}}^{\Delta_{S_{f}}})=\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\eta\varepsilon_{0})}$ where $T$ follows the distribution $P_{T}(t;\Delta_{S_{f}}/\varepsilon_{0})$ and $\eta=\frac{\Delta_{f}}{\Delta_{S_{f}}}$. For this case, we do not assume the knowledge of the sensitivity of the query. Using the empirical estimation presented in Section 4.2, if we choose the sampled sensitivity for privacy at risk $\gamma_{2}=1$, we obtain an approximation for $\eta$. ###### Lemma 27 For a given value of accuracy parameter $\rho$, $\frac{\Delta_{f}}{\Delta_{S_{f}}^{*}}=1+\mathcal{O}\left(\frac{\rho}{\Delta_{S_{f}}^{*}}\right)$ where $\Delta_{S_{f}}^{*}=F_{n}^{-1}(1)$. $\mathcal{O}\left(\frac{\rho}{\Delta_{S_{f}}^{*}}\right)$ denotes order of $\frac{\rho}{\Delta_{S_{f}}^{*}}$, i.e., $\mathcal{O}\left(\frac{\rho}{\Delta_{S_{f}}^{*}}\right)=k\frac{\rho}{\Delta_{S_{f}}^{*}}$ for some $k\geq 1$. Proof For a given value of accuracy parameter $\rho$ and any $\Delta>0$, $F_{n}(\Delta)-F(\Delta)\leq\rho$ Since above inequality is true for any value of $\Delta$, let $\Delta=F^{-1}(1)$. Therefore, $\displaystyle F_{n}(F^{-1}(1))-F(F^{-1}(1))$ $\displaystyle\leq\rho$ $\displaystyle F_{n}(F^{-1}(1))$ $\displaystyle\leq 1+\rho$ (24) Since a cumulative distribution function is $1$-Lipschitz [(Papoulis and Pillai, 2002)], $\displaystyle|F_{n}(F_{n}^{-1}(1))-F_{n}(F^{-1}(1))|$ $\displaystyle\leq|F_{n}^{-1}(1)-F^{-1}(1)|$ $\displaystyle|F_{n}(F_{n}^{-1}(1))-F_{n}(F^{-1}(1))|$ $\displaystyle\leq|\Delta_{S_{f}}^{*}-\Delta_{f}|$ $\displaystyle\rho$ $\displaystyle\leq\Delta_{f}-\Delta_{S_{f}}^{*}$ $\displaystyle 1+\frac{\rho}{\Delta_{S_{f}}^{*}}$ $\displaystyle\leq\frac{\Delta_{f}}{\Delta_{S_{f}}^{*}}$ where we used result from Equation 24 in step 3. Introducing $\mathcal{O}\left(\frac{\rho}{\Delta_{S_{f}}^{*}}\right)$ completes the proof. ###### Lemma 28 For Laplace Mechanism $\mathcal{L}_{\varepsilon_{0}}^{\Delta_{S_{f}}}$ with sampled sensitivity $\Delta_{S_{f}}$ of a query $f:\mathcal{D}\rightarrow\mathbb{R}^{k}$ and for any $Z\subseteq Range(\mathcal{L}_{\varepsilon}^{\Delta_{S_{f}}})$, $\mathbb{P}\left[\ln{\left|\frac{\mathbb{P}(\mathcal{L}_{\varepsilon_{0}}(x)\in Z)}{\mathbb{P}(\mathcal{L}_{\varepsilon_{0}}(y)\in Z)}\right|}\leq\varepsilon\right]\geq\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\eta\varepsilon_{0})}\gamma_{2}(1-2e^{-2\rho^{2}n})$ where $n$ is the number of samples used to find sampled sensitivity, $\rho\in[0,1]$ is a accuracy parameter and $\eta=\frac{\Delta_{f}}{\Delta_{S_{f}}}$. The outer probability is calculated with respect to support of the data-generation distribution $\mathcal{G}$. Proof The proof follows from the proof of Lemma 24 and Lemma 28. Consider, $\displaystyle\mathbb{P}(A_{\varepsilon_{0}}^{\Delta_{S_{f}}})$ $\displaystyle\geq\mathbb{P}(A_{\varepsilon_{0}}^{\Delta_{S_{f}}}|B^{\Delta_{S_{f}}})\mathbb{P}(B^{\Delta_{S_{f}}}|C_{\rho}^{\Delta_{S_{f}}})\mathbb{P}(C_{\rho}^{\Delta_{S_{f}}})$ $\displaystyle\geq\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\eta\varepsilon_{0})}\cdot\gamma_{2}\cdot(1-2e^{-2\rho^{2}n})$ (25) The first term in the final step of Equation 25 follows from the result in Corollary 26 where $T$ follows $\mathrm{BesselK}(k,\frac{\Delta_{S_{f}}}{\varepsilon_{0}})$. It is the probability with which the Laplace mechanism $\mathcal{L}_{\varepsilon_{0}}^{\Delta_{S_{f}}}$ satisfies $\varepsilon$-differential privacy for a given value of sampled sensitivity. Probability of occurrence of event $A_{\varepsilon_{0}}^{\Delta_{S_{f}}}$ calculated by accounting for both explicit and implicit sources of randomness gives the risk for privacy level $\varepsilon$. Thus, the proof of Lemma 28 completes the proof for Theorem 15. Comparing the equations in Theorem 15 and Lemma 28, we observe that $\gamma_{3}\triangleq\frac{\mathbb{P}(T\leq\varepsilon)}{\mathbb{P}(T\leq\eta\varepsilon_{0})}\cdot\gamma_{2}$ (26) The privacy at risk, as defined in Equation 26, is free from the term that accounts for the accuracy of sampled estimate. If we know cumulative distribution of the sensitivity, we do not suffer from the uncertainty of introduced by sampling from the empirical distribution.

2024-09-04T02:54:56.470816

2003.01025

arxiv-papers

# Dynamic Queue-Jump Lane for Emergency Vehicles under Partially Connected Settings: A Multi-Agent Deep Reinforcement Learning Approach Haoran Su Kejian Shi Joseph. Y.J. Chow Li Jin<EMAIL_ADDRESS>Tandon School of Engineering, New York University, Brooklyn, NY, 11201 ###### Abstract Emergency vehicle (EMV) service is a key function of cities and is exceedingly challenging due to urban traffic congestion. A main reason behind EMV service delay is the lack of communication and cooperation between vehicles blocking EMVs. In this paper, we study the improvement of EMV service under V2X connectivity. We consider the establishment of dynamic queue jump lanes (DQJLs) based on real-time coordination of connected vehicles in the presence of non-connected human driven vehicles. We develop a novel Markov decision process formulation for the DQJL coordination strategies, which explicitly accounts for the uncertainty of drivers’ yielding pattern to approaching EMVs. Based on pairs of neural networks representing actors and critics for agent vehicles, we develop a multi-agent actor-critic deep reinforcement learning algorithm which handles varying number of vehicles and random proportion of connected vehicles in the traffic. Approaching the optimal coordination strategies via indirect and direct reinforcement learning, we present two schemata to address multi-agent reinforcement learning on this connected vehicle application. Both approaches are validated, on a micro-simulation testbed SUMO, to establish a DQJL fast and safely. Validation results reveal that, with DQJL coordination strategies, it saves up to 30% time for EMVs to pass a link-level intelligent urban roadway than the baseline scenario. ###### keywords: Connected Vehicles, Multi-agent System , Reinforcement Learning, Emergency Vehicles, V2X ††journal: Transportation Research Part C ## 1 Introduction Increasing population and urbanization have made it exceedingly challenging to operate urban emergency services efficiently. For example, historical data from New York City, USA [1] shows that the number of annual emergency vehicle (EMV) incidents has grown from 1,114,693 in 2004 to 1,352,766 in 2014, with corresponding average response time of 7:53 min and 9:23 min, respectively [2]. This means an approximately 20% increase in response time in ten years. In the case of cardiac arrest, every minute until defibrillation reduces survival chances by 7% to 10%, and after 8 minutes there is little chance of survival [3]. Cities are less resilient with worsening response time from EMVs (ambulances, fire trucks, police cars), mainly due to traffic congestion. The performance of these EMV service systems in congested traffic can be improved with technology. As a core of modern ITSs, wireless vehicle-to- everything (V2X) connectivity, such as 5G-cellular network recently, provide significant opportunities for improving urban emergency response. On the one hand, wireless connectivity provides EMVs the traffic conditions on possible routes between the station (hospital, fire station, police station, etc.) and the call, which enables more efficient dispatch and routing. Through V2I communication with the EMVs on duty, traffic managers can broadcast the planned route of EMVs to non-EMVs that may be affected, and non-EMVs can cooperate to form alternative queue-jump lanes for approaching EMVs. In this study, we introduce a novel concept of dynamic queue-jump lanes (DQJL). A queue jump lane (QJL) [4, 5] is a lane type used for bus operation so that buses can bypass long queues prior to intersections. In the context of EMVs, we define a DQJL to be a dynamic lane that is formed while an EMV is en- route to clear the downstream space. This is achieved by using V2X technology to inform downstream vehicles of the temporary change in the lane type so that vehicles would shift out of the lane. Doing so should significantly reduce the response time for EMVs on the urban roadway. By monitoring and managing the connected components of the roadway, we are able to produce real-time coordination strategies to clear a path for the approaching EMVs. DQJL involves several uncertainties that impact the its implementation. One is how far downstream should the V2X set the lane type? Setting the DQJL too far downstream would negatively impact the background traffic. Setting it too short might not give vehicles enough time to comply. Secondly, a DQJL strategy can be implemented with an allocation recommendation to inform downstream vehicles of which open spaces to pull over to. Open spaces are finite and limited, and their usage depends on drivers’ compliance. Poorly coordinated allocations can lead to low compliance due to difficult maneuvering requirements. The latter decision is called ”DQJL with coordination” while DQJL without coordination indicates no allocation recommendation given. Consider the stochastic driving behaviors and other uncertainties in the road environment, we formulate the problem into a partially observable Markov decision process (MDP). Aiming for real-time decision making when coordinating, we adopt the multi-agent actor-critic deep reinforcement learning to approach optimal coordination strategies. The methodology considers partial connected settings, indicating our frameworks are flexible with varying numbers of vehicles and connected vehicles’ penetration rates. The main contributions of the presented study are as follows: we introduce the concept of DQJL and solve the objectives of DQJL applications under stochastic road environment; we develop a multi-agent actor-critic deep reinforcement learning algorithm which copes with varying numbers of agent vehicles and penetration rates; addressing the real-time optimal coordination strategies via indirect and direct approach, we validate the DQJL coordinated system performs significantly better than the baseline scenario without any coordination. The rest of the paper is organized in the following sections. In Sec. 2, we overview related studies on queue-jump lanes for EMVs and actor-critic reinforcement learning methods in similar ITS applications. In Sec. 3, we elaborate the DQJL problem formulation as a Markov decision process. In Sec. 4, we demonstrate our methodology, including the indirect and direct multi- agent reinforcement learning approaches. We elaborate our experimental design and synthesize a simulation dataset for the training in Sec. 5. We validate the proposed methodology and provide insights on the results in Sec. 6. At last, conclusions are made in Sec. 7. Figure 1: DQJL Coordination for an emergency vehicle. ## 2 Literature Review In this section, we discuss literature available on Queue jump lane in Subsec. 2.1, and existing multi-agent deep reinforcement learning techniques on similar connected vehicle applications in Subsec. 2.2. ### 2.1 Queue jump lane for emergency vehicles Queue jump lane (DQJ) has never been used for EMV deployment and is considered a novel operation strategy to apply this technology for EMV deployment with the aid of connected vehicle technologies. Although QJL is a relatively new technology, literature has already demonstrates the positive effects they have in reducing travel time variability, especially when used in conjunction with transit signal priority (TSP). However, they are all based on moving- bottleneck models for buses [4, 5]; we are borrowing this bus operation strategy for EMV deployment in our setting, since EMVs typically move faster than non-EMVs and since EMVs can “preempt” non-EMV traffic because of their priority. At the same time, with only siren technologies, other vehicles often do not get enough warning time from the EMVs. Even then, there is a lack of clarity in the direction of the route to avoid. The confusion, particularly under highly congested scenarios, leads to increased delays as mentioned above, and also to 4 to 17 times higher accident rates [6] and increased severity of collisions [7], which can further lead to increased response time. A study by Savolainen et al. [8] using data from Detroit, Michigan, verified this sensitivity of driver behaviors to different ITS communication methods for EMV route information. Hannoun et al. [9] use mixed integer linear programming to develop a static path clearance control strategy for approaching EMVs while QJLs have not been studied as a dynamic control strategy and there’s an urgent need for capturing uncertainties in realistic traffic conditions, especially in events under non- deterministic setting such as yielding to an approaching EMV. To apply dynamic control strategy for vehicle’s motion planning for QJL establishment, we introduce the concept of dynamic queue jump lane (DQJL), see Fig. 1. During the process of clearing a QJL for the passing EMV, non-EMVs are constantly monitored and instructed with actions so that DQJLs are established quickly and safely. In the presented work, we also investigate the DQJL applications under a partially connected scenario, under which some vehicles are equipped with communication devices to establish complete communication channels between themselves and the infrastructure but others do not. ### 2.2 Actor-critic method in ITS applications Many state-of-the-art ITS applications are adopting deep reinforcement learning techniques to realize connected vehicle applications. Among reinforcement learning schemes, the actor critic method, which combines the Q-learning and policy gradient, has been favored by extensive connected vehicle applications on cooperative traffic signal controls [10, 11, 12], connected vehicle platooning [13], or trajectory planning [14, 15]. Extending the actor critic algorithm from a single ”super-agent” perspective to the multi-agent setting allows us to concentrate more on microscopic traffic management problems under connected settings. Under the multi-agent setting, it is assumed, by the essential definition of multi-agent systems that each vehicle doesn’t have the full knowledge of the road environment, but can partially observe its surroundings. The local observation contains information obtained through physical surroundings, such as neighboring vehicles’ positions, as well as messages through communication channels among connected vehicles and the infrastructure. Different actor critic architectures are used by these connected vehicle applications for various application purposes. The fully centralized setting describes that all information is processed in a centralized controlled system. Namely, the policy and value networks for all vehicles, connected or autonomous, are stored in the centralized controller. All vehicles report their own observation spaces to the controller and, as a result, the controller has the full knowledge of the road environment. Decisions for each connected vehicles are then made by the centralized controller. The fully centralized setting based on actor critic methods appear in vehicle communication establishment [16], coordination through intersections [17], and a previous study on DQJL coordination under fully connected scenario [18]. However, the fully centralized setting does not accommodate this application due to its limited usage under partially connected road environment, restricting the application’s practicality within the short to medium future. The centralized setting also raises concerns on the high latency and synchronization cost, making real-time decision extremely difficult among vehicles. At the same time, the fully decentralized settings are broadly approved in ITS applications. Each vehicle has its own policy network and actor network deployed on board. Each vehicle interacts with the environment independently. Methodologies addressing the multi-agent vehicle applications under this setting originate from the independent actor-critic (IAC) [19]. Plenty of robotic motion planning vehicle applications [20, 21] are based on the fully decentralized setting. The centralized controller, under this scheme, is no longer needed. Communications among vehicles are significantly reduced. While treating other vehicles as parts of the road environment faces huge risk of not converging as the environment becomes non-stationary. Vehicles also have difficulties in identifying their roles in cooperative tasks. Each individual vehicle is required to equipped advanced sensors, such as Lidar and cameras, to fully observe the road environment, significantly increasing the establishment cost of the system. Proposed by [22, 23], the centralized training with decentralized execution framework overcomes challenges in previous settings. Under this framework, each vehicle has its own policy network deployed on board, while the corresponding value network is stored in the centralized controller. During the training stage, the centralized controller updates critic through TD learning with all agents’ observation and their actions. The policy network then outputs the optimal action based on the vehicle’s local observation. After the training stage, the value networks are completely abandoned and vehicles are able to operate based on their own observation spaces of the road environment and their policy networks, eliminating the concerns on synchronization. Since the centralized controller train value networks together, the road environment becomes stationary because the joint action is known, even though policy networks may change during the training stage. The comparisons between these frameworks are summarized in Table. 1. Denoting the local observation and action for a vehicle as $o^{i}_{t}$ and $a^{i}_{t}$ at step $t$, the joint observation is then denoted as $\mathbf{o}_{t}$ and joint action as $\mathbf{a}_{t}$. The actors and critics indicate the input information are either local or global. The latency column indicates whether the low latency and synchronization cost are highly required during execution stage or not, while the sensor column indicates if the vehicles need to equip the Lidar/cameras under the corresponding framework. Settings | Actor | Critic | Latency | Sensors ---|---|---|---|--- Centralized | $\pi(a^{i}_{t}|\mathbf{o}_{t};\theta_{i})$ | $Q(\mathbf{o}_{t},\mathbf{a}_{t};w_{i})$ | + | - Decentralized | $\pi(a^{i}_{t}|o^{i}_{t};\theta_{i})$ | $Q(o^{i}_{t},a^{i}_{t};w_{i})$ | - | + CTDE | $\pi(a^{i}_{t}|o^{i}_{t};\theta_{i})$ | $Q(\mathbf{o}_{t},\mathbf{a}_{t};w_{i})$ | - | - Table 1: Summary of actor-critic based frameworks for connected vehicles. Multi-agent actor critic algorithms based on CTDE architecture, such as MADDPG, are applied in vehicle motion planning through signalized intersection application [24], traffic flow optimization [25], and vehicle network resources allocation [26]. The existing literature view vehicles as the agents for cooperative or competitive traffic tasks. While these existing literature neither handle the heterogeneity of vehicle groups under partially connected setting, nor do they consider for varying size of the agents, i.e. vehicles. They also do not explicitly elaborate how to cope with discrete action space using multi-agent actor critic algorithm. ## 3 Model Formulation In this section, we elaborate how we model the DQJL application as a partially-observed Markov decision process. In Subsec. 3.1, we reveal the structure of the DQJL coordination framework. The partially connected setting is explained in Subsec. 3.2, and components of the proposed MDP are illustrated in 3.3, 3.4, 3.5, 3.6 respectively. The objective function for the proposed MDP is presented in Subsec. 3.7. ### 3.1 DQJL coordination framework In order to model the establishment of DQJL for an emergency vehicle (EMV), let us take a look at a typical urban road segment. An urban road segment consists of two lanes facing the same direction. When an EMV on duty approaches this road segment, the centralized control system based on cellular network will send out real time yielding coordination instructions to connected vehicles. With the cellular-assisted network prevailing in V2X communication [27, 28, 29] nowadays, connected vehicles’ information including positions, velocity, acceleration/deceleration and vehicle unique attributes such as vehicle length and most comfortable deceleration are communicated directly among connected vehicles and the infrastructure. Figure 2: DQJL coordination framework under cellular-assisted vehicle networks. Approaching EMVs remain equipped with siren technologies to provide acoustic and visual warnings to vehicles, bikers and pedestrians. At the same time, EMVs’ VANET-based [7] onboard warning systems are able to broadcast emergency vehicles’ kinematic information to connected vehicles on the roadway. Sensors and detectors are deployed along the roadway to capture necessary features of non-connected vehicles. Vehicles’ kinematic information including real time velocities and positions on the roadway are collected via technologies such as inductive and magnetic loop sensors [30, 31]. Along the roadway, one or more roadside units (RSU) monitor and identify whether the approaching vehicles are connected or not. They also serve to synchronize data and coordinate when dealing with emergency cases. Information from the connected vehicles via the cellular-assisted vehicle network and data collected from sensors are processed together in RSUs. The RSUs then send the processed data to the centralized controller and wait for real-time coordination strategies as feedback. The RSUs then broadcast coordination instructions to the corresponding vehicles. The overall coordination framework is summarized in Fig. 2. At the same time, the communication framework has a strict demand for low latency and synchronization cost in communication to ensure the compatibility of real-time coordination. To ensure the approaching EMV can pass the road segment as fast as possible, it is assumed the EMV does not perform lane change during passing so its average velocity can be maximized. The lane on which EMV is operating is named as the passing lane for referring convenience. All vehicles originally on the passing lane will pull over onto the other lane, which is referred as the neighboring lane. Stochastic road dynamics originate from stochastic driving behaviors. For example, the uncertainties in drivers’ perception-reaction time result in different reaction distance as shown in Fig. 3. Distinct braking abilities of different vehicles bring significantly different braking distances even when they start to decelerate at the same velocity. Stochastic driving behaviors also includes the randomness in lane-changing pattern when performing pull- over and $etc.$ Figure 3: Braking process of a non-EMV after receiving a yielding instruction. Due to the uncertainties mentioned above, it is beneficial to picture the controlled system in a Markov decision process (MDP) framework so that the randomness can be addressed properly. Table. 2 provides notations for variables used for describing the controlled system. Table 2: Roadway Environment Notations Notation | Meaning ---|--- $x$ | front bump position of the vehicle $v$ | velocity of the vehicle $l$ | length of the vehicle $b^{*}$ | baseline acceleration/deceleration of the vehicle $y$ | lane position of the vehicle $\xi$ | the yielding status of the vehicle $\phi$ | vehicle category: {CV, HV or trivial vehicle} $L$ | length of the road segment $d$ | minimum safety gap between two vehicle ### 3.2 The partially connected mixed traffic According to the description of the controlled system above, the stream of vehicles has two categories of vehicles: the first one is the traditional human-driven vehicles (HV) with no advanced communication equipment and automation technologies; The second type is the connected vehicles (CV), which are highly connected within the framework and are able to share real-time information to the infrastructure or other CVs. For this particular application, vehicular automated technologies that operate the wheeling or acceleration of the vehicle are not considered, which means all vehicles are essentially operated by human drivers. Inspired by control strategies for partially automated traffic [32, 33], the study develops a set of control strategies for the partially connected traffic. On the one hand, HVs and CVs both contribute to the stochastic road dynamics due to the fact that they are operated by human drivers. All vehicles are unique with respect to the vehicles length and their deceleration rate as a result of their distinct braking abilities. All drivers are unique with respect to their perception reaction time, lane-changing behavior and other driving styles. These uncertainties constitute the stochastic traffic dynamics in this link-level segment. Moreover, CVs with explicit communication channels can be monitored and ”managed” collectively to achieve cooperative tasks, even in the presence of HVs. The cooperation scheme is supposed to handle with the stochastic road dynamics mentioned above. At the same time, although HVs are not spontaneously sharing information with CVs or the intelligent infrastructure, their state information is constantly monitored by the infrastructure and observed by their surrounding CVs. ### 3.3 Agents and state In this MDP, the agents have two categories: HVs and CVs. The CVs are referred as active agents since their behaviors are controlled by the learning process, and HVs as passive agents since their behaviors are not controlled by the learning process. At step $t$, the state representation of the an non-EMV, CV or HV, is expressed by the following features: $s^{i}_{t}=[x^{i}_{t},y^{i}_{t},v^{i}_{t},\xi^{i}_{t},l^{i},b^{*}_{i},\phi^{i}],$ where $\xi^{i}_{t}$ indicates whether or not a non-EMV is in the yielding process for the approaching EMV, and $\phi^{i}$ helps identify whether or not the $i$th non-EMV is a CV, HV or trivial vehicle. Therefore, the ground truth state of the whole controlled system at step $t$ is characterized as $\mathbf{s}_{t}=[s^{0}_{t},s^{1}_{t},s^{2}_{t},\dots,s^{n}_{t}],$ where $s^{0}_{t}$ is the state representation of the EMV and there are $n$ non-EMVs on the roadway when coordination process begins. Notice that all non-EMVs and their drivers are unique and non-exchangeable. Non-EMVs are differentiated by their vehicle length $l$ and their baseline acceleration/deceleration $b^{*}$ due to their unique braking ability. The drivers are also differentiable due to nature of everyone’s unique driving style and behavior. Thus, the agents are heterogeneous in this MDP framework. ### 3.4 Observation space Each non-EMV only observes partial road segment. To define local observation of anon-EMV, the four closest adjacent non-EMVs are selected, which are the ego vehicle’s leading vehicle, its following vehicle and the leading vehicle and the following vehicles on the neighboring lane. At the same time, the approaching EMV can broadcast its real time kinematic information to all CVs on this road segment, and any non-EMV would include the EMV’s information in its own local observation. The adjacent vehicles and the approaching EMV imposes significant impacts on the ego vehicle’s motion planning decisions. Fig. 4 demonstrates the local observation captured by any non-EMV on this road segment. Figure 4: Example of the local observation captured by any non-EMV on the road segment. For the $i$th non-EMV on this road segment, its local observation is described as $o^{i}_{t}=[s^{0}_{t},s^{i}_{t},s^{ij}_{t},s^{ik}_{t},s^{il}_{t},s^{im}_{t}],$ where $ij,ik,il,im$ represents $i$th vehicle’s adjacent vehicles at step $t$ respectively, and $s^{0}_{t}$ stands for the state representation of the EMV at step $t$. The dimension of any non-EMV’s observation space should be fixed. If a non-EMV does not have a complete observation space, such as the leading vehicle of a lane, trivial vehicles are introduced to fill up the observation space. A trivial vehicle contains insignificant vehicle features and it is labeled during the learning process, which helps the controller to ignore its interaction with other vehicles. ### 3.5 Action For HVs, the actions are primarily determined by the distance between the EMV and itself when drivers hear the sirens or have visual confirmation of the vehicle. This distance-based action function with w.r.t to step is described as $a^{i}_{t}=\begin{cases}0,&\text{if }x^{i}_{t}-x^{i}_{t}\geq L_{HV}.\\\ 1,&\text{otherwise}.\\\ \end{cases}$ (1) For CVs, the agents are human drivers operating the connected vehicles, so the action instructed to each human driver is to yield or not. Thus, the discrete action space for agent $i$ at step $t$ is represented as $a^{i}_{t}\in\\{0,1\\}$. Notice that both the yielding and the moving forward instructions are high- level motion planning representations, and they don’t determine the vehicles’ absolute velocities as well as accelerations. With yielding or moving forward instructions, human drivers drive according to the road environment dynamics in addition to their unique and stochastic driving styles. Defining the action in such way circumvents the difficulty of explicitly indicating the specific kinematics attributes as well as the stochastic driving behaviors associated. The joint action of $n$ non-EMVs at step $t$ is then represented as $\mathbf{a}_{t}=[a^{1}_{t},a^{2}_{t},\dots,a^{n}_{t}].$ ### 3.6 Reward For the DQJL application, three reward functions are named to approach the purpose of establishing a DQJL safely as fast as possible. First of all, to avoid vehicle collision, a significant negative reward $P_{collision}$ for overlapping of any two neighboring vehicles’ positions is imposed: $r^{collision}_{t}=\begin{cases}P_{collision},&\text{ if }x^{i}_{t}+d<x^{j}_{t}-l^{j},\forall i,j\text{ when }y^{i}_{t}=y^{j}_{t}.\\\ 0,&\text{otherwise.}\\\ \end{cases}$ To motivate non-EMVs to clear the path, the second reward function penalizes every step elapsed, $r^{elapsed}_{t}=\begin{cases}-1,&\text{ if }x^{0}_{t}-l^{0}<L.\\\ 0,&\text{otherwise,}\\\ \end{cases}$ where $l^{0}$ represents the vehicle length of the approach EMV. The third reward function emphasizes the urgency of yielding for each non-EMV in the way of the EMV. Namely, $r^{i}_{t}=\begin{cases}-\frac{P_{priority}}{x^{i}_{t}-x^{0}_{t}},&\text{if }x^{i}_{t}\leq L\text{ and }y^{i}_{t}=0,\\\ 0,&\text{otherwise,}\\\ \end{cases}$ where $P$ is a penalty constant. This definition prioritizes the yielding of non-EMVs which are closer to the EMV that are downstream from it. To summarize, the immediate step reward at step $t$ is $R(t)=r^{collision}_{t}+r^{elapsed}_{t}+\sum_{i=1}^{n}r^{i}_{t}(1-y_{i}^{t}).$ Since the application is cooperative, the reward received by each agent is identical to the team reward, which is $R^{1}_{t}=R^{2}_{t}=\dots=R^{n}_{t}=R_{t}.$ ### 3.7 MDP objective function Defining the long-term return for the team as $U_{t}$ at step $t$, the objective function for the DQJL coordination can be represented as $\max U_{0}=\sum^{\infty}_{k=0}\gamma^{k}R_{k}.$ (2) Solving (2) not only returns the optimal coordination strategy but also reveals the minimum amount of time for DQJL establishment. Notice that since the immediate step rewards are affected by the stochastic road environment, the expected return reflects the uncertainties during the DQJL coordination process. ## 4 Multi-Agent Actor-critic Method for Coordination Strategies In this section, we introduce the our approach for DQJL coordination strategies. Starting with Subsec. 4.1 and 4.2, we revisit the preliminaries of RL and the actor critic method. In Subsec. 4.3, we show the step-by-step derivation of the multi-agent actor-critic method for DQJL coordination. In Subsec. 4.5, we elaborate how to construct a road dynamic model reflecting the realistic traffic condition, and then employ the multi-agent actor critic algorithms under centralized training with decentralized execution framework as the indirect approach. In Subsec. 4.6, we demonstrate the pipeline for the embedding of the MARL algorithm on Simulation of Urban Mobility (SUMO) as the direct approach. The deep neural network structures supporting the presented algorithm as function approximators are presented in Subsec. 4.4. ### 4.1 Bellman optimality In the fully observable MDP, an agent has the full knowledge of the environment $s\in S$ at step $t$, and select an action $a$ according to its policy $\pi(a|s)$. The state transition occurs as $s_{t}\sim s_{t+1}$ and an immediate step reward is received in the form of $R_{t}=R(s_{t},a_{t},s_{t+1})$. If the MDP has infinite-horizon, the sampled expected return under some policy is calculated as $R^{\pi}_{t}=\sum_{\tau=t}^{\infty}\gamma^{\tau-t}r_{t}$, where $\tau\in[0,1)$ is the discount factor. From there, Q-function under some policy is defined as $Q^{\pi}(s,a)=\operatorname{\mathbb{E}}{}[R^{\pi}_{t}|s_{t}=s,a_{t}=a].$ (3) Hence, the optimal Q-function is obtained from $Q^{*}=\max_{\pi}Q^{\pi}$, yielding the optimal policy $\pi^{*}$. Under $\pi^{*}$, $a\in\arg\max_{a_{t+1}}Q^{*}(s,a_{t+1})$. According to the Bellman’s Equation [34], the optimal Q-function is solved by $Q^{*}(s,a)=R_{t}+\gamma\sum_{s_{t+1}\in S}p(s_{t+1}|s_{t},a_{t})\max_{a_{t+1}\in A}Q(s_{t},a_{t+1}).$ (4) Since $R_{t}$ and $p(s_{t+1}|s_{t},a_{t})$ are unknown to the agent, the agent has to approach the optimality by purely data-driven dynamic programming based on the collected experience $(s_{t},a_{t},R_{t},s_{t+1})$. ### 4.2 Single-agent actor-critic method Actor-critic method, introduced by [35], combines the policy gradient algorithm with value-based learning method. State-of-the-art actor-critic based methods, such as DDPG [36] and soft actor-critic method [37], exhibit outstanding performance in a variety of control problems in stochastic processes, especially in the motion planning control of connected and autonomous vehicles. Consider an intelligent vehicle as the agent with a discrete action space, e.g. driving forward and backing up. The vehicle’s state-value function under some policy $\pi$ is calculated as $V_{\pi}(s)=\sum_{a\in A}\pi(a|s)Q_{\pi}(a,s),$ where $\pi(a|s)$ represents the policy function and $Q_{\pi}(a,s)$ refers to the state-action function. $V_{\pi}(s)$ estimates the vehicle’s standing in the application. With the approximation power of neural networks, the policy function is parameterized with a set of trainable parameters as $\pi(a|s;\bm{\theta})$, and the state-action function is parameterized as $Q_{\pi}(a,s;\mathbf{w})$. The state-value function is then parameterized as $V_{\pi}(s;\bm{\theta},\mathbf{w})$. By sampling an action through the vehicle’s current policy $a\sim\pi(\cdot|s;\bm{\theta})$, the corresponding next state $s_{t+1}$ and immediate step reward $R_{t}$ are observed. The value network, i.e the critic, is trained via the temporal difference (TD) learning. First, $Q(s,a;\mathbf{w})$ and $Q(s_{t+1},a_{t+1};\mathbf{w})$ are computed and a TD target is set as $y_{t}=R_{t}+\gamma Q(s_{t+1},a_{t+1};\mathbf{w}).$ (5) The purpose of setting up (5) is to minimize the difference between the $y_{t}$ and $Q(s_{t},a_{t};\mathbf{w})$. Mathematically, $\min L(\mathbf{w})=\frac{1}{2}[y_{t}-Q(s_{t},a_{t};\mathbf{w})]^{2},$ and $\mathbf{w}$ is updated by the gradient ascent method as $\mathbf{w}_{t+1}=\mathbf{w}_{t}+\alpha\frac{\partial L(\mathbf{w})}{\partial\mathbf{w}}|_{\mathbf{w}=\mathbf{w_{t}}},$ (6) in which $\alpha$ is the learning rate for the value network. The policy network, i.e. the actor, is updated through the traditional policy gradient algorithm. Denoting the policy gradient of an action $a_{t}$ as $\mathbf{g}(a_{t},\bm{\theta})=\frac{\partial\log\pi(a|s;\bm{\theta}))}{\partial\bm{\theta}}Q(s_{t},a;\mathbf{w})$, it is easy to derive the policy gradient with respect to the state-value function as $\frac{\partial V(s;\bm{\theta},\mathbf{w}_{t})}{\partial\bm{\theta}}=\operatorname{\mathbb{E}}_{A}[\mathbf{g}(A,\bm{\theta})].$ (7) By randomly sampling an action through Monte Carlo simulation $a_{t}\sim\pi(\cdot|s_{t};\bm{\theta}_{t})$, $\mathbf{g}(a_{t})$ is guaranteed unbiased. $\bm{\theta}$ is then updated through the stochastic gradient descent as $\bm{\theta}_{t+1}=\bm{\theta}_{t}+\beta\mathbf{g}(a,\bm{\theta}_{t}),$ (8) where $\beta$ is the learning rate of the policy network. By setting up the applicable reward functions, a vehicle’s critic will improve its rating ability with regard to the ground truth estimation. The actor, on the other hand, will make the optimal decision by the critic’s standard, guiding the vehicle achieve the application purpose. ### 4.3 Multi-agent actor-critic method in discrete action space Under the CTDE framework, each vehicle in the system, HV or CV, has a local policy network, and the corresponding is stored in the centralized controller. The focus of the learning process is on the policy networks of the CVs since HVs are not controlled by the learning process and their actors are fixed. To maintain the training consistency, HVs are viewed as agents. The main idea behind CTDE is that the road environment is considered stationary after knowing the joint actions of all non-EMVs even when some actors are updated in the training. Assuming there are $J$ CVs and total $N$ non-EMVs in the controlled system, the CTDE architecture is summarized in Fig. 5. Figure 5: The CTDE architecture on the DQJL application. The controlled system is described as a fixed length $L$ road environment with $n$ non-EMVs. These non-EMVs have parameterized policies with $\bm{\theta}=\\{\theta_{0},\theta_{1},\dots,\theta_{n}\\}$ and their policies are summarized as $\bm{\pi}=\\{\pi_{1},\pi_{1},\dots,\pi_{n}\\}$. $\theta$ for HVs are constant and they are not updated through training. Another set of parameters $\mathbf{w}=\\{w_{1},w_{2},\dots,w_{n}\\}$ is selected to denote the value networks stored in the centralized controller. #### 4.3.1 Experience replay initialization Suppose at some step $t$, the joint observation of all vehicles is $\mathbf{s}_{t}$. A joint action $a_{t}$ is taken and the joint next state $\mathbf{s}_{t+1}$ is observed. An experience replay bank $D=[\mathbf{s}_{t},\mathbf{s}_{t+1},\mathbf{a}_{t},R^{1}_{t},R^{2}_{t},\dots,R^{n}_{t}]$ to store the joint state, joint next state, joint action and joint reward so that centralized the training stage can have a more efficient use of the previous experience. #### 4.3.2 Centralized critics update Similar to the employment of the TD learning shown in (5) for single-agent RL, a TD target for the multi-agent scenario is named as $y_{t}=R_{t}+\gamma Q^{\bm{\theta}^{\prime}}_{i}(\mathbf{s}_{t+1},\mathbf{a}_{t+1};\mathbf{w}_{i})|_{a^{j}_{t+1}=\theta_{j}^{\prime}(o^{j}_{t+1})},$ (9) and the target is to minimize the error between $y_{t}$ and $Q^{\bm{\theta}}_{i}(\mathbf{s}_{t},\mathbf{a}_{t})$ as $\min L(\mathbf{w}_{i})=\frac{1}{2}\operatorname{\mathbb{E}}_{(\mathbf{s}_{t},a_{t},R_{t},\mathbf{s}_{t+1})\in D}[(Q^{\bm{\theta}}_{i}(\mathbf{s}_{t},\mathbf{a}_{t})-y_{t})^{2}].$ (10) In (9), $\bm{\theta}^{\prime}=\\{\theta_{1}^{\prime},\theta_{2}^{\prime},\dots,\theta_{n}^{\prime}\\}$ represents the target policy set with delayed parameters. Through the gradient ascent algorithm as in (6), the value network considering for all agents’ states and actions are updated. #### 4.3.3 Decentralized actors update Actor deployed on each agent vehicle is updated through policy gradient. The gradient of expected return for $i$th non-EMV in the discrete action space is described as $\nabla_{\theta_{i}}J(\theta_{i})=\operatorname{\mathbb{E}}_{\mathbf{s},a_{i}\sim\pi_{i}}[\nabla_{\theta_{i}}\pi_{i}(a^{i}_{t}|o^{i}_{t})\nabla_{a^{i}_{t}}Q^{\bm{\pi}}_{i}(\mathbf{s}_{t},\mathbf{a}_{t})],$ (11) where $Q^{\bm{\pi}}_{i}(\mathbf{s}_{t},\mathbf{a}_{t})$ refers to the action- value function stored in the centralized critic shown in (10). The centralized critic value considers all actions from all agents, making the environment stationary in training. Since the action space is discrete, a Gumbel-softmax function [38] is employed to produce differentiable action samples from the policy when performing the Monte Carlo simulation. The differentiable sampled actions help to train the policy without using the log derivative trick as used in DDPG [22]. Notice in (10) that even though the experience replay doesn’t explicitly memorize the joint next action, the centralized critic utilizes all agents’ target actors to predict the joint next action and improve the stability of the learning during the update. The training of the centralized critics enhance the core idea of CTDE that, knowing the joint action, the road environment is stationary even though some policies change. Following the idea of multi-agent actor critic method in discrete action space, we expand and summarize the proposed MARL algorithm in Algo. 1 for the optimal DQJL coordination strategy with dynamic amount of non-EMVs in the presence of HVs. Algorithm 1 Multi-Agent Actor Critic for DQJL Coordination 1: Initialize $M$ pairs of policy networks and value networks 2: Initialize the experience history $D$ with mini-batch size 3: for each coordination training episode do 4: reset road environment for the initial state $\mathbf{s}$ 5: insert trivial vehicles until there are $M$ non-EMVs 6: for each coordination training step do 7: sample an action $a^{i}_{t}$ for each CV as $a^{i}_{t}=\pi(\cdot|o^{i}_{t})$ 8: sample an action $a^{i}_{t}$ for each HV based on (1) 9: execute $\mathbf{a}_{t}$ and observe $\mathbf{s}_{t+1}$ and $R_{t}$ 10: store $(\mathbf{s}_{t},\mathbf{a}_{t},R_{t},\mathbf{s}_{t+1})$ into $D$ 11: update state $\mathbf{s}_{t}\xleftarrow{}\mathbf{s}_{t+1}$ 12: for non-EMV = $1$ to $M$ do 13: if the vehicle is non-trivial then 14: draw a mini-batch sample $(\mathbf{s}_{t},\mathbf{a}_{t},R_{t},\mathbf{s}_{t+1})$ from $D$ 15: set $y_{i}$ and update the critic according to (10) 16: if the vehicle is a CV then 17: update the actor according to (11) 18: end if 19: end if 20: end for 21: update target networks by $\theta_{i}^{\prime}\xleftarrow{}(1-\tau)\theta_{i}^{\prime}+\tau\theta_{i}$ 22: end for 23: end for ### 4.4 Deep neural network architecture Vehicle trajectories are temporal-sensitive data, and vehicles maintain action consistency if they are yielding to an approaching EMV, i.e. a yielding vehicle decelerates and changes lane until it parks into the neighboring lane. Utilizing a long short-term memory (LSTM) layer as the output layer for the actor, the network can further reduce non-stationarity. The action space for this application is binary, so the LSTM layer is followed by a Gumbel softmax function to output the probabilities of yielding or driving forward respectively. As introduced in Algo. 1, the proposed methodology can handle various number of vehicles in the system by fixing the dimension of state representations. Adopting a similar technique in a previous study [18], trivial vehicles are added into the controlled system until there are $M$ pairs of networks. Although some networks are idle through the process, this technique circumvents the challenge in many connected vehicles applications where the number of agents is varying. This technique ensures the proposed methodology’s engineering practicality. See Fig. 6 for the detail neural network architectures and dimensions of the policy network and value network. The policy network takes input of the vehicle’s local observation, and processed through a double layer multi-layer perceptron (MLP). Each layer of the MLP consists of a normalization layer, a linear layer as well as a ReLU function. A LSTM layer is then attached, and two probabilities are resulted from the Gumbel-softmax function, standing for the vehicle’s probabilities of yielding or not at this step respectively. The value network takes the input of the road environment state representation and the joint action at this step. After concatenating them together, the network feeds the processed vector through a three layer MLP. The result from the value network is a scalar, representing the centralized state-action value for this particular agent. Adam Optimizer is selected to optimize losses for both networks. For the policy network, the input is the flatten local observation with dimension of $6\times 7$, linear layer 1 has dimension of $42\times 64$ and linear layer 2 has dimension of $64\times 128$. The LSTM layer takes input dimension of 128 and output 128. Linear layer 3 has dimension of $128\times 2$. For the value network, the concatenation layer has a dimension of $(7+1)M\times 256$, linear layer 4 has a dimension of $256\times 256$, and linear layer 5 has a dimension of $256\times 512$. Linear layer 6 is a fully connected layer which outputs the state-value function. Since the agents are heterogeneous in this application, they are trained in separate networks and no parameter sharing involves as all agents are non- exchangeable. Figure 6: Deep neural network architectures for a pair of policy network and value network. ### 4.5 Indirect MARL approach The transition probabilities between states needs to be provided to complete the MDP setting. The state transition probabilities, for DQJL application, are the collective result of the longitudinal and latitudinal dynamics, stochastic driving behavior and the coordination strategy. The real road dynamics for partially connected settings are usually complicated and difficult to interpret. Hence, some fundamental models are incorporated to picture the road dynamics. The road dynamics can be categorized in the longitudinal and latitudinal direction taking account of the stochastic driving behavior as well as DQJL coordination. #### 4.5.1 Longitudinal dynamics Along the direction of the traffic flow, it is natural to employ a car- following model to capture the longitudinal dynamics. In this study, we modify and adopt the discrete version of the intelligent driver model (IDM) [39] so that vehicles’ positions, velocities and accelerations can be computed across the discrete time horizon, i.e. MDP steps. To avoid denotation confusion, $u$ is used to represent acceleration. For a non-EMV not in the yielding process, the discrete IDM model states for the next step $\displaystyle v^{i}_{t+1}$ $\displaystyle=v^{i}_{t}+u^{i}_{t}\Delta t,$ $\displaystyle x^{i}_{t+1}$ $\displaystyle=x^{i}_{t}+v^{i}_{t}\Delta t+\frac{u^{i}_{t}\Delta t}{2},$ in which the acceleration of the vehicle at step $t$ can be determined by the ego vehicle $i$ and its leading vehicle $j$ as $u^{i}_{t}=u_{0}[1-(\frac{v^{i}_{t}}{v_{0}})^{4}-(\frac{s^{*}(v^{i}_{t},v^{j}_{t}-v^{i}_{t})}{x^{j}_{t}-x^{i}_{j}-l^{j}})^{2}],$ where $s^{*}(v^{i}_{t},v^{j}_{t}-v^{i}_{t})$ stands for the desired dynamic distance of two neighboring vehicles and can be approximated by $s^{*}(v^{i}_{t},v^{j}_{t}-v^{i}_{t})=d+v^{i}_{t}T+\frac{v^{i}_{t}(v^{j}_{t}-v^{i}_{t})}{2\sqrt{u_{0}b_{0}}}.$ In the equations above, $u_{0},b_{0}$ represents the IDM acceleration and deceleration baseline correspondingly, $T$ stands for the safety headway and $d$ represents the minimum safety gap between two neighboring vehicles. For a non-EMV in the yielding process, its longitudinal dynamics involves mainly deceleration. As illustrated in Fig. 3, there are two longitudinal uncertainties in a vehicle’s braking motion. The first one represents drivers’ perception-reaction time, $t_{r}$, which stands for a delay between human drivers’ perception and execution. As suggested in [40] on drivers’ perception abilities, the perception-reaction time follows the normal distribution of $t_{r}\sim\mathcal{N}(2.25s,(0.5s)^{2})$ is selected for this study. Decision time is also consider included in the perception-reaction time in this application as most decisions are made by the centralized controller. To realize the perception-reaction time, additional data structure is used to count the MDP step until the drivers start to decelerate. The second type of uncertainty in the longitudinal direction is the deceleration of vehicles when they are yielding. Each vehicle has its unique baseline deceleration rate and the human driver has its own unique braking behavior. The deceleration of the vehicle is captured as a white noise in addition to the baseline deceleration rate $b^{*}$. Mathematically speaking, the deceleration, not counting for car-following model, for a vehicle after receiving an yielding instruction at time $T_{i}$ is: $\displaystyle b^{i}_{t}=\begin{cases}0,&t\in[T_{i},T_{i}+t_{r}).\\\ {b}^{*}_{i}+\epsilon_{decel},&t\geq T_{i}+t_{r}.\\\ \end{cases}$ Thus, the overall longitudinal dynamics for a non-EMV is described as a function of an vehicle’s velocity, position, as well as its leading velocity and position: $\displaystyle v^{i}_{t+1}=f(v^{i}_{t},x^{i}_{t},v^{ij}_{t},x^{ij}_{t})=\begin{cases}v^{i}_{t}+u^{i}_{t}\Delta t,&\text{if }\xi^{i}_{t}=0,\\\ v^{i}_{t}-{b}^{i}_{t}\Delta t,&\text{ otherwise},\\\ \end{cases}$ where $ij$ denotes the leading vehicle of vehicle $i$. Notice that the EMV strictly follows the longitudinal dynamics. Its front bump position, velocity, and acceleration are determined by the distance between itself and its leading vehicle as well as the difference between their speeds. Since the maximum allowable speed of the EMV is much larger than that of a non-EMV, it is expected that the EMV travels faster when it doesn’t have any leading vehicles, i.e. DQJL established. #### 4.5.2 Latitudinal dynamics Inspired by the benchmark lane-changing model [41], the dynamic lane position of vehicles in a DQJL process are viewed to be the result of stochastic driving behaviors and coordination strategies. If an EMV is approaching on the passing lane of this road segment, the centralized controller aims to form a vehicle platoon on the neighboring lane. If a non-EMV on the passing lane starts to yield, it no longer follows the car-following model and begins to decelerate until it pulls over onto the neighboring lane, at which step this non-EMV stops yielding. Meanwhile, a vehicle on the neighboring lane will also decelerate temporarily. The yielding process for this decelerating non-EMV is considered completed when it has a new leading vehicle in the platoon. All vehicles will not necessarily brake until stop. Thus, the yielding status $\xi^{i}$ can be tracked as $\xi^{i}_{t+1}=g(\xi^{i}_{t},a^{i}_{t})=\begin{cases}1,\text{if }a^{i}_{t}=1\text{ or }\xi^{i}_{t}=1\\\ 0,\text{otherwise}.\end{cases}$ After the perception-reaction time, the yielding non-EMV attempts to pull over onto the neighboring lane, the probability of successfully lane changing, without considering for collisions, is captured as a geometric distribution parameter $p$. If the average time for successfully lane changing is $t_{lc}$ and temporal step length is $\Delta$, it is easy to derive that $p=\frac{1}{t_{lc}/\Delta t}\Rightarrow p=\frac{\Delta t}{t_{lc}},$ and the lane position $y$ of a pulling-over vehicle for the next step can be determined as $y^{i}_{t+1}=h(y^{i}_{t},\xi^{i}_{t})=\begin{cases}0,&t\in[T_{i},T_{i}+t_{r}),\\\ Y\sim G(\frac{\Delta t}{t_{lc}}),&t\geq t_{r}.\\\ \end{cases}$ #### 4.5.3 State transition for indirect MARL Adopting both longitudinal and latitudinal dynamics with stochastic driving behavior, it is easy to mathematically describe the next state $\mathbf{s}_{t+1}\sim\mathbf{s}_{t}$ with some joint action $\mathbf{a}$ to be $\mathbf{s}_{t+1}=\begin{bmatrix}x^{1}_{t}+v^{1}_{t}\Delta t&h(y^{1}_{t},\xi^{1}_{t})&f(v^{1}_{t},x^{1}_{t},v^{1j}_{t},x^{1j}_{t})&g(\xi^{i}_{t},a^{i}_{t})&l^{1}&b^{*}_{1}&\phi^{1}\\\ x^{2}_{t}+v^{2}_{t}\Delta t&h(y^{2}_{t},\xi^{2}_{t})&f(v^{2}_{t},x^{2}_{t},v^{2j}_{t},x^{2j}_{t})&g(\xi^{i}_{t},a^{i}_{t})&l^{2}&b^{*}_{2}&\phi^{2}\\\ \vdots&&&&&&\vdots\\\ x^{n}_{t}+v^{n}_{t}\Delta t&h(y^{n}_{t},\xi^{n}_{t})&f(v^{n}_{t},x^{n}_{t},v^{nj}_{t},x^{nj}_{t})&g(\xi^{i}_{t},a^{n}_{t})&l^{n}&b^{*}_{n}&\phi^{n}\\\ \end{bmatrix},$ (12) where $ij$ represents $i$th vehicle’s leading vehicle. ### 4.6 Direct MARL approach However, the real traffic conditions are much more complicated than the longitudinal and latitudinal dynamics. Human drivers are obeying car-following models, lane changing models, and other explicit or implicit driving dynamics with uncertainties. It is inefficient to capture all potential dynamics and develop a kinematics models to process through the proposed MARL framework. Thus, the proposed methodology should not assume a differentiable model of the road environment dynamics. Data-driven reinforcement learning is able to detect and understand the comprehensive road environment dynamics and the noise distributions, which represent stochastic driving behavior from human drivers. Furthermore, the proposed framework is compatible with various types of communication structures among the vehicles and the infrastructure. The CTDE framework cannot only be adopted to realize cooperative connected vehicles applications such as establishing DQJLs for EMVs, but also can be utilized for other connected vehicles applications, cooperative or competitive, particularly in those with stochastic dynamics incurred by human drivers. ## 5 Simulation Test Bed Experimental Design Since it is impossible to perform in-field vehicle coordination training due to safety and expense, we employ Simulation of Urban Mobility (SUMO) [42] as the test bed. In this section, we demonstrate the experimental design with generating simulation data in Subsec. 5.1, designing and implementing the simulation in Subsec. 5.2. Lastly, the model parameters and training hyperparameters are shown in Subsec. 5.3. Figure 7: EMV is entering the road section from the left. ### 5.1 Data synthesis Providing realistic numerical values to simulate the realistic traffic conditions on a two-lane urban roadway is crucial for the reproduciblility of application in field. Therefore, a dataset containing randomly generate starting conditions, including different number of vehicles and different penetration ratio, is prepared to serve as the training set for both indirect and direct MARL approaches. To better compare the quantitative results under different penetration ratios, different number of non-EMVs are selected to further generate a grouped test set to reflect the difference under the chosen road environment configurations.111The dataset is available at shorturl.at/PRT45. As shown in Table. 3, we generate the vehicle features based on listed literature source. The statistics of some vehicle features are presented in Fig. 8. Table 3: Randomly generated vehicle features Vehicle Feature | Value | Source ---|---|--- Vehicle Length $l$ | $\mathcal{N}(4.5m,(1m)^{2})$ | [43] Vehicle Baseline Deceleration $b^{*}$ | $\mathcal{N}(-2m/s^{2},(1m/s^{2})^{2})$ | [44, 45] Standard Deviation of Deceleration $\epsilon_{decel}$ | $0.5m/s^{2}$ | [44, 45] The road segment studied is an arterial class III with LOS category C according to Arterial LOS standards [46]. The nominal value for road environment configuration applied to both indirect and direct MARL training is listed in Table. 4. Table 4: System configuration parameters System Configuration Parameters | Value ---|--- Temporal Step Length $\Delta t$ | 0.5s Length of the Road segment $L$ | 200m Minimum Safety Gap $d$ | 0.5m Length of Reaction Distance $L_{HV}$ | 75m Non-EMV Starting Velocity $v_{0}^{1}$ | 4.5m/s EMV Starting Velocity $v_{0}^{0}$ | 8m/s EMV Maximum Allowable Velocity $v_{max}^{0}$ | 12m/s (a) Generated starting positions (b) Generated vehicle lengths Figure 8: The generated vehicle starting positions and average vehicle lengths. ### 5.2 RL-SUMO embedding For the indirect MARL approach, customized IDM parameters selected for the training process is listed in Table. 5. The trained coordination strategy is then validated on the test set on SUMO. Table 5: Indirect MARL IDM model parameters Modified IDM model parameters | Value | Source ---|---|--- Baseline Acceleration $u_{0}$ | $3m/s^{2}$ | [47] Baseline Deceleration $b_{0}$ | $-2m/s^{2}$ | [47] Desired Traffic Speed $v^{*}$ | $10m/s$ | [47, 48] Desired Headway $T_{0}$ | $1.5s$ | [47, 48] For the direct MARL approach, the state transitions are completely determined by the SUMO dynamics. Instead of developing an accurate road dynamic model when yielding to approach EMVs, this approach does not assume an differentiable model and is able to handle all potential state transitions. Inspired by Flow [49], the embedding on SUMO is achieved via socket programming and the TraCI package in SUMO. The complete multi-agent reinforcement learning pipeline on SUMO is presented in Fig. 9. Figure 9: SUMO-based multi-agent reinforcement learning training pipeline for DQJL problem. The pipeline combines the multi-agent road environment with the SUMO test bed. The proposed pipeline processes and manages the observations, actions, and rewards as input or output to the SUMO platform. Coordinated by TraCI, the pipeline can initialize, load and reset experiment road configurations. Instead of adopting the blue-light device, in SUMO sub-lane modulo, which provide a virtual passage between lanes [50, 51], we initialize another vehicle starting upstream to serve as the EMV in the simulation. The EMV’s initial velocity and maximum allowable velocity are set according to Table. 4 for the fast passage through the segment. The other EMV vehicle features, such as vehicle length, are trivial as they do not influence the learning process. In the direct MARL approach, built-in vehicle lane-changing models are overwritten so that non-EMVs only change lane when instructed, or forced if they are HVs. see Table. 6. Table 6: Configuration parameters for SUMO vehicle lane-changing Lane Changing Parameters | Value ---|--- collision.mingap-factor | 0 speed mode | 0 lane change mode | 0 lsStrategic | 0 lsCooperative | 0.5 Similar to the noise perturbation adopted in the indirect MARL approach, identical distributions of noises reflecting stochastic driving behaviors are added to each vehicle during the training stage for the indirect MARL. Through TraCI, additional data structures are initialized to track and execute the randomness. For instance, if an drivers’ perception reaction time is sampled as $2.5s$, a step counter is employed to make the driver move at $\frac{2.5s}{\Delta t}=5$ steps delayed according to Table. 4. ### 5.3 Training hyperparameters To maintain the consistency in training under indirect and direct MARL approaches, the training hyperparameters presented in Table. 7 are selected for the MARL training stage. Table 7: RL training hyperparameters Model Parameters and Hyper-parameters | Value ---|--- Priority Penalize Constant $P_{priority}$ | 0.5 Collision Penalize Constant $P_{collision}$ | -1000 Discount Factor $\gamma$ | 0.99 Minibatch Size | 64 Actor Learning Rate $\alpha$ | $10^{-4}$ Critic Learning Rate $\beta$ | $10^{-3}$ Replay Memory Size $D$ | 10000 Initial Epsilon $\epsilon$ | 0.99 Epsilon decay | $10^{-3}$ Loss Optimizer | Adam ## 6 Validation Results and Discussion In this section, we analyze the test bed validation results from the designed experiments. Training performance for both indirect and direct MARL approaches with the proposed neural network structure are compared in Subsec. 6.1. The validated EMV passing time resulted from both approaches are evaluated in Subsec. 6.2, and their corresponding training time are assessed in Subsec. 6.3. The real-time coordination compatibility is also justified in Subsec. 6.4. ### 6.1 Training performance comparison The learning performance with multiple independent runs for the indirect and direct reinforcement learning framework is presented in Fig. 10. (a) Indirect MARL results after 15000 episodes. (b) Direct MARL results after 15000 episodes. Figure 10: The training performance for Fig. 10(a)-indirect MARL and Fig. 10(b)-direct MARL. The dark lines highlights the mean value of these runs and the shaded area stands for one standard deviation. According to the learning behaviors of the indirect and direct MARL approaches, the indirect MARL approach converges much faster w.r.t number of episodes. The indirect MARL approach achieves optimal average reward within approximately 3000 episodes, while the direct MARL approach converges at an approximately equivalent level within around 8000 episodes. The fast convergence of the indirect MARL approach takes advantage of the significant reduction in exploration time due to establishment of road environment dynamics beforehand. Both learning curves become stable onward, and the fluctuations representing the effects of randomness in road environment initialization and stochastic road dynamics. ### 6.2 Emergency vehicle passing time validation To assess the DQJL coordination strategy from the proposed approaches, different groups of experiments are set up to compare the EMV passing time on a 200 meters road segment. The results under the indirect reinforcement learning framework are shown in Fig. 11 grouped by different connectivity penetration rate. Within the same group of experiments, i.e. same number of non-EMVs, four scenarios with different penetration rates are separated but all other vehicles’ features are kept consistent, including their starting positions.222An EMV passing demo is available at shorturl.at/djsx4 for qualitative analysis. According to Fig. 11, the EMV passing time exhibits monotonically decreasing pattern when the proportion of connected vehicles increases. The all HVs scenario is referred as the baseline scenario without any DQJL coordination. When the road is more crowded, it is evident from the data that distinctions of EMV passing time between all CVs and other scenarios broaden. The reason is the approaching EMV may be blocked by one more HV in the coordination process, significantly reducing its velocity meanwhile. As a result, the system performance of the DQJL coordination may be bottle-necked in the presence of HVs. Comparing the grouped results along the numbers of non-EMVs in the system, it appears that EMV passing time increments with the number of vehicles in the system. The increasing patterns differentiate, however, with different penetration rates. With all HVs in the traffic, the EMV passing time grows with an increasing manner when the road becomes more congested, whereas the EMV passing time with 100% penetration rate grows marginally. It is then not difficult to imply that the difference will scale when the target road segment becomes increasing congested. Considering the results, with all connected vehicles, the proposed approach saves approximately 30% of EMV passing time compared with the baseline scenario. The validation process is based on a Intel CPU i9 with 3.6 GHz with a NVIDIA GeForce 2080 Ti GPU, with an average decision time of 5.3 ms. Figure 11: EMV passing time results from indirect reinforcement learning approach. Figure 12: Comparison of EMV passing time between two approaches. Correlating the EMV passing time with coordination strategies from the indirect MARL approach and direct MARL approach, the coordination strategies from these two approaches can be quantitatively assessed. As shown in Fig. 12 comparing the EMV passing time under coordination strategies from both approaches, it is straightforward to see that the coordination strategy from the direct MARL approach is slightly better than that from the indirect MARL approach as it results in shorter time of DQJL establishment, which can be explained by the fact that the direct MARL approach grasps the real testbed state transition better. Since the difference in EMV passing time between these two approaches is insignificant across densities, it is considered that both MARL approaches yield the optimal DQJL coordination strategies, especially when their learning curves converge at a comparable level. ### 6.3 Training time comparison The training time reflects the cost of the proposed reinforcement learning approaches. Although the proposed methodology is capable of handling various numbers of vehicles in the controlled system, a set of densities, representing a group of numbers of non-EMVs, is selected for the examination of the training time with both approaches. Instead of allowing both approaches converge at optimal reward level, training time for 5000 episodes are recorded. Figure 13: MARL training time for 5,000 episodes against different road densities. Notice that the proposed methodology can handle various quantities of non-EMVs in the system and these densities are only chosen for experimenting the scalablity of the proposed methodologies. All training runs are conducted on a Intel CPU i9 with 3.6 GHz with a NVIDIA GeForce 2080 Ti GPU. It is evident from the training time results shown in Fig.13 that the direct reinforcement learning approach requires approximately 4 times of training time to finish 5000 episodes of training. The result is equivalent to say, the pipeline presented in Fig. 9 of allocating, initializing, and exchanging information takes additional of 3 times of training time besides performing numerical computations. It is also noticeable that the training time is increasing with an accelerating pace, which implies the ”curse of dimensionality” of the proposed algorithm. When the number of non-EMVs in the controlled system increases, the training time to reach optimal coordination tends to grow exponentially. ### 6.4 Real-time compatibility validation The average decision time based on the validation configuration is 5.3ms. According to the 3GPP vehicle-to-everything communication standards [52], the minimum requirement for an collision-free message sending interval is 10ms. Thus, the proposed framework is compatible for generating real-time coordination strategies with additional time for the support of functional expansions. ## 7 Conclusion and Future Directions In this study, we propose a multi-agent actor-critic based deep reinforcement algorithm to address the fast and safe DQJL establishment under the partially connected setting. By utilizing the centralized training with decentralized execution framework, the proposed methodology is scalable and extensible for multi-agent vehicular motion planning control problems in connected vehicle applications. In this study, two ways approaching the optimal coordination strategies are presented-indirect MARL approach based on an modified car- following model and stochastic driving behavior, and direct MARL approach based on the traffic simulation software. Both approaches are validated on the SUMO testbed, eventually, with the baseline scenario in which the traffic consists of all HVs and no coordination strategy is provided. The validation results suggest, with the real time coordination instructions informing the connected component in the traffic, the EMV passing time can be reduced by approximately 30%, providing a promising direction for facilitating emergency service vehicles in intelligent urban road environment. Finally, we evaluate the training cost of both approaches and validate the real-time compatibility with the proposed connectivity schema. To summarize the contributions of the presented study, 1. 1. we introduced the concept of dynamic queue-jump lane (DQJL) to reduce EMVs passing time on intelligent urban roadways; 2. 2. we developed a multi-agent actor-critic based deep reinforcement learning algorithm which addresses varying number of agent vehicles with discrete space for action representations for DQJL applications; 3. 3. we proposed a connected vehicle application framework which deals with partially connected traffic and stochastic driving behaviors; 4. 4. we presented two approaches, direct and indirect, when applying multi-agent reinforcement learning on connected vehicle applications and we validated the proposed coordination significantly improves emergency vehicle performances on connected urban roadway. The study can be extended into a few directions for the realization of the DQJL coordination in the field. First, regarding the privacy concern of a connected vehicle’s policy through vehicular communications, policy inference networks can be additionally deployed onto each agent vehicle. The policy interference network allows each agent to learn other vehicles’ policies without explicitly asking through communication channel. Second, the problem setting can be extended into signalized or unsignalized traffic intersections to recognize the multi-agent coordination pattern in a more complicated road environment. Realizing the intersection coordination and link-level coordination based on the presented work, we are able to present the network- level DQJL coordination, which will further improve the emergency vehicles’ performance. ## 8 Acknowledgment This research was supported by the C2SMART University Transportation Center at New York University, and Dwight David Eisenhower Transportation Fellowship Program (DDETFP) from the United States Department of Transportation. The authors thank Dr. Shuseng Wang for inspirations and advice on the methodology. The authors also thank Hengfeng Lu for preparing graphs. ## 9 Declaration of Competing Interest The authors declare that they have no competing financial interests or personal relationships, to the best of their knowledge, that could have appeared to influence the work reported in this paper. ## References * Government [2020a] N. Y. C. Government, New York End-To-End Response Times, 2019 (accessed November 22, 2020)a. 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2024-09-04T02:54:56.487299

2003.01032

arxiv-papers

# A universal scheme for robust self-testing in the prepare-and-measure scenario Nikolai Miklin<EMAIL_ADDRESS>Institute of Theoretical Physics and Astrophysics, National Quantum Information Center, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-306 Gdańsk, Poland International Centre for Theory of Quantum Technologies (ICTQT), University of Gdansk, 80-308 Gdańsk, Poland Michał Oszmaniec<EMAIL_ADDRESS>Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland ###### Abstract We consider the problem of certification of arbitrary ensembles of pure states and projective measurements solely from the experimental statistics in the prepare-and-measure scenario assuming the upper bound on the dimension of the Hilbert space. To this aim we propose a universal and intuitive scheme based on establishing perfect correlations between target states and suitably-chosen projective measurements. The method works in all finite dimensions and allows for robust certification of the overlaps between arbitrary preparation states and between the corresponding measurement operators. Finally, we prove that for qubits our technique can be used to robustly self-test arbitrary configurations of pure quantum states and projective measurements. These results pave the way towards practical application of the prepare-and-measure paradigm to certification of quantum devices. ## 1 Introduction Quantum devices are becoming more and more complex and the possibilities of their precise control and manipulation keep increasing. Recently reported demonstration of quantum computational advantage by Google [1] is only an intermediate milestone and quantum technologies have a potential real-life applications in fields such as quantum sensing [2], simulation of quantum systems [3], efficient computation [4] and machine learning [5, 6]. With the increasing complexity of quantum systems, there is a growing need for certification and verification of their performance. This task is usually realized via the combination of quantum tomography and various benchmarking schemes (see [7] for a recent review). However, these methods, despite being powerful and universally applicable, depend on the assumptions about the inner workings of quantum systems, such as perfect measurements or uncorrelated and independent errors. In contrast to these approaches _self-testing_ is a method which aims at proving the uniqueness of the implemented states or measurements based solely on the observed statistics and under minimal physical assumptions. The paradigm of self-testing was first introduced in the context of quantum cryptography [8], with the aim to obtain trust in cryptographic devices (see [9] for a recent review). Initially, it was applied to correlations observed in the Bell scenario [10] (see e.g., [8, 11, 12, 13]). The most know result in this area is certification of the singlet state in the case of maximal violation of the Clauser-Horne-Shimony-Holt Bell inequality [14]. With the growing number of results on self-testing, more and more attention is being drawn to prepare-and-measure scenarios, that are more experimentally appealing as compared to the one of Bell (see e.g., [15, 16, 17, 18]). Therein, one does not need to ensure space-like separation of the measurement events by two parties; in contrast, one party, Alice, communicates some of her states to Bob, who measures them in some basis of his choice. In order to get meaningful certification results further assumptions are needed. In the most commonly studied _semi-device-independent_ (SDI) scenario [19], one assumes that the dimension of the quantum system used for transmitting information is bounded from above. There exist, however, alternative approaches based on other constraints like minimal overlap [16], mean energy constraint [20] or entropy constraint [21]. Let us briefly recap on what has been done so far in the area of SDI self- testing. First, self-testing results were proven for mutually unbiased bases (MUBs) in dimension $2$, for both state ensembles and measurements [22]. This was further generalised to SDI certification of mutual unbiasedness of pairs of bases in an arbitrary dimension in Ref. [23]. Methods for self-testing of extremal qubit positive operator-valued measures (POVMs) were proposed in [24, 25] and further extended to symmetric-informationally complete (SIC) POVMs [26]. Importantly, all of the above results either rely on numerical approaches, for general state preparations and POVMs, or work only for special scenarios that exhibit many symmetries. In this work we propose a simple analytical method allowing to certify overlaps between preparations of arbitrary pure states and arbitrary projective measurements in _qudit_ systems. The scheme relies on establishing perfect correlations between preparation states and outcomes of suitably- chosen projective measurements. The method is universally applicable and robust to experimental noise. We prove that for qubits our SDI certification method can be used to obtain a robust self-testing result for arbitrary preparations of pure qubit states and corresponding projective measurements. While for higher dimensions we do not show self-testing, our scheme allows for SDI certification of numerous inherently quantum properties. The examples include, but are not limited to: _arbitrary_ overlaps between any number of measurement bases, MUB conditions, information-completeness of measurements, and SIC relations among measurement effects. We believe that our findings greatly extend the applicability of the paradigm of self-testing in the SDI setting. They will likely have application for certification of near-term quantum computers [7], especially since our scheme is not sensitive to state-preparation and measurement errors, which is one of the major problems in certification of quantum devices [27]. We also expect possible cryptographic applications as our setup is very similar to the one of textbook quantum key distribution schemes [28, 29]. The perfect correlations can be utilized for generation of the secret key, while the rest can be used to estimate the security. Thus, our methods can be directly applied for certification of quantum devices implementing protocols such as BB84 [28], which is normally achieved by introducing additional preparation states or measurement bases [30]. Before we proceed, let us first fix some of the notations used in this paper. Let $X$ be a linear operator acting on a finite-dimensional Hilbert space $\mathcal{H}$. Throughout the paper we use $\left|\left|\hskip 1.0ptX\hskip 1.0pt\right|\right|$, $\left|\left|\hskip 1.0ptX\hskip 1.0pt\right|\right|_{F}$ to denote operator norm and Frobenius norm of $X$. We will also use $|\mathbf{n}|$ to denote the Euclidean norm of $\mathbf{n}\in\mathbb{R}^{3}$, and $[n]$ to denote an $n$-element set $\\{1,2,\dots,n\\}$. ## 2 Description of the scenario We consider the prepare-and-measure scenario in which in each run of the experiment Alice prepares a quantum $d$-level system in one of the states from a finite set of preparations $\varrho_{a}^{x}$, for which we use two indexes $x\in[n]$ and $a\in[d]$. Subsequently, Bob performs a measurement on this state with a finite choice of measurement settings $y\in[n]$ having the possible outcomes $b\in[d]$. We assume that measurement process performed by Bob is described by quantum mechanics and furthermore that the parties do not communicate in any other way and do not have access to any entangled states or shared randomness [31] (the role of this assumption is discussed in detail later in the text). This implies that the observed statistics $p(b|a,x,y)$ are given by the Born rule i.e., $p(b|a,x,y)={\mathrm{tr}}(\varrho^{x}_{a}M^{y}_{b})$, where $\mathbf{M}^{y}=(M^{y}_{1},M^{y}_{2},\dots,M^{y}_{d})$ is a quantum measurement (POVM) performed by Bob upon the choice of the setting $y$. The goal of SDI certification is then to identify the preparation states and the measurements, or their properties, based solely on the observed statistics $p(b|a,x,y)$ _assuming_ the upper bound on the dimension $d$ and the validity of Born’s rule. We say that certain states $\varrho^{x}_{a}$ and measurements $\mathbf{M}^{y}$ can be _self-tested_ if the observed statistics specify these objects uniquely up to a unitary transformation and, perhaps, a global transposition. Figure 1: The idea of the certification scheme for the overlaps between states. Alice chooses her inputs $a,x$ and Bob his input $y$. Alice sends states $\varrho^{x}_{a}$ to Bob, who produces his outcome $b$. After establishing perfect correlations between preparation states and measurements for $y=x$, the rest of the statistics can be used to compute the overlaps between the preparation states. This can be used to directly certify presence of coherences between Alice’s input states and between effects of measurements implemented by Bob. ## 3 Certification of overlaps We start with a presentation of our scheme for certifying pairwise overlaps between _pure qudit_ preparation states and between the corresponding _projective_ measurements. By “corresponding" we mean that in our scheme we set these states and measurements to be “equal" in a sense that $\varrho^{x}_{a}=(\varrho^{x}_{a})^{2}=M^{x}_{a}$ for all $a\in[d]$ and $x\in[n]$. In what follows we will refer to these objects as target pure states and target measurements respectively. Their experimental counterparts we denote as $\tilde{\varrho}_{a}^{x}$, and $\mathbf{\tilde{M}}^{y}$ respectively. By “experimental" we mean any states and measurements defined over the same Hilbert space as the target ones and that reproduce the _observed statistics_ i.e., $\tilde{p}(b|a,x,y)={\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{M}^{y}_{b})$. Clearly, we _do not_ assume that the experimental states and measurement have to be “equal". The idea of our certification scheme is very intuitive, yet powerful (see Fig. 1). Assume that Alice and Bob prepared their devices in a way that $\tilde{p}(b|a,x,y)=1$, whenever $y=x$ and $b=a$. In other words, outcomes of Bob’s measurement are _perfectly correlated_ with the preparations of Alice (whenever $x=y$). Since the quantum dimension is upper bounded by $d$, we can easily conclude that $\tilde{\varrho}^{x}_{a}=\tilde{M}^{x}_{a}$, for all $a\in[d]$ and $x\in[n]$. Clearly, after these perfect correlations are established, the “cross-terms", can be used to compute the overlaps between the preparation states and between measurement operators: $\tilde{p}(b|a,x,y)={\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{M}^{y}_{b})={\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{\varrho}^{y}_{b})={\mathrm{tr}}(\tilde{M}^{x}_{a}\tilde{M}^{y}_{b})$. Therefore, if the experimental statistics $\tilde{p}(b|a,x,y)$ match the target statistics $p(b|a,x,y)$, we can certify that overlaps between experimental states match those of the target states. The same holds for the corresponding measurement operators. Our method can be also applied when experimental statistics do not match exactly the target ones. ###### Theorem 1 (Robust SDI certification of overlaps). Consider pure target qudit preparation states $\varrho^{x}_{a}$ and target projective measurements $\mathbf{M}^{y}$, where $a\in[d]$ and $x,y\in[n]$. Assume that $\varrho^{x}_{a}=M^{x}_{a}$ for all $a,x$ and furthermore that experimental states $\tilde{\varrho}^{x}_{a}$ and measurements $\mathbf{\tilde{M}}^{y}$ act on Hilbert space of dimension at most $d$ and generate statistics $\tilde{p}(b|a,x,y)={\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{M}^{y}_{b})$ such that $|\tilde{p}(b|a,x,y)-{\mathrm{tr}}(\varrho^{x}_{a}M^{y}_{b})|\leq\varepsilon$, for all $a,b,x,y$. Then, input states $\tilde{\varrho}^{x}_{a}$ are almost pure and measurements $\mathbf{\tilde{M}}^{y}$ are almost projective in the sense that $\displaystyle\text{for all $x$\ \ }\sum_{a=1}^{d}\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}\hskip 1.0pt\right|\right|\geq d(1-2\varepsilon)\ ,$ (1) $\displaystyle\text{for all $y$\ \ }\sum_{b=1}^{d}\left|\left|\hskip 1.0pt\tilde{M}^{y}_{b}\hskip 1.0pt\right|\right|\geq d(1-\varepsilon)\ .$ (2) Moreover, for all $x\neq x^{\prime}$, $a\neq a^{\prime}$, $y\neq y^{\prime}$, and $b\neq b^{\prime}$, we have $\displaystyle|{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})-{\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})|\leq\varepsilon+\sqrt{2\varepsilon+d^{2}\varepsilon^{2}},$ (3) $\displaystyle|{\mathrm{tr}}(\tilde{M}^{y}_{b}\tilde{M}^{y^{\prime}}_{b^{\prime}})-{\mathrm{tr}}(M^{y}_{b}M^{y^{\prime}}_{b^{\prime}})|\leq\varepsilon+(1+d\varepsilon)\sqrt{2\varepsilon+d^{2}\varepsilon^{2}}\ .$ ###### Proof. We start with a straightforward proof of Eq.(2). Since $\left|\left|\hskip 1.0pt\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|\geq{\mathrm{tr}}(\tilde{M}^{x}_{a}\varrho)$ for any state $\varrho$, the relation ${\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{M}^{x}_{a})\geq 1-\varepsilon$, valid for all $a,x$, implies $\left|\left|\hskip 1.0pt\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|\geq 1-\varepsilon$, for all $a,x$, and hence for all $x$ we have $\sum_{a=1}^{d}\left|\left|\hskip 1.0pt\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|\geq d-d\varepsilon$. To prove the bounds on the norms of experimental states in Eq. (1), we fix a setting $x$ and use a decomposition: $\tilde{M}^{x}_{a}=\lambda_{a}|\phi_{a}\rangle\langle\phi_{a}|+\text{Rest}_{a}\ ,$ (4) where $\lambda_{a}$ is the largest eigenvalue of $\tilde{M}^{x}_{a}$, $|\phi_{a}\rangle\langle\phi_{a}|$ is the corresponding eigenvector and, $\text{Rest}_{a}$ is a positive-semi-definite operator satisfying $\langle\phi_{a}|\text{Rest}_{a}|\phi_{a}\rangle=0$ (to keep the derivations legible, we omit the index $x$). Using the fact that observed statistics are $\varepsilon$ close to the target ones we get: $\displaystyle 1-\varepsilon\leq{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{M}^{x}_{a})=\lambda_{a}\langle\phi_{a}|\tilde{\varrho}^{x}_{a}|\phi_{a}\rangle+{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\text{Rest}_{a})\leq\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}\hskip 1.0pt\right|\right|+{\mathrm{tr}}(\text{Rest}_{a})\ .$ (5) By taking the sum over $a$ we obtain $\sum_{a=1}^{d}\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}\hskip 1.0pt\right|\right|\geq d(1-\varepsilon)-\sum_{a=1}^{d}{\mathrm{tr}}(\text{Rest}_{a})$. We can now use the identity $d=\sum_{a=1}^{d}{\mathrm{tr}}(\tilde{M}^{x}_{a})$, which follows form that fact that operators $M^{x}_{a}$ form a POVM in $\mathbb{C}^{d}$. This equality allows us to give a bound: $d=\sum_{a=1}^{d}\lambda_{a}+\sum_{a=1}^{d}{\mathrm{tr}}(\text{Rest}_{a})\leq d(1-\varepsilon)+\sum_{a=1}^{d}{\mathrm{tr}}(\text{Rest}_{a})\ ,$ (6) which is equivalent to $\sum_{a=1}^{d}{\mathrm{tr}}(\text{Rest}_{a})\leq d\varepsilon$. Inserting this to $\sum_{a=1}^{d}\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}\hskip 1.0pt\right|\right|\geq d(1-\varepsilon)-\sum_{a=1}^{d}{\mathrm{tr}}(\text{Rest}_{a})$ we obtain Eq. (1). We now proceed to the proof of Eqs. (3). We start with the one for the overlaps between preparation states. The proof is given by the following sequence of inequalities which hold for every $a\neq a^{\prime}$, $x\neq x^{\prime}$: $\displaystyle|{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})-{\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})|\leq|{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})-{\mathrm{tr}}(\tilde{M}^{x}_{a}\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})|+|{\mathrm{tr}}(\tilde{M}^{x}_{a}\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})-{\mathrm{tr}}(M^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})|$ $\displaystyle\leq\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}-\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|+\varepsilon\leq\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}-\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|_{F}+\varepsilon=\sqrt{{\mathrm{tr}}((\tilde{\varrho}^{x}_{a})^{2})+{\mathrm{tr}}((\tilde{M}^{x}_{a})^{2})-2{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{M}^{x}_{a})}+\varepsilon$ $\displaystyle\leq\sqrt{1+(1+d^{2}\varepsilon^{2})-2(1-\varepsilon)}=\varepsilon+\sqrt{2\varepsilon+d^{2}\varepsilon^{2}}.$ All of the above inequalities are pretty straightforward apart from ${\mathrm{tr}}((\tilde{M}^{x}_{a})^{2})\leq 1+d^{2}\varepsilon^{2}$ that we prove below. For this we again use the partial spectral decomposition form Eq. (4) (without writing the superscript $x$ as above) which implies: $\displaystyle{\mathrm{tr}}((\tilde{M}^{x}_{a})^{2})=\lambda_{a}^{2}+{\mathrm{tr}}(\text{Rest}_{a}^{2})\leq 1+({\mathrm{tr}}(\text{Rest}_{a}))^{2}\leq 1+d^{2}\varepsilon^{2}.$ (7) The proof for the overlaps of POVM effects is very similar to the one for states with the only difference being the following inequality: $\displaystyle|{\mathrm{tr}}(\tilde{M}^{x}_{a}\tilde{M}^{x^{\prime}}_{a^{\prime}})-{\mathrm{tr}}(\tilde{M}^{x}_{a}\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})|\leq(1+d\varepsilon)\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}-\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|,$ (8) that is used in the second step in the proof in Eq. (3). One can easily verify the validity of this inequality by writing once more the decomposition $\tilde{M}^{x^{\prime}}_{a^{\prime}}=\lambda_{a}|\phi_{a^{\prime}}\rangle\langle\phi_{a^{\prime}}|+\text{Rest}_{a^{\prime}}$ and remembering that ${\mathrm{tr}}(\text{Rest}_{a})\leq d\varepsilon$. ∎ The above result states that if the statistics observed in our certification scheme vary just a little bit form the target ones, the overlaps of the experimental states are also close to the overlaps between target states (and analogously for measurements). To our best knowledge analogous results have been previously known only for very special symmetric target states and measurements forming MUBs [22, 23]. In Appendix B (Lemma 1) we improve the above bounds for the case of qubit systems. Moreover, in Appendix A we prove, by giving explicit examples, that the bounds in Eq. (3) are tight in the first orders in $\sqrt{\varepsilon}$ and $d$. ## 4 Self-testing of qubits We now show that certification of overlaps allows to prove robust self-testing result for _arbitrary_ pure qubit preparations and projective measurements appearing in our certification scheme. ###### Theorem 2 (Robust self-testing of qubit systems). Consider target pure qubit states $\varrho^{x}_{a}$ and projective measurements $\mathbf{M}^{y}$, where $a=1,2$, $x,y\in[n]$, and $\varrho^{x}_{a}=M^{x}_{a}$ for all $a,x$. Assume that experimental qubit states $\tilde{\varrho}^{x}_{a}$ and measurements $\mathbf{\tilde{M}}^{y}$ generate statistics $\tilde{p}(b|a,x,y)=\mathrm{tr}(\tilde{\varrho}^{x}_{a}\tilde{M}^{y}_{b})$ such that $|\tilde{p}(b|a,x,y)-{\mathrm{tr}}(\varrho^{x}_{a}M^{y}_{b})|\leq\varepsilon$, for all $a,b=1,2$ and $x,y\in[n]$. Then, there exist $\varepsilon_{0}$ such that for $\varepsilon\leq\varepsilon_{0}$ there exist a qubit unitary $U$ such that $\displaystyle\frac{1}{2n}\sum_{a,x}{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{a})^{(T)}U^{\dagger}\varrho^{x}_{a})\geq 1-f(\varepsilon)\ ,$ $\displaystyle\frac{1}{2n}\sum_{b,y}{\mathrm{tr}}(U(\tilde{M}^{y}_{b})^{(T)}U^{\dagger}M^{y}_{b})\geq 1-g(\varepsilon)\ ,$ (9) where $(\cdot)^{(T)}$ is the transposition with respect to a fixed basis in $\mathbb{C}^{2}$ that may have to be applied to all experimental states and measurements at the same time. Moreover, functions $f,g:[0,\varepsilon_{0})\rightarrow\mathbb{R}_{+}$ depend solely on the target states and measurements and, for small $\varepsilon$, have the asymptotics $f(\varepsilon)\propto\varepsilon,\ g(\varepsilon)\propto\varepsilon$. In the above we used the fidelity $F(\varrho,\sigma)={\mathrm{tr}}(\varrho\sigma)$ to indicate the closeness between _rotated_ experimental states and target pure states (and analogously for measurements), following existing literature. The case $\varepsilon=0$ corresponds to ideal reconstruction of target states and measurements after applying a suitable isometry. We remark that we allow only unitary operations (and possible transposition), as opposed to general channels [22, 25], to be applied to the experimental states in order to approximate the target states _as well as possible_. In Appendix B we give a formal version of the above result and its proof. Moreover, we present there robustness bounds expressed in terms of the trace distance and its analogue for measurements [32]. We remark that the functions $f,g$ become unbounded once Bloch vectors of target qubit states become singular, i.e., once the target states are close to being aligned in a space of smaller dimension. ###### Remark. Certification of overlaps between pure states in general does not allow for their self-testing in higher-dimensional systems. This is e.g., due to the existence of unitary inequivalent sets of SIC-POVMs for $d=3$ [33] and MUBs for $d=4$ [34] (even if we allow for complex conjugation). ###### Sketch of the proof. We give a full proof for the ideal case ($\varepsilon=0$) below. From Theorem 1 it follows that for all $x,x^{\prime},a,a^{\prime}$ we have $\tilde{\varrho^{x}_{a}}=\tilde{M}^{x}_{a}$ and ${\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})={\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})$. Using the Bloch representation, $\varrho^{x}_{a}=\frac{1}{2}(\mathbbm{1}+\mathbf{n}^{x}_{a}\cdot\bm{\sigma})$, we can conclude that also $\mathbf{\tilde{n}}^{x}_{a}\cdot\mathbf{\tilde{n}}^{x^{\prime}}_{a^{\prime}}=\mathbf{n}^{x}_{a}\cdot\mathbf{n}^{x^{\prime}}_{a^{\prime}}$, where $\mathbf{n}^{x}_{a}$ and $\tilde{\mathbf{n}}^{x}_{a}$ are Bloch vectors of $\varrho^{x}_{a}$ and $\tilde{\varrho}^{x}_{a}$ respectively. Assume now that the vectors $\mathbf{n}^{1}_{1},\mathbf{n}^{2}_{1},\mathbf{n}^{3}_{1}$ are linear independent. Let $O$ be the linear transformation defined by $O\mathbf{n}^{x}_{1}=\mathbf{\tilde{n}}^{x}_{1}$, $x=1,2,3$, and let $L$ be the matrix whose rows are the vectors $\mathbf{n}^{x}_{1}$, $x=1,2,3$. Then, we have $LO^{T}OL^{T}=LL^{T}$, and consequently, since $L$ is invertible by the construction, $O^{T}O=\mathbbm{1}_{3}$, i.e., $O$ is an orthogonal transformation in $\mathbb{R}^{3}$. It is well-known [35] that if $\det(O)=1$, there exist a unitary matrix $U$ such that ${\mathrm{tr}}(U(\tilde{\varrho}^{x}_{a})U^{\dagger}\varrho^{x}_{a})=1$ for $x=1,2,3$ and $a=1$. By our assumption all remaining states $\varrho^{x}_{a}$ can be decomposed in the basis $\\{\mathbbm{1},\varrho^{1}_{1},\varrho^{2}_{1},\varrho^{3}_{1}\\}$, with the coefficients depending solely on the overlaps ${\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})$. The, the same $U$ that maps $\tilde{\varrho}^{x}_{1}$ to $\varrho^{x}_{1}$, for $x=1,2,3$, also connects the remaining pairs of states. Finally, if $\det(O)=-1$, the transformation $O$ corresponds to application of the transposition in the standard basis of $\mathbb{C}^{2}$ followed by application of a unitary operation $U$, determined by $O$ [36]. For the general case $\varepsilon>0$ we consider Cholesky factorisations of Gram matrices, denoted by $\Gamma$ and $\tilde{\Gamma}$, of Bloch vectors $\mathbf{n}^{1}_{1},\mathbf{n}^{2}_{1},\mathbf{n}^{3}_{1}$ and their experimental counterparts respectively. Theorem 1 is then used to bound $\left|\left|\hskip 1.0pt\Gamma-\tilde{\Gamma}\hskip 1.0pt\right|\right|_{F}$ and utilize results of Ref. [37] to gauge how much the Cholesky decompositions of $\Gamma$ and $\tilde{\Gamma}$ differ in the Frobenius norm. The latter can be connected to the average fidelity between the selected target and experimental states. The robustness for the remaining states follows from the fact that they can be decomposed (as operators) using the three initially chosen target states and the identity. If vectors from the set $\\{{\mathbf{n}^{x}_{a}\\}}$ span two dimensional space then the same arguments for both $\varepsilon=0$ and $\varepsilon>0$ can be repeated for the considered subspace. Importantly, in this case additional transposition is not necessary. ∎ Figure 2: Lower bounds for the average fidelity between the experimental and target states. Part (a) presents results for $n=2,3$ qubit MUBs. Part (b) presents results for two qubit bases for different degree of bias $\alpha$. Value $\alpha=0$ corresponds to two MUBs while $\alpha=1$ gives two identical bases. ## 5 Examples We now apply the quantitative formulation of Theorem 2 to lower-bound average fidelities for different configurations of target states as a function of the allowed error $\varepsilon$. In Figure 2 we present results for eigenstates of $n=2$ and $n=3$ Pauli matrices (i.e., states forming $n=2$ and $n=3$ qubit MUBs), and states belonging to two biased projective measurements satisfying ${\mathrm{tr}}{(\varrho^{1}_{a}\varrho^{2}_{a})}=\frac{1+\alpha}{2}$, $a=1,2$ where we take $\alpha\in[0,1]$. For $n=2$ MUBs we compare our results with Ref. [22] that aimed at self- testing of qubit MUBs. The results of Ref. [22] give the upper bound on average fidelity equal $0.75$ for the deviation of $\simeq 0.1$ in the figure of merit. In our scheme this happens for $\varepsilon\simeq 0.033$ as shown on Fig. 2. To test the versatility of our scheme, we also applied it to $n=3$ MUBs, trine and tetrahedral configurations of qubit states. Quantitative results concerning these examples are listed in Table 1, while detailed derivations are given in Appendix D. For trine and tetrahedral configurations the robustness is obtained via the so-called Procrustes problem, described in Appendix C, based on Ref. [38]. For cases other than two MUBs we cannot make any comparison with the existing literature since, to our best knowledge, these case have not been studied previously. Configuration | $\varepsilon_{0}$ | $C$ ---|---|--- 2 MUBs | $\approx 0.062$ | $\frac{7}{2}+\sqrt{2}$ 3 MUBs | $\approx 0.030$ | $6$ 2 biased bases | $\lesssim\frac{4-3\alpha-\sqrt{7-6\alpha}}{18}$ | $2+\left(1+\frac{\sqrt{1+\alpha}}{\sqrt{2}(1-\alpha)}\right)^{2}$ Trine | $\approx 0.058$ | $\frac{19}{3}$ Tetrahedron | $\approx 0.037$ | $10$ Table 1: Results of quantitative variant of Theorem 2 applied to different configurations of target quantum states. The threshold $\varepsilon_{0}$ sets the maximal noise level which is tolerated by our scheme. The constant $C$ is defined via the relation $f(\varepsilon)\stackrel{{\scriptstyle\varepsilon\rightarrow 0}}{{\approx}}C\varepsilon$, where $1-f$ is the lower bound on the average fidelity from Eq. (2). ## 6 Shared randomness Throughout the article we assumed that preparation and measurement devices are uncorrelated, which is often true in practice. However, one may also consider a situation in which Alice and Bob share a random variable $\lambda$, that they can use to coordinate their actions in a given round of the experiment. The most general statistics that can be generated in such a scenario can be expressed as $p(b|a,x,y)=\int d\lambda p(\lambda){\mathrm{tr}}(\varrho^{x}_{a}(\lambda)M^{y}_{b}(\lambda))$, where $p(\lambda)$ denotes the probability distribution of $\lambda$. Interestingly, the presence of shared randomness makes our main results, as stated by Theorems 1 and 2, inapplicable. The easiest way to see it is to consider the following simple example of $n=d=2$, one bit of shared randomness $\lambda\in\\{1,2\\}$, and $\tilde{\varrho}^{1,2}_{a}=|a\rangle\langle a|$, $\tilde{M}^{1,2}_{b}=|b\rangle\langle b|$. This clearly satisfies the requirement on $\tilde{p}(a|a,x,x)=1$. Now, Alice’s and Bob’s devices can decide to “flip" their preparations and measurement operators whenever $x=2$, $y=2$ and $\lambda=2$ (note that $\lambda$ can be distributed in an arbitrary way). This procedure does not affect the correlations $\tilde{p}(a|a,x,y=x)$, but it can be used to set $\tilde{p}(a|a,1,2)$ to an arbitrary value. Additionally, we believe that in the presence of shared randomness, the usual notion of self-testing in the SDI setting has to be reconsidered. Specifically, for _any_ quantum realisation giving the statistics $p(b|a,x,y)={\mathrm{tr}}(\varrho^{x}_{a}M^{y}_{b})$, one can always consider a strategy in which with probability $p(\lambda)$ Alice prepares states $\varrho^{x}_{a}(\lambda)=U_{\lambda}\varrho^{x}_{a}U^{\dagger}_{\lambda}$ and Bob implements measurements $M^{y}_{b}(\lambda)=U_{\lambda}M^{y}_{b}U^{\dagger}_{\lambda}$, where $U_{\lambda}$ is some arbitrary unitary transformation depending on $\lambda$. Clearly, such strategy reproduces the original statistics, and makes it impossible to find a single unitary that connects the target and experimental states (or measurements). While, we do not consider the assumption on no shared randomness strong, below we propose a modification of our SDI certification scheme that allows to certify overlaps between arbitrary pure states even in the presence of shared randomness. The idea is to introduce, for every pair of non-orthogonal states, a suitable _intermediate state_ [39], that enforces fixed overlaps between experimental states in every round of the experiment. Recall that in our certification scheme we consider pure target qudit states $\varrho^{x}_{a}$ and target projective measurements $\mathbf{M}^{y}$, where $a\in[d]$ and $x,y\in[n]$. We extend this scheme by introducing an additional intermediate target state $\varrho_{z}$ for every two pairs of Alice’s input $(x,a)$ , $(x^{\prime},a^{\prime})$, where $a\neq a^{\prime}$ and $x\neq x^{\prime}$. The state $\varrho_{z}$ is chosen as the unique state satisfying: ${\mathrm{tr}}\left(\varrho_{z}(\varrho^{x}_{a}+\varrho^{x^{\prime}}_{a^{\prime}})\right)=1+{\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})^{\frac{1}{2}}\ .$ (10) Let $\tilde{p}(b|z,y)$ denote the experimental statistics, corresponding to inputs $z,y$ and outcome $b$. In the ideal scenario, in which experimental statistics satisfy assumptions of Theorem 1 with $\varepsilon=0$, assume additionally that $\tilde{p}(a|z,x)+\tilde{p}(a^{\prime}|z,x^{\prime})=1+\sqrt{\tilde{p}(a^{\prime}|x,a,x^{\prime})}\ .$ Below we prove that under the above assumptions for all $\lambda$ we have ${\mathrm{tr}}(\tilde{\varrho}^{x}_{a}(\lambda)\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}(\lambda))={\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})$. Since in the ideal case overlaps between preparation states _do not depend_ on $\lambda$ and match the target value, our modified certification scheme works also in the presence of shared randomness. In the noisy scenario one can generally expect to be able to upper-bound the probability (over the choice of $\lambda$) of the deviations from perfect correlations between states and measurements, as a function of the error $\varepsilon$. We postpone the discussion of this interesting point to the future work. ###### Proof of soundness of certification in the presence of shared randomness. Consider a general physical realisation of experimental statistics $\tilde{p}(b|x,a,y)$, $\tilde{p}(b|z,y)$ via classically correlated preparations and measurements on $\mathbb{C}^{d}$: $\displaystyle\tilde{p}(b|x,a,y)$ $\displaystyle=\int d\lambda p(\lambda){\mathrm{tr}}(\tilde{\varrho}^{x}_{a}(\lambda)\tilde{M}^{y}_{b}(\lambda))\ ,$ (11) $\displaystyle\tilde{p}(b|z,y)$ $\displaystyle=\int d\lambda p(\lambda){\mathrm{tr}}(\tilde{\varrho}_{z}(\lambda)\tilde{M}^{y}_{b}(\lambda))\ ,$ (12) where $x,y\in[n]$, $a,b\in[d]$ and the variable (input) $z$ labels elements in the set of unordered pairs $\\{{(x,a),(x^{\prime},a^{\prime})\\}}$, where $x\neq x^{\prime}$ and $a\neq a^{\prime}$. Assume now that the above statistics match _exactly_ the ones required by our scheme and are compatible with Eq. (10) in the sense that: $\displaystyle\tilde{p}(b|x,a,y)$ $\displaystyle=$ $\displaystyle{\mathrm{tr}}(\varrho^{x}_{a}\varrho^{y}_{b})\ ,$ (13) $\displaystyle\tilde{p}(a|z,x)+\tilde{p}(a^{\prime}|z,x^{\prime})$ $\displaystyle=$ $\displaystyle 1+\sqrt{\tilde{p}(a^{\prime}|x,a,x^{\prime})}\ .$ (14) In what follows we prove that if the above constraints are satisfied, then for all111Technically speaking this equality holds for _almost all_ $\lambda$.: $\displaystyle\tilde{\varrho}^{x}_{a}(\lambda)$ $\displaystyle=$ $\displaystyle\tilde{M}^{x}_{a}(\lambda)\ ,$ (15) $\displaystyle{\mathrm{tr}}(\varrho^{x}_{a}\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})$ $\displaystyle=$ $\displaystyle{\mathrm{tr}}(\varrho^{x}_{a}(\lambda)\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}(\lambda))\ ,$ (16) where $x,x^{\prime}\in[n]$, $a,a^{\prime}\in[d]$. The above equation means that the overlaps between preparation sates (and measurements) do not depend on on the value of the shared random variable $\lambda$. The proof of Eq. (15) is straightforward. Namely, form Eq. (13) if follows that: $\int d\lambda p(\lambda){\mathrm{tr}}\left(\tilde{\varrho}^{x}_{a}(\lambda)\tilde{M}^{x}_{a}(\lambda)\right)=1\ .$ (17) Since ${\mathrm{tr}}(\tilde{\varrho}^{x}_{a}(\lambda)\tilde{M}^{x}_{a}(\lambda))\leq 1$, we get that for all $\lambda$ we have ${\mathrm{tr}}(\tilde{\varrho}^{x}_{a}(\lambda)\tilde{M}^{x}_{a}(\lambda))=1$. Since this reasoning can be repeated for all $a\in[d]$ (for the fixed value of $x$) and since operators $\tilde{M}^{x}_{a}$ form a POVM on $\mathbb{C}^{d}$ , we finally get Eq. (15). The proof of Eq. (16) is more involved and relies on both Eqs. (13) and (14). Specifically, from Eq. (13) and the already established identity $\tilde{\varrho}^{x}_{a}=\tilde{M}^{x}_{a}$, it follows that Eq. (14) is equivalent to: $\int d\lambda p(\lambda){\mathrm{tr}}\left(\tilde{\varrho}_{z}(\lambda)[\tilde{\varrho}^{x}_{a}(\lambda)+\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}(\lambda)]\right)=1+\sqrt{{\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})}\ ,$ (18) where moreover: $\int d\lambda p(\lambda){\mathrm{tr}}\left(\tilde{\varrho}^{x}_{a}(\lambda)\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}(\lambda)\right)={\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})\ .$ (19) Using Bloch representation of qubit states (states $\tilde{\varrho}^{x}_{a}(\lambda),\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}(\lambda)$ are pure and hence span a two dimensional subspace of $\mathbb{C}^{d}$) it is straightforward to obtain the bound: ${\mathrm{tr}}\left(\tilde{\varrho}_{z}(\lambda)[\tilde{\varrho}^{x}_{a}(\lambda)+\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}(\lambda)]\right)\leq 1+\sqrt{{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}(\lambda)\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}(\lambda))}\ .$ (20) After setting $g(\lambda)=\sqrt{{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}(\lambda)\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}(\lambda))}$ and using Eq. (19) we obtain: $\int d\lambda p(\lambda)g(\lambda)\geq\sqrt{\int d\lambda p(\lambda)g(\lambda)^{2}}\ .$ (21) Using Chauchy-Schwartz inequality for the left hand side of the above inequality we finally get: $\int d\lambda p(\lambda)g(\lambda)=\sqrt{\int d\lambda p(\lambda)g(\lambda)^{2}}\ ,$ (22) which is equivalent to saying that the variance of the random variable $g(\lambda)$ vanishes. Therefore, $g(\lambda)=\alpha$, where $\alpha$ is some numerical constant. We conclude the proof by noting that from Eq. (19) it follows that $\alpha^{2}={\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})$ which implies Eq. (16). ∎ ## 7 Extension of the scheme to SDI characterization of rank-1 non-projective measurements In our work we focus on self-testing of arbitrary configurations of preparation states and projective measurements. In this section we discuss possible extension of our results to certification of properties of general rank-1 measurements in arbitrary dimension $d$. To that end, we introduce additional states coming from suitably-chosen orthonormal bases per every effect of the measurement we want to certify. With the help of these extra states (and corresponding projective measurements) we can e.g., certify the following properties of POVMs in arbitrary dimension $d$: (i) information completeness, (ii) extremality of rank-1 POVMs, and (iii) SIC-property. The main idea of the extension is the following. Suppose we would like to self-test an $m$-outcome POVM $\mathbf{N}=(N_{1},\dots,N_{m})$ acting on a Hilbert space of dimension $d<m$ with each effect $N_{b}$ being of rank $1$ i.e., $N_{b}=\alpha_{b}\sigma_{b}$, for some pure states $\sigma_{b}$ and positive scalars $\alpha_{b}$. We consider $m$ sets of preparation states $\\{\varrho^{x}_{a}\\}$, $a\in[d]$, ${x\in[m]}$ and the corresponding projective measurements $\mathbf{M}^{y}$, $y\in[m]$. Assume now that in the idealized case we observe statistics satisfying: $\displaystyle{\mathrm{tr}}(\varrho^{b}_{a}N_{b})=0,\quad\forall b\in[m],\,a\neq 1,$ $\displaystyle{\mathrm{tr}}(\varrho^{x}_{a}M^{x}_{b})=\delta_{a,b},\quad\forall x\in[m]\ .$ (23) From the second set of conditions we conclude that $\\{\varrho^{x}_{a}\\}_{a}$ forms, as before, a basis of pure states for each $x$. From the first condition in Eq. (7) we get that $\mathbf{N}$ has to be a rank-1 POVM and, moreover, that $\varrho^{b}_{1}=\sigma_{b}$ ($N_{b}$ is orthogonal to all of the states except the one with $a=1$). Consequently, we have ${\mathrm{tr}}(\varrho^{b}_{1}N_{b})=\alpha_{b}$ for all $b\in[m]$. This fact together with the measured overlaps ${\mathrm{tr}}(\varrho^{b}_{1}\varrho^{b^{\prime}}_{1})={\mathrm{tr}}(\varrho^{b}_{1}M^{b^{\prime}}_{1})$ gives the information about the overlaps ${\mathrm{tr}}(N_{b}N_{b^{\prime}})$ between effects of the POVM $\mathbf{N}$. The knowledge of all overlaps ${\mathrm{tr}}(N_{b}N_{b^{\prime}})$ (including $b^{\prime}=b$) and the fact that $\mathbf{N}$ is a rank-1 POVM allow to certify many interesting properties of $\mathbf{N}$. First, as long as $m\leq d^{2}$ we can certify extremality of $\mathbf{N}$. This follows from the fact that rank-1 POVMs are extremal if and only if their effects are linearly independent (see e.g., Ref. [40]). Linear independence can be directly inferred from the experimentally accessible moment matrix $\Gamma_{b,b^{\prime}}={\mathrm{tr}}(N_{b}N_{b^{\prime}})$. Specifically, $\Gamma$ is non-singular if and only if operators $\\{N_{b}\\}_{b}$ are linear independent. Now, if $m=d^{2}$ we can use the same reasoning to infer information completeness of $\mathbf{N}$. Finally, from the moment matrix $\Gamma$ we can directly verify the SIC property i.e., the condition ${\mathrm{tr}}(N_{b}N_{b^{\prime}})=\frac{1+\delta_{b,b^{\prime}}}{d^{2}(d+1)},\forall b,b^{\prime}$. ## 8 Discussion We have presented a systematic analytical scheme for noise-resilient certification of overlaps between arbitrary configurations of pure quantum states and rank-$1$ projective measurements in the prepare-and-measure scenario. For qubits our scheme can be used to robustly self-test general ensembles of pure quantum sates and the corresponding projective measurements. We believe that these findings pave the way towards systematic certification of general quantum systems in the semi-device-independent paradigm. This is supported by the concrete qubit results from Table 1 and by the universality of our protocol for certifying overlaps of quantum sates in arbitrary dimension. There are many exciting research directions that stem from our work. One of them is a systematic SDI characterization of generalized measurements beyond qubits. Another research direction is the robustness analysis of our scheme in the presence of shared randomness. An important contribution to the filed of self-testing would be a generalization of our self-testing results for qubit to higher-dimensional systems. Finally, it is interesting to combine our methods with recent findings relating state discrimination games and resource theories of measurements and channels [41, 42, 43]. We would also like to draw reader’s attention to a related work [44]. ## Acknowledgements We thank Marcin Pawłowski, Jedrzej Kaniewski and Tanmoy Biswas for interesting discussions and comments. NM acknowledges the financial support by First TEAM Grant No. 2016-1/5. We acknowledge partial support by the Foundation for Polish Science (IRAP project, ICTQT, contract no. 2018/MAB/5, co-financed by EU within Smart Growth Operational Programme). We acknowledge the support from International Research Agendas Programme of the Foundation for Polish Science funded by the European Regional Development Fund. MO acknowledges the financial support by TEAM-NET project (contract no. POIR.04.04.00-00-17C1/18-00). ## Appendix In this Appendix we provide technical details that were omitted in the main text. First, in Appendix A we prove statements regarding saturation of the bounds in Theorem 1. Second, in Appendix B we formulate and prove a quantitative version of Theorem 2, which concerns robustness of our self- testing protocol expressed in terms of average fidelities. We also prove there Theorem 3 that gives analogous results expressed in terms of trace distance. In Appendix C we provide alternative robustness analysis based on orthogonal Procrustes problem. In Appendix D we provide technical details about the considered examples in Table 1. Finally, in Appendix E we give proofs of two auxiliary Lemmas needed in the proof of the technical version of Theorem 2. ## Appendix A Explicit form of states and measurements saturating the bounds in Theorem 1 We start by giving an example of four states and two qubit measurements ($n=2$, $d=2$), for which the bound in Eq. (3) on the deviation of the overlaps scales like $\sqrt{\varepsilon}$. Their explicit form is given below: $\displaystyle\varrho^{1}_{1}=|\tilde{0}\rangle\langle\tilde{0}|,\quad|\tilde{0}\rangle=\sqrt{1-\varepsilon}|0\rangle+\sqrt{\varepsilon}|1\rangle,\quad\varrho^{2}_{1}=|+\rangle\langle+|,$ (24) where $|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$, and as required by our scheme $\varrho^{x}_{2}=\mathbbm{1}-\varrho^{x}_{1},\;x=1,2$ and $M^{x}_{a}=\varrho^{x}_{a}$, for all $x,a$. Let the experimental states and measurements be the following: $\displaystyle\tilde{\varrho}^{1}_{1}=|0\rangle\langle 0|,\quad\tilde{\varrho}^{1}_{2}=|1\rangle\langle 1|,\quad\tilde{\varrho}^{2}_{1}=|+\rangle\langle+|,\quad\tilde{\varrho}^{2}_{2}=|-\rangle\langle-|,$ $\displaystyle\tilde{M}^{1}_{1}=|\tilde{0}\rangle\langle\tilde{0}|,\quad\tilde{M}^{2}_{1}=|\tilde{+}\rangle\langle\tilde{+}|,\quad|\tilde{+}\rangle=\sqrt{1-\varepsilon}|+\rangle+\sqrt{\varepsilon}|-\rangle\ .$ (25) First, we need to show that the experimental statistics is deviated from the target one by at most $\varepsilon$, i.e., $|{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{M}^{y}_{b})-{\mathrm{tr}}(\varrho^{x}_{a}\varrho^{y}_{b})|\leq\varepsilon,\forall a,b,x,y$. Since $\tilde{\varrho}^{x}_{1}+\tilde{\varrho}^{x}_{2}=\mathbbm{1}$ for both $x=1,2$, it is sufficient to consider the cases of $a=1,b=1$. Calculating the statistics we can see that ${\mathrm{tr}}(\tilde{\varrho}^{1}_{1}\tilde{M}^{1}_{1})={\mathrm{tr}}(\tilde{\varrho}^{2}_{1}\tilde{M}^{2}_{1})=1-\varepsilon$, and ${\mathrm{tr}}(\tilde{\varrho}^{1}_{1}\tilde{M}^{2}_{1})={\mathrm{tr}}(\tilde{\varrho}^{2}_{1}\tilde{M}^{1}_{1})=\frac{1}{2}+\sqrt{\varepsilon-\varepsilon^{2}}$ and the latter is just the same as the target probability, ${\mathrm{tr}}(\varrho^{1}_{1}\varrho^{2}_{1})$. Finally, calculation of the overlap between the states yields: $\displaystyle|{\mathrm{tr}}(\varrho^{1}_{1}\varrho^{2}_{1})-{\mathrm{tr}}(\tilde{\varrho}^{1}_{1}\tilde{\varrho}^{2}_{1})|=\sqrt{\varepsilon-\varepsilon^{2}},$ (26) which confirms the scaling of the first order of $\sqrt{\varepsilon}$. Now, let us show that the bounds from Eq. (3) are also tight in the first order of $d$. We still consider the case of $n=2$, but now $d$ is arbitrary. Let us consider the following POVM: $M_{1}=|1\rangle\langle 1|+\varepsilon(d-1)|+\rangle\langle+|\ ,\ M_{a}=(|a\rangle-\delta|+\rangle)(\langle a|-\delta\langle+|)\ a=2,3,\dots,d\ ,$ (27) where $\delta=\frac{1}{\sqrt{d-1}}+\sqrt{\frac{1}{d-1}-\varepsilon}$, $\\{|a\rangle\\}_{a=1}^{d}$ is the computational basis of $\mathbb{C}^{d}$ and $|+\rangle=\frac{1}{\sqrt{d-1}}\sum_{a=2}^{d}|a\rangle$ is the "maximally coherent" state in the subspace spanned by $\\{|i\rangle\\}_{i=2}^{d}$. In the proof of Theorem 1 the quantity $|{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})-{\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})|$ is upper-bounded by $\varepsilon+\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}-\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|$, for which there exist a state $\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}$ and an effect $\tilde{M}^{x^{\prime}}_{a^{\prime}}$ reaching the bound. Now if we take $\mathbf{\tilde{M}}^{x}$ to be the POVM we just introduced, and $\tilde{\varrho}^{x}_{1}=|1\rangle\langle 1|$, $\tilde{\varrho}^{x}_{a}=\frac{M_{a}}{{\mathrm{tr}}(M_{a})}$, $a=2,3,\dots,d$, the conditions of the Theorem 1 will be satisfied and the resulting bound will be $d\varepsilon$. ## Appendix B Quantitative statement of the robust self-testing In this part we first formulate Theorem 3 in which we present robustness of our self-testing scheme in terms of trace distance222Recall that the trace distance is defined via $\mathrm{d}_{\mathrm{tr}}(\sigma,\varrho)=\frac{1}{2}\left|\left|\hskip 1.0pt\sigma-\varrho\hskip 1.0pt\right|\right|_{1}=\frac{1}{2}{\mathrm{tr}}\sqrt{(\sigma-\varrho)^{\dagger}(\tilde{\varrho}-\varrho)}$ and it has a neat operational interpretation in term of optimal success probability $p$ of distinguishing $\varrho$ and $\sigma$ via the most general quantum measurements: $p=\frac{1}{2}(1+\mathrm{d}_{\mathrm{tr}}(\sigma,\varrho))$. for states and operator norm for measurements. Then, we give a quantitative statement of Theorem 2, that concerns robustness of our self-testing scheme in terms of average fidelities. We then proceed with proofs of both results. In what follows we will need the following definition. ###### Definition 1. Let $\\{\mathbf{n}_{i}\\}_{i\in[n]}$ be a set of $n$ vectors in $\mathbb{R}^{l}$. Matrix $\Gamma\in\mathbb{R}^{n\times n}$ is called the Gram matrix of the set $\\{\mathbf{n}_{i}\\}_{i\in[n]}$, if its elements are given by $\Gamma_{i,j}=\mathbf{n}_{i}\cdot\mathbf{n}_{j}$, $i,j\in[n]$. ###### Theorem 3 (Robust self-testing for qubits via trace distance and operator norm). Consider pure target qubit preparation states $\varrho^{x}_{a}$ and target projective measurements $\mathbf{M}^{y}$, where $a=1,2$ and $x,y\in[n]$. Assume that $\varrho^{x}_{a}=M^{x}_{a}$ for all $a,x$ and, furthermore, that experimental states $\tilde{\varrho}^{x}_{a}$ and measurements $\mathbf{\tilde{M}}^{y}$ act on Hilbert space of dimension at most $d$ and generate statistics $\tilde{p}(b|a,x,y)={\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{M}^{y}_{b})$ such that $|\tilde{p}(b|a,x,y)-{\mathrm{tr}}(\varrho^{x}_{a}M^{y}_{b})|\leq\varepsilon$, for all $a,b,x,y$. Let $k\in\\{{2,3\\}}$ be the cardinality of the maximal set of linearly independent Bloch vectors of states $\varrho^{x}_{1}$. Fix a set $S\subset[n]$ of $k$ linearly independent vectors $\\{\mathbf{n}^{x}_{1}\\}_{x\in S}$, construct their Gram matrix $\Gamma_{S}$, and let $L_{S}$ be a Cholesky factor of $\Gamma_{S}$ (i.e., $\Gamma_{S}=L_{S}L^{T}_{S}$ and $L_{S}$ is lower triangular). For every $x\in[n]\setminus S$ and both $a=1,2$ let $\mathbf{c}^{x,a}_{S}$, denote the coefficients of decomposition of $\mathbf{n}^{x}_{a}$ as a linear combination of $\\{\mathbf{n}^{x}_{1}\\}_{x\in S}$. Finally, let us define three auxiliary functions $\displaystyle F_{k}(\varepsilon)\coloneqq\sqrt{\varepsilon}\sqrt{4k(k-1)}\sqrt{1+2\sqrt{\varepsilon}+\frac{k+3}{k-1}\varepsilon}\ ,$ $\displaystyle O_{k}(\varepsilon)\coloneqq 2((k-1)\sqrt{\varepsilon}+(k+1)\varepsilon)\ ,$ (28) $\displaystyle E_{S,k}(\varepsilon)\coloneqq\frac{1}{2\sqrt{2}}\frac{\left|\left|\hskip 1.0pt\Gamma_{S}^{-1}\hskip 1.0pt\right|\right|F_{k}(\varepsilon)}{\sqrt{1-\left|\left|\hskip 1.0pt\Gamma_{S}^{-1}\hskip 1.0pt\right|\right|O_{k}(\varepsilon)}}\min\left[\frac{\left|\left|\hskip 1.0pt\Gamma_{S}\hskip 1.0pt\right|\right|}{\left|\left|\hskip 1.0pt\Gamma_{S}\hskip 1.0pt\right|\right|_{F}},\frac{\left|\left|\hskip 1.0ptL_{S}\hskip 1.0pt\right|\right|}{\sqrt{k}}\right]\ .$ Then, there is a region of $\varepsilon\in[0,\varepsilon_{0})$ determined by $\displaystyle\left|\left|\hskip 1.0pt\Gamma_{S}^{-1}\hskip 1.0pt\right|\right|O_{k}(\varepsilon)\leq 1,$ (29) for which there exist a qubit unitary matrix $U$ such that $\displaystyle\frac{1}{k}\sum_{x\in S}\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{1})^{(T)}U^{\dagger},\varrho^{x}_{1})\leq E_{S,k}(\varepsilon)\ ,$ (30a) $\displaystyle\frac{1}{k}\sum_{x\in S}\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{2})^{(T)}U^{\dagger},\varrho^{x}_{2})\leq E_{S,k}(\varepsilon)+2\sqrt{\varepsilon}\ ,$ (30b) $\displaystyle\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{a})^{(T)}U^{\dagger},\varrho^{x}_{a})\leq\sqrt{k}{|\mathbf{c}^{a,x}_{S}|E_{S,k}(\varepsilon)}+\frac{\sqrt{k}}{2}\left(|\mathbf{c}^{a,x}_{S}|+\frac{\sqrt{k}}{k-1}\right)\frac{\left|\left|\hskip 1.0pt\Gamma_{S}^{-1}\hskip 1.0pt\right|\right|O_{k}(\varepsilon)}{1-\left|\left|\hskip 1.0pt\Gamma_{S}^{-1}\hskip 1.0pt\right|\right|O_{k}(\varepsilon)},$ (30c) $\displaystyle\text{for}\;x\notin S,\ a=1,2\ ,$ $\displaystyle\frac{1}{k}\sum_{y\in S}\left|\left|\hskip 1.0ptU(\tilde{M}^{y}_{1})^{(T)}U^{\dagger}-M^{y}_{1}\hskip 1.0pt\right|\right|\leq E_{S,k}(\varepsilon)+\sqrt{\varepsilon}\ ,$ (30d) $\displaystyle\left|\left|\hskip 1.0ptU(\tilde{M}^{y}_{1})^{(T)}U^{\dagger}-M^{y}_{1}\hskip 1.0pt\right|\right|\leq\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{y}_{1})^{(T)}U^{\dagger},\varrho^{y}_{1})+\sqrt{\varepsilon},\quad\text{for}\;y\notin S\ ,$ (30e) where $(\cdot)^{(T)}$ is the transposition (with respect a fixed basis in $\mathbb{C}^{2}$) that may have to be applied to all experimental states and measurements simultaneously. The reason for such a formulation of the theorem lays in the proof techniques that were used to derive it. Namely, we use results on the stability of the Cholesky factorization. Since the result employed by us (Ref. [37]) is valid for positive-definite matrices we cannot apply it to the Gram matrix of all of the vectors $\mathbf{n}^{x}_{a}$ due to their linear dependence. As a result, our bounds depend on the particular choice of the set $S$ of states whose Bloch vectors are linearly independent. In what follows, without the loss of generality we take $S=\\{1,2\\}$ for $k=2$ and $S=\\{1,2,3\\}$ for $k=3$ in the proof. In practice, however, one should consider all subsets $S$ of states that give non-singular Gram matrices having in mind a specific application. _Outline of the proof.–_ In what follows we will consider a particular subset $\\{\mathbf{n}^{x}_{1}\\}_{x\in S}$ of $k$ linearly independent Bloch vectors. For simplify we will omit the subscript $S$ whenever possible. The main idea of the proof is to use a Cholesky factors of $k\times k$ the Gram matrices $\Gamma$ and $\tilde{\Gamma}$ of target ($\\{\mathbf{n}^{x}_{1}\\}_{x\in S}$) and experimental ($\\{\mathbf{\tilde{n}}^{x}_{1}\\}_{x\in S}$) Bloch vectors respectively. We then make use of the result of Ref. [37] on the stability of Cholesky factorization which, in simple terms, states that if $\Gamma$ and $\tilde{\Gamma}$ are close to each other, so are their Cholesky factors $L$ and $\tilde{L}$. Specifically, this result, which we quote bellow (see Theorem [Sun 1991]), sets an upper bound on the Frobenius norm of $\Delta L=\tilde{L}-L$ in terms of the Frobenius norm of the perturbation $\Delta\Gamma=\tilde{\Gamma}-\Gamma$. The Frobeniuous norm of the perturbation $\Delta L$ can be connected to the trace distance between states $\varrho^{x}_{1}$ and the rotated states $\tilde{\varrho}^{x}_{1}$ in the selected subset $S$. On the other hand, the bound on $\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|_{F}$ can be estimated from our assumption: $|p(b|a,x,y)-\tilde{p}(b|a,x,y)|\leq\varepsilon$. This bound follows directly from the results of Theorem 1, but here we use an improved qubit stability bounds, which are given by Lemma 1. This, in turn, leads to stronger estimates for the norms of $\Delta\Gamma$ in Lemma 2 bellow. Combining all these results produces bounds in Eq. (30a). In the next part of the proof we determine the trace distance between states $\varrho^{x}_{2}$ and $\tilde{\varrho}^{x}_{2}$ for $x\in S$ based solely on the fact that $\varrho^{x}_{2}=\mathbbm{1}-\varrho^{x}_{1}$ and $\tilde{\varrho}^{x}_{2}$ are _close to_ $\mathbbm{1}-\tilde{\varrho}^{x}_{1}$. This gives the bounds in Eq. (30b). For states $x\notin S$ we use the fact that they can be decomposed in the basis of the states in $S$. Since this linear decomposition can be different for target and experimental states, we use the result on stability of linear systems (see Theorem [Higham 2002], which we also quote below). The bounds for the states $x\notin S$ are given by Eq. (30c). Finally, we connect the bounds for the distance between the target and experimental measurements and the distance between the corresponding states resulting in Eq. (30d) and (30e). ###### Theorem 2 (Quantitative formulation of robust self-testing for qubits). Consider pure target qubit preparation states $\varrho^{x}_{a}$ and target projective measurements $\mathbf{M}^{y}$, where $a=1,2$ and $x,y\in[n]$. Assume that $\varrho^{x}_{a}=M^{x}_{a}$ for all $a,x$ and, furthermore, that experimental states $\tilde{\varrho}^{x}_{a}$ and measurements $\mathbf{\tilde{M}}^{y}$ act on Hilbert space of dimension at most $d$ and generate statistics $\tilde{p}(b|a,x,y)={\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{M}^{y}_{b})$ such that $|\tilde{p}(b|a,x,y)-{\mathrm{tr}}(\varrho^{x}_{a}M^{y}_{b})|\leq\varepsilon$, for all $a,b,x,y$. Let $k\in\\{{2,3\\}}$ be the cardinality of the maximal set of linearly independent Bloch vectors of states $\varrho^{x}_{1}$. Fix a set $S\subset[n]$ of $k$ linearly independent vectors $\\{\mathbf{n}^{x}_{1}\\}_{x\in S}$, construct their Gram matrix $\Gamma_{S}$, and let $L_{S}$ be a Cholesky factor of $\Gamma_{S}$ (i.e., $\Gamma_{S}=L_{S}L^{T}_{S}$ and $L_{S}$ is lower triangular). For every $x\in[n]\setminus S$ and both $a=1,2$ let $\mathbf{c}^{x,a}_{S}$, denote the coefficients of decomposition of $\mathbf{n}^{x}_{a}$ as a linear combination of $\\{\mathbf{n}^{x}_{1}\\}_{x\in S}$. Finally, let us define three auxiliary functions $\displaystyle F_{k}(\varepsilon)\coloneqq\sqrt{\varepsilon}\sqrt{4k(k-1)}\sqrt{1+2\sqrt{\varepsilon}+\frac{k+3}{k-1}\varepsilon}\ ,$ $\displaystyle O_{k}(\varepsilon)\coloneqq 2((k-1)\sqrt{\varepsilon}+(k+1)\varepsilon)\ ,$ $\displaystyle E_{S,k}(\varepsilon)\coloneqq\frac{1}{2\sqrt{2}}\frac{\left|\left|\hskip 1.0pt\Gamma_{S}^{-1}\hskip 1.0pt\right|\right|F_{k}(\varepsilon)}{\sqrt{1-\left|\left|\hskip 1.0pt\Gamma_{S}^{-1}\hskip 1.0pt\right|\right|O_{k}(\varepsilon)}}\min\left[\frac{\left|\left|\hskip 1.0pt\Gamma_{S}\hskip 1.0pt\right|\right|}{\left|\left|\hskip 1.0pt\Gamma_{S}\hskip 1.0pt\right|\right|_{F}},\frac{\left|\left|\hskip 1.0ptL_{S}\hskip 1.0pt\right|\right|}{\sqrt{k}}\right]\ .$ (31) Then, there is a region of $\varepsilon\in[0,\varepsilon_{0})$ determined by $\displaystyle\left|\left|\hskip 1.0pt\Gamma_{S}^{-1}\hskip 1.0pt\right|\right|O_{k}(\varepsilon)\leq 1,$ (32) for which there exist a qubit unitary matrix $U$ such that $\displaystyle\frac{1}{k}\sum_{x\in S}{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{1})^{(T)}U^{\dagger}\varrho^{x}_{1})\geq 1-\frac{\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}-E_{S,k}(\varepsilon)^{2}\ ,$ (33a) $\displaystyle\frac{1}{k}\sum_{x\in S}{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{2})^{(T)}U^{\dagger}\varrho^{x}_{2})\geq 1-\frac{\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}-(E_{S,k}(\varepsilon)+2\sqrt{\varepsilon})^{2}\ ,$ (33b) $\displaystyle{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{a})^{(T)}U^{\dagger}\varrho^{x}_{a})\geq 1-\frac{\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}-\left(\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{a})^{(T)}U^{\dagger},\varrho^{x}_{a})\right)^{2},\quad\text{for}\;x\notin S,\ a=1,2\ ,$ (33c) $\displaystyle\frac{1}{k}\sum_{y\in S}{\mathrm{tr}}(U(\tilde{M}^{y}_{1})^{(T)}U^{\dagger}M^{y}_{1})\geq 1-\frac{5}{2}\varepsilon-\left(\sqrt{2}E_{S,k}(\varepsilon)+\sqrt{\varepsilon}\right)^{2}\ ,$ (33d) $\displaystyle{\mathrm{tr}}(U(\tilde{M}^{y}_{1})^{(T)}U^{\dagger}M^{y}_{1})\geq 1-\varepsilon-\left(\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{y}_{1})^{(T)}U^{\dagger},\varrho^{y}_{1})+\sqrt{\varepsilon}\right)^{2},\quad\text{for}\;y\notin S\ ,$ (33e) where $(\cdot)^{(T)}$ is the transposition (with respect a fixed basis in $\mathbb{C}^{2}$) that may have to be applied to all experimental states and measurements simultaneously. Note that in formulas (33c) and (30e) we have used trace distance $\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{a})^{(T)}U^{\dagger},\varrho^{x}_{a})$ in order to simplify the resulting formulas. The bound on this quantity is given in Eq. (30c). ###### Remark. Results of Theorem 2 are obtained using a similar reasoning to that given in the outline of the proof of Theorem 3. However, the proof steps are supplemented by bounds connecting fidelity to the trace distance (for states) and operator norm (for measurement operators). The proof of Theorem 2 is given at the end of this part of the Appendix. Before we proceed we state two theorems from the literature that are needed for our proof. First, we repeat the statement of the Theorem 1.4 from Ref. [37] for real-valued matrices. Second, we state a result concerning stability of systems of linear equation, which we borrow from Ref. [45] (Theorem 7.2). We changed the notation used in these papers in accordance with our paper. ###### Theorem (Sun 1991 - Stability of Cholesky factorisation). Let $\Gamma$ be an $k\times k$ positive definite matrix and $\Gamma=LL^{T}$ its Cholesky factorization. If $\Delta\Gamma$ is an $k\times k$ symmetric matrix satisfying $\displaystyle\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right|\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|\leq 1,$ (34) then there is a unique Cholesky factorization $\displaystyle\Gamma+\Delta\Gamma=(L+\Delta L)(L+\Delta L)^{T},$ and $\displaystyle\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}\leq\frac{\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right|\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|_{F}}{\sqrt{2(1-\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right|\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|})}\min\left[\frac{\left|\left|\hskip 1.0ptL\hskip 1.0pt\right|\right|_{F}\left|\left|\hskip 1.0pt\Gamma\hskip 1.0pt\right|\right|}{\left|\left|\hskip 1.0pt\Gamma\hskip 1.0pt\right|\right|_{F}},\left|\left|\hskip 1.0ptL\hskip 1.0pt\right|\right|\right].$ (35) ###### Remark. This theorem dates back to 1991, and, of course there have been attempts to improve this result. However, the recent review [46] on this topic suggests that the bound given in the Theorem above is the most appropriate in our case (Remark 3.2, Ref. [46]). ###### Theorem (Higham 2002 - Stability of systems of linear equations). Let $\mathbf{c}$ be a solution of a system of linear equations $\Gamma\mathbf{c}=\mathbf{g}$, where $\mathbf{c},\mathbf{g}\in\mathbb{R}^{k}$ and $\Gamma\in\mathrm{Mat}_{k\times k}(\mathbb{R})$. Let now $\mathbf{\tilde{c}}$ be a solution to $(\Gamma+\Delta\Gamma)\mathbf{\tilde{c}}=\mathbf{g}+\mathbf{\Delta g}$, where $\Delta\Gamma\in\mathrm{Mat}_{k\times k}(\mathbb{R})$, $\mathbf{\Delta g}\in\mathbb{R}^{k}$. Assume that there exits $\delta^{\prime}>0$ such that $\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|\leq\delta^{\prime}\left|\left|\hskip 1.0ptE\hskip 1.0pt\right|\right|$ and $|\mathbf{\Delta g}|\leq\delta^{\prime}|\mathbf{f}|$ (for some $E\in\mathrm{Mat}_{k\times k}(\mathbb{R})$, $\mathbf{f}\in\mathbb{R}^{k}$), and that $\delta^{\prime}\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right|\left|\left|\hskip 1.0ptE\hskip 1.0pt\right|\right|\leq 1$. Then, we have $\displaystyle\frac{|\mathbf{c}-\mathbf{\tilde{c}}|}{|\mathbf{c}\,|}\leq\frac{\delta^{\prime}}{1-\delta^{\prime}\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right|\left|\left|\hskip 1.0ptE\hskip 1.0pt\right|\right|}\left(\frac{\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right||\mathbf{f}|}{|\mathbf{c}\,|}+\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right|\left|\left|\hskip 1.0ptE\hskip 1.0pt\right|\right|\right)\ .$ (36) ###### Lemma 1. Under the conditions of Theorem 1 qubit states and measurements $\\{\varrho^{x}_{a}\\}_{a,x}$, $\\{\mathbf{M}^{y}\\}_{y}$ and $\\{\tilde{\varrho}^{x}_{a}\\}_{a,x}$, $\\{\mathbf{\tilde{M}}^{y}\\}_{y}$ satisfy $\displaystyle\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}\hskip 1.0pt\right|\right|\geq\frac{1-2\varepsilon}{1-\varepsilon},\;\forall a,x,$ $\displaystyle|{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})-{\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})|\leq\varepsilon+\sqrt{\varepsilon},$ $\displaystyle|{\mathrm{tr}}(\tilde{M}^{y}_{b}\tilde{M}^{y^{\prime}}_{b^{\prime}})-{\mathrm{tr}}(M^{y}_{b}M^{y^{\prime}}_{b^{\prime}})|\leq\varepsilon+(1+\varepsilon)\sqrt{\varepsilon},$ $\forall x\neq x^{\prime},a\neq a^{\prime}$ and $\forall y\neq y^{\prime},b\neq b^{\prime}$ respectively, and $\varepsilon\leq\frac{1}{3}$. ###### Lemma 2. Let $\Gamma$ be the Gram matrix of Bloch vectors of target states $\\{\varrho^{x}_{1}\\}_{x\in S}$. Likewise, let $\tilde{\Gamma}$ be the Gram matrix of Bloch vectors of experimental states $\\{\tilde{\varrho}^{x}_{1}\\}_{x\in S}$. Let $|p(b|a,x,y)-\tilde{p}(b|a,x,y)|\leq\varepsilon$ for all $b,a,x,y$ and let $k=|S|$ be the cardinality of the set $S$. Then, we have the following bounds on norms of $\Delta\Gamma=\tilde{\Gamma}-\Gamma$ $\displaystyle\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|_{F}\leq F_{k}(\varepsilon)\ ,$ $\displaystyle\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|\leq O_{k}(\varepsilon)\ ,$ (37) where $\displaystyle F_{k}(\varepsilon)=\sqrt{\varepsilon}\sqrt{4k(k-1)}\sqrt{1+2\sqrt{\varepsilon}+\frac{k+3}{k-1}\varepsilon}\ ,$ $\displaystyle O_{k}(\varepsilon)=2((k-1)\sqrt{\varepsilon}+(k+1)\varepsilon)\ .$ ###### Proof of Theorem 3. Let $\tilde{\Gamma}$ be the Gram matrix of Bloch vectors $\\{\mathbf{\tilde{n}}^{x}_{1}\\}_{x\in S}$ of the experimental states for $x\in S$. Let us for now assume that the Cholesky factorization $\tilde{\Gamma}=\tilde{L}\tilde{L}^{T}$ exists and let $\Delta L=\tilde{L}-L$. We start our _first part of the proof_ regarding the states $x\in S$, $a=1$ by connecting the square of the norm $\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}$ with trace distance between the states $\\{\tilde{\varrho}^{x}_{1}\\}_{x\in S}$ and $\\{\varrho^{x}_{1}\\}_{x\in S}$. From the sketch of the proof of Theorem 3 it follows that the Bloch vectors $\\{\mathbf{n}^{x}_{1}\\}_{x\in S}$ of the target states and the vectors $\\{\mathbf{l}^{x}\\}_{x\in S}$, transpose of which are rows of $L$, are connected via an orthogonal transformation, which we denote as $O$, i.e., $\mathbf{l}^{x}=O\mathbf{n}^{x}_{1}$, $x\in S$. Similarly, for the Bloch vectors $\\{\mathbf{\tilde{n}}^{x}_{1}\\}_{x\in S}$ of the experimental states and the vectors $\\{\mathbf{\tilde{l}}^{x}\\}_{x\in S}$ that form $\tilde{L}$ there exists an orthogonal transformation $\tilde{O}$, such that $\mathbf{\tilde{l}}^{x}=\tilde{O}\mathbf{\tilde{n}}^{x}_{1}$, $x\in S$. Let us define states $\tau^{x}=\frac{1}{2}(\mathbbm{1}+\mathbf{l}^{x}\cdot\bm{\sigma})$, and $\tilde{\tau}^{x}=\frac{1}{2}(\mathbbm{1}+\mathbf{\tilde{l}}^{x}\cdot\bm{\sigma})$, $x\in S$. We know that there exist unitary transformations $V$ and $\tilde{V}$ such that: $\tau^{x}=V(\varrho^{x}_{1})^{(T)}V^{\dagger}$, and $\tilde{\tau}^{x}=\tilde{V}(\tilde{\varrho}^{x}_{1})^{(T)}\tilde{V}^{\dagger}$ for $x\in S$ (the optional transposition $(\cdot)^{(T)}$ is reserved for the cases $\det(O)=-1$ and $\det(\tilde{O})=-1$). We have the following sequence of equalities valid for $x\in S$: $\displaystyle\mathrm{d}_{\mathrm{tr}}(\tilde{\tau}^{x},\tau^{x})$ $\displaystyle=$ $\displaystyle\mathrm{d}_{\mathrm{tr}}(\tilde{V}(\tilde{\varrho}^{x}_{1})^{(T)}\tilde{V}^{\dagger},V(\varrho^{x}_{1})^{(T)}V^{\dagger})=\mathrm{d}_{\mathrm{tr}}(V^{\dagger}\tilde{V}(\tilde{\varrho}^{x}_{1})^{(T)}\tilde{V}^{\dagger}V,(\varrho^{x}_{1})^{(T)})$ (38) $\displaystyle=$ $\displaystyle\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{1})^{(T)}U^{\dagger},\varrho^{x}_{1})\ ,$ where we used the invariance of trace distancee under unitary evolution and transposition, and also denoted the resulting unitary $V^{\dagger}\tilde{V}$ as $U$. It is clear that one transposition in the final formula in Eq. (38) is enough as the case $\det(\tilde{O})=\det(O)=-1$ is equivalent to having $\det(\tilde{O})=\det(O)=1$ and changing $U$ to $U^{T}$. We need to show now that this unitary $U$ satisfies the claim of Theorem 2 for the states $x\in S$. For qubits, trace distance can be expressed directly via Eucleedian distance between Bloch vectors: $\displaystyle\mathrm{d}_{\mathrm{tr}}(\tilde{\tau}^{x},\tau^{x})=\frac{1}{2}|\mathbf{\tilde{l}}^{x}-\mathbf{l}^{x}|\ .$ (39) Using this fact and Eq. (38), it is straightforward to derive the following upper bound: $\displaystyle\frac{1}{k}\sum_{x=1}^{k}\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{1})^{(T)}U^{\dagger},\varrho^{x}_{1})\leq\frac{1}{2\sqrt{k}}\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}\ .$ (40) We now use Theorem [Sun 1991] (Ref. [37]) to upper-bound $\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}$. Specifically, we apply it to the Gram matrix of the Bloch vectors of states $\\{\varrho^{x}_{1}\\}_{x\in S}$. We have $\left|\left|\hskip 1.0ptL\hskip 1.0pt\right|\right|_{F}=\sqrt{k}$, since the target states are assumed to be pure. The direct substitution of the bound in Eq. (35) into Eq. (40) gives the bound in Eq. (30a), and also the condition in Eq. (29) when the Theorem 3 (and also Theorem 2) applies. In the beginning of the proof we assumed that the Cholesky factorization $\tilde{\Gamma}=\tilde{L}\tilde{L}^{T}$ exists. The condition in Eq. (29) gives the sufficient condition for this to hold. We conclude this part of the proof by noting that a similar statement in terms of fidelity (Theorem 2, Eq. (33a)) follows from Eq. (40) by connecting fidelity to the trace distance as described in Eq. (54). In _the second part of the proof_ we derive upper bounds on the trace distances between the states corresponding to $x\in S$ and $a=2$. As we will see, those can be connected to the bounds for the states with $x\in S$ and $a=1$. Indeed, we can write the following for every $x\in S$: $\displaystyle 2\mathrm{d}_{\mathrm{tr}}(U\tilde{\varrho}^{x}_{2}U^{\dagger},\varrho^{x}_{2})$ $\displaystyle=\left|\left|\hskip 1.0pt\varrho^{x}_{2}-U\tilde{\varrho}^{x}_{2}U^{\dagger}\hskip 1.0pt\right|\right|_{1}\leq\left|\left|\hskip 1.0pt\mathbbm{1}-\varrho^{x}_{1}-U(\mathbbm{1}-\tilde{\varrho}^{x}_{1})U^{\dagger}\hskip 1.0pt\right|\right|_{1}+\left|\left|\hskip 1.0ptU(\mathbbm{1}-\tilde{\varrho}^{x}_{1}-\tilde{\varrho}^{x}_{2})U^{\dagger}\hskip 1.0pt\right|\right|_{1}$ $\displaystyle\leq\left|\left|\hskip 1.0pt\varrho^{x}_{1}-U\tilde{\varrho}^{x}_{1}U^{\dagger}\hskip 1.0pt\right|\right|_{1}+\left|\left|\hskip 1.0pt\mathbbm{1}-\tilde{\varrho}^{x}_{1}+\tilde{M}^{x}_{2}\hskip 1.0pt\right|\right|_{1}-\left|\left|\hskip 1.0pt\tilde{M}^{x}_{2}-\tilde{\varrho}^{x}_{2}\hskip 1.0pt\right|\right|_{1}$ $\displaystyle=2\mathrm{d}_{\mathrm{tr}}(U\tilde{\varrho}^{x}_{1}U^{\dagger},\varrho^{x}_{1})+\sum_{a=1,2}\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}-\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|_{1}.$ Exactly the same reasoning can be applied to upper bound the trace distance $\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{2})^{T}U^{\dagger},\varrho^{x}_{2})$, in which case the transposition will propagate to $\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{1})^{T}U^{\dagger},\varrho^{x}_{1})$ but would not affect the term $\sum_{a=1,2}\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}-\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|_{1}$ (as we can take $(\tilde{M}^{x}_{a})^{T}$). To finish the calculations we need to upper-bound the second summand in Eq. (B). We do it by writing $\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}-\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|_{1}\leq 2\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}-\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|\leq 2\sqrt{\varepsilon}$, $\forall a,x$, where the last inequality is proven in Lemma 1 (see Eq. (77)). From here, it is easy to obtain the bound in Eq. (30b). The reason why we do not simply apply the same reasoning to the states corresponding to $a=2$ and $x\in S$ as for the states with $a=1$ is because we want the isometry $U$ to be the same for all of the states. In _the third part of the proof_ we derive the bounds for the preparations states corresponding to $x\notin S$ and both $a=1,2$. We start by reminding ourselves that the set of states $\\{\varrho^{x}_{1}\\}_{x\in S}$ is assumed to be tomographically complete. Is is equivalent to the assumption that the vectors $\\{\mathbf{l}^{x}\\}_{x\in S}$, defined above, are linearly independent and span $\mathbb{R}^{3}$ for $k=3$, or the considered subspace for $k=2$. The condition in Eq. (29) of the Theorem 3 (also stated in Eq. (34)) ensures that the same holds for the vectors $\\{\mathbf{\tilde{l}}^{x}\\}_{x\in S}$. If so, let us expand the Bloch vectors of the states $\varrho^{x^{\prime}}_{a^{\prime}}$ and $U\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}U^{\dagger}$, $x^{\prime}\notin S$ in terms of $\\{\mathbf{l}^{x}\\}_{x\in S}$ and $\\{\mathbf{\tilde{l}}^{x}\\}_{x\in S}$ respectively. Let us denote the coefficients of these linear expansions as $\\{c^{a^{\prime},x^{\prime}}_{x}\\}_{x\in S}$ and $\\{\tilde{c}^{a^{\prime},x^{\prime}}_{x}\\}_{x\in S}$ respectively. These coefficients will, of course, depend on $x^{\prime}$ and $a^{\prime}$ as well as on the choice of the set $S$. However, for simplicity of the derivations we are going omit the subscripts $x^{\prime}$ and $a^{\prime}$ until we present the final result. We will also assume, without loss of generality, that $S=\\{1,2,3\\}$ (or $S=\\{1,2\\}$ for $k=2$). It is clear that the coefficients $\\{c_{x}\\}_{x\in S}$ and $\\{\tilde{c}_{x}\\}_{x\in S}$ satisfy the following respective systems of linear equations: $\displaystyle\Gamma\mathbf{c}=\mathbf{g},\quad\tilde{\Gamma}\mathbf{\tilde{c}}=\mathbf{\tilde{g}},$ (42) where $\mathbf{c}=(c_{1},c_{2},c_{3})$, $\mathbf{g}=(2{\mathrm{tr}}(\varrho^{x^{\prime}}_{a^{\prime}}\varrho^{1}_{1})-1,2{\mathrm{tr}}(\varrho^{x^{\prime}}_{a^{\prime}}\varrho^{2}_{1})-1,2{\mathrm{tr}}(\varrho^{x^{\prime}}_{a^{\prime}}\varrho^{3}_{1})-1)$ and analogously $\mathbf{\tilde{c}}=(\tilde{c}_{1},\tilde{c}_{2},\tilde{c}_{3})$, $\mathbf{g}=(2{\mathrm{tr}}(\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}\tilde{\varrho}^{1}_{1})-1,2{\mathrm{tr}}(\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}\tilde{\varrho}^{2}_{1})-1,2{\mathrm{tr}}(\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}\tilde{\varrho}^{3}_{1})-1)$ whenever the cardinality $k$ of the set $S$ is $3$. For $k=2$ and $\mathbf{c},\mathbf{\tilde{c}},\mathbf{g},\mathbf{\tilde{g}}\in\mathbb{R}^{2}$ their definition is analogous. We again omitted the subscripts $a^{\prime},x^{\prime}$ for the vectors $\mathbf{g}$ and $\mathbf{\tilde{g}}$ for simplicity. The matrices $\Gamma$ and $\tilde{\Gamma}$ are still the moment matrices for the states $x\in S$, $a=1$. In complete analogy to Eq. (39) we can write: $\displaystyle\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})^{(T)}U^{\dagger},\varrho^{x^{\prime}}_{a^{\prime}})=\frac{1}{2}\Big{|}\sum_{x\in S}(\tilde{c}_{x}\mathbf{\tilde{l}}^{x}-c_{x}\mathbf{l}^{x})\Big{|}.$ (43) We then can upper-bound the latter norm as follows: $\displaystyle\Big{|}\sum_{x\in S}(\tilde{c}_{x}\mathbf{\tilde{l}}^{x}-c_{x}\mathbf{l}^{x})\Big{|}\leq\Big{|}\sum_{x\in S}c_{x}(\mathbf{\tilde{l}}^{x}-\mathbf{l}^{x})\Big{|}+\Big{|}\sum_{x\in S}(\tilde{c}_{x}-c_{x})\mathbf{\tilde{l}}^{x}\Big{|}\leq|\mathbf{c}\,|\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}+\sum_{x\in S}|\tilde{c}_{x}-c_{x}||\mathbf{\tilde{l}}^{x}|.$ In the equation above we used the following relation: $\displaystyle|\sum_{x\in S}c_{x}(\mathbf{\tilde{l}}^{x}-\mathbf{l}^{x})|=\sqrt{\sum_{i=1}^{3}\left(\sum_{x\in S}c_{x}(\tilde{l}^{x}_{i}-l^{x}_{i})\right)^{2}}\leq\sqrt{\sum_{x^{\prime}\in S}c^{2}_{x^{\prime}}\sum_{i=1}^{3}\sum_{x\in S}(\tilde{l}^{x}_{i}-l^{x}_{i})^{2}}=|\mathbf{c}\,|\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}.$ (45) Also, since the norms $|\mathbf{\tilde{l}}^{x}|$ can be upper-bounded by $1$ for all $x$, the resulting upper bound can be simplified further to be $|\mathbf{c}\,|\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}+\sum_{x\in S}|\tilde{c}_{x}-c_{x}|$. To estimate the deviation $|\tilde{c}_{x}-c_{x}|$ we apply Theorem [Higham 2002] by taking $\Delta\Gamma=\tilde{\Gamma}-\Gamma$ and $\mathbf{\Delta g}=\mathbf{\tilde{g}}-\mathbf{g}$. The bound on the operator norm $\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|$ is given by Lemma 2. At the same time $|\mathbf{\Delta g}|=2\sqrt{\sum_{x\in S}({\mathrm{tr}}(\varrho^{x^{\prime}}_{a^{\prime}}\varrho^{x}_{1})-{\mathrm{tr}}(\tilde{\varrho}^{x^{\prime}}_{a^{\prime}}\tilde{\varrho}^{x}_{1}))^{2}}\leq 2\sqrt{k}(\sqrt{\varepsilon}+\varepsilon)$. If we take $\delta^{\prime}=2((k-1)\sqrt{\varepsilon}+(k+1)\varepsilon)$, some matrix $E$ with $\left|\left|\hskip 1.0ptE\hskip 1.0pt\right|\right|=1$, vector $\mathbf{f}$ such that $|\mathbf{f}|=\frac{\sqrt{k}}{k-1}$, we satisfy the conditions of the Theorem [Higham 2002] and get the following bound: $\displaystyle|\mathbf{c}-\mathbf{\tilde{c}}|\leq\left(|\mathbf{c}|+\frac{\sqrt{k}}{k-1}\right)\frac{\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right|\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|}{1-\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right|\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|}\ ,$ (46) As the final step we need to connect the bound in Eq. (B) with the one in Eq. (46) by the relation between $1$-norm and the Euclidean norm in $\mathbb{R}^{k}$, which effectively adds a factor of $\sqrt{k}$ to the bound in Eq. (46). Combining everything together we obtain the following: $\displaystyle\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})^{(T)}U^{\dagger},\varrho^{x^{\prime}}_{a^{\prime}})\leq\frac{1}{2}|\mathbf{c}\,|\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}+\frac{\sqrt{k}}{2}\left(|\mathbf{c}|+\frac{\sqrt{k}}{k-1}\right)\frac{\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right|\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|}{1-\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right|\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|},$ $\displaystyle x^{\prime}\notin S,a^{\prime}=1,2.$ (47) The above bound is given in Eq. (30c) in terms of the quantity $E_{S,k}(\varepsilon)$ (Eq. (28)), where we took $\left|\left|\hskip 1.0pt\Delta L_{S}\hskip 1.0pt\right|\right|_{F}=2\sqrt{k}E_{S,k}(\varepsilon)$ and brought back all the necessary subscripts. Again, using the relation in Eq. (55) we can derive bounds on the fidelity between the states $\varrho^{x}_{a}$ and $\tilde{\varrho}^{x}_{a}$ for $x\notin S$. In _the fourth, final, part of the proof_ we derive the bounds for the measurements. We do it by connecting the distance between the experimental measurements and the distance between the corresponding states. Indeed, we can write the following: $\displaystyle\left|\left|\hskip 1.0ptU(\tilde{M}^{y}_{1})^{(T)}U^{\dagger}-M^{y}_{1}\hskip 1.0pt\right|\right|\leq\left|\left|\hskip 1.0ptU(\tilde{\varrho}^{y}_{1})^{(T)}U^{\dagger}-\varrho^{y}_{1}\hskip 1.0pt\right|\right|+\left|\left|\hskip 1.0pt\tilde{\varrho}^{y}_{1}-\tilde{M}^{y}_{1}\hskip 1.0pt\right|\right|,\quad\forall y,$ (48) where we used the triangle inequality and the invariance of the norm $\left|\left|\hskip 1.0pt\tilde{\varrho}^{y}_{1}-\tilde{M}^{y}_{1}\hskip 1.0pt\right|\right|$ under unitary transformations. Remembering that $\left|\left|\hskip 1.0pt\tilde{\varrho}^{y}_{1}-\tilde{M}^{y}_{1}\hskip 1.0pt\right|\right|\leq\sqrt{\varepsilon}$ (Lemma 1, Eq. (77)) and $\left|\left|\hskip 1.0ptU(\tilde{\varrho}^{y}_{1})^{(T)}U^{\dagger}-\varrho^{y}_{1}\hskip 1.0pt\right|\right|=\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{y}_{1})^{(T)}U^{\dagger},\varrho^{y}_{1})$, we produce the bounds in Eqs. (30d,30e). ∎ Finally, we proceed to the proof of Theorem 2, which gives a quantitative robustness analysis expressed in terms of average fidelity. ###### Proof of Theorem 2. Let us first state a useful identity between Frobenius distance and the fidelity, valid for an arbitrary pure states $\varrho$ and a Hermitian operator $X$: $\left|\left|\hskip 1.0ptX-\varrho\hskip 1.0pt\right|\right|^{2}_{F}=1+{\mathrm{tr}}(X^{2})-2{\mathrm{tr}}(X\varrho)\ .$ (49) We follow exactly the same steps that were given in the proof of Theorem 3. We assume that the reader is familiar with the notation introduced there. First, in order to derive the bound in Eq. (33a) we repeat the reasoning preceding Eq. (38). Analogously to the trace distance discussed there, we use the invariance of the fidelity under unitary transformations and transposition which gives us for all $x\in S=[k]$: ${\mathrm{tr}}(\tilde{\tau}^{x}\tau^{x})={\mathrm{tr}}(U(\tilde{\varrho}^{x}_{1})^{(T)}U^{\dagger}\varrho^{x}_{1})\ ,$ (50) where $U=V^{\dagger}\tilde{V}$. Using the above formula and standard algebra involving Pauli matrices we obtain: $\frac{1}{k}\sum_{x=1}^{k}{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{1})^{(T)}U^{\dagger}\varrho^{x}_{1})=\frac{1}{2}+\frac{1}{2k}\sum_{x=1}^{k}\mathbf{l}^{x}\cdot\mathbf{\tilde{l}}^{x}\ .$ (51) On the other hand, we have the following identity: $\displaystyle\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|^{2}_{F}=\sum_{x=1}^{k}|\mathbf{l}^{x}|^{2}+\sum_{x=1}^{k}|\mathbf{\tilde{l}}^{x}|^{2}-2\sum_{x=1}^{k}\mathbf{l}^{x}\cdot\mathbf{\tilde{l}}^{x}\ ,$ (52) where $|\cdot|$ is a standard Euclidean norm in $\mathbb{R}^{3}$. Using the identity $|\mathbf{\tilde{l}}^{x}|^{2}=2{\mathrm{tr}}((\tilde{\varrho}^{x}_{1})^{2})-1$ and the fact that target states are pure (which implies $|\mathbf{l}^{x}|=1$) we obtain: $\displaystyle\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|^{2}_{F}=2\sum_{x=1}^{k}{\mathrm{tr}}((\tilde{\varrho}^{x}_{1})^{2})-2\sum_{x=1}^{k}\mathbf{l}^{x}\cdot\mathbf{\tilde{l}}^{x}\ .$ (53) Inserting the above into Eq. (51) and using the fact that ${\mathrm{tr}}{(\tilde{\varrho}^{x}_{1})^{2}}\geq 1-\frac{2\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}$ (this follows straightforwardly from Lemma 2), we finally obtain: $\frac{1}{k}\sum_{x=1}^{k}{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{1})^{(T)}U^{\dagger}\varrho^{x}_{1})\geq 1-\frac{\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}-\frac{\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|^{2}_{F}}{4k}\ .$ (54) We complete the proof of Eq. (33a) by again employing, exactly as before, Theorem [Sun 1991] (Ref. [37]) in order to upper-bound $\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}$. Now, to derive Eq. (33b), which gives the bounds for the fidelity of states for $x\in[k]$ and $a=2$, we can use the following inequality which can be derived from Eq. (49) and from the relation ${\mathrm{tr}}{(\tilde{\varrho}^{x}_{1})^{2}}\geq 1-\frac{2\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}$: $\displaystyle{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{a})^{(T)}U^{\dagger}\varrho^{x}_{a})\geq 1-\frac{\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}-(\mathrm{d}_{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{a})^{(T)}U^{\dagger},\varrho^{x}_{a}))^{2}.$ (55) From Eqs. (40,B) it follows that: $\displaystyle\frac{1}{k}\sum_{x=1}^{k}(\mathrm{d}_{\mathrm{tr}}(\tilde{\varrho}^{x}_{2})^{(T)}U^{\dagger},\varrho^{x}_{2}))^{2}\leq\frac{1}{k}\sum_{x=1}^{k}(\mathrm{d}_{\mathrm{tr}}(\tilde{\varrho}^{x}_{1})^{(T)}U^{\dagger},\varrho^{x}_{1})+2\sqrt{\varepsilon})^{2}$ (56) $\displaystyle\leq\frac{\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|^{2}_{F}}{4k}+\frac{2\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}}{\sqrt{k}}\sqrt{\varepsilon}+4\varepsilon=\left(\frac{\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}}{2\sqrt{k}}+2\sqrt{\varepsilon}\right)^{2},$ which gives the desired bound form Eq. (33b): $\displaystyle\frac{1}{k}\sum_{x=1}^{k}{\mathrm{tr}}(U(\tilde{\varrho}^{x}_{2})^{(T)}U^{\dagger}\varrho^{x}_{2})\geq 1-\frac{\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}-\left(\frac{\left|\left|\hskip 1.0pt\Delta L\hskip 1.0pt\right|\right|_{F}}{2\sqrt{k}}+2\sqrt{\varepsilon}\right)^{2}\ .$ (57) The proof of the remaining bounds in Eqs. (33c,33d,33e) is straightforward and follows directly form the formula in Eq. (49). In particular, in order to prove Eq. (33c) we set $X=\tilde{\varrho}^{x}_{a}$ and $\varrho=\varrho^{x}_{a}$ in Eq. (49), use the inequality ${\mathrm{tr}}{(\tilde{\varrho}^{x}_{a})^{2}}\geq 1-\frac{2\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}$ and the relation $\left|\left|\hskip 1.0pt\sigma-\varrho\hskip 1.0pt\right|\right|^{2}_{F}=2\mathrm{d}_{\mathrm{tr}}(\varrho,\sigma)^{2}$, with latter being true for arbitrary qubits states $\varrho$ and $\sigma$. Derivation of Eq. (33d) is completely analogous to the one of Eq. (33b). Finally, for Eq. (33e), the reasoning is again analogous to Eq. (33c), where, after setting $X=\tilde{M}^{y}_{1}$ and $\varrho=\varrho^{y}_{1}$, it is additionally necessary to use a simple lower bound: ${\mathrm{tr}}((M^{y}_{1})^{2})\geq(1-\varepsilon)^{2}$, which follows from the conditions of Theorem 2. ∎ ## Appendix C Alternative bounds from Procrustes ###### Lemma 3 (Robust self-testing for qubits from Procrustes). Consider pure target qubit preparation states $\varrho^{x}_{a}$ and target projective measurements $\mathbf{M}^{y}$, where $a=1,2$ and $x,y\in[n]$. Assume that $\varrho^{x}_{a}=M^{x}_{a}$ for all $a,x$ and, furthermore, that experimental states $\tilde{\varrho}^{x}_{a}$ and measurements $\mathbf{\tilde{M}}^{y}$ act on Hilbert space of dimension at most $d$ and generate statistics $\tilde{p}(b|a,x,y)={\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{M}^{y}_{b})$ such that $|\tilde{p}(b|a,x,y)-{\mathrm{tr}}(\varrho^{x}_{a}M^{y}_{b})|\leq\varepsilon$, for all $a,b,x,y$. Let $\\{\varrho_{i}\\}_{i=1}^{m}$ be a subset of $m$ considered states among $\varrho^{x}_{a}$. Let $L$ be a matrix whose rows are the Bloch vectors of states $\varrho_{i}$, $i\in[m]$, and let $k\in\\{{2,3\\}}$ be its rank ($m\geq k$). Assume, without loss of generality, that for $k=2$ the third component of the Bloch vectors is $0$. In that case, truncate $L$ to the first two columns. Let us define two auxiliary functions: $\displaystyle P_{m}(\varepsilon,L)=\begin{cases}\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|F_{m}(\varepsilon)+\min\left[\frac{\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|F_{m}(\varepsilon)}{\sqrt{1-\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|^{2}F_{m}(\varepsilon)}},\sqrt{\sqrt{k}F_{m}(\varepsilon)}\right],&\text{if}\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|\sqrt{F_{m}(\varepsilon)}<1\\\ \left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|F_{m}(\varepsilon)+\sqrt{\sqrt{k}F_{m}(\varepsilon)}&\text{otherwise,}\end{cases}$ (58) and $\displaystyle F_{m}(\varepsilon)=\sqrt{4m(m-1)\varepsilon\left(1+2\sqrt{\varepsilon}+\frac{m+3}{m-1}\varepsilon\right)},$ (59) where $L^{\ddagger}=(LL^{T})^{-1}L^{T}$. There exist a unitary matrix $U$ such that: $\displaystyle\frac{1}{m}\sum_{i=1}^{m}{\mathrm{tr}}(\varrho_{i}U\tilde{\varrho}_{i}U^{\dagger})\geq 1-\frac{\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}-\frac{1}{4m}P^{2}_{m}(\varepsilon,L).$ (60) In this section we derive alternative bounds for the fidelity between preparation states that follows from the bounds on the so-called orthogonal Procrustes problem [47]. The problem itself can be formulated as follows. Given two sets of vectors $\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{m}$ and $\mathbf{y}_{1},\mathbf{y}_{2},\dots,\mathbf{y}_{m}$ in $\mathbb{R}^{d}$ find an orthogonal transformation $O\in\mathrm{O}(d)$ in $\mathbb{R}^{d}$ that minimizes $\sum_{i=1}^{m}|\mathbf{x}_{i}-O\mathbf{y}_{i}|$. This problem has a clear relevance to our task. Indeed, if we take $\mathbf{x}_{i}$ to be the Bloch vectors of the target qubit preparation states and $\mathbf{y}_{i}$ the Bloch vectors of the experimental states, then minimization over $\mathrm{O}(3)$ is the same as the problem of finding a unitary transformation that connects those qubit states. In Ref. [38] the bounds on Procrustes problem were derived. We give formulation of Theorem 1 from Ref. [38] below, where we change the notation according to our problem. ###### Theorem (Arias-Castro et.al. 2020 - A perturbation bound for Procrustes). Given two tall matrices $L$ and $\tilde{L}$ of same size with $L$ having full rank, and set $\delta^{2}=\left|\left|\hskip 1.0pt\tilde{L}\tilde{L}^{T}-LL^{T}\hskip 1.0pt\right|\right|_{F}$. Then we have $\displaystyle\min_{O\in\mathrm{O}(k)}\left|\left|\hskip 1.0ptL-O\tilde{L}\hskip 1.0pt\right|\right|_{F}\leq\begin{cases}\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|\delta^{2}+\min\left[\frac{\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|\delta^{2}}{\sqrt{1-\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|^{2}\delta^{2}}},k^{\frac{1}{4}}\delta\right],&\text{if}\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|\delta<1\\\ \left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|\delta^{2}+k^{\frac{1}{4}}\delta&\text{otherwise.}\end{cases}$ (61) In the formulation of the above theorem $L^{\ddagger}$ stands for the Moore–Penrose inverse, which can be defined as $L^{\ddagger}=(LL^{T})^{-1}L^{T}$ for tall matrices of full rank. ###### Proof of Lemma 3. . Unitizing the results of the Theorem [Arias-Castro et.al. 2020] is rather straightforward. Let $L$ be a matrix which rows are the Bloch vectors of all target preparation states $\\{\varrho^{x}_{a}\\}_{a,x}$. Let $\tilde{L}$ in turn be the matrix of Bloch vectors of the experimental states $\\{\tilde{\varrho}^{x}_{a}\\}_{a,x}$. The matrices $LL^{T}$ and $\tilde{L}\tilde{L}^{T}$ are then, of course, the full Gram matrices $\Gamma$ and $\tilde{\Gamma}$. By “full" we mean that now we do not select a subset $S$ of linearly independent vectors among the Bloch vectors of $\varrho^{x}_{a}$. Instead, $\Gamma$ and $\tilde{\Gamma}$ are formed by all the considered states, which can still be a subset of the $2n$ states ($x\in[n]$, $a\in[2]$). We will be using $m$ to denote the number of the considered states, and a simple one-indexed set $\\{\varrho_{i}\\}_{i}$ to denote the states themselves. Estimating $\delta^{2}$ from the formulation of the Theorem [Arias-Castro et.al. 2020] is a direct application of the bound on $\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|_{F}$ from Lemma 2, where now instead of $k$ one should put the number $m$ of the considered preparation states. As for the left-hand side of Eq. (61), we can write the following: $\displaystyle\left|\left|\hskip 1.0ptL-O\tilde{L}\hskip 1.0pt\right|\right|_{F}^{2}={\mathrm{tr}}(LL^{T})+{\mathrm{tr}}(\tilde{L}\tilde{L}^{T})-2{\mathrm{tr}}(L^{T}O\tilde{L})$ $\displaystyle\geq m+m-m\frac{4\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}-4\sum_{i=1}^{m}{\mathrm{tr}}(\varrho_{i}U\tilde{\varrho}_{i}U^{\dagger})+2m$ (62) where we use the following identity: $\displaystyle{\mathrm{tr}}(L^{T}O\tilde{L})=\sum_{i=1}^{m}\mathbf{n}_{i}\cdot O\mathbf{\tilde{n}}_{i}=2\sum_{i=1}^{m}{\mathrm{tr}}(\varrho_{i}U\tilde{\varrho}_{i}U^{\dagger})-m,$ (63) and $U$ is the unitary transformation in $\mathrm{SU}(2)$ corresponding to the orthogonal transformation $O$ of the Bloch vectors. The bound on the average fidelity between $m$ target preparation states and the corresponding experimental states is then simply: $\displaystyle\frac{1}{m}\sum_{i=1}^{m}{\mathrm{tr}}(\varrho_{i}U\tilde{\varrho}_{i}U^{\dagger})\geq 1-\frac{\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}-\frac{1}{4m}P^{2}_{m}(\varepsilon,L)$ (64) where $\displaystyle P_{m}(\varepsilon,L)=\begin{cases}\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|F_{m}(\varepsilon)+\min\left[\frac{\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|F_{m}(\varepsilon)}{\sqrt{1-\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|^{2}F_{m}(\varepsilon)}},\sqrt{\sqrt{k}F_{m}(\varepsilon)}\right],&\text{if}\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|\sqrt{F_{m}(\varepsilon)}<1\\\ \left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|F_{m}(\varepsilon)+\sqrt{\sqrt{k}F_{m}(\varepsilon)}&\text{otherwise,}\end{cases}$ (65) and $\displaystyle F_{m}(\varepsilon)=\sqrt{4m(m-1)\varepsilon\left(1+2\sqrt{\varepsilon}+\frac{m+3}{m-1}\varepsilon\right)}.$ (66) ∎ ## Appendix D Examples Below we provide detailed derivations of the results presented in Table 1. The first example concerns $n=2,3$ MUBs in $d=2$. Since for MUBs ${\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})=\frac{1}{2}$, for $x\neq x^{\prime},\forall a,a^{\prime}$, it follows that $\Gamma$ is an identity matrix in $\mathbb{R}^{n}$ ($n\in\\{2,3\\}$). Hence, in Theorem 2 (see Appendix B) we should take $\left|\left|\hskip 1.0pt\Gamma_{S}\hskip 1.0pt\right|\right|=\left|\left|\hskip 1.0pt\Gamma^{-1}_{S}\hskip 1.0pt\right|\right|=\left|\left|\hskip 1.0ptL_{S}\hskip 1.0pt\right|\right|=1$, and $\left|\left|\hskip 1.0pt\Gamma\hskip 1.0pt\right|\right|_{F}=\sqrt{n}$. The resulting bound is the average between expressions in Eq. (30a) and Eq. (30b) with the function $E_{S,k}(\varepsilon)$ being simply: $\displaystyle E_{S,k}(\varepsilon)=\frac{1}{2\sqrt{2n}}\frac{F_{k}(\varepsilon)}{\sqrt{1-O_{k}(\varepsilon)}}.$ (67) The leading linear term is given in Table 1 for both $n=2,3$. The second example is a little less straightforward. From the condition ${\mathrm{tr}}(\varrho^{1}_{1}\varrho^{2}_{1})=\frac{1+\alpha}{2}$, $\alpha\in(-1,1)$ we obtain that $\Gamma=\left(\begin{smallmatrix}1&\alpha\\\ \alpha&1\end{smallmatrix}\right)$, and hence $\left|\left|\hskip 1.0pt\Gamma\hskip 1.0pt\right|\right|=1+|\alpha|$, $\left|\left|\hskip 1.0pt\Gamma^{-1}\hskip 1.0pt\right|\right|=\frac{1}{1-|\alpha|}$, and $\left|\left|\hskip 1.0pt\Gamma\hskip 1.0pt\right|\right|_{F}=\sqrt{2+2\alpha^{2}}$. The output $L$ of the Cholesky factorization is $L=\left(\begin{smallmatrix}1&0\\\ \alpha&\sqrt{1-\alpha^{2}}\end{smallmatrix}\right)$, which leads to $\left|\left|\hskip 1.0ptL\hskip 1.0pt\right|\right|=\sqrt{1+|a|}$. This also determines the minimum in Eq. (28) to be $\frac{\sqrt{1+|a|}}{\sqrt{2}}$. Plugging this values in Eq. (28) gives: $\displaystyle E_{\\{1,2\\},2}(\varepsilon)=\frac{1}{4}\frac{\sqrt{1+|\alpha|}F_{k}(\varepsilon)}{\sqrt{1-|\alpha|-O_{k}(\varepsilon)}}.$ (68) The final bound is again the average of the bounds in Eq. (30a) and Eq. (30b). The first order in $\varepsilon$ for this bound is given in Table 1. The applicability of the above bound is determined by the inequality $1-|\alpha|-O_{2}(\varepsilon)\geq 0$. Importantly, the latter condition gives nonempty region $\varepsilon\in[0,\varepsilon_{0})$ whenever $|\alpha|>0$. The third example is a trine ensemble of states $(\varrho^{1}_{1},\varrho^{2}_{2},\varrho^{3}_{1})$, with $\varrho^{x}_{1}=\frac{\mathbbm{1}}{2}+\frac{1}{2}\mathbf{n}_{x}\cdot\bm{\sigma}$, $x=1,2,3$, and where $\mathbf{n}_{1}=(1,0,0)$, $\mathbf{n}_{2}=\left(-\frac{1}{2},\frac{\sqrt{3}}{2},0\right)$, and $\mathbf{n}_{3}=\left(-\frac{1}{2},-\frac{\sqrt{3}}{2},0\right)$. For this configuration of the preparation states the alternative robustness analysis via Procrustes (see Appendix C) gives better bounds. Given the vectors $\mathbf{n}_{i}$, $i=1,2,3$ we can directly compute that $\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|=\sqrt{\frac{2}{3}}$. Inserting this value to Lemma 3 produces the results given in Table 1. The fourth example is the tetrahedron, with $\mathbf{n}_{1}=(0,0,1)$, $\mathbf{n}_{2}=\left(\sqrt{\frac{8}{9}},0,-\frac{1}{3}\right)$, $\mathbf{n}_{3}=\left(-\sqrt{\frac{2}{9}},\sqrt{\frac{2}{3}},-\frac{1}{3}\right)$, $\mathbf{n}_{4}=\left(-\sqrt{\frac{2}{9}},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right)$, and $\varrho^{x}_{1}=\frac{\mathbbm{1}}{2}+\frac{1}{2}\mathbf{n}_{x}\cdot\bm{\sigma}$, $x=1,2,3,4$ as before. In this case, we also employ the bounds from Procrustes. For the above configuration of states we have that $\left|\left|\hskip 1.0ptL^{\ddagger}\hskip 1.0pt\right|\right|=\frac{\sqrt{3}}{2}$, which leads to the results in Table 1. ## Appendix E Proofs of auxiliary results ###### Proof of Lemma 1. First of all, we improve the bound on the norm of each of the experimental states. From $\left|\left|\hskip 1.0pt\tilde{M}^{y}_{b}\hskip 1.0pt\right|\right|\geq 1-\varepsilon$, for $d=2$ it follows immediately that the second (second largest) eigenvalue of each of the effects $\tilde{M}^{y}_{b}$ cannot exceed $\varepsilon$. Hence, we can conclude that ${\mathrm{tr}}(\tilde{M}^{y}_{b})\leq 1+\varepsilon,\forall y,b$. Secondly, we can improve the bound on the norm of each of the experimental states. For that let us write the spectral decomposition of each experimental state and POVM effect: $\displaystyle\tilde{\varrho}^{x}_{a}$ $\displaystyle=\eta(\mathbbm{1}-|\psi\rangle\langle\psi|)+(1-\eta)|\psi\rangle\langle\psi|=\eta\mathbbm{1}+(1-2\eta)|\psi\rangle\langle\psi|$ (69) $\displaystyle\tilde{M}^{x}_{a}$ $\displaystyle=\lambda_{0}|\phi\rangle\langle\phi|+\lambda_{1}(\mathbbm{1}-|\phi\rangle\langle\phi|),$ where we assume that $\lambda_{0}\geq\lambda_{1}$. We omitted the indices $x,a$ for $\eta$, $\lambda_{0},\lambda_{1}$ and $\psi$,$\phi$ for simplicity. We can assume, without loss of generality, that $\eta\geq\frac{1}{2}$, and let us also assume for now that $|\langle\phi|\psi\rangle|^{2}\leq\frac{1}{2}$. From the condition ${\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{M}^{x}_{a})\geq 1-\varepsilon$, it then follows that: $\displaystyle\eta(\lambda_{0}+\lambda_{1})+(1-2\eta)(\lambda_{0}|\langle\phi|\psi\rangle|^{2}+\lambda_{1}(1-|\langle\phi|\psi\rangle|^{2}))\geq 1-\varepsilon,$ (70) from where we can obtain a lower bound on $\eta$, namely: $\displaystyle\eta\geq\frac{1}{1-2|\langle\phi|\psi\rangle|^{2}}\left(\frac{1-\varepsilon-\lambda_{1}}{\lambda_{0}-\lambda_{1}}-\frac{1}{2}\right)+\frac{1}{2}.$ (71) The expression on the right-hand side of the above inequality is maximal when $|\langle\phi|\psi\rangle|^{2}=0$ and $\lambda_{0}=1$, $\lambda_{1}=\varepsilon$, which returns the bound $\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}\hskip 1.0pt\right|\right|\geq\frac{1-2\varepsilon}{1-\varepsilon}$. Now let us return to our assumption $|\langle\phi|\psi\rangle|^{2}\leq\frac{1}{2}$, for which the above bound is valid. We can upper-bound $\eta$ by $1$ in Eq. (70), which returns nontrivial upper bound on $|\langle\phi|\psi\rangle|^{2}$ that happens to be $1-\frac{1-2\varepsilon}{1-\varepsilon}$. This function is below $\frac{1}{2}$ for $\varepsilon\leq\frac{1}{3}$, i.e., for $\varepsilon\leq\frac{1}{3}$ our newly-derived bound on $\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}\hskip 1.0pt\right|\right|$ is valid. The region $\varepsilon\in[0,\frac{1}{3}]$ is significantly larger than the resulting region in which our self-testing argument are valid, so this assumption does not affect our final results. Let us now try to improve the bounds for the overlaps. In the proof of Theorem 1 (see Eq. (3)) we have already established that: $\displaystyle|{\mathrm{tr}}(\tilde{\varrho}^{x}_{a}\tilde{\varrho}^{x^{\prime}}_{a^{\prime}})-{\mathrm{tr}}(\varrho^{x}_{a}\varrho^{x^{\prime}}_{a^{\prime}})|\leq\varepsilon+\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}-\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|,\quad\forall x\neq x^{\prime},\forall a,a^{\prime}.$ (72) Let us now refine the bound on $\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}-\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|$. For simplicity, we present this result below for a pair of operators $\varrho=(1-\eta)\mathbbm{1}+(2\eta-1)|\psi\rangle\langle\psi|$ and $M=\lambda_{1}\mathbbm{1}+(\lambda_{0}-\lambda_{1})|\phi\rangle\langle\phi|$, that satisfy the conditions ${\mathrm{tr}}(\varrho M)\geq 1-\varepsilon$, and $1-\varepsilon\leq\lambda_{0}\leq 1$, $0\leq\lambda_{1}\leq\varepsilon$. To compute the norm, we look for the eigenvector $|\xi\rangle$ of the operator $(\varrho-M)$, i.e., $(\varrho-M)|\xi\rangle=\Lambda|\xi\rangle$, corresponding to the maximal eigenvalue. We have the following quadratic equation for the eigenvalue $\Lambda$: $\displaystyle\Lambda^{2}-\Lambda(1-\lambda_{0}-\lambda_{1})+\eta(1-\eta)-\lambda_{1}+\lambda_{0}\lambda_{1}-\eta(\lambda_{0}-\lambda_{1})+(2\eta-1)(\lambda_{0}-\lambda_{1})|\langle\phi|\psi\rangle|^{2}=0.$ The sum of the roots of this equation is equal to $1-\lambda_{0}-\lambda_{1}$. Since we know that $|1-\lambda_{0}-\lambda_{1}|\leq\varepsilon$, then either both roots are of the same sign, in which case the largest eigenvalue could only be $\varepsilon$, or they are of the opposite sign. In the latter case, it is evident that the absolute values of both roots are maximal whenever the free term in Eq. (E) is minimal. We then can upper-bound the solutions to Eq. (E) by lower-bounding the free term using the condition ${\mathrm{tr}}(\varrho M)\geq 1-\varepsilon$. Indeed, from ${\mathrm{tr}}(\varrho M)\geq 1-\varepsilon$ we infer immediately that: $\displaystyle{\mathrm{tr}}(\varrho M)=\lambda_{0}-\eta(\lambda_{0}-\lambda_{1})+(2\eta-1)(\lambda_{0}-\lambda_{1})|\langle\phi|\psi\rangle|^{2}\geq 1-\varepsilon,$ (74) and thus we reduce Eq. (E) to: $\displaystyle\Lambda^{2}-\Lambda(1-\lambda_{0}-\lambda_{1})+\eta(1-\eta)-\lambda_{0}-\lambda_{1}+\lambda_{0}\lambda_{1}+1-\varepsilon=0.$ (75) Using the same argument we can set $\eta=1$, because $\eta(1-\eta)\geq 0$. The positive root of Eq. (75) is equal to: $\displaystyle\Lambda=\frac{1-\lambda_{0}-\lambda_{1}}{2}+\sqrt{\left(\frac{1-\lambda_{0}-\lambda_{1}}{2}\right)^{2}-(1-\lambda_{0})(1-\lambda_{1})+\varepsilon}.$ (76) It is easy to check that the above expression does not have any local maxima w.r.t. $\lambda_{0}$,$\lambda_{1}$ on the domain $1-\varepsilon\leq\lambda_{0}\leq 1$, $0\leq\lambda_{1}\leq\varepsilon$, whenever $\varepsilon<\frac{1}{2}$, which we assume to be the case. Thus, we conclude that the maximal value of $\Lambda$ corresponds to the boundary of the region of $(\lambda_{0},\lambda_{1})$. By considering this boundary we find that this maximal value corresponds to the case of $\lambda_{0}=1$ and $\lambda_{1}=0$ which yields $\Lambda=\sqrt{\varepsilon}$. From the above argument we finally conclude that: $\displaystyle\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}-\tilde{M}^{x}_{a}\hskip 1.0pt\right|\right|\leq\sqrt{\varepsilon},\quad\forall a,$ (77) which completes the proof for the state overlaps. From Eq. (77) and the fact that ${\mathrm{tr}}(\tilde{M}^{y}_{b})\leq 1+\varepsilon$, it is easy to obtain the improved bound on the overlaps between measurement effect. ∎ ###### Proof of Lemma 2. Let us start by deriving the bound on the Frobenius norm $\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|_{F}$: $\displaystyle\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|^{2}_{F}=\sum_{x=1}^{k}(|\Gamma_{x,x}-\tilde{\Gamma}_{x,x}|^{2}+\sum_{x^{\prime}\neq x}|\Gamma_{x,x^{\prime}}-\tilde{\Gamma}_{x,x^{\prime}}|^{2})$ $\displaystyle=4\sum_{x=1}^{k}\Big{(}(1-{\mathrm{tr}}(\tilde{\varrho}^{x}_{1})^{2})^{2}+\sum_{x^{\prime}\neq x}|{\mathrm{tr}}(\varrho^{x}_{1}\varrho^{x^{\prime}}_{1})-{\mathrm{tr}}(\tilde{\varrho}^{x}_{1}\tilde{\varrho}^{x^{\prime}}_{1})|^{2}\Big{)}.$ (78) We have already established the bound on $|{\mathrm{tr}}(\varrho^{x}_{1}\varrho^{x^{\prime}}_{1})-{\mathrm{tr}}(\tilde{\varrho}^{x}_{1}\tilde{\varrho}^{x^{\prime}}_{1})|$ in Lemma 1. The bound on $(1-{\mathrm{tr}}(\tilde{\varrho}^{x}_{1})^{2})^{2}$ can be obtained from the bound on the norm of each $\tilde{\varrho}^{x}_{1}$. Namely, from the condition $\left|\left|\hskip 1.0pt\tilde{\varrho}^{x}_{a}\hskip 1.0pt\right|\right|\geq\frac{1-2\varepsilon}{1-\varepsilon}$ we can immediately conclude that: $\displaystyle 1-{\mathrm{tr}}(\tilde{\varrho}^{x}_{1})^{2}\leq\frac{2\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}.$ (79) From here, it is easy to get to the final bound on $\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|^{2}_{F}$, which reads: $\displaystyle\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|^{2}_{F}\leq 4k(k-1)(\varepsilon+\sqrt{\varepsilon})^{2}+\frac{16k\varepsilon^{2}(1-2\varepsilon)^{2}}{(1-\varepsilon)^{4}}\leq 4k(k-1)\varepsilon\left(1+2\sqrt{\varepsilon}+\frac{k+3}{k-1}\varepsilon\right),$ (80) where we made some approximations to simplify the result. Now, let us derive the bound on $\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|$. In principle, we know that $\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|\leq\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|_{F}$, but we can derive a better bound based on the fact that the diagonal entries of $\Delta\Gamma$ are much less than the off-diagonal entries. First of all, due to the triangle inequality, we can write $\left|\left|\hskip 1.0pt\Delta\Gamma\hskip 1.0pt\right|\right|\leq\left|\left|\hskip 1.0pt\mathrm{diag}(\Delta\Gamma)\hskip 1.0pt\right|\right|+\left|\left|\hskip 1.0pt\mathrm{offdiag}(\Delta\Gamma)\hskip 1.0pt\right|\right|$, where we split $\Delta\Gamma$ on the diagonal and off-diagonal parts. The first term, $\left|\left|\hskip 1.0pt\mathrm{diag}(\Delta\Gamma)\hskip 1.0pt\right|\right|$, can be easily bounded as follows: $\displaystyle\left|\left|\hskip 1.0pt\mathrm{diag}(\Delta\Gamma)\hskip 1.0pt\right|\right|=2\max_{x}(1-{\mathrm{tr}}(\tilde{\varrho}^{x}_{1})^{2})\leq\frac{4\varepsilon(1-2\varepsilon)}{(1-\varepsilon)^{2}}\leq 4\varepsilon.$ (81) As for the off-diagonal part, we give the proof for two cases $k=2$ and $k=3$ separately. For $k=2$, $\left|\left|\hskip 1.0pt\mathrm{offdiag}(\Delta\Gamma)\hskip 1.0pt\right|\right|=2|{\mathrm{tr}}(\varrho^{1}_{1}\varrho^{2}_{1})-{\mathrm{tr}}(\tilde{\varrho}^{1}_{1}\tilde{\varrho}^{2}_{1})|\leq 2\sqrt{\varepsilon}+2\varepsilon$, where we used the results of Lemma 1. As for $k=3$, we will need some intermediate result, namely the following relation: $\displaystyle\left|\left|\hskip 1.0ptA\hskip 1.0pt\right|\right|\leq\sqrt{\frac{k-1}{k}}\left|\left|\hskip 1.0ptA\hskip 1.0pt\right|\right|_{F},$ (82) where $k$ is the size of the matrix $A$ with ${\mathrm{tr}}(A)=0$. We give a proof of this below. Let us assume that $\\{\lambda_{i}\\}_{i=1}^{k}$ are the eigenvalues of matrix $A$, hence we know that $\sum_{i=1}^{k}\lambda_{i}=0.$ Let $\lambda_{1}$ be the largest eigenvalue, i.e., the operator norm of $A$, if $\lambda_{1}\geq 0$. If we wish to maximize the Frobenius norm of $A$ for fixed $\lambda_{1}$, the following lower-bound has to be satisfied: $\displaystyle\left|\left|\hskip 1.0ptA\hskip 1.0pt\right|\right|_{F}^{2}=\lambda_{1}^{2}+\sum_{i=2}^{k}\lambda_{i}^{2}\geq\lambda_{1}^{2}+\frac{1}{k-1}\left(\sum_{i=2}^{k}|\lambda_{i}|\right)^{2}\geq\lambda_{1}^{2}+\frac{1}{k-1}\left(\sum_{i=2}^{k}\lambda_{i}\right)^{2}=\lambda_{1}^{2}\frac{k}{k-1},$ (83) which proves the bound in Eq. (82). Using the above result, we obtain the following bound: $\displaystyle\left|\left|\hskip 1.0pt\mathrm{offdiag}(\Delta\Gamma)\hskip 1.0pt\right|\right|\leq\sqrt{\frac{2}{3}}\left|\left|\hskip 1.0pt\mathrm{offdiag}(\Delta\Gamma)\hskip 1.0pt\right|\right|_{F}=2\sqrt{\frac{2}{3}}\sqrt{\sum_{x\neq x^{\prime}}|{\mathrm{tr}}(\varrho^{x}_{1}\varrho^{x^{\prime}}_{1})-{\mathrm{tr}}(\tilde{\varrho}^{x}_{1}\tilde{\varrho}^{x^{\prime}}_{1})|^{2}}\leq 4(\sqrt{\varepsilon}+\varepsilon),$ which completes out proof. ∎ ## References * Arute et al. 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2024-09-04T02:54:56.504392

2003.01037

arxiv-papers

# One or Two Frequencies? The Scattering Transform Answers ††thanks: This work is supported by the NSF award 1633259 (BIRDVOX). Vincent Lostanlen Music and Audio Research Lab New York University New York, NY, USA <EMAIL_ADDRESS>Alice Cohen-Hadria IRIT University of Toulouse — CNRS Toulouse, France <EMAIL_ADDRESS>Juan Pablo Bello Music and Audio Research Lab New York University New York, NY, USA <EMAIL_ADDRESS> ###### Abstract With the aim of constructing a biologically plausible model of machine listening, we study the representation of a multicomponent stationary signal by a wavelet scattering network. First, we show that renormalizing second- order nodes by their first-order parents gives a simple numerical criterion to assess whether two neighboring components will interfere psychoacoustically. Secondly, we run a manifold learning algorithm (Isomap) on scattering coefficients to visualize the similarity space underlying parametric additive synthesis. Thirdly, we generalize the “one or two components” framework to three sine waves or more, and prove that the effective scattering depth of a Fourier series grows in logarithmic proportion to its bandwidth. ###### Index Terms: Audio systems, Amplitude modulation, Continuous wavelet transform, Fourier series, Multi-layer neural network. ## I Introduction In the mammalian auditory system, cochlear hair cells operate like band-pass filters whose equivalent rectangular bandwidth (ERB) grows in proportion to their center frequency. Given two sine waves $t\mapsto\bm{y_{1}}(t)=a_{1}\cos(f_{1}t+\varphi_{1})$ and $t\mapsto\bm{y_{2}}(t)=a_{2}\cos(f_{2}t+\varphi_{2})$ of respective frequencies $f_{1}>0$ and $f_{2}>0$, we perceive their mixture as a musical chord insofar as $\bm{y_{1}}$ and $\bm{y_{2}}$ belong to disjoint critical bands. However, if $a_{2}\ll a_{1}$ or $f_{2}\approx f_{1}$, then the tone $\bm{y_{2}}$ is said to be _masked_ by $\bm{y_{1}}$. In lieu of two pure tones, we hear a “beating tone”: i.e., a locally sinusoidal wave whose carrier frequency is $\frac{1}{2}(f_{1}+f_{2})$ and whose modulation frequency is $\frac{1}{2}|f_{1}-f_{2}|$. In humans, the resolution of beating tones involves physiological processes beyond the cochlea, i.e., in the primary auditory cortex. The scattering transform ($\mathbf{S}$) is a deep convolutional operator which alternates constant-$Q$ wavelet decompositions and the application of pointwise complex modulus, up to some time scale $T$. Broadly speaking, its first two layers ($\mathbf{S_{1}}$ and $\mathbf{S_{2}}$) resemble the functioning of the cochlea and the primary auditory cortex, respectively. In the context of audio classification, scattering transforms have been succesfully employed to represent speech [2], environmental sounds [13], urban sounds [20], musical instruments [10], rhythms [8], and playing techniques [24]. Therefore, the scattering transform simultaneously enjoys a diverse range of practical motivations, a firm rooting in wavelet theory, and a plausible correspondence with neurophysiology. This article discusses the response of the scattering transform operator to a complex tone input $\bm{y}:t\mapsto\bm{y_{1}}(t)+\bm{y_{2}}(t)$, depending on the sinusoidal parameters of $\bm{y_{1}}$ and $\bm{y_{2}}$. In this respect, we follow a well-established methodology in nonstationary signal processing, colloquially known as: “One or two frequencies? The X Answers”, where X is the nonlinear operator of interest. The key idea is to identify transitional regimes in the response of X with respect to variations in relative amplitude ($\frac{a_{2}}{a_{1}}$), relative frequency ($\frac{f_{2}}{f_{1}}$), and relative phase ($\varphi_{2}-\varphi_{1}$). Prior publications have done so for X being the empirical mode decomposition [19], the synchrosqueezing transform [25], and the singular spectrum analysis operator [9]. We extend this line of research to the case where X is the scattering transform in dimension one. ## II Wavelet-based recursive interferometry Let $\bm{\psi}\in\mathbf{L}^{2}(\mathbb{R},\mathbb{C})$ a Hilbert-analytic filter with null average, unit center frequency, and an ERB equal to $1/Q$. We define a constant-$Q$ wavelet filterbank as the family $\bm{\psi}_{\lambda}:t\mapsto\lambda\bm{\psi}(\lambda t)$. Each wavelet $\bm{\psi}_{\lambda}$ has a center frequency of $\lambda$, an ERB of $\lambda/Q$, and an effective receptive field of $(2\pi Q/\lambda)$ in the time domain. In practice, the frequency variable $\lambda$ gets discretized according to a geometric progression of common ratio $2^{\frac{1}{Q}}$. Consequently, every continuous signal $\bm{y}$ that is bandlimited to $[f_{\min},f_{\max}]$ activates a number of $Q\log_{2}({\frac{f_{\max}}{f_{\min}}})$ wavelets $\bm{\psi_{\lambda}}$ at most. We define the scalogram of $\bm{y}$ as the squared complex modulus of its constant-$Q$ transform (CQT): $\mathbf{U_{1}}\bm{y}:(t,\lambda_{1})\longmapsto\left|\int_{-\infty}^{+\infty}\bm{y}(t^{\prime})\bm{\psi_{\lambda_{1}}}(t-t^{\prime})\;\mathrm{d}t^{\prime}\right|^{2}.$ (1) Likewise, we define a second layer of nonlinear transformation for $\bm{y}$ as the “scalogram of its scalogram”: $\mathbf{U_{2}}\bm{y}:(t,\lambda_{1},\lambda_{2})\longmapsto\Big{|}\big{|}\bm{y}\ast\bm{\psi_{\lambda_{1}}}\big{|}^{2}\ast\bm{\psi_{\lambda_{2}}}\Big{|}^{2}(t),$ (2) where the asterisk denotes a convolution product. This construct may be iterated for every integer $m$ by “scattering” the multivariate signal $\mathbf{U_{m}}\bm{y}$ into all wavelet subbands $\lambda_{m}<\lambda_{m-1}$: $\mathbf{U_{m+1}}\bm{y}:(t,\lambda_{1}\ldots\lambda_{m+1})\longmapsto\\\ \Big{|}\mathbf{U_{m}}\bm{y}(t,\lambda_{1}\ldots\lambda_{m})\ast\bm{\psi}_{\lambda_{m}}\Big{|}^{2}(t,\lambda_{1}\ldots\lambda_{m}).$ (3) Note that the original definition of the scattering transform adopts the complex modulus ($|z|=\sqrt{z\bar{z}}$) rather its square ($|z|^{2}=z\bar{z}$) as its activation function. This is to ensure that $\mathbf{U_{m}}$ is a non- expansive map in terms of Lipschitz regularity. However, to simplify our calculation and spare an intermediate stage of linearization of the square root, we choose to employ a measure of power rather than amplitude. This idea was initially proposed by [6] in the context of marine bioacoustics. Every layer $m$ in this deep convolutional network composes an invariant linear system (namely, the CQT) and a pointwise operation (the squared complex modulus). Thus, by recurrence over the depth variable $m$, every tensor $\mathbf{U_{m}}\bm{y}$ is equivariant to the action of delay operators. In order to replace this equivariance property by an invariance property, we integrate each $\mathbf{U_{m}}$ over some predefined time scale $T$, yielding the invariant scattering transform: $\displaystyle\mathbf{S_{m}}\bm{y}:(t,p)\longmapsto\int_{-\infty}^{+\infty}\mathbf{U_{m}}(t^{\prime},p)\bm{\phi}_{T}(t-t^{\prime})\;\mathrm{d}t^{\prime},$ (4) where the $m$-tuple $p=(\lambda_{1}\ldots\lambda_{m})$ is a scattering path and the signal $\bm{\phi}_{T}$ is a real-valued low-pass filter of time scale $T$. ## III Auditory masking in a scattering network Figure 1: Superimposed heatmaps of second-order masking coefficients $\mathbf{\widetilde{S}_{2}}\bm{y}$ after a scattering transform of two sine waves $\bm{y_{1}}$ and $\bm{y_{2}}$, measured around the frequency $f_{1}$, as a function of relative amplitude $\frac{a_{2}}{a_{1}}$ and relative frequency difference $\frac{|f_{2}-f_{1}|}{f_{1}}$. The color of each blot denotes the resolution $\lambda_{2}$ at the second layer. Wavelets have an asymmetric profile (Gammatone wavelets) and a quality factor $Q=4$. The second layer covers an interval of nine octaves below $f_{1}$. For the sake of clarity, we only display one interference pattern per octave. Given $n\in\\{1,2\\}$, the convolution between every sine wave $\bm{y_{n}}$ and every wavelet $\bm{\psi_{\lambda_{1}}}$ writes as a multiplication in the Fourier domain. Because $\bm{\psi_{\lambda_{1}}}$ is Hilbert-analytic, only the analytic part $\bm{y_{n}^{\mathrm{a}}}=\bm{y_{n}}+\mathrm{i}\mathcal{H}\\{\mathbf{y_{n}}\\}=a_{n}\exp(\mathrm{i}(f_{n}t+\varphi_{n}))$ of the real signal $\bm{y_{n}}$ is preserved in the CQT: $\left(\bm{y_{n}}\ast\bm{\psi_{\lambda_{1}}}\right)(t)=\dfrac{1}{2}\bm{\widehat{\psi}_{\lambda_{1}}}(f_{n})\bm{y_{n}^{\mathrm{a}}}(t).$ (5) By linearity of the CQT, we expand the interference between $\bm{y_{1}}$ and $\bm{y_{2}}$ by heterodyning: $\displaystyle\Big{|}(\bm{y_{1}}+\bm{y_{2}})\ast\bm{\psi_{\lambda_{1}}}\Big{|}^{2}(t)=\frac{1}{2}\Big{|}\bm{\widehat{\psi}}\big{(}\frac{f_{1}}{\lambda_{1}}\big{)}\Big{|}^{2}a_{1}^{2}+\frac{1}{2}\Big{|}\bm{\widehat{\psi}}\big{(}\frac{f_{2}}{\lambda_{1}}\big{)}\Big{|}^{2}a_{2}^{2}$ $\displaystyle+\mathfrak{R}\Bigg{(}\\!\bm{\widehat{\psi}}\Big{(}\frac{f_{1}}{\lambda_{1}}\Big{)}\bm{\widehat{\psi}^{\ast}}\Big{(}\frac{f_{2}}{\lambda_{1}}\Big{)}\\!\Bigg{)}a_{1}a_{2}\cos\big{(}(f_{2}-f_{1})t+(\varphi_{2}-\varphi_{1})\big{)}.$ (6) Because the wavelet $\bm{\psi}$ has a null average, the two constant terms in the equation above are absorbed by the first layer of the scattering network, and disappear at deeper layers. However, the cross term, proportional to $a_{1}a_{2}$, is a “difference tone” of fundamental frequency $\Delta\\!f=|f_{2}-f_{1}|$. The authors of a previous publication [3] have remarked that this difference tone elicits a peak in second-order scattering coefficients for the path $p=(\lambda_{1},\lambda_{2})=(f_{1},|f_{2}-f_{1}|)$. In the following, we generalize their study to include the effect of the relative amplitude $\frac{a_{2}}{a_{1}}$, the wavelet shape $\bm{\psi}$, the quality factor $Q$, and the time scale of local stationarity $T$. Equation 6 illustrates how the scalogram operator $\mathbf{U_{1}}$ converts a complex tone (two frequencies $f_{1}$ and $f_{2}$) into a simple tone (one frequency $|f_{2}-f_{1}|$). For this simple tone to carry a nonnegligible amplitude in $\mathbf{U_{2}}$, three conditions must be satisfied. First, the rectangular term $a_{1}a_{2}$ must be nonnegligible in comparison to the square terms $a_{1}^{2}$ and $a_{2}^{2}$. Secondly, there must exist a wavelet $\bm{\psi}_{\lambda_{1}}$ whose spectrum encompasses both frequencies $f_{1}$ and $f_{2}$. Said otherwise, $\lambda_{1}$ must satisfy the inequalities $|\frac{f_{n}}{\lambda_{1}}-1|\ll\frac{1}{Q}$, both for $f_{n}=f_{1}$ and for $f_{n}=f_{2}$. Thirdly, the frequency difference $|f_{2}-f_{1}|$ must belong to the passband of some second-order wavelet $\bm{\psi_{\lambda_{2}}}$. Yet, in practice, to guarantee the temporal localization of scattering coefficients and restrict the filterbank to a finite number of octaves, the scaling factor of every $\bm{\psi_{\lambda_{m}}}$ is upper-bounded by the temporal constant $T$. Therefore, the period $\frac{2\pi}{|f_{2}-f_{1}|}$ of the difference tone should be under the pseudo-period of the wavelet with support $T$; i.e., a pseudo-period of $QT$. Hence the third condition: $|f_{2}-f_{1}|\ll\frac{2\pi Q}{T}$. One simple way of quantifying the amount of mutual interference between signals $\bm{y_{1}}$ and $\bm{y_{2}}$ is to renormalize second-order coefficients by their first-order “parent” coefficients: $\mathbf{\widetilde{S}_{2}}\bm{y}(t,\lambda_{1},\lambda_{2})=\dfrac{\mathbf{S_{2}}\bm{y}(t,\lambda_{1},\lambda_{2})}{\mathbf{S_{1}}\bm{y}(t,\lambda_{1})}$ (7) This operation, initially proposed by [2], is conceptually analogous to classical methods in adaptive gain control, notably per-channel energy normalization (PCEN) [14]. In accordance with the “one or two frequencies” methodology, Figure 1 illustrates the value of this ratio of energies in the subband $\lambda_{1}=f_{1}$, for different values of relative amplitude $\frac{a_{2}}{a_{1}}$ and relative frequency difference $\frac{|f_{2}-f_{1}|}{f_{1}}$. We fixed $f_{2}<f_{1}$ without loss of generality. As expected, we observe that, for $a_{2}\approx a_{1}$ and a relative frequency difference between $\frac{Q}{f_{1}T}$ and $\frac{1}{Q}$, second-layer wavelets $\bm{\psi_{\lambda_{2}}}$ resonate with the difference tone as a result of the interference between signals $\bm{y_{1}}$ and $\bm{y_{2}}$. ## IV Application to manifold learning To demonstrate the ability of the scattering transform to characterize auditory masking, we build a dataset of complex tones according to the following additive synthesis model: $\bm{y}_{\alpha,r}(t)=\sum_{n=1}^{N}\dfrac{1+(-1)^{n}r}{n^{\alpha}}\cos(nf_{1}t)\bm{\phi}_{T}(t),$ (8) where $\bm{\phi}_{T}$ is a Hann window of duration $T$. This additive synthesis model depends upon two parameters: the Fourier decay $\alpha$ and the relative odd-to-even amplitude difference $r$. Figure 2 displays the CQT log-magnitude spectrum of $\bm{y}_{\alpha,r}$ for different values of $\alpha$ and $r$. In practice, we set $T$ to $1024$ samples, $N$ to $32$ harmonics, and $f_{1}$ between $12$ and $24$ cycles. Figure 2: Log-magnitudes of synthetic musical tones as a function of wavelet log-frequency ($\log\lambda$). Ticks of the vertical (resp. horizontal) axis denote relative amplitude (resp. frequency) intervals of 10 dB (resp. one octave). Parameters $\alpha$ and $r$ denote the Fourier decay exponent and the relative odd-to-even amplitude difference $r$ respectively. See Equation 8 for details. Our synthetic dataset comprises $2500$ audio signals in total, corresponding to $50$ values of $\alpha$ between $0$ and $2$ and $50$ values of $r$ between $0$ and $1$, while $f_{1}$ is an integer chosen uniformly at random between $12$ and $24$. We extract the scattering transform of each signal $\bm{y}_{\alpha,r}$ up to order $M=2$, with $Q=1$ and $J=8$, by means of the Kymatio Python package [4]. Concatenating $QJ$ first-order coefficients with $\frac{1}{2}Q^{2}J(J-1)$ second-order coefficients yields a representation in dimension $37$. For visualization purposes, we bring the $37$-dimensional space of scattering coefficients to the dimension three by means of the Isomap algorithm for unsupervised manifold learning [22]. The appeal behind Isomap is that pairwise Euclidean distances in the 3-D point cloud approximate the corresponding geodesic distances over the $K$-nearest neighbor graph associated to the dataset. Throughout this paper, we set the number of neighbors to $K=100$ and measure neighboring relationships by comparing high-dimensional $\ell^{2}$ distances. Crucially, in the case of the scattering transform, these $\ell^{2}$ distances are provably stable (i.e., Lipschitz-continuous) to the action of diffeomorphisms [16, Theorem 2.12]. Scattering transform embedding Audiovisual correspondence (Open-L3) embedding Mel-frequency cepstral coefficient (MFCC) embedding Color: $f_{1}\qquad\qquad$ Color: $\alpha\quad\qquad\qquad$ Color: $r\qquad$ Figure 3: Isomap embedding of synthetic musical notes, as described by their scattering transform coefficients (top); their Open-L3 coefficients (center); and their mel-frequency cepstral coefficients (MFCC, bottom). The color of a dot, ranging from red to blue via white, denotes the fundamental frequency $f_{1}$ (left), the Fourier decay exponent $\alpha$ (center), and the relative odd-to-even amplitude difference $r$ (right) respectively. Note that all methods are unsupervised: triplets ($f_{1}$, $\alpha$, $r$) are not directly supplied to the models, but only serve for color grading post hoc. See Section IV for details. Figure 3 (top) illustrates our findings. We observe that, after scattering transform and Isomap dimensionality reduction, the dataset appears as a 3-D Cartesian mesh whose principal components align with $f_{1}$, $\alpha$, and $r$ respectively. This result demonstrates that the scattering transform is capable of disentangling and linearizing multiple factors of variability in the spectral envelope of periodic signals, even if those factors are not directly amenable to diffeomorphisms. As a point of comparison, Figure 3 presents the outcome of Isomap on alternative feature representations: Open-L3 embedding (center) and mel- frequency cepstral coefficients (MFCC, bottom). The former results from training a deep convolutional network (convnet) on a self-supervised task of audiovisual correspondence, and yields $6177$ coefficients [7]. The latter resuts from a log-mel-spectrogram representation, followed by a discrete cosine transform (DCT) over the mel-frequency axis, and yields $12$ coefficients. We compute MFCC with librosa v0.7 [18] default parameters. We observe that Open-L3 embeddings correctly disentangles boundary conditions ($r$) from fundamental frequency ($f_{1}$), but fails to disentangle Fourier decay ($\alpha$) from $f_{1}$. Instead, correlations between $r$ and $f_{1}$ are positive for low-pitched sounds ($12$ to $16$ cycles) and negative for high-pitched sounds ($16$ to $24$ cycles). Although this failure deserves a more formal inquiry, we hypothesize that this it stems from the small convolutional receptive field of the $L^{3}$-Net: $24$ mel subbands, i.e., roughly half an octave around $1$ kHz. Figure 4: Energy decay as a function of wavelet scattering depth $m$, for mixtures of $N$ components with equal amplitudes, equal phases, and evenly spaced frequencies. The color of each line plot denotes the integer part of $\log_{2}N$. In this experiment, wavelets have a sine cardinal profile (Shannon wavelets) and a quality factor equal to $Q=1$. Each filterbank covers seven octaves. Moreover, in the case of MFCC, we find that the variability in fundamental frequency ($f_{1}$) dominates the variability in spectral shape parameters ($\alpha$ and $r$), thus yielding a rectilinear embedding (top). This observation is in line with a previous publication [11], which showed statistically that MFCCs are overly sensitive to frequency transposition in complex tones. From this qualitative benchmark, it appears that the scattering transform is a more interpretable representation of periodic signals than Open-L3, while incurring a smaller computational cost. However, in the presence of aperiodic signals such as environmental sounds, Open-L3 outperforms the scattering transform in terms of classification accuracy with linear support vector machines [5]. To remain competitive, the scattering transform must not only capture heterodyne interference, but also joint spectrotemporal modulations [1]. In this context, future work will strive to combine insights from multiresolution analysis and deep self-supervised learning. ## V Beyond pairwise interference: full-depth scattering networks In speech and music processing, pitched sounds are rarely approximable as a mixture of merely two components. More often than not, they contain ten components or more, and span across multiple octaves in the Fourier domain. Thus, computing the masking coefficient at the second layer only provides a crude description of the timbral content within each critical band. Indeed, $\mathbf{S_{2}}$ encodes pairwise interference between sinusoidal components but fails to characterize more intricate structures in the spectral envelope of $\bm{y}$. To address this issue, we propose to study the scattering transform beyond order two, thus encompassing heterodyne structures of greater multiplicity. For the sake of mathematical tractability, we consider the following mother wavelet, hereafter called “complex Shannon wavelet” after [15, Section 7.2.2]: $\bm{\psi}:t\longmapsto\dfrac{\exp(2\mathrm{i}t)-\exp(\mathrm{i}t)}{2\pi\mathrm{i}t}$ (9) The definition of a scattering transform with complex Shannon wavelets requires to resort to the theory of tempered distributions. We refer to [21] for further mathematical details. The following theorem, proven in the Appendix, describes the response of a deep scattering network in the important particular case of a periodic signal with finite bandwidth. ###### Theorem V.1. Let $\bm{y}\in\mathcal{C}^{\infty}(\mathbb{R})$ a periodic signal of fundamental frequency $f_{1}$. Let $\bm{\psi}$ the complex Shannon wavelet as in Equation 9 and $\mathbf{U_{1}}$ its associated scalogram operator as in Equation 1. If $\bm{y}$ has a finite bandwidth of $M$ octaves, then its scattering coefficients $\mathbf{U_{m}}\bm{y}$ are zero for any $m>M$. This result is in agreement with the theorem of exponential decay of scattering coefficients [23]. Note, however, that [23] expresses an upper bound on the energy at fixed depth for integrable signals, while we express an upper bound on the depth at fixed bandwidth for periodic signals. We apply the theorem above to the case of a signal containing $N$ components of equal amplitudes, equal phases, and evenly spaced frequencies: $\bm{y}:t\mapsto\sum_{n=1}^{N}a_{1}\cos(nf_{1}t+\varphi_{1})$. Figure 4 illustrates the decay of scatterered energy as a function of depth. The conceptual analogy between depth and scale was originally proposed by [17] in a theoretical effort to clarify the role of hierarchical symmetries in convnets. Although our findings support this analogy, we note that computing a scattering transform with $M=\log_{2}T$ layers is often impractical. However, if the Fourier series in $\bm{y}$ satisfies a self-similarity assumption, it is possible to match the representational capacity of a full-depth scattering network while keeping the depth to $M=2$. Indeed, spiral scattering performs wavelet convolutions over time, over log-frequency, and across octaves, thereby capturing the spectrotemporal periodicity of Shepard tones and Shepard-Risset glissandos [12]. Further research is needed to integrate broadband demodulation into deep convolutional architectures for machine listening. ## VI Conclusion In this article, we have studied the role of every layer in a scattering network by means of a well-established methodology, colloquially known as “one or two components” [19]. We have come up with a numerical criterion of psychoacoustic masking; demonstrated that the scattering transform disentangles multiple factors of variability in the spectral envelope; and proven that the effective scattered depth of Fourier series is bounded by the logarithm of its bandwidth, thus emphasizing the importance of capturing geometric regularity across temporal scales. ## References * [1] Joakim Andén, Vincent Lostanlen and Stéphane Mallat “Joint time–frequency scattering” In _IEEE Transactions on Signal Processing_ 67.14 IEEE, 2019, pp. 3704–3718 * [2] Joakim Andén and Stéphane Mallat “Deep scattering spectrum” In _IEEE Trans. Signal Process._ 62.16 IEEE, 2014, pp. 4114–4128 * [3] Joakim Andén and Stéphane Mallat “Scattering representation of modulated sounds” In _Proc. DAFx_ , 2012 * [4] Mathieu Andreux et al. “Kymatio: Scattering transforms in Python” In _JMLR_ 21.60, 2020, pp. 1–6 * [5] Relja Arandjelovic and Andrew Zisserman “Look, listen and learn” In _Proceedings of the IEEE International Conference on Computer Vision_ , 2017, pp. 609–617 * [6] Randall Balestriero and Hervé Glotin “Linear time complexity deep Fourier scattering network and extension to nonlinear invariants” In _arXiv_ 1707.05841, 2017 * [7] Jason Cramer, Ho-Hsiang Wu, Justin Salamon and Juan Pablo Bello “Look, listen, and learn more: Design choices for deep audio embeddings” In _Proc. ICASSP_ , 2019, pp. 3852–3856 IEEE * [8] Daniel Haider and Peter Balazs “Extraction of Rhythmical Features with the Gabor Scattering Transform” In _Proc. CMMR_ , 2019 * [9] Jinane Harmouche, Dominique Fourer, Pierre Auger and Patrick Flandrin “Une ou deux composantes: la réponse de l’analyse spectrale singulière” In _Actes du colloque GRETSI_ , 2015 * [10] Vincent Lostanlen, Joakim Andén and Mathieu Lagrange “Extended playing techniques: The next milestone in musical instrument recognition” In _Proc. DLfM_ , 2018 * [11] Vincent Lostanlen and Carmine-Emanuele Cella “Deep convolutional networks on the pitch spiral for musical instrument recognition” In _Proc. ISMIR_ , 2016 * [12] Vincent Lostanlen and Stéphane Mallat “Wavelet scattering on the pitch spiral” In _Proc. DAFX_ , 2016 * [13] Vincent Lostanlen, Grégoire Lafay, Joakim Andén and Mathieu Lagrange “Relevance-based quantization of scattering features for unsupervised mining of environmental audio” In _EURASIP J. Audio Speech Mus. Process._ 2018.1 Springer, 2018, pp. 15 * [14] V. Lostanlen et al. “Per-Channel Energy Normalization: Why and How” In _IEEE Signal Proc. Let._ 26.1, 2019, pp. 39–43 * [15] Stéphane Mallat “A wavelet tour of signal processing: The sparse way” Associated Press, 2008 * [16] Stéphane Mallat “Group invariant scattering” In _Comm. Pure Appl. Math._ 65.10 Wiley Online Library, 2012, pp. 1331–1398 * [17] Stéphane Mallat “Understanding deep convolutional networks” In _Phil. Trans. R. Soc._ 374.2065 The Royal Society Publishing, 2016, pp. 20150203 * [18] Brian McFee et al. “librosa: 0.7.2”, 2020 DOI: 10.5281/zenodo.3606573 * [19] Gabriel Rilling and Patrick Flandrin “One or two frequencies? The empirical mode decomposition answers” In _IEEE Trans. Signal Process._ 56.1 IEEE, 2008, pp. 85–95 * [20] Justin Salamon and Juan Pablo Bello “Feature learning with deep scattering for urban sound analysis” In _Proc. EUSIPCO_ , 2015, pp. 724–728 IEEE * [21] Robert S Strichartz “A guide to distribution theory and Fourier transforms” World Scientific Publishing Company, 2003 * [22] Joshua B Tenenbaum, Vin De Silva and John C Langford “A global geometric framework for nonlinear dimensionality reduction” In _science_ 290.5500 American Association for the Advancement of Science, 2000, pp. 2319–2323 * [23] Irène Waldspurger “Exponential decay of scattering coefficients” In _Proc. SampTA_ , 2017, pp. 143–146 IEEE * [24] Changhong Wang, Emmanouil Benetos, Vincent Lostanlen and Elaine Chew “Adaptive time–frequency scattering for periodic modulation recognition in music signals” In _Proc. ISMIR_ , 2019 * [25] Hau-Tieng Wu, Patrick Flandrin and Ingrid Daubechies “One or two frequencies? The synchrosqueezing answers” In _Adv. Adapt. Data Anal._ 3.01–02 World Scientific, 2011, pp. 29–39 ## Appendix: proof of Theorem V.1 ###### Proof. We reason by induction over the depth variable $M$. The base case ($M=1$) leads to $\mathbf{U_{1}}\bm{y}(t,\lambda)=1$ if $\lambda<f_{1}\leq 2\lambda$ and zero otherwise. Because $\bm{\psi}$ has one vanishing moment, it follows that $\mathbf{U_{2}}\bm{y}$ is zero, and likewise at deeper layers. To prove the induction step at depth $M$, to decompose $\bm{y}$ into a low-pass approximation $(\bm{y}\ast\bm{g_{M}})$ spanning the subband $[0;2^{M}f_{1}[$ and a high-pass detail $(\bm{y}\ast\bm{h_{M}})$ spanning the subband $[2^{M}f_{1};2^{(M+1)}f_{1}[$. Denoting by $c_{n}$ the complex-valued Fourier coefficients of $\bm{y}$, we have at every time $t\in\mathbb{R}$: $\displaystyle\bm{y}(t)$ $\displaystyle=(\bm{y}\ast\bm{g_{M}})(t)$ $\displaystyle+$ $\displaystyle(\bm{y}\ast\bm{h_{M}})(t)$ $\displaystyle=\sum_{|n|\leq 2^{M}}c_{n}\exp(\mathrm{i}nf_{1}t)$ $\displaystyle+$ $\displaystyle\sum_{|n|>2^{M}}c_{n}\exp(\mathrm{i}nf_{1}t)$ (10) On one hand, the coarse term $(\bm{y}\ast\bm{g_{M}})$ has a bandwidth of $M$ octaves. Therefore, by the induction hypothesis, we have $\mathbf{U_{m}}(\bm{y}\ast\bm{g_{M}})=0$ for $m>M$, and _a fortiori_ for $m>(M+1)$. On the other hand, we consider the complex Shannon scalogram of $(\bm{y}\ast\bm{h_{M}})$ in some subband $\lambda>0$: $\big{|}\bm{y}\ast\bm{h_{M}}\ast\bm{\psi}_{\lambda}\big{|}^{2}(t)\leq\big{|}\bm{y}\ast\bm{h_{M}}\ast\bm{\psi}_{2^{M}}\big{|}^{2}(t)\hfill\\\ \hfill=\sum_{n=(1+2^{M})}^{2^{M+1}}\sum_{k=(1+2^{M})}^{2^{M+1}}c_{n}c_{k}^{\ast}\exp\big{(}\mathrm{i}(n-k)f_{1}t\big{)}$ (11) In the double sum above, all integer differences of the form $(n-k)$ range between $-(2^{M}-1)$ and $(2^{M}-1)$. Thus, $\big{|}\bm{y}\ast\bm{h_{M}}\ast\bm{\psi}_{2^{M}}\big{|}^{2}$ is a periodic signal of fundamental frequency $f_{1}$ spanning $M$ octaves. Furthermore, because $\bm{h_{M}}=\bm{\psi}_{2^{M}}$, $\big{|}\bm{y}\ast\bm{h_{M}}\ast\bm{\psi}_{\lambda}\big{|}^{2}$ has a smaller bandwidth than $\big{|}\bm{y}\ast\bm{h_{M}}\ast\bm{\psi}_{2^{M}}\big{|}^{2}$; i.e., $M$ octaves or less. By the induction hypothesis, we have: $\forall\lambda,\;\mathbf{U_{M+1}}\big{(}|\bm{y}\ast\bm{h_{M}}\ast\bm{\psi}_{\lambda}|^{2}\big{)}(\lambda_{1},\ldots,\lambda_{m+1})=0.$ (12) In the equation above, we recognize the scattering path $p=(\lambda,\lambda_{1},\ldots,\lambda_{M+1})$ of $\mathbf{U_{M+2}}$. Finally, because the scattering transform is a nonexpansive operator [16, Prop. 2.5], we have the inequality: $\big{\|}\mathbf{U_{M+2}}(\bm{y})\big{\|}\leq\big{\|}\mathbf{U_{M+2}}(\bm{y}\ast\bm{g_{M}})\big{\|}+\big{\|}\mathbf{U_{M+2}}(\bm{y}\ast\bm{h_{M}})\big{\|}=0,$ (13) which implies $\mathbf{U_{M+2}}\bm{y}=0$, and likewise at deeper layers. We conclude by induction that the theorem holds for any $M$. ∎

2024-09-04T02:54:56.514197

2003.01040

arxiv-papers

# Scaling Up Multiagent Reinforcement Learning for Robotic Systems: Learn an Adaptive Sparse Communication Graph Chuangchuang Sun, Macheng Shen, and Jonathan P. How ∗Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139. Emails: {ccsun1, macshen<EMAIL_ADDRESS> ###### Abstract The complexity of multiagent reinforcement learning (MARL) in multiagent systems increases exponentially with respect to the agent number. This scalability issue prevents MARL from being applied in large-scale multiagent systems. However, one critical feature in MARL that is often neglected is that the interactions between agents are quite sparse. Without exploiting this sparsity structure, existing works aggregate information from all of the agents and thus have a high sample complexity. To address this issue, we propose an adaptive sparse attention mechanism by generalizing a sparsity- inducing activation function. Then a sparse communication graph in MARL is learned by graph neural networks based on this new attention mechanism. Through this sparsity structure, the agents can communicate in an effective as well as efficient way via only selectively attending to agents that matter the most and thus the scale of the MARL problem is reduced with little optimality compromised. Comparative results show that our algorithm can learn an interpretable sparse structure and outperforms previous works by a significant margin on applications involving a large-scale multiagent system. ## I INTRODUCTION Reinforcement Learning (RL) has achieved enormous successes in robotics [1] and gaming [2] in both single and multiagent settings. For example, deep reinforcement learning (DRL) achieved super-human performance in the two- player game Go, which has a very high-dimensional state-action space [3, 4]. However, in multiagent scenarios, the sizes of the state space, joint action space, and joint observation space grow exponentially with the number of agents. As a result of this high dimensionality, existing multiagent reinforcement learning (MARL) algorithms require significant computational resources to learn an optimal policy, which impedes the application of MARL to systems such as swarm robotics [5]. Thus, improving the scalability of MARL is a necessary step towards building large-scale multiagent learning systems for real-world applications. In MARL, the increase of complexity of finding an optimal joint policy, with respect to the number of agents, is a result of coupled interactions between agents [6]. However, in many multiagent scenarios, the interactions between agents are quite sparse. For example, in a soccer game, an agent typically only needs to pay attention to other nearby agents when dribbling because agents far away are not able to intercept. The existence of such sparsity structures of the state transition dynamics (or the state-action-reward relationships) suggests that an agent may only need to attend to information from a small subset of the agents for near-optimal decision-making. Note that the other players that require attention might not be nearby, such as the receiver of a long pass in soccer. In such cases, the agent only needs to selectively attend to agents that “matter the most”. As a result, the agent can spatially and temporally reduce the scale of the planning problem. In large-scale MARL, sample complexity is a bottleneck of scalability [7]. To reduce the sample complexity, another feature we can exploit is the interchangeability of homogeneous agents: switching two agents’ state/action will not make any difference to the environment. This interchangeability implies permutation-invariance of the multiagent state-action value function (a.k.a. the centralized $Q$-function) as well as interchangeability of agent policies. However, many MARL algorithms such as MADDPG [8], VDN [9], QMIX [10] do not exploit this symmetry and thus have to learn this interchangeability from experience, which increases the sample complexity unnecessarily. Graph neural network (GNN) is a specific neural network architecture in which permutation-invariance features can be embedded via graph pooling operations, so this approach has been applied in MARL [11, 12, 13] to exploit the interchangeability. As MARL is a non-structural scenario where the links/connections between the nodes/agents are ambiguous to decide, a graph has to be created in advance to apply GNN for MARL. Refs. [11, 12, 13], apply ad-hoc methods, such as $k$-nearest neighbors, hard threshold, and random dropout to obtain a graph structure. However, these methods require handcrafted metrics to measure the closeness between agents, which are scenario-specific and thus not general/principled. Inappropriately selecting neighbors based on a poorly designed closeness metric could lead to the failure of learning a useful policy. While attention mechanisms [14] could be applied to learn the strength of the connections between a pair of agents (i.e., closeness metric) in a general and principled way, such strengths are often dense, leading to a nearly-complete computation graph that does not benefit scalability. The dense attention mechanism results from that the softmax activation function operated on the raw attention logits generates a probability distribution with full support. One solution to enforce a sparse graph is top $k$ thresholding [15], which keeps the $k$-largest attention scores and truncates the rest to zero. However, this truncation is a non-differentiable operation that may cause problems for gradient-based optimization algorithms, such as those used in end-to-end training. Therefore, a sparse attention mechanism that preserves the gradient flow necessary for gradient-based training is required. To address the non-differentiability issue in sparse attention mechanisms, we generalize sparsemax [16] and obtain a sparsity mechanism whose pattern is adaptive to the environment states. This sparsity mechanism can reduce the complexity of both the forward pass and the back-propagation of the policy and value networks, as well as preserving the end-to-end trainability in contrast to hard thresholding. With the introduction of GNN and generalized sparsemax, which can preserve permutation invariance and promote sparsity respectively, the scalability of MARL is improved. The discussion so far was restricted to homogeneous agents and thus permutation-invariance is desirable. However, in heterogeneous multiagent systems or competitive environments, permutation invariance and interchangeability are no longer valid. For example, in soccer, switching positions of two players from different sides can make a difference to the game. To address this heterogeneity, GNN-based MARL must distinguish the different semantic meanings of the connections between different agent pairs (e.g. friend/friend relationship versus friend/foe relationship). We address this requirement by multi-relational graph convolution network [17] to pass messages using different graph convolution layers on graph edge connections with different semantic meanings. To summarize, we propose to learn an adaptive sparse communication graph within the GNN-based framework to improve the scalability of MARL, which applies to both homogeneous and heterogeneous multiagent systems in mixed cooperative-competitive scenarios. ### I-A Related Work One of the existing works exploiting the structure in MARL is the mean-field reinforcement learning (MFRL) [18] algorithm, which takes as input the observation and the mean action of neighboring agents to make the decision, and neglects the actions of all the other agents. This simplification leads to good scalability. However, the mean action cannot distinguish the difference among neighboring agents and the locality approximations fail to capture information from a far but important agent for optimal decision-making, which leads to sub-optimal policies. Multi-Actor-Attention-Critic (MAAC) is proposed in [19] to aggregate information using attention mechanism from all the other agents. Similarly, [11, 13, 20] also employ the attention mechanism to learn a representation for the action-value function. However, the communication graphs used there are either dense or ad-hoc ($k$ nearest neighbors), which makes the learning difficult. Sparse attention mechanisms were first studied by the natural language processing community in [16], where sparsemax was proposed as a sparse alternative to the activation function softmax. The basic idea is to project the attention logits onto the probability simplex, which can generate zero entries once the projection hits the boundary of the simplex. While generalized sparse attention mechanisms were further studied in [21, 22, 23], they are not adaptive to the state in the context of MARL, in terms of the sparsity pattern. Given this state of the art, the contributions of this paper are twofold. First, we propose a new adaptive sparse attention mechanism in MARL to learn a sparse communication graph, which improves the scalability of MARL by lowering the sample complexity. Second, we extend our GNN-based MARL to heterogeneous systems in mixed cooperative-competitive settings using multi-relational GNN. The evaluations show that our algorithm significantly outperforms previous approaches on applications involving a large number of agents. This technique can be applied to empower large-scale autonomous systems such as swarm robotics. ## II PRELIMINARIES ### II-A Multiagent Reinforcement Learning As a multiagent extension of Markov decision processes (MDPs), a Markov game is defined as a tuple $\langle N,S,\\{O_{i}\\}_{i\in N},\\{A_{i}\\}_{i\in N},\\{r_{i}\\}_{i\in N},\gamma\rangle$, where $N=[1,\ldots,n]$ is a set of agent indices, $S$ is the set of state, $\\{O_{i}\\}_{i\in N}$ and $\\{A_{i}\\}_{i\in N}$ are the joint observation and joint action sets, respectively. The $i$th agent chooses actions via a stochastic policy $\pi_{\theta_{i}}:O_{i}\times A_{i}\to[0,1]$, which leads to the next state according to the state transition function $\mathcal{T}:{S}\texttimes A_{1}\texttimes\ldots\texttimes A_{n}\to S$. The $i$th agent also obtains a reward as a function of the state and agent’s action $r_{i}:S\texttimes\\{A_{i}\\}_{i\in N}\to\mbox{$\mathbb{R}$}$, and receives a private observation correlated with the state $o_{i}:S\times\\{A_{i}\\}_{i\in N}\to O_{i}$. The initial states are determined by a distribution $\rho:S\to[0,1]$. The $i$th agent aims to maximize its own total expected return $R_{i}=\sum_{t=1}^{T}\gamma^{t}r_{i}^{t}$, with discount factor $\gamma$ and time horizon $T$. ### II-B Multi-head attention The scaled dot-product attention mechanism was first proposed in [14] for natural language processing. An attention function maps the query and a set of key-value pairs to the output, which is the weighted sum of the values. The weight assigned to the each value calculated via a compatibility function of the query and the corresponding key. In the context of MARL, let $h_{i},i\in N$ be the representation of the agents. Key, query and value of agent $i$ is defined as $K_{i}^{l}=W_{K}h_{i}^{l}\in\mathbb{R}^{d_{K}}$, $Q_{i}^{l}=W_{Q}h_{i}^{l}$ and $V_{i}^{l}=W_{V}h_{i}^{l}$, respectively with $W_{K},W_{Q}$ and $W_{V}$ are parameter matrices. The output for agent $i$ is then $\text{Att}_{i}(h)=\sum_{j}w_{ij}V_{j},$ (1) where $w_{i\bullet}\in\mathbb{R}^{n}$, the $i$-th row of the weight matrix $w$, is defined as $w_{i\bullet}=\sigma_{a}\Big{(}\frac{(K_{i})^{T}Q}{\sqrt{d_{K}}}\Big{)}$ (2) with $\sigma_{a}$ being the softmax function in previous works of GNN-based MARL. The weight $w_{i\bullet}$ is dense as $\operatorname*{softmax}_{i}(z)\neq 0$ for any vector $z$ and $i$. To increase the expressiveness, multi-head attention is applied here via simply concatenating the outputs from a single attention function [14]. ### II-C Relational GNN In heterogeneous multiagent systems, different agent pair can have different relations, such as friend or foe in a two-party zero-sum game. As a result, information aggregation from agents with different relations should have different parameters. Work in [17] proposed relational graph convolutional network to model multi-relational data. The forward-pass update of agent $i$ in a multi-relational graph is as follows $h_{i}^{(l+1)}=\sigma\Big{(}\sum_{r\in\mathcal{R}}\sum_{j\in\mathcal{N}_{i}^{r}}\frac{1}{c_{i,r}}W_{r}^{(l)}h_{j}^{(l)}+W_{0}^{(l)}h_{i}^{(l)}\Big{)},$ (3) where $\mathcal{N}_{i}^{r}$ denotes the set of neighbor indices of agent $i$ under relation $r\in\mathcal{R}$ and $c_{i,r}$ is a normalization constant. To distinguish the heterogeneity in MARL, similar to this convolution-based multi-relational GNN, we apply different attention heads on agent pairs with different relations. ## III APPROACH In this section, we present our approach to exploit the sparsity in MARL by generalizing the dense soft-max attention to adaptive sparse attention. Moreover, our approach to apply multi-relational attention mechanism for heterogeneous games involving competitive agents is also introduced. ### III-A Learning a communication graph via adaptive sparse attention The scaled dot-product attention is applied to learn the communication graph in MARL. If an attention weight between a pair of agents is zero, then there is no communication/message passing between them. Thus, the normalization function $\sigma_{a}(\bullet)$ in (2) is critical to learn a communication graph. As usually used in the attention mechanism [14] or classifications, $\sigma_{a}(\bullet)$ is usually set to be softmax, which cannot induce sparsity. We propose an adaptive sparse activation function as an alternative to softmax. Let $x\in\mathbb{R}^{d}$ be the raw attention logits and $y$ be normalized attention strength in the ($d-1$)-dimensional probability simplex defined as $\Delta^{d}:=\\{y\in\mathbb{R}^{d}|y\geq 0,\textbf{1}^{T}y=1\\}$. We are interested in the mapping from $x\in\mathbb{R}^{d}$ to $y\in\Delta^{d}$. In other words, such a mapping can transform real weights to a probability distribution, i.e., the normalized attention strength between a pair of agents. The classical softmax, used in most attention mechanisms, is defined component-wisely as $y_{i}=\operatorname*{softmax}_{i}(x)=\frac{e^{x_{i}}}{\sum_{i=1}^{d}e^{x_{i}}}.$ (4) A limitation of the softmax transformation is that the resulting probability distribution always has full support, which makes the communication graph dense, resulting in high complexity. In order to reduce the complexity, our idea is to replace the softmax activation function with a generalized activation function, which could adaptively be dense or sparse based on the state. To investigate alternative activation functions to softmax, consider the max operator defined as $\max(x):=\max_{i\in[d]}(x_{i})=\sup_{y\in\Delta^{d}}y^{T}x,$ (5) where $[d]=\\{1,\ldots,d\\}$. The second equality comes from that the supremum of the linear form over a simplex is always achieved at a vertex, i.e., one of the standard basis vector $\\{e_{i}\\}_{i\in[d]}$. As a result, the max operator puts all the probability mass onto a single element, or in other words, only one entry of $y$ is nonzero corresponding to the largest entry of $x$. For example, with $x=[0,t]\in\mbox{$\mathbb{R}$}^{2}$, the probability distribution w.r.t. the logit $t$, i.e., $(\operatorname*{arg\,sup}_{y\in\Delta^{d}}y^{T}x)_{2}$, is a step function, as $(\operatorname*{arg\,sup}_{y\in\Delta^{d}}y^{T}x)_{2}$ equals 1 if $t>0$ and $0$ otherwise. This discontinuity at $t=0$ of the step function is not amenable to gradient-based optimization algorithms for training deep neural networks. One solution to the discontinuity issue encountered in (6) is to add a regularized $\Omega(y)$ in the max operator as $\Pi_{\Omega}(x)=\operatorname*{arg\,max}_{y\in\Delta^{d}}y^{T}x+\gamma\Omega(y)$ (6) Different regularizers $\Omega(y)$ produce different mappings with distinct properties (see summary in Table I). Note that with $\Omega(y)$ as the Shannon entropy, $\Pi_{\Omega}(x)$ recovers softmax. With the states/observations evolving, the ideal profile of $\Pi_{\Omega}(x)$ should be able to adapt the sparsity extent (controlled via $\gamma$) and the pattern (controlled via the selection of $\Omega(y)$) accordingly. TABLE I: List of different regularizers and their corresponding mappings $y=\Pi_{\Omega}(x)$, where $x$ is the raw attention logits and $y$ is the probability distribution in $\Delta^{d}$. Entropy | $\Omega(y)$ | $\Pi_{\Omega}(x)$ | Ref. ---|---|---|--- Shannon | $\sum_{i}y_{i}\log(y_{i})$ | $\operatorname*{softmax}_{i}(x)=\frac{e^{x_{i}}}{\sum_{i=1}^{d}e^{x_{i}}}$ | [22] $l_{2}$ norm | $-\frac{1}{2}\sum_{i}y_{i}^{2}$ | $\arg\min_{y\in\Delta^{d}}\|y-x\|^{2}$ | [16] Tsallis | $\left\\{\begin{array}[]{ll}\frac{\sum_{i}(y_{i}-y_{i}^{\alpha})}{\alpha(\alpha-1)},&\alpha\neq 1\\\ \sum_{i}y_{i}\log(y_{i}),&\alpha=1\end{array}\right.$ | No closed-form | [24] Generalized | $\displaystyle\frac{1}{q}\sum_{i}(y_{i}-\frac{\displaystyle e^{qy_{i}}-1}{\displaystyle e^{q}-1})$ | No closed-form | [25] TABLE II: List of different $G(x)$ and their resulting mappings $\Pi_{\Omega}(x)$ $\gamma G_{i}(x)$ | $\frac{e^{x_{i}}}{\sum_{i}e^{x_{i}}}$ | $\frac{{x_{i}^{2}}}{\sum_{i}{x_{i}^{2}}}$ | $x_{i}$ ---|---|---|--- $\Pi_{\Omega}(x)$ | softmax | softmax | sparsemax Property | Translation invariance $\Pi_{\Omega}(x)=\Pi_{\Omega}(x+c\textbf{1})$ | Scaling invariance $\Pi_{\Omega}(x)=\Pi_{\Omega}(cx)$ | Translation invariance $\Pi_{\Omega}(x)=\Pi_{\Omega}(x+c\textbf{1})$ Example | $\Pi_{\Omega}([100,101])=\Pi_{\Omega}([0,1])$ | $\Pi_{\Omega}([1,2])=\Pi_{\Omega}([1,2]\times 10^{-3})$ | $\Pi_{\Omega}([100,101])=\Pi_{\Omega}([0,1])$ Note that the Tsallis entropy and the generalized entropy in Table I do not have closed-form solutions [22], which will increase the computational burden since iterative numerical algorithms will have to be employed. Sparsemax has a closed-form solution and can induce sparsity, but sparsemax is not adaptive and lacks flexibility as it is unable to switch from one sparsity pattern to another when necessary. We aim to combine the advantages and avoid the disadvantages using this new formulation $\Pi_{\Omega}(x)=\operatorname*{arg\,min}_{y\in\Delta^{d}}||y-\gamma G(x)||^{2},$ (7) with $G(x):\mbox{$\mathbb{R}$}^{d}\to\mbox{$\mathbb{R}$}^{d}$ and $\gamma$ being a learnable neural network and a scalar, respectively. By choosing different $G(x)$, $\Pi_{\Omega}(x)$ can exhibit different sparsity patterns including softmax and sparsemax. With $G(x)$ fixed, the parameter $\gamma$ can control how sparse the output could be, similar to the temperature parameter in softmax. The summary in Table II shows that (7) will lead to a general mapping and can combine properties such as translation and scaling invariance adaptively. Work in [23] proposed sparse-hourglass that can adjust the trade- off between translation and scaling invariance via tunable parameters. However, it is unclear under which circumstances one property is more desirable than the other, so there is little to no prior knowledge on how to tune such parameters. In contrast, our formulation in (7) can balance such trade-off via learning $G(x)$ and $\gamma$ while work in [23] is based on a fixed form of $G(x)$ with tunable parameters. While we can let the neural network learn $G(x):\mbox{$\mathbb{R}$}^{d}\to\mbox{$\mathbb{R}$}^{d}$ without any restrictions, there is indeed prior knowledge that we can apply, e.g., monotonicity. It is desired to keep the monotonicity of $\Pi_{\Omega}(x)$, i.e., $\forall x_{i}>x_{j},(\Pi_{\Omega}(x))_{i}>(\Pi_{\Omega}(x)_{j}$, as larger attention logit should be mapped into larger attention strength. As sparsemax is monotonic, this requires that $\forall x_{i}>x_{j},G_{i}(x)>G_{j}(x)$, or in other words, the order of the input of $G(x)$ coincides with that of the output. To keep this property, $G(x)$ is designed component-wisely as $G_{i}(x)=\psi(\phi_{1}(x_{i}),\sum_{i}\phi_{2}(x_{i}))$, with $\psi:\mbox{$\mathbb{R}$}^{2}\to\mbox{$\mathbb{R}$}^{1},\phi_{1},\phi_{2}:\mbox{$\mathbb{R}$}^{1}\to\mbox{$\mathbb{R}$}^{1}$ are neural networks with hidden layers. Note that $G_{i}(x)$ should be coupled with all of the entries of $x$ instead of be a univariate function only depending on $x_{i}$, as demonstrated in Table II. As the second argument of $\psi$ (i.e., $\sum_{i}\phi_{2}(x_{i})$) is invariant to $G_{i}(x),\forall i\in[d]$, the order preserving of $G(x):\mbox{$\mathbb{R}$}^{d}\to\mbox{$\mathbb{R}$}^{d}$ is equivalent to the monotonicity of $\psi(\bullet)$ and $\phi_{1}(\bullet)$. In order to keep this monotonicity, we enforce all the weights of the networks $\psi$ and $\phi_{1}$ to be positive [26], by applying an absolute value function on the weights. This architecture can accelerate the learning process with extra prior knowledge, as it is monotonic by design. ### III-B Message passing in MARL via GNN We will present how the information is aggregated to learn a representation for per-agent value/policy network using a graph neural network. The scaled dot-product attention mechanism (Section II-B) with our generalized sparsemax as the activation function, denoted as sparse-Att, is applied to learn a communication graph and pass messages through the connections in the graph. We start with homogeneous multiagent system, where the relation between any agent pair is identical. A graph is defined as $\mathcal{G}:=(\mathcal{V},\mathcal{E})$, where $v_{i}\in\mathcal{V}$ represent an agent and the cardinality of $\mathcal{V}$ is $|\mathcal{V}|$. Moreover, $e_{ij}\in\mathcal{E}$ is $1$ if agent $i$ and $j$ can communicate directly (or agent $j$ is observable to agent $i$), and $0$ otherwise. This is a restriction on the communication graph and $\mathcal{E}$ is the set of all possible edges. Then sparse-Att aims to learn a subset of $\mathcal{E}$ via induced sparsity without compromising much optimality. For agent $i$, let $U_{i}=f_{a}(X_{i})$ and $E_{i}$ be its observation and entity encoding respectively, where $X_{i},i\in\mathcal{V}$ is the local state and $f_{a}$ is a learnable agent encoder network. Then the initial observation embedding of agent $i$, denoted as $h_{i}^{(1)}$, is $h_{i}^{(1)}=f_{mp}(U_{i}\|E_{i}),$ (8) where $f_{mp}$ is another learnable network and the operator $\|$ denotes concatenation. Then at hop $l$ ($l$-th round of message passing), agent $i$ aggregates information from its possible neighbors belonging to the set $\mathcal{N}=\\{j\in\mathcal{V}|e_{ij}=1\\}$ as follows $h_{i}^{(l+1)}=f_{mp}\Big{(}h_{i}^{(l)}\|\text{sparse- Att}_{i}^{\mathcal{N}}(h^{(l)})\Big{)}.$ (9) With $l\geq 2$, the multi-hop message passing can enable the agent to obtain information from beyond its immediate neighbors. In the message aggregation from all of the agents $\mathcal{N}$, identical parameters are used in $\text{sparse-Att}_{i}^{\mathcal{N}}$, which enforces the permutation- invariance. This property is desirable because homogeneous agents are interchangeable. However, interchangeability is no longer applicable to heterogeneous systems or mixed cooperative-competitive environment. For example, with $\mathcal{V}_{1},\mathcal{V}_{2}\subseteq\mathcal{V}$ being a two-team partition of $\mathcal{V}$, agents cooperate with other agents from the same team but compete against agents from the other team. For agent $i\in\mathcal{V}_{1}$, its teammate neighborhood and enemy neighborhood are $\mathcal{N_{+}}=\\{j\in\mathcal{V}_{1}|e_{ij}=1\\}$ and $\mathcal{N_{-}}=\\{j\in\mathcal{V}_{2}|e_{ij}=1\\}$, respectively. The edges connecting teammates and enemies are called positive and negative edges. Then based on multi-relational GNN, agent $i$ aggregates information at hop $l$ in the following way $h_{i}^{(l+1)}=f_{mp}\Big{(}h_{i}^{(l)}\|\text{sparse- Att}_{i}^{\mathcal{N}_{+}}(h^{(l)})\|\text{sparse- Att}_{i}^{\mathcal{N}_{-}}(h^{(l)})\Big{)},$ where $\text{sparse-Att}_{i}^{\mathcal{N}_{+}}$ and $\text{sparse- Att}_{i}^{\mathcal{N}_{-}}$ are different attention heads. Additionally, balance theory [27] suggests that “the teammate of my teammate is my teammate” and “the enemy of my enemy is my teammate.” In a two-team competitive game, any walk (a sequence of nodes and edges of a graph) between an agent pair in the communication graph, comprising of both positive and negative edges, will lead to the same relation between the agent pair [28]. This property eliminates the ambiguity that the information aggregated from the same agent (but different walk) might have a different teammate/enemy property. Figure 1: Our sparse-Att framework consists of three modules: encoder, multi- relational sparse attention mechanism, and value/policy network, with homogeneous agents sharing all parameters. Agents employ different attention heads to aggregate information alongside connections with different semantic meanings, followed by a concatenation. $L$ is the number of the message- passing rounds; see (9). “concat” denotes the concatenation operation. Here only two classes (shown in red and blue) of heterogeneous agents are shown for simplicity. The proposed algorithmic framework is illustrated in Fig. 1. After $L$ rounds of message passing, each agent has an updated encoding $h_{i}^{(L+1)}$. This encoding is then fed into the value network and the policy network, which estimate the state value and a probability distribution over all possible actions, respectively. As homogeneous agents are interchangeable, they share all of the parameters, including entity encoding, policy, value and message passing. Proximal policy gradient (PPO, [29]) is employed to train the model in an end-to-end manner. As only local information is required, the proposed approach is decentralized. Moreover, our approach maintains the transferability of GNN-based approaches as all the network dimensions are invariant to agent/entity number in the system. ## IV EXPERIMENTS ### IV-A Task description The proposed algorithm is evaluated in three swarm robotics tasks: Coverage, Formation, and ParticleSoccer [30], first two of which are cooperative and the third is competitive. The tasks are simulated in the Multiagent Particle Environment111https://github.com/openai/multiagent-particle-envs(MAPE [8]). The agents in MAPE can move in a 2-dimensional space following a double integrator dynamic model. The action space of the agents is discretized, with each agent can accelerate/decelerate in both $X$ and $Y$ direction. The three tasks are briefly introduced as follows. Coverage: There are $n_{A}$ agents (light purple) and $n_{L}$ landmarks (black) in the environment (see illustration in Fig. 2(a)). The objective for the agents is to cover the landmarks with the smallest possible number of timesteps. Agents are not assigned to reach a certain landmark, but instead, have to figure out the assignment via communication such that the task can be finished optimally. Formation: There are $n_{A}$ agents (blue) and $1$ landmarks (black) in the environment (see illustration in Fig. 2(b)), with $n_{A}$ being an even natural number. The agents need to split into two sub-teams of equal size, with each of them building a formation of a regular pentagon. The two regular pentagons with different sizes are both centered at the landmark. ParticleSoccer: There are $n_{A}$ agents and 3 landmarks in the environment (see illustration in Fig. 2(c)), with the bigger landmark as a movable ball and the two smaller ones as a fixed landmark. A team wins the game via pushing the black ball to the opponent team’s goal. The goal color of the light blue (red, resp.) team is blue (red, resp.). (a) Coverage (b) Formation (c) ParticleSoccer Figure 2: Three different simulation tasks used in this work. ### IV-B Implementation specifications The agent encoder $f_{a}(\bullet)$ and the entity encoder take input the $4$-dimensional agent states and $2$-dimensional entity states, respectively. The queries, keys, and values in all of the sparse attention mechanism are $128$-dimensional. The communication hop is $L=2$. All neural networks are fully connected with the ReLU activation function. In the sparsity-promoting function (7), $\phi_{1},\phi_{2}$ and $\psi$ all have one hidden layer with dimensions being $16$, $16$ and $64$, respectively. The absolute value function is used to keep the weights of the monotonicity-preserving neural network positive. Evaluation is performed every $320$ episodes and PPO update is executed for $4$ epochs after collecting experience of $4096$ timesteps. ### IV-C Results In the cooperative scenarios i.e., Coverage and Formation, two metrics are used to evaluate the algorithms. The first is the average reward per step and the second is the task success rate. Higher means better performance for both metrics. We compare our algorithms with two baselines: GNN-based MARL with dense attention mechanism [11] and MAAC [19]. These two algorithms are considered to be strong baselines as they reported advantageous results against algorithms including MADDPG [8], COMA [31], VDN [32] and QMIX [10]. Public repositories222https://github.com/sumitsk/matrl.git333 https://github.com/shariqiqbal2810/MAAC are used for comparison. As both repositories also apply their algorithms on MAPE, the default hyperparameters are used for comparison. In simulation, we set $n_{A}=30$ and $n_{A}=20$ for Coverage and Formation, respectively. Fig. 4 and Fig. 4 demonstrated that our algorithm can achieve higher rewards than the two baselines with fewer episodes. This validates that sparse-Att can accelerate the learning process via aggregating information from agents that matter the most. Moreover, in terms of the second metric, i.e., success rate, our algorithm consistently outperforms the two baselines by a significant margin (with a much smaller variance), as shown in Fig. 5. The evaluations of both metrics for two scenarios provide strong support for the advantages of our algorithm. Figure 3: Reward comparison of our algorithm against two baselines for the Coverage task. Figure 4: Reward comparison of our algorithm against two baselines for the Formation task. (a) Coverage (b) Formation Figure 5: Performance comparison of three algorithm on two scenarios. Multiple policies learned from each algorithm are evaluated and the mean/standard deviation are plotted. For the competitive ParticleSoccer task, we set $n_{A}=20$ with both red team and blue team of size $\frac{n_{A}}{2}=10$. As this task is competitive, the above two metrics are no longer applicable. Instead, we let the red (blue, resp.) play against a blue (red, resp.) team from another algorithm. Table III presents the results of the inter-algorithm competition. The overall score of each algorithm equals the sum of the winning evaluation episodes of its red team and blue team playing against blue and red team respectively from other algorithms. The overall scores in Table III show that our algorithm can learn strong policies. TABLE III: Evaluation of three algorithms in the competitive ParticleSoccer task. Each pair is evaluated for $50$ episodes and the $(\bullet,\bullet,\bullet)$ in each cell denotes the number of red team winning episodes, blue team wining episodes and the draw episodes. A draw means that neither team scores within a given episode length. $\text{win}_{\text{red}}$ and $\text{win}_{\text{blue}}$ are the winning episodes of the red and blue team, respectively when competing against blue and red team from other algorithms. Red Blue | sparse-Att (ours) | dense-Att | MAAC | $\text{win}_{\text{red}}$ ---|---|---|---|--- sparse-Att (ours) | blue!25$(48,0,2)$ | $(15,0,35)$ | $(26,0,24)$ | $41$ dense-Att | $(9,1,40)$ | blue!25$(5,0,45)$ | $(3,0,47)$ | $11$ MAAC | (7,0,43) | (2,0,48) | blue!25$(3,0,47)$ | $9$ $\text{win}_{\text{blue}}$ | $-15$ | $-17$ | $-29$ | N/A | sparse-Att (ours) | dense-Att | MAAC | overall scores: $\text{win}_{\text{red}}+\text{win}_{\text{blue}}$ | $\mathbf{26}$ | $-6$ | $-20$ | ### IV-D Interpretability of the sparse communication graph Let us proceed by considering the inherent sparity in Formation and ParticleSoccer. As mentioned in the description of the Formation scenario, the formation of each pentagon is related to half of the agents, while the sub- team assignments need to be learned. In the implementation, the reward is set to require that the first $\frac{n_{A}}{2}$ agents closest to the landmark build the formations of the inner pentagon and the remaining $\frac{n_{A}}{2}$ agents to build the formations of the outer pentagon. With the convergence of the learning algorithm, once a sub-team partition is learned to complete the two sub-tasks, the learned agent indexing of each team should not vary due to the distance sorting and the two pentagons are relatively far away. As a result, the reward to complete each sub-task is only related to the corresponding sub-team and hence the two sub-teams are decoupled from each other. The adjacency matrix of the learned communication graph shown in Fig. 6(a) validates that the inter-team communication is very sparse. This adjacency matrix is up to row/column permutation as indexing of each sub-team is learned without being known as a prior. Moreover, in a sub-team, the algorithm learns a communication graph similar to a star-graph. It can be understood that each sub-team selects a leader. As a star-graph is a connected graph with possibly minimum edges, this communication protocol is both effective and efficient. Also, the length of the path between any agent pair in a star graph is no greater than $2$, which echos the two-hop communication ($L=2$) we used in the simulation. That is because due to the two-hop message- passing, the agents can eventually communicate with agents as far as two edges away, which includes all of the agents in a star graph. Note that the sparsity on the diagonal entries of the communication graph does not mean that the agent’s own information is neglected, as it is separately concatenated; see (9). Also, in the ParticleSoccer scenario, from each team’s perspective, agents need to coordinate tightly within the team to greedily push the ball to the other team’s goal while only attending to a small number of agents from the other team. This leads to dense intra-team communication but relatively sparse inter-team communication. This is validated by the approximately block- diagonal adjacency matrix of the learned communication graph in Fig. 6(b). (a) Coverage (b) Formation Figure 6: Sparse communication graph for two scenarios. For the Coverage, our sparse-Att learns to split into two sub-team as desired and the learned sparse star-like communication graph makes communication both effective and efficient. In the ParticleSoccer, sparse-Att learn to pay more attention to teammates and a necessary subset of enemies. ## V CONCLUSIONS and FUTURE WORK This paper exploits sparsity to scale up Multi-Agent Reinforcement Learning (MARL), which is motivated by the fact that interactions are often sparse in multiagent systems. We propose a new general and adaptive sparsity-inducing activation function to empower an attention mechanism, which can learn a sparse communication graph among agents. The sparse communication graph can make the message-passing both effective and efficient such that the scalability of MARL is improved without compromising optimality. Our algorithm outperforms two baselines by a significant margin on three tasks. Moreover, for scenarios with inherent sparsity, it is shown that the sparsity of the learned communication graph is interpretable. 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